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2,869,038,156,328 | arxiv | \section{Introduction}
Modern communications systems increasingly utilize \ac{ml} algorithms to meet the compound requirements of high-dimensional \ac{ce} in massive \ac{MIMO} \ac{OFDM} applications \cite{8052521}.
The channel characteristics of the whole propagation environment of a \ac{BS} cell can be described by means of a \ac{PDF} $f_{\bm{h}}$.
This \ac{PDF} \( f_{\bm{h}} \) describes the stochastic nature of all channels in the whole coverage area of a \ac{BS} and therefore captures ambient information.
Every channel of any \ac{MT} within the \ac{BS} cell is a realization of a random variable with \ac{PDF} \( f_{\bm{h}} \).
The main problem is that this \ac{PDF} is typically not available analytically.
For this reason, many classical channel estimation approaches cannot be applied or this ambient information is ignored and replaced with Gaussian assumptions which may only hold locally around a given user.
In this setting, \ac{ml} approaches play an increasingly important role.
These aim to (implicitly) learn the underlying \ac{PDF} from data samples such that the ambient information is taken into account in \ac{ml} channel estimation algorithms \cite{9789120,KoFeTuUt22, KoFeTuUt21J}.
According to this development, many new channel models have been designed to capture the complex channel characteristics of a whole \ac{BS} environment.
Modern channel simulators based on ray tracing or stochastic-geometric models allow for the generation of large synthetic datasets that can be used for training and testing, e.g., \cite{QuaDRiGa1, Alkhateeb2019, Remcom}.
Even though such increasingly complex simulators generate ever more realistic \ac{BS} scenarios, it is crucial to also evaluate the performance of \ac{phy} algorithms on real-world data, i.e., on data collected in a measurement campaign.
This is especially important and interesting for \ac{ml} algorithms which mainly depend on the underlying data (\ac{PDF}), e.g., \cite{9789120,KoFeTuUt21J,TuKoFeBaXuUt}.
An evaluation of an \ac{ml}-based \ac{ce} algorithm on measurement data was done in \cite{HeDeWeKoUt19}, where a neural network-based estimator is analyzed.
However, the estimator in~\cite{HeDeWeKoUt19} was originally derived via assumptions on the channel model and on the antenna configuration which might not hold in practice.
In this work, we evaluate the recently proposed \ac{GMM}-based channel estimator from \cite{KoFeTuUt21J} on data originating from the same measurement campaign.
The estimator first approximates the \ac{PDF} \( f_{\bm{h}} \) of the whole radio propagation environment with a \ac{GMM}.
This is done offline and only once.
Thereafter, the estimator utilizes this ambient information for \ac{ce} in the online phase.
The estimator is proven to asymptotically converge to the optimal \ac{CME} (which would be calculated using the unknown \ac{PDF} \( f_{\bm{h}} \)) but so far was only evaluated on synthetic data.
Our experiments indicate that the \ac{GMM} estimator captures the ambient information well
because it outperforms state-of-the-art \ac{ce} algorithms evaluated (and trained) on the same measurement data.
In particular, we achieve lower \acp{MSE} and higher spectral efficiencies.
In addition, we generate synthetic channel data using a state-of-the-art channel simulator and train the \ac{GMM} estimator once on these and once on the measurement data, and we apply the estimator once to the synthetic and once to the measurement data for evaluation.
We observe that providing suitable ambient information in the training phase beneficially impacts the \ac{ce} performance.
The remainder of this work is organized as follows.
In \Cref{sec:ce_gmm}, the \ac{GMM} channel estimator is explained and in \Cref{sec:campaign}, the measurement campaign is described and a channel simulator is introduced for comparison.
\Cref{sec:sim_results} provides simulation results and \Cref{sec:conlusion} concludes this work.
\section{Gaussian Mixture Model Channel Estimator}
\label{sec:ce_gmm}
We consider \ac{ce} in the uplink from a single-antenna \ac{MT} located within the cell to a \ac{BS}.
The \ac{BS} is equipped with $N$ antennas.
After correlating with the commonly known pilot sequence, we obtain the noisy observation
\begin{equation}
\bm{y} = \bm{h} + \bm{n} \in \mathbb{C}^{N},
\end{equation}
where $\bm{h} \in \mathbb{C}^{N}$ is the uplink-channel of a certain \ac{MT} located within the coverage area of the \ac{BS} and $\bm{n} \sim \mathcal{N}_\mathbb{C}(\mathbf{0}, \bm{C}_{\bm{n}} = \sigma^2 \bm{I})$ denotes the \ac{AWGN}.
The goal is then to estimate the channel $\bm{h}$ given $\bm{y}$, i.e., to denoise the observation $\bm{y}$.
The stochastic nature of all channels in the whole coverage area of the \ac{BS} is assumed to be described by means of a continuous \ac{PDF} \( f_{\bm{h}} \).
Every channel \( \bm{h} \) of any \ac{MT} located within the \ac{BS} cell is a realization of a random variable with \ac{PDF} \( f_{\bm{h}} \).
For such a system model, the \ac{MSE}-optimal channel estimator is given by the \ac{CME}
\begin{equation}\label{eq:conditional_mean}
\hat{\mbh} = \expec[\bm{h} \mid \bm{y}] = \int \bm{h} f_{\bm{h}\mid\bm{y}}(\bm{h}\mid\bm{y}) d \bm{h},
\end{equation}
which can generally not be computed analytically.
Further, \( f_{\bm{h}} \) is typically not available in an analytic form.
However, in \cite{KoFeTuUt21J} a method to approximate~\eqref{eq:conditional_mean} with the help of \acp{GMM} was proposed.
To this end, assuming to have access to a set \( \mathcal{H}_M = \{ \bm{h}_m \}_{m=1}^M \) of training channel samples, which represent the radio propagation environment (ambient information), and motivated by universal approximation properties of \acp{GMM}~\cite{NgNgChMc20}, we fit a \ac{GMM} \( f_{\bm{h}}^{(K)} \) with \( K \) components to \( \mathcal{H}_M \) in order to approximate the unknown channel \ac{PDF} \( f_{\bm{h}} \).
A \ac{GMM} is a \ac{PDF} of the form~\cite{bookBi06}
\begin{equation}\label{eq:gmm_of_h}
f^{(K)}_{\bm{h}}(\bm{h}) = \sum_{k=1}^K p(k) \mathcal{N}_{\mathbb{C}}(\bm{h}; {\bm{\mu}}_k, \bm{C}_k),
\end{equation}
where every summand is one of its \( K \) components.
It is characterized by the means ${\bm{\mu}}_k \in \mathbb{C}^N$, the covariances $\bm{C}_k \in \mathbb{C}^{N\times N}$, and the mixing coefficients $p(k)$. Maximum likelihood estimates of these parameters can be computed using an \ac{EM} algorithm and the training data set \( \mc{H}_M \), see~\cite{bookBi06}.
The idea in~\cite{KoFeTuUt21J} is to compute the \ac{MSE}-optimal estimator \( \hat{\mbh}_{\text{GMM}}^{(K)} \) for channels distributed according to \( f_{\bm{h}}^{(K)} \) and to use it to estimate the channels distributed according to \( f_{\bm{h}} \).
This estimator \( \hat{\mbh}_{\text{GMM}}^{(K)} \) converges pointwise to the optimal estimator \( \hat{\mbh} \) from~\eqref{eq:conditional_mean} as \( K \to \infty \), cf.~\cite{KoFeTuUt21J}.
Once the (offline) \ac{GMM} fitting process is done,
the (online) channel estimates
can be computed in closed form:
\begin{equation}\label{eq:gmm_estimator_closed_form}
\hat{\mbh}_{\text{GMM}}^{(K)}(\bm{y}) = \sum_{k=1}^K p(k \mid \bm{y}) \hat{\mbh}_{\text{LMMSE},k}(\bm{y}),
\end{equation}
with the responsibilities
\begin{equation}\label{eq:responsibilities}
p(k \mid \bm{y}) = \frac{p(k) \mathcal{N}_{\mathbb{C}}(\bm{y}; \mbmu_k, \mbC_k + \bm{C}_{\bm{n}})}{\sum_{i=1}^K p(i) \mathcal{N}_{\mathbb{C}}(\bm{y}; \mbmu_i, \mbC_i + \bm{C}_{\bm{n}}) },
\end{equation}
and
\begin{equation}\label{eq:lmmse_formula}
\hat{\mbh}_{\text{LMMSE},k}(\bm{y}) =
\mbC_k ( \mbC_k + \bm{C}_{\bm{n}})^{-1} (\bm{y} - \mbmu_k) + \mbmu_k.
\end{equation}
The weights \( p(k \mid \bm{y}) \) are the probabilities that component \( k \) generated the current observation \( \bm{y} \), cf.~\cite{KoFeTuUt21J}.
\subsection{Complexity Analysis and Low Cost Adaptations}
The inverse in \eqref{eq:lmmse_formula} can be precomputed offline for various \acp{SNR} because the \ac{GMM} covariance matrices \( \mbC_k \) do not change once the \ac{GMM} fitting process is done.
Accordingly, evaluating \eqref{eq:lmmse_formula} online is dominated by matrix-vector multiplications and has a complexity of \( \mc{O}(N^2) \).
It remains to calculate the responsibilities~\eqref{eq:responsibilities} by evaluating Gaussian densities.
A Gaussian density with mean \( {\bm{\mu}} \in \mathbb{C}^N \) and covariance matrix \( \bm{C} \in \mathbb{C}^{N\times N} \) can be written as
\begin{equation}\label{eq:gaussian_density}
\mathcal{N}_{\mathbb{C}}(\bm{h}; {\bm{\mu}}, \bm{C}) = \frac{\exp(-(\bm{h} - {\bm{\mu}})^{\operatorname{H}} \bm{C}^{-1} (\bm{h} - {\bm{\mu}}))}{\pi^N \det(\bm{C})}.
\end{equation}
Again, since the \ac{GMM} covariance matrices and mean vectors do not change between observations, the inverses and the determinants of the densities can be pre-computed offline.
Therefore, the online evaluation is also in this case dominated by matrix-vector multiplications and has a complexity of \( \mathcal{O}(N^2) \).
Overall, evaluating \eqref{eq:gmm_estimator_closed_form} has a complexity of \( \mathcal{O}(K N^2) \) \cite{KoFeTuUt21J}.
Since \( \hat{\mbh}_{\text{GMM}}^{(K)} \) converges pointwise to the \ac{MSE}-optimal estimator $\hat{\bm{h}}$ form \eqref{eq:conditional_mean} as \( K \to \infty \),
a trade-off between the performance of the estimator and the complexity can be achieved by adjusting the number \( K \) of \ac{GMM} components.
The complexity of the estimator from \eqref{eq:gmm_estimator_closed_form} can be reduced by introducing structural constraints to the \ac{GMM} covariance matrices \( \mbC_k \).
For example, in case of a \ac{ULA} employed at the \ac{BS}, it is common to assume Toeplitz covariance matrices, see, e.g., \cite{NeWiUt18}.
Further, for large numbers of antenna elements, a Toeplitz matrix is well approximated by a circulant matrix~\cite{Gr06}.
Motivated by these common assumptions we enforce structural constraints onto the \ac{GMM} covariances.
Since we consider exclusively an environment, where at the \ac{BS} a \ac{URA} with $N_{\mathrm{v}}$ vertical and $N_{\mathrm{h}}$ horizontal ($N=N_{\mathrm{v}} \times N_{\mathrm{h}}$) elements is employed, the structural assumptions result in block-Toeplitz matrices with Toeplitz blocks, or block-circulant matrices with circulant blocks, respectively \cite{KoHeUt19}.
In general the structured covariances can be expressed as
\begin{equation}
\mbC_k = \bm{Q}^{\operatorname{H}} \diag(\bm{c}_k) \bm{Q},
\end{equation}
where on the one hand, when assuming a Toeplitz structure, $\bm{Q} = \bm{Q}_{N_{\mathrm{v}}} \otimes \bm{Q}_{N_{\mathrm{h}}}$, where $\bm{Q}_J$ contains the first $J$ colums of a $2J\times 2J$ \ac{DFT} matrix, and $\bm{c}_k \in \mathbb{R}_{+}^{4\NhN_{\mathrm{v}}}$ \cite{KoHeUt19, St86}.
On the other hand, when assuming circular structure, we have $\bm{Q} = \bm{F}_{N_{\mathrm{v}}} \otimes \bm{F}_{N_{\mathrm{h}}}$, where $\bm{F}_J$ is the $J\times J$ DFT-matrix, and $\bm{c}_k \in \mathbb{R}_{+}^{\NhN_{\mathrm{v}}}$.
In both cases, the structural constraints allow to store only the $\bm{c}_k$'s of a \ac{GMM} which drastically reduces the memory overhead and the number of parameters to be learned, similar as in \cite{KoFeTuUt21J, FeJoHuKoTuUt22}.
Further, in case of circular covariances, the complexity of evaluating \eqref{eq:gmm_estimator_closed_form} reduces to \( \mathcal{O}(K N \log(N)) \), where 2D-DFT transforms are exploited when evaluating~\eqref{eq:responsibilities} and~\eqref{eq:lmmse_formula}, cf. \cite{KoFeTuUt21J}.
\section{Measurement Campaign and Synthetic Data}
\label{sec:campaign}
The measurement campaign was conducted at the Nokia campus in Stuttgart, Germany, in October/November 2017.
As can be seen in \Cref{fig:meas_campaign}, the receive antenna with a down-tilt of $\SI{10}{\degree}$ was mounted on a rooftop about $\SI{20}{m}$ above the ground and comprises a \ac{URA} with $N_{\mathrm{v}}=4$ vertical and $N_{\mathrm{h}}=16$ horizontal single polarized patch antennas.
The horizontal spacing is $\lambda$ and the vertical spacing equals $\lambda/2$, where the geometry of the \ac{BS} antenna array was adapted to the \ac{UMi} propagation scenario, which exhibited a larger horizontal than vertical angular spread.
The carrier frequency is $\SI{2.18}{\giga\hertz}$.
The \ac{BS} transmitted time-frequency orthogonal pilots using $\SI{10}{\mega\hertz}$ \ac{OFDM} waveforms.
In particular, $600$ sub-carriers with $\SI{15}{\kilo\hertz}$ spacing were used, which resembles typical \ac{LTE} numerology.
The pilots were sent continuously with a periodicity of $\SI{0.5}{ms}$ and were arranged in $50$ separate subbands, with $12$ consecutive subcarriers each, for channel sounding purposes.
For the duration of one pilot burst the propagation channel was assumed to remain constant.
The single monopole receive antenna, which mimics the \ac{MT}, was mounted on top of a moving vehicle at a height of $\SI{1.5}{m}$.
The maximum speed was $\SI{25}{kmph}$.
Synchronization between the transmitter and receiver was achieved using GPS.
The data was collected by a TSMW receiver and stored on a Rohde \& Schwarz IQR hard disk recorder.
In a post-processing step, by the correlation of the received signal with the pilot sequence a channel realization vector with $N=N_{\mathrm{v}} \times N_{\mathrm{h}}$ coefficients per subband was extracted.
The measurement was conducted at a high \ac{SNR}, which ranged from $\SI{20}{dB}$ to $\SI{30}{dB}$.
Thus, the measured channels are regarded as ground truth.
Further, we assume fully calibrated antennas and thus channel reciprocity is assumed.
In this work, we will therefore consider a system, where we artificially corrupt the measured channels with \ac{AWGN} at specific \acp{SNR} and thereby obtain noisy observations $\bm{y} = \bm{h} + \bm{n}$.
The task is then to denoise the observations and obtain an estimated channel $\hat{\bm{h}}$.
We want to highlight that we investigate a single-snapshot scenario, i.e., the
coherence interval of the covariance matrix and of the channel is identical (the channel covariance matrix changes at the
same time scale as the channel).
\begin{figure}
\centering
\begin{tikzpicture}
\draw (0,0) node[below right] {\includegraphics[width=0.975\columnwidth]{fcell_map.jpg}};
\draw (0,0) node[below right] {\includegraphics[width=0.975\columnwidth]{fcell_nlos_los.pdf}};
\end{tikzpicture}
\caption{Measurement setup on the Nokia campus in Stuttgart, Germany.}
\label{fig:meas_campaign}
\end{figure}
\subsection{Synthetic Data Generation using QuaDRiGa}
Version $2.6.1$ of the QuaDRiGa channel simulator \cite{QuaDRiGa1, QuaDRiGa2} was used to generate \ac{CSI} in a \ac{UMi} scenario.
The environment for which synthetic data is generated was adapted as closely as possible to the circumstances of the measurement environment.
For this reason, the carrier frequency was set to $\SI{2.18}{\giga\hertz}$.
The base station is placed at a height of 20 meters.
The minimum and maximum distances between \acp{MT} and the \ac{BS} are $\SI{35}{\meter}$ and $\SI{315}{\meter}$, respectively.
The \acp{MT} are located outdoors at a height of $\SI{1.5}{\meter}$.
A QuaDRiGa channel is given by \( \bm{h} = \sum_{\ell=1}^{L} \bm{g}_{\ell} e^{-2\pi j f_c \tau_{\ell}} \)
with $\ell$ the path number, $L$ the number of \acp{MPC}, $f_c$ the carrier frequency, and $\tau_{\ell}$ the \( \ell \)th path delay.
The number \( L \) depends on whether there is \ac{LOS} or \ac{NLOS} propagation, cf. \cite{QuaDRiGa2}.
The coefficients vector \( \bm{g}_{\ell} \) consists of one complex entry for each antenna pair and comprises the attenuation of a path, the antenna radiation pattern weighting, and the polarization.
As described in the QuaDRiGa manual~\cite{QuaDRiGa2}, the generated channels are post-processed to remove the path gain.
\section{Experiments and Results}
\label{sec:sim_results}
Normalizing the data so that $\expec[\|\bm{h}\|^2] = N$ allows us to define an \ac{SNR} in our simulations as \( \frac{1}{\sigma^2} \).
Given $T$ test samples \( \{ \bm{h}_t \}_{t=1}^T \) and obtaining corresponding channel estimates \( \{ \hat{\mbh}_t \}_{t=1}^T \), we use the normalized \ac{MSE} \( \frac{1}{NT} \sum_{t=1}^T \| \bm{h}_t - \hat{\mbh}_t \|^2\) as performance measure.
The following baseline estimators are considered.
In the system at hand, the least squares estimate is simply given by the noisy observations \( \hat{\mbh}_{\text{LS}} = \bm{y} \).
Another baseline is the sample covariance matrix based approach, where we construct a sample covariance matrix $\bm{C}_s = \frac{1}{M} \sum_{m=1}^M \bm{h}_m \bm{h}_m^{\operatorname{H}}$ given a set of training samples and calculate \ac{LMMSE} channel estimates $\hat{\mbh}_{\text{s-cov}} = \bm{C}_s (\bm{C}_s + \bm{C}_{\bm{n}})^{-1} \bm{y}$.
Compressive sensing approaches commonly assume that the channel exhibits a certain structure: $\bm{h} \approx \bm{D} \bm{t}$, where $\bm{D} \in \mathbb{C}^{N\times L}$ is a dictionary. We used an oversampled \ac{DFT} matrix as dictionary, with $L=4N$ (cf., e.g., \cite{AlLeHe15}).
A compressive sensing algorithm like \ac{OMP}~\cite{Gharavi} can then be used to obtain a sparse vector \( \bm{t} \), and the estimated channel is calculated as $\hat{\mbh}_{\text{OMP}} = \bm{D} \bm{t}$. Since the sparsity order of the channel is not known but the algorithm's performance crucially depends on it, we use a genie-aided approach to obtain a bound on the performance of the algorithm. In particular, we use the true channel to choose the optimal sparsity order.
We further compare to a \ac{CNN}-based channel estimator, which was introduced in~\cite{NeWiUt18}.
There, the authors exploit assumptions which stem from a spatial channel model (3GPP, cf.~\cite{3GPP_scm}) in order to derive the \ac{CNN} architecture. The \ac{CNN} is then trained on measurement data to compensate the mismatch of the assumptions and the real world data. We use the rectified linear unit as activation function and the input transform is based on the \( 2N \times 2N \) \ac{DFT} matrix, cf. \cite[Equation (43)]{NeWiUt18}.
\input{simresultstex/simomeasURAOctoverSNR}
In \Cref{fig:meas_URA_over_snr_NMSE}, we use $T=10{,}000$ channel samples stemming from the measurement campaign for evaluating the performances of the different channel estimators.
In particular, we compare the GMM estimator with full covariances, denoted by ``GMM'', and block Toeplitz (``Toep. GMM'') or block circulant (``circ. GMM'') covariances, to the state-of-the-art estimators described above.
In total we use $M=300{,}000$ channel samples stemming from the measurement campaign as training data in the fitting process of the \ac{GMM} approaches, each with $K=64$ components.
Also the learning process of the \ac{CNN} estimator (``CNN'') and the construction of a sample covariance matrix for the sample covariance \ac{LMMSE} estimation approach (``sample cov.'') use these \( M \) samples.
The \ac{GMM} estimator with full covariance matrices performs best over the whole SNR range from $\SI{-15}{dB}$ to $\SI{20}{dB}$, followed by the ``Toep. GMM'' and the ``circ. GMM'' approaches.
As expected, the GMM estimator’s performance suffers from introducing structural covariance constraints but it still outperforms the other channel estimation approaches.
With ``GMM (QuaDRiGa)'' we depict the \ac{GMM} approach where synthetic training data ($M=300{,}000$ samples) is used to fit the \ac{GMM}.
Despite using synthetic data of an environment, which was adapted as closely as possible to the circumstances of the measurement campaign's environment, we can observe a severe performance degradation in the estimation performance of the \ac{GMM} estimator.
In \Cref{fig:quad_URA_over_snr_NMSE}, we replace the test data and use $T=10{,}000$ synthetic channel samples for comparing the ``GMM'' estimator (fitted with measurement data), the ``CNN'' approach (trained on measurement data), the ``sample cov.'' approach (sample covariance obtained using measurement data) with the ``GMM (QuaDRiGa)'' (fitted with synthetic data) approach.
We can observe that the ``GMM (QuaDRiGa)'' approach now performs best since the learned ambient information now matches the synthetic test data on which the estimator is evaluated.
We conclude that using synthetic channel data is not suitable to replicate the
ambient information of the campus where the measurement campaign was conducted, and vice versa.
Thus, this validates the claim that ambient information is learnt by the \ac{GMM} when provided suitable training data.
We further want to highlight that the \ac{CNN} estimator which constitutes a data based approach as well, should also be able to capture the underlying ambient information of the considered propagation environment.
Up to some extent this seems to be the case since the \ac{CNN} approach exhibits the best performance right after the \ac{GMM} approaches.
Nevertheless, even the \ac{GMM} approach with block circulant covariances, which has the same order of complexity as the \ac{CNN} approach, yields a better overall estimation performance.
\input{simresultstex/simonquadURAOctoberSNR}
In \Cref{fig:meas_URA_over_snr_DR} we consider the same simulation parameters as in \Cref{fig:meas_URA_over_snr_NMSE}, and analyze a performance upper bound for the achievable spectral efficiency \begin{equation}
\bar{r} = E\left[\log_2\left(1 + \dfrac{|\hat{\bm{h}}^H\bm{h}|^2}{\sigma^2||\hat{\bm{h}}||^2}\right)\right],
\end{equation}
when applying a matched filter $\frac{\hat{\bm{h}}^H}{||\hat{\bm{h}}||}$ in the uplink \cite{NeWiUt18}, which may also be interpreted as a measure of the accuracy of the estimated channel subspace \cite{NeWiUt18}.
Note, that there is no one-by-one relation between the spectral efficiency and the \ac{MSE} in general, cf. \cite{HeDeWeKoUt19}.
In essence, we can observe that the \ac{GMM} yields the best performance while there is only a minor degradation when using structured covariances.
\input{simresultstex/simDRonmeasURAOctoberSNR}
\input{simresultstex/simCompareSamples}
\input{simresultstex/simCompareSamplesToepCirc}
In \Cref{fig:comp_samples}, we depict the behavior of the \ac{GMM} estimator with full covariances for a varying number of components $K$ and a varying number of training data used to fit the \ac{GMM}.
The \ac{SNR} is $\SI{10}{dB}$.
The number of parameters of a \ac{GMM} increases with an increasing $K$, which requires more training data to achieve a good fit.
As the figure suggests, as long as there are enough training data,
increasing $K$ leads to a better performance.
In contrast, in \Cref{fig:compare_samples_toep_circ}, where we consider \acp{GMM} with fewer parameters by assuming either block Toeplitz (top) or block circulant (bottom) covariances, we can observe that the estimator already achieves a good performance with a low to moderate number of available training samples.
In particular, with structured covariances, increasing $K$ leads to a better performance already with more than $100{,}000$ available training samples.
Overall, in \Cref{fig:comp_samples} and \Cref{fig:compare_samples_toep_circ}, even a small to moderate number of components ($K=16$ or $K=32$) performs well. Accordingly, a suitable number of components $K$ should be determined based on the amount of available training data and the desired overall estimation complexity.
\input{simresultstex/simomeas_comps_valmeas}
\input{simresultstex/simomeas_comps_valquad}
In \Cref{fig:avg_resps_meas} and \Cref{fig:avg_resps_quad}, we aim to investigate the differences in the distributions of the synthetic and the measurement data from a different perspective:
We plot the average responsibilities, cf. \eqref{eq:responsibilities}, of the \ac{GMM} when fitted on either synthetic (``GMM (QuaDRiGa)'') or on measurement (``GMM'') data.
To this end, we evaluate \( p(k \mid \bm{y}) \) for each observation $\bm{y}$ and average these responsibilities with respect to the samples in the evaluation set with $T=10{,}000$ samples and an SNR of $\SI{10}{dB}$.
Afterwards, we sort the components from highest to lowest average responsibility.
In \Cref{fig:avg_resps_meas}, the evaluation set contains only measurement data.
It can be observed that the average responsibilities of the \ac{GMM} fitted on synthetic or on measurement data are similar up to some extent.
That is, the channel simulator is able to capture general information about the underlying \ac{UMi} scenario---but not the details of the measurement environment in its full extent.
In contrast, in \Cref{fig:avg_resps_quad}, the same \acp{GMM} are evaluated with synthetic data.
The average responsibilities can be clearly distinguished since a large mismatch can be observed.
A possible explanation for this observation is that the \ac{GMM} fitted onto the measurement data is specifically designed for the environment of the measurement campaign with unique immanent characteristics.
In contrast to the channel simulator with a stochastic nature (hence aiming to model general \ac{UMi} scenarios), this \ac{GMM} does not generalize to different \ac{UMi} scenarios.
This behavior is desirable since the distinctive design allows for performance gains as discussed earlier.
\Cref{fig:avg_resps_meas} and \Cref{fig:avg_resps_quad} show above all that the attempt to represent channel data with the false \ac{GMM} only works to a limited extent, which can be seen, among other performance metrics, from the fact that fewer components of the false \ac{GMM} are identified as representative than would be the case with the correct \ac{GMM}.
This underlines the importance of an evaluation with measurement data.
\section{Conclusion and Outlook}
\label{sec:conlusion}
In this work, we used real-world data stemming from a measurement campaign in order to evaluate and validate a recently introduced \ac{GMM}-based algorithm for uplink channel estimation.
Our experiments suggest that the \ac{GMM} estimator learns the intrinsic characteristics of a given base station's whole radio propagation environment.
To validate the claim that ambient information is learnt, we conducted experiments, where we used test data either stemming from the measurement campaign or synthetic data.
We observed that providing suitable ambient information, which is implicitly contained within the data, in the training phase (offline), beneficially impacts the channel estimation performance in the online phase.
We further showed that structurally constrained covariances of the GMM, which are motivated by model-based insights, also work well when using real-world data.
In particular, one can drastically reduce the computational complexity and memory overhead with only small performance losses.
An immediate additional advantage is that less training data is needed due to the lower number of \ac{GMM} parameters, which need to be fitted, when assuming structural constraints.
Future work might consider a more accurate and involved emulation of the propagation environment using a digital twin. For example, a digital representative of the propagation environment can be generated using a ray tracing tool, where the measurement campus with all of the buildings and streets, which are characteristic for certain propagation properties, is recreated virtually.
Given the digital twin of the propagation environment, the performance of the data based channel estimators might be evaluated under these more accurate digital representatives.
\balance
\bibliographystyle{IEEEtran}
|
2,869,038,156,329 | arxiv | \section{Introduction}\subsection{Statement of the main
results}\subsubsection{Unitarily invariant measures on spaces of infinite complex matrices} Let $\Mat$ be the
space of all infinite matrices whose rows and columns are indexed by natural numbers and whose entries are
complex:
\[
\Mat \;=\; \left\{ z=(z_{ij})_{i,j \in \mathbb N}, \; z_{ij}\in \mathbb C \right\}.
\]
Let $U(\infty)$ be the infinite unitary group: an infinite matrix $u=(u_{ij})_{i,j\in {\mathbb N}}$ belongs to
$U(\infty)$ if there exists a natural number $n_0$ such that the matrix
$$
(u_{ij})_{i,j\in [1,n_0]}
$$
is unitary, while $u_{ii}=1$ if $i>n_0$ and $u_{ij}=0$ if $i\neq j$, $\max(i,j)>n_0$.
The group $U(\infty) \times U(\infty)$ acts on $\Mat$ by multiplication on both sides:
\[
T_{(u_1,u_2)}z \;=\; u_1zu_2^{-1}.
\]
Recall that a $U(\infty)\times U(\infty)$-invariant measure on $\Mat$, finite or infinite, is called
\textit{ergodic} if any $U(\infty)\times U(\infty)$-invariant Borel set either has measure zero or has complement
of measure zero. Finite ergodic $U(\infty)\times U(\infty)$-invariant measures on $\Mat$ have been classified by
Pickrell \cite{pickrell}. The first main result of this paper is that,
under natural assumptions, an ergodic $U(\infty)\times U(\infty)$-invariant measure on $\Mat$ must be finite.
Precisely, let $m\in\mathbb N$ and let $\classFNC$ denote the space of Borel measures $\nu$ on $\Mat$ such that for any $R>0$ we have
$$
\nu\left(\left\{z\in\Mat: \max_{i,j\leqslant m}|z_{ij}|<R\right\}\right)<+\infty.
$$
\begin{theorem} \label{mainz}
If a $\UxU$-invariant Borel measure from the class $\classFNC$ is ergodic then it is finite.
\end{theorem}
A measure $\nu\in\classFNC$ is automatically sigma-finite,
clearly satisfies all assumptions of the ergodic decomposition theorem of \cite{Bufetov} and therefore admits a
decomposition into ergodic components. By definition, almost all ergodic components of a
measure $\nu\in\classFNC$ must themselves lie in the class $\classFNC$.
Let ${\mathfrak M}_{erg}(\Mat)$
stand for the set of $\UxU$-invariant ergodic Borel probability measures on $\Mat$; the set ${\mathfrak
M}_{erg}(\Mat)$ is a Borel subset of the space of all Borel probability measures on $\Mat$ (see, e.g.,
\cite{Bufetov}, where the claim is proved for all measurable Borel actions of inductively compact groups). Theorem
\ref{mainz} and the ergodic decomposition theorem of \cite{Bufetov} now implies the following
\begin{corollary}\label{corz}
For any $\UxU$-invariant Borel measure
$$
\nu\in \classFNC
$$
there exists a unique sigma-finite Borel measure
${\tilde\nu}$ on ${\mathfrak M}_{erg}(\Mat)$ such that
\begin{equation}
\label{ergdecfmz} \nu=\int\limits_{{\mathfrak M}_{erg}(\Mat)}\eta d{\tilde \nu}(\eta).
\end{equation}
\end{corollary}
The integral in \eqref{ergdecfmz} is understood in the usual weak sense: for every Borel subset $A\subset\Mat$ we
have
$$
\nu(A)=\int\limits_{{\mathfrak M}_{erg}(\Mat)}\eta(A) d{\tilde \nu}(\eta).
$$
\subsubsection{Unitarily invariant measures on spaces of infinite
Hermitian matrices}
Now let $H\subset \Mat$ be the space of infinite Hermitian matrices:
$$
H=\{h=(h_{ij})_{i,j\in {\mathbb N}}, h_{ij}={\overline h_{ji}}\}.
$$
The group $U(\infty)$ naturally acts on the space $H$ by conjugation. Finite ergodic $U(\infty)$-invariant
measures on $H$ have also been classified by Pickrell \cite{pickrell} (see also Olshanski and Vershik \cite{OV}).
An analogue of Theorem \ref{mainz} holds
in this case as well.
Precisely, a Borel measure $\nu$ on $H$ is said to belong to the class $\mathfrak F(m, H)$ if for any $R>0$ we
have
$$
\nu(\{h\in H: \max\limits_{i\leq m,j\leq m} |h_{ij}|\leq R\})<\infty.
$$
\begin{theorem}\label{mainh}
If a $U(\infty)$-invariant measure from the class $\mathfrak F(m, H)$ is ergodic, then it is finite.
\end{theorem}
As before, let ${\mathfrak M}_{erg}(H)$ stand for the set of $U(\infty)$-invariant ergodic Borel probability
measures on $H$; the set ${\mathfrak M}_{erg}(H)$ is a Borel subset of the space of all Borel probability measures
on $H$. Theorem \ref{mainh} now implies
\begin{corollary}\label{corh}
For any $U(\infty)$-invariant Borel measure $\nu\in \mathfrak F(m, H)$ there exists a unique sigma-finite Borel
measure ${\tilde\nu}$ on ${\mathfrak M}_{erg}(H)$ such that
\begin{equation}
\label{ergdecfmh} \nu=\int\limits_{{\mathfrak M}_{erg}(H)}\eta d{\tilde \nu}(\eta).
\end{equation}
\end{corollary}
The integral in \eqref{ergdecfmh} is again understood in the weak sense.
One expects similar results to hold for all the $10$ series of homogeneous spaces (see, e,.g., \cite{O1, O2}).
\subsubsection{Infinite Hua-Pickrell measures}
A natural example of measures lying in the class ${\mathfrak F}(m,H)$ is given by infinite Hua-Pickrell measures
introduced by Borodin and Olshanski \cite{BO}, Section 8, Subsection ``Infinite measures''. In fact, for any $m\in
{\mathbb N}$, Borodin and Olshanski give explicit examples of measures lying in the class ${\mathfrak F}(m,H)$ but not in
the class ${\mathfrak F}(m-1,H)$. Starting from the Pickrell measures \cite{pickrell3}, a similar construction can
be carried out to obtain infinite $\UxU$-invariant measures on $\Mat$ lying in the class ${\mathfrak F}(m,\Mat)$
but not in the class ${\mathfrak F}(m-1,\Mat)$ for any $m\in {\mathbb N}$. Corollaries \ref{corz}, \ref{corh} show
now that ergodic components of infinite Hua-Pickrell measures are finite.
\subsection{Outline of the proofs of Theorems \ref{mainz}, \ref{mainh}.}
Olshanski and Vershik \cite{OV} gave a completely different proof for Pickrell's Classification Theorem of
$U(\infty)$-invariant ergodic measures on $H$, and their method has been adapted to
ergodic $\UxU$-invariant measures on $\Mat$ by Rabaoui \cite{rabaoui1}, \cite{rabaoui2}.
The proof of Theorems \ref{mainz}, \ref{mainh} is based on the Olshanski-Vershik approach.
First, following Vershik \cite{V1}, to each infinite matrix we assign its sequence of {\it orbital measures}
obtained by averaging over exhausting sequences of compact subgroups in our infinite-dimensional unitary groups. A
simple general argument shows that precompactness of the family of orbital measures for almost all points implies
finiteness of an ergodic measure. Using the work of Olshanski and Vershik \cite{OV} and Rabaoui
\cite{rabaoui1}, \cite{rabaoui2}, we give a sufficient condition, called ``radial boundedness'' of a matrix, for weak
precompactness of its family of orbital measures: namely, it is shown that the sequence of orbital measures is
weakly precompact as soon as the norms (and, in case of $H$, also the traces)
of $n\times n$ ``corners'' of our matrix do not grow too fast as
$n\to\infty$. To complete the proof of Theorem \ref{mainh}, it remains to show that with respect to any measure in the class ${\mathfrak F}(m, H)$, almost all
matrices are indeed radially bounded (the same statement, with the same proof, also holds for $\classFNC$). This
is done in two steps: first, it is shown that if a measure from the class ${\mathfrak F}(m, H)$ is
$U(\infty)$-invariant, then its suitably averaged conditional measures yield a {\it finite} $U(\infty)$-invariant
measure --- with respect to which almost all points must then be radially bounded; second, applying a finite
permutation of columns and rows, one deduces radial boundedness for the initial matrix and completes the proof.
\subsection{Projections and conditional measures}
For $n\in \mathbb N$, let ${\rm Mat}(n, {\mathbb C})$ be the space of all $n\times n$ complex matrices.
Introduce a map
\[
\textstyle \Pi_{[1,n]}: \Mat\rightarrow {\rm Mat}(n, {\mathbb C})
\]
by the formula
\[
\textstyle \Pi_{[1,n]}z \;=\; (z_{ij})_{i,j=1,\ldots,n}, \ z \in \Mat.
\]
If a measure $\nu$ on $\Mat$ is infinite, then the projection $\left(\Pi_{[1,n]}\right)_*\nu$ may fail to be
well-defined. The class $\classFNC$ consists precisely of those measures $\nu$ for which the projection
$\left(\Pi_{[1,m]}\right)_*\nu$ (and, consequently, all projections $\left(\Pi_{[1,n]}\right)_*\nu$ for
$n\geqslant m$) are indeed well-defined. Equivalently, by Rohlin's Theorem on existence of conditional measures, a
measure $\nu$ belongs to the class $\classFNC$ if and only if:
\begin{enumerate}
\item there exists a measure
$\overline{\nu}$ on the space ${\rm Mat}(m, {\mathbb C})$ assigning finite weight
to every compact set;
\item for
$\overline{\nu}$-almost every $z^{(m)}\in {\rm Mat}(m, {\mathbb C})$ there exists a Borel probability measure
$\nu_{z^{(m)}}$ on $\Mat$ supported on the set $\left(\Pi_{[1,m]}\right)^{-1}z^{(m)}$ such that for every Borel
subset $A\subset\Mat$ the map
$$
z^{(m)}\to\nu_{z^{(m)}}(A)
$$
is $\overline{\nu}$-measurable and that we have a decomposition
\begin{equation}
\label{ergdecz} \nu=\int\limits_{{\rm Mat}(m,\mathbb{C})} \nu_{z^{(m)}}\,d\overline{\nu}\left(z^{(m)}\right)
\end{equation}
again understood in the weak sense.
\end{enumerate}
A similar description can be given for measures in the class $\classF$: a Borel measure $\nu$ on $H$ belongs to
the class $\classF$ if and only if there exists a measure $\overline{\nu}$ on the space $H(m)$ of $m\times
m$-Hermitian matrices which assigns finite weight to every compact set and, for $\overline{\nu}$-almost every
$h^{(m)}\in H(m)$ there exists a Borel probability measure $\nu_{h^{(m)}}$ such that
\begin{equation}
\label{ergdech} \nu=\int_{H(m)}\nu_{h^{(m)}}\,d\overline{\nu}\left(h^{(m)}\right),
\end{equation}
where the decomposition (\ref{ergdech}) is understood in the same way as the decomposition (\ref{ergdecz}).
{\bf Acknowledgements.} Grigori Olshanski posed the problem to me, and I am greatly indebted to him. I am deeply
grateful to Sevak Mkrtchyan and Konstantin Tolmachov for helpful discussions. I am deeply grateful to Lisa
Rebrova, Nikita Kozin and Nikita Medyankin for typesetting parts of the manuscript.
Part of this work was done when I was visiting Victoria University Wellington, the Joint Institute for
Nuclear Research in Dubna and Kungliga Tekniska H{\"o}gskolan in Stockholm. I am deeply grateful to these institutions
for their warm hospitality.
This work was supported in part by an Alfred P. Sloan Research Fellowship, by the Grant MK-4893.2010.1 of the
President of the Russian Federation, by the Programme on Mathematical Control Theory of the Presidium of the
Russian Academy of Sciences, by the Programme 2.1.1/5328 of the Russian Ministry of Education and Research, by the
RFBR-CNRS grant 10-01-93115, by the Edgar Odell Lovett Fund at Rice University and by the National Science
Foundation under grant DMS~0604386.
\section{Weak recurrence}
The proof is based on the following simple general observation. Let $X$ be a complete metric space, and let $G$ be
an inductively compact group, in other words,
$$
G=\bigcup\limits_{n=1}^{\infty}K(n), \ \ K(n)\subset K(n+1)
$$
where the groups $K(n)$, $n\in\mathbb{N}$, are compact and metrizable. Let $\mathfrak{T}$ be a continuous action
of $G$ on $X$ (continuity is here understood with respect to the totality of the variables).
Each group $K(n)$ is endowed with the Haar measure $\mu_{K(n)}$, and to each point $x\in X$ we assign, following
Vershik \cite{V1}, the corresponding sequence of \emph{orbital measures} $\mu_{K(n)}^x$ on $X$ given by the
formula
$$
\int_X f(y)\,d\mu_{K(n)}^x(y)=\int_{K(n)}f\left(T_gx\right)\mu_{K(n)}(g),
$$
valid for any bounded continuous function $f$ on $X$. Given a family ${\mathfrak A}$ of Borel probability
measures on $X$, we say that the family $\mathfrak A$ is \emph{weakly recurrent} if for any positive bounded
continuous function $f$ on $X$ we have
$$
\inf_{\nu\in {\mathfrak A}}\int f\,d\nu>0.
$$
\begin{proposition}
\label{weakrecimplfin} Let $\nu$ be an ergodic $\mathfrak{T}$-invariant measure on $X$ that assigns finite weight
to every ball and admits a set $B$, $\nu(B)>0$, such that for every $x\in B$ the sequence of orbital measures
$\mu_{K(n)}^x$ is weakly recurrent. Then $\nu$ is finite.
\end{proposition}
\begin{proof} Consider the space $L_2(X, \nu)$; for $n\in\mathbb{N}$,
let $L_2(X, \nu)^{K(n)}$ be the subspace of $K(n)$-invariant functions, and let $P_n:L_2(X, \nu)\to L_2(X,
\nu)^{K(n)}$ be the corresponding orthogonal projection.
If the measure $\nu$ is ergodic and infinite, then
\begin{equation}
\label{emptyinter} \bigcap\limits_{n=1}^{\infty} L_2(X,\nu)^{K(n)}=0.
\end{equation}
Indeed, let $L_2(X, \nu)^{G}$ be the subspace of $G$-invariant square-integrable functions. By definition, we have
\begin{equation}
\bigcap\limits_{n=1}^{\infty} L_2(X,\nu)^{K(n)}=L_2(X, \nu)^{G}.
\end{equation}
Now, if the measure $\nu$ is ergodic and assigns finite weight to every ball, then, by results of \cite{Bufetov},
it is also indecomposable in the sense that any Borel set $A\subset X$ such that for any $g\in G$ we have
$\nu(T_gA\Delta A)=0$ must satisfy either $\nu(A)=0$ or $\nu(X\setminus A)=0$. It follows that $L_2(X,
\nu)^{G}=0$, and \eqref{emptyinter} is proved.
For any $f\in L_2(X,\nu)$ we thus have $P_nf\to 0$ in $L_2(X,\nu)$ as $n\to\infty$. Along a subsequence we then
also have $P_{n_k}f\to 0$ almost surely with the respect to the measure $\nu$.
If $f$ is continuous and square-integrable, then the equality
$$
P_n f(x)=\int\limits_X f(y)\,d\mu_{K(n)}^x(y)
$$
holds for $\nu$-almost all $x$.
Take, therefore, $f$ to be a positive, continuous, square-integrable function on $X$ (the existence of such a
function follows from the fact that the measure $\nu$ assigns finite weight to balls: indeed, taking $x_0\in X$,
letting $d$ be the distance on $X$, and setting $f(x)=\psi(d(x_0,x))$, where $\psi:\mathbb{R}\to\mathbb{R}$ is
positive, continuous, and decaying rapidly enough at infinity, we obtain the desired function).
If $\nu$ is ergodic and infinite, then, from the above, for almost all $x\in X$ we have
$$
\lim\limits_{n\to\infty}\int f\,d\mu_{K(n)}^x=0.
$$
In particular, for $\nu$-almost all $x\in X$, the sequence of orbital measures is not weakly recurrent, which
contradicts the assumptions of the proposition.
\end{proof}
{\bf Remark.} The argument above, combined with the ergodic decomposition theorem of \cite{Bufetov}, yields a slightly
stronger statement: if a $\mathfrak{T}$-invariant measure $\nu$ on $X$ that
assigns finite weight to every ball is such that for $\nu$-almost every every $x\in X$ the sequence of orbital
measures $\mu_{K(n)}^x$ is weakly recurrent, then the ergodic components of $\nu$ are almost surely finite.
It remains to derive
Theorems \ref{mainz}, \ref{mainh} from Proposition \ref{weakrecimplfin}. We start with Theorem \ref{mainh}.
\section{Proof of Theorem \ref{mainh}}
\subsection{Radial boundedness}
A matrix $h\in H$ will be called \emph{radially bounded} in $H$ if
$$
\sup_{n\in\mathbb{N}}\frac{|\tr\left(\Pi_{[1,n]}h\right)|}{n}<+\infty, \
\sup_{n\in\mathbb{N}}\frac{\tr\left(\Pi_{[1,n]}h\right)^2}{n^2}<+\infty.
$$
We shall now see that if $h\in H$ is radially bounded in $H$, then the family of orbital measures $\mu_n^h$, $n\in
{\mathbb N}$, is precompact in the weak topology on $H$, and, consequently, weakly recurrent.
Recall that if $X$ is a complete separable metric space, $\mathfrak M(X)$ the space of Borel probability measures
on $X$, then the {\it weak topology} on ${\mathfrak M}(X)$ is defined as follows. Let $ f_1,\ldots, f_k: \; X
\longrightarrow \mathbb R $ be bounded continuous functions on $X$, let $ \varepsilon_1,\ldots, \varepsilon_k
\;>\;0, $ let $\nu_0 \in \mathfrak M(X)$ and consider the set
\begin{equation} \label{bwt}
\left\{ \nu \in \mathfrak M(X): \; \left| \int f_i \, d\nu - \int f_i \, d \nu_0 \right| < \varepsilon_i, \; i=1,
\ldots, k\right\}
\end{equation}
Sets of the form \eqref{bwt} form the basis of the weak topology on $\mathfrak M(X)$. Our assumptions on $X$ imply
that the space $\mathfrak M(X)$ endowed with the weak topology is itself metrizable and separable; for instance,
the L\'evy-Prohorov metric or the Kantorovich-Rubinstein metric induce the weak topology on $X$ (see, e.g.,
\cite{Bogachev}, Section 8.3). The symbol $\Rightarrow$ will denote weak convergence in the space $\mathfrak
M(X)$.
It is clear that weak precompactness of a family of probability measures implies weak recurrence.
\begin{proposition}
\label{radbddprecomph} If a matrix $h\in H$ is radially bounded then the sequence
$\left\{\mu_n^h\right\}_{n\in\mathbb{N}}$ of orbital measures corresponding to $h$ is weakly precompact.
\end{proposition}
This Proposition is an immediate Corollary of Theorem 4.1 in Olshanski-Vershik \cite{OV}. Indeed, let $h\in H$ be
radially bounded, let
\[
h(n) = \Pi_{[1,n]}h=(h_{ij})_{i,j=1,\ldots,n},
\]
let
\[
\lambda_1^{(n)} \ge \ldots \ge \lambda_{k_n}^{(n)} \ge 0
\]
be the nonnegative eigenvalues of $h(n)$ arranged in decreasing order, and let
\[
\widetilde\lambda_1^{(n)} \le \widetilde\lambda_2^{(n)} \le \ldots \le \widetilde\lambda_{l_n}^{(n)} <0
\]
be the negative eigenvalues of $h(n)$ arranged in increasing order. Set
\[
x_i^{(n)} = \frac{\lambda_i^{(n)}}{n}, \qquad \widetilde x_i^{(n)} = \frac{\widetilde \lambda_i^{(n)}}{n};
\]
\[
\gamma_1^{(n)} = \frac{\tr h(n)}{n}, \qquad \gamma_2^{(n)} = \frac{\tr h^2(n)}{n^2}.
\]
Let $h$ is radially bounded, and let positive constants $C_1, C_2$ be such that for all $n\in {\mathbb N}$ we have
$$
{|\tr\left(\Pi_{[1,n]}h\right)|}\leq C_1 n, \ \
\tr\left(\Pi_{[1,n]}h\right)^2\leq C_2n^2.
$$
We clearly have
$$
|\gamma_1^{(n)}|\leq C_1, \ \ 0\leq \gamma_2^{(n)}\leq C_2,
$$
and, for all $i= 1, \dots, n$, we have
$$
|x_i^{(n)}|, \ |{\tilde x}_i^{(n)}| \leq C_2.
$$
Therefore, any infinite set of natural numbers contains a subsequence $n_r$ such that
sequences $\gamma_1^{(n_r)}, \gamma_2^{(n_r)}$, as well as the sequences $x_i^{(n_r)}, \widetilde x_i^{(n_r)}$ for
all $i=1,2,\ldots$ converge to a finite limit as $r \to \infty$. By the Olshanski-Vershik Theorem (Theorem 4.1 in
\cite{OV}), in this case the sequence $\mu_{n_r}^h$ of orbital measures weakly converges (in fact, to an ergodic
$U(\infty)$-invariant probability measure) as $r \to \infty$. The Proposition is proved completely.
{\bf Remark.} The converse claim (which, however, we do not need for our argument) also holds: if the sequence
of orbital measures for a matrix $h\in H$ is weakly pecompact, then the matrix $h$ is radially bounded.
This immediately follows from claim
(ii) of Theorem 4.1 of Olshanski and Vershik \cite{OV}. Note that, while claim (ii) in \cite{OV} is only
formulated for the full sequence of orbital measures, the same result, with the identical proof, is valid for any
infinite subsequence of orbital measures.
Observe that Theorem 4.1 in Olshanski-Vershik \cite{OV} as well as the Ergodic Decomposition Theorem of
Borodin-Olshanski \cite{BO} immediately imply the following
\begin{proposition}
\label{finradbdd} If $\nu$ is a finite Borel $U(\infty)$-invariant measure on $H$, then $\nu$-almost every $h\in
H$ is radially bounded.
\end{proposition}
\begin{proof}
Indeed, if $\nu$ is an ergodic probability measure, then the claim is part of the statement of the
Olshanski-Vershik Theorem: in this case, for $\nu$-almost all $h\in H$, the sequence of orbital measures $\mu_n^h$
weakly converges to $\nu$. For a general finite measure, the result follows from the Ergodic Decomposition Theorem
of Borodin and Olshanski \cite{BO}.
\end{proof}
To complete the proof of Theorem \ref{mainh}, it remains to establish
\begin{proposition}\label{radbdd}
If a $U(\infty)$-invariant measure $\nu$ belongs to the class $\classF$ for some $m\in\mathbb{N}$, then $\nu$-almost every $h\in H$ is radially bounded.
\end{proposition}
\subsection{Proof of Proposition \ref{radbdd}}
For a matrix $z\in\Mat$, $n\in {\mathbb N}$, denote
$$
\Pi_{[n,\infty)}z=(z_{ij})_{i,j=n, n+1, \dots}.
$$
We start by showing that, under the assumptions of the proposition,
for $\nu$-almost every $h\in H$ the matrix $\Pi_{[m, \infty)}h$ is radially bounded.
Take a measure $\nu \in \mathfrak F(m; H)$ and consider the corresponding canonical decomposition (\ref{ergdech})
into conditional measures.
\begin{proposition}
\label{uinv} Let $\nu \in \mathfrak F(m; H)$ be $U(\infty)$-invariant. Then for ${\overline \nu}$-almost every
$h^{(m)} \in H(m)$ the probability measure
\[
\textstyle \left( \Pi_{[m,\infty)} \right)_* \nu_{h^{(m)}}
\]
on $H$ is also $U(\infty)$-invariant.
\end{proposition}
\begin{proof}
Let $U_m(\infty) \subset U(\infty)$ be the subgroup of matrices $u = (u_{ij})$ satisfying the conditions:
\begin{enumerate}
\item if $\min (i,j) \le m, \; i\ne j$, then $u_{ij}=0$
\item if $i \le m$, then $u_{ii}=1$
\end{enumerate}
It follows from the definitions that if $u \in U_m(\infty)$, then $\Pi_{[m,\infty)}u \in U(\infty)$, and that the
map
\[
\Pi_{[m, \infty)}: \, U_m(\infty) \longrightarrow U(\infty)
\]
is a group isomorphism.
For $u \in U(\infty)$ let ${\mathfrak t}_u: \, H\to H$ be given by the formula
\[
\mathfrak t_u (h) \;=\; u^{-1}hu.
\]
Let $U'_m(\infty) \subset U_m(\infty)$ be a countable subgroup such that any Borel probability measure $\eta$ on
$H$ satisfying $ (\mathfrak t_u)_*\eta = \eta$ for all $u \in U'_m(\infty)$ must be invariant under the whole
group $U_m(\infty)$.
Uniqueness of Rohlin's system of conditional measures implies that for $\overline \nu$-almost every $h^{(m)} \in
H(m)$ and every $u \in U'_m(\infty)$ we have
\begin{equation} \label{uuinv}
\nu_{h^{(m)}} \;=\; (\mathfrak t_u)_*\nu_{h^{(m)}}.
\end{equation}
By definition of the subgroup $U'_m(\infty)$, the equality \eqref{uinv} also holds for all $u \in U_m(\infty)$.
Now let $A$ be a measurable subset of $H$, and let
\[
\textstyle \widetilde A_{h^{(m)}} \;=\; \left\{ h \in H: \, \Pi_{[1,m]} h=h^{(m)}, \, \Pi_{[m,\infty)}h \in A
\right\}.
\]
Let $u \in U(\infty)$ and let $\widetilde u \in U_m(\infty)$ be defined by the formula
\[
\textstyle \Pi_{[m,\infty)} \widetilde u \;=\; u.
\]
From the definitions it follows:
\[
\mathfrak t_{\widetilde u} (\widetilde A_{h^{(m)}}) \;=\; \textstyle \left\{ h \in H: \, \Pi_{[1,m]} h = h^{(m)},
\, \Pi_{[m,\infty)} h \in \mathfrak t_u(A). \right\}
\]
Since
\[
\nu_{h^{(m)}} ( \widetilde A_{h^{(m)}} ) \; = \; \nu_{h^{(m)}} ( \mathfrak t_{\widetilde u} (\widetilde
A_{h^{(m)}} ) ),
\]
we have
\[
\textstyle \left( \Pi_{[m, \infty)} \right)_* \nu_{h^{(m)}} (A) \;=\; \left( \Pi_{[m, \infty)} \right)_*
\nu_{h^{(m)}} \left( \mathfrak t_u (A) \right),
\]
and the proposition is proved.
\end{proof}
\begin{corollary}
\label{projradbdd} If $\nu \in \mathfrak F(m; H)$ is $U(\infty)$-invariant, then for $\nu$-almost every $h\in H$
the the matrix $\Pi_{[m,\infty)} (h)$ is radially bounded.
\end{corollary}
We proceed with the proof of Proposition \ref{radbdd}.
Let $\check u \in U(\infty)$ be defined as follows:
\begin{align}
\label{uhatdef}
\check u_{i,m+i} = \check u_{m+i,i} = 1 & \qquad i=1,\ldots, m\\
\check u_{2m+i,2m+i}=1 & \qquad i\in \mathbb N\\
\check u_{ij} = 0 & \qquad\text{otherwise}.
\end{align}
\begin{proposition}\label{radest}
Let $h \in H$. If $\Pi_{[m,\infty)}(h)$ and $\Pi_{[m,\infty)}(\check u^{-1}h\check u)$ are radially bounded, then
$h$ is also radially bounded.
\end{proposition}
\begin{proof}
If $\Pi_{[m,\infty)}(h)$ is radially bounded, then
$$
\sup_{n\in\mathbb{N}}\frac{|\tr\left(\Pi_{[1,n]}(\Pi_{[m,\infty)}h)\right)|}{n}<+\infty,
$$
and, since for $n>m$ we have
$$
\tr\left(\Pi_{[1,n]}(h)\right)=\tr\left(\Pi_{[1,n]}(\Pi_{[m,\infty)}h)\right)+\tr\left(\Pi_{[1,m]}(h)\right),
$$
it follows that
$$
\sup_{n\in\mathbb{N}}\frac{|\tr\left(\Pi_{[1,n]}h\right)|}{n}<+\infty.
$$
It remains to show that
$$
\sup_{n\in\mathbb{N}}\frac{\tr\left(\Pi_{[1,n]}h\right)^2}{n^2}<+\infty.
$$
Let $\pi$ be a permutation of $\mathbb N$ defined as follows:
\[
\pi(i) =
\begin{cases}
m+i &i=1,\ldots, m;\\
i-m &i=m+1, \ldots, 2m;\\
i &i>2m.
\end{cases}
\]
By definition, for any $h\in H$ we have
\[
(\check u^{-1}h\check u)_{ij} \;=\; \check h_{\pi(i)\pi(j)}.
\]
Consequently, for any $N \in \mathbb N$ we have
\[
\sum_{i,j=1}^N \left| h_{ij} \right|^2 \;\le\; \sum_{i,j=m+1}^N \left| h_{ij} \right|^2 + \sum_{i,j=m+1}^N \left|
(\check u^{-1}h\check u)_{ij} \right|^2 + \sum_{i,j=1}^{2m} \left| h_{ij} \right|^2.
\]
\end{proof}
Proposition \ref{radbdd} is now immediate from Corollary \ref{projradbdd} and Propositions \ref{uinv},
\ref{radest}.
Theorem \ref{mainh} is proved completely.
\section{Proof of Theorem \ref{mainz}.}
The proof is similar (and simpler) in this case.
Again, a matrix $z\in\Mat$ will be called \emph{radially bounded} if
$$
\sup_{n\in \mathbb{N}}\frac{\tr\left(\Pi_{[1,n]}z\right)^{*}\left(\Pi_{[1,n]}z\right)}{n^2}<+\infty
$$
(here, as usual, the symbol $z^*$ stands for the transpose conjugate of a matrix $z$). As before, we assign to a matrix
$z\in\Mat$ the sequence $\mu_n^z$ of orbital measures corresponding to the sequence of compact subgroups
$U(n)\times U(n)$, $n\in\mathbb{N}$,
and
say that a matrix $z\in\Mat$ is \emph{weakly recurrent} if for any bounded positive continuous function $f$ on
$\Mat$ we have
$$
\inf_{n\in\mathbb{N}}\int_{\Mat}f\,d\mu_n^z > 0
$$
Again
we have the following
\begin{proposition}
\label{radbddprecompz} If a matrix $z\in \Mat$ is radially bounded then the sequence of orbital measures
$\mu_n^z$ is weakly precompact. In particular, if $z$ is radially bounded, then $z$ is also weakly recurrent.
\end{proposition}
{\bf Remark.} As before, the converse statement also holds: if the sequence of orbital measures is weakly precompact,
then $z$ is radially bounded.
\textbf{Proof.} This, again, follows from Rabaoui's work \cite{rabaoui1}, \cite{rabaoui2}. Indeed, let $z\in\Mat$, let
$$
z(n)=\Pi_{[1,n]}z,
$$
let
$$\lambda_1^{(n)}\geqslant\dots\geqslant\lambda_n^{(n)}\geqslant 0$$
be the eigenvalues of the matrix $(z(n))^{*}z(n)$ arranged in decreasing order, and set
$$
x_i^{(n)}=\frac{\lambda_i^{(n)}}{n^2},
\ \ \gamma^{(n)}=\frac{\tr\left(z(n)^{*}z(n)\right)}{n^2}.
$$
If $z$ is radially bounded, then any infinite set of natural numbers contains a subsequence $n_r$ such that the
sequence $\gamma^{(n^r)}$ as well as all the sequences $x_i^{n_r}$, $i=1,\dots,$ converge (to a finite limit) as
$r\to\infty$. In this case, by Rabaoui's theorem \cite{rabaoui1}, \cite{rabaoui2}, the sequence of orbital measures $\mu_{n_r}^z$
weakly converges to a probability measure as $r\to\infty$; weak precompactness is thus established.
To conclude the proof of the Theorem, it therefore remains to establish the following
\begin{proposition}
\label{radbddz} Let $m\in\mathbb{N}$ and let $\nu\in\classFNC$. Then $\nu$-almost every $z\in\Mat$ is radially
bounded.
\end{proposition}
The proof follows the same pattern as that of Proposition \ref{radbdd}. Again, using Pickrell's classification of
ergodic probability measures as well as the ergodic decomposition theorem of \cite{Bufetov}, we have
\begin{proposition}
\label{radbddzfin} Let $\nu$ be a $U(\infty)\times U(\infty)$-invariant probability measure on $\Mat$. Then
$\nu$-almost every $z\in\Mat$ is radially bounded.
\end{proposition}
Given $\nu \in \mathfrak F(m, \Mat)$, we consider, again, the decomposition
\[
\nu \;=\; \int\limits_{{\rm Mat}(m, \mathbb C)} \nu_{z^{(m)}} \, d{\overline\nu} (z^{(m)}).
\]
Here ${\rm Mat}(m, \mathbb C)$ stands for the space of all $m\times m$-matrices with complex entries; the measure
$\overline\nu$ is the projection of $\nu$ onto ${\rm Mat}(m, \mathbb C)$ which is well-defined by definition of
the class $\mathfrak F(m, \Mat)$; and, for ${\overline \nu}$-almost every point $z^{(m)}\in {\rm Mat}(m, \mathbb
C)$ the measure $\nu_{z^{(m)}}$ is the canonical conditional probability measure given by Rohlin's Theorem. Again,
we have the following
\begin{proposition}
\label{zinv} If $\nu \in \mathfrak F(m, \Mat)$ is $U(\infty) \times U(\infty)$-invariant, then, for ${\overline
\nu}$-almost all $z^{(m)} \in {\rm Mat }(m, \mathbb C)$, the measure
\[
\textstyle \left( \Pi_{[m, \infty)} \right)_* \nu_{z^{(m)}}
\]
is also $U(\infty)\times U(\infty)$-invariant.
\end{proposition}
\begin{proof}
The proof of this Proposition is exactly the same as that of Proposition \ref{uinv}.
\end{proof}
It follows from Proposition \ref{zinv} that for $\nu$-almost every $z$, the matrix $\Pi_{[m, \infty)}z$ is radially bounded.
To obtain boundedness for the matrix $z$ itself, we again apply a permutation of rows and columns.
Denote
\[
\textstyle \tau_n (z) \;=\;\displaystyle \tr\left( \left( \Pi_{[1,n]}z \right)^* \Pi_{[1,n]}z
\right) \;=\; \sum_{i,j=1}^n \left| z_{ij} \right|^2.
\]
Let the matrix $\check u\in U(\infty)$ be defined by (\ref{uhatdef}).
The following clear inequality that holds for any $z \in \Mat$ and all $n>3m$:
\[
\textstyle \tau_n (z) \;\le\; \tau_{2m}(z)+
\tau_n \left(\Pi_{[m, \infty)}z\right) + \tau_n \left(\Pi_{[m, \infty)}(
{\check u}^{-1} z{\check u}\right).
\]
Consequently, if $\nu \in \mathfrak F(m, \Mat)$ is $U(\infty) \times U(\infty)$-invariant, then
$\nu$-almost
every $z \in \Mat$
is radially bounded, and
Theorem \ref{mainz} is proved completely.
|
2,869,038,156,330 | arxiv | \section{Introduction} \label{sec:intro}
Supervised machine learning algorithms rely on the ability to acquire an abundance of labeled data, or data with known labels (i.e., classifications). While unlabeled data---data {\it without} known labels---is ubiquitous in most applications of interest, obtaining labels for such training data can be costly.
Semi-supervised learning (SSL) methods leverage unlabeled data to achieve an accurate classification with significantly fewer training points. Simultaneously, the choice of training points can significantly affect classifier performance, especially due to the limited size of the training set of labeled data in the case of SSL.
Active learning seeks to judiciously select a limited number of {\it query points} from the unlabeled data that will inform the machine learning task at hand.
These points are then labeled by an expert, or human in the loop, with the aim of significantly improving the performance of the classifier.
While there are various paradigms for active learning \cite{settles_active_2012}, we focus on {\it pool-based} active learning wherein an unlabeled pool of data is available at each iteration of the active learning process from which query points may be selected. This paradigm is the natural fit for applying active learning in conjunction with semi-supervised learning since the unlabeled pool is also used by the the underlying semi-supervised learner. These query points are selected by optimizing an {\it acquisition function} over the discrete set of points available in the unlabeled pool of data. That is, if $\mathcal U \subset \mathcal X$ is the set of currently unlabeled points in a pool of data inputs $\mathcal X \subset \mathbb R^d$, then the active learning process at each iteration selects the next query point $x^\ast \in \mathcal U$ to be the minimizer of a real-valued acquisition function
\[
x^\ast = \argmin_{x \in \mathcal U} \ \mathcal A(x),
\]
where $\mathcal A$ can depend on the current state of labeled information (i.e., the labeled data $\mathcal L = \mathcal X - \mathcal U$ and corresponding labels for points in $\mathcal L$).
The above process (policy) for selecting query points is \textit{sequential} as only a single unlabeled point is chosen to be labeled at each iteration, as opposed to the \textit{batch} active learning paradigm. In batch active learning, a set of query points $\mathcal Q \subset \mathcal U$ is chosen at each iteration. While this is an important extension of the sequential paradigm and is an active area of current research in the literature \cite{sener_active_2018, gal_deep_2017, vahidian_coresets_2020, miller_model-change_2021}, we focus on the sequential case in this work.
Acquisition functions for active learning have been introduced for various machine learning models, especially support vector machines \cite{tong_support_2001, jiang_minimum-margin_2019, balcan_margin_2007}, deep neural networks\cite{gal_deep_2017, sener_active_2018, kushnir_diffusion-based_2020, shui_deep_2020, simeoni_rethinking_2021}, and graph-based classifiers \cite{zhu_combining_2003, ji_variance_2012, ma_sigma_2013, qiao_uncertainty_2019, miller_model-change_2021, murphy_unsupervised_2019}. We focus on graph-based classifiers for our underlying semi-supervised learning model due to their ability to capture clustering structure in data and their superior performance in the {\it low-label rate regime}---wherein the labeled data constitutes a very small fraction of the total amount of data.
Most active learning methods for deep learning assume a moderate to large amount of initially labeled data to start the active learning process. While there is exciting progress in improving the low-label rate performance of deep semi-supervised learning \cite{sohn2020fixmatch, sellars2022Laplacenet, zheng2022Simmatch} and few-shot learning \cite{zhang2022differentiable, he2022attribute}, we restrict the focus of this paper to well-established graph-based paradigms for this setting.
An important aspect in the application of active learning in real-world datasets is the inherent tradeoff between using active learning queries to either explore the given dataset or exploit the current classifier's inferred decision boundaries. This tradeoff is reminiscent of the similarly named ``exploration versus exploitation'' tradeoff in reinforcement learning. In active learning, it is important to thoroughly explore the dataset in the early stages, and exploit the classifier's information in later stages. Algorithms that exploit too quickly can fail to properly explore the dataset, potentially missing important information, while algorithms that fail to exploit the classifier in later stages can miss out on some of the main benefits of active learning.
In this work, we provide a simple, yet effective, acquisition function for use in graph-based active learning in the low-label rate regime that provides a natural transition between exploration and exploitation summarized in a single hyperparameter. We demonstrate through both numerical experiments and theoretical results that this acquisition function explores prior to exploitation. We prove theoretical guarantees on our method through analyzing the continuum limit partial differential eqauation (PDE) that serves as a proxy for the discrete, graph-based operator. This is a novel approach to providing sampling guarantees in graph-based active learning. We also provide experiments on a toy problem that illustrates our theoretical results, and the importance of the exploration versus exploitation hyperparameter in our method.
\subsection{Previous work} \label{subsec:prev-work}
The theoretical foundations in active learning have mainly focused on proving sample-efficiency results for linearly-separable datasets---oftentimes restricted to the unit sphere \cite{balcan2009agnostic, dasgupta_coarse_2006, hanneke_bound_2007}---for low-complexity function classes using disagreement or margin-based acquisition functions \cite{hanneke_theory_2014, hanneke_minimax_2015, balcan2009agnostic, balcan_margin_2007}. These provide convenient bounds on the number of active learning choices necessary for the associated classifier to achieve (near) perfect classification on these datasets with simple geometry.
In contrast, much of the focus for theoretical work in graph-based active learning leverages assumptions on the clustering structure of the data that is assumed to be captured in the graph structure \cite{murphy_unsupervised_2019, dasarathy_s2_2015}, which sometimes is assumed to be hierarchical \cite{dasgupta_hierarchical_2008, dasgupta_two_2011, cloninger_cautious_2021}. A central priority in this line of inquiry establishes guarantees that, given assumptions about the clustering structure of the observed dataset $\mathcal X$, the active learning method in question will query points from \textit{all} clusters (i.e., ensure exploration). The low-label rate regime of active learning---the focus of of this current work---is the natural setting for establishing such theoretical guarantees.
Laplacian Learning \cite{zhu_semi-supervised_2003} has been a common graph-based semi-supervised learning model for a number of graph-based active learning methods \cite{zhu_combining_2003, ji_variance_2012, ma_sigma_2013, jun_graph-based_2016}. However, little work has been done to provide theoretical guarantees for these methods, possibly due to the inherent difficulty in proving meaningful estimates on the solutions of discrete graph equations. Other important works in active learning have focused primarily on improving the performance of deep neural networks via active learning with either (1) moderate to large amounts of labeled data available to the classifier \cite{gal_deep_2017, zhu_robust_2019} or (2) coreset methods that are agnostic to the observed labels of the labeled data seen throughout the active learning process \cite{sener_active_2018, vahidian_coresets_2020}. Our current work is focused on the \emph{low-label rate regime}, which is an arguably more fitting regime for semi-supervised and active learning. Furthermore, in contrast to coreset methods, our acquisition function directly depends on the observed classes of the labeled data.
Graph neural networks (GNN) \cite{welling2016semi, zhou2018graph} are an important area of graph-based methods for machine learning, and various methods for active learning have been proposed \cite{hu2020policy, ijcai2018p296, cai2017active, Zhang_Tong_Xia_Zhu_Chi_Ying_2022}. GNNs consider network graphs whose connectivity is a priori determined via metadata relevant to the task (e.g., co-authorship in citation networks) and then use the node-level features to learn representations and transformations of features for the learning task. In contrast, we consider similarity graphs where the connectivity structure is determined only by the node-level features and directly learn a node function on this graph structure.
Continuum limit analysis of graph-based methods has been an active area of research for providing rigorous analysis of graph-based learning \cite{calder_consistency_2019, calder2022improved, calder_poisson_2020, calder2020properly, calder2018game, slepcev2019analysis,dunlop2020large}. In this analysis, a discrete graph is viewed as a random geometric graph that is sampled from a density $\rho: \mathbb R^d \rightarrow \mathbb R_+$ defined in a high-dimensional space (possibly constrained to a manifold $\mathcal M \subset \mathbb R^d$ therein). The graph Laplacian matrix can be analyzed via its continuum-limit counterpart, which is a second order density weighted diffusion operator (or a weighted Laplace-Beltrami operator on the manifold). An important development relevant to the current work is the Properly Weighted Graph Laplacian \cite{calder2020properly}, which reweights the graph in the Laplacian learning model of \cite{zhu_semi-supervised_2003} to correct for the degenerate behavior of Laplacian learning in the extremely low-label rate regime. This provides the setting for a well-defined, properly scaled graph-based semi-supervised learning model that we use in our current work to provide rigorous bounds on the acquisition function values to control the exploration versus exploitation tradeoff.
In order to apply active learning in practice, it is essential to design computationally efficient acquisition functions.
Much of the current literature has sought to design more sophisticated methods that often have higher computational complexity (e.g., requiring the full inversion of the graph Laplacian matrix).
Uncertainty sampling \cite{settles_active_2012} is an example of a computationally efficient acquisition function since it only requires the output of the classifier on the unlabeled data. However, uncertainty sampling methods will often mainly select query points that concentrate along decision boundaries while ignoring large regions of the dataset that are distant from any labeled points. Phrased in the terminology of the exploration versus exploitation tradeoff in reinforcement learning, uncertainty sampling is often overly ``exploitative'' and often achieves poor overall accuracy in empirical experiments \cite{ji_variance_2012}.
In contrast, methods such as variance optimization (VOpt) \cite{ji_variance_2012}, $\Sigma$-Opt \cite{ma_sigma_2013}, Coresets \cite{sener_active_2018}, LAND \cite{murphy_unsupervised_2019}, and CAL \cite{cloninger_cautious_2021} could be characterized as primarily ``explorative'' methods. Oftentimes, however, such explorative methods, or other methods that are designed to both explore and exploit \cite{miller_model-change_2021, karzand_maximin_2020, zhu_combining_2003, gal_deep_2017} are more expensive to compute than simply using uncertainty sampling. For example, VOpt \cite{ji_variance_2012} and $\Sigma$-Opt \cite{ma_sigma_2013} require the computation and storage of a dense $N \times N$ covariance matrix that must be updated each active learning iteration. The work of \cite{miller_model-change_2021} proposed a computationally efficient adaptation of these methods via a projection onto a subset of the graph Laplacian's eigenvalues and eigenvectors.As a consequence of sometimes significantly poor performance from this spectral truncation method in our experiments, we provide a ``full'' computation of V-Opt and $\Sigma$-Opt in certain experiments by restricting the computation to only a subset of unlabeled data which allows us to bypass the need to invert the graph Laplacian matrix (Section \ref{sec:larger-datasets}). This heuristic, however, is still very expensive to compute at each active learning iteration making it not a viable option for moderate to large datasets in practice.
In this work, we show that uncertainty sampling, \textit{when properly designed for the graph-based semi-supervised learning model} can both explore and exploit, and outperforms existing methods in terms of computational complexity, overall accuracy, and exploration rates.
\subsection{Overview of paper} \label{sec:overview-contents}
The rest of the paper continues as follows. We begin in Section \ref{sec:model-setup} with a description of the Properly Weighted Laplacian Learning model from \cite{calder2020properly} that will be the underlying graph-based semi-supervised learning model for our proposed active learning method. We also introduce the minimum norm acquisition function in this section, along with other useful preliminaries for the rest of the paper. In Section \ref{sec:results}, we begin with illustrative experiments in two-dimensions to illustrate the delicate balance between exploration and exploitation in graph-based active learning.
Section \ref{sec:larger-datasets} compares our proposed active learning method to other acquisition functions on larger, more ``real-world'' datasets that have been adapted to provide an experimental setup wherein exploration is essential for success in the active learning task. Thereafter, we present theoretical guarantees for the minimum norm acquisition function in the continuum limit setting in Section \ref{sec:theory}, along with an extended look at the theory in one dimension in Section \ref{sec:1d-theory}.
\subsection{Notation} \label{subsec:notation}
Let $\|\cdot\|_2$ denote the standard Euclidean norm where the space is inferred from the input. We let $|\cdot|$ denote either the absolute value of a scalar in $\mathbb R$ or the cardinality of a set, where from context the intended usage should be clear. We denote the set of points $x \in \mathcal X$ with $x \not\in \mathcal U$ as $\mathcal X \setminus \mathcal U$.
\section{Model setup and acquisition function introduction} \label{sec:model-setup}
Let $\mathcal X = \{x_1, x_2, \ldots, x_N\} \subset \mathbb R^d$ be a set of inputs for which we assume each $x \in \mathcal X$ belongs to one of $C$ classes. Suppose that we have access to a subset $\mathcal L \subset \mathcal X$ of labeled inputs (\textit{labeled data}) for which we have observed the ground-truth classification $y(x) \in \{1, \ldots, C\}$ for each $x \in \mathcal L$.
The rest of the inputs, $\mathcal U := \mathcal X \setminus \mathcal L$, are termed the \textit{unlabeled data} as no explicit observation of the underlying classification have been seen for $x \in \mathcal U$. The semi-supervised learning task is to use both $\mathcal L$ and $\mathcal U$, with the associated labels $\{y(x)\}_{x \in \mathcal L}$, to infer the classification of the points in $\mathcal U$.
Sequential active learning extends semi-supervised learning by selecting a sequence of \textit{query points} $x_1^\ast, x_2^\ast, \ldots$ as part of an iterative process that alternates between (1) calculating the semi-supervised classifier given the current labeled information and (2) selecting and subsequently labeling an unlabeled query point $x_n^\ast \in \mathcal U_n$, where $\mathcal U_n = \mathcal X - \mathcal L_n = \mathcal X - (\mathcal L \cup \{x_1^\ast, x_2^\ast, \ldots, x_{n-1}^\ast\})$. Labeling a query point $x_i^\ast$ consists of obtaining the corresponding label $y(x_n^\ast)$ and then adding $x_n^\ast$ to the set of labeled data from the current iteration, $\mathcal L_{n} = \mathcal L_{n-1} \cup \{x_n^\ast\}$. To avoid this cumbersome notation, however, we will drop the explicit dependence of $\mathcal U_i, \mathcal L_i$ on the iteration $i$ and simply refer to the unlabeled and labeled data at the \textit{current} iteration as respectively $\mathcal U$ and $\mathcal L$.
Returning to the underlying semi-supervised learning problem, graph Laplacians have often been used to propagate labeled information from $\mathcal L$ to $\mathcal U$ \cite{zhu_semi-supervised_2003, bertozzi_diffuse_2016, calder_poisson_2020, calder2020properly, shi2017weighted, bertozzi2019graph, calder2018game,welling2016semi}. From the set of feature vectors $\mathcal X$, consider a similarity graph $G(\mathcal X, W)$ with weight matrix $w_{ij} = \kappa(x_i,x_j)$ that captures the similarity between inputs $x_i,x_j$ for each pair of points in $\mathcal X$. We use $\mathcal X$ to denote both the set of feature vectors as well as the node set for the graph $G$ to avoid introducing more notation. Laplacian learning \cite{zhu_semi-supervised_2003} is an important graph-based semi-supervised learning model for both this current work and many previous graph-based active learning works, and solves the constrained problem of identifying a graph function $u :\mathcal X \rightarrow \mathbb R^C$ via the minimization of
\begin{align}\label{eq:lap-learning}
\min_{u: \mathcal X \rightarrow \mathbb R^d}\ &\sum_{x_i,x_j \in \mathcal X} w_{ij} \|u(x_i) - u(x_j)\|_2^2 \\
\text{subject to }& u(x) = e_{y(x)} \text{ for } x \in \mathcal L. \nonumber
\end{align}
The vector $e_{y(x)} \in \mathbb R^C$ is the standard Euclidean basis vector in $\mathbb R^C$ whose entries are all $0$ except the entry corresponding to the label $y(x)\in \{1, \ldots, C\}$.
The learned function $u$ that minimizes \eqref{eq:lap-learning} constitutes a harmonic extension of the given labels in $\mathcal L$ to the unlabeled data since $u$ is a harmonic function on the graph. For the classification task, the inferred classification of $x \in \mathcal U$ is then obtained by thresholding on the learned function's output at $x$, $u(x) \in \mathbb R^C$. That is, the inferred classification $\hat{y}(x)$ for $x \in \mathcal U$ is given by
\[
\hat{y}(x) = \argmax_{c=1,2, \ldots, C} \ u_c(x),
\]
where $u_c(x)$ denotes the $c^{th}$ entry of $u(x)$.
Various previous works \cite{calder_rates_2020,calder2020properly, shi2017weighted, nadler2009infiniteunlabelled,flores2022analysis,calder_poisson_2020} have shown that when the amount of labeled information is small compared to the size of the graph (i.e., the \textit{low-label rate regime}), the performance of minimizers of \eqref{eq:lap-learning} degrades substantially. The solution $u$ becomes roughly constant with sharp spikes near the labeled set, and the classification tends to predict the same label for most datapoints. Of particular interest to the current work is the Properly Weighted Laplacian learning work in \cite{calder2020properly}, wherein a weighting $\gamma: \mathcal X \rightarrow \mathbb R_+$ that scales like $\operatorname{dist}(x, \mathcal L)^{-\alpha}$ for $\alpha > d-2$ is used to reweight the edges in the graph to correct the singular behavior of solutions to \eqref{eq:lap-learning}. We use an improvement to the Properly Weighted Laplacian that is called Poisson ReWeighted Laplace Learning (PWLL) and will be described in detail in another paper \cite{calder2022poisson}. PWLL performs semi-supervised learning by solving the problem
\begin{align}\label{eq:rw-lap-learning}
\min_{u: \mathcal X \rightarrow \mathbb R^d}\ &\sum_{x_i,x_j \in \mathcal X} \gamma(x_i) \gamma(x_j)w_{ij} \|u(x_i) - u(x_j)\|_2^2 \\
\text{subject to }& u(x) = e_{y(x)} \text{ for } x \in \mathcal L, \nonumber
\end{align}
where the reweighting function $\gamma$ is computed by solving the graph Poisson equation
\begin{equation}\label{eq:gamma_eq_discrete}
\sum_{x_j \in \mathcal X} w_{ij}(\gamma(x_i) - \gamma(x_j)) = \sum_{x_k \in \mathcal L}\left( \delta_{ik} - \tfrac{1}{N}\right) \ \ \ \text{for all } x_i\in \mathcal X.
\end{equation}
In the previous work on the Properly Weighted Laplacian \cite{calder2020properly}, the weight $\gamma$ was explicitly chosen to satisfy $\gamma(x)\sim\operatorname{dist}(x, \mathcal L)^{-\alpha}$, while in the PWLL, $\gamma$ is learned from the data, making the method more adaptive with fewer hyperparameters. The motivation for the Poisson equation \eqref{eq:gamma_eq_discrete} is that the continuum version of this equation is related to the fundamental solution of Laplace's equation, which produces the correct scaling in $\gamma$ near the labeled set.
The reason for using PWLL is that minimizers of \eqref{eq:rw-lap-learning} have a well-defined continuum limit in the case when the amount of labeled data is fixed and the number of nodes $|\mathcal X| = N \rightarrow \infty$. This will allow us to analyze the behavior of our proposed minimum norm acquisition function applied to the PWLL model in the continuum limit setting in Section \ref{sec:theory}.
\subsection{Solution decay parameter} \label{subsec:tau-decay}
We introduce an adaptation of \eqref{eq:rw-lap-learning} that increases the decay rate of the corresponding solutions away from labeled points via a type of Tikhonoff regularization in the variational problem. Controlling this decay will prove to be crucial for ensuring that query points selected via our minimum norm acquisition function (Section \ref{subsec:min-norm-af}) will explore the extent of the dataset prior to exploiting current classifier decision boundaries. Given $\tau \ge 0$, we consider solutions of the following variational problem
\begin{align}\label{eq:rw-lap-learning-tau}
\min_{u: \mathcal X \rightarrow \mathbb R^d}\ &\sum_{x_i,x_j \in \mathcal X} \gamma(x_i) \gamma(x_j)w_{ij} \|u(x_i) - u(x_j)\|_2^2 \ \ +\ \ \tau \sum_{x_i \in \mathcal U} \|u(x_i)\|_2^2\\
\text{subject to }& u(x) = e_{y(x)} \text{ for } x \in \mathcal L. \nonumber
\end{align}
It is straightforward to see that for $\tau > 0$ the additional term in \eqref{eq:rw-lap-learning-tau} encourages the solution $u$ to have \textit{smaller} values away from the labeled data, where the values are fixed. When $\tau = 0$, we recover \eqref{eq:rw-lap-learning}. We will refer to this graph-based semi-supervised learning model as Poisson ReWeighted Laplace Learning with $\tau$-Regularization (PWLL-$\tau$).
To illustrate the role of the decay parameter, let us consider a simple one dimensional version of this problem in the continuum of the form
\[\min_{u} \int_{a}^b u'(x)^2 + \tau u(x)^2\, dx,\]
where $[a,b]$ is the domain and the minimization would be restricted by some boundary conditions on $u$ (i.e., on the labeled set). Minimizers of this problem satisfy the ordinary differential equation (i.e., the Euler-Lagrange equation) $\tau u - u'' = 0$, which has two linearly independent solutions $e^{\pm \sqrt{\tau}x}$. Since the solution we are interested in is bounded, the exponentially growing one can be discarded, and we are left with exponential decay in the solutions with rate $ \sqrt{\tau}$ away from the labeled set. Thus, at least in this simple example, we can see how the introduction of the diagonal perturbation $\tau$ in PWLL leads to exponential decay of solutions, which is essential for the method to properly \emph{explore} the dataset. We postpone developing this theory further until Section \ref{sec:theory}.
\subsection{Minimum norm acquisition function} \label{subsec:min-norm-af}
We now introduce the acquisition function that we propose to properly balance exploration and exploitation in graph-based active learning in the PWLL-$\tau$ model. We simply use the Euclidean norm of the output vector at each unlabeled point, $x \in \mathcal U$:
\begin{equation}\label{eq:min-norm-af}
\mathcal A(x) = \|u(x)\|_2 = \sqrt{u_1^2(x) + u_2^2(x) + \ldots + u_C^2(x)}.
\end{equation}
Due to the solution decay resulting from the $\tau$-regularization term in \eqref{eq:rw-lap-learning-tau}, unlabeled points that are far from all labeled points will have small Euclidean norm for their corresponding output vector. In the low-label rate regime, this property encourages query points selected by \eqref{eq:min-norm-af} to be spread out over the extent of the dataset, until a sufficient number of points have been labeled to ``cover'' the dataset. After this has been achieved in the active learning process, the learned functions for \eqref{eq:rw-lap-learning-tau} will have smaller norms in regions between labeled points of differing classes due to the rapid decay in solutions near the transition between classes. This described behavior reflects the desired properties for balancing exploration prior to exploitation in active learning. Through both numerical experiments and theoretical results, we demonstrate this acquisition function's utility for this purpose.
The acquisition function \eqref{eq:min-norm-af} is a novel type of uncertainty sampling \cite{settles_active_2012}, wherein only the values of the learned function $u$ at each active learning iteration are used to determine the selection of query points. Indeed, one may interpret the small Euclidean norm of the learned function at an unlabeled node, $\|u(x)\|_2$, to reflect uncertainty about the resulting inferred classification, $\hat{y}(x)$. Other uncertainty sampling methods, such as \textit{smallest margin sampling} \cite{settles_active_2012}, also compute the uncertainty of the learned model at an unlabeled point via properties of the output vector $u(x) \in \mathbb R^C$. However, these criterion often either (1) only compare 2 entries of the vector to compute a measure of margin uncertainty or (2) normalize the output vector to lie on the simplex to be interpreted as class probabilities. In both cases, these measures of uncertainty in the classification of unlabeled points in unexplored regions of the dataset might not be as emphasized by the acquisition function compared to points that lie near the decision boundaries of the learned classifier. In other words, most previous uncertainty sampling methods can often be characterized as solely exploitative and lack explorative behavior, with the results being decreased overall performance of the classifier on the dataset. Our minimum norm acquisition function \eqref{eq:min-norm-af}, however, is designed to prioritize the selection of query points in unexplored regions of the dataset which is properly reflected in the decay of the learned functions in the PWLL-$\tau$ model \eqref{eq:rw-lap-learning-tau}. In this sense, we are able to ensure exploration prior to exploitation in the active learning process using the minimum norm acquisition function \eqref{eq:min-norm-af} in the PWLL-$\tau$ model.
\begin{remark}[Decay Schedule for $\tau$]
As we demonstrate through some toy experiments in Section \ref{sec:toy-experiments}, there is benefit to decreasing the value of $\tau \ge 0$ as the active learning process progresses in order to more effectively transition from explorative to exploitative queries. While there are various ways to design this, we simply identify a constant $\mu \in (0,1)$ so that the decreasing sequence of hyperparameter values $\tau_{n+1} = \mu \tau_n$ with initial value $\tau_0 > 0$ satisfies that $\tau_{2K} \le \epsilon$, where $\epsilon$ is chosen to be $\epsilon = 10^{-9}$. For our experiments, we set $K$ to be the number of clusters, which in the case of our tests is known a priori. In practice, this choice of $K$ would be a user-defined choice to control the ``aggressiveness'' of the decay schedule of $\tau$. For $n \ge 2K$, we set $\tau_n = 0$. Thus, we calculate
\[
\mu = \left( \frac{\epsilon }{\tau_0}\right)^{\frac{1}{2K}} \in (0,1)
\]
which ensures a decaying sequence of $\tau$ values as desired. We note that an interesting line of inquiry for future research would be to investigate a more rigorous understanding of how to adaptively select $\tau \ge 0$ during the active learning process. We leave this question for future research and simply use the proposed decay schedule above.
\end{remark}
In Table \ref{table:unc-sampling}, we introduce the abbreviations for and other useful information pertaining to the uncertainty sampling acquisition functions that we will consider in the current work---smallest margin, minimum norm, and minimum norm with $\tau$-decay uncertainty sampling.
\newcommand{\ra}[1]{\renewcommand{\arraystretch}{#1}}
\begin{table*}[h!]\centering \label{table:unc-sampling}
\ra{1.3}
\begin{scriptsize}
\begin{tabular}{@{}lccc@{}}\toprule
\textbf{Full Name} & \textbf{Abbreviation} & $\mathcal A(x)$ & \textbf{Underlying Classifier} \\
\midrule
Smallest Margin Unc.~Sampling & Unc.~(SM) & $u_{c_1^\ast}(x) - u_{c_2^\ast}(x)$ & PWLL \\
Minimum Norm Unc.~Sampling & Unc.~(Norm) & $\|u(x)\|_2$ & PWLL-$\tau$, fixed $\tau > 0$ \\
\multirow{2}{15em}{Minimum Norm Unc.~Sampling with $\tau$-decay} & Unc.~(Norm, $\tau \rightarrow 0$) & $\|u(x)\|_2$ & PWLL-$\tau$, decay $\tau \rightarrow 0$ \\
& & & \\
\bottomrule
\end{tabular}
\end{scriptsize}
\caption{Description of uncertainty sampling acquisition functions that will be compared throughout the experiments in the following sections. Unc.~(SM) considers the difference between the largest and second largest entries of the output vector $u(x)$, denoted by $c_1^\ast$ and $c_2^\ast$ respectively.}
\end{table*}
\section{Results}\label{sec:results}
In this section, we present numerical examples to demonstrate our claim that our proposed Unc.~(Norm) and Unc.~(Norm $\tau \rightarrow 0$) acquisition functions in the PWLL-$\tau$ model \eqref{eq:rw-lap-learning-tau} are effective at both exploration and exploitation. We begin in Section \ref{sec:toy-experiments} with a set of toy examples in 2-dimensions to facilitate visualizing the choices of query points during the active learning process and highlight the efficacy of implementing the $\tau$-decay in Unc.~(Norm, $\tau \rightarrow 0$) for balancing exploration and exploitation. In Section \ref{sec:isolet-results}, we recreate an experiment from \cite{ji_variance_2012} on the Isolet dataset \cite{uci} that highlighted the superior performance of the VOpt criterion compared to Unc.~(SM). We demonstrate that our proposed uncertainty sampling method Unc.~(Norm, $\tau \rightarrow 0$) achieves results comparable to VOpt on this dataset, essentially correcting the behavior of uncertainty sampling in this empirical study.
In Section \ref{sec:larger-datasets}, we perform active learning experiments on larger, more ``real-world'' datasets. We use the \textbf{MNIST} \cite{lecun-mnisthandwrittendigit-2010}, \textbf{FASHIONMNIST} \cite{xiao2017fashionmnist}, and \textbf{EMNIST} \cite{cohen2017emnist} datasets, and we interpret the original ground-truth classes (e.g. digits 0-9 in \textbf{MNIST}) as \textit{clusters} on which we impose a different classification structure by grouping many clusters into a single class. This creates an experimental setting that necessitates exploration of initially unlabeled ``clusters'' in order to achieve high overall accuracy. We include similar experiments in Section \ref{smsec:imbalanced-results} of the Supplemental Material to verify the performance of the proposed method in the presence of disparate class and cluster sizes.
While most previous work in the active learning literature (both graph-based and neural network classifiers) demonstrate acquisition function performance with only accuracy plots, we suggest another useful quantity for comparing performances. In the larger experiments of Sections \ref{sec:larger-datasets} and \ref{smsec:imbalanced-results}, we plot \textit{the proportion of clusters that have been queried} as a function of active learning iteration. These plots reflect how efficiently an acquisition function explores the clustering structure of the dataset, as captured by how quickly the proportionality curve increases toward 1.0. These cluster exploration plots are especially insightful for assessing performance in low label-rate active learning. An acquisition function that properly and consistently explores the clustering structure of the dataset will achieve an average cluster proportion of 1.0 faster than other acquisition functions and within a reasonable number of active learning queries.
\subsection{Comment regarding comparison to other methods}
We comment here on a few notable active learning methods that are left out of our numerical comparisons. The LAND (Learning by Active Non-linear Diffusion) \cite{murphy_unsupervised_2019} and CAL (Cautious Active Learning) \cite{cloninger_cautious_2021} methods are important works in geometry-inspired active learning. In the LAND algorithm, Murphy and Maggioni use diffusion distances from a random walk interpretation of a similarity graph to select diverse sets of query points that are located in dense regions of the graph. Adjusting a model hyperparameter in the diffusion distances can reveal hierarchical clustering structure in the dataset which can encourage query points to be chosen at different resolution levels of the clustering structure.
In a similar vein, the CAL algorithm by Cloninger and Mhaskar \cite{cloninger_cautious_2021} uses hierarchical clustering structure in the dataset to guide the query set selection process. By constructing a highly localized similarity kernel via Hermite polynomials, query points are selected at various resolution levels. Both the LAND and CAL algorithms have been shown to be effective at selecting query points in pixel classification for hyperspectral imagery applications. We, however, found that the current implementations of these algorithms were unable to scale to our larger experiments.
Furthermore, we suggest that these methods may be more appropriately identified as ``coreset'' selection methods. Such methods leverage the geometry of the underlying dataset (e.g., the diffusion distances as captured by the similarity graph in LAND), but not the set of labels observed at labeled points during the active learning process. This is similar to other coreset methods that have been presented in both coreset and data summarization literature \cite{sener_active_2018, vahidian_coresets_2020, mirzasoleiman2017big}. In contrast, our uncertainty-based criterion in this work combines both geometric information about the data as captured by the similarity graph structure and the observed labels at each labeled point via the output classification at each iteration. This makes our method more similar to the primary flavor of active learning methods. For these two reasons, we omit direct numerical comparison with these other methods.
\subsection{Toy examples} \label{sec:toy-experiments}
We first illustrate our claim regarding our minimum norm uncertainty sampling criterion for graph-based active learning with synthetic datasets that are directly visualizable (i.e., the data lies in only two dimensions). The first experiment---which we refer to as the \textbf{Blobs} experiment---illustrates how a non-zero value for $\tau$ in the initial phase of active learning is crucial for ensuring exploration of the dataset. The second experiment---which we refer to as the \textbf{Box} experiment---illustrates the need to decrease the value of $\tau$ to ensure the transition from exploration to exploitation. These experiments also allow us to directly observe the qualitative characteristics of the active learning query choices in uncertainty sampling.
\subsubsection{Blobs experiment} \label{sec:blobs-experiment}
\begin{figure}
\centering
\subfigure[Ground Truth]{\includegraphics[clip=True,trim=120 60 120 60,width=0.4\textwidth]{imgs/toy_figures/blobs/groundtruth.jpg}}
\subfigure[Accuracy Results]{\includegraphics[width=0.5\textwidth]{imgs/toy_figures/blobs_rwll_accplot.jpg}}
\caption{Ground Truth (a) and Accuracy Results (b) for \textbf{Blobs} experiment. Notice that Unc.~(SM) achieves very poor overall accuracy. We show in Figure \ref{fig:unc-blobs-combined} that this is due to premature exploitation.}
\label{fig:gt-blobs}
\end{figure}
The \textbf{Blobs} dataset is comprised of eight Gaussian clusters, each of equivalent size (300) and variance ($\sigma^2 = 0.17^2$), whose centers (i.e., means) lie evenly spaced apart on the unit circle. That is, each cluster $\Omega_i$ is defined by randomly sampling 300 points from a Gaussian with mean $\mu_i = (\cos(\pi i /4), \sin(\pi i/4))^T \in \mathbb R^2$ and standard deviation $\sigma_i= \sigma = 0.17$. The classification structure of the clusters is then assigned in an alternating fashion, as shown in Figure \ref{fig:gt-blobs}(a). In each individual run of the experiment, one initially labeled point per \textit{class} combine to be the starting labeled set, and then 100 active learning query points are selected sequentially via a specified acquisition function. Different acquisition functions then define different runs of the experiment.
For each acquisition function, we ran 10 experiments with different initially labeled points. The average accuracy at each iteration of an experiment is plotted is Figure \ref{fig:gt-blobs}(b). The main purpose of this experiment is to compare and contrast the characteristics of the query points selected by Unc.~(SM), Unc.~(Norm), and Unc.~(Norm, $\tau \rightarrow 0$). For comparison and reference in these toy experiments, we include the results of using the VOpt\cite{ji_variance_2012} acquisition function as well as Random sampling (i.e., select $x_i^\ast \in \mathcal U$ with uniform probability over $\mathcal U$ at each iteration).
\begin{figure}
\centering
\subfigure[Unc.~(SM), Initial]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/unc_rwll_afvals_0.jpg}}
\subfigure[Unc.~(SM), Iter 9]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/unc_rwll_afvals_8.jpg}}
\subfigure[Unc.~(SM), Iter 100]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/unc_rwll_afvals_99.jpg}} \\
\subfigure[Unc.~(Norm), Initial]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/uncnorm_rwll0010_afvals_0.jpg}}
\subfigure[Unc.~(Norm), Iter 9]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/uncnorm_rwll0010_afvals_8.jpg}}
\subfigure[Unc.~(Norm), Iter 100]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/uncnorm_rwll0010_afvals_99.jpg}} \\
\subfigure[Unc.~(Norm, $\tau \rightarrow 0$), Initial]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/uncnormdecaytau_rwll0010_afvals_0.jpg}}
\subfigure[Unc.~(Norm, $\tau \rightarrow 0$), Iter 9]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/uncnormdecaytau_rwll0010_afvals_8.jpg}}
\subfigure[Unc.~(Norm, $\tau \rightarrow 0$), Iter 100]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/uncnormdecaytau_rwll0010_afvals_99.jpg}}
\caption{Acquisition Function Values for Unc.~(SM), Unc.~(Norm), and Unc.~(Norm, $\tau \rightarrow 0$) at different stages of the \textbf{Blobs} experiment. Labeled points are marked as red stars and brighter regions of the heatmap indicate higher acquisition function values.}
\label{fig:unc-blobs-combined}
\end{figure}
The main observation from this experiment is how poorly Unc.~(SM) performs, as it only attains an overall accuracy of roughly 62\% as the average over the trials. In Figure \ref{fig:unc-blobs-combined}(a-c), we show one trial's acquisition function values heatmap at three different stages of the active learning process using Unc.~(SM). We observe that the active learning queries have been primarily focused on the boundaries between a few clusters, while missing other clusters completely. At each iteration, the heatmap of acquisition function values has only focused on the current classifier's decision boundary which can lead to missing such clusters. In essence, we would qualify the behavior here as ``premature exploitation'', prior to proper exploration of the dataset.
In contrast, Figures \ref{fig:unc-blobs-combined} (d-i) demonstrate how the ``minimum norm'' uncertainty acquisition functions properly explore the extent of the geometric clustering structure. Both have sampled from every cluster in the ring. It is instructive to further see though that Unc.~(Norm)---which employs a fixed value of $\tau > 0$ at every iteration---has not sampled more frequently \textit{between} clusters by the end of the trial. We may characterize this behavior as not transitioning to proper exploitation of cluster boundaries. On the other hand, in Figure \ref{fig:unc-blobs-combined}(i), we see that by using this minimum norm uncertainty sampling \textit{with decaying values of $\tau \rightarrow 0$} we more frequently sample at the proper cluster boundaries after having sampled from each cluster.
\subsubsection{Box experiment} \label{sec:box-experiment}
\begin{figure}
\centering
\subfigure[Ground Truth]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/groundtruth.jpg}}
\hspace{8mm}
\subfigure[Accuracy Results]{\includegraphics[width=0.5\textwidth]{imgs/toy_figures/box_rwll_accplot.jpg}}
\caption{Ground Truth (a) and Accuracy Results (b) for \textbf{Box} experiment. Notice that Unc.~(Norm) achieves suboptimal overall accuracy. We show in Figure \ref{fig:unc-box-combined}(f) that the distribution of query points later in the active learning process reflect a lack of transition to exploitation.}
\label{fig:gt-box}
\end{figure}
The \textbf{Box} dataset is simply a 65 $\times$ 65 lattice of points on the unit square, with removing points that lie within a thin, vertical band centered at $x = 0.3$ which also defines the class boundary line (Figure \ref{fig:gt-box}). In contrast to the \textbf{Blobs} experiment, the \textbf{Box} experiment illustrates the need to transition from exploration to exploitation, and how this is accomplished by decreasing $\tau \rightarrow 0$. In the accuracy plot (Figure \ref{fig:gt-box}(b)), notice how the accuracy achieved by Unc.~(Norm) levels off at a \textit{lower} overall accuracy than both Unc.~(SM) and Unc.~(Norm $\tau \rightarrow 0$). Figure \ref{fig:unc-box-combined} demonstrates that this is due to ``over exploration'' of the dataset instead of transitioning to refining the decision boundary between classes. Active learning seeks to balance exploration versus exploitation while still being sample efficient, making as few active learning queries as possible.
\begin{figure}
\centering
\subfigure[Unc.~(SM), Initial]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/unc_rwll_afvals_0.jpg}}
\hspace{2mm}
\subfigure[Unc.~(SM), Iter 15]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/unc_rwll_afvals_15.jpg}}
\hspace{2mm}
\subfigure[Unc.~(SM), Iter 50]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/unc_rwll_afvals_50.jpg}} \\
\subfigure[Unc.~(Norm), Initial]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/uncnorm_rwll0010_afvals_0.jpg}}
\hspace{2mm}
\subfigure[Unc.~(Norm), Iter 15]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/uncnorm_rwll0010_afvals_15.jpg}}
\hspace{2mm}
\subfigure[Unc.~(Norm), Iter 50]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/uncnorm_rwll0010_afvals_50.jpg}} \\
\subfigure[Unc.~(Norm, $\tau \rightarrow 0$), Initial]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/uncnormdecaytau_rwll0010_afvals_0.jpg}}
\hspace{2mm}
\subfigure[Unc.~(Norm, $\tau \rightarrow 0$), Iter 15]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/uncnormdecaytau_rwll0010_afvals_15.jpg}}
\hspace{2mm}
\subfigure[Unc.~(Norm, $\tau \rightarrow 0$), Iter 50]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/uncnormdecaytau_rwll0010_afvals_50.jpg}}
\caption{Acquisition Function Values for Unc.~(SM), Unc.~(Norm), and Unc.~(Norm, $\tau \rightarrow 0$) at different stages of the \textbf{Box} experiment. Labeled points are marked as red stars and brighter regions of the heatmap indicate higher acquisition function values.}
\label{fig:unc-box-combined}
\end{figure}
As shown in Figures \ref{fig:unc-box-combined} (a-f), both Unc.~(SM) and Unc.~(Norm, $\tau \rightarrow 0$) more efficiently sample the decision boundary between the two classes in this \textbf{Box} dataset. Due to the very simple structure of the dataset, purely exploiting decision boundary information---as done by Unc.~(SM)---is optimal. In contrast, Unc.~(Norm, $\tau \rightarrow 0$) ensures to sparsely explore the extent of the right side of the box \textit{prior to} exploiting the decision boundary. This is due to the decreasing value of $\tau$ over the iterations, and allows for a straightforward transition between exploration and exploitation. We set the value of $K=8$ for the $tau$-decay schedule so that by 8 active learning queries we have transitioned to exploitation.
\subsubsection{Overall observations}
From the toy experiments presented in Sections \ref{sec:blobs-experiment} and \ref{sec:box-experiment}, we see that the minimum norm uncertainy sampling \textit{with decaying values of $\tau$} has the desired behavior for a sample-efficient criterion that both explores and exploits during the active learning process. Ensuring this behavior in uncertainty sampling is also desirable because of the relatively light computational complexity that uncertainty sampling incurs. We now demonstrate on more complicated, ``real-world'' datasets the effectiveness of minimum norm uncertainty sampling in graph-based active learning.
\subsection{Isolet example} \label{sec:isolet-results}
We demonstrate in this section that minimum norm uncertainty sampling in the PWLL-$\tau$ model overcomes the previously negative results that have characterized uncertainty sampling. In \cite{ji_variance_2012}, the authors introduced the Variance Optimization (i.e., VOpt) acquisition function which quantifies how much unlabeled points would decrease the variance of the conditional distribution over Laplace learning node functions. They showcased this acquisition function on the Isolet spoken letter dataset\footnote{Accessed via \url{https://archive.ics.uci.edu/ml/datasets/isolet}.} from the UCI repository \cite{uci}, which contains 26 different classes. They compared against smallest margin uncertainty sampling (Unc.~(SM)) among other acquisition functions. Of particular interest to the current work is how poorly Unc.~(SM) performed on this task, resulting in significantly worse accuracies than even random sampling.\footnote{We refer the reader to original paper \cite{ji_variance_2012} for more details.} We demonstrate that similar---even superior---performance can be attained on this task by simply using this minimum norm uncertainty sampling (Unc.~(Norm)) that is more appropriate for low-label rate active learning.
\begin{figure}[!h]
\centering
\subfigure[Accuracy]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures/isolet_laplace_accplot.jpg}}
\subfigure[Cluster Proportion]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures/isolet_rwll0100_clusterplot.jpg}}
\caption{Accuracy Results (a) and Cluster Proportion (b) plots for \textbf{ISOLET} dataset. Accuracies shown here are in the original Laplace learning model \cite{zhu_semi-supervised_2003} for a more direct comparison with the results from \cite{ji_variance_2012}.}
\label{fig:isolet-results}
\end{figure}
In Figure \ref{fig:isolet-results}(a), we plot the accuracy results of an active learning test that mimics the setup of the full Isolet dataset (with 26 classes) as described in \cite{ji_variance_2012}. In addition to recreating the results of their test, we have added the results of using Unc.~(Norm, $\tau \rightarrow 0$) for comparison. Please note that the accuracies reported here are in the original Laplace learning model of \cite{zhu_semi-supervised_2003}, not the reweighted Laplace learning \cite{calder2020properly} model that is the focus of the rest of the paper and experimental results. We only add the result of Unc.~(Norm, $\tau \rightarrow 0$) to allow for clearer plots, as Unc.~(Norm) performed nearly identically to Unc.~(Norm, $\tau \rightarrow 0$).
Each trial (out of 10 total trials) for an acquisition function begins with only a single initially labeled point and 100 query points are thereafter selected sequentially. Thus, only one class has been sampled from at the start of each trial. In Figure \ref{fig:isolet-results}(b), we report the average fraction of ``clusters'' that have been sampled by each iteration of the active learning process. In this case, we treat each individual class as a different cluster. Such a plot demonstrates the explorative capabilities of the acquisition functions as applied to this dataset.
Similar to the test reported in \cite{ji_variance_2012}, smallest margin uncertainty sampling (Unc.~(SM)) performs very poorly at this task, both in terms of accuracy and cluster exploration. Our proposed minimium norm uncertainty sampling (Unc.~(Norm, $\tau \rightarrow 0$)), however, outperforms VOpt in terms of cluster exploration and provides very similar accuracy results. As another point of comparison, the calculation of VOpt requires either an eigendecomposition or a full inversion of the graph Laplacian matrix, whereas Unc.~(Norm, $\tau \rightarrow 0$) merely requires the current output of the reweighted Laplace learning model. These results provide encouraging evidence for the utility of the proposed method of uncertainty sampling in this current work.
\subsection{Larger datasets} \label{sec:larger-datasets}
In this section, we present the results of active learning experiments for multiclass classification problems derived from the \textbf{MNIST} \cite{lecun-mnisthandwrittendigit-2010}, \textbf{FASHIONMNIST} \cite{xiao2017fashionmnist}, and \textbf{EMNIST} datasets \cite{cohen2017emnist}. We construct similarity graphs for each of these datasets by first embedding the points via the use of variational autoencoders (VAE) \cite{kingma2013auto, kingma_introduction_2019} that were previously trained\footnote{The representations for \textbf{MNIST} and \textbf{FASHIONMNIST} are available in the GraphLearning package \cite{graphlearning}, while the code used to train the VAE for \textbf{EMNIST} is available in our Github repo \url{https://github.com/millerk22/rwll_active_learning}.} in an unsupervised fashion, similar to \cite{calder_poisson_2020}.
Since a main crux of the present work is to ensure \textit{both} exploration of clusters in a dataset and exploitation of cluster boundaries, we adapt the classification structure of the above datasets to require both. That is, we take the ``true'' class labelings $y_i \in \{0, 1, \ldots, C\}$ (e.g. digits 0-9 for \textbf{MNIST}) and reassign them to one of $k < C$ classes by taking $y_i^{new} \equiv y_i \operatorname{mod} k$; see Table \ref{table:mod-classes} below.
\begin{table*}[h!]\centering \label{table:mod-classes}
\ra{1.3}
\begin{tabular}{@{}cccccc@{}}\toprule
Resulting Mod Class & 0 & 1 & 2 & 3 & 4 \\
\midrule
\textbf{MNIST} & 0,3,6,9 & 1,4,7 & 2,5,8 & - & -\\
\textbf{FASHIONMNIST} & 0,3,6,9 & 1,4,7 & 2,5,8 & - & -\\
\textbf{EMNIST} & 0,5,\ldots,45 & 1,6,\ldots,46 & 2,7,\ldots,42 & 3,8,\ldots,43 & 4,9,\ldots,44\\
\bottomrule
\end{tabular}
\caption{Mapping of ground truth class label to $\operatorname{mod} k$ labeling for experiments of Section \ref{sec:larger-datasets}. Each ground truth class, is interpreted as a different ``cluster'' and the resulting class structure for the experiments have multiple clusters per class. For \textbf{MNIST} and \textbf{FASHIONMNIST}, there 10 total ground truth classes and we take labels modulo $k=3$. For \textbf{EMNIST}, there are 47 total ground truth classes and we take labels modulo $k=5$.}
\end{table*}
For each trial of an acquisition function, we select one initially labeled point per \textit{``modulo''} class; therefore, only a subset of ``clusters'' (i.e., the original true classes) has an initially labeled point. In order to perform active learning successfully in these experiments, query points chosen by the acquisition function over the trial must sample from each cluster. In this way, we have created an experimental setup with high-dimensional datasets with potentially more complicated clustering structures wherein we test and compare the following acquisition functions: Uncertainty Sampling (SM), Unc.~(Norm), Unc.~(Norm, $\tau \rightarrow 0$), Random, VOpt \cite{ji_variance_2012} (see Remark \ref{remark:vopt}), $\Sigma$-Opt \cite{ma_sigma_2013} (also see Remark \ref{remark:vopt}), and MCVOpt \cite{miller_efficient_2020}. We perform 10 trials for each acquisition function, where each trial begins with a different initially labeled subset. To clarify, trials begin with only 3 labeled points in the \textbf{MNIST} and \textbf{FASHIONMNIST} experiments and with only 5 labeled points in the \textbf{EMNIST} experiments.
In the left panel of Figures \ref{fig:mnist-results}-\ref{fig:emnist-results}, we show the accuracy performance of each acquisition function averaged over the 10 trials. The right panels of each of these figures display the average proportion of clusters that have been sampled by the acquisition functions at each iteration of the active learning process. We refer to these plots as ``Cluster Exploration'' plots since they directly assess the explorative capabilities of the acquisition functions in question.
We observe that across these experiments, both Unc.~(Norm) and Unc.~(Norm, $\tau \rightarrow 0$) consistently achieve the best accuracy and cluster exploration results. It is somewhat surprising that without decaying $\tau$, the Unc.~(Norm) acquisition function seems to perform the best even after each cluster has been explored. The experiments in Section \ref{sec:toy-experiments} suggest that the optimal performance in the exploitation phase of active learning would require taking $\tau \rightarrow 0$. We hypothesize that the clustering structure of high-dimensional data---like these datasets---is much more complicated than our intuition would suggest from analyzing toy and other visualizable (i.e., 1D, 2D, or 3D) datasets. Regardless, we see that minimum norm uncertainty acquisition function consistently outperforms other acquisition functions in these low-label rate active learning experiments. We emphasize again here that the computational cost of uncertainty sampling acquisition functions make them especially useful for active learning.
\begin{remark} \label{remark:vopt}
Due to the large nature of these datasets, computing the original VOpt and $\Sigma$-Opt criterions are inefficient (and often intractable) since it requires computing the inverse of a perturbed graph Laplacian matrix; this inverse is dense and burdensome to store in memory. We initially used an approximate criterion that utilizes a subset of eigenvalues and eigenvectors of the graph Laplacian, similar to what was done in \cite{miller_model-change_2021}. However, we noticed significantly poor results on the \textbf{MNIST} and \textbf{FASHIONMNIST} experiments seemingly due to the spectral truncation with a resulting oversampling of a single cluster during the active learning process.
As an alternative to the spectral truncation, we performed a full calculation of these acquisition functions on a small, random subset of $500$ unlabeled points at each active learning iteration. This performed significantly better than the spectral truncation in these two experiments, and so we report their performance in this section. In Figures \ref{fig:mnist-results} and \ref{fig:fashionmnist-results} we refer to this adaptation with the suffix ``(Full)''; e.g., we name its results by ``VOpt (Full)''. The small choice of unlabeled points on which to evaluate the acquisition function in the full setting is due to the burdensome computation needed at each step that scales with the size of this subset; at this reported choice of $500$ points each active learning iteration already takes roughly 6 minutes to complete. Due to its even greater size, we do not perform this computation on the \textbf{EMNIST} dataset, and furthermore the performance of the approximate VOpt already achieves comparable accuracy to the other reported methods in this dataset.
\end{remark}
\begin{figure}
\centering
\subfigure[Accuracy]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures_full/mnist-mod3_rwll_accplot_voptfull.jpg}}
\subfigure[Cluster Proportion]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures_full/mnist_clusterplot_voptfull.jpg}}
\caption{Accuracy Results (a) and Cluster Proportion (b) plots for \textbf{MNIST} dataset.}
\label{fig:mnist-results}
\end{figure}
\begin{figure}
\centering
\subfigure[Accuracy]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures_full/fashionmnist-mod3_rwll_accplot_voptfull.jpg}}
\subfigure[Cluster Proportion]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures_full/fashionmnist_clusterplot_voptfull.jpg}}
\caption{Accuracy Results (a) and Cluster Proportion (b) plots for \textbf{FASHIONMNIST} dataset. }
\label{fig:fashionmnist-results}
\end{figure}
\begin{figure}
\centering
\subfigure[Accuracy]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures_full/emnist-mod5_rwll_accplot_voptfull.jpg}}
\subfigure[Cluster Proportion]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures_full/emnist_clusterplot_voptfull.jpg}}
\caption{Accuracy Results (a) and Cluster Proportion (b) plots for \textbf{EMNIST} dataset. }
\label{fig:emnist-results}
\end{figure}
\section{Continuum analysis of active learning} \label{sec:theory}
We now study our active learning approach rigorously through its continuum limit. As was shown in \cite{calder2020properly}, the continuum limit of \eqref{eq:rw-lap-learning} is the family of singularly weighted elliptic equations
\begin{equation}\label{eq:rw_lap_continuum}
\left\{
\begin{aligned}
\tau u_i - \rho^{-1}\div\left(\gamma\rho^2 \nabla u_i \right) &= 0,&& \text{in } \Omega \setminus \L \\
u_i &= 1,&& \text{on } \L_i\\
u_i&= 0,&& \text{on } \L\setminus \L_i,
\end{aligned}
\right.
\end{equation}
where $\rho$ is the density of the datapoints, $\gamma$ is the singular reweighting, described in more detail below, $\L_i\subset \Omega$ are the labeled points in the $i^{\rm th}$ class, and $\L = \cup_{i=1}^C \L_i$ the locations of all labeled points. The notation $\nabla$ refers to the gradient vector and $\div$ is the divergence. The solutions $u_i$ also satisfy the homogeneous Neumann boundary condition $\nabla u \cdot \nu = 0$ on $\partial \Omega$, where $\nu$ is the outward unit normal vector to $\Omega$, but we omit writing this as it is not directly used in any of our arguments. We assume the sets $\L_i$ are all finite collections of points. The classification decision for any point $x\not\in \L$ is given by
\[\ell(x) = \argmax_{1 \leq i \leq C} u_i(x).\]
The continuum version of the uncertainty sampling acquisition function is then given by
\begin{equation}\label{eq:acq_cont}
\mathcal{A}(x) = \sqrt{u_1(x)^2 + u_2(x)^2 + \cdots + u_C(x)^2}.
\end{equation}
The aim in this section is to use continuum PDE analysis to rigorously establish the exploration versus exploitation tradeoff in uncertainty norm sampling, and illustrate how it depends on the choice of the decay parameter $\tau$.
\subsection{Illustrative 1D continuum analysis} \label{sec:1d-theory}
We proceed at first with an analysis of the continuum equations \eqref{eq:rw_lap_continuum} in the one dimensional setting, where the equations are ordinary differential equations (ODEs). The analysis is straightforward and the reweighting \eqref{eq:gamma} is no longer necessary for well-defined continuum equations with finite labeled data. The conclusions are insightful for the subsequent generalization to higher dimensions in Section \ref{sec:exploration}.
Consider an interval $\Omega = (x_{min}, x_{max}) \subset \mathbb R$ with density $0 < \rho_{min} \le \rho(x) \le \rho_{max} < +\infty$.
Assume a binary classification structure on this dataset, and further assume we have been given at least one labeled point per class. Let the pairs $\{(x_i, y_i)\}_{i=1}^\ell \subset \Omega \times \{0,1\}$ be the input-class values for the currently labeled points. Without loss of generality, let us assume that the indexing on these labeled points reflects their ordering in the reals; namely, $x_i < x_{i+1}$ for each $i \le \ell -1$. For ease in our discussion, we also assume that $x_1 = x_{min}$ and $x_\ell = x_{max}$, the endpoints of the domain (see Figure \ref{fig:1d-init}).
\begin{figure}[h]
\centering
\includegraphics[width=0.9\textwidth]{imgs/1d-init-labeled.pdf}
\vspace{-1em}
\caption{Visualization of 1D continuum example setup. The density $\rho(x)$ is plotted in gray, while the labeled points $x_1, x_2, x_3, x_4$ are plotted where the corresponding label is denoted by $\times$ or a solid dot. $\R_s$ marks the length between two similarly labeled points, while $\R_o$ marks the length between two oppositely labeled points.}
\label{fig:1d-init}
\end{figure}
Solving the PWLL-$\tau$ equation\footnote{Without the reweighting \eqref{eq:gamma} due to the simple geometry in one dimension.} \eqref{eq:rw_lap_continuum} on $\Omega$ can be broken into a number of subproblems defined on the intervals $(x_1, x_2), \ldots, (x_{\ell-1}, x_{\ell}) \subset \mathbb R$, with boundary conditions determined by the corresponding labels of the endpoints $x_i$. There are three separate kinds of sub-problems that need to be solved, as determined by these boundary conditions, that we will term (1) the \textit{oppositely labeled problem} (when $y_i \not= y_{i+1}$) and (2) the \textit{similarly labeled problem} (when $y_i = y_{i+1}$).
Given the current state of the labeled data, the active learning process selects a new query point $x^\ast = \argmin_{x \in \Omega} \ \mathcal A(x)$ via the minimum norm acquisition function \eqref{eq:acq_cont}.
We can quantify the explorative behavior of our acquisition function \eqref{eq:acq_cont} by comparing the minimizers of $\mathcal A(x)$ in the different subintervals $(x_i, x_{i+1})$. Due to the simple geometry of the problems in one dimension, our analysis reduces to a pairwise comparison of $\mathcal A(x)$ on (i) an interval of length $\R_o$ between \textit{oppositely labeled points} and (ii) an interval of length $\R_s$ between \textit{similarly labeled points}. Defining $\mathcal A_s $ and $\mathcal A_o$ as the acquisition function on the respective oppositely and similarly labeled problem subintervals, we compare the values $\min \mathcal A_o(x)$ and $\min \mathcal A_s(x)$ on said subintervals.
In this simple one-dimensional problem, we may characterize ``explorative'' query points as residing in relatively \textit{large} intervals between labeled points, regardless of the labels of the endpoints. Conversely, we characterize ``exploitative'' query points as residing between \textit{oppositely labeled points that are close together}. In Figure \ref{fig:1d-init}, exploration would correspond to sampling in $(x_1,x_2)$ or $(x_3,x_4)$, while exploitation would correspond to sampling in $(x_2, x_3)$.
The acquisition function \eqref{eq:acq_cont} is directly a function of the magnitudes of the solutions to \eqref{eq:rw_lap_continuum} with the corresponding boundary conditions. Due to the boundary conditions intervals between oppositely labeled points, the solutions to \eqref{eq:rw_lap_continuum} necessarily interpolate between $0$ and $1$ along the interval (see Figure \ref{fig:solutions-acqfuncs}(a)). In the oppositely labeled problem, however, there is only decay in the solution $v_0$ that solves \eqref{eq:rw_lap_continuum} with labels $y(x_i) = y(x_{i+1}) = 1$ when $\tau >0$, and the extent of this decay is controlled by the size of $\tau$, the length of the interval $\R_s$, and the density $\rho$ on the interval. As such, we identify how $\tau$ must be chosen in order to produce small acquisition function values between similarly labeled points in relatively large regions as compared to large values in relatively small regions between oppositely labeled points.
\subsubsection{Exploration guarantee in one dimension}
\begin{figure}[t]
\centering
\subfigure[Oppositely Labeled]{\includegraphics[height=12em]{imgs/1d_opp_plot2.jpg}}
\hspace{3em}
\subfigure[Similarly Labeled]{\includegraphics[height=12em]{imgs/1d_same_plot.jpg}}
\caption{Visualization of one-dimensional solutions to \eqref{eq:rw_lap_continuum} in oppositely (a) and similarly (b) labeled regions. Solutions $u_0,u_1$ (blue, green lines) to \eqref{eq:rw_lap_continuum} are shown in panel (a) when endpoints have opposite labels, while $v_0,v_1$ (red, orange lines) to \eqref{eq:rw_lap_continuum} are shown in panel (b) when endpoints have the same labels. The background density $\rho$ in each respective region is shown in gray, and the acquisition function value at the midpoint of the interval is shown as a black dot. The minimum acquisition function value occurs at the midpoint if the density is symmetric, which we show here for simpler presentation. }
\label{fig:solutions-acqfuncs}
\end{figure}
In Supplemental Material Section \ref{smsec:warmup-const-density}, we first derive an explicit condition in the case that the density $\rho(x) \equiv \rho$ is constant to guarantee that $\min \mathcal A_s(x) < \min \mathcal A_o(x)$. As long as the length between oppositely labeled points ($\R_o$) is \textit{small} enough compared to the length between similarly labeled points ($\R_o$),
then this gives the condition that the quantity $\frac{\tau\R^2_s}{\rho}$ must be relatively \textit{large}. We can then generalize this result to cases when the density $\rho(x)$ is no longer constant, but rather obeys some mild assumptions. Namely, we give the mild assumption that the density $\rho(x)$ (i) is sufficiently smooth, (ii) is \textit{symmetric about the midpoint of the interval} (see Assumption \ref{assumption:symmetry}) between similarly labeled points, and (iii) obeys \textit{a bounded derivative condition at the ends of the interval} (see Assumption \ref{assumption:end-intervals}) between oppositely labeled points. Under these mild assumptions we give the following simplified guarantee on exploration, which we prove rigorously in Section \ref{smsec:compare-1d-bounds}.
\begin{proposition}[Simplified version of Proposition \ref{smprop:1d-result}] \label{prop:1d-result}
Suppose that the density $\rho(x)$ satisfies Assumption \ref{assumption:end-intervals} in the oppositely labeled problem region and Assumption \ref{assumption:symmetry} in the similarly labeled problem region. Let the interval length $\R_o$ be relatively small compared to $\R_s$; i.e., $\R_o = \beta \R_s$ for some $\beta \le \frac{1}{4}$. Then we are ensured that
\[
\min_{x}\ \mathcal A_s(x) < \min_{x } \ \mathcal A_o(x)
\]
as long as $\tau > 0$ and $\R_s$ jointly satisfy the following inequality
\begin{equation} \label{eq:tau-condition-messy-simplifieid}
\R_s^2 \left( C_0(\rho_s) \sqrt{\tau} - C_1(\rho_o) \beta^2 \tau \right) \ge 8\ln 2,
\end{equation}
where $C_0(\rho_s)$ and $C_1(\rho_o)$ are constants that depend on the density $\rho$ on the similarly and oppositely labeled intervals, respectively denoted $\rho_s$ and $\rho_o$.
\end{proposition}
Figure \ref{fig:solutions-acqfuncs} illustrates the main idea of Proposition \ref{prop:1d-result}. As long the similarly labeled region (Figure \ref{fig:solutions-acqfuncs}(b) has significantly large regions where the density $\rho(x)$ is sufficiently small compared to the oppositely labeled region (Figure \ref{fig:solutions-acqfuncs}(a)), then we can be assured that choosing $\tau > 0$ large enough will result in query points between similarly labeled points that are relatively far from each other.
This relationship is also summarized in the inequality \eqref{eq:tau-condition-messy-simplifieid}, which highlights that larger $\tau$ and interval length $\R_s$ are necessary in order to satisfy said inequality. This inequality simply quantifies how $\tau>0$ must be chosen in order to select query points between similarly labeled points that are relatively far away from each other; i.e., to ensure exploration of such regions that would otherwise be missed if $\tau$ we not sufficiently large. The effect of the relative ratio of the intervals, $\beta$, is also highlighted in \eqref{eq:tau-condition-messy-simplifieid}; namely, the smaller the region between oppositely labeled points the easier it is to satisfy this inequality. Intuitively one can see that if $\beta$ is not small, then the region between oppositely labeled points is relatively large and it will be more difficult to satisfy the stated inequality. However, in this case querying between oppositely labeld points that are relatively distant from each other is desirable and would be characterized as explorative.
\subsection{Exploration bounds in arbitrary dimensions}
\label{sec:exploration}
In this section, we show how larger values for $\tau$ lead to explorative behaviour in higher dimensional problems. In particular, we show that the acquisition function $\mathcal{A}(x)$ is small on unexplored clusters, and large on sufficiently well-explored clusters. This ensures that adequate exploration occurs before exploitation.
Let us remark that the reweighting term $\gamma$ must be sufficiently singular near the labels $\L$ in order to ensure that \eqref{eq:rw_lap_continuum} is well-posed. We recall from \cite{calder2020properly} that we require that $\gamma$ has the form
\begin{equation}\label{eq:gamma}
\gamma(x) = 1 + \dist(x,\L)^{-\alpha},
\end{equation}
where $\alpha > d-2$. In practice, we choose $\gamma$ as the solution of the graph Poisson equation \eqref{eq:gamma_eq_discrete} introduced earlier. To make the analysis in this section tractable, we assume here that $\gamma$ satisfies \eqref{eq:gamma}, as was assumed in \cite{calder2020properly}.
We emphasize here that without the singular reweighting $\gamma$, the equation \eqref{eq:rw_lap_continuum} is ill-posed when the label set $\L$ is finite, and as such, there is no continuum version of active learning for us to study.
For an open set $A\subset \R^d$ and $r>0$ we define the nonlocal boundary $\partial_r A$ as
\[\partial_r A = \overline{(A + B_r)} \setminus A.\]
The nonlocal boundary is essentially a tube of radius $r$ surrounding the set $A$. The usual boundary is obtained by taking $r=0$, so $\partial A=\partial_0 A$.
Our first result concerns upper bounds on the acquisition function in an unexplored cluster.
\begin{theorem}\label{thm:explore_new_cluster}
Let $\tau\geq 0$, $s,R>0$ and $\mathcal{D} \subset \Omega$ with $\partial_{2s} \mathcal{D} \subset \Omega$ and $\L\cap (\mathcal{D}+B_{R+2s})=\varnothing$. Let
\[\delta = \max_{\partial_{2s} \mathcal{D}}\rho.\]
Assume that
\begin{equation}\label{eq:explore_cond}
\sqrt{\frac{\tau}{\delta}} \geq 3\left(\tfrac{d}{s} + 2\|\nabla \log \rho\|_{L^\infty(\partial_s \mathcal{D})}\right)(1+R^{-\alpha}) + 3R^{-\alpha-1}.
\end{equation}
Then it holds that
\begin{equation}\label{eq:acq_upper_bound}
\sup_{\mathcal{D}}\mathcal{A} \leq \sqrt{C}\exp\left(-\frac{s}{4}\sqrt{\frac{\tau}{\delta}}\right).
\end{equation}
\end{theorem}
\begin{remark}\label{rem:upper_bound}
Theorem \ref{thm:explore_new_cluster} shows that the acquisition function $\mathcal{A}$ is exponentially small on an unexplored cluster $\mathcal{D}$ provided there is a thin surrounding set $\partial_s \mathcal{D}$ of the cluster on which the density is small (less than $\delta$), relatively smooth (so $\nabla \log \rho$ is not too large), and relatively far away from other labeled datapoints (so that $R$ is not too large). All of these smallness assumptions are relative to the size of the ratio $\tau/\delta$ as expressed in \eqref{eq:explore_cond}.
\end{remark}
To ensure that new clusters are explored, we also need to lower bound the acquisition function nearby the existing labeled set. To do this, we need to introduce a model for the clusterability of the dataset. Let $\Omega_1,\Omega_2,\dots,\Omega_C\subset \Omega$ be disjoint sets representing each of the $C$ classes in the dataset. We assume that the labels are chosen from the corresponding class sets, so that $\L_i \subset \Omega_i$ for each $i$. We assume there is a positive separation between the classes, measured by the quantity
\begin{equation}\label{eq:cluster_sep}
\mathcal{S} := \min_{i\neq j}\dist(\Omega_i,\Omega_j).
\end{equation}
The definition of $\mathcal{S}$ implies that $(\Omega_i + B_\mathcal{S})\cap \Omega_j = \varnothing$ for all $i\neq j$. We define the union of the classes as $\Omega'= \cup_{i=1}^C \Omega_i$. We note that we do not have $\Omega'=\Omega$, and it is important that there is room in the background $\Omega\setminus \Omega'$, which provides a separation between classes. The background $\Omega\setminus \Omega'$ may have low density (though we do not assume this below), and can consist of outliers or datapoints that have characteristics of multiple classes and may be hard to classify.
\begin{theorem}\label{thm:explore_labels}
Let $\tau\geq 0$ and $\alpha>d-2$. Assume that $\L_i\subset \Omega_i$ for $i=1,\dots,C$, and let $r>0$ be small enough so that $r \leq \tfrac{1}{4}\mathcal{S}$,
\begin{equation}\label{eq:rtau2}
\tau r^d \leq \frac{1}{2^d9}(\alpha+2-d)^2\inf_{\Omega' + B_{2r}}\rho,
\end{equation}
and
\begin{equation}\label{eq:rcond2}
4\|\nabla \log \rho\|_{L^\infty(\Omega'+B_{2r})}(1 + 2^\alpha r^{\alpha})r + \alpha 2^\alpha r^{\alpha} \leq \tfrac{1}{4}(\alpha + 2-d).
\end{equation}
Then we have
\begin{equation}\label{eq:acq_lower}
\inf_{\L + B_r}\mathcal{A} \geq 1 - 2^{-\frac{1}{2}(\alpha + 2-d)}.
\end{equation}
\end{theorem}
\begin{figure}[!t]
\centering
\includegraphics[width=0.7\textwidth]{imgs/clusters}
\caption{Illustration of the implications of Theorems \ref{thm:explore_new_cluster} and \ref{thm:explore_labels}, and the discussion in Remark \ref{rem:explanation}. The gray regions are the 4 clusters of high density in the dataset, and the density is small $\rho \leq \delta$ between clusters. The current labeled set are the points at the centers of the blue balls. Theorems \ref{thm:explore_new_cluster} and \ref{thm:explore_labels} guarantee that the next labeled point cannot lie in any of the blue balls, which correspond to the dilated label set $\L + B_r$. Once the dilated labels cover the existing clusters, the algorithm is guaranteed to select a point from the unexplored cluster $\mathcal{D}$. The number of labeled points selected from a given cluster during exploration is bounded by its $\frac{r}{2}$-packing number, as explained in Remark \ref{rem:explanation}. }
\label{fig:clusters}
\end{figure}
\begin{remark}\label{rem:explanation}
Let us make some remarks on the applications of Theorems \ref{thm:explore_new_cluster} and \ref{thm:explore_labels}. First, we note that the choice of $s$ in Theorem \ref{thm:explore_new_cluster} can be made proportional to the separation between clusters $\mathcal{S}$ defined in \eqref{eq:cluster_sep}. We can then choose $\tau$ to ensure \eqref{eq:explore_cond} holds in Theorem \ref{thm:explore_new_cluster}, and choose $r>0$ to satisfy the conditions in Theorem \ref{thm:explore_labels}. These choices are all dependent on the domain, the clusterability assumption, and the density, but are independent of the choices of labeled points $\L_i$. Now, combining Theorems \ref{thm:explore_new_cluster} and \ref{thm:explore_labels} we see the condition
\begin{equation}\label{eq:acq_cond}
\sqrt{C}\exp\left(-\frac{\mathcal{S}}{4}\sqrt{\frac{\tau}{\delta}}\right)\leq 1 - 2^{-\frac{1}{2}(\alpha + 2-d)}
\end{equation}
is important. Whenever \eqref{eq:acq_cond} holds, the region $\mathcal{D}$ will be explored \emph{before} a new labeled point is chosen within distance $r$ of any existing labeled point. This is exactly the \emph{exploration} property that we desire in an active learning algorithm. In the early stages, the algorithm should seek to explore new clusters, or continue to sufficiently explore existing clusters. The algorithm will not choose another labeled point within distance $r$ of an existing label until the entire dataset is thoroughly explored, at which point the active learning algorithm should switch to exploitation.
In fact, we can make the statements above a little more precise. Whenever $\L_i + B_r \supset \Omega_i$, we have from Theorem \ref{thm:explore_labels} that
\[\inf_{\Omega_i} \mathcal{A} \geq 1 - 2^{-\frac{1}{2}(\alpha + 2-d)}.\]
In this case, provided \eqref{eq:acq_cond} holds, the algorithm will not select another point from $\Omega_i$ until \emph{all} other cluster have been explored. Since the algorithm also cannot choose a new point within distance $r$ of existing points, the set $\L_i$ is a $r$-net of $\Omega_i$. In particular, the balls $B_{\frac{r}{2}}(z)$ for $z\in \L_i$ are disjoint, so $\L_i+B_{\frac{r}{2}}$ is a $\tfrac{r}{2}$-\emph{packing} of $\Omega_i$. We define an $\epsilon$-\emph{packing} of $\Omega_i$ as a disjoint union of $\epsilon$-balls that are centered at points in $\Omega_i$. Therefore, the maximum number of points in $\L_i$ is given by the $\epsilon$-\emph{packing number} of $\Omega_i$ with $\epsilon=\tfrac{r}{2}$, which is defined as
\[M(\Omega_i,\epsilon) = \max\left\{m \, : \, \text{there exists an } \epsilon \text{-packing of }\Omega_i \text{ with }m\text{ balls.} \right\}.\]
Thus, Theorems \ref{thm:explore_new_cluster} and \ref{thm:explore_labels} show that our uncertainty norm sampling active learning algorithm, in the continuum, cannot select more than the packing number $M(\Omega_i,\tfrac{r}{2})$ of points from $\Omega_i$ until \emph{all} clusters have been explored. The packing number $M(\Omega_i,\tfrac{r}{2})$ depends on the geometry of the cluster $\Omega_i$ and can be large for clusters that are not spherical (e.g., clusters that are ``thin'' and ``long'' in certain directions). These results are illustrated in Figure \ref{fig:clusters}.
The reader may have observed there is an implicit assumption made throughout this remark that there are no labeled points selected from the background region $\Omega\setminus \Omega'$. Indeed, if such outlying datapoints are selected as labeled points, then our results do not hold. In practice, one can perform sampling proportional to a density estimation, or simply remove outliers, to avoid such an issue. We discuss how this can be done in Section \ref{sec:kde}, and we have performed experiments with this. We have found that the results are similar with and without the outlier removal, so we see this as an extra step that one has the option of performing in practice, to maximally align the algorithm with the theory, but we do not see it as a necessary step in practice.
\end{remark}
\section{Conclusion}
We have demonstrated that uncertainty sampling is sufficient for exploration in graph-based active learning by using the norm of the output node function of the PWLL-$\tau$ model as an acquisition function. We provide rigorous mathematical guarantees on the explorative behavior of the proposed acquisition function. This is made possible by the well-posedness of the corresponding continuum limit PDE of the PWLL-$\tau$ model. Our analysis elucidates how the choice of hyperparamter $\tau >0$ directly influences these guarantees; in the one dimensional case this effect is most clearly illustrated. In addition, we provide numerical experiments that further illustrate the effect of both our acquisition function and the hyperparameter $\tau$ on the sequence of active learning query points. Other numerical experiments confirm our theoretical guarantees and demonstrate favorable performance in terms of both accuracy and cluster exploration.
\bibliographystyle{siamplain}
\section{Introduction} \label{sec:intro}
Supervised machine learning algorithms rely on the ability to acquire an abundance of labeled data, or data with known labels (i.e., classifications). While unlabeled data---data {\it without} known labels---is ubiquitous in most applications of interest, obtaining labels for such training data can be costly.
Semi-supervised learning (SSL) methods leverage unlabeled data to achieve an accurate classification with significantly fewer training points. Simultaneously, the choice of training points can significantly affect classifier performance, especially due to the limited size of the training set of labeled data in the case of SSL.
Active learning seeks to judiciously select a limited number of {\it query points} from the unlabeled data that will inform the machine learning task at hand.
These points are then labeled by an expert, or human in the loop, with the aim of significantly improving the performance of the classifier.
While there are various paradigms for active learning \cite{settles_active_2012}, we focus on {\it pool-based} active learning wherein an unlabeled pool of data is available at each iteration of the active learning process from which query points may be selected. This paradigm is the natural fit for applying active learning in conjunction with semi-supervised learning since the unlabeled pool is also used by the the underlying semi-supervised learner. These query points are selected by optimizing an {\it acquisition function} over the discrete set of points available in the unlabeled pool of data. That is, if $\mathcal U \subset \mathcal X$ is the set of currently unlabeled points in a pool of data inputs $\mathcal X \subset \mathbb R^d$, then the active learning process at each iteration selects the next query point $x^\ast \in \mathcal U$ to be the minimizer of a real-valued acquisition function
\[
x^\ast = \argmin_{x \in \mathcal U} \ \mathcal A(x),
\]
where $\mathcal A$ can depend on the current state of labeled information (i.e., the labeled data $\mathcal L = \mathcal X - \mathcal U$ and corresponding labels for points in $\mathcal L$).
The above process (policy) for selecting query points is \textit{sequential} as only a single unlabeled point is chosen to be labeled at each iteration, as opposed to the \textit{batch} active learning paradigm. In batch active learning, a set of query points $\mathcal Q \subset \mathcal U$ is chosen at each iteration. While this is an important extension of the sequential paradigm and is an active area of current research in the literature \cite{sener_active_2018, gal_deep_2017, vahidian_coresets_2020, miller_model-change_2021}, we focus on the sequential case in this work.
Acquisition functions for active learning have been introduced for various machine learning models, especially support vector machines \cite{tong_support_2001, jiang_minimum-margin_2019, balcan_margin_2007}, deep neural networks\cite{gal_deep_2017, sener_active_2018, kushnir_diffusion-based_2020, shui_deep_2020, simeoni_rethinking_2021}, and graph-based classifiers \cite{zhu_combining_2003, ji_variance_2012, ma_sigma_2013, qiao_uncertainty_2019, miller_model-change_2021, murphy_unsupervised_2019}. We focus on graph-based classifiers for our underlying semi-supervised learning model due to their ability to capture clustering structure in data and their superior performance in the {\it low-label rate regime}---wherein the labeled data constitutes a very small fraction of the total amount of data.
Most active learning methods for deep learning assume a moderate to large amount of initially labeled data to start the active learning process. While there is exciting progress in improving the low-label rate performance of deep semi-supervised learning \cite{sohn2020fixmatch, sellars2022Laplacenet, zheng2022Simmatch} and few-shot learning \cite{zhang2022differentiable, he2022attribute}, we restrict the focus of this paper to well-established graph-based paradigms for this setting.
An important aspect in the application of active learning in real-world datasets is the inherent tradeoff between using active learning queries to either explore the given dataset or exploit the current classifier's inferred decision boundaries. This tradeoff is reminiscent of the similarly named ``exploration versus exploitation'' tradeoff in reinforcement learning. In active learning, it is important to thoroughly explore the dataset in the early stages, and exploit the classifier's information in later stages. Algorithms that exploit too quickly can fail to properly explore the dataset, potentially missing important information, while algorithms that fail to exploit the classifier in later stages can miss out on some of the main benefits of active learning.
In this work, we provide a simple, yet effective, acquisition function for use in graph-based active learning in the low-label rate regime that provides a natural transition between exploration and exploitation summarized in a single hyperparameter. We demonstrate through both numerical experiments and theoretical results that this acquisition function explores prior to exploitation. We prove theoretical guarantees on our method through analyzing the continuum limit partial differential eqauation (PDE) that serves as a proxy for the discrete, graph-based operator. This is a novel approach to providing sampling guarantees in graph-based active learning. We also provide experiments on a toy problem that illustrates our theoretical results, and the importance of the exploration versus exploitation hyperparameter in our method.
\subsection{Previous work} \label{subsec:prev-work}
The theoretical foundations in active learning have mainly focused on proving sample-efficiency results for linearly-separable datasets---oftentimes restricted to the unit sphere \cite{balcan2009agnostic, dasgupta_coarse_2006, hanneke_bound_2007}---for low-complexity function classes using disagreement or margin-based acquisition functions \cite{hanneke_theory_2014, hanneke_minimax_2015, balcan2009agnostic, balcan_margin_2007}. These provide convenient bounds on the number of active learning choices necessary for the associated classifier to achieve (near) perfect classification on these datasets with simple geometry.
In contrast, much of the focus for theoretical work in graph-based active learning leverages assumptions on the clustering structure of the data that is assumed to be captured in the graph structure \cite{murphy_unsupervised_2019, dasarathy_s2_2015}, which sometimes is assumed to be hierarchical \cite{dasgupta_hierarchical_2008, dasgupta_two_2011, cloninger_cautious_2021}. A central priority in this line of inquiry establishes guarantees that, given assumptions about the clustering structure of the observed dataset $\mathcal X$, the active learning method in question will query points from \textit{all} clusters (i.e., ensure exploration). The low-label rate regime of active learning---the focus of of this current work---is the natural setting for establishing such theoretical guarantees.
Laplacian Learning \cite{zhu_semi-supervised_2003} has been a common graph-based semi-supervised learning model for a number of graph-based active learning methods \cite{zhu_combining_2003, ji_variance_2012, ma_sigma_2013, jun_graph-based_2016}. However, little work has been done to provide theoretical guarantees for these methods, possibly due to the inherent difficulty in proving meaningful estimates on the solutions of discrete graph equations. Other important works in active learning have focused primarily on improving the performance of deep neural networks via active learning with either (1) moderate to large amounts of labeled data available to the classifier \cite{gal_deep_2017, zhu_robust_2019} or (2) coreset methods that are agnostic to the observed labels of the labeled data seen throughout the active learning process \cite{sener_active_2018, vahidian_coresets_2020}. Our current work is focused on the \emph{low-label rate regime}, which is an arguably more fitting regime for semi-supervised and active learning. Furthermore, in contrast to coreset methods, our acquisition function directly depends on the observed classes of the labeled data.
Graph neural networks (GNN) \cite{welling2016semi, zhou2018graph} are an important area of graph-based methods for machine learning, and various methods for active learning have been proposed \cite{hu2020policy, ijcai2018p296, cai2017active, Zhang_Tong_Xia_Zhu_Chi_Ying_2022}. GNNs consider network graphs whose connectivity is a priori determined via metadata relevant to the task (e.g., co-authorship in citation networks) and then use the node-level features to learn representations and transformations of features for the learning task. In contrast, we consider similarity graphs where the connectivity structure is determined only by the node-level features and directly learn a node function on this graph structure.
Continuum limit analysis of graph-based methods has been an active area of research for providing rigorous analysis of graph-based learning \cite{calder_consistency_2019, calder2022improved, calder_poisson_2020, calder2020properly, calder2018game, slepcev2019analysis,dunlop2020large}. In this analysis, a discrete graph is viewed as a random geometric graph that is sampled from a density $\rho: \mathbb R^d \rightarrow \mathbb R_+$ defined in a high-dimensional space (possibly constrained to a manifold $\mathcal M \subset \mathbb R^d$ therein). The graph Laplacian matrix can be analyzed via its continuum-limit counterpart, which is a second order density weighted diffusion operator (or a weighted Laplace-Beltrami operator on the manifold). An important development relevant to the current work is the Properly Weighted Graph Laplacian \cite{calder2020properly}, which reweights the graph in the Laplacian learning model of \cite{zhu_semi-supervised_2003} to correct for the degenerate behavior of Laplacian learning in the extremely low-label rate regime. This provides the setting for a well-defined, properly scaled graph-based semi-supervised learning model that we use in our current work to provide rigorous bounds on the acquisition function values to control the exploration versus exploitation tradeoff.
In order to apply active learning in practice, it is essential to design computationally efficient acquisition functions.
Much of the current literature has sought to design more sophisticated methods that often have higher computational complexity (e.g., requiring the full inversion of the graph Laplacian matrix).
Uncertainty sampling \cite{settles_active_2012} is an example of a computationally efficient acquisition function since it only requires the output of the classifier on the unlabeled data. However, uncertainty sampling methods will often mainly select query points that concentrate along decision boundaries while ignoring large regions of the dataset that are distant from any labeled points. Phrased in the terminology of the exploration versus exploitation tradeoff in reinforcement learning, uncertainty sampling is often overly ``exploitative'' and often achieves poor overall accuracy in empirical experiments \cite{ji_variance_2012}.
In contrast, methods such as variance optimization (VOpt) \cite{ji_variance_2012}, $\Sigma$-Opt \cite{ma_sigma_2013}, Coresets \cite{sener_active_2018}, LAND \cite{murphy_unsupervised_2019}, and CAL \cite{cloninger_cautious_2021} could be characterized as primarily ``explorative'' methods. Oftentimes, however, such explorative methods, or other methods that are designed to both explore and exploit \cite{miller_model-change_2021, karzand_maximin_2020, zhu_combining_2003, gal_deep_2017} are more expensive to compute than simply using uncertainty sampling. For example, VOpt \cite{ji_variance_2012} and $\Sigma$-Opt \cite{ma_sigma_2013} require the computation and storage of a dense $N \times N$ covariance matrix that must be updated each active learning iteration. The work of \cite{miller_model-change_2021} proposed a computationally efficient adaptation of these methods via a projection onto a subset of the graph Laplacian's eigenvalues and eigenvectors.As a consequence of sometimes significantly poor performance from this spectral truncation method in our experiments, we provide a ``full'' computation of V-Opt and $\Sigma$-Opt in certain experiments by restricting the computation to only a subset of unlabeled data which allows us to bypass the need to invert the graph Laplacian matrix (Section \ref{sec:larger-datasets}). This heuristic, however, is still very expensive to compute at each active learning iteration making it not a viable option for moderate to large datasets in practice.
In this work, we show that uncertainty sampling, \textit{when properly designed for the graph-based semi-supervised learning model} can both explore and exploit, and outperforms existing methods in terms of computational complexity, overall accuracy, and exploration rates.
\subsection{Overview of paper} \label{sec:overview-contents}
The rest of the paper continues as follows. We begin in Section \ref{sec:model-setup} with a description of the Properly Weighted Laplacian Learning model from \cite{calder2020properly} that will be the underlying graph-based semi-supervised learning model for our proposed active learning method. We also introduce the minimum norm acquisition function in this section, along with other useful preliminaries for the rest of the paper. In Section \ref{sec:results}, we begin with illustrative experiments in two-dimensions to illustrate the delicate balance between exploration and exploitation in graph-based active learning.
Section \ref{sec:larger-datasets} compares our proposed active learning method to other acquisition functions on larger, more ``real-world'' datasets that have been adapted to provide an experimental setup wherein exploration is essential for success in the active learning task. Thereafter, we present theoretical guarantees for the minimum norm acquisition function in the continuum limit setting in Section \ref{sec:theory}, along with an extended look at the theory in one dimension in Section \ref{sec:1d-theory}.
\subsection{Notation} \label{subsec:notation}
Let $\|\cdot\|_2$ denote the standard Euclidean norm where the space is inferred from the input. We let $|\cdot|$ denote either the absolute value of a scalar in $\mathbb R$ or the cardinality of a set, where from context the intended usage should be clear. We denote the set of points $x \in \mathcal X$ with $x \not\in \mathcal U$ as $\mathcal X \setminus \mathcal U$.
\section{Model setup and acquisition function introduction} \label{sec:model-setup}
Let $\mathcal X = \{x_1, x_2, \ldots, x_N\} \subset \mathbb R^d$ be a set of inputs for which we assume each $x \in \mathcal X$ belongs to one of $C$ classes. Suppose that we have access to a subset $\mathcal L \subset \mathcal X$ of labeled inputs (\textit{labeled data}) for which we have observed the ground-truth classification $y(x) \in \{1, \ldots, C\}$ for each $x \in \mathcal L$.
The rest of the inputs, $\mathcal U := \mathcal X \setminus \mathcal L$, are termed the \textit{unlabeled data} as no explicit observation of the underlying classification have been seen for $x \in \mathcal U$. The semi-supervised learning task is to use both $\mathcal L$ and $\mathcal U$, with the associated labels $\{y(x)\}_{x \in \mathcal L}$, to infer the classification of the points in $\mathcal U$.
Sequential active learning extends semi-supervised learning by selecting a sequence of \textit{query points} $x_1^\ast, x_2^\ast, \ldots$ as part of an iterative process that alternates between (1) calculating the semi-supervised classifier given the current labeled information and (2) selecting and subsequently labeling an unlabeled query point $x_n^\ast \in \mathcal U_n$, where $\mathcal U_n = \mathcal X - \mathcal L_n = \mathcal X - (\mathcal L \cup \{x_1^\ast, x_2^\ast, \ldots, x_{n-1}^\ast\})$. Labeling a query point $x_i^\ast$ consists of obtaining the corresponding label $y(x_n^\ast)$ and then adding $x_n^\ast$ to the set of labeled data from the current iteration, $\mathcal L_{n} = \mathcal L_{n-1} \cup \{x_n^\ast\}$. To avoid this cumbersome notation, however, we will drop the explicit dependence of $\mathcal U_i, \mathcal L_i$ on the iteration $i$ and simply refer to the unlabeled and labeled data at the \textit{current} iteration as respectively $\mathcal U$ and $\mathcal L$.
Returning to the underlying semi-supervised learning problem, graph Laplacians have often been used to propagate labeled information from $\mathcal L$ to $\mathcal U$ \cite{zhu_semi-supervised_2003, bertozzi_diffuse_2016, calder_poisson_2020, calder2020properly, shi2017weighted, bertozzi2019graph, calder2018game,welling2016semi}. From the set of feature vectors $\mathcal X$, consider a similarity graph $G(\mathcal X, W)$ with weight matrix $w_{ij} = \kappa(x_i,x_j)$ that captures the similarity between inputs $x_i,x_j$ for each pair of points in $\mathcal X$. We use $\mathcal X$ to denote both the set of feature vectors as well as the node set for the graph $G$ to avoid introducing more notation. Laplacian learning \cite{zhu_semi-supervised_2003} is an important graph-based semi-supervised learning model for both this current work and many previous graph-based active learning works, and solves the constrained problem of identifying a graph function $u :\mathcal X \rightarrow \mathbb R^C$ via the minimization of
\begin{align}\label{eq:lap-learning}
\min_{u: \mathcal X \rightarrow \mathbb R^d}\ &\sum_{x_i,x_j \in \mathcal X} w_{ij} \|u(x_i) - u(x_j)\|_2^2 \\
\text{subject to }& u(x) = e_{y(x)} \text{ for } x \in \mathcal L. \nonumber
\end{align}
The vector $e_{y(x)} \in \mathbb R^C$ is the standard Euclidean basis vector in $\mathbb R^C$ whose entries are all $0$ except the entry corresponding to the label $y(x)\in \{1, \ldots, C\}$.
The learned function $u$ that minimizes \eqref{eq:lap-learning} constitutes a harmonic extension of the given labels in $\mathcal L$ to the unlabeled data since $u$ is a harmonic function on the graph. For the classification task, the inferred classification of $x \in \mathcal U$ is then obtained by thresholding on the learned function's output at $x$, $u(x) \in \mathbb R^C$. That is, the inferred classification $\hat{y}(x)$ for $x \in \mathcal U$ is given by
\[
\hat{y}(x) = \argmax_{c=1,2, \ldots, C} \ u_c(x),
\]
where $u_c(x)$ denotes the $c^{th}$ entry of $u(x)$.
Various previous works \cite{calder_rates_2020,calder2020properly, shi2017weighted, nadler2009infiniteunlabelled,flores2022analysis,calder_poisson_2020} have shown that when the amount of labeled information is small compared to the size of the graph (i.e., the \textit{low-label rate regime}), the performance of minimizers of \eqref{eq:lap-learning} degrades substantially. The solution $u$ becomes roughly constant with sharp spikes near the labeled set, and the classification tends to predict the same label for most datapoints. Of particular interest to the current work is the Properly Weighted Laplacian learning work in \cite{calder2020properly}, wherein a weighting $\gamma: \mathcal X \rightarrow \mathbb R_+$ that scales like $\operatorname{dist}(x, \mathcal L)^{-\alpha}$ for $\alpha > d-2$ is used to reweight the edges in the graph to correct the singular behavior of solutions to \eqref{eq:lap-learning}. We use an improvement to the Properly Weighted Laplacian that is called Poisson ReWeighted Laplace Learning (PWLL) and will be described in detail in another paper \cite{calder2022poisson}. PWLL performs semi-supervised learning by solving the problem
\begin{align}\label{eq:rw-lap-learning}
\min_{u: \mathcal X \rightarrow \mathbb R^d}\ &\sum_{x_i,x_j \in \mathcal X} \gamma(x_i) \gamma(x_j)w_{ij} \|u(x_i) - u(x_j)\|_2^2 \\
\text{subject to }& u(x) = e_{y(x)} \text{ for } x \in \mathcal L, \nonumber
\end{align}
where the reweighting function $\gamma$ is computed by solving the graph Poisson equation
\begin{equation}\label{eq:gamma_eq_discrete}
\sum_{x_j \in \mathcal X} w_{ij}(\gamma(x_i) - \gamma(x_j)) = \sum_{x_k \in \mathcal L}\left( \delta_{ik} - \tfrac{1}{N}\right) \ \ \ \text{for all } x_i\in \mathcal X.
\end{equation}
In the previous work on the Properly Weighted Laplacian \cite{calder2020properly}, the weight $\gamma$ was explicitly chosen to satisfy $\gamma(x)\sim\operatorname{dist}(x, \mathcal L)^{-\alpha}$, while in the PWLL, $\gamma$ is learned from the data, making the method more adaptive with fewer hyperparameters. The motivation for the Poisson equation \eqref{eq:gamma_eq_discrete} is that the continuum version of this equation is related to the fundamental solution of Laplace's equation, which produces the correct scaling in $\gamma$ near the labeled set.
The reason for using PWLL is that minimizers of \eqref{eq:rw-lap-learning} have a well-defined continuum limit in the case when the amount of labeled data is fixed and the number of nodes $|\mathcal X| = N \rightarrow \infty$. This will allow us to analyze the behavior of our proposed minimum norm acquisition function applied to the PWLL model in the continuum limit setting in Section \ref{sec:theory}.
\subsection{Solution decay parameter} \label{subsec:tau-decay}
We introduce an adaptation of \eqref{eq:rw-lap-learning} that increases the decay rate of the corresponding solutions away from labeled points via a type of Tikhonoff regularization in the variational problem. Controlling this decay will prove to be crucial for ensuring that query points selected via our minimum norm acquisition function (Section \ref{subsec:min-norm-af}) will explore the extent of the dataset prior to exploiting current classifier decision boundaries. Given $\tau \ge 0$, we consider solutions of the following variational problem
\begin{align}\label{eq:rw-lap-learning-tau}
\min_{u: \mathcal X \rightarrow \mathbb R^d}\ &\sum_{x_i,x_j \in \mathcal X} \gamma(x_i) \gamma(x_j)w_{ij} \|u(x_i) - u(x_j)\|_2^2 \ \ +\ \ \tau \sum_{x_i \in \mathcal U} \|u(x_i)\|_2^2\\
\text{subject to }& u(x) = e_{y(x)} \text{ for } x \in \mathcal L. \nonumber
\end{align}
It is straightforward to see that for $\tau > 0$ the additional term in \eqref{eq:rw-lap-learning-tau} encourages the solution $u$ to have \textit{smaller} values away from the labeled data, where the values are fixed. When $\tau = 0$, we recover \eqref{eq:rw-lap-learning}. We will refer to this graph-based semi-supervised learning model as Poisson ReWeighted Laplace Learning with $\tau$-Regularization (PWLL-$\tau$).
To illustrate the role of the decay parameter, let us consider a simple one dimensional version of this problem in the continuum of the form
\[\min_{u} \int_{a}^b u'(x)^2 + \tau u(x)^2\, dx,\]
where $[a,b]$ is the domain and the minimization would be restricted by some boundary conditions on $u$ (i.e., on the labeled set). Minimizers of this problem satisfy the ordinary differential equation (i.e., the Euler-Lagrange equation) $\tau u - u'' = 0$, which has two linearly independent solutions $e^{\pm \sqrt{\tau}x}$. Since the solution we are interested in is bounded, the exponentially growing one can be discarded, and we are left with exponential decay in the solutions with rate $ \sqrt{\tau}$ away from the labeled set. Thus, at least in this simple example, we can see how the introduction of the diagonal perturbation $\tau$ in PWLL leads to exponential decay of solutions, which is essential for the method to properly \emph{explore} the dataset. We postpone developing this theory further until Section \ref{sec:theory}.
\subsection{Minimum norm acquisition function} \label{subsec:min-norm-af}
We now introduce the acquisition function that we propose to properly balance exploration and exploitation in graph-based active learning in the PWLL-$\tau$ model. We simply use the Euclidean norm of the output vector at each unlabeled point, $x \in \mathcal U$:
\begin{equation}\label{eq:min-norm-af}
\mathcal A(x) = \|u(x)\|_2 = \sqrt{u_1^2(x) + u_2^2(x) + \ldots + u_C^2(x)}.
\end{equation}
Due to the solution decay resulting from the $\tau$-regularization term in \eqref{eq:rw-lap-learning-tau}, unlabeled points that are far from all labeled points will have small Euclidean norm for their corresponding output vector. In the low-label rate regime, this property encourages query points selected by \eqref{eq:min-norm-af} to be spread out over the extent of the dataset, until a sufficient number of points have been labeled to ``cover'' the dataset. After this has been achieved in the active learning process, the learned functions for \eqref{eq:rw-lap-learning-tau} will have smaller norms in regions between labeled points of differing classes due to the rapid decay in solutions near the transition between classes. This described behavior reflects the desired properties for balancing exploration prior to exploitation in active learning. Through both numerical experiments and theoretical results, we demonstrate this acquisition function's utility for this purpose.
The acquisition function \eqref{eq:min-norm-af} is a novel type of uncertainty sampling \cite{settles_active_2012}, wherein only the values of the learned function $u$ at each active learning iteration are used to determine the selection of query points. Indeed, one may interpret the small Euclidean norm of the learned function at an unlabeled node, $\|u(x)\|_2$, to reflect uncertainty about the resulting inferred classification, $\hat{y}(x)$. Other uncertainty sampling methods, such as \textit{smallest margin sampling} \cite{settles_active_2012}, also compute the uncertainty of the learned model at an unlabeled point via properties of the output vector $u(x) \in \mathbb R^C$. However, these criterion often either (1) only compare 2 entries of the vector to compute a measure of margin uncertainty or (2) normalize the output vector to lie on the simplex to be interpreted as class probabilities. In both cases, these measures of uncertainty in the classification of unlabeled points in unexplored regions of the dataset might not be as emphasized by the acquisition function compared to points that lie near the decision boundaries of the learned classifier. In other words, most previous uncertainty sampling methods can often be characterized as solely exploitative and lack explorative behavior, with the results being decreased overall performance of the classifier on the dataset. Our minimum norm acquisition function \eqref{eq:min-norm-af}, however, is designed to prioritize the selection of query points in unexplored regions of the dataset which is properly reflected in the decay of the learned functions in the PWLL-$\tau$ model \eqref{eq:rw-lap-learning-tau}. In this sense, we are able to ensure exploration prior to exploitation in the active learning process using the minimum norm acquisition function \eqref{eq:min-norm-af} in the PWLL-$\tau$ model.
\begin{remark}[Decay Schedule for $\tau$]
As we demonstrate through some toy experiments in Section \ref{sec:toy-experiments}, there is benefit to decreasing the value of $\tau \ge 0$ as the active learning process progresses in order to more effectively transition from explorative to exploitative queries. While there are various ways to design this, we simply identify a constant $\mu \in (0,1)$ so that the decreasing sequence of hyperparameter values $\tau_{n+1} = \mu \tau_n$ with initial value $\tau_0 > 0$ satisfies that $\tau_{2K} \le \epsilon$, where $\epsilon$ is chosen to be $\epsilon = 10^{-9}$. For our experiments, we set $K$ to be the number of clusters, which in the case of our tests is known a priori. In practice, this choice of $K$ would be a user-defined choice to control the ``aggressiveness'' of the decay schedule of $\tau$. For $n \ge 2K$, we set $\tau_n = 0$. Thus, we calculate
\[
\mu = \left( \frac{\epsilon }{\tau_0}\right)^{\frac{1}{2K}} \in (0,1)
\]
which ensures a decaying sequence of $\tau$ values as desired. We note that an interesting line of inquiry for future research would be to investigate a more rigorous understanding of how to adaptively select $\tau \ge 0$ during the active learning process. We leave this question for future research and simply use the proposed decay schedule above.
\end{remark}
In Table \ref{table:unc-sampling}, we introduce the abbreviations for and other useful information pertaining to the uncertainty sampling acquisition functions that we will consider in the current work---smallest margin, minimum norm, and minimum norm with $\tau$-decay uncertainty sampling.
\newcommand{\ra}[1]{\renewcommand{\arraystretch}{#1}}
\begin{table*}[h!]\centering \label{table:unc-sampling}
\ra{1.3}
\begin{scriptsize}
\begin{tabular}{@{}lccc@{}}\toprule
\textbf{Full Name} & \textbf{Abbreviation} & $\mathcal A(x)$ & \textbf{Underlying Classifier} \\
\midrule
Smallest Margin Unc.~Sampling & Unc.~(SM) & $u_{c_1^\ast}(x) - u_{c_2^\ast}(x)$ & PWLL \\
Minimum Norm Unc.~Sampling & Unc.~(Norm) & $\|u(x)\|_2$ & PWLL-$\tau$, fixed $\tau > 0$ \\
\multirow{2}{15em}{Minimum Norm Unc.~Sampling with $\tau$-decay} & Unc.~(Norm, $\tau \rightarrow 0$) & $\|u(x)\|_2$ & PWLL-$\tau$, decay $\tau \rightarrow 0$ \\
& & & \\
\bottomrule
\end{tabular}
\end{scriptsize}
\caption{Description of uncertainty sampling acquisition functions that will be compared throughout the experiments in the following sections. Unc.~(SM) considers the difference between the largest and second largest entries of the output vector $u(x)$, denoted by $c_1^\ast$ and $c_2^\ast$ respectively.}
\end{table*}
\section{Results}\label{sec:results}
In this section, we present numerical examples to demonstrate our claim that our proposed Unc.~(Norm) and Unc.~(Norm $\tau \rightarrow 0$) acquisition functions in the PWLL-$\tau$ model \eqref{eq:rw-lap-learning-tau} are effective at both exploration and exploitation. We begin in Section \ref{sec:toy-experiments} with a set of toy examples in 2-dimensions to facilitate visualizing the choices of query points during the active learning process and highlight the efficacy of implementing the $\tau$-decay in Unc.~(Norm, $\tau \rightarrow 0$) for balancing exploration and exploitation. In Section \ref{sec:isolet-results}, we recreate an experiment from \cite{ji_variance_2012} on the Isolet dataset \cite{uci} that highlighted the superior performance of the VOpt criterion compared to Unc.~(SM). We demonstrate that our proposed uncertainty sampling method Unc.~(Norm, $\tau \rightarrow 0$) achieves results comparable to VOpt on this dataset, essentially correcting the behavior of uncertainty sampling in this empirical study.
In Section \ref{sec:larger-datasets}, we perform active learning experiments on larger, more ``real-world'' datasets. We use the \textbf{MNIST} \cite{lecun-mnisthandwrittendigit-2010}, \textbf{FASHIONMNIST} \cite{xiao2017fashionmnist}, and \textbf{EMNIST} \cite{cohen2017emnist} datasets, and we interpret the original ground-truth classes (e.g. digits 0-9 in \textbf{MNIST}) as \textit{clusters} on which we impose a different classification structure by grouping many clusters into a single class. This creates an experimental setting that necessitates exploration of initially unlabeled ``clusters'' in order to achieve high overall accuracy. We include similar experiments in Section \ref{smsec:imbalanced-results} of the Supplemental Material to verify the performance of the proposed method in the presence of disparate class and cluster sizes.
While most previous work in the active learning literature (both graph-based and neural network classifiers) demonstrate acquisition function performance with only accuracy plots, we suggest another useful quantity for comparing performances. In the larger experiments of Sections \ref{sec:larger-datasets} and \ref{smsec:imbalanced-results}, we plot \textit{the proportion of clusters that have been queried} as a function of active learning iteration. These plots reflect how efficiently an acquisition function explores the clustering structure of the dataset, as captured by how quickly the proportionality curve increases toward 1.0. These cluster exploration plots are especially insightful for assessing performance in low label-rate active learning. An acquisition function that properly and consistently explores the clustering structure of the dataset will achieve an average cluster proportion of 1.0 faster than other acquisition functions and within a reasonable number of active learning queries.
\subsection{Comment regarding comparison to other methods}
We comment here on a few notable active learning methods that are left out of our numerical comparisons. The LAND (Learning by Active Non-linear Diffusion) \cite{murphy_unsupervised_2019} and CAL (Cautious Active Learning) \cite{cloninger_cautious_2021} methods are important works in geometry-inspired active learning. In the LAND algorithm, Murphy and Maggioni use diffusion distances from a random walk interpretation of a similarity graph to select diverse sets of query points that are located in dense regions of the graph. Adjusting a model hyperparameter in the diffusion distances can reveal hierarchical clustering structure in the dataset which can encourage query points to be chosen at different resolution levels of the clustering structure.
In a similar vein, the CAL algorithm by Cloninger and Mhaskar \cite{cloninger_cautious_2021} uses hierarchical clustering structure in the dataset to guide the query set selection process. By constructing a highly localized similarity kernel via Hermite polynomials, query points are selected at various resolution levels. Both the LAND and CAL algorithms have been shown to be effective at selecting query points in pixel classification for hyperspectral imagery applications. We, however, found that the current implementations of these algorithms were unable to scale to our larger experiments.
Furthermore, we suggest that these methods may be more appropriately identified as ``coreset'' selection methods. Such methods leverage the geometry of the underlying dataset (e.g., the diffusion distances as captured by the similarity graph in LAND), but not the set of labels observed at labeled points during the active learning process. This is similar to other coreset methods that have been presented in both coreset and data summarization literature \cite{sener_active_2018, vahidian_coresets_2020, mirzasoleiman2017big}. In contrast, our uncertainty-based criterion in this work combines both geometric information about the data as captured by the similarity graph structure and the observed labels at each labeled point via the output classification at each iteration. This makes our method more similar to the primary flavor of active learning methods. For these two reasons, we omit direct numerical comparison with these other methods.
\subsection{Toy examples} \label{sec:toy-experiments}
We first illustrate our claim regarding our minimum norm uncertainty sampling criterion for graph-based active learning with synthetic datasets that are directly visualizable (i.e., the data lies in only two dimensions). The first experiment---which we refer to as the \textbf{Blobs} experiment---illustrates how a non-zero value for $\tau$ in the initial phase of active learning is crucial for ensuring exploration of the dataset. The second experiment---which we refer to as the \textbf{Box} experiment---illustrates the need to decrease the value of $\tau$ to ensure the transition from exploration to exploitation. These experiments also allow us to directly observe the qualitative characteristics of the active learning query choices in uncertainty sampling.
\subsubsection{Blobs experiment} \label{sec:blobs-experiment}
\begin{figure}
\centering
\subfigure[Ground Truth]{\includegraphics[clip=True,trim=120 60 120 60,width=0.4\textwidth]{imgs/toy_figures/blobs/groundtruth.jpg}}
\subfigure[Accuracy Results]{\includegraphics[width=0.5\textwidth]{imgs/toy_figures/blobs_rwll_accplot.jpg}}
\caption{Ground Truth (a) and Accuracy Results (b) for \textbf{Blobs} experiment. Notice that Unc.~(SM) achieves very poor overall accuracy. We show in Figure \ref{fig:unc-blobs-combined} that this is due to premature exploitation.}
\label{fig:gt-blobs}
\end{figure}
The \textbf{Blobs} dataset is comprised of eight Gaussian clusters, each of equivalent size (300) and variance ($\sigma^2 = 0.17^2$), whose centers (i.e., means) lie evenly spaced apart on the unit circle. That is, each cluster $\Omega_i$ is defined by randomly sampling 300 points from a Gaussian with mean $\mu_i = (\cos(\pi i /4), \sin(\pi i/4))^T \in \mathbb R^2$ and standard deviation $\sigma_i= \sigma = 0.17$. The classification structure of the clusters is then assigned in an alternating fashion, as shown in Figure \ref{fig:gt-blobs}(a). In each individual run of the experiment, one initially labeled point per \textit{class} combine to be the starting labeled set, and then 100 active learning query points are selected sequentially via a specified acquisition function. Different acquisition functions then define different runs of the experiment.
For each acquisition function, we ran 10 experiments with different initially labeled points. The average accuracy at each iteration of an experiment is plotted is Figure \ref{fig:gt-blobs}(b). The main purpose of this experiment is to compare and contrast the characteristics of the query points selected by Unc.~(SM), Unc.~(Norm), and Unc.~(Norm, $\tau \rightarrow 0$). For comparison and reference in these toy experiments, we include the results of using the VOpt\cite{ji_variance_2012} acquisition function as well as Random sampling (i.e., select $x_i^\ast \in \mathcal U$ with uniform probability over $\mathcal U$ at each iteration).
\begin{figure}
\centering
\subfigure[Unc.~(SM), Initial]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/unc_rwll_afvals_0.jpg}}
\subfigure[Unc.~(SM), Iter 9]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/unc_rwll_afvals_8.jpg}}
\subfigure[Unc.~(SM), Iter 100]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/unc_rwll_afvals_99.jpg}} \\
\subfigure[Unc.~(Norm), Initial]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/uncnorm_rwll0010_afvals_0.jpg}}
\subfigure[Unc.~(Norm), Iter 9]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/uncnorm_rwll0010_afvals_8.jpg}}
\subfigure[Unc.~(Norm), Iter 100]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/uncnorm_rwll0010_afvals_99.jpg}} \\
\subfigure[Unc.~(Norm, $\tau \rightarrow 0$), Initial]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/uncnormdecaytau_rwll0010_afvals_0.jpg}}
\subfigure[Unc.~(Norm, $\tau \rightarrow 0$), Iter 9]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/uncnormdecaytau_rwll0010_afvals_8.jpg}}
\subfigure[Unc.~(Norm, $\tau \rightarrow 0$), Iter 100]{\includegraphics[clip=True,trim=120 60 100 60,width=0.32\textwidth]{imgs/toy_figures/blobs/uncnormdecaytau_rwll0010_afvals_99.jpg}}
\caption{Acquisition Function Values for Unc.~(SM), Unc.~(Norm), and Unc.~(Norm, $\tau \rightarrow 0$) at different stages of the \textbf{Blobs} experiment. Labeled points are marked as red stars and brighter regions of the heatmap indicate higher acquisition function values.}
\label{fig:unc-blobs-combined}
\end{figure}
The main observation from this experiment is how poorly Unc.~(SM) performs, as it only attains an overall accuracy of roughly 62\% as the average over the trials. In Figure \ref{fig:unc-blobs-combined}(a-c), we show one trial's acquisition function values heatmap at three different stages of the active learning process using Unc.~(SM). We observe that the active learning queries have been primarily focused on the boundaries between a few clusters, while missing other clusters completely. At each iteration, the heatmap of acquisition function values has only focused on the current classifier's decision boundary which can lead to missing such clusters. In essence, we would qualify the behavior here as ``premature exploitation'', prior to proper exploration of the dataset.
In contrast, Figures \ref{fig:unc-blobs-combined} (d-i) demonstrate how the ``minimum norm'' uncertainty acquisition functions properly explore the extent of the geometric clustering structure. Both have sampled from every cluster in the ring. It is instructive to further see though that Unc.~(Norm)---which employs a fixed value of $\tau > 0$ at every iteration---has not sampled more frequently \textit{between} clusters by the end of the trial. We may characterize this behavior as not transitioning to proper exploitation of cluster boundaries. On the other hand, in Figure \ref{fig:unc-blobs-combined}(i), we see that by using this minimum norm uncertainty sampling \textit{with decaying values of $\tau \rightarrow 0$} we more frequently sample at the proper cluster boundaries after having sampled from each cluster.
\subsubsection{Box experiment} \label{sec:box-experiment}
\begin{figure}
\centering
\subfigure[Ground Truth]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/groundtruth.jpg}}
\hspace{8mm}
\subfigure[Accuracy Results]{\includegraphics[width=0.5\textwidth]{imgs/toy_figures/box_rwll_accplot.jpg}}
\caption{Ground Truth (a) and Accuracy Results (b) for \textbf{Box} experiment. Notice that Unc.~(Norm) achieves suboptimal overall accuracy. We show in Figure \ref{fig:unc-box-combined}(f) that the distribution of query points later in the active learning process reflect a lack of transition to exploitation.}
\label{fig:gt-box}
\end{figure}
The \textbf{Box} dataset is simply a 65 $\times$ 65 lattice of points on the unit square, with removing points that lie within a thin, vertical band centered at $x = 0.3$ which also defines the class boundary line (Figure \ref{fig:gt-box}). In contrast to the \textbf{Blobs} experiment, the \textbf{Box} experiment illustrates the need to transition from exploration to exploitation, and how this is accomplished by decreasing $\tau \rightarrow 0$. In the accuracy plot (Figure \ref{fig:gt-box}(b)), notice how the accuracy achieved by Unc.~(Norm) levels off at a \textit{lower} overall accuracy than both Unc.~(SM) and Unc.~(Norm $\tau \rightarrow 0$). Figure \ref{fig:unc-box-combined} demonstrates that this is due to ``over exploration'' of the dataset instead of transitioning to refining the decision boundary between classes. Active learning seeks to balance exploration versus exploitation while still being sample efficient, making as few active learning queries as possible.
\begin{figure}
\centering
\subfigure[Unc.~(SM), Initial]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/unc_rwll_afvals_0.jpg}}
\hspace{2mm}
\subfigure[Unc.~(SM), Iter 15]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/unc_rwll_afvals_15.jpg}}
\hspace{2mm}
\subfigure[Unc.~(SM), Iter 50]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/unc_rwll_afvals_50.jpg}} \\
\subfigure[Unc.~(Norm), Initial]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/uncnorm_rwll0010_afvals_0.jpg}}
\hspace{2mm}
\subfigure[Unc.~(Norm), Iter 15]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/uncnorm_rwll0010_afvals_15.jpg}}
\hspace{2mm}
\subfigure[Unc.~(Norm), Iter 50]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/uncnorm_rwll0010_afvals_50.jpg}} \\
\subfigure[Unc.~(Norm, $\tau \rightarrow 0$), Initial]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/uncnormdecaytau_rwll0010_afvals_0.jpg}}
\hspace{2mm}
\subfigure[Unc.~(Norm, $\tau \rightarrow 0$), Iter 15]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/uncnormdecaytau_rwll0010_afvals_15.jpg}}
\hspace{2mm}
\subfigure[Unc.~(Norm, $\tau \rightarrow 0$), Iter 50]{\includegraphics[clip=true,trim= 120 60 120 60,width=0.3\textwidth]{imgs/toy_figures/box/uncnormdecaytau_rwll0010_afvals_50.jpg}}
\caption{Acquisition Function Values for Unc.~(SM), Unc.~(Norm), and Unc.~(Norm, $\tau \rightarrow 0$) at different stages of the \textbf{Box} experiment. Labeled points are marked as red stars and brighter regions of the heatmap indicate higher acquisition function values.}
\label{fig:unc-box-combined}
\end{figure}
As shown in Figures \ref{fig:unc-box-combined} (a-f), both Unc.~(SM) and Unc.~(Norm, $\tau \rightarrow 0$) more efficiently sample the decision boundary between the two classes in this \textbf{Box} dataset. Due to the very simple structure of the dataset, purely exploiting decision boundary information---as done by Unc.~(SM)---is optimal. In contrast, Unc.~(Norm, $\tau \rightarrow 0$) ensures to sparsely explore the extent of the right side of the box \textit{prior to} exploiting the decision boundary. This is due to the decreasing value of $\tau$ over the iterations, and allows for a straightforward transition between exploration and exploitation. We set the value of $K=8$ for the $tau$-decay schedule so that by 8 active learning queries we have transitioned to exploitation.
\subsubsection{Overall observations}
From the toy experiments presented in Sections \ref{sec:blobs-experiment} and \ref{sec:box-experiment}, we see that the minimum norm uncertainy sampling \textit{with decaying values of $\tau$} has the desired behavior for a sample-efficient criterion that both explores and exploits during the active learning process. Ensuring this behavior in uncertainty sampling is also desirable because of the relatively light computational complexity that uncertainty sampling incurs. We now demonstrate on more complicated, ``real-world'' datasets the effectiveness of minimum norm uncertainty sampling in graph-based active learning.
\subsection{Isolet example} \label{sec:isolet-results}
We demonstrate in this section that minimum norm uncertainty sampling in the PWLL-$\tau$ model overcomes the previously negative results that have characterized uncertainty sampling. In \cite{ji_variance_2012}, the authors introduced the Variance Optimization (i.e., VOpt) acquisition function which quantifies how much unlabeled points would decrease the variance of the conditional distribution over Laplace learning node functions. They showcased this acquisition function on the Isolet spoken letter dataset\footnote{Accessed via \url{https://archive.ics.uci.edu/ml/datasets/isolet}.} from the UCI repository \cite{uci}, which contains 26 different classes. They compared against smallest margin uncertainty sampling (Unc.~(SM)) among other acquisition functions. Of particular interest to the current work is how poorly Unc.~(SM) performed on this task, resulting in significantly worse accuracies than even random sampling.\footnote{We refer the reader to original paper \cite{ji_variance_2012} for more details.} We demonstrate that similar---even superior---performance can be attained on this task by simply using this minimum norm uncertainty sampling (Unc.~(Norm)) that is more appropriate for low-label rate active learning.
\begin{figure}[!h]
\centering
\subfigure[Accuracy]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures/isolet_laplace_accplot.jpg}}
\subfigure[Cluster Proportion]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures/isolet_rwll0100_clusterplot.jpg}}
\caption{Accuracy Results (a) and Cluster Proportion (b) plots for \textbf{ISOLET} dataset. Accuracies shown here are in the original Laplace learning model \cite{zhu_semi-supervised_2003} for a more direct comparison with the results from \cite{ji_variance_2012}.}
\label{fig:isolet-results}
\end{figure}
In Figure \ref{fig:isolet-results}(a), we plot the accuracy results of an active learning test that mimics the setup of the full Isolet dataset (with 26 classes) as described in \cite{ji_variance_2012}. In addition to recreating the results of their test, we have added the results of using Unc.~(Norm, $\tau \rightarrow 0$) for comparison. Please note that the accuracies reported here are in the original Laplace learning model of \cite{zhu_semi-supervised_2003}, not the reweighted Laplace learning \cite{calder2020properly} model that is the focus of the rest of the paper and experimental results. We only add the result of Unc.~(Norm, $\tau \rightarrow 0$) to allow for clearer plots, as Unc.~(Norm) performed nearly identically to Unc.~(Norm, $\tau \rightarrow 0$).
Each trial (out of 10 total trials) for an acquisition function begins with only a single initially labeled point and 100 query points are thereafter selected sequentially. Thus, only one class has been sampled from at the start of each trial. In Figure \ref{fig:isolet-results}(b), we report the average fraction of ``clusters'' that have been sampled by each iteration of the active learning process. In this case, we treat each individual class as a different cluster. Such a plot demonstrates the explorative capabilities of the acquisition functions as applied to this dataset.
Similar to the test reported in \cite{ji_variance_2012}, smallest margin uncertainty sampling (Unc.~(SM)) performs very poorly at this task, both in terms of accuracy and cluster exploration. Our proposed minimium norm uncertainty sampling (Unc.~(Norm, $\tau \rightarrow 0$)), however, outperforms VOpt in terms of cluster exploration and provides very similar accuracy results. As another point of comparison, the calculation of VOpt requires either an eigendecomposition or a full inversion of the graph Laplacian matrix, whereas Unc.~(Norm, $\tau \rightarrow 0$) merely requires the current output of the reweighted Laplace learning model. These results provide encouraging evidence for the utility of the proposed method of uncertainty sampling in this current work.
\subsection{Larger datasets} \label{sec:larger-datasets}
In this section, we present the results of active learning experiments for multiclass classification problems derived from the \textbf{MNIST} \cite{lecun-mnisthandwrittendigit-2010}, \textbf{FASHIONMNIST} \cite{xiao2017fashionmnist}, and \textbf{EMNIST} datasets \cite{cohen2017emnist}. We construct similarity graphs for each of these datasets by first embedding the points via the use of variational autoencoders (VAE) \cite{kingma2013auto, kingma_introduction_2019} that were previously trained\footnote{The representations for \textbf{MNIST} and \textbf{FASHIONMNIST} are available in the GraphLearning package \cite{graphlearning}, while the code used to train the VAE for \textbf{EMNIST} is available in our Github repo \url{https://github.com/millerk22/rwll_active_learning}.} in an unsupervised fashion, similar to \cite{calder_poisson_2020}.
Since a main crux of the present work is to ensure \textit{both} exploration of clusters in a dataset and exploitation of cluster boundaries, we adapt the classification structure of the above datasets to require both. That is, we take the ``true'' class labelings $y_i \in \{0, 1, \ldots, C\}$ (e.g. digits 0-9 for \textbf{MNIST}) and reassign them to one of $k < C$ classes by taking $y_i^{new} \equiv y_i \operatorname{mod} k$; see Table \ref{table:mod-classes} below.
\begin{table*}[h!]\centering \label{table:mod-classes}
\ra{1.3}
\begin{tabular}{@{}cccccc@{}}\toprule
Resulting Mod Class & 0 & 1 & 2 & 3 & 4 \\
\midrule
\textbf{MNIST} & 0,3,6,9 & 1,4,7 & 2,5,8 & - & -\\
\textbf{FASHIONMNIST} & 0,3,6,9 & 1,4,7 & 2,5,8 & - & -\\
\textbf{EMNIST} & 0,5,\ldots,45 & 1,6,\ldots,46 & 2,7,\ldots,42 & 3,8,\ldots,43 & 4,9,\ldots,44\\
\bottomrule
\end{tabular}
\caption{Mapping of ground truth class label to $\operatorname{mod} k$ labeling for experiments of Section \ref{sec:larger-datasets}. Each ground truth class, is interpreted as a different ``cluster'' and the resulting class structure for the experiments have multiple clusters per class. For \textbf{MNIST} and \textbf{FASHIONMNIST}, there 10 total ground truth classes and we take labels modulo $k=3$. For \textbf{EMNIST}, there are 47 total ground truth classes and we take labels modulo $k=5$.}
\end{table*}
For each trial of an acquisition function, we select one initially labeled point per \textit{``modulo''} class; therefore, only a subset of ``clusters'' (i.e., the original true classes) has an initially labeled point. In order to perform active learning successfully in these experiments, query points chosen by the acquisition function over the trial must sample from each cluster. In this way, we have created an experimental setup with high-dimensional datasets with potentially more complicated clustering structures wherein we test and compare the following acquisition functions: Uncertainty Sampling (SM), Unc.~(Norm), Unc.~(Norm, $\tau \rightarrow 0$), Random, VOpt \cite{ji_variance_2012} (see Remark \ref{remark:vopt}), $\Sigma$-Opt \cite{ma_sigma_2013} (also see Remark \ref{remark:vopt}), and MCVOpt \cite{miller_efficient_2020}. We perform 10 trials for each acquisition function, where each trial begins with a different initially labeled subset. To clarify, trials begin with only 3 labeled points in the \textbf{MNIST} and \textbf{FASHIONMNIST} experiments and with only 5 labeled points in the \textbf{EMNIST} experiments.
In the left panel of Figures \ref{fig:mnist-results}-\ref{fig:emnist-results}, we show the accuracy performance of each acquisition function averaged over the 10 trials. The right panels of each of these figures display the average proportion of clusters that have been sampled by the acquisition functions at each iteration of the active learning process. We refer to these plots as ``Cluster Exploration'' plots since they directly assess the explorative capabilities of the acquisition functions in question.
We observe that across these experiments, both Unc.~(Norm) and Unc.~(Norm, $\tau \rightarrow 0$) consistently achieve the best accuracy and cluster exploration results. It is somewhat surprising that without decaying $\tau$, the Unc.~(Norm) acquisition function seems to perform the best even after each cluster has been explored. The experiments in Section \ref{sec:toy-experiments} suggest that the optimal performance in the exploitation phase of active learning would require taking $\tau \rightarrow 0$. We hypothesize that the clustering structure of high-dimensional data---like these datasets---is much more complicated than our intuition would suggest from analyzing toy and other visualizable (i.e., 1D, 2D, or 3D) datasets. Regardless, we see that minimum norm uncertainty acquisition function consistently outperforms other acquisition functions in these low-label rate active learning experiments. We emphasize again here that the computational cost of uncertainty sampling acquisition functions make them especially useful for active learning.
\begin{remark} \label{remark:vopt}
Due to the large nature of these datasets, computing the original VOpt and $\Sigma$-Opt criterions are inefficient (and often intractable) since it requires computing the inverse of a perturbed graph Laplacian matrix; this inverse is dense and burdensome to store in memory. We initially used an approximate criterion that utilizes a subset of eigenvalues and eigenvectors of the graph Laplacian, similar to what was done in \cite{miller_model-change_2021}. However, we noticed significantly poor results on the \textbf{MNIST} and \textbf{FASHIONMNIST} experiments seemingly due to the spectral truncation with a resulting oversampling of a single cluster during the active learning process.
As an alternative to the spectral truncation, we performed a full calculation of these acquisition functions on a small, random subset of $500$ unlabeled points at each active learning iteration. This performed significantly better than the spectral truncation in these two experiments, and so we report their performance in this section. In Figures \ref{fig:mnist-results} and \ref{fig:fashionmnist-results} we refer to this adaptation with the suffix ``(Full)''; e.g., we name its results by ``VOpt (Full)''. The small choice of unlabeled points on which to evaluate the acquisition function in the full setting is due to the burdensome computation needed at each step that scales with the size of this subset; at this reported choice of $500$ points each active learning iteration already takes roughly 6 minutes to complete. Due to its even greater size, we do not perform this computation on the \textbf{EMNIST} dataset, and furthermore the performance of the approximate VOpt already achieves comparable accuracy to the other reported methods in this dataset.
\end{remark}
\begin{figure}
\centering
\subfigure[Accuracy]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures_full/mnist-mod3_rwll_accplot_voptfull.jpg}}
\subfigure[Cluster Proportion]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures_full/mnist_clusterplot_voptfull.jpg}}
\caption{Accuracy Results (a) and Cluster Proportion (b) plots for \textbf{MNIST} dataset.}
\label{fig:mnist-results}
\end{figure}
\begin{figure}
\centering
\subfigure[Accuracy]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures_full/fashionmnist-mod3_rwll_accplot_voptfull.jpg}}
\subfigure[Cluster Proportion]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures_full/fashionmnist_clusterplot_voptfull.jpg}}
\caption{Accuracy Results (a) and Cluster Proportion (b) plots for \textbf{FASHIONMNIST} dataset. }
\label{fig:fashionmnist-results}
\end{figure}
\begin{figure}
\centering
\subfigure[Accuracy]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures_full/emnist-mod5_rwll_accplot_voptfull.jpg}}
\subfigure[Cluster Proportion]{\includegraphics[width=0.49\textwidth]{imgs/large_dataset_figures_full/emnist_clusterplot_voptfull.jpg}}
\caption{Accuracy Results (a) and Cluster Proportion (b) plots for \textbf{EMNIST} dataset. }
\label{fig:emnist-results}
\end{figure}
\section{Continuum analysis of active learning} \label{sec:theory}
We now study our active learning approach rigorously through its continuum limit. As was shown in \cite{calder2020properly}, the continuum limit of \eqref{eq:rw-lap-learning} is the family of singularly weighted elliptic equations
\begin{equation}\label{eq:rw_lap_continuum}
\left\{
\begin{aligned}
\tau u_i - \rho^{-1}\div\left(\gamma\rho^2 \nabla u_i \right) &= 0,&& \text{in } \Omega \setminus \L \\
u_i &= 1,&& \text{on } \L_i\\
u_i&= 0,&& \text{on } \L\setminus \L_i,
\end{aligned}
\right.
\end{equation}
where $\rho$ is the density of the datapoints, $\gamma$ is the singular reweighting, described in more detail below, $\L_i\subset \Omega$ are the labeled points in the $i^{\rm th}$ class, and $\L = \cup_{i=1}^C \L_i$ the locations of all labeled points. The notation $\nabla$ refers to the gradient vector and $\div$ is the divergence. The solutions $u_i$ also satisfy the homogeneous Neumann boundary condition $\nabla u \cdot \nu = 0$ on $\partial \Omega$, where $\nu$ is the outward unit normal vector to $\Omega$, but we omit writing this as it is not directly used in any of our arguments. We assume the sets $\L_i$ are all finite collections of points. The classification decision for any point $x\not\in \L$ is given by
\[\ell(x) = \argmax_{1 \leq i \leq C} u_i(x).\]
The continuum version of the uncertainty sampling acquisition function is then given by
\begin{equation}\label{eq:acq_cont}
\mathcal{A}(x) = \sqrt{u_1(x)^2 + u_2(x)^2 + \cdots + u_C(x)^2}.
\end{equation}
The aim in this section is to use continuum PDE analysis to rigorously establish the exploration versus exploitation tradeoff in uncertainty norm sampling, and illustrate how it depends on the choice of the decay parameter $\tau$.
\subsection{Illustrative 1D continuum analysis} \label{sec:1d-theory}
We proceed at first with an analysis of the continuum equations \eqref{eq:rw_lap_continuum} in the one dimensional setting, where the equations are ordinary differential equations (ODEs). The analysis is straightforward and the reweighting \eqref{eq:gamma} is no longer necessary for well-defined continuum equations with finite labeled data. The conclusions are insightful for the subsequent generalization to higher dimensions in Section \ref{sec:exploration}.
Consider an interval $\Omega = (x_{min}, x_{max}) \subset \mathbb R$ with density $0 < \rho_{min} \le \rho(x) \le \rho_{max} < +\infty$.
Assume a binary classification structure on this dataset, and further assume we have been given at least one labeled point per class. Let the pairs $\{(x_i, y_i)\}_{i=1}^\ell \subset \Omega \times \{0,1\}$ be the input-class values for the currently labeled points. Without loss of generality, let us assume that the indexing on these labeled points reflects their ordering in the reals; namely, $x_i < x_{i+1}$ for each $i \le \ell -1$. For ease in our discussion, we also assume that $x_1 = x_{min}$ and $x_\ell = x_{max}$, the endpoints of the domain (see Figure \ref{fig:1d-init}).
\begin{figure}[h]
\centering
\includegraphics[width=0.9\textwidth]{imgs/1d-init-labeled.pdf}
\vspace{-1em}
\caption{Visualization of 1D continuum example setup. The density $\rho(x)$ is plotted in gray, while the labeled points $x_1, x_2, x_3, x_4$ are plotted where the corresponding label is denoted by $\times$ or a solid dot. $\R_s$ marks the length between two similarly labeled points, while $\R_o$ marks the length between two oppositely labeled points.}
\label{fig:1d-init}
\end{figure}
Solving the PWLL-$\tau$ equation\footnote{Without the reweighting \eqref{eq:gamma} due to the simple geometry in one dimension.} \eqref{eq:rw_lap_continuum} on $\Omega$ can be broken into a number of subproblems defined on the intervals $(x_1, x_2), \ldots, (x_{\ell-1}, x_{\ell}) \subset \mathbb R$, with boundary conditions determined by the corresponding labels of the endpoints $x_i$. There are three separate kinds of sub-problems that need to be solved, as determined by these boundary conditions, that we will term (1) the \textit{oppositely labeled problem} (when $y_i \not= y_{i+1}$) and (2) the \textit{similarly labeled problem} (when $y_i = y_{i+1}$).
Given the current state of the labeled data, the active learning process selects a new query point $x^\ast = \argmin_{x \in \Omega} \ \mathcal A(x)$ via the minimum norm acquisition function \eqref{eq:acq_cont}.
We can quantify the explorative behavior of our acquisition function \eqref{eq:acq_cont} by comparing the minimizers of $\mathcal A(x)$ in the different subintervals $(x_i, x_{i+1})$. Due to the simple geometry of the problems in one dimension, our analysis reduces to a pairwise comparison of $\mathcal A(x)$ on (i) an interval of length $\R_o$ between \textit{oppositely labeled points} and (ii) an interval of length $\R_s$ between \textit{similarly labeled points}. Defining $\mathcal A_s $ and $\mathcal A_o$ as the acquisition function on the respective oppositely and similarly labeled problem subintervals, we compare the values $\min \mathcal A_o(x)$ and $\min \mathcal A_s(x)$ on said subintervals.
In this simple one-dimensional problem, we may characterize ``explorative'' query points as residing in relatively \textit{large} intervals between labeled points, regardless of the labels of the endpoints. Conversely, we characterize ``exploitative'' query points as residing between \textit{oppositely labeled points that are close together}. In Figure \ref{fig:1d-init}, exploration would correspond to sampling in $(x_1,x_2)$ or $(x_3,x_4)$, while exploitation would correspond to sampling in $(x_2, x_3)$.
The acquisition function \eqref{eq:acq_cont} is directly a function of the magnitudes of the solutions to \eqref{eq:rw_lap_continuum} with the corresponding boundary conditions. Due to the boundary conditions intervals between oppositely labeled points, the solutions to \eqref{eq:rw_lap_continuum} necessarily interpolate between $0$ and $1$ along the interval (see Figure \ref{fig:solutions-acqfuncs}(a)). In the oppositely labeled problem, however, there is only decay in the solution $v_0$ that solves \eqref{eq:rw_lap_continuum} with labels $y(x_i) = y(x_{i+1}) = 1$ when $\tau >0$, and the extent of this decay is controlled by the size of $\tau$, the length of the interval $\R_s$, and the density $\rho$ on the interval. As such, we identify how $\tau$ must be chosen in order to produce small acquisition function values between similarly labeled points in relatively large regions as compared to large values in relatively small regions between oppositely labeled points.
\subsubsection{Exploration guarantee in one dimension}
\begin{figure}[t]
\centering
\subfigure[Oppositely Labeled]{\includegraphics[height=12em]{imgs/1d_opp_plot2.jpg}}
\hspace{3em}
\subfigure[Similarly Labeled]{\includegraphics[height=12em]{imgs/1d_same_plot.jpg}}
\caption{Visualization of one-dimensional solutions to \eqref{eq:rw_lap_continuum} in oppositely (a) and similarly (b) labeled regions. Solutions $u_0,u_1$ (blue, green lines) to \eqref{eq:rw_lap_continuum} are shown in panel (a) when endpoints have opposite labels, while $v_0,v_1$ (red, orange lines) to \eqref{eq:rw_lap_continuum} are shown in panel (b) when endpoints have the same labels. The background density $\rho$ in each respective region is shown in gray, and the acquisition function value at the midpoint of the interval is shown as a black dot. The minimum acquisition function value occurs at the midpoint if the density is symmetric, which we show here for simpler presentation. }
\label{fig:solutions-acqfuncs}
\end{figure}
In Supplemental Material Section \ref{smsec:warmup-const-density}, we first derive an explicit condition in the case that the density $\rho(x) \equiv \rho$ is constant to guarantee that $\min \mathcal A_s(x) < \min \mathcal A_o(x)$. As long as the length between oppositely labeled points ($\R_o$) is \textit{small} enough compared to the length between similarly labeled points ($\R_o$),
then this gives the condition that the quantity $\frac{\tau\R^2_s}{\rho}$ must be relatively \textit{large}. We can then generalize this result to cases when the density $\rho(x)$ is no longer constant, but rather obeys some mild assumptions. Namely, we give the mild assumption that the density $\rho(x)$ (i) is sufficiently smooth, (ii) is \textit{symmetric about the midpoint of the interval} (see Assumption \ref{assumption:symmetry}) between similarly labeled points, and (iii) obeys \textit{a bounded derivative condition at the ends of the interval} (see Assumption \ref{assumption:end-intervals}) between oppositely labeled points. Under these mild assumptions we give the following simplified guarantee on exploration, which we prove rigorously in Section \ref{smsec:compare-1d-bounds}.
\begin{proposition}[Simplified version of Proposition \ref{smprop:1d-result}] \label{prop:1d-result}
Suppose that the density $\rho(x)$ satisfies Assumption \ref{assumption:end-intervals} in the oppositely labeled problem region and Assumption \ref{assumption:symmetry} in the similarly labeled problem region. Let the interval length $\R_o$ be relatively small compared to $\R_s$; i.e., $\R_o = \beta \R_s$ for some $\beta \le \frac{1}{4}$. Then we are ensured that
\[
\min_{x}\ \mathcal A_s(x) < \min_{x } \ \mathcal A_o(x)
\]
as long as $\tau > 0$ and $\R_s$ jointly satisfy the following inequality
\begin{equation} \label{eq:tau-condition-messy-simplifieid}
\R_s^2 \left( C_0(\rho_s) \sqrt{\tau} - C_1(\rho_o) \beta^2 \tau \right) \ge 8\ln 2,
\end{equation}
where $C_0(\rho_s)$ and $C_1(\rho_o)$ are constants that depend on the density $\rho$ on the similarly and oppositely labeled intervals, respectively denoted $\rho_s$ and $\rho_o$.
\end{proposition}
Figure \ref{fig:solutions-acqfuncs} illustrates the main idea of Proposition \ref{prop:1d-result}. As long the similarly labeled region (Figure \ref{fig:solutions-acqfuncs}(b) has significantly large regions where the density $\rho(x)$ is sufficiently small compared to the oppositely labeled region (Figure \ref{fig:solutions-acqfuncs}(a)), then we can be assured that choosing $\tau > 0$ large enough will result in query points between similarly labeled points that are relatively far from each other.
This relationship is also summarized in the inequality \eqref{eq:tau-condition-messy-simplifieid}, which highlights that larger $\tau$ and interval length $\R_s$ are necessary in order to satisfy said inequality. This inequality simply quantifies how $\tau>0$ must be chosen in order to select query points between similarly labeled points that are relatively far away from each other; i.e., to ensure exploration of such regions that would otherwise be missed if $\tau$ we not sufficiently large. The effect of the relative ratio of the intervals, $\beta$, is also highlighted in \eqref{eq:tau-condition-messy-simplifieid}; namely, the smaller the region between oppositely labeled points the easier it is to satisfy this inequality. Intuitively one can see that if $\beta$ is not small, then the region between oppositely labeled points is relatively large and it will be more difficult to satisfy the stated inequality. However, in this case querying between oppositely labeld points that are relatively distant from each other is desirable and would be characterized as explorative.
\subsection{Exploration bounds in arbitrary dimensions}
\label{sec:exploration}
In this section, we show how larger values for $\tau$ lead to explorative behaviour in higher dimensional problems. In particular, we show that the acquisition function $\mathcal{A}(x)$ is small on unexplored clusters, and large on sufficiently well-explored clusters. This ensures that adequate exploration occurs before exploitation.
Let us remark that the reweighting term $\gamma$ must be sufficiently singular near the labels $\L$ in order to ensure that \eqref{eq:rw_lap_continuum} is well-posed. We recall from \cite{calder2020properly} that we require that $\gamma$ has the form
\begin{equation}\label{eq:gamma}
\gamma(x) = 1 + \dist(x,\L)^{-\alpha},
\end{equation}
where $\alpha > d-2$. In practice, we choose $\gamma$ as the solution of the graph Poisson equation \eqref{eq:gamma_eq_discrete} introduced earlier. To make the analysis in this section tractable, we assume here that $\gamma$ satisfies \eqref{eq:gamma}, as was assumed in \cite{calder2020properly}.
We emphasize here that without the singular reweighting $\gamma$, the equation \eqref{eq:rw_lap_continuum} is ill-posed when the label set $\L$ is finite, and as such, there is no continuum version of active learning for us to study.
For an open set $A\subset \R^d$ and $r>0$ we define the nonlocal boundary $\partial_r A$ as
\[\partial_r A = \overline{(A + B_r)} \setminus A.\]
The nonlocal boundary is essentially a tube of radius $r$ surrounding the set $A$. The usual boundary is obtained by taking $r=0$, so $\partial A=\partial_0 A$.
Our first result concerns upper bounds on the acquisition function in an unexplored cluster.
\begin{theorem}\label{thm:explore_new_cluster}
Let $\tau\geq 0$, $s,R>0$ and $\mathcal{D} \subset \Omega$ with $\partial_{2s} \mathcal{D} \subset \Omega$ and $\L\cap (\mathcal{D}+B_{R+2s})=\varnothing$. Let
\[\delta = \max_{\partial_{2s} \mathcal{D}}\rho.\]
Assume that
\begin{equation}\label{eq:explore_cond}
\sqrt{\frac{\tau}{\delta}} \geq 3\left(\tfrac{d}{s} + 2\|\nabla \log \rho\|_{L^\infty(\partial_s \mathcal{D})}\right)(1+R^{-\alpha}) + 3R^{-\alpha-1}.
\end{equation}
Then it holds that
\begin{equation}\label{eq:acq_upper_bound}
\sup_{\mathcal{D}}\mathcal{A} \leq \sqrt{C}\exp\left(-\frac{s}{4}\sqrt{\frac{\tau}{\delta}}\right).
\end{equation}
\end{theorem}
\begin{remark}\label{rem:upper_bound}
Theorem \ref{thm:explore_new_cluster} shows that the acquisition function $\mathcal{A}$ is exponentially small on an unexplored cluster $\mathcal{D}$ provided there is a thin surrounding set $\partial_s \mathcal{D}$ of the cluster on which the density is small (less than $\delta$), relatively smooth (so $\nabla \log \rho$ is not too large), and relatively far away from other labeled datapoints (so that $R$ is not too large). All of these smallness assumptions are relative to the size of the ratio $\tau/\delta$ as expressed in \eqref{eq:explore_cond}.
\end{remark}
To ensure that new clusters are explored, we also need to lower bound the acquisition function nearby the existing labeled set. To do this, we need to introduce a model for the clusterability of the dataset. Let $\Omega_1,\Omega_2,\dots,\Omega_C\subset \Omega$ be disjoint sets representing each of the $C$ classes in the dataset. We assume that the labels are chosen from the corresponding class sets, so that $\L_i \subset \Omega_i$ for each $i$. We assume there is a positive separation between the classes, measured by the quantity
\begin{equation}\label{eq:cluster_sep}
\mathcal{S} := \min_{i\neq j}\dist(\Omega_i,\Omega_j).
\end{equation}
The definition of $\mathcal{S}$ implies that $(\Omega_i + B_\mathcal{S})\cap \Omega_j = \varnothing$ for all $i\neq j$. We define the union of the classes as $\Omega'= \cup_{i=1}^C \Omega_i$. We note that we do not have $\Omega'=\Omega$, and it is important that there is room in the background $\Omega\setminus \Omega'$, which provides a separation between classes. The background $\Omega\setminus \Omega'$ may have low density (though we do not assume this below), and can consist of outliers or datapoints that have characteristics of multiple classes and may be hard to classify.
\begin{theorem}\label{thm:explore_labels}
Let $\tau\geq 0$ and $\alpha>d-2$. Assume that $\L_i\subset \Omega_i$ for $i=1,\dots,C$, and let $r>0$ be small enough so that $r \leq \tfrac{1}{4}\mathcal{S}$,
\begin{equation}\label{eq:rtau2}
\tau r^d \leq \frac{1}{2^d9}(\alpha+2-d)^2\inf_{\Omega' + B_{2r}}\rho,
\end{equation}
and
\begin{equation}\label{eq:rcond2}
4\|\nabla \log \rho\|_{L^\infty(\Omega'+B_{2r})}(1 + 2^\alpha r^{\alpha})r + \alpha 2^\alpha r^{\alpha} \leq \tfrac{1}{4}(\alpha + 2-d).
\end{equation}
Then we have
\begin{equation}\label{eq:acq_lower}
\inf_{\L + B_r}\mathcal{A} \geq 1 - 2^{-\frac{1}{2}(\alpha + 2-d)}.
\end{equation}
\end{theorem}
\begin{figure}[!t]
\centering
\includegraphics[width=0.7\textwidth]{imgs/clusters}
\caption{Illustration of the implications of Theorems \ref{thm:explore_new_cluster} and \ref{thm:explore_labels}, and the discussion in Remark \ref{rem:explanation}. The gray regions are the 4 clusters of high density in the dataset, and the density is small $\rho \leq \delta$ between clusters. The current labeled set are the points at the centers of the blue balls. Theorems \ref{thm:explore_new_cluster} and \ref{thm:explore_labels} guarantee that the next labeled point cannot lie in any of the blue balls, which correspond to the dilated label set $\L + B_r$. Once the dilated labels cover the existing clusters, the algorithm is guaranteed to select a point from the unexplored cluster $\mathcal{D}$. The number of labeled points selected from a given cluster during exploration is bounded by its $\frac{r}{2}$-packing number, as explained in Remark \ref{rem:explanation}. }
\label{fig:clusters}
\end{figure}
\begin{remark}\label{rem:explanation}
Let us make some remarks on the applications of Theorems \ref{thm:explore_new_cluster} and \ref{thm:explore_labels}. First, we note that the choice of $s$ in Theorem \ref{thm:explore_new_cluster} can be made proportional to the separation between clusters $\mathcal{S}$ defined in \eqref{eq:cluster_sep}. We can then choose $\tau$ to ensure \eqref{eq:explore_cond} holds in Theorem \ref{thm:explore_new_cluster}, and choose $r>0$ to satisfy the conditions in Theorem \ref{thm:explore_labels}. These choices are all dependent on the domain, the clusterability assumption, and the density, but are independent of the choices of labeled points $\L_i$. Now, combining Theorems \ref{thm:explore_new_cluster} and \ref{thm:explore_labels} we see the condition
\begin{equation}\label{eq:acq_cond}
\sqrt{C}\exp\left(-\frac{\mathcal{S}}{4}\sqrt{\frac{\tau}{\delta}}\right)\leq 1 - 2^{-\frac{1}{2}(\alpha + 2-d)}
\end{equation}
is important. Whenever \eqref{eq:acq_cond} holds, the region $\mathcal{D}$ will be explored \emph{before} a new labeled point is chosen within distance $r$ of any existing labeled point. This is exactly the \emph{exploration} property that we desire in an active learning algorithm. In the early stages, the algorithm should seek to explore new clusters, or continue to sufficiently explore existing clusters. The algorithm will not choose another labeled point within distance $r$ of an existing label until the entire dataset is thoroughly explored, at which point the active learning algorithm should switch to exploitation.
In fact, we can make the statements above a little more precise. Whenever $\L_i + B_r \supset \Omega_i$, we have from Theorem \ref{thm:explore_labels} that
\[\inf_{\Omega_i} \mathcal{A} \geq 1 - 2^{-\frac{1}{2}(\alpha + 2-d)}.\]
In this case, provided \eqref{eq:acq_cond} holds, the algorithm will not select another point from $\Omega_i$ until \emph{all} other cluster have been explored. Since the algorithm also cannot choose a new point within distance $r$ of existing points, the set $\L_i$ is a $r$-net of $\Omega_i$. In particular, the balls $B_{\frac{r}{2}}(z)$ for $z\in \L_i$ are disjoint, so $\L_i+B_{\frac{r}{2}}$ is a $\tfrac{r}{2}$-\emph{packing} of $\Omega_i$. We define an $\epsilon$-\emph{packing} of $\Omega_i$ as a disjoint union of $\epsilon$-balls that are centered at points in $\Omega_i$. Therefore, the maximum number of points in $\L_i$ is given by the $\epsilon$-\emph{packing number} of $\Omega_i$ with $\epsilon=\tfrac{r}{2}$, which is defined as
\[M(\Omega_i,\epsilon) = \max\left\{m \, : \, \text{there exists an } \epsilon \text{-packing of }\Omega_i \text{ with }m\text{ balls.} \right\}.\]
Thus, Theorems \ref{thm:explore_new_cluster} and \ref{thm:explore_labels} show that our uncertainty norm sampling active learning algorithm, in the continuum, cannot select more than the packing number $M(\Omega_i,\tfrac{r}{2})$ of points from $\Omega_i$ until \emph{all} clusters have been explored. The packing number $M(\Omega_i,\tfrac{r}{2})$ depends on the geometry of the cluster $\Omega_i$ and can be large for clusters that are not spherical (e.g., clusters that are ``thin'' and ``long'' in certain directions). These results are illustrated in Figure \ref{fig:clusters}.
The reader may have observed there is an implicit assumption made throughout this remark that there are no labeled points selected from the background region $\Omega\setminus \Omega'$. Indeed, if such outlying datapoints are selected as labeled points, then our results do not hold. In practice, one can perform sampling proportional to a density estimation, or simply remove outliers, to avoid such an issue. We discuss how this can be done in Section \ref{sec:kde}, and we have performed experiments with this. We have found that the results are similar with and without the outlier removal, so we see this as an extra step that one has the option of performing in practice, to maximally align the algorithm with the theory, but we do not see it as a necessary step in practice.
\end{remark}
\section{Conclusion}
We have demonstrated that uncertainty sampling is sufficient for exploration in graph-based active learning by using the norm of the output node function of the PWLL-$\tau$ model as an acquisition function. We provide rigorous mathematical guarantees on the explorative behavior of the proposed acquisition function. This is made possible by the well-posedness of the corresponding continuum limit PDE of the PWLL-$\tau$ model. Our analysis elucidates how the choice of hyperparamter $\tau >0$ directly influences these guarantees; in the one dimensional case this effect is most clearly illustrated. In addition, we provide numerical experiments that further illustrate the effect of both our acquisition function and the hyperparameter $\tau$ on the sequence of active learning query points. Other numerical experiments confirm our theoretical guarantees and demonstrate favorable performance in terms of both accuracy and cluster exploration.
\bibliographystyle{siamplain}
|
2,869,038,156,331 | arxiv | \section{Introduction}
\noindent
In a series of papers \cite{Belinschi-Nica-Eta,Belinschi-Nica-B_t,Belinschi-Nica-Free-BM}, Belinschi and Nica introduced and investigated a family of transformations $\mf{B}_t$. These transformations operate on measures, or more generally on ``non-commutative joint distributions'', i.e. states on non-commutative polynomial algebras. $\mf{B}_t$ are defined using a mixture of free and Boolean convolutions (see Section~\ref{Subsec:Convolutions}). They showed that these transformations form a semigroup, and for $t=1$, $\mf{B}_1$ is exactly the (Bercovici-Pata) bijection $\mf{B}$ between the infinitely divisible distributions in the free and Boolean probability theories. These transformations also have a remarkable relation to the free multiplicative convolution. On the other hand, Belinschi and Nica also proved that for a general state $\psi$,
\[
\State{\psi \boxplus \gamma_t} = \mf{B}_t[\State{\psi}],
\]
where $\boxplus$ is the free (additive) convolution and $\gamma_t$ is the free convolution semigroup of semicircular distributions, the free version of the heat semigroup. The map $\Phi$, which intertwines the actions of $\mf{B}_t$ and the semicircular evolution, can be described in the one-variable case as follows: for a measure $\nu$, $\State{\nu}$ is the measure whose Boolean cumulant generating function is
\[
\eta^{\State{\nu}}(z) = \int_{\mf{R}} \left( \frac{1}{1 - x z} - 1 - x z \right) \frac{1}{x^2} \,d\nu(x).
\]
The formula above is the Boolean version of the Kolmogorov representation (see Theorem~8.5 in \cite{Dur}). $\Phi$ is also a shift on the Jacobi parameters of the measure, see \cite{AnsAppell3}.
\medskip\noindent
Free probability \cite{VDN,Nica-Speicher-book} and Boolean probability theories are two of only three natural non-commutative probability theories (in addition to the usual probability theory; the third theory, monotone probability, also appears in the paper as an auxiliary device). Both the free and the Boolean setting are in fact particular cases of a more general construction for a space with \emph{two} expectations, or more precisely an algebra with two states $(\mc{A}, \phi, \psi)$. This theory is usually called the conditionally free or c-free theory. The objects in it are best described using their two-state cumulants $R^{\phi, \psi}$, which restrict to Boolean cumulants $\eta^\phi$ for $\psi = \delta_0$ and to free cumulants $R^\phi$ for $\psi = \phi$. It is not true that $R^{\phi, \psi}$ are always the free cumulants of some state. However, conversely, if $\psi$ is arbitrary and $\rho$ is a freely infinitely divisible state, then $R^\rho = R^{\phi, \psi}$ for some $\phi$. So we can define the map $\phi = \State{\rho, \psi}$. It is easy to see that $\State{\gamma, \cdot} = \Phi$, where $\gamma$ is the free product of standard semicircular distributions. Moreover, we show that, now for general $\rho_t$,
\[
\State{\rho, \psi \boxplus \rho_t} = \mf{B}_t[\State{\rho, \psi}].
\]
This equation remains somewhat mysterious. However, we also exhibit an operator representation which provides a realization for it. This requires obtaining operator representations for all the ingredient maps, a result which may be of independent interest.
\medskip\noindent
Our original motivation for the study of the maps $\mf{B}_t$ and $\Phi$ came from their connection to the free Meixner distributions and states. Free Meixner distributions were originally defined as measures whose orthogonal polynomials have a special (free Sheffer) form, and in \cite{AnsBoolean} we showed that they have exactly the same characterization in terms of Boolean Sheffer families. In fact, as explained in \cite{AnsAppell3}, the natural home of the free Meixner distributions is in the c-free theory: they are those distributions $\phi$ which, for some $\psi$, have orthogonal c-free Appell (rather than the more general Sheffer) polynomials.
\medskip\noindent
It was noted by Belinschi and Nica and also in \cite{AnsBoolean} that the operations $\mf{B}_t$ take the class of free Meixner states to itself. We introduce a new two-parameter family $\set{\mf{B}_{\alpha, t}}$, which also form a semigroup. For $\alpha = 0$, these are exactly the Belinschi-Nica maps, while for $t=0$ these maps relate the Boolean convolution with the so-called Fermi convolution. These maps also preserve the free Meixner class, and moreover, in one variable using them the class can be generated from a single distribution $\frac{1}{2} (\delta_{-1} + \delta_1)$. In addition,
\[
\State{\rho, \psi \boxplus \rho_t \boxplus \delta_\alpha} = \mf{B}_{\alpha, t}[\State{\rho, \psi}],
\]
The actions on the free Meixner distributions of the operations $\boxplus$, $\mf{B}_{\alpha, t}$, and $\State{\rho, \cdot}$ can be computed explicitly, and lead to the more general results.
\medskip\noindent
The paper is organized as follows. Section~\ref{Section:Preliminaries} introduces the basic notions, in particular the two-state cumulants. Section~\ref{Section:Meixner} describes the appearance of the free Meixner distributions in the c-free theory, and defines $\State{\rho, \psi}$ and $\mf{B}_{\mb{a}, t}$. Section~\ref{Section:Transformations} contains the main results: semigroup property of $\mf{B}_{\mb{a}, t}$, positivity, fixed point, and image descriptions for $\State{\rho, \psi}$, and combinatorial and operator proofs of the equation which relates them.
\medskip\noindent
\textbf{Acknowledgements.}
I would like to thank Andu Nica and Serban Belinschi for a number of discussions which contributed to the development of this paper, and for explaining their work to me. I also thank Laura Matusevich for useful comments.
\section{Preliminaries}
\label{Section:Preliminaries}
\noindent
We will freely use the notions and notation from the Preliminaries section of \cite{AnsBoolean}; here we list the highlights.
\subsection{Polynomials and power series}
Let $\mf{C}\langle \mb{x} \rangle = \mf{C}\langle x_1, x_2, \ldots, x_d \rangle$ be all the polynomials with complex coefficients in $d$ non-com\-mu\-ting variables. They form a unital $\ast$-algebra.
\medskip\noindent
For a non-commutative power series $G$ in
\[
\mb{z} = (z_1, z_2, \ldots, z_d)
\]
define the left non-commutative partial derivative $D_i G$ by a linear extension of $D_i(1) = 0$,
\[
D_i z_{\vec{u}} = \delta_{i u(1)} z_{u(2)} \ldots z_{u(n)}.
\]
\subsection{Algebras and states}
Algebras $\mc{A}$ in this paper will always be complex $\ast$-algebras and, unless stated otherwise, unital. If the algebra is non-unital, one can always form its unitization $\mf{C} 1 \oplus \mc{A}$; if $\mc{A}$ was a $C^\ast$-algebra, its unitization can be made into one as well.
\medskip\noindent
Functionals $\mc{A} \rightarrow \mf{C}$ will always be linear, unital, and $\ast$-compatible. A state is a functional which in addition is positive definite, that is
\[
\state{X^\ast X} \geq 0
\]
(zero value for non-zero $X$ is allowed). A functional on $\mf{C} 1 \oplus \mc{A}$ is \emph{conditionally positive definite} if its restriction to $\mc{A}$ is positive definite. In particular, a functional on $\mf{C} \langle \mb{x} \rangle$ is conditionally positive definite if it is positive definite on polynomials without constant term.
\medskip\noindent
Most of the time we will be working with states on $\mf{C} \langle \mb{x} \rangle$ arising as joint distributions. For
\[
X_1, X_2, \ldots, X_d \in \mc{A}^{sa},
\]
their joint distribution with respect to $\psi$ is a state on $\mf{C} \langle \mb{x} \rangle$ determined by
\[
\state{P(\mb{x})} = \psi^{X_1, X_2, \ldots, X_d}\left[P(x_1, x_2, \ldots, x_d)\right] = \psi \left[P(X_1, X_2, \ldots, X_d)\right].
\]
The numbers $\state{x_{\vec{u}}}$ are the moments of $\phi$. More generally, for $d$ non-commuting indeterminates $\mb{z} = (z_1, \ldots, z_d)$, the series
\[
M(\mb{z}) = \sum_{\vec{u}} \state{x_{\vec{u}}} z_{\vec{u}}
\]
is the moment generating function of $\phi$.
\medskip\noindent
For a probability measure $\mu$ on $\mf{R}$ all of whose moments are finite, its monic orthogonal polynomials $\set{P_n}$ satisfy three-term recursion relations
\begin{equation}
\label{Three-term-recursion}
x P_n(x) = P_{n+1}(x) + \beta_n P_n(x) + \gamma_n P_{n-1}(x),
\end{equation}
with initial conditions $P_{-1} = 0$, $P_0 = 1$. We will call the parameter sequences
\[
(\beta_0, \beta_1, \beta_2, \ldots), (\gamma_1, \gamma_2, \gamma_3, \ldots)
\]
the Jacobi parameter sequences for $\mu$. Generalizations of such parameters for states with MOPS were found in \cite{AnsMulti-Sheffer}.
\subsection{Partitions}
We will denote the lattice of non-crossing partitions of $n$ elements by $\NC(n)$, the corresponding lattice of interval partitions by $\Int(n)$. A class $B \in \pi$ of a non-crossing partition is inner if for $j \in B$,
\[
\exists \ i \stackrel{\pi}{\sim} k \stackrel{\pi}{\not \sim} j: i < j < k,
\]
otherwise $B$ is outer. The collection of all the inner classes of $\pi$ will be denoted $\Inner(\pi)$, and similarly for $\Outer(\pi)$.
\medskip\noindent
From \cite{AnsFree-Meixner}, we take the following notation: for a set $V$,
\[
\NC'(V) = \set{\pi \in \NC(V): \min(V) \stackrel{\pi}{\sim} \max(V)}.
\]
For $\pi \in \NC'(V)$, we will denote $B^o(\pi)$ the unique outer class of $\pi$.
\medskip\noindent
For comparison, we also recall the notation from \cite{Belinschi-Nica-Eta}: for $\pi, \sigma \in \NC(n)$, denote $\pi \ll \sigma$ if
\[
\pi \leq \sigma, \text{ i.e. } i \stackrel{\pi}{\sim} j \Rightarrow i \stackrel{\sigma}{\sim} j
\]
but
\[
\forall B \in \sigma, \min(B) \stackrel{\pi}{\sim} \max(B).
\]
In particular, $\NC'(n) = \set{\pi \in \NC(n): \pi \ll \hat{1}_n}$.
\subsection{Cumulants}
For a state $\phi$, its Boolean cumulant generating function is defined by
\[
\eta^\phi(\mb{z}) = 1 - (1 + M^\phi(\mb{z}))^{-1},
\]
and its coefficients are the Boolean cumulants of $\phi$. They can also be expressed in terms of moments of $\phi$ using the lattice of interval partitions. Similarly, the free cumulant generating function of a state $\psi$ is defined by the implicit equation
\begin{equation}
\label{R-M-w}
M^\psi(\mb{w}) = \CumFun{\psi}{(1 + M^\psi(\mb{w})) \mb{w}},
\end{equation}
where
\[
(1 + M^\psi(\mb{w})) \mb{w} = \Bigl( (1 + M^\psi(\mb{w})) w_1, (1 + M^\psi(\mb{w})) w_2, \ldots, (1 + M^\psi(\mb{w})) w_d \Bigr).
\]
We will frequently, sometimes without comment, use the change of variables
\begin{equation}
\label{z-M-w}
z_i = (1 + M^\psi(\mb{w})) w_i , \qquad w_i = (1 + \CumFun{\psi}{\mb{z}})^{-1} z_i.
\end{equation}
The coefficients of $\CumFun{\psi}{\mb{z}}$ are the free cumulants of $\psi$, and can also be expressed in terms of the moments of $\psi$ using the lattice of non-crossing partitions:
\begin{equation}
\label{R-cumulant-moment}
\psi[x_1 \ldots x_n] = \sum_{\pi \in NC(n)} \prod_{B \in \pi} \Cum{\psi}{x_i : i \in B}.
\end{equation}
\subsection{Two-state cumulants}
The typical setting in this paper will be a triple $(\mc{A}, \phi, \psi)$, where $\mc{A}$ is an algebra and $\phi, \psi$ are functionals on it. By rotation, we can assume without loss of generality that $\phi$ is normalized to have zero means and identity covariance. Then no such assumptions can be made on $\psi$.
\medskip\noindent
We define the conditionally free cumulants of the pair $(\phi, \psi)$ via
\[
\state{x_1 \ldots x_n} = \sum_{\pi \in \NC(n)} \prod_{B \in \Outer(\pi)} \Cum{\phi, \psi}{\prod_{i \in B} x_i} \prod_{C \in \Inner(\pi)} \Cum{\psi}{\prod_{j \in C} x_j}.
\]
Their generating function is
\[
\CumFun{\phi, \psi}{\mb{z}} = \sum_{\vec{u}} \Cum{\phi, \psi}{x_{\vec{u}}} z_{\vec{u}}.
\]
Equivalently (up to changes of variables, this is Theorem~5.1 of \cite{BLS96}), we could have defined the conditionally free cumulant generating function via the condition
\begin{equation}
\label{eta-M-R}
\eta^\phi(\mb{w}) = (1 + M^\psi(\mb{w}))^{-1} \CumFun{\phi, \psi}{(1 + M^\psi(\mb{w})) \mb{w}}.
\end{equation}
\medskip\noindent
For elements $X_1, X_2, \ldots, X_n \in \mc{A}^{sa}$, we will denote their joint cumulants
\[
\Cum{\phi, \psi}{X_1, X_2, \ldots, X_n} = \Cum{\phi^{X_1, X_2, \ldots, X_n}, \psi^{X_1, X_2, \ldots, X_n}}{x_1, x_2, \ldots, x_n}
\]
to be the corresponding joint cumulants with respect to their joint distributions.
\begin{Defn}
\label{Defn:c-free}
Let $(\mc{A}, \phi, \psi)$ be an algebra with two states.
\begin{enumerate}
\item
Subalgebras $\mc{A}_1, \ldots, \mc{A}_d \subset \mc{A}$ are conditionally free, or c-free, with respect to $(\phi, \psi)$ if for any $n \geq 2$,
\[
a_i \in \mc{A}_{u(i)}, \quad i = 1, 2, \ldots, n, \qquad u(1) \neq u(2) \neq \ldots \neq u(n),
\]
the relation
\[
\psi[a_1] = \psi[a_2] = \ldots = \psi[a_n] = 0
\]
implies
\begin{equation}
\label{centered-product}
\state{a_1 a_2 \ldots a_n} = \state{a_1} \state{a_2} \ldots \state{a_n}.
\end{equation}
\item
The subalgebras are $(\phi | \psi)$ free if for $a_1, a_2, \ldots, a_n \in \bigcup_{j=1}^d \mc{A}_j$,
\[
\Cum{\phi, \psi}{a_1, a_2, \ldots, a_n} = 0
\]
unless all $a_i$ lie in the same subalgebra.
\end{enumerate}
\end{Defn}
\noindent
As pointed out in \cite{Boz-Bryc-Two-states}, these properties are \emph{not} equivalent, however they become equivalent under the extra requirement that the subalgebras are $\psi$-freely independent. In any case, throughout the paper we will be working with cumulants and will not actually encounter conditional freeness.
\begin{Example}
If $a, c$ are c-free from $b$, then (Lemma 2.1 of \cite{BLS96})
\[
\state{a b} = \state{a} \state{b},
\]
\[
\state{a b c}
= \state{a} \state{b} \state{c} + \left( \state{a c} - \state{a} \state{c} \right) \psi[b].
\]
\end{Example}
\begin{Example}
\label{Example:Conditional-freeness}
The following are important particular cases of conditional freeness.
\begin{enumerate}
\item
If $\phi = \psi$, so that $(\mc{A}, \phi)$ is an algebra with a single state, conditional freeness with respect to $(\phi, \phi)$ is the same as free independence with respect to $\phi$. Moreover, $R^{\phi, \phi} = R^\phi$.
\item
If $\mc{A}$ is a non-unital algebra, define a state $\delta_0$ on its unitization $\mf{C} 1 \oplus \mc{A}$ by $\delta_0[1] = 1$, $\delta_0[\mc{A}] = 0$. Then conditional freeness of subalgebras $(\mf{C} 1 \oplus \mc{A}_1), \ldots, (\mf{C} 1 \oplus \mc{A}_d)$ with respect to $(\phi, \delta_0)$ is the same as Boolean independence of subalgebras $\mc{A}_1, \ldots, \mc{A}_d$ with respect to $\phi$. Moreover, $R^{\phi, \delta_0} = \eta^\phi$.
\item
Specializing the preceding example, if $\mc{A} = \mf{C} \langle \mb{x} \rangle$, it is a unitization of the algebra of polynomials without constant term, and $\delta_0[P]$ is the constant term of a polynomial, so that we denote, even for non-commuting polynomials,
\[
\delta_0[P] = P(0).
\]
\item
More generally, for any $\mb{a} \in \mf{R}^d$, we can define a state on $\mf{C} \langle \mb{x} \rangle$ by
\[
\delta_{\mb{a}}[P] = P(a_1, a_2, \ldots, a_d).
\]
These are exactly the multiplicative $\ast$-linear functionals on $\mf{C} \langle \mb{x} \rangle$. Note that $\delta_{\mb{a}}$ is the free product of $\delta_{a_i}$, so it is indeed a state, and here and in all the preceding examples, the two parts of Definition~\ref{Defn:c-free} coincide. This particular case of the c-free theory is related to the objects in \cite{Krystek-Yoshida-t}, and also to Fermi independence (see Lemma~\ref{Lemma:B_t-semigroup}).
\end{enumerate}
\end{Example}
\noindent
See the references for other particular cases and generalizations of conditional freeness; the appearance of the free Meixner laws (Section~\ref{Subsec:Meixner}) in related contexts has been observed even more widely. Note also that the only other natural product, the monotone product of Muraki, can also to some degree be handled in the c-free language \cite{Franz-Multiplicative-monotone}: the monotone product of $\phi_1$ with $\phi_2$ is their conditionally free product for the second state $\delta_0 \ast \phi_2$. However, while the monotone product is associative, the triple product is apparently not a conditionally free product. See also Remark~\ref{Remark:Monotone} and Lemma~\ref{Lemma:Monotone-Phi}.
\subsection{Convolutions}
\label{Subsec:Convolutions}
If $\phi, \psi$ are two unital linear functionals on $\mf{C} \langle \mb{x} \rangle$, then $\phi \boxplus \psi$ is their free convolution, that is a unital linear functional on $\mf{C} \langle \mb{x} \rangle$ determined by
\begin{equation}
\label{Free-convolution}
\CumFun{\phi}{\mb{z}} + \CumFun{\psi}{\mb{z}} = \CumFun{\phi \boxplus \psi}{\mb{z}}.
\end{equation}
Similarly, $\phi \uplus \psi$, their Boolean convolution, is a unital linear functional on $\mf{C} \langle \mb{x} \rangle$ determined by
\[
\eta^\phi(\mb{z}) + \eta^\psi(\mb{z}) = \eta^{\phi \uplus \psi}(\mb{z}).
\]
See Lecture~12 of \cite{Nica-Speicher-book} for the relation between free convolution and free independence; the relation in the Boolean case is similar.
\medskip\noindent
The functionals $\rho^{\boxplus t}$ form a free convolution semigroup if $\rho^{\boxplus t} \boxplus \rho^{\boxplus s} = \rho^{\boxplus (t+s)}$. If $\rho^{\boxplus t}$ are \emph{states} for all $t \geq 0$, we say that $\rho$ is freely infinitely divisible, and will denote the corresponding semigroup by $\set{\rho_t}$. The functionals $\phi^{\uplus t}$ form a Boolean convolution semigroup if $\phi^{\uplus t} \uplus \phi^{\uplus s} = \phi^{\uplus (t+s)}$. Any state is infinitely divisible in the Boolean sense.
\subsection{Free Meixner distributions and states}
\label{Subsec:Meixner}
The semicircular distribution with mean $\alpha$ and variance $\beta$ is
\[
d\SC(\alpha, \beta)(x) = \frac{1}{2 \pi \beta} \sqrt{(4 \beta - (x - \alpha)^2} \chf{[-2 \sqrt{\beta}, 2 \sqrt{\beta}]}(x) \,dx
\]
For every $\alpha, \beta$, $\SC(\alpha t, \beta t)$ form a free convolution semigroup with respect to $t$.
\medskip\noindent
For $b \in \mf{R}$, $1 + c \geq 0$, the free Meixner distributions, normalized to have mean zero and variance one, are
\[
d\mu_{b,c}(x) = \frac{1}{2 \pi} \frac{\sqrt{4 (1 + c) - (x - b)^2}}{1 + b x + c x^2} \,dx + \text{ zero, one, or two atoms}.
\]
They are characterized by their Jacobi parameter sequences having the special form
\[
(0, b, b, b, \ldots), (1, 1+c, 1+c, 1+c, \ldots),
\]
or by the special form of the generating function of their orthogonal polynomials. In particular, $\mu_{0,0} = \SC(0,1)$ is the standard semicircular distribution, $\mu_{b,0}$ are the centered free Poisson distributions, and $\mu_{b,-1}$ are the normalized Bernoulli distributions. Moreover,
\[
\mu_{b,c} = \SC(b,1+c)^{\uplus 1/(1+c)} \uplus \delta_{-b}.
\]
Un-normalized free Meixner distributions, of mean $\alpha$ and variance $t$, are $\mu_{b,c}^{\boxplus t} \boxplus \delta_{\alpha}$ or $\mu_{\beta, \gamma}^{\uplus t} \uplus \delta_\alpha$, see Lemma~\ref{Lemma:Meixner}.
\medskip\noindent
More generally, free Meixner states are states on $\mf{C} \langle \mb{x} \rangle$, characterized by a number of equivalent conditions (see \cite{AnsFree-Meixner}), among them the equations
\begin{equation}
\label{free-quadratic-PDE}
D_i D_j \CumFun{\phi}{\mb{z}} = \delta_{ij} + \sum_{k=1}^d B_{ij}^k D_k \CumFun{\phi}{\mb{z}} + C_{ij} D_i \CumFun{\phi}{\mb{z}} D_j \CumFun{\phi}{\mb{z}}.
\end{equation}
for certain $\set{B_{ij}^k, C_{ij}}$. In \cite{AnsBoolean}, these equations were shown to be equivalent to
\begin{equation}
\label{Boolean-quadratic-PDE}
D_i D_j \eta^\phi(\mb{z}) = \delta_{ij} + \sum_{k=1}^d B_{ij}^k D_k \eta^\phi(\mb{z}) + (1 + C_{ij}) D_i \eta^\phi(\mb{z}) D_j \eta^\phi(\mb{z}).
\end{equation}
\subsection{Orthogonality of the c-free Appell polynomials}
The c-free version of the Appell polynomials were investigated in \cite{AnsAppell3}. A result which motivated the investigation of the free Meixner distributions below is the following lemma.
\begin{Lemma}[Lemma~5, Theorem~6 of \cite{AnsAppell3}]
The c-free Appell polynomials in $(\mc{A}, \phi, \psi)$ are orthogonal if and only if $\psi$ is a free product of semicircular distributions $\SC(b_i, 1 + c_i)$, and $\phi = \State{\psi}$. In this case, $\CumFun{\phi, \psi}{\mb{z}} = \sum_{i=1}^d z_i^2$, and $\phi$ is a free Meixner state.
\end{Lemma}
\noindent
Here $\Phi$ is a map defined by Belinschi and Nica in \cite{Belinschi-Nica-Free-BM} via
\[
\eta^{\State{\psi}}(\mb{w}) = \sum_{i=1}^d w_i (1 + M^\psi(\mb{w})) w_i,
\]
see \cite{AnsAppell3} for other descriptions of it.
\section{Free Meixner distributions and conditional freeness}
\label{Section:Meixner}
\begin{Lemma}
\label{Lemma:boxplus-uplus}
For two functionals $\phi, \psi$, $\CumFun{\phi, \psi}{\mb{z}} = \sum_{i=1}^d a_i z_i + \beta \CumFun{\psi}{\mb{z}}$ if and only if
\[
\phi = \delta_{\mb{a}} \uplus \psi^{\uplus \beta}.
\]
Equivalently, if $\rho$ is a (not necessarily positive) functional with $\CumFun{\rho}{\mb{z}} = \CumFun{\phi, \psi}{\mb{z}}$ and $\CumFun{\psi}{\mb{z}} = \sum_{i=1}^d a_i z_i + t \CumFun{\rho}{\mb{z}}$, then
\[
\psi = \delta_{\mb{a}} \boxplus \rho^{\boxplus t} = \delta_{\mb{a}} \uplus \phi^{\uplus t}.
\]
\end{Lemma}
\begin{proof}
If $\CumFun{\phi, \psi}{\mb{z}} = \sum_{i=1}^d a_i z_i + \beta \CumFun{\psi}{\mb{z}}$, then
\[
\begin{split}
\eta^\phi(\mb{w})
& = (1 + M^\psi(\mb{w}))^{-1} \CumFun{\phi, \psi}{(1 + M^\psi(\mb{w})) \mb{w}} \\
& = \sum_{i=1}^d a_i w_i + \beta (1 + M^\psi(\mb{w}))^{-1} \CumFun{\psi}{(1 + M^\psi(\mb{w})) \mb{w}} \\
& = \sum_{i=1}^d a_i w_i + \beta (1 + M^\psi(\mb{w}))^{-1} M^\psi(\mb{w})
= \sum_{i=1}^d a_i w_i + \beta \eta^\psi(\mb{w})
= \eta^{\delta_{\mb{a}}} + \eta^{\psi^{\uplus \beta}},
\end{split}
\]
so
\[
\phi = \delta_{\mb{a}} \uplus \psi^{\uplus \beta}.
\]
Reversing the argument gives the reverse implication. The proof of the second statement is similar.
\end{proof}
\begin{Defn-Remark}
For $\rho, \psi$ linear functionals, define the map
\[
(\rho, \psi) \mapsto \State{\rho, \psi}
\]
via
\[
\eta^{\State{\rho, \psi}}(\mb{w}) = (1 + M^\psi(\mb{w}))^{-1} \CumFun{\rho}{(1 + M^\psi(\mb{w})) \mb{w}}.
\]
In other words, for
\[
\phi = \State{\rho, \psi}
\]
we have
\[
\CumFun{\rho}{\mb{z}} = \CumFun{\phi, \psi}{\mb{z}}.
\]
Note that $\CumFun{\SC(0,1)}{\mb{z}} = \sum_{i=1}^d z_i^2$, so that
\[
\eta^{\State{\SC(0,1), \psi}}(\mb{w}) = (1 + M^\psi(\mb{w}))^{-1} \sum_{i=1}^d (1 + M^\psi(\mb{w})) w_i (1 + M^\psi(\mb{w})) w_i
\]
and
\[
\State{\SC(0,1), \psi} = \State{\psi}.
\]
On the other hand, from $R^{\phi, \phi} = R^\phi$ and $R^{\phi, \delta_0} = \eta^\phi$ it follows that for any $\rho$, $\State{\rho, \rho} = \rho$ and $\mf{B}[\State{\rho, \delta_0}] = \rho$, where the map $\mf{B}$ is defined in the next remark. See also Remark~\ref{Remark:Lenczewski}.
\medskip\noindent
We will discuss in the next section under what conditions $\State{\rho, \psi}$ is a state.
\end{Defn-Remark}
\begin{Defn-Remark}
For $\mb{a} \in \mf{R}^d$, define the transformation $\mf{B}_{\mb{a}, t}$ by
\[
\mf{B}_{\mb{a}, t}[\rho] = \Bigl((\rho^{\boxplus (1 + t)} \boxplus \delta_{\mb{a}}) \uplus \delta_{-\mb{a}} \Bigr)^{\uplus 1/(1+t)}.
\]
$\mf{B}_{\mb{a}, t}$ maps functionals to functionals; for $t \geq 0$, it maps states to states. For $t > -1$, $\mf{B}_{\mb{a}, t}$ maps freely infinitely divisible states to states; for smaller $t$, the domains of $\mf{B}_t$ as a map from states to states get progressively smaller. For $\mb{a} = 0$,
\[
\mf{B}_{0, t}[\rho] = \mf{B}_t[\rho] = (\rho^{\boxplus (1+t)})^{\uplus (1/(1+t))}
\]
is the Belinschi-Nica transformation \cite{Belinschi-Nica-B_t,Belinschi-Nica-Free-BM}. In particular, $\mf{B}_{0,1} = \mf{B}$, the Boolean-to-free version of the Bercovici-Pata bijection. On the other hand, we will show below that for each $\rho$,
\[
\mf{B}_{(\rho[x_1], \ldots, \rho[x_d]), 0}[\rho]
\]
is the image of $\rho$ under the Boolean-to-Fermi version of the Bercovici-Pata bijection, in the sense of \cite{Oravecz-Fermi}.
\medskip\noindent
The preceding lemma states that if $\CumFun{\rho}{\mb{z}} = \CumFun{\phi, \psi}{\mb{z}}$ and $\CumFun{\psi}{\mb{z}} = \sum_{i=1}^d a_i z_i + (1 + t) \CumFun{\rho}{\mb{z}}$, then $\phi = \mf{B}_{\mb{a}, t}[\rho]$. In other words,
\begin{equation}
\label{Phi-rho-1+t}
\phi = \mf{B}_{\mb{a}, t}[\rho] = \State{\rho, \rho^{\boxplus (1+t)} \boxplus \delta_{\mb{a}}} = \State{\rho, \rho \boxplus \rho^{\boxplus t} \boxplus \delta_{\mb{a}}}.
\end{equation}
We will generalize this result in the next section. Note also that in this case, $\psi = U_{\mb{t}}(\phi)$ in the sense of \cite{Krystek-Yoshida-t}.
\end{Defn-Remark}
\noindent
The following lemma extends Proposition~8 from \cite{AnsBoolean} and Example~4.5 and Remark~4.6 from \cite{Belinschi-Nica-B_t}.
\begin{Lemma}
\label{Lemma:Meixner}
\medskip\noindent
\begin{enumerate}
\item
For $\mu_{b,c}$ a free Meixner distribution,
\[
\mf{B}_{\alpha, t}[\mu_{b,c}] = \mu_{b + \alpha, c + t}
\]
is also a free Meixner distribution.
\item
More generally, for $\phi_{\set{T_i}, C}$ a free Meixner state,
\[
\mf{B}_{\mb{a}, t}[\phi_{\set{T_i}, C}] = \phi_{\set{a_i I + T_i}, tI + C}
\]
is also a free Meixner state.
\item
Any free Meixner distribution can be obtained from the Bernoulli distribution
\[
\mu_{0,-1} = \frac{1}{2} \delta_{-1} + \frac{1}{2} \delta_1
\]
by the application of the appropriate $\mf{B}_{\alpha, t}$.
\item
For $\rho = \mu_{b,c}$ a free Meixner distribution,
\[
\State{\mu_{b,c}, \mu_{b,c}^{\boxplus (1+t)} \boxplus \delta_\alpha} = \mf{B}_{\alpha, t}[\mu_{b,c}] = \mu_{b + \alpha, c + t}.
\]
A partial converse to this statement is given in Proposition~\ref{Prop:Meixner-characterization}.
\end{enumerate}
\end{Lemma}
\begin{proof}
For part (a), it suffices to show that
\[
\delta_{\beta - b} \boxplus \mu_{b, c}^{\boxplus (1 + \gamma - c)} = \delta_{\beta - b} \uplus \mu_{\beta, \gamma}^{\uplus (1 + \gamma - c)}.
\]
$\mu_{b,c}$ has Jacobi parameter sequences
\[
\set{(0, b, b, b, \ldots,), (1, 1+c, 1+c, 1+c, \ldots)},
\]
As a particular case of the results in Section 3.1 of \cite{AnsFree-Meixner}, it follows that $\delta_{\beta - b} \boxplus \mu_{b, c}^{\boxplus (1 + \gamma - c)}$ has Jacobi parameter sequences
\[
\set{(\beta - b, \beta, \beta, \beta, \ldots,), (1 + \gamma - c, 1 + \gamma, 1 + \gamma, 1 + \gamma, \ldots)}.
\]
On the other hand, as observed in \cite{Boz-Wys} or the Appendix of \cite{AnsBoolean}, for any measure $\mu$ with Jacobi parameters
\[
\set{(\beta_0, \beta_1, \beta_2, \ldots), (\gamma_1, \gamma_2, \gamma_3, \ldots)},
\]
the measure $\delta_\alpha \uplus \mu^{\uplus t}$ has Jacobi parameters
\[
\set{(\alpha + t \beta_0, \beta_1, \beta_2, \ldots), (t \gamma_1, \gamma_2, \gamma_3, \ldots)}.
\]
The result follows. The proof for part (b) is similar. For part (c),
\[
\mu_{b,c} = \mf{B}_{b,1+c}[\mu_{0,-1}].
\]
Part (d) follows by combining part (a) with equation~\eqref{Phi-rho-1+t}.
\end{proof}
\begin{Example}
In the c-free central limit theorem (Theorem~4.3 of \cite{BLS96}), for the limiting distribution $\CumFun{\rho}{z} = \CumFun{\phi, \psi}{z} = z^2$, so that $\rho = \mu_{0,0}$ is the semicircular distribution and
\[
\phi = \State{\psi}
\]
If moreover $\CumFun{\psi}{z} = t z^2$, that is $\psi = \rho^{\boxplus t}$, then
\[
\phi = \State{\mu_{0,0}^{\boxplus t}} = \mu_{0, -1 + t}
\]
so that $\phi$ can be any symmetric free Meixner distribution.
\medskip\noindent
In the c-free Poisson limit theorem (centered version of Theorem~4.4 of \cite{BLS96}), for the limiting distribution $\CumFun{\rho}{z} = \CumFun{\phi, \psi}{z} = \frac{z^2}{1 - b z}$, so $\rho = \mu_{b,0}$ is the centered free Poisson distribution. Using Lemma~\ref{Lemma:Monotone-Phi}, it follows that
\[
\phi = \State{\mu_{b,0}, \psi} = \State{\delta_b \rhd \psi} = \State{\delta_b \uplus \psi}.
\]
If moreover $\CumFun{\psi}{z} = t \frac{z^2}{1 - a z}$ for $a = b$, then $\psi = \rho^{\boxplus t}$ and
\[
\phi = \State{\rho, \psi} = \State{\mu_{b,0}, \mu_{b,0}^{\boxplus t}} = \mu_{b, -1 + t},
\]
so that $\phi$ can be any free Meixner distribution. If $a \neq b$, then
\[
\phi = \State{\delta_b \uplus \mu_{a,0}^{\boxplus t}} = \State{\delta_b \uplus \mf{B}_{t-1}[\mu_{a,0}]^{\uplus t}} = \State{\delta_b \uplus \mu_{a,-1+t}^{\uplus t}} = \State{\delta_{b-a} \uplus \SC(a,t)},
\]
which is a distribution whose Jacobi parameter sequences are constant after step three, cf.\ Theorems 11 and 12 of \cite{Kry-Woj-Associative}.
\medskip\noindent
Similar arguments explain the appearance of the free Meixner distributions in various contexts which can be derived from the c-free formalism.
\medskip\noindent
Finally, if $\phi = \mu_{\beta, \gamma}$ and $\rho = \mu_{b, c}$ are both free Meixner distributions, as long as
\[
(c \leq 0, \gamma \geq c) \text{ or } 1 + \gamma \geq c > 0,
\]
$\CumFun{\phi, \psi}{z} = \CumFun{\rho}{z}$ for $\psi = \mu_{b, c}^{\boxplus (1 + \gamma - c)} \boxplus \delta_{\beta - b}$. Moreover, for arbitrary free Meixner distributions, $\mu_{\beta, \gamma} = \mf{B}_{\beta - b, \gamma - c}[\mu_{b, c}]$. In this case $1 + \gamma - c$ may be negative, however $\mu_{b, c}$ is in the range of $\mf{B}_{\alpha, 1 + c}$ (its $\boxplus$-divisibility indicator in the sense of equation~1.7 of \cite{Belinschi-Nica-B_t} is $1+c$), and so in the domain of $\mf{B}_{\alpha, t}$ for $t \geq - (1 + c)$.
\medskip\noindent
Similar results hold in the multivariate case. See also Remark~\ref{Remark:Lenczewski}.
\end{Example}
\begin{Prop}
If $\phi$ is a free Meixner state, then
\[
\begin{split}
D_i D_j \CumFun{\phi, \psi}{\mb{z}}
& = \delta_{ij} + \sum_k B_{ij}^k D_k \CumFun{\phi, \psi}{\mb{z}} + (1 + C_{ij}) D_i \CumFun{\phi, \psi}{\mb{z}} D_j \CumFun{\phi, \psi}{\mb{z}} \\
&\quad - D_i \CumFun{\psi}{\mb{z}} D_j \CumFun{\phi, \psi}{\mb{z}}.
\end{split}
\]
Note that this reduces to equation~\eqref{free-quadratic-PDE} for $\psi = \phi$, and to equation~\eqref{Boolean-quadratic-PDE} for $\psi = \delta_0$.
\end{Prop}
\begin{proof}
From relation~\eqref{eta-M-R}, it follows that
\[
D_j \eta^{\phi}(\mb{w})
= (D_j R^{\phi, \psi}) \left((1 + M^\psi(\mb{w})) \mb{w}\right)
\]
and
\[
\begin{split}
(D_i D_j R^{\phi, \psi}) \left((1 + M^\psi(\mb{w})) \mb{w}\right)
& = D_i \left[ (1 + M^\psi(\mb{w}))^{-1} (D_j R^{\phi, \psi}) \left((1 + M^\psi(\mb{w})) \mb{w}\right) \right] \\
& = D_i \left[ (1 + M^\psi(\mb{w}))^{-1} D_j \eta^\phi(\mb{w}) \right] \\
& = D_i (1 + M^\psi(\mb{w}))^{-1} D_j \eta^\phi(\mb{w}) + D_i D_j \eta^\phi(\mb{w}) \\
& = - D_i \eta^\psi(\mb{w}) D_j \eta^\phi(\mb{w}) + D_i D_j \eta^\phi(\mb{w}).
\end{split}
\]
From equation~\eqref{Boolean-quadratic-PDE} for $\eta^\phi$ we get
\[
\begin{split}
D_i D_j \CumFun{\phi, \psi}{\mb{z}}
& = \delta_{ij} + \sum_k B_{ij}^k D_k \CumFun{\phi, \psi}{\mb{z}} + (1 + C_{ij}) D_i \CumFun{\phi, \psi}{\mb{z}} D_j \CumFun{\phi, \psi}{\mb{z}} \\
& \quad - D_i \eta^{\psi}(\mb{w}) D_j \CumFun{\phi, \psi}{\mb{z}}
\end{split}
\]
Since
\begin{equation}
\label{D-eta-D-R}
\begin{split}
D_i \eta^\psi(\mb{w})
& = D_i \left[(1 + M^\psi(\mb{w}))^{-1} M^\psi(\mb{w}) \right] \\
& = D_i \left[(1 + M^\psi(\mb{w}))^{-1} \CumFun{\psi}{(1 + M^\psi(\mb{w})) \mb{w}} \right]
= (D_i R^\psi) \left( (1 + M^\psi(\mb{w})) \mb{w} \right)
\end{split}
\end{equation}
we finally get
\[
\begin{split}
D_i D_j \CumFun{\phi, \psi}{\mb{z}}
& = \delta_{ij} + \sum_k B_{ij}^k D_k \CumFun{\phi, \psi}{\mb{z}} + (1 + C_{ij}) D_i \CumFun{\phi, \psi}{\mb{z}} D_j \CumFun{\phi, \psi}{\mb{z}} \\
&\quad - D_i \CumFun{\psi}{\mb{z}} D_j \CumFun{\phi, \psi}{\mb{z}}. \qedhere
\end{split}
\]
\end{proof}
\begin{Remark}[Laha-Lukacs relations]
In \cite{Laha-Lukacs}, Laha and Lukacs proved that the Meixner distributions are characterized by a certain property involving linear conditional expectations and quadratic conditional variances. In \cite{Boz-Bryc}, Bo{\.z}ejko and Bryc proved that an identical characterization holds, in the free setting, for the free Meixner distributions. In \cite{Boz-Bryc-Two-states}, they further characterized all distributions having the Laha-Lukacs property in the two-state setting. In our notation, and with some extra assumptions, their result states that in this case,
\[
\CumFun{\phi, \psi}{z} = \frac{z^2}{1 - b z - c \CumFun{\psi}{z}},
\]
or equivalently
\[
D^2 \CumFun{\phi, \psi}{z} = 1 + b D \CumFun{\phi, \psi}{z} + c D \CumFun{\psi}{z} D \CumFun{\phi, \psi}{z}.
\]
Thus for $b=c=0$, one gets $\CumFun{\phi, \psi}{z} = z^2$ and $\phi = \State{\psi}$, while for $c = 0$, one gets $\CumFun{\phi, \psi}{z} = \frac{z^2}{1 - b z}$ and the appropriate analog of the Poisson law. On the other hand, for general $b,c$, $\CumFun{\phi, \psi}{z}$ does not itself satisfy a quadratic equation for general $\psi$, so the general two-state Laha-Lukacs distributions are not directly related to the free Meixner states. This was already observed in Section 4.4 of \cite{AnsBoolean}, where the Boolean Laha-Lukacs distributions (a particular case of the result in \cite{Boz-Bryc-Two-states}) were found to be the Bernoulli distributions, which include the Boolean analogs of the normal and Poisson laws, but do not include all the free Meixner distributions. On the other hand, if $\CumFun{\phi, \psi}{z} = \CumFun{\psi}{z}$, in which case also $\phi = \psi$, one recovers a quadratic equation, all free Meixner distributions, and free probability.
\end{Remark}
\section{The transformations $\mf{B}_{\mb{a}, t}$ and $\Phi[\rho, \psi]$}
\label{Section:Transformations}
\subsection{Properties of $\mf{B}_{\mb{a}, t}$}
\begin{Lemma}
\label{Lemma:B_t-semigroup}
The maps $\mf{B}_{\mb{a}, t}$ form a semigroup:
\[
\mf{B}_{\mb{a}, t} \circ \mf{B}_{\mb{b}, s} = \mf{B}_{\mb{a} + \mb{b}, t + s}.
\]
In particular, $\mf{B}_t$ and $\mf{B}_{\mb{a}, 0}$ commute, and
\[
\mf{B}_{\mb{a}, t}
= \mf{B}_t \circ \mf{B}_{\mb{a}, 0}.
\]
Moreover, for each $\rho$,
\[
\mf{B}_{(\rho[x_1], \ldots, \rho[x_d]), 0}[\rho]
\]
is the image of $\rho$ under the Boolean-to-Fermi version of the Bercovici-Pata bijection, in the sense of \cite{Oravecz-Fermi}.
\end{Lemma}
\begin{proof}
The representation
\[
\eta^{\mf{B}_{\mb{a}, t}[\rho]}[x_i] = \eta^\rho[x_i] = \rho[x_i]
\]
and for $n > 1$, $\abs{\vec{u}} = n$,
\[
\eta^{\mf{B}_{\mb{a}, t}[\rho]}[x_{u(1)}, x_{u(2)}, \ldots, x_{u(n)}] = \sum_{\pi \in \NC'(n)} \sum_{S \subset \Sing(\pi)} \Bigl( \prod_{i \in S} a_{u(i)} \Bigr) t^{\abs{S^c} - 1} \prod_{B \in S^c} \eta^\rho[x_{u(i)}: i \in B]
\]
is obtained by the same methods as in \cite{Belinschi-Nica-Eta}. In particular,
\[
\eta^{\mf{B}_{0, t}[\rho]}[x_{u(1)}, x_{u(2)}, \ldots, x_{u(n)}] = \sum_{\pi \in \NC'(n)} t^{\abs{\pi} - 1} \prod_{B \in \pi} \eta^\rho[x_{u(i)}: i \in B]
\]
and
\begin{equation}
\label{Formula-B_a}
\eta^{\mf{B}_{\mb{a}, 0}[\rho]}[x_{u(1)}, x_{u(2)}, \ldots, x_{u(n)}] = \sum_{\set{1,n} \subset \Lambda \subset \set{1, 2, \ldots, n}} \Bigl( \prod_{i \not \in \Lambda} a_{u(i)} \Bigr) \eta^\rho[x_{u(i)}: i \in \Lambda]
\end{equation}
which are easily seen to commute, and which proves the semigroup property.
\medskip\noindent
Moreover, denoting $\overrightarrow{\rho[x]} = (\rho[x_1], \rho[x_2], \ldots, \rho[x_d])$,
\[
\begin{split}
\eta^{\mf{B}_{\overrightarrow{\rho[x]}, 0}[\rho]}[x_{u(1)}, x_{u(2)}, \ldots, x_{u(n)}]
& = \sum_{\set{1,n} \subset \Lambda \subset \set{1, 2, \ldots, n}} \Bigl( \prod_{i \not \in \Lambda} \rho[x_{u(i)}] \Bigr) \eta^{\rho} [x_{u(i)}: i \in \Lambda] \\
& = \sum_{\substack{\pi \in \NC'(n) \\ \Inner(\pi) \subset \Sing(\pi)}} \Bigl( \prod_{\set{i} \in \Inner(\pi)} \rho[x_{u(i)}] \Bigr) \eta^{\rho} [x_{u(i)}: i \in B^o(\pi)].
\end{split}
\]
Using Definition~2.2 from \cite{Oravecz-Fermi}, it now easily follows that the Boolean cumulants of $\rho$ are the Fermi cumulants of $\mf{B}_{\overrightarrow{\rho[x]}, 0}[\rho]$. In other words, for each $\rho$, $\mf{B}_{\overrightarrow{\rho[x]}, 0}[\rho]$ is the image of $\rho$ under the Boolean-to-Fermi version of the Bercovici-Pata bijection.
\end{proof}
\subsection{Properties of $\State{\cdot, \cdot}$}
\begin{Thm}
\label{Thm:c-free-representation}
On $\mf{C} \langle x_1, x_2, \ldots, x_d \rangle$, let $\psi$ be a positive definite functional and $\mu$ a conditionally positive definite functional. Define a functional $\eta$ via
\[
M^\eta(\mb{w}) = (1 + M^\psi(\mb{w}))^{-1} M^\mu((1 + M^\psi(\mb{w})) \mb{w}) .
\]
Then $\eta$ is conditionally positive definite.
\end{Thm}
\noindent
The proof is delayed until Section~\ref{Subsec:Operator-models}.
\begin{Cor}
\label{Cor:cpd-one-variable}
Let $f, g$ be power series whose coefficient sequences are positive definite. Then the coefficient sequence of
\[
f(z g(z)) g(z)
\]
is also positive definite.
\end{Cor}
\begin{proof}
In a single variable, a sequence $\set{m_0, m_1, m_2, \ldots}$ is conditionally positive definite if and only if the sequence $\set{m_2, m_3, m_4, \ldots}$ is positive definite. In particular, the coefficient sequence of $z^2 f(z)$ is conditionally positive definite. So by Theorem~\ref{Thm:c-free-representation}, the coefficient sequence of
\[
g(z)^{-1} (z g(z))^2 f(z g(z)) = z^2 f(z g(z) g(z)
\]
is conditionally positive definite, and therefore the coefficient sequence of $f(z g(z)) g(z)$ is positive definite.
\end{proof}
\noindent
The following corollary was proved in Lemma 6.1 of \cite{Krystek-Conditional}, in the one-variable compactly supported case, by complex-analytic methods.
\begin{Cor}
\label{Cor:Phi-positive-definite}
Let $\psi$ be a state. For any conditionally positive functional $\mu$, there exists a state $\phi$ such that $R^{\phi, \psi} = M^\mu$. Equivalently, for any freely infinitely divisible state $\rho$, there exists a state $\phi$ such that $R^{\phi, \psi} = R^\rho$.
\end{Cor}
\begin{proof}
The conditionally positive functional $\eta$ obtained in Theorem~\ref{Thm:c-free-representation} is necessarily a Boolean cumulant functional of a state $\phi$. The second statement follows from the fact that the free cumulant functional of a freely infinitely divisible state is conditionally positive definite.
\end{proof}
\begin{Remark}[Schoenberg correspondence]
\label{Remark:Schoenberg}
Another interpretation of this result is in terms of the Schoenberg correspondence, see Corollary~3.6 of \cite{Franz-Unification} or \cite{SchurCondPos}. Let $\psi$ be any freely infinitely divisible state. A functional $\mu = R^\rho$ is conditionally positive if and only if it a generator of a c-free convolution semigroup of pairs of states $(\phi(t), \psi(t))$, in the sense that
\[
\mu = \frac{d}{dt}\Bigl|_{t=0}\Bigr. \phi(t),
\]
with $\psi(1) = \psi$. Indeed, for $\psi(t) = \psi^{\boxplus t}$, and $\phi(t)$ chosen so that
\[
R^{\phi(t), \psi^{\boxplus t}} = t M^\mu = t R^\rho = R^{\rho^{\boxplus t}},
\]
in other words for $\phi(t) = \State{\rho^{\boxplus t}, \psi^{\boxplus t}}$, the two properties above hold. The converse is standard. Note that we recover the Boolean version of the correspondence for $\psi = \delta_0$, and the free version for $\psi = \rho$.
\end{Remark}
\begin{Defn-Remark}
\label{DEfn-Remark-Monotone}
For two functionals $\tau, \psi$, the monotone convolution $\tau \rhd \psi$ of Muraki \cite{Franz-Muraki-Markov-monotone} is determined by
\begin{equation}
\label{Monotone-convolution}
(1 + M^{\tau \rhd \psi}(\mb{w})) = \Bigl(1 + M^\tau \bigl((1 + M^\psi(\mb{w})) \mb{w} \bigr) \Bigr) (1 + M^\psi(\mb{w})).
\end{equation}
So far, this operation has apparently been considered only in one variable, in which case this equation is equivalent to the condition
\[
F_{\tau \rhd \psi}(w) = F_{\tau}(F_\psi(w))
\]
on the reciprocal Cauchy transforms of the corresponding measures. It follows from Remark~\ref{Remark:Monotone} that the monotone convolution of states is a state.
\end{Defn-Remark}
\begin{Lemma}
\label{Lemma:Monotone-Phi}
Let $\psi$ be a state.
\begin{enumerate}
\item
Suppose that $R^\rho$ has the special form
\begin{equation}
\label{free-Phi}
\CumFun{\rho}{\mb{z}} = \sum_{i=1}^d z_i (1 + M^\tau(\mb{z})) z_i,
\end{equation}
which we will denote by $\rho = \Phi_{\text{free}}[\tau]$, where $\tau$ is a state. Note that $\Phi_{\text{free}}[\tau] = \mf{B}[\State{\tau}]$. Then
\begin{equation}
\label{Phi-monotone}
\State{\tau \rhd \psi} = \State{\Phi_{\text{free}}[\tau], \psi}.
\end{equation}
\item $\delta_{\mb{a}} \rhd \psi = \delta_{\mb{a}} \uplus \psi$.
\end{enumerate}
\end{Lemma}
\begin{proof}
Since $\tau$ is a state, by Lemmas 12 and 13 of \cite{AnsMulti-Sheffer} $R^\rho$ is conditionally positive definite and so $\rho$ is freely infinitely divisible. Then
\[
(1 + M^\psi(\mb{w}))^{-1} \CumFun{\rho}{(1 + M^\psi(\mb{w})) \mb{w}}
= \sum_{i=1}^d w_i \Bigl(1 + M^\tau \bigl((1 + M^\psi(\mb{w})) \mb{w} \bigr) \Bigr) (1 + M^\psi(\mb{w})) w_i
\]
so that for $\phi = \State{\rho, \psi}$,
\[
\eta^\phi(\mb{w}) = \sum_{i=1}^d w_i (1 + M^{\tau \rhd \psi}(\mb{w})) w_i.
\]
Equation~\eqref{Phi-monotone} follows.
\medskip\noindent
For part (b), by definition
\[
\begin{split}
(1 + M^{\delta_{\mb{a}} \rhd \psi}(\mb{w}))
& = \Bigl(1 + M^{\delta_{\mb{a}}} \bigl((1 + M^\psi(\mb{w})) \mb{w} \bigr) \Bigr) (1 + M^\psi(\mb{w})) \\
& = \Bigl( 1 - \sum_{i=1}^d a_i (1 + M^\psi(\mb{w})) w_i \Bigr)^{-1} (1 + M^\psi(\mb{w})),
\end{split}
\]
so
\[
\begin{split}
\eta^{\delta_{\mb{a}} \rhd \psi}(\mb{w})
& = 1 - \Bigl(1 + M^{\delta_{\mb{a}} \rhd \psi}(\mb{w}) \Bigr)^{-1} \\
& = 1 - (1 + M^\psi(\mb{w}))^{-1} \Bigl( 1 - \sum_{i=1}^d a_i (1 + M^\psi(\mb{w})) w_i \Bigr) \\
& = 1 - (1 + M^\psi(\mb{w}))^{-1} + \sum_{i=1}^d a_i w_i
= \eta^{\delta_{\mb{a}} \uplus \psi}(\mb{w}). \qedhere
\end{split}
\]
\end{proof}
\begin{Thm}
\label{Thm:Phi-properties}
\medskip\noindent
\begin{enumerate}
\item
For $\phi = \State{\rho, \psi}$, any two of the functionals $(\phi, \rho, \psi)$ uniquely determine the third.
\item
$\State{\cdot, \cdot}$ is a well-defined map
\[
(\rho = \text{ freely infinitely divisible state, } \psi = \text{ state}) \mapsto (\phi = \text{ state}).
\]
From now on, unless stated otherwise, we will consider $\State{\cdot, \cdot}$ with these domain and range. Note that $\phi$ has the same mean and covariance as $\rho$.
\item
$\rho$ is the unique fixed point of the maps $\State{\rho, \cdot}$ and $\State{\cdot, \rho}$. In particular, a free product of standard semicircular distributions is the unique fixed point of $\Phi$.
\item
For fixed $\rho$ with mean zero and identity covariance, the image of the single-variable map $\State{\rho, \cdot}$ consists of all states with the same mean and covariance as $\rho$ if and only if $\rho = \mu_{0,0}$ is semicircular, in which case $\State{\mu_{0,0}, \psi} = \State{\psi}$, or more generally if $\rho = \mu_{b,0}$ is centered free Poisson, in which case
\[
\State{\mu_{b,0}, \psi} = \State{\psi \uplus \delta_b}.
\]
In several variables, $\State{\rho, \cdot}$ is never onto.
\item
For fixed $\psi$, the map $\State{\cdot, \psi}$ is onto if and only if $\psi = \delta_0$, in which case $\State{\rho, \delta_0} = \mf{B}^{-1}[\rho]$, or more generally if $\psi = \delta_{\mb{a}}$, in which case
\[
\State{\rho, \delta_{\mb{a}}} = \mf{B}_{\mb{a}, -1}[\rho] = \mf{B}_{\mb{a}, 0} \circ \mf{B}^{-1}[\rho].
\]
\end{enumerate}
\end{Thm}
\begin{proof}
For part (a), we note that $R^\rho = R^{\phi, \psi}$, $\phi = \State{\rho, \psi}$, and $\psi$ is determined by
\begin{equation}
\label{psi-determined}
\Bigl((1 + M^\psi(\mb{w})) w_1, \ldots, (1 + M^\psi(\mb{w})) w_d \Bigr) = (\mb{D} R^\rho)^{\langle -1 \rangle} \Bigl( \mb{D} \eta^\phi(\mb{w}) \Bigr),
\end{equation}
where $(\mb{D} R^\rho)^{\langle -1 \rangle}(\mb{z})$ is the inverse of the $d$-tuple of power series
\[
\Bigl( D_1 \CumFun{\rho}{z_1, \ldots, z_d}, \ldots, D_d \CumFun{\rho}{z_1, \ldots, z_d} \Bigr)
\]
with respect to composition.
\medskip\noindent
Part (b) is a re-formulation of Corollary~\ref{Cor:Phi-positive-definite}.
\medskip\noindent
Since $\CumFun{\psi, \psi}{\mb{z}} = \CumFun{\psi}{\mb{z}}$, $\State{\psi, \psi} = \psi$. So by part (a), $\State{\rho, \psi} = \psi = \State{\psi, \psi}$ if and only if $\psi = \rho$. Similarly, $\State{\rho, \psi} = \rho = \State{\rho, \rho}$ if and only if $\rho = \psi$.
\medskip\noindent
If the map $\State{\cdot, \cdot}$ is onto, in particular the Bernoulli distribution, with $\eta^\phi(\mb{w}) = \sum_{i=1}^d w_i^2$, is in the image. In this case, using equation~\eqref{D-eta-D-R},
\[
(D_i R^{\rho}) \left((1 + M^\psi(\mb{w})) \mb{w} \right) = D_i \eta^\phi(\mb{w}) = w_i,
\]
so that
\[
D_i \CumFun{\rho}{\mb{z}} = \Bigl(1 + \CumFun{\psi}{\mb{z}} \Bigr)^{-1} z_i
\]
and
\[
\CumFun{\rho}{\mb{z}} = \sum_{i=1}^d z_i \Bigl(1 + \CumFun{\psi}{\mb{z}} \Bigr)^{-1} z_i.
\]
Thus in the notation of the preceding Lemma, $\rho = \Phi_{\text{free}}[\tau]$, where
\begin{equation}
\label{M-R-inverse}
1 + M^\tau(\mb{z}) = \Bigl(1 + \CumFun{\psi}{\mb{z}} \Bigr)^{-1}
\end{equation}
Since $\rho$ is freely infinitely divisible, $R^\rho$ is conditionally positive definite, and therefore $\tau$ is positive definite. Indeed, for any $i$
\[
\tau[P(\mb{x})^\ast P(\mb{x})]
= \Cum{\rho}{x_i P(\mb{x})^\ast P(\mb{x}) x_i}
= \Cum{\rho}{(P(\mb{x}) x_i)^\ast (P(\mb{x}) x_i)} \geq 0.
\]
On the other hand, from equation \eqref{M-R-inverse}
\begin{multline*}
1 + \sum_i \tau[x_i] z_i + \sum_{i,j} \tau[x_i x_j] z_i z_j + \ldots \\
= 1 - \sum_i \Cum{\psi}{x_i} z_i - \sum_{i,j} \Cum{\psi}{x_i x_j} z_i z_j + \Bigl( \sum_i \Cum{\psi}{x_i} z_i \Bigr) \Bigl( \sum_j \Cum{\psi}{x_j} z_j \Bigr),
\end{multline*}
so that $\tau[x_i] = - \psi[x_i]$ and
\[
\Cum{\tau}{x_i x_j} = \tau[x_i x_j] - \tau[x_i] \tau[x_j] = - \Cum{\psi}{x_i x_j}.
\]
Thus the covariance matrices of $\psi, \tau$ differ by a sign. On the other hand, since $\tau, \psi$ are positive definite, so are their covariance matrices. It follows that these matrices are both zero. Therefore both $\tau$ and $\psi$ are multiplicative linear functionals, and so delta measures.
\medskip\noindent
If $\tau = \delta_{\mb{a}}$, then $1 + M^\tau(\mb{z}) = (1 - \sum_{i=1}^d a_i z_i)^{-1}$ and $\CumFun{\rho}{\mb{z}} = \sum_{j=1}^d z_j (1 - \sum_{i=1}^d a_i z_i)^{-1} z_j$, so that
\[
D_i D_j \CumFun{\rho}{\mb{z}} = \delta_{ij} + a_i D_j \CumFun{\rho}{\mb{z}}
\]
and $\rho$ is the free Meixner state $\phi_{\set{a_i I}, 0}$, the free product of centered free Poisson distributions. $\rho$ has the special form in the equation~\eqref{free-Phi}, so by the preceding Lemma,
\[
\State{\Phi_{\text{free}}[\delta_{\mb{a}}], \psi}
= \State{\delta_{\mb{a}} \rhd \psi}
= \State{\delta_{\mb{a}} \uplus \psi}.
\]
In several variables, $\Phi$ is not onto all the states with mean zero and identity covariance, for example it follows from the proof of Theorem~6 of \cite{AnsAppell3} that most of the free Meixner states are not in its image. Therefore in this case, $\State{\rho, \cdot}$ is never onto. In one variable, by the arguments used in the proof of Corollary~\ref{Cor:cpd-one-variable}, $\Phi$ is onto. Finally,
\[
\State{\cdot, \delta_{\mb{a}}} = \mf{B}_{\mb{a}, 0}[\State{\cdot, \delta_0}] = \mf{B}_{\mb{a}, 0} \circ \mf{B}^{-1}
\]
are all bijections from freely infinitely divisible states onto all states.
\end{proof}
\begin{Thm}
\label{Thm:Evolution}
Let $\rho$ be a freely infinitely divisible state with the free convolution semigroup $\set{\rho_t}$.
\begin{enumerate}
\item
\[
\State{\rho^{\boxplus t} \boxplus \delta_{\mb{a}}, \psi} = \State{\rho, \psi}^{\uplus t} \uplus \delta_{\mb{a}}.
\]
\item
\[
\State{\rho, \psi \boxplus \rho_t} = \mf{B}_t[\State{\rho, \psi}].
\]
Equivalently, if $R^{\phi, \psi} = R^\rho$, then $R^{\mf{B}_t[\phi], \psi \boxplus \rho_t} = R^{\phi, \psi} = R^\rho$ does not depend on $t$. More generally,
\[
\State{\rho, \psi \boxplus \rho^{\boxplus t} \boxplus \delta_{\mb{a}}}
= \mf{B}_{\mb{a}, t}[\State{\rho, \psi}]
= \mf{B}_t[\State{\rho, \psi \boxplus \delta_\mb{a}}]
= \mf{B}_{\mb{a}, 0}[\State{\rho, \psi \boxplus \rho_t}].
\]
\end{enumerate}
\end{Thm}
\noindent
We will present two proofs of this theorem. The combinatorial proof has all the details and works for general functionals. On the other hand, the operator-representation proof in the next section may be more illuminating.
\begin{proof}[Combinatorial proof]
Part (a) follows by exactly the same method as Lemma~\ref{Lemma:boxplus-uplus}.
\medskip\noindent
For part (b), on one hand, by expanding the defining relation~\eqref{eta-M-R},
\[
\begin{split}
& \eta^{\State{\rho, \psi \boxplus \rho_t}}[x_1, x_2, \ldots, x_n] \\
&\qquad = \sum_{\substack{\Lambda \subset \set{1, \ldots, n} \\ \Lambda = \set{u(0) = 1, u(1), u(2), \ldots, u(k) = n}}} \Cum{\rho}{x_i : i \in \Lambda} \prod_{j=1}^k (\psi \boxplus \rho_t)[x_i: u(j-1)+1 \leq i \leq u(j)-1].
\end{split}
\]
Combining this with the definition~\eqref{Free-convolution} of free convolution and expansion~\eqref{R-cumulant-moment}, we get
\begin{multline}
\label{Phi-psi-rho}
\eta^{\State{\rho, \psi \boxplus \rho_t}}[x_1, x_2, \ldots, x_n] \\
= \sum_{\pi \in \NC'(n)} \sum_{V: B^o(\pi) \in V \subset \pi} t^{\abs{V} - 1} \prod_{B \in V} \Cum{\rho}{x_i: i \in B} \prod_{C \in \pi \backslash V} \psi[x_i: i \in C].
\end{multline}
On the other hand,
\[
\begin{split}
& \eta^{\State{\rho, \psi}}[x_1, x_2, \ldots, x_n] \\
&\qquad = \sum_{\substack{\Lambda \subset \set{1, \ldots, n} \\ \Lambda = \set{u(0) = 1, u(1), u(2), \ldots, u(k) = n}}} \Cum{\rho}{x_i : i \in \Lambda} \prod_{j=1}^k \psi[x_i: u(j-1)+1 \leq i \leq u(j)-1] \\
&\qquad = \sum_{\sigma \in \NC'(n)} \Cum{\rho}{x_i: i \in B^o(\sigma)} \prod_{C \in \Inner(\pi)} \Cum{\psi}{x_i: i \in C}.
\end{split}
\]
Also, by Remark 4.4 of \cite{Belinschi-Nica-Free-BM},
\[
\eta^{\mf{B}_t[\phi]}[x_1, x_2, \ldots, x_n]
= \sum_{\omega \in \NC'(n)} t^{\abs{\omega} - 1} \prod_{B \in \omega} \eta^\phi[x_i: i \in B].
\]
Thus
\begin{equation}
\label{B-rho-psi}
\begin{split}
& \eta^{\mf{B}_t[\State{\rho, \psi}]}[x_1, x_2, \ldots, x_n] \\
&\qquad = \sum_{\omega \in \NC'(n)} t^{\abs{\omega} - 1} \prod_{B \in \omega} \sum_{\sigma \in \NC'(B)} \Cum{\rho}{x_i: i \in B^o(\sigma)} \prod_{C \in \Inner(\pi)} \Cum{\psi}{x_i: i \in C}.
\end{split}
\end{equation}
Clearly every term of the sum \eqref{B-rho-psi} appears in the sum \eqref{Phi-psi-rho} for some pair $(\pi, V)$. It remains to show the converse, namely that each such pair corresponds to the unique collection
\[
\set{\omega = (B_1, B_2, \ldots, B_k) \in \NC'(n), \set{\sigma_i \in \NC'(B_i)}}
\]
with
\begin{equation}
\label{pi-sigma}
i \stackrel{\pi}{\sim} j \Leftrightarrow i \stackrel{\sigma_s}{\sim} j \text{ for some } s
\end{equation}
and
\begin{equation}
\label{V-union}
V = \cup_{i=1}^k B_o(\sigma_i).
\end{equation}
This correspondence is very closely related to Lemma 3 of \cite{AnsFree-Meixner} and Remark 6.3 of \cite{Belinschi-Nica-Free-BM}. Namely, define $\omega$ as follows: for any $j$, let $j \stackrel{\omega}{\sim} i$ for $i$ the largest element with the property that
\[
i, i' \in B \in V, \quad i \leq j < i'.
\]
Note that since $B^o(\pi) \in V$, such an $i$ always exists. Clearly $\omega \in \NC'(n)$ and $\pi \leq \omega$, so we can define each $\sigma_s$ via equation \eqref{pi-sigma}. For each $B \in \omega$, $\min(B)$ and $\max(B)$ lie in the same class of $\pi$ which in fact belongs to $V$. Relation \eqref{V-union} and $\sigma_i \in \NC'(B_i)$ follow.
\medskip\noindent
For the second equation in part (b), it suffices to show that
\[
\mf{B}_{\mb{a}, 0}[\State{\rho, \psi}] = \State{\rho, \psi \boxplus \delta_\mb{a}},
\]
which follows by very similar methods using representation~\eqref{Formula-B_a}.
\end{proof}
\begin{Prop}
\label{Prop:Meixner-characterization}
Let $\rho, \psi$ be freely infinitely divisible distributions, so that $\phi = \State{\rho, \psi}$, by definition, is a c-freely infinitely divisible distribution. Suppose $\phi$ is a free Meixner distribution. Then all the distributions in its semigroup
\[
\phi(t) = \State{\rho^{\boxplus t}, \psi^{\boxplus t}}
\]
from Remark~\ref{Remark:Schoenberg} are (un-normalized) free Meixner if and only if $\rho$ is free Meixner and $\psi = \rho^{\boxplus (1+s)} \boxplus \delta_{\alpha}$. Thus, while the Boolean ($\psi = \delta_0$) and free ($\psi = \rho$) evolutions preserve the Meixner class, general c-free evolution does not.
\end{Prop}
\begin{proof}
If $\rho = \mu_{b,c}$ is freely infinitely divisible, so that $c \geq 0$, then
\[
\phi(t) = \State{\mu_{b,c}^{\boxplus t}, \mu_{b,c}^{\boxplus (1+s) t} \boxplus \delta_{\alpha t}}
= \State{\mu_{b,c}, \mu_{b,c}^{\boxplus (1+s) t} \boxplus \delta_{\alpha t}}^{\uplus t}
= \mu_{b + \alpha t, -1 + c + (1+s) t}^{\uplus t}
\]
is a free Meixner distribution, with $-1 + c + (1+s) t \geq -1$. For the converse, suppose that all $\phi(t)$ are free Meixner. Normalize $\phi$ to have mean zero and variance $1$. Then
\begin{equation}
\label{PDE-one-variable}
D^2 \eta^{\phi(t)}(z) = t + b(t) D \eta^{\phi(t)} (z) + c(t) \Bigl( D \eta^{\phi(t)}(z) \Bigr)^2.
\end{equation}
To simplify notation, we write $x_{u(1)}, \ldots, x_{u(n)}$, although in the single-variable context all of these are equal to $x$. We know that
\[
\begin{split}
\eta^{\phi}[x_{u(1)}, x_{u(2)}, \ldots, x_{u(n)}]
& = \sum_{\pi \in \NC'(n)} \Cum{\rho}{x_i: i \in B^o(\pi)} \prod_{C \in \Inner(\pi)} \Cum{\psi}{x_i: i \in C} \\
& = \Cum{\rho}{x_{u(1)}, x_{u(2)}, \ldots, x_{u(n)}} + \text{ products of lower order terms}.
\end{split}
\]
Therefore given $\phi$, each $\Cum{\rho}{\cdot}$ is uniquely determined by the lower order $\Cum{\psi}{\cdot}$. Thus
\[
\eta^{\phi(t)}[x_{u(1)}, x_{u(2)}, \ldots, x_{u(n)}] = \sum_{\pi \in \NC'(n)} t^{\abs{\pi}} \Cum{\rho}{x_i: i \in B^o(\pi)} \prod_{C \in \Inner(\pi)} \Cum{\psi}{x_i: i \in C}.
\]
can be expressed in terms of $\eta^{\phi}[\cdot]$ and $\Cum{\psi}{\cdot}$ of order $n$ and lower. On the other hand, from equation~\eqref{PDE-one-variable} we get
\[
\begin{split}
\eta^{\phi(t)}[x_j, x_i, x_{u(1)}, \ldots, x_{u(n)}]
& = b(t) \eta^{\phi(t)}[x_k, x_{u(1)}, \ldots, x_{u(n)}] \\
&\quad + \sum_{s=0}^n c(t) \eta^{\phi(t)}[x_i, x_{u(1)}, \ldots, x_{u(s)}] \eta^{\phi(t)}[x_j,x_{u(s+1)}, \ldots, x_{u(n)}].
\end{split}
\]
These equations for $n = 1, 2$ determine $b(t)$ and $c(t)$ uniquely in terms of $\eta^{\phi}[\cdot]$ and $\Cum{\psi}{\cdot}$; in fact,
\[
b(t) = b(1) + (t-1) \Cum{\psi}{x}, \qquad c(t) = \frac{1}{t} \left( c(1) + (t-1) \Cum{\psi}{x,x} \right).
\]
Therefore these equations for larger $n$ determine all the $\psi$-cumulants in terms of $\eta^{\phi}[\cdot]$ and the $\psi$-cumulants of order $1$ and $2$.
If the mean of $\psi$ is $\alpha$ and variance $1+s$, then $\psi = \rho^{\boxplus (1+s)} \boxplus \delta_{\alpha}$ has the correct first two cumulants, and therefore is the correct state. Finally,
\[
\phi = \State{\rho, \rho^{\boxplus (1+s)} \boxplus \alpha} = \mf{B}_{\alpha, s}[\rho]
\]
and so $\rho = \mf{B}_{-\alpha, -s}[\phi]$; since $\rho$ is a state, this expression is well defined. Since $\phi$ is free Meixner, it follows that $\rho$ is a free Meixner distribution.
\end{proof}
\begin{Remark}
\label{Remark:Lenczewski}
In addition to monotone convolution, another operation related to $\Phi$ is the orthogonal convolution $\vdash$ of Lenczewski. In fact, in the single variable case,
\[
\State{\mf{B}[\tau], \psi} = \tau \vdash \psi.
\]
Some results in this paper can be reformulated as multivariate versions of the results in \cite{Lenczewski-Decompositions-convolution}. For example, our equation~\eqref{Phi-monotone} in Lemma~\ref{Lemma:Monotone-Phi} can be extended to
\[
\State{\tau \rhd \psi} = \State{\mf{B}[\State{\tau}], \psi} = \State{\tau} \vdash \psi,
\]
closely related to Corollary~6.4 of \cite{Lenczewski-Decompositions-convolution}. In Theorem~\ref{Thm:Phi-properties}, the positivity of $\Phi$ in part (b) corresponds to the positivity of the orthogonal convolution; this also explains why the first argument of $\Phi$ is naturally taken to be freely infinitely divisible. Parts (d) and (e) of that theorem imply that
\[
\mu_{b, -1} \vdash \psi = \mf{B}_{b,0}[\State{\psi}]
\]
and
\[
\tau \vdash \delta_a = \mf{B}_{a,0}[\tau],
\]
which are Examples 6.2 and 6.1 of \cite{Lenczewski-Decompositions-convolution}, respectively. Similarly, Theorem~\ref{Thm:Evolution} can be re-formulated in terms of the orthogonal convolution. In fact, some results in \cite{Lenczewski-Decompositions-convolution} are obtained by complex-analytic methods which apply to measures with possibly infinite moments, and can be used to obtain extensions of our results in the single-variable context.
\end{Remark}
\subsection{Operator models}
\label{Subsec:Operator-models}
\begin{Lemma}
\label{Lemma:Basic-operator-representations}
For a Hilbert space $\mc{H}$, by its Boolean Fock space we mean the Hilbert space $\mf{C} \Omega \oplus \mc{H}$ and by its full Fock space the Hilbert space $\mc{F}(\mc{H}) = \mf{C} \Omega \oplus \bigoplus_{n=1}^\infty \mc{H}^{\otimes n}$. For $\zeta \in \mc{H}$, its Boolean creation and annihilation operators on the Boolean Fock space are
\begin{align*}
a_\zeta^{b,+}(\varepsilon) &= \ip{\Omega}{\varepsilon} \zeta, \\
a_\zeta^{b,-}(\varepsilon) &= \ip{\zeta}{\varepsilon} \Omega,
\end{align*}
and its free creation and annihilation operators on the full Fock space are
\begin{align*}
a_{\zeta}^{f,+}(\varepsilon_1 \otimes \ldots \otimes \varepsilon_n) &= \zeta \otimes \varepsilon_1 \otimes \ldots \otimes \varepsilon_n, \\
a_{\zeta}^{f,-}(\varepsilon_1 \otimes \ldots \otimes \varepsilon_n) &= \ip{\zeta}{\varepsilon_1} \varepsilon_2 \otimes \ldots \otimes \varepsilon_n.
\end{align*}
For $H \in \mc{L}(\mc{H})$, it acts on the Boolean Fock space by $H \Omega = 0$, and its gauge operator on the full Fock space is
\[
p(H) (\varepsilon_1 \otimes \ldots \otimes \varepsilon_n) = (H \varepsilon_1) \otimes \varepsilon_2 \otimes \ldots \otimes \varepsilon_n.
\]
Finally, denote by $P_\Omega$ the projection on $\Omega$.
\begin{enumerate}
\item
A state $\psi$ corresponds to a collection of data
\[
(\mc{K}, \xi \in \mc{K}, K_i \in \mc{L}(\mc{K})),
\]
where $\mc{K}$ is a Hilbert space, $\xi \in \mc{K}$ a unit vector, and $\set{K_1, K_2, \ldots, K_d} \in \mc{L}(\mc{K})$ a $d$-tuple of symmetric operators with a common invariant dense domain containing $\xi$, via
\[
\psi[x_{\vec{u}}] = \ip{\xi}{K_{\vec{u}} \xi}.
\]
Conversely, any such collection always gives a state. (We will omit the last comment and the conditions on the operators in subsequent constructions.)
\item
A state $\phi$ also corresponds to a collection of data
\[
(\mc{K}, \varepsilon_i \in \mc{K}, S_i \in \mc{L}(\mc{K}), \alpha_i \in \mf{R})
\]
as follows: on the Boolean Fock space $\mf{C} \Omega \oplus \mc{K}$,
\[
\phi[x_{\vec{u}}] = \ip{\Omega}{\Bigl(a_{\varepsilon_i}^{b,+} + a_{\varepsilon_i}^{b,-} + S_i + \alpha_i P_\Omega \Bigr)_{\vec{u}} \Omega}.
\]
Here $\ip{\varepsilon_i}{S_{\vec{u}} \varepsilon_j}$ are the Boolean cumulants of $\phi$.
\item
A conditionally positive definite functional $\mu$ corresponds to a collection of data
\[
(\mc{H}, \zeta_i \in \mc{H}, H_i \in \mc{L}(\mc{H}), \lambda_i \in \mf{R})
\]
via
\[
\mu \left[ x_i x_{\vec{u}} x_j \right] = \ip{\zeta_i}{ H_{\vec{u}} \zeta_j}, \qquad \mu[x_i] = \lambda_i.
\]
\item
A freely infinitely divisible state $\rho$ corresponds to a collection of data
\[
(\mc{H}, \zeta_i \in \mc{H}, H_i \in \mc{L}(\mc{H}), \lambda_i \in \mf{R})
\]
as follows: on the full Fock space $\mc{F}(\mc{H})$,
\[
\rho[x_{\vec{u}}] = \ip{\Omega}{\Bigl(a_{\zeta_i}^{f,+} + a_{\zeta_i}^{f,-} + p(H_i) + \lambda_i I \Bigr)_{\vec{u}} \Omega}.
\]
Here $\ip{\zeta_i}{H_{\vec{u}} \zeta_j}$ are the free cumulants of $\rho$.
\end{enumerate}
\end{Lemma}
\begin{proof}
Part (a) is standard; briefly, take $\mc{K}$ to be the completion of the quotient of $\mf{C} \langle \mb{x} \rangle$ with respect to the $\psi$-norm, $\xi$ to be the vector image of $1$, and $K_i$ to be the operator image of $x_i$. Part (b) is Proposition~15 of \cite{AnsBoolean}. Part (c) is also standard, see Proposition 3.2 of \cite{SchurCondPos}; briefly, take $\mc{H}$ to be the completion of the quotient of
\[
\set{P \in \mf{C} \langle \mb{x} \rangle | P(0) = 0}
\]
with respect to the $\mu$-norm, $\zeta_i$ to be the vector image of $x_i$, and $K_i$ to be the operator image of $x_i$. Finally, since a state $\rho$ is freely infinitely divisible if and only if $R^\rho$ is conditionally positive definite, part (d) follows from part (c).
\end{proof}
\begin{proof}[Proof of Theorem~\ref{Thm:c-free-representation}]
For $\psi, \mu$ represented as in Lemma~\ref{Lemma:Basic-operator-representations}, on the Hilbert space
\[
\mc{K} \otimes \mc{H},
\]
consider the operators
\[
K_i \otimes I + P_\xi \otimes H_i,
\]
where $P_\xi$ is the projection onto $\xi$ in $\mc{K}$. We will show that
\begin{equation}
\label{eta-cpd}
\eta \left[ x_i \prod_{s=1}^n x_{u(s)} x_j \right] = \ip{\xi \otimes \zeta_i}{ \prod_{s=1}^n \Bigl(K_{u(s)} \otimes I + P_\xi \otimes H_{u(s)} \Bigr) (\xi \otimes \zeta_j)}.
\end{equation}
The operators $K_i \otimes I + P_\xi \otimes H_i$ are symmetric, therefore it will follow that $\eta$ is conditionally positive definite.
\medskip\noindent
To prove \eqref{eta-cpd}, we note that
\[
\begin{split}
& \ip{\xi \otimes \zeta_i}{ \prod_{s=1}^k \Bigl(K_{u(s)} \otimes I + P_\xi \otimes H_{u(s)} \Bigr) (\xi \otimes \zeta_j)} \\
&\quad = \sum_{\substack{\Lambda \subset \set{1, 2, \ldots, n} \\ \Lambda = \set{v(1), v(2), \ldots, v(l)}}} \ip{\xi}{K_{u(1)} \ldots K_{u(v(1)-1)} P_\xi K_{u(v(1)+1)} \ldots K_{u(v(2)-1)} P_\xi \ldots K_{u(n)} \xi} \\
&\qquad\qquad\qquad\qquad\qquad \ip{\zeta_i}{I \ldots I H_{u(v(1))} I \ldots I H_{u(v(2))} I \ldots I H_{u(v(l))} I \ldots I \zeta_j} \\
&\quad = \sum_{\substack{\Lambda \subset \set{1, 2, \ldots, n} \\ \Lambda = \set{v(1), v(2), \ldots, v(l)}}}
\ip{\zeta_i}{\prod_{s=1}^l H_{u(v(s))} \zeta_j} \prod_{r=0}^l \ip{\xi}{\prod_{s = v(r)+1}^{v(r+1)-1} K_{u(s)} \xi} \\
&\quad = \sum_{\substack{\Lambda \subset \set{1, 2, \ldots, n} \\ \Lambda = \set{v(1), v(2), \ldots, v(l)}}} \mu \left[ x_i \prod_{s=1}^l x_{u(v(s))} x_j \right] \prod_{r=0}^l \psi \left[ \prod_{s = v(r)+1}^{v(r+1)-1} x_{u(s)} \right].
\end{split}
\]
It remains to note that
\[
\begin{split}
& 1 + \sum_{n=1}^\infty \sum_{\abs{\vec{u}} = n} \sum_{\substack{\Lambda \subset \set{1, 2, \ldots, n} \\ \Lambda = \set{1, v(1), v(2), \ldots, v(l), n}}} \mu \left[ x_{u(1)} \prod_{s=1}^l x_{u(v(s))} x_{u(n)} \right] \prod_{r=0}^l \psi \left[ \prod_{s = v(r)+1}^{v(r+1)-1} x_{u(s)} \right] w_{\vec{u}} \\
&\quad = 1 + \sum_{n=1}^\infty \sum_{\abs{\vec{u}} = n} \mu \left[ x_{u(1)} x_{u(2)} \ldots x_{u(n)} \right] w_{u(1)} (1 + M^\psi(\mb{w})) w_{u(2)} (1 + M^\psi(\mb{w})) w_{u(3)} \\
&\qquad\qquad\qquad\qquad \ldots (1 + M^\psi(\mb{w})) w_{u(n)} \\
&\quad = 1 + (1 + M^\psi(\mb{w}))^{-1} M^\mu((1 + M^\psi(\mb{w})) \mb{w})
= 1 + M^\eta(\mb{w}). \qedhere
\end{split}
\]
\end{proof}
\begin{Remark}[Relation to monotone probability]
\label{Remark:Monotone}
Lenczewski used a similar construction in Boo\-le\-an probability, and Franz and Muraki \cite{Franz-Muraki-Markov-monotone} in monotone probability: if $\psi$ is a joint distribution of the operators $\set{K_i}$ with respect to the vector state of $\xi$, and $\phi$ is the joint distribution of the operators $\set{H_i}$ with respect to the vector state of $\zeta$, then the joint distribution of the operators $\set{K_i \otimes I + P_\xi \otimes H_i}$ with respect to the vector state of $\xi \otimes \zeta$ is the monotone convolution $\phi \rhd \psi$. This provides an operator representation proof of Lemma~\ref{Lemma:Monotone-Phi}.
\end{Remark}
\begin{Lemma}
\label{Lemma:B-Phi-representations}
We use the notation of Lemma~\ref{Lemma:Basic-operator-representations}.
\begin{enumerate}
\item
For a freely infinitely divisible state $\rho$ and a state $\psi$, the state $\State{\rho, \psi}$ is represented on the Boolean Fock space
\[
\mf{C} \Omega \oplus (\mc{K} \otimes \mc{H})
\]
as
\[
\State{\rho, \psi}[x_{\vec{u}}] = \ip{\Omega}{\Bigl(a_{\xi \otimes \zeta_i}^{b,+} + a_{\xi \otimes \zeta_i}^{b,-} + K_i \otimes I + P_\xi \otimes H_i + \lambda_i P_\Omega \Bigr)_{\vec{u}} \Omega}.
\]
\item
For a state $\phi$, the freely infinitely divisible state $\mf{B}[\phi]$ is represented on the full Fock space $\mc{F}(\mc{K})$ as
\[
\mf{B}[\phi][x_{\vec{u}}] = \ip{\Omega}{\Bigl(a_{\varepsilon_i}^{f,+} + a_{\varepsilon_i}^{f,-} + p(S_i) + \alpha_i I \Bigr)_{\vec{u}} \Omega}.
\]
\item
For $\rho$, $\psi$ as above, the state $\psi \boxplus \rho$ is represented on the space
\[
\mc{F}(\mc{K} \otimes \mc{H}) \otimes \mc{K} \simeq \mc{K} \oplus \Bigl( \bigoplus_{n=1}^\infty (\mc{K} \otimes \mc{H})^{\otimes n} \otimes \mc{K} \Bigr)
\]
as
\[
(\psi \boxplus \rho)[x_{\vec{u}}] = \ip{\xi}{ \Bigl(a_{\xi \otimes \zeta_i}^{f,+} \otimes I + a_{\xi \otimes \zeta_i}^{f,-} \otimes I + p(K_i \otimes I) \otimes I + p(P_\xi \otimes H_i) \otimes I + \lambda_i I \Bigr)_{\vec{u}} \xi}.
\]
\end{enumerate}
\end{Lemma}
\begin{proof}
Part (a) follows by combining Theorem~\ref{Thm:c-free-representation} with part (b) of Lemma~\ref{Lemma:Basic-operator-representations}. Part (b) follows from the definition that
\[
\CumFun{\mf{B}[\phi]}{\mb{z}} = \eta^{\phi}(\mb{z})
\]
and parts (b), (d) of Lemma~\ref{Lemma:Basic-operator-representations}. Part (c) follows from the fact that
\[
(\psi \boxplus \rho)[x_1, x_2, \ldots, x_n] = \sum_{\substack{B_1, B_2, \ldots, B_k \subset \set{1, \ldots, n} \\ B_i \cap B_j = \emptyset \text{ for } i \neq j}} \prod_{i=1}^k \Cum{\rho}{x_j: j \in B_i} \prod_{C \in (B_1, B_2, \ldots, B_k)^c} \psi[x_j: j \in C],
\]
where
\[
(B_1, B_2, \ldots, B_k)^c = (C_1, C_2, \ldots, C_l)
\]
is the smallest collection of disjoint subsets of $\set{1, \ldots, n}$ such that
\[
(B_1, \ldots, B_k, C_1, \ldots, C_l) \in \NC(n). \qedhere
\]
\end{proof}
\begin{Remark}[Operator representation proof of Theorem~\ref{Thm:Evolution}]
Combining parts (a), (b) of the preceding Lemma, the freely infinitely divisible state $\mf{B}[\State{\rho, \psi}]$ is represented on the full Fock space
\[
\mc{F}(\mc{K} \otimes \mc{H}) = \mf{C} \Omega \oplus \bigoplus_{n=1}^\infty (\mc{K} \otimes \mc{H})^{\otimes n}
\]
as
\[
\mf{B}[\State{\rho, \psi}][x_{\vec{u}}] = \ip{\Omega}{\Bigl(a_{\xi \otimes \zeta_i}^{f,+} + a_{\xi \otimes \zeta_i}^{f,-} + p(K_i \otimes I + P_\xi \otimes H_i) + \lambda_i I \Bigr)_{\vec{u}} \Omega}.
\]
\medskip\noindent
On the other hand, combining parts (a), (c), the state $\State{\rho, \psi \boxplus \rho}$ is represented on the space
\[
\mf{C} \Omega \oplus \mc{F}(\mc{K} \otimes \mc{H}) \otimes \mc{K} \otimes \mc{H} \simeq \mc{F}(\mc{K} \otimes \mc{H})
\]
as
\[
\begin{split}
\State{\rho, \psi \boxplus \rho}[x_{\vec{u}}] & = \Bigl\langle \Omega, \Bigl(a_{\xi \otimes \zeta_i}^{b,+} + a_{\xi \otimes \zeta_i}^{b,-} \\
&\qquad + \bigl(a_{\xi \otimes \zeta_i}^{f,+} \otimes I + a_{\xi \otimes \zeta_i}^{f,-} \otimes I + p(K_i \otimes I) \otimes I + p(P_\xi \otimes H_i) \otimes I + \lambda_i I \bigr) \otimes I \\
&\qquad + I \otimes P_\xi \otimes H_i + \lambda_i P_\Omega \Bigr)_{\vec{u}} \Omega \Bigr\rangle \\
& = \ip{\Omega}{\Bigl(a_{\xi \otimes \zeta_i}^{f,+} + a_{\xi \otimes \zeta_i}^{f,-} + p(K_i \otimes I) + p(P_\xi \otimes H_i) + \lambda_i I \Bigr)_{\vec{u}} \Omega}.
\end{split}
\]
Thus $\mf{B}[\State{\rho, \psi}] = \State{\rho, \psi \boxplus \rho}$. The more general equation involving $\mf{B}_t$ follows from this by using
\[
\mf{B}_t[\phi] = \mf{B}[\phi^{\uplus t}]^{\uplus (1/t)},
\]
which in turn can be deduced from the semigroup property of $\set{\mf{B}_t}$.
\medskip\noindent
There are proofs by similar methods of other results in the paper, such as the remaining parts of Theorem~\ref{Thm:Evolution} and Lemma~\ref{Lemma:B_t-semigroup}; they are left to an interested reader.
\end{Remark}
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2,869,038,156,332 | arxiv | \section{Introduction}
Finite speed planar random motions are a very natural class of stochastic processes to describe real movements on a two-dimensional space. We can imagine a particle randomly moving in $\mathbb{R}^2$ with $n\in \mathbb{N}$ velocities $v_k = (v_{kx}, v_{ky})\in \mathbb{R}^2,\ k=1,\dots,n$, changing at random times according to different chance mechanisms. In general, from the velocity $v_j$ it is possible to switch to an arbitrary velocity $v_k$ with probability $p_{jk}\in [0,1],\ j,k=1,\dots, n$. The changes of velocity can be cyclic, that is a deterministic law is assumed to pass from $v_k$ to $v_{k+1}$, with $v_{k+n}=v_k$, or probabilistic, when from the velocity $v_k$ one can switch to the other ones according to some random law. The switches are usually governed by a Poisson process (homogeneous as well as non-homogeneous, as in the present paper) or by some more general renewal processes, see Di Crescenzo \cite{Dc2002}.
The main goal of the investigation for planar random motions is the probability distribution of the vector process $\bigl(X(t),Y(t)\bigr)$, which describes the position, at time $t\ge0$, of the moving particle. The study of finite speed planar random processes in continuous time have been first undertaken by probabilistic and physicists such as Orsingher \cite{O1986} and Masoliver \textit{et al.} \cite{MPW1993}. In Kolesnik and Turbin \cite{KT1998} was proved the interesting connection between planar random motions with $n$ different directions and $n$-th order hyperbolic partial differential equations.
Cyclic planar motions with three directions were treated by Di Crescenzo \cite{Dc2002}, Orsingher \cite{O2002}, Leorato and Orsingher \cite{LO2004} with displacements of random length and with different kinds of mechanism governing the switches of directions.
In the last decades several papers also dealt with multidimensional evolutions, see Samoilenko \cite{S2001}. Lachal \textit{et al.} \cite{LLO2006} studied minimal cyclic random motions, i.e. motions in $\mathbb{R}^d$ with $d+1$ directions forming a regular hyperpolyhedreon, with the technique based on order statistics. The distributions obtained involve Bessel functions the prototype of which is
$$ I_{0,n}(x) = \sum_{k=0}^\infty \Bigl(\frac{x}{n}\Bigr)^{2k}\frac{1}{k!^n},$$
with $n\in \mathbb{N}$. These results were extended by Lachal \cite{L2006}. Here the author provides a general integral formula for the distribution of a cyclic motion in $\mathbb{R}^d$ with a finite number of velocities. The study of motions in multidimensional spaces with an infinite number of directions has been carried out, for example, by Kolesnik and Orsingher \cite{KO2005}, concerning a planar evolution, and by Orsingher and De Gregorio \cite{ODg2007}, regarding higher spaces.
\\
In this paper we focus on planar random motions with orthogonal directions $d_k=v_k/|v_k|=\bigl(\cos(k\pi/2), \sin(k\pi/2)\bigr),\ |v_k|=c>0$ with $k=0,1,2,3$ (clearly $d_{k+n}=d_k$ for natural $n$) can be classified into five categories. Cyclic motion (see above), standard orthogonal motion (from direction $d_k$ the particle can switch either to $d_{k-1}$ or $d_{k+1}$), standard motion with Bernoulli trials (where the particle can also skip the change of direction), orthogonal motion with reflection (where the particle can switch either to one of the orthogonal directions or bounce back to $d_{k+2}$) and uniformly orthogonal motion (where the new direction is chosen among all the four possibile ones). The process governing the changes of direction is assumed here as a non-homogeneous Poisson process with rate $\lambda(t)$. This is a substantial difference with all the previous researches in this field. The standard orthogonal motion with $\lambda(t)=\lambda$ was examined by Orsingher and Kolesnik \cite{OK1996} and Orsingher \cite{O2000}, while the motion with reflection was considered in Kolesnik and Orsingher \cite{KO2001}. On the other hand Orsingher \textit{et al.} \cite{OGZ2020} studied the cyclic version, also in $\mathbb{R}^3$.
Some recent papers in the physical literature also treat multidimensional finite speed random motions on both the plane and higher spaces, see Mertens \textit{et al.} \cite{MetAl2012}, Elgeti and Gompper \cite{EG2015}, Hartmann \textit{et al.} \cite{HetAl2020}, Mori \textit{et al.} and Sevilla \cite{S2020}. We also recall the work of Santra \textit{et al.} \cite{SBS2020} where a section is also devoted to an orthogonal planar motion. However, explicit general laws of the distribution of the current position $\bigl(X(t),Y(t)\bigr)$ are not derived.
\begin{figure}
\begin{minipage}{0.5\textwidth}
\centering
\begin{tikzpicture}[scale = 0.71]
\draw[dashed, gray] (4,0) -- (0,4) node[above right, black, scale = 0.9]{$ct$};
\draw[dashed, gray] (0,4) -- (-4,0) node[above left, black, scale = 0.9]{$-ct$};
\draw[dashed, gray] (-4,0) -- (0,-4) node[below left, black, scale = 0.9]{$-ct$};
\draw[dashed, gray] (0,-4) -- (4,0) node[above right, black, scale = 0.9]{$ct$};
\draw[->, thick, gray] (-5,0) -- (5,0) node[below, scale = 1, black]{$\pmb{X(t)}$};
\draw[->, thick, gray] (0,-5) -- (0,5) node[left, scale = 1, black]{ $\pmb{Y(t)}$};
\draw (0,0)--(0.5,0)--(0.5,1)--(-0.3,1)--(-0.3,2)--(-1,2)--(-1,2.2);
\filldraw (0,0) circle (0.8pt); \filldraw (0.5,0) circle (0.8pt); \filldraw (0.5,1) circle (1pt); \filldraw (-0.3,1) circle (0.8pt); \filldraw (-0.3,2) circle (0.8pt); \filldraw (-1,2) circle (0.8pt);
\draw (0,0)--(0,0.4)--(-0.5,0.4)--(-0.5,1.6)--(-2.4,1.6);
\filldraw (0,0) circle (0.8pt); \filldraw(0,0.4) circle (0.8pt); \filldraw (-0.5,0.4) circle (0.8pt); \filldraw (-0.5,1.6) circle(0.8pt);
\draw (0,0)--(-0.8,0)--(-0.8,-2.2)--(-0.6,-2.2)--(-0.6,-2.6)--(-0.2,-2.6);
\filldraw(-0.8,0) circle (0.8pt); \filldraw (-0.8,-2.2) circle (0.8pt); \filldraw(-0.6,-2.2) circle(0.8pt); \filldraw(-0.6,-2.6) circle (0.8pt);
\draw (0,0)--(0,-0.8)--(1,-0.8)--(1,-0.9)--(1.4,-0.9)--(1.8,-0.9)--(1.8, 0.2);
\filldraw(0,-0.8)circle (0.8pt); \filldraw(1,-0.8)circle (0.8pt); \filldraw(1,-0.9)circle (0.8pt); \filldraw(1.4,-0.9)circle (0.8pt); \filldraw(1.8,-0.9)circle (0.8pt);
\end{tikzpicture}
\caption{\small Sample paths of a standard \newline orthogonal planar motion.}\label{MPS_1}
\end{minipage}\hfill
\begin{minipage}{0.5\textwidth}
\centering
\begin{tikzpicture}[scale = 0.71]
\draw[dashed, gray] (4,0) -- (0,4) node[above right, black, scale = 0.9]{$ct$};
\draw[dashed, gray] (0,4) -- (-4,0) node[above left, black, scale = 0.9]{$-ct$};
\draw[dashed, gray] (-4,0) -- (0,-4) node[below left, black, scale = 0.9]{$-ct$};
\draw[dashed, gray] (0,-4) -- (4,0) node[above right, black, scale = 0.9]{$ct$};
\draw[->, thick, gray] (-5,0) -- (5,0) node[below, scale = 1, black]{$\pmb{X(t)}$};
\draw[->, thick, gray] (0,-5) -- (0,5) node[left, scale = 1, black]{ $\pmb{Y(t)}$};
\draw (0,0)--(0.2,0)--(0.2,1.5)--(0.2,1.2)--(1.2,1.2)--(1.2,0.95)--(1.2,1.05)--(1.6,1.05);
\filldraw (0,0)circle (0.8pt); \filldraw(0.2,0)circle (0.8pt); \filldraw(0.2,1.5)circle (0.8pt); \filldraw(0.2,1.2)circle (0.8pt); \filldraw(1.2,1.2)circle (0.8pt); \filldraw(1.2,0.95)circle (0.8pt); \filldraw(1.2,1.05)circle (0.8pt);
\draw (0,0)--(0,0.4)--(-0.2,0.4)--(-0.2, 0.55)--(-0.2,0.25)--(-0.2,0.55)--(-1,0.55)--(-1, 0.4)--(-2.7,0.4);
\filldraw(0,0.4)circle (0.8pt); \filldraw(-0.2,0.4)circle (0.8pt); \filldraw(-0.2, 0.55);\filldraw(-0.2,0.25)circle (0.8pt); \filldraw(-0.2,0.55)circle (0.8pt); \filldraw(-1,0.55)circle (0.8pt); \filldraw(-1, 0.4)circle (0.8pt);
\draw (0,0)--(-0.8,0)--(-0.8,-2)--(-0.8,-1.3)--(-1.1,-1.3);
\filldraw(-0.8,0)circle (0.8pt); \filldraw(-0.8,-2)circle (0.8pt); \filldraw(-0.8,-1.3)circle (0.8pt);
\draw (0,0)--(0,-0.8)--(1,-0.8)--(1,-1)--(1.7,-1)--(1.4,-1)--(1.4, -1.4)--(1.4,-1.2)--(2,-1.2);
\filldraw(0,-0.8)circle (0.8pt); \filldraw(1,-0.8)circle (0.8pt); \filldraw(1,-1)circle (0.8pt); \filldraw(1.7,-1)circle (0.8pt); \filldraw(1.4,-1)circle (0.8pt); \filldraw(1.4, -1.4)circle (0.8pt); \filldraw(1.4,-1.2)circle (0.8pt);
\end{tikzpicture}
\caption{\small Sample paths of a reflecting orthogonal planar motion.}\label{MPS_2}
\end{minipage}
\end{figure}
In the orthogonal case we are able to obtain general laws for the vector process $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$ including the case where the switches of directions are governed by a non-homogeneous Poisson process. In this case we are able to prove that the current position $\bigl(X(t),Y(t)\bigr)$, for the standard case, can be represented as
\begin{equation}\label{decomposizioneIntro}
\begin{cases}
X(t) = U(t) + V(t)\\
Y(t) = U(t) - V(t)
\end{cases}
\end{equation}
where $U=\{U(t)\}_{t\ge0}$ and $V=\{V(t)\}_{t\ge0}$ are independent one-dimensional telegraph process. Their absolutely continuous components $p=p_U(x,t) = p_V(x,t)$ are solution to
\begin{equation}\label{equazioneTelegrafoIntroduzione}
\frac{\partial^2p}{\partial t^2} +\lambda(t)\frac{\partial p}{\partial t} = \frac{c^2}{4} \frac{\partial^2 p}{\partial x^2}.
\end{equation}
This means that $U$ and $V$ are (independent) telegraph processes with rate function $\lambda(t)/2$ and velocity $c/2$. This extends to the non-homogeneous case a previous result for the homogeneous Poisson process governing the switches of direction, see Orsingher \cite{O2000}. For all cases where the telegraph equation (\ref{equazioneTelegrafoIntroduzione}) can be treated, it is possible to arrive at the general law
\begin{equation}
P\{X(t)\in \dif x, \, Y(t)\in \dif y\}/(\dif x\dif y) = p(x,y,t) = \frac{1}{2}p_U\Bigl(\frac{x+y}{2}\Bigr)p_V\Bigl(\frac{x-y}{2}\Bigr),
\end{equation}
for $(x,y)\in S_{ct} = \{(x,y)\in\mathbb{R}^2\,:\,|x|+|y|\le ct\}$.
If $\Lambda(t) = \int_0^t\lambda(s)\dif s<\infty,\ t\ge0,$ the particle can reach the edge $\partial S_{ct}$ of its support and in this case we have also a direct derivation of the probability law, for example, on the side of $\partial S_{ct}$ belonging to the first quadrant, $\partial S_{ct}\cap (0,\infty)\times(0,\infty)$,
$$f(\eta, t)\dif \eta = P\{X(t)+Y(t)=ct,\, X(t)-Y(t)\in \dif \eta\},\ \ \ |\eta|<ct,$$
This probability satisfies the following second-order differential system
\begin{equation}\label{sistemaFrontieraAltoDx}
\begin{cases}
\frac{\partial^2 f}{\partial t^2} +2\lambda(t)\frac{\partial f}{\partial t} +\frac{1}{2}\Bigl(\frac{3}{2}\lambda(t)^2+\lambda'(t)\Bigr)f =c^2\frac{\partial^2 f}{\partial \eta^2},\ \ \ |\eta|<ct,\\
f(\eta,t)\ge0,\\
\int_{-ct}^{ct}f(\eta,t)\dif \eta = \frac{1}{2}(e^{-\Lambda(t)/2}-e^{-\Lambda(t)}).
\end{cases}
\end{equation} We note that
$$P\{\bigl(X(t),Y(t)\bigr)\in \partial S_{ct}\} = 2\Bigl(e^{-\frac{1}{2}\Lambda(t)}-e^{-\Lambda(t)}\Bigr)+e^{-\Lambda(t)},$$
where the first component pertains the sides of $\partial S_{ct}$ and the last term is the probability of reaching the vertexes of $S_{ct}$.
Another important probabilistic information concerns the distribution on squares of half-diagonal $0\le z\le ct$, which reads
\[P\{|X(t)|+|Y(t)|\in \dif z\} = 4 p_U\Bigl(\frac{z}{2}, t \Bigr)\int_0^{\frac{z}{2}} p_V(w,t)\dif w,\]
this can be interpreted as the distribution of the $L_1$-distance of the standard orthogonal process.
A complete picture of the motion is achieved thanks to the analysis of the marginal components. We give the third-order partial differential equation governing the projections $X(t)$ and $Y(t)$ and the characteristics of the one-dimensional motion they describe. The distribution $P\{X(t)\in \dif x\}$, with $|x|<ct$, is sometimes a hard technical problem which we tackle in some special cases.
Finally, we are able to extend the results of the standard orthogonal motion to the motion with Bernoulli trials and with different velocities along the two axes, thus producing some asymmetry.
The third section of the paper concerns the planar motion with reflection. This is substantially different from the standard one because at each switch of direction the particle can either deviate orthogonally or bounce back. This makes the distribution of the position process more complicated because of the appearance of an additional singularity along the diagonals of the support $S_{ct}$. One of the main consequences of the possible reflection of the particle is that a decomposition of the form (\ref{decomposizioneIntro}) makes the processes $U$ and $V$ dependent. Also here these are one-dimensional telegraph processes and we are able to describes their relationship.
When the rate function is constant, $\lambda(t)=\lambda>0\ \forall \ t\ge0$, Kolesnik and Orsingher \cite{KO2001} obtained the distribution on both the edge and the diagonals of $S_{ct}$. Here we extend these results to the case of a non-homogeneous Poisson process governing the switches. In particular, we are again able to connect the probabilities of the planar motion to those of one-dimensional telegraph processes.
At last, we extend the results of the reflecting planar motion to the uniformly orthogonal motion and to a wider class of orthogonal processes.
\section{Standard orthogonal planar random motion}
In this section we consider planar motions with directions $d_k = v_k/|v_k| = (\cos(k\pi/2), \sin(k\pi/2)), \\ k=0,1,2,3$ where $|v_k|=c>0$. The Poisson process governing the changes of direction is non-homogeneous with rate function $\lambda(t)\ge0, \ t\ge0$. At each Poisson event the moving particle can switch to one of the orthogonal directions with probability $1/2$.
\\We denote by $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0},$ the current position of the moving particle and $\{N(t\}_{t\ge0}$ the number of changes of direction recorded up to time $t$.
\subsection{The governing partial differential equation}\label{sottoSezioneEquazioneStandard}
\begin{theorem}\label{teoremaEquazioneStandard}
The absolutely continuous component $p = p(x,y,t)\in C^4\bigl(\mathbb{R}^2\times[0,\infty), [0,\infty)\bigr)$ of the distribution of the standard orthogonal process $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$ satisfies the following fourth-order differential equation with time-varying coefficients
\begin{equation}\label{equazioneStandard}
\biggl(\frac{\partial^2 }{\partial t^2}+2\lambda\frac{\partial}{\partial t} +\lambda^2+\lambda'\biggr)\biggl(\frac{\partial^2}{\partial t^2} +2\lambda\frac{\partial}{\partial t} -c^2\Bigl(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\Bigr)\biggr)p +c^4\frac{\partial^4 p}{\partial x^2\partial y^2} = \lambda'\frac{\partial^2 p}{\partial t^2}+(\lambda''+\lambda\lambda')\frac{\partial p}{\partial t}
\end{equation}
where $\lambda=\lambda(t)\in C^2\bigl((0,\infty),[0,\infty)\bigl)$ denotes the rate function of the non-homogeneous Poisson process governing the changes of direction.
\end{theorem}
\begin{proof}
We use the following notations
\begin{equation}\label{notazione}
f_k(x,y,t) \dif x\dif y = P\{X(t)\in \dif x, Y(t)\in \dif y,D(t) = d_k\},\ \ \ k=0,1,2,3,
\end{equation}
and $\{D(t)\}_{t>0}$ is the process taking the four directions $d_k$. We remark that at Poisson times the particle moving with direction $d_k$ can pass to $d_{k+1},d_{k-1}$ ($d_{k+4} = d_k = d_{k-4}$) with equal probability. We observe that, for $t\ge0$ and $(x,y)\in S_{c(t+\dif t)}$
\[f_0(x,y,t+\dif t) = f_0(x-c\dif t, t)\bigl(1-\lambda(t)\dif t \bigr) + \bigl[f_1(x,y-c\dif t,t )+f_3(x,y+c\dif t,t) \bigr]\lambda(t)\dif t+ o(\dif t) \]
and similarly for $f_1,f_2$ and $f_3$. With this at hand, we obtain the differential system governing the probability densities (\ref{notazione}),
\begin{equation}\label{sistemaStandard}
\begin{cases}
\frac{\partial f_0}{\partial t} = -c\frac{\partial f_0}{\partial x}+\frac{\lambda(t)}{2}(f_1+f_3-2f_0),\\
\frac{\partial f_1}{\partial t} = -c\frac{\partial f_1}{\partial y}+\frac{\lambda(t)}{2}(f_2+f_0-2f_1),\\
\frac{\partial f_2}{\partial t} = c\frac{\partial f_2}{\partial x}+\frac{\lambda(t)}{2}(f_1+f_3-2f_2),\\
\frac{\partial f_3}{\partial t} = c\frac{\partial f_3}{\partial y}+\frac{\lambda(t)}{2}(f_2+f_0-2f_3),
\end{cases} \text{and then\ \ \ }
\begin{cases}
\frac{\partial g_1}{\partial t} = -c\frac{\partial g_2}{\partial x}+\lambda(t) (g_3-g_1),\\
\frac{\partial g_2}{\partial t} = -c\frac{\partial g_1}{\partial x}-\lambda(t) g_2,\\
\frac{\partial g_3}{\partial t} = -c\frac{\partial g_4}{\partial y}+\lambda(t) (g_1-g_3),\\
\frac{\partial g_4}{\partial t} = -c\frac{\partial g_3}{\partial y}-\lambda(t) g_4.
\end{cases}
\end{equation}
where we used the following transformation to obtain the second system appearing in (\ref{sistemaStandard})
$$f_0+f_2 =g_1, \ \ \ f_0-f_2 = g_2, \ \ \ f_1+ f_3=g_3, \ \ \ f_1-f_3 = g_4.$$
Put $\lambda = \lambda(t)$. By differentions and substitutions we pass from the system of four first-order equations (\ref{sistemaStandard}) to the following system of second-order equations with two unknown functions $g_1$ and $g_3$
\begin{equation}\label{sistemaSecondoOrdineStandard}
\begin{cases}
\frac{\partial^2 g_1}{\partial t^2} = c^2 \frac{\partial^2 g_1}{\partial x^2} -2\lambda\frac{\partial g_1}{\partial t} +\lambda\frac{\partial g_3}{\partial t}+ (\lambda^2 +\lambda')(g_3-g_1),\\
\frac{\partial^2 g_3}{\partial t^2} = c^2 \frac{\partial^2 g_3}{\partial y^2} -2\lambda\frac{\partial g_3}{\partial t} +\lambda\frac{\partial g_1}{\partial t}+ (\lambda^2 +\lambda')(g_1-g_3).
\end{cases}
\end{equation}
By summing up and subtracting equations (\ref{sistemaSecondoOrdineStandard}) we obtain
\begin{equation}\label{sistemapwStandard}
\begin{cases}
\frac{\partial^2 p}{\partial t^2} = \frac{c^2}{2} \biggl[\Bigl(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\Bigr)p+\Bigl(\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}\Bigr)w\biggr]-\lambda\frac{\partial p}{\partial t},\\[8pt]
\frac{\partial^2 w}{\partial t^2} = \frac{c^2}{2} \biggl[\Bigl(\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}\Bigr)p+\Bigl(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\Bigr)w\biggr]-3\lambda\frac{\partial w}{\partial t}-2(\lambda^2+\lambda')w,
\end{cases}
\end{equation}
where $p=g_1+g_3$ and $w = g_1-g_3$. We outline the method to pass from (\ref{sistemapwStandard}) to (\ref{equazioneStandard}). By means of the following differential operators
\begin{equation*}
\begin{array}{l}
\Delta^+ = \frac{c^2}{2}\Bigl(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\Bigr),\ \ \ F = \Delta^+ -\lambda\frac{\partial }{\partial t},
\\
\Delta^- = \frac{c^2}{2}\Bigl(\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}\Bigr), \ \ \ G =\Delta^+ -3\lambda\frac{\partial }{\partial t}-2(\lambda^2+\lambda'),
\end{array}
\end{equation*}
the system (\ref{sistemapwStandard}) reads
\begin{equation}\label{sistemapwStandardSemplificato}
\begin{cases}
\frac{\partial^2 p}{\partial t^2} = Fp+\Delta^-w,\\
\frac{\partial^2 w}{\partial t^2} =\Delta^-p +Gw.
\end{cases}
\end{equation}
The elimination of $w$ from (\ref{sistemapwStandardSemplificato}) is performed by taking the second-order time derivative and by considering the commutativity of the differential operators
\begin{align}\label{}
\frac{\partial^4p}{\partial t^4}&=\frac{\partial^2 }{\partial t^2}Fp+\Delta^-\frac{\partial^2 }{\partial t^2}w = \frac{\partial^2 }{\partial t^2}Fp+\Delta^-\bigl(\Delta^-p +Gw\bigr) \nonumber\\
&= \frac{\partial^2 }{\partial t^2}Fp + (\Delta^-)^2p +G(\Delta^-w )= \frac{\partial^2 }{\partial t^2}Fp + (\Delta^-)^2p +G\Bigl(\frac{\partial^2 p}{\partial t^2}-Fp\Bigr).\nonumber
\end{align}
After some calculation, we obtain
\begin{align}\label{equazioneStandardEsplicita}
\frac{\partial^4p}{\partial t^4} + 4\lambda \frac{\partial^3p}{\partial t^3}+(5\lambda+4\lambda')\frac{\partial^2p}{\partial t^2}+ (2\lambda^2+5\lambda\lambda'+\lambda'')\frac{\partial p}{\partial t} + c^4\frac{\partial^4 p}{\partial x^2\partial y^2}\\
=c^2\Bigl(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\Bigr)&\Bigl(\frac{\partial^2p}{\partial t^2}+2\lambda\frac{\partial p}{\partial t} + (\lambda^2+\lambda')p\Bigr)\nonumber
\end{align}
and this coincides with the claimed result (\ref{equazioneStandard}).
\end{proof}
If $\lambda(t) = \lambda>0\ \forall\ t$, equation (\ref{equazioneStandard}) reduces to
\begin{equation}\label{equazioneStandardCasoCostante}
\Bigl(\frac{\partial^2 }{\partial t^2}+2\lambda\frac{\partial}{\partial t} +\lambda^2\Bigr)\biggl(\frac{\partial^2}{\partial t^2} +2\lambda\frac{\partial}{\partial t} -c^2\Bigl(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\Bigr)\biggr)p +c^4\frac{\partial^4 p}{\partial x^2\partial y^2} = 0
\end{equation}
which coincides with (3.6) of Orsingher and Kolesnik \cite{OK1996}..
It is well-known that under Kac's conditions, i.e. if $\lambda,c\longrightarrow\infty$ such that $\lambda/c^2\longrightarrow \sigma^2$, the standard orthogonal planar motion converges to a planar Brownian motion with diffusivity $\sigma^2$ (it is sufficient to divide by $\lambda^3$ equation (\ref{equazioneStandardCasoCostante})).
\begin{remark}
Let $\lambda=\lambda(t)\in C^3\bigl((0,\infty),[0,\infty)\bigl)$. By means of the transformation $p(x,y,t) = e^{-\int_0^t\lambda(s)\dif s}u(x,y,t)$ with $\int_0^t\lambda(s)\dif s<\infty$, we obtain the fourth order partial differential equation in $u$
\begin{equation*}\label{}
\biggl(\frac{\partial^2}{\partial t^2} -\lambda^2 -c^2\Bigl(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\Bigr)\biggr)\frac{\partial^2u}{\partial t^2}+c^4\frac{\partial^4 u}{\partial x^2\partial y^2} = 2\lambda'\frac{\partial^2u}{\partial t^2}+3(\lambda''+\lambda')\frac{\partial u}{\partial t}+ (\lambda'^2+\lambda\lambda'' +\lambda''')u.
\end{equation*}
\hfill$\diamond$
\end{remark}
\begin{remark}[Standard motion with Bernoulli trials]\label{remarkStandardBernoulliEquazione}
Let us consider a planar orthogonal random motion that behaves as the standard one with the upgrade that, at the Poisson times, it can continue to move along the same direction with probability $1-q\in[0,1)$. This means that at each occurrence of the Poisson events, the particle deviates on each orthogonal direction with probability $q/2$. This extension permits us to consider a refracting behavior for the particle. We call it \textit{$q$-standard motion}, $q\in(0,1]$ (or Standard motion with Bernoulli trials).
By proceeding as above in the case of the standard orthogonal motion, it can be proved that the absolutely continuous component of this generalized motion satisfies equation (\ref{equazioneStandard}) (or equivalently (\ref{equazioneStandardEsplicita})) with rate function $\lambda_q(t) = q\lambda(t)$ instead of $\lambda(t)$. This can be proved by calculating the differential system corresponding to (\ref{sistemaStandard}). Here the rate function $q\lambda(t)$ replaces $\lambda(t)$.
Under Kac's conditions, the $q$-standard motion converges to planar Brownian motion with diffusivity $\sigma^2/q$.\hfill$\diamond$
\end{remark}
\begin{remark}[Asymmetric motion]\label{remarkEquazioneAsimmetrico}
If we assume that the particle can move on the $x$-axis with velocity $c_X=|v_1|=|v_3|>0$ and on the $y$-axis with velocity $c_Y = |v_2|=|v_3|>0$ slight changes are needed. We can ``symmetrize'' the velocities by considering the new space coordinates $(x',y')$ which are related to the original ones by means of the scaling
$$x' = x/c_X,\ \ \ y' = y/c_Y.$$
These assumptions permit us to consider an asymmetric behavior of the particle. The support of the asymmetric motion $\bigl(X(t),Y(t)\bigr)$ is the rhombus
\begin{equation*}
R_{t} = \Big\{(x,y)\in \mathbb{R}^2\,:\,\frac{|x|}{c_X}+\frac{|y|}{c_Y}\le t\Big\}.
\end{equation*}
The fourth-order differential equation governing the density of the absolutely continuous component of the asymmetric motion is obtained by performing the following substitutions in (\ref{equazioneStandard})
\[ c^2\Bigl(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\Bigr) \rightarrow c^2_X\frac{\partial^2}{\partial x^2}+c^2_Y\frac{\partial^2}{\partial y^2}, \ \ \ \ c^4\frac{\partial^4 u}{\partial x^2\partial y^2}\rightarrow c_X^2c_Y^2\frac{\partial^4 u}{\partial x^2\partial y^2}. \]
\\Note that all these considerations can be applied to the standard motion with Bernoulli trials as well.
\hfill$\diamond$
\end{remark}
\subsection{Explicit representation of the probability distribution}
As we previously observed, the particle performing the standard planar random motion $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$ is located at time $t>0$ in the square
\begin{equation}\label{supportoQuadrato}
S_{ct} = \{(x,y)\in \mathbb{R}^2\,:\,|x|+|y|\le ct\}.
\end{equation}
Furthermore, at time $t$ it lies on the border $\partial S_{ct}$ with probability
\begin{align}\label{massaFrontieraCasoStandard}
P\big\{\bigl(X(t),Y(t)\bigr)\in \partial S_{ct}\big\} &= P\{N(t) = 0\}+\sum_{k=1}^\infty \frac{1}{2^{k-1}}P\{N(t) = k\} \\
&=2e^{-\frac{1}{2}\int_0^t\lambda(s)\dif s} -e^{-\int_0^t\lambda(s)\dif s},\nonumber
\end{align}
provided that $\int_0^t\lambda(s)\dif s<\infty$. Note that the probability of being on the vertices, $V_{ct} = \{(0,\pm ct),(\pm ct,0)\}$, is equal to $P\{N(t)=0\}$.
\\This means that if $\int_0^t\lambda(s)\dif s=\infty$ the moving particle will be inside the square $S_{ct}$ with probability one and $\bigl(X(t),Y(t)\bigr)$ is an absolutely continuous random vector for all $t>0$.
On each side of $\partial S_{ct}$ the moving particle performs a telegraph process (see below for further explanations).
We note that the absolutely continuous part of the distribution inside $S_{ct}$ and on the edge $\partial S_{ct}$ coincide at the instant $t^*>0$ such that
\[\int_0^{t^*}\lambda(s)\dif s = -2 \ln\Bigl(1-\frac{1}{\sqrt{2}}\Bigr).\]
Finally, we observe that in order to have only a singular component on the front edge $\partial S_{ct}$ we must assume that the particle can choose with probability $1/4$ one of the couples of direction $(d_k, d_{k+1})$, with $k=0,1,2,3$ and $d_4 = d_0$, and then move alternatively with the directions of the chosen couple and with switches occurring at Poisson times.
In the following proposition we study the distribution of the motion on the border $\partial S_{ct}$. In this case we deal with a particle that is continuously forced to reach a set which is itself moving outwards. This is the main source of difficulty of the next statement.
\begin{proposition}\label{propFrontieraStandard}
Let $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$ be a standard orthogonal planar random motion with changes of direction paced by a Poisson process with rate $\lambda(t)\in C^1\bigl((0,\infty),[0,\infty)\bigl)$ such that $\Lambda(t) = \int_0^t\lambda(s)\dif s<\infty$ for all $t>0$. Then, for $|\eta|<ct$
\begin{equation}\label{leggeFrontieraAltoDx}
f(\eta, t)\dif \eta =P\{X(t)+Y(t) = ct, X(t)-Y(t)\in \dif \eta\}
\end{equation}
satisfies the differential problem
\begin{equation}\label{sistemaFrontieraAltoDx}
\begin{cases}
\frac{\partial^2 f}{\partial t^2} +2\lambda(t)\frac{\partial f}{\partial t} +\frac{1}{2}\Bigl(\frac{3}{2}\lambda(t)^2+\lambda'(t)\Bigr)f =c^2\frac{\partial^2 f}{\partial \eta^2},\\
f(\eta,t)\ge0,\\
\int_{-ct}^{ct}f(\eta,t)\dif \eta = \frac{1}{2}(e^{-\Lambda(t)/2}-e^{-\Lambda(t)}).
\end{cases}
\end{equation}
\end{proposition}
Note that (\ref{leggeFrontieraAltoDx}) is the probability that the motion lies on the border of its support in the first quadrant, that is
\begin{equation}\label{insiemeFrentieraAltoDx}
\partial F_{ct} = \partial S_{ct}^{(1)}\setminus V_{ct} = \{(x,y)\in \mathbb{R}^2\,:\, x+y = ct, |x-y|<ct\}
\end{equation}
where $V_{ct} = \{(ct,0),(0,ct),(-ct,0),(0,-ct)\}$.
Clearly, equivalent results hold for the other components of the border $\partial S_{ct}$.
\begin{proof}
In order that the particle reaches at time $t>0$ a point on the set $\partial F_{ct}$ it must alternate between rightward ($d_0$) and upward ($d_1$) displacements for the whole time interval $[0,t]$.
\\In order to describe the probabilistic behavior on this border, we need
\begin{equation}\label{notazioneFrontieraStandard}
\begin{cases}
f_0(\eta, t) \dif \eta = P\{X(t)+Y(t) = ct,\ X(t)-Y(t)\in \dif \eta,\ D(t) = d_0\},\\
f_1(\eta, t) \dif \eta = P\{X(t)+Y(t) = ct,\ X(t)-Y(t)\in \dif \eta,\ D(t) = d_1\}.
\end{cases}
\end{equation}
We note that if the particle at time $t$ is moving rightward and was on $\partial F_{ct}$, that is $x+y=ct$, then at time $t+\dif t$ it will be on $\partial F_{c(t+\dif t)}$ with coordinates $(x+c\dif t, y)$ that is $X(t+\dif t)-Y(t+\dif t) =x+c\dif t-y = \eta+\dif\eta\ (\dif \eta = c\dif t)$ and similarly for the upward movements. As a consequence we have that
\[f_0(\eta, t+\dif t) = f_0(\eta-c\dif t,t )(1-\lambda(t)\dif t)+ f_1(\eta +c\dif t,t)\frac{1}{2}\lambda(t)\dif t.\]
By means of simple calculation we obtain
\begin{equation}
\begin{cases}
\frac{\partial f_0}{\partial t} = -c\frac{\partial f_0}{\partial \eta}+\frac{\lambda(t)}{2}(f_1-2f_0),\\
\frac{\partial f_1}{\partial t} = c\frac{\partial f_1}{\partial \eta}+\frac{\lambda(t)}{2}(f_0-2f_1),
\end{cases}\text{and}\ \ \
\begin{cases}
\frac{\partial f}{\partial t} = -c\frac{\partial w}{\partial \eta}-\frac{\lambda(t)}{2}f,\\
\frac{\partial w}{\partial t} = -c\frac{\partial f}{\partial \eta}-\frac{3}{2}\lambda(t) w,
\end{cases}
\end{equation}
where $f=f_0+f_1$ (it coincides with (\ref{leggeFrontieraAltoDx})) and $w = f_0-f_1$. From the above system we extract the second-order differential equation appearing in (\ref{sistemaFrontieraAltoDx}).
The integral condition of (\ref{sistemaFrontieraAltoDx}) follows from the fact that
\begin{align*}
\int_{-ct}^{ct}f(\eta,t)\dif \eta = P\big\{\bigl(X(t),Y(t)\bigr)\in \partial F_{ct} \big\} &= \frac{1}{4}P\big\{\bigl(X(t),Y(t)\bigr)\in \partial S_{ct}\setminus V_{ct} \big\}\\
&=\frac{1}{2}(e^{-\Lambda(t)/2}-e^{-\Lambda(t)})
\end{align*}
which can be easily derived from formula (\ref{massaFrontieraCasoStandard}).
\end{proof}
\begin{remark}
In the special case where $\lambda(t)=\lambda\ \forall \ t$, the system (\ref{sistemaFrontieraAltoDx}) reduces to
\begin{equation}\label{sistemaFrontieraAltoDxCostante}
\begin{cases}
\frac{\partial^2 f}{\partial t^2} +2\lambda\frac{\partial f}{\partial t} +\frac{3}{4}\lambda^2f =\frac{c^2}{4}\frac{\partial^2f}{\partial \eta^2},\\
f(\eta,t)\ge 0,\\
\int_{-ct}^{ct}f(\eta,t)\dif \eta = \frac{1}{2}(e^{-\frac{\lambda t}{2}}-e^{-\lambda t}).
\end{cases}
\end{equation}
The differential equation above can be further reduced to the Klein-Gordon equation by means of the transformation $f(\eta, t) = e^{-\lambda t}q(\eta, t)$,
\[\frac{\partial^2 q}{\partial t^2} -\frac{c^2}{4}\frac{\partial^2 q}{\partial \eta^2}=\frac{\lambda^2}{4}q,\ \ \ \ |\eta|<ct.\]
Hence, the solution of (\ref{sistemaFrontieraAltoDxCostante}) is
\[f(\eta, t)=\frac{e^{-\lambda t}}{4c}\Biggl[\frac{\lambda}{2} I_0\Bigl(\frac{\lambda}{2c}\sqrt{c^2t^2-\eta^2} \Bigr)+\frac{\partial}{\partial t}I_0\Bigl(\frac{\lambda}{2c}\sqrt{c^2t^2-\eta^2} \Bigr)\Biggr]. \]
From this result, we can also extract the conditional distributions with respect to the number of switches, $N(t)$. For integer $k\ge0$,
\begin{align}
P\{X(t)+Y(t)& = ct, X(t)-Y(t)\in \dif \eta\,|\,N(t)=2k+1\} \nonumber\\
&=2P\{X(t)+Y(t) = ct, X(t)-Y(t)\in \dif \eta\,|\,N(t)=2k+2\}\nonumber\\
&=\frac{(2k+1)!}{2\,k!^2}\frac{(c^2t^2-\eta^2)^k}{(4ct)^{2k+1}}\dif \eta \nonumber
\end{align}
where $|\eta|<ct$.\hfill$\diamond$
\end{remark}
Our main result is stated in the following theorem.
\begin{theorem}\label{teoremaDecomposizioneXYUV}
The standard planar orthogonal random motion $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$, with rate function $\lambda(t)\in C^2\bigl((0,\infty),[0,\infty)\bigl)$, is equal in distribution to the linear transformation of two independent one-dimensional telegraph processes $\{U(t)\}_{t\ge0},\ \{V(t)\}_{t\ge0}$ with parameters $(c/2, \lambda(t)/2)$
\begin{equation}\label{decomposizioneXYUV}
\begin{cases}
X (t)= U(t)+V(t),\\
Y(t) = U(t)-V(t).
\end{cases}
\end{equation}
\end{theorem}
The statement (\ref{decomposizioneXYUV}) shows that $\bigl(X(t),Y(t)\bigr)$ is a rotation of $45^\text{o}$ of the vector with independent components $\bigl(U(t),V(t)\bigr)$.
Note that the absolutely continuous component $p(x,y,t)\dif x\dif y \\= P\{X(t)\in \dif x,Y(t)\in \dif y\}$ is given by
\begin{equation}\label{}
p(x,y,t)=\frac{1}{2}p_U\Bigl(\frac{x+y}{2},t\Bigr)p_V\Bigl(\frac{x-y}{2},t\Bigr)
\end{equation}
where $p_U$ and $p_V$ are the densities of one-dimensional telegraph processes, i.e. they are solutions of the Cauchy problem
\begin{equation}\label{CauchyTelegrafo}
\begin{cases}
\frac{\partial^2 u}{\partial t^2} +\lambda(t)\frac{\partial u}{\partial t} =\frac{c^2}{4}\frac{\partial^2 u}{\partial x^2}, \\
u(x,0) = \delta(x),\ \ \ \frac{\partial u}{\partial t}(x,t)\big|_{t=0} = 0,
\end{cases}
\end{equation}
where $\delta$ is the Dirac delta function centered in $x=0$.
\begin{proof}
If $\Lambda(t) =\int_0^t\lambda(s)\dif s<\infty$ the distribution of $\bigl(X(t),Y(t)\bigr),\ t\ge0,$ has a positive probability mass on $\partial S_{ct}$. It is trivial to show that, for $|\eta|<ct$,
\[f(\eta,t) = \frac{e^{-\frac{\Lambda(t)}{2}}}{2}\,p_V\Bigl(\frac{\eta}{2}, t\Bigr)\]
satisfies (\ref{sistemaFrontieraAltoDx}). The probability on the vertices of the border $\partial S_{ct}$ easily follows as well.
For the absolutely continuous component of the distribution, it is sufficient to prove that $p = p(x,y,t)$ satisfies (\ref{equazioneStandard}). In order to spare calculation we pass to variables
\begin{equation}\label{cambioVariabiliLeggeEsplicita}
u = \frac{x+y}{2},\ \ \ v = \frac{x-y}{2}
\end{equation}
and prove that
\begin{equation}\label{p_uv}
q(u,v,t)=\frac{1}{2}p_U(u,t)p_V(v,t)=p(u+v,u-v,t)
\end{equation}
(disregarding the factor $1/2$) satisfies, with $\lambda=\lambda(t)$,
\begin{equation}\label{equazioneStandard_uv}
\biggl(\frac{\partial^2 }{\partial t^2}+2\lambda\frac{\partial}{\partial t} +\lambda^2+\lambda'\biggr)\biggl(\frac{\partial^2}{\partial t^2} +2\lambda\frac{\partial}{\partial t} -\frac{c^2}{2}\Bigl(\frac{\partial^2}{\partial u^2}+\frac{\partial^2}{\partial v^2}\Bigr)\biggr)q +\frac{c^4}{16}\Bigl(\frac{\partial^2}{\partial u^2}-\frac{\partial^2}{\partial v^2}\Bigr)^2q = \lambda'\frac{\partial^2 q}{\partial t^2}+(\lambda''+\lambda\lambda')\frac{\partial q}{\partial t}
\end{equation}
where $p_U$ and $p_V$ satisfy (\ref{CauchyTelegrafo}). Equation (\ref{equazioneStandard_uv}) is obtained from (\ref{equazioneStandard}) by means of the change of variables (\ref{cambioVariabiliLeggeEsplicita}).
\\From (\ref{p_uv}) we have that
\begin{equation}\label{derivata2quv0}
\frac{\partial^2 q}{\partial t^2}=\frac{c^2}{4}\Bigl(\frac{\partial^2}{\partial u^2}+\frac{\partial^2}{\partial v^2}\Bigr)q - \lambda\frac{\partial q}{\partial t}+2\frac{\partial p_U}{\partial t}\frac{\partial p_V}{\partial t}
\end{equation}
and then, by multiplying by $2$ both members of (\ref{derivata2quv0}) we obtain
\begin{equation}\label{derivata2quv}
\frac{\partial^2 q}{\partial t^2}+2\lambda\frac{\partial q}{\partial t}-\frac{c^2}{2}\Bigl(\frac{\partial^2}{\partial u^2}+\frac{\partial^2}{\partial v^2}\Bigr)q =4\frac{\partial p_U}{\partial t}\frac{\partial p_V}{\partial t}-\frac{\partial^2 q}{\partial t^2}
\end{equation}
where the second-order planar telegraph different operator appearing in (\ref{equazioneStandard_uv}) emerges. By multiplying both members of (\ref{derivata2quv}) by the operator $\frac{\partial^2 }{\partial t^2}+2\lambda(t)\frac{\partial}{\partial t} +\lambda^2(t)$, we have
\begin{align}\label{operatore1_quv}
\Bigl(\frac{\partial^2 }{\partial t^2}+2\lambda\frac{\partial}{\partial t} +&\lambda^2\Bigr) \biggl( \frac{\partial^2 q}{\partial t^2}+2\lambda\frac{\partial q}{\partial t}-\frac{c^2}{2}\Bigl(\frac{\partial^2}{\partial u^2}+\frac{\partial^2}{\partial v^2}\Bigr)q \biggr) \\
&=\Bigl(\frac{\partial^2 }{\partial t^2}+2\lambda\frac{\partial}{\partial t} +\lambda^2\Bigr) \Bigl(4\frac{\partial p_U}{\partial t}\frac{\partial p_V}{\partial t}-\frac{\partial^2 q}{\partial t^2} \Bigr)\nonumber\\
& = \lambda^2 \Bigl(4\frac{\partial p_U}{\partial t}\frac{\partial p_V}{\partial t}-\frac{\partial^2 q}{\partial t^2} \Bigr)+ 2\lambda \Bigl(4\frac{\partial^2 p_U}{\partial t^2}\frac{\partial p_V}{\partial t}+4\frac{\partial p_U}{\partial t}\frac{\partial^2 p_V}{\partial t^2}-\frac{\partial^3 q}{\partial t^3} \Bigr)\nonumber\\
&\ \ \ + \Bigl(4\frac{\partial^3 p_U}{\partial t^3}\frac{\partial p_V}{\partial t}+8\frac{\partial^2 p_U}{\partial t^2}\frac{\partial^2 p_V}{\partial t^2}+4\frac{\partial p_U}{\partial t}\frac{\partial^3 p_V}{\partial t^3}-\frac{\partial^4 q}{\partial t^4} \Bigr)\nonumber
\end{align}
where simple derivations are involved. In view of (\ref{CauchyTelegrafo}) applied successively and by convenient substitutions we produce the identity
\begin{align}\label{operatore2_quv}
\frac{c^4}{16}\Bigl(\frac{\partial^2}{\partial u^2}-\frac{\partial^2}{\partial v^2}\Bigr)^2q & = p_V\Bigl(\frac{\partial^4 p_U}{\partial t^4} +2\lambda\frac{\partial^3 p_U}{\partial t^3} +(\lambda^2+2\lambda')\frac{\partial^2 p_U}{\partial t^2}+(\lambda\lambda' +\lambda'')\frac{\partial p_U}{\partial t}\Bigr)\\
&\ \ \ + p_U\Bigl(\frac{\partial^4 p_V}{\partial t^4} +2\lambda\frac{\partial^3 p_V}{\partial t^3} +(\lambda^2+2\lambda')\frac{\partial^2 p_V}{\partial t^2}+(\lambda\lambda' +\lambda'')\frac{\partial p_V}{\partial t}\Bigr)\nonumber\\
&\ \ \ - 2\biggl( \frac{\partial^2 p_U}{\partial t^2}\frac{\partial^2 p_V}{\partial t^2}+\lambda^2\frac{\partial p_U}{\partial t}\frac{\partial p_V}{\partial t}+\lambda\Bigl(\frac{\partial^2 p_U}{\partial t^2}\frac{\partial p_V}{\partial t}+\frac{\partial p_U}{\partial t}\frac{\partial^2 p_V}{\partial t^2} \Bigr) \biggr)\nonumber.
\end{align}
By summing up (\ref{operatore1_quv}) and (\ref{operatore2_quv}) and by keeping in mind (\ref{derivata2quv0}), the terms with time-varying coefficients $\lambda^0,\lambda$ and $\lambda^2$ cancel out and we obtain
\begin{align}
\Bigl(\frac{\partial^2 }{\partial t^2}+2\lambda\frac{\partial}{\partial t} &+\lambda^2\Bigr) \biggl( \frac{\partial^2 q}{\partial t^2}+\lambda\frac{\partial q}{\partial t}-\frac{c^2}{2}\Bigl(\frac{\partial^2}{\partial u^2}+\frac{\partial^2}{\partial v^2}\Bigr)q \biggr) +\frac{c^4}{16}\Bigl(\frac{\partial^2}{\partial u^2}-\frac{\partial^2}{\partial v^2}\Bigr)^2q \label{termineDimostrazioneDecomposizione}\\
& = (\lambda\lambda'+\lambda'')\Bigl(\frac{\partial p_U}{\partial t} p_V+p_U
\frac{\partial p_V}{\partial t}\Bigr) + 2\lambda'\Bigl(\frac{\partial^2 p_U}{\partial t^2} p_V+p_U\frac{\partial^2 p_V}{\partial t^2}\Bigr) \nonumber \\
&=(\lambda\lambda'+\lambda'')\frac{\partial q}{\partial t}+ 2\lambda'\Bigl(\frac{\partial^2q}{\partial t^2}- 2\frac{\partial p_U}{\partial t}\frac{\partial p_V}{\partial t} \Bigr) \nonumber\\
& =(\lambda\lambda'+\lambda'')\frac{\partial q}{\partial t}+ 2\lambda'\frac{\partial^2q}{\partial t^2}-2\lambda'\biggl( \frac{\partial^2 q}{\partial t^2}+\lambda\frac{\partial q}{\partial t}-\frac{c^2}{4}\Bigl(\frac{\partial^2}{\partial u^2}+\frac{\partial^2}{\partial v^2}\Bigr)q \biggr) \nonumber
\end{align}
where in the last equality we suitably used (\ref{derivata2quv0}). Equation (\ref{termineDimostrazioneDecomposizione}) coincides with (\ref{equazioneStandard_uv}) and this completes the proof concerning the absolutely continuous component.
\end{proof}
In force of (\ref{decomposizioneXYUV}), assuming that $X(t)+Y(t)=z\in [-ct,ct]$, the distribution of the standard orthogonal motion coincides with that of a one-dimensional telegraph process with rate $\lambda(t)/2$ and velocities $\pm c$. It similarly happens if we assume $X(t)-Y(t)=z\in [-ct,ct]$. We point out that the displayed conditions imply that the motion lies on a segment that is parallel to some sides of $\partial S_{ct}$.
\\
Representation (\ref{decomposizioneXYUV}) permits us to focus on the study of one-dimensional processes only. We thus know the explicit distribution of $\bigl(X(t),Y(t)\bigr)$ when the rate function $\lambda(t),\ t>0,$ has one of the following forms, with $\lambda >0$,
\[\lambda(t)=\lambda,\ \ \ \ \ \lambda(t)=\frac{\lambda}{t} \ \text{(Foong and Van Kolck \cite{FVk1992})},\]
\[\lambda(t)=\lambda\text{th}(\lambda t) \ \text{(Iacus \cite{I2001})},\ \ \ \lambda(t)=\lambda\text{coth}(\lambda t) \ \text{(Garra and Orsingher \cite{GO2016})}.\]
For $\lambda(t) =\lambda/t,\: \lambda\text{coth}(\lambda t)$, the singular component of the distribution is absent because $\int_0^t \lambda(s)\dif s = \infty$.
\begin{remark}[$L_1$-distance
Let $\{Z(t)\}_{t\ge0}$ be the stochastic process describing the Manhattan distance of the particle $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$ from the origin of the coordinate axes, i.e. $Z(t)= |X(t)|+|Y(t)|$. Let $\{U(t)\}_{t\ge0}$ and $\{V(t)\}_{t\ge0}$ be two independent one-dimensional telegraph processes with parameters $(c/2, \lambda(t)/2)$. By means of Theorem \ref{teoremaDecomposizioneXYUV} we have that, for $z\in [0,ct)$
\begin{align}\label{}
P\{Z(t)\in \dif z\}/\dif z &= \frac{\partial }{\partial z}P\{|X(t)|+|Y(t)|\le z\} =4\frac{\partial }{\partial z}\int_0^z\dif x\int_0^{z-x}p(x,y,t)\dif y \nonumber\\
&=2\frac{\partial }{\partial z}\int_0^z\dif x\int_0^{z-x}p_U\Bigl(\frac{x+y}{2},t\Bigr)p_V\Bigl(\frac{x-y}{2},t\Bigr)\dif y\nonumber\\
&= 2\,p_U\Bigl(\frac{z}{2},t\Bigr)\int_0^z p_V\Bigl(\frac{2x-z}{2},t\Bigr)\dif x=4\,p_U\Bigl(\frac{z}{2},t\Bigr)\int_0^{\frac{z}{2}} p_V(w,t)\dif w\nonumber.
\end{align}
We can also express the maximum $L_1$-distance of $\bigl(X(t),Y(t)\bigr)$ from the origin in terms of the distribution of a functional of a one-dimensional telegraph process. Let $\beta\in [0,ct]$, with (\ref{decomposizioneXYUV}) at hand, we obtain that
\begin{align}\label{}
P\Big\{&\max_{0\le s\le t}Z(s)\le \beta\Big\} = P\Big\{\max_{0\le s\le t}(\,|X(s)|+|Y(s)|\,)\le\beta\Big\} \nonumber\\
&=P\Big\{\max_{0\le s\le t}\big\{\,X(s)+Y(s),\,-X(s)-Y(s),\,X(s)-Y(s),\,-X(s)+Y(s)\,\big\}\le\beta\Big\}\nonumber \\
&=P\Big\{\max_{0\le s\le t}\{2|U(s)|,\,2|V(s)|\}\le\beta\Big\}=P\Big\{\max_{0\le s\le t} |U(s)|\le\beta/2\Big\}^2.
\nonumber
\end{align}
As far as we know, the distribution of $\max_{0\le s\le t} |U(s)|$ is unknown for all rate functions $\lambda(t)$. The interested reader can find a detailed study of the (one-sided) maximum of the constant rate one-dimensional telegraph process in Cinque and Orsingher \cite{CO2020}.
\hfill $\diamond$
\end{remark}
\begin{figure}
\begin{minipage}{0.5\textwidth}
\centering
\begin{tikzpicture}[scale = 0.71]
\draw[dashed, gray] (4,0) -- (0,4) node[above right, black, scale = 0.9]{$ct$};
\draw[dashed, gray] (0,4) -- (-4,0) node[above left, black, scale = 0.9]{$-ct$};
\draw[dashed, gray] (-4,0) -- (0,-4) node[below left, black, scale = 0.9]{$-ct$};
\draw[dashed, gray] (0,-4) -- (4,0) node[above right, black, scale = 0.9]{$ct$};
\draw (2.5,0) -- (0,2.5) node[right, black, scale = 0.8]{$z$};
\draw(0,2.5) -- (-2.5,0) node[above, black, scale = 0.8]{$-z\ \ $};
\draw (-2.5,0) -- (0,-2.5) node[left, black, scale = 0.8]{$-z$};
\draw (0,-2.5) -- (2.5,0) node[above, black, scale = 0.8]{$z$};
\draw[->, thick, gray] (-5,0) -- (5,0) node[below, scale = 1, black]{$\pmb{X(t)}$};
\draw[->, thick, gray] (0,-5) -- (0,5) node[left, scale = 1, black]{ $\pmb{Y(t)}$};
\draw (0,0)--(0.5,0)--(0.5,1)--(-0.3,1)--(-0.3,1.9)--(-1,1.9)--(-1,1.5);
\filldraw (0,0) circle (0.8pt); \filldraw (0.5,0) circle (0.8pt); \filldraw (0.5,1) circle (1pt); \filldraw (-0.3,1) circle (0.8pt); \filldraw (-0.3,1.9) circle (0.8pt); \filldraw (-1,1.9) circle (0.8pt);
\draw (0,0)--(-0.8,0)--(-0.8,-2.2)--(-1.2,-2.2)--(-1.2,-2.8);
\filldraw (-0.8,0) circle (1pt); \filldraw (-0.8,-2.2) circle (0.8pt); \filldraw (-1.2,-2.2) circle (0.8pt); \filldraw (-1.2,-2.2) circle (0.8pt);
\draw (0,0)--(0,-0.8)--(1,-0.8)--(1,-0.9)--(1.6,-0.9)--(1.6, 0.4);
\filldraw(0,-0.8)circle (0.8pt);\filldraw(1,-0.8)circle (0.8pt);\filldraw(1,-0.9)circle (0.8pt);\filldraw(1.6,-0.9)circle (0.8pt);
\end{tikzpicture}
\caption{\small The trajectory starting with $d_0$ has $L_1$-distance from the origin equal to $z\in$$(0,ct)$\newline at time $t$. The trajectory starting with $d_3$ \newline has maximum $L_1$-distance from the origin \newline equal to $z$.}\label{L1dist}
\end{minipage}\hfill
\begin{minipage}{0.5\textwidth}
\centering
\begin{tikzpicture}[scale = 0.71]
\draw[dashed, gray] (3,0) -- (0,4.5) node[right, black, scale = 0.9]{$c_Yt$};
\draw[dashed, gray] (0,4.5) -- (-3,0) node[above left, black, scale = 0.9]{$-c_Xt$};
\draw[dashed, gray] (-3,0) -- (0,-4.5) node[below left, black, scale = 0.9]{$-c_Yt$};
\draw[dashed, gray] (0,-4.5) -- (3,0) node[above right, black, scale = 0.9]{$c_Xt$};
\draw[->, thick, gray] (-5,0) -- (5,0) node[below, scale = 1, black]{$\pmb{X(t)}$};
\draw[->, thick, gray] (0,-5.2) -- (0,5.2) node[left, scale = 1, black]{ $\pmb{Y(t)}$};
\draw (0,0)--(-0.5,0)--(-0.5,1)--(-0.5,2.25)--(-1.5,2.25);
\filldraw (-0.5,0) circle (1pt); \filldraw (-0.5,1) circle (0.8pt); \filldraw (-0.5,2.25) circle (0.8pt);
\draw (0,0)--(0,0.3)--(1.2,0.3)--(1.2,0.6)--(1.2,1.18)--(0.8,1.18)--(0.5,1.18)--(0.15,1.18);
\filldraw (0,0.3) circle (1pt); \filldraw (1.2,0.3) circle (0.8pt); \filldraw (1.2,0.6) circle (0.8pt); \filldraw (1.2,1.18) circle (0.8pt);\filldraw (0.8,1.18) circle (0.8pt); \filldraw (0.5,1.18) circle (0.8pt);
\draw (0,0)--(0.2,0)--(0.2,-1.3)--(0.2,-1.5)--(0.8,-1.5)--(0.8,-1.1)--(0.6,-1.1)--(0.6,-2.1);
\filldraw (0.2,0) circle (1pt); \filldraw (0.2,-1.3) circle (0.8pt); \filldraw (0.2,-1.5) circle (0.8pt); \filldraw (0.8,-1.5) circle (0.8pt);\filldraw (0.8,-1.1) circle (0.8pt); \filldraw (0.6,-1.1) circle (0.8pt);
\draw (0,0)--(0,-0.4)--(-0.8,-0.4)--(-0.8,-1)--(-0.5,-1)--(-0.5,-1.5)--(-0.5,-2.4)--(-0.65,-2.4)--(-0.8,-2.4);
\filldraw (0,-0.4) circle (0.8pt); \filldraw (-0.8,-0.4) circle (0.8pt); \filldraw (-0.8,-1) circle (0.8pt);\filldraw (-0.5,-1) circle (1pt); \filldraw (-0.5,-1.5) circle (0.8pt); \filldraw (-0.5,-2.4) circle (0.8pt); \filldraw (-0.65,-2.4) circle (0.8pt);
\end{tikzpicture}
\caption{\small Sample paths of an asymmetric ($c_X<c_Y$) standard orthogonal planar motion with Bernoulli trials.}\label{AsimmetricoBernStandard}
\end{minipage}
\end{figure}
\begin{remark}[Standard motion with Bernoulli trials]\label{remarkStandardBernoulliLegge}
Let us consider the $q$-standard motion $\{\bigl(X_q(t),Y_q(t)\bigr)\}_{t\ge0}$, with $q\in(0,1]$ being the probability that the particle changes direction at Poisson times (see Remark \ref{remarkStandardBernoulliEquazione}). In this case, the particle, at time $t\ge0$, will be located in the square $S_{ct}$ defined in (\ref{supportoQuadrato}). However, if $\int_0^t \lambda(s)\dif s<\infty$, it can reach also the vertices of the square whatever the number of Poisson events in $[0,t]$ is,
\[P\big\{\bigl(X_q(t),Y_q(t)\bigr)\in V_{ct}\big\} = \sum_{n=0}^\infty(1-q)^nP\{N(t) = n\} = e^{-q\int_0^t \lambda(s)\dif s}.\]
It is very interesting to observe that, by keeping in mind Remark \ref{remarkStandardBernoulliEquazione} and by proceeding as above, also this motion can be expressed by means of the representation (\ref{decomposizioneXYUV}), but in this case the one-dimensional telegraph processes are such that at all Poisson events, the change of direction occurs with probability $q\in(0,1]$. Now, by observing that such a telegraph process is equal in distribution to a telegraph process of rate $q\lambda(t)$, we obtain the following statement:
\\\textit{the standard motion with Bernoulli trials (with probability of changes equal to $q\in(0,1]$) is equal in distribution to a standard orthogonal motion with rate function $\lambda_q(t) = q\lambda(t)$}.
\end{remark}
\begin{remark}[Asymmetric motion]\label{remarkAsimmetricoLegge}
Let $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$ be the stochastic motion introduced in Remark \ref{remarkEquazioneAsimmetrico}, i.e. describing the movement of a particle running on the $x$-axis with velocity $c_X=|v_1|=|v_3|>0$ and on the $y$-axis with velocity $c_Y = |v_2|=|v_3|>0$. Then the following representation holds (in distribution)
\begin{equation}\label{decomposizioneXYUVasimmetrica}
\begin{cases}
X (t)= c_X\bigl(U(t)+V(t)\bigr)\\
Y(t) = c_Y\bigl(U(t)-V(t)\bigr)
\end{cases}
\end{equation}
where $\{U(t)\}_{t\ge0}$ and $\{V(t)\}_{t\ge0}$ are two independent one-dimensional telegraph processes with parameters $(1/2, \lambda(t)/2)$, with $\lambda(t)\in C^2\bigl((0,\infty),[0,\infty)\bigl)$.
Remark \ref{remarkEquazioneAsimmetrico} and Theorem \ref{teoremaDecomposizioneXYUV} lead to (\ref{decomposizioneXYUVasimmetrica}) concerning an asymmetric standard orthogonal planar random motion.
In view of Remark \ref{remarkStandardBernoulliLegge}, all these considerations holds for the standard motion with Bernoulli trials as well.\hfill$\diamond$
\end{remark}
\subsection{The marginal component of the planar motion}
Thanks to decomposition (\ref{decomposizioneXYUV}), we can obtain the law of the marginal components of the vector process $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$, that is we can infer the distribution of the projection $X(t)$ (or equivalently $Y(t)$). The distribution of $X(t)$ follows from the convolution of two one-dimensional telegraph processes $U(t)$ and $V(t)$ with rates $(\lambda(t)/2, c/2)$. For $|x|<ct$
\begin{equation*}
P\{X(t)\in \dif x\} = p(x,t)\dif x = \frac{\dif x}{2}\int_{\max\{-\frac{ct}{2}+x, -\frac{ct}{2}\}}^{\min\{\frac{ct}{2}, \frac{ct}{2}+x\}} p_U(y,t)p_V(x-y,t)\dif y.
\end{equation*}
Alternatively, from (\ref{decomposizioneXYUV}) we can write
\begin{equation*}
P\{X(t)\in \dif x\} = \frac{\dif x}{2}\int_{|x|-ct}^{-|x|+ct} p_U\Bigl(\frac{x+y}{2},t\Bigr)p_V\Bigl(\frac{x-y}{2},t\Bigr)\dif y.
\end{equation*}
We recall that the investigation of the sum of two independent telegraph process with constant rate function, $\lambda(t) = \lambda>0 \ \forall\ t$, has been carried out by Kolesnik \cite{K2014}.
\\
\begin{remark}
It is well-known that in the cases of $\lambda(t) = \lambda, \lambda\text{th}(\lambda t), \lambda\text{coth}(\lambda t)$, the distribution of the one-dimensional telegraph process is given in terms of the modified Bessel functions of order $0$ and $1$. It is interesting to show the following convolution
\begin{align}
&\int_{\max\{-\frac{ct}{2}+x, -\frac{ct}{2}\}}^{\min\{\frac{ct}{2}, \frac{ct}{2}+x\}} I_0\Bigl(\frac{\lambda}{c}\sqrt{\frac{c^2t^2}{4}-y^2}\Bigr) I_0\Bigl(\frac{\lambda}{c}\sqrt{\frac{c^2t^2}{4}-(y-x)^2}\Bigr)\Bigg|_{x = 0}\dif y\nonumber\\
&\ \ \ \ = \sum_{h=0}^\infty\sum_{k=0}^\infty \Bigl(\frac{\lambda}{2c}\Bigr)^{2h+2k} \frac{1}{h!^2k!^2}\int_{-\frac{ct}{2}}^{\frac{ct}{2}}\Bigl(\frac{c^2t^2}{4}-y^2\Bigr)^{h+k}\dif y \nonumber\\
&\ \ \ \ = \sum_{h=0}^\infty\sum_{k=0}^\infty \Bigl(\frac{\lambda}{2c}\Bigr)^{2h+2k} \frac{1}{h!^2k!^2} \Bigl(\frac{c t}{2}\Bigr)^{2h+2k+1}\frac{\Gamma(h+k+1)\Gamma(1/2)}{\Gamma(h+k+1+1/2)} \label{passaggioCalcoloConvoluzioneI0in0}\\
&\ \ \ \ = c \sum_{h=0}^\infty\sum_{k=0}^\infty \Bigl(\frac{\lambda t}{2}\Bigr)^{2h+2k} \frac{t\, (h+k)!^2}{h!^2k!^2(2h+2k+1)!} = c \sum_{h=0}^\infty\sum_{l=h}^\infty \Bigl(\frac{\lambda t}{2}\Bigr)^{2l} \frac{t \,l!^2}{h!^2(l-h)!^2(2l+1)!} \nonumber\\
&\ \ \ \ = c \sum_{l=0}^\infty\Bigl(\frac{\lambda t}{2}\Bigr)^{2l} \frac{t }{(2l+1)!}\sum_{h=0}^l \binom{l}{h}^{2} = c \sum_{l=0}^\infty\Bigl(\frac{\lambda t}{2}\Bigr)^{2l} \frac{t }{(2l+1)!}\binom{2l}{l}\label{passaggio2CalcoloConvoluzioneI0in0} \\
&\ \ \ \ =c \sum_{l=0}^\infty\frac{1}{l!^2}\Bigl(\frac{\lambda }{2}\Bigr)^{2l} \int_0^t s^{2l}\dif s = c\int_0^t I_0(\lambda s) \dif s \nonumber
\end{align}
where in step (\ref{passaggioCalcoloConvoluzioneI0in0}) we used the duplication formula of the Gamma function and in (\ref{passaggio2CalcoloConvoluzioneI0in0}) we applied the Vandermonde identity.
Similarly, we obtain
$$ \int_{\max\{-\frac{ct}{2}+x, -\frac{ct}{2}\}}^{\min\{\frac{ct}{2}, \frac{ct}{2}+x\}} I_0\Bigl(\frac{\lambda}{c}\sqrt{\frac{c^2t^2}{4}-y^2}\Bigr) \frac{\partial }{\partial t}I_0\Bigl(\frac{\lambda}{c}\sqrt{\frac{c^2t^2}{4}-(y-x)^2}\Bigr)\Bigg|_{x = 0}\dif y =\frac{c}{2}\Bigl( I_0(\lambda t)-1\Bigl)$$
and
$$ \int_{\max\{-\frac{ct}{2}+x, -\frac{ct}{2}\}}^{\min\{\frac{ct}{2}, \frac{ct}{2}+x\}} \frac{\partial }{\partial t}I_0\Bigl(\frac{\lambda}{c}\sqrt{\frac{c^2t^2}{4}-y^2}\Bigr) \frac{\partial }{\partial t}I_0\Bigl(\frac{\lambda}{c}\sqrt{\frac{c^2t^2}{4}-(y-x)^2}\Bigr)\Bigg|_{x = 0}\dif y =\frac{\lambda c}{4}\Bigl(I_1(\lambda t)-\frac{\lambda t}{2}\Bigr).$$
We note that the density of the marginal $X(t)$ of the planar standard motion attains its maximum at $x = 0$.
\hfill$\diamond$
\end{remark}
Let $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$ be a standard orthogonal planar motion. We note that when the vector process moves horizontally, that is when $X(t)$ is active, the vertical motion is suspended up to the occurrence of the next Poisson event. When the vector proceeds along the vertical direction and a Poisson event occurs, the particle uniformly switches to either the rightward or leftward direction.
\\
Below we rigorously prove that the marginal component of the standard orthogonal planar motion is distributed as a one-dimensional telegraph-type process with three velocities, $\pm c$ ($c>0$) and $0$ and such that it changes speed at Poisson paced times with the following rule: if the current speed is $c$ or $-c$, then it stops, meaning that it changes to speed $0$; otherwise, if the velocity is $0$, it uniformly selects the next velocity between $c$ and $-c$.
\begin{theorem}\label{teoremaMotoConStop}
Let $\{N(t)\}_{t\ge0}$ be a Poisson process with rate function $\lambda(t)\in C^1\bigl((0,\infty),[0,\infty)\bigl)$, $V_0$ a r.v., independent of $N(t)$, and $\{V(t)\}_{t\ge0}$ such that $V(0) = V_0$,
\[V_0 = \begin{cases}\begin{array}{r l}
+c, & w.p.\ 1/4,\\0, & w.p.\ 1/2,\\-c, & w.p.\ 1/4,
\end{array}\end{cases} \text{and} \ \ \begin{cases}
P\{V(t+\dif t)=c\,|\,V(t) = 0,\,N(t,t+\dif t]=1\}=1/2,\\
P\{V(t+\dif t)=-c\,|\,V(t) = 0,\,N(t,t+\dif t]=1\}=1/2,\\
P\{V(t+\dif t)=0\,|\,V(t) = \pm c,\,N(t,t+\dif t]=1\}=1.
\end{cases} \]
Let $\{\mathcal{T}(t)\}_{t\ge0}$ be a one-dimensional stochastic motion such that
$\mathcal{T}(t) = \int_0^t V(s)\dif s$, then the transition density $p(x,t)\dif x = P\{\mathcal{T}(t)\in \dif x\}$ satisfies the following third-order partial differential equation
\begin{equation}\label{equazioneLeggeMarginaleStandard}
\frac{\partial^3p}{\partial t^3} +3\lambda(t)\frac{\partial p^2}{\partial t^2} +\bigl(2\lambda^2(t)+\lambda'(t) \bigr) \frac{\partial p}{\partial t}= c^2\frac{\partial^3 p}{\partial t\partial x^2}+c^2 \lambda(t)\frac{\partial^2 p}{\partial x^2}.
\end{equation}
Finally, if $\int_0^t\lambda(s)\dif s<\infty$,
\begin{equation}\label{singolaritaMarginale}
P\{\mathcal{T}(t) = ct\}=P\{\mathcal{T}(t) = -ct\}=\frac{1}{2}P\{\mathcal{T}(t) = 0\}=\frac{e^{-\int_0^t\lambda(s)\dif s}}{4}.
\end{equation}
\end{theorem}
We say that $\mathcal{T}$ has velocity $c$ and rate function $\lambda(t)$ (both positive).
\begin{proof}
Let $v_0\in\{-c,0,c\}$, then $P\{\mathcal{T}(t) = v_0t\} = P\{V_0 = v_0,\ N(t) = 0\}$ and this proves (\ref{singolaritaMarginale}).
To prove (\ref{equazioneLeggeMarginaleStandard}) we use the following probability functions
\begin{equation}
\begin{split}\label{leggiMotoUnidimensionaleCongiuntoV}
f_1(x,t)\dif x=P\{\mathcal{T}&(t)\in \dif x,\,V(t) = c\},\ \ \ f_2(x,t)\dif x =P\{\mathcal{T}(t)\in \dif x,\, V(t) = -c\},\\
& f_0(x,t)\dif x=P\{\mathcal{T}(t)\in \dif x, \,V(t)=0\}.
\end{split}\end{equation}
The probabilities (\ref{leggiMotoUnidimensionaleCongiuntoV}) are related by the following differential system of differential equations
\begin{equation}\label{sistemaMotoUnidimensionaleCongiuntoV}
\begin{cases}
\frac{\partial f_1}{\partial t} = -c\frac{\partial f_1}{\partial x}+\frac{\lambda(t)}{2}(f_0-2f_1)\\
\frac{\partial f_0}{\partial t} =\lambda(t)(f_1+f_2-f_0)\\
\frac{\partial f_2}{\partial t} = c\frac{\partial f_2}{\partial x}+\frac{\lambda(t)}{2}(f_0-2f_2)
\end{cases} \text{and then}\ \ \ \begin{cases}
\frac{\partial g_1}{\partial t} = -c\frac{\partial g_2}{\partial x}+\lambda(t) (f_0-g_1)\\
\frac{\partial g_2}{\partial t} = -c\frac{\partial g_1}{\partial x}-\lambda(t) g_2\\
\frac{\partial f_0}{\partial t} = \lambda(t)(g_1-f_0)
\end{cases}\end{equation}
where we simplified the first differential system by means of the auxiliary functions $g_1 = f_1+f_2,\ g_2 = f_1-f_2$. By suitably using the equations of the second system of (\ref{sistemaMotoUnidimensionaleCongiuntoV}) we pass to the differential system
\begin{equation}\label{sistemaMotoUnidimensionaleG2}
\begin{cases}
\frac{\partial^2 g_1}{\partial t^2} = c^2\frac{\partial^2 g_1}{\partial x^2}- \lambda(t)\frac{\partial g_1}{\partial t}+\lambda^2(t) (f_0-g_1)+\frac{\partial }{\partial t}\bigl( \lambda(t) (f_0-g_1)\bigr),\\
\frac{\partial f_0}{\partial t} = \lambda(t)(g_1-f_0).
\end{cases}
\end{equation}
By deriving the second equation with respect to $x$ and by considering the functions $p= g_1+f_0$ (this is the probability density of the motion) and $w = g_1-f_0$, after some calculation, we obtain
\begin{equation}\label{sistemaMotoUnidimensionaleP}
\begin{cases}
2\frac{\partial^2 p}{\partial t^2} = c^2\frac{\partial^2 p}{\partial x^2}+c^2\frac{\partial^2 w}{\partial x^2}- \lambda(t)\frac{\partial p}{\partial t}- \lambda(t)\frac{\partial w}{\partial t}-2\lambda^2(t)w = c^2\frac{\partial^2 p}{\partial x^2}+c^2\frac{\partial^2 w}{\partial x^2}- 2\lambda(t)\frac{\partial p}{\partial t}, \\
\frac{\partial p}{\partial t} = \frac{\partial w}{\partial t}+2\lambda(t)w.
\end{cases}
\end{equation}
Finally, by deriving twice with respect to $t$ the second equation of (\ref{sistemaMotoUnidimensionaleP}) and substituting $\frac{\partial^2 w}{\partial x^2}$ from the first equation, we obtain the third order differential equation (\ref{equazioneLeggeMarginaleStandard}).
\end{proof}
\begin{theorem}\label{teoremaSommaTelegrafi}
Let $\{U(t)\}_{t\ge0}$, $\{V(t)\}_{t\ge0}$ be two independent one-dimensional telegraph processes with parameters $(c/2,\lambda(t)/2)$, with $\lambda(t)\in C^1\bigl((0,\infty),[0,\infty)\bigl)$. The process $X(t) = U(t)+V(t)$, $t\ge0$, is equal in distribution to the one-dimensional process $\mathcal{T}(t),\ t\ge0$, defined in Theorem \ref{teoremaMotoConStop}.
\end{theorem}
\begin{proof}
If $\int_0^t\lambda(s)\dif s<\infty$, it is straightforward to show that the probability mass of the discrete component of $X(t)=U(t)+V(t)$ coincides with (\ref{singolaritaMarginale}).
We now show that the absolutely continuous component of $X(t)$ satisfies the third-order partial differential equation (\ref{equazioneLeggeMarginaleStandard}).
$p_U(u,t)\dif u = P\{U(t)\in \dif u\}$ and $p_U(v,t)\dif v = P\{V(t)\in \dif v\}$ are both solutions of (\ref{CauchyTelegrafo}). By writing $H_X(\gamma, t) =\mathbb{E}\bigl[e^{i\gamma X(t)}\bigr]$ and $H_U(\gamma, t)=\mathbb{E}\bigl[e^{i\gamma U(t)}\bigr]$, $\gamma \in \mathbb{R}$, we readily obtain
\[ H_X(\gamma, t) = H_U^2(\gamma, t)\ \ \ \text{and}\ \ \ \frac{\partial^2 H_U}{\partial t^2}+ \lambda(t)\frac{\partial H_U}{\partial t} = -\frac{\gamma^2c^2}{4}H_U.\]
Thus,
\begin{equation*}\label{}
\frac{\partial H_X}{\partial t}=2H_U(\gamma, t)\frac{\partial H_U}{\partial t},\ \ \ \frac{\partial^2 H_X}{\partial t^2}=2 \Bigl(\frac{\partial H_U}{\partial t}\Bigr)^2-\lambda(t)\frac{\partial H_X}{\partial t} -\frac{\gamma^2c^2}{2}H_X
\end{equation*}
and
\begin{align*}
\frac{\partial^3 H_X}{\partial t^3}&=4\frac{\partial H_U}{\partial t}\frac{\partial^2 H_U}{\partial t^2}-\lambda'(t)\frac{\partial H_X}{\partial t}-\lambda(t)\frac{\partial^2 H_X}{\partial t^2}-\frac{\gamma^2c^2}{2}\frac{\partial H_X}{\partial t}\nonumber\\
&=-4\lambda(t) \Bigl(\frac{\partial H_U}{\partial t}\Bigr)^2-\gamma^2c^2H_U\frac{\partial H_U}{\partial t}-\lambda'(t)\frac{\partial H_X}{\partial t}-\lambda(t)\frac{\partial^2 H_X}{\partial t^2}-\frac{\gamma^2c^2}{2}\frac{\partial H_X}{\partial t} \nonumber\\
&=-2\lambda(t)\Bigl(\frac{\partial^2 H_X}{\partial t^2}+ \lambda(t)\frac{\partial H_X}{\partial t} +\frac{\gamma^2c^2}{2}H_X\Bigr)-\gamma^2c^2\frac{\partial H_X}{\partial t} -\lambda'(t)\frac{\partial H_X}{\partial t}-\lambda(t)\frac{\partial^2 H_X}{\partial t^2}\nonumber\\
&=-3\lambda(t)\frac{\partial^2 H_X}{\partial t^2}-\bigl(2\lambda^2(t)+\lambda'(t)\bigr)\frac{\partial H_X}{\partial t}-\gamma^2c^2\frac{\partial H_X}{\partial t}-\lambda(t)\gamma^2c^2H_X
\end{align*}
and the inverse Fourier transform straightforwardly yields equation (\ref{equazioneLeggeMarginaleStandard}).
\end{proof}
By taking into account representation (\ref{decomposizioneXYUV}) the next statement follows as a consequence of the previous theorem.
\begin{corollary}
Let $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$ be a standard orthogonal planar motion with rate function $\lambda(t)\in C^2\bigl((0,\infty),[0,\infty)\bigl)$. The marginal processes $X(t)$ and $Y(t)$ are equal in distribution to the one-dimensional process $\mathcal{T}(t),\ t\ge0$, defined in Theorem \ref{teoremaMotoConStop}.
\end{corollary}
By suitably manipulating (\ref{equazioneLeggeMarginaleStandard}), we obtain that the absolutely continuous component of the distribution of the marginal component of the standard orthogonal motion $X(t) = U(t)+V(t)$ and $Y(t) = U(t)-V(t),\ t\ge0,$ satisfies
\begin{equation}\label{equazioneLeggeMarginaleStandardCostante}
\Bigl(\frac{\partial }{\partial t} +\lambda(t) \Bigr) \Bigl(\frac{\partial^2}{\partial t^2}+2\lambda(t)\frac{\partial}{\partial t}-c^2\frac{\partial^2}{\partial x^2} \Bigr)p=\lambda'(t)\frac{\partial p}{\partial t}.
\end{equation}
Equation (\ref{equazioneLeggeMarginaleStandardCostante}) reduces to formula (4.7) of Kolesnik \cite{K2014} if $\lambda(t) = \lambda \ \forall\ t$ (we recall that we are considering $U$ and $V$ independent telegraph processes with velocity $c/2$ and rate function $\lambda(t)/2$).
\begin{remark}[Asymmetric motion]
By keeping in mind formula (\ref{decomposizioneXYUVasimmetrica}), thanks to Theorem \ref{teoremaSommaTelegrafi}, we obtain that the marginal components $X(t)$ and $Y(t)$, $t\ge0$, of the asymmetric standard planar motion are equal in distribution to the process $\mathcal{T}(t),\ t\ge0$, defined in Theorem \ref{teoremaMotoConStop}, with velocities $c_X$ and $c_Y$ respectively. \hfill$\diamond$
\end{remark}
\section{Reflecting orthogonal planar random motions}\label{Sezione3}
We consider $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$ a \textit{reflecting orthogonal planar random motion} with changes of direction paced by a non-homogeneous Poisson process $\{N(t)\}_{t\ge0}$ with rate $\lambda(t)\in C^1\bigl((0,\infty),[0,\infty)\bigl)$. $\bigl(X(t),Y(t)\bigr)$ is the stochastic vector process describing the position, at time $t$, of a particle moving with the rules of the standard orthogonal motion (see above), but with the possibility to bounce back. Therefore, at all Poisson times it can uniformly switch to one of the available directions, except for the current one. This motion has been studied by Kolesnik and Orsingher \cite{KO2001} in the case of a constant rate function $\lambda(t) = \lambda>0\ \forall \ t$.
\\
At time $t\ge0$, the reflecting orthogonal motion is located in the square $S_{ct}$ defined in (\ref{supportoQuadrato}). If the rate function is such that $\Lambda(t) = \int_0^t\lambda(s) \dif s<\infty$, the distribution of the motion has two singular components, the border of the square $S_{ct}$ and its diagonals. Let $V_{ct} = \{(0,\pm ct),(\pm ct,0)\}$ being the set of the vertices of $S_{ct}$ and $\partial D_{ct} = \{(x,y)\in S_{ct}\,:\, x=0\ \text{or}\ y=0\}$ being the diagonals of $S_{ct}$, then $P\{\bigl(X(t),Y(t)\bigr) \in V_{ct}\}=e^{-\Lambda(t)}$,
\begin{equation}\label{riflessioneProbFrontiera}
P\{\bigl(X(t),Y(t)\bigr) \in \partial S_{ct}\setminus V_{ct}\}=\frac{2}{3}\sum_{n=1}^\infty P\{N(t) = n\}\frac{1}{3^{n-1}} = 2\Bigl(e^{-\frac{2\Lambda(t)}{3}}-e^{-\Lambda(t)}\Bigr),
\end{equation}
because, in order to reach the edge $\partial S_{ct}$, the particle can not bounce back and it must always move towards the ``selected'' side of the border. Finally,
\begin{equation}\label{riflessioneProbCroce}
P\{\bigl(X(t),Y(t)\bigr) \in \partial D_{ct}\setminus V_{ct}\}=\sum_{n=1}^\infty P\{N(t) = n\}\frac{1}{3^{n}} = e^{-\frac{2\Lambda(t)}{3}}-e^{-\Lambda(t)},
\end{equation}
because the particle must always move back and forth.
We now present a general result concerning the probability density inside the square $S_{ct}$, that is $p(x,y,t)\dif x\dif y = P\{X(t)\in \dif x,\ Y(t)\in \dif y\}$, for $(x,y)\in S_{ct}$.
\begin{theorem}
\label{teoremaEquazioneRiflessione}
The absolutely continuous component $p = p(x,y,t)\in C^4\bigl(\mathbb{R}^2\times[0,\infty), [0,\infty)\bigr)$ of the distribution of the reflecting orthogonal process $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$ satisfies the following fourth-order differential system
\begin{equation}\label{equazioneRiflessione}
\begin{cases}
\frac{\partial^4p}{\partial t^4} + 4\lambda \frac{\partial^3p}{\partial t^3}+\Bigl(\frac{16}{3}\lambda^2+4\lambda'\Bigr)\frac{\partial^2p}{\partial t^2}+ \Bigl(\frac{64}{27}\lambda^3+\frac{16}{3}\lambda\lambda'+\frac{4}{3}\lambda''\Bigr)\frac{\partial p}{\partial t} + c^4\frac{\partial^4 p}{\partial x^2\partial y^2}\\
\hspace{5cm}=c^2\Bigl(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\Bigr)\Bigl(\frac{\partial^2p}{\partial t^2}+2\lambda\frac{\partial p}{\partial t} + \bigl(\frac{8}{9}\lambda^2+\frac{2}{3}\lambda'\bigr)p\Bigr),\\
p(x,y,t)\ge0,\\
\int_{S_{ct}}p(x,y,t)\dif x\dif y = 1- 3e^{-\frac{2}{3}\int_0^t\lambda(s)\dif s}+2e^{-\int_0^t\lambda(s)\dif s},
\end{cases}\end{equation}
where $\lambda=\lambda(t)\in C^{2}\bigl((0,\infty),[0,\infty)\bigl)$ denotes the rate function of the non-homogeneous Poisson process governing the changes of direction.
\end{theorem}
\begin{proof}
By means of notation (\ref{notazione}), we obtain the differential system
\begin{equation}\label{sistemaRiflessione}
\begin{cases}
\frac{\partial f_0}{\partial t} = -c\frac{\partial f_0}{\partial x}+\frac{\lambda(t)}{3}(f_1+f_2+f_3-3f_0),\\
\frac{\partial f_1}{\partial t} = -c\frac{\partial f_1}{\partial y}+\frac{\lambda(t)}{3}(f_0+f_2+f_3-3f_1),\\
\frac{\partial f_2}{\partial t} = c\frac{\partial f_2}{\partial x}+\frac{\lambda(t)}{3}(f_0+f_1+f_3-3f_2),\\
\frac{\partial f_3}{\partial t} = c\frac{\partial f_3}{\partial y}+\frac{\lambda(t)}{3}(f_0+f_1+f_2-3f_3).
\end{cases}\end{equation}
The claimed result follows by proceeding as in Section \ref{sottoSezioneEquazioneStandard} in order to proof Theorem \ref{teoremaEquazioneStandard}.
\end{proof}
By dividing the differential equation (\ref{equazioneRiflessione}) by $\lambda^3$, it is easy to observe that the reflecting orthogonal planar motion converges to Brownian motion with diffusivity $3\sigma^2/4$ under Kac's conditions.
\\
We now consider $\int_0^t\lambda(s)\dif s<\infty$ and we study the density on the singular components, i.e. on the border and the diagonals of the square $S_{ct}$.
\begin{proposition}\label{proposizioneFrontieraRiflessione}
Let $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$ be a reflecting orthogonal planar random motion with changes of direction paced by a Poisson process with rate $\lambda(t)\in C^1\bigl((0,\infty),[0,\infty)\bigr)$ such that $\Lambda(t) = \int_0^t\lambda(s)\dif s<\infty$ for all $t>0$. Then, $f(\eta, t)\dif \eta =P\{X(t)+Y(t) = ct, X(t)-Y(t)\in \dif \eta\}$, for $|\eta|<ct$, satisfies the differential problem
\begin{equation}\label{sistemaFrontieraAltoDxRiflessione}
\begin{cases}
\frac{\partial^2 f}{\partial t^2} +2\lambda(t)\frac{\partial f}{\partial t} +\frac{2}{3}\Bigl(\frac{4}{3}\lambda(t)^2+\lambda'(t)\Bigr)f =c^2\frac{\partial^2 f}{\partial \eta^2},\\
f(\eta,t)\ge0,\\
\int_{-ct}^{ct}f(\eta,t)\dif \eta = \frac{1}{2}(e^{-2\Lambda(t)/3}-e^{-\Lambda(t)}).
\end{cases}
\end{equation}
The function
\begin{equation} \label{formulaGeneraleFrontieraRiflessione}
f(\eta, t) = \frac{e^{-2\Lambda(t)/3}}{4} \bar p\Bigl(\frac{\eta}{2},t\Bigr) ,
\end{equation}
solves (\ref{sistemaFrontieraAltoDxRiflessione}). In (\ref{formulaGeneraleFrontieraRiflessione}), $\bar p(x,t)$ is the absolutely continuous component of the distribution of a telegraph process with rate function $\lambda(t)/3$ and velocity $c/2$.
\end{proposition}
The proposition refers to the probability that the motion lies on the border of its support in the first quadrant, defined in (\ref{insiemeFrentieraAltoDx}). Equivalent results hold for the other three sides of $\partial S_{ct}$.
\begin{proof}
In order to obtain the differential equation in (\ref{sistemaFrontieraAltoDxRiflessione}) we proceed as in Proposition \ref{propFrontieraStandard}. By using notation (\ref{notazioneFrontieraStandard}) we can write
\begin{equation}\label{sistemiFrontieraRiflessioneG}
\begin{cases}
\frac{\partial f_0}{\partial t} = -c\frac{\partial f_0}{\partial \eta}+\frac{\lambda(t)}{3}(f_1-3f_0)\\
\frac{\partial f_1}{\partial t} = c\frac{\partial f_1}{\partial \eta}+\frac{\lambda(t)}{3}(f_0-3f_1)
\end{cases}\text{and}\ \ \
\begin{cases}
\frac{\partial f}{\partial t} = -c\frac{\partial w}{\partial \eta}-\frac{2\lambda(t)}{3}f\\
\frac{\partial w}{\partial t} = -c\frac{\partial f}{\partial \eta}-\frac{4\lambda(t)}{3}w
\end{cases}
\end{equation}
where we performed the change of variables $f = f_0+f_1,\ w=f_0-f_1$. Now, it is easy to obtain the first formula of system (\ref{sistemaFrontieraAltoDxRiflessione}).
To prove the second part of the statement, we proceed as follows. Let $\{\mathcal{T}_c(t)\}_{t\ge0}$, $c>0$, denotes a one-dimensional telegraph process with rate function $\lambda(t)/3$ and velocities $\pm c$. From the definition of the telegraph motion we obtain that $\mathcal{T}_c(t)/2 \stackrel{d}{=} \mathcal{T}_{c/2}(t)\ \forall\ t$. Now, we express (\ref{formulaGeneraleFrontieraRiflessione}) in terms of the density $q$ of $\mathcal{T}_c(t
)$,
$$q(\eta, t)\dif \eta = P\{\mathcal{T}_c(t)\in \dif \eta\} = P\Big\{\mathcal{T}_{c/2}(t)\in \frac{\dif \eta}{2}\Big\} = \frac{\dif \eta}{2}\bar p\Bigl(\frac{\eta}{2},t\Bigr) $$
thus $f(\eta,t) = q(\eta, t)e^{-2\Lambda(t)/3}$. By keeping in mind that $q$ satisfies the generalized telegraph equation
$$\frac{\partial^2 q}{\partial t^2} +\frac{2\lambda(t)}{3}\frac{\partial q}{\partial t} = c^2\frac{\partial^2 q}{\partial \eta^2},\ \text{ with condition }\ \int_{-ct}^{ct}q(\eta, t)\dif \eta = 1-e^{-\frac{\Lambda(t)}{3}},$$
it is easy to show that (\ref{formulaGeneraleFrontieraRiflessione}) satisfies (\ref{sistemaFrontieraAltoDxRiflessione}).
\end{proof}
Proposition \ref{proposizioneFrontieraRiflessione} concerns only the side $\partial S^{(1)}_{ct} = \{(x,y)\in \mathbb{R}^2\,:\, x+y = ct,\\ |x-y|\le ct\}$, but it equivalently holds on all the other components of the border $\partial S_{ct}$.
We can interpret the above proposition thanks to the following reasoning. Inspired by the results of Theorem \ref{teoremaDecomposizioneXYUV} we consider the rotated process $\{\bigl(U(t),V(t)\bigr)\}_{t\ge0}$ where
\begin{equation}\label{decomposizioneXYUVRiflessione}
U(t) = \frac{X (t)+Y(t)}{2},\ \ \ V(t) = \frac{X (t)-Y(t)}{2}.
\end{equation}
By means of a direct investigation of the marginal process $U$ (and equivalently $V$) we observe that it describes a telegraph motion with velocity $c/2$ and rate function $2\lambda(t)/3$. In fact, at each Poisson event, the projection $U(t)$ changes direction with probability $2/3$, due to the switch of the motion $\bigl(X(t),Y(t)\bigr)$ (for instance, if $\bigl(X(t),Y(t)\bigr)$ moves with velocity $D(t)=d_0$, $U(t)$ moves with speed $+c/2$ and it will change velocity if $\bigl(X(t),Y(t)\bigr)$ takes either direction $d_1$ or $d_2$).
Therefore, $U$ and $V$ are identical, but dependent, telegraph processes with rate function $2\lambda(t)/3$ and velocity $c/2$.
Now, in view of Proposition \ref{proposizioneFrontieraRiflessione} and (\ref{decomposizioneXYUVRiflessione}), we obtain that
\begin{align*}
f(\eta,t)\dif \eta &= P\{X(t)+Y(t)=ct,\ X(t)-Y(t)\in \dif \eta\} \\
&= P\Big\{U(t)=\frac{ct}{2}\Big\}P\Big\{ V(t)\in \frac{\dif \eta}{2}\,\Big|\,U(t) = \frac{ct}{2} \Big\} = \frac{e^{-2\Lambda(t)/3}}{2} \frac{\bar p\bigl(\frac{\eta}{2},t\bigr)}{2},
\end{align*}
meaning that $V(t)$, knowing that $U(t)= ct/2$, is distributed as a telegraph process with rate function $\lambda(t)/3$ and velocity $c/2$.
\begin{remark}
In the particular case $\lambda(t) = \lambda\ \forall\ t$, we can solve system (\ref{sistemaFrontieraAltoDxRiflessione}) by using the transformation $f(\eta,t)=e^{-\lambda t}q(\eta, t)$. This leads to the Klein-Gordon equation $\frac{\partial^2 q}{\partial t^2} -\frac{c^2}{4}\frac{\partial^2 q}{\partial \eta^2}=\frac{\lambda^2}{9}q$, $|\eta|<ct$. Therefore, the solution of (\ref{sistemaFrontieraAltoDxRiflessione}) is
\begin{align}
f(\eta, t)&=\frac{e^{-\lambda t}}{4c}\Biggl[\frac{\lambda}{3} I_0\Bigl(\frac{\lambda}{3c}\sqrt{c^2t^2-\eta^2} \Bigr)+\frac{\partial}{\partial t}I_0\Bigl(\frac{\lambda}{3c}\sqrt{c^2t^2-\eta^2} \Bigr)\Biggr]\label{frontieraRiflessioneCostante} \\
&=\frac{e^{-\frac{2\lambda t}{3}}}{2}\,\frac{e^{-\frac{\lambda t}{3}}}{2c}\Biggl[\frac{\lambda}{3} I_0\Bigl(\frac{\lambda}{3c}\sqrt{c^2t^2-\eta^2} \Bigr)+\frac{\partial}{\partial t}I_0\Bigl(\frac{\lambda}{3c}\sqrt{c^2t^2-\eta^2} \Bigr)\Biggr]. \nonumber
\end{align}
which coincides with (\ref{formulaGeneraleFrontieraRiflessione}). Probability (\ref{frontieraRiflessioneCostante}) first appeared in Kolesnik and Orsingher (2001) (formula (3.3)). Furthermore, we derive the distribution conditionally on the exact number of changes of direction, for integer $k\ge0$,
\begin{align}
&P\{X(t)+Y(t) = ct, X(t)-Y(t)\in \dif \eta\,|\,N(t)=2k+1\} \nonumber\\
&=3P\{X(t)+Y(t) = ct, X(t)-Y(t)\in \dif \eta\,|\,N(t)=2k+2\}=\frac{(2k+1)!}{2\,k!^2}\frac{(c^2t^2-\eta^2)^k}{(6ct)^{2k+1}}\dif \eta \nonumber
\end{align}
where $|\eta|<ct$.\hfill $\diamond$
\end{remark}
\begin{proposition}\label{propCroceRiflessione}
Let $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$ be a reflecting orthogonal planar random motion with changes of direction paced by a Poisson process with rate $\lambda(t)$ such that $\Lambda(t) = \int_0^t\lambda(s)\dif s<\infty$ for all $t>0$. Then, for $|x|<ct$, $f(x,t) = P\{X(t)\in \dif x, Y(t)=0\}$, satisfies the differential problem (\ref{sistemaFrontieraAltoDxRiflessione}) with $x$ instead of $\eta$ and it is given by (\ref{formulaGeneraleFrontieraRiflessione}).
\end{proposition}
The proposition concerns the distribution on the horizontal diagonal of $S_{ct}$, but it equivalently holds on the vertical one $\{(x,y)\in S_{ct}\,:\,x=0\}$.
\begin{proof}
The probabilities
\begin{equation*}
\begin{cases}
g_0(x, t) \dif \eta = P\{X(t)\in \dif x,\ Y(t)=0,\ D(t) = d_0\}\\
g_2(x, t) \dif \eta = P\{X(t)\in \dif x,\ Y(t)=0,\ D(t) = d_2\}
\end{cases}
\end{equation*}
satisfy the systems appearing in (\ref{sistemiFrontieraRiflessioneG}) with $f_0\rightarrow g_0,\,f_1\rightarrow g_2$ and $f = g_0+g_2,\, w = g_0-g_2$.
\end{proof}
We now focus on the marginal component of the reflecting motion.
\begin{theorem}
Let $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$ be a reflecting orthogonal planar motion whose changes of direction are governed by a Poisson process $\{N(t)\}_{t\ge0}$ with rate function $\lambda(t)\in C^1\bigl((0,\infty),[0,\infty)\bigl)$. The absolutely continuous component of the distribution of the marginal processes $X(t)$ (and equivalently $Y(t)$) satisfies the following third-order partial differential equation
\begin{equation}\label{equazioneLeggeMarginaleRiflessione}
\frac{\partial^3p}{\partial t^3} +\frac{8\lambda(t)}{3}\frac{\partial p^2}{\partial t^2} +\frac{4}{3}\Bigl(\frac{4}{3}\lambda^2(t)+\lambda'(t) \Bigr) \frac{\partial p}{\partial t}= c^2\frac{\partial^3 p}{\partial t\partial x^2}+\frac{2c^2 \lambda(t)}{3}\frac{\partial^2 p}{\partial x^2}.
\end{equation}
Finally, if $\int_0^t\lambda(s)\dif s<\infty$,
\begin{equation*}\label{}
P\{X(t) = ct\}=P\{X(t) = -ct\}=\frac{e^{-\int_0^t\lambda(s)\dif s}}{4},\ \ \ \ P\{X(t) = 0\} = \frac{e^{-\frac{2}{3}\int_0^t\lambda(s)\dif s}}{2}.
\end{equation*}
\end{theorem}
\begin{proof}
In order to derive the absolutely continuous component of the distribution of the projection process $\{X(t)\}_{t\ge0}$ we use probabilities (\ref{leggiMotoUnidimensionaleCongiuntoV}) with $X$ instead of $\mathcal{T}$. The system governing the functions reads
\begin{equation}\label{sistemaMarginaleRiflessione}
\begin{cases}
\frac{\partial f_1}{\partial t} = -c\frac{\partial f_1}{\partial x}+\frac{\lambda(t)}{3}(f_0+f_2-3f_1),\\
\frac{\partial f_0}{\partial t} =\frac{2}{3}\lambda(t)(f_1+f_2-f_0),\\
\frac{\partial f_2}{\partial t} = c\frac{\partial f_2}{\partial x}+\frac{\lambda(t)}{3}(f_0+f_1-3f_2).
\end{cases}
\end{equation}
Some explanation is needed for the second equation of (\ref{sistemaMarginaleRiflessione}). It follows by writing
\begin{equation}\label{equazioneEstesaF0}
f_0(x,t+\dif t) = f_0(x,t)(1-\lambda(t)\dif t)+f_0(x,t)\frac{1}{3}\lambda(t)\dif t+\bigl(f_1(x,t)+f_2(x,t)\bigr)\frac{2}{3}\lambda(t)\dif t + o(\dif t).
\end{equation}
The second term of the second member of (\ref{equazioneEstesaF0}) must be interpreted by considering that if $X(t)$ is stopped at $x$, the planar motion is moving vertically and if a Poisson event occurs, with probability $2/3$ the particle starts moving horizontally and with probability $1/3$ reflects and thus the $x$-coordinate does not change. By $f_1(x,t)2\lambda(t)\dif t /3$ we represent the probability that the particle is running rightward and the new direction is either vertical upwards or vertical downwards, the same explanation holds for the term with $f_2$.
\\Now, the claimed result follows by proceeding as in the proof of Theorem \ref{teoremaMotoConStop}.
\end{proof}
We observe that the marginal processes $X(t)$ and $Y(t)$ behave like a one-dimensional process $\{\mathcal{T}(t)\}_{t\ge0}$ whose changes of direction are paced by $N(t)$, and that is defined as $\mathcal{T}(t) = \int_0^t V(s)\dif s$, where $\{V(t)\}_{t\ge0}$ describes the random velocity such that $V(0) = V_0$,
\[V_0 = \begin{cases}\begin{array}{r l}
+c, & w.p.\ 1/4,\\0, & w.p.\ 1/2,\\-c ,& w.p.\ 1/4,
\end{array}\end{cases}\text{and} \ \ \begin{cases}
P\{V(t+\dif t)=0\,|\,V(t) = 0,\,N(t,t+\dif t]=1\}=1/3,\\
P\{V(t+\dif t)=c\,|\,V(t) = 0,\,N(t,t+\dif t]=1\}=1/3,\\
P\{V(t+\dif t)=0\,|\,V(t) = c,\,N(t,t+\dif t]=1\}=2/3,\\
P\{V(t+\dif t)=-c\,|\,V(t) = c,\,N(t,t+\dif t]=1\}=1/3,
\end{cases}\]
and equivalently by replacing $-c$ with $c$ and vice versa.
\subsubsection*{Explicit representation of the reflecting planar motion.}
Let us assume $\Lambda(t) = \int_0^t\lambda(s)\dif s<\infty$. Here we explore the rotation (\ref{decomposizioneXYUVRiflessione}) of the reflecting orthogonal motion $\bigl(X(t),Y(t)\bigr),\ t\ge0$. We previously explained that the processes $\{U(t)\}_{t\ge0}$ and $\{V(t)\}_{t\ge0}$ appearing in (\ref{decomposizioneXYUVRiflessione}) are two dependent one-dimensional telegraph processes with rate function $2\lambda(t)/3$ and velocity $c/2$. We now describe their dependence in terms of the Poisson processes, of rate $2\lambda(t)/3$, governing the changes of velocities of $U$ and $V$, denoted by $\{N^{(U)}(t)\}_{t\ge0}$ and $\{N^{(V)}(t)\}_{t\ge0}$ respectively. For $t\ge0$, we write that
\begin{equation}\label{decomposizioneProcessiPoisson}
N^{(U)}(t) = N_U(t)+\tilde N(t),\ \ \ \ N^{(V)}(t) = N_V(t)+\tilde N(t),
\end{equation}
where $\{N_U(t)\}_{t\ge0},\ \{N_V(t)\}_{t\ge0}$ and $\{\tilde N(t)\}_{t\ge0}$ are three independent Poisson processes with rate function $\lambda(t)/3$. Clearly, the presence of $\tilde N$ in both $N^{(U)}$ and $N^{(V)}$ makes $U$ and $V$ dependent.
\\In order to explain expression (\ref{decomposizioneProcessiPoisson}), we recall that the process $\{N(t)\}_{t\ge0}$, denotes the Poisson process, with rate $\lambda(t)$, governing the changes of direction of the reflecting planar motion. Now, when $N$ indicates the occurrence of an event, the particle can:
\begin{itemize}
\item[($i$)] switch to an orthogonal direction and therefore only one among $U=(X+Y)/2$ and $V=(X-Y)/2$ changes velocity (for instance, let the particle moving with direction $d_0$. This means that $X$ increases and therefore both $U$ and $V$ move with positive speed. If the particle deviates to the orthogonal direction $d_1$, $X$ stops and $Y$ increases, thus $U$ continues moving with positive velocity while $V$ starts moving negatively oriented. This also implies that $N_V$ perceives the event. On the other hand, if the particle deviates from $d_0$ to $d_3$, that is $Y$ decreases, $U$ starts moving backward and $V$ continues moving forward, meaning that $N_U$ makes a jump);
\item[($ii$)] reflect to the opposite direction and therefore both $U$ and $V$ change velocity (meaning that $\tilde N$ perceives the event).
\end{itemize}
Thus, at each event counted by $N$, the process $U$ changes velocity and $V$ does not with probability $1/3$, $U$ does not change speed and $V$ does with probability $1/3$ and they both invert their velocity with probability $1/3$. This argument justifies the form (\ref{decomposizioneProcessiPoisson}) of the Poisson processes connected to $U$ and $V$. In particular, the process $\tilde N$ counts the simultaneous switches of $U$ and $V$.
Clearly, the total number of switches up to time $t\ge0,$ is given by $N(t) = N_U(t)+N_V(t) + \tilde N(t)\ a.s.$. In view of (\ref{decomposizioneXYUVRiflessione}) and (\ref{decomposizioneProcessiPoisson}) we easily obtain result (\ref{formulaGeneraleFrontieraRiflessione}). In fact, for $|\eta|<ct$
\begin{align*}
P\{&X(t)+Y(t)=ct,\, X(t)-Y(t)\in \dif \eta\} = P\Big\{U(t) = \frac{ct}{2},\,V(t)\in \frac{\dif \eta}{2}\Big\}\\
&= P\Big\{U(t) = \frac{ct}{2}\Big\}P\Big\{V(t)\in \frac{\dif \eta}{2}\,\Big|\,U(t) = \frac{ct}{2}\Big\}=P\Big\{U(t) = \frac{ct}{2}\Big\}P\Big\{V(t)\in \frac{\dif \eta}{2}\,\Big|\,\tilde N(t)=0\Big\}
\end{align*}
which coincides with (\ref{formulaGeneraleFrontieraRiflessione}) since, conditionally on $\tilde N(t)=0$, $V(t)$ is a telegraph process with changes of velocity governed by $N^{(V)}(t) = N_V(t)\sim Poisson(\Lambda(t)/3)$.
Finally, we give also the exact form of the general density of the position at time $t\ge0$. For $(x,y)\in S_{ct}\setminus \partial S_{ct}$,
\begin{align}
&P\{X(t)\in\dif x,\, Y(t)\in \dif y\} = P\Big\{U(t) \in \frac{\dif x+y}{2},\, V(t)\in \frac{x-\dif y}{2}\Big\} \nonumber\\
& = \sum_{\substack{n_U,n_V,n =0\\ n_U+n_V+n\ge1}}^\infty P\Big\{U(t) \in \frac{\dif x+y}{2},\, V(t)\in \frac{x-\dif y}{2}\,\Big|\,N_U(t)=n_U,\,N_V(t)=n_V,\,\tilde N(t)=n\Big\}\nonumber\\
&\ \ \ \times P\{N_U(t)=n_U,\,N_V(t)=n_V,\,\tilde N(t)=n\}\nonumber\\
& = \sum_{\substack{n_U,n_V,n =0\\ n_U+n_V+n\ge1}}^\infty P\Big\{U(t) \in \frac{\dif x+y}{2}\,\Big|N^{(U)}(t)=n_U+n\Big\}P\Big\{V(t)\in \frac{x-\dif y}{2}\,\Big|N^{(V)}(t)=n_V+n\Big\}\nonumber\\
&\ \ \ \times P\{N_U(t)=n_U\}P\{N_V(t)=n_V\}P\{\tilde N(t)=n\}\nonumber\\
& = \sum_{h=0}^\infty P\Big\{U(t) \in \frac{\dif x+y}{2}\,\Big|N^{(U)}(t)=h\Big\}\sum_{k=0}^\infty P\Big\{V(t)\in \frac{x-\dif y}{2}\,\Big|N^{(V)}(t)=k\Big\}\label{leggeRiflessioneDecomposizione}\\
&\ \ \ \times \sum_{\substack{n =0\\ h+k\ge1}}^{\min\{h,k\}}P\{N_U(t)=h-n\}P\{N_V(t)=k-n\}P\{\tilde N(t)=n\}\nonumber.
\end{align}
The probabilities appearing in the last sum of (\ref{leggeRiflessioneDecomposizione}) are well-known, while the conditional distributions in the first two sums are known in the case of $\lambda(t)=\lambda>0\ \forall \ t$.
\\
We note that in the case of the standard orthogonal motion representation (\ref{decomposizioneProcessiPoisson}) holds with $\tilde N(t)=0\ \forall \ t\ a.s.$ and $N_U(t),N_V(t)\sim Poisson(\Lambda(t)/2)$, because at each event recorded by $N$ the one-dimensional process $U$ revers its speed and $V$ does not with probability $1/2$ and vice versa. The crucial fact is that $U$ and $V$ can not switch simultaneously.
\begin{remark}[Conjecture on the $L_1$-distance]
Here we conjecture a connection between the probabilities of the reflecting motion and the standard one. Let $S_u = \{(x,y)\in\mathbb{R}^2\,:\, |x|+|y|<u\}$, with $0\le u\le ct$, $p(x,y,t)$ be the absolutely continuous component of the reflecting orthogonal motion $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$ and $f(x,t)$ be the distribution on the $x$-diagonal of $S_{ct}$ (that coincides with the distribution on the $y$-diagonal, see Proposition \ref{propCroceRiflessione}). Now, we can write, by also assuming $Z(t)=|X(t)|+|Y(t)|,\ t\ge0$, (look at Figure \ref{L1dist})
\begin{align}
P\{Z(t)< u\} &= P\big\{\bigl(X(t),Y(t)\bigr)\in S_{u}\big\}\nonumber\\
& =4\int_0^u\dif x\int_0^{u-x}p(x,y,t)\dif y+2\int_{-u}^uf(x,t)\dif x.
\end{align}
If $\Lambda(t) = \int_0^t\lambda(s)\dif s<\infty$, for $u=ct$, thanks to (\ref{riflessioneProbFrontiera}) and the probability that the motion reaches one of the vertexes, i.e. $e^{-\Lambda(t)}$, we have that
\begin{equation}\label{CongetturaDistanza2} P\{Z(t)< ct\} =P\big\{\bigl(X(t),Y(t)\bigr)\in S_{ct}\big\} = \Bigl(1-e^{-\frac{2}{3}\Lambda(t)}\Bigr)^2+e^{-\Lambda(t)}\Bigl(1-e^{-\frac{\Lambda(t)}{3}}\Bigr).
\end{equation}
The first term of the right-hand-side of (\ref{CongetturaDistanza2}) is related to the singular part of a standard orthogonal motion $\{\bigl(X_S(t),Y_S(t)\bigr)\}_{t\ge0}$ with rate function $2\lambda(t)/3$ and velocity $c$. The second part can be interpreted as
$$P\{N(t)= 0\}\int_{-ct}^{ct}P\{\mathcal{T}(t)\in \dif x\} = e^{-\Lambda(t)}\Bigl(1-e^{-\frac{\Lambda(t)}{3}}\Bigr)$$
where $\mathcal{T}$ is a one-dimensional telegraph process with parameters $(\lambda(t)/3, c/2)$, independent of the Poisson process.
Since this decomposition is true on the border we can imagine to extend it to a general square $S_u$. Therefore, we may conjecture that, for $0\le u\le ct$ and by considering the above notation,
\begin{align}
P&\{Z(t)<u\} = P\big\{\bigl(X(t),Y(t)\bigr)\in S_{u}\big\}\label{congetturaDistanzaRiflessione} \\
&=4\int_0^u\dif x\int_0^{u-x} P\{X_S(t)\in\dif x,\, Y_S(t)\in\dif y\}+P\{N(t)=0\}\int_{-u}^{u} P\{\mathcal{T}(t)\in \dif x\}.\nonumber
\end{align}
Thanks to Theorem \ref{teoremaDecomposizioneXYUV} and the known literature concerning the telegraph motion, all the distributions appearing in the second term of (\ref{congetturaDistanzaRiflessione}) are known in the case where the rate function is either $\lambda(t) = \lambda$ or $\lambda(t)= \lambda\,\text{th}(\lambda t)$, with $\lambda>0$.
\end{remark}
\subsection{Reflecting motion with Bernoulli trials}
Here, we study a reflecting orthogonal planar random motion $\{\bigl(X(t),Y(t)\bigr)\}_{t\ge0}$ which can skip the change of direction with probability $1-q\in[0,1)$. This means that when a Poisson event occurs, the particle continues with the same direction with probability $1-q$ and it switches to each of the other possible directions with probability $q/3$. We call this process \textit{$q$-reflecting orthogonal motion}, $q\in(0,1]$ (or reflecting orthogonal motion with Bernoulli trials).
\begin{theorem}
The $q$-reflecting orthogonal planar motion $\{\bigl(X_q(t),Y_q(t)\bigr)\}_{t\ge0}$, with $q\in(0,1]$ and rate function $\lambda(t)\in C^2\bigl((0,\infty),[0,\infty)\bigl)$ is equal in distribution to a reflecting orthogonal planar motion with rate function $q\lambda(t)$.
\end{theorem}
\begin{proof}
Clearly, at time $t\ge0$, the set of possible positions of the moving particle is the square $S_{ct}$, defined in (\ref{supportoQuadrato}).
If $\Lambda(t) = \int_0^t\lambda(s)\dif s<\infty$, the singular component of the distribution is composed of the border $\partial S_{ct}$ and the diagonals of the square, $\partial D_{ct}$, defined above (see the beginning of Section \ref{Sezione3}). Let $V_{ct}=\{(0,\pm ct),(\pm ct,0)\}$, then
\begin{equation*}
P\{\bigl(X_q(t),Y_q(t)\bigr) \in V_{ct}\} = \sum_{n=0}^\infty P\{N(t)=n\}(1-q)^n = e^{-q\Lambda(t)},
\end{equation*}
because the particle reaches a vertex if it never changes direction.
\begin{align*}
P\{\bigl(X_q(t),Y_q(t)\bigr) \in \partial S_{ct}\setminus V_{ct}\} &= \sum_{n=1}^\infty P\{N(t)=n\} \sum_{k=0}^{n-1}(1-q)^k\frac{2q}{3}\Bigl(\frac{q}{3}+(1-q)\Bigr)^{n-k-1} \\
&= 2\Bigl(e^{-\frac{2q\Lambda(t)}{3}}-e^{-q\Lambda(t)}\Bigr),\nonumber
\end{align*}
where the second sum represents all the possible steps where the particle continues along the initial direction for $k$ times (with probability $(1-q)^k$), then it changes to one of the two orthogonal directions ($2q/3$) and finally it keeps moving towards the edge by choosing between this last direction and the starting one for the remaining $n-k-1$ displacements.
\\The probability of remaining on the diagonals is instead equal to
\begin{align*}
P\{\bigl(X_q(t),Y_q(t)\bigr) \in\partial D_{ct}\setminus V_{ct}\} &= \sum_{n=0}^\infty P\{N(t)=n\}\sum_{k=0}^{n-1}\binom{n}{k}(1-q)^k\Bigl(\frac{q}{3}\Bigr)^{n-k} \\
&=e^{-\frac{2q\Lambda(t)}{3}}-e^{-q\Lambda(t)},
\end{align*}
where the second sum represents all the possible sequences of directions containing only the initial one and the opposite one, which appears at least once.
By applying notation (\ref{notazioneFrontieraStandard}) for the $q$-reflecting motion, it is easy to obtain that $f_0,f_1$ satisfy the differential system (\ref{sistemiFrontieraRiflessioneG}) with $q\lambda(t)$ replacing $\lambda(t)$. Therefore, the density on the border $\partial S_{ct}^{(1)}$ satisfies system (\ref{sistemaFrontieraAltoDxRiflessione}) with $q\lambda(t)$ in place of $\lambda(t)$. This is sufficient to prove that the stated equality in distribution holds on the border of the square $S_{ct}$.
Similarly, with Proposition \ref{propCroceRiflessione} at hand, we obtain that the equality in distribution holds on the diagonals of the support by observing that $f(x,t) \dif x= P\{X_q(t)\in \dif x, Y_q(t)=0\}$ satisfies system (\ref{sistemaFrontieraAltoDxRiflessione}) with $q\lambda(t)$ replacing $\lambda(t)$.
For the absolutely continuous component we consider notation (\ref{notazione}). For $(x,y)\in S_{c(t+\dif t)}$
\begin{align*}
f_0(x,y,& t+\dif t) = f_0(x-c\dif t,y, t)\bigl(1-\lambda(t)\dif t\bigr) + f_0(x-c\dif t,y, t)\lambda(t)\dif t(1-q)\\
&+\Bigl(f_1(x,y-c\dif t,t) +f_2(x+c\dif t,y,t)+ f_3(x,y+c\dif t,t)\Bigr)\frac{q}{3}\lambda(t)\dif t +o(\dif t)
\end{align*}
and similarly for $f_1,f_2$ and $f_3$. These relationships yield system (\ref{sistemaRiflessione}) with $q\lambda(t)$ instead of $\lambda(t)$. This is sufficient to conclude the proof of the theorem.
\end{proof}
Clearly, under Kac's conditions, the $q$-reflecting orthogonal motion converge to planar Brownian motion with diffusivity $3\sigma^2/(4q)$.
\begin{remark}[Uniform orthogonal planar motion]
A particular case of the $q$-reflecting motion is the uniform orthogonal random motion, that is the process describing the position of a particle that, at each Poisson event, uniformly chooses the new direction among all the four possible directions. We obtain this motion if $q =3/4$. Note that, if $\lambda(t) = \lambda>0,\,t\ge 0$, the motion replicates at each Poisson event independently on the previous displacements and velocities. \hfill$\diamond$
\end{remark}
\begin{remark}[General random motion with Bernoulli trials]
It is interesting to observe that the behavior presented for the motions with Bernoulli trials, meaning with a positive probability to skip the change of direction, can be easily extended. In particular, a $q$-standard or $q$-reflecting motion with rate function $\lambda(t)>0,\ t\ge0$, and a time-varying probability of change of direction $q(t)\in (0,1]$, is equal in distribution to the ``basic'' motion with rate function $\lambda_q(t) = q(t)\lambda(t)$.
More generally, we can state the following.
\\\textit{Let $X_q = \{X_q(t)\}_{t\ge0}$ be a random motion in $\mathbb{R}^d$ moving with $n$ different velocities, $d,n\in \mathbb{N}$. The changes of direction of $X_q$ are paced by a Poisson process $\{N(t)\}_{t\ge0}$ with rate function $\lambda(t)\in C^n\bigl((0,\infty),[0,\infty) \bigr)$. Assume that $X_q$, at any Poisson event, skips the change of direction with probability $q(t)\in C^n\bigl((0,\infty),[0,1] \bigr)$. Then, $X_q$ is equal in distribution to a motion $\{X(t)\}_{t\ge0}$ that behaves like $X_q$, but it can not skip the switch of direction and has rate function $\lambda_q(t) = q(t)\lambda(t)$.}
The statement can be proved by observing that the point process governing the changes of direction of $X_q$ is equal in distribution to a Poisson process with rate function $\lambda_q(t)$.
\hfill$\diamond$
\end{remark}
\footnotesize{
|
2,869,038,156,333 | arxiv | \section{Introduction}
\textbf{Background and historical notes.} We consider the large time behavior of solutions to the one-dimensional Schr\"{o}dinger equation
\begin{eqnarray}\label{NLS}
\begin{cases}
i\partial_t u+\frac{1}{2}\partial_x^2u+\lambda |u|^\alpha u=0,\ t>0,\ x\in \mathbb{R},\\
u(0,x)=u_0(x),\ x\in \mathbb{R},
\end{cases}
\end{eqnarray}
where $\alpha >0$, $\lambda \in\mathbb{C}$, and $u:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{C}$ is a complex-valued function. From the physical point of view, (\ref{NLS}) is said to be a governing equation of the light traveling through optical fibers, in which $|u(t,x)|$ describes the amplitude of the electric field, $t$ denotes the position along the fiber and $x$ stands for the temporal parameter expressing a form of pulse. As for the nonlinear coefficient, $\text{Re}\lambda $ denotes the magnitude of the nonlinear Kerr effect and $\text{Im} \lambda $ implies the magnitude of dissipation due to nonlinear Ohm's law (see e.g. \cite{GP}). The equation (\ref{NLS}) is also a particular case of the more general complex Ginzburg-Landau equation on $\mathbb{R}^N$
\begin{eqnarray}
\partial_{t} u=e^{i \theta} \Delta u+\zeta|u|^{\alpha} u\notag
\end{eqnarray}
where $|\theta| \leq \frac{\pi}{2}$ and $\zeta \in \mathbb{C}$, which is a generic modulation equation that describes the nonlinear evolution of patterns at near-critical conditions. See for instance \cite{Cr,Mie,Ste}.
There are many papers that studied the global well-posedness problem, decay and the asymptotic behavior of the solution (see \cite{CaHan,CaJFA,DZ03,Hayashi,Kita,Kita2,Ozawa,Zhang} and references therein). We use the following classification with respect to the value $\alpha $: the values $\alpha >2$ we call the super-critical in the scattering problem, the value $\alpha =2$ is the critical one and $0<\alpha <2$ we refer to the sub-critical.
Let us recall some known large time asymptotics of (\ref{NLS}) with $\lambda \in \mathbb{R}$. Concerning the super-critical case $\alpha >2$, it is well-known that the solution $u(t)$ behaves like a free solution $\exp \left((it/2)\partial_ x^2\right)\phi$ for $t$ sufficiently large (\cite{Ginibre0, Strauss2, Tsutsumi}). The strategy for this free asymptotic profile largely relies on the rapid decay of the nonlinearity. More precisely, since $\int_{1}^{\infty }|u(t)|^\alpha \mathrm{d}t\approx\int_{1}^{\infty }t^{-\alpha /2} \mathrm{d}t<\infty $ by expecting that $u(t)$ decays like a free solution, the nonlinearity can be regarded as negligible in the long time dynamics. As for the sub-critical and critical case $0<\alpha \le2$, the situation changes. The nonexistence of usual scattering states was obtained \cite{Ba,Strauss} by making use of the time decay estimate of solutions obtained from pseudo-conformal conservation law. In the case $\alpha =2$, Ozawa \cite{Ozawa} constructed modified wave operators to the equation (\ref{NLS}) for small scattering states, and Hayashi-Naumkin \cite{Hayashi} proved the time decay and the large time asymptotics of $u(t)$ for small initial data. According to their results, the small solution $u(t)$ asymptotically tends to a modified free solution. More precisely, there are $\mathbb{C}$-valued $W(x)\in L^\infty \cap L^2$ and $\mathbb{R}$-valued $\Phi(x)\in L^\infty $ such that as $t\rightarrow \infty $
\begin{equation}
u(t,x)= \frac{1}{\sqrt {it}}W(\frac{x}{t})\exp \left(i\frac{x^2}{2t}+i\lambda |W(\frac{x}{t})|^2\log t+i\Phi(\frac{x}{t})\right)+O_{L^\infty _x}(t^{-1/2})\notag
\end{equation}
While the subcritical case $0<\alpha <2$ seems to be completely open. To our knowledge, there are no results about the precise behavior or any kind of modified scattering of $u$ for large time.
Many works have also dealt with the complex coefficient case. We can only expect the large time asymptotics of (\ref{NLS}) for the case $\text{Im} \lambda >0$, since it is proved in \cite{Ca2} that, under the assumptions $\text{Im} \lambda <0,\ 0<\alpha <\infty $, there exists a class of blowup solutions to (\ref{NLS}). On the other hand, it is easy to see that
\begin{equation}
\|u(t)\|_{L^2}^2+2\text{Im} \lambda \int _0^t \|u(\tau)\|_{L^{\alpha +2}}^{\alpha +2}d\tau =\|u_0\|_{L^2}^2,\label{141}
\end{equation}
which suggests a dissipative structure for $\text{Im} \lambda >0$. In what follows, we concentrate our attention on the sub-critical and critical case: $0<\alpha \le2$ (For the super-critical case, we refer to the aforementioned works \cite{Ginibre0, Strauss2, Tsutsumi}, where a super-critical real $\lambda $ were studied, and the ideals are still applicable to a complex $\lambda $ with $\text{Im} \lambda >0$). The critical case $\alpha =2$ has been studied in \cite{CPDE}, in which the positivity of $\text{Im} \lambda $ visibly affects the decay rate of $\left\|u(t,x)\right\|_{L^\infty _x}$ and, actually, it decays like $(t\log t)^{-1/2}$. Since the nontrivial free solution only decays like $O(t^{-1/2})$, this gain of additional logarithmic time decay reflects a dissipative character. This result is then extended in \cite{Kita,Kita2} to the subcritical case. For $\alpha <2$ is sufficiently close to $2$, Kita-Shimomura \cite{Kita} established the time decay estimates
\begin{equation}
\|u(t,x)\|_{L^\infty _x}\lesssim t^{-1/\alpha },\qquad \text{for }t\ge1 \label{dsa}
\end{equation}
and the asymptotic formula of the solutions. In addition, it is proved in \cite{Kita2} that,
under the large dissipative assumption
\begin{eqnarray}\label{lambda1}
\lambda _2\ge\frac{\alpha\left|\lambda _1\right| }{2\sqrt{ \alpha +1}},\qquad \lambda =\lambda _1+i\lambda _2,
\end{eqnarray}
all solutions with initial value in $H^1(\mathbb{R})\cap L^2 (\mathbb{R}, |x|^2dx)$ satisfy the $L^\infty $ decay estimate (\ref{dsa}) when $\frac{1+\sqrt {33}}{4}<\alpha <2$, and possess a large time asymptotic state when $\frac{9+\sqrt {177}}{12}<\alpha <2$.
The strategy used in \cite{Kita, Kita2,CPDE} is to apply the operator $\mathcal{F} U(-t)$ to the equation (\ref{NLS}), where $U(t)=e^{it/2 \Delta }$ is the Schr\"odinger operator. Using the factorization technique of the Schr\"odinger operator $U(t)$, they obtain an ODE for $\mathcal{F} U(-t)u(t)$
\begin{equation}
i\partial_t \mathcal{F} U(-t)u(t)=\lambda t^{-\alpha /2}|\mathcal{F} U(-t)u(t)|^\alpha \mathcal{F} U(-t)u(t)+O_{L^\infty_x } (t^{-\alpha /2-\mu}),\ 0<\mu<1/4.,\label{191}
\end{equation}
from which, they deduce the large time asymptotics of $\mathcal{F} U(-t)u(t,x)$ and then in the solution $u(t,x)$.
The present work aims to complete the previous results on the time asymptotic behavior of the solutions obtained in \cite{Kita,Kita2,CPDE}. More precisely, for arbitrary large initial data, we present the uniform time decay estimates when $4/3<\alpha <2$, and the large time asymptotics of the solution when $\frac{7+\sqrt{145}}{12}<\alpha <2$.
\vspace{0.5cm}
\noindent \textbf{Notation and function spaces.} To state our result precisely, we now give some notations. Throughout the paper, $F(\xi)$ denotes the second order constant coefficients classical elliptic symbol, which has an expansion
\begin{eqnarray}
\label{1.3}F(\xi)=c_2\xi^2+c_1\xi+c_0
\end{eqnarray}
with $c_2>0,\ c_1,\ c_0\in \mathbb{R}$. We introduce the notations $D_t=\frac{\partial_t}{i}$, $D=\frac{\partial_x}{i}$ and the vector field
\begin{equation}
\mathcal{L} =x+tF'(D).\label{5161}
\end{equation}
For $\psi \in L^1(\mathbb{R})$, $\mathcal{F} \psi$ is represented as $\mathcal{F} \psi (\xi)=(2\pi)^{-1/2}\int_{ \mathbb{R}}\psi(x) e^{-ix\xi}dx$. $[A,B]$ denotes the commutator $AB-BA$. Different positive constants we denote by the same letter $C$. We introduce some function spaces. $\mathcal {S} (\mathbb{R}^2) $ denotes the usual two-dimensional Schwarz space. $L^p=L^p(\mathbb{R})$ denotes the usual Lebesgue space with the norm $\|\phi\|_{L^p}=(\int _{\mathbb{R}}|\phi(x)|^p dx)^{1/p}$ if $1\le p<\infty $ and $\|\phi\|_{L^\infty }=\text{ess. sup }\left\{|\phi(x)|;x\in \mathbb{R}\right\} $. The weighted Sobolev space is defined by $ H^{0,m}=L^2(\mathbb{R})\cap L^2(\mathbb{R},|x|^{2m}dx)$.
\vspace{0.5cm}
\noindent \textbf{Main results.} We are now ready to state the main result.
\begin{thm}\label{T1.1}
Assume that $u_0\in H^{0,1}$, $0<\alpha <2$, $\lambda $ satisfies the condition (\ref{lambda1}) and $\mathcal{L} $ is the vector filed defined in (\ref{5161}). Then there exists a unique global solution $u\in C\left([0,\infty ),\ L^2 \right)$ to the Cauchy problem
\begin{equation}\left\{
\begin{array}{ll}
(D_t -F(D))u=\lambda|u|^\alpha u,&t>0,x\in\mathbb{R}
\\ u(x,0)= u_0(x),
\end{array}\label{1.1}
\right.
\end{equation}
satisfying
\begin{eqnarray}\label{3192}
\left\|u(t,x)\right\|_{L_x^2}+\left\|\mathcal{L}u(t,x)\right\|_{L_x^2}\le C\left\|u_0\right\|_{ H^{0,1}},\ t\ge0
\end{eqnarray}
and
\begin{equation}
\label{ldecay}
\|u(t,x)\|_{L^\infty _x}\le C\|u_0\|_{H^{0,1}}t^{-1/2},\ t>1.
\end{equation}
Furthermore, if $u_0\in H^{0,2}$ and $\alpha \ge1$, then
\begin{eqnarray}\label{491}
\left\|\mathcal{L} ^2u(t,x)\right\|_{L_x^2}\le C(\left\|u_0\right\|_{H^{0,2}}+\left\|u_0\right\|_{H^{0,2}}^{2\alpha +1})t^{2-\alpha }, \ t\ge0.
\end{eqnarray}
\end{thm}
\begin{rem}
Applying the operator $\mathcal{L} $ to the equation (\ref{1.1}), we obtain easily the energy inequality
\begin{equation}
\|\mathcal{L} u(t,\cdot)\|_{L^2}\lesssim \|\mathcal{L} u(1,\cdot)\|_{L^2}+\int_{1}^{t}\|u(\tau,\cdot)\|_{L^\infty }^\alpha \|\mathcal{L} u(t,\cdot)\|_{L^2} \mathrm{d}\tau .\label{energy}
\end{equation}
Using Gronwall's inequality and a priori estimate $\|u(\tau,\cdot)\|_{L^\infty }\lesssim \varepsilon \tau^{-1/2} $, Hayashi and Naumkin \cite{Hayashi} obtained a moderate growth rate of $\|\mathcal{L} u\|_{L^2}$ in the critical case $\alpha =2$, which is essential to close the bootstrap assumption on $\|u(t,x)\|_{L^\infty _x}.$
While the limit in [Theorem \ref{T1.3}, part (d)] demonstrates that one can not hope to deduce this from (\ref{energy}) for $\alpha <2$ as in \cite{Hayashi}, even for the small initial data. So we assume the large dissipative condition (\ref{lambda1}) as in \cite{Kita2}, which is used in (\ref{151}) to derive the uniform bound (\ref{3192}).
\end{rem}
Next, we derive the time decay rate of the global solution obtained in Theorem \ref{T1.1}.
\begin{thm}
\label{T1.2}
Assume that $u_0\in H^{0,1}$, $4/3<\alpha<2$, $\lambda $ satisfies the condition (\ref{lambda1}) and $u$ is the global solution obtained in Theorem \ref{T1.1}. There exists a constant $C>0$ such that for all $t\ge1$,
\begin{eqnarray}\label{decay}
\left\|u(t,x)\right\|_{L_x^\infty }\le Ct^{-1/\alpha }.
\end{eqnarray}
\end{thm}
\begin{rem}
A similar time decay estimate as in (\ref{decay}) was obtained in \cite{Kita2} under the assumptions $\frac{1+\sqrt{33}}{4}<\alpha \le2$, which was then extended to the case $\frac{7+\sqrt{145}}{12}<\alpha \le2$ in \cite{Jin}. We note that $\frac{4}{3}<\frac{1+\sqrt{33}}{4}\approx 1.686$ and $\frac{4}{3}<\frac{7+\sqrt{145}}{12}\approx1.586$. Therefore, Theorem \ref{T1.2} is an improvement of the corresponding results in \cite{Jin,Kita2}.
\end{rem}
\begin{rem}
The additional assumption $\alpha >4/3$ ensures that the remainder term $v_{\Lambda ^c}$ decays faster than $v$ (see (\ref{1511}) and (\ref{1510})).
\end{rem}
\begin{rem}
Theorem \ref{T1.2} is valid without any smallness conditions on the initial data. Moreover, the solution decays faster than the free solution. Recall that in one space dimension, the free solution decays like $t^{-1/2}$.
\end{rem}
Finally, we give a large time asymptotic formula for the solutions and show the existence of modified scattering states for a certain range of the exponent in the nonlinear term.
\begin{thm}\label{T1.3}
Suppose that the assumptions in Theorem \ref{T1.2} are satisfied and
\begin{equation}
u_0\in H^{0,1},\ \frac{1+\sqrt{33}}{4}<\alpha <2,\notag
\end{equation}
or
\begin{equation}
u_0\in H^{0,2},\ \frac{7+\sqrt{145}}{12}<\alpha <2,\notag
\end{equation}
then the followings hold:\\
(a) Let
\begin{equation}\label{eq1}
\Phi(t,x)=\int_{1}^{t}s^{-\alpha /2}\left|v_\Lambda(s,x) \right| ^\alpha \mathrm{d}s,
\end{equation}
where $v_{\Lambda }$ is the function defined in (\ref{1219}). There exists a unique complex valued function $z_+(x)\in L^\infty _x\cap L^2_x$ such that for some $\kappa>0$,
\begin{eqnarray}
\left\|v_{\Lambda }(t,x)\exp\left(-i(w(x)t+\lambda \Phi(t,x))\right)-z_+(x)\right\|_{L^\infty _x\cap L^2_x}=O(t^{-\kappa})\notag
\end{eqnarray}
holds as $t\rightarrow \infty $, where $w(x)=:-(x+c_1)^2/(4c_2)+c_0.$ \\
(b) Let
\begin{eqnarray}\label{3301}
K(t,x)=1+\frac{2\alpha \lambda _2}{2-\alpha }\left|z_+(x)\right|^\alpha \left(t^{(2-\alpha )/2}-1\right),
\end{eqnarray}
\begin{eqnarray}
\psi_+(x)=\alpha \lambda_2 \int_{1}^{\infty }s^{-\alpha /2}\left(\left|v_{\Lambda }(s,x)\right|^\alpha\exp \left( {\alpha \lambda _2\Phi(s,x)}\right) -\left|z_+(x)\right|^\alpha \right) \mathrm{d}s,\label{a2}
\end{eqnarray}
and
\begin{equation}\label{eq2}
S(t,x)=\frac{1}{\alpha \lambda _2}\log \left(K(t,x)+\psi_+(x)\right).
\end{equation}
The asymptotic formula
\begin{eqnarray}\label{3302}
u(t,x)=\frac{1}{\sqrt t}e^{i\left(w(\frac{x}{t})t+\lambda S(t,\frac{x}{t})\right)}z_+(\frac{x}{t})+O_{L^\infty _x}(t^{-1/2-\kappa})\cap O_{L^2_x}(t^{-\kappa})
\end{eqnarray}
holds as $t\rightarrow \infty $, where $\kappa$ is the same constant as in part (a). \\
(c) Let $u_+(x)=\frac{1}{\sqrt {4\pi c_2}} e^{-i\frac{\pi}{4}}e^{-i\frac{c_1x}{2c_2}} (\mathcal{F} z_+)(\frac{x}{2c_2})$, we have the modified linear scattering
\begin{equation}
\lim_{t\rightarrow \infty }\left\|u(t,x)-e^{i\lambda S(t,\frac{x}{t})}e^{iF(D)t}u_+(x)\right\|_{L_x^2}=0.\label{123}
\end{equation}\\
(d) If $u_0\neq0$, then the limit
\begin{eqnarray}
\lim_{t\rightarrow \infty } t^{\frac{1}{\alpha }}\|u(t,x)\|_{L^\infty _x}=\left(\frac{2-\alpha }{2\alpha \lambda _2}\right)^{\frac{1}{\alpha }},\qquad \text{when }\alpha _0<\alpha <2,\label{qwe2}
\end{eqnarray}
exists and is independent of the initial value, where $\alpha _0=\frac{5+\sqrt {89}}{8} \approx 1.804$.
\end{thm}
\begin{rem}
A similar large time asymptotic formula of the solutions is obtained in \cite{Kita2} in the case $\frac{9+\sqrt{177}}{12}<\alpha <2$.
Since $\frac{7+\sqrt{145}}{12}\approx 1.587< \frac{1+\sqrt{33}}{4}\approx 1.686<\frac{9+\sqrt{177}}{12}\approx 1.859$, we see that Theorem \ref{T1.3} generates the result of \cite{Kita2} in the range of $\alpha $.
\end{rem}
\begin{rem}
The assumption on the lower bound of $\alpha $ ensures the convergence of the integral (\ref{3222}). It can be extended to $\alpha >\frac{2}{3}$ if we assume some nonvanishing conditions on the initial values. See e.g. \cite{CaHan,CaJFA}.
\end{rem}
\begin{rem}
According to the asymptotic formulas (\ref{3302}) and (\ref{123}), we see that the solution $u$ is not asymptotically free. By the definition (\ref{eq2}) of $S(t,x)$, we can write the modification factor $e^{i\lambda S(t,x)}$ explicitly:
\begin{equation}
e^{i\lambda S(t,x)}= \frac{\exp \left\{\frac{i\lambda _1}{\alpha \lambda _2}\log \left\{1+\frac{2\alpha \lambda _2}{2-\alpha }|z_+(x)|^\alpha (t^{(2-\alpha )/2}-1)+\psi_+(x)\right\} \right\} }{(1+\frac{2\alpha \lambda _2}{2-\alpha }|z_+(x)|^\alpha (t^{(2-\alpha )/2}-1)+\psi_+(x))^{1/\alpha }}.\label{652}
\end{equation}
\end{rem}
\vspace{0.5cm}
\noindent \textbf{Strategy of the proof.} We briefly sketch the strategy used to derive the decay estimate (\ref{decay}), which is the key to establishing the large time asymptotics of the solution. We adapt the semiclassical analysis method introduced by Delort \cite{Delort}, see also \cite{S,Zhang} which are more close to the problem we are considering. We make first a semiclassical change of variables
\begin{equation}
u(t,x)=\frac{1}{\sqrt{t}}v(t,\frac{x}{t}),\label{uv1}
\end{equation}
for some new unknown function $v$, that allows to rewrite the equation (\ref{1.1}) as
\begin{equation}\label{1.15}
(D_t-G_h^w(x\xi+F(\xi)))v=\lambda h^{\alpha /2}|v|^\alpha v,
\end{equation}
where the semiclassical parameter $h=\frac1t$, and the Weyl quantization of a symbol $a$ is given by
\begin{equation}
G_h^w(a)u(x)=\frac{1}{2\pi h}\iint e^{\frac{i}{h}(x-y)\xi}a(\frac{x+y}{2},\xi)u(y)dyd\xi. \notag
\end{equation}
By (\ref{uv1}), the decay estimate (\ref{decay}) is equivalent to
\begin{equation}
\|v(t,x)\|_{L^\infty _x}\le Ct^{1/2-1/\alpha }.\label{1511}
\end{equation}
If we develop the symbol $x\xi+F(\xi)$ as follows by using (\ref{1.3})
\begin{equation}
x\xi+F(\xi)=w(x)+\frac{(x+F'(\xi))^2}{4c_2}\qquad \text{with } w(x)=-\frac{(x+c_1)^2}{4c_2}+c_0,\label{15100}
\end{equation}
we deduce from (\ref{1.15}) an ODE for $v$:
\begin{equation}
D_tv=w(x)v+\lambda h^{\alpha /2}|v|^\alpha v+\frac{1}{4c_2}G_h^w((x+F'(\xi))^2)v\label{e123}
\end{equation}
By semiclassical Sobolev inequality, $\|G_h^w((x+F'(\xi))^2)v\|_{L^\infty _x}$ is controlled by some energy norm of $v$ that contains the spatial derivative of order three. While deducing this norm from the equation (\ref{1.15}) via the standard energy method requires $\alpha \ge 2$. Instead, we use the operators whose symbols are localized in a neighbourhood of $M=:\{(x,\xi)\in \mathbb{R}^2:\ x+F'(\xi)=0\}$ of size $O(\sqrt h)$.
In that way we can apply Proposition \ref{P2.6} to pass uniform norms of the remainders to the $L^2$ norm losing only a power $h^{-1/4}$.
More precisely, we set \begin{eqnarray}
v_{\Lambda }=G_h^w(\gamma(\frac{x+F'(\xi)}{\sqrt h}))v,\label{1219}
\end{eqnarray}
where $\gamma\in C_0^\infty (\mathbb{R})$ satisfying $\gamma=1$ in a neighbourhood of zero. In Lemma \ref{l4.3-00}, we will show that
$v_{\Lambda ^c}=:G_h^w(1-\gamma(\frac{x+F'(\xi)}{\sqrt {h}}))v$ satisfies the uniform estimate
\begin{equation}
\|v_{\Lambda ^c}(t,x)\|_{L^\infty _x}\lesssim t^{-1/4}.\label{1510}
\end{equation}
We see that it sufficies to prove the estimate
\begin{equation}
\|v_{\Lambda }(t,x)\|_{L^\infty _x} \le Ct^{1/2-1/\alpha },\notag
\end{equation}
since $v_{\Lambda ^c}$ decays faster than $v$ by the assumption $\alpha >4/3$.
Applying $G_h^w(\gamma(\frac{x+F'(\xi)}{\sqrt{h}}))$ to (\ref{e123}) and using (\ref{15100}) we obtain the ODE for $v_{\Lambda }$
\begin{equation}
D_tv_\Lambda=w(x)v_{\Lambda }+\lambda h^{\alpha /2} |v_\Lambda|^\alpha v_\Lambda+ R(v)\label{552}
\end{equation}
where the remainder
\begin{eqnarray}
R(v)&=&[D_t-G^w_h(x\xi+F(\xi)),G^w_h(\gamma(\frac{x+F'(\xi)}{\sqrt{h}}))]v+\frac{1}{4c_2}G_h^w((x+F'(\xi))^2)v_{\Lambda }\notag\\
&&-\lambda h^{\alpha /2} G^w_h(1-\gamma(\frac{x+F'(\xi)}{\sqrt{h}}))(|v|^\alpha v)+\lambda h^{\alpha /2}\left( |v|^\alpha v-|v_\Lambda|^\alpha v_\Lambda\right)\notag
\end{eqnarray}
satisfies the estimate (see Lemmas \ref{l4.3}--\ref{l5.6})
\begin{equation}
\|R(v)\|_{L^\infty _x}\lesssim t^{-5/4}+(\|v_{\Lambda }\|_{L^\infty _x}^\alpha +\|v\|_{L^\infty _x}^\alpha )t^{-\alpha /2-1/4}.\notag
\end{equation}
Note that $R(v)$ decays faster than the remainder in (\ref{191}) when $\alpha <2$. Performing a bootstrap and a contradiction argument, one finally deduce from the ODE (\ref{552}) the desired $L^\infty $ estimate for $v_{\Lambda }$, and then in the solution $u$.
The details can be found in Subsection \ref{sub2}.
\vspace{0.5cm}
\noindent \textbf{Outline.}
The framework of this paper is organized as follows.
In Section \ref{S2}, we present the definitions and some useful properties of Semiclassical pseudo-differential operators. In Section \ref{S3}, we establish the global existence and uniqueness of the solution to (\ref{1.1}). In Section \ref{S5}, we prove the decay estimates as stated in Theorem \ref{T1.2}, combining the bootstrap and the contradiction argument. Finally, in Section \ref{S6}, we establish the asymptotic formulas in Theorem \ref{T1.3}.
\section{Semiclassical pseudo-differential operators}\label{S2}
The proof of the main theorem will rely on the use of the semiclassical pseudo-differential calculus. For simplicity, we give only the definitions and properties of the operators we shall use. For more properties about semiclassical pseudo-differential operators, we refer to Chapter 7 of the book of Dimassi-Sj\"{o}strand \cite{D-S} and Chapter 4 of the book of Zworski \cite{Zworski}.
\begin{defn}
Let $a(x,\xi)\in \mathcal{S} (\mathbb{R}^2)$ and $h\in ( 0,1] $. Define the Weyl quantization to be the operator $G^w_h(a)$ acting on $u\in \mathcal{S}(\mathbb{R})$ by the formula
\begin{equation}
G^w_h(a)u=\frac{1}{2\pi h}\int_{\mathbb{R}}\int_{\mathbb{R}}
e^{\frac{i}{h}(x-y)\xi}a(\frac{x+y}{2},\xi)u(y)dyd\xi.\notag
\end{equation}
\end{defn}
We have the following boundedness for Weyl quantization.
\begin{prop}[Proposition 2.7 in \cite{Zhang}]\label{P2.6}
Let $a(\xi)$ be a smooth function satisfying $|\partial_\xi^\alpha a(\xi)|$ $\leq C_\alpha <\xi>^{-1-\alpha}$ for any $\alpha \in \mathbb{N}$. Then for $h\in ( 0,1] $
\begin{equation}
\|G^w_h(a(\frac{x+F'(\xi)}{\sqrt{h}}))\|_{\mathcal{L}(L^2,L^\infty)}=O(h^{-\frac{1}{4}}),
\ \|G^w_h(a(\frac{x+F'(\xi)}{\sqrt{h}}))\|_{\mathcal{L}(L^2,L^2)}=O(1).\notag
\end{equation}
\end{prop}
Next, we introduce some useful composition properties for Weyl quantization.
\begin{prop}[Theorem 7.3 in \cite{D-S}]\label{ab}
Suppose that $a,b\in\mathcal{S}(\mathbb{R}^2)$. Then
$$
G_h^w(a\sharp b)=G_h^w(a)\circ G_h^w(b),
$$
where
\begin{equation}
a\sharp b(x,\xi):=\frac{1}{(\pi h)^{2}}\int_\mathbb{R}\int_\mathbb{R}\int_\mathbb{R}\int_\mathbb{R}
e^{\frac{2i}{h}(\eta z- y\zeta)}a(x+z,\xi+\zeta)b(x+y,\xi+\eta)dyd\eta dz d\zeta. \notag
\end{equation}
\end{prop}
\begin{prop}[Proposition 2.4 in \cite{Zhang}]\label{P2.3-0}
Suppose that $a,b\in\mathcal{S}(\mathbb{R}^2)$. Then
\begin{equation}
a\sharp b=ab+\frac{ih}{2} (\partial_x a\partial_\xi b-\partial_\xi a\partial_x b)+R, \notag
\end{equation}
where
\begin{eqnarray}
R&=&\frac{1}{(2\pi )^{2}}\int_\mathbb{R}\int_\mathbb{R}\int_\mathbb{R}\int_\mathbb{R}
e^{\frac{2i}{h}(\eta z- y\zeta)}
\left\{-
\int^1_0\partial_x^2a(x+tz,\xi)(1-t)dt
\partial_\eta^2 b (x+y,\xi+\eta)\right.\nonumber\\
&&\left.+\int^1_0\int^1_0\partial_x\partial_\xi a(x+sz,\xi+t\zeta) dsdt\partial_\eta\partial_y b (x+y,\xi+\eta)
\right.\nonumber\\
&&\left. -\int^1_0\partial_\xi^2a(x,\xi+t\zeta)(1-t) dt\partial_y^2 b (x+y,\xi+\eta)\right\}
dyd\eta dz d\zeta.\notag
\end{eqnarray}
\end{prop}
\begin{lem}\label{l0}
Assume that $\Gamma_{0}(\xi)$ is a smooth function, satisfying $|\partial^\alpha \Gamma_{0}(\xi)|
\le C_\alpha <\xi>^{-1-\alpha}$ for any $\alpha \in \mathbb{N}$. Then we have
\begin{equation}\label{4131}
\Gamma_{0}(\frac{x+F'(\xi)}{\sqrt{h}})\sharp\frac{x+F'(\xi)}{\sqrt{h}}= \frac{x+F'(\xi)}{\sqrt{h}}\sharp \Gamma_{0}(\frac{x+F'(\xi)}{\sqrt{h}})=\Gamma_{0}(\frac{x+F'(\xi)}{\sqrt{h}})\frac{x+F'(\xi)}{\sqrt{h}}.
\end{equation}
In addition, if $|\partial^\alpha (\xi\Gamma_0(\xi))|
\le C_\alpha <\xi>^{-1-\alpha}$ for any $\alpha \in \mathbb{N}$, then we have
\begin{equation}\label{4132}
\left( \Gamma_{0}(\frac{x+F'(\xi)}{\sqrt{h}})\sharp\frac{x+F'(\xi)}{\sqrt{h}}\right)\sharp \frac{x+F'(\xi)}{\sqrt{h}}=\Gamma_{0}(\frac{x+F'(\xi)}{\sqrt{h}})(\frac{x+F'(\xi)}{\sqrt{h}})^2.
\end{equation}
\end{lem}
\begin{proof}
An application of Proposition \ref{P2.3-0} yields
\begin{eqnarray}
&& \Gamma_{0}(\frac{x+F'(\xi)}{\sqrt{h}})\sharp \frac{x+F'(\xi)}{\sqrt{h}}\notag\\
&=& \Gamma_{0}(\frac{x+F'(\xi)}{\sqrt{h}}) \frac{x+F'(\xi)}{\sqrt{h}}+\frac{ih}{2}\left(\partial_x\Gamma_{0}(\frac{x+F'(\xi)}{\sqrt{h}})\partial_\xi (\frac{x+F'(\xi)}{\sqrt{h}})\right. \notag\\
&&\left.\qquad-\partial_\xi\Gamma_{0}(\frac{x+F'(\xi)}{\sqrt{h}})\partial_x (\frac{x+F'(\xi)}{\sqrt{h}})\right)+R\notag\\
&=& \Gamma_{0}(\frac{x+F'(\xi)}{\sqrt{h}}) (\frac{x+F'(\xi)}{\sqrt{h}}),\notag
\end{eqnarray}
where $R=0$, since $\partial_{\xi\xi }(\frac{x+F'(\xi)}{\sqrt{h}})=\partial_{xx} (\frac{x+F'(\xi)}{\sqrt{h}})=\partial_{x\xi} (\frac{x+F'(\xi)}{\sqrt{h}})=0$.
Similarly, we can show that
\begin{eqnarray}
\frac{x+F'(\xi)}{\sqrt{h}}\sharp \Gamma_{0}(\frac{x+F'(\xi)}{\sqrt{h}})=\Gamma_{0}(\frac{x+F'(\xi)}{\sqrt{h}}) (\frac{x+F'(\xi)}{\sqrt{h}}).\notag
\end{eqnarray}This completes the proof of (\ref{4131}). Finally, (\ref{4132}) follows easily from (\ref{4131}):
\begin{eqnarray}
\Gamma_0(\frac{x+F'(\xi)}{\sqrt h})(\frac{x+F'(\xi)}{\sqrt h})^2&=&\left(\Gamma_{0}(\frac{x+F'(\xi)}{\sqrt h})\frac{x+F'(\xi)}{\sqrt h}\right)\sharp \frac{x+F'(\xi)}{\sqrt h}\notag\\
&=&\left(\Gamma_{0}(\frac{x+F'(\xi)}{\sqrt h})\sharp \frac{x+F'(\xi)}{\sqrt h}\right)\sharp \frac{x+F'(\xi)}{\sqrt h}.\notag
\end{eqnarray}
\end{proof}
\section{Proof of Theorem \ref{T1.1}}\label{S3}
In this section, we prove the global existence and uniqueness results in Theorem \ref{T1.1}.
We will use the following lemmas.
\begin{lem}\label{ll1}
Assume $f:[1,T]\times \mathbb{R}\rightarrow \mathbb{C},\ T>1$ is a smooth function, there exists a positive constant $C$ independent of $T, f$ such that for all $t\in[1,T]$
\begin{equation}
\left\|f(t,x)\right\|_{L^\infty _x}\le Ct^{-1/2}\left\|f(t,x)\right\|_{L^2 _x}^{1/2}\left\|\mathcal{L}f(t,x)\right\|_{L^2 _x}^{1/2},\label{4133}
\end{equation}
\begin{equation}
\|\mathcal{L} f(t,x)\|_{L^4_x}\le C\|f(t,x)\|_{L^\infty _x}^{1/2}\|\mathcal{L} ^2f(t,x)\|_{L^2_x}^{1/2}.\label{4134}
\end{equation}
\end{lem}
\begin{proof}
To begin with, it is useful to introduce a certain phase function. Let
\begin{equation}
\phi(t,x)=\frac{x^2+2c_1tx}{4c_2t}.\notag
\end{equation}
Since
\begin{equation}
\mathcal{L}=x+tF'(D)=x+c_1t-2c_2it\partial_x,\label{1234}
\end{equation}
it is straightforward to check that
\begin{equation}
-2c_2it \partial_x (f(t,x)e^{i\phi(t,x)})=e^{i\phi(t,x)}\mathcal{L} f(t,x), \label{4135}
\end{equation}
\begin{equation}
-4c_2^2t^2 \partial_{xx} (f(t,x)e^{i\phi(t,x)})=e^{i\phi(t,x)}\mathcal{L}^2 f(t,x).\label{4136}
\end{equation}
From (\ref{4135}), we have, for $t\in [1,T]$,
\begin{eqnarray}\label{45200}
\left\|\partial_x (f(t,x)e^{i\phi(t,x)})\right\|_{L_x^2}\le \frac{C}{t}\left\|\mathcal{L}f(t,x)\right\|_{L_x^2}.
\end{eqnarray}
On the other hand, using Gagliardo-Nirenberg's inequality, we obtain
\begin{eqnarray}
\left\|f(t,x)\right\|_{L^\infty _x}=\left\|f(t,x)e^{i\phi(t,x)}\right\|_{L^\infty _x}\le C\left\|f(t,x)e^{i\phi(t,x)}\right\|_{L_x^2}^{1/2}\left\|\partial_x (f(t,x)e^{i\phi(t,x)})\right\|_{L_x^2}^{1/2}.\notag
\end{eqnarray}
This together with (\ref{45200}) yields (\ref{4133}).
Similarly, it follows from (\ref{4135}) and Gagliardo-Nirenberg's inequality that
\begin{eqnarray}
\|\mathcal{L} f(t,x)\|_{L_x^4}\le Ct \|f(t,x)e^{i\phi(t,x)}\|_{L^\infty }^{1/2}\|\partial_{xx}(f(t,x)e^{i\phi(t,x)})\|_{L^2}^{1/2},\notag
\end{eqnarray}
which together with (\ref{4136}) gives the desired estimate (\ref{4134}).
\end{proof}
Using the classical energy estimate method, we obtain the following lemma easily and omit the details.
\begin{lem}\label{l4.1}
Assume $\text{Im} \lambda >0$, and $u\in C([0,T]; L^2)$ is a solution of (\ref{1.1}), then we have
\begin{equation}
\|u(t,\cdot)\|_{L^2}\le\|u_0\|_{L^2}, \ t\in[0,T].\notag
\end{equation}
\end{lem}
\begin{proof}[\textbf{Proof of Theorem \ref{T1.1}}]
For the initial datum $u_0\in H^{0,1}$, the existence and uniqueness of a local strong $L^2$ solution to the Cauchy problem (\ref{1.1}) easily follow from Strichartz's estimate
\begin{eqnarray}
\left\|u\right\|_{L^\infty L^2\cap L^4 L^\infty }\le C\left\|u_0\right\|_{L^2}+C\left\||u|^\alpha u\right\|_{L^1L^2}, \notag
\end{eqnarray}
and a standard contraction argument. Moreover, by applying Lemma \ref{l4.1}, we can extend this local solution to $[0,\infty ) $ easily and omit the details.
In what follows, we prove the estimate (\ref{3192})--(\ref{491}) and thus completing the proof of Theorem \ref{T1.1}.
\noindent \textbf{Estimate of $\left\|\mathcal{L}u\right\|_{L^2}$.}
Applying the operator ${\mathcal{L}}$ to (\ref{1.1}), and using the fundamental commutation property $
[D_t-F(D),\mathcal{L}]=0$, we get
\begin{equation}
(D_t-F(D))\mathcal{L}u=\lambda \mathcal{L}(\left|u\right|^\alpha u).\notag
\end{equation}
This implies
\begin{equation}
\frac{1}{2} \frac{d}{dt}\|{\mathcal{L}} u\|_{L^2}^2=-\text{Im} \int_{\mathbb{R}} \lambda {\mathcal{L}}(\left|u\right|^\alpha u)\overline{{\mathcal{L}}u} \mathrm{d}x.\notag
\end{equation}
On the other hand, from the expressions of $\mathcal{L} $ in (\ref{1234}), it is straightforward to check that
\begin{eqnarray}\label{z1}
\mathcal{L}(\left|u\right|^\alpha u)=\frac{\alpha +2}{2}\left|u\right|^{\alpha } \mathcal{L}u-\frac{\alpha }{2}\left|u\right|^{\alpha -2}u^2\overline{ \mathcal{L} u};
\end{eqnarray}
and that for $\alpha \ge1$,
\begin{eqnarray}\label{z2}
\mathcal{L}^2(|u|^\alpha u)&=&\frac{\alpha +2}{2}|u|^\alpha \mathcal{L}^2u+\frac{\alpha }{2}|u|^{\alpha -2}u^2\overline{\mathcal{L}^2u}+\frac{(\alpha +2)\alpha }{2}|u|^{\alpha -2}\text{Im} (\mathcal{L}u \overline{u})\mathcal{L}u \notag\\
&&-\frac{\alpha (\alpha -2)}{2}|u|^{\alpha -4}u^2 \text{Im} (\mathcal{L}u \overline{u})\overline{\mathcal{L}u}-\alpha |u|^{\alpha -2}u|\mathcal{L}u|^2.
\end{eqnarray}
From (\ref{z1}) and the large dissipative condition (\ref{lambda1}), we have for all $t\ge0$,
\begin{eqnarray}
&&-\text{Im} \left(\lambda {\mathcal{L}}(\left|u\right|^\alpha u)\overline{{\mathcal{L}}u}\right)\notag\\
&=&-\text{Im} \lambda \frac{\alpha +2}{2}\left|u\right|^\alpha \left| {\mathcal{L} }u\right|^2+\text{Im} (\lambda \frac{\alpha }{2}\left|u\right|^{\alpha -2}u^2\overline{{\mathcal{L} }u}^2)\notag\\
&\le& (-\frac{\alpha +2}{2}\lambda _2+\frac{\alpha }{2}|\lambda |)\left|u\right|^\alpha |{\mathcal{L} }u|^2\le0.\label{151}
\end{eqnarray}
Therefore, for all $t\ge0$, we have
\begin{equation}\label{3234}
\|{\mathcal{L}}u(t)\|_{L^2}\le \left\|{{\mathcal L}}u(0)\right\|_{L^2}=\left\|xu_0\right\|_{L^2},
\end{equation}
which together with Lemma \ref{l4.1} yields (\ref{3192}).
\noindent\textbf{Estimate of $\left\|\mathcal{L}^2u\right\|_{L^2}$. } Using the commutation relation $[D_t-F(D),\mathcal{L}^2]=0$, we get
\begin{eqnarray}
\frac{1}{2}\frac{d}{dt}\left\|\mathcal{L}^2u\right\|_{L^2}^2=-\text{Im} \int_{\mathbb{R}}\lambda \mathcal{L}^2(|u|^\alpha u)\cdot \overline{\mathcal{L}^2u} \mathrm{d}x.\label{432}
\end{eqnarray}
From (\ref{z2}) and the large dissipative condition (\ref{lambda1}), we have that for all $t\ge0$,
\begin{eqnarray}
&&-\text{Im} \int_{\mathbb{R}}\lambda \mathcal{L}^2(|u|^\alpha u)\cdot \overline{\mathcal{L}^2u} \mathrm{d}x\notag\\
&\le& -\lambda _2\frac{\alpha +2}{2}\int_{\mathbb{R}}|u|^\alpha |\mathcal{L}^2u|^2 \mathrm{d}x+\frac{\alpha }{2}|\lambda | \lambda \int_{\mathbb{R}}|u|^{\alpha }|\mathcal{L} ^2 u|^2 \mathrm{d}x\notag\\
&&+C\int_{\mathbb{R}}|u|^{\alpha -1}|\mathcal{L}u|^2|\mathcal{L}^2u| \mathrm{d}x\notag\\
&\le&C\left\|u\right\|_{L^\infty }^{\alpha -1}\left\|\mathcal{L}u\right\|_{L^\infty }\left\|\mathcal{L}u\right\|_{L^2}\left\|\mathcal{L}^2u\right\|_{L^2}.\notag
\end{eqnarray}
An application of Lemma \ref{ll1} then yields
\begin{eqnarray}
&&-\text{Im} \int_{\mathbb{R}}\lambda\mathcal{L}^2(|u|^\alpha u)\cdot \overline{\mathcal{L}^2u} \mathrm{d}x\notag\\
&\le& Ct^{-\alpha /2}\left\|u\right\|_{L^2}^{\frac{\alpha -1}{2}}\left\|\mathcal{L}u\right\|_{L^2}^{\frac{\alpha +2}{2}}\left\|\mathcal{L}^2u\right\|_{L^2}^{\frac{3}{2}}\notag\\
&\le& Ct^{-\alpha /2}\left\|u_0\right\|_{H^{0,1}}^{\alpha +\frac{1}{2}}\left\|\mathcal{L}^2u\right\|_{L^2}^{\frac{3}{2}},\label{433}
\end{eqnarray}
where the last inequality holds by applying (\ref{3234}) and Lemma \ref{l4.1}. Substituting (\ref{433}) into (\ref{432}), we get
\begin{eqnarray}
\frac{d}{dt}\left\|\mathcal{L}^2u\right\|_{L^2}^{1/2}\le Ct^{-\alpha /2}\left\|u_0\right\|_{H^{0,1}}^{\alpha +1/2}.\notag
\end{eqnarray}
Hence
\begin{eqnarray}
\left\|\mathcal{L}^2u\right\|_{L^2}\le \left\|u_0\right\|_{H^{0,2}}+Ct^{2-\alpha }\left\|u_0\right\|_{H^{0,1}}^{2\alpha +1},\label{0451}
\end{eqnarray}
which yields (\ref{491}).
Finally, from Lemma \ref{ll1}, Lemma \ref{l4.1} and (\ref{3234}), the desired decay estimate (\ref{ldecay}) follows.
\end{proof}
\section{The proof of Theorem \ref{T1.2}}\label{S5}
This section is devoted to proving Theorem \ref{T1.2}. It is organized in two subsections.
In the first one, we make first a semiclassical change of variables (\ref{4.2}) and then prove in Lemma \ref{l4.3-00} that $v_{\Lambda ^c}$ can be considered as a remainder. In addition, we derive the ODE (\ref{4.34}) for $v_{\Lambda }$ and then estimate the remainders $R_1(v),\ R_2(v)$ in the rest of this subsection.
In the second one, we use the ODE (\ref{4.34}) to derive the uniform for $v_{\Lambda }$ and then in the solution $u$, combining the bootstrap and the contradiction argument.
\subsection{Semiclassical reduction of the problem}\label{sub1}
We rewrite the problem in the semiclassical framework. Set
\begin{equation}\label{4.2}
u(t,x)=\frac{1}{\sqrt{t}}v(t,\frac{x}{t}),\ h=\frac{1}{t}.
\end{equation}
Then the equation (\ref{1.1}) is rewritten as
\begin{equation}
(D_t-G_h^w(x\xi+F(\xi)))v=\lambda t^{-\alpha /2}|v|^\alpha v.\label{3.4-000}
\end{equation}
At the same time, set
\begin{equation}
\widetilde{\mathcal{L}}=\frac{1}{h}G_h^w(x+F'(\xi)).\label{3.4}
\end{equation}
Indeed, since $F'(\xi)=2c_2\xi+c_1$, we have that $ \widetilde{\mathcal{L}}=(x+c_1)t+2c_2D_x$ so that
\begin{equation}\label{4103}
\mathcal{L} u(t,x)=\frac{1}{\sqrt t}(\widetilde{\mathcal{L}}v)(t,\frac{x}{t}),\qquad \mathcal{L} ^2u(t,x)=\frac{1}{ \sqrt {t}}(\widetilde{\mathcal{L} }^2v)(t,\frac{x}{t}).
\end{equation}
Moreover, one has
\begin{equation}
\|u(t,\cdot)\|_{L^2}=\|v(t,\cdot)\|_{L^2},\
\ \|u(t,\cdot)\|_{L^\infty}=t^{-1/2}\|v(t,\cdot)\|_{L^\infty}, \label{3.4-0}
\end{equation}
and
\begin{equation}\label{4104}
\|\mathcal{L}u(t,\cdot)\|_{L^2}=\|\widetilde{\mathcal{L}}v(t,\cdot)\|_{L^2}, \ \left\|\mathcal{L} ^2u(t,\cdot)\right\|_{L^2}=\|\widetilde{\mathcal{L} }^2v(t,\cdot)\|_{L^2}.
\end{equation}
Therefore the proof of Theorem \ref{T1.2} reduces to estimate $\left\|v(t,\cdot)\right\|_{L^\infty }$. To do so, we decompose $v=v_\Lambda+v_{\Lambda^c}$ with
\begin{equation}\label{3194}
v_\Lambda=G^w_h(\Gamma) v,
\end{equation}
where $\Gamma(x,\xi)=\gamma(\frac{x+F'(\xi)}{\sqrt{h}})$ with $\gamma\in C^\infty_0(\mathbb{R})$ satisfying that $\gamma\equiv1$ in a neighbourhood of zero.
We have the following $L^\infty-$estimates for $v_{\Lambda^c}$, which shows that $v_{\Lambda }$ can be considered as a small perturbation of $v$.
\begin{lem}\label{l4.3-00}
Assume $u_0\in H^{0,1}$. Then for all $t\ge1$,
\begin{equation}
\|v_{\Lambda^c}(t,\cdot)\|_{L^\infty}\leq C\|u_0\|_{H^{0,1}}t^{-1/4} ,\ \|v_{\Lambda^c}(t,\cdot)\|_{L^2}\leq C\|u_0\|_{H^{0,1}}t^{-1/2}. \notag
\end{equation}
\end{lem}
\begin{proof}
Let $\Gamma_{-1}(\xi)=\frac{1-\gamma(\xi)}{\xi},$ satisfying $|\partial^\alpha \Gamma_{-1}(\xi)|
\le C_\alpha <\xi>^{-1-\alpha}$ for any $\alpha\in \mathbb{N}$. Then we can write
\begin{eqnarray}
1-\Gamma(x,\xi)
&=&\sqrt h\Gamma_{-1}(\frac{x+F'(\xi)}{\sqrt{h}})(\frac{x+F'(\xi)}{{h}}). \notag
\end{eqnarray}
From Lemma \ref{l0}, and the definition of $\widetilde{\mathcal{L}}$ in (\ref{3.4}), we get
\begin{eqnarray}
&&G_h^w(1-\Gamma)v=\sqrt h G_h^w(\Gamma_{-1}(\frac{x+F'(\xi)}{\sqrt h}))\circ (\widetilde{\mathcal{L}}v).\notag
\end{eqnarray}
An application of Proposition \ref{P2.6} yields, for all $t\ge1$,
\begin{equation}
\label{b1}
\left\|G_h^w(1-\Gamma)v\right\|_{L^\infty }\le C\|\widetilde{\mathcal{L}}v\|_{L^2}h^{1/4},\ \left\|G_h^w(1-\Gamma)v\right\|_{L^2 }\le C\|\widetilde{\mathcal{L}}v\|_{L^2}h^{1/2}.
\end{equation}
Since $\|\widetilde{\mathcal{L} }v\|_{L^2}\le C\|u_0\|_{H^{0,1}}$ by (\ref{4104}) and (\ref{3192}), we obtain the desired estimate in Lemma \ref{l4.3-00}.
\end{proof}
To get the $L^\infty -$ estimates for $v_\Lambda $ in large time, we deduce an ODE from the PDE system (\ref{3.4-000}):
\begin{equation}
D_tv_\Lambda=w(x)v_{\Lambda }+\lambda t^{-\alpha /2} |v_\Lambda|^\alpha v_\Lambda+ t^{-\alpha /2} \left(R_1(v)+R_2(v)\right)\label{4.34}
\end{equation}
where $w(x)=-(x+c_1)^2/(4c_2)+c_0$ and
\begin{eqnarray}
R_1(v)&=&t^{\alpha /2}[D_t-G^w_h(x\xi+F(\xi)),G^w_h(\Gamma)]v\notag\\
&&+t^{\alpha /2}\left(G_h^w(x\xi+F(\xi))-w(x)\right)v_{\Lambda},\notag
\end{eqnarray}
\begin{equation}
R_2(v)=-\lambda G^w_h(1-\Gamma)(|v|^\alpha v)+\lambda \left( |v|^\alpha v-|v_\Lambda|^\alpha v_\Lambda\right). \notag
\end{equation}
The rest of this subsection is devoted to proving the following estimates for the remainders $R_1(v)$ and $R_2(v)$.
\begin{prop}
\label{lp}
For any $t\ge1$, we have \\
(a) when $u_0\in H^{0,1}$:
\begin{equation}
\|R_1(v)\|_{L^\infty}\leq C \left\|u_0\right\| _{H^{0,1}}t^{-5/4+\alpha /2},\notag
\end{equation}
\begin{equation}
\|R_1(v)\|_{L^2}\leq C\left\|u_0\right\| _{ H^{0,1}}t^{-3/2+\alpha /2},\notag
\end{equation}
\begin{equation}
\|R_2(v)\|_{L^\infty }\leq C \left\|u_0\right\| _{H^{0,1}}(\left\|v\right\|_{L^\infty }^\alpha+\left\|v_{\Lambda }\right\|_{L^\infty }^\alpha ) t^{-1/4},\notag
\end{equation}
\begin{equation}
\|R_2(v)\|_{L^2}\leq C \left\|u_0\right\| _{H^{0,1}}(\left\|v\right\|_{L^\infty }^\alpha+\left\|v_{\Lambda }\right\|_{L^\infty }^\alpha ) t^{-1/2}.\notag
\end{equation}
(b) when $u_0\in H^{0,2}$:
\begin{equation}
\|R_1(v)\|_{L^\infty}\leq C (\|u_0\|_{H^{0,2}}+\|u_0\|_{H^{0,2}}^{2\alpha +1})t^{1/4-\alpha/2 },\notag
\end{equation}
\begin{equation}
\|R_1(v)\|_{L^2}\leq C(\|u_0\|_{H^{0,2}}+\|u_0\|_{H^{0,2}}^{2\alpha +1})t^{-\alpha/2 },\notag
\end{equation}
\begin{equation}\notag
\|R_2(v)\|_{L^\infty }\leq C (\|u_0\|_{H^{0,2}}+\|u_0\|_{H^{0,2}}^{2\alpha +1})(\left\|v\right\|_{L^\infty }^\alpha+\left\|v_{\Lambda }\right\|_{L^\infty }^\alpha ) t^{5/4-\alpha },
\end{equation}
\begin{equation}\notag
\|R_2(v)\|_{L^2}\leq C (\|u_0\|_{H^{0,2}}+\|u_0\|_{H^{0,2}}^{2\alpha +1})(\left\|v\right\|_{L^\infty }^\alpha+\left\|v_{\Lambda }\right\|_{L^\infty }^\alpha ) t^{1-\alpha }.
\end{equation}
\end{prop}
The proof of Proposition \ref{lp} is split into Lemmas \ref{l4.3}--\ref{l5.6} below.
\begin{lem}\label{l4.3}
If $u_0\in H^{0,1}$, then for all $t\ge1$,
\begin{eqnarray}
\|[D_t-G^w_h(x\xi+F(\xi)),G^w_h(\Gamma)]v\|_{L^\infty}\leq C\|\widetilde{\mathcal{L} }v\|_{L^2}t^{-5/4},\notag\\ \|[D_t-G^w_h(x\xi+F(\xi)),G^w_h(\Gamma)]v\|_{L^2} \le C \|\widetilde{\mathcal{L} }v\|_{L^2}t^{-3/2}.\notag
\end{eqnarray}
Furthermore, if $u_0\in H^{0,2}$ and $\alpha \ge1$, then for all $t\ge1$,
\begin{eqnarray}
\|[D_t-G^w_h(x\xi+F(\xi)),G^w_h(\Gamma)]v\|_{L^\infty}\leq C\|\widetilde{\mathcal{L} }^2v\|_{L^2}t^{-7/4},\notag\\ \|[D_t-G^w_h(x\xi+F(\xi)),G^w_h(\Gamma)]v\|_{L^2} \le C\|\widetilde{\mathcal{L} }^2v\|_{L^2}t^{-2}. \notag
\end{eqnarray}
\end{lem}
\begin{proof}
Since $h=t^{-1}$, by a direct computation, we have
\begin{eqnarray}\label{3291}
&& [D_t,G^w_h(\Gamma)]f\notag\\
&=&-hi G^w_h(\Gamma)f+
\frac{1}{2\pi h}\int_{\mathbb{R}}\int_{\mathbb{R}} e^{i(x-y)\frac{\xi}{h}}(x-y)\xi \gamma(\frac{\frac{x+y}{2}+F'(\xi)}{\sqrt h})f(t,y)dyd\xi\nonumber\\
&&+\frac{1}{2\pi h}\int_{\mathbb{R}}\int_{\mathbb{R}} e^{i(x-y)\frac{\xi}{h}} \gamma'(\frac{\frac{x+y}{2}+F'(\xi)}{\sqrt{h}})
(\frac{x+y}{2}+F'(\xi))\frac{\sqrt{h}}{2i}f(t,y)dyd\xi\nonumber\\
&=& ih G^w_h[\gamma'(\frac{x+F'(\xi)}{\sqrt{h}})(\xi\frac{F''(\xi)}{\sqrt{h}}-\frac{x+F'(\xi)}{2\sqrt{h}})]f, \label{4.12}
\end{eqnarray}
where we used the fact that
\begin{eqnarray}
&& \frac{1}{2\pi h}\int_{\mathbb{R}}\int_{\mathbb{R}} e^{i(x-y)\frac{\xi}{h}}(x-y)\xi \gamma(\frac{\frac{x+y}{2}+F'(\xi)}{\sqrt h})f(t,y)dyd\xi\nonumber\\
&=&
\frac{1}{2\pi h}\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{h}{i} \partial_\xi e^{i(x-y)\frac{\xi}{h}} \xi \gamma(\frac{\frac{x+y}{2}+F'(\xi)}{\sqrt h})f(t,y)dyd\xi\nonumber\\
&=& hi G^w_h(\Gamma)f+\frac{1}{2\pi h}\int_{\mathbb{R}}\int_{\mathbb{R}} e^{i(x-y)\frac{\xi}{h}}\xi \gamma'(\frac{\frac{x+y}{2}+F'(\xi)}{\sqrt{h}})
F''(\xi)i\sqrt{h}f(t,y)dyd\xi.\notag
\end{eqnarray}
Then using Propositions \ref{ab} and \ref{P2.3-0} we write
\begin{equation}
[G^w_h(x\xi+F(\xi)),G^w_h(\Gamma)]=ih G^w_h(\gamma'(\frac{x+F'(\xi)}{\sqrt{h}})(\xi\frac{F''(\xi)}{\sqrt{h}}-\frac{x+F'(\xi)}{\sqrt{h}}))
+r_1-r_2,\label{4.14}
\end{equation}
where
\begin{eqnarray}
r_1
&=&\frac{1}{(2\pi)^2}\int_\mathbb{R}\int_\mathbb{R}\int_\mathbb{R}\int_\mathbb{R}
e^{\frac{2i}{h} (\eta z-y \zeta)}\frac{1}{h}\gamma''(\frac{x+y+F'(\xi+\eta)}{\sqrt h})\left\{F''(\xi+\eta)\phantom{\int_0^1}\right. \notag\\
&&\left.-\int _{0}^1F''(\xi+t\zeta)(1-t)dt\right\}
dyd\eta dz d\zeta,\notag
\end{eqnarray}
\begin{eqnarray}
r_2
&=&\frac{1}{(2\pi)^2}\int_\mathbb{R}\int_\mathbb{R}\int_\mathbb{R}\int_\mathbb{R}
e^{\frac{2i}{h} (\eta z-y \zeta)} \left\{-\frac{1}{h}\int_0^1 \gamma''(\frac{x+tz+F'(\xi)}{\sqrt h})(1-t)dtF''(\xi+\eta)\right. \notag\\
&&\left.+\int_0^1\int_0^1 \frac{1}{h}\gamma''(\frac{x+sz+F'(\xi+t\zeta)}{\sqrt h})F''(\xi+t\zeta)dsdt\right\}
dyd\eta dz d\zeta.\notag
\end{eqnarray}
Since $F''=2c_2$ and $
\int_{ \mathbb{R}}e^{\frac{2i\eta z}{h}}dz=\delta(\eta)\pi h$, $\int_{ \mathbb{R}}e^{-\frac{2iy\zeta}{h}}d\zeta=\delta(y)\pi h$, we have $r_1=\frac{c_2h}{4}\gamma''(\frac{x+F'(\xi)}{\sqrt h})$. Similarly, we have $r_2=\frac{c_2h}{4}\gamma''(\frac{x+F'(\xi)}{\sqrt h})$. So we deduce from (\ref{3291}) and (\ref{4.14}) that
\begin{eqnarray}\label{451}
[D_t-G^w_h(x\xi+F(\xi)),G^w_h(\Gamma)]v=\frac{ih}{2}G_h^w(\gamma'(\frac{x+F'(\xi)}{\sqrt h})\frac{x+F'(\xi)}{\sqrt{h}})v.
\end{eqnarray}
On the other hand, using Lemma \ref{l0}, we can rewrite (\ref{451}) as
\begin{eqnarray}
[D_t-G^w_h(x\xi+F(\xi)),G^w_h(\Gamma)]v=\frac{ih^{3/2}}{2} G_h^w(\gamma'(\frac{x+F'(\xi)}{\sqrt h}))\circ (\widetilde{\mathcal{L}}v),\ \text{when } u_0\in H^{0,1}\notag
\end{eqnarray}
\begin{eqnarray}
[D_t-G^w_h(x\xi+F(\xi)),G^w_h(\Gamma)]v =\frac{ih^2}{2} G_h^w(\Gamma_{-2}(\frac{x+F'(\xi)}{\sqrt h}))\circ (\widetilde{\mathcal{L}}^2v),\ \text{when } u_0\in H^{0,2}\notag
\end{eqnarray}
where $\Gamma_{-2}(\xi)=\frac{\gamma'(\xi)}{\xi}$ satisfying $|\partial^\alpha \Gamma_{-2}(\xi)|
\le C_\alpha <\xi>^{-1-\alpha}$ for any $\alpha\in \mathbb{N}$.
The above identities and an application of Proposition \ref{P2.6} yields the desired estimates in Lemma \ref{l4.3}.
\end{proof}
\begin{lem}\label{l5}
If $u_0\in H^{0,1}$, then for all $t\ge1$,
\begin{equation}
\|\left(G_h^w(x\xi+F(\xi))-w(x)\right)v_{\Lambda}\|_{L^\infty}
\leq C\|\widetilde{\mathcal{L} }v\|_{L^2}t^{-5/4},\notag
\end{equation}
\begin{equation}
\|\left(G_h^w(x\xi+F(\xi))-w(x)\right)v_{\Lambda}\|_{L^2}\le C\|\widetilde{\mathcal{L} }v\|_{L^2}t^{-3/2}.\notag
\end{equation}
Furthermore, if $u_0\in H^{0,2}$ and $\alpha \ge1$, then for all $t\ge1$,
\begin{equation}
\|\left(G_h^w(x\xi+F(\xi))-w(x)\right)v_{\Lambda}\|_{L^\infty}
\leq C\|\widetilde{\mathcal{L} }^2v\|_{L^2}t^{-7/4}\notag
\end{equation}
\begin{equation}
\|\left(G_h^w(x\xi+F(\xi))-w(x)\right)v_{\Lambda}\|_{L^2}\le C\|\widetilde{\mathcal{L} }^2v\|_{L^2}t^{-2}.\notag
\end{equation}
\end{lem}
\begin{proof}
Applying Lemma \ref{l0} to (\ref{1510}) we get
\begin{eqnarray}
&&\left(G_h^w(x\xi+F(\xi))-w(x)\right)v_{\Lambda}=\frac{1}{4c_2}G_h^w((x+F'(\xi))^2\gamma (\frac{x+F'(\xi)}{\sqrt h}))v\notag\\
&&= \frac{1}{4c_2}\begin{cases}
h^{3/2}G_h^w(\Gamma_{-3}(\frac{x+F'(\xi)}{\sqrt{h}})(\widetilde{\mathcal{L}}v),\ u_0\in H^{0,1} \\
h^2G_h^w(\gamma(\frac{x+F'(\xi)}{\sqrt{h}})(\widetilde{\mathcal{L}}^2v),\ u_0\in H^{0,2}
\end{cases}\notag
\end{eqnarray}
where $\Gamma_{-3}(\xi)=\xi \gamma(\xi)$ satisfying $|\partial^\alpha \Gamma_{-3}(\xi)|
\le C_\alpha <\xi>^{-1-\alpha}$ for any $\alpha\in \mathbb{N}$.
The above identities and an application of Proposition \ref{P2.6} yields the desired estimates in Lemma \ref{l5}.
\end{proof}
\begin{lem}\label{l4.5}
If $u_0\in H^{0,1}$, then for all $t\ge1$,
\begin{equation}
\|G^w_h(1-\Gamma)(|v|^\alpha v)\|_{L^\infty}\le C\|v\|_{L^\infty }^\alpha \|\widetilde{\mathcal{L} }v\|_{L^2}t^{-1/4},\notag
\end{equation}
\begin{equation}
\|G^w_h(1-\Gamma)(|v|^\alpha v)\|_{L^2}\le C\|v\|_{L^\infty }^\alpha \|\widetilde{\mathcal{L}}v\|_{L^2}t^{-1/2}.\notag
\end{equation}
Furthermore, if $u_0\in H^{0,2}$ and $\alpha \ge1$, then for all $t\ge1$,
\begin{equation}
\|G^w_h(1-\Gamma)(|v|^\alpha v)\|_{L^\infty}\le C\|v\|_{L^\infty }^\alpha \|\widetilde{\mathcal{L} }^2v\|_{L^2}t^{-3/4}, \notag
\end{equation}
\begin{equation}
\|G^w_h(1-\Gamma)(|v|^\alpha v)\|_{L^2}\le C\|v\|_{L^\infty }^\alpha\|\widetilde{\mathcal{L} }^2v\|_{L^2}t^{-1}. \notag
\end{equation}
\end{lem}
\begin{proof}
We first claim that
\begin{equation}
\|\widetilde{\mathcal{L} }(|v|^\alpha v)\|_{L^2}\le C\|v\|_{L^\infty }^\alpha \|\widetilde{\mathcal{L} }v\|_{L^2},\label{asd1}
\end{equation}
and for $\alpha \ge1$,
\begin{equation}
\|\widetilde{\mathcal{L} }^2(|v|^\alpha v)\|_{L^2}\le C\|v\|_{L^\infty }^\alpha \|\widetilde{\mathcal{L} }^2v\|_{L^2}.\label{asd2}
\end{equation}
We only give the proof of (\ref{asd2}) as (\ref{asd1}) can be proved in a similar way. Since
\begin{equation}
\mathcal{L} ^2(|u|^\alpha u)=t^{-\frac{\alpha +1}{2}}\widetilde{\mathcal{L} }^2(|v|^\alpha v)(t,\frac{x}{t})\notag
\end{equation}
by the second formula in (\ref{4103}), it follows from Lemma \ref{ll1} and (\ref{z2}) that
\begin{eqnarray}
\|\widetilde{\mathcal{L} }^2(|v|^\alpha v)\|_{L^2}&\lesssim & t^{\alpha /2} \left(\|u\|_{L^\infty }^\alpha \|\mathcal{L} ^2u\|_{L^2}+\|u\|_{L^\infty }^{\alpha -1}\|\mathcal{L} u\|_{L^4}^2 \right)\notag\\
&\lesssim & t^{\alpha /2} \|u\|_{L^\infty } ^\alpha \|\mathcal{L} ^2u\|_{L^2},\notag
\end{eqnarray}
which together with (\ref{4103})--(\ref{3.4-0}) yields the desired estimate (\ref{asd2}).
We now resume the proof of Lemma \ref{l4.5}. Using the same method as that used to derive (\ref{b1}), one obtains
\begin{equation}
\left\|G_h^w(1-\Gamma)(|v|^\alpha v)\right\|_{L^\infty }\le Ch^{1/4}\|\widetilde{\mathcal{L}}(|v|^\alpha v)\|_{L^2}\notag
\end{equation}
\begin{equation}
\left\|G_h^w(1-\Gamma)(|v|^\alpha v)\right\|_{L^2 }\le Ch^{1/2}\|\widetilde{\mathcal{L}}(|v|^\alpha v)\|_{L^2},\notag
\end{equation}
which together with (\ref{asd1}) proves the first part of Lemma \ref{l4.5}.
When $u_0\in H^{0,2}$, we write
\begin{equation}
1-\Gamma(x,\xi)=h\Gamma_{-4}(\frac{x+F'(\xi)}{\sqrt h})(\frac{x+F'(\xi)}{ h})^2,\notag
\end{equation}
where $\Gamma_{-4}(\xi)=\frac{1-\gamma(\xi)}{\xi^2}$, satisfying $|\partial_\xi^\alpha \Gamma_{-4}(\xi)|\le C_\alpha <\xi>^{-\alpha }$ for any $\alpha \in \mathbb{N}$. Then using Lemma \ref{l0}, and the definition of $\widetilde{\mathcal{L}}$ in (\ref{3.4}), one gets
\begin{eqnarray}
&&G_h^w(1-\Gamma)(|v|^\alpha v)= h G_h^w(\Gamma_{-4}(\frac{x+F'(\xi)}{\sqrt h}))\circ \widetilde{\mathcal{L}}^2(|v|^\alpha v).\notag
\end{eqnarray}
An application of Proposition \ref{P2.6} yields, for all $t\ge1$,
\begin{equation}
\left\|G_h^w(1-\Gamma)(|v|^\alpha v)\right\|_{L^\infty }\le Ch^{3/4}\|\widetilde{\mathcal{L} }^2(|v|^\alpha v)\|_{L^2},\notag
\end{equation}
\begin{equation}
\left\|G_h^w(1-\Gamma)(|v|^\alpha v)\right\|_{L^2 }\le C h\|\widetilde{\mathcal{L} }^2(|v|^\alpha v)\|_{L^2}.\notag
\end{equation}
This together with (\ref{asd2}) proves the second part of Lemma \ref{l4.5}.
\end{proof}
Using similar argument as in the proof of Lemma \ref{l4.5}, we obtain the following lemma easily and omit the details.
\begin{lem}\label{l5.6}
If $u_0\in H^{0,1}$, then for all $t\ge1$,
\begin{equation}
\||v|^\alpha v-|v_\Lambda|^\alpha v_\Lambda\|_{L^\infty}\le C (\left\|v\right\|_{L^\infty }^\alpha+\left\|v_{\Lambda }\right\|_{L^\infty }^\alpha )\|\widetilde{\mathcal{L}}v\|_{L^2}t^{-1/4},\notag
\end{equation}
\begin{equation}
\||v|^\alpha v-|v_\Lambda|^\alpha v_\Lambda\|_{L^2} \le C (\left\|v\right\|_{L^\infty }^\alpha+\left\|v_{\Lambda }\right\|_{L^\infty }^\alpha ) \|\widetilde{\mathcal{L}}v\|_{L^2}t^{-1/2}. \notag
\end{equation}
Furthermore, if $u_0\in H^{0,2}$ and $\alpha \ge1$, then for all $t\ge1$, \begin{equation}
\||v|^\alpha v-|v_\Lambda|^\alpha v_\Lambda\|_{L^\infty}\le C (\left\|v\right\|_{L^\infty }^\alpha+\left\|v_{\Lambda }\right\|_{L^\infty }^\alpha )\|\widetilde{\mathcal{L} }^2v\|_{L^2}t^{-3/4},\notag
\end{equation}
\begin{equation}
\||v|^\alpha v-|v_\Lambda|^\alpha v_\Lambda\|_{L^2} \le C (\left\|v\right\|_{L^\infty }^\alpha+\left\|v_{\Lambda }\right\|_{L^\infty }^\alpha ) \|\widetilde{\mathcal{L} }^2v\|_{L^2}t^{-1}. \notag
\end{equation}
\end{lem}
\begin{proof}
[\textbf{Proof of Proposition \ref{lp} }]
It follows from (\ref{3234}), (\ref{0451}) and (\ref{4104}) that for all $t\ge1$,
\begin{equation}
\label{521}
\|\widetilde{\mathcal{L} }v(t,\cdot)\|_{L^2}\le \|u_0\|_{H^{0,1}},
\end{equation}
\begin{equation}
\label{4105}
\|\widetilde{\mathcal{L} }^2v(t,\cdot)\|_{L^2}\le C(\left\|u_0\right\|_{H^{0,2}}+\|u_0\|_{H^{0,2}}^{2\alpha +1})t^{2-\alpha }.
\end{equation}
Proposition \ref{lp} then follows from (\ref{521})-(\ref{4105}) and Lemmas \ref{l4.3}--\ref{l5.6}.
\end{proof}
\subsection{The rough $L^\infty $ estimate for $v_{\Lambda }$}\label{sub2}
The goal of this subsection is to derive the $L^\infty $ estimate for $v_{\Lambda }$ from the ODE (\ref{3.4-000}) and therefore completing the proof of Theorem \ref{T1.2}.
Notice that the estimate of $R_2(v)$ depends on $\|v_{\Lambda }\|_{L^\infty }$ and the initial value is large, we apply the bootstrap argument to derive the decay estimate (\ref{d1}) with $K$ large.
Let $K$ be a sufficiently large constant such that
\begin{eqnarray}
\label{4121}
K>\max \left\{2^{1/2+1/\alpha },\ 1\right\}
\end{eqnarray}
\begin{eqnarray}\label{3201}
\frac{2-\alpha }{2}K^{-\alpha }+C_2\alpha 2^{\alpha +2}K^{-1} < \alpha \lambda _2,
\end{eqnarray}
where $C_1,\ C_2$ are constants depending on $u_0$ that appear in (\ref{e1}), (\ref{e5}) respectively.
We assume that $v$ satisfies a bootstrap hypotheses on $t\in[1,T_1]$:
\begin{eqnarray}\label{bass}
\left\|v_{\Lambda}(t,x)\right\|_{L_x^\infty }\le 2 K t^{1/2-1/\alpha }.
\end{eqnarray}
From Lemma \ref{l4.3-00}, (\ref{3.4-0}) and the local decay estimate (\ref{ldecay}), we see that, for $t\in[1,2]$,
\begin{eqnarray}\label{e1}
\left\|v_{\Lambda }(t,x)\right\|_{L_x^\infty }&\le& \left\|v(t,x)\right\|_{L_x^\infty }+ \left\|v_{\Lambda^c }(t,x)\right\|_{L_x^\infty }\notag\\
&\le&C_1(1+t^{-1/4}) < K t^{1/2-1/\alpha }.
\end{eqnarray}
This implies that $T_1>2$. Moreover, it follows from (\ref{bass}) and Proposition \ref{lp} that $\lambda t^{-\alpha /2} |v_\Lambda|^\alpha v_\Lambda+ t^{-\alpha /2} \left(R_1(v)+R_2(v)\right)$ is integrable on $(1,T_1 )$; so that $v_{\Lambda}(t,x)\in $ $C((1,T_1),$ $L^\infty )$ by the equation (\ref{4.34}).
The following lemma is crucial to close the bootstrap hypotheses (\ref{bass}).
\begin{lem}\label{l4.8}
Under the assumptions (\ref{bass}) and $4/3<\alpha <2$, we have that, for all $t\in (1,T_1)$,
\begin{equation}
\left\|v_{\Lambda }(t,x)\right\|_{L_x^\infty }\le K t^{1/2-1/\alpha }.\notag
\end{equation}
\end{lem}
\begin{proof}The proof is inspired by Lemma 2.3 of \cite{Kita}. We prove it by contradiction argument. Assume there exists some $(t_0,\xi_0)\in(1,T_1)\times \mathbb{R}$ such that $\left|v_{\Lambda }(t_0,\xi_0)\right|>Kt_0^{1/2-1/\alpha }$.
From (\ref{e1}) and the continuity of $v_{\Lambda }(t,\xi_0)$, we can find $t_*\in(1,t_0)$ such that
\begin{equation}
\label{492}
\left|v_{\Lambda }(t,\xi_0)\right|>Kt^{1/2-1/\alpha },\qquad \text{holds for all } t_*<t\le t_0
\end{equation}
and furthermore $\left|v_{\Lambda }(t_*,\xi_0)\right|=Kt_*^{1/2-1/\alpha }$. Multiplying $\left|v_{\Lambda }(t,\xi_0)\right|^{-(\alpha +2)}\overline{v_{\Lambda }(t,\xi_0)}$ on both hand sides of (\ref{4.34}) and taking the imaginary part, we obtain
\begin{equation}\label{e2}
-\frac{1}{\alpha }\frac{d}{dt}\left|v_{\Lambda }\right|^{-\alpha }
=- \lambda_2 t^{-\alpha /2}-\text{Im} \left(t^{-\alpha /2}\left(R_1(v)+R_2(v)\right)\overline{v_{\Lambda }}\right)\left|v_{\Lambda }\right|^{-(\alpha +2)}.
\end{equation}
On the other hand, from Proposition \ref{lp}, we have
\begin{eqnarray}\label{e4}
\left\|R_1(v)+R_2(v)\right\|_{L^\infty }&\le& C\left(t^{-5/4+\alpha /2}+(\left\|v_{\Lambda}\right\|_{L^\infty }^\alpha +\left\|v_{\Lambda^c}\right\|_{L^\infty }^\alpha)t^{-1/4}\right)\notag\\
&\le& C2^{\alpha +2}K^\alpha t^{-5/4+\alpha /2},
\end{eqnarray}
for all $t\in (t_*,t_0)$, where we used (\ref{4121}), Lemma \ref{l4.3-00} and the bootstrap hypotheses (\ref{bass}) to bound $\|v_{\Lambda }\|_{L^\infty }^\alpha +\|v_{\Lambda ^c}\|_{L^\infty }^\alpha $. It then follows from (\ref{492})--(\ref{e4}) that there exists $C_2>0$ such that for $t_*<t<t_0$
\begin{eqnarray} \label{e5}
-\frac{1}{\alpha }\frac{d}{dt}\left|v_{\Lambda }(t,\xi_0)\right|^{-\alpha }\le -\lambda _2t^{-\alpha /2}+C_22^{\alpha +2}K^{-1} t^{-3/4-\alpha /2+1/\alpha }.
\end{eqnarray}
Integrating (\ref{e5}) from $t_*$ to $t$, we get
\begin{eqnarray}
&&\left|v_{\Lambda }(t,\xi_0)\right|^{-\alpha }-\left|v_{\Lambda }(t_*,\xi_0)\right|^{-\alpha }\notag\\
&\ge&\frac{2\alpha \lambda _2}{2-\alpha }\left(t^{1-\alpha /2}-t_*^{1-\alpha /2}\right)-C_2\alpha 2^{\alpha +2}K^{-1} \int_{t^*}^{t}s^{-3/4-\alpha /2+1/\alpha } \mathrm{d}s .\notag
\end{eqnarray}
This inequality together with $\left|v_{\Lambda }(t_*,\xi_0)\right|=Kt_*^{1/2-1/\alpha }$ implies that
\begin{eqnarray}
\left(t^{1/\alpha -1/2}\left|v_{\Lambda }(t,\xi_0)\right|\right)^{-\alpha }
&\ge& \left(\frac{t_*}{t}\right)^{1-\alpha /2}K^{-\alpha} +\frac{2\alpha \lambda _2}{2-\alpha }\left(1-\left(\frac{t_*}{t}\right)^{1-\alpha /2}\right)\notag\\
&&-C_2\alpha 2^{\alpha +2}K^{-1} t^{-1+\alpha /2}\int_{t^*}^{t}s^{-3/4-\alpha /2+1/\alpha } \mathrm{d}s\notag\\
&=:&f(t).\notag
\end{eqnarray}
Note that $f(t_*)=K^{-\alpha }$ and $f(t)$ is monotone increasing around $t=t_*$. Indeed, by (\ref{3201})
\begin{equation}
f'(t_*)=\left(\frac{\alpha -2}{2}K^{-\alpha }+\alpha \lambda _2-C_2\alpha 2^{\alpha +2}K^{-1} t_*^{-3/4+1/\alpha }\right)t_*^{-1}>0. \notag
\end{equation}
Thus, if $t$ is slightly larger than $t_*$, then $\left(t^{1/\alpha -1/2}\left|v_{\Lambda }(t,\xi_0)\right|\right)^{-\alpha }>K^{-\alpha }$, which contradicts (\ref{492}), from which Lemma \ref{l4.8} follows.
\end{proof}
\begin{proof}
[\textbf{Proof of Theorem \ref{T1.2}}]
Lemma \ref{l4.8} and a standard continuation argument imply $T_1=\infty $ and that for all $t\ge1$,
\begin{equation}
\label{d1}
\left\|v_{\Lambda}(t,x)\right\|_{L_x^\infty }\le 2 K t^{1/2-1/\alpha }.
\end{equation}
This together with (\ref{3.4-0}) and Lemma \ref{l4.3-00} yields $\|u(t,x)\|_{L^\infty _x}\lesssim t^{-1/\alpha }$, from which Theorem \ref{T1.2} follows.
\end{proof}
\section{The proof of Theorem \ref{T1.3}} \label{S6}
In this section, we prove Theorem \ref{T1.3}. We only consider the nontrivial case $u_0\neq0$. To establish the asymptotic formula for the solutions for a wider range of $\alpha $, we first need to refine the $L^\infty $ estimate of $v_{\Lambda}$.
\subsection{The refined $L^\infty $ estimate for $v_{\Lambda }$}
In this subsection, we show how to obtain the refined $L^\infty $ estimate for $v_{\Lambda }$ (Lemma \ref{l5.1}). This will be essential in proving the large time asymptotics of the solutions in Subsection \ref{sub3}.
Substituting the rough $L^\infty $ bound (\ref{d1}) of $v_{\Lambda }$ into Proposition \ref{lp}, we can rewrite the equation (\ref{4.34}) as
\begin{equation}
D_tv_\Lambda=w(x)v_{\Lambda }+\lambda t^{-\alpha /2} |v_\Lambda|^\alpha v_\Lambda+ t^{-\alpha /2} R(v), \label{4.340}
\end{equation}
where $R(v)$ satisfies, for all $t\ge1$, \\
(a) $u_0\in H^{0,1}$:
\begin{equation}
\|R(v)\|_{L^\infty}\leq Ct^{-\frac{5}{4}+\alpha /2},\label{r2}
\end{equation}
\begin{equation}
\|R(v)\|_{L^2}\leq Ct^{-3/2+\alpha /2}.\label{r1}
\end{equation}
(b) $u_0\in H^{0,2}$:
\begin{equation}
\|R(v)\|_{L^\infty}\leq Ct^{1/4-\alpha/2 },\label{r3}
\end{equation}
\begin{equation}
\|R(v)\|_{L^2}\leq Ct^{-\alpha/2 }.\label{r4}
\end{equation}
For $0<\varepsilon<2\alpha \lambda _2$, we define
\begin{eqnarray}
\begin{cases}
h_1(\varepsilon ):=\frac{3}{4}-\frac{\alpha}{2} +\lambda _2\frac{2-\alpha }{2\alpha\lambda _2-\varepsilon },\ \text{when }u_0\in H^{0,1}, \frac{1+\sqrt{33}}{4}<\alpha <2,\\
h_2(\varepsilon ):=\frac{9}{4}-\frac{3\alpha}{2} +\lambda _2\frac{2-\alpha }{2\alpha\lambda _2-\varepsilon },\ \text{when }u_0\in H^{0,2}, \frac{7+\sqrt{145}}{12}<\alpha <2.
\end{cases}\notag
\end{eqnarray}
Then by direct computation, we have
\begin{eqnarray}
\begin{cases}
h_1(0)=\frac{3}{4}-\frac{\alpha}{2} +\frac{2-\alpha }{2\alpha }<0,\ \text{when }u_0\in H^{0,1}, \frac{1+\sqrt{33}}{4}<\alpha <2,\\
h_2(0)=\frac{9}{4}-\frac{3\alpha}{2} +\frac{2-\alpha }{2\alpha }<0, \ \text{when }u_0\in H^{0,2}, \frac{7+\sqrt{145}}{12}<\alpha <2.
\end{cases}
\end{eqnarray}
By the continuity of $h_1,\ h_2$, we can find $0<\varepsilon _1<2\alpha \lambda _2$ such that for any $0<\varepsilon _0<\varepsilon _1$
\begin{equation}
\frac{3}{4}-\frac{\alpha}{2} +\lambda _2\frac{2-\alpha }{2\alpha \lambda _2-\varepsilon_0 }<0,\ \text{when }u_0\in H^{0,1}, \frac{1+\sqrt{33}}{4}<\alpha <2,\label{3221}
\end{equation}
\begin{equation}
\frac{9}{4}-\frac{3\alpha}{2} +\lambda _2\frac{2-\alpha }{2\alpha \lambda _2-\varepsilon_0 }<0,\ \text{when }u_0\in H^{0,2}, \frac{7+\sqrt{145}}{12}<\alpha <2.\label{4241}
\end{equation}
For $0<\varepsilon _0<\varepsilon _1$, we define
\begin{eqnarray}\label{3211}
K_0=\left(\frac{2-\alpha }{2\alpha \lambda _2-\varepsilon_0 }\right)^{1/\alpha },\qquad T_0=\max \left\{\left(\frac{2C_3\alpha }{\varepsilon _0K_0^{\alpha +1} }\right)^{\frac{4\alpha }{3\alpha -4}},\ e\right\} ,
\end{eqnarray}
where $ C_3>0$ is the constant in (\ref{e3}) that depends on $u_0$.
The following refined $L^\infty $ estimate for $v_{\Lambda }$ is true:
\begin{lem}\label{l5.1}
Assume $u_0\in H^{0,1},\ \frac{1+\sqrt{33}}{4}<\alpha <2$ or $u_0\in H^{0,2},\ \frac{7+\sqrt{145}}{12}<\alpha <2$. There exists $T^*(\varepsilon _0)>T_0$, such that for all $t>T^*(\varepsilon _0)$
\begin{eqnarray}
\left\|v_{\Lambda }(t,x)\right\|_{L_x^\infty }\le K_0t^{1/2-1/\alpha }.\notag
\end{eqnarray}
\end{lem}
\begin{proof}
In what follows, we prove Lemma \ref{l5.1} in the case $u_0\in H^{0,1},\ \frac{1+\sqrt{33}}{4}<\alpha <2$ by the contradiction argument. The case $u_0\in H^{0,2}$, $\frac{7+\sqrt{145}}{12}<\alpha <2$ follows from a similar argument, but using (\ref{r3})--(\ref{r4}), (\ref{4241}) instead of (\ref{r2})--(\ref{r1}), (\ref{3221}).
Assume by contradiction that there exist $\{(t_n,\xi_n)\}_{n=1}^\infty \subset (T_0,\infty )\times \mathbb{R}$ with $t_n$ monotone increases to $\infty $ such that
\begin{equation}\label{4111}
\left|v_{\Lambda }(t_n,\xi_n)\right|>K_0t_n^{1/2-1/\alpha },\qquad \text{for all }n\ge1.
\end{equation}
We first claim that, for every fixed $n\ge1$, we have
\begin{eqnarray}\label{3212}
|v_{\Lambda }(t,\xi_n)|>
K_0t^{1/2-1/\alpha }, \qquad \text{for all } t\in (T_0,t_n).
\end{eqnarray}
In fact, if this claim is not true, there would exist some $t^*_n\in(T_0,t_n)$ such that
\begin{equation}\label{4112}
\left|v_{\Lambda }(t,\xi_n)\right|>K_0t^{1/2-1/\alpha }, \qquad \text{holds for all } t\in (t^*_n,t_n),
\end{equation}
and furthermore $\left|v_{\Lambda }(t^*_n,\xi_n)\right|=K_0(t^*_n)^{1/2-1/\alpha }$. Multiplying $\left|v_{\Lambda }(t,\xi_n)\right|^{-(\alpha +2)}\overline{v_{\Lambda }(t,\xi_n)}$ on both hand sides of (\ref{4.340}) and taking the imaginary part, we obtain, for all $t\in (t_n^*,t_n)$,
\begin{eqnarray}
-\frac{1}{\alpha }\frac{d}{dt}\left|v_{\Lambda }(t,\xi_n)\right|^{-\alpha }&=&- \lambda_2 t^{-\alpha /2}-\text{Im} \left(t^{-\alpha /2}R(v)\overline{v_{\Lambda }}\right)\left|v_{\Lambda }\right|^{-(\alpha +2)}.\label{13}
\end{eqnarray}
Using (\ref{r2}) and (\ref{4112}) to bound the third term in (\ref{13}), we deduce that there exists $C_3>0$ such that for all $t\in (t_n^*,t_n)$,
\begin{equation}
-\frac{1}{\alpha }\frac{d}{dt}\left|v_{\Lambda }(t,\xi_n)\right|^{-\alpha }\le -\lambda _2 t^{-\alpha /2}+C_3 K_0^{-(\alpha +1)}t^{-3/4-\alpha /2+1/\alpha }.\label{e3}
\end{equation}
Integrating the above inequality from $t_n^*$ to $t$ and using $\left|v_{\Lambda }(t_n^*,\xi_0)\right|=K_0(t_n^*)^{1/2-1/\alpha }$, we obtain
\begin{eqnarray}
\left(t^{1/\alpha -1/2}\left|v_{\Lambda }(t,\xi_n)\right|\right)^{-\alpha }&\ge& \left(\frac{t_n^*}{t}\right)^{1-\alpha /2}K_0^{-\alpha} +\frac{2\alpha \lambda _2}{2-\alpha }\left(1-\left(\frac{t_n^*}{t}\right)^{1-\alpha /2}\right)\notag\\
&&-C_3 \alpha K_0^{-(\alpha +1)}t^{-1 +\alpha /2 }\int_{t^*}^{t}s^{-3/4-\alpha /2+1/\alpha } \mathrm{d}s\notag\\
&=:&g(t).\notag
\end{eqnarray}
We see that $g(t_n^*)=K_0^{-\alpha }$ and $g(t)$ is monotone increasing around $t=t_n^*$. Indeed,
\begin{eqnarray}
g'(t_n^*)=\left(\frac{\alpha -2}{2}K_0^{-\alpha }+\alpha \lambda _2-C_3 \alpha K_0^{-(\alpha +1)}(t_n^*)^{-3/4+1/\alpha }\right)(t_n^*)^{-1}>0,\notag
\end{eqnarray}
where we used (\ref{3211}).
Thus, if $t$ is slightly lager than $t_n^*$, then $\left|t^{1/\alpha -1/2}v_{\Lambda}(t,\xi_n)\right|^{-\alpha }>K_0^{-\alpha }$, which contradicts (\ref{4112}). Thus we finish the proof of Claim (\ref{3212}).
In what follows, we use Claim (\ref{3212}) to derive a contradiction to (\ref{4111}). Notice that (\ref{e3}) holds for all $t\in (T_0,t_n)$ by a similar argument used before, but using Claim (\ref{3212}) instead of (\ref{4112}). Integrating (\ref{e3}) from $T_0$ to $t_n$, using (\ref{r2}) and Claim (\ref{3212}), we have, for every $n\ge1$,
\begin{eqnarray}\label{3261}
\left(t_n^{1/\alpha -1/2}\left|v_{\Lambda }(t_n,\xi_n)\right|\right)^{-\alpha }
&\ge& \frac{|v_\Lambda (T_0,\xi_n)|^{-\alpha }}{t_n^{1-\alpha /2}} +\frac{2\alpha \lambda _2}{2-\alpha }\left(1-\left(\frac{T_0}{t_n}\right)^{1-\alpha /2}\right)\notag\\
&&-C_3\alpha K_0^{-(\alpha +1)}t_n^{-1 +\alpha /2 }\int_{T_0}^{t_n}s^{-3/4-\alpha /2+1/\alpha } \mathrm{d}s.
\end{eqnarray}
Since $K_0^{-\alpha }\ge (t_n^{1/\alpha -1/2}\left|v_{\Lambda }(t_n,\xi_n)\right|)^{-\alpha }$ by (\ref{4111}), and $\frac{|v_{\Lambda}(T_0,\xi_n)|^{-\alpha }}{t_n^{1-\alpha /2}}\rightarrow 0$ as $n\rightarrow \infty $ by Claim (\ref{3212}), we can let $n\rightarrow \infty $ in (\ref{3261}) and obtain
\begin{eqnarray}
K_0^{-\alpha }\ge \frac{2\alpha \lambda _2}{2-\alpha }.\notag
\end{eqnarray}
This contradicts the definition of $K_0$ in (\ref{3211}), and thus completing the proof of Lemma \ref{l5.1}.
\end{proof}
\subsection{The proof of Theorem \ref{T1.3}}\label{sub3}
We are now in a position to prove Theorem \ref{T1.3}. We give only the proof for the case $u_0\in H^{0,1}$, $ \frac{1+\sqrt{33}}{4}<\alpha <2$. The case $u_0\in H^{0,2}$, $\frac{7+\sqrt{145}}{12}<\alpha <2$ follows from a similar argument, but using (\ref{r3})--(\ref{r4}), (\ref{4241}) instead of (\ref{r2})--(\ref{r1}), (\ref{3221}). The proof is inspired by Section 5 of \cite{CPDE}.
\begin{proof}
[\textbf{Proof of part (a)}]
Assume $T^*(\varepsilon _0)$ is the constant in Lemma \ref{l5.1}, $\Phi(t,x)$, $ K(t,x)$, $\psi_+(x)$, $S(t,x)$ are the functions defined in (\ref{eq1})--(\ref{eq2}), respectively.
From the definition of $\Phi(t,x)$ in (\ref{eq1}) and Lemma \ref{l5.1}, we have
\begin{eqnarray}\label{3224}
\left\|\Phi(t,x)\right\|_{L_x^\infty }&\le& \int_{1}^{T^*(\varepsilon _0)}s^{-\alpha /2}\|v_{\Lambda }(s,x)\|_{L^\infty _x}^\alpha \mathrm{d}s+ K_0^\alpha \int_{T^*(\varepsilon _0)}^{t}s^{-1} \mathrm{d}s\notag\\
&\le& CT^*(\varepsilon _0)+ K_0^\alpha \log t,\label{3195}
\end{eqnarray}
for all $t\ge1$, where we used (\ref{d1}) to estimate the first integral.
Set
\begin{equation}\label{4107}
z(t,x)=v_\Lambda(t,x) e^{-i(w(x)t+\lambda \Phi(t,x))}, \qquad t>1.
\end{equation}
From the equation (\ref{4.340}) and (\ref{4107}), we have that for $t>1$,
\begin{eqnarray}
\partial_t z(t,x)=\frac{iR(v)}{t^{\alpha /2}}e^{-i(w(x)t+\lambda\Phi(t,x))};\notag
\end{eqnarray}
so that for all $t_2>t_1>1$,
\begin{eqnarray}
z(t_2,x)-z(t_1,x)=i\int_{t_1}^{t_2}s^{-\alpha /2}R(v)e^{-i(w(x)s+\lambda\Phi(s,x))} \mathrm{d}s. \label{7}
\end{eqnarray}
Since $-1/4+\lambda _2K_0^\alpha <0$ by (\ref{3221}) and (\ref{3211}), it follows from (\ref{r2})--(\ref{r1}), (\ref{3195}) and (\ref{7}) that
\begin{eqnarray}\label{3303}
\left\|z(t_2,x)-z(t_1,x)\right\|_{L_x^\infty \cap L_x^2}&\lesssim& \int_{t_1}^{t_2}s^{-\alpha /2}\left\|R(v)\right\| _{ L_x^\infty \cap L_x^2}e^{\lambda _2\left\|\Phi(s,x)\right\|_{L_x^\infty }}\mathrm{d}s \notag\\
&\lesssim_{\varepsilon _0} & \int_{t_1}^{t_2}s^{-5/4+\lambda _2K_0^\alpha } \mathrm{d}s\notag\\
&\lesssim_{\varepsilon _0} & t_1^{-1/4+\lambda _2K_0^\alpha },
\end{eqnarray}
holds for any $t_2>t_1>1$.
Thus there exists $z_+(x)\in L_x^2\cap L_x^\infty $ such that
\begin{eqnarray}\label{3196}
\left\|z(t,x)-z_+(x)\right\|_{L_x^2\cap L_x^\infty }\lesssim_{\varepsilon _0} t^{-1/4+\lambda _2K_0^\alpha }.\label{1.12}
\end{eqnarray}
This finishes the proof of part (a).
\end{proof}
\begin{proof}
[\textbf{Proof of part (b)}] The first step is to derive the asymptotic formula (\ref{3222}) for $\Phi(t,x)$. Note that
\begin{eqnarray}
\partial_t \Phi(t,x)=t^{-\alpha /2}\left|v_\Lambda(t,x) \right|^\alpha =t^{-\alpha /2}\left|z(t,x)\right|^\alpha e^{-\alpha\lambda _2 \Phi(t,x)}\notag
\end{eqnarray}
for all $t>1$ by the definition of $z(t,x)$ in (\ref{4107}); so that
\begin{eqnarray}
\partial_t e^{\alpha \lambda _2\Phi(t,x)}=\alpha \lambda _2t^{-\alpha /2}\left|z(t,x)\right|^{\alpha }.\notag
\end{eqnarray}
Integrating the above equation from $1$ to $t$, we get
\begin{eqnarray}\label{4}
e^{\alpha \lambda _2\Phi(t,x)}=1+\alpha \lambda _2\int_{1}^{t}s^{-\alpha /2}\left|z(s,x)\right| ^{\alpha }\mathrm{d}s.
\end{eqnarray}
From (\ref{3301}), (\ref{a2}) and (\ref{4}), we have
\begin{eqnarray}\label{3197}
e^{\alpha \lambda _2\Phi(t,x)}-K(t,x)-\psi_+(x)=- \alpha \lambda_2 \int_{t}^{\infty }s^{-\alpha /2}\left(\left|z(s,x)\right|^\alpha -\left|z_+(x)\right|^\alpha \right) \mathrm{d}s.
\end{eqnarray}
Since $\left|\left|u\right|^\alpha -\left|v\right|^\alpha \right|\lesssim (\left|u\right|^{\alpha -1}+\left|v\right|^{\alpha -1})\left|u-v\right|$, and $z(s,x),\ z_+(x)\in L_x^\infty $, we deduce from (\ref{3196}) and (\ref{3197}) that, for all $t\ge 1$,
\begin{eqnarray}\label{3222}
\left\|e^{\alpha \lambda _2\Phi(t,x)}-K(t,x)-\psi_+(x)\right\|_{L_x^\infty \cap L_x^2}\lesssim_{\varepsilon _0} \int_{t}^{\infty }s^{-1/4-\alpha/2 +\lambda _2K_0^\alpha } \mathrm{d}s\lesssim t^{-\beta},
\end{eqnarray}
where $\beta=-(3/4-\alpha/2 +\lambda _2K_0^\alpha )>0$ by (\ref{3221}) and (\ref{3211}).
We now prove the asymptotic formula (\ref{3222}). Since $|e^{iw(x)t+i\lambda \Phi(t,x)}|=e^{-\lambda _2\Phi(t,x)}\le1$, we have
\begin{eqnarray}\label{3304}
&&\|e^{i(w(x)t+\lambda S(t,x))}z_+(x)-v_{\Lambda }(t,x)\|_{L_x^\infty\cap L_x^2 }\notag\\
&\le& \|e^{iw(x)t}(e^{i\lambda S(t,x)}-e^{i\lambda \Phi(t,x)})z_+\|_{L_x^\infty\cap L_x^2 }+\|e^{iw(x)t}e^{i\lambda \Phi(t,x)}z_+-v_{\Lambda }(t,x)\|_{L_x^\infty\cap L_x^2 }\notag\\
&\le&\|e^{i\lambda _1S(t,x)}(e^{-\lambda _2S(t,x)}-e^{-\lambda _2\Phi(t,x)})z_+(x)\|_{L_x^\infty\cap L_x^2 }\notag\\
&&+\|(e^{i\lambda _1S(t,x)}-e^{i\lambda _1\Phi(t,x)})e^{-\lambda _2\Phi(t,x)}z_+(x)\|_{L_x^\infty \cap L_x^2}\notag\\
&&\qquad +\|z_+(x)-z(t,x)\|_{L_x^\infty\cap L_x^2 }.
\end{eqnarray}
Note that $K(t,x)+\psi_+(x)\ge1/2$ for $t$ sufficiently large by (\ref{3222}); so that
\begin{eqnarray}\label{3225}
&&\|e^{i\lambda _1S(t,x)}(e^{-\lambda _2S(t,x)}-e^{-\lambda _2\Phi(t,x)})z_+(x)\|_{L_x^\infty\cap L_x^2 }\notag\\
&=& \|(K(t,x)+\psi_+(x))^{-1/\alpha }-e^{-\lambda _2\Phi(t,x)}\|_{L_x^\infty\cap L_x^2}\notag\\
&\lesssim &\|e^{\lambda _2\Phi(t,x)}-(K(t,x)+\psi_+(x))^{1/\alpha }\| _{L_x^\infty \cap L_x^2}\notag\\
&\lesssim & \|e^{\alpha \lambda _2\Phi(t,x)}-(K(t,x)+\psi_+(x))\|^{1/\alpha } _{L_x^\infty\cap L_x^2 }\lesssim_{\varepsilon _0} t^{-\beta /\alpha }.
\end{eqnarray}
Similarly, we have
\begin{eqnarray}\label{3305}
&&\|(e^{i\lambda _1S(t,x)}-e^{i\lambda _1\Phi(t,x)})e^{-\lambda _2\Phi(t,x)}z_+(x)\|_{L_x^\infty \cap L_x^2}\notag\\
&\lesssim & \|S(t,x)-\Phi(t,x)\|_{L_x^\infty \cap L_x^2}\notag\\
&\lesssim & \|e^{\lambda _2S(t,x)}-e^{\lambda _2\Phi(t,x)}\|_{L_x^\infty \cap L_x^2}\notag\\
&= & \|e^{\lambda _2\Phi(t,x)}-(K(t,x)+\psi_+(x))^{1/\alpha }\| _{L_x^\infty \cap L_x^2}
\lesssim_{\varepsilon _0} t^{-\beta /\alpha }.
\end{eqnarray}
Substituting (\ref{3303}), (\ref{3225})--(\ref{3305}) into (\ref{3304}), we get
\begin{eqnarray}\label{3306}
\|e^{i(w(x)t+\lambda S(t,x))}z_+(x)-v_{\Lambda }(t,x)\|_{L_x^\infty\cap L_x^2 }\lesssim_{\varepsilon _0} t^{-\gamma},
\end{eqnarray}
for $t$ sufficiently large, where $0<\gamma:=\min \{1/4-\lambda _2K_0^\alpha, \ \beta/\alpha \}<1/4$. It then follows from Lemma \ref{l4.3-00}, (\ref{4.2}) and (\ref{3306}) that the asymptotic formula (\ref{3302}) holds.
\end{proof}
\begin{proof}
[\textbf{Proof of part (c)}] From the asymptotic formula (\ref{3302}), we have
\begin{eqnarray}
&&e^{-iF(D)t}e^{-i\lambda S(t,\frac{x}{t})}u(t,x)\notag\\
&=&e^{-iF(D)t}\frac{1}{\sqrt t}z_+(\frac{x}{t})e^{iw(\frac{x}{t})t}+O_{L^2}(t^{-\gamma})\notag\\
&=& \frac{1}{2\pi} \iint e^{i(x-y)\xi}e^{-iF(\xi)t}\frac{1}{\sqrt t}z_+(\frac{y}{t})e^{iw(\frac{y}{t})t}dyd\xi+O_{L^2}(t^{-\gamma})\notag\\
&=& \frac{\sqrt t}{2\pi} \iint e^{ix\xi-it(y\xi+F(\xi))}z_+(y)e^{itw(y)}dyd\xi+O_{L^2}(t^{-\gamma})\notag\\
&=& \frac{\sqrt t}{2\pi }\iint e^{ix\xi-itc_2(\xi+\frac{y+c_1}{2c_2})^2}z_+(y)dyd\xi+O_{L^2}(t^{-\gamma})\label{4252}
\end{eqnarray}
as $t\rightarrow \infty $, where we use $y\xi+F(\xi)=w(y)+c_2(\xi+\frac{y+c_1}{2c_2})^2$ in the last step.
Moreover, making a change of variables and then using the dominated convergence theorem, we obtain
\begin{eqnarray}
&&\lim_{t\rightarrow \infty }\frac{\sqrt t}{2\pi }\iint e^{ix\xi-itc_2(\xi+\frac{y+c_1}{2c_2})^2}z_+(y)dyd\xi\notag\\
&=& \lim_{t\rightarrow \infty } \frac{1}{2\pi} \iint e^{ix(\frac{\zeta}{\sqrt t}-\frac{y+c_1}{2c_2})}e^{-ic_2\zeta^2}z_+(y)dyd\zeta \notag\\
&=& \frac{1}{2\pi} \iint e^{-ix\frac{y+c_1}{2c_2}}e^{-ic_2\zeta^2}z_+(y)dyd\zeta \notag\\
&=&\frac{1}{\sqrt {4\pi c_2}}e^{-i\frac{\pi}{4}}e^{-i\frac{c_1x}{2c_2}} (\mathcal{F} z_+)(\frac{x}{2c_2})\qquad \text{ in } L^2.\label{4253}
\end{eqnarray}
The modified scattering formula (\ref{123}) is now an immediate consequence of (\ref{4252}) and (\ref{4253}).
\end{proof}
\begin{proof}
[\textbf{Proof of part (d)}]
From (\ref{4.2}), Lemma \ref{l4.3-00} and Lemma \ref{l5.1}, we have
\begin{eqnarray}
t^{1/\alpha }\|u(t,x)\|_{L^\infty _x}&\le & t^{1/\alpha -1/2} \left(\|v_{\Lambda }\|_{L^\infty }+\|v_{\Lambda ^c}\|_{L^\infty }\right)\notag\\
&\le& \left(\frac{2-\alpha }{2\alpha \lambda _2-\varepsilon _0}\right)^{1/\alpha }+Ct^{1/\alpha -3/4},\ \text{for all } t>T^*(\varepsilon _0).\notag
\end{eqnarray}
Since $\varepsilon _0$ can be chosen to be arbitrarily small, we obtain
\begin{eqnarray}
\limsup_{t\rightarrow \infty }t^{1/\alpha }\|u(t,x)\|_{L^\infty _x}\le\left(\frac{2-\alpha }{2\alpha \lambda _2}\right)^{1/\alpha }.\notag
\end{eqnarray}
Therefore, the proof of part (d) reduces to show that
\begin{eqnarray}
\liminf _{t\rightarrow \infty }t^{1/\alpha }\|u(t,x)\|_{L^\infty _x}\ge \left(\frac{2-\alpha }{2\alpha \lambda _2}\right)^{1/\alpha }. \label{z651}
\end{eqnarray}
Assume for a while that we have proved
\begin{claim}\label{c123}
If the limit function $z_+(x)$ in (\ref{3196}) satisfies $z_+(x)=0$ for a.e. $x\in \mathbb{R}$, then we must have $u_0=0$. \end{claim}
\noindent Then since $u_0\neq 0$, there exists $x_0\in \mathbb{R}$ such that $z_+(x_0)\neq0$. So by (\ref{3306}) and (\ref{652}), we have
\begin{eqnarray}
t^{1/\alpha-1/2 }\|v_{\Lambda }(t,x)\|_{L^\infty _x}&\ge& \frac{t^{1/\alpha-1/2 }|z_+(x_0)|}{(1+\frac{2\alpha \lambda _2}{2-\alpha }|z_+(x_0)|^\alpha (t^{(2-\alpha )/2}-1)+\psi_+(x_0))^{1/\alpha }}\notag\\
&&+O_{\varepsilon_0 }(t^{1/\alpha -1/2-\gamma(\varepsilon _0)}),\label{121}
\end{eqnarray}
where $\gamma(\varepsilon _0)=\min \left\{1/4-\lambda _2 K_0^\alpha ,\ \beta/\alpha \right\} $ with $\beta=-(3/4-\alpha /2+\lambda _2 K_0^\alpha )$ (see (\ref{3222}) and (\ref{3306})). Since
$\alpha >\frac{5+\sqrt {89}}{8}$ and $\lambda _2 K_0^\alpha =\frac{(2-\alpha )\lambda _2}{2\alpha \lambda _2-\varepsilon _0}$, we have by direct calculation
\begin{equation}
\lim _{\varepsilon _0 \rightarrow0}\left(\frac{1}{\alpha }-\frac{1}{2}-\gamma(\varepsilon _0)\right)=\frac{1}{\alpha }-\frac{1}{2}-\min \left\{\frac{1}{4}-\frac{2-\alpha }{2\alpha },\frac{2\alpha ^2-\alpha -4}{4\alpha ^2}\right\} <0.\notag
\end{equation}
Therefore, taking $\varepsilon _0>0$ sufficiently small, we deduce from (\ref{121}) that
\begin{equation}
\liminf_{t\rightarrow \infty }t^{1/\alpha-1/2 }\|v_{\Lambda }(t,x)\|_{L^\infty _x}\ge \left(\frac{2-\alpha }{2\alpha \lambda _2}\right)^{1/\alpha }. \notag
\end{equation}
This together with (\ref{4.2}) and Lemma \ref{l4.3-00} yields the limit (\ref{z651}).
\end{proof}
\begin{proof}[\textbf{Proof of Claim \ref{c123}}]
The key observation is that the solution decays faster when $z_+=0$:
\begin{eqnarray}
\|u(t,x)\|_{L^\infty _x}\lesssim t^{-3/4},\ \|u(t,x)\|_{L^2 _x}\lesssim t^{-1/2}\qquad \text{for }t>T^*(\varepsilon _0).\label{655}
\end{eqnarray}
In fact, since $z_+=0$, it follows from (\ref{4107})--(\ref{7}) that
\begin{eqnarray}
v_{\Lambda }(t,x)=-i \int_{t}^{\infty }e^{iw(x)(t-s)+i\lambda \left(\Phi(t,x)-\Phi(s,x)\right)}s^{-\alpha /2} R(v)(s)\mathrm{d}s.\label{653}
\end{eqnarray}
On the other hand, using (\ref{eq1}) and Lemma \ref{l5.1}, we have, for $s>t>T^*(\varepsilon _0)$
\begin{eqnarray}
\|\Phi(t,x)-\Phi(s,x)\|_{L^\infty _x}\le \int_{t}^{s}\tau^{-\alpha /2}\|v_{\Lambda }(\tau,x)\|_{L^\infty _x}^{\alpha } \mathrm{d}\tau\le K_0^\alpha \log \frac{s}{t}. \label{654}
\end{eqnarray}
Substituting (\ref{654}) to (\ref{653}), and using the $L^\infty $ bound of $R(v)$ in (\ref{r2}), we get
\begin{eqnarray}
\|v_{\Lambda }(t,x)\|_{L^\infty _x}\lesssim \int_{t}^{\infty } \left(\frac{s}{t}\right)^{\lambda _2K_0^\alpha }s^{-\alpha /2}s^{-5/4+\alpha /2}\mathrm{d}s\lesssim t^{-1/4},\ t>T^*(\varepsilon _0). \notag
\end{eqnarray}
Similarly, we have
\begin{eqnarray}
\|v_{\Lambda }(t,x)\|_{L^2 _x}\lesssim t^{-1/2},\ t>T^*(\varepsilon _0). \notag
\end{eqnarray}
The above two inequalities together with (\ref{4.2}) and Lemma \ref{l4.3-00} yield (\ref{655}).
Next, we apply the decay estimates (\ref{655}) to prove that $u_0=0$. Since $z_+=0$, it follows from the equation (\ref{1.1}) and the asymptotic formula (\ref{3302}) that
\begin{eqnarray}
u(t,x)=\lambda \int_{t}^{\infty }e^{iF(D)(t-s)} (|u|^\alpha u)(s)\mathrm{d}s.\notag
\end{eqnarray}
Then applying Strichartz's estimate and H\"older's inequality, we get
\begin{eqnarray}
\|u\|_{L^4([T,\infty ),L_x^\infty) }&\lesssim& \int_{T}^{\infty }\|u(s,x)\|_{L^\infty _x}^\alpha \|u(s,x)\|_{L^2_x} \mathrm{d}s\notag\\
&\lesssim & \|u\|_{L^4([T,\infty ),L_x^\infty) }\left(\int_{T}^{\infty }\left(\|u(s,x)\|_{L^\infty _x}^{\alpha -1}\|u(s,x)\|_{L^2}\right)^{4/3} \mathrm{d}s\right)^{3/4}\notag\\
&\le& C\|u\|_{L^4([T,\infty ),L_x^\infty) } T^{1-3\alpha /4},\notag
\end{eqnarray}
where we use (\ref{655}) in the last step. Since $\alpha >4/3$, we can choose $T>T^*(\varepsilon _0)$ sufficiently large such that $CT^{1-3\alpha /4}\le \frac{1}{2}$; so that $\|u\|_{L^4([T,\infty ),L_x^\infty) }=0$. This together with the uniqueness of the solutions implies $u\equiv0$, from which Claim \ref{c123} follows.
\end{proof}
\section*{Acknowledgements}
This work is
partially supported by NSF of
China under Grants 11771389, 11931010 and 11621101.
|
2,869,038,156,334 | arxiv | \section{Introduction}
The interest of differential geometric techniques in the analysis of systems of ordinary differential equations (ODEs) is undeniable. Lie point symmetries, integrating factors and their generalizations are just several examples of geometric tools which have been successfully applied in the study of systems of ODEs and their related mathematical and physical problems.
In this work, we survey the geometric theory of Jacobi multipliers \cite{Ja44a,Cl09} and non-local symmetries \cite{KKV04,KV89} to study a family of relevant nonlinear oscillators that have been attracting some attention in recent years \cite{BGS11,Ga09,GB11,MR12,CRS04,BEHR08,BEHRR11}. For instance, it was proved that they can be understood as oscillators in manifolds of constant curvature \cite{CRSS04}, they admit compatible bi-Hamiltonian structures, and their properties can also be analyzed by means of coalgebra techniques \cite{BEHR08,BEHRR11}. Some of their properties have also been obtained by means of the so-called $\lambda$-symmetries \cite{MR12}.
First, we use Jacobi multipliers \cite{Ja44a,Cl09} to go over the above-mentioned oscillators from a geometrical point of view. This allows us to obtain some of their constants of motion. We also obtain known and new Lagrangian and Hamiltonian functions for these oscillators. It is worth noting that the new derived Lagrangian functions are of a non-mechanical type, i.e. they do not possess a kinetic term given by a $2$-contravariant tensor field.
Subsequently, we review a `new method' to obtain non-local symmetries developed by Gandarias and coworkers \cite{BGS11,Ga09,GB11}. We show that their procedure is a consequence of the non-local symmetry idea formalised by Krasil'shchik and Vinogradov several years before \cite{V99,Vi89,Ca07}. Despite this, Gandarias and coworkers' applications of this method are still relevant, as they illustrate that certain systems of differential equations with no classical point symmetries can admit non-local symmetries that lead to unveil their properties.
The study of non-local symmetries demands the use of infinite-dimensional jet bundles \cite{Krasilshchik2011}. We illustrate how this geometric approach can easily be applied to study our family of oscillators. Indeed, the calculation of such non-local symmetries in the problem under study is very similar to the case of a finite-dimensional jet bundle $J^p\pi$. It is essentially the geometrical interpretation what differs. Additionally, our techniques provide very simple and relevant examples of finite-dimensional diffieties describing ODEs, which is not the usual approach as they are mainly concerned with infinite-dimensional manifolds describing PDEs.
The use of infinite dimensional jet bundles provides other advantages. Many structures of $J^p\pi$, e.g. the Cartan distribution, become simpler when defined on the infinite-dimensional jet bundle $J^\infty\pi$. Moreover, we can define geometric structures on this latter space that cannot properly be defined on $J^p\pi$, e.g. the total derivative. Moreover, $J^\infty\pi$ possesses a geometric structure richer than that of $J^p\pi$, e.g. it admits Lie symmetries than cannot be described in terms of Lie symmetries defined on $p$-order jet bundles \cite{Ca07}.
Apart from showing the usefulness of infinite-dimensional jet bundles and reviewing Gandarias' results, we devise a new idea to easily determine non-local symmetries for certain systems of higher-order differential equations. As an application, we retrieve in a natural and rigorous way a result given by Gandarias as an ansatz for the oscillators of this work \cite{BGS11}.
The paper goes as follows. Section 2 is devoted to surveying the theory of Jacobi multipliers and its relation with Lagrangians and constants of motion. In Section 3 we apply Jacobi multipliers to study relevant types of oscillators. In Section 4, non-local symmetries of differential equations are briefly presented. We show that the method developed by
Gandarias and coworkers \cite{BGS11,Ga09} reduces to the non-local symmetry concept developed by Krasil'hinski and Vinogradov and we apply this idea to the mentioned oscillators in Section 5. A method to improve the derivation of such non-local symmetries is provided in Section 6 and we summarise our results and comment on our future work in Section 7.
\section{Jacobi multipliers, Lagrangians and constants of motion}
Let $M$ stand for an oriented manifold, i.e. $M$ is equipped with a volume form $\Omega$. Given a vector field $X$ on $M$, we call {\it divergence} of $X$ relative to $\Omega$ the unique function ${\rm div}\, X:M\rightarrow\mathbb{R}$ satisfying \cite{Ja44a,Cl09}:
$$
\mathcal{L}_{X}\Omega=({\rm div} X)\,\Omega\,.
$$
A {\it Jacobi multiplier} for $X$ is a non-vanishing function $\mu:M\rightarrow\mathbb{R}$ satisfying that $\mu( i(X)\Omega)$ is a closed form, or equivalently $\mathcal{L}_X(\mu\,\Omega)=0$. In other words, $\mu$ is such that
$$
{\rm div}(\mu X)=0,\qquad \mu(x)\neq 0,\qquad \forall x\in M\,.
$$
{}From $\mathcal{L}_X(f\,\Omega)=(Xf+f\,\mathrm{div}\,X)\,\Omega$, for every $f\in C^\infty(M)$, we see that a function $\mu$ is a Jacobi multiplier for $X$ if and only if $\mu$ does not vanish and satisfies
\begin{equation}
\mu \,{\rm div}X + X\mu = 0\,.\label{JLMcond}
\end{equation}
A function $I:M\rightarrow \mathbb{R}$ is a
first-integral of $X$, i.e. $XI=0$, if and only if ${\rm d}I$ annihilates the generalized distribution $\mathcal{D}=\{v\in {\rm T}M\,|\, \exists \,p\in M, v=X_p\}$ generated by $X$, i.e. $({\rm d}I)_p(X_p)=0$ at each $p\in M$. Let us restrict ourselves to an open subset $U=\{p\in M: X_p\neq 0\,,\, ({\rm d}I)_p\neq 0\}$ of a two-dimensional manifold $M$. In this case, $i(X)\Omega$ defines a one-dimensional codistribution annihilating $\mathcal{D}$ and hence $X$ admits a Jacobi multiplier $\mu$ such that $\mu\, i(X)\Omega={\rm d}I$. Using that $(i(X)\Omega)\wedge {\rm d}f=-(i(X){\rm d}f)\wedge \Omega=-(Xf)\Omega$, for each $f\in C^\infty(U)$, we find that
${\rm d}I\wedge {\rm d}f=-\mu \,(Xf)\,\Omega$.
This expression also shows that $f$ is a first-integral of $X$ if and only if ${\rm d}I\wedge {\rm d}f=0$, i.e. $f$ is a function of $I$ and $f=\varphi(I)$ for a certain real function $\varphi:\mathbb{R}\rightarrow \mathbb{R}$.
Jacobi multipliers have many applications \cite{Cl09,NL08a,NT08b,NL08c,GN15,BGM14,MR14}. In particular, we are interested in their use to construct Lagrangians and constants of motion for second-order differential equations \cite{NL08b,NT08a,MR09,CGK}. Let us briefly survey this topic.
Assume hereafter that ${\rm T}\mathbb{R}$ is endowed with the volume form $\Omega={\rm d}x\wedge {\rm d}v$. Consider a second-order differential equation
\begin{equation}\label{SODE}
\frac{{\rm d}^ 2x}{{\rm d}t^2}=F\left(x,\frac{{\rm d}x}{{\rm d}t}\right),\qquad x\in\mathbb{R}\,,
\end{equation}
with $F:{\rm T}\mathbb{R}\rightarrow \mathbb{R}$ being an arbitrary function. Adding a new variable $v\equiv {\rm d}x/{\rm d}t$, we see that (\ref{SODE}) can be rewritten as
\begin{equation}\label{FODE}
\left\{\begin{aligned}
\frac{{\rm d}x}{{\rm d}t}&=v\,,\\
\frac{{\rm d}v}{{\rm d}t}&=F\left(x,v\right)\,,
\end{aligned}\right.
\end{equation}
whose particular solutions are integral curves of the vector field on ${\rm T}\mathbb{R}$ given by
\begin{equation}
\Gamma=v\frac{\partial}{\partial x}+F(x,v)\frac{\partial}{\partial v}.\label{aSODEvf}
\end{equation}
A first-integral of $\Gamma$ is usually called a constant of motion for $\Gamma$ or, equivalently, for system (\ref{FODE}). By substituting $v$ by ${\rm d}x/{\rm d}t$, this first-integral gives rise to a constant of motion to (\ref{SODE}).
As $\textrm{div\,} \Gamma=\partial F/\partial v$ in this case, then the Jacobi's multiplier condition (\ref{JLMcond}) amounts to
\begin{equation}
v\pd \mu x+\pd{(\mu\,F)}v=0\,. \label{multcond1}
\end{equation}
The Jacobi multiplier $\mu$ satisfying this condition is also called a Jacobi multiplier
for the second-order differential equation (\ref{SODE}).
\begin{theorem}\label{Main1}
The differential equation determining the solutions of the
Euler--Lagrange equation defined by a regular Lagrangian function $L(x,v)$ possesses, when written in its normal form (\ref{SODE}),
a Jacobi multiplier given by the function
\begin{equation}
\mu=\frac{\partial^2L}{\partial v^2}.\label{multL}
\end{equation}
Conversely, if $\mu$ is a Jacobi multiplier for a second-order differential
equation (\ref{SODE}), then (\ref{SODE}) admits a regular Lagrangian $L(x,v)$ satisfying (\ref{multL}).
\end{theorem}
\begin{proof}
Assume $L$ to be a regular Lagrangian for (\ref{SODE}) and define the non-vanishing function $\mu$ by (\ref{multL}). Note that then $F$ is given by
$$F(x,v)=\frac 1\mu\left(\pd Lx-v\pd{^2L}{x\partial v}\right)\,,
$$
and, using this, we see that
$$
\begin{aligned}
v\pd \mu x+\pd{}v\left(\mu F\right)&=v\pd \mu x+\pd{}v\left(\pd Lx-v\pd{^2L}{x\partial
v}\right)=v\pd{^3L}{v^2\partial x}+\pd{^2L}{x\partial
v}-\pd{^2L}{x\partial v}-v\pd{^3L}{v^2\partial x}=0\,.
\end{aligned}
$$
Therefore, $\mu$ given by (\ref{multL}) satisfies the Jacobi multiplier equation (\ref{multcond1}). Since $L$ is assumed to be regular, then the function
$\mu$ given by (\ref{multL}) does not vanish and becomes a Jacobi multiplier of (\ref{SODE}).
Conversely, if $\mu$ is a Jacobi multiplier
for (\ref{aSODEvf}), then the functions $L$ satisfying (\ref{multL}) are of the form
\begin{equation}
L(x,v)=\int^v {\rm d}v'\int^{v'}\mu(x,\zeta) \,{\rm d}\zeta+ \phi_1(x)\,v+\phi_2(x)\,\label{posL}
\end{equation}
for arbitrary functions $\phi_1,\phi_2:\mathbb{R}\rightarrow \mathbb{R}$. The term $\phi_1(x)\,v$ is a gauge term which
can be fixed equal to zero, i.e. $L$ can be assumed to be of the form
\begin{equation}
L(x,v)=\int^v {\rm d}v'\int^{v'}\mu(x,\zeta) \,{\rm d}\zeta+\phi_2(x)\,.\label{posLsingauge}
\end{equation}
and the function $\phi_2$ can be chosen in a unique way\cite{CGR09}, up to a constant, so that the Euler--Lagrange equation reproduces the equation for the integral curves for the
given vector field (\ref{aSODEvf}). Indeed, using (\ref{posLsingauge}) we see that
$$
\frac{\partial L}{\partial x}=\int^v{\rm d}v'\int^{v'}\frac{\partial \mu}{\partial x}(x,\zeta)\, {\rm d}\zeta+ \frac{{\rm d} \phi_2}{{\rm d} x}(x)\,,
$$ and in order to the Euler--Lagrange equation for the Lagrangian (\ref{posLsingauge}) to give the dynamics we should have:
\begin{equation*}
\int^v\!\!{\rm d}v'\int^{v'}\pd{\mu}x(x,\zeta) \,
{\rm d}\zeta+\frac{{\rm d}\phi_2}{{\rm d}x}(x)=v
\int^v\pd{\mu}x(x,\zeta) \, {\rm d}\zeta+\mu(x,v)F(x,v)
\,.
\end{equation*}
But note that
\begin{multline}
{\displaystyle{\pd{}v}}\left(v{\displaystyle{\int^v}}\,{\displaystyle{\pd{\mu}x}}(x,\zeta) \,
{\rm d}\zeta+\mu(x,v)F(x,v)- {\displaystyle{\int^v{\rm d}v'\int^{v'}}}\,{\displaystyle{\!\!\!\pd{\mu}x}}(x,\zeta){\rm d}\zeta\right)\\
=v{\displaystyle{\pd
\mu x}}(x,v)+F(x,v){\displaystyle{\pd \mu v(x,v)}}+\mu(x,v){\displaystyle{\pd F v(x,v)}}\,,
\end{multline}
which vanishes because of the multiplier condition (\ref{multcond1}). Consequently, the
function $\phi_2$ exists and is uniquely determined, up to a constant, by
\begin{equation}
\label{phidos}
\frac{{\rm d}\phi_2}{{\rm d}x}(x)=v\int^v\,\pd{\mu}x(x,\zeta) \,
{\rm d}\zeta+\mu(x,v)F(x,v)-\int^v{\rm d}v'\int^{v'}\,\pd{\mu}x(x,\zeta)\,{\rm d}\zeta.
\end{equation}
An integration by parts in the double integral leads to
\begin{equation*}
\int^v{\rm d}v'\int^{v'} \pd \mu x(x,\zeta)\,{\rm d}\zeta=
\\v\int^v \pd \mu x(x,\zeta)\,{\rm d}\zeta -\int^vv'\, \pd \mu x(x,v')\,{\rm d}v',
\end{equation*}
that when substituted in (\ref{phidos}) gives
$$
\frac{{\rm d}\phi_2}{{\rm d}x}(x)=\mu(x,v)\, F(x,v)+\int^v\zeta\,\pd{\mu}x(x,\zeta)\,{\rm d}\zeta,
$$
which using the Jacobi multiplier condition (\ref{multcond1}) reduces to
$$
\frac{{\rm d}\phi_2}{{\rm d}x}(x)=\mu(x,v)F(x,v)-\int_{v_0}^v \pd{(\mu\,F)}{\zeta}(x,\zeta)\,{\rm d}\zeta,
$$
and therefore $\phi_2$ is given by:
\begin{equation}
\phi_2(x)=\int^x (\mu\,F)(\zeta,v_0)\,{\rm d}\zeta\,.\label{phidos2}
\end{equation}
Additionally, since $\mu$ is non-vanishing and in view of (\ref{multL}), we obtain that $L$ is regular.
\end{proof}
Another remarkable result concerning the inverse problem is given in the following Proposition \cite{Le81,Lo96a}.
\begin{proposition}\label{LagInt}
If $I$ is a constant of motion for the vector field $\Gamma$
at a point $\xi\in {\rm T}\mathbb{R}$ where $\Gamma_\xi\neq 0$ and $({\rm d}I)_\xi\neq 0$, then
\begin{equation}\label{Lsol}
L(x,v)=v\int^v\frac{I(x,\zeta)}{\zeta^2} {\rm d}\zeta
\end{equation}
is a Lagrangian for the given vector field around a neighborhood of $\xi$.
\end{proposition}
\begin{proof}
Since $\Gamma$ and ${\rm d}I$ do not vanish at $\xi$, there exists around this point a Jacobi multiplier $\mu$ for $\Gamma$ relative to $\Omega={\rm d}x\wedge {\rm d}v$ such that
$$
\mu\, i(\Gamma)\Omega={\rm d}I\ \Longleftrightarrow\
\mu \bigl(v\, {\rm d}v-F(x,v)\, {\rm d}x \bigr) = {\rm d}I.
$$
Therefore,
$$
\mu=\frac{1}{v}\, \pd Iv,\qquad -\mu F=\frac{\partial I}{\partial x}.
$$
In view of Theorem \ref{Main1}, there exists a Lagrangian $L$ such that
$$\pd{^2L}{v^2}=\frac 1v \, \pd Iv,
$$
from where, an integration by parts leads to
$$
\begin{aligned}
\pd Lv&= \int^v\frac 1\zeta \pd I{\zeta}\,{\rm d}\zeta=\frac {I(x,v)}{v}+\int^v\frac{I(x,\zeta)}{\zeta^2}\,{\rm d}\zeta=\pd{}{v}\left( v\int^v\frac{I(x,\zeta)}{\zeta^2}\,{\rm d}\zeta\right).
\end{aligned}
$$
This shows that $L$ must be given by
$$
L(x,v)=v\int^v\frac{I(x,\zeta)}{\zeta^2}{\rm d}\zeta+\phi(x)
$$
for a certain $\phi:x\in \mathbb{R}\mapsto \phi(x)\in \mathbb{R}$. Imposing $L$ to be a Lagrangian for $\Gamma$ and recalling that $\partial I/\partial x=-\mu F$, we obtain
$$
\frac{\partial L}{\partial x}-\frac{\rm d}{{\rm d}t}\frac{\partial L}{\partial v}=0\Rightarrow - v\int^v\frac{[\mu F](x,\zeta)}{\zeta^2}{\rm d}\zeta+\frac{{\rm d}\phi}{{\rm d}x}(x)-\frac{\rm d}{{\rm d}t}\left(\frac {I(x,v)}{v}+\int^v\frac{I(x,\zeta)}{\zeta^2}\,{\rm d}\zeta\right)=0.
$$
Hence,
$$
- v\int^v\frac{[\mu F](x,\zeta)}{\zeta^2}{\rm d}\zeta+\frac{{\rm d}\phi}{{\rm d}x}-\frac{\partial I}{\partial x}-\frac{F}{v}\frac{\partial I}{\partial v}+\frac{F}{v^2}I+v\int^v\frac{[\mu F](x,\zeta)}{\zeta^2}{\rm d}\zeta-\frac{FI}{v^2}=\frac{{\rm d}\phi}{{\rm d}x}=0.
$$
Hence, $\phi$ is an irrelevant constant and we obtain that $L$ is essentially given, up to an also irrelevant gauge term, by (\ref{Lsol}).
\end{proof}
Apart from providing a variational description for second-order differential equations (SODEs) as (\ref{SODE}), Jacobi multipliers can also be employed to derive their
$t$-independent constants of motion, namely first-integrals for the associated $\Gamma$. More specifically, given two Jacobi multipliers $\mu_1$ and $\mu_2$ of the vector field $\Gamma$, the function
$$
\varphi = \frac{\mu_1}{\mu_2}
$$
is a constant of motion for (\ref{FODE}) and, by substituting $v$ by ${\rm d}x/{\rm d}t$, we obtain a $t$-independent constant of motion for (\ref{SODE}). Indeed, as ${\rm div}(\mu_i \Gamma)=\Gamma\mu_i+\mu_i\, {\rm div}\Gamma=0$, for $i=1,2$, it follows
$$
\Gamma\varphi =\frac{\mu_2\Gamma\mu_1-\mu_1\Gamma \mu_2}{\mu_2^2}=0.
$$
Consequently, the non-uniqueness of such a
Lagrangian function, i.e. the existence of alternative Lagrangians
can be used to determine constants of the
motion as it was proved in \cite{CS66} for the one-dimensional case and generalized in \cite{HH81} for the
multidimensional case (see also \cite{CI83} for a geometric approach).
In addition, given a non-vanishing $t$-independent constant of motion $\varphi$ for $\Gamma$, then $\mu_1\varphi$ is a new Jacobi multiplier for $\Gamma$. This shows that given a fixed Jacobi multiplier $\mu_1$ for $\Gamma$, any other Jacobi multiplier
$\mu$ for $\Gamma$ arises as the product of $\mu_1$ times a non-vanishing function $G\in C^\infty(\mathbb{R})$ of a given nontrivial constant of motion $\varphi_1$ for $\Gamma$, i.e.
$\mu=G(\varphi_1)\mu_1$.
\section{Jacobi multipliers and nonlinear oscillators}
Let us now use the above results to analyse the nonlinear oscillators
\begin{equation}\label{NonL1}
\frac{{\rm d}^2x}{{\rm d}t^2}-\frac{kx}{1+kx^2}\left(\frac{{\rm d}x}{{\rm d}t}\right)^2+\frac{\alpha^2x}{1+kx^2}=0,\qquad \alpha\in \mathbb{R},
\end{equation}
and
\begin{equation}\label{NonL2}
\frac{{\rm d}^2x}{{\rm d}t^2}+\frac{kx}{1+kx^2}\left(\frac{{\rm d}x}{{\rm d}t}\right)^2
+ \frac{\alpha^2x}{(1+kx^2)^3}=0, {\qquad} \alpha\in \mathbb{R},
\end{equation}
which have recently been drawing some attention \cite{BGS11,CGK,CRSS04,ML74,BEHR08}. For instance, the Hamiltonian description of the quantum analogues of both systems led to suggest a Lagrangian description for nonlinear oscillators\cite{ML74} . Here, $x\in \mathbb{R}$ when $k\geq0$ but $|x|\neq 1/\sqrt{|k|}$ for $k<0$. For simplicity, we study the bounded motions with $x\in (-1/\sqrt{-k},1/\sqrt{-k })$ when $k<0$. Some generalizations of these results to higher-order dimensions
for (\ref{NonL1}) were devised in \cite{CRSS04} and some of the non-local symmetries for (\ref{NonL1}) and (\ref{NonL2}) were described in \cite{BGS11}.
The second oscillator (\ref{NonL2}) is the one-dimensional case of the Hamiltonian superintegrable system studied in \cite{BEHR08}. Apart from its superintegrability,
this system has attracted some attention due to the fact that it can be investigated through an $\mathfrak{sl}(2,\mathbb{R})$--Poisson coalgebra (cf. \cite{BEHR08}).
The straightforward generalization to $n$-dimensions of both oscillators leads to oscillators of constant and variable curvature \cite{BEHR08}.
\subsection{First nonlinear oscillator }
We can write (\ref{NonL1}) as a first-order system:
\begin{equation}\label{AsoNonL}
\left\{
\begin{aligned}
\frac{{\rm d}x}{{\rm d}t}&=v,\\
\frac{{\rm d}v}{{\rm d}t}&=\frac{(kv^2-\alpha^2)x}{1+kx^2}.
\end{aligned}\right.
\end{equation}
System (\ref{AsoNonL}) describes the integral curves of the vector field
$$
\Gamma=v\frac{\partial}{\partial x}+\frac{x(kv^2-\alpha^2 )}{1+kx^2}\frac{\partial}{\partial v},
$$
and as
$\textrm{div\,}\Gamma = 2kxv/(1+kx^2)$,
its Jacobi multipliers, $\mu:{\rm T}\mathbb{R}\rightarrow \mathbb{R}$, are the non-vanishing solutions of the equation
$$
\textrm{div\,}(\mu \Gamma)=\Gamma\mu+\mu\,\textrm{div\,}\Gamma=0,
$$
that in this case can be written as
\begin{equation}
v\frac{\partial \mu}{\partial x}+\frac{x(kv^2-\alpha^2)}{1+kx^2}\frac{\partial \mu}{\partial v}+\frac{2\mu kxv}{1+kx^2}=0.\label{PDEs}
\end{equation}
This equation can explicitly be solved by the {\it method of characteristics}. This method reduces solving the above PDE to determining the integral curves of a vector field: the so-called {\it characteristic curves}. When characteristics are determined, solutions for the PDE are obtained by gluing them together giving rise to a hypersurface (see \cite{Olver,Me05} for details). The characteristic curves of (\ref{PDEs}) can be described by means of the referred to as {\it characteristic system} \cite{Olver,Me05}, namely
\begin{equation}\label{CharSys}
\frac{{\rm d}x}{v}=\frac{(1+kx^2){\rm d}v}{x(kv^2-\alpha^2)}
=-\frac{(1+kx^2){\rm d}\mu}{2 kxv\mu}.
\end{equation}
Let us solve these equations for $k=0$, i.e. the harmonic oscillator. In this case, we have ${\rm div}\,\Gamma=0$ and
Jacobi multipliers become mere first-integrals of $\Gamma$. The characteristic curves are given by
$$
\mu =\Upsilon_1, \qquad x^2+v^2=\Upsilon_2,
$$
where $\Upsilon_1, \Upsilon_2$ are real constants. This means that Jacobi multipliers are non-vanishing functions of the form
$\mu=\mu(x^2+v^2)$.
By integrating the characteristic equations for $k\neq 0$, we find that the characteristic curves are given by
$$
\mu (1+k x^2)=\Upsilon_1, \qquad\qquad \frac{1+kx^2}{kv^2-\alpha^2}=\Upsilon_2,
$$
where $\Upsilon_1, \Upsilon_2$ are real constants. We know that any surface $(x,y,\mu(x,v))$ obtained by gluing characteristic curves is a solution to (\ref{PDEs}), namely any subset $(x,y,\mu)$ of
points of $\mathbb{R}^3$ satisfying
$$
K({(1+kx^2)}/{(kv^2-\alpha^2)},\mu (1+k x^2))=0,
$$
for a fixed function $K:\mathbb{R}^2\rightarrow \mathbb{R}$, with $\partial_2K\neq 0$ (observe that this amounts to $\mu=\mu(x,y)$). If we impose the boundary conditions $\mu_1(0,v)=1$ and $\mu_2(0,v)=1/(kv^2-\alpha^2)$,
we easily obtain, respectively, the Jacobi multipliers
$$
\mu_1 = \frac{1}{1+kx^2},\qquad\qquad \mu_2=\frac1{kv^2-\alpha^2}.
$$
Both Jacobi multipliers lead to the existence of a constant of motion for $\Gamma$ of the form
\begin{equation}\label{Integral1}
I=\frac{\mu_2}{\mu_1}=\frac{1+kx^2}{kv^2-\alpha^2}.
\end{equation}
Note that, as previously remarked, a constant of motion $I$ for (\ref{SODE}) and a Jacobi multiplier $\mu_1$
enable us to recover all the Jacobi multipliers for $X$ as $\mu=G(I) \mu_1 $ with $G(I)$ being an appropriately non-vanishing function.
Let us now turn to working out a Lagrangian for oscillator (\ref{NonL1}) by using the method of Jacobi multipliers and $\mu_1$ (observe that $\mu_1$ is also a Jacobi multiplier of (\ref{NonL1}) for $k=0$). Recall that this method states that (\ref{NonL1}) admits a Lagrangian $L_1:{\rm T}\mathbb{R}\rightarrow\mathbb{R}$ satisfying
$$
\frac{\partial^2 L_1}{\partial v^2}=\mu_1=\frac{1}{1+kx^2}.
$$
This yields
$$
L_1(x,v) = \frac{1}{2} \frac{v^2}{(1+kx^2)}+\phi_1(x)\,v+\phi_2(x) \,,
$$
for certain functions $\phi_1, \phi_2:\mathbb{R}\rightarrow \mathbb{R}$. We can set the gauge term $\phi_1(x)\,v$ equal to zero while $\phi_2$ is to be determined using (\ref{phidos2}). More specifically, choosing $v_0=0$ in Theorem \ref{Main1}, we obtain that the function $\phi_2 $ is given, up to the addition of a constant, by
$$
\begin{aligned}
\phi_2(x) &=\int_0^x (\mu\,F)(\zeta,0)\,{\rm d}\zeta=\int_0^x\frac{1}{1+k\zeta^2}\,\frac{(-\alpha^2 )\zeta}{1+k\zeta^2}{\rm d}\zeta=\left[\frac{\alpha^2}{2k}\,\frac{1}{1+k\zeta^2}\right]_0^x=-\frac{\alpha^2\,x^2}{2(1+kx^2)}\,.
\end{aligned}
$$
Therefore the Lagrangian for (\ref{NonL1}) is given, up to addition of a gauge term, by
\begin{equation*}
L_1(x,v)= \frac{1}{2} \frac{v^2-\alpha^2 x^2}{1+kx^2} \,,
\end{equation*}
and the corresponding momentum and Hamiltonian read
\begin{equation*}
p=\frac{\partial L_1}{\partial x}=\frac{v}{1+kx^2}\qquad \Rightarrow \qquad H_1(x,p) = \frac{1}{2} (1+kx^2)p^2+\frac{1}2\frac{\alpha^2\,x^2}{1+kx^2} \,.
\end{equation*}
We note that the function $L_1$ coincides with the Lagrangian obtained in \cite{ML74} by direct approach. It is remarkable that $L_1$ is a standard mechanical Lagrangian: it is given by a kinetic term quadratic in the velocities minus a potential term. Moreover,
we can also prove that $L_1$ is, up to an irrelevant additive constant, the Lagrangian function $L$ for $\Gamma$ obtained by using Proposition \ref{LagInt} and the constant of motion $I=\mu_1/(2k\mu_2)$:
$$
L(x,v)=v\int^v\frac{I(x,\zeta)}{\zeta^2}{\rm d}\zeta=v\int^v\frac{k\zeta^2-\alpha^2}{2k(1+kx^2)\zeta^2}{\rm d}\zeta=\frac{kv^2+\alpha^2}{2k(1+kx^2)}=L_1+\frac{\alpha^2}{2k}.
$$
Meanwhile, the second Jacobi multiplier, $\mu_2=(kv^2-\alpha^2)^{-1}$, gives rise to a non-mechanical Lagrangian. Indeed, if we assume $k>0$ and $\partial^2L_2/\partial v^2=\mu_2$, we obtain that, up to irrelevant gauge terms, the corresponding Lagrangian reads
$$
L_2(x,v)=-\frac{v}{\sqrt{k}\alpha}{\rm arcth}\left(\frac{\sqrt{k}v}{\alpha}\right)-\frac{1}{2k}\ln |\alpha^2-kv^2|+\phi_2(x),
$$
where
$$
\phi_2(x)=\int^x_0(\mu F)(\zeta,0){\rm d}\zeta=\int^x_0\,\frac{\zeta {\rm d}\zeta}{1+k\zeta^2}=\frac{\ln (1+kx^2)}{2k}.
$$
Hence,
$$
L_2(x,v)=-\frac{v}{\sqrt{k}\alpha}{\rm arcth}\left(\frac{\sqrt{k}v}{\alpha}\right)+\frac1{2k}{\ln \frac{1+kx^2}{|\alpha^2-kv^2|}}.
$$
In consequence,
$$
p=-\frac{{\rm arcth}(\sqrt{k}v/\alpha)}{\sqrt{k}\,\alpha}\Rightarrow
H_2(x,p)=\frac{1}{2k}\ln\left(\frac{\alpha^2\left[1-{\rm th}(\sqrt{k}\,p\,\alpha)^2\right]}{1+kx^2}\right).
$$
The case $k<0$ leads to a similar result.
\subsection{Second nonlinear oscillator }
We now apply Jacobi multipliers to nonlinear oscillators (\ref{NonL2}). Proceeding as before, we consider such systems as first-order systems by adding a new variable $v\equiv {\rm d}x/{\rm d}t$ to obtain
\begin{equation}\label{AsoNonL2}
\left\{
\begin{aligned}
\frac{{\rm d}x}{{\rm d}t}&=v\,,\\
\frac{{\rm d}v}{{\rm d}t}&=-\frac{kxv^2}{1+kx^2}-\frac{\alpha^2x}{(1+kx^2)^3}\,.
\end{aligned}\right.
\end{equation}
We drop the case $k=0$ as it leads to the standard harmonic oscillator which was analysed in previous section. So, we now focus upon the case $k\neq 0$.
The multiplier equation (\ref{multcond1}) for the vector field associated to the above system reads
$$
v\frac{\partial \mu}{\partial x}-\left(\frac{kxv^2}{1+kx^2}+\frac{\alpha^2x}{(1+kx^2)^3}\right)\frac{\partial \mu}{\partial v}-\frac{2\mu kxv}{1+kx^2}=0\,,
$$
whose {\it characteristic system} is
$$
\frac{{\rm d}x}{v}=-\frac{(1+kx^2)^3\, {\rm d}v}{kxv^2(1+kx^2)^2+\alpha^2x}=\frac{(1+kx^2)\,{\rm d}\mu}{2 kxv\mu}\,.
$$
The equality between the first and the last term shows that
$$
\frac{{\rm d}x}{1+kx^2}=\frac{{\rm d}\mu}{2kx\mu}\Rightarrow \frac{\mu}{1+kx^2}=\Upsilon_1
$$
for a real constant $\Upsilon_1$. Meanwhile, the integration of the equation
$$
\frac{{\rm d}x}{v}=-\frac{(1+kx^2)^3\, {\rm d}v}{kxv^2(1+kx^2)^2+\alpha^2x}
$$
goes as follows. Rewrite the above equation as
$$
-\frac{kv^2(1+kx^2)^2+\alpha^2}{(1+kx^2)^3}=\frac{v\,{\rm d}v}{x\, {\rm d}x}=\frac{{\rm d}v^2}{{\rm d}x^2}\,.
$$
The local change of variables $w\equiv v^2$, $z\equiv x^2$ transforms the above equation into
$$
\frac{{\rm d}w}{{\rm d}z}=-\frac{k}{1+kz}w-\frac{\alpha^2}{(1+kz)^3}\,,
$$
which finally gives
$$
\frac{k(1+kx^2)^2v^2-\alpha^2}{k(1+kx^2)}=\Upsilon_2\,,
$$
for a certain real constant $\Upsilon_2$. Resuming, we obtain the characteristic curves
$$
\frac{\mu}{1+kx^2}=\Upsilon_1,\qquad \frac{k(1+kx^2)^2v^2-\alpha^2}{k(1+kx^2)}=\Upsilon_2\,.
$$
Imposing, for instance, $\mu(0,v)=1$ or $\mu(0,v)=(kv^2-\alpha^2)/k$, we obtain
$$
\mu_1=1+kx^2,\qquad \mu_2={k(1+kx^2)^2v^2-\alpha^2}\,,
$$
which enable us to define a constant of motion
\begin{equation}\label{Integral2}
I=\frac{\mu_2}{\mu_1}=\frac{k(1+kx^2)^2v^2-\alpha^2}{1+kx^2}\,.
\end{equation}
One of the reasons of the interest of our results, in particular of the first-integrals (\ref{Integral1}) and (\ref{Integral2}), is that they provide a new geometric approach to results provided previously in \cite{BGS11}. Additionally, along with Proposition \ref{LagInt}, it enables us to obtain new Lagrangian descriptions of the oscillators under study.
Let us work out a Lagrangian for (\ref{NonL2}) via $\mu_1$. From equation
$$
\frac{\partial^2 L_1}{\partial v^2}=\mu_1=1+kx^2,
$$
we obtain
$$
L_1(x,v) = \frac{1}{2} (1+kx^2)\,v^2 + \bar \phi_1(x)v+\bar \phi_2(x),
$$
for certain functions $\bar \phi_1$ and $\bar \phi_2$. The gauge term $\bar{\phi}_1(x)v$ can be set to zero
and $\bar \phi_2(x) $ is determined by (\ref{phidos2}) where we choose $v_0=0$, i.e.
$$
\begin{aligned}
\bar \phi_2(x) &=\int_0^x (\mu\,F)(\zeta,0)\,{\rm d}\zeta=\int_0^x\frac{(-\alpha^2\,\zeta )}{(1+k\zeta^2)^2}\,{\rm d}\zeta=\left[\frac{\alpha^2}{2k(1+kx^2)}\right]_0^x=-\frac{\alpha^2\,x^2}{2(1+kx^2)}\,.
\end{aligned}
$$
Then, the Lagrangian for (\ref{NonL2}) is given, up to addition of a gauge term, by
\begin{equation*}
L_1(x,v)=\frac{1}{2}(1+kx^2)\,v^2 - \frac{1}{2}\frac{\alpha^2\,x^2}{1+kx^2} \,,
\end{equation*}
with corresponding Hamiltonian
\begin{equation*}
H_1(x,p) = \frac{1}{2}\frac{p^2}{1+kx^2}+\frac{1}{2}\frac{\alpha^2\,x^2}{1+kx^2} \,.
\end{equation*}
Meanwhile, if we make use of the second Jacobi multiplier, $\mu_2=k(1+kx^2)^2v^2-\alpha^2$, we obtain a non-standard Lagrangian, because
$$
\frac{\partial^2 L_2}{\partial v^2}=\mu_2\Longrightarrow L_2(x,v)=\frac{k}{12}v^4(1+kx^2)^2-\frac{v^2\alpha^2}{2}+\bar{\phi}_2(x),
$$
with
$$
\bar{\phi}_2(x)=\int_0^x (\mu\,F)(\zeta,0)\,{\rm d}\zeta=\int^x_0\frac{\alpha^4\zeta}{(1+k\zeta^2)^3}{\rm d}\zeta=\frac{\alpha x^2(2+k x^2)}{4(1+kx^2)^2}.
$$
Hence, up to an irrelevant gauge term, we obtain that
$$
L_2(x,v)=\frac{k}{12}v^4(1+kx^2)^2-\frac{v^2\alpha^2}{2}+\frac{\alpha x^2(2+k x^2)}{4(1+kx^2)^2}.
$$
Consequently,
$$
p=\frac k3(1+kx^2)^2v^3-\alpha^2v\Rightarrow v=\pm\frac{\sqrt{p+\alpha^2}}{\sqrt{k+2k^2x^2+k^3x^4}}
$$
and
$$
H_2(x,p)=\frac{p^2+4\alpha^2+\alpha(-kx^2(2+kx^2)+3\alpha^3-4\alpha\sqrt{k(1+kx^2)^2(p+\alpha^2)}}{4k(1+kx^2)^2}.
$$
\section{Jet bundle formulation of symmetries of differential equations}
Systems of differential equations and their symmetries admit an alternative geometric approach: instead of considering them as vector fields, we describe them as geometric structures in jet bundles.
We now recall the basic ingredients of this formulation before passing to study non-local symmetries of differential equations.
Let $(E,\mathbb{R},\pi)$ be a fiber bundle with total space $E\equiv \mathbb{R}\times N$, base $\mathbb{R}$ and submersion $\pi:(t,x)\in E\mapsto t \in \mathbb{R}$. Given local coordinates $\{x^j\}_{j=1,\ldots,n}$ on $N$ and $t$ on $\mathbb{R}$, we can naturally define a coordinate system $\{t,x^j_{0)}\equiv x^j\}_{j=1,\ldots,n}$ on $E$. Given a section $\sigma:\mathbb{R}\rightarrow E$ of $(E,\mathbb{R},\pi)$, we write $j^k_{t_0}\sigma=(t_0,x_{0)},x_{1)},\ldots,x_{k)})$ for the $k$-order {\it jet prolongation} of $\sigma$ at $t_0$, i.e. the equivalence class of sections $\gamma:t\in\mathbb{R}\mapsto (t,\gamma^1(t),\ldots,\gamma^n(t))\in E$ such that
$$
\gamma^j(t_0)=x^j_{0)},\,\,\,\quad \frac{{\rm d}^i\gamma^j}{{\rm d}t^i}(t_0)=x^j_{i)},\,\,\, \quad\qquad i=1,\ldots,k,\quad\,\,\, j=1,\ldots,n\,.
$$
We denote by $J^k\pi$, for $k\geq 0$, the space of $k$-order jet prolongations ($k$-jets) of the fiber bundle $(E,\mathbb{R},\pi)$ and we define $J^0\pi\equiv E$. Alternatively, we write $J^k(\mathbb{R},E)$ for $J^k\pi$ when $\pi$ is understood from the knowledge of $E$ and $\mathbb{R}$. The space $J^k\pi$ is a finite-dimensional manifold with local coordinates $\{t,x^j_{i)}\}_{\stackrel{i=0,\ldots,k}{j=1,\ldots,n}}$ of the form
$$
t(j^k_{t_0}\gamma)=t_0,\qquad
x^j_{i)}(j^k_{t_0}\gamma)=\frac{{\rm d}^i\gamma^j}{{\rm d}t^i}(t_0),\qquad\qquad i=0,\ldots,k,\quad\,\,\, j=1,\ldots,n\,,
$$
with ${\rm d}^0\gamma^j/{\rm d}t^0(t_0)\equiv \gamma^j(t_0)$.
It is well known that $(J^k\pi,\mathbb{R},\pi_k:J^k\pi\rightarrow \mathbb{R})$ is a fiber bundle: the {\it $k$-order jet bundle} associated with $(E,\mathbb{R},\pi)$. The sections of the $k$-order jet bundle being the prolongation of a section of $E$ are called ($k$-order) {\it holonomic sections}\cite{DS89,KS08}.
Consider the $C^\infty(J^k\pi)$-module of 1-forms $\theta\in \Lambda^1(J^k\pi)$ satisfying that $(j^k\sigma)^*\theta=0$ for every section $\sigma:\mathbb{R}\rightarrow E$. The elements of this module are called {\it contact forms} on $J^k\pi$. It can be proved that this module is a locally free-module generated by the contact forms $\theta^j_{i)}\equiv{\rm d}x^j_{i)}-x^{j}_{i+1)}\,{\rm d}t$, with $j=1,\ldots,n$ and $i=0,\ldots, k-1$.
We can endow $J^k\pi$ with the distribution $\mathcal{C}^k$ spanned by vector fields annihilating all contact forms on $J^k\pi$, the referred to as {\it contact or Cartan distribution} of $J^k\pi$. In particular, tangent vectors to graphs of $k$-order jet prolongations belong to $\mathcal{C}^k$. More generally, the Cartan distribution of $J^k\pi$ is spanned by
\begin{equation}\label{gen}
D_{k)}=\frac{\partial}{\partial t}+\sum_{j=1}^n\sum_{i=0}^{k-1} x^{j}_{i+1)}\frac{\partial}{\partial x^j_{i)}},\quad\,\,\, D_{j}=\frac{\partial}{\partial x_{k)}^j},\,\,\, \qquad j=1,\ldots,n.
\end{equation}
Note that $\mathcal{C}^{k}$ is not involutive for finite $k$ and have dimension $n+1$.
We call {\it Lie symmetries}\cite{Ca07} of $\mathcal{C}^{k}$ the infinitesimal symmetries of $\mathcal{C}^k$, i.e. the vector fields $Y$ on $J^k\pi$ satisfying that $[Y,X]$ takes values in $\mathcal{C}^k$ for every vector field $X$ taking values in $\mathcal{C}^k$. In other words, Lie symmetries of $\mathcal{C}^k$ are those vector fields whose flows give rise to transformations mapping $k$-order holonomic sections into $k$-order holonomic sections.
Given a vector field $X$ on $E$, its {\it prolongation} to $J^k\pi$ is the unique Lie symmetry, $X^{(k)}$, of $\mathcal{C}^k$ whose holonomic integral curves are
the prolongations to $J^k\pi$ of integral curves of $X$. Equivalently, $X^{(k)}$ is the unique Lie symmetry of $\mathcal{C}^k$ projecting onto $X$ under $\pi_{k,0}:j^k_t\sigma\in J^k\pi\mapsto \sigma(t)\in E$.
{\it Lie point symmetries} of $\mathcal{C}^k$ are Lie symmetries of $\mathcal{C}^k$ that are prolongations to $J^k\pi$ of vector fields on $E$.
The Lie--B\"acklund theorem states that not all Lie symmetries of $\mathcal{C}^k$ are Lie point symmetries. Non-Lie point symmetries, the referred to as {\it contact Lie symmetries}, can always
be considered as liftings of a uniquely defined Lie symmetry on $J^1\pi$ (see \cite{Ca07}).
In the above framework, a $k$-order system of differential equations is a closed embedded submanifold $\mathcal{E}\subset J^k\pi$ and its particular solutions are sections of $\pi$
whose $k$-order prolongations belong to $\mathcal{E}$. We say that a system of $k$-order differential equations is in normal form when the natural projection
$\pi_{k,k-1}:j^k\sigma\in\mathcal{E}\mapsto j^{k-1}\sigma\in J^{k-1}\pi$ is a submersion, e.g. $\mathcal{E}=\{j^2_tx\in J^2(\mathbb{R},\mathbb{R}^3):x^1_{2)}-x^2_{2)}=0\}$.
We say that $\mathcal{E}$ is in normal form and not underdetermined when $\pi_{k,k-1}:j^k\sigma\in\mathcal{E}\mapsto j^{k-1}\sigma\in J^{k-1}\pi$ is a diffeomorphism,
e.g. $\mathcal{E}=\{j^2_tx\in J^2(\mathbb{R},\mathbb{R}^3)\mid x^j_{2)}=F^j(t,x,x_{1)}),j=1,2\}$ for arbitrary functions $F^1,F^2:J^1(\mathbb{R},\mathbb{R}^3)\rightarrow \mathbb{R}$.
When $\mathcal{E}\subset J^k\pi$ is in normal form and not underdetermined, there exists a vectorial mapping $\Delta:J^k\pi\rightarrow \mathbb{R}^n$ allowing us to write that $\mathcal{E}=\Delta^{-1}(0)$.
A Lie symmetry of $\mathcal{C}^k$ that is tangent to $\mathcal{E}$ is called a {\it classical symmetry of $\mathcal{E}$} \cite{Ca07}. On the other hand, the term classical symmetry of $\mathcal{E}$ has also being employed \cite{Vi89} to refer to a vector field on $E$ giving rise to a uniparametric group of transformations mapping particular solutions to $\mathcal{E}$ to particular solutions to $\mathcal{E}$. In this work we will mainly use the first definition. Nevertheless, if a classical symmetry $Y$ is a Lie point symmetry, then we also call classical symmetry the unique vector field on $E$ whose prolongation to $J^k\pi$ is $Y$. More specifically, we say that $Y$ is a {\it classical point symmetry} of $\mathcal{E}$.
The projections $\pi_{h,k}:J^h\pi\rightarrow J^k\pi$, with $h>k$, enable us to define the bundle of infinite jets $(J^{\infty}\pi,\mathbb{R},\pi_\infty:J^{\infty}\pi\rightarrow \mathbb{R})$ as the inverse limit of the projections
$$
\mathbb{R}\leftarrow E\leftarrow J^1\pi\leftarrow J^2\pi\leftarrow\ldots
$$
The commutative ring of differentiable functions over $J^\infty\pi$ is defined by $\mathcal{F}(\pi)\equiv\bigcup_{l=0}^\infty C^{\infty}(J^l\pi)$. Note that each element of $\mathcal{F}(\pi)$ depends on a finite subset of variables of $\{t,x^j_{i)}\}_{\stackrel{i\in \mathbb{N}\cup \{0\}}{j=1,\ldots,n}}$. When there exists a natural injection between two manifolds, e.g. $J^k\pi\hookrightarrow J^{k'}\pi$ for $k'>k$, we may consider each function on the first manifold as a function on the second, e.g. an element of $C^\infty(J^k\pi)$ as an element of $C^\infty(J^{k'}\pi)$, so as to simplify the notation.
Given a section $\sigma:\mathbb{R}\rightarrow J^k\pi$, we call {\it infinite prolongation} of $\sigma$ the section $j^\infty\sigma:t\in\mathbb{R}\mapsto j^\infty_t\sigma\in J^\infty\pi$ given in coordinates by
$$
j_t^\infty\sigma\equiv\left(t,\sigma(t),\frac{{\rm d}\sigma}{{\rm d}t}(t),\frac{{\rm d}^2\sigma}{{\rm d}t^2}(t),\ldots,\right).
$$
Similarly to finite-dimensional manifolds, vector fields on $J^\infty\pi$ are defined as derivations of the commutative ring $\mathcal{F}(\pi)$ and the $\mathcal{F}(\pi)$-module of vector fields on $J^\infty\pi$ becomes a Lie algebra with respect to the commutator of derivations. The
tangent vectors to infinite prolongations of sections of $E$ span a distribution on $J^\infty \pi$ spanned by the derivation on $\mathcal{F}(\pi)$ given by
$$
D=\frac{\partial}{\partial t}+\sum_{j=1}^n\sum_{i=0}^{\infty} x^{j}_{i+1)}\frac{\partial}{\partial x^j_{i)}}.
$$
Although $D$ depends on an infinite number of variables, it induces a well-defined derivation on $\mathcal{F}(\pi)$ due to the fact that every function in $\mathcal{F}(\pi)$ only depends on a finite number of variables. The distribution $\mathcal{C}$ spanned by $D$ is the referred to as {\it contact or Cartan distribution} of $J^\infty\pi$, which is one-dimensional and therefore involutive. We cannot ensure that $D$ is integrable as the Frobenius Theorem does not apply to distributions on infinite-dimensional manifolds.
We call Lie symmetries of $\mathcal{C}$ the infinitesimal symmetries of $\mathcal{C}$. As these Lie symmetries of $\mathcal{C}$ are defined on an infinite-dimensional manifold, we cannot ensure that they are associated to uni-parametric groups of diffeomorphisms on $J^\infty\pi$.
It is interesting to note that many structures on the jet spaces $J^k\pi$ become simpler when passing to $J^\infty\pi$, e.g. the Cartan distribution becomes one-dimensional and involutive. Moreover, $J^\infty\pi$ is geometrically richer than finite-order jet bundles. For instance, Lie symmetries of $\mathcal{C}$ need not be lifts neither of vector fields on $E$ nor of vector fields on $J^1\pi$ due to the fact that the Lie--B\"acklund theorem does not apply to such Lie symmetries \cite{Ca07}.
Given a $k$-order system of differential equations $\mathcal{E}\subset J^k\pi$, the {\it $l$-prolongation of $\mathcal{E}$} is the set of points $\mathcal{E}^{(l)}\equiv\{j^{k+l}_t\sigma\,|\,j^k\sigma\subset \mathcal{E}\}\subset J^{k+l}\pi$ (see \cite{Ca07,Vi89} for details). Further, we call {\it infinite prolongation of $\mathcal{E}$} the set $\mathcal{E}^{\infty}\equiv\{j^{\infty}_t\sigma\,|\,j^k\sigma\subset \mathcal{E}\}\subset J^{\infty}\pi$. If $\mathcal{E}$ is in normal form and not underdetermined, then $\mathcal{E}=\Delta^{-1}(0)$ for a certain mapping $\Delta:j^k_tx\in J^k\pi\mapsto (x^1_{k)}-F^1(j_t^{k-1}x),\ldots,x^n_{k)}-F^n(j_t^{k-1}x))\in \mathbb{R}^n$ and functions $F^j:J^{k-1}\pi\rightarrow \mathbb{R}$. In this case, $\mathcal{E}^\infty$ is a finite-dimensional manifold locally determined by the infinite set of conditions
\begin{equation}\label{conCon}
\Delta=0,\qquad D^s\Delta=\stackrel{s-{\rm times}}{\overbrace{D(D(\ldots D(D}}\Delta)\ldots))=0\,,
\end{equation}
with $s=1,2,\ldots$ The above conditions determine all derivatives of particular solutions of $\mathcal{E}$ out of the value of the first $(k-1)$-derivatives.
Hence, a local set of coordinates for $\mathcal{E}$ can be considered in a natural way as elements of $\mathcal{F}(\pi)$ giving rise to a coordinate system on $\mathcal{E}^\infty$, which becomes a finite-dimensional manifold. In view of (\ref{conCon}), the restriction of $D$ to $\mathcal{E}^\infty$ is tangent to $\mathcal{E}^\infty$. This allows us to endow $\mathcal{E}^\infty$ with a one-dimensional distribution $\mathcal{C}|_{\mathcal{E}^\infty}$. The pair given by $\mathcal{E}^\infty$ and $\mathcal{C}|_{\mathcal{E}^\infty}$ becomes what is called a one-dimensional {\it diffiety}. We define $\mathcal{F}(\mathcal{E}^\infty)$ to be the restriction to $\mathcal{E}^\infty$ of functions of $\mathcal{F}(\pi)$. The Lie symmetries of $\mathcal{C}$ that are tangent to $\mathcal{E}^\infty$ are called {\it higher symmetries} of $\mathcal{E}$.
\begin{definition}
Let $\mathcal{E}$ be a $k$-order system of ODEs, we say that a bundle $(\mathcal{E}^c,\mathcal{E}^\infty,\kappa:\mathcal{E}^c\rightarrow\mathcal{E}^\infty)$ is a {\it covering} for $\mathcal{E}$ if the bundle ${\mathcal{E}}^c$ can be endowed with a one-dimensional distribution
$$
\mathcal{C}^c=\{\mathcal{C}^c_p\}_{p\in \widetilde{\mathcal{E}}}
$$
in such a way that $(\kappa_*)_p:{\mathcal{C}^c_p}\rightarrow \mathcal{C}_{\kappa(p)}$ is a linear isomorphism for each $p\in{\mathcal{E}^c}$.
\end{definition}
\begin{definition} Let $\mathcal{E}$ be a $k$-order system of ODEs and let ${\mathcal{E}^c}$ be a covering for $\mathcal{E}^\infty$. We call {\it non-local symmetry} for $\mathcal{E}$ an infinitesimal symmetry of the distribution ${\mathcal{C}^c}$.
\end{definition}
We call dimension of the covering, $\dim(\kappa)$, the dimension of the fiber of the bundle $\kappa$.
Given a covering $\kappa$, integral manifolds $\mathcal{S}$ of ${\mathcal{C}^c}$ project under $\kappa$ onto integral manifolds of $\mathcal{C}|_{\mathcal{E}^\infty}$, i.e. onto prolongations of
particular solutions of $\mathcal{E}$. It follows that, in this picture, symmetries
of ${\mathcal{C}^c}$ shuffle integral manifolds of ${\mathcal{C}^c}$ and, projecting under $\kappa$, we can obtain particular solutions to $\mathcal{E}$.
\section{Extended formalism and non-local symmetries}
Recently Gandarias and coworkers have studied a method for obtaining non-local symmetries for certain particular systems of ODEs \cite{BGS11, Ga09, GB11}. The main idea is to add a new additional equation to the original one in such a way that the new higher-dimensional system can be endowed with a classical symmetry.
In this section, we study this technique from a geometric perspective.
The starting point is that this procedure must be considered, in geometric terms, as a very particular case of an extended formalism where the covering has a total space given by a one-dimensional diffiety. We also mention the relation of this approach with some results previously obtained by Krasil'shchi, Vinogradov and coworkers \cite{V99,Ca07,KKV04}.
We begin with a system $\mathcal{E}$ of $k$-order ODEs
given by $\mathcal{E}=\Delta^{-1}(0)\subset J^k{\tau_{n+1}}$ for a mapping $\Delta:J^k\tau_{n+1}\rightarrow \mathbb{R}^p$ with $(\mathbb{R}^{n+1},\mathbb{R},{\tau}_{n+1}:(t,x)\in\mathbb{R}\times\mathbb{R}^{n}\mapsto t\in \mathbb{R})$ and coordinates
\begin{equation}\label{original}
\Delta^i\left(t,x,\frac{{\rm d}x}{{\rm d}t},\ldots,\frac{{\rm d}^kx}{{\rm d}t^k}\right)=0,\quad i=1,\ldots,p,\quad x\in\mathbb{R}^n
\end{equation}
and we assume that $\mathcal{E}$ is underdetermined and in normal form, i.e. $n=p$ and $\pi_{k,k-1}:\mathcal{E}\subset J^k\pi\rightarrow J^{k-1}\pi$ is a diffeomorphism.
A classical point symmetry of $\mathcal{E}$, represented by an $\epsilon$-parametric group of transformations given infinitesimally by
$$
\left\{\begin{aligned}
\bar t& = t+\epsilon\,\xi(t,x) \,,\\
\bar{x}^j& = x^j+\epsilon\,\eta^j(t,x) \,,\\
\end{aligned}\right.\qquad j=1,\ldots,n,
$$
preserves the set of solutions of the equation, that is, transforms particular solutions of (\ref{original}) into particular solutions of the same equation. Unfortunately, many systems of differential equations do not possess classical point symmetries. In that case, the method of non-local symmetries (applied to some particular cases in \cite{BGS11, Ga09, GB11} and related to some questions discussed in \cite{V99,Ca07,KKV04}) can be of a great usefulness.
A {classical point symmetry} for the system (\ref{original}) is a vector field $Y$ on $J^k{\tau}_{n+1}$ such that (i) is the lift of a vector field $Y_0$ on $J^0{\tau}_{n+1}\simeq \mathbb{R}^{n+1}$, and (ii) is tangent to the submanifold $\mathcal{E}$.
In coordinates if $Y_0$ takes the form
$$
Y_0(t,x)=\xi(t,x)\frac{\partial}{\partial t}+\sum_{j=1}^n\eta^j(t,x)\frac{\partial}{\partial x^j},
$$
then $Y$ is given by
\begin{equation}\label{prol}
Y=\xi\frac{\partial}{\partial t}+\sum_{j=1}^n\left(\eta^j\frac{\partial}{\partial x^j} + \sum_{i=1}^k\varphi^{(i)}_{\xi,\eta}\frac{\partial}{\partial x^j_{i)}}\right),
\end{equation}
where the functions
$\varphi^{(i)}_{\xi,\eta}:J^k\tau_{n+1}\subset J^\infty \tau_{n+1}\rightarrow\mathbb{R}$
can be obtained from the functions $\xi$ and $\eta^j$, with $j=1,\ldots,n$ (for a detailed explanation see\cite{St89}).
Suppose that the system (\ref{original}) is given. Let us construct a new system containing (\ref{original}) as a particular part. Consider the new jet bundles associated to the bundle $(\mathbb{R}^{n+2},\mathbb{R},\tau_{n+2}:(t,\bar x)\in\mathbb{R}^{n+2}\mapsto t\in \mathbb{R})$, with $\bar x\equiv (x,w)\in \mathbb{R}^{n+1}$. For a certain fixed function $H:J^1\tau_{n+1}\rightarrow \mathbb{R}$,
we can define a new system
$\widetilde{\mathcal{E}}\subset J^k\tau_{n+2}$ as follows
\begin{equation}\label{cov}
\Delta\left(t,x,\frac{{\rm d}x}{{\rm d}t},\ldots,\frac{{\rm d}^{k}x}{{\rm d}t^k}\right)=0,\quad
\frac{{\rm d} w}{{\rm d}t}=H, \quad\frac{{\rm d}^2 w}{{\rm d}t^2}=D_{k)}H,\quad\ldots,\quad\frac{{\rm d}^k w}{{\rm d}t^k}=D^{k-1}_{k)}H,
\end{equation}
containing as a particular part the initial system (\ref{original}) and where $D_{k)}$ is one of the generators of the Cartan distribution of $J^k\tau_{n+2}$ given in (\ref{gen}). Observe that although $\mathcal{E}$ was in normal form, the new system (\ref{cov}) is not in normal form as well: the derivatives ${\rm d}^kw/{\rm d}t^k$ can be expressed in terms of the lower derivatives, but such lower derivatives must hold several additional conditions. More geometrically, we cannot consider (\ref{cov}) straightforwardly as a submanifold $\widetilde{\mathcal{E}}\subset J^k\tau_{n+2}$ in such a way that $\tau_{k,k-1}:\widetilde{\mathcal{E}}\subset J^{k}\tau_{n+2}\rightarrow J^{k-1}\tau_{n+2}$ is epijective.
In any case, we can consider the prolongations $\mathcal{E}^\infty$ and $\widetilde{\mathcal{E}}^\infty$ of $\mathcal{E}$ and $\widetilde {\mathcal{E}}$, respectively. Since the system $\mathcal{E}$ is in normal form and in view of the definition of $\widetilde{\mathcal{E}}$, we see that the values of each derivative ${\rm d}^{\bar p}x/{\rm d}t^{\bar p}$, with $\bar{p}>k$, and ${\rm d}^{\bar p}w/{\rm d}t^{\bar p}$, with $\bar p>1$, of each particular solution of $\mathcal{E}$ and $\widetilde{\mathcal{E}}$ can be determined from the previous derivatives. Thus, $\mathcal{E}^\infty$ and $\widetilde{\mathcal{E}}^\infty$ are finite-dimensional manifolds and
$$
\dim\widetilde{\mathcal{E}}^\infty = \dim\widetilde{\mathcal{E}} \,,\qquad
\dim\mathcal{E}^\infty = \dim\mathcal{E}.
$$
Hence, a local coordinate system on $\mathcal{E}$ or $\widetilde{\mathcal{E}}$ induces a local coordinate system on their infinite prolongations. This means that expressions in coordinates on $\mathcal{E}$ and $\widetilde{\mathcal{E}}$ can be understood as expressions on $\mathcal{E}^\infty$ or $\widetilde{\mathcal{E}}^\infty$ indistinctly. This property is important: it allows us to identify $\mathcal{E}^\infty$ with $\mathcal{E}$ and $\widetilde{\mathcal{E}}^\infty$ with $\widetilde{\mathcal{E}}$. Hence, calculations in infinite-dimensional jet bundles are as difficult as for finite-dimensional jet bundles and the whole procedure is properly defined in a rigorous more powerful geometrical manner. For instance, $D$, which has no sense on $J^k\tau_{n+1}$, can be however correctly considered when restricted to $\mathcal{E}^\infty$.
Since $D$ is tangent to $\mathcal{E}^\infty$, we can define the restriction $D|_{\mathcal{E}^\infty}$ of this operator to $\mathcal{E}^\infty$. If $\widetilde{D}$ is the analogue of $D$ on $J^\infty\tau_{n+2}$, this operator is also tangent to $\widetilde{\mathcal{E}}^\infty$ and we can also define its restriction, $\widetilde D|_{\widetilde{\mathcal{E}}^\infty}$, to $\widetilde{\mathcal{E}}^\infty$. The vector field $\widetilde{D}$ induces a one-dimensional distribution $\widetilde{\mathcal{C}}$ on $J^\infty\tau_{n+2}$ and
$\widetilde D|_{\widetilde{\mathcal{E}}^\infty}$ spans a one-dimensional distribution $\widetilde{\mathcal{C}}|_{\widetilde{\mathcal{E}}^\infty}$, turning the pair $(\widetilde{\mathcal{E}}^\infty,\widetilde{\mathcal{C}}|_{\widetilde{\mathcal{E}}^\infty})$ into what is called a diffiety of dimension one: the dimension one refers to the fact that we have defined a one-dimensional distribution $\widetilde{\mathcal{C}}|_{\widetilde{\mathcal{E}}^\infty}$ on $\widetilde{\mathcal{E}}^\infty$. Moreover, we have the following property.
\begin{proposition}
Let $\widetilde{D}$ be the vector field on $J^\infty\tau_{n+2}$ given by
\begin{equation}\label{restrc}
\widetilde{D} = D+\sum_{i=0}^\infty w_{i+1)}\frac{\partial}{\partial w_{i)}}, \qquad w_0\equiv w.
\end{equation}
Then, we have locally
\begin{equation}\label{restrc2}
\widetilde D|_{\widetilde{\mathcal{E}}^\infty}= D|_{\mathcal{E}^\infty}+H\frac{\partial}{\partial w} \,=\sum_{j=1}^n\left[\sum_{p=0}^{k-2}x^j_{p+1)}\frac{\partial}{\partial x^j_{p)}}+F^j(j^{k-1}_tx)\frac{\partial}{\partial x^j_{k-1)}}\right]+H(j^1_tx)\frac{\partial}{\partial w},
\end{equation}
for certain functions $F^1,\ldots,F^n:\widetilde{\mathcal{E}}^\infty\rightarrow \mathbb{R}$.
\end{proposition}
\begin{proof} Expression (\ref{restrc}) is trivially the generator of the Cartan distribution of $J^\infty\tau_{n+2}$.
As we assume $\mathcal{E}$ to be in normal form and not underdetermined, the higher-order derivatives of the particular solutions to $\mathcal{E}$, namely ${\rm d}^kx^j/{\rm d}t^k$, can be locally written as a function of the previous derivatives. So, ${\rm d}^kx^j/{\rm d}t^k=F^j(j^{k-1}_tx)$ for $j=1,\ldots,n$ and certain functions $F^j: J^{k-1}\tau_{n+1}\rightarrow \mathbb{R}$ which is understood in the natural way as a function on $\widetilde{\mathcal{E}}^\infty\subset J^\infty\tau_{n+2}$.
Using this, recalling that $\{t,x_{i)},w\}_{i=0,\ldots,k-1}$ forms a coordinate system for $\widetilde{\mathcal{E}}^\infty$ and restricting $\widetilde{D}$ from $J^\infty\tau_{n+2}$ to $\widetilde{\mathcal{E}}^\infty$, we obtain that the expression (\ref{restrc2}) follows from (\ref{restrc}).
\end{proof}
The natural projection $\Pi:(t,x,w)\in\mathbb{R}^{n+2}\mapsto (t,x)\in\mathbb{R}^{n+1}$ lifts to
a projection $J^\infty \Pi:J^\infty \tau_{n+2}\rightarrow J^\infty \tau_{n+1}$ satisfying
$$
J^\infty \Pi(j^\infty_t \bar x)=J^\infty \Pi(j^\infty_t (x,w))=j^\infty_t x \,.
$$
This projection induces a map $J^\infty \Pi|_{\widetilde {\mathcal{E}}^\infty}:\widetilde{\mathcal{E}}^\infty\rightarrow\mathcal{E}^\infty$ obeying that
$$
\left(J^\infty \Pi|_{\widetilde{\mathcal{E}}^\infty}\right)_*(\widetilde D|_{\widetilde{\mathcal{E}}^\infty})=D|_{\mathcal{E}^\infty}.
$$
As a consequence, $J^\infty \Pi$ induces an isomorphism $(J^\infty \Pi|_{\widetilde{\mathcal{E}}^\infty})_{*\xi}:\widetilde{\mathcal{C}}_\xi\rightarrow \mathcal{C}_{J^\infty \Pi(\xi)}$ for every $\xi \in \widetilde{\mathcal{E}}^\infty$ and therefore a covering
$\kappa_*\equiv (J^\infty \Pi|_{\widetilde{\mathcal{E}}^\infty})_*:\widetilde{\mathcal{C}}|_{\widetilde{\mathcal{E}}^\infty}\rightarrow \mathcal{C}|_{{\mathcal{E}}^\infty}$.
Hence, if the system $\widetilde{\mathcal{E}}$ admits a classical symmetry $Y$, e.g. (\ref{prol}), then $Y$ can be lift to a Lie symmetry $Y^\infty$ of $\widetilde{\mathcal{C}}$, namely a higher symmetry for $\widetilde{\mathcal{E}}$. Both vector fields, $Y$ and $Y^\infty$, are tangent to $\widetilde{\mathcal{E}}$ and $\widetilde{\mathcal{E}}^\infty$, respectively. It is worth noting that due to our assumptions, the coordinate expression in $\mathcal{\widetilde{E}}$ of $Y|_\mathcal{\widetilde{E}}$ and the coordinate expression of $Y^\infty|_{\widetilde{\mathcal{E}}^\infty}$ in $\widetilde{\mathcal{E}}^\infty$ are the same. Moreover, as $Y^\infty$ is a higher symmetry, it leaves invariant $\widetilde{\mathcal{C}}$ and it becomes a non-local symmetry of $\mathcal{E}^\infty$ when restricted to $\widetilde{\mathcal{E}}^\infty$.
If $\widetilde{\mathcal{E}}$ admits a classical point symmetry $Y$, then we have a one-parametric group of diffeomorphisms given, in infinitesimal form, as
$$
\left\{\begin{aligned}
t^*& = t + \epsilon\,\xi(t,x,w) \,,\\
x^*&= x + \epsilon\,\phi(t,x,w) \,,\\
w^*&= w + \epsilon\,\eta(t,x,w) \,,
\end{aligned}\right.
$$
transforming solutions to the system $\widetilde{\mathcal{E}}$ into solutions of $\widetilde{\mathcal{E}}$. Hence, the set of transformations
$$
\left\{\begin{aligned}
t^*&= t + \epsilon\,\xi(t,x,w(t)) \,,\\
x^*&= x + \epsilon\,\phi(t,x,w(t)) \,,
\end{aligned}\right.
$$
enables us to map particular solutions to $\mathcal{E}$ into solutions of $\mathcal{E}$ by means of the curves $w(t)$ corresponding to particular
solutions of $\widetilde{\mathcal{E}}$.
We can summarise the main results proved in this section as follows.
We have proved we can embed a given system $\mathcal{E}$ into a bigger one whose infinity prolongation has the structure of a diffiety. This structure gives rise to a covering for the initial system.
In \cite{Ca07} another covering for the initial system is constructed so as to study it through non-local symmetries. Nevertheless, that covering cannot be considered neither as a diffiety nor a submanifold of a jet bundle without additional constructions. So, our interpretation is more powerful as we can straightforwardly use classical symmetries of $\widetilde{\mathcal{E}}$ to construct non-local symmetries of $\mathcal{E}$.
Gandarias developed slightly modifications of her method, but all of them can be retrieved as particular cases of the above geometric approach.
\section{Extended formalism for the nonlinear oscillators}
In this section, we provide a geometric method to construct non-local symmetries for second-order autonomous differential equations. This method is based upon considering our initial second-order differential equations as part of an extended system whose form can be determined out of the initial one. In following subsections we show that this procedure retrieves as a particular case the results given by Gandarias and coworkers.
\begin{theorem}\label{Th} Every system $\mathcal{E}$ corresponding to
\begin{equation}\label{ExtSysLem}
\begin{aligned}
\frac{{\rm d}^2x}{{\rm d}t^2}&= F(x,v),
\end{aligned
\end{equation}
can be extended to a larger system $\widetilde{\mathcal{E}}$ with the additional equation ${\rm d}w/{\rm d}t=H(x,v)$ in such a way that $Y=gX_H$, where $X_H=v\partial_x+F\partial_v+H\partial_w$ is an infinitesimal symmetry of the distribution $\widetilde{\mathcal{C}}|_{\widetilde{\mathcal{E}}^\infty}$ generated by $\partial_t+X_H$ and $g\in \mathcal{F}(\widetilde{\mathcal{E}}^\infty)$ is a first-integral of $X_H,\partial_x,\partial_t$ (as vector fields on $\widetilde{\mathcal{E}}^\infty$). In consequence, $Y$ is a non-local symmetry of (\ref{ExtSysLem}).
\end{theorem}
\begin{proof} We have that $[gX_H,\partial_t+X_H]=-(\partial_tg+X_Hg)X_H$. So, $gX_H$ is an infinitesimal symmetry of $\widetilde{\mathcal{C}}|_{\widetilde{\mathcal{E}}^\infty}$ if and only if $\partial_tg+X_Hg=0$. Under the assumed conditions for $g$, we obtain that $gX_H$ is an infinitesimal symmetry of $\widetilde{\mathcal{C}}|_{\widetilde{\mathcal{E}}^\infty}$. Let us prove that there exists a nonconstant function $g$ satisfying the above conditions.
Set $H(x,v)\equiv F(x,v)h(v)$ for a certain function $h(v)\neq 0$. Since $\partial g/\partial x=0$, then
$$
0=X_Hg=F(x,v)\left(\frac{\partial g}{\partial v}+h(v)\frac{\partial g}{\partial w}\right).
$$
If $F\neq 0$ and we require $g$ to be non-constant, then our definition of $H$ ensures the existence of a non-trivial $g$ depending only on $v$ and $w$. If $F=0$, we can choose any $g$ with the required properties of our theorem.
\end{proof}
\begin{corollary} A classical infinitesimal symmetry for the system $\widetilde{\mathcal{E}}$ gives rise to a non-local symmetry of $\mathcal{E}$.
\end{corollary}
\begin{proof} Every classical symmetry for $\widetilde{\mathcal{E}}$ can be extended in view of the Lie--B\"acklund Theorem to a higher symmetry $Y^\infty$ of $\mathcal{\widetilde{E}}$. This higher symmetry is tangent to $\widetilde{\mathcal{E}}^\infty$ and a Lie symmetry of $\widetilde{\mathcal{C}}$, which is, by construction of $\widetilde{\mathcal{E}}^\infty$, tangent to $\widetilde{\mathcal{E}}^\infty$. Hence, $Y^\infty|_{\widetilde{\mathcal{E}}^\infty}$ is a symmetry of $\widetilde{\mathcal{C}}|_{\widetilde{\mathcal{E}}^\infty}$ and it becomes a non-local symmetry for $\mathcal{E}$.
\end{proof}
\subsection{First nonlinear oscillator }
Let us review the approach given by Gandarias to study the nonlinear oscillator (\ref{NonL1}). The first-order system (\ref{AsoNonL}) associated to (\ref{NonL1}) is embedded into a new one on ${\rm T}\mathbb{R}\times\mathbb{R}$ of the form
\begin{equation}\label{ExtSys}
\left\{
\begin{aligned}
\frac{{\rm d}x}{{\rm d}t}&= v,\\
\frac{{\rm d}v}{{\rm d}t}&= \frac{(kv^2-\alpha^2)x}{1+kx^2},\\
\frac{{\rm d}w}{{\rm d}t}&=H(x,v),
\end{aligned}\right.
\end{equation}
where $H(x,v)$ is a, undetermined for the moment, function, to be fixed later on. Let us study the Lie point symmetries of this system.
Particular solutions to system (\ref{ExtSys}) are in a one-to-one correspondence with the integral curves $t\mapsto (t,x(t),v(t),w(t))$ of the vector field
$$
\bar X_H \equiv \frac{\partial}{\partial t}+X_H\equiv \frac{\partial}{\partial t}+ v\frac{\partial}{\partial x}+ \frac{(kv^2-\alpha^2)x}{1+kx^2}\frac{\partial}{\partial v}+H(x,v)\frac{\partial}{\partial w} \,.
$$
Given a vector field $Y=\xi\partial/\partial t+\phi\partial/\partial x+\psi\partial/\partial v+\eta\partial/\partial w$ on $\mathbb{R}^2\times{\rm T}\mathbb{R}$, where we include the time variable $t$, we know that $Y$ determines a Lie point symmetry of this system if $[Y,\partial_t+X_H]=f(\partial_t+X_H)$, where we recall that $t,x,v,w$ are considered as coordinates on $\mathbb{R}^2\times{\rm T}\mathbb{R}$ and $f\in C^\infty(\mathbb{R}^2\times{\rm T}\mathbb{R})$. Equivalently $Y$ is a classical point symmetry of this system
if its four coefficients satisfy the following equations
\begin{multline}\label{complsys}
(Y^{(1)}\Delta^1)_{\Delta=0}=v(kx^2+ 1)\xi_t + vH( kx^2 + 1)\xi_w +xv(kv^2-\alpha^2)\xi_v
+v^2(kx^2+ 1)\xi_x\\
-x(kv^2-\alpha^2)\phi_v-(kx^2+1)v\phi_x
- (k x^2+1)\phi_t- H (kx^2+1)\phi_w+ v^2(kx^2+1)\psi = 0, \\
\qquad\qquad\qquad\qquad- H_xk\phi x^2 + H (kx^2+1)\eta_w +\eta_tkx^2-\alpha^2x\eta_v- H_v\psi- H_x\phi +\eta_x = 0, \\
(Y^{(1)}\Delta^2)_{\Delta=0}=(kx^2+1)(\alpha^2-kv^2)x\xi_t- H( kx^2 +1)x [kv^2-\alpha^2]\xi_w-x^2(kv^2-\alpha^2)^2\xi_v\qquad\qquad\qquad\qquad\\
+ vx[1+kx^2](-kv^2 +\alpha^2)\xi_x
+(k x^2 +1)(kv^2-\alpha^2)x\psi_v +(kx^2-1) (kv^2-\alpha^2)\phi\\
\qquad\qquad \qquad+(kx^2+ 1)^2v\psi_x - 2kxv(1+kx^2)\psi + H(kx^2+ 1)^2\psi_w+ (k x^2+1)^2\phi_t = 0\\
(Y^{(1)}\Delta^3)_{\Delta=0}=-H(kx^2+1)\xi_t- H^2(kx^2+ 1)\xi_w- Hx(kv^2-\alpha^2 )\xi_v{\qquad\qquad\qquad\qquad\qquad}\\
- H(kx^2+1)v\xi_x- H_x\xi(kx^2 +1) + kxv^2\eta_v+(kx^2 + 1)v\eta_x- H_vk\psi x^2\\
\end{multline}
for $\Delta^1=\dot x-v$, $\Delta^2= \dot v-(kv^2-\alpha^2)x/(1+kx^2)$, $\Delta^3=\dot w-H$ and $Y^{(1)}$ being the prolongation to $J^1\tau_4$, with $\tau_4:(t,x,v,w)\in \mathbb{R}^4\mapsto t\in \mathbb{R}$, of the vector field $Y$ on $\mathbb{R}^4$.
We include expressions (\ref{complsys}) to solve several minor typos and mistakes in the previous literature.
This is a quite difficult system to be solved, which suggests us to assume some kind of simplification. This was done in \cite{BGS11}, whose authors considered as an ansatz a particular form for $\xi,\phi,\psi, \eta$. Now we reconsider this whole approach in a more geometrical and rigorous way.
Equivalently, the differential equation (\ref{NonL1}) can be considered along with the equation ${\rm d}w/{\rm d}t=H(x,v)$. As commented in the latter section, this system can be understood as a submanifold $\widetilde{\mathcal{E}}$ of $J^2\tau_3$ with $\tau_3:(t,x,w)\in\mathbb{R}^3\mapsto t\in\mathbb{R}$.
Let us use Theorem \ref{Th} to study the infinitesimal symmetries of $\widetilde{\mathcal{C}}|_{\mathcal{E}^\infty}$. Recall that this amounts to a non-local symmetry for $\mathcal{E}$.
We can construct a non-local symmetry by assuming $Y=gX_H$ with $H(x,v)=(X_Hv)(x,v)h(v)$ and $g$ being a first-integral of $X_H$ independent of $t$ and $x$, namely, such that
$$
X_Hg=\frac{(kv^2-\alpha^2)x}{1+kx^2}\left(\,\frac{\partial g}{\partial v}+ h(v)\frac{\partial g}{\partial w}\right)=0,
$$
where we fixed according to Theorem \ref{Th}
$$
H(x,v)=\frac{(kv^2-\alpha^2)x}{1+kx^2}h(v).
$$
By assuming $h(v)=1/v$, we obtain a simple first-integral for $X_H$ of the form $g=e^w/v$. Hence,
$$
Y=e^w\left(\frac{\partial}{\partial x}+H\frac{\partial}{\partial v}+\frac{H}{v}\frac{\partial}{\partial w}\right).
$$
Indeed, observe that $[Y,X_H]=0$.
As $\{t,x,v,w\}$ can be understood as coordinates of $\mathbb{R}^2\times {\rm T}\mathbb{R}$ and $\widetilde{\mathcal{E}}$, the vector field $Y$ can also be considered as a vector field on $\mathbb{R}^2\times {\rm T}\mathbb{R}$. In this way, $Y$ is the same Lie symmetry provided in \cite{BGS11}, where it was obtained by the derivation of a particular solution of (\ref{complsys}) using an {\it ad hoc} ansatz for $Y$ and $H$. Meanwhile, we here use a covering to show that Gandarias' and coworkers ansatz corresponds to choose a certain $H$ so that a first-integral for $X_H$ independent of $x,t$ can be obtained. This immediately leads to their same final result.
Note also that we could in principle choose another function $H$ which could potentially lead to different non-local symmetries of $\mathcal{E}$. Nevertheless, the form chosen
in this work makes computations easier in many cases.
\subsection{Second nonlinear oscillator }
We can now apply the above method to equations (\ref{NonL2}) to recover the same result provided in \cite{BGS11}. In this new case, the first-order system (\ref{AsoNonL2}) associated to (\ref{NonL2}) is embedded into one
\begin{equation}\label{ExtSys2}
\left\{
\begin{aligned}
\frac{{\rm d}x}{{\rm d}t}&= v,\\
\frac{{\rm d}v}{{\rm d}t}&= -\frac{kxv^2}{1+kx^2}-\frac{\alpha^2x}{(1+kx^2)^3},\\
\frac{{\rm d}w}{{\rm d}t}&=H(x,v),
\end{aligned}\right.
\end{equation}
on $\mathbb{R}^3$, where $H(x,v)$ is a function to be fixed next. Additionally, we can consider this system as a submanifold $\mathcal{E}\subset J^2\tau_3$ with $\tau_3:(t,x,w)\in\mathbb{R}^3\mapsto t\in\mathbb{R}$. This system describes the integral curves $t\rightarrow (t,x(t),v(t),w(t))$ of the vector field on $\mathbb{R}^2\times{\rm T}\mathbb{R}\simeq \mathcal{\widetilde{E}}$ of the form
$$
\bar X_H\equiv \frac{\partial}{\partial t}+X_H\equiv v\frac{\partial}{\partial x}-\left(\frac{kxv^2}{1+kx^2}+\frac{\alpha^2x}{(1+kx^2)^3}\right)\frac{\partial}{\partial v}+H\frac{\partial}{\partial w}.
$$
We fix $H$ to be of the previously commented form, i.e.
$$
H(x,v)=-\left(\frac{kxv^2}{1+kx^2}+\frac{\alpha^2x}{(1+kx^2)^3}\right)\frac{1}{v}.
$$
Hence, $X_H$ admits a locally defined first-integral $g$ that does not depend neither on $x$ nor on $t$, namely, such that
$$
X_Hg=-\left(\frac{kxv^2}{1+kx^2}+\frac{\alpha^2x}{(1+kx^2)^3}\right)\frac{\partial g}{\partial v}+H\frac{\partial g}{\partial w}=0.
$$
This leads to a simple first-integral for $X_H$ of the form $g=e^w/v$. We can now obtain a Lie symmetry of the system by choosing $Y=gX_H$, which reads
$$
Y=e^w\left(\frac{\partial}{\partial x}+H\frac{\partial}{\partial v}+\frac{H}v\frac{\partial}{\partial w}\right),
$$
which is again the same classical symmetry provided in \cite{BGS11} but we here understand it as a symmetry of $\widetilde{\mathcal{C}}$ on $\widetilde{\mathcal{E}}^\infty$, i.e. a non-local symmetry of $\mathcal{E}$.
\section{Final comments }
This paper has been mainly concerned with the following two points: Jacobi multipliers and non-local symmetries.
\begin{itemize}
\item The Jacobi multipliers have been first considered in relation with the inverse problem of the Lagrangian formalism and then applied to the study of two particular nonlinear oscillators.
\item The theory of non-local symmetries has been studied by making use of a geometric approach. We prove that the extended formalism can be a very interesting procedure for obtaining symmetries of nonlinear systems.
\item We have shown that the use of infinite-dimensional jet manifolds does not complicate the description of non-local symmetries of systems and allows us to develop a more rigorous theoretical approach. In addition, certain structures are now naturally defined.
\item In the future we aim to apply the theory of non-local symmetries to a generalisation of the nonlinear oscillators studied in this work that contain an isotopic term. This will describe as a particular case the non-linear oscillators detailed in \cite{BEHR08} on a one-dimensional manifold.
\item Diffieties are mainly used in the study of systems of partial differential equations. Nevertheless, we aim to show that these structures may play a r\^ole also for the study of relevant systems of first-order differential equations.
\end{itemize}
\section{Acknowledgments}
Research of J. de Lucas founded by the Polish National Science Centre grant MAESTRO under the contract number DEC-2012/06/A/ST1/00256.
Partial financial support by research projects MTM2012-33575, MTM2011-15725-E and E24/1 (DGA)
are acknowledged. J. de Lucas also acknowledges a stay at the University of Zaragoza supported by Gobierno de Arag\'on (FMI43/10).
|
2,869,038,156,335 | arxiv | \section{Introduction}
\par Let $A=\{{\bf a}_1,\ldots,{\bf a}_m\}\subseteq \mathbb{N}^n$
be a vector configuration in $\mathbb{Q}^n$ and
$\mathbb{N}A:=\{l_1{\bf a}_1+\cdots+l_m{\bf a}_m \ | \ l_i \in
\mathbb{N}\}$ the corresponding affine semigroup. We grade the
polynomial ring $K[x_1,\ldots,x_m]$ over any field $K$ by the
semigroup $\mathbb{N}A$ setting $\deg_{A}(x_i)={\bf a}_i$ for
$i=1,\ldots,m$. For ${\bf u}=(u_1,\ldots,u_m) \in \mathbb{N}^m$,
we define the $A$-{\em degree} of the monomial ${\bf x}^{{\bf
u}}:=x_1^{u_1} \cdots x_m^{u_m}$ to be \[ \deg_{A}({\bf x}^{{\bf
u}}):=u_1{\bf a}_1+\cdots+u_m{\bf a}_m \in \mathbb{N}A.\] The
{\em toric ideal} $I_{A}$ associated to $A$ is the prime ideal
generated by all the binomials ${\bf x}^{{\bf u}}- {\bf x}^{{\bf
v}}$ such that $\deg_{A}({\bf x}^{{\bf u}})=\deg_{A}({\bf x}^{{\bf
v}})$, see \cite{St}. For such binomials, we define $\deg_A({\bf
x}^{{\bf u}}- {\bf x}^{{\bf v}}):=\deg_{A}({\bf x}^{{\bf u}})$.
Toric ideals have a large number of applications in several areas
including: algebraic statistics, biology, computer algebra,
computer aided geometric design, dynamical systems,
hypergeometric differential equations, integer programming, mirror
symmetry, toric geometry and graph theory, see \cite{ATY, DS, E-S, MS, St}.
In graph theory there are several monomial or binomial ideals associated to a
graph, see
\cite{CoN, FF, HH, H, NP, SVV1, SVV, SS, Vi1, Vi, V}, depending on the properties one wishes to study. One of them is the toric
ideal of a graph which has been extensively studied over the last
years, see \cite{CoN, FF, G, G2, K, Ka, OH1, OH, OH2, Hi-O, OH-Ram, VV,
Vi2, Vi}.
The toric ideals are {\em
binomial ideals}, i.e. polynomial ideals generated by binomials. There
are certain binomials in a toric ideal, such as minimal, indispensable, primitive, circuit
and fundamental binomials provide crucial
information about the ideal and therefore they have been studied
in more detail.\\
A binomial $B\in I_A$ is called {\em minimal} if it belongs to at
least one minimal system of generators of $I_A$. The minimal
binomials, up to scalar multiple, are finitely many.
Their number is computed in \cite{ChTh} in terms of combinatorial
invariants of a simplicial complex associated to the toric
ideal. The minimal binomials are characterized as the binomials that can not
be written as a combination of binomials of smaller $A$-degree, see \cite{ChTh,PS}.\\
A binomial $B\in I_A$ is called {\em indispensable} if there exists a nonzero
constant multiple of it to every
minimal system of generators of $I_A$.
A recent problem arising from Algebraic Statistics
is when a toric ideal have a unique minimal
system of binomial generators, see \cite{ChKT, AT}. To study this problem Ohsugi and
Hibi introduced in \cite{Hi-O} the notion of indispensable
binomials and they gave necessary and sufficient conditions for toric
ideals associated with certain finite
graphs to possess unique minimal systems of binomial generators.\\
An irreducible binomial $x^{{\bf u}^+}- x^{{\bf u}^-}$ in $I_{A}$
is called {\em primitive} if there exists no other binomial $
x^{{\bf v}^+}- x^{{\bf v}^-} \in I_{A}$ such that $ x^{{\bf v}^+}$
divides $ x^{{\bf u}^+}$ and $ x^{{\bf v}^-}$ divides $
x^{{\bf u}^-}$. The set of all primitive binomials forms the Graver
basis of the toric ideal, see \cite{St}. It follows from the
definition that a non primitive binomial can be written as a sum of products of
monomials times binomials of $I_A$ of smaller
$A$-degree therefore minimal binomials must be primitive, see also Lemma 3.1 of \cite{OH}.
\\
The support of a monomial $x^{\bf{u}}$ of $K[x_{1},\dots,x_{m}]$
is $supp(x^{\bf{u}}):=\{i\ | \ x_{i}\ divides\ x^{\bf{u}}\}$ and
the support of a binomial $B=x^{\bf{u}}-x^{\bf{v}}$ is
$supp(B):=supp(x^{\bf{u}})\cup supp(x^{\bf{v}})$. An irreducible
binomial $B$ belonging to $I_{A}$ is called a \emph{circuit} of
$I_{A}$ if there is no binomial $B'\in I_{A}$ such that
$supp(B')\subsetneqq supp(B)$. A binomial $B\in I_{A}$ is a
circuit of $I_{A}$ if and only if $I_{A}\cap K[x_{i}\ | i\in
supp(B)]$ is generated by $B$.\\
For a vector ${\bf b}=(b_1,\dots ,b_n)\in \mathbb{N}^n$ we define
$supp({\bf b})=\{i | b_i\neq 0\}$. For a semigroup $\mathbb{N}A$
we denote $K[\mathbb{N}A]$ the semigroup ring of $\mathbb{N}A$.
The semigroup ring $K[\mathbb{N}A]$ is isomorphic to the quotient
$K[x_1,\ldots,x_m]/I_A$, see \cite{MS}. Let ${F}$ be a subset
of $\{1,\dots ,n\}$ then $A_{{F}}$ is the set
$\{{\bf a}_i| supp({\bf a}_i)\subset {{F}}\}$. The semigroup
ring $K[\mathbb{N}A_{{F}}]$ is called combinatorial pure
subring of $K[\mathbb{N}A]$, see \cite{OHH} and for a
generalization, see \cite{O}. A binomial $B\in I_A$ is called {\em
fundamental} if there exists a combinatorial pure subring
$K[\mathbb{N}A_{{F}}]$ such that $K[x_i|{\bf a}_i\in A_{{F}}]\cap
I_A=I_{A_{{F}}}=<B>$.
These kinds of binomials are related to each other. The indispensable
binomials are always minimal and the minimal are always primitive.
Also the fundamental binomials are circuits and indispensable, while the
circuits are also primitive. The toric ideals of graphs is the best kind of toric ideals in order to understand
how circuits, fundamentals,
primitive, minimal and indispensable binomials are related, see
Theorems \ref{circuit}, \ref{primitive}, \ref{fundam},
\ref{minimal}, \ref{indispen}, and to show that the
above relations are strict, see Example \ref{example}.
Actually the toric ideal of a graph gives a way to `view' the
ideal through the graph, but also to construct toric ideals with desired properties.
In the case of the toric ideal of a graph there were several articles in the literature that characterize
these kinds of binomials, most of them for particular cases of graphs, see \cite{G,Ka,OH,OH2,Hi-O,OH-Ram,Vi,V}.
The aim of this article is to
characterize primitive, minimal, indispensable and fundamental binomials of a toric ideal of a graph
for a general graph and thus understanding better the toric ideal. These characterizations maybe useful
to solve problems in the theory of toric ideals of graphs.
The results in
this paper are inspired and guided by the work of Oshugi and Hibi
\cite {OH,Hi-O}. In section 2 we present some
terminology, notations and results about the toric ideals of
graphs. In section 3 we provide the converse of the
characterization of Ohsugi and Hibi \cite{OH} of the primitive
elements of toric ideals of graphs. In section 4 we characterize
the minimal, the indispensable and the fundamental binomials of the toric
ideal of a graph and we give an example that explains the relations
between fundamental, primitive, circuit, minimal and indispensable
binomials. At the end we remark that although the results in the
article are stated and proved for simple graphs, they are also valid with
small adjustments for general graphs with loops and multiple
edges, see Remark \ref{remark}.
\section{Toric ideals of graphs}
\par
In the next chapters, $G$ will be a finite simple connected graph on the vertex set
$V(G)=\{v_{1},\ldots,v_{n}\}$, except at the final remark \ref{remark} where the graph $G$ may have
multiple edges and loops. Let $E(G)=\{e_{1},\ldots,e_{m}\}$ be
the set of edges of $G$ and $\mathbb{K}[e_{1},\ldots,e_{m}]$
the polynomial ring in the $m$ variables $e_{1},\ldots,e_{m}$ over a field $\mathbb{K}$. We
will associate each edge $e=\{v_{i},v_{j}\}\in E(G)$ with
$a_{e}=v_{i}+v_{j}$ in the free abelian group generated by the
vertices and let $A_{G}=\{a_{e}\ | \ e\in E(G)\}$. With $I_{G}$ we denote
the toric ideal $I_{A_{G}}$ in
$\mathbb{K}[e_{1},\ldots,e_{m}]$.
\par
A \emph{walk} connecting $v_{1}\in V(G)$ and
$v_{q+1}\in V(G)$ is a finite sequence of the form
$$w=(\{v_{i_1},v_{i_2}\},\{v_{i_2},v_{i_3}\},\ldots,\{v_{i_q},v_{i_{q+1}}\})$$
with each $e_{i_j}=\{v_{i_j},v_{i_{j+1}}\}\in E(G)$. We call a walk
$w'=(e_{j_{1}},\dots,e_{j_{t}})$ a \emph{subwalk} of $w$ if
$e_{j_1}\cdots e_{j_t}| e_{i_1}\cdots e_{i_q}.$ An edge $e=\{v,u\}$ of a walk $w$
may be denoted also by $(v,u)$ to emphasize the order that the vertices $v, u$ appear
in the walk $w$.
\emph{Length}
of the walk $w$ is called the number $q$ of edges of the walk. An
even (respectively odd) walk is a walk of \emph{even} (respectively odd) length.
A walk
$w=(\{v_{i_1},v_{i_2}\},\{v_{i_2},v_{i_3}\},\ldots,\{v_{i_q},v_{i_{q+1}}\})$
is called \emph{closed} if $v_{i_{q+1}}=v_{i_1}$. A \emph{cycle}
is a closed walk
$$(\{v_{i_1},v_{i_2}\},\{v_{i_2},v_{i_3}\},\ldots,\{v_{i_q},v_{i_{1}}\})$$ with
$v_{i_k}\neq v_{i_j},$ for every $ 1\leq k < j \leq q$. Depending
on the property of the walk that we want to emphasize we may
denote a walk $w$ by a sequence of vertices and edges $(v_{i_1},
e_{i_1}, v_{i_2}, \ldots ,v_{i_q}, e_{i_q}, v_{i_{q+1}})$ or only
vertices $(v_{i_1},v_{i_2},v_{i_3},\ldots,v_{i_{q+1}})$ or only
edges $(e_{i_1},\ldots ,e_{i_q})$ or the edges and vertices that
we want to emphasize and sometimes we separate the walk into
subwalks. For a walk $w=(e_{i_{1}},\dots,e_{i_{s}})$ we denote by
$-w$ the walk $(e_{i_{s}},\dots,e_{i_{1}})$. Note that, although the graph $G$ has no multiple edges, the
same edge $e$ may appear more than once in a walk. In this case $e$ is
called {\em multiple edge of the walk $w$}. If $w'$ is a subwalk
of $w$ then it follows from the definition of a subwalk that the
multiplicity of an edge in $w'$ is less than or equal to the
multiplicity of the same edge in $w$.
Given an even closed walk $$w =(e_{i_1}, e_{i_2},\cdots,
e_{i_{2q}})$$ of the graph $G$ we denote by
$$E^+(w)=\prod _{k=1}^{q} e_{i_{2k-1}},\ E^-(w)=\prod _{k=1}^{q} e_{i_{2k}}$$
and by $B_w$ the binomial
$$B_w=\prod _{k=1}^{q} e_{i_{2k-1}}-\prod _{k=1}^{q} e_{i_{2k}}$$
belonging to the toric ideal $I_G$. Actually the toric ideal $I_G$
is generated by binomials of this form, see \cite{Vi}. The same
walk can be written in different ways but the corresponding
binomials may differ only in the sign. Note also that different
walks may correspond to the same binomial. For example both walks
$(e_1, e_2, e_3, e_4, e_5, e_6, e_7, e_8, e_9, e_{10})$ and $(e_1,
e_2, e_9, e_8, e_5, e_6, e_7, e_4, e_3, e_{10})$ of the graph $b$
in figure 1 correspond to the same binomial
$e_1e_3e_5e_7e_9-e_2e_4e_6e_8e_{10}$. For convenience by $\bf{w}$
we denote the subgraph of $G$ with vertices the vertices of the
walk and edges the edges of the walk $w$.
If $W$ is a subset of the vertex set $V(G)$ of $G$ then the {\em
induced subgraph} of $G$ on $W$ is the subgraph of $G$ whose
vertex set is $W$ and whose edge set is $\{\{v, u\}\in E(G)|v,u\in
W\}$. When $w$ is a closed walk we denote by $G_w$ the induced
graph of $G$ on the set of vertices $V({\bf
w})$ of ${\bf w}$. An even
closed walk $w=(e_{i_1}, e_{i_2},\cdots, e_{i_{2q}})$ is said to
be primitive if there exists no even closed subwalk $\xi$ of $w$ of smaller
length such that $E^+(\xi)| E^+(w)$ and $E^-(\xi)| E^-(w)$. The walk $w$
is primitive if and only if the binomial $B_w$ is primitive.
\begin{center}
\psfrag{A}{$e_{1}$}\psfrag{B}{$e_{2}$}\psfrag{C}{$e_{3}$}\psfrag{D}{$e_{4}$}\psfrag{E}{$e_{5}$}
\psfrag{F}{$e_{6}$}\psfrag{G}{$e_{8}$}\psfrag{H}{$e_{7}$}\psfrag{I}{$a$}
\psfrag{K}{$e_{1}$}\psfrag{L}{$e_{2}$}\psfrag{M}{$e_{3}$}\psfrag{N}{$e_{4}$}\psfrag{O}{$e_{5}$}
\psfrag{P}{$e_{6}$}\psfrag{Q}{$e_{7}$}\psfrag{R}{$e_{8}$}\psfrag{S}{$e_{9}$}\psfrag{T}{$e_{10}$}\psfrag{U}{$b$}
\includegraphics{ena.eps}\\
{Figure 1.}
\end{center}
The walk $w=(e_1, e_2, e_3, e_4, e_5, e_6, e_7, e_8)$ of the graph
in Figure $1a$ is not primitive, since there exists a closed even
subwalk of $w$, for example $(e_1, e_2, e_7, e_8)$ such that $e_1
e_7| e_1e_3e_5e_7$ and $e_2e_8|e_2e_4e_6e_8$. While the walk in Figure 1b
$(e_1, e_2, e_3,\\ e_4, e_5, e_6, e_7, e_8, e_9, e_{10})$ is
primitive, although there exists an even closed subwalk $(e_3, e_4,
e_8, e_9)$ but neither $e_3e_8$ divides $e_1e_3e_5e_7e_9$ nor
$e_4e_9$ divides $e_1e_3e_5e_7e_9$.
A necessary characterization of the primitive elements were given by Ohsugi
and Hibi in \cite[Lemma 2.1]{OH}:
\begin{thm1}\label{prim} Let $G$ be a finite connected graph.
If $B\in I_{G}$ is primitive, then we have $B=B_{w}$ where $w$ is
one of the following even closed walks:
\begin{enumerate}
\item $w$ is an even cycle of $G$
\item $w=(c_{1},c_{2})$, where $c_{1}$ and $c_{2}$ are odd cycles of $G$ having exactly one common vertex
\item $w=(c_{1},w_{1},c_{2},w_{2})$, where $c_{1}$ and $c_{2}$ are odd cycles of $G$ having no
common vertex and where $w_{1}$ and $w_{2}$ are walks of $G$ both of which combine a vertex
$v_{1}$ of $c_{1}$ and a vertex $v_{2}$ of $c_{2}$.
\end{enumerate}
\end{thm1}
It is easy to see that any binomial in the first two cases is always
primitive but this is not true in the third case. Theorem
\ref{primitive} characterizes completely all primitive binomials.
We will finish this section with a necessary and sufficient
characterization of circuits that was given by Villarreal in
\cite[Proposition 4.2]{Vi}:
\begin{thm1}\label{circuit}Let $G$ be a finite connected graph. The binomial $B\in I_{G}$ is circuit if and only if $B=B_{w}$ where
\begin{enumerate}
\item $w$ is an even cycle or
\item two odd cycles intersecting in exactly one vertex or
\item two vertex disjoint odd cycles joined by a path.
\end{enumerate}
\end{thm1}
\section{Primitive walks of graphs}
The aim of this chapter is to determine the form of primitive
walks by making more precise the corresponding result by
Ohsugi-Hibi, see Theorem \ref{prim} or \cite[Lemma 2.1]{OH}.
\\A {\em cut edge} (respectively {\em cut vertex}) is an edge (respectively vertex) of
the graph whose removal increases the number of connected
components of the remaining subgraph. A graph is called {\em
biconnected} if it is connected and does not contain a cut
vertex. A {\em block} is a maximal biconnected subgraph of a given
graph $G$.
\\Every even primitive walk $w=(e_{i_1},\ldots,e_{i_{2k}})$
partitions the set of edges in the two sets $w^+= \{e_{i_j}|j \
{\it odd}\}, w^-=\{e_{i_j}|j \ {\it even}\}$, otherwise the
binomial $B_w$ is not irreducible.\\
The edges of $w^+$ are called odd edges of the walk and those of
$w^-$ even. Note that for a closed even walk whether an edge is even or
odd depends only on the edge that you start counting from. So it is
not important to identify whether an edge is even or odd but to separate the
edges in the two disjoint classes. \emph{Sink} of a block $B$ is a
common vertex of two odd or two even edges of the walk $w$ which
belong to the block $B$. In particular if $e$ is a cut edge of a
primitive walk then $e$ appears at least twice in the walk and
belongs either to $w^+$ or $w^-$. Therefore both vertices of $e$
are sinks. Sink is a property of the walk $w$ and not of the
underlying graph $\bf{w}$. For example in Figure 1a the walk
$(e_{1},e_{2},e_{7},e_{8})$ has no sink, while in the walk
$(e_{1},e_{2},e_{7},e_{8},e_{1},e_{2},e_{7},e_{8})$ all four
vertices are sinks. Note also that the walk $(e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7},e_{8})$
in Figure 1a
has one cut
vertex which is not a sink of either block. The walk
$(e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7},e_{8},e_{9},e_{10})$
in Figure 1b has two cut vertices which are both sinks of all of
their blocks. Theorem \ref{primitive} explains that this is
because the first one is not primitive while the second is.
\begin{center}
\psfrag{D}{$B$}
\includegraphics{four.eps}
\\
{Figure 2.}
\end{center}
\begin{thm1} \label{primitive}
Let $G$ a graph and $w$ an even closed walk of $G$. The walk $w$
is primitive if and only if
\begin{enumerate}
\item every block of $\bf{w}$ is a cycle or a cut edge,
\item every multiple edge of the walk $w$ is a double edge of the walk and a cut edge of $\bf{w}$,
\item every cut vertex of $\bf{w}$ belongs to exactly two blocks and it is a sink of both.
\end{enumerate}
\end{thm1}
\textbf{Proof.} Let $w$ be an even primitive closed walk and let B
be a block of ${\bf w}$ which is not a cut edge. We will prove
that it is a cycle. Suppose not. Let $w=(e_{i_1},
\ldots , e_{i_{2s}})$ and $w_B=(e_{i_{j_1}}, \ldots , e_{i_{j_q}})$
the closed subwalk of
$w$ such that the graph of ${\bf w}_B$ is the block $B$, where
$e_{i_{j_1}}, \ldots , e_{i_{j_q}}$ are all the edges of the walk $w$ that belong to the the block $B$ and
$j_1<j_2<\dots <j_q$. This is a
closed walk since two blocks intersect in at most one point which is a cut vertex of the graph ${\bf w}$. Since
$B$ is not a cycle, there must be at least one vertex of the walk
$w_B$ which appears twice in $w_B$. If it was exactly one
vertex like that then it should be a cut vertex of $B$
contradicting the biconnectivity of the block $B$. Therefore, there
must exist at least two vertices $v,u$ of the block $B$ that
appear at least twice in the walk $w_B$ and so $w$ can be written
$w=(v,w_{1},u,w_{2},v,w_{3},u,w_{4})$, where
$w_{1},w_{2},w_{3},w_{4}$ are subwalks of $w$. Note that the
vertices $v,u$ are in this order in $w$ since otherwise $v$ or $u$
will be a cut vertex of $B$. The walk $w$ is primitive, therefore
the closed walk $(v,w_{1},u,w_{2},v)$ is odd, one of the lengths of the subwalks
$w_{1},w_{2}$ has the same parity as the length of $w_{3}$, and both the first edge of
$w_{1}$ and the last of $w_{2}$ belong to $w^+$. Combining
all these, exactly one of the two closed walks
$\xi_1=(v,w_{1},u,-w_{3},v)$ or $\xi_2=(v,w_{3},u,w_{2},v)$ is a
closed even subwalk of $w$ such that either $E^+(\xi_1)| E^+(w)$ and
$E^-(\xi_1)| E^-(w)$ or $E^+(\xi_2)| E^-(w)$ and $E^-(\xi_2)| E^+(w)$.
This contradicts the primitiveness of $w$. So every block is a cycle
or
a cut edge. \\
Let $e=\{u,v\}$ be a multiple edge of $w$. Whenever $e$ appears is either in $w^+$ or $w^-$, since $w$ is a primitive walk.
The edge $e$ may appear in the walk $w$ in two different
ways, as $(\dots, u,
e, v, \dots)$ or $(\dots, v, e, u, \dots)$. There are two cases.
First case: At least two times the edge appears in the same
way $(\dots, u, e, v, \dots)$ (or $(\dots, v, e, u, \dots)$).
Then the walk $w$ can be written in the form $(u, e, v, w_1, u, e,
v, \dots)$. Since $w$ is primitive and $e$ is written as the first edge of $w$, all the times that $e$ appears
is in $w^+$. Therefore the walk $w_1$ is odd, which means that
$\xi=(u, e, v, w_1, u)$ is an even closed walk, $E^+(\xi)|E^+(w)$
and $E^-(\xi)|E^-(w)$. This contradicts the primitiveness of the walk
$w$. \\
Second case: The edge $e$ appears exactly twice in the walk
and in the two different ways, so $w=(u, e, v, w_1, v, e, u, w_2,
u)$. As before the walks $w_1$, $w_2$ are odd, therefore the first
and the last edges of $w_1$ and $w_2$ all belong to $w^-$.
Suppose that $e$ is not a cut edge of $\bf{w}$ then the $w_1$,
$w_2$ have at least one common vertex $y$. We rewrite $w$ as
$(u,e,v,w_1',y,w_1'',v,e,u,w_2',y,w_2'',u)$. Since $w_2$ is an odd
walk, one of $w_2', w_2''$ is odd and the other is even. Therefore
exactly one of the two walks $(u,e,v,w_1',y,w_2'',u)$,
$(u,e,v,w_1',y,-w_2',u)$ is an even closed walk $\xi$ such that
$E^+(\xi)|E^+(w)$ and $E^-(\xi)|E^-(w)$, contradicting the
primitiveness of the walk $w$. We conclude that $e$ is a double
edge of the walk $w$ and a cut edge of $\bf{w}$.\\
Let $v$ be a cut vertex, then it belongs to at least two blocks.
Since $v$ is a cut vertex $w$ can be written as $w=(v, e_1,\dots
,e_s, v, e_{s+1}, \dots, e_t, v, \dots )$. where $e_{1}, e_{s}$
are in the same block $B$ and $\{e_i|1\leq i\leq s\}\cap \{e_i|s+1\leq i\leq t\}=\emptyset$. Then $e_{1},e_{s}$ are both in
$w^+$. Otherwise $(v, e_{1},\dots,e_{s},v)$ is an even
closed subwalk of $w$, contradicting the primitiveness of the walk
$w$. So $v$ is a sink and the subwalk $(v,e_{1},\dots,e_{s},v)$ is odd. Similarly
the walk $(v,e_{s+1},\dots,e_{t},v)$ is odd and $e_{s+1}, e_t$ are both in $w^-$.
Then $w'=(v, e_1,\dots ,e_s, v, e_{s+1}, \dots, e_t, v)$ is
an even subwalk of $w$ such that $E^+(w')|E^+(w)$ and $E^-(w')|E^-(w)$
and since $w$ is primitive $w'=w$. We
conclude that $v$ belongs to exactly two blocks of ${\bf w}$ and
it is a sink of both.
\newline
Conversely let $w$ be an even closed walk satisfying the three
conditions of the Theorem which is not primitive. Then there exists
a primitive subwalk $w'$ of $w$ of smaller length than $w$, such
that $E^+(w')|E^+(w)$ and $E^-(w')|E^-(w)$. From the first part of
the proof we know that also $w'$ satisfies the three conditions
of the Theorem \ref{primitive}. We claim that the graphs ${\bf w}$ and ${\bf w}'$
have exactly the same blocks.
Let $B_{w'}$ be a block of $w'$ then there exists a block $B_w$ of
$w$ such that $B_{w'}\subset B_w$. From the first condition
$B_{w'}$ is a cut edge or a cycle. Suppose that $B_{w'}=\{e\}$ is
a cut edge of
${\bf w}'$ then $e$ must be double edge of $w'$.
Since $E^+(w')|E^+(w)$ and $E^-(w')|E^-(w)$ the edge
$e$ is a multiple edge of $w$ and therefore from the second
condition a cut edge of ${\bf w}$, thus a block of ${\bf w}$. In
the case that $B_{w'}$ is a cycle obviously $B_w$ is the same cycle and
therefore $B_{w'}= B_w$. So all blocks of $\mathbf{w}'$ are blocks
of $\mathbf{w}$. Conversely suppose that there exist a block of $\mathbf{w}$
which is not a block of $\mathbf{w}'$. Since ${\bf w}$ is connected
there must be at least
one block of $\mathbf{w}$ which is not a block of $\mathbf{w}'$
and has a contact point with $\mathbf{w}'$. Then this point should
be a sink of both since $E(w')^+| E(w)^+$ and $E(w')^-| E(w)^-$.
But if it is a sink of $w'$ then it should belong to exactly two
blocks of ${\bf w}'$. This implies that it should belong to at least three
blocks of ${\bf w}$, a contradiction to the third property of $w$.\\ Therefore the graphs ${\bf w}$ and ${\bf
w}'$ are identical and every simple edge of the walk $w'$ is a
simple edge of $w$ and every double edge (cut edge) of the walk
$w'$ is
a double edge of $w$. Therefore $E^+(w')=E^+(w)$ and
$E^-(w')=E^-(w)$. Therefore they have the same length,
a contradiction. We conclude that $w$ is primitive. \hfill $\square$
From Theorem \ref{primitive} easily follows the following corrolary that describes the underlying graph
of a primitive walk.
\begin{cor1} \label{primitive-graph}
Let $G$ a graph and $W$ a subgraph of $G$. The subgraph $W$ is the graph ${\bf w}$ of a primitive walk $w$
if and only if
\begin{enumerate}
\item every block of $W$ is a cycle or a cut edge and
\item every cut vertex of $W$ belongs to exactly two blocks and separates the graph in two parts, the total number of edges
of the cyclic blocks in each part is odd.
\end{enumerate}
\end{cor1}
\section{Minimal and indispensable binomials of graphs }
The first aim of this section is to characterize the walks $w$ of the graph $G$ such that the binomial $B_w$ belongs to a minimal system
of generators of the ideal $I_G$. Certainly the walk has to be primitive, but this is not enouph. The walk must have more properties,
the first one it depends on the graph ${\bf w}$ and the rest on the induced graph $G_w$ of $w$, see Proposition \ref{sprim} and Theorem \ref{minimal}.
\begin{def1}
A binomial $B\in I_{G}$ is called minimal if it belongs to a
minimal system of binomial generators of $I_{G}$.
\end{def1}
\begin{def1} We call strongly primitive walk a primitive walk that has
not two sinks with distance one in any cyclic block.
\end{def1}
\begin{prop1}\label{sprim} Let $w$ be an even closed walk such that the binomial $B_w$ is minimal
then the walk $w$ is strongly primitive.
\end{prop1}
\textbf{Proof.}
\begin{center}
\psfrag{A}{$\xi_1$}\psfrag{B}{$\xi_3$}\psfrag{D}{$\xi_2$}\psfrag{C}{$e$}\psfrag{E}{$u$}\psfrag{F}{$v$}
\includegraphics{neo.eps}\\
{Figure 3.}
\end{center}
The binomial $B_{w}$ is minimal therefore the walk $w$ is primitive.
Suppose that $w$ is not strongly primitive, then there exist two
sinks $v, u$ of the same block $B$ with distance one. We will call
$e$ the edge $\{v,u\}$. Then $w$ can be written as
$(v,\xi_{1},v,e,u,\xi_{2},u,\xi_{3},v)$ for some walks $\xi_1, \xi_2, \xi_3$, where at least the first and the
last edge of $\xi_{3}$ are in the block $B$. The walks $\xi_1,\ \xi_2$ are
closed walks and since $w$ is primitive they are necessarily odd.
Therefore $\xi_3$ is also odd. So the first and the last edge of
$\xi_{3}$, as also $e$ are in $w^-$. Since $v,u$ are sinks the closed
walks $w_{1}=(\xi_{1},e,\xi_{2},e)$ and $w_{2}=(e,\xi_{3})$ are
both even and the binomial
$B_{w}=B_{w_{1}}\frac{E^+(w_{2})}{e}+B_{w_{2}}\frac{E^-(w_{1})}{e}$
is not minimal, a contradiction. Note that $E^+(w_2)/e\neq 1\neq
E^-(w_1)/e$, otherwise the even closed walk $w_1$ or $w_2$ has length 2,
which is impossible since $G$ has no multiple edges. \hfill $\square$
While the property of a walk to be primitive depends only on the
graph ${\bf w}$, the property of the walk to be minimal or
indispensable depends also on the induced graph $G_w$. An edge $f$
of the graph $G$ is called a \emph{chord} of the walk $w$ if the vertices of the edge
$f$ belong to $V(\bf{w})$ and $f\notin E (\bf{w})$. In other words an
edge is called chord of the walk $w$ if it belongs to $E(G_w)$ but not in $E(\textbf{w})$. Let $w$ be an
even closed walk
$((v_{1},v_{2}),(v_{2},v_{3}),\ldots,(v_{2k},v_{1}))$ and
$f=\{v_{i},v_{j}\}$ a chord of $w$. Then $f$ \emph{breaks} $w$ in two
walks:
$$w_{1}=(e_{1},\ldots,e_{i-1}, f, e_{j},\ldots,e_{2k})$$ and
$$w_{2}=(e_{i},\ldots,e_{j-1},f),$$ where $e_{s}=(v_{s},v_{s+1}),\ 1\leq s\leq 2k$ and $e_{2k}=(v_{2k},v_{1}).$
The two walks are both even or both odd.
In the next definition we are interested in chords of the walk. We
partition the set of chords of a primitive even walk in three
parts: bridges, even chords and odd chords.
\begin{def1} A chord $f=\{v_{1},v_{2}\}$
is called bridge of a primitive walk $w$ if
there exist two different blocks $B_{1},B_{2}$ of $\bf{w}$ such that
$v_{1}\in B_{1}$ and $v_{2}\in B_{2}$. A chord is called even
(respectively odd) if it is not a bridge and breaks the walk in
two even walks (respectively in two odd walks).
\end{def1}
In the walk of Figure 4, there are three chords which are bridges
of $w$, those marked by $b$ and there is one chord which is even,
it is marked by $c$. In the walks of Figure 5, all chords
are odd. Note that the two vertices of a bridge may also belong to the same block,
for example that happens in one of the three bridges in Figure 4.
\begin{center}
\psfrag{A}{$i$}\psfrag{B}{$j$}\psfrag{C}{$c$}\psfrag{D}{$B_{s}$}\psfrag{E}{$b$}\psfrag{F}{$b$}\psfrag{G}{$b$}
\includegraphics{six.eps}\\
{Figure 4.}
\end{center}
The next definition generalizes the corresponding definitions of
Ohsugi and Hibi, see \cite{Hi-O}.
\begin{def1}\label{creffec} Let
$w=((v_{i_{1}},v_{i_{2}}), (v_{i_{2}},v_{i_{3}}),\cdots ,
(v_{i_{2q}},v_{i_{1}}))$ be a primitive walk. Let
$f=\{v_{i_{s}},v_{i_{j}}\}$ and $f'=\{v_{i_{s'}},v_{i_{j'}}\}$ be two
odd chords (that means not bridges and $j-s,j'-s'$ are even) with $1\leq
s<j\leq 2q$ and $1\leq s'<j'\leq 2q$. We say that $f$ and $f'$
cross effectively in $w$ if $s'-s$ is odd (then necessarily $j-s',
j'-j, j'-s$ are odd) and either $s<s'<j<j'$ or $s'<s<j'<j$.
\end{def1}
\begin{def1}\label{F_4}
We call an $F_4$ of the walk $w$ a cycle $(e, f, e', f')$ of
length four which consists of two edges $e,e'$ of the walk $w$
both odd or both even, and two odd chords $f$ and $f'$ which
cross effectively in $w$.
\end{def1}
In Figure 5 there are two cyclic blocks of primitive walks and in
each one exactly two odd chords which cross effectively. In the
first block they form an $F_{4}$, while in the second they do not.
Combining Definitions \ref{creffec} and \ref{F_4} two odd chords
are part of an $F_4$ if $i'-j=\pm 1$ and $j'-i=\pm 1$, or
$i'-i=\pm 1$ and $j'-j=\pm 1$.
\begin{center}
\psfrag{A}{$f$}\psfrag{B}{$f'$}\psfrag{C}{$B_{s}$}\psfrag{D}{$F_{4}$}
\psfrag{E}{$Cross \ effectively\ odd \ chords$}\psfrag{F}{$f'$}\psfrag{G}{$f$}
\includegraphics{seven.eps}\\
{Figure 5.}
\end{center}
\begin{def1}\label{stronglycross}
Let $w$ be a primitive walk and $f, f'$ be two odd chords. We say that $f, f'$ cross strongly effectively
in $w$ if they cross effectively and they do not form an $F_4$ in $w$.
\end{def1}
\begin{prop1} \label{minimal1} Let $w$ be a primitive walk. If $B_{w}$ is a minimal binomial then
all the chords of $w$ are
odd and there are not two of them which cross strongly effectively.
\end{prop1}
\textbf{Proof.} Let $w=(e_1, e_2, \dots ,e_{2s})$ be a
primitive walk. If $B_{w}$ is a minimal binomial, then from
Proposition \ref{sprim} it follows that $w$ is strongly primitive.
Let $e=\{v_1, v_{2l-1}\}$ be an even chord of $w$, and let
$w_1=(e_1, e_2, \dots ,e_{2l-1},e)$, $w_2=(e, e_{2l},\dots
,e_{2s})$ be the two even walks that $e$ breaks $w$. Then
$B_{w}=B_{w_{1}}\frac{E^+(w_{2})}{e}-B_{w_{2}}\frac{E^-(w_{1})}{e}$,
so $B_w$ is not minimal. Note that $E^+(w_2)/e\neq 1\neq
E^-(w_1)/e$, since $G$ has no multiple edges. \\Suppose that a
minimal binomial $B_w$ has a bridge $e=\{v_1, v_2\}$. Since $v_1,
v_2$ belong to different blocks there must be at least one cut
vertex $v$ such that the walk $w$ can be written $(v, w_1, v_1,
w_2, v, w_3, v_2, w_4, v)$. Note that if $v=v_{1}$ or $v=v_{2}$
one of the walks $w_{1}$, $w_{4}$ is empty. The closed walks $(v, w_1,
v_1, w_2, v)$ and $(v, w_3, v_2, w_4, v)$ are both odd, otherwise
$B_w$ is not primitive. Therefore one of the $w_1, w_2$ has to be
odd and the other even. Similarly for $w_3, w_4$. Note also that the four
walks $(v,w_{1},v_{1},w_{2},v,w_{3},v_{2},w_{4},v)$ ,
$(v,w_{1},v_{1},w_{2},v,-w_{4},v_{2},-w_{3},v)$,
$(v,-w_{2},v_{1},-w_{1},v,w_{3},v_{2},w_{4},v)$ and
$(v,-w_{2},v_{1},-w_{1},v,-w_{4},v_{2},-w_{3},v)$ give the same
binomial. Therefore we can assume that $w_{1},w_{3}$ are odd and
$w_{2},w_{4}$ are even. Then the two closed walks
$\zeta_{1}=(w_{2},w_{3},e)$ and $\zeta_{2}=(w_{4},w_{1},e)$ are
even and
$B_{w}=B_{\zeta_{1}}\frac{E^+(\zeta_{2})}{e}-B_{\zeta_{2}}\frac{E^-(\zeta_{1})}{e}$
is not minimal, a contradiction. Note that
$\frac{E^+(\zeta_{2})}{e}\neq 1 \neq
\frac{E^-(\zeta_{1})}{e}$, since $G$ has no multiple edges.\\
Suppose now that $w$ has two odd chords $f=\{v_1, v_2\} ,f'=\{u_1, u_2\}$ which cross strongly
effectively in
$w$. Then $w$ is in the form
$(v_1, w_1, u_1, w_2, v_2, w_3, u_2, w_4, v_1)$.
We have that the walks $\xi_{1}=(w_{1},f',-w_{3},f)$, and
$\xi_{2}=(w_{2},f,-w_{4},f')$ are even, since the walks
$(w_{1},w_{2},f),(w_{2},w_{3},f'),(w_{3},w_{4},f), (w_{4},w_{1},f')$ are
odd. Then
$B_{w}=B_{\xi_{1}}\frac{E^+(\xi_{2})}{ff'}-B_{\xi_{2}}\frac{E^-(\xi_{1})}{ff'}$
is not minimal, a contradiction. Note that since the odd chords $f, f'$ do not form an
$F_{4}$, $\frac{E^+(\xi_{2})}{ff'}\neq 1 \neq \frac{E^-(\xi_{1})}{ff'}$.
\hfill $\square$
\begin{def1} Two primitive walks $w, w'$ differ by an
$F_4$, $\xi=(e_1,f_1,e_2,f_2)$, if $w=(w_1, e_1, w_2, e_2)$ and $w'=(w_1,
f_1,-w_2, f_2)$, where both $w_1, w_2$ are odd walks. Two even closed walks $w, w'$
are $F_4$-equivalent if either $w=w'$ or there exists a series of walks $w_1=w, w_2, \dots, w_{n-1}, w_n=w'$ such that
$w_i$ and $w_{i+1}$ differ by an $F_4$, where $1\leq i\leq n-1$.
\end{def1}
Note that if $w$ and $w'$ are $F_4$-equivalent then the induced graphs $G_w$ and $G_{w'}$ are
equal. We denote by $L_w$ the equivalence class of $w$ under the $F_4$-equivalent relation.
\begin{prop1} \label{minimal2} If the primitive walks $w$ and $w'$
are $F_4$-equivalent then $B_w$ is minimal if and only if $B_{w'}$ is minimal.
\end{prop1}
\textbf{Proof.} Suppose that $w=(w_1, e_1, w_2, e_2)$ and $w'=(w_1,
f_1,-w_2, f_2)$ are even closed walks which differ by an $F_{4}$, where $F_{4}$ is $\xi=(e_1,f_1,e_2,f_2)$.
Then $B_{w}=B_{w'}-\frac{E^-(w)}{e_{1}e_{2}}B_{\xi}$ and the result follows.\hfill $\square$
The $F_4$ separates the vertices of ${\bf w}$ in two parts $V({\bf w}_1), V({\bf w}_2)$, since both edges $e_1, e_2$ of an $F_4$, $(e_1, f_1, e_2, f_2)$, belong to the same block of $w=(w_1, e_1, w_2, e_2)$,
\begin{def1} We say that an odd chord $f$ of a primitive walk $w=(w_1, e_1, w_2, e_2)$ crosses an $F_4$, $(e_1, f_1, e_2, f_2)$,
if one of the vertices of $f$ is in $V({\bf w}_1)$, the other in $V({\bf w}_2)$ and $f$ is different from $f_1, f_2$.
\end{def1}
\begin{prop1} \label{minimal3} Let $w$ be a primitive walk. If $B_{w}$ is a minimal binomial, then
no odd chord crosses an $F_4$ of the walk $w$.
\end{prop1}
\textbf{Proof.} Let $B_w$ be a minimal binomial. Suppose that there exists an odd chord $f=\{v_1, v_2\}$ that crosses the $F_4$, $(e_1, f_1, e_2, f_2)$, of the walk
$w=(w_1, e_1, w_2, e_2)$. Then $w$ can be written in the form $(w_1',v_1,w_1'', e_1, w_2',v_2, w_2'', e_2)$. The chord $f$ is odd
therefore the walks $(f, w_2'',e_2, w_1')$ and $(f, w_1'',e_1, w_2')$ are both odd. Also, since $(e_1, f_1, e_2, f_2)$ is an $F_4$,
the walks $w_1$ and $w_2$ are both odd. Therefore $(w_1'', f_1, -w_2'',f)$ and $(w_1', f, -w_2',f_2)$ are both even.
So, from the definition, $f$ is an even chord of $w'=(w_1, f_1, w_2, f_2)$. Note that $f$ is not a bridge of $w'$ since it is not a bridge
of $w$. Therefore from Proposition \ref{minimal1} $B_{w'}$ is not minimal and from Proposition \ref{minimal2} $B_w$ is not minimal, a contradiction.
\hfill $\square$
In fact, for a primitive walk $w$ the walks in $L_w$ are primitive, an $F_4$ of $w$ is an $F_4$ for all walks in $L_w$, although sometimes chords and edges change role.
A bridge of $w$ is a bridge for every walk in $L_w$ and odd chords of $w$ (respectively even chords)
are odd chords (respectively even chords) for every walk in $L_w$, except if they cross an $F_4$. In the last case they may change parity, it depends on how many $F_4$
they cross.
\begin{thm1} \label{minimal}
Let $w$ be an even closed walk. $B_{w}$ is a minimal binomial if
and only if $w$ is strongly primitive, all the chords of $w$ are
odd, there are not two of them which cross strongly effectively and no odd chord crosses an $F_4$ of the walk $w$.
\end{thm1}
\textbf{Proof.} The one direction follows from Propositions \ref{minimal1} and \ref{minimal3}. \newline For the converse, let $w$ be an
even closed walk such that all the chords of $w$ are odd, there
are not two of them which cross strongly effectively and no odd chord crosses an $F_4$ of the walk $w$.
Suppose that $B_{w}$ is not minimal. Then there exists a minimal
walk $\delta$ such that $E^+(\delta) | E^+(w)$ and
$E^+(\delta) \not= E^+(w)$,
thus edges of $\delta^{+}$ are edges of $w^{+}$.
We have
$\deg_{A_G}(E^-(\delta))=\deg_{A_G}(E^+(\delta))<\deg_{A_G}(E^+(w))=\deg_{A_G}(E^-(w))$. This means that the vertices of $\delta^{-}$ are in ${\bf w}$
and so edges of
$\delta^{-}$ are edges or chords of $w$, which means actually they are odd
chords by hypothesis.\\ We claim that every such $\delta $ is an $F_4$ of $w$. Suppose not, then among all those
walks $\delta $ which are not $F_4$ of $w$ and $E^+(\delta) | E^+(w)$ and
$E^+(\delta) \neq E^+(w)$, we choose one, $\gamma$, such that $\gamma $ has the fewest possible chords of $w$.\\
First case: The walk $\gamma $ does not have any chords of $w$, then all edges of $\gamma^{-}$ are edges of $w$, so
$\gamma^{+}\subset w^{+}$ and $\gamma^{-}\subset w$ and since $w$
is primitive then there exists at least one $e \in \gamma^{-}\cap
w^{+}$. Therefore $\gamma=(\dots, e_{1},e,e_{2},\dots)$, where all edges
$e_{1},e,e_{2}$ are in $w^{+}$. Note that whenever
there are two blocks joined by a cut vertex, the adjoining edges in the two different blocks have different parity, since the walk $w$ is primitive.
Thus all the edges $e_{1},e,e_{2}$ are in one block
of $w$, which necessarily is a cycle and then the two vertices in
between are sinks of $w$. A contradiction to strongly
primitiveness. Note that if two of $e_{1},e,e_{2}$ are the same edge, then
this edge will be a double edge of $w$ and therefore a cut edge of ${\bf w}$, so the edges $e_{1},e,e_{2}$ are in two blocks, a contradiction.\\
Second case: $\gamma^{-}$ has at least one chord of $w$. Then
$\gamma=(w_{1},f_{1},w_{2},f_{2},\dots,w_{s},f_{s})$ where
$w_{1},\dots,w_{s}$ are subwalks of $w$ and $f_{1},\dots,f_{s}$
are odd chords of $w$ satisfying the hypotheses and $s$ is minimal. Both vertices of
an odd chord $f$ of $w$ are in the same cyclic block, thus $f$ divides
$w$ into two regions $w^{+}(f),w^{-}(f)$. There must exist at
least one chord $f_{i}$ such that the region $w^{+}(f_{i})$ does
not contain a chord.
The last edge of $w_i$ and the first of $w_{i+1}$ are in $\gamma^+\subset w^+$. The chord $f_i$ is odd, therefore the
one of these two edges is in $w^+(f_i)$ and the other in $w^-(f_i)$. Without loss of generality we can suppose that the first edge of $w_{i+1}$
is in $w^+(f_i)$. The walk $\gamma$ is closed and none of the vertices of $f_i$ is a cut vertex of $\gamma $, since $f_i$ is not a bridge of $w$, therefore
there must be a chord which has a
vertex in $w^{+}(f_{i})$ and a vertex in $w^{-}(f_{i})$. This chord is the $f_{i+1}$ since $w^+(f_i)$ does not contain a chord.
Let $f_i=\{v_{i_{s}},v_{i_{j}}\}$ and $f_{i+1}=\{v_{i_{s'}},v_{i_{j'}}\}$.
Since the
first and the last edge of $w_{i+1}$ are in $\gamma^{+}\subset
w^{+}$, $s'-j$ (the number of edges of $w_{i+1}$) is odd. But from
the hypothesis $f_{i},f_{i+1}$ can not cross effectively except if
they form an $F_{4}$, which means that either $|s'-j|=|j'-s|=1$ or
$|s'-s|=|j'-j|=1$. In the first case $w$ is an $F_4$, the $(e_{i_s}, f_i, e_{i_{s'}}, f_{i+1})$.
In the second case there exists an even minimal walk
$\gamma'=(w_{1},f_{1}, \dots ,w_{i},e_{i_{s+1}},-w_{i+1},e_{i_{s+1}},w_{i+2},\dots)$
with two less chords, a contradiction to the minimality of the chords of $\gamma$.
We conclude that if for a walk $\delta $ we have $E^+(\delta) | E^+(w)$ and
$E^+(\delta) \neq E^+(w)$, then $\delta$ is an $F_4$ of $w$. Remark that the conditions of Theorem \ref{minimal} if they are satisfied by
the walk $w$, then they are also satisfied by any other walk in $L_w$. We fix a minimal set $\{B_{w_1},\cdots ,B_{w_t}\}$ of binomial generators for the ideal $I_{A_G}$,
for some even closed walks $w_1, \dots ,w_t$. For a
$w' \in { L}_w$ we define
$$
r(w')=min\{\sum_{l=1}^t\vert g_l \vert \mid B_{w'}=\sum_{i=1}^k
g_l B_{w_l} \}
$$
and $\vert g_l \vert$ is the number of monomials of $g_l$. We take a walk $v \in
{ L}_w$ such that $r(v)$ is minimal. We claim that $B_v$ is one of the minimal generators $B_{w_1},\cdots ,B_{w_t}$.
Suppose not,
then it is written in the form
$B_{v}=E^+(v)-E^-(v)=\sum_{r=1}^q g_{i_r}B_{w_{i_r}}$ and without loss of generality we can suppose that
$E^+(w_{i_1})|E^+(v)$ and
$E^+(v)/E^+(w_{i_1})\not=1 $ is a monomial in $g_1$. Then $w_{i_1} $ is necessarily an $F_4$, $(e_1, f_1, e_2, f_2)$, of $v=(v_1, e_1, v_2, e_2)$ and $e_1e_2=E^+(w_{i_1})|E^+(v)$.
Consider $v'=(v_1, f_1, v_2, f_2)$, then $v'\in L_w$ and $$B_{v'}=E^+(v')-E^-(v')=\frac{E^+(v)}{e_{1}e_{2}}f_1f_2-E^-(v)=(g_1-\frac{E^+(v)}{e_{1}e_{2}})B_{w_{i_1}}+\sum_{r=2}^q
g_{i_r}B_{w_{i_r}}. $$ But then $r(v')<r(v)$ which is a contradiction. Note that the coefficients of the monomials in $g_i$
are $1$ or $-1$, see \cite{ChKT, DS, St} for more information about the generation of a toric ideal.\\ Therefore $B_v$ is minimal and
from Proposition \ref{minimal2} $B_w$ is
minimal. \hfill $\square$
Note that in the cases in Theorem \ref{minimal} where we have
more than one $F_4$, two $F_4$ of the walk cannot have a common edge
and they cannot cross, since in all these cases we get
an odd chord which crosses an $F_4$.
\begin{thm1}\label{indispen}
Let $w$ be an even closed walk. $B_{w}$ is an
indispensable binomial if and only if $w$ is a strongly primitive walk, all the chords of $w$ are
odd and there are not two of them which cross effectively.
\end{thm1}
\textbf{Proof.} From Proposition \ref{minimal2} if $w$ has an $F_4$ then $B_w$ is not indispensable, since it can
be replaced by $B_{w'}$. So $w$ has not an $F_4$, then the result follows from Theorem \ref{minimal}.
\newline
Conversely, let $w$ be a strongly primitive walk, all the chords of $w$ are
odd and there are not two of them which cross effectively.
Suppose that $B_{w}$ is not indispensable. Then there exists a minimal
walk $\delta \neq w$ such that $E^+(\delta) | E^+(w)$,
thus edges of $\delta^{+}$ are edges of $w^{+}$.
We have
$\deg_{A_G}(E^-(\delta))=\deg_{A_G}(E^+(\delta))\leq \deg_{A_G}(E^+(w))=\deg_{A_G}(E^-(w))$. This means that the vertices of $\delta^{-}$ are in ${\bf w}$
and so edges of
$\delta^{-}$ are edges or chords of $w$, which means actually they are odd
chords by hypothesis.\\
First case: The walk $\delta $ does not have any chords of $w$. In that case the proof is exactly the same as in the corresponding
part in the proof of Theorem \ref{minimal}.\\
Second case: $\delta^{-}$ has at least one chord of $w$. Then
$\delta=(w_{1},f_{1},w_{2},f_{2},\dots,w_{s},f_{s})$ where
$w_{1},\dots,w_{s}$ are subwalks of $w$ and $f_{1},\dots,f_{s}$
are odd chords of $w$ satisfying the hypotheses. There must exist at
least one chord $f_{i}$ such that the region $w^{+}(f_{i})$ does
not contain a chord.
The last edge of $w_i$ and the first of $w_{i+1}$ are in $\delta^+\subset w^+$. The chord $f_i$ is odd, therefore the
one of these two edges is in $w^+(f_i)$ and the other in $w^-(f_i)$. Without loss of generality
we can suppose that the first edge of $w_{i+1}$
is in $w^+(f_i)$. The walk $\delta$ is closed and none of the vertices of $f_i$ is a cut vertex of $\delta $, since $f_i$ is not a bridge of $w$, so
there must be a chord of $w$ which has a
vertex in $w^{+}(f_{i})$ and a vertex in $w^{-}(f_{i})$. This chord is the $f_{i+1}$ since $w^+(f_i)$ does not contain a chord.
Let $f_i=\{v_{i_{s}},v_{i_{j}}\}$ and $f_{i+1}=\{v_{i_{s'}},v_{i_{j'}}\}$.
Since the
first and the last edge of $w_{i+1}$ are in $\delta^{+}\subset
w^{+}$, the number of edges $(s'-j)$ of $w_{i+1}$ is odd. Therefore $f_i$ and $f_{i+1}$ cross effectively, a contradiction.
\\ Therefore $B_w$ is indispensable. \hfill $\square$
Remark that combining Theorem \ref{minimal}, Proposition \ref{minimal2} and Theorem \ref{indispen},
we have that if $B_{w}$ is indispensable then $w$ has no $F_{4}$ and if $B_{w}$ is minimal but not
indispensable then $B_{w}$ has at least one $F_{4}$. If no minimal generator has an $F_4$ then the toric ideal
is generated by indispensable binomials, so the ideal $I_G$ has a unique system of
binomial generators and conversely.
An even closed walk $w$ of a graph $G$ is called {\em
fundamental} if for every even closed walk $w'$ of the induced
subgraph of $G_{w}$ it holds $B_{w'}\in <B_w>$. A binomial
$B_w$ is fundamental if $w$ is fundamental, see \cite{Hi-O}.
\begin{thm1}\label{fundam} If $w$ is an even closed walk, then the binomial
$B_w$ is fundamental if and only if $w$ is a circuit and
has no chords except in the case that it is a cycle with no even chords and at most one odd chord.
\end{thm1}
\textbf{Proof.} Let $w$ be an even closed walk such that the binomial
$B_{w}$ is fundamental. From \cite[Theorem 1.1.]{OH2} we know that
$B_{w}$ is a circuit and $B_{w}$ is an indispensable binomial.
Since $B_{w}$ is a circuit, from Proposition \ref{circuit} there
are three cases. If $w$ is a cycle the result follows from
\cite[Lemma 4.2.]{OH2}. In the other two cases, $w$ is a circuit
with no even chords and bridges, since $B_{w}$ is indispensable.
Suppose that $w$ has an odd chord. The odd chord necessarily is a
chord of one of the two odd cycles. Every chord of an odd cycle
breaks the cycle in two cycles, one of which is odd and the other even. The
even cycle gives a binomial in $I_{G_w}$ which is not in $<B_w>$.
A contradiction arises since $B_{w}$ is fundamental.
\newline Conversely if $w$ is a cycle with no even chords and at
most one odd chord, the result follows from \cite[Lemma
4.2.]{OH2}. On the other hand if $w$ is not a cycle then it is a
circuit with no chords. Therefore $w$ has no even cycles and
$B_{w}$ is fundamental. \hfill $\square$
\begin{ex1}\label{example}{\rm The simplest possible graph which
shows that the relations between fundamental, primitive,
indispensable, minimal binomials and circuits are strict is the
following: let $G$ be the graph with 10 vertices and 14 edges of
figure 6.
\begin{center}
\psfrag{A}{$e_{1}$}\psfrag{B}{$e_{2}$}\psfrag{C}{$e_{3}$}\psfrag{D}{$e_{4}$}\psfrag{E}{$e_{5}$}\psfrag{F}{$e_{6}$}\psfrag{G}{$e_{7}$}
\psfrag{H}{$e_{8}$}\psfrag{I}{$e_{9}$}\psfrag{J}{$e_{10}$}\psfrag{K}{$e_{11}$}\psfrag{L}{$e_{12}$}\psfrag{M}{$e_{13}$}\psfrag{N}{$e_{14}$}
\includegraphics{nine.eps}\\
{Figure 6.}
\end{center}
The Graver basis has twenty two elements:
$B_1=e_2e_{12}-e_{13}e_{14}$, $B_2=e_2e_{11}-e_3e_{13}$,
$B_3=e_3e_{12}-e_{11}e_{14}$, $B_4=e_4e_9-e_5e_{10}$,
$B_5=e_{14}e_{2}e_{4}^2e_{6}e_{8}-e_{1}e_{3}^2e_{5}^2e_{7}$,
$B_6=e_{14}e_{2}e_{10}^2e_{6}e_{8}-e_{1}e_{3}^2e_{9}^2e_{7}$,
$B_7=e_{13}e_{12}e_{4}^2e_{6}e_{8}-e_{1}e_{11}^2e_{5}^2e_{7}$,
$B_8=e_{13}e_{12}e_{10}^2e_{6}e_{8}-e_{1}e_{11}^2e_{9}^2e_{7}$,
$B_9=e_{14}e_{2}e_{4}e_{6}e_{8}e_{10}-e_{1}e_{3}^2e_{5}e_{7}e_{9}$,
$B_{10}=e_{13}e_{12}e_{4}e_{6}e_{8}e_{10}-e_{1}e_{11}^2e_{5}e_{7}e_{9}$,
$B_{11}=e_{3}e_{1}e_{11}e_{5}^2e_{7}-e_{2}e_{12}e_{4}^2e_{6}e_{8}$,
$B_{12}=e_{3}e_{1}e_{11}e_{9}^2e_{7}-e_{2}e_{12}e_{10}^2e_{6}e_{8}$,
$B_{13}=e_{3}e_{1}e_{11}e_{5}e_{9}e_{7}-e_{2}e_{12}e_{4}e_{10}e_{6}e_{8}$,
$B_{14}=e_{3}e_{1}e_{11}e_{5}^2e_{7}-e_{14}e_{13}e_{4}^2e_{6}e_{8}$,
$B_{15}=e_{3}e_{1}e_{11}e_{9}^2e_{7}-e_{14}e_{13}e_{10}^2e_{6}e_{8}$,
$B_{16}=e_{3}e_{1}e_{11}e_{5}e_{9}e_{7}-e_{14}e_{13}e_{4}e_{10}e_{6}e_{8}$,
$B_{17}=e_{14}e_{1}e_{11}^2e_{9}^2e_{7}-e_{2}e_{12}^2e_{10}^2e_{6}e_{8}$,
$B_{18}=e_{14}e_{1}e_{11}^2e_{5}^2e_{7}-e_{2}e_{12}^2e_{4}^2e_{6}e_{8}$,
$B_{19}=e_{14}e_{1}e_{11}^2e_{9}e_{5}e_{7}-e_{2}e_{12}^2e_{4}e_{10}e_{6}e_{8}$,
$B_{20}=e_{13}e_{1}e_{3}^2e_{9}^2e_{7}-e_{12}e_{2}^2e_{10}^2e_{6}e_{8}$,
$B_{21}=e_{13}e_{1}e_{3}^2e_{5}^2e_{7}-e_{12}e_{2}^2e_{4}^2e_{6}e_{8}$,
$B_{22}=e_{13}e_{1}e_{3}^2e_{9}e_{5}e_{7}-e_{12}e_{2}^2e_{4}e_{10}e_{6}e_{8}$.
The first eight of them are fundamental binomials. The first ten
are indispensable binomials and the first sixteen binomials are
minimal. Note that the number of minimal generators $\mu(I_G)$ is
$13$ and there are $8$ different, up to non zero constants,
minimal systems of binomial generators. The cause of the
dispensability of the binomials $ B_{11},\ldots,B_{16}$ is the
existence of an $F_4$, $(e_2,e_{13},e_{12}, e_{14})$. The cause of
the primitive elements $B_{17},B_{18},B_{19}$ and
$B_{20},B_{21},B_{22}$ not to be minimal is the existence of
bridges: $e_3$ in the first three and $e_{11}$ in the last three. Finally
all of them are circuits except the binomials
$B_{9},B_{10},B_{13},B_{16},B_{19},B_{22}$. }
\end{ex1}
\begin{rem1}\label{remark}{\rm
For simplicity of the statements and the proofs we assumed that
the graphs are simple. But actually most of the results are valid
with small adjustments for graphs with loops and multiple edges.
Theorem \ref{primitive} about primitive walks is true exactly as
it is stated, but note that you may have cycles with one edge, a
loop, and cycles with two edges, in the case that you have
multiple edges between two vertices. The property of a walk to
give a minimal binomial depends on the induced graph and one may
have chords which are loops or multiple edges. In this case in
Theorem \ref{minimal}, which describes the even closed walks that
determine minimal generators, chords which are multiple edges are
also permitted and loops where the vertex of the loop is not a cut
vertex of ${\bf w}$. While in Theorem \ref{indispen} chords which
are multiple edges are not permitted, but chords which are loops such that the
vertices of the loops are not cut vertices of ${\bf w}$ are permitted.
}
\end{rem1}
|
2,869,038,156,336 | arxiv | \section{Introduction}
The ever-increasing number of exoplanet discoveries has enabled the characterization of the underlying population of planets in our galaxy. Planet frequencies have been determined by multiple detection methods: RV \citep[e.g.][]{2005ApJ...622.1102F,2008PASP..120..531C,2008A&A...487..373S,2009A&A...493..639M,2010PASP..122..905J,2010Sci...330..653H,2011arXiv1109.2497M,2013A&A...549A.109B}, transits \citep{2006ApJ...644L..37G,2011ApJ...736...19B,2011ApJ...742...38Y,2011ApJ...738..151C,2012ApJS..201...15H,2012ApJ...745...20T,2013ApJ...764..105S,2013ApJ...767...95D,2013ApJ...766...81F}, microlensing \citep{2002ApJ...566..463G,2010ApJ...720.1073G,2010ApJ...710.1641S,2011Natur.473..349S,2012Natur.481..167C}, and direct imaging \citep{2010ApJ...717..878N,2011ApJ...733..126C,2012A&A...541A.133Q}. These studies have provided interesting results, but, individually, are constrained to limited regions of parameter space (i.e. some given intervals of planet mass and period). Synthesizing detection results from multiple methods to derive planet occurrences that cover larger regions of parameter space would provide much more powerful constraints on demographics of exoplanets than is provided by individual techniques. Such synthesized data sets will better inform formation and migration models of exoplanets.
Perhaps surprisingly, M dwarf hosts are the best characterized sample in terms of exoplanet demographics. RV surveys are most sensitive to planets on orbits smaller than a few AU (ultimately depending on the duration and cadence of a given survey). At large separations, from $\sim 10$ to $100~$AU, direct imaging is currently the only technique with the capability to provide information, and then, only for young stars. The only method capable of deriving constraints on the demographics of exoplanets in the intermediate regime of separations from a few to $\sim 10~$AU is microlensing. However, for a range of lens distances, $dD_l$, the contribution to the rate of microlensing events scales as $\propto n\left(D_l\right)M_l^{1/2}$, where $n\left(D_l\right)$ is the number density of lenses and $M_l$ is the lens mass. Thus, the integrated microlensing event rate is explicitly dependent on the mass function of lenses. The slope of the mass function for $M_l\lesssim M_{\odot}$ is such that there are roughly equal numbers of lens stars per logarithmic interval in mass. Thus, lower mass objects are more numerous and more often act as lenses in a microlensing event. Indeed, \citet{2010ApJ...720.1073G} (hereafter GA10) report the typical mass in their sample of microlensing events to be $\sim 0.5~M_{\odot}$. This means that constraints on exoplanet demographics at ``intermediate'' separations (few to $\sim 10~$ AU) exist primarily for M dwarfs, as that is the population best probed by microlensing.
The low giant planet frequencies around M dwarfs inferred from RV surveys have been been heralded as a victory for the core accretion theory of planet formation, which makes the generic prediction that giant planets should be rare around such stars \citep{2004ApJ...612L..73L,2005ApJ...626.1045I,2008ApJ...673..502K}. However, microlensing has found an occurrence rate of giant planets, albeit planets that are somewhat less massive than those found by RV (but nevertheless still giant planets), that is more than an order of magnitude larger than that inferred from RV. On the other hand, microlensing is sensitive to larger separations than RV, typically detecting planets beyond the ice line. If the microlensing results are correct, they imply that giant planets do form relatively frequently around low mass stars, but do not migrate, perhaps posing a challenge to core accretion theory.
Table~\ref{tab:freq_constraints} lists the constraints on giant planet occurrence rates around M dwarfs from the microlensing survey of GA10 and the RV surveys of \citet{2010PASP..122..905J} (hereafter JJ10) and \citet{2013A&A...549A.109B} (hereafter BX13), including the planetary mass and orbital period intervals over which the frequency measurements are valid.
\begin{table*}
\centering
\caption{\label{tab:freq_constraints} Planet frequency around M dwarfs from microlensing and RV surveys. The mass and period intervals for the microlensing measurement were estimated using the typical lens mass of $M_l\sim 0.5~M_{\odot}$ and the typical mass ratio $q\sim 5\times10^{-4}$. The mass limit for the CPS sample assumes a $0.5~M_{\odot}$ host at an orbital separation of 1~AU. See \S~\ref{sec:sample_properties} for details.}
\begin{tabular}{@{\extracolsep{0pt}}lcccc@{\extracolsep{0pt}}}
\hline
\hline
\rule{0pt}{2.6ex}\rule[-1.8ex]{0pt}{0pt} & $\frac{d^2N}{d\log{\left(m_p\sin{i}\right)}d\log{\left(a\right)}}$ $\left[{\rm dex^{-2}}\right]$& Period Interval [days] & Mass Interval $\left[M_{\oplus}\right]$ & Reference \\ \hline
\hline
Microlensing & $0.36\pm 0.15$ & $560\lesssim P \lesssim 5600$ & $10\lesssim m_p\sin{i} \lesssim 3000$ & GA10 \\ \hline
HARPS (RV) & $0.0080^{+0.0077}_{-0.0043}$ & $P \lesssim 2000$ & $m_p\sin{i}\gtrsim100$ & BX13 \\ \hline
CPS (RV) & $0.0085\pm 0.0041$ & $P \lesssim 2000$ & $m_p\sin{i}\gtrsim150$ & JJ10 \\ \hline
\end{tabular}
\end{table*}
There are several potential reasons for this large difference in inferred giant planet frequency. The properties and demographics of the observed sample of host stars observed with microlensing could well be different from the targeted (local) M dwarfs monitored with RV. RV studies have shown a clear trend of planet occurrence with metallicity \citep{2005ApJ...622.1102F,2010PASP..122..905J,2013A&A...551A..36N,2014ApJ...781...28M} and the slope of the Galactic metallicity gradient \citep[see e.g.][and references therein]{2012ApJ...746..149C,2013arXiv1311.4569H} suggests that the metallicity distribution of local M dwarfs is systematically lower than that of the GA10 microlensing sample. Furthermore, some of the lenses in the GA10 microlensing sample could be K or G dwarfs, or even stellar remnants, although the fraction of events with such lenses to all events is expected to be relatively low \citep[e.g.][]{2000ApJ...535..928G}. It could also be that the population of planets orbiting local M dwarfs differs from the population orbiting M dwarfs in other parts of the galaxy, and in particular, planets orbiting stars in the Galactic bulge.
However, perhaps the simplest potential explanation for the large discrepancy in the observed giant planet frequency around M dwarfs is the different ranges of orbital period and planet mass probed by the two discovery methods. Indeed, \citet{clanton_gaudi14a} suggest that the slope of the planetary mass function is sufficiently steep that even a small difference in the minimum detectable planet mass can lead to a large change in the inferred frequency of planetary companions.
Thus, motivated by the order-of-magnitude difference in the frequency of giant planets orbiting M dwarfs inferred by microlensing and RV surveys, we have developed in a companion paper \citep{clanton_gaudi14a} the methodology necessary to statistically compare the constraints on exoplanet demographics inferred independently from these two very different discovery methods. We also justify the need for a careful statistical comparison between these two datasets by showing an order of magnitude estimate of the velocity semi-amplitude, $K$, and the period, $P$, of the ``typical'' microlensing planet, which we define as one residing in the peak region of sensitivity for the GA10 microlensing sample. This typical planet has a host star mass of $M_l\sim 0.5~M_{\odot}$, a planet-to-star mass ratio of $q\sim 5\times10^{-4}$, and a projected separation of $r_{\perp}\sim 2.5~$AU, corresponding to a planet mass of $m_p \sim 0.26~M_{\rm Jup} \sim{\rm M_{\rm Sat}}$. We find that for $\sin{i}\approx 0.866$ (the median value for randomly distributed orbits) and a circular orbit, the typical microlensing planet will have a period of about 7~years and produce a radial velocity semi-amplitude of $5~{\rm m~s^{-1}}$. We further demonstrate that for a fiducial RV survey with $N=30$ epochs, measurement uncertainties of $\sigma = 4~{\rm m~s^{-1}}$, and a time baseline of $T=10~$years, the typical microlensing planet would then be marginally detectable with a signal-to-noise ratio (SNR) of 5. This suggests that there is at least some degree of overlap in the planet parameter space probed by RV and microlensing surveys.
In \citet{clanton_gaudi14a}, we then predict the joint probability distribution of RV observables for the whole planet population inferred from microlensing surveys. We find that the population has a median period of $P_{\rm med} \approx 9.4~$yr with a 68\% interval of $3.35\leq P/{\rm yr}\leq 23.7$ and a median RV semi-amplitude of $K_{\rm med}\approx 0.24~{\rm m~s^{-1}}$ with a 68\% interval of $0.0944\leq K/{\rm m~s^{-1}}\leq 1.33$. The California Planet Survey (CPS) includes a sample of 111 M dwarfs \citep{2014ApJ...781...28M} (hereafter MB14) which have been monitored for a median time baseline of over 10~years. The RV survey of HARPS includes 102 M dwarfs (BX13) that have been monitored for longer than 4~years. Thus, at least in terms of orbital period, these surveys should be sensitive to a significant fraction of the planet population inferred from microlensing. However, the fact that a majority of these planets produce radial velocities $K\lesssim 1~{\rm m~s^{-1}}$ means that many will remain undetectable by current generation RV surveys; this is primarily due to the steeply declining planetary mass function inferred by microlensing, $dN/d\log{q}\propto q^{-0.68\pm 0.20}$ \citep{2010ApJ...710.1641S}.
The results of \citet{clanton_gaudi14a} thus, qualitatively, indicate that the constraints on giant planet occurrence around M dwarfs inferred independently from microlensing and RV surveys are consistent. However, because the planetary mass function inferred by microlensing is so steep, the level of consistency is, quantitatively, very sensitive to the actual detection limits of a given RV survey. The primary aim of this paper is then to make an actual quantitative comparison of the planet detection results from microlensing and RVs. We start with a simulated population of microlensing-detected planets, the properties and occurrence rates of which are consistent with the actual population inferred from microlensing surveys for exoplanets \citep[GA10;][]{2010ApJ...710.1641S}, and map these into a population of analogous planets orbiting host stars monitored with RV. We next use the detection limits reported by BX13 for the HARPS M dwarf sample to predict the number of planets they should detect and compare this with the number of detections they report. We perform the same comparison with the CPS sample (MB14), but because they have yet to fully characterize the detection limits for each of their stars, this comparison is not as robust. For both comparisons, we also predict the number and magnitude of long-term RV trends that should be found and compare with the reported values. In doing so, we show that microlensing predicts that RV surveys should see a handful of giant planets around M dwarfs at the very longest periods to which they are sensitive. These planets have indeed been found. Because the detection results of these two discovery techniques are consistent, we are able to synthesize their independent constraints on the demographics of planets around M dwarfs to determine planet frequencies across a very wide region of parameter space, covering the mass interval $1<m_p\sin{i}/M_{\oplus}<10^4$ and period interval $1<P/{\rm days}<10^5$. We quote integrated planet frequencies over the period range $1<P/{\rm days}<10^4$ since our statistics are more robust in this interval.
Readers who are mainly interested in our results, but not necessarily the details, need only refer to figure~\ref{fig:freq_plot} and read the summary and discussion in \S~\ref{sec:discussion}. The full paper is organized as follows. We begin with a discussion of what exactly we mean by the term ``giant planet'' in \S~\ref{sec:giant_planet_def}. In \S~\ref{sec:sample_properties} we describe the sample properties of the microlensing and RV surveys we compare. We summarize the methodology developed in \citet{clanton_gaudi14a} to map the observable parameters of a planet detected by microlensing to the observable parameters of an analogous planet orbiting a star monitored with RV and describe the application of this methodology to this paper in \S~\ref{sec:methods}. We present our results, comparing our predicted numbers of detections and trends with the reported values of RV surveys in \S~\ref{sec:results}. \S~\ref{sec:uncertainties} details sources of uncertainty in our analysis. We derive combined constraints on the planet frequency around M dwarfs from RV and microlensing surveys in \S~\ref{sec:synthesis} and conclude with a discussion of our results in \S~\ref{sec:discussion}. Finally, we examine the properties of the planets accessible by both techniques in the Appendix.
\section{Definition of a ``Giant Planet''}
\label{sec:giant_planet_def}
At this point, it is worth discussing what we mean by a ``giant planet.'' This has not been precisely defined in the literature (to the best of our knowledge), but because microlensing surveys infer a steep planetary mass function, the precise definition is important. Giant planets, unlike terrestrial planets, should have significant hydrogen and helium atmospheres, and thus must form within the short timescales for gas dispersal in protoplanetary disks of $\sim 1-10~$Myr \citep[e.g.][]{1995Natur.373..494Z,2006ApJ...651.1177P}. Terrestrial planets and the cores of giant planets are believed to be formed via coagulation of planetesimals, initially tens of kilometers in size, growing through phases of both runaway and oligarchic growth \citep{safronov1969,1980ARA&A..18...77W,1985prpl.conf.1100H,1988Icar...74..542S,1989Icar...77..330W,1998Icar..131..171K}. Cores with masses of just $\sim 0.1~M_{\oplus}$ can attract gaseous envelopes, which are held up against gravity by pressure gradients maintained by the release of energy from planetesimals actively accreting onto the core. Further growth in core mass enables the attraction of still more nebular gas, such that the core accretion of planetesimals can no longer supply enough energy to support the increasingly massive envelope. The gaseous envelope contracts in response, increasing the rates of attraction of planetesimals and gas. Cores that reach a critical (or crossover) mass, such that the mass of the envelope is equal to the mass of the core ($M_{\rm env}\sim M_{\rm core}$), will accrete gas at a rate that increases exponentially with time, while the timescale for core accretion remains roughly constant. Various calculations have found that the critical mass should be somewhere in the range of $5-20~M_{\oplus}$, and is a function of the grain opacity and the rate of core accretion \citep{1980PThPh..64..544M,1986Icar...67..391B,1996Icar..124...62P,2000ApJ...537.1013I,2006ApJ...648..666R}.
Thus, a nascent planet with a core that reaches this critical mass before depletion of the nebular gas will ultimately be primarily composed of hydrogen and helium --- a giant planet. The final masses of giant planets then depends on the amount of gas they can accrete after this point, which is limited by available reservoir of gas that will eventually run out either because the planet opens a gap in the disk (assuming no gap-crossing accretion streams) or because the disk gas disperses before gap opening due to processes such as viscous dissipation, photoevaporation, and the like \citep[see e.g.][]{2007ApJ...667..557T}. The final masses of giant planets should then be upwards of some tens of Earth masses.
We define giant planets as having $>50\%$ hydrogen and helium by mass, which, in the core accretion paradigm, would imply that their cores must have reached the critical mass before the complete dispersal of disk gases. We choose to define a ``minimum'' giant planet mass of $0.1~M_{\rm Jup}\sim 30~M_{\oplus}$. We believe this to be a reasonable threshold because planets with $m_p\gtrsim 0.1~M_{\rm Jup}$ are likely composed of $>50\%$ hydrogen and helium by mass, unless their protoplanetary disk was very massive (and thus the isolation mass was large) or the heavy element content was $\gg 10\%$ \footnote{We note that there may exist counterexamples. For example, HD 149026b, originally discovered by \citet{2005ApJ...633..465S}, is believed to have a highly metal-enriched composition, probably $>50\%$ heavy elements by mass. HD 149026b has a mass of $m_p=0.37~M_{\rm Jup}\sim118~M_{\oplus}$, a radius of $R_p=0.8~R_{\rm Jup}$, and an orbital period of $P=2.9~$days \citep{2009ApJ...696..241C}. \citet{2009ApJ...696..241C} estimate this planet to have a core made up of elements heavier than hydrogen and helium with a mass in the range of $45-70~M_{\oplus}$, depending on the assumed stellar age and core density.}. For perspective, Jupiter and Saturn ($\sim 0.3~M_{\rm Jup}$) are primarily composed of hydrogen and helium, while Neptune ($\sim 0.05~M_{\rm Jup}$) and Uranus ($\sim 0.05~M_{\rm Jup}$) contain roughly 5-15\% hydrogen and helium, 25\% rocks, and 60-70\% ices, by mass, assuming the ice-to-rock ratio is protosolar \citep{1991uran.book...29P,1995P&SS...43.1517P,1995netr.conf..109H,2005AREPS..33..493G}.
\section{Microlensing and RV Sample Properties}
\label{sec:sample_properties}
\subsection{Microlensing Sample}
\label{subsec:gould_sample}
The microlensing sample of GA10 is an unbiased sample composed of 13 high-magnification events, fitting specific criteria that is described in detail in their paper. Unlike RV surveys, not much is known about the host (lens) stars in the microlensing sample. Nothing is known about the metallicity of the lens stars and there are estimates of, or upper limits on, the lens mass only for a subset of the sample. They report the lens stars (those with and without planets) to have a mass distribution centered around $0.5~M_{\odot}$ and thus adopt a typical lens mass for the sample of $M_l\sim 0.5~M_{\odot}$. As for the planet/host-star mass ratio and Einstein radius, they find typical values of $q\sim5\times10^{-4}$ and $R_E = 3.5$~AU$\left(M_{\star}/M_{\odot}\right)^{1/2}$, respectively. Using this sample, GA10 found the observed frequency of ice and gas giant planets (in the mass-ratio interval $-4.5 < \log{q} < -2$) around low-mass stars to be
\begin{equation}
\frac{d^2N_{\rm pl}}{d\log{q}~d\log{s}} = \left(0.36 \pm 0.15\right)~{\rm dex}^{-2}
\end{equation}
at the mean mass ratio $q_0 = 5\times 10^{-4}$ and sensitive to a wide range of projected separations, $s_{\rm max}^{-1}R_E \lesssim r_{\perp} \lesssim s_{\rm max}R_E$, where $R_E = 3.5~{\rm AU}~\left(M_{\star}/{\rm M_{\odot}}\right)^{1/2}$ and $s_{\rm max} \sim \left(q/10^{-4.3}\right)^{1/3}$, corresponding to deprojected separations of a few times larger than the position of the snow line in these systems. In order to better compare this frequency measurement with those from RV samples, we use the typical $M_l$ and $q$, along with the median value of $\sin{i}\approx 0.866$ and the median relation $a\sim r_{\perp}/0.866$ for randomly distributed orbits (see \citet{clanton_gaudi14a}), to estimate the frequency in terms of RV parameters,
\begin{equation}
\frac{d^2N}{d\log{\left(m_p\sin{i}\right)}d\log{\left(a\right)}} = \left(0.36 \pm 0.15\right)~{\rm dex^{-2}}\; ,
\end{equation}
over the planetary mass interval $10\lesssim m_p\sin{i}/M_{\oplus}\lesssim 3\times10^3$ and the period interval $6\times10^2\lesssim P/{\rm days}\lesssim 6\times10^3$. Additionally, GA10 report no significant deviation from a flat distribution in $\log{s}$ for the events included in their analysis.
GA10 measure a normalization, but are unable to determine the slope of the planetary mass function. \citet{2010ApJ...710.1641S} assume a power-law form for the planetary mass-ratio function (also assuming planets follow a flat distribution in $\log{s}$) and measure the slope using the mass ratios of 10 microlensing-detected planets and their estimated detection efficiencies for each event, finding $dN/d\log{q}\propto q^{-0.68\pm 0.20}$.
\citet{2012Natur.481..167C} use a few new microlensing-detected planets along with the previous constraints on the normalization by GA10 and the slope by \citet{2010ApJ...710.1641S} to measure the cool-planet mass function over an orbital range of $0.5-10~$AU, finding $d^2N/(d\log{m_p}\; d\log{a})=0.24^{+0.16}_{-0.10}\left(m_p/M_{\rm Sat}\right)^{-0.73\pm 0.17}$. In this paper, we choose to adopt the independent measurements of GA10 and \citet{2010ApJ...710.1641S} to construct our own planetary mass-ratio function, rather than adopt that of \citet{2012Natur.481..167C} (although, as we later show, the form we derive is consistent with that of \citet{2012Natur.481..167C}). We choose to do this because the measurements of GA10 and \cite{2010ApJ...710.1641S} are more closely related to the observable quantities we use as a starting point in this study.
\subsection{HARPS M Dwarf Sample}
\label{subsec:bonfils_sample}
The stellar sample of BX13 is a volume limited collection of 102 M dwarfs closer than 11~pc and brighter than $V = 14$~mag, with declinations $\delta < +20^{\circ}$ and with projected rotational velocities $v\sin{i} \lesssim 6.5$~m~s$^{-1}$. Known spectroscopic binaries and visual pairs with separations $< 5$'' were removed from the sample. The brightness range for this sample is $V = 7.3$~mag to 14~mag, with a median brightness of $V = 11.43~$mag. The stellar masses range between 0.09 to $0.6~M_{\odot}$, with a median mass of $0.27~M_{\odot}$. \citet{2013A&A...551A..36N} determine the metallicities of the stars in this sample, reporting [Fe/H] values ranging from -0.88~dex to 0.32~dex, with mean and median values of -0.13~dex and -0.11~dex, respectively. RV observations of this sample were made using the HARPS instrument \citep{2003Msngr.114...20M, 2004A&A...423..385P}. BX13 quote a precision of $\sigma \sim 80$~cm~s$^{-1}$ for $V = 7 - 10$ stars and $\sigma \sim 2.5^{\left(10 - V\right)/2}~{\rm m~s^{-1}}$ for $V = 10 - 14$ stars, which includes instrumental errors in addition to the photon noise. Their actual errors are larger, due to stellar jitter.
BX13 report planet frequencies in several bins of $m_p\sin{i}$ and period. In order to better compare with the microlensing constraint, we have combined and transformed their detections into bins of $\log{\left(m_p\sin{i}\right)}$ and $\log{\left(a\right)}$, using the sample median stellar mass of $0.27~M_{\odot}$, to give a frequency
\begin{equation}
\frac{d^2N}{d\log{\left(m_p\sin{i}\right)}d\log{\left(a\right)}} = \left(0.0057 \pm 0.0029\right)~{\rm dex^{-2}}\; ,
\end{equation}
for planets with $10 < m_p\sin{i}/M_{\oplus} < 10^4$ and $1 < P/{\rm days} < 10^3$. In the above calculation we did not include their period ranges of $10^3$--$10^4$ days, where the sensitivity of their survey rapidly declines. If we include the entire period range from $1 < P/{\rm days} < 10^4$, this frequency becomes
\begin{equation}
\frac{d^2N}{d\log{\left(m_p\sin{i}\right)}d\log{\left(a\right)}} = \left(0.0088 \pm 0.0039\right)~{\rm dex^{-2}}\; .
\end{equation}
\subsection{CPS M Dwarf Sample}
\label{subsec:johnson_sample}
The stellar sample of the RV study conducted by JJ10 included about 120 M dwarfs brighter than $V=11.5$ monitored by the CPS team with HIRES \citep{1994SPIE.2198..362V} at Keck Observatory, and are reported to have masses between $M_{\star} < 0.6~M_{\odot}$ and a wide range of metallicites between $-0.6 < {\rm \left[Fe/H\right]} < 0.6$. Their analysis consisted of planets with semi-major axes $a < 2.5$~AU and systems with velocity semi-amplitudes of $K > 20~{\rm m~s^{-1}}$. Using this sample, JJ10 found the observed frequency of giant planets around low-mass stars, corrected for the average stellar metallicity, to be $2.5 \pm 1.2\%$. For comparison with the microlensing results, we convert this into units of dex$^{-2}$ by dividing by the area it covers in the $\log{\left(m_p\sin{i}\right)}$--$\log{\left(a\right)}$ plane. This non-rectangular area is bound by the above mentioned constraints, imposed by the set of planets included in the analysis. This yields a frequency of
\begin{equation}
\frac{d^2N_{\rm pl}}{d\log{\left(m_p\sin{i}\right)}~d\log{\left(a\right)}} = \left(0.0085 \pm 0.0041\right)~{\rm dex}^{-2}\; ,\label{eqn:jj_freq_estimate}
\end{equation}
for masses $m_p\sin{i}\gtrsim150~M_{\oplus} \left(M/0.5~{M_{\odot}}\right)^{1/2}\left(a/{\rm AU}\right)^{1/2}$ and periods $P \lesssim 2\times10^3~{\rm days}\left(M/0.5~M_{\odot}\right)^{-1/2}$, where we have chosen a characteristic host mass of $M\sim 0.5~M_{\odot}$ to transform to similar parameters as the RV survey of BX13 and the microlensing survey of GA10.
The CPS continues to monitor these stars and, since the study of JJ10, has extended their sample to M dwarfs (which they define as having $B-V>1.44$) brighter than $V=13.5$, bringing their M dwarf sample to a total of 131 stars with no known stellar companions within two arcseconds and all closer than 16~pc. MB14 further refine this sample by excluding stars with known, nearby stellar binary companions. The final sample, which we will refer to as the ``CPS sample'' throughout this paper, consists of 111 M dwarfs with a median time baseline of $11.8~$yr, a median of $29$ epochs per star, and typical Doppler precisions of a couple meters per second. They also estimate $\sim 3-6~{\rm m~s^{-1}}$ of stellar jitter for the majority of their stars. In this study, we will compare the numbers of detections and trends the CPS have discovered from this M dwarf sample (of which the sample of JJ10 is a subset) to the amount we predict they should find based on the microlensing measurements of planet frequency around low mass stars.
\section{Methods}
\label{sec:methods}
In \citet{clanton_gaudi14a}, we developed the methodology to map the observable parameters of a planet detected by microlensing to the observable parameters of an analogous planet orbiting a star monitored with RV, i.e. $\left(q,s\right)\rightarrow \left(K,P\right)$, where $K$ and $P$ are the velocity semi-amplitude and orbital period, respectively. We then used this procedure to show that a fiducial RV survey with a precision of $\sigma=4~{\rm m~s^{-1}}$, an average number of epochs per star of $N=30$, a duration of $T=10~$years, and monitoring 100 stars uniformly (in log space) covering the mass interval $0.07~M_{\odot}\leq M_{\star}\leq 1.0~M_{\odot}$, should on average detect $4.9^{+4.6}_{-2.6}$ planets and identify $2.4^{+2.4}_{-1.4}$ long-term RV trends resulting from planets at a SNR of at least 5, motivating a more rigorous comparison to actual RV surveys.
In this section, we first provide a brief account of the methods and results presented in \citet{clanton_gaudi14a}, followed by a description of how we apply this methodology to directly compare planet detection results from the microlensing survey of GA10 to those from the HARPS (BX13) and CPS (MB14) RV surveys of M dwarfs. For more details on the methodology, refer to \citet{clanton_gaudi14a}.
\subsection{Mapping Analogs of Planets Found by Microlensing into RV Observables}
\label{subsec:mapping_summary}
The general procedure detailed in \citet{clanton_gaudi14a} is comprised of a two steps. The first step is the mapping $\left(q,s\right) \rightarrow \left(m_p,r_{\perp}\right)$ using a Galactic model. Here, $q$ and $s$ are the planet-to-star mass ratio and the planet-star projected separation in units of the Einstein radius ($\theta_E$), respectively, and are the quantities measured in a microlensing planet detection. The mapping between these measurements and the true planet mass, $m_p=qM_l$, and the projected separation in physical units, $r_{\perp}=sD_l\theta_E$, requires a Galactic model because the precise forms of the distributions of physical parameters of microlensing systems are unknown. In particular, we do not know the true distribution of lens masses, $M_l$, or distances, $D_l$, nor do we know with certainty whether the lens lies in the disk or the bulge in a given microlensing event. We account for this by drawing these parameters from basic priors and weighting by the corresponding microlensing event rate, $\Gamma$, assuming a Galactic model.
The second step is the mapping $\left(m_p,r_{\perp}\right)\rightarrow \left(K,P\right)$, where $K$ and $P$ are the velocity semi-amplitude and the orbital period, respectively. This is accomplished by adopting priors on, and marginalizing over, the Keplerian orbital parameters (i.e. inclination, eccentricity, mean anomaly, and argument of periastron) of the microlensing-detected systems to get a distribution of semimajor axes, which then immediately gives the $P$ distribution by way of Kepler's third law. Combining the period distribution with $m_p$ and the distribution of inclinations, we are able to derive the distribution of $K$.
Figure \ref{fig:marg_dist} shows the resultant joint distribution of $K$ and $P$ for a population of planets analogous to that inferred from microlensing, marginalized over all planet and host star properties inferred from microlensing, as well as all orbital parameters \citep{clanton_gaudi14a}. The median values we found are $P_{\rm med} \approx 9.4~$yr and $K_{\rm med}\approx 0.24~{\rm m~s^{-1}}$. The 68\% intervals in $P$ and $K$ are $3.35\leq P/{\rm yr}\leq 23.7$ and $0.0944\leq K/{\rm m~s^{-1}}\leq 1.33$, respectively, and their 95\% intervals are $1.50\leq P/{\rm yr}\leq 94.4$ and $0.0422\leq K/{\rm m~s^{-1}}\leq 16.8$, respectively. In \citet{clanton_gaudi14a}, we demonstrated how to compute the expected number of planets an RV survey should detect, as well as the number of long-term RV trends (due to planets) that should be seen, by parameterizing RV detection limits in terms of a SNR threshold.
\begin{figure*}
\epsscale{0.9}
\plotone{fig1.eps}
\caption{Mapping of microlensing planets into RV observables, from \citet{clanton_gaudi14a}. Shown in greyscale are contours of the probability density of $K$ and $P$, marginalized over the entire microlensing parameter space. The contour levels, going from grey to black, are $1\%$, $10\%$, $25\%$, $50\%$ and $80\%$ of the peak density. The filled yellow circle represents where the typical microlensing planet lies in this parameter space at the median inclination and mean anomaly and on a circular orbit ($K_{\rm typ}\sim 5~{\rm m~s^{-1}}$, $P_{\rm typ}\sim 7~$yr). The blue and red colored line represent the median RV detection limit curves for the surveys of BX13 and JJ10, respectively. Planets that lie above these lines and have periods less than the duration of the RV survey are detectable, while those with longer periods might show up as long-term RV trends. The colored histograms represent the the total numbers of detections plus trends for the HARPS sample (blue curve) and the CPS sample (red curve) as a function of $P$ (top panel) and $K$ (right panel). It is clear from these colored histograms that RV surveys are beginning to sample the full period distribution of the planet population inferred from microlensing, but are only able to catch the tail of the $K$ distribution towards higher values, or equivalently, the high-mass end of this planet population.
\label{fig:marg_dist}}
\end{figure*}
We showed that the phase-averaged SNR, which we designate as $\mathcal{Q}$, assuming uniform and continuous sampling of the RV curve, is
\begin{align}
\mathcal{Q} = & {} \left(\frac{N}{2}\right)^{1/2}\left(\frac{K}{\sigma}\right) \nonumber \\
& {} \times \left\{1-\frac{1}{\pi^2}\left(\frac{P}{T}\right)^2\sin^2{\left(\frac{\pi T}{P}\right)}\right\}^{1/2}\; , \label{eqn:snr_full}
\end{align}
where $N$ is the average number of epochs per star, $\sigma$ is the average RV precision and $T$ is the time baseline of the RV survey. In the limit where the period is much less than the time baseline, $P\ll T$, this reduces to
\begin{equation}
\mathcal{Q} \approx \left(N/2\right)^{1/2}\left(K/\sigma\right)\; , \label{eqn:snr_small_p}
\end{equation}
which is also a good approximation for periods up to $P\sim T$ when approaching from $T$ from small $P$. We assume an effective sensitivity for our fiducial RV survey by assuming a SNR threshold, $\mathcal{Q}_{\rm min}$, above which planets can be detected. Solving equation (\ref{eqn:snr_full}) for $K$ in terms of $P$, we find a sensitivity of
\begin{align}
K_{\rm min} = & {} \mathcal{Q}_{\rm min}\sigma\left(\frac{2}{N}\right)^{1/2} \nonumber \\
& {} \times \left\{1-\frac{1}{\pi^2}\left(\frac{P}{T}\right)^2\sin^2{\left(\frac{\pi T}{P}\right)}\right\}^{-1/2} \; , \label{eqn:k_sens_first}
\end{align}
meaning that the RV survey will be sensitive to planets that produce velocity semi-amplitudes greater than or equal to $K_{\rm min}$ at SNRs of $\mathcal{Q}_{\rm min}$ or greater. We further make the approximation that only planets with periods $P\leq T$ will be detected, whereas planets with periods $P>T$ can possibly be identified as long-term RV trends.
In this study, to compute the expected number of detections and long-term trends for the RV survey of BX13, we approximate the detection limit curves they provide for each star in their sample by fitting equation (\ref{eqn:k_sens_first}) to their curves with $\mathcal{Q}_{\rm min}$ as a free parameter. We provide more information on our approximation of the detection limits of both the HARPS and CPS samples in \S~\ref{subsec:method_application} and \S~\ref{subsubsec:bonfils_detailed_comparison}.
Figure \ref{fig:marg_dist} shows a couple examples of such a sensitivity curve, given by equation (\ref{eqn:k_sens_first}), over-plotted on top of our joint distribution of $K$ and $P$. The blue curve represents the median detection limit as a function of period for the HARPS sample (BX13), which has the median values $N_{\rm med}=8$, $\sigma_{\rm med}\approx 4.2~{\rm m~s^{-1}}$, $T_{\rm med}\approx 4.1~$yr, $M_{\star, {\rm med}}=0.27~M_{\odot}$, and $\mathcal{Q}_{\rm min, med}\approx 8.9$, and the red curve is that of the CPS sample (MB14), which has the median values $N_{\rm med}=28$, $\sigma_{\rm med}\approx 4.1~{\rm m~s^{-1}}$, $T_{\rm med}\approx 11.1~$yr, $M_{\star, {\rm med}}=0.43~M_{\odot}$, and $\mathcal{Q}_{\rm min, med}\approx 8.3$.
We can rewrite equation (\ref{eqn:k_sens_first}) in terms of a minimum $m_p\sin{i}$ by substituting the velocity semi-amplitude equation for $K$ and solving, to yield an equivalent sensitivity in terms of planetary mass
\begin{align}
\left.m_p\sin{i}\right|_{\rm min} = & {} \; \mathcal{Q}_{\rm min}\sigma M_{\star}^{2/3}\left(\frac{2}{N}\right)^{1/2}\left(\frac{P}{2\pi G}\right)^{1/3} \nonumber \\
& {} \times \left\{1-\frac{1}{\pi^2}\left(\frac{P}{T}\right)^2\sin^2{\left(\frac{\pi T}{P}\right)}\right\}^{-1/2} \label{eqn:mpsini_sens_first}
\end{align}
which evaluates to
\begin{align}
\left.m_p\sin{i}\right|_{\rm min} \approx & {}\; 69~M_{\oplus}\left(\frac{P}{\rm 7~yr}\right)^{1/3}\left(\frac{M_{\star}}{0.5~M_{\odot}}\right)^{2/3} \nonumber \\
& {} \times \left(\frac{\mathcal{Q}_{\rm min}}{5}\right)\left(\frac{\sigma}{4~{\rm m~s^{-1}}}\right) \nonumber \\
& {} \times \left(\frac{N}{30}\right)^{-1/2}
\end{align}
in the approximation $P\ll T$.
Also plotted in the top and right panels of figure~\ref{fig:marg_dist} are colored histograms representing the total numbers of detections plus trends for the HARPS sample (blue curve) and the CPS sample (red curve) as a function of $P$ (top panel) and $K$ (right panel). It is clear from these colored histograms that RV surveys are beginning to sample the full period distribution of the planet population inferred from microlensing, but are only able to catch the tail of the $K$ distribution towards higher values, or equivalently, the high-mass end of this planet population.
\subsection{Application: Comparing with Real RV Surveys}
\label{subsec:method_application}
The application of this methodology to compare microlensing detections to those reported by real RV surveys is a little more involved than our description above. In that simple estimate, we assumed each star had the same number of epochs, the same measurement uncertainties at each epoch, and that each star was observed over the same time baseline. The reality is that RV surveys have varying sensitivities for each of their monitored stars which need to be included in a direct comparison. We must also take care to construct a microlensing sample that is consistent with that of real RV surveys, i.e. one with the same distribution of host star masses. In this section, we describe how we do this in order to perform independent statistical comparisons of planet detection results from microlensing with each of the RV surveys of HARPS and CPS.
When comparing with the HARPS survey, we begin with an ensemble of microlensing events for a sample of planet-hosting stars in the mass interval $0.07\leq M_l/M_{\odot}\leq 1.0$ for which we have numerically determined the joint distributions of the RV observables $K$ and $P$. In order to force the microlensing sample to be consistent with that of HARPS, we consider only microlensing detections around lenses with $\left|M_l-M_{\star}\right|\leq \sigma_{M_{\star}}$ for each star in the RV sample, where $M_l$ is the lens mass for a given microlensing event, $M_{\star}$ is the mass of the RV monitored star, and $\sigma_{M_{\star}}$ is the uncertainty on the measurement of $M_{\star}$. This yields a set of distributions of $K$ and $P$, each corresponding to a particular microlensing planet detection that has been mapped into these observables. We then sum up all the joint $K$ and $P$ distributions for each set of events with lens star masses within $\pm \sigma_{M_{\star}}$ of $M_{\star}$. The summation and weighting of these distributions is done in exactly same manner as described in \S~\ref{subsec:mapping_summary} (and in more detail in \citet{clanton_gaudi14a}), except that now, rather than marginalizing over the entire mass interval $0.07\leq M_l/M_{\odot}\leq 1.0$, we have instead marginalized over all lens masses within $\pm \sigma_{M_{\star}}$. We are left with a single distribution, $d^2N_{\rm pl}/(dKdP)$, for each star in the RV sample. We note that by matching the host mass distribution of our simulated sample to that of HARPS, we are implicitly assuming that the microlensing planet distribution is independent of host mass, $M_{\star}$. This is unavoidable because the microlensing sample is not large enough to subdivide and determine the planet frequency dependence on host mass.
In order to compute the expected number of detections and trends for each star in the HARPS RV sample, we must first model the sensitivity of their survey for each star, in terms of $K$ and $P$. For each star in their sample, BX13 graphically provide detection limit curves, i.e. the minimum $m_p\sin{i}$ to which they are sensitive as a function of $P$. They generate these detection limits by systematically injecting known (fictitious) planetary signals into their data and determining the subset of these signals that are detectable (see \S~6 of BX13 for a more detailed explanation). We approximately reproduce these detection limits by parameterizing in terms of a minimum SNR. We use the values of $\sigma$, $M_{\star}$, $T$, and $N$ for each star provided by BX13, including $\mathcal{Q}_{\rm min}$ as a free parameter, to match (by eye) equation (\ref{eqn:mpsini_sens_first}) to the detection limit curves for each star. We describe the RV measurement uncertainties we adopt in \S~\ref{subsubsec:bonfils_detailed_comparison}. Many of these curves are quite noisy (see figure 18 of BX13), so we match to the approximate mean of the noise in these curves by eye. This parameterization of their detection limits can be interpreted as computing the minimum SNR to which the survey can detect a planet or identify a long-term RV trend. The distribution of $\mathcal{Q}_{\rm min}$ we find for the HARPS sample is shown in figure \ref{fig:sn_fig}. The fact that $\mathcal{Q}_{\rm min}$ varies from star to star is a reflection of the non-uniformity of the HARPS M dwarf sample, i.e. each star has a different number of epochs, and spans a different time baseline, resulting in differing detection limits within the sample. The four stars with $\mathcal{Q}_{\rm min}\geq 50$ shown in figure~\ref{fig:sn_fig} are from stars with just four epochs that span relatively short time baselines.
These SNR values are used in conjunction with equation (\ref{eqn:k_sens_first}) to compute the number of detections and trends we expect the HARPS M dwarf survey to find in the same manner as described in \S~{\ref{subsec:mapping_summary}} and illustrated in figure \ref{fig:marg_dist}. These expected numbers of detections and trends are then compared with the actual numbers reported by BX13. The results and comparison is presented in \S~\ref{subsec:bonfils_comparison}.
\begin{figure}[t!]
\epsscale{1.2}
\plotone{fig2.eps}
\caption{Distribution of SNR thresholds ($\mathcal{Q}_{\rm min}$) we find for the HARPS and CPS M dwarf samples. The median values for these surveys are 8.9 and 8.3, respectively. These values represent the minimum SNRs to which a given RV survey can detect a planet or identify a long-term RV trend, and are used to approximate the detection sensitivities of these two RV surveys for each star in their samples.
\label{fig:sn_fig}}
\end{figure}
We follow an identical procedure for computing the expected numbers of detections and trends for the CPS survey, except for the way in which we estimate their detection limits. The CPS team has not yet determined the individual detection sensitivities for their sample, so to roughly estimate their detection limits (in terms of $\mathcal{Q}_{\rm min}$) we assume the sensitivities of their stars are similar to those of stars with similar systematics in the HARPS sample. We compute values of $\sigma_i/N^{1/2}$ for all stars in both RV samples, where $\sigma_i$ is the RV measurement precision (not including ``external'' noise sources, e.g. stellar jitter) and $N$ is the number of epochs. Each star in the CPS sample is ``matched'' to the star in the HARPS sample with the nearest value of $\sigma_i/N^{1/2}$. We assume the matched pairs of stars have similar sensitivities, and assign the stars in the CPS sample the same sensitivities (i.e. the same minimum SNR, $\mathcal{Q}_{\rm min}$) as that of the star in the HARPS sample to which they are matched. Since the CPS team reports stellar jitter values of $3-6~{\rm m~s^{-1}}$ for all stars in their sample, we only ``match'' them to stars in the HARPS sample which have consistent ``external'' errors of $\sigma_e\leq6~{\rm m~s^{-1}}$. The resultant distribution of $\mathcal{Q}_{\rm min}$ we obtain for the CPS sample is displayed against that of the BX13 in figure \ref{fig:sn_fig}, and has a median value of 8.3. The expected number of planet detections and long-term RV trends is calculated in the same manner as those for the HARPS survey. Our results and comparison with the CPS sample is presented in \S~\ref{subsec:johnson_comparison}. Ideally, we would like to do this comparison more accurately once the CPS determines their detection limits for their sample.
In \citet{clanton_gaudi14a}, we derive the planetary mass-ratio and projected separation function
\begin{align}
\frac{d^2N_{\rm pl}}{d\log{s}~d\log{q}} = & {} \left(0.23\pm 0.10\right)~{\rm dex^{-2}} \nonumber \\
& {} \times \left(\frac{q}{q_0}\right)^{-0.68\pm 0.20} \; , \label{eqn:planetary_mass_function}
\end{align}
where $q_0=5\times10^{-4}$. We adopt the slope of the planetary mass-ratio function $dN_{\rm pl}/d\log{q} \propto q^{p}$, where $p=-0.68\pm 0.20$, from \citet{2010ApJ...710.1641S} and normalize it using the integrated frequency measurement of $d^2N_{\rm pl}/(d\log{q}~d\log{s})\equiv \mathcal{G}=(0.36\pm0.15)~{\rm dex}^{-2}$ by GA10. We assume planets are uniformly distributed in $\log{s}$ since the distribution of projected separations from the sample of GA10 is consistent with such a distribution. As we will show in \S~\ref{sec:uncertainties}, the main uncertainties in our results arise from the uncertainties in $p$ and $\mathcal{G}$.
Mathematically, the total number of planet detections we expect a RV sample to yield for a given realization $i$ in our simulation (corresponding to given values of $p_i$ and $\mathcal{G}_i$) is
\begin{equation}
N_{\rm det, i}=\displaystyle \sum_k N_{\rm det, i, k}\; ,
\end{equation}
where $N_{\rm det, i, k}$ is the number of expected planet detections for a given star $k$,
\begin{align}
N_{\rm det, i, k} = & {} \displaystyle \int dM_l \int dD_l \int d\log{q} \int d\log{s} \nonumber \\
& {} \times \int dK \int dP \left.\frac{d^6N_{\rm pl}}{dKdPdM_ldD_ld\log{q}~d\log{s}}\right|_{i} \nonumber \\
& {} \times \Phi_{{\rm det}, k}\left(\mathcal{Q}\right)\Phi_{{\rm det}, k}\left(P\right)\Phi_k\left(M_l\right)\; , \label{eqn:n_det_analytic}
\end{align}
where $\Phi_{{\rm det},k}\left(\mathcal{Q}\right)$ and $\Phi_{{\rm det},k}\left(P\right)$ are selection functions on a given star constraining the detections to those planets which have SNRs larger than the threshold value (i.e. $\mathcal{Q}_{\rm min}$) and periods smaller than the time baseline of observations, $T$, for that particular star in the RV sample with which we are comparing. The functional forms of these are $\Phi_{{\rm det},k}\left(\mathcal{Q}\right)=\Theta\left(\mathcal{Q} - \mathcal{Q}_{{\rm min},k}\right)$ and $\Phi_{{\rm det},k}\left(P\right)=\Theta\left(T_k-P\right)$, respectively, where $\Theta$ is the Heaviside step function. In equation (\ref{eqn:n_det_analytic}), $\Phi_k\left(M_l\right)$ is the selection function on lens masses that we employ to force our microlensing sample to have the same stellar mass distribution as the RV survey to which we are comparing, having the functional form $\Phi_k\left(M_l\right)=\Theta\left[M_l - \left(M_{\star, k}-\sigma_{M_{\star, k}}\right)\right]\Theta\left[\left(M_{\star, k}+\sigma_{M_{\star, k}}\right)-M_l\right]$.
The integrand of equation (\ref{eqn:n_det_analytic}) (not including the selection functions) represents the distribution of $K$ and $P$ for a single system, i.e. only one $M_l$, $D_l$, $\log{q}$, and $\log{s}$, marginalized over all possible orbital configurations. Integrating this distribution marginalizes over all planet and host star properties inferred from microlensing. Multiplying this distribution by selection functions of RV detectability and on the host star mass, as in equation (\ref{eqn:n_det_analytic}), and integrating yields the number of RV detectable planets for a given host star mass. As we showed in \citet{clanton_gaudi14a}, the distribution function is given formally as
\begin{align}
& {} \left.\frac{d^6N_{\rm pl}}{dKdPdM_ldD_ld\log{q}d\log{s}}\right|_{i} = \mathcal{F}_i\displaystyle \int_{\left\{\alpha\right\}}d\left\{\alpha\right\} \nonumber \\
& {} \hspace{0.8in} \times \frac{d^n{\rm N_{pl}}}{d\left\{\alpha\right\}}\delta\left(K\left(m_p, i, M_l, a\right)-K'\right)\nonumber \\
& {} \hspace{0.8in} \times \delta\left(P\left(M_l, m_p, a\right)-P'\right)\delta\left(M_l-M_l'\right) \nonumber \\
& {} \hspace{0.8in} \times \delta\left(D_l-D_l'\right)\delta\left(q-q'\right)\delta\left(s-s'\right)\; ,\label{eqn:planet_dist_function}
\end{align}
where $\left\{\alpha\right\}$ is the set of all $n$ intrinsic, physical parameters on which the frequency of planets fundamentally depends. We assume the form
\begin{align}
\frac{d^n{\rm N_{pl}}}{d\left\{\alpha\right\}} = & {} \frac{d{\rm N_{pl}}}{di}\frac{d{\rm N_{pl}}}{da}\frac{d{\rm N_{pl}}}{dM_0}\frac{d^2{\rm N_{pl}}}{d\log{q}~d\log{s}} \nonumber \\
& {} \times \frac{d{\rm N_{pl}}}{dM_l}\frac{d{\rm N_{pl}}}{dD_l}\frac{d{\rm N_{pl}}}{d\omega}\frac{d{\rm N_{pl}}}{de}\; , \label{eqn:orb_marg_function}
\end{align}
and we note that
\begin{equation}
\frac{dN_{\rm pl}}{dM_l}\frac{dN_{\rm pl}}{dD_l} \propto \displaystyle \int \int \frac{d^4d\Gamma}{dD_ldM_ld^2\boldsymbol{\mu}}\Phi\left(t_E\right)d^2\boldsymbol{\mu}\; ,
\end{equation}
where $d^4d\Gamma/(dD_ldM_ld^2\boldsymbol{\mu})$ is the event rate of a given microlensing event, $\Phi\left(t_E\right)=\Theta\left(t_E/{\rm days} - 10\right)$ is a selection function on the event timescale, $t_E$, and $\boldsymbol{\mu}$ is the lens-source relative proper motion. Finally, the $\mathcal{F}_i$ in equation (\ref{eqn:planet_dist_function}) represents the effective number of planets per star in the area over which our simulated planetary microlensing evetns are sampled, i.e., the integral over that area weighted by the joint distribution function $d^2N_{\rm pl}/(d\log{q}~d\log{s})$,
\begin{equation}
\mathcal{F}_i = \mathcal{A}_i\displaystyle \int_{\log{0.5}}^{\log{2.5}} \int_{-5}^{-2}\left(\frac{q}{q_0}\right)^{p_i}d\log{q}~d\log{s}\; .\label{eqn:f_a_n}
\end{equation}
We find a mean value and 68\% confidence interval of $\mathcal{F}=1.5\pm 0.6$. For our final results, we adopt the mean value of the number of detections from all realizations (i.e the expectation value) and the 68\% confidence intervals to represent our errors. Uncertainties in $p$ and $\mathcal{G}$ are numerically propagated through our simulations and are responsible for the uncertainties in our final results.
Similarly, the total number of expected long-term RV trends per star for an RV survey is given by equation (\ref{eqn:n_det_analytic}), but with the new selection function $\Phi_{{\rm det}, k}\left(P\right)\rightarrow\Phi_{{\rm tr}, k}\left(P\right)=\Theta\left(P-T_k\right)$, such that only planets with periods larger than the time baseline of observations for a given star are counted as trends. Refer to \citet{clanton_gaudi14a} for a more complete description of the mathematical formalism presented here.
\section{Results}
\label{sec:results}
We compare the numbers of planet detections and long-term trends reported for the HARPS (BX13) and CPS (MB14) M dwarf surveys to the amount we predict they should find by assuming a population of planets analogous to that inferred from microlensing surveys. Since BX13 provide detection limits for each of the stars in the HARPS M dwarf sample, we primarily focus on the comparison with their survey, first performing an order of magnitude comparison for the number of predicted planet detections before doing a more detailed analysis. We then compare with the CPS sample by assuming their detection sensitivities are similar to that of BX13 for stars with similar RV uncertainties between the two surveys, as described above.
\subsection{Comparison with HARPS Planet Detections}
\label{subsec:bonfils_comparison}
\subsubsection{Order of Magnitude Comparison}
\label{subsubsec:bonfils_OoM_comparison}
In order to better understand the result of our detailed calculation, we first derive an order of magnitude estimate of the number of RV-detectable planets in the HARPS sample by assuming their survey is uniformly sensitive to planets over a given range of mass ratios and projected separations. We then estimate the planet frequency at the median mass ratio and projected separation in this range, which we designate as
\begin{equation}
f_{\rm med}=\left.\frac{d^2N_{\rm pl}}{d\log{q}d\log{s}}\right|_{q=q_{\rm med}, s=s_{\rm med}}\; ,
\end{equation}
and make the approximation that this does not change over the entire parameter space to which BX13 is sensitive. Multiplying this by the sample size of HARPS and the area in $\log{q}-\log{s}$ space over which we assume they are sensitive yields a rough estimate of the number of expected planet detections
\begin{align}
N_{\rm pl} \sim & {} \; N_{\star}\left(\log{q_{\rm max}}-\log{q_{\rm min}}\right) \nonumber \\
& {} \; \times \left(\log{s_{\rm max}}-\log{s_{\rm min}}\right)f_{\rm med}\; . \label{eqn:n_det_estimate}
\end{align}
We assume BX13 is sensitive to the higher end of the range of mass ratios to which microlensing is sensitive, so that $q_{\rm max}=10^{-2}$. To estimate $q_{\rm min}$, we roughly compute their average sensitivity limit by using representative values of $M_{\star}$, $N$, $\sigma$, and the median $\mathcal{Q}_{\rm min}$ (see \S~\ref{subsubsec:bonfils_detailed_comparison} and figure \ref{fig:sn_fig}). Substituting for $K$ in equation (\ref{eqn:snr_small_p}) using the standard velocity semi-amplitude equation for a circular orbit, solving for $m_p\sin{i}$ and dividing both sides by $M_{\star}$, we obtain an expression for the minimum mass ratio, to which an RV survey will be sensitive (in the limit $P\ll T$),
\begin{equation}
q_{\rm min} \sim \left(\frac{P_{\rm typ}}{2\pi G}\right)^{1/3}M_{\star}^{-1/3}\mathcal{Q}_{\rm min}\sigma\sqrt{\frac{2}{N}}\; , \label{eqn:q_sensitivity_bonfils}
\end{equation}
where $P_{\rm typ}\approx 7~$yr is the period for the typical microlensing planet found in \citet{clanton_gaudi14a}. Using the median values reported by BX13 for the HARPS sample of $N_{\rm med}=8$, $\sigma_{\rm med}=4.2~{\rm m~s^{-1}}$, $M_{\rm \star, med}=0.27~M_{\odot}$, and ${\rm \mathcal{Q}_{\rm min, med}}\sim 10$, we estimate the ``average'' minimum mass ratio to which they are sensitive to be $\log{q_{\rm min}}\approx -2.7$.
We then assume that BX13 can efficiently detect planets at the lower end of the range of projected separations to which microlensing is also sensitive, which sets $s_{\rm min}=0.5$. We approximate the largest projected separation to which BX13 are sensitive as $s_{\rm max}\sim \left(T_{\rm med}/P_{\rm typ}\right)^{2/3}\approx 0.69$, where $P_{\rm typ}\approx 7~$yr is the period of the typical microlensing planet and $T_{\rm med}=4.1~$years is the median time baseline for the HARPS M dwarfs. For a $0.5~M_{\odot}$ star, these ranges roughly correspond to planet masses between $\sim 1-5~M_{\rm Jup}$ and projected separations between $\sim 1-2~$AU (for $D_l/D_s = 1/2$). The median log values are then $\log{q_{\rm med}} \approx \left(\log{q_{\rm max}}+\log{q_{\rm min}}\right)/2\approx -2.35$ and $\log{s_{\rm med}} \approx \left(\log{s_{\rm max}}+\log{s_{\rm min}}\right)/2\approx -0.2$. We find a mean and 68\% confidence interval of $f_{\rm med}= 0.064^{+0.042}_{-0.043}$ using equation (\ref{eqn:planetary_mass_function}) and these median values.
Using these values and equation (\ref{eqn:n_det_estimate}), we expect BX13 to detect $N_{\rm pl} = 0.63^{+0.41}_{-0.42}$ planets from the $N_{\star}\sim 100$ stars they monitor, where the errors on this estimate come from uncertainties in the normalization (GA10) and exponent \citep{2010ApJ...710.1641S} of the planetary mass-ratio function given by equation (\ref{eqn:planetary_mass_function}). This answer is within a factor of $\sim 2$ of the result we obtain from the detailed calculation in the next section.
\subsubsection{Detailed Comparison}
\label{subsubsec:bonfils_detailed_comparison}
BX13 monitor a total of 102 stars. We discard the four stars with less than four epochs. We also eliminate Gl 803 from the sample. The mass they report for this star is $0.75~M_{\odot}$, which is derived from the empirical mass-luminosity relationship of \citet{2000A&A...364..217D} in conjunction with parallax information and K-band photometry. They note in a footnote below their Table 3 that Gl 803 (AU Mic) is a $\sim 20~$Myr star with a circumstellar disk and so the calibration for determining its mass may not be valid given its age. To keep their mass estimations consistent, they chose not to adopt the mass found by \citet{2004Sci...303.1990K} for this star of $0.5~M_{\odot}$. We argue that Gl 803 should not be included in their sample on the grounds that it is not an M dwarf given the mass estimate they choose to adopt. We note that there are no known planets around this star, although it does show variation of a couple hundred meters per second. However, with only four epochs, we cannot say anything about the source of this variation. Thus, the refined HARPS M dwarf sample we consider includes 97 M dwarfs with four or more epochs.
We obtain data on each of these 97 stars in the HARPS sample from Tables 3 and 4 of their paper. We use the number of epochs per star, $N$, the overall uncertainties ($\sigma_{\rm tot}$) for each, including both ``internal'' ($\sigma_i$) and ``external'' ($\sigma_e$) errors, $\sigma_{\rm tot} \equiv \sqrt{\sigma_i^2+\sigma_e^2}$, and the mass of each star, $M_{\star}$ (see BX13 for a discussion of their uncertainties). We also obtain the time baseline for observations, $T$, for each star from the plots in their Figure 18. Since they do not report uncertainties in the host star mass estimates, we turn to the original reference for the method they use to compute the masses. \citet{2000A&A...364..217D} required that the stars they used to calibrate their mass-luminosity relationships have a mass accuracy of $\lesssim 10\%$, so we adopt uncertainties in the mass of the stars in the HARPS sample to be $10\%$. We use these data and the detection limits in figure 18 of BX13 to estimate their sensitivities and compute the expected number of planet detections and long-term RV trends as described in \S~\ref{subsec:method_application}.
We find the total expected number of planet detections by BX13 to be
$N_{\rm det} = 1.4\pm 0.8$ and a lower limit on the number of trends they should see to be $N_{\rm t} = 2.1^{+1.2}_{-1.4}$, where the errors on these quantities are due to the uncertainties in the slope and normalization of our planetary mass function (see \S~\ref{subsec:method_application}). Our estimate of the number of trends is a lower limit because we are considering only populations of planets, whereas the RV survey could also be seeing trends due to distant stellar or brown dwarf companions. We bin the number of expected detections, trends, and total planets in decades of $m_p\sin{i}$ and $P$, similar to Table 11 in BX13, which we report in table \ref{tab:predicted_dets_trends}. For comparison, we also include the values reported by BX13.
In \citet{clanton_gaudi14a}, we determined that a fiducial RV survey (with $N=30$, $\sigma=4~{\rm m~s^{-1}}$, $T=10~$yr) should on average detect $0.049^{+0.46}_{-0.26}$ planets per star at a SNR of 5 or higher. If the sensitivities of BX13 for each star were equal to those of the fiducial survey, and if their sample covered the mass interval $0.07\leq M_{\star}/M_{\odot}\leq1.0$ in a log-uniform fashion (as was the case for our fiducial RV survey), we would have predicted that BX13 should have detected $4.9^{+4.6}_{-2.6}$ planets since their sample size is nearly $N_{\star}\sim 100~$ stars. This number is a factor $\sim 3.5$ larger than our final, detailed estimate. The difference arises from the fact that $\mathcal{Q}_{\rm min}=5$ for our fiducial survey, whereas the median value for HARPS is $\mathcal{Q}_{\rm min}\sim 10$, meaning our fiducial survey is overall more sensitive than HARPS. Our order of magnitude estimates turn out to be good enough to yield the right answer to within a factor of a few, but highlights the importance of understanding the detailed detection sensitivities of an entire sample to obtain accurate statistics.
\begin{table*}
\caption{\label{tab:predicted_dets_trends} Predicted detections and trends for the HARPS M dwarf survey (BX13), binned in $m_{\rm p}\sin{i}-P$ space. In each bin, $N_d$ is the number of predicted detections, $N_t$ is the number of predicted trends and $f$ is the derived planet frequency. The bold numbers are our results, while the unbolded values are those reported by BX13. There are no trend values for BX13 because it is not clear in which bins their reported trends lie (with the exception of Gl 832b, which we have included as a trend rather than a detection; see text). Uncertainties in our results are due to uncertainties in both the normalization and slope of the planetary mass function we adopt from the measurements by GA10 and \citet{2010ApJ...710.1641S}, respectively.}
\begin{tabular}{l|rrrrr}
\hline \hline
\multicolumn{1}{c|}{$m_{\rm p}\sin{i}$} & \multicolumn{5}{c}{Orbital Period [day]} \\
\multicolumn{1}{c|}{[M$_{\oplus}$]} & 1$-$10 & $10-10^2$ & $10^2-10^3$ & $10^3-10^4$ & $10^4-10^5$\\
\hline
& $N_d = \mathbf{0.0}, 0$& $N_d = \mathbf{0.0}, 0$& $N_d = \mathbf{(9.3^{+9.4}_{-9.28})E-3}, 0$& $N_d = \mathbf{0.013^{+0.011}_{-0.0126}}, 0$& $N_d = \mathbf{0.0d}, -$\\
$10^3-10^4$& $N_t = \mathbf{0.0}, -$& $N_t = \mathbf{0.0}, -$& $N_t = \mathbf{(4.5^{+5.3}_{-4.47})E-4}, -$& $N_t = \mathbf{0.093^{+0.066}_{-0.080}}, -$& $N_t = \mathbf{0.016^{+0.014}_{-0.0155}}, -$\\
& $f = \mathbf{-}, <0.01$& $f = \mathbf{-}, <0.01$& $f = \mathbf{(1.0^{+1.1}_{-0.98})E-4}, <0.01$& $f = \mathbf{(1.2^{+8.4}_{-1.0})E-3}, <0.01$& $f = \mathbf{(3.6^{+3.4}_{-3.5})E-4}, -$\\
\hline
& $N_d = \mathbf{0.0}, 0$& $N_d = \mathbf{0.0}, 2$& $N_d = \mathbf{0.32^{+0.21}_{-0.24}}, 0$& $N_d = \mathbf{0.41^{+0.28}_{-0.29}}, 1$& $N_d = \mathbf{0.0}, -$\\
$10^2-10^3$& $N_t = \mathbf{0.0}, -$& $N_t = \mathbf{0.0}, -$& $N_t = \mathbf{0.012^{+0.011}_{-0.010}}, -$& $N_t = \mathbf{1.5^{+0.89}_{-1.0}}, 1$& $N_t = \mathbf{0.060^{+0.040}_{-0.046}}, -$\\
& $f = \mathbf{-}, <0.01$& $f = \mathbf{-}, 0.02^{+0.03}_{-0.01}$& $f = \mathbf{(4.7^{+3.1}_{-3.4})E-3}, <0.01$& $f = \mathbf{0.038^{+0.023}_{-0.026}}, 0.019^{+0.043}_{-0.015}$& $f = \mathbf{(7.9^{+4.8}_{-5.4})E-3}, -$\\
\hline
& $N_d = \mathbf{0.0}, 2$& $N_d = \mathbf{(1.7^{+1.7}_{-1.6})E-4}, 0$& $N_d = \mathbf{0.28^{+0.14}_{-0.16}}, 0$& $N_d = \mathbf{0.31\pm 0.17}, 0$& $N_d = \mathbf{0.0}, -$\\
$10-10^2$& $N_t = \mathbf{0.0}, -$& $N_t = \mathbf{0.0}, -$& $N_t = \mathbf{0.023^{+0.013}_{-0.014}}, -$& $N_t = \mathbf{0.45\pm 0.24}, -$& $N_t = \mathbf{(1.0^{+1.3}_{-0.99})E-3}, -$\\
& $f = \mathbf{-}, 0.03^{+0.04}_{-0.01}$& $f = \mathbf{-}, <0.02$& $f = \mathbf{0.020\pm 0.009}, <0.04$& $f = \mathbf{0.16^{+0.068}_{-0.072}}, <0.12$& $f = \mathbf{0.032^{+0.012}_{-0.014}}, -$\\
\hline
& $N_d = \mathbf{0.0}, 5$& $N_d = \mathbf{(2.9^{+2.9}_{-2.8})E-5}, 3$& $N_d = \mathbf{0.010\pm 0.007}, 0$& $N_d = \mathbf{(2.6^{+2.0}_{-2.3})E-3}, 0$& $N_d = \mathbf{0.0}, -$\\
$1-10$& $N_t = \mathbf{0.0}, -$& $N_t = \mathbf{0.0}, -$& $N_t = \mathbf{(3.7^{+1.9}_{-3.67})E-4}, -$& $N_t = \mathbf{(1.4^{+1.2}_{-1.39})E-3}, -$& $N_t = \mathbf{0.0}, -$\\
& $f = \mathbf{-}, 0.36^{+0.24}_{-0.10}$& $f = \mathbf{-}, 0.52^{+0.50}_{-0.16}$& $f = \mathbf{0.080\pm 0.031}, -$& $f = \mathbf{0.64^{+0.25}_{-0.26}}, -$& $f = \mathbf{0.12^{+0.051}_{-0.049}}, -$\\
\hline\hline
\end{tabular}
\end{table*}
{\bf Detections: } Before we directly compare our predicted detections with the values reported by BX13, we first examine their reported detections. In the bin corresponding to $10^2 \leq m\sin{i}/ M_{\oplus} \leq 10^3$ and $10^3 \leq P/{\rm days} \leq 10^4$, they report the detection of two planets, Gl 832b and Gl 849b. They describe their data on these two planets in their \S~5.1. In the case of Gl 832b, they report that the HARPS data indicate a long-period RV variation at high confidence level, but with their data alone, they cannot uniquely determine the Keplerian orbit and thus are unable to confirm the planetary nature of Gl 832b. Only when they combine the HARPS data with the AAT data, are they able to refine the orbit of the planet and confirm its planetary nature. Thus, we argue that the HARPS survey sample should not include the detection of Gl 832b when determining planet frequencies from their survey. In the case of Gl 849b, the HARPS data confirms it as a Jupiter-mass companion. When they combine Keck RVs for this planet, they report that a single planet is not enough to explain the RV variation, but since they are able to identify the companion as a planet with HARPS data alone, this planet should be included in the sample. Therefore, the number of detections in this bin of Table 11 of BX13 should to be one, rather than two, and the planet frequency here should be $f=0.019^{+0.043}_{-0.015}$. However, since the HARPS data alone confirm long-term variation, we include Gl 832b as an identified trend by their survey.
In particular, we focus on comparing our predictions for planet detections and trends with the actual numbers reported by BX13 for orbital periods longer than $\sim 100~$days. Microlensing surveys have little or no sensitivity to shorter orbital periods and thus we are unable to compare with in these regions where there is no overlap between microlensing and RV. We predict that BX13 should detect a total of $N_{\rm det} = 1.4\pm 0.8$ planets. The majority of these predicted planet detections for HARPS lie in four bins (see table~\ref{tab:predicted_dets_trends}). The largest amount of predicted planet detections, with $N_d=0.41^{+0.38}_{-0.18}$, lie in the $10^2\leq m_p\sin{i}/M_{\rm Jup}\leq 10^3$ and $10^3\leq P/{\rm days}\leq 10^4$ bin. The only reported planet detection by BX13 falls into this bin (Gl 849b) with a mass of $m_p\sin{i}=372\pm 19~M_{\oplus}$ and an orbital period of $P=2165\pm 132~$days. The other three bins within which we predict a significant amount of planet detections include $10^2\leq m_p\sin{i}/M_{\rm Jup}\leq 10^3$ and $10^2\leq P/{\rm days}\leq 10^3$ with $N_d=0.32^{+0.29}_{-0.16}$, $10\leq m_p\sin{i}/M_{\rm Jup}\leq 10^2$ and $10^3\leq P/{\rm days}\leq 10^4$ with $N_d=0.31^{+0.21}_{-0.13}$, and finally $10\leq m_p\sin{i}/M_{\rm Jup}\leq 10^2$ and $10^2\leq P/{\rm days}\leq 10^3$ with $N_d=0.28^{+0.18}_{-0.12}$. The fact that BX13 do not report any planet detections in these three bins is consistent with our predictions since the Poisson probabilities of detecting zero planets, assuming the predicted number of detections is equal to the mean number of planets residing in these bins such that $P(0)=e^{-N_d}$, are $0.74\pm 0.12$, $0.75^{+0.16}_{-0.17}$, and $0.76^{+0.12}_{-0.11}$, respectively.
In summary, we predict that the HARPS survey should find about one planet with a period right at the edge of the survey duration and indeed BX13 report the detection of such a planet (Gl 849b). Thus, consistency between microlensing and radial velocity surveys in the region of planet parameter space in which they overlap implies that the giant planet frequencies inferred from the two types of surveys are in fact consistent. We conclude that RV surveys are detecting only the high-mass end of the population of giant planets inferred by microlensing, leading to their underestimate of the total giant planet frequency around M dwarfs.
{\bf Trends: } In our approximation, we expect planets to be identified as long-term RV drifts when they have periods greater than the time baseline of observations of their host star, i.e. $P>T$, and produce detectable signals, i.e. lying on or above the detection limit curve for their host star (as exemplified in figure \ref{fig:marg_dist}). In the limit $P\gg T$, the RV trends will be basic, linear accelerations, the slope of which depends on the phase covered by the actual observations. However, when $P$ is just larger than $T$, by our approximation such a planet will also be considered as a trend, but will exhibit more complex variation than a linear trend. We compute the RV accelerations for our predicted trend-producing planets by multiplying the maximum possible slope, $2\pi K/P$, by a factor $\cos{\phi}$, where $\phi$ is the phase angle at the time of observation, randomly and uniformly drawn between $[0,2\pi)$. We ignore the eccentricity in computing the slopes and make the approximation $P\gg T$.
Under these assumptions, we predict that the HARPS M dwarf survey should find at least one or two trends ($N_{\rm t} = 2.1^{+1.2}_{-1.4}$), with a median RV acceleration and 68\% confidence interval of $7.9_{-5.8}^{+19.}~{\rm m~s^{-1}~yr^{-1}}$, most likely in the bin with $10^3\leq P/{\rm days}\leq 10^4$ and $10^2\leq m_p\sin{i}/M_{\oplus}\leq 10^3$ (there is expected to be $1.5^{+1.3}_{-0.6}$ RV trends due to planets in this bin as shown in table~\ref{tab:predicted_dets_trends}). As discussed above, Gl 832b falls in this bin with a reported acceleration of $5.198~{\rm m~s^{-1}}$. Indeed, the RV time series for this star (shown in figure 3 of BX13) does exhibit more complex variability than a simple linear trend. BX13 report additional long-term RV trends in their sample. The largest, statistically significant RV acceleration (i.e. with a false alarm probability (FAP) less than 0.01) reported by BX13 is $-9.616~{\rm m~s^{-1}~yr^{-1}}$ from the star Gl 849 (MB14 also detect RV acceleration of this star). They report a total of 15 stars to have RV slopes with FAP$<0.01$, with a median magnitude of $2.65~{\rm m~s^{-1}~yr^{-1}}$. Of these 15 stars, the report only five of them to have ``smooth'' RV drifts, namely LP 771-95A, Gl 367, Gl 618A, Gl 680, and Gl 880, while the rest exhibit more complex variability. The median magnitude of these ``smooth'' RV accelerations is $3.20~{\rm m~s^{-1}~yr^{-1}}$.
Figure \ref{fig:trends} shows the histograms and CDFs of all trends and the smooth trends reported by BX13, along with the distribution of drifts that we predict. We perform a two-sample Kolmogorov-Smirnov (K-S) test between our predicted distribution of RV trends and that of all 15 significant trends from HARPS and find a $D$-statistic of 0.52 with probability $P\left(D\right)=2.8\times10^{-3}$, demonstrating that the two distributions are inconsistent. We also perform a two-sample K-S test between our predicted distribution and that of just the 5 significant, smooth trends found in the HARPS sample, which yields $D=0.52$ with probability $P\left(D\right)=0.084$.
\begin{figure}[h!]
\epsscale{1.1}
\plotone{fig3.eps}
\caption{The top panel shows the relative number of long-term RV trends for the actual HARPS sample and our predicted sample. The blue dot-dashed lines include the stars BX13 report to have significant RV trends (with FAP$<0.01$) and the red dashed lines are a subset of these stars for which BX13 report smooth RV variation. The black lines are our predicted trends which are computed as $2\pi K\cos{\phi}/P$ for the systems we expect to show up with trends. The bottom panel shows the cumulative distribution functions of these distributions. We perform K-S tests and find that our predicted distribution of trends is inconsistent with the distribution of all RV trends (yet not necessarily the subset of smooth trends), suggesting that a majority of the trends identified in the HARPS sample, if arising from companions, are due to more distant and more massive stellar or brown dwarf companions, or planets to which microlensing is not sensitive.
\label{fig:trends}}
\end{figure}
We can explain the RV accelerations BX13 detect from Gl 832b and Gl 849c as arising from planetary companions predicted by microlensing. In the next section, we discuss how MB14 are able to constrain the mass of Gl 849c to be $m_p\sin{i}=0.70\pm 0.31~M_{\rm Jup}$ and its orbital period to be $19.3^{+17.1}_{-5.9}$~years by measuring the rate of change in RV acceleration, or the ``jerk.'' This most likely places Gl 849c into the same bin of mass and period as Gl 832b, where we predict $1.5^{+1.3}_{-0.6}$. However, the remaining 13 RV drifts are inconsistent with the hypothesis that they are caused by planetary companions analogous to the population inferred from microlensing. MB14 suggest that at least two of the trends detected by BX13, those of Gl 250B and Gl 618B, can be attributed to long-period binary companions. It is unclear if the remainder of the RV trends are due to planets beyond the sensitivity of current microlensing surveys, stellar or brown dwarf binary companions, or even magnetic activity \citep[e.g.][]{1988lsla.book.....G,2012A&A...541A...9G}.
We can assess the plausibility that the measured trends are due to planetary mass companions that are at periods outside those for which microlensing is sensitive. If we let $a_{\rm t}$ be the magnitude of a given trend measured by BX13, then setting $2\pi K\cos{\phi}/P=a_{\rm t}$, substituting for $K$ using the standard velocity semi-amplitude equation, and solving for $m_p\sin{i}$ yields the minimum companion mass required to produce the observed trend as a function of orbital period
\begin{align}
m_p\sin{i} = & {} \left(\frac{P}{2\pi}\right)^{4/3}G^{-1/3}M_{\star}^{2/3}\frac{a_{\rm t}}{\cos{\phi}} \\
= & {}\; 0.44~M_{\rm Jup} \left(\frac{a_{\rm t}}{1~{\rm m~s^{-1}~yr^{-1}}}\right)\left(\frac{P}{30~{\rm yr}}\right)^{4/3} \nonumber \\
& {} \times\left(\frac{M_{\star}}{0.5~M_{\odot}}\right)^{2/3}\left(\frac{1}{\cos{\phi}}\right) \; . \label{eqn:min_trend_mass}
\end{align}
Using equation~\ref{eqn:min_trend_mass}, we plot the minimum required companion mass to yield the measured RV accelerations reported by BX13 for the 13 unexplained trends in figure~\ref{fig:possible_trends_masses}. We assume that $\cos{\phi}=1$ in our calculations because the exact orbital phase during observations is unknown; any other value of $\cos{\phi}$ would serve to increase the required companion mass, so this assumption assures we are indeed estimating the minimum required companion mass. We plot these values assuming the companions are at orbital periods of 30, 50, and 100~years. As we have previously shown, a planet with an orbital period of about $P\sim 30~$years, which corresponds to a projected separation of roughly 2.5 times the Einstein radius of the typical lens, is just beyond the sensitivity of microlensing surveys. The minimum required companion masses at all periods are consistent with giant planets ($m_p\sin{i}>0.1~M_{\rm Jup}$), with just one exception. BX13 report the measurement of a $0.206~{\rm m~s^{-1}~yr^{-1}}$ RV acceleration of Gl 431.1, which has a minimum required companion mass of roughly $27~M_{\oplus}$ if it orbits at a period of 30~years. Thus, if giant planets are common at orbital periods beyond $\sim 30~$years, it is plausible that these are the source of the majority of the long-term RV trends measured by BX13 in the HARPS M dwarf sample. However, we note that there are significantly less trends reported by MB14 for the CPS M dwarfs despite having a larger sample size than HARPS.
\begin{figure}[h!]
\epsscale{1.1}
\plotone{fig4.eps}
\caption{The minimum companion masses required to produce the 13 unexplained long-term RV trends observed by BX13 in the HARPS M dwarf sample assuming the source of these trends is at a given period (see equation~\ref{eqn:min_trend_mass}). The black histogram represents the minimum required masses if the companions have orbital periods of 30~years. The blue, dashed histogram and the red, dot-dashed histogram represents these companions at orbital periods of 50 and 100~years, respectively.
\label{fig:possible_trends_masses}}
\end{figure}
\subsection{Comparison with CPS Planet Detections}
\label{subsec:johnson_comparison}
MB14 provide basic parameters for each of the 111 M dwarfs in their sample (which we describe in \S~\ref{subsec:johnson_sample}), including the stellar mass, number of RV measurements, time baseline of observations, and average RV precision. Since the CPS team has not yet determined individual RV detection sensitivities for each of their stars, we make a very rough estimate of their sensitivity by matching CPS stars with those from the HARPS sample with similar systematics as described in \S~\ref{subsec:method_application}. We then determine the expected number of planet detections and long-term RV trends the CPS team should see in the same manner as we did for the HARPS sample.
{\bf Detections: }We predict a total of $N_{\rm det}= 4.7^{+2.5}_{-2.8}$ detected planets with periods longer than $10^2~$days from the CPS M dwarf sample, and indeed this sample has yielded 4 such planets. We expect $2.2^{+1.4}_{-1.5}$ of our predicted planet detections to have a mass between $10^2-10^3~$M$_{\oplus}$ and a period between $10^3-10^4~$days. Two of the CPS detections, Gl 179b \citep{2010ApJ...721.1467H} and Gl 849 \citep{2006PASP..118.1685B}, lie in this bin. The other two CPS detections, Gl 317b \citep{2007ApJ...670..833J} and Gl 649b \citep{2010PASP..122..149J}, lie in the mass range $10^2\leq m_p\sin{i}/M_{\oplus}\leq 10^3$ and the period range $10^2\leq P/{\rm days}\leq 10^3$. Of our predicted planets, we expect $0.46^{+0.30}_{-0.34}$ to lie in this bin. If this number is indeed true number of planets in this bin, then the Poisson probability of detecting two planets is $0.19^{+0.07}_{-0.08}$, which we consider to be marginally significant. However, as we discuss in \S~\ref{sec:synthesis}, the sensitivity of microlensing falls off towards shorter periods in this bin, while the sensitivity of RV surveys decreases towards longer periods. We therefore expect the planet frequency in this bin to be larger than the value we predict from microlensing in this paper, so it is not surprising that we under-predict the number of planet detections in this period range. Of the remaining predicted planet detections, we expect $1.4^{+0.72}_{-0.74}$ planet detections with $10\leq m_p\sin{i}/M_{\oplus}\leq 10^2$ and $10^3\leq P/{\rm days}\leq 10^4$, $0.44^{+0.21}_{-0.23}$ detections with $10\leq m_p\sin{i}/M_{\oplus}\leq 10^2$ and $10^2\leq P/{\rm days}\leq 10^3$, and $0.12^{+0.09}_{-0.10}$ detections with $10^3\leq m_p\sin{i}/M_{\oplus}\leq 10^4$ and $10^3\leq P/{\rm days}\leq 10^4$. There are no CPS detections in these bins, the Poisson probabilities for which are $0.31\pm 0.20$, $0.67^{+0.15}_{-0.14}$, and $0.90^{+0.11}_{-0.09}$, respectively, assuming that the true number of planets in these bins are the predicted values.
{\bf Trends: }We predict that the CPS M dwarf sample should see a total of $N_{\rm t}= 1.8^{+1.1}_{-1.2}$ long-term RV drifts due to giant planets on long-period orbits. Of these, we predict $1.1^{+0.69}_{-0.75}$ will be due to a giant planet with $0.31\lesssim m_p\sin{i}/M_{\rm Jup}\lesssim 3.1$ and $2.7\lesssim P/{\rm yr}\lesssim 27$. There are four other bins that we predict to harbor a significant source of RV trends in the CPS sample: between $0.31\leq m_p\sin{i}/M_{\rm Jup}\leq 3.1$ and $27\leq P/{\rm yr}\leq 270$ we predict $0.30^{+0.19}_{-0.22}$ trends, between $0.031\leq m_p\sin{i}/M_{\rm Jup}\leq 0.31$ and $2.7\leq P/{\rm yr}\leq 27$ we predict $0.28\pm 0.15$ trends, between $3.1\leq m_p\sin{i}/M_{\rm Jup}\leq 31$ and $2.7\leq P/{\rm yr}\leq 27$ we predict $0.081^{+0.065}_{-0.073}$ trends, and between $3.1\leq m_p\sin{i}/M_{\rm Jup}\leq 31$ and $27\leq P/{\rm yr}\leq 270$ we predict $0.042^{+0.038}_{-0.040}$ trends.
MB14 report a total of four measured RV accelerations. Of these, that of Gl 849 exhibits significant curvature (or ``jerk''), allowing for constraints on the mass and period of the long-period companion. They find a median minimum mass of $m_p\sin{i}=0.70\pm 0.31~M_{\rm Jup}$ and a median period of $19.3^{+17.1}_{-5.9}~$years. Although MB14 are able to place constraints on the companion properties for this measured trend (and imaging rules out stellar mass companions), in our simulation this planet would be counted as a trend. The mass and period most likely place it in the bin we predict the most trends to lie. The weak constraints on the orbital period could scatter this trend into the next higher period bin, which happens to be another bin for which we predict a significant number of trends.
The remaining three other stars for which MB14 measure significant RV accelerations are Gl 317 ($2.51\pm 0.62~{\rm m~s^{-1}~yr^{-1}}$), Gl 179 ($-1.17\pm 0.29~{\rm m~s^{-1}~yr^{-1}}$), and Hip 57050 ($1.39\pm 0.39~{\rm m~s^{-1}~yr^{-1}}$). Imaging with NIRC2 (instrument PI: Keith Matthews) using the AO system at the W. M. Keck Observatory \citep{2000PASP..112..315W} in the $K'$ or $K_s$ filters rule out most stellar-mass companions and some brown dwarfs as the source of these trends. The low inferred brown dwarf frequency around M dwarfs from \citet{2012AJ....144...64D} and similarly low frequency of brown dwarf companions to FGK stars from \citet{2009ApJS..181...62M} lead MB14 to conclude that these trends are probably due to giant planets. However, they do mention that their imaging of Hip 57050 is only complete at separations smaller than 1 arcsecond ($r_{\perp}\approx 11~$AU), leaving some parameter space for a low mass M dwarf companion to be the cause of the RV acceleration.
We predict a trend that is consistent with that caused by Gl 849, but overall our numbers seem to be marginally consistent with the four observed RV accelerations by MB14, if they are indeed due to planetary companions. The Poisson probability of detecting four trends when the true mean is $N_t=1.8^{+1.1}_{-1.2}$ is $0.036^{+0.11}_{-0.033}$. If MB14 have misclassified one of their detected trends, and turns out to be due to a brown dwarf companion rather than a planetary companion, then the Poisson probability of detecting 3 trends if the true mean is $N_t=1.8^{+1.1}_{-1.2}$ increases to $0.10^{+0.11}_{-0.08}$.
As we did for the BX13 trends, we can compute the minimum companion mass required to produce the trends MB14 measure for Gl 317, Gl 179, and Hip 57050 using equation~\ref{eqn:min_trend_mass}. At a period of 30~years, the minimum required companion mass for these stars is $1.0~M_{\rm Jup}$, $0.47~M_{\rm Jup}$, and $0.55~M_{\rm Jup}$, respectively. At a period of 50~years, we calculate $2.0~M_{\rm Jup}$, $0.92~M_{\rm Jup}$, and $1.1~M_{\rm Jup}$, respectively. At 100~years, we find $5.0~M_{\rm Jup}$, $2.3~M_{\rm Jup}$, and $2.8~M_{\rm Jup}$, respectively. The companions responsible for producing these long-term trends MB14 measure could be giant planet planets and would be beyond the sensitivity of current microlensing surveys. We note that although our inferred frequency is consistent with that of MB14, it is nevertheless a median factor of 2.2 ($0.22-8.8$ at 95\% confidence) times smaller, potentially due to the fact that microlensing is missing such a population of very long-period super-Jupiters, which is being inferred by MB14 by these trends that microlensing does not predict. In fact, if MB14 were to ignore these three trends, we expect they would infer a frequency nearly identical to ours.
One caveat with this comparison is that we do not have the actual detection limits for each star in the CPS sample so we are forced to estimate them by matching to stars in the HARPS M dwarf sample with similar systematics. Due to the steep planetary mass function inferred from microlensing, the numbers of predicted detections and trends are very sensitive to the detection limits. The right panel in figure \ref{fig:marg_dist} showing the distribution of $K$ marginalized over all $P$ reflects this steep mass function. In order to make more robust predictions for the CPS M dwarf sample examined by MB14, we would need more accurate sensitivity estimates. In order to illustrate this point, we assume that the overall distribution of $\mathcal{Q}_{\rm min}$ remains the same, but multiply each by a constant SNR scale factor, $\mathcal{C}$, to determine the detection limits for the CPS M dwarf sample. We then calculate the total predicted numbers of planet detections and trends. We plot $N_d$ and $N_t$ as a function of $\mathcal{C}$ in figure~\ref{fig:nd_nt_scaling}. Increasing $\mathcal{Q}_{\rm min}$ by a factor of 2 results in roughly 1.8 fewer detections and 0.9 fewer trends, while decreasing $\mathcal{Q}_{\rm min}$ by a factor of 2 results in roughly 3 more detections and 1.6 more trends.
\begin{figure}[h!]
\epsscale{1.1}
\plotone{fig5.eps}
\caption{The numbers of predicted detections and trends as a function of the SNR scale factor, $\mathcal{C}$. Since we do not have detailed detection sensitivities for each star in the CPS sample, we estimate their detection limits by assuming that stars in the CPS sample with similar systematics to stars in the HARPS sample have the same sensitivities (see \S~\ref{subsec:method_application}). This plot shows that our predicted number of detections and trends depends sensitively on the detection limits we assume due to the steeply declining mass function inferred from microlensing surveys.
\label{fig:nd_nt_scaling}}
\end{figure}
\subsection{Additional Simulations and Results}
\label{sec:additional_simulations}
We ran additional simulations where we altered the systematics of the HARPS and CPS surveys in order to see how the numbers of predicted detections and trends would change if the time baselines for each star were increased, or if they were able to reduce both internal and external noise sources. We ran three different tests where we 1) doubled the time baseline, $T$, of observations for each star, 2) fixed measurement errors at $\sigma = \left(\sigma_i^2+\sigma_e^2\right)^{1/2}=1~{\rm m~s^{-1}}$, and 3) both doubled $T$ and fixed $\sigma = 1~{\rm m~s^{-1}}$. The results from each of these simulations for the HARPS survey are as follows: 1) $N_{\rm det} = 2.0\pm 1.4$, $N_{\rm t} = 1.2\pm 0.9$, 2) $N_{\rm det} = 3.7\pm 2.7$, $N_{\rm t} = 6.3\pm 4.5$, and 3) $N_{\rm det} = 6.7\pm 4.8$, $N_{\rm t} = 4.5\pm 3.2$. As expected, we find that increasing the duration of observations and reducing the uncertainties increases the numbers of predicted detections and trends. For the CPS sample, we find the results: 1) $N_{\rm det} = 4.9\pm 3.5$, $N_{\rm t} = 0.60\pm 0.43$, 2) $N_{\rm det} = 11.\pm 8$, $N_{\rm t} = 4.6\pm 3.3$, and 3) $N_{\rm det} = 14.\pm 10$, $N_{\rm t} = 2.1\pm 1.5$. Doubling observation times does not double the expected number of detections, as the median time baseline for the CPS sample is over 10 years, whereas the median period for planets found by microlensing surveys is about $9.4~$years (see Figure \ref{fig:marg_dist}).
At least in terms of orbital period, the CPS survey is sensitive to a majority of the population of planets inferred from microlensing surveys. In the case of both RV surveys, decreasing measurement uncertainties greatly increases the number of expected RV planet detections at a range of orbital periods, but peaking near the edge of their survey durations. Thus, if RV surveys hope to detect the entire population of giant planets inferred by microlensing, rather than just the high-mass end, the typical measurement uncertainties need to be reduced by a factor of a few to cut further into the steep planetary mass function.
\subsection{Properties of Planets Accessible to Microlensing and RV}
\label{subsec:planet_props}
In addition to computing the number of planet analogs to which RV surveys are sensitive, we are also interested in the properties of such planets. In the appendix, we examine the distributions of microlensing and orbital parameters for the planets we predict will show up as detections and long-term RV trends in the HARPS sample to determine if there is a subset of the planet population inferred from microlensing towards which RV surveys are particularly sensitive.
Not surprisingly, we find that the planets we predict HARPS will detect is sensitive to the distribution of projected separations, $s$, preferring small values of $s$, and preferring higher values of the planet to host star mass ratio, $q$. We also find that predicted detections have a slight bias against lens distances at and near the halfway point between the Earth and the source, where the Einstein radius is maximized. This is a reflection of the fact that the RV signal decreases with increasing orbital separation, as well as the fact that the median time baseline for stars in the HARPS sample is shorter than the median period of the entire population of microlensing planets by a factor of $\sim 2$. Additionally, there is a preference for planet detections around more massive hosts, even at fixed $q$. However, we find that there is no significant preference for RV planet detections of analogs to the planets found around bulge or disk lenses by microlensing (assuming planets are equally common around all stars regardless of their location in the Galaxy). See the appendix for additional discussion.
\section{Uncertainties}
\label{sec:uncertainties}
\subsection{Normalization and Slope of the Microlensing Mass-Ratio Function}
\label{subsec:norm_slope_unc}
The main sources of uncertainty in our calculations are the uncertainties in the microlensing measurements of the normalization and slope of the planetary mass function \citep[GA10;][]{2010ApJ...710.1641S}. The quoted uncertainties of our results throughout this paper are due to these sources. See \S~\ref{subsec:method_application} and \citet{clanton_gaudi14a} for a description of how these uncertainties are propagated.
There is another source of uncertainty, which we mention here but do not explicitly include in our final results. This stems from our assumption that the distribution function of companions $d^2N_{\rm pl}/(d\log{q}~d\log{s})$ is invariant in mass ratio, $q$, rather than planet mass, $m_p$. Microlensing surveys are currently unable to distringuish between these two assumptions. Therefore, the planet frequency we infer for planets of a given $m_p$ depends on the primary mass. We have adopted a typical primary mass for the microlensing sample of $M_l\sim 0.5~M_{\odot}$. On the other hand, the median stellar mass of the HARPS M dwarf sample (BX13) is $0.3~M_{\odot}$ and that of the CPS M dwarf sample (MB14) is $0.41~M_{\odot}$. Therefore, our assumption of a fixed distribution function in $q$ means that we are assigning a lower planet frequency at fixed planet mass for the BX13 and MB14 samples than for the microlensing sample. However, the mass distribution and typical mass of the microlensing sample is uncertain, and values as low as $0.3~M_{\odot}$ are possible. Had we adopted lower values, our inferred frequencies for the HARPS and CPS sample would be higher. To estimate the level of this effect, we integrate our planetary mass-ratio function over the mass interval $1\leq m_p/M_{\rm Jup}\leq 13$ assuming a host mass of $M_{\star} = 0.5~M_{\odot}$ and divide by the mean value of this same integral calculated for the host star masses of each of the stars in the HARPS sample. We then repeat this exercise for the CPS sample. We find frequencies that are factors of 1.4 and 1.2 times higher, respectively, indicating that the actual frequencies we infer could be up to $\sim 40\%$ higher for the HARPS sample and up to $\sim 20\%$ for the CPS sample. Given the fact that the uncertainties on our final results due to the slope and normalization of our planetary mass-ratio function are typically around the $\sim 50\%$ level, there are some cases where this effect could be significant.
\subsection{Galactic Model and Microlensing Parameter Distribution}
\label{subsec:gal_model_mlens_param_unc}
There is also some degree of unquantified uncertainties due to our choice of priors on the planet and host star properties of our simulated sample (e.g. priors on lens masses and distances, planetary orbital parameters). However, we expect any such errors to be subdominant because we are mostly able to reproduce the observed distributions of host star parameters from the actual microlensing sample by appropriately weighting by the event rate, with the single possible exception of the distribution of lens distances. We describe in great detail the comparison of the distribution of such parameters between our simulated sample and the actual microlensing sample of GA10 in \citet{clanton_gaudi14a}.
\subsection{Contamination from FGK Stars and Remnants}
\label{subsec:contam_unc}
There could be unquantified sources of error in our analysis related to differences between the microlensing and RV samples. For example, microlensing is only able to measure lens masses for a subset of all events. While each of the planet-hosting lenses in the GA10 sample have mass measurements (or at least mass upper limits), it could be the case that a fraction of the lenses included in the GA10 sample are not actually M dwarfs, but are instead stellar remnants (white dwarfs, neutron stars, or black holes) or even K and G stars. \citet{2000ApJ...535..928G} estimated that $\sim 20\%$ of detected microlensing events are due to remnants that are completely unrecognizable from their timescale distribution. Consequently, we expect the resultant uncertainty to be small in comparison to the Poisson error on the number of planet detections included in the GA10 study and thus not a significant source of error in our analysis.
\subsection{Differences in the Metallicity Distribution of RV and Microlensing Hosts}
\label{subsec:metallicity_unc}
In general, any Galactic gradient of properties that affect planet frequency could affect our results. The most obvious of such properties is the Galactic metallicity gradient \citep[see e.g.][]{2012ApJ...746..149C,2013arXiv1311.4569H}. While RV surveys of M dwarfs are limited to targets within tens of parsecs, microlensing probes stellar hosts much further away and towards the Galactic center, at distances of a few to several kiloparsecs. Microlensing also probes stars in the Galactic bulge, which may not form giant planets \citep[e.g.][]{2013MNRAS.431...63T}. The metallicities of the disk stars in microlensing samples are therefore expected to be enhanced relative to those monitored by RV. RV surveys have shown a strong correlation between metallicity and planet frequency over a wide range of metallicities \citep[e.g.][JJ10, MB14]{2005ApJ...622.1102F}, and thus the Galactic metallicity gradient has been hypothesized to be the cause of the difference in inferred giant planet frequency around M dwarfs between microlensing and RV surveys. JJ10 found the empirical relation between giant planet occurrence, stellar mass, and metallicity
\begin{align}
f\left(M_{\star}, {\rm [Fe/H]}\right) = & {} \left(0.07\pm 0.01\right)\left(M_{\star}/M_{\odot}\right)^{1.0\pm 0.3} \nonumber \\
& {} \times 10^{(1.2\pm 0.2){\rm [Fe/H]}}\;\label{eqn:jj_metallicity_corr}
\end{align}
for giant planets ($K>20~{\rm m~s^{-1}}$) on orbits within $a<2.5~$AU by analyzing the full CPS sample, which includes 1194 stars in the mass interval $0.2<M_{\star}/M_{\odot}\lesssim 2.0$ and the metallicity interval $-1.0<{\rm [Fe/H]}<+0.55$. Examining just the CPS M dwarfs, MB14 find the relation
\begin{align}
f\left(M_{\star}, {\rm [Fe/H]}\right) = & {} 0.039^{+0.056}_{-0.028}\left(M_{\star}/M_{\odot}\right)^{0.8^{+1.1}_{-0.9}} \nonumber \\
& {} \times 10^{(3.8\pm 1.2){\rm [Fe/H]}}\;,\label{eqn:mb_metallicity_corr}
\end{align}
for planets with masses $1<m_p\sin{i}/M_{\rm Jup}<13$ on orbits within $a<20~$AU, which has a significantly steeper scaling with metallicity than the JJ10 result. This implies that the dependence of the frequency of Jupiter and super-Jupiter mass planets on host metallicity is much steeper for M dwarfs than for higher mass stars. On the other hand, \citet{2013A&A...551A..36N} find a more shallow metallicity dependence for Jovian hosts of
\begin{equation}
f({\rm [Fe/H]}) = (0.02\pm 0.02)\times 10^{(1.97\pm 1.25){\rm [Fe/H]}}\; \label{eqn:harps_metallicity_corr}
\end{equation}
by examining the HARPS M dwarf sample.\footnote{Although \citet{2013A&A...551A..36N} do not specify a period range over which this relation is valid, we can reasonably assume it holds for periods less than a couple thousand days, which is roughly the median time baseline of observations for the HARPS M dwarf sample (BX13).} These authors also analyze a combined HARPS and CPS M dwarf data set and report
\begin{equation}
f({\rm [Fe/H]}) = (0.03\pm 0.02)\times 10^{(2.94\pm 1.03){\rm [Fe/H]}}\; \label{eqn:harps_cps_metallicity_corr}
\end{equation}
for Jovian hosts from the combined data set. MB14 acknowledge the shallower dependence on metallicity reported by the \citet{2013A&A...551A..36N} study and attribute it to their inclusion of a sub-Jupiter mass planet in their sample of Jovian hosts, which happens to orbit a star with a metallicity of ${\rm [Fe/H]=-0.19\pm 0.08}$. MB14 further emphasize the fact that there are no planets with $m_p\sin{i}>1~M_{\rm Jup}$ orbiting M dwarfs with measured metallicities below $+0.08~$dex in either the HARPS or CPS samples.
We would like to know what these relations between planet frequency and metallicity imply for the frequency of giant planets expected from the microlensing sample. We therefore apply these relations to a simulated microlensing host star sample with a stellar mass distribution similar to that expected for actual microlensing samples, covering the range $0.07\leq M_l/M_{\odot}\leq 1.0$ in a log-uniform fashion (see \citealt{clanton_gaudi14a} for details on creating such a sample). The remaining task is to determine the metallicities of the stars in our simulated sample.
Actual microlensing samples are mostly comprised of low-mass and distant (and thus faint, typically with $V\gtrsim 18$) stars, the light from which is often blended with that of the source and perhaps also nearby stars due to crowded fields and limited seeing from ground-based observations. Metallicity measurements are therefore out of reach with current technology and the metallicity distribution of the microlensing sample remains unknown. Instead, we estimate the metallicity distribution of our simulated sample using the recent Galactic metallicity maps from the SDSS-III APOGEE experiment \citep{2013arXiv1311.4569H} for our disk lenses, and the bulge metallicity distribution function (MDF) derived from a sample of microlensed dwarfs and subgiants \citep{2013A&A...549A.147B} for our bulge lenses. As we demonstrate in \citet{clanton_gaudi14a}, the parameter distributions (e.g. $M_l$, $t_E$) of our simulated sample basically match those of the GA10 sample (except for lens distances --- we will come back to this later in the section), and thus we expect the metallicity distribution for the actual microlensing sample to be roughly similar to that which we derive here.
We determine the metallicities of our simulated disk lenses as follows. Table 2 of \citet{2013arXiv1311.4569H} lists the parameters of their fits to the measured metallicities as a function of height above the plane, $z$, and Galactocentric radius, $R$. We model the median metallicities of disk stars as a function of $R$ (in several bins of $|z|$, as in \citealt{2013arXiv1311.4569H}) using these linear fits. At a given $R$, we assume the distribution in metallicity about these median values is a Gaussian with a standard deviation of $0.2~$dex, which is equal the measured spread these authors report. We then assign our disk lenses a random metallicity drawn from a Gaussian constructed in the above manner. The Galactocentric radius of a given disk lens, with distance $D_l$ from Earth and at Galactic longitude and latitude $(l,b)$, is $R=(x^2+y^2)^{1/2}$, where $x=R_0-D_l\cos{l}\cos{b}$ and $y=D_l\sin{l}\cos{b}$, and where we take $R_0=8~$kpc as the Solar radius. The height above the Galactic disk of a given lens is given by $z=D_l\sin{b}$. We set a maximum possible metallicity of ${\rm [M/H]}=0.6~$dex as there are no measurements of stellar metallicities larger than this value in \citet{2013arXiv1311.4569H}.
We assign each of our bulge lenses a random metallicity from the MDF shown in figure 12a of \citet{2013A&A...549A.147B}, regardless of the location of the event, $(l, b)$, since \citet{2013A&A...549A.147B} do not find statistically significant differences in the metallicity distributions of stars closer to ($\left|b\right|\leq 3^{\circ}$) or farther from ($\left|b\right|> 3^{\circ}$) the Galactic plane nor in the metallicity distributions of stars closer to ($\left|l\right|\leq 2^{\circ}$) or farther from ($\left|l\right|> 2^{\circ}$) the Galactic center.
The resultant metallicity distribution for our simulated microlensing sample is shown in figure~\ref{fig:mlens_metals}. The blue and red lines represent the metallicities of the bulge and disk lenses, respectively, while the black line shows the distribution of the full sample. The median metallicity of the full sample is 0.17~dex with a 68\% confidence interval of $-0.23<{\rm [M/H]/dex}<0.41$ and a 95\% confidence interval of $-1.0<{\rm [M/H]/dex}<0.54$. While not strictly true, we assume that ${\rm [M/H]}$ traces ${\rm [Fe/H]}$ and adopt these values as the ${\rm [Fe/H]}$ values for our simulated microlensing sample. For comparison, the median metallicity of both the HARPS and CPS M dwarf samples is about ${\rm [Fe/H]}_{\rm med}=-0.1~$dex \citep[][MB14]{2013A&A...551A..36N}. As expected, we find that the distribution of metallicities for our simulated microlensing sample is systematically higher than that of RV surveys.
\begin{figure}[h!]
\epsscale{1.1}
\plotone{fig6.eps}
\caption{The top panel shows the relative number of lens star metallicities for our simulated microlensing sample. The blue and red lines represent the metallicities of the bulge and disk lenses, respectively, while the black line represents the full sample. The metallicities of our bulge lenses were estimated using the MDF measured by \citet{2013A&A...549A.147B} from microlensed bulge dwarfs and we estimate the metallicities of our disk lenses using the Galactic metallicity gradients measured by \citet{2013arXiv1311.4569H} from the SDSS-III APOGEE experiment. The bottom panel shows the cumulative fraction of metallicities for the bulge and disk lenses, as well as for the full sample. We find a median metallicity for our simulated sample of 0.17~dex with a 68\% confidence interval of $-0.23<{\rm [M/H]/dex}<0.41$.
\label{fig:mlens_metals}}
\end{figure}
Now that we have metallicities for the microlensing sample, we compute the frequency of Jupiters and super-Jupiters on orbits within $a<20~$AU implied from the CPS results (MB14) given by equation~(\ref{eqn:mb_metallicity_corr}) using the median values of the fit parameters (i.e. the median normalization and scalings with host mass and metallicity reported by MB14). Figure \ref{fig:rv_mlens_freqs} shows the resultant distributions and cumulative distribution functions of the implied frequency of planets with $1<m_p\sin{i}/M_{\rm Jup}<13$ for our simulated microlensing sample and the CPS sample.
We find a mean occurrence rate of Jupiters and super-Jupiters of $0.36$ is expected for our simulated microlensing sample from the MB14 relation. This expected frequency is discrepant by a median factor of 13 ($4.4-44$ at 95\% confidence) from the actual value. In \S~\ref{sec:synthesis} we derive planet frequencies from the combined constraints of real microlensing surveys and the HARPS RV survey, and from these combined constraints, we find a frequency of planets with masses $1<m_p\sin{i}/M_{\rm Jup}<13$ and periods $1\leq P/{\rm days}\leq 10^5$ of $0.032^{+0.014}_{-0.017}$, consistent with the value reported by MB14 for the CPS M dwarfs of $0.065\pm 0.030$ but still a median factor 2.3 ($0.22-8.8$ at 95\% confidence) times smaller (see previous section for discussion). We will discuss a few possible reasons for the inconsistency in the frequencies implied by the MB14 relation for our simulated microlensing sample and the actual value we find from the combined constraints of microlensing and RV surveys.
First, we pose a question. What if giant planets do not form around bulge stars \citep[e.g.][]{2013MNRAS.431...63T}? To investigate this possibility, we repeat the calculation described above for our simulated microlensing sample, except we set the frequency of giant planets for bulge hosts to zero. We find a mean occurrence rate of 0.25 implied by equation~\ref{eqn:mb_metallicity_corr}, which is discrepant from the actual value by a median factor of 9.1 ($3.0-30$ at 95\% confidence). Thus, while this hypothesis shifts the implied frequency for the microlensing sample in the right direction, it does not seem to be enough to cause agreement. On the other hand, this idea is attractive for another reason. It could also explain the difference in the lens distance, $D_l$, distributions between our simulated sample (which assumes planets are equally common around stars regardless of their location) and that of the actual GA10 microlensing sample. GA10 find a median lens distance of 3.4~kpc, while our simulated microlensing sample yields a median value of 6.7~kpc. If there are no planets in the bulge, then the median distance to planet hosting lenses in the disk is 5.8~kpc. Thus, the idea that planets do not form around bulge stars could help to explain the shorter lens distances inferred from the GA10 sample relative to that inferred from our simulated sample, while leaving the distributions of the other microlensing parameters for these samples in agreement (see \S~5.3.3 of \citet{clanton_gaudi14a} for more information on the properties of our simulated sample and how they compare with those of the GA10 sample).
Next, we examine the possibility that the MB14 relation is not correct. We also compute the implied occurrence rates from the relations between planet frequency and mass and metallicity derived in JJ10 and \citet{2013A&A...551A..36N}. The mean occurrence rates implied by equation~\ref{eqn:jj_metallicity_corr} (JJ10) for our simulated microlensing sample is 0.058, a median factor of just 2.1 ($0.70-7.0$ at 95\% confidence) larger. The mean occurrence rate implied by equation~\ref{eqn:harps_metallicity_corr} \citep{2013A&A...551A..36N} is 0.065, while the value from equation~\ref{eqn:harps_cps_metallicity_corr} \citep{2013A&A...551A..36N} is 0.23. These implied frequencies are median factors of 2.4 ($0.79-7.9$ at 95\% confidence) and 8.3 ($2.8-28$ at 95\% confidence), respectively, larger than the actual value. The more shallow scalings of these other relations do bring the RV-expected planet frequencies closer to agreement. Perhaps surprisingly, the JJ10 relation provides the best agreement between the CPS sample and our simulated microlensing sample, even though their stellar sample included higher mass (FGK) stars whereas the MB14 and \citet{2013A&A...551A..36N} samples include only M dwarfs.
Given the fact that the median metallicity of our simulated microlensing sample is about 0.17~dex (assuming the distribution we derive is correct), while that of the CPS M dwarfs is -0.1~dex, then it seems that the frequency of giant planets must have a weaker dependence on ${\rm [Fe/H]}$ then implied by MB14. Another possibility is that the metallicity dependence saturates at some value, with (e.g.) a flat distribution for metallicities above the saturation value. It could even be a combination of the various effects we discuss, i.e. a suppression of the formation of giant planets in the bulge, a slightly weaker scaling with metallicity, and a saturation of the giant planet frequency above some threshold ${\rm [Fe/H]}$.
Finally, we also note that it is unlikely the MB14 relation between the frequency of planets with masses $1\lesssim m_p\sin{i}/M_{\rm Jup}\lesssim 13$ on orbits with $a<20~$AU extends down to giant planets with masses between $0.1\lesssim m_p\sin{i}/M_{\rm Jup}\lesssim 1$ given the fact that RV surveys generally do not detect the bulk of this planet population due to the steep planetary mass function inferred from microlensing \citet{2010ApJ...710.1641S}. We show in the next section that the frequency of giant planets with masses between $0.1\lesssim m_p\sin{i}/M_{\rm Jup}\lesssim 30$ and periods between $1\leq P/{\rm days}\leq 10^5$ is $f_{\rm G}=0.17^{+0.07}_{-0.08}$, which would seem to suggest that the scaling of giant planet frequency with host metallicity is also a function of planetary mass. This is supported by the results of \citet{2013A&A...551A..36N}, which demonstrate that the scaling of planet frequency with host metallicity for Neptunian hosts (as opposed to Jovian hosts) is not only more shallow, but that it possibly even works in the opposite direction (i.e. that planet frequency of Neptunes is anti-correlated with ${\rm [Fe/H]}$), although the latter is not statistically significant when compared against a constant functional form. These authors therefore determine that a constant functional form of $f=0.03\pm 0.01$ is preferred for Neptunian hosts, which is quite different from the relations they find for Jovian hosts given by equations~(\ref{eqn:harps_metallicity_corr}) and (\ref{eqn:harps_cps_metallicity_corr}).
In the end, the scaling of the frequency of giant planets (in particular of Jupiters and super-Jupiters) with stellar metallicity among M dwarfs remains a puzzle. However, we predict that if future RV surveys can begin detecting the bulk of the giant planet population inferred from microlensing, which typically have $K\sim 1~{\rm m~s^{-1}}$ and $P\sim 9~$yr, these planets will be detected around more metal-poor stars. The frequency of Jupiters and super-Jupiters around metal-rich stars is already found to be very high from RV surveys, which implies that the large population of giant planets with $0.1\lesssim m_p\sin{i}/M_{\rm Jup}\lesssim 1$ inferred from microlensing (and not currently detected by RV surveys) would either be detected around stars with lower metallicities or in multi-planet systems around the metal-rich M dwarfs.
\begin{figure}[t!]
\epsscale{1.1}
\plotone{fig7.eps}
\caption{The top panel shows the relative number of stars with a frequency of planets with masses $1<m_p\sin{i}/M_{\rm Jup}<13$ on orbits within $a<20~$AU implied by equation~(\ref{eqn:mb_metallicity_corr}) given their mass and metallicity. The blue and black lines represent the planet frequencies for the CPS M dwarf sample and our simulated microlensing sample (which has the same stellar, i.e. lens, mass distribution), respectively. The bottom panel shows the cumulative fraction of systems with a given planet frequency for both samples. We find a mean giant planet occurrence rate of 0.36 for our simulated microlensing sample implied by the MB14 relation, which is a median factor of 13 ($4.4-44$ at 95\% confidence) larger than the value we derive from microlensing and RV constraints from HARPS in \S~\ref{sec:synthesis}.
\label{fig:rv_mlens_freqs}}
\end{figure}
\section{Synthesizing Planet Detection Results from Multiple Detection Methods}
\label{sec:synthesis}
We have demonstrated that microlensing predicts consistent numbers of planet detections for the HARPS and CPS M dwarf surveys in the regions of planet parameter space for which there is some overlap. This enables us to synthesize the detection results, i.e. combine the individual constraints, from microlensing and RV surveys to determine planet frequencies across a very wide region of parameter space. We choose to combine the constraints from the microlensing results \citep[GA10;][]{2010ApJ...710.1641S} with those from the RV survey of HARPS (BX13) since their detection limits have been carefully characterized.
Table \ref{tab:synthesized_fs} and figure \ref{fig:freq_plot} display these combined constraints in bins of $\log{P}$ and $m_p\sin{i}$. Our methods for combining these results are as follows.
\begin{itemize}
\item $1\leq P/{\rm days}\leq 10^2$: Since microlensing has basically no sensitivity to planets with periods $\lesssim 10^2~$days, we use the constraints on the planet frequencies for these periods from the HARPS survey alone.
\item $10^2\leq P/{\rm days}\leq 10^3$: We include constraints from both microlensing and RV within this period range. However, in these bins, BX13 measure only upper limits on the planet frequency. The frequencies we derive in this paper from microlensing are consistent with these upper limits, however they serve as lower limits on the planet frequency in these bins since microlensing surveys are incomplete for these periods. Therefore, the true frequency is likely somewhere between the RV and microlensing estimates in these bins.
\item $10^2\leq P/{\rm days}\leq 10^3$: Although there is some overlap for these periods, this parameter space is dominated by microlensing, and so we adopt the microlensing estimates.
\item $10^4\leq P/{\rm days} \leq 10^5$: The HARPS survey has no sensitivity to these orbital periods (other than trends), while the sensitivity of microlensing surveys cuts off near the short end of this range for higher planet masses. Thus, we adopt the microlensing estimates in these bins but note that, due to the rapidly declining sensitivity of microlensing surveys in this period range (and especially for the lower-mass bins at these periods), the frequencies in these bins are really lower limits.
\end{itemize}
The microlensing results constrain the frequency of planets with projected separations near the Einstein ring for masses down to about $\sim 5~M_{\oplus}$, so the planet frequencies we derive in the mass range $1\leq m_p\sin{i}/M_{\oplus}\leq 10$ from microlensing requires an extrapolation of our mass function (see equation~\ref{eqn:planetary_mass_function}). However, the required extrapolation is only about 0.7~dex in $\log{q}$ for a primary mass of $M_l\sim 0.5~M_{\odot}$.
The planet frequency as a function of $\log{(m_p\sin{i}/M_{\oplus})}$ and $\log{(P/{\rm days})}$ that we derive with the above rules are displayed in figure~\ref{fig:freq_plot}. The cells are color coded according to the synthesized planet frequency in the corresponding area. In cells where we have a lower limit from microlensing, the color represents this lower limit, whereas the cells that have only upper limits are given colors equal to the quoted 1$\sigma$ upper limits.
\begin{table*}
\centering
\caption{\label{tab:synthesized_fs} Planet frequency as measured by RV, $f_{\rm RV}$, and microlensing, $f_{\rm \mu lens}$, surveys. The quantity $f_{\rm syn}$ is the planet frequency derived from constraints on either one or both of RV and microlensing detection results, depending on the sensitivity of these techniques in a given mass and period bin (see text for more details).}
\begin{tabular}{l|rrrrr}
\hline \hline
\multicolumn{1}{c|}{$m_{\rm p}\sin{i}$} & \multicolumn{5}{c}{Orbital Period [days]} \\
\multicolumn{1}{c|}{[M$_{\oplus}$]} & 1$-$10 & $10-10^2$ & $10^2-10^3$ & $10^3-10^4$ & $10^4-10^5$\\
\hline
& $f_{\rm RV} < 0.01$& $f_{\rm RV} < 0.01$& $f_{\rm RV} < 0.01$& $f_{\rm RV} < 0.01$& $-$\\
$10^3-10^4$& $-$& $-$& $f_{\rm \mu lens} = (1.0^{+1.1}_{-1.0})E-4$& $f_{\rm \mu lens} = (1.2^{+8.4}_{-1.0})E-3$& $f_{\rm \mu lens} = (3.6^{+3.4}_{-3.5})E-4$\\
& $f_{\rm syn} < 0.01$& $f_{\rm syn} < 0.01$& $(1.0^{+1.1}_{-1.0})E-4 \leq f_{\rm syn} < 0.01$& $f_{\rm syn} = (1.2^{+8.4}_{-1.0})E-3$& $f_{\rm syn} \geq (3.6^{+3.4}_{-3.5})E-4$\\
\hline
& $f_{\rm RV} < 0.01$& $f_{\rm RV} = 0.02_{-0.01}^{+0.03}$& $f_{\rm RV} < 0.01$& $f_{\rm RV} = 0.019_{-0.015}^{+0.043}$& $-$\\
$10^2-10^3$& $-$& $-$& $f_{\rm \mu lens} = (4.7^{+3.1}_{-3.4})E-3$& $f_{\rm \mu lens} = 0.038^{+0.023}_{-0.026}$& $f_{\rm \mu lens} = (7.9^{+4.8}_{-5.4})E-3$\\
& $f_{\rm syn} < 0.01$& $f_{\rm syn} = 0.02_{-0.01}^{+0.03}$& $(4.7^{+3.1}_{-3.4})E-3\leq f_{\rm syn} < 0.01$& $f_{\rm syn} = 0.038^{+0.023}_{-0.026}$& $f_{\rm syn} \geq (7.9^{+4.8}_{-5.4})E-3$\\
\hline
& $f_{\rm RV} = 0.03_{-0.01}^{+0.04}$& $f_{\rm RV} < 0.02$& $f_{\rm RV} < 0.04$& $f_{\rm RV} < 0.12$& $-$\\
$10-10^2$& $-$& $-$& $f_{\rm \mu lens} = 0.020\pm 0.009$& $f_{\rm \mu lens} = 0.16^{+0.068}_{-0.072}$& $f_{\rm \mu lens} = 0.032^{+0.012}_{-0.014}$\\
& $f_{\rm syn} = 0.03_{-0.01}^{+0.04}$& $f_{\rm syn} = < 0.02$& $0.020\pm 0.009 \leq f_{\rm syn} < 0.04$& $f_{\rm syn} = 0.16^{+0.068}_{-0.072}$& $f_{\rm syn} \geq 0.032^{+0.012}_{-0.014}$\\
\hline
& $f_{\rm RV} = 0.36_{-0.10}^{+0.24}$& $f_{\rm RV} = 0.52_{-0.16}^{+0.50}$& $-$& $-$& $-$\\
$1-10$& $-$& $-$& $f_{\rm \mu lens} = 0.080\pm 0.031$& $f_{\rm \mu lens} = 0.64^{+0.25}_{-0.26}$& $f_{\rm \mu lens} = 0.12^{+0.051}_{-0.049}$\\
& $f_{\rm syn} = 0.36_{-0.10}^{+0.24}$& $f_{\rm syn} = 0.52_{-0.16}^{+0.50}$& $f_{\rm syn} \geq 0.080\pm 0.031$& $f_{\rm syn} = 0.64^{+0.25}_{-0.26}$& $f_{\rm syn} \geq 0.12^{+0.051}_{-0.049}$\\
\hline\hline
\end{tabular}
\end{table*}
\begin{figure*}[h!]
\epsscale{1.1}
\plotone{fig8.eps}
\caption{Planet frequency as a function of $\log{(P/{\rm day})}$ and $\log{(m_p\sin{i}/M_{\oplus})}$. The numbers displayed in the upper right of each cell are the planet frequencies (or upper limits) derived by BX13 from the HARPS M dwarf sample, while those just under these are the planet frequencies we derive in this study from microlensing. The values in the lower right corner of each cell are the synthesized planet frequencies from both the RV and microlensing constraints (see text for an explanation of how we combine the statistics). The cells are color coded according to the synthesized planet frequency in the corresponding area. In cells where we have a lower limit from microlensing, the color represents this lower limit, whereas the cells that have only upper limits are given colors equal to the quoted 1$\sigma$ upper limits. The uncertainties on all these values are listed in table~\ref{tab:synthesized_fs}.
\label{fig:freq_plot}}
\end{figure*}
We also derive the integrated frequencies of various populations of planets. In order to compute the contribution due to microlensing constraints, we bin the output of our simulations over the appropriate area of planet mass and period, and add the result with the contribution from the RV constraints of the HARPS survey in accordance with the rules we list above. However, because BX13 report the planet frequencies in decade bins of mass and period, we cannot robustly determine true planet frequencies from their survey for planet populations with mass cutoffs that are not equal to 1, 10, 100, or 1000 $M_{\oplus}$. To determine their frequencies, BX13 compute an effective number of stars whose detection limits confidently exclude the existence of planets with similar masses and periods. This effective number of stars is only determined in specific bins and we do not know how these are distributed within a given bin. Thus, we must make approximations and we compute the frequencies in the following manner. We assume the mean number of planets of a given population (e.g. giant planets) is equal to the actual number BX13 detect, drawing a value from a Poisson distribution with such a mean. We then divide by our own effective number of stars which we calculate by normalizing the frequency of the population to a specific value at the actual number of detections. This specific value is an approximation of the planet frequency which we compute by assuming the planet frequency is evenly distributed throughout the BX13 decade bins. We numerically determine uncertainties for the HARPS constraints simply from the Poisson error on the number of detections, combining them (numerically) with the uncertainties from the microlensing contribution.
We find the frequency of giant planets with $30\lesssim m_p\sin{i}/M_{\oplus}\lesssim 10^4$ and $1\leq P/{\rm days}\leq 10^4$ to be $f_{\rm G}=0.15^{+0.06}_{-0.07}$, or $f_{\rm G}=0.17^{+0.07}_{-0.08}$ over the period interval $1\leq P/{\rm days}\leq 10^5$. A more conservative definition of giant planets ($50\lesssim m_p\sin{i}/M_{\oplus}\lesssim 10^4$), yields a frequency of $f_{\rm G'}=0.11\pm 0.05$ ($1\leq P/{\rm days}\leq 10^4$) or $f_{\rm G'}=0.13\pm 0.06$ ($1\leq P/{\rm days}\leq 10^5$). The frequency of Jupiters and super-Jupiters ($1\lesssim m_p\sin{i}/M_{\rm Jup}\lesssim 13$) with periods $1\leq P/{\rm days}\leq 10^4$ is $f_{\rm J}=0.029^{+0.013}_{-0.014}$ or $f_{\rm J}=0.032^{+0.014}_{-0.017}$ over the period interval $1\leq P/{\rm days}\leq 10^5$, consistent within $1\sigma$ of the measurement by MB14 from the CPS M dwarfs of $f_{\rm J}=0.065\pm 0.030$. As we mentioned in \S~{\ref{subsec:johnson_comparison}}, although this frequency is consistent with that of MB14, it is nevertheless a median factor of 2.3 ($0.22-8.8$ at 95\% confidence) times smaller, potentially due to the fact that microlensing is missing a population of very long-period super-Jupiters that is being inferred by MB14. Integrating over the entire mass range, we find the frequency of all planets with $1\leq m_p\sin{i}/M_{\oplus}\leq 10^4$ and $1\leq P/{\rm days}\leq 10^4$ to be $f_p=1.9\pm 0.5$ or $f_p=2.0\pm 0.5$ over the period interval $1\leq P/{\rm days}\leq 10^5$.
Microlensing surveys are sensitive to planets at projected separations out to roughly $s=2.5$. For the typical lens star ($M_l\sim 0.5~M_{\odot}$), this corresponds to $r_{\perp}\sim sR_E\sim 7$~AU, or $P\sim 10^4~$days (assuming $D_l/D_s=1/2$ and $a=r_{\perp}$). Thus, while there is some sensitivity to planets beyond $10^4~$days, we are not able to derive strong constraints on planet frequencies for periods beyond $10^4$ days. In this paper, we restrict our integrated estimates of planet frequency to periods $P\leq 10^4~$days. In a future paper, we plan to more accurately characterize the giant planet frequency at longer periods by including constraints from direct imaging surveys.
We note that the median number of epochs for stars in the HARPS M dwarf sample (BX13) is 8, with only 14 of their total 97 stars having more than 40 total epochs. Of these 14 stars, seven are planet hosts (Gl 176, Gl 433, Gl 581, Gl 667C, Gl 674, Gl 832, Gl 876), three show periodic variability that BX13 show to correlate with stellar activity (Gl 205, Gl 388, Gl 479), three are shown to have statistically significant long-term RV trends (Gl 1, Gl 273, Gl 887), and one is a bright M2 star of which BX13 take exposures to construct a numerical weighted mask to cross-correlate their spectra and compute RVs (Gl 877). There is only one planet host in their sample with less than 40 total epochs (Gl 849 with $N=35$), but which was previously known to host a giant planet \citet{2006PASP..118.1685B}. If it is the case that the planet hosts were specifically targeted for additional observations as a result of the presence of a planet, and these additional observations were not excluded when quantifying the planet sensitivity, then the planet frequencies inferred from these data are biased. However, we are unable to quantify the magnitude, or even the sign, of this bias.
\subsection{Comparison of Combined Constraints with Other Measurements of Planet Frequency}
\label{subsec:freq_comparisons}
Now that we have derived the planet frequency around M dwarfs across a very wide region of planet parameter space, we can compare with other measurements of planet frequency. In particular, we make rough comparisons with frequencies from a sample of M dwarfs from \emph{Kepler} by \citet{2013ApJ...767...95D} and by \citet{2013ApJ...764..105S}, as well as a measurement of the giant planet ($0.3\leq m_p\sin{i}/M_{\rm Jup}\leq 15$) frequency around F, G, and K dwarfs by \citet{2008PASP..120..531C}.
\subsubsection{\emph{Kepler} M Dwarfs}
\label{subsubsec:kep_comparison}
\citet{2013ApJ...767...95D} refine the stellar parameters of a sample of M dwarfs from \emph{Kepler} and compute planet frequencies as a function of orbital period and planetary radius. We perform a rough comparison with their results in the bins corresponding to planet masses between $1\leq m_p\sin{i}/M_{\oplus}\leq 10$ and orbital periods between $1\leq P/{\rm days}\leq 10^2$. The empirical mass-radius relations derived in \citet{2014ApJ...783L...6W} tell us that a planetary radius of $\approx 4~R_{\oplus}$ corresponds to a mass of $\approx 10~M_{\oplus}$. Assuming planetary densities identical to that of the Earth, the \citet{2014ApJ...783L...6W} relations say $1~R_{\oplus}$ corresponds to $1~M_{\oplus}$. Thus, in terms of radius, we compare the frequencies derived by \citet{2013ApJ...767...95D} between $1\leq R_p/R_{\oplus}\leq 4$ for periods between $1\leq P/{\rm days}\leq 10^2$ (ignoring the $\sin{i}$ factor). We choose to compare results in these specific bins of mass (radius) and orbital period because these are the regions of this parameter space for which we expect the most overlap between \emph{Kepler} and the HARPS RV survey, from which we derive our constraints on planet frequency in these bins.
We rebin the data in figure~15 of \citet{2013ApJ...767...95D}, adding up their planet frequencies in the bins between $1\leq R_p/R_{\oplus}\leq 4$ and $1\leq P/{\rm days}\leq 10$ (and multiplying by the appropriate fractions for the cells that are not fully contained in this range, assuming a uniform distribution in $\log{R}$ and $\log{P}$). This roughly yields the frequency of planets with masses between $1\leq m_p\sin{i}/M_{\oplus}\leq 10$ in the same period range. We find this frequency to be $0.23\pm 0.03$, which is nearly consistent with the frequency inferred from the HARPS survey by BX13 of $0.36^{+0.50}_{-0.10}$. In this same mass interval but for orbital periods of $10\leq P/{\rm days}\leq 10^2$, we compute a frequency from the \citet{2013ApJ...767...95D} results of $0.51\pm 0.10$. The frequency in the corresponding bin measured by BX13 is $0.52^{+0.50}_{-0.16}$, consistent with our calculation from the \emph{Kepler} M dwarfs.
According to the \citet{2014ApJ...783L...6W} mass-radius relation, a $4~R_{\oplus}$ planet will have a mass of about $25~M_{\oplus}$. As we discussed in \S~\ref{sec:giant_planet_def}, the lowest mass of a ``giant planet'' is uncertain, and likely encompasses a range of masses. Even more uncertain, then, is the transition radius between rocky, icy, and giant planets. However, if we make the simple assumption that all the planets with radii $R_p>4~R_{\oplus}$ in the \citet{2013ApJ...767...95D} M dwarf sample from \emph{Kepler} are giant planets, then we calculate the frequency of giant planets with masses $m_p\sin{i}\gtrsim 30~M_{\oplus}$ and periods $1\leq P/{\rm days}\leq 10$ to be $0.014\pm 0.007$. This is nearly consistent with the frequency of planets in the same mass and period ranges measured BX13 of $0.043\pm 0.021$.
\citet{2013ApJ...764..105S}, assuming the five planets of Kepler-32 \citep{2012ApJ...750..114F} are representative of the full ensemble of planet candidates orbiting the \emph{Kepler} M dwarfs, infer a planet occurrence rate of $1.0\pm 0.1$ planet per star. While \citet{2013ApJ...764..105S} do not explicitly state the planetary radius and orbital period intervals over which this measurement is integrated, examining their figure~6 seems to indicate intervals of $m_p\gtrsim1~M_{\oplus}$ and $P\lesssim150~{\rm days}$, where they have adopted the planetary mass-radius relation of \citet{2011ApJS..197....8L} which takes the form $m_p\propto R_p^{2.06}$. In these same intervals, we find an occurrence rate of $0.94^{+0.35}_{-0.26}$, consistent with \citet{2013ApJ...764..105S}. \citet{2013ApJ...764..105S} also calculate the occurrence rate of planets with $R_p>2~R_{\oplus}$ (corresponding to a mass of $\approx 5~M_{\oplus}$ according to the mass-radius relation of \citealt{2014ApJ...783L...6W}) and $P<50~{\rm days}$ to be $0.26\pm 0.05$. In these same intervals, we again find a consistent planet frequency of $0.37^{+0.18}_{-0.13}$.
\subsubsection{Planet Frequency Around FGK Dwarfs}
\label{subsubsec:kep_comparison}
\citet{2008PASP..120..531C} analyzed a sample of RV-monitored FGK stars and measured the occurrence rate of planets with masses between $0.3\leq m_p\sin{i}/M_{\rm Jup}\leq 10$ and orbital periods $P<5.2~$yr to be $0.085\pm 0.013$. They extrapolate to find the frequency of such planets with orbital semimajor axes $a<20~$AU, assuming either a flat distribution in $P$ beyond $2000~$days or a power-law distribution ($\propto P^{0.26}$), to be $0.17\pm0.03$ and $0.19\pm 0.03$, respectively. Around a Solar-type star, 20~AU is roughly 7.4 times the location of the ice line, assuming $a_{\rm ice}=2.7~{\rm AU}(M/M_{\odot})^{2/3}$, where we have adopted the scaling found by \citet{2008ApJ...673..502K} for mass accretion rates that are proportional to stellar mass, $\dot{M}\propto M_{\star}$. In order to compare planet frequencies between FGK and M dwarf populations, we want to examine orbital separations that probe similar formation environments, so we compute the frequency from our combined constraints over the same range of planetary masses, but for orbital separations that are within 7.4 times the location of the ice line, which is about 12.5~AU for the typical microlensing star of $0.5~M_{\odot}$. Thus, we find for masses in the range $0.3\leq m_p\sin{i}/M_{\rm Jup}\leq 10$ and $a<12.5~$AU ($P<62.5~$yr) a frequency of $0.072^{+0.034}_{-0.038}$, which is a median factor of 2.8 ($0.81-9.5$ at 95\% confidence) to 3.1 ($0.89-9.9$ at 95\% confidence) times smaller than, and thus marginally inconsistent at the $2\sigma$ level with, the values found by \citet{2008PASP..120..531C} for FGK stars.
If we extrapolate the \citet{2008PASP..120..531C} planetary mass function to include all giant planets ($0.1\leq m_p\sin{i}/M_{\rm Jup}\leq 10$) within $a<20~$AU, we find a frequency of $0.31\pm 0.07$. Our combined constraints give a giant planet frequency for $0.1\leq m_p\sin{i}/M_{\rm Jup}\leq 10$ and $P<62.5~$yr of $0.16\pm 0.07$. This is a median factor of 2.2 ($0.73-5.9$ at 95\% confidence) times smaller than that which we calculate by extrapolating the \citep{2008PASP..120..531C} result. Thus, while giant planets are not intrinsically rare around M dwarfs, they are rarer than the population observed around FGK dwarfs at $1\sigma$ and marginally inconsistent at the $2\sigma$ level.
\section{Summary and Discussion}
\label{sec:discussion}
In this paper, we map the observable parameters ($q,s$) of the population of planets inferred from microlensing into the observables ($K,P$) of an analogous population of planets orbiting a stellar sample monitored with RV. We derive joint distributions of these RV observables for simulated samples of microlensing systems with similar stellar mass distributions as the M dwarf RV surveys of HARPS (BX13) and CPS (MB14). We then apply the actual RV detection limits reported by BX13 to predict the number of planet detections and long-term RV trends we expect the HARPS survey to find, and we apply roughly estimated detection limits to make predictions for the CPS sample. Comparing our predictions with the actual numbers reported by these RV surveys, we find consistency. We predict that HARPS should find $N_{\rm det}=1.4\pm 0.8$ planets right at the edge of their survey limit, and indeed, they find one such planet around Gl 849 (BX13). This star also appears in the CPS sample, where this very same planet was originally discovered \citep{2006PASP..118.1685B}. We expect the CPS survey to detect $N_{\rm det}=4.7^{+2.5}_{-2.8}$ planets with periods $P\gtrsim 100~$days and masses $m_p\sin{i}\gtrsim 10^2~M_{\oplus}$. The number of such planets they actually detect is four, around the stars Gl 179, Gl 317, Gl 649, and Gl 849 \citep{2010ApJ...721.1467H,2007ApJ...670..833J,2010PASP..122..149J,2006PASP..118.1685B}.
The fact that our predicted numbers of detections and the actual numbers are consistent implies that microlensing and RV surveys are largely disjoint, with only a small amount of overlap for orbital periods between roughly $100-10^3~$days and planetary masses larger than about a Jupiter mass. This limited overlap is such that, due to the steeply declining planetary mass function, RV surveys infer low giant planet frequencies around M dwarfs, detecting only the high-mass end of the giant planet population ($m_p\gtrsim M_{\rm Jup}$) inferred by microlensing. For RV surveys to be sensitive to the majority of this population, measurement precisions of $\sim 1~{\rm m~s^{-1}}$ (including instrumental errors and stellar jitter) over time baselines of $\sim 10~$years are required. The frequency of Jupiters and super-Jupiters around metal-rich stars is already found to be very high from current RV surveys, which implies that the large population of giant planets with $0.1\lesssim m_p\sin{i}/M_{\rm Jup}\lesssim 1$ inferred from microlensing (and not currently detected by RV surveys) would either be detected by future, more sensitive RV surveys around stars with lower metallicities or in multi-planet systems around the metal-rich M dwarfs.
However, we are left with a puzzle concerning the scaling of the frequency of Jovian planets ($1\lesssim m_p\sin{i}/M_{\rm Jup}\lesssim 13$) with stellar metallicity inferred from the CPS M dwarf sample \citep{2014ApJ...781...28M}. We estimate the metallicity distribution of our simulated microlensing sample using the bulge MDF of \citet{2013A&A...549A.147B} and the Galactic metallicity gradients from \citet{2013arXiv1311.4569H} and find a median metallicity of ${\rm [M/H]} = 0.17~$dex with a 68\% confidence interval of $-0.23<{\rm [M/H]}/{\rm dec}<0.41$. Using this metallicity distribution, we find that the occurrence rate implied by the scaling inferred by MB14 is over-predicted by a median factor of 13 ($4.4-44$ at 95\% confidence) relative to the actual frequency found by microlensing surveys. This could suggest that the MB14 relation is incorrect or perhaps incomplete. A significantly shallower scaling with metallicity seems to be required for agreement (more in line with that reported by \citealt{2010PASP..122..905J} or perhaps \citealt{2013A&A...551A..36N}), or perhaps the metallicity dependence saturates at some value, with (e.g.) a flat distribution for metallicities above the saturation value. We also investigate another possibility. What if giant planets do not form around bulge stars \citep[e.g.][]{2013MNRAS.431...63T}? We show that if this were true, the occurrence rate for the microlensing sample implied by the MB14 relation moves closer to agreement with the measured value (a median factor of 9.1, or $3.0-30$ at 95\% confidence, discrepant), but probably does not account for the full difference. This solution would also be attractive because it could partially explain the difference in the lens distance distributions between our simulated microlensing sample and the GA10 sample.
We also point out that it seems unlikely the relations between planet frequency and metallicity hold for giant planets with masses $0.1\lesssim m_p\sin{i}/M_{\rm Jup}\lesssim 1$ given the fact that RV surveys are not sensitive to the bulk of the giant planet population inferred from microlensing surveys. This suggests that the scaling of giant planet frequency with host metallicity is a function of planetary mass. This hypothesis is supported by the results of \citet{2013A&A...551A..36N}, which suggest that the scaling of planet frequency with host metallicity is significantly different between Jovian and Neptunian hosts.
Finally, since we have demonstrated that the giant planet frequencies measured by microlensing and RV surveys are actually consistent, we are able to combine their constraints to determine planet frequencies across a very wide region of parameter space. The combined constraints on the giant planet occurrence rate around M dwarfs as a function of orbital period and planet mass are summarized in table~\ref{tab:synthesized_fs} and plotted in figure~\ref{fig:freq_plot}. We also show that the planet frequencies in the mass range $1\leq m_p\sin{i}/M_{\oplus}\leq 10$ and period range $1\leq P/{\rm days}\leq 10^2$ are consistent with the detection results from the \emph{Kepler} M dwarf sample reported by \citet{2013ApJ...767...95D} and \citet{2013ApJ...764..105S}. We can integrate over various regions of this plane to compute total planet frequencies.
We find the frequency of giant planets with $30\lesssim m_p\sin{i}/M_{\oplus}\lesssim 10^4$ and $1\leq P/{\rm days}\leq 10^4$ to be $f_{\rm G}=0.15^{+0.06}_{-0.07}$. For a more conservative definition of giant planets ($50\lesssim m_p\sin{i}/M_{\oplus}\lesssim 10^4$), we find $f_{\rm G'}=0.11\pm 0.05$. The frequency of Jupiters and super-Jupiters ($1\lesssim m_p\sin{i}/M_{\rm Jup}\lesssim 13$) with periods $1\leq P/{\rm days}\leq 10^4$ is $f_{\rm J}=0.029^{+0.013}_{-0.015}$, consistent with the measurement by MB14 of $f_{\rm J}=0.065\pm 0.030$. We find the frequency of all planets with $1\leq m_p\sin{i}/M_{\oplus}\leq 10^4$ and $1\leq P/{\rm days}\leq10^4$ to be $f_p=1.9\pm 0.5$. These planet frequencies are closer to lower limits on the planet frequency, because our combined constraints on the planet frequency include the lower limits in the period range $10^2-10^3~$days, where the sensitivity of microlensing surveys declines.
This is a very broad result, covering four orders of magnitude in planetary mass and four orders of magnitude in orbital period. But perhaps more importantly, it demonstrates that it is possible to get a more complete picture of the demographics of exoplanets by including constraints from multiple discovery methods. In a future paper, we plan to compare and synthesize the planet detection results found here with those from direct imaging surveys.
\acknowledgments
This research has made use of NASA's Astrophysics Data System and was partially supported by NSF CAREER Grant AST-1056524. We thank John Johnson and Benjamin Montet for helpful comments and conversations.
|
2,869,038,156,337 | arxiv | \section{Supplementary Materials}
\beginsupplement
\begin{center}
\renewcommand{\arraystretch}{0.5}
\setlength{\tabcolsep}{12pt}
\begin{table}[htbp]
\caption{\label{SCXRD} Crystallographic parameters of YbIr$_3$Ge$_7$ single crystals at $T$ = 299 K ($R\bar{3}c$)}
\begin{tabular}{c|c}
\hline
$a$ (\AA) & 7.8062(10) \\
$c$ (\AA) & 20.621(5) \\
$V$ (\AA$ ^{3}$) & 1088.2(4) \\
crystal dimensions (mm$^{3}$) & 0.02 x 0.04 x 0.06 \\
$\theta$ range ($^{\circ}$) & 3.6 - 30.4 \\
extinction coefficient & 0.000107(13) \\
absorption coefficient (mm$^{-1}$) & 94.87 \\
measured reflections & 7380 \\
independent reflections & 374 \\
R$_{int}$ & 0.046 \\
goodness-of-fit on F$^2$ & 1.20 \\
$R_1(F)$ for ${F^2}_o \textgreater 2\sigma ({F^2}_o)^a$ & 0.017 \\
$wR_2({F^2}_o)^b$ & 0.033 \\ \hline
\end{tabular}
$^{a}R_1 = \sum\mid\mid F_o\mid - \mid F_c\mid \mid / \sum \mid F_o \mid$
$^bwR_2 = [\sum[w({F_o}^2 - {F_c}^2)^2]/ \sum[w({F_o}^2)^2]]^{1/2}$
\end{table}
\end{center}
\begin{center}
\renewcommand{\arraystretch}{1.2}
\setlength{\tabcolsep}{10pt}
\begin{table*}[h!]
\caption{\label{Atomic} Atomic positions, $U_{eq}$ values, and occupancies for single crystals of YbIr$_3$Ge$_7$}
\begin{tabular}{ c|c|c|c|c|c}
\hline
Atom & x & y & z & $U_{eq}$ (\AA$^2$)$^a$ & Occupancy\\ \hline
Yb & 0 & 0 & 0 & 0.00652(14) & 1\\
Ir & 0.31885(3) & 0 & $\frac{1}{4}$ & 0.00321(11) & 1\\
Ge1 & 0.54369(8) & 0.68054(8) & 0.03056(2) & 0.0050(2) & 0.970(3)\\
Ge2 & 0 & 0 & $\frac{1}{4}$ & 0.0048(4) & 0.911(6)\\ \hline
\end{tabular}
$^a$ $U_{eq}$ is defined as one-third of the trace of the orthogonalized $U_{ij}$ tensor.
\end{table*}
\end{center}
\begin{figure}[h!]
\includegraphics[width=\columnwidth,clip]{XRD.eps}
\caption{\label{XRD} Room temperature powder x-ray diffraction pattern for YbIr$_3$Ir$_7$ (black symbols) together with the calculated pattern (red line), the their difference (violet line) and calculated peak positions (blue vertical lines) using space group $R\bar{3}c$.}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=\columnwidth,clip]{TC.eps}
\caption{\label{TC}The ordering temperature $T_{\rm C}$ (vertical dashed line) for YbIr$_3$Ge$_7$ determined from peaks in C$_p$ ($H$ = 0, black circles, left axis), a minimum in d$M$/dT ($\mu_0H$ = 0.1 T, red triangles, right axis), and d$\rho$/dT ($H$ = 0, blue squares, far left axis)}
\end{figure}
\end{document}
|
2,869,038,156,338 | arxiv | \section*{Introduction} Let $k$ be a field. In the paper~\cite{K-R}, the authors associated a Hopf algebra $H_\mathcal{D}$ over $k$, with each data $\mathcal{D}:=(G,\chi,z,\lambda)$, consisting of:
\begin{itemize}
\smallskip
\item[-] a finite group $G$,
\smallskip
\item[-] a character $\chi$ of $G$ with values in $k$,
\smallskip
\item[-] a central element $z$ of $G$,
\smallskip
\item[-] an element $\lambda\in k$,
\smallskip
\end{itemize}
such that $\chi^n=1$ or $\lambda(z^n-1_G)=0$, where $n$ is the order of $\chi(z)$. As an algebra $H_{\mathcal{D}}$ is generated by $G$ and $x$, subject to the group relations of $G$,
$$
xg=\chi(g)gx\quad\text{for all $g\in G$}\qquad\text{and}\qquad x^n=\lambda(z^n-1_G).
$$
The coalgebra structure of $H_{\mathcal{D}}$ is determined by
\begin{align*}
&\Delta(g):=g\otimes g\quad\text{for $g\in G$,}&& \Delta(x):=1\otimes x + x\otimes z,\\
&\epsilon(g):=1\quad \text{for $g\in G$,} &&\epsilon(x):=0,
\end{align*}
and its antipode is given by
$$
S(g)=g^{-1}\quad\text{and}\quad S(x)=-xz^{-1}.
$$
As was point out in~\cite{K-R}, as a vector space $H_{\mathcal{D}}$ has basis $\{gx^m| g \in G, 0 \le m < n\}$. Consequently $\dim H_{\mathcal{D}}=n|G|$.
The algebras $H_{\mathcal{D}}$ are called {\em rank~$1$ Hopf algebras}. The simplest examples are the Taft algebras $H_{n^2}$, which are the rank~$1$ Hopf algebra obtained taking $\mathcal{D}:=(C_n,\chi,g,1)$, where $C_n = \langle g \rangle$ is the cyclic group of order $n>1$ and $\chi(g)$ is a primitive $n$-th root of~$1$.
In~\cite{M} Masuoka studied the cleft extensions of the Taft algebras, giving a very elegant description of its crossed product systems. In \cite{D-T} the Masuoka description was reproduced with simplified proofs, and studying this description were derived several interesting consequences. Motivated by these works, in this paper we introduce a family of braided Hopf algebras that generalize the algebras $H_{\mathcal{D}}$, and we study their cleft extensions.
\smallskip
The paper is organized in the following way:
In Section~1 we recall the well know notions of Gaussian binomial coefficients and braided Hopf algebra, and we make a quick review of some results obtained in \cite{G-G}. The unique new result in this section are the formulas obtained in Theorem~\ref{equiv entre cleft, H-Galois con base normal e isomorfo a un producto cruzado}(5). In Section~2 we associate a braided Hopf algebra with each data $\mathcal{D}=(G,\chi,z,\lambda,q)$ (where $G$ is a finite group, $\chi$ is a character of $G$, $z$ is a central element of $G$ and $q,\lambda$ are scalars) that satisfies suitable conditions. The main result is Corollary~\ref{algebras de Hopf trenzadas de K-R}, in which the algebras $H_{\mathcal{D}}$ are introduced. When $q=1$, we recover the rank~$1$ Hopf algebras defined by Krop y Radford. In Section~3 we describy the $H_{\mathcal{D}}$-space structures, the $H_{\mathcal{D}}$-comodule structures and the $H_{\mathcal{D}}$-comodule algebra structures (in Proposition~\ref{estructuras trenzadas de HsubD}, Corollary~\ref{segunda caracterizacion de HsubD comodules} and Theorem~\ref{caracterizacion de HsubD comodule algebras}, respectively). In Section~4 we characterize the $H_{\mathcal{D}}$ cleft extensions, and, finally, in Section~5 we study two particular examples.
\section{Preliminaries} In this paper $k$ is a field, we work in the category of $k$-vector spaces, and consequently all the maps are $k$-linear maps. Moreover we let $U\otimes V$ denote the tensor product $U\otimes_k V$ of each pair of vector spaces $U$ and $V$. We assume that the algebras are associative unitary and the coalgebras are coassociative counitary. For an algebra $A$ and a coalgebra $C$, we let $\mu\colon A\otimes A\to A$, $\eta\colon k\to A$, $\Delta\colon C\to C\otimes C$ and $\eta\colon C\to k$ denote the multiplication map, the unit, the comultiplication map and the counit, respectively, specified with a subscript if necessary.
\subsection{Gaussian binomial coefficients} Let $q$ be a variable. For any $j\in \mathds{N}_0$ set
$$
(j)_q := \sum_{i=0}^{j-1} q^i = \frac{q^j-1}{q-1}\quad\text{and}\quad (j)!_q := (1)_q(2)_q\cdots (j)_q = \frac{(q-1)(q^2-1) \cdots (q^j-1)}{(q-1)^j}.
$$
The Gauss binomials are the rational functions in $q$ defined by
\begin{equation}
\binom{i}{j}_q :=\frac{(i)!_q}{(j)!_q (i-j)!_q}\qquad\text{for $0\le j\le i$.}\label{eq3}
\end{equation}
A direct computation shows that
$$
\binom{r}{0}_q = \binom{i}{i}_q = 1\quad\text{and}\quad \binom{i}{j}_q = q^{i-j}\binom{i-1}{j-1}_q + \binom{i-1}{j}_q \quad\text{for $0<j<i$.}
$$
From these equalities it follows easily that the Gauss binomials are actually polynomials. The Gauss binomials can be evaluated in arbitrary elements of $k$, but the equality~\eqref{eq3} only makes sense for $q=1$ and for $q\ne 1$ such that $q^l\ne 1$ for all $l\le \max(j,i-j)$. We will need the following well known result ($q$-binomial formula). Let $B$ be a $k$-algebra and $q\in k$. If $x,y\in B$ satisfy $yx = qxy$, then
\begin{equation}
(x+y)^i = \sum_{j=0}^i \binom{i}{j}_q x^jy^{i-j}\qquad\text{for each $i\ge 0$.}\label{eq12}
\end{equation}
Let $i,j\ge 0$ and let $0\le l\le i+j$. Using the equality~\eqref{eq12} to compute $(x+y)^i(x+y)^j$ in two different ways and comparing coefficients we obtain that
\begin{equation}
\binom{i+j}{l}_q=\sum_{\substack{ 0\le s \le i\\ 0\le t \le j\\ s+t=l }} q^{(i-s)t}\binom{i}{s}_q\binom{j}{t}_q.\label{eq13}
\end{equation}
\subsection{Braided Hopf algebras} Let $V$, $W$ be vector spaces and let $c\colon V\otimes W \to W\otimes V$ be a map. Recall that:
\begin{itemize}
\smallskip
\item[-] If $V$ is an algebra, then $c$ is compatible with the algebra structure of $V$ if
$$
\quad c \hs \circ \hs(\eta\otimes W)= W\otimes \eta\quad\text{and}\quad c \hs \circ \hs (\mu\otimes W)= (W\otimes \mu)\hs \circ \hs(c\otimes V) \hs \circ \hs (V\otimes c).
$$
\smallskip
\item[-] If $V$ is a coalgebra, then $c$ is compatible with the coalgebra structure of $V$ if
$$
\quad (W\otimes \epsilon)\hs \circ \hs c = \epsilon\otimes W\quad\text{and}\quad (W\otimes \Delta) \hs \circ \hs c = (c\otimes V)\hs \circ \hs (V\otimes c)\hs \circ \hs (\Delta \otimes W).
$$
\smallskip
\end{itemize}
More precisely, the first equality in the first item says that $c$ is {\em compatible with the unit of $V$} and the second one says that it is {\em compatible with the multiplication map of $V$}, while the first equality in the second item says that $c$ is {\em compatible with the counit of $V$} and the second one says that it is {\em compatible with the comultiplication map of $V$}. Of course, there are similar compatibilities when $W$ is an algebra or a coalgebra.
\smallskip
\begin{definition}\label{def: braided bialgebra} A {\em braided bialgebra} is a vector space $H$ endowed with an algebra structure, a coalgebra structure and a braiding operator $c\in \Aut_k(H\otimes H)$, called the {\em braid} of $H$, such that $c$ is compatible with the algebra and coalgebra structures of $H$, $\eta$ is a coalgebra morphism, $\epsilon$ is an algebra morphism and
$$
\Delta\hs \circ \hs\mu = (\mu\otimes \mu)\hs \circ \hs(H\otimes c \otimes H)\hs \circ \hs(\Delta \otimes \Delta).
$$
Furthermore, if there exists a map $S\colon H\to H$, which is the inverse of the identity map for the convolution product, then we say that $H$ is a {\em braided Hopf algebra} and we call $S$ the {\em antipode} of~$H$.
\end{definition}
Usually $H$ denotes a braided bialgebra, understanding the structure maps, and $c$ denotes its braid. If necessary, we will write $c_H$ instead of $c$.
\smallskip
Let $A$ and $B$ be algebras. It is well known that if a map $c \colon B\otimes A \longrightarrow A\otimes B$ is compatible with the algebra structures of $A$ and $B$, then $A\otimes B$ endowed with the multiplication map
$$
\mu := (\mu_A\otimes \mu_B)\hs \circ \hs (A\otimes c\otimes B),
$$
is an associative algebra with unit $1\otimes 1$, which is called {\em the twisted tensor product of $A$ with $B$ associated with $c$} and denoted $A\otimes_c B$. Similarly, if $C$ and $D$ are coalgebras and $c\colon C\otimes D \longrightarrow D\otimes C$ is a map compatible with the coalgebra structures of $C$ and $D$, then $C\otimes D$ endowed with the comultiplication map
$$
\Delta := (C\otimes c \otimes D)\hs \circ \hs (\Delta_C\otimes \Delta_D),
$$
is a coassociative coalgebra with counit $\epsilon\otimes \epsilon$, which is called {\em the twisted tensor product of $C$ with $D$ associated with $c$} and denoted $C\otimes^c D$.
\begin{remark} Let $H$ be a vector space which is both an algebra and a coalgebra, and let
$$
c\colon H\otimes H\longrightarrow H\otimes H
$$
be a braiding operator. Assume that $c$ is compatible with the algebra and the coalgebra structures of $H$. Then $H$ is a braided bialgebra iff its comultiplication map $\Delta \colon H\to H\otimes_c H$ and its counit $\epsilon \colon H\to k$ are algebra maps.
\end{remark}
\subsection{Left $\bm{H}$-spaces, left $\bm{H}$-algebras and left $\bm{H}$-coalgebras}
\begin{definition}\label{def: H-braided space} Let $H$ be a braided bialgebra. A {\em left $H$-space} $(V,s)$ is a vector space $V$, endowed with a bijective map $s\colon H\otimes V \longrightarrow V\otimes H$, called the {\em left transposition of $H$ on $V$}, which is compatible with the bialgebra structure of $H$ and satisfies
$$
(s\otimes H)\hs \circ \hs (H\otimes s)\hs \circ \hs (c\otimes V) = (V\otimes c)\hs \circ \hs (s\otimes H)\hs \circ \hs (H\otimes s)
$$
(compatibility of $s$ with the braid). Sometimes, when it is evident, the map $s$ is not explicitly specified. In these cases we will say that $V$ is a left $H$-braided space in order to point out that there is a left transposition involved in the definition. We adopt a similar convention for all the definitions below. Let $(V',s')$ be another left $H$-space. A $k$-linear map $f\colon V \to V'$ is said to be a {\em morphism of left $H$-spaces}, from $(V,s)$ to $(V',s')$, if $(f\otimes H)\hs \circ \hs s = s' \hs \circ \hs (H\otimes f)$.
\end{definition}
\begin{remark}\label{basta verificar sobre generadores} Let $s\colon H\otimes V \longrightarrow V\otimes H$ be a map compatible with the unit, the mul\-tiplication map and the braid of $H$ and let $X\subseteq H$ be a set that generates $H$ as an algebra. In order to show that $s$ is a left transposition it suffices to check the compatibility of $s$ with the counit and the comultiplication map of $H$ on simple tensors $h\otimes v$ with $h\in X$ and $v\in V$.
\end{remark}
We let $\mathcal{LHB}$ denote the category of all left $H$-braided spaces. It is easy to check that this is a monoidal category with
\begin{itemize}
\smallskip
\item[-] unit $(k,\tau)$, where $\tau\colon H\otimes k\to k\otimes H$ is the flip,
\smallskip
\item[-] tensor product
$$
\qquad\quad (U,s_U)\otimes(V,s_V) := (U\otimes V, s_{U\otimes V}),
$$
where $s_{U\otimes V}$ is the map $s_{U\otimes V}:= (U\otimes s_V)\hs \circ \hs(s_U\otimes V)$,
\smallskip
\item[-] the usual associativity and unit constraints.
\smallskip
\end{itemize}
\begin{definition}\label{def: braided alg} A {\em left $H$-algebra} $(A,s)$ is an algebra in $\mathcal{LHB}$.
\end{definition}
\begin{definition}\label{def: transp alg} A {\em left transposition of $H$ on an algebra $A$} is a bijective map $s\colon H\otimes A\longrightarrow A\otimes H$, satisfying
\smallskip
\begin{enumerate}
\item $(A,s)$ is a left $H$-space,
\smallskip
\item $s$ is compatible with the algebra structure of $A$.
\end{enumerate}
\end{definition}
\begin{remark}\label{re: alg in LB_H} A left $H$-algebra is nothing but a pair $(A,s)$ consisting of an algebra $A$ and a left transposition $s\colon H\otimes A \longrightarrow A\otimes H$. Let $(A',s')$ be another left \mbox{$H$-algebra}. A map $f\colon A\to A'$ is a morphism of left $H$-algebras, from $(A,s)$ to $(A',s')$, iff it is a morphism of standard algebras and $(f\otimes H) \hs \circ \hs s = s' \hs \circ \hs (H\otimes f)$.
\end{remark}
\begin{definition}\label{def: braided coalg} A {\em left $H$-coalgebra} $(C,s)$ is a coalgebra in $\mathcal{LHB}$.
\end{definition}
\begin{definition}\label{def: transp coalg} A {\em left transposition of $H$ on a coalgebra $C$} is a bijective map $s\colon H\otimes C\longrightarrow C\otimes H$, satisfying
\begin{enumerate}
\smallskip
\item $(C,s)$ is a left $H$-space,
\smallskip
\item $s$ is compatible with the coalgebra structure of $C$.
\smallskip
\end{enumerate}
\end{definition}
\begin{remark}\label{re: coalg in LB_H} A left $H$-coalgebra is nothing but a pair $(C,s)$ consisting of a coalgebra $C$ and a left transposition $s\colon H\otimes C \longrightarrow C\otimes H$. Let $(C',s')$ be another left $H$-co\-algebra. A map $f\colon C\to C'$ is a morphism of left $H$-coalgebras, from $(C,s)$ to $(C',s')$, iff it is a morphism of standard coalgebras and $(f\otimes H) \hs \circ \hs s = s' \hs \circ \hs (H\otimes f)$.
\end{remark}
Since $(H,c)$ is an algebra and a coalgebra in $\mathcal{LHB}$, it makes sense to consider $(H,c)$-modules and $(H,c)$-comodules in this monoidal category.
\subsection{Left $\bm{H}$-modules and left $\bm{H}$-module algebras}
\begin{definition}\label{def: H-braided module} We will say that $(V,s)$ is a {\em left $H$-module} to mean that it is a left $(H,c)$-module in $\mathcal{LHB}$. Notice that the classical left $H$-modules can be identified with the left $H$-modules $(V,s)$ in which $s$ is the flip.
\end{definition}
\begin{remark}\label{re: H-braided module} A left $H$-space $(V,s)$ is a left $H$-module iff $V$ is a standard left $H$-module and
$$
s\hs \circ \hs (H\otimes \rho) = (\rho\otimes H) \hs \circ \hs (H\otimes s) \hs \circ \hs (c\otimes V),
$$
where $\rho$ denotes the action of $H$ on $V$. Let $(V',s')$ be another left $H$-module. A map $f\colon V\to V'$ is a {\em morphism of left $H$-modules}, from $(V,s)$ to $(V',s')$, iff it is an $H$-linear map and the equality $(f\otimes H)\hs \circ \hs s = s'\hs \circ \hs (H\otimes f)$ holds. We let ${}_H\mathcal{LHB}$ denote the category of left $H$-modules in $\mathcal{LHB}$.
\end{remark}
\begin{proposition}[\cite{G-G}*{Proposition~5.6}]\label{prop: _H sub LB H es monoidal} The category ${}_H\mathcal{LHB}$ is monoidal. Its unit is $(k,\tau)$ endowed with the trivial left $H$-module structure, and the tensor product of the left $H$-modules $(U,s_U)$ and $(V,s_V)$, with actions $\rho_U$ and $\rho_V$ respectively, is the left $H$-space $(U,s_U)\otimes (V,s_V)$, endowed with the left \mbox{$H$-module} action given by
$$
\rho_{U\otimes V}:= (\rho_U\otimes \rho_V)\hs \circ \hs(H\otimes s_U\otimes V)\hs \circ \hs (\Delta_H\otimes U\otimes V).
$$
The associativity and unit constraints are the usual ones.
\end{proposition}
\begin{definition}\label{def: H-braided mod alg} We say that $(A,s)$ is a {\em left $H$-module algebra} if it is an algebra in ${}_H \mathcal{LHB}$.
\end{definition}
\begin{remark}\label{re: car de H-braid mod alg} $(A,s)$ is a left $H$-module algebra iff the following facts hold:
\begin{enumerate}
\smallskip
\item $A$ is an algebra and a standard left $H$-module,
\smallskip
\item $s$ is a left transposition of $H$ on $A$,
\smallskip
\item $s\hs \circ \hs (H\otimes \rho) = (\rho\otimes H) \hs \circ \hs (H\otimes s) \hs \circ \hs (c\otimes A)$,
\smallskip
\item $\rho \hs \circ \hs (H\otimes\mu_A) = \mu_A\hs \circ \hs (\rho\otimes \rho)\hs \circ \hs (H\otimes s\otimes A)\hs \circ \hs (\Delta_H \otimes A\otimes A)$,
\smallskip
\item $\rho(h\otimes 1_A) = \epsilon(h)1_A$ for all $h\in H$,
\smallskip
\end{enumerate}
where $\rho$ denotes the action of $H$ on $A$.
\smallskip
Let $(A',s')$ be another left $H$-module algebra. A map $f\colon A\to A'$ is a {\em morphism of left \mbox{$H$-module} algebras}, from $(A,s)$ to $(A',s')$, iff it is an $H$-linear morphism of standard algebras that satisfies $(f\otimes H)\hs \circ \hs s= s'\hs \circ \hs (H\otimes f)$.
\end{remark}
\subsection{Right $\bm{H}$-comodules and right $\bm{H}$-comodule algebras}
\begin{definition}\label{def: H-braided comodule} We will say that $(V,s)$ is a {\em right $H$-comodule} if it is a right $(H,c)$-comodule in $\mathcal{LHB}$.
\end{definition}
\begin{remark}\label{re: H-braided comodule} A left $H$-space $(V,s)$ is a right $H$-comodule iff $V$ is a standard right $H$-comodule and
\begin{equation}
(\nu\otimes H)\hs \circ \hs s = (V\otimes c)\hs \circ \hs (s\otimes H)\hs \circ \hs (H\otimes \nu),\label{eq6}
\end{equation}
where $\nu$ denotes the coaction of $H$ on $V$. Let $(V',s')$ be another right $H$-comodule. A map $f\colon V\to V'$ is a {\em morphism of right $H$-comodules}, from $(V,s)$ to $(V',s')$, iff it is an $H$-colinear map and $(f\otimes H)\hs \circ \hs s = s'\hs \circ \hs (H\otimes f)$. We let $\mathcal{LHB}^H$ denote the category of right $H$-comodules in $\mathcal{LHB}$.
\end{remark}
\begin{definition}\label{coinvariantes} Let $(V,s)$ be a right $H$-comodule. An element $v\in V$ is said to be {\em coinvariant} if $\nu(v)=v\otimes 1_H$.
\end{definition}
\begin{remark} For each right $H$-comodule $(V,s)$, the set $V^{\coH}$, of coinvariant elements of $V$, is a vector subspace of $V$. Furthermore, $s(H\otimes V^{\coH})=V^{\coH}\otimes H$, and the pair $(V^{\coH},s_{V^{\coH}})$, where $s_{V^{\coH}}\colon H\otimes V^{\coH}\to V^{\coH}\otimes H$ is the restriction of $s$, is a left $H$-space.
\end{remark}
\begin{proposition}[\cite{G-G}*{Proposition 5.2}]\label{prop: LHB spu H es monoidal} The category $\mathcal{LHB}^H$ is monoidal. Its unit is $(k,\tau)$, en\-do\-wed with the trivial right $H$-comodule structure, and the tensor product of the right $H$-comodu\-les $(U,s_U)$ and $(V,s_V)$, with coactions $\nu_U$ and $\nu_V$ respectively, is $(U,s_U)\otimes (V,s_V)$, endowed with the right $H$-comodule coaction
$$
\nu_{U\otimes V}:= (U\otimes V\otimes \mu_H)\hs \circ \hs(U\otimes s_V\otimes H) \hs \circ \hs (\nu_U\otimes \nu_V).
$$
The associativity and unit constraints are the usual ones.
\end{proposition}
\begin{definition}\label{def: H-braided comod alg} We say that $(A,s)$ is a {\em right $H$-comodule algebra} if it is an algebra in $\mathcal{LHB}^H$.
\end{definition}
\begin{remark}\label{re: H-braided comod alg} $(A,s)$ is a right $H$-comodule algebra iff the following facts hold:
\begin{enumerate}
\smallskip
\item $A$ is an algebra and a standard right $H$-comodule,
\smallskip
\item $s$ is a left transposition of $H$ on $A$,
\smallskip
\item $(\nu\otimes H)\hs \circ \hs s = (A\otimes c) \hs \circ \hs (s\otimes H) \hs \circ \hs (H\otimes \nu)$,
\smallskip
\item $\nu\hs \circ \hs \mu_A = (\mu_A\otimes \mu_H)\hs \circ \hs (A\otimes s\otimes H)\hs \circ \hs (\nu \otimes \nu)$,
\smallskip
\item $\nu(1_A) = 1_A\otimes 1_H$,
\smallskip
\end{enumerate}
where $\nu$ denotes the coaction of $H$ on $A$.
\smallskip
Let $(A',s')$ be another right $H$-comodule algebra. A map $f\colon A\to A'$ is a {\em morphism of right $H$-comodule algebras}, from $(A,s)$ to $(A',s')$, iff it is an $H$-colinear morphism of standard algebras that satisfies $(f\otimes H) \hs \circ \hs s = s' \hs \circ \hs (H\otimes f)$.
\end{remark}
Recall that $A\otimes_s H$ denote the algebra with underlying vector space $A\otimes H$, multiplication map
$$
\mu_{A\otimes_s H}:= (\mu_A\otimes \mu_H)\hs \circ \hs (A\otimes s\otimes H)
$$
and unit $1_A\otimes 1_H$. Conditions~(4) and~(5) of Remark~\ref{re: H-braided comod alg} say that $\nu\colon A\to A\otimes_s H$ is a morphism of algebras.
\subsection{Hopf crossed products and $\bm{H}$-extensions}
\begin{definition}\label{def: weak H-modules} A left $H$-space $(V,s)$, endowed with a map $\rho\colon H\otimes V\to V$, is said to be a {\em weak left $H$-module} if
\begin{enumerate}
\smallskip
\item $\rho(1_H\otimes v) = v$, for all $v\in V$,
\smallskip
\item $s\hs \circ \hs (H\otimes\rho) = (\rho\otimes H)\hs \circ \hs (H\otimes s)\hs \circ \hs (c\otimes V)$.
\end{enumerate}
\end{definition}
The category ${}_{wH}\mathcal{LHB}$, of weak left $H$-modules in $\mathcal{LHB}$, becomes a monoidal category in the same way that ${}_H\mathcal{LHB}$ does. A {\em weak left $H$-module algebra} $(A,s)$ is, by definition, an algebra in~${}_{wH}\mathcal{LHB}$.
\begin{remark}\label{re: weak H-modulo algebra} $(A,s)$ is a left weak $H$-module algebra iff $A$ is an usual algebra, $s$ is a left transposition of $H$ on $A$ and the structure map $\rho$ satisfies the following conditions:
\begin{enumerate}
\smallskip
\item $\rho(1_H\otimes a) = a$, for all $a\in A$,
\smallskip
\item $s\hs \circ \hs (H\otimes\rho) = (\rho\otimes H)\hs \circ \hs (H\otimes s)\hs \circ \hs (c\otimes A)$,
\smallskip
\item $\rho\hs \circ \hs (H\otimes\mu_A) =\mu_A\hs \circ \hs(\rho\otimes\rho)\hs \circ \hs (H\otimes s\otimes A)\hs \circ \hs (\Delta_H\otimes A\otimes A)$,
\smallskip
\item $\rho(h\otimes 1_A)= \epsilon(h)1_A$, for all $h\in H$.
\end{enumerate}
\end{remark}
Let $A$ be an algebra and $s\colon H\otimes A \longrightarrow A\otimes H$ a left transposition. A map $\rho\colon H\otimes A\to A$ is said to be a {\em weak action} of $H$ on $(A,s)$ or a {\em weak $s$-action} of $H$ on $A$, if it satisfies the conditions of the above remark.
\begin{definition}\label{def: normal, cociclo, condicion de modulo torcido} Let $A$ be an algebra, $s\colon H\otimes A\longrightarrow A\otimes H$ a left transposition and $\rho\colon H\otimes A\to A$ a weak action of $H$ on $(A,s)$. Let $\sigma\colon H\otimes H \to A$ be a map. We say that $\sigma$ is {\em normal} if
$$
\sigma(1_H\otimes h)=\sigma(h\otimes 1_H)= \epsilon(h)\qquad\text{for all $h\in H$,}
$$
and that $\sigma$ is a {\em cocycle that satisfies the twisted module condition} if
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\comult(12,0)[1];\comult(14,0)[1];\draw (15.5,0) -- (15.5,-2); \braid(12.5,-1)[1]; \draw (11.5,-1) -- (11.5,-2); \draw (14.5,-1) -- (14.5,-2); \transposition(14.5,-2)[1]; \draw (13.5,-2) -- (13.5,-3); \draw (15.5,-3) -- (15.5,-4); \transposition(13.5,-3)[1]; \cocycle(14.5,-4)[1]; \draw (12.5,-2) -- (12.5,-4); \laction(12.5,-4)[0,1]; \draw (11.5,-1) -- (11.5,-3.5); \draw (11.5,-3.5) .. controls (11.5,-3.8) and (12.5,-4.7) .. (12.5,-5); \laction(12.5,-5)[0,1]; \mult(13.5,-6)[1]; \draw (15,-5) .. controls (15,-5.3) and (14.5,-5.7) .. (14.5,-6);
\end{scope}
\begin{scope}[xshift=0.3cm,yshift=0cm]
\node at (17,-3.5){=};
\end{scope}
\begin{scope}[xshift=13.4cm,yshift=-0.5cm]
\comult(5,0)[1];\comult(7,0)[1]; \braid(5.5,-1)[1]; \draw (8,0) -- (8,-4); \draw (4.5,-1) -- (4.5,-2); \draw (7.5,-1) -- (7.5,-2); \cocycle(4.5,-2)[1]; \mult(6.5,-2)[1]; \laction(7,-3)[0,1];
\draw (8,-4) .. controls (8,-4.3) and (7.8,-4.2) .. (7.8,-4.5);
\draw (5,-3).. controls (5,-3.3) and (6.3,-4.2) .. (6.3,-4.5); \mult(6.3,-4.5)[1.5];
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\begin{scope}[xshift=16cm,yshift=0cm]
\node at (5.8,-3.5){,};
\end{scope}
\begin{scope}[xshift=18.2cm,yshift=0cm]
\node at (5.8,-3.4){where};
\end{scope}
\begin{scope}[xshift=18.2cm,yshift=0cm]
\cocycle(7,-3)[1];
\end{scope}
\begin{scope}[xshift=19.7cm,yshift=0cm]
\node at (7.5,-3.5){$= \sigma$.};
\end{scope}
\end{tikzpicture}
\end{center}
More precisely, the first equality is the cocycle condition and the second one is the twisted module condition. Finally we say that $\sigma$ is {\em compatible with $s$} if it is a map in $\mathcal{LHB}$. In other words, if
$$
(\sigma\otimes H)\hs \circ \hs(H\otimes c)\hs \circ \hs(c\otimes H) = s\hs \circ \hs(H\otimes \sigma).
$$
\end{definition}
Let $s\colon H\otimes A\to A\otimes H$ be a left transposition, $\rho\colon H\otimes A\to A$ a weak $s$-action and $\sigma\colon H\otimes H\to A$ a normal cocycle compatible with $s$, that satisfies the twisted module condition. Consider the maps $\chi\colon H\otimes A\longrightarrow A\otimes H$ and $\mathcal{F}\colon H\otimes H\longrightarrow A\otimes H$ defined by
$$
\chi := (\rho\otimes H)\hs \circ \hs (H\otimes s)\hs \circ \hs(\Delta\otimes A)\quad\text{and}\quad\mathcal{F}:= (\sigma\otimes \mu_H) \hs \circ \hs (H\otimes c \otimes H)\hs \circ \hs (\Delta\otimes \Delta).
$$
\begin{definition}\label{def de producto cruzado} The {\em crossed product associated with $(s,\rho,\sigma)$} is the $k$-algebra $A\#_{\rho,\sigma}^s H$, whose underlying $k$-vector space is $A\otimes H$ and whose multiplication map is
$$
\mu:= (\mu_A\otimes H)\hs \circ \hs(\mu_A\otimes \mathcal{F}) \hs \circ \hs (A\otimes \chi \otimes H).
$$
\end{definition}
From now on, a simple tensor $a\otimes h$ of $A\#_{\rho,\sigma}^s H$ will usually be written $a\# h$.
\begin{theorem}[\cite{G-G}*{Theorems 2.3, 6.3 and 9.3}]\label{propiedades basicas de productos cruzados1} The algebra $A\#_{\rho,\sigma}^s H$ is associative and has unity $1_A\# 1_H$.
\end{theorem}
\begin{theorem}[\cite{G-G}*{Propositions 10.3 and 10.4}]\label{propiedades basicas de productos cruzados2} The map
$$
\widehat{s}: H\otimes A\#_{\rho,\sigma}^s H \longrightarrow A\#_{\rho,\sigma}^s H \otimes H,
$$
defined by $\widehat{s}:= (A\otimes c)\hs \circ \hs (s\otimes H)$ is a left transposition of $H$ on $A\#_{\rho,\sigma}^s H$ and the pair $(A\#_{\rho,\sigma}^s H,\widehat{s})$, endowed with the coaction $\nu_{A\#_{\rho,\sigma}^s H}:=A\otimes \Delta$, is a right $H$-comodule algebra.
\end{theorem}
\begin{definition}\label{def: H extension} Let $(B,s)$ be a right $H$-comodule algebra and let $i\colon A\hookrightarrow B$ be an algebra inclusion. We say that $(i\colon A\hookrightarrow B,s)$ is an {\em $H$-extension} of $A$ if $i(A) = B^{\coH}$. Let $(i'\colon A\hookrightarrow B',s')$ be another $H$-extension of $A$. We say that $(i\colon A\hookrightarrow B,s)$ and $(i'\colon A\hookrightarrow B',s')$ are {\em equivalent} if there is a right $H$-comodule algebra isomorphism $f\colon (B,s) \to (B',s')$, which is also a left $A$-module homomorphism.
\end{definition}
\begin{remark} For each $H$-extension $(i\colon A\hookrightarrow B,s)$ of $A$, the map $s_A\colon H\otimes A\longrightarrow A\otimes H$, induced by $s$, is a left transposition (in other words, $(A,s_A)$ is a left $H$-algebra).
\end{remark}
\begin{example}\label{Los productos cruzados son H extensiones} $(i\colon A\hookrightarrow A\#_{\rho,\sigma}^s H,\widehat{s})$, where $i(a):= a\# 1_H$, is an $H$-extension of $A$.
\end{example}
\begin{definition}\label{def: cleft, de Galois y normal} Let $(i\colon A\hookrightarrow B,s)$ be an $H$-extension. We say that:
\begin{enumerate}
\smallskip
\item $(i,s)$ is {\em cleft} if there is a convolution invertible right $H$-comodule map $\gamma\colon (H,c)\to (B,s)$,
\smallskip
\item $(i,s)$ is {\em $H$-Galois} if the map $\beta_B\colon B\otimes_A B \longrightarrow B\otimes H$, defined by $\beta_B(b\otimes b')= (b\otimes 1_H)\nu(b')$, where $\nu$ denotes the coaction of $B$, is bijective,
\smallskip
\item $(i,s)$ has the {\em normal basis property} if there exists a left $A$-linear and right $H$-colinear isomorphism $\phi\colon (A\otimes H,\widehat{s_A}) \longrightarrow (B,s)$, where the coaction of $A\otimes H$ is $A\otimes \Delta$ and $\widehat{s_A}= (A\otimes c)\hs \circ \hs (s_A\otimes H)$.
\end{enumerate}
\end{definition}
\begin{definition}\label{definicion de aplicacion cleft} Let $(i\colon A\hookrightarrow B,s)$ be an $H$-extension of $A$. If $(i,s)$ is cleft, then each one of the maps $\gamma$ satisfying the conditions required in item~(1) of Definition~\ref{def: cleft, de Galois y normal} is called a {\em cleft map} of $(i,s)$, and if $(i,s)$ has the normal basis property, then each one of the left $A$-linear right and $H$-colinear isomorphism $\phi\colon (A\otimes H,\widehat{s_A})\longrightarrow (B,s)$ is called a {\em normal basis} of $B$.
\end{definition}
\begin{remark}[\cite{G-G}*{Section 10}] If $\gamma$ is a {\em cleft map} of $(i\colon A\hookrightarrow B,s)$, then $\gamma(1_H)\in B^{\times}$ and the map $\gamma':=\gamma(1_H)^{-1}\gamma$ is a cleft map that satisfies $\gamma'(1_H)=1_B$.
\end{remark}
\begin{lemma}\label{cae en A} Let $H$ be a braided Hopf algebra and let $(i\colon A\hookrightarrow B,s)$ be a cleft $H$-extension, with a cleft map $\gamma$. The map $f\colon H\otimes A\to B$, defined by
$$
f:=\mu_B\hs \circ \hs (\mu_B\otimes B)\hs \circ \hs (\gamma\otimes i \otimes \gamma^{-1})\hs \circ \hs(H\otimes s_A)\hs \circ \hs (\Delta\otimes A)
$$
takes its values in $i(A)$.
\end{lemma}
\begin{proof} Let $\lambda_r^X\colon X\to X\otimes k$ be the canonical map. We must prove that
$$
\nu\hs \circ \hs f = (f\otimes\eta )\hs \circ \hs (H\otimes\lambda_r^A).
$$
A direct computation shows that
\begin{align*}
\nu\hs \circ \hs f & = \nu\hs \circ \hs \mu_B\hs \circ \hs (\mu_B\otimes B)\hs \circ \hs (\gamma\otimes i \otimes \gamma^{-1})\hs \circ \hs(H\otimes s_A)\hs \circ \hs (\Delta\otimes A)\\
&= (\mu_B\otimes\mu_H)\hs \circ \hs (B\otimes s\otimes H)\hs \circ \hs (\nu\otimes \nu)\hs \circ \hs (\mu_B\otimes B)\hs \circ \hs (\gamma\otimes i \otimes \gamma^{-1})\hs \circ \hs(H\otimes s_A)\hs \circ \hs (\Delta\otimes A)\\
&= (\mu_B\otimes \mu_H)\hs \circ \hs (\mu_B\otimes s \otimes H)\hs \circ \hs (B\otimes s \otimes \nu)\hs \circ \hs(\nu\otimes i \otimes \gamma^{-1}) \hs \circ \hs(\gamma\otimes s_A)\hs \circ \hs (\Delta\otimes A)\\
&= (\mu_B\otimes \mu_H) \hs \circ \hs (\mu_B\otimes s \otimes H) \hs \circ \hs (B\otimes i\otimes H \otimes \nu)\hs \circ \hs (\gamma\otimes s_A \otimes \gamma^{-1})\hs \circ \hs (\Delta \otimes s_A)\hs \circ \hs (\Delta\otimes A)\\
&= (\mu_B\otimes\mu_H)\hs \circ \hs (B\otimes s\otimes H)\hs \circ \hs (\mu_B\otimes H\otimes \nu\hs \circ \hs \gamma^{-1}) \hs \circ \hs (B\otimes i\otimes\Delta)\hs \circ \hs (\gamma\otimes s_A) \hs \circ \hs(\Delta\otimes A)\\
& = (\mu_B\otimes H)\hs \circ \hs (\mu_B\otimes L) \hs \circ \hs (B\otimes i \otimes H)\hs \circ \hs (\gamma\otimes s_A) \hs \circ \hs(\Delta\otimes A),
\end{align*}
where
$$
L:= (B\otimes\mu_H)\hs \circ \hs (s\otimes H)\hs \circ \hs (H\otimes\nu\hs \circ \hs \gamma^{-1})\hs \circ \hs \Delta.
$$
Since, by \cite{G-G}*{Lemma 10.7},
\begin{align*}
L &=(B\otimes \mu_H)\hs \circ \hs(s\otimes H)\hs \circ \hs (H\otimes\gamma^{-1}\otimes S)\hs \circ \hs (H\otimes c\hs \circ \hs \Delta)\hs \circ \hs \Delta\\
&=(\gamma^{-1}\otimes \mu_H)\hs \circ \hs (c\otimes S)\hs \circ \hs (H\otimes c) \hs \circ \hs (H\otimes\Delta)\hs \circ \hs \Delta\\
&=(\gamma^{-1}\otimes \mu_H)\hs \circ \hs (c\otimes S)\hs \circ \hs (H\otimes c) \hs \circ \hs (\Delta\otimes H)\hs \circ \hs \Delta\\
&=(\gamma^{-1}\otimes H)\hs \circ \hs c\hs \circ \hs (\mu_H\otimes H)\hs \circ \hs (H\otimes S\otimes\!H) \hs \circ \hs (\Delta\otimes H)\hs \circ \hs \Delta\\
&=(\gamma^{-1}\otimes H)\hs \circ \hs c\hs \circ \hs (\eta\hs \circ \hs\epsilon\otimes H)\hs \circ \hs \Delta\\
&= \gamma^{-1}\otimes \eta,
\end{align*}
we have
$$
\nu\hs \circ \hs f = (\mu_B\otimes H)\hs \circ \hs (B\otimes \gamma^{-1}\otimes\eta)\hs \circ \hs (\mu_B\otimes \lambda_r^H)\hs \circ \hs (B\otimes i\otimes H) \hs \circ \hs (\gamma\otimes s_A) \hs \circ \hs(\Delta\otimes A) = (f\otimes\eta )\hs \circ \hs (H\otimes\lambda_r^A),
$$
as desired.
\end{proof}
\begin{theorem}\label{equiv entre cleft, H-Galois con base normal e isomorfo a un producto cruzado} Let $H$ be a braided Hopf algebra and let $(i\colon A\hookrightarrow B,s)$ be an $H$-extension. The following assertions are equivalent:
\smallskip
\begin{enumerate}
\item $(i,s)$ is cleft.
\smallskip
\item $(i,s)$ is $H$-Galois with a normal basis.
\smallskip
\item There is a crossed product $A\#_{\rho,\sigma}^{s_A} H$, with convolution invertible cocycle $\sigma\colon H\otimes^c H\to A$, and a right $H$-comodule algebra isomorphism
$$
(B,s)\longrightarrow (A\#_{\rho,\sigma}^{s_A} H,\widehat{s_A}),
$$
which is also left $A$-linear.
\smallskip
\end{enumerate}
Furthermore, if $\gamma$ is a cleft map of $(i,s)$ with $\gamma(1_H) = 1_B$, then
\begin{enumerate}[resume]
\smallskip
\item The map $\phi\colon (A\otimes H,\widehat{s_A})\longrightarrow (B,s)$, defined by $\phi(a\otimes h):= i(a)\gamma(h)$, is a normal basis of $B$.
\smallskip
\item The weak action $\rho$ and the cocycle $\sigma$ are given by
\begin{align}
& i\hs \circ \hs \rho= \mu_B\hs \circ \hs (\mu_B\otimes B)\hs \circ \hs(\gamma\otimes i\otimes \gamma^{-1})\hs \circ \hs (H\otimes s_A)\hs \circ \hs(\Delta \otimes A)\label{ecua1}
\shortintertext{and}
& i\hs \circ \hs \sigma=\mu_B\hs \circ \hs (\mu_B\otimes \gamma^{-1})\hs \circ \hs(\gamma\otimes\gamma\otimes \mu_H) \hs \circ \hs \Delta_{H\otimes^c H}.\label{ecua2}
\end{align}
\end{enumerate}
\end{theorem}
\begin{proof} The equivalence between the first three items is \cite{G-G}*{Theorem~10.6}, and the fourth item was proved in the proof of that Theorem. It remains to check the last one. By item~(4), the discussion below~\cite{G-G}*{Definition 10.5} and the proof of Theorem 10.6 of~\cite{G-G}, we know that $\phi$ is bijective, that
$$
(i\otimes H)\hs \circ \hs\phi^{-1}(b) = b_{(0)}\gamma^{-1}(b_{(1)})\otimes b_{(2)},
$$
and that the maps $\rho\colon H\otimes A\to A$ and $\sigma\colon H \otimes H \to A$, are given by
$$
\rho(h\otimes a):=(A\otimes \epsilon)\hs \circ \hs \phi^{-1}(\gamma(h)i(a))\quad\text{and}\quad \sigma(h\otimes l):= (A\otimes \epsilon)\hs \circ \hs \phi^{-1}(\gamma(h)\gamma(l)).
$$
We must check that $\rho$ and $\sigma$ satisfy~\eqref{ecua1} and~\eqref{ecua2}, respectively. Let $f$ be as in Lemma~\ref{cae en A} and let $i^{-1}$ be the compositional inverse of $i\colon A\to i(A)$. Since
\begin{align*}
\mu_B\hs \circ \hs (\gamma\otimes i) & = \mu_B\hs \circ \hs (\mu_B\otimes \eta_B\hs \circ \hs \epsilon)\hs \circ \hs (B\otimes i\otimes H)\hs \circ \hs (\gamma\otimes s_A)\hs \circ \hs (\Delta\otimes A)\\
& = \mu_B\hs \circ \hs (B\otimes\mu_B)\hs \circ \hs (B\otimes \gamma^{-1}\otimes \gamma)\hs \circ \hs (\mu_B\otimes \Delta) \hs \circ \hs (B\otimes i\otimes H) \hs \circ \hs (\gamma\otimes s_A)\hs \circ \hs (\Delta\otimes A)\\
& = \mu_B\hs \circ \hs (\mu_B\otimes\mu_B)\hs \circ \hs (\gamma\otimes i\otimes \gamma^{-1}\otimes \gamma)\hs \circ \hs (H\otimes s_A\otimes H)\hs \circ \hs (\Delta\otimes s_A)\hs \circ \hs (\Delta\otimes A)\\
& = \mu_B\hs \circ \hs (f\otimes\gamma)\hs \circ \hs (H\otimes s_A)\hs \circ \hs (\Delta\otimes A),
\end{align*}
and, by Lemma~\ref{cae en A},
$$
\mu_B\hs \circ \hs (f\otimes \gamma) = \phi\hs \circ \hs (i^{-1}\hs \circ \hs f\otimes H),
$$
we have
\begin{align*}
i\hs \circ \hs \rho &= (i\otimes \epsilon)\hs \circ \hs \phi^{-1}\hs \circ \hs \mu_B\hs \circ \hs (\gamma\otimes i)\\
& =(i\otimes \epsilon)\hs \circ \hs \phi^{-1}\hs \circ \hs \mu_B\hs \circ \hs (f\otimes \gamma)\hs \circ \hs (H\otimes s_A) \hs \circ \hs (\Delta\otimes A)\\
& =(i\otimes \epsilon)\hs \circ \hs (i^{-1}\hs \circ \hs f\otimes H)\hs \circ \hs (H\otimes s_A) \hs \circ \hs (\Delta\otimes A)\\
& = \mu_B\hs \circ \hs (\mu_B\otimes B)\hs \circ \hs(\gamma\otimes i\otimes \gamma^{-1})\hs \circ \hs (H\otimes s_A)\hs \circ \hs(\Delta \otimes A).
\end{align*}
Finally,
\begin{align*}
i\hs \circ \hs \sigma & = (i\otimes \epsilon)\hs \circ \hs \phi^{-1} \hs \circ \hs \mu_B\hs \circ \hs (\gamma\otimes \gamma)\\
& =\mu_B\hs \circ \hs(B\otimes \gamma^{-1})\hs \circ \hs \nu \hs \circ \hs \mu_B\hs \circ \hs (\gamma\otimes \gamma)\\
& =\mu_B\hs \circ \hs (B\otimes \gamma^{-1})\hs \circ \hs(\mu_B\otimes \mu_H)\hs \circ \hs (B\otimes s \otimes B)\hs \circ \hs(\nu\otimes \nu)\hs \circ \hs (\gamma\otimes \gamma)\\
&= \mu_B\hs \circ \hs (\mu_B\otimes \gamma^{-1})\hs \circ \hs(\gamma\otimes\gamma\otimes \mu_H) \hs \circ \hs \Delta_{H\otimes^c H},
\end{align*}
as desired.
\end{proof}
\begin{remark} In the proof of Theorem 10.6 of~\cite{G-G} was also shown that $\phi\colon A\#_{\rho,\sigma}^{s_A} H \to B$ is an algebra isomorphism.
\end{remark}
\section{A family of braided Hopf algebras}
Let $G$ be a finite group, $\chi\colon G\to k^{\times}$ a character, $n>1$ in $\mathds{N}$, and $U\in kG$, where~$kG$ denotes the group algebra of $G$ with coefficients in $k$. Set $\mathcal{E}:=(G,\chi,U,n)$ and write $U:=\sum_{g\in G}\lambda_g g$.
\begin{proposition}\label{estructura de algebra de B_D} There exists an associative algebra $B_{\mathcal{E}}$ such that
\begin{itemize}
\smallskip
\item[-] $B_{\mathcal{E}}$ is generated by $G$ and an element $x\in B_{\mathcal{E}}\setminus kG$,
\smallskip
\item[-] $\mathcal{B}:=\{gx^i: g\in G \text{ and } 0\le i < n\}$ is a basis of $B_{\mathcal{E}}$ as a $k$-vector space,
\smallskip
\item[-] the multiplication of elements of $\mathcal{B}$ is given by:
$$
\quad gx^ihx^j:= \begin{cases}
\chi^i(h)gh x^{i+j} &\text{if $i+j<n$,}\\
\chi^i(h)gh U x^{i+j-n} &\text{if $i+j\ge n$,}
\end{cases}
$$
\smallskip
\end{itemize}
iff $\lambda_{hgh^{-1}}=\chi^n(h)\lambda_g$ for all $h,g\in G$, and $\chi(g)=1$ for all $g\in G$ such that $\lambda_{g}\ne 0$.
\end{proposition}
\begin{proof} Let $V:=kx_0\oplus\cdots\oplus kx_{n-1}$, where $x_0,\dots, x_{n-1}$ are indeterminate. We will prove the result by showing that there is an associative and unitary algebra $kG\# V$, with underlying vector space $kG\otimes V$, whose multiplication map satisfies
\begin{align*}
&(g\otimes x_i)(g'\otimes x_0)=\chi^i(g')gg'\otimes x_i \hspace{7pt} \text{ for all $i$ and all $g\in G$}
\shortintertext{and}
&(1_G\otimes x_i)(1_G\otimes x_j)= \begin{cases} 1_G\otimes x_{i+j} & \text{if $i+j<n$,}\\ U \otimes x_{i+j-n} &\text{if $i+j\ge n$,}\end{cases}
\end{align*}
iff
\begin{enumerate}
\smallskip
\item $\chi(g)=1$ for all $g\in G$ such that $\lambda_{g}\ne 0$,
\smallskip
\item $\lambda_{hgh^{-1}}=\chi^n(h)\lambda_g$ for all $h,g\in G$.
\smallskip
\end{enumerate}
By the theory of general crossed products developed in~\cite{Br}, for this it suffices to check that the maps
$$
\Phi\colon V\otimes k{G}\longrightarrow kG\otimes V\quad\text{and}\quad \mathcal{F}\colon V\otimes V \longrightarrow kG\otimes V,
$$
given by
$$
\Phi(x_i\otimes g):= \chi^i(g)g\otimes x_i\quad\text{and}\quad \mathcal{F}(x_i\otimes x_j):= \begin{cases} 1_G\otimes x_{i+j} & \text{if $i+j<n$,}\\ U \otimes x_{i+j-n} &\text{if $i+j\ge n$,}\end{cases}
$$
satisfy
$$
\Phi(x_i\otimes 1_G) = 1_G\otimes x_i,\quad \Phi(x_0\otimes g) = g\otimes x_0,\quad \mathcal{F}(x_0\otimes x_i)=\mathcal{F}(x_i\otimes x_0) = 1_G\otimes x_i,
$$
$$
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\begin{scope}[yshift=-0.5cm]
\node at (0,0.3){$\scriptstyle V$}; \node at (1,0.3){$\scriptstyle kG$}; \node at (2,0.3){$\scriptstyle kG$}; \twisting(0,-0)[1]; \draw (2,0) -- (2,-1); \draw (0,-1) -- (0,-2); \twisting(1,-1)[1]; \mult(0,-2)[1]; \draw (2,-2) -- (1.5,-3);
\end{scope}
\begin{scope}[xshift=0.1cm,yshift=0cm]
\node at (2.5,-1.8){=};
\end{scope}
\begin{scope}[xshift=0.2cm,yshift=-0.75cm]
\node at (3,0.3){$\scriptstyle V$}; \node at (4,0.3){$\scriptstyle kG$}; \node at (5,0.3){$\scriptstyle kG$};\mult(4,0)[1]; \twisting(3,-1)[1.5]; \draw (3,0) -- (3,-1);
\end{scope}
\begin{scope}[xshift=3.2cm,yshift=0cm]
\node at (2.5,-1.8){,};
\end{scope}
\begin{scope}[xshift=7.7cm, yshift=-0.5cm]
\node at (0,0.3){$\scriptstyle V$}; \node at (1,0.3){$\scriptstyle V$}; \node at (2,0.3){$\scriptstyle kG$}; \doublemap(0,0)[\scriptstyle \mathcal{F}]; \draw (2,0) -- (2,-1); \draw (0,-1) -- (0,-2); \twisting(1,-1)[1]; \mult(0,-2)[1]; \draw (2,-2) -- (1.5,-3);
\end{scope}
\begin{scope}[xshift=7.7cm,yshift=0cm]
\node at (2.5,-1.8){=};
\end{scope}
\begin{scope}[xshift=7.8cm]
\node at (3,0.3){$\scriptstyle V$}; \node at (4,0.3){$\scriptstyle V$}; \node at (5,0.3){$\scriptstyle kG$}; \draw (3,0) -- (3,-1);\twisting(4,0)[1]; \twisting(3,-1)[1]; \draw (5,-1) -- (5,-2); \draw (3,-2) -- (3,-3); \doublemap(4,-2)[\scriptstyle \mathcal{F}];\mult(3,-3)[1]; \draw (5,-3) -- (4.5,-4);
\end{scope}
\begin{scope}[xshift=7.7cm,yshift=0cm]
\node at (7.3,-1.7){and};
\end{scope}
\begin{scope}[xshift=17cm, yshift=-0.5cm]
\node at (0,0.3){$\scriptstyle V$}; \node at (1,0.3){$\scriptstyle V$}; \node at (2,0.3){$\scriptstyle V$};\doublemap(0,0)[\scriptstyle \mathcal{F}]; \draw (2,0) -- (2,-1); \draw (0,-1) -- (0,-2); \doublemap(1,-1)[\scriptstyle \mathcal{F}]; \mult(0,-2)[1]; \draw (2,-2) -- (1.5,-3);
\end{scope}
\begin{scope}[xshift=17.4cm,yshift=0cm]
\node at (2.5,-1.8){=};
\end{scope}
\begin{scope}[xshift=17.5cm,yshift=0cm]
\node at (3,0.3){$\scriptstyle V$}; \node at (4,0.3){$\scriptstyle V$}; \node at (5,0.3){$\scriptstyle V$}; \draw (3,0) -- (3,-1);\doublemap(4,0)[\scriptstyle \mathcal{F}]; \twisting(3,-1)[1]; \draw (5,-1) -- (5,-2); \draw (3,-2) -- (3,-3); \doublemap(4,-2)[\scriptstyle \mathcal{F}];\mult(3,-3)[1]; \draw (5,-3) -- (4.5,-4);
\end{scope}
\begin{scope}[xshift=20.cm,yshift=0cm]
\node at (2.5,-1.8){,};
\end{scope}
\end{tikzpicture}
\label{Brtwistmodcond}
$$
where \begin{tikzpicture}[scale=0.25]
\def\twisting(#1,#2)[#3]{\draw (#1+#3,#2) .. controls (#1+#3,#2-0.05*#3) and (#1+0.96*#3,#2-0.15*#3).. (#1+0.9*#3,#2-0.2*#3) (#1,#2-1*#3) .. controls (#1,#2-0.95*#3) and (#1+0.04*#3,#2-0.85*#3).. (#1+0.1*#3,#2-0.8*#3) (#1+0.1*#3,#2-0.8*#3) ..controls (#1+0.25*#3,#2-0.8*#3) and (#1+0.45*#3,#2-0.69*#3) .. (#1+0.50*#3,#2-0.66*#3) (#1+0.1*#3,#2-0.8*#3) ..controls (#1+0.1*#3,#2-0.65*#3) and (#1+0.22*#3,#2-0.54*#3) .. (#1+0.275*#3,#2-0.505*#3) (#1+0.72*#3,#2-0.50*#3) .. controls (#1+0.75*#3,#2-0.47*#3) and (#1+0.9*#3,#2-0.4*#3).. (#1+0.9*#3,#2-0.2*#3) (#1,#2) .. controls (#1,#2-0.05*#3) and (#1+0.04*#3,#2-0.15*#3).. (#1+0.1*#3,#2-0.2*#3) (#1+0.5*#3,#2-0.34*#3) .. controls (#1+0.6*#3,#2-0.27*#3) and (#1+0.65*#3,#2-0.2*#3).. (#1+0.9*#3,#2-0.2*#3) (#1+#3,#2-#3) .. controls (#1+#3,#2-0.95*#3) and (#1+0.96*#3,#2-0.85*#3).. (#1+0.9*#3,#2-0.8*#3) (#1+#3,#2) .. controls (#1+#3,#2-0.05*#3) and (#1+0.96*#3,#2-0.15*#3).. (#1+0.9*#3,#2-0.2*#3) (#1+0.1*#3,#2-0.2*#3) .. controls (#1+0.3*#3,#2-0.2*#3) and (#1+0.46*#3,#2-0.31*#3) .. (#1+0.5*#3,#2-0.34*#3) (#1+0.1*#3,#2-0.2*#3) .. controls (#1+0.1*#3,#2-0.38*#3) and (#1+0.256*#3,#2-0.49*#3) .. (#1+0.275*#3,#2-0.505*#3) (#1+0.50*#3,#2-0.66*#3) .. controls (#1+0.548*#3,#2-0.686*#3) and (#1+0.70*#3,#2-0.8*#3)..(#1+0.9*#3,#2-0.8*#3) (#1+#3,#2-1*#3) .. controls (#1+#3,#2-0.95*#3) and (#1+0.96*#3,#2-0.85*#3).. (#1+0.9*#3,#2-0.8*#3) (#1+0.72*#3,#2-0.50*#3) .. controls (#1+0.80*#3,#2-0.56*#3) and (#1+0.9*#3,#2-0.73*#3)..(#1+0.9*#3,#2-0.8*#3)(#1+0.72*#3,#2-0.50*#3) -- (#1+0.50*#3,#2-0.66*#3) -- (#1+0.275*#3,#2-0.505*#3) -- (#1+0.5*#3,#2-0.34*#3) -- (#1+0.72*#3,#2-0.50*#3)}
\begin{scope}
\twisting(0,0)[1];
\end{scope}
\end{tikzpicture}
stands for $\Phi$, iff conditions~(1) and~(2) are fulfilled.
\smallskip
\noindent By the very definitions of $\Phi$ and $\mathcal{F}$, the first four conditions always hold. Assume that the other ones hold. Evaluating the fifth one in $x_1\otimes x_{n-1}\otimes h$ we see that
$$
\sum_{g\in G}\lambda_g g h\otimes x_0=\sum_{g\in G} \chi^n(h) \lambda_g h g\otimes x_0\quad\text{for all $g,h\in G$},
$$
or equivalently,
$$
\lambda_{hgh^{-1}}=\chi^n(h) \lambda_g\quad\text{for all $g,h\in G$,}
$$
and evaluating the sixth one in $x_1\otimes x_{n-1}\otimes x_1$ we see that
$$
\chi(g)=1\quad\text{for all $g\in G$ with $\lambda_g\ne 0$.}
$$
Conversely, a direct computation proves that if these facts are true, then the equalities in the last two diagrams hold.
\end{proof}
\begin{corollary}\label{si lambda_g ne 0 par un g central, chi^n = 0} If there is an algebra $B_{\mathcal{E}}$ satisfying the conditions required in Propositions~\ref{estructura de algebra de B_D}, and there exists $g$ in the center $\Z G$ of $G$ with $\lambda_{g} \ne 0$, then $\chi^n=1$.
\end{corollary}
\begin{remark} It is clear that if there exists, then $B_{\mathcal{E}}$ is a $k$-algebra unitary with unit $1_Gx^0$, that $kG$ is a subalgebra of $B_{\mathcal{E}}$ and that $B_{\mathcal{E}}$ is unique up to isomorphism.
\end{remark}
\begin{remark}\label{gen y red} Using that $B_{\mathcal{E}}$ has dimension $n|G|$ it is easy to see that it is canonically isomorphic to the algebra generated by the group $G$ and the element $x$ subject to the relations $x^n = U$ and $xg = \chi(g)gx$ for all $g\in G$.
\end{remark}
Given $q\in k^{\times}$, let
$$
c_q\colon B_{\mathcal{E}}\otimes B_{\mathcal{E}} \longrightarrow B_{\mathcal{E}}\otimes B_{\mathcal{E}}
$$
be the $k$-linear map defined by $c_q(gx^i\otimes hx^j):= q^{ij}\, hx^j \otimes gx^i$. It is easy to check that $c_q$ is a braiding operator that is compatible with the unit of $B_{\mathcal{E}}$. Furthermore,
\begin{itemize}
\smallskip
\item[-] a direct computation shows that $c_q$ is compatible with the multiplication map of $B_{\mathcal{E}}$ iff $U=0$ or $q^n=1$,
\smallskip
\item[-] by Remark~\ref{gen y red} there exists an algebra map $\epsilon\colon B_{\mathcal{E}}\to k$ such that $\epsilon(x)=0$ and $\epsilon(g)=1$ for all $g\in G$ iff $\sum_{g\in G} \lambda_g=0$. Moreover, in this case, $c_q$ is compatible with $\epsilon$.
\end{itemize}
\begin{proposition}\label{estructura de bialgebra de B sub D} Let $\mathcal{E}$ be as at the beginning of this section, $z\!\in\! G$ and $q\!\in\! k^{\times}$. Assume that $B_{\mathcal{E}}$ exists. Then, the algebra $B_{\mathcal{E}}$ is a braided bialgebra with braid $c_q$ and comultiplication map $\Delta$ defined by
\begin{equation}
\Delta(x):=1\otimes x+x\otimes z \qquad\text{and}\qquad\Delta(g) := g\otimes g\quad\text{for $g\in G$}\label{eqa1}
\end{equation}
iff
\begin{enumerate}
\smallskip
\item $\binom{n}{j}_{q\chi(z)} = 0$ for all $0<j<n$,
\smallskip
\item $z\in \Z G$ and $U=\lambda(z^n-1_G)$ for some $\lambda\in k$, where $\lambda=0$ if $z^n\ne 1_G$ and $q^n\ne 1$.
\end{enumerate}
\end{proposition}
\begin{proof} Since $B_{\mathcal{E}}$ is generated by the group $G$ and the element $x$ subject to the relations $x^n = U$ and $xg = \chi(g)gx$ for all $g\in G$, there exists an algebra map $\Delta\colon B_{\mathcal{E}} \longrightarrow B_{\mathcal{E}} \otimes_{c_q} B_{\mathcal{E}}$ such that~\eqref{eqa1} is satisfied iff the equalities
\begin{align}
& (h\otimes h)(g\otimes g) = hg\otimes hg,\notag\\
&(1\otimes x+x\otimes z)(g\otimes g) = \chi(g) (g\otimes g)(1\otimes x+x\otimes z)\notag
\shortintertext{and}
& (1\otimes x+x\otimes z)^n = \sum_{l\in G} \lambda_l\, l\otimes l\label{latercera}
\end{align}
hold in $B_{\mathcal{E}} \otimes_{c_q} B_{\mathcal{E}}$ for all $h,g\!\in\! G$. The first equality is trivial, while the second one is e\-qui\-valent to
\begin{equation*}
\chi(g)(g\otimes gx + gx\otimes zg) = \chi(g)(g\otimes gx+gx\otimes gz)\qquad\text{for all $g\in G$,}
\end{equation*}
and so it is fulfilled iff $z$ is in the center of $G$. In order to deal with the last one, we note that, in $B_{\mathcal{E}} \otimes_{c_q} B_{\mathcal{E}}$,
$$
(1\otimes x)(x\otimes z)= q\chi(z)\, x\otimes zx = q\chi(z)(x\otimes z)(1\otimes x),
$$
and so, by formula~\eqref{eq12},
\begin{equation*}
(1\otimes x+x\otimes z)^n = \sum_{j=0}^n \binom{n}{j}_{q\chi(z)}(x\otimes z)^j(1\otimes x)^{n-j} = \sum_{j=0}^n \binom{n}{j}_{q\chi(z)} x^j\otimes z^jx^{n-j}.
\end{equation*}
Hence, equality~\eqref{latercera} holds iff
$$
\sum_{j=0}^n \binom{n}{j}_{q\chi(z)} x^j\otimes z^jx^{n-j} = \sum_{l\in G} \lambda_l\, l\otimes l,
$$
which is clearly equivalent to
\begin{align*}
& \binom{n}{j}_{q\chi(z)} = 0\qquad\text{for all $0<j<n$,}
\shortintertext{and}
& \sum_{l\in G} \lambda_l\, l\otimes l = 1\otimes x^n+x^n\otimes z^n = 1\otimes U + U\otimes z^n = \sum_{l\in G} 1\otimes \lambda_l l + \sum_{l\in G} \lambda_l l \otimes z^n.
\end{align*}
If $z^n = 1_G$ this happens iff $\lambda_l = 0$ for all $l\in G$, while if $z^n \ne 1_G$, this happens iff $\lambda_l = 0$ for all $l\ne 1_G,z^n$ and if $\lambda_{z^n}=-\lambda_{1_G}$. By the way, this computation shows that if $\Delta$ exists, then the augmentation $\epsilon$ introduced above, is well defined. Moreover, by formula~\eqref{eq12},
\begin{equation}
\Delta(gx^i):= \sum_{j=0}^i \binom{i}{j}_{q\chi(z)} (g\otimes g)(x\otimes z)^j(1\otimes x)^{i-j}= \sum_{j=0}^i \binom{i}{j}_{q\chi(z)} gx^j\otimes gz^j x^{i-j}\label{comultiplication}
\end{equation}
for all $g\in G$ and $i\ge 0$. Using this it is easy to see that $c_q$ is compatible with $\Delta$. Since we already know that $c_q$ is compatible with $1_{B_{\mathcal{E}}}$, the multiplication map of $B_{\mathcal{E}}$ and $\epsilon$, in order to finish the proof we only must check that $\Delta$ is coassociative and that $\epsilon$ is its counit. But, since $c_q$ is compatible with $\Delta$ and $\Delta$ is an algebra map, it suffices to verify these facts on $x$ and $g\in G$, which is trivial.
\end{proof}
\begin{remark} Let $\mathcal{E}$ be as at the beginning of this section. If $U = \lambda(z^n-1_G)$ with $z\in \Z G$, $z^n\ne 1_G$ and $\lambda\in k^{\times}$, then the hypothesis of Proposition~\ref{estructura de algebra de B_D} are equivalent to $\chi^n = 1$, while if $U=0$, then the hypothesis of Proposition~\ref{estructura de algebra de B_D} are automatically satisfied.
\end{remark}
\begin{remark} It is easy to see that $\binom{n}{1}_{q\chi(z)} = 0$ implies $(q\chi(z))^n = 1$ and that if $q\chi(z)$ is an $n$-th primitive root of unit, then $\binom{n}{j}_{q\chi(z)} = 0$ for all $0<j<n$.
\end{remark}
\begin{corollary}\label{algebras de Hopf trenzadas de K-R} Each data $\mathcal{D}=(G,\chi,z,\lambda,q)$ consisting of:
\begin{itemize}
\smallskip
\item[-] a finite group $G$,
\smallskip
\item[-] a character $\chi$ of $G$ with values in $k$,
\smallskip
\item[-] a central element $z$ of $G$,
\smallskip
\item[-] elements $q\in k^{\times}$ and $\lambda\in k$,
\smallskip
\end{itemize}
such that
\begin{itemize}
\smallskip
\item[-] $q\chi(z)$ is a root of $1$ of order $n$ greater than $1$,
\smallskip
\item[-] if $\lambda(z^n-1_G)\ne 0$, then $\chi^n=1$,
\end{itemize}
has associated a braided Hopf algebra $H_{\mathcal{D}}$. As an algebra, $H_{\mathcal{D}}$ is generated by the group $G$ and the element $x$ subject to the relations $x^n=\lambda(z^n-1_G)$ and $xg=\chi(g)gx$ for all $g\in G$, the coalgebra structure of $H_{\mathcal{D}}$ is determined by
\begin{align*}
&\Delta(g):=g\otimes g\,\,\text{ for $g\in G$,}&& \Delta(x):=1\otimes x + x\otimes z,\\
&\epsilon(g):=1\,\,\text{ for $g\in G$,} &&\epsilon(x):=0,
\end{align*}
the braid $c_q$ of $H_{\mathcal{D}}$ is defined by
\begin{equation}
c_q(gx^i\otimes hx^j):= q^{ij}\,hx^j \otimes gx^i,\label{def braid}
\end{equation}
and its antipode is given by
\begin{equation}
S(gx^i) := (-1)^i(q\chi(z))^{\frac{i(i-1)}{2}}x^iz^{-i}g^{-1}.\label{eqa4}
\end{equation}
Furthermore, as a vector space $H_{\mathcal{D}}$ has basis
$$
\{gx^i:g \in G\text{ and } 0 \le i < n\},
$$
and consequently, $\dim \bigl(H_{\mathcal{D}}\bigr) = n|G|$.
\end{corollary}
\begin{proof} Let $\mathcal{E}:=\bigl(G,\chi,\lambda(z^n-1_G),n\bigr)$ and let $B_{\mathcal{E}}$ be the algebra obtained applying Proposition~\ref{estructura de algebra de B_D}. Now note that if $\lambda(z^n-1_G)$, then $\chi^n=1$ and so $q^n = q^n\chi(z)^n = 1$. Hence, we can apply Pro\-po\-si\-tion~\ref{estructura de bialgebra de B sub D}, which implies that $B_{\mathcal{E}}$ has a braided bialgebra structure with comultiplication map, counit and braid as in its statement. Let $H_{\mathcal{D}}$ denote this bialgebra. It remains to check that the map $S$ given by~\eqref{eqa4} is the antipode of $H_{\mathcal{D}}$. Since
$$
S\hs \circ \hs \mu(gx^i\otimes hx^j) = \mu \hs \circ \hs (S\otimes S)\hs \circ \hs c_q(gx^i\otimes hx^j),
$$
for this it suffices to verify that
$$
S(x) + xS(z)=S(1)x+S(x)z=0\qquad\text{and}\qquad S(g)g=gS(g)=1\quad\text{for all $g\in G$,}
$$
which is evident.
\end{proof}
\begin{comment}
\begin{align*}
\mu \hs \circ \hs (S\otimes S)\hs \circ \hs c (gx^i\otimes hx^j ) & = q^{ij}\mu \hs \circ \hs (S\otimes S)(hx^j\otimes gx^i )\\
& = (-1)^{i+j}q^{ij}(q\chi(z))^{\frac{i(i-1)+j(j-1)}{2}}\mu \bigl(x^jz^{-j}h^{-1}\otimes x^iz^{-i}g^{-1}\bigr)\\
& = (-1)^{i+j}q^{ij}(q\chi(z))^{\frac{i(i-1)+j(j-1)}{2}}\chi(z)^{ij}\chi(h)^i x^{i+j} z^{-i-j}h^{-1}g^{-1}\\
& = (-1)^{i+j}\chi(h)^i (q\chi(z))^{\frac{(i+j)(i+j-1)}{2}}x^{i+j}z^{-i-j}h^{-1}g^{-1}
\end{align*}
\begin{align*}
S \hs \circ \hs \mu (gx^i\otimes hx^j ) & = \chi(h)^i S(ghx^{i+j})\\
& = (-1)^{i+j}\chi(h)^i (q\chi(z))^{\frac{(i+j)(i+j-1)}{2}}x^{i+j}z^{-i-j}h^{-1}g^{-1}
\end{align*}
Let $H_{\mathcal{D}}$ denote this bialgebra and let $H_{\mathcal{D}}^{\op}$ be the braided opposite algebra of $H_{\mathcal{D}}$. So $H_{\mathcal{D}}^{\op}$ has the same underlying vector space as $H_{\mathcal{D}}$ and its multiplication map is given by $\mu_{H_{\mathcal{D}}^{\op}}:= \mu_{H_{\mathcal{D}}}\hs \circ \hs c$. For each $a\in H_{\mathcal{D}}$ we let $\overline{a}$ denote $a$ considered as an element of $H_{\mathcal{D}}^{\op}$. Clearly
$$
\overline{x^ig}\, \overline{x^jh} = q^{ij} \overline{x^j h x^i g} = q^{ij}\chi^{-i}(h) \overline{x^{i+j}hg}\qquad\text{for all $h,g\in G$ and $i\ge 0$,}
$$
and so
\begin{align}
&\overline{g^{-1}}\,\overline{h^{-1}} = \overline{h^{-1}g^{-1}} = \overline{(gh)^{-1}},\label{ep1}\\
&\overline{xz^{-1}}\,\overline{g^{-1}} = \chi^{-1}(g^{-1}) \overline{xg^{-1}z^{-1}} = \chi(g)\overline{xz^{-1}g^{-1}} = \chi(g) \overline{g^{-1}}\,\overline{xz^{-1}}\label{ep2}
\shortintertext{and}
&\overline{x^iz^{-i}}\,\overline{xz^{-1}} = q^i\chi^{-i}(z^{-1})\overline{x^{i+1}z^{-(i+1)}} = (q\chi(z))^i \overline{x^{i+1}z^{-(i+1)}}.\label{ep3}
\end{align}
Note that the last equality implies
\begin{equation}
\overline{xz^{-1}}^{\,n} = (q\chi(z))^{\frac{n(n-1)}{2}}\overline{x^nz^{-n}} = (-1)^{n-1}\lambda\overline{(z^n-1_G)z^{-n}} = (-1)^n\lambda(\overline{z^{-n}-1_G)}.\label{ep4}
\end{equation}
Since $H_{\mathcal{D}}$ is the algebra generated by the group $G$ and the element $x$ subject to the relations $x^n = \lambda(z^n-1_G)$ and $xg = \chi(g)gx$ for all $g\in G$ it follows from equalities~\eqref{ep1}, \eqref{ep2} and~\eqref{ep4} that there exists an algebra morphism
$$
\widetilde{S}\colon H_{\mathcal{D}}\longrightarrow H_{\mathcal{D}}^{\op},
$$
such that $\widetilde{S}(x) = -\overline{xz^{-1}}$ and $\widetilde{S}(g) = \overline{g^{-1}}$. Furthermore, by~\eqref{ep3}
$$
\widetilde{S}(gx^i) = (-1)^i \overline{g^{-1}}\,\overline{xz^{-1}}^{\,i} = (-1)^i(q\chi(z))^{\frac{i(i-1)}{2}} \overline{x^iz^{-i}g^{-1}},
$$
and consequently $S\hs \circ \hs \mu= \mu \hs \circ \hs (S\otimes S)\hs \circ \hs c$.
\end{comment}
\begin{remark} If $\lambda(z^n-1_G)= 0$, then we can assume without lost of generality (and we do it), that $\lambda = 0$.
\end{remark}
\begin{remark} Assume that $n>1$. The previous corollary also holds if the hypothesis that $q\chi(z)$ is a root of $1$ of order $n$ is replaced by $\binom{n}{j}_{q\chi(z)} = 0$ for all $0<j<n$. However, from now on we will consider that $q\chi(z)$ is a root of $1$ of order $n$.
\end{remark}
\section{Right $\bm{H_{\mathcal{D}}}$-comodule algebras}\label{H_D comodule algebras}
Let $G$ be a group, $V$ be a $k$-vector space and $s\colon k[G]\otimes V\to V \otimes k[G]$ a $k$-linear map. Evidently, there is a unique family of maps $(\alpha_x^y\colon V\to V)_{x,y\in G}$, such that
$$
s(x\otimes v)=\sum_{y\in G} \alpha_x^y(v)\otimes y.
$$
\begin{proposition}\label{primer resultado sobre estructuras trenzadas de grupos} The pair $(V,s)$ is a left $k[G]$-space iff $s$ is a bijective map and the fo\-llo\-wing conditions hold:
\begin{enumerate}
\smallskip
\item $(\alpha_x^y)_{y\in G}$ is a complete family of orthogonal idempotents, for all $x\in G$,
\smallskip
\item $\alpha_1^1=\ide$,
\smallskip
\item $\alpha_{xy}^z=\sum_{uw=z}\alpha_x^u\hs \circ \hs \alpha_y^w$, for all $x,y,z\in G$.
\end{enumerate}
\end{proposition}
\begin{proof} Mimic the proof of \cite{G-G}*{Proposition 4.10}.
\end{proof}
For $x,y\in G$, let $V_x^y:=\{v\in V:s(x\otimes v)=v\otimes y\}$.
\begin{proposition}\label{segundo resultado sobre estructuras trenzadas de grupos} The pair $(V,s)$ is a left $k[G]$-space iff:
\begin{enumerate}
\smallskip
\item $\bigoplus_{z\in G} V_x^z = V = \bigoplus_{z\in G} V_z^x$, for all $x\in G$,
\smallskip
\item $V_1^1=V$,
\smallskip
\item $V_{xy}^z=\bigoplus_{uw=z} V_x^u\cap V_y^w$, for all $x,y,z\in G$.
\end{enumerate}
\end{proposition}
\begin{proof} Mimic the proof of \cite{G-G}*{Propositions 4.11 and 4.13}.
\end{proof}
\begin{theorem}\label{tercer resultado sobre estructuras trenzadas de grupos} If $G$ is a finitely generated group, then each left $k[G]$-space $(V,s)$ determines an $\Aut(G)$-gradation
$$
V=\bigoplus_{\zeta\in \Aut(G)} V_{\zeta}
$$
on $V$, by
$$
V_{\zeta}:=\bigcap_{x\in G} V_x^{\zeta(x)}=\{v\in V: s(x\otimes v)=v\otimes \zeta(x) \text{ for all $x\in G$}\}.
$$
Moreover, the correspondence that each left $k[G]$-space $(V,s)$, with underlying vector space $V$, assigns the $\Aut(G)$-gradation of $V$ obtained as above, is bijective.
\end{theorem}
\begin{proof} Mimic the proof of~\cite{G-G}*{Theorem 4.14}.
\end{proof}
In the sequel $\mathcal{D}:=(G,\chi,z,\lambda,q)$ and $H_{\mathcal{D}}$ are as in Corollary~\ref{algebras de Hopf trenzadas de K-R} and we will freely use the notations and properties established there. Furthermore, to abbreviate expressions we set $p:=\chi(z)$. We now begin with the study of the right $H_{\mathcal{D}}$-braided comodule algebras. We let $\Aut_{\chi,z}(G)$ denote the subgroup of $\Aut(G)$ consisting of all the automorphism $\phi$ such that $\phi(z)=z$ and $\chi\hs \circ \hs \phi=\chi$.
\begin{proposition}\label{estructuras trenzadas de HsubD (primer resultado)} If $(p,q)\ne (1,-1)$, then for all left $H_{\mathcal{D}}$-space $(V,s)$ it is true that
\begin{align}
& s(kG\otimes V)=V\otimes kG,\nonumber\\
& s(z\otimes v)=v\otimes z\quad\text{for all $v\in V$,}\nonumber
\shortintertext{and there exists $\alpha\in \Aut(V)$ such that}
&s(x\otimes v)=\alpha(v)\otimes x\quad\text{for all $v\in V$.}\label{eqq1}
\end{align}
\end{proposition}
\begin{proof} Write
$$
s(gx^i\otimes v) = \sum_{\substack{h\in G\\ 0\le j<n}} \beta^{g,i}_{h,j}(v) \otimes hx^j.
$$
Since $S^2(gx^i)=q^{i(i-1)}p^{-i}gx^i$, we have
\begin{align*}
q^{i(i-1)}p^{-i} \sum_{\substack{h\in G\\ 0\le j<n}} \beta^{g,i}_{h,j}(v) \otimes hx^j &=q^{i(i-1)}p^{-i} s(gx^i\otimes v)\\
&= s\hs \circ \hs (S^2\otimes V)(gx^i\otimes v)\\
&= (V\otimes S^2)\hs \circ \hs s(gx^i\otimes v)\\
&= \sum_{\substack{h\in G\\ 0\le j<n}} \beta^{g,i}_{h,j}(v) \otimes S^2(hx^j)\\
&= \sum_{\substack{h\in G\\ 0\le j<n}} q^{j(j-1)}p^{-j}\beta^{g,i}_{h,j}(v) \otimes hx^j,
\end{align*}
and consequently,
\begin{equation}\label{equu1}
\beta^{g,i}_{h,j}\ne 0 \Longrightarrow q^{j(j-1)-i(i-1)} = p^{j-i}.
\end{equation}
Using now that $s$ is compatible with $\Delta$, we obtain that
\begin{equation}
\begin{aligned}
\sum_{\substack{h\in G\\ 0\le i<n}}\sum_{j=0}^i\binom{i}{j}_{qp}\beta^{g,0}_{h,i}(v)\otimes hx^j\otimes hz^jx^{i-j} &= (V\otimes \Delta)\hs \circ \hs s(g\otimes v)\\
&= (s\otimes H_{\mathcal{D}}) \hs \circ \hs (H_{\mathcal{D}}\otimes s)\hs \circ \hs (\Delta\otimes V)(g\otimes v)\\
&= \adjustlimits\sum_{\substack{h\in G\\ 0\le i<n}} \sum_{\substack{h'\in G\\ 0\le i'<n}} \beta^{g,0}_{h,i} \hs \circ \hs \beta^{g,0}_{h',i'}(v) \otimes hx^i\otimes h'x^{i'}.
\end{aligned}\label{eqq2}
\end{equation}
Hence,
\begin{equation}
\beta_{h,i}^{g,0}\hs \circ \hs \beta_{h',i'}^{g,0} = \begin{cases} \binom{i+i'}{i}_{qp} \beta_{h,i+i'}^{g,0} &\text{if $h' = hz^i$ and $i+i' < n$,}\\ 0 &\text{otherwise.} \end{cases}\label{ee1}
\end{equation}
Combining this with~\eqref{equu1} we obtain that
$$
\beta_{h,i}^{g,0}\ne 0\Longrightarrow \beta_{h,j}^{g,0}\ne 0\text{ for all $j\le i$ } \Longrightarrow q^{j(j-1)} = p^j \text{ for $j\le i$.}
$$
Consequently, if $\beta_{h,i}^{g,0}\ne 0$ for some $g\in G$ and $i\ge 1$, then $p = q^0 = 1$. Hence if $p\ne 1$, then $\beta_{h,i}^{g,0}=0$ for all $g\in G$ and $i\ge 1$. Assume that $p = 1$. If $\beta_{h,i}^{g,0}\ne 0$ for some $g\in G$ and $i\ge 2$, then $q^2 = p^2 = 1$. But this is impossible, since it implies that $n:=\ord(qp)=\ord(q)\le 2$, which contradicts that $i<n$. Therefore
\begin{equation}
s(g\otimes v) = \begin{cases}\sum_{h\in G}\beta^{g,0}_{h,0}(v)\otimes h &\text{if $p\ne 1$,}\\ \sum_{h\in G}\beta^{g,0}_{h,0}(v)\otimes h + \sum_{h\in G}\beta^{g,0}_{h,1}(v)\otimes hx &\text{if $p = 1$.}\end{cases}\label{ee2}
\end{equation}
On the other hand, due to $s$ is compatible with the counit of $H_{\mathcal{D}}$, we get
$$
\sum_{h\in G} \beta_{h,0}^{g,0}=\ide\quad\text{for all $g\in G$,}
$$
which, combined with the particular case of~\eqref{ee1} obtained by taken $i=i'=0$, shows that
\begin{equation}
\bigl(\beta_{h,0}^{g,0}\bigr)_{h\in G}\,\text{ is a complete family of orthogonal idempotents for all $g\in G$.}\label{ee3}
\end{equation}
Equality~\eqref{ee2} shows that if $p\ne 1$, then $s(kG\otimes V) \subseteq V\otimes kG$. Assume now that $p =1$ and $q\ne -1$ (which implies $n>2$). Using that $s$ is compatible with the multiplication map of $H_{\mathcal{D}}$ we get that
\begin{equation}
\begin{aligned}
v\otimes 1&= s(g^{-1}g\otimes v)\\
&=(v\otimes\mu)\hs \circ \hs (s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)(g^{-1}\otimes g\otimes v)\\
& = \sum_{h\in G}\sum_{l\in G}\beta^{g^{-1},0}_{h,0}\hs \circ \hs\beta_{1,0}^{g,0}(v)\otimes hl + \sum_{h\in G}\sum_{l\in G}\chi(l)\beta_{h,1}^{g^{-1},0}\hs \circ \hs\beta_{l,0}^{g,0}(v)\otimes hlx \\
& + \sum_{h\in G}\sum_{l\in G}\beta_{h,0}^{g^{-1},0}\hs \circ \hs\beta_{l,1}^{g,0}(v)\otimes hlx + \sum_{h\in G}\sum_{l\in G}\chi(l)\beta_{h,1}^{g^{-1},0}\hs \circ \hs\beta_{l,1}^{g,0}(v)\otimes hl x^2.
\end{aligned}\label{eqq3}
\end{equation}
Consequently,
$$
\sum_{h\in G}\beta_{h^{-1},0}^{g^{-1},0}\hs \circ \hs\beta_{g,0}^{h,0} = \ide_V,
$$
which by~\eqref{ee3} implies that
$$
\beta_{g^{-1},0}^{h^{-1},0}(v)=v\quad\text{for all $v\in \ima\bigl(\beta_{h,0}^{g,0} \bigr)$ and $h,g\in G$.}
$$
Since $\bigl(\beta_{g,0}^{h,0}\bigr)_{h\in G}$ and $\bigl(\beta^{g^{-1},0}_{h,0} \bigr)_{h\in G}$ are complete families of orthogonal idempotents, from this it follows that
$$
\beta^{g^{-1},0})_{h^{-1},0}=\beta^{g,0}_{h,0}\quad\text{for all $h,g\in G$.}
$$
Combining this with~\eqref{eqq3}, we conclude that
\begin{equation}
\begin{aligned}
0 &=\sum_{h\in G}\sum_{l\in G}\chi(l)\beta_{h,1}^{g^{-1},0}\hs \circ \hs\beta^{g,0}_{l,0}(v)\otimes hlx + \sum_{h\in G}\sum_{l\in G}\beta^{g^{-1},0}_{h,0}\hs \circ \hs\beta_{l,1}^{g,0}(v)\otimes hlx\\
&= \sum_{h\in G}\sum_{l\in G}\chi(l)\beta_{h,1}^{g^{-1},0}\hs \circ \hs\beta^{g^{-1},0}_{ l^{-1},0}(v)\otimes hlx + \sum_{h\in G}\sum_{l\in G}\beta^{g,0}_{h^{-1},0}\hs \circ \hs\beta_{l,1}^{g,0}(v) \otimes hlx\\
& = \sum_{h\in G}\chi(z^{-1}h^{-1})\beta_{h,1}^{g^{-1},0}(v)\otimes z^{-1}x + \sum_{h\in G} \beta_{h,1}^{g,0}(v)\otimes x,
\end{aligned}\label{ecua3}
\end{equation}
where the last equality follows from the fact that by~\eqref{ee1}
\begin{equation}
\beta_{h,1}^{g,0}\hs \circ \hs \beta_{h',0}^{g,0} = \begin{cases}\beta_{h,1}^{g,0} &\text{if $h'=hz$,}\\ 0 &\text{otherwise,}\end{cases}\quad\text{and} \quad\beta_{h,0}^{g,0} \hs \circ \hs\beta_{h',1}^{g,0} = \begin{cases}\beta_{h,1}^{g,0} &\text{if $h' = h$,}\\ 0 & \text{otherwise.}\end{cases}
\label{ee4}
\end{equation}
Note that by~\eqref{ee3} and the second equality in~\eqref{ee4}, the images of the maps $\beta_{h,1}^{g,0}$ are in direct sum, for each $g\in G$. Hence, from~\eqref{ecua3} it follows that if $z\ne 1$, then $\beta_{h,1}^{g,0}=0$ for all $g,h\in G$, which by equality~\eqref{ee2} implies that $s(kG\otimes V) \subseteq V\otimes kG$. We now assume additionally that $z=1$. Then $\Prim(H_{\mathcal{D}})= kx$ and so, by~\cite{G-G}*{Proposition 4.4} there exists an automorphism $\alpha$ of $V$ such that $s(x\otimes v)=\alpha(v)\otimes x$ for all $v\in V$. Furthermore, by the compatibility of $s$ with $c_q$,
\begin{align*}
\sum_{h\in G} \beta_{h,0}^{g,0}\hs \circ \hs \alpha(v)\otimes x\otimes h & + q \sum_{h\in G} \beta_{h,1}^{g,0}\hs \circ \hs \alpha(v)\otimes x\otimes hx \\
&= (V\otimes c_q)\hs \circ \hs (s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)(g\otimes x\otimes v)\\
&= (s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)\hs \circ \hs (c_q \otimes V)(g\otimes x\otimes v)\\
&= \sum_{h\in G}\alpha \hs \circ \hs \beta_{h,0}^{g,0}(v)\otimes x\otimes h + \sum_{h\in G} \alpha\hs \circ \hs \beta_{h,1}^{g,0} (v)\otimes x\otimes hx
\end{align*}
for all $g\in G$, and therefore
\begin{equation}
\alpha\hs \circ \hs \beta_{h,0}^{g,0}= \beta_{h,0}^{g,0}\hs \circ \hs \alpha\quad\text{and}\quad\alpha \hs \circ \hs \beta_{h,1}^{g,0}= q\beta_{h,1}^{g,0}\hs \circ \hs \alpha\quad \text{for all $h,g\in G$.} \label{eqq7}
\end{equation}
Using now that $s$ is compatible with the multiplication map of $H_\mathcal{D}$, we obtain that
\begin{align*}
\chi(g)\sum_{h\in G}\beta_{h,0}^{g,0}\hs \circ \hs\alpha(v)\otimes hx & + \chi(g)\sum_{h\in G} \beta_{h,1}^{g,0}\hs \circ \hs \alpha(v)\otimes hx^2 \\
&=(V\otimes\mu)\hs \circ \hs(s\otimes H_{\mathcal{D}})\hs \circ \hs(H_{\mathcal{D}}\otimes s)(\chi(g)g\otimes x\otimes v)\\
&=(s\otimes H_{\mathcal{D}})\hs \circ \hs(H_{\mathcal{D}}\otimes s)\hs \circ \hs(\mu\otimes V)(\chi(g)g\otimes x\otimes v)\\
&=(s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)\hs \circ \hs (\mu \otimes V)(x\otimes g\otimes v)\\
&=(V\otimes \mu)\hs \circ \hs (s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)(x\otimes g\otimes v)\\
&=\sum_{h\in G}\chi(h)\,\alpha\hs \circ \hs\beta_{h,0}^{g,0}(v)\otimes hx + \sum_{h\in G}\chi(h)\, \alpha\hs \circ \hs \beta_{h,1}^{g,0}(v)\otimes hx^2
\end{align*}
for all $g\in G$, which combined with~\eqref{eqq7} gives
\begin{align}
&\chi(g)\,\beta_{h,0}^{g,0}\hs \circ \hs\alpha = \chi(h)\,\alpha\hs \circ \hs\beta_{h,0}^{g,0} = \chi(h)\,\beta_{h,0}^{g,0}\hs \circ \hs\alpha\label{eqq8}
\shortintertext{and}
& \chi(g)\,\beta_{h,1}^{g,0}\hs \circ \hs \alpha = \chi(h)\,\alpha\hs \circ \hs\beta_{h,1}^{g,0} = \chi(h)q\,\beta_{h,1}^{g,0}\hs \circ \hs\alpha\label{eqq9}
\end{align}
for all $g,h\in G$. Since $\alpha$ is bijective, from~\eqref{eqq8} it follows that if $\beta_{h,0}^{g,0}\ne 0$, then $\chi(g)=\chi(h)$. Combining this with~\eqref{ee4}, we see that $\beta_{h,1}^{g,0}\ne 0 \Rightarrow \beta_{h,0}^{g,0}\ne 0\Rightarrow \chi(g)=\chi(h)$. Therefore, from~\eqref{eqq9} it follows that if there exist $g,h\in G$ such that $\beta_{h,1}^{g,0}\ne 0$, then $q=1$, which is false. So, $\beta_{h,1}^{g,0}= 0$ for all $g,h\in G$. This concludes the proof that $s(kG\otimes V)\subseteq V\otimes kG$. But then a similar computation with $s$ replaced by $s^{-1}$ shows that $s(kG\otimes V)\supseteq V \otimes kG$, and so the equality holds.
\smallskip
We now return to the general case and we claim that
\begin{enumerate}
\smallskip
\item $\beta_{h,j}^{1,1}= 0$ for all $h\in G$ and $j\ge 2$,
\smallskip
\item $\beta_{h,1}^{1,1}= 0$ for $h\ne 1_G$,
\smallskip
\item $\beta_{1,1}^{1,1}$ is bijective,
\smallskip
\item $\beta_{z,0}^{z,0} = \ide$ and $\beta_{h,0}^{z,0} = 0$ for $h\ne z$,
\smallskip
\item $\beta_{h,0}^{1,1} = 0$ for $h\notin\{1_G,z\}$,
\smallskip
\item If $z\ne 1$, then $\beta_{z,0}^{1,1} = -\beta_{1,0}^{1,1}$ while if $z = 1_G$, then $\beta_{1,0}^{1,1} = 0$,
\smallskip
\end{enumerate}
In fact, $s(kG\otimes V) \subseteq V\otimes kG$ means that $\beta^{g,0}_{h,j}=0$ for all $g,h\in G$ and $j>0$. Hence, by the compatibility of $s$ with $\Delta$,
\begin{equation}
\begin{aligned}
\sum_{\substack{h\in G\\ 0\le i<n}}\sum_{j=0}^i\binom{i}{j}_{qp}\beta^{1,1}_{h,i}(v)\otimes hx^j \otimes hz^jx^{i-j} & = (V\otimes \Delta)\hs \circ \hs s (x\otimes v)\\
& = (s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)\hs \circ \hs (\Delta\otimes V)(x\otimes v)\\
& = \sum_{\substack{h\in G\\ 0\le i<n}} \beta^{1,1}_{h,i}(v)\otimes 1\otimes hx^i\\
& + \sum_{\substack{h,l\in G\\ 0\le i<n}}\beta^{1,1}_{l,i}\hs \circ \hs \beta^{z,0}_{h,0}(v)\otimes lx^i\otimes h.
\end{aligned}\label{eqq5}
\end{equation}
This implies that items~(1) and (2) are true and that
\begin{itemize}
\smallskip
\item[(8)] $\beta_{1,1}^{1,1}=\beta_{1,1}^{1,1}\hs \circ \hs \beta_{z,0}^{z,0}$ and $\beta_{1,1}^{1,1}\hs \circ \hs \beta_{h,0}^{z,0} = 0$ for all $h\in G\setminus \{z\}$,
\smallskip
\item[(9)] $\beta_{h,0}^{1,1}=\beta_{h,0}^{1,1}\hs \circ \hs \beta_{h,0}^{z,0}$ and $\beta_{h,0}^{1,1}=-\beta_{1,0}^{1,1}\hs \circ \hs \beta_{h,0}^{z,0}$ for all $h \in G\setminus\{1_G\}$,
\smallskip
\item[(10)] $\beta_{h,0}^{1,1}\hs \circ \hs \beta_{1,0}^{z,0} = 0$ for all $h\in G$.
\smallskip
\end{itemize}
By items~(1) and~(2) and condition~\eqref{equu1},
\begin{equation}
s(x\otimes v) = \begin{cases} \beta_{1,1}^{1,1}(v)\otimes x &\text{if $p \ne 1$,}\\ \beta_{1,1}^{1,1}(v)\otimes x + \sum_{h\in G} \beta_{h,0}^{1,1}(v)\otimes h &\text{if $p = 1$.} \end{cases}\label{ecua4}
\end{equation}
This immediately implies that $\beta_{1,1}^{1,1}$ is injective. In fact, if $\beta_{1,1}^{1,1}(v) = 0$, then $s(x\otimes v)\in s(kG\otimes V)$, and so $v=0$ since $s\colon H_{\mathcal{D}}\otimes V\to V\otimes H_{\mathcal{D}}$ is injective. Item~(4) follows from item~(8) and the injectivity of $\beta_{1,1}^{1,1}$. Hence, by item~(9), we have $\beta_{h,0}^{1,1}=\beta_{h,0}^{1,1}\hs \circ \hs \beta_{h,0}^{z,0} = 0$ for all $h\in G\setminus \{1_G,z\}$, proving item~(5). Note also that by items~(4), (9) and (10),
$$
\beta_{z,0}^{1,1}=\begin{cases}
-\beta_{1,0}^{1,1}\hs \circ \hs \beta_{z,0}^{z,0} = -\beta_{1,0}^{1,1} &\text{if $z\ne 1_G$,}\\
\beta_{1,0}^{1,1}\hs \circ \hs \beta_{1,0}^{1,0} = 0 &\text{if $z = 1_G$,}
\end{cases}
$$
which proves item~(6). Combining item~(4) with the fact that $\beta^{z,0}_{h,j}=0$ for all $h\in G$ and $j>0$, we deduce that
$$
s(z\otimes v)=v\otimes z\qquad\text{for all $v\in V$.}
$$
Furthermore, by items~(5) and~(6), equality~\eqref{ecua4} becomes
\begin{equation}\label{ecua5}
s(x\otimes v) = \begin{cases} \beta_{1,1}^{1,1}(v)\otimes x &\text{if $p\ne 1$ or $z = 1_G$,}\\ \beta_{1,1}^{1,1}(v)\otimes x + \beta_{1,0}^{1,1}(v)\otimes 1_{H_{\mathcal{D}}} - \beta_{1,0}^{1,1}(v)\otimes z &\text{otherwise.}\end{cases}
\end{equation}
Next we prove that if $p=1$ and $q\ne -1$, then $\beta_{1,0}^{1,1} = 0$. If $z=1_G$ this was checked above. So we can assume that $z\ne 1_G$. To abbreviate expressions we set $\alpha:=\beta_{1,1}^{1,1}$ and $\beta:=\beta_{1,0}^{1,1}$. Evaluating
$$
(s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)\hs \circ \hs (c_q\otimes V) \qquad\text{and}\qquad (V\otimes c_q)\hs \circ \hs(s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)
$$
in $x\otimes x\otimes v$ for all $v\in V$, and using~\eqref{ecua5} and that these maps coincide, we see that
\begin{equation*}
q \beta\hs \circ \hs \alpha = \alpha\hs \circ \hs \beta\quad\text{and}\quad q\alpha\hs \circ \hs \beta = \beta \hs \circ \hs \alpha.
\end{equation*}
Then $q^2\alpha\circ \beta = \alpha\circ \beta$, and so $\beta = 0$, since $q^2\ne 1$ and $\alpha$ is injective. Hence~\eqref{ecua5} becomes
$$
s(x\otimes v) = \alpha (v)\otimes v \quad\text{for all $v\in V$.}
$$
Consequently $s(x\otimes V)\subseteq V\otimes x$ and a similar computation with $s$ replaced by $s^{-1}$ shows that $s(x\otimes V)\supseteq V\otimes x$, which immediately proves that $\alpha$ is a surjective map.
\begin{comment}
We now consider two cases:
\smallskip
\noindent $\bm{z\ne 1}$.\enspace Since $z^{-1}\ne 1$ and by equality~\eqref{eqq4} the images of the maps $\beta_{h,1}^{g,0}$ are in direct sum, for each $g\in G$, it follows from~\eqref{ecua3} that $\beta_{h,1}^{g,0}=0$ for all $g,h\in G$. Therefore, by equality~\eqref{eqq6} we have
$s(kG\otimes V)\subseteq V\otimes kG$, and a similar computation with $s$ replaced by $s^{-1}$ shows that $s(kG\otimes V) \supseteq V\otimes kG$.
Let $\alpha:=\beta_{1,1}^{1,1}$ and $\beta:=\beta_{1,0}^{1,1}$. Evaluating
$$
(s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)\hs \circ \hs (c\otimes V) \qquad\text{and}\qquad (V\otimes c)\hs \circ \hs(s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)
$$
in $x\otimes x\otimes v$ for all $v\in V$, and using~\eqref{ecua5} and that these maps coincide, we see that
\begin{equation*}
q \beta\hs \circ \hs \alpha = \alpha\hs \circ \hs \beta\quad\text{and}\quad q\alpha\hs \circ \hs \beta = \beta \hs \circ \hs \alpha.
\end{equation*}
Then $q^2\alpha\circ \beta = \alpha\circ \beta$, and so $\beta = 0$, since $q^2\ne 0$ and $\alpha$ is injective.
\smallskip
\noindent $\bm{z=1}$.\enspace Since $\Prim(H_{\mathcal{D}})= kx$, by \cite{G-G}*{Proposition 4.4} there exists an automorphism $\alpha$ of $V$ such that $s(x\otimes v)=\alpha(v)\otimes x$ for all $v\in V$. In particular, $s$ satisfies condition~\eqref{eqq1}. Furthermore, by the compatibility of $s$ with $c$,
\begin{align*}
\sum_{h\in G} \beta_{h,0}^{g,0}\hs \circ \hs \alpha(v)\otimes x\otimes h & + q \sum_{h\in G} \beta_{h,1}^{g,0}\hs \circ \hs \alpha(v)\otimes x\otimes hx \\
&= (V\otimes c)\hs \circ \hs (s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)(g\otimes x\otimes v)\\
&= (s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)\hs \circ \hs (c \otimes V)(g\otimes x\otimes v)\\
&=\sum_{h\in G}\alpha \hs \circ \hs \beta_{h,0}^{g,0}(v)\otimes x\otimes h + \sum_{h\in G} \alpha\hs \circ \hs \beta_{h,1}^{g,0} (v)\otimes x\otimes hx
\end{align*}
for all $g\in G$. Consequently
\begin{equation}
\alpha\hs \circ \hs \beta_{h,0}^{g,0}= \beta_{h,0}^{g,0}\hs \circ \hs \alpha\quad\text{and}\quad \alpha\hs \circ \hs \beta_{h,1}^{g,0}= q\beta_{h,1}^{g,0}\hs \circ \hs \alpha\quad \text{for all $g,h\in G$.}\label{eqq7}
\end{equation}
Using now that $s$ is compatible with the multiplication map of $H_\mathcal{D}$, we obtain that
\begin{align*}
\chi(g) \sum_{h\in G} \beta_{h,0}^{g,0}\hs \circ \hs \alpha(v)\otimes hx & + \chi(g) \sum_{h\in G} \beta_{h,1}^{g,0}\hs \circ \hs \alpha(v)\otimes hx^2 \\
&= (V\otimes \mu)\hs \circ \hs (s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)(\chi(g)g\otimes x\otimes v)\\
&= (s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)\hs \circ \hs (\mu \otimes V)(\chi(g)g\otimes x\otimes v)\\
&=(s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)\hs \circ \hs (\mu \otimes V)(x\otimes g\otimes v)\\
&= (V\otimes \mu)\hs \circ \hs (s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes s)(x\otimes g\otimes v)\\
&= \sum_{h\in G} \chi(h)\, \alpha\hs \circ \hs \beta_{h,0}^{g,0}(v)\otimes hx + \sum_{h\in G} \chi(h)\, \alpha\hs \circ \hs \beta_{h,1}^{g,0}\otimes hx^2
\end{align*}
for all $g\in G$. Hence
\begin{align}
&\chi(g)\, \beta_{h,0}^{g,0}\hs \circ \hs \alpha = \chi(h)\, \alpha \hs \circ \hs \beta_{h,0}^{g,0} =\chi(h)\, \beta_{h,0}^{g,0}\hs \circ \hs \alpha\label{eqq8}
\shortintertext{and}
& \chi(g)\,\beta_{h,1}^{g,0}\hs \circ \hs \alpha = \chi(h)\, \alpha \hs \circ \hs \beta_{h,1}^{g,0} =\chi(h)q\, \beta_{h,1}^{g,0}\hs \circ \hs \alpha\label{eqq9}
\end{align}
for all $g,h\in G$. Since $\alpha$ is bijective, from~\eqref{eqq8} it follows that if $\beta_{h,0}^{g,0}\ne 0$, then $\chi(g)=\chi(h)$. Combining~\eqref{eqq8} with~\eqref{eqq4}, we see that $\beta_{h,1}^{g,0}\ne 0 \Rightarrow \beta_{h,0}^{g,0}\ne 0\Rightarrow \chi(g)=\chi(h)$. Therefore, from~\eqref{eqq9} it follows that if there exist $g,h\in G$ such that $\beta_{h,1}^{g,0}\ne 0$, then $q=1$, which is false. So, $\beta_{h,1}^{g,0}= 0$ for all $g,h\in G$.
\end{comment}
\end{proof}
\begin{comment} definición de $H$-space regular (por si vamos a considerar el caso (p,q)=(1,-1))
Given a braided Hopf algebra $H$ and grouplike elements $g,h$ of $H$, we let $H_{g,h}$ denote the subvector space of $H$ form by its $(g,h)$-primitive elements.
\begin{definition}\label{reg transposition} Let $H$ be a braided Hopf algebra and let $H_G:=\bigoplus_{g,h\in G} H_{g,h}$, where $G$ the set of grouplike elements of $H$. We say that a left $H$-space $(V,s)$ is {\em regular} if $s(kG\otimes V)=V\otimes kG$ and $s(H_G\otimes V)=V\otimes H_G$. In this case we also say that $s$ is {\em regular}.
\end{definition}
By Proposition~\ref{estructuras trenzadas de HsubD (primer resultado)}, each left $H_{\mathcal{D}}$-space $(V,s)$ is regular when $(q,p)\ne (-1,1)$. This is not long true if $(q,p)=(-1,1)$ and $z\ne 1$. For instance, consider the $4$-dimensional braided Hopf algebra $H$ generated by a grouplike element $z$ and a $(1,z)$-primitive element~$x$, subjects to the relations $z^2=1$, $x^2=0$, and $xz=zx$, with braid given by
$$
c(z^ix^j\otimes z^lx^m):=(-1)^{jm} z^lx^m\otimes z^ix^j,
$$
which is the braided Hopf algebra $H_{\mathcal{D}}$ associated to the data $\mathcal{D}:=(G,\chi,z,0,-1)$, where $G$ is the order cyclic group $\{1,z\}$ and $\chi$ is the trivial character. Then, given a vector space $V$, an automorphism $\alpha$ of $V$ and an endomorphism $\beta$ of $V$ such that $\beta^2=0$ and $\alpha\hs \circ \hs \beta=-\beta\hs \circ \hs \alpha$, the map $s\colon H_{\mathcal{D}}\otimes V\longrightarrow V\otimes H_{\mathcal{D}}$, defined by
$$
s(1\otimes v):=v\otimes 1,\quad s(z\otimes v):=v\otimes z \quad\text{and}\quad s(x\otimes v):= \alpha(v)\otimes x + \beta(v)\otimes z - \beta(v)\otimes 1,
$$
is a left transposition.
\end{comment}
In the rest of the paper we assume that $(p,q)\ne (1,-1)$.
\begin{proposition}\label{estructuras trenzadas de HsubD} Let $V$ be a $k$-vector space endowed with an $\Aut_{\chi,z}(G)$-gradation
$$
V = \bigoplus_{\zeta\in \Aut_{\chi,z}(G)} V_{\zeta}
$$
and an automorphism $\alpha\colon V\to V$ fulfilling
\begin{itemize}
\smallskip
\item[-] $\alpha(V_{\zeta})=V_{\zeta}$ for all $\zeta\in\Aut_{\chi,z}(G)$,
\smallskip
\item[-] $\alpha^n=\ide$ if $\lambda(z^n-1_G)\ne 0$.
\smallskip
\end{itemize}
Then the pair $(V,s)$, where $s\colon H_{\mathcal{D}}\otimes V\longrightarrow V\otimes H_{\mathcal{D}}$ is the map defined by
\begin{equation}
s(gx^i\otimes v):=\alpha^i(v)\otimes \zeta(g)x^i\qquad \text{for all $v\in V_{\zeta}$,}\label{eqrankone1}
\end{equation}
is a left $H_{\mathcal{D}}$-space. Furthermore, all the left $H_{\mathcal{D}}$-spaces with underlying $k$-vector space $V$ have this form.
\end{proposition}
\begin{proof} It is easy to check that the map $s$ defined by~\eqref{eqrankone1} is compatible with the unit, the counit, the multiplication map and the braid of $H_{\mathcal{D}}$. So, by Remark~\ref{basta verificar sobre generadores}, in order to check that $s$ is a left transposition it suffices to verify that
$$
(s\otimes H_{\mathcal{D}})\hs \circ \hs(H_{\mathcal{D}}\otimes s)\hs \circ \hs(\Delta\otimes V)(x\otimes v)= (V\otimes \Delta)\hs \circ \hs s(x\otimes v)
$$
and
$$
(s\otimes H_{\mathcal{D}})\hs \circ \hs(H_{\mathcal{D}}\otimes s)\hs \circ \hs(\Delta\otimes V)(g\otimes v)= (V\otimes \Delta)\hs \circ \hs s(g\otimes v)\quad\text{for $g\in G$,}
$$
which is clear.
\smallskip
Conversely, assume that $(V,s)$ is a left $H_{\mathcal{D}}$-space. By Proposition~\ref{estructuras trenzadas de HsubD (primer resultado)} and Theorem~\ref{tercer resultado sobre estructuras trenzadas de grupos}, we know that there exist an automorphism $\alpha$ of $V$ and a gradation
$$
V=\bigoplus_{\zeta\in \Aut(G)} V_{\zeta}
$$
of $V$, such that $s(g\otimes v)= v\otimes \zeta(g)$ and $s(x\otimes v)=\alpha(v)\otimes x$ for all $g\in G$ and $v\in V_{\zeta}$. Again by Proposition~\ref{estructuras trenzadas de HsubD (primer resultado)}, we also know that $s(z\otimes v)=v\otimes z$ for all $v\in V$. Therefore, if $V_{\zeta}\ne 0$, then $\zeta(z)=z$. Now, let $g\in G$ and $v\in V_{\zeta}\setminus \{0\}$. A direct computation shows that
\begin{align*}
\alpha(v) \otimes x\zeta(g)& =s(xg\otimes v)\\
&=s\bigl(\chi(g)gx \otimes v\bigr)\\
&=\sum_{\phi\in \Aut(G)} \alpha(v)_{\phi}\otimes \chi(g)\phi(g)x\\
&= \sum_{\phi\in \Aut(G)} \alpha(v)_{\phi}\otimes \chi(g)\chi(\phi(g))^{-1}x\phi(g).
\end{align*}
Since $g$ is arbitrary, from this it follows that $\alpha(v)_{\phi}=0$ for $\phi\ne \zeta$ and that $\chi(\zeta(g)) = \chi(g)$. So
$$
\alpha(V_{\zeta})=V_{\zeta}\qquad\text{and}\qquad \chi\hs \circ \hs \zeta=\chi.
$$
Lastly, suppose that $\lambda(z^n-1_G)\ne 0$. Then
$$
v\otimes \lambda(z^n-1_G)=s\bigl(\lambda(z^n-1_G)\otimes v\bigr)=s (x^n\otimes v)= \alpha^n(v)\otimes x^n= \alpha^n(v)\otimes \lambda(z^n-1_G),
$$
for each $v\in V$. This shows that $\alpha^n=\ide$ and finishes the proof.
\end{proof}
Our next aim is to characterize the right $H_{\mathcal{D}}$-braided comodule structures. Let $(V,s)$ be a left $H_{\mathcal{D}}$-space and let
$$
V=\bigoplus_{\zeta\in \Aut_{\chi,z}(G)} V_{\zeta}\qquad\text{and}\qquad \alpha\colon V\longrightarrow V
$$
be the decomposition and the automorphism associated with the left transposition $s$. Each map
$$
\nu\colon V\longrightarrow V\otimes H_{\mathcal{D}}
$$
determines and it is determined by a family of maps
\begin{equation}
\bigl(U^g_i\colon V\longrightarrow V\bigr)_{g\in G,\, 0\le i<n}\label{defUgm}
\end{equation}
via
\begin{equation}
\nu(v):= \sum_{\cramped{\substack{g\in G\\ 0\le i<n}}} U^g_i(v)\otimes gx^i.\label{defnu}
\end{equation}
\begin{proposition}\label{caracterizacion de Hsub D comodules} The pair $(V,s)$ is a right $H_{\mathcal{D}}$-comodule via $\nu$ iff
\begin{enumerate}
\smallskip
\item $U^g_i(V_{\zeta})\subseteq V_{\zeta}$ for all $g\in G$, $\zeta \in \Aut_{\chi,z}(G)$ and $i\in \{0,1\}$,
\smallskip
\item $(U^g_0)_{g\in G}$ is a complete family of orthogonal idempotents,
\smallskip
\item $U^g_1=U^g_0\hs \circ \hs U^g_1=U^g_1\hs \circ \hs U^{gz}_0$ for all $g\in G$,
\smallskip
\item $U^g_i=\frac{1}{(i)!_{qp}}\, U^g_1\hs \circ \hs U^{gz}_1\hs \circ \hs\cdots\hs \circ \hs U^{gz^{i-1}}_1$ for all $g\in G$ and $1\le i<n$,
\smallskip
\item $U^g_1\hs \circ \hs U^{gz}_1\hs \circ \hs\cdots\hs \circ \hs U^{gz^{n-1}}_1 =0$ for all $g\in G$,
\smallskip
\item $\alpha\hs \circ \hs U^g_0 = U^g_0\hs \circ \hs \alpha$ and $q\,\alpha\hs \circ \hs U^g_1 = U^g_1\hs \circ \hs \alpha$ for all $g\in G$.
\end{enumerate}
\end{proposition}
\begin{proof} For each $v\in V_{\zeta}$, $h\in G$ and $0\le j<n$, write
$$
U^h_j(v)=\sum_{\phi\in \Aut_{\chi,z}(G)} U^h_j(v)_{\phi}\qquad\text{with $U^h_j(v)_{\phi}\in V_{\phi}$}.
$$
Since
$$
(\nu\otimes H_{\mathcal{D}})\hs \circ \hs s(gx^i\otimes v) = \sum_{\cramped{\substack{h\in G\\ 0\le j<n}}} U^h_j\bigl(\alpha^i(v)\bigr)\otimes hx^j\otimes \zeta(g)x^i
$$
and
$$
(V\otimes c_q)\hs \circ \hs (s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes \nu)(gx^i\otimes v)= \sum_{\cramped{\substack{h\in G\\ 0\le j<n}}}\sum_{\phi\in \Aut_{\chi,z}(G)}\! q^{ij}\,\alpha^i\bigl(U^h_j(v)_{\phi}\bigr)\otimes hx^j \otimes \phi(g)x^i
$$
the map $\nu$ satisfies condition~\eqref{eq6} in Remark~\ref{re: H-braided comodule} iff
$$
\sum_{\cramped{\substack{h\in G\\ 0\le j<n}}} U^h_j\bigl(\alpha^i(v)\bigr)\otimes hx^j\otimes \zeta(g)x^i = \sum_{\cramped{\substack{h\in G\\ 0\le j<n}}}\sum_{\phi\in \Aut_{\chi,z}(G)}\! q^{ij}\,\alpha^i\bigl(U^h_j(v)_{\phi}\bigr)\otimes hx^j \otimes \phi(g)x^i,
$$
for all $\zeta \in \Aut_{\chi,z}(G)$, $v\in V_{\zeta}$, $g\in G$ and $0\le i<n$. Since $\zeta$, $v$ and $g$ are arbitrary, $\alpha(V_{\phi})=V_{\phi}$ for all $\phi\in\Aut_{\chi,z}(G)$, and $\alpha$ is bijective, this happens iff
\begin{equation}
U^h_j(V_{\zeta})\subseteq V_{\zeta}\quad\text{and}\quad q^j\,\alpha\hs \circ \hs U^h_j= U^h_j\hs \circ \hs \alpha,\label{eqrankone4}
\end{equation}
for all $h$, $j$, and $\zeta$. On the other hand, since $\epsilon(gx^i)=\delta_{0i}$, the map $\nu$ is counitary iff
\begin{equation}
\sum_{g\in G} U^g_0=\ide,
\label{eqrankone5}
\end{equation}
and since
\begin{align*}
&(V\otimes \Delta)\hs \circ \hs \nu(v)= \sum_{\cramped{\substack{g\in G\\ 0\le i<n}}} U^g_i(v) \otimes \Delta(gx^i) =\sum_{\cramped{\substack{g\in G\\ 0\le i<n}}} \sum_{j=0}^i \binom{i}{j}_{qp} U^g_i(v) \otimes gx^j \otimes gz^jx^{i-j}
\shortintertext{and}
& (\nu\otimes H_{\mathcal{D}})\hs \circ \hs \nu(v) = \sum_{\cramped{\substack{h\in G\\ 0\le l<n}}} \nu\bigl(U^h_l(v)\bigr)\otimes hx^l = \sum_{\cramped{\substack{h\in G\\ 0\le l<n}}} \sum_{\cramped{\substack{g\in G\\ 0\le j<n}}} U^g_j\bigl(U^h_l(v)\bigr)\otimes gx^j \otimes hx^l,
\end{align*}
it is coassociative iff
\begin{equation}\label{eqrankone6}
U^g_j\hs \circ \hs U^h_l=\begin{cases}
\binom{j+l}{j}_{qp} U^g_{j+l} &\text{if $h=gz^j$ and $j+l<n$,}\\
0 &\text{otherwise.}
\end{cases}
\end{equation}
Thus, in order to prove this proposition we must show items~(1)--(6) are equivalent to conditions~\eqref{eqrankone4}, \eqref{eqrankone5} and~\eqref{eqrankone6}. It is evident that~\eqref{eqrankone4} implies items~(1) and~(6), while items~(2) and~(3) follow from~\eqref{eqrankone5} and~\eqref{eqrankone6}. Finally, using~\eqref{eqrankone6} again it is easy to prove by induction on $j$ that items~(4) and~(5) are also satisfied. Conversely, assume that the maps $U^g_i$ satisfy items~(1)--(6). It is clear that item~(2) implies condition~\eqref{eqrankone5}, and equality~\eqref{eqrankone4} follows from items~(1), (4) and~(6). It remains to prove equality~\eqref{eqrankone6}. We claim that
\begin{equation}
U^f_i\hs \circ \hs U^{fz^i}_j = \begin{cases}
\binom{i+j}{i}_{qp} U^f_{i+j} &\text{if $i+j<n$,}\\
0 &\text{if $i+j\ge n$.}
\end{cases}\label{eqrankone7}
\end{equation}
By item~(2) this is true if $j=i=0$. In order to check it when $j>0$ and $i=0$ or $j=0$ and $i>0$, it suffices to note that by items~(3) and~(4),
$$
U^f_0\hs \circ \hs U^f_i= \frac{1}{(i)!_{qp}}\, U^f_0 \hs \circ \hs U^f_1\hs \circ \hs\cdots\hs \circ \hs U^{fz^{i-1}}_1= \frac{1}{(i)!_{qp}}\, U^f_1\hs \circ \hs\cdots\hs \circ \hs U^{fz^{i-1}}_1= U^f_i
$$
and
$$
U^f_i\hs \circ \hs U^{fz^i}_0= \frac{1}{(i)!_{qp}}\,U^f_1\hs \circ \hs\cdots\hs \circ \hs U^{fz^{i-1}}_1\hs \circ \hs U^{fz^i}_0=\frac{1}{(i)!_{qp}}\,U^f_1\hs \circ \hs\cdots\hs \circ \hs U^{fz^{i-1}}_1= U^f_i,
$$
respectively. Assume now that $j>0$ and $i>0$. Then, by item~(4),
$$
U^f_i\hs \circ \hs U^{fz^i}_j =\frac{1}{(i)!_{qp}}\frac{1}{(j)!_{qp}}\, U^f_1\hs \circ \hs\cdots\hs \circ \hs U^{fz^{i-1}}_1\hs \circ \hs U^{fz^i}_1\hs \circ \hs\cdots\hs \circ \hs U^{fz^{i+j-1}}_1,
$$
and the claim follows immediately from items~(4) and~(5). Note now that~\eqref{eqrankone7} implies that
$$
U^f_i\hs \circ \hs U^h_j=U^f_i\hs \circ \hs U^{fz^i}_0\hs \circ \hs U^h_0\hs \circ \hs U^h_j
$$
which, combined with item~(2), shows that
$$
U^f_i\hs \circ \hs U^h_j=0 \quad\text{if $h\ne fz^i$,}
$$
finishing the proof of~\eqref{eqrankone6}.
\end{proof}
\begin{corollary}\label{segunda caracterizacion de HsubD comodules} Let $V$ be a $k$-vector space. Each data consisting of
\begin{itemize}
\smallskip
\item[-] a $G\times \Aut_{\chi,z}(G)$-gradation $\displaystyle{V=\bigoplus_{(g,\zeta)\in G\times \Aut_{\chi,z}(G)} V_{g,\zeta}}$ of $V$,
\smallskip
\item[-] an automorphism $\alpha\colon V\to V$ of $V$ such that
$$
\qquad \alpha^n=\ide\text{ if }\lambda(z^n-1_G)\ne 0\quad\text{and}\quad \alpha(V_{g,\zeta})=V_{g,\zeta}\text{ for all $(g,\zeta)\in G\times \Aut_{\chi,z}(G)$,}
$$
\smallskip
\item[-] a map $U\colon V \to V$, such that
$$
\qquad U\hs \circ \hs \alpha = q\,\alpha\hs \circ \hs U,\quad U^n = 0\quad\text{and}\quad U(V_{g,\zeta})\subseteq V_{gz^{-1},\zeta} \text{ for all $(g,\zeta)\in G\times \Aut_{\chi,z}(G)$},
$$
\smallskip
\end{itemize}
determines univocally a right $H_{\mathcal{D}}$-comodule $(V,s)$, in which
\begin{itemize}
\smallskip
\item[-] $s\colon H_{\mathcal{D}}\otimes V \longrightarrow V\otimes H_{\mathcal{D}}$ is the left transposition of $H_{\mathcal{D}}$ on $V$ associated as in~\eqref{eqrankone1} with the map $\alpha$ and the $\Aut_{\chi,z}(G)$-gradation of $V$
$$
V=\bigoplus_{\zeta\in \Aut_{\chi,z}(G)}V_{\zeta},\qquad\text{where } V_{\zeta}:=\bigoplus_{g\in G} V_{g,\zeta},
$$
\smallskip
\item[-] the coaction $\nu\colon V\to V\otimes H_{\mathcal{D}}$ of $(V,s)$ is defined by
$$
\nu(v):=\sum_{j=0}^{n-1} \frac{1}{(j)!_{qp}} U^j(v)\otimes z^{-j}gx^j\qquad\text{for all $v\in V_g$,}
$$
where, for all $g\in G$,
$$
V_g:=\bigoplus_{\zeta\in \Aut_{\chi,z}(G)} V_{g,\zeta}.
$$
\smallskip
\end{itemize}
Furthermore, all the right $H_{\mathcal{D}}$-braided comodules with underlying $k$-vector space $V$ have this form.
\end{corollary}
\begin{proof} Assume we have a data as in the statement. Then we define a family of maps as in~\eqref{defUgm}, by
\begin{itemize}
\smallskip
\item[-] $U^g_0(v) := \pi_g(v)$, where $\pi_g\colon V\to V_g$ is the projection onto $V_g$ along $\bigoplus_{h\in G\setminus\{g\}} V_h$,
\smallskip
\item[-] $U^g_1 := U^g_0\hs \circ \hs U\hs \circ \hs U^{gz}_0$,
\smallskip
\item[-] $U^g_j=\frac{1}{(j)!_{qp}}\, U^g_1\hs \circ \hs U^{gz}_1\hs \circ \hs\cdots\hs \circ \hs U^{gz^{j-1}}_1$ for all $1<j<n$.
\smallskip
\end{itemize}
We must check that these maps satisfy the conditions required in Proposition~\ref{caracterizacion de Hsub D comodules}. Item~(1) is fulfilled since $U(V_{\zeta})\subseteq V_{\zeta}$ and $U_0^g(V_{\zeta})\subseteq V_{\zeta}$ for all $g\in G$, items~(2)--(4) hold by the very definition of the maps $U_i^g$, and item~(6) is fulfilled since $\alpha(V_g)=V_g$ for all $g\in G$ and $U\hs \circ \hs \alpha = q\,\alpha\hs \circ \hs U$. We next prove item~(5). Since $U^n = 0$, this trivially follows if we prove that, for all $j\ge 1$,
$$
U^g_1\hs \circ \hs U^{gz}_1\hs \circ \hs\cdots\hs \circ \hs U^{gz^{j-1}}_1(v) = \begin{cases} U^j(v) &\text{if $v\in V_{gz^j}$,}\\ 0& \text{if $v\in V_h$ with $h\ne gz^j$.}\end{cases}
$$
Clearly if $v\in V_h$ with $h\ne gz^j$, then $U_0^{gz^j}(v) = 0$, and so
$$
U^g_1\hs \circ \hs U^{gz}_1\hs \circ \hs\cdots\hs \circ \hs U^{gz^{j-1}}_1(v) = U^g_1\hs \circ \hs\cdots\hs \circ \hs U^{gz^{j-1}}_1\hs \circ \hs U_0^{gz^j}(v) = 0.
$$
It remains to consider the case $v\in V_{gz^j,\zeta}$. We proceed by induction on $j$. If $j=1$, then
$$
U_1^g(v) = U_0^g\hs \circ \hs U\hs \circ \hs U_0^{gz}(v) = U_0^g\hs \circ \hs U(v) = U(v),
$$
because $U(v)\in V_g$. Assume now $j>1$ and the result is valid for $j-1$. Then
$$
U^g_1\hs \circ \hs U^{gz}_1\hs \circ \hs\cdots\hs \circ \hs U^{gz^{j-1}}_1(v) = U^g_1\hs \circ \hs\cdots\hs \circ \hs U^{gz^{j-2}}_1 \hs \circ \hs U^{gz^{j-1}}_1(v) = U^g_1\hs \circ \hs\cdots\hs \circ \hs U^{gz^{j-2}}_1 \hs \circ \hs U(v) = U^j(v),
$$
where the last equality follows from the inductive hypothesis and the fact that $U(v)\in V_{gz^{j-1}}$.
\smallskip
Conversely assume that $(V,s)$ is a right $H_{\mathcal{D}}$-comodule via a coaction $\nu\colon V\to V\otimes H_{\mathcal{D}}$. Let
$$
V=\bigoplus_{\zeta\in \Aut_{\chi,z}(G)} V_{\zeta}\qquad\text{and}\qquad \alpha\colon V\longrightarrow V
$$
be the decomposition and the automorphism associated with $s$ (see Proposition~\ref{estructuras trenzadas de HsubD}). By items~(1) and~(2) of Proposition~\ref{caracterizacion de Hsub D comodules}, we know that, for each $\zeta\in \Aut_{\chi,z}(G)$, the maps $U^g_0$'s determine by restriction a complete family $(U^g_0\colon V_{\zeta}\to V_{\zeta})_{g\in G}$ of orthogonal idempotents. Let
$$
V_{\zeta} = \bigoplus_{g\in G} V_{g,\zeta}
$$
be the decomposition associated with this family. Clearly
$$
V = \bigoplus_{(g,\zeta)\in G\times \Aut_{\chi,z}(G)} V_{g,\zeta}.
$$
By item~(6) of Proposition~\ref{caracterizacion de Hsub D comodules}, we have $\alpha\hs \circ \hs U^g_0 = U^g_0\hs \circ \hs \alpha$ for all $g\in G$. Since, by Proposition~\ref{estructuras trenzadas de HsubD} we know that $\alpha(V_{\zeta}) = V_{\zeta}$ for all $\zeta\in \Aut(G)$, this implies that
$$
\alpha(V_{g,\zeta})=V_{g,\zeta}\qquad \text{ for all $(g,\zeta)\in G\times \Aut_{\chi,z}(G)$}.
$$
We now define a map $U\colon V\to V$ by
$$
U(v) = U^g_1(v)\qquad\text{for all $v\in V_{gz,\zeta}$.}
$$
From items~(3) and (5) of Proposition~\ref{caracterizacion de Hsub D comodules}, it follows that $U^n = 0$, and using the second equality in item~(6) of the same proposition, we obtain that $\alpha\hs \circ \hs U =q\, U\hs \circ \hs\alpha$. Finally, by items~(1) and~(3) of Proposition~\ref{caracterizacion de Hsub D comodules}, we have $U(V_{g,\zeta})\subseteq V_{z^{-1}g,\zeta}$ for all $(g,\zeta)\in G\times \Aut_{\chi,z}(G)$.
\smallskip
We leave the reader the task to prove that the construction given in the two parts of this proof are reciprocal one of each other.
\end{proof}
\begin{remark} Assume that $q=1$, or equivalently, that $H_{\mathcal{D}}$ is a Krop-Radford Hopf algebra. In this case $(V,s)$ is a standard $H_{\mathcal{D}}$-comodule (that is, $s$ is the flip) iff $V_{g,\zeta}= 0$ for $\zeta \ne \ide$ and $\alpha$ is the identity map. Hence, in order to obtain the standard $H_{\mathcal{D}}$-comodule structures, the conditions that we need verify (given in Corollary~\ref{segunda caracterizacion de HsubD comodules}) are considerably simplified.
\end{remark}
\begin{corollary}\label{coinvarianes de un HsubD comodulo} With the notations of the previous corollary, $V^{\coH}=V_{1_G}\cap \ker(U)$.
\end{corollary}
\begin{proof} This is an immediate consequence of Corollary~\ref{segunda caracterizacion de HsubD comodules}.
\end{proof}
\begin{proposition}\label{estructura de transposiciones de HsubD} Let $B$ be an algebra. If
\begin{equation}
B = \bigoplus_{\zeta\in \Aut_{\chi,z} (G)^{\op}} B_{\zeta}\label{eqrankone8}
\end{equation}
is an $\Aut_{\chi,z}(G)^{\op}$-gradation of $B$ as an algebra and $\alpha\colon B\to B$ an automorphism of algebras that satisfies
\begin{itemize}
\smallskip
\item[-] $\alpha(B_{\zeta}) = B_{\zeta}$ for all $\zeta\in\Aut_{\chi,z}(G)$,
\smallskip
\item[-] $\alpha^n=\ide$ if $\lambda(z^n-1_G)\ne 0$,
\smallskip
\end{itemize}
then, the map $s\colon H_{\mathcal{D}}\otimes B\longrightarrow B\otimes H_{\mathcal{D}}$, given by
\begin{equation}
s(gx^i\otimes b)=\alpha^i(b)\otimes \zeta(g)x^i\quad\text{for all $b\in B_{\zeta}$,}\label{eq2}
\end{equation}
is a left transposition of $H_{\mathcal{D}}$ on the algebra $B$. Furthermore, all the left transpositions of $H_{\mathcal{D}}$ on $B$ have this form.
\end{proposition}
\begin{proof} By Proposition~\ref{estructuras trenzadas de HsubD} in order to prove this it suffices to check that the formula~\eqref{eq2} defines a map compatible with the unit and the multiplication map of $B$ iff~\eqref{eqrankone8} is a gradation of $B$ as an algebra and $\alpha$ is an automorphism of algebras. We left this task to the reader.
\end{proof}
The group $\Aut_{\chi,z}(G)$ acts on $G^{\op}$ via $\zeta\cdot g:= \zeta(g)$. So, it makes sense to consider the semidirect product $G(\chi,z):= G^{\op}\rtimes \Aut_{\chi,z}(G)$.
\begin{definition}\label{graduacion compatible con D} Let $\mathcal{D}=(G,\chi,z,\lambda,q)$ be as in Corollary~\ref{algebras de Hopf trenzadas de K-R} and let $B$ be an algebra endowed with an algebra automorphism $\alpha\colon B\to B$, a map $U\colon B\to B$ and a $G(\chi,z)^{\op}$-gradation
\begin{equation}
B=\bigoplus_{(g,\zeta)\in G(\chi,z)^{\op}} B_{g,\zeta},\label{eeqq1}
\end{equation}
of $B$ as a vector space. We say that the decomposition~\eqref{eeqq1} of $B$ is {\em compatible with $\mathcal{D}$} if one of the following conditions is fulfilled:
\begin{enumerate}
\smallskip
\item $\lambda(z^n-1_G)=0$ and $\eqref{eeqq1}$ is a gradation of $B$ as an algebra.
\smallskip
\item $\lambda(z^n-1_G)\ne 0$, $1_B\in B_{1_G,\ide}$,
\begin{equation}\label{compa'}
\qquad\qquad\quad B_{g,\zeta}B_{h,\phi} \subseteq B_{\phi(g)h,\phi\hs \circ \hs \zeta}\oplus B_{z^{-n}\phi(g)h,\phi\hs \circ \hs \zeta}\quad\text{for all $(g,\zeta),(h,\phi) \in G(\chi,z)^{\op}$,}
\end{equation}
and, for each $b\in B_{g,\zeta}$ and $c\in B_{h,\phi}$, the homogeneous component $(bc)_{z^{-n}\phi(g)h,\phi\hs \circ \hs \zeta}$ of $bc$ of degree $(z^{-n}\phi(g)h,\phi\hs \circ \hs \zeta)$ is given by
\begin{equation}\label{compa}
\qquad (bc)_{z^{-n}\phi(g)h,\phi\hs \circ \hs \zeta}:= -\lambda \sum_{j=1}^{n-1}\frac{p^{j^2}\chi(h)^j}{(j)!_{qp} (n-j)!_{qp}} U^j(b)\alpha^j \bigl(U^{n-j}(c)\bigr).
\end{equation}
\end{enumerate}
\end{definition}
\begin{theorem}\label{caracterizacion de HsubD comodule algebras} Let $B$ be an algebra. Each data consisting of
\begin{itemize}
\smallskip
\item[-] a $G(\chi,z)^{\op}$-gradation
\begin{equation}
\qquad B=\bigoplus_{(g,\zeta)\in G(\chi,z)^{\op}} B_{g,\zeta},\label{eeeq1}
\end{equation}
of $B$ as a vector space,
\smallskip
\item[-] an algebra automorphism $\alpha\colon B\to B$ of $B$ such that
$$
\qquad \alpha^n=\ide\text{ if }\lambda(z^n-1_G)\ne 0\quad\text{and}\quad \alpha(B_{g,\zeta})=B_{g,\zeta}\text{ for all $(g,\zeta)\in G(\chi,z)^{\op}$,}
$$
\smallskip
\item[-] a map $U\colon B \to B$ such that
\begin{align}
&\quad\text{the decomposition~\eqref{eeeq1} is compatible with $\mathcal{D}$,}\notag\\
&\quad U\hs \circ \hs \alpha = q\,\alpha\hs \circ \hs U,\notag\\
&\quad U^n = 0,\notag\\
&\quad U(B_{g,\zeta})\subseteq B_{z^{-1}g,\zeta} \quad\text{for all $(g,\zeta)\in G(\chi,z)^{\op}$}\notag
\shortintertext{and}
&\quad U(bc) = bU(c) + \chi(h)U(b)\alpha(c)\quad\text{for all $b\in B$ and $c\in B_h$,}\label{eqrankone8*}
\end{align}
where
$$
\qquad B_h:=\bigoplus_{\zeta\in \Aut_{\chi,z}(G)} B_{h,\zeta}\qquad\text{for all $h\in G$,}
$$
\smallskip
\end{itemize}
determines a right $H_{\mathcal{D}}$-comodule algebra $(B,s)$, in which $s\colon H_{\mathcal{D}}\otimes B \longrightarrow B\otimes H_{\mathcal{D}}$ is the left transposition of $H_{\mathcal{D}}$ on $B$ associated with the map $\alpha$ and the $\Aut_{\chi,z}(G)^{\op}$-gradation of $B$
\begin{equation}
B=\bigoplus_{\zeta\in \Aut_{\chi,z}(G)^{\op}}B_{\zeta},\label{eqqqq}
\end{equation}
where $B_{\zeta}:=\bigoplus_{g\in G} B_{g,\zeta}$. The coaction $\nu\colon B\longrightarrow B\otimes H_{\mathcal{D}}$ of $(B,s)$ is given by
\begin{equation}\label{eeeq2}
\nu(b):=\sum_{i=0}^{n-1} \frac{1}{(i)!_{qp}} U^i(b)\otimes z^{-i} gx^i\qquad\text{for all $b\in B_g$.}
\end{equation}
Furthermore, all the right $H_{\mathcal{D}}$-braided comodule algebra structures with underlying algebra $B$ are obtained in this way.
\end{theorem}
\begin{proof} Let $(B,s)$ be a right $H_{\mathcal{D}}$-comodule, with $s$ a left transposition of $H_{\mathcal{D}}$ on the algebra $B$. Consider the subspaces $B_{g,\zeta}$ of $B$ and the maps $\alpha$ and $U$ associated with $(B,s)$ as in Corollary~\ref{segunda caracterizacion de HsubD comodules}. By that corollary, Proposition~\ref{estructura de transposiciones de HsubD}, and Remarks~\ref{re: H-braided comodule} and~\ref{re: H-braided comod alg}, in order to finish the proof it suffices to show that the coaction $\nu$ of $(B,s)$ satisfy
\begin{equation}
\nu(1_B)=1_B\otimes 1_{H_{\mathcal{D}}}\quad\text{and}\quad\nu\hs \circ \hs \mu_B = (\mu_B\otimes \mu_{H_{\mathcal{D}}})\hs \circ \hs (B\otimes s\otimes H_{\mathcal{D}})\hs \circ \hs (\nu \otimes \nu) \label{eqa}
\end{equation}
iff the decomposition
\begin{equation}
B=\bigoplus_{(g,\zeta)\in G(\chi,z)^{\op}} B_{g,\zeta},\label{eeqq4}
\end{equation}
of $B$, is compatible with $\mathcal{D}$ and $U$ satisfies condition~\eqref{eqrankone8*}. First we make some remarks. Let $b\in B_{g,\zeta}$ and $c\in B_{h,\phi}$. By the definition of $\nu$,
\begin{equation}
\nu(bc) = \sum_{i=0}^{n-1} \sum_{f\in G} \frac{1}{(i)!_{qp}} U^i\bigl((bc)_f\bigr)\otimes z^{-i}fx^i.\label{eeqq5}
\end{equation}
On the other hand, a direct computation shows that
\begin{equation}
\begin{aligned}
F(b,c) &=\sum_{j=0}^{n-1} \sum_{i=0}^{n-1} \frac{p^{-ij}\chi(h)^j}{(j)!_{qp}(i)!_{qp}} U^j(b)\alpha^j\bigl(U^i(c)\bigr)\otimes z^{-i-j}\phi(g) h x^{i+j}\\
&=\sum_{u=0}^{2n-2}\sum_{\substack{j=0\\0\le u-j<n}}^{n-1} \frac{p^{-(u-j)j}\chi(h)^j}{(j)!_{qp}(u-j)!_{qp}} U^j(b)\alpha^j\bigl(U^{u-j}(c)\bigr)\otimes z^{-u}\phi(g) h x^u,
\end{aligned}\label{eeqq6'}
\end{equation}
where to abbreviate expressions we write
$$
F(b,c):=(\mu_B\otimes\mu_{H_{\mathcal{D}}})\hs \circ \hs (B\otimes s \otimes H_{\mathcal{D}})(\nu(b)\otimes\nu(c)).
$$
Set
$$
A_u^j(b,c):= \frac{p^{(j-u)j}\chi(h)^j}{(j)!_{qp}(u-j)!_{qp}} U^j(b)\alpha^j\bigl(U^{u-j}(c)\bigr).
$$
Since $x^n=\lambda(z^n-1_G)$, equality~\eqref{eeqq6'} becomes
\begin{equation}
\begin{aligned}
F(b,c)&=\sum_{i=0}^{n-1} \sum_{j=0}^i A_i^j(b,c) \otimes z^{-i}\phi(g) hx^i +\lambda \sum_{i=0}^{n-1} \sum_{j=i+1}^{n-i} A_{i+n}^j(b,c) \otimes z^{-n-i}\phi(g) h (z^n-1_G)x^i\\
&=\sum_{i=0}^{n-1}\biggl(\sum_{j=0}^i A_i^j(b,c)+\lambda \sum_{j=i+1}^{n-1} A_{i+n}^j(b,c)\biggr) \otimes z^{-i}\phi(g) hx^i\\
& -\sum_{i=0}^{n-1} \sum_{j=i+1}^{n-1} \lambda A_{i+n}^j(b,c) \otimes z^{-n-i}\phi(g) h x^i.
\end{aligned}\label{eeqq6}
\end{equation}
Next we prove the part $\Rightarrow$). We assume that $\lambda(z^n-1_G)\ne0$ and leave the case $\lambda(z^n-1_G)=0$, which is easier, to the reader. To begin with note that by the first equality in~\eqref{eqa},
$$
1_B\in B_{1_G}=\bigoplus_{\zeta\in \Aut_{\chi,z}(G)} B_{1_G,\zeta}.
$$
Since, on the other hand,~\eqref{eqqqq} is a gradation of $B$ as an algebra, necessarily
$$
1_B\in B_{\ide}=\bigoplus_{g\in G} B_{g,\ide},
$$
and so $1_B\in B_{1_G,\ide}$. Recall that $b\in B_{g,\zeta}$ and $c\in B_{h,\phi}$. Since by the second equality in~\eqref{eqa}, equations~\eqref{eeqq5} and~\eqref{eeqq6} coincide, we have
\begin{equation}\label{eqqqq1}
(bc)_f = \begin{cases} A_0^0(b,c)+\lambda \sum_{j=1}^{n-1} A_n^j(b,c)&\text{if $f = \phi(g)h$,}\\- \lambda \sum_{j=1}^{n-1} A_n^j(b,c)&\text{if $f = z^{-n}\phi(g)h$,}\\0&\text{otherwise,}\end{cases}
\end{equation}
and
\begin{equation}\label{eqqqq2}
U\bigl((bc)_f\bigr) = \begin{cases}\sum_{j=0}^1A_1^j(b,c)+\lambda \sum_{j=2}^{n-1} A_{1+n}^j(b,c)&\text{if $f = \phi(g)h$,}\\ - \lambda\sum_{j=2}^{n-1} A_{1+n}^j(b,c)&\text{if $f = z^{-n}\phi(g)h$,}\\ 0&\text{otherwise.}\end{cases}
\end{equation}
Since, by Proposition~\ref{estructura de transposiciones de HsubD}, we know that $bc \in B_{\phi\circ \zeta}$, from equality~\eqref{eqqqq1} it follows easily that the decomposition~\eqref{eeeq1} is compatible with $\mathcal{D}$ (recall that if $\lambda(z^n-1_G)\ne 0$, then $p^n=1$). Finally, by~\eqref{eqqqq2}
$$
U(bc) = \sum_f U\bigl((bc)_f\bigr) = \sum_{j=0}^1 A_1^j(b,c)+\lambda \sum_{j=2}^{n-1} A_{1+n}^j(b,c) -\lambda \sum_{j=2}^{n-1} A_{1+n}^j(b,c) = A_1^0(b,c) + A_1^1(b,c),
$$
and so, \eqref{eqrankone8*} is true.
\smallskip
We now prove the part $\Leftarrow$). So we assume that the decomposition~\eqref{eeqq4} is compatible with $\mathcal{D}$ and that $U$ satisfies~\eqref{eqrankone8*}. Again we consider the case $\lambda(z^n-1_G)\ne0$ and leave the case $\lambda(z^n-1_G)=0$, which is easier, to the reader. To begin with note that $\nu(1_B) = 1_B\otimes 1_{H_{\mathcal{D}}}$, because
$$
1_B\in B_{1_G}\qquad\text{and}\qquad U(1_B)=1_BU(1_B)+U(1_B)1_B \Rightarrow U(1_B)=0.
$$
So, we are reduce to prove that the second condition in~\eqref{eqa} is fulfilled. By equalities~\eqref{eeqq5} and~\eqref{eeqq6}, this is equivalent to prove that for all $0\le i < n$ and $f\in G$,
\begin{equation}\label{eqqqq3}
\frac{1}{(i)!_{qp}} U^i\bigl((bc)_f\bigr) = \begin{cases}\sum_{j=0}^i A_i^j(b,c)+\lambda \sum_{j=i+1}^{n-1} A_{i+n}^j(b,c)&\text{if $f = \phi(g)h$,}\\ - \lambda\sum_{j=i+1}^{n-1} A_{i+n}^j(b,c)&\text{if $f = z^{-n}\phi(g)h$,}\\ 0&\text{otherwise.}\end{cases}
\end{equation}
For $i=0$ this follows easily from the fact that equality~\eqref{compa} holds and
$$
A_0^0(b,c) = bc = (bc)_{\phi(g)h,\phi\hs \circ \hs \zeta} + (bc)_{z^{-n}\phi(g)h,\phi\hs \circ \hs \zeta},
$$
since the decomposition~\eqref{eeqq4} is compatible with $\mathcal{D}$. Assume by inductive hypothesis that equality~\eqref{compa} is true for $i$ and that $i<n-1$. This implies that
\begin{equation*}
\frac{1}{(i)!_{qp}}U^{i+1}\bigl((bc)_f\bigr) = \begin{cases}\sum_{j=0}^i U\bigl(A_i^j(b,c)\bigr)+\lambda \sum_{j=i+1}^{n-1} U\bigl(A_{i+n}^j(b,c) \bigr)&\text{if $f = \phi(g)h$,}\\ - \lambda\sum_{j=i+1}^{n-1} U\bigl(A_{i+n}^j(b,c)\bigr)&\text{if $f = z^{-n}\phi(g)h$,}\\ 0&\text{otherwise.}\end{cases}
\end{equation*}
So, we must prove that
\begin{equation}\label{pepe1}
\frac{1}{(i+1)_{qp}} \sum_{j=i+1}^{n-1} U\bigl(A_{i+n}^j(b,c)\bigr) = \sum_{j=i+2}^{n-1} A_{i+1+n}^j(b,c)
\end{equation}
and
\begin{equation}\label{pepe2}
\frac{1}{(i+1)_{qp}} \sum_{j=0}^i U\bigl(A_i^j(b,c)\bigr) = \sum_{j=0}^{i+1} A_{i+1}^j(b,c).
\end{equation}
Recall again that $b\in B_{g,\zeta}$ and $c\in B_{h,\phi}$. Using the equality~\eqref{eqrankone8*} and the facts that $U\hs \circ \hs \alpha = q \alpha\hs \circ \hs U$, $U^u(c) \in B_{z^{-u}h,\phi}$ for all $u\in \mathds{N}$, and $p^n = 1$, we obtain
\begin{align*}
U\bigl(A_i^j(b,c)\bigr) &= \frac{p^{(j-i)j}\chi(h)^j}{(j)!_{qp}(i-j)!_{qp}} U\bigl(U^j(b)\alpha^j\bigl(U^{i-j}(c)\bigr)\bigr)\\
& = \frac{p^{(j-i)j}\chi(h)^j}{(j)!_{qp}(i-j)!_{qp}} \Bigl(q^j U^j(b)\alpha^j\bigl(U^{i+1-j}(c)\bigr) + p^{j-i} \chi(h) U^{j+1}(b)\alpha^{j+1}\bigl(U^{i-j}(c)\bigr)\Bigr)
\end{align*}
and
\begin{align*}
U\bigl(A_{i+n}^j&(b,c)\bigr) = \frac{p^{(j-i)j}\chi(h)^j}{(j)!_{qp}(i+n-j)!_{qp}} U\bigl(U^j(b)\alpha^j\bigl(U^{i+n-j}(c)\bigr)\bigr)\\
& = \frac{p^{(j-i)j}\chi(h)^j}{(j)!_{qp}(i+n-j)!_{qp}} \Bigl(q^j U^j(b)\alpha^j\bigl(U^{j+1+n-j}(c)\bigr) +
p^{j-i} \chi(h) U^{j+1}(b)\alpha^{j+1}\bigl(U^{i+n-j}(c)\bigr)\Bigr).
\end{align*}
Since $U^n = 0$, this implies that
\begin{align*}
\sum_{j=0}^i U\bigl(A_i^j(b,c)\bigr) & = \sum_{j=0}^i \frac{p^{(j-i)j}\chi(h)^jq^j} {(j)!_{qp}(i-j)!_{qp}} U^j(b)\alpha^j\bigl(U^{i+1-j}(c)\bigr)\\
& + \sum_{j=0}^i\frac{p^{(j-i)(j+1)}\chi(h)^{j+1}}{(j)!_{qp}(i-j)!_{qp}} U^{j+1}(b)\alpha^{j+1}\bigl(U^{i-j}(c)\bigr)\\
& = \sum_{j=0}^i \frac{p^{(j-i)j}\chi(h)^jq^j} {(j)!_{qp}(i-j)!_{qp}} U^j(b)\alpha^j\bigl(U^{i+1-j}(c)\bigr)\\
& + \sum_{j=1}^{i+1}\frac{p^{(j-i-1)j}\chi(h)^j}{(j-1)!_{qp}(i+1-j)!_{qp}} U^j(b)\alpha^j\bigl(U^{i+1-j}(c)\bigr)
\end{align*}
and
\begin{align*}
\sum_{j=i+1}^{n-1} U\bigl(A_{i+n}^j(b,c)\bigr) & = \sum_{j=i+1}^{n-1} \frac{p^{(j-i)j}\chi(h)^jq^j} {(j)!_{qp}(i+n-j)!_{qp}} U^j(b)\alpha^j\bigl(U^{i+1+n-j}(c)\bigr)\\
& + \sum_{j=i+1}^{n-1}\frac{p^{(j-i)(j+1)}\chi(h)^{j+1}}{(j)!_{qp}(i+n-j)!_{qp}} U^{j+1}(b)\alpha^{j+1}\bigl(U^{i+n-j}(c)\bigr)\\
& = \sum_{j=i+2}^{n-1} \frac{p^{(j-i)j}\chi(h)^jq^j} {(j)!_{qp}(i+n-j)!_{qp}} U^j(b)\alpha^j\bigl(U^{i+1+n-j}(c)\bigr)\\
& + \sum_{j=i+2}^{n-1}\frac{p^{(j-i-1)j}\chi(h)^j}{(j-1)!_{qp}(i+1+n-j)!_{qp}} U^j(b)\alpha^j\bigl(U^{i+1+n-j}(c)\bigr).
\end{align*}
Consequently in order to finish the proof of equalities~\eqref{pepe1} and~\eqref{pepe2} it suffices to see that
\begin{align*}
& \frac{(i+1)_{qp}} {(i+1)!_{qp}} = \frac{1} {(i)!_{qp}},\\
&\frac{(i+1)_{qp} \chi(h)^{i+1}} {(i+1)!_{qp}} = \frac{ \chi(h)^{i+1}}{(i)!_{qp}},\\
&\frac{(i+1)_{qp} p^{(j-i-1)j}\chi(h)^j} {(j)!_{qp}(i+1-j)!_{qp}} = \frac{p^{(j-i)j}\chi(h)^jq^j} {(j)!_{qp}(i-j)!_{qp}} + \frac{p^{(j-i-1)j} \chi(h)^j}{(j-1)!_{qp}(i+1-j)!_{qp}} &&\text{for $1\le j\le i$}
\intertext{and}
& \frac{(i+1)_{qp} p^{(j-i-1)j}\chi(h)^j} {(j)!_{qp}(i+1+n-j)!_{qp}} = \frac{p^{(j-i)j}\chi(h)^jq^j} {(j)!_{qp}(i+n-j)!_{qp}} + \frac{p^{(j-i-1)j} \chi(h)^j}{(j-1)!_{qp}(i+1+n-j)!_{qp}}&&\text{for $i+2\le j< n$.}
\end{align*}
But the first two equalities are trivial and the last ones are equivalent to
\begin{align*}
&(i+1)_{qp} = p^j q^j(i+1-j)_{qp} + (j)_{qp} &&\text{for $1\le j\le i$}
\intertext{and}
&(i+1)_{qp} = p^j q^j(i+1+n-j)_{qp} + (j)_{qp} &&\text{for $i+2\le j< n$,}
\end{align*}
which can be easily checked.
\end{proof}
\begin{remark} Assume that $q=1$, or equivalently, that $H_{\mathcal{D}}$ is a Krop-Radford Hopf algebra. In this case $(B,s)$ is a standard $H_{\mathcal{D}}$-comodule algebra (that is, $s$ is the flip) iff $B_{g,\zeta}= 0$ for $\zeta \ne \ide$ and $\alpha$ is the identity map. Hence, in order to obtain the standard $H_{\mathcal{D}}$-comodule algebra structures, the conditions that we need verify (given in Theorem~\ref{caracterizacion de HsubD comodule algebras}) are considerably simplified.
\end{remark}
\begin{remark}\label{estructura inducida de kG-modulo algebra} Let $(B,s)$ be a right $H_{\mathcal{D}}$-comodule algebra and let $B_{G}:=\{b\in B: \nu(b)\in B\otimes kG\}$. Note that
$$
B_G=\ker(U)= \bigoplus_{(g,\zeta)\in G(\chi,z)^{\op}} B_{g,\zeta}\cap \ker(U).
$$
Moreover,
$$
\nu(B_G)\subseteq B_G\otimes kG,
$$
because $(\nu\otimes H_{\mathcal{D}})\hs \circ \hs \nu=B\otimes \Delta)\hs \circ \hs \nu$. Consequently, since
$$
(B\otimes c_q)\hs \circ \hs (s\otimes H_{\mathcal{D}})\hs \circ \hs (H_{\mathcal{D}}\otimes \nu)= (\nu\otimes H_{\mathcal{D}})\hs \circ \hs s,
$$
we have $s(H_{\mathcal{D}}\otimes B_G)\subseteq B_G\otimes H_{\mathcal{D}}$. Similarly $s^{-1}(B_G\otimes H_{\mathcal{D}})\subseteq H_{\mathcal{D}}\otimes B_G$, and so,
$$
s(H_{\mathcal{D}}\otimes B_G) = B_G\otimes H_{\mathcal{D}}.
$$
Furthermore $B_G$ is a subalgebra of $B$ because
$$
\nu\hs \circ \hs \mu = (\mu_B\otimes \mu_{H_{\mathcal{D}}})\hs \circ \hs (B \otimes s \otimes H_{\mathcal{D}}) \hs \circ \hs (\nu \otimes \nu).
$$
Clearly $s$ induce by restriction a left transposition $\tilde{s}$ of $kG$ on $B_G$. From the previous discussion it follows that $(B_G,\tilde{s})$ is a right $kG$-comodule algebra.
\end{remark}
\section{$\bm{H_{\mathcal{D}}}$ cleft extensions} Throughout this section we use freely the notations introduced in Section~\ref{H_D comodule algebras} and the characterization of right $H_{\mathcal{D}}$-comodule algebras obtained in Theorem~\ref{caracterizacion de HsubD comodule algebras}. Let $(B,s)$ be a right $H_{\mathcal{D}}$-comodule algebra and let $C:=B^{\coH_{\mathcal{D}}}$. Recall that, by Corollary~\ref{coinvarianes de un HsubD comodulo},
$$
C=B_{1_G}\cap \ker(U)= \bigoplus_{\zeta\in \Aut_{\chi,z}(G)} B_{1_G,\zeta}\cap \ker(U).
$$
\begin{theorem}\label{caracterizacion de cleft} The extension $(C \hookrightarrow B,s)$ is cleft iff there exists $b_x\in B$ and a family $(b_g)_{g\in G}$ of elements of $B^{\times}$, such that
\begin{enumerate}[label=\emph{(\alph*)}]
\smallskip
\item $b_g\in B_{g,\ide}\cap \ker(U)$ for all $g\in G$,
\smallskip
\item $b_x\in B_{z,\ide}\cap U^{-1}(1_B)$,
\smallskip
\item $\alpha(b_x)=qb_x$,
\smallskip
\item $\alpha(b_g)=b_g$ for all $g\in G$.
\smallskip
\end{enumerate}
If this is the case, then the map $\gamma\colon H_{\mathcal{D}} \to B$, defined by $\gamma(gx^i):=b_gb_x^i$, is a cleft map, and its convolution inverse is given by
$$
\qquad \gamma^{-1}(gx^i) = (-1)^i (qp)^{\frac{i(i-1)}{2}}b_x^ib_{gz^i}^{-1}.
$$
\end{theorem}
\begin{proof} Assume that $(C \hookrightarrow B,s)$ is a cleft extension and fix a cleft map $\gamma \colon H_{\mathcal{D}} \to B$ such that $\gamma(1)=1$. For every $g\in G$ and $0\le i< n$, set $b_{gx^i}:= \gamma(gx^i)$. Since $\gamma$ is a right comodule map,
$$
\nu(b_g)=b_g\otimes g \quad\text{and}\quad \nu(b_x)=1_B\otimes x + b_x\otimes z,
$$
which, by formula~\eqref{eeeq2}, is equivalent to
$$
b_g\in B_g \cap \ker(U)\quad\text{and}\qquad b_x\in B_z\cap U^{-1}(1).
$$
Moreover $b_g$ is invertible for each $g\in G$, because $\gamma$ is convolution invertible. On the other hand evaluating the equality $(\gamma\otimes H_{\mathcal{D}})\hs \circ \hs c_q = s\hs \circ \hs (H_{\mathcal{D}}\otimes \gamma)$ in $h\otimes x$, $x\otimes x$, $h\otimes g$ and $x\otimes g$, where $h\in G$ is arbitrary, we obtain that
$$
b_x\in B_{\ide},\quad \alpha(b_x)=qb_x,\quad b_g \in B_{\ide}\quad\text{and}\quad \alpha(b_g)=b_g,
$$
for all $g\in G$. Thus, items~\mbox{(a)--(d)} hold. Conversely, assume that there exists $b_x\in B$ and a family $(b_g)_{g\in G}$ of elements of $B^{\times}$ satisfying statements~(a)--(d). We are going to prove that $(C \hookrightarrow B,s)$ is cleft and the map $\gamma\colon H_{\mathcal{D}} \to B$, defined by $\gamma(gx^i):=b_gb_x^i$, is a cleft map. First note that
$$
(\gamma\otimes H_{\mathcal{D}})\hs \circ \hs c_q (gx^i\otimes hx^j) = q^{ij} b_hb_x^j\otimes gx^i= s(gx^i\otimes b_hb_x^j)= s\hs \circ \hs (H_{\mathcal{D}}\otimes \gamma) (gx^i\otimes hx^j),
$$
for all $h,g\in G$ and $0\le i,j<n$. So we must only check that $\gamma$ is convolution invertible and
\begin{equation}\label{eeeq3}
\nu\hs \circ \hs \gamma(gx^i)= (\gamma\otimes H_{\mathcal{D}})\hs \circ \hs \Delta(gx^i) \quad\text{for all $g\in G$ and $0\le i < n$}.
\end{equation}
For $g=1$ and $i=0$ it is evident that this is true. Assume it is true for $g=1$ and $i=i_0$, and that $i_0<n-1$. Then
\begin{align*}
\nu\hs \circ \hs \gamma(x^{i_0+1}) &=\nu\bigl(\gamma(x^{i_0}) b_x\bigr) \\
&=(\mu_{B}\otimes \mu_{H_{\mathcal{D}}})\hs \circ \hs (B\otimes s \otimes H_{\mathcal{D}})\bigl(\nu(\gamma(x^{i_0})) \otimes \nu(b_x)\bigr)\\
&= \sum_{j=0}^{i_0}\binom{i_0}{j}_{qp}(\mu_{B}\otimes \mu_{H_{\mathcal{D}}})\hs \circ \hs (B\otimes s \otimes H_{\mathcal{D}})(b_x^j\otimes z^jx^{i_0-j}\otimes 1_B\otimes x)\\
&+ \sum_{j=0}^{i_0}\binom{i_0}{j}_{qp}(\mu_{B}\otimes \mu_{H_{\mathcal{D}}})\hs \circ \hs (B\otimes s \otimes H_{\mathcal{D}})(b_x^j\otimes z^jx^{i_0-j}\otimes b_x\otimes z)\\
&= \sum_{j=0}^{i_0}\binom{i_0}{j}_{qp} b_x^j\otimes z^jx^{i_0+1-j} + \sum_{j=0}^{i_0}\binom{i_0}{j}_{qp} b_x^j \alpha^{i_0-j}(b_x)\otimes z^jx^{i_0-j}z\\
&= \sum_{j=0}^{i_0}\binom{i_0}{j}_{qp} b_x^j\otimes z^jx^{i_0+1-j} + \sum_{j=1}^{i_0+1}\binom{i_0}{j-1}_{qp} q^{i_0+1-j}p^{i_0+1-j}b_x^j \otimes z^jx^{i_0+1-j}\\
&=\sum_{j=0}^{i_0}\binom{i_0+1}{j}_{qp} b_x^j\otimes z^jx^{i_0+1-j}.
\end{align*}
So, equality~\eqref{eeeq3} holds when $g=1_G$. But then
\begin{align*}
\nu\hs \circ \hs \gamma(gx^i) &= \nu\bigl(b_g\gamma(x^i)\bigr)\\
&= \sum_{j=0}^i\binom{i}{j}_{qp} (\mu_B\otimes \mu_{H_{\mathcal{D}}})\hs \circ \hs (B\otimes s\otimes H_{\mathcal{D}})(b_g\otimes g \otimes b_x^j \otimes z^jx^{i-j})\\
&= \sum_{j=0}^i\binom{i}{j}_{qp} b_gb_x^j\otimes gz^jx^{i-j}.
\end{align*}
It remains to check that $\gamma$ is convolution invertible. As was noted in \cite{D-T}*{Section 3},
\begin{equation}
\sum_{j=0}^i (-1)^j(qp)^{\frac{j(j-1)}{2}}\binom{i}{j}_{qp}=\begin{cases} 1 & \text{if $i=0$,}\\ 0 &\text{if $0<i<n$.}\end{cases} \label{formula}
\end{equation}
Using this it is easy to prove that $\gamma$ is invertible with
$$
\gamma^{-1}(gx^i) = (-1)^i (qp)^{\frac{i(i-1)}{2}}b_x^ib_{gz^i}^{-1},
$$
which finishes the proof.
\end{proof}
In the previous theorem we can assume without loose of generality that $b_1=1_B$.
\begin{comment}
\begin{lemma} We have:
\begin{enumerate}
\smallskip
\item $b_g^i\in B_{g^i}\cap \ker(U)$ for all $i\ge 0$,
\smallskip
\item $U(b_x^i) = (i)_{qp}$ for all $i\ge 0$,
\smallskip
\item $b_x^i\in B_{z^i}$ for all $0\le i< n$,
\smallskip
\item If $\lambda(z^n-1_G)=0$, then $b_x^n\in B_1$ and if $\lambda(z^n-1_G)\ne 0$, then $b_x^n\in B_{z^n}\oplus B_1$ and $(b_x^n)_1 = -\lambda$,
\end{enumerate}
\end{lemma}
\begin{proof} In the proof of all items it is convenient to consider by separate the cases $\lambda(z^n-1_G)\ne 0$ and $\lambda(z^n-1_G) = 0$. We will consider the first case which is more difficult.
\smallskip
\noindent (1)\enspace This follows easily by induction on $i$ using~\eqref{compa'},~\eqref{compa} and~\eqref{eqrankone8*}.
\smallskip
\noindent (2)\enspace This follows easily by induction on $i$ using~\eqref{eqrankone8*}.
\smallskip
\noindent (3)\enspace This follows easily by induction on $i$ using~\eqref{compa'} and~\eqref{compa}.
\smallskip
\noindent (4)\enspace This follows easily using item~(3), \eqref{compa'} and~\eqref{compa}.
\end{proof}
\end{comment}
\begin{theorem}\label{algunas propiedades de las extensiones cleft} Assume that $(C \hookrightarrow B,s)$ is cleft. Take $b_x\in B$ and a family $(b_g)_{g\in G}$ of elements of $B^{\times}$ with $b_1=1_B$, in such a way that conditions~(a)--(d) of Theorem~\ref{caracterizacion de cleft} are fulfilled. Then
\begin{enumerate}
\smallskip
\item $B$ is a free left $C$-module with basis $\{b_gb_x^i : g\in G \text{ and } 0\le i< n\}$.
\smallskip
\item Set $\mathfrak{b}:=b_x^nb_z^{|z|-n}$ and for all $g\in G$ set $\mathfrak{a}_g:=b_g^{|g|}$ and $\mathfrak{c}_g:= (b_xb_g- \chi(g) b_gb_x)b_g^{-1}b_z^{-1}$. Then $\mathfrak{a_g}\in C^{\times}$, $\mathfrak{c}_g\in C$, and if $x^n=0$, then $\mathfrak{b}\in C$.
\smallskip
\item The weak action of $H_{\mathcal{D}}$ on $C$ associated with $\gamma$ according to item~(5) of Theorem~\ref{equiv entre cleft, H-Galois con base normal e isomorfo a un producto cruzado}, is given by
$$
\qquad\quad gx^i \rightharpoonup c=\sum_{j=0}^i (-1)^j(qp)^{\frac{j(j-1)}{2}}\binom{i}{j}_{qp} b_gb_x^{i-j} \alpha^j(c)b_x^j b_{\zeta(g)z^i}^{-1} \quad\text{for $c\in B_{1_G,\zeta}\cap \ker(U)$.}
$$
\smallskip
\item The two cocycle $\sigma\colon H_{\mathcal{D}}\otimes H_{\mathcal{D}} \to C$ associated with $\gamma$ according to item~(5) of Theorem~\ref{equiv entre cleft, H-Galois con base normal e isomorfo a un producto cruzado}, is given by
\begin{align*}
\qquad\quad\,\,\, \sigma(gx^s\otimes hx^r) & = \sum_{\substack{0\le i\le s \\ 0\le j\le r \\ \xi_{ij}<n}}\! (-1)^{\xi_{ij}} \binom{s}{i}_{qp}\binom{r}{j}_{qp} (qp)^{\frac{\xi_{ij}(\xi_{ij}-1)}{2}+sj-ij}\chi(h)^{s-i} b_gb_x^i b_h b_x^{s+r-i} b_{ghz^{s+r}}^{-1}\\
& + \lambda \!\!\sum_{\substack{0\le i\le s \\ 0\le j\le r \\ \xi_{ij}\ge n}}\! (-1)^{\xi'_{ij}} \binom{s}{i}_{qp}\binom{r}{j}_{qp} (qp)^{\frac{\xi'_{ij}(\xi'_{ij}-1)}{2}+sj-ij}\chi(h)^{s-i} b_gb_x^i b_h b_x^{\xi_{in}}b_{ghz^{s+r}}^{-1}\\
& - \lambda\!\! \sum_{\substack{0\le i\le s \\ 0\le j\le r \\ \xi_{ij}\ge n}}\! (-1)^{\xi'_{ij}}\binom{s}{i}_{qp}\binom{r}{j}_{qp} (qp)^{\frac{\xi'_{ij}(\xi'_{ij}-1)}{2}+sj-ij}\chi(h)^{s-i} b_gb_x^i b_h b_x^{\xi_{in}} b_{ghz^{s+r-n}}^{-1},
\end{align*}
where $\xi_{ij}:=s+r-i-j$ and $\xi'_{ij}:=\xi_{ij}-n$.
\end{enumerate}
\end{theorem}
\begin{proof} (1)\enspace By item~(4) of Theorem~\ref{equiv entre cleft, H-Galois con base normal e isomorfo a un producto cruzado}, the map $\phi\colon C\otimes H_{\mathcal{D}}\to B$, given by $\phi(c\otimes y):= c\gamma(y)$, is a normal basis. Item~(1) is an immediately consequence of this fact.
\smallskip
\noindent (2)\enspace Using item~(4) of Remark~\ref{re: H-braided comod alg} it is easy to check by induction on $i$, that
\begin{equation}
\nu(b_g^i)= b_g^i\otimes g^i\quad\text{and}\quad \nu(b_x^i)=\sum_{j=0}^i \binom{i}{j}_{qp} b_x^j\otimes z^jx^{i-j}\quad\text{for all $g\in G$ and $i\ge 0$}.\label{pa}
\end{equation}
By the first equality
$$
\nu(\mathfrak{a}_g)=\mathfrak{a}_g \otimes 1_{H_{\mathcal{D}}} \quad\text{for all $g\in G$},
$$
and so $\mathfrak{a}_g\in C$. Since $\nu\colon B\to B\otimes_s H$ is an algebra map, we have
\begin{equation}
\nu(\mathfrak{a}_g^{-1})=\mathfrak{a}_g^{-1} \otimes 1_{H_{\mathcal{D}}}\quad\text{and}\quad \nu(b_g^{-1}) = b_g^{-1}\otimes g^{-1} \quad\text{for all $g\in G$,}\label{pa1}
\end{equation}
which implies in particular that $\mathfrak{a}_g\in C^{\times}$. Note also that, by the second equality in~\eqref{pa},
$$
\nu(b_x^n)= 1_B\otimes x^n+ b_x^n \otimes z^n.
$$
because $\binom{n}{j}_{qp}=0$ for $0<j<n$. Consequently,
\begin{align*}
\nu(\mathfrak{b}) & = \nu(b_x^nb_z^{|z|-n})\\
& =(\mu_B\otimes \mu_{H_{\mathcal{D}}})\hs \circ \hs (B\otimes s\otimes H_{\mathcal{D}})(1_B\otimes x^n\otimes b_z^{|z|-n}\otimes z^{-n} + b_x^n \otimes z^n\otimes b_z^{|z|-n}\otimes z^{-n})\\
& = (\mu_B\otimes \mu_{H_{\mathcal{D}}})(1_B\otimes b_z^{|z|-n} \otimes x^n \otimes z^{-n} + b_x^n \otimes b_z^{|z|-n}\otimes z^n\otimes z^{-n})\\
& = b_z^{|z|-n} \otimes x^nz^{-n} + \mathfrak{b}\otimes 1_{H_{\mathcal{D}}},
\end{align*}
which implies that, if $x^n = 0$, then $\mathfrak{b}\in C$. Furthermore, for all $g\in G$,
\begin{align*}
&\nu(b_xb_g) = (\mu_B\otimes \mu_{H_{\mathcal{D}}})\hs \circ \hs (B\otimes s\otimes H_{\mathcal{D}})(1_B\otimes x \otimes b_g \otimes g + b_x\otimes z \otimes b_g \otimes g)\\
&\phantom{\nu(b_xb_g)} = (\mu_B\otimes \mu_{H_{\mathcal{D}}})(1_B\otimes b_g \otimes x \otimes g + b_x\otimes b_g \otimes z \otimes g)\\
&\phantom{\nu(b_xb_g)}= \chi(g)\,b_g\otimes gx + b_xb_g\otimes zg
\shortintertext{and}
&\nu(\chi(g)\,b_gb_x)= (\mu_B\otimes \mu_{H_{\mathcal{D}}})\hs \circ \hs (B\otimes s\otimes H_{\mathcal{D}})(\chi(g)\,b_g\otimes g\otimes 1_B\otimes x + \chi(g)\,b_g\otimes g\otimes b_x\otimes z)\\
&\phantom{\nu(\chi(g)\,b_xb_g)}=(\mu_B\otimes \mu_{H_{\mathcal{D}}})(\chi(g)\,b_g\otimes 1_B\otimes g\otimes x + \chi(g)\,b_g\otimes b_x\otimes g\otimes z)\\
&\phantom{\nu(\chi(g)\,b_xb_g)}=\chi(g)\,b_g\otimes gx + \chi(g)\,b_gb_x\otimes zg.
\end{align*}
Combining this with the second equality in~\eqref{pa1}, we obtain
\begin{align*}
\nu(\mathfrak{c}_g) & = \nu\bigl((b_xb_g-\chi(g)\,b_gb_x)b_g^{-1}b_z^{-1}\bigr)\\
& = (\mu_B\otimes \mu_{H_{\mathcal{D}}})\hs \circ \hs (B\otimes s\otimes H_{\mathcal{D}})\bigl((b_xb_g -\chi(g)\,b_gb_x)\otimes zg \otimes b_g^{-1}b_z^{-1}\otimes g^{-1}z^{-1})\\
& = (\mu_B\otimes \mu_{H_{\mathcal{D}}})\bigl((b_xb_g -\chi(g)\,b_gb_x)\otimes b_g^{-1}b_z^{-1}\otimes zg \otimes g^{-1}z^{-1})\\
& = (b_xb_g -\chi(g)\,b_gb_x)b_g^{-1}b_z^{-1}\otimes 1_{H_{\mathcal{D}}},
\end{align*}
and so $\mathfrak{c}_g\in C$, as desired.
\smallskip
\noindent (3)\enspace This follows by a direct computation from item~(5) of Theorem~\ref{equiv entre cleft, H-Galois con base normal e isomorfo a un producto cruzado}, using equalities~\eqref{comultiplication} and~\eqref{eqrankone1}, and the formulas for $\gamma$ and $\gamma^{-1}$ that appears in Theorem~\ref{caracterizacion de cleft}.
\smallskip
\noindent (4)\enspace This follows by a direct computation from item~(5) of Theorem~\ref{equiv entre cleft, H-Galois con base normal e isomorfo a un producto cruzado}, using equalities~\eqref{comultiplication} and~\eqref{def braid}, and the formulas for $\gamma$ and $\gamma^{-1}$ that appears in Theorem~\ref{caracterizacion de cleft}.
\end{proof}
The Proposition~\ref{simplificacion} below is useful to simplify the computation of the first sum in the right side of the equality in Theorem~\ref{algunas propiedades de las extensiones cleft}(4).
\begin{lemma} With the notations of the previous result, we have
$$
\sum_{j=0}^r (-1)^{\xi_{ij}} \binom{r}{j}_{qp} (qp)^{\frac{\xi_{ij}(\xi_{ij}-1)}{2}+sj-ij} = \begin{cases} (-1)^{s-i}(qp)^{\frac{(s-i)(s-i-1)}{2}} & \text{if $r=0$,}\\ 0 &\text{if $0<r<n$.}\end{cases}
$$
\end{lemma}
\begin{proof} Let $a:= s-i$ and $b:=r-j$. Since $\xi_{ij} = a+b$, we have
\begin{align*}
\sum_{j=0}^r (-1)^{\xi_{ij}} \binom{r}{j}_{qp} (qp)^{\frac{\xi_{ij}(\xi_{ij}-1)}{2}+sj-ij}& = \sum_{b=0}^r (-1)^{a+b} \binom{r}{b}_{qp} (qp)^{\frac{(a+b)(a+b-1)}{2}+ar-ab}\\
& = (-1)^a (qp)^{ar + \frac{a(a-1)}{2}} \sum_{b=0}^r (-1)^b \binom{r}{b}_{qp} (qp)^{\frac{b(b-1)}{2}},
\end{align*}
which combined with~\eqref{formula} gives the desired result.
\end{proof}
\begin{proposition}\label{simplificacion} Let $r,s,i\ge 0$ with $0\le i\le s$. The following assertions holds
\begin{enumerate}
\smallskip
\item If $r = 0$, then
$$
\qquad \sum_{\substack{0\le j\le r \\ \xi_{ij}<n}} (-1)^{\xi_{ij}} \binom{r}{j}_{qp} (qp)^{\frac{\xi_{ij}(\xi_{ij}-1)}{2}+sj-ij} = (-1)^{s-i}(qp)^{\frac{(s-i)(s-i-1)}{2}}.
$$
\smallskip
\item If $r > 0$, then
$$
\qquad \sum_{\substack{0\le j\le r \\ \xi_{ij}<n}} (-1)^{\xi_{ij}} \binom{r}{j}_{qp} (qp)^{\frac{\xi_{ij}(\xi_{ij}-1)}{2}+sj-ij} = \sum_{\substack{0\le j\le r \\ \xi_{ij}\ge n}} (-1)^{\xi_{ij}+1} \binom{r}{j}_{qp} (qp)^{\frac{\xi_{ij}(\xi_{ij}-1)}{2}+sj-ij},
$$
where $\xi_{ij}:=s+r-i-j$
\end{enumerate}
\end{proposition}
\begin{proof} By the previous lemma.
\end{proof}
\begin{corollary} Let $r,s,i\ge 0$ with $0\le i\le s$. If $0<r< n-s+i$, then
$$
\sum_{\substack{0\le j\le r \\ \xi_{ij}<n}} (-1)^{\xi_{ij}} \binom{r}{j}_{qp} (qp)^{\frac{\xi_{ij}(\xi_{ij}-1)}{2}+sj-ij} = 0.
$$
\end{corollary}
\begin{proof} By Proposition~\ref{simplificacion}.
\end{proof}
\section{Examples} In this section we consider two examples of the braided Hopf algebras $H_{\mathcal{D}}$ defined in Corollary~\ref{algebras de Hopf trenzadas de K-R} and we apply the results obtained in the previous section in order to determine their cleft extensions.
\subsection{First example} Consider the datum $\mathcal{D}=(C_2\times C_2\times C_2,\chi,z,\lambda,q)$, where:
\begin{itemize}
\smallskip
\item[-] $C_2=\{1,g\}$ is the multiplicative group of order $2$,
\smallskip
\item[-] $\chi\colon C_2\times C_2\times C_2\longrightarrow \mathds{C}$ is the character given by $\chi(g^{i_1},g^{i_2},g^{i_3}):= (-1)^{i_1+i_2+i_3}$,
\smallskip
\item[-] $z:=(g,g,g)$,
\smallskip
\item[-] $q=1$ and $\lambda=1$.
\smallskip
\end{itemize}
In this case $p:=\chi(z)=-1$, $n=2$ and the Hopf braided $\mathds{C}$-algebra $H_{\mathcal{D}}$ of Corollary~\ref{algebras de Hopf trenzadas de K-R} is the $\mathds{C}$-algebra generated by the group $G:= C_2\times C_2\times C_2$ and an element $x$ subject to the relations
$$
x^2=z^2-1_G=0\quad \text{and}\quad x(g^{i_1},g^{i_2},g^{i_3})=(-1)^{i_1+i_2+i_3}(g^{i_1},g^{i_2},g^{i_3})x,
$$
endowed with the standard Hopf algebra structure with comultiplication map $\Delta$, counit $\epsilon$ and antipode $S$, given by
\begin{align*}
&\Delta(\mathbf{g}):= \mathbf{g}\otimes \mathbf{g}, && \Delta(x):= 1\otimes x+x\otimes z,\\
& \epsilon(\mathbf{g}):=1, && \epsilon(x):=0\\
& S(\mathbf{g}):=\mathbf{g}^{-1}, && S(\mathbf{g}x):=-xz^{-1}\mathbf{g}^{-1},
\end{align*}
where $\mathbf{g}$ denotes an arbitrary element of $G$. Let $S_3$ be the symmetric group in $\{1,2,3\}$. It is easy to check that the map
$$
\theta\colon S_3^{\op}\to \Aut_{\chi,z}(G),
$$
defined by $\theta(\sigma) (g^{i_1}, g^{i_2}, g^{i_3}):= (g^{i_{\sigma(1)}},g^{i_{\sigma(2)}},g^{i_{\sigma(3)}})$, is an isomorphism.
\subsubsection{$\bm{H_{\mathcal{D}}}$-spaces}\label{ex1p1} Let $V$ be a $\mathds{C}$-vector space. By Proposition~\ref{estructuras trenzadas de HsubD} to have a left $H_{\mathcal{D}}$-space structure with underlying vector space $V$ is ``the same'' that to have a gradation
$$
V=\bigoplus_{\sigma\in S_3} V_{\sigma}
$$
and an automorphism $\alpha\colon V\to V$ such that $\alpha(V_\sigma)=V_{\sigma}$ for all $\sigma\in S_3$. The structure map
$$
s\colon H_{\mathcal{D}}\otimes V\longrightarrow V\otimes H_{\mathcal{D}},
$$
constructed from these data, is given by
\begin{align*}
& s\bigl((g^{i_1},g^{i_2},g^{i_3})\otimes v\bigr) := v\otimes (g^{i_{\sigma(1)}}, g^{i_{\sigma(2)}}, g^{i_{\sigma(3)}})
\shortintertext{and}
& s\bigl((g^{i_1},g^{i_2},g^{i_3})x\otimes v\bigr) := \alpha(v)\otimes (g^{i_{\sigma(1)}}, g^{i_{\sigma(2)}}, g^{i_{\sigma(3)}})x,
\end{align*}
for each $v\in V_{\sigma}$.
\subsubsection{$\bm{H_{\mathcal{D}}}$-comodules} Let $V$ be a $\mathds{C}$-vector space. By Corollary~\ref{segunda caracterizacion de HsubD comodules} each right $H_{\mathcal{D}}$-comodule structure $(V,s)$ with underlying vector space $V$ is univocally determined by the following data:
\begin{enumerate}[label=\emph{(\alph*)}]
\smallskip
\item A decomposition
$$
\quad\qquad V=\bigoplus_{(\mathbf{g},\sigma)\in G\times S_3^{\op}} V_{\mathbf{g},\sigma},
$$
\smallskip
\item An automorphism $\alpha\colon V\to V$ that satisfies $\alpha\bigl(V_{\mathbf{g},\sigma}\bigr) = V_{\mathbf{g},\sigma}$ for all $(\mathbf{g},\sigma)\in G\times S_3^{\op}$,
\smallskip
\item A map $U\colon V\to V$ such that
$$
\quad\qquad U\circ \alpha= \alpha\circ U, \qquad U^2=0\qquad\text{and}\qquad U\bigl(V_{\mathbf{g},\sigma}\bigr)\subseteq V_{\mathbf{g}z,\sigma}\quad \text{for all $(\mathbf{g},\sigma)\in G\times S_3^{\op}$.}
$$
\smallskip
\end{enumerate}
The formula for the transposition $s$ of $H_{\mathcal{D}}$ on $V$ is the one obtained in Subsection~\ref{ex1p1} (where we take $V_{\sigma}:=\bigoplus_{\mathbf{g}\in G} V_{\mathbf{g},\sigma}$ for each $\sigma\in S_3$), while the $H_{\mathcal{D}}$-coaction $\nu$ is given by
$$
\nu(v) = v\otimes (g^{i_1},g^{i_2},g^{i_3}) + U(v)\otimes (g^{i_1+1},g^{i_2+1},g^{i_3+1})x,
$$
for $v\in \bigoplus_{\sigma\in S_3} V_{\mathbf{g},\sigma}$ with $\mathbf{g} = (g^{i_1},g^{i_2},g^{i_3})$.
\smallskip
Next given a decomposition as in item~(a), we give a proceeding to construct an automorphism $\alpha\colon V\to V$ and a map $U\colon V\to V$ satisfying the conditions required in items~(b) and~(c): First we decompose each space $V_{\mathbf{g},\sigma}$ as a direct sum
$$
V_{\mathbf{g},\sigma}=V_{\mathbf{g},\sigma}^0\oplus V_{\mathbf{g},\sigma}^1,
$$
in such a way that $\dim_{\mathds{C}}(V_{\mathbf{g},\sigma}^1)\le \dim_{\mathds{C}}(V_{\mathbf{g}z,\sigma}^0)$, and we fix an injective morphisms
$$
U_{\mathbf{g},\sigma}\colon V_{\mathbf{g},\sigma}^1\longrightarrow V_{\mathbf{g}z,\sigma}^0,
$$
for each $(\mathbf{g},\sigma)\in G\times S_3^{\op}$. Then we define $U$ on $V_{\mathbf{g},\sigma}$, by
$$
U(v) = \begin{cases} U_{\mathbf{g},\sigma}(v) & \text{if $v\in V_{\mathbf{g},\sigma}^1$,}\\ 0 & \text{if $v\in V_{\mathbf{g},\sigma}^0$.}\end{cases}
$$
It remains to construct $\alpha$. Let $(\mathbf{g},\sigma)\in G\times S_3^{\op}$ arbitrary. Since $U\circ \alpha =\alpha\circ U$, $\alpha(V_{\mathbf{g},\sigma})\subseteq V_{\mathbf{g},\sigma}$ and $U_{\mathbf{g},\sigma}$ is injective, there exists morphisms
$$
\alpha^0_{\mathbf{g},\sigma}\colon V_{\mathbf{g},\sigma}^0 \longrightarrow V_{\mathbf{g},\sigma}^0,\qquad \alpha^1_{\mathbf{g},\sigma}\colon V_{\mathbf{g},\sigma}^1 \longrightarrow V_{\mathbf{g},\sigma}^1 \qquad\text{and}\qquad \alpha^{10}_{\mathbf{g},\sigma}\colon V_{\mathbf{g},\sigma}^1 \longrightarrow V_{\mathbf{g},\sigma}^0,
$$
such that
$$
\alpha(v_0,v_1)= \bigl(\alpha^0_{\mathbf{g},\sigma}(v_0)+\alpha^{10}_{\mathbf{g},\sigma}(v_1),\alpha^1_{\mathbf{g},\sigma}(v_1)\bigr)\qquad\text{for all $(v_0,v_1)\in V_{\mathbf{g},\sigma}^0\oplus V_{\mathbf{g},\sigma}^1$.}
$$
Moreover, since $\alpha$ is an automorphism, the maps $\alpha^0_{\mathbf{g},\sigma}$ and $\alpha^1_{\mathbf{g},\sigma}$ are also automorphisms. All these maps can be constructed as follows: For each $(\mathbf{g},\sigma)\in G\times S_3^{\op}$ we take an arbitrary automorphism $\alpha^1_{\mathbf{g},\sigma}$ of $V_{\mathbf{g},\sigma}^1$. Then, for each $(\mathbf{g},\sigma)\in G\times S_3^{\op}$, we choose $\alpha^0_{\mathbf{g},\sigma}$ as an automorphism of $V_{\mathbf{g},\sigma}^0$ such that
$$
\alpha^0_{\mathbf{g},\sigma}\bigl(U_{\mathbf{g}z,\sigma}(v)\bigr) = U_{\mathbf{g},\sigma}\bigl(\alpha^1_{\mathbf{g}z,\sigma}(v)\bigr)\qquad\text{for all $v\in V^1_{\mathbf{g}z,\sigma}$}
$$
(which is forced by the condition $U\circ \alpha = \alpha \circ U$). Finally, we take $\alpha^{10}_{\mathbf{g},\sigma}$ as an arbitrary automorphism.
\begin{remark}\label{coinvariante ejemplo 1} By Corollary~\ref{coinvarianes de un HsubD comodulo} we know that $V^{\coH_{\mathcal{D}}}= V_{1_G}\cap \ker (U)$, where $V_{1_G}=\bigoplus_{\sigma\in S_3} V_{1_G,\sigma}$.
\end{remark}
\begin{remark}\label{caso clasico comodulos ejemplo 1} We are in the classical case (i.e. $s$ is the flip) iff $V_{\mathbf{g},\sigma}=0$ for $\sigma \ne \ide$ and $\alpha$ is the identity map. So, in this case the decomposition in item a) above have at most eight nonzero summands, item b) becomes trivial and the first condition in item c) also becomes trivial.
\end{remark}
\subsubsection{Transpositions of $\bm{H_{\mathcal{D}}}$ on an algebra} By Proposition~\ref{estructura de transposiciones de HsubD}, for each $\mathds{C}$-algebra $B$, to have a transposition $s\colon H_{\mathcal{D}}\otimes B \longrightarrow B\otimes H_{\mathcal{D}}$ is equivalent to have an algebra gradation
$$
B=\bigoplus_{\sigma\in S_3^{\op}} B_{\sigma}
$$
and an automorphism of algebras $\alpha \colon B\to B$ such that $\alpha(B_{\sigma})= B_{\sigma}$ for all $\sigma\in S_3$. The structure map $s\colon H_{\mathcal{D}}\otimes B\longrightarrow B\otimes H_{\mathcal{D}}$, constructed from these data, is the same as in Subsection~\ref{ex1p1}.
\subsubsection{Right $\bm{H_{\mathcal{D}}}$-comodule algebras} By the discussion above Definition~\ref{graduacion compatible con D} we know that the group $S_3^{\op}$ acts on $G$ via
$$
\sigma\cdot (g^{i_1},g^{i_2},g^{i_3}):= (g^{i_{\sigma(1)}},g^{i_{\sigma(2)}},g^{i_{\sigma(3)}}).
$$
Consider the semidirect product $G(\chi,z):= G\rtimes S_3^{\op}$. We are going to work with $G(\chi,z)^{\op}$. Its underlying set is $C_2\times C_2 \times C_2\times S_3$ and its product is given by
$$
(g^{i_1},g^{i_2},g^{i_3},\sigma) (g^{j_1},g^{j_2},g^{j_3},\tau)= (g^{j_1+i_{\tau(1)}},g^{j_2+i_{\tau(2)}},g^{j_3+i_{\tau(3)}},\sigma\circ \tau).
$$
Let $B$ be a $\mathds{C}$-algebra. By Theorem~\ref{caracterizacion de HsubD comodule algebras} to have a right $H_{\mathcal{D}}$-comodule algebra $(B,s)$ is equivalent to have
\begin{enumerate}[label=\emph{(\alph*)}]
\smallskip
\item a $G(\chi,z)^{\op}$-gradation
$$
\qquad\quad B= \bigoplus_{(\mathbf{g},\sigma)\in G(\chi,z)^{\op}} B_{\mathbf{g},\sigma}
$$
of $B$ as an algebra,
\smallskip
\item an automorphism of algebras $\alpha\colon B\to B$ such that
$$
\qquad\quad \alpha\bigl(B_{\mathbf{g},\sigma}\bigr) \subseteq B_{\mathbf{g},\sigma}\quad \text{for all $(\mathbf{g},\sigma)\in G(\chi,z)^{\op}$,}
$$
\smallskip
\item a map $U\colon B\to B$ such that
\begin{itemize}
\smallskip
\item[-] $U\circ \alpha=\alpha\circ U$,
\smallskip
\item[-] $U^2=0$,
\smallskip
\item[-] $U\bigl(B_{\mathbf{g},\sigma}\bigr)\subseteq B_{\mathbf{g}z,\sigma}$\quad \text{for all $(\mathbf{g},\sigma)\in G(\chi,z)^{\op}$,}
\smallskip
\item[-] the equality
$$
\quad\qquad\qquad U(bc)=bU(c) + (-1)^{i_1+i_2+i_3}U(b)\alpha(c)
$$
holds for all $b\in B$ and $c\in B_{(g^{i_1},g^{i_2},g^{i_3})}:= \bigoplus_{\sigma\in S_3} B_{(g^{i_1},g^{i_2},g^{i_3}),\sigma}$.
\end{itemize}
\end{enumerate}
\begin{remark}\label{caso clasico comodulo algebras ejemplo 1} We are in the classical case (i.e. $s$ is the flip) iff $B_{\mathbf{g},\sigma}=0$ for $\sigma \ne \ide$ and $\alpha$ is the identity map. So, in this case the gradation in item~(a) is a $G$-gradation, item~(b) is trivial, and item~(c) is considerably simplified.
\end{remark}
\subsubsection{Right $\bm{H_{\mathcal{D}}}$-cleft extensions} Let $C:=B^{\coH_{\mathcal{D}}}$. By Corollary~\ref{coinvarianes de un HsubD comodulo} we know that
$$
C=B_{1_G}\cap \ker(U)= \bigoplus_{\sigma\in S_3} B_{1_G,\sigma}\cap \ker(U).
$$
By Theorem~\ref{caracterizacion de cleft} and the comment bellow that result, the extension $(C\hookrightarrow B,s)$ is cleft iff there exists $b_x\in B$ and a family $(b_\mathbf{g})_{\mathbf{g}\in G}$ of elements of $B^{\times}$, such that
\begin{enumerate}[label=\emph{(\alph*)}]
\smallskip
\item $b_{1_G}=1$,
\smallskip
\item $b_{\mathbf{g}}\in B_{\mathbf{g},\ide}\cap \ker(U)$ for all $\mathbf{g}\in G$,
\smallskip
\item $b_x\in B_{(g,g,g),\ide}\cap U^{-1}(1)$,
\smallskip
\item $\alpha(b_x)=b_x$,
\smallskip
\item $\alpha(b_{\mathbf{g}})=b_{\mathbf{g}}$ for all $\mathbf{g}\in G$.
\smallskip
\end{enumerate}
By Theorem~\ref{algunas propiedades de las extensiones cleft} we know that
\begin{enumerate}
\smallskip
\item $B$ is a free left $C$-module with basis $\{b_gb_x^i : g\in G \text{ and } 0\le i\le 1\}$.
\smallskip
\item By Theorem~\ref{algunas propiedades de las extensiones cleft}(3), the weak action of $H_{\mathcal{D}}$ on $C$ associated with $\gamma$ according to item~(5) of Theorem~\ref{equiv entre cleft, H-Galois con base normal e isomorfo a un producto cruzado}, is given by
\begin{align*}
&\quad (g^{i_1}, g^{i_2}, g^{i_3})\rightharpoonup c = b_{(g^{i_1}, g^{i_2}, g^{i_3})} c b^{-1}_{(g^{i_{\sigma(1)}}, g^{i_{\sigma(2)}} g^{i_{\sigma(3)}})}
\shortintertext{and}
&\quad (g^{i_1}, g^{i_2}, g^{i_3})x\rightharpoonup c = b_{(g^{i_1}, g^{i_2}, g^{i_3})}\bigl(b_x c - \alpha(c) b_x\bigr) b^{-1}_{(g^{i_{\sigma(1)}+1}, g^{i_{\sigma(2)}+1}, g^{i_{\sigma(3)}+1})}
\end{align*}
for $c\in B_{1_G,\sigma}\cap \ker(U)$.
\smallskip
\item By Theorem~\ref{algunas propiedades de las extensiones cleft}(3), the two cocycle $\sigma\colon H_{\mathcal{D}}\otimes H_{\mathcal{D}} \to C$, associated with $\gamma$ according to item~(5) of Theorem~\ref{equiv entre cleft, H-Galois con base normal e isomorfo a un producto cruzado}, is given by
\begin{align*}
&\quad\qquad \sigma(\mathbf{g}\otimes \mathbf{h}) = b_{\mathbf{g}}b_{\mathbf{h}} b_{\mathbf{g}\mathbf{h}}^{-1},\\
&\quad\qquad \sigma(\mathbf{g}x \otimes \mathbf{h}) = - \chi(\mathbf{h}) b_{\mathbf{g}}b_{\mathbf{h}} b_x b_{\mathbf{g}\mathbf{h}(g,g,g)}^{-1} + b_{\mathbf{g}} b_x b_{\mathbf{h}} b_{\mathbf{g}\mathbf{h}(g,g,g)}^{-1},\\
&\quad\qquad \sigma(\mathbf{g} \otimes \mathbf{h}x) = 0
\shortintertext{and}
&\quad\qquad \sigma(\mathbf{g}x \otimes \mathbf{h}x) = \chi(h) b_{\mathbf{g}}b_{\mathbf{h}} b_x^2 b_{\mathbf{g}\mathbf{h}}^{-1},
\end{align*}
for $\mathbf{g},\mathbf{h}\in G$.
\end{enumerate}
\begin{remark} It is clear that once choiced $b^{(1)}_g:=b_{(g,1,1)}$, $b^{(2)}_g:=b_{(1,g,1)}$ and $b^{(3)}_g:=b_{(1,1,g)}$, one can take $b_{(g,g,1)}:=b^{(1)}_gb^{(2)}_g$, $b_{(g,1,g)}:=b^{(1)}_gb^{(3)}_g$, $b_{(1,g,g)}:=b^{(2)}_gb^{(3)}_g$ and $b_{(g,g,g)}:=b^{(1)}_gb^{(2)}_gb^{(3)}_g$.
\end{remark}
\subsection{Second example} Consider the datum $\mathcal{D}=(C_6,\chi,z,\lambda,q)$, where:
\begin{itemize}
\smallskip
\item[-] $C_6=\{1,g,g^2,g^3,g^4,g^5\}$ is the multiplicative cyclic group of order~$6$,
\smallskip
\item[-] $\chi\colon C_6\longrightarrow \mathds{C}$ is the character given by $\chi(g^i):=\xi^i$, where $\xi$ is a root of order~$3$ of~$1$,
\smallskip
\item[-] $z:=g$,
\smallskip
\item[-] $q=\xi$ and $\lambda=1$.
\smallskip
\end{itemize}
In this case $p:=\chi(z)=\xi$, $n=3$ and the Hopf braided $\mathds{C}$-algebra $H_{\mathcal{D}}$ of Corollary~\ref{algebras de Hopf trenzadas de K-R} is the $\mathds{C}$-algebra generated by the group $C_6$ and an element $x$ subject to the relations
$$
x^3=g^3-1 = -2\quad \text{and}\quad xg=\xi gx,
$$
endowed with the braided Hopf algebra structure with comultiplication map $\Delta$, counit $\epsilon$, antipode $S$ and braid $c_{\xi}$, given by
\begin{align*}
&\Delta(g^i):= g^i\otimes g^i, && \Delta(x):= 1\otimes x+x\otimes g,\\
& \epsilon(g^i):=1, && \epsilon(x):=0\\
& S(g^ix^j):=(-1)^j\xi^{j(j-1)}x^j g^{-j-i}, \\
&c_{\xi}(g^ix^j\otimes g^kx^l)=\xi^{jl} g^kx^l\otimes g^ix^j.
\end{align*}
It is clear that $\Aut_{\chi,z}(C_6)=\{\ide\}$.
\subsubsection{$\bm{H_{\mathcal{D}}}$-spaces}\label{ex2p1} Let $V$ be a $\mathds{C}$-vector space. By Proposition~\ref{estructuras trenzadas de HsubD} we know that to have an $H_{\mathcal{D}}$-space structure with underlying vector space $V$ is equivalent to have an automorphism~$\alpha\colon V\to V$ such that $\alpha^3=\ide$. The structure map $s\colon H_{\mathcal{D}}\otimes V\longrightarrow V\otimes H_{\mathcal{D}}$ construct from these data is given by
$$
s\bigl(g^ix^j\otimes v\bigr) := \alpha^j(v)\otimes g^ix^j.
$$
\subsubsection{$\bm{H_{\mathcal{D}}}$-comodules} Let $V$ be a $\mathds{C}$-vector space. By Corollary~\ref{segunda caracterizacion de HsubD comodules} the right $H_{\mathcal{D}}$-comodule structures $(V,s)$ with underlying vector space $V$ are univocally determined by the following data:
\begin{enumerate}[label=\emph{(\alph*)}]
\smallskip
\item a decomposition
$$
\quad\qquad V=\bigoplus_{g^i\in C_6} V_{g^i}=V_1\oplus V_g \oplus V_{g^2}\oplus V_{g^3} \oplus V_{g^4}\oplus V_{g^5},
$$
\smallskip
\item an automorphism $\alpha\colon V\to V$ that satisfies $\alpha^3=\ide$ and $\alpha\bigl(V_{g^i}\bigr) = V_{g^i}$ for all $i$,
\smallskip
\item a map $U\colon V\to V$ such that
$$
\quad\qquad U\circ \alpha= \xi\alpha\circ U, \quad U^3=0\quad\text{and}\quad U\bigl(V_{g^i}\bigr)\subseteq V_{g^{i-1}}\quad\text{for all $i$.}
$$
\smallskip
\end{enumerate}
The formula for the transposition $s$ of $H_{\mathcal{D}}$ on $V$ is the one obtained in Subsection~\ref{ex2p1}, while the $H_{\mathcal{D}}$-coaction $\nu$ is given by
$$
\nu(v) = v\otimes g^i + U(v)\otimes g^{i-1}x - \xi U^2(v)\otimes g^{i-2}x^2 \qquad\text{for all $v\in V_{g^i}$.}
$$
\subsubsection{Transpositions of $\bm{H_{\mathcal{D}}}$ on an algebra} By Proposition~\ref{estructura de transposiciones de HsubD}, for each $\mathds{C}$-algebra $B$, to have a transposition $s\colon H_{\mathcal{D}}\otimes B \longrightarrow B\otimes H_{\mathcal{D}}$ is equivalent to have an automorphism of algebras $\alpha\colon B\to B$ such that $\alpha^3=\ide$. The structure map $s\colon H_{\mathcal{D}}\otimes B\longrightarrow B\otimes H_{\mathcal{D}}$, constructed from these data, is the same as in Subsection~\ref{ex2p1}.
\subsubsection{Right $\bm{H_{\mathcal{D}}}$-comodule algebras} Let $B$ be a $\mathds{C}$-algebra. By Theorem~\ref{caracterizacion de HsubD comodule algebras} to have a right $H_{\mathcal{D}}$-comodule algebra $(B,s)$ is equivalent to have
\begin{enumerate}[label=\emph{(\alph*)}]
\smallskip
\item a $C_6$-gradation
$$
\qquad\quad B= B_1\oplus B_g \oplus B_{g^2}\oplus B_{g^3} \oplus B_{g^4}\oplus B_{g^5},
$$
of $B$ as a vector space such that $1_B\in B_1$ and $B_{g^i}B_{g^j}\subseteq B_{g^{i+j}}\oplus B_{g^{i+j-3}}$ for all $i,j$.
\smallskip
\item an automorphism of algebras $\alpha\colon B\to B$ such that
$$
\qquad\quad \alpha^3=\ide\quad\text{and}\quad\alpha\bigl(B_{g^i}\bigr) \subseteq B_{g^i}\quad \text{for all $i$,}
$$
\smallskip
\item a map $U\colon B\to B$ such that
\begin{itemize}
\smallskip
\item[-] $U\circ \alpha=\xi \alpha\circ U$,
\smallskip
\item[-] $U^3=0$,
\smallskip
\item[-] $U\bigl(B_{g^i}\bigr)\subseteq B_{g^{i-1}}$\quad \text{for all $i$,}
\smallskip
\item[-] the equality
$$
\quad\qquad\qquad U(bc)=bU(c) + \xi^i U(b)\alpha(c)
$$
holds for all $b\in B$ and $c\in B_{g^i}$,
\smallskip
\item[-] For $b\in B_{g^i}$ and $c\in B_{g^j}$, the component $(bc)_{g^{i+j-3}}\in B_{g^{i+j-3}}$ of $bc$ is given by
$$
\quad\qquad (bc)_{g^{i+j-3}}= \xi^j U(b) \alpha(U^2(c)) + \xi^{2j} U^2(b) \alpha^2(U(c)).
$$
\end{itemize}
\end{enumerate}
\subsubsection{Right $\bm{H_{\mathcal{D}}}$-cleft extensions}
Let $C:=B^{\coH_{\mathcal{D}}}$. By Corollary~\ref{coinvarianes de un HsubD comodulo} we know that
$$
C=B_{1}\cap \ker(U).
$$
By Theorem~\ref{caracterizacion de cleft} and the comment bellow that result, the extension $(C\hookrightarrow B,s)$ is cleft iff there exists $b_x\in B$ and a family $(b_{g^i})_{g^i\in C_6}$ of elements of $B^{\times}$, such that
\begin{enumerate}[label=\emph{(\alph*)}]
\smallskip
\item $b_1=1$,
\smallskip
\item $b_{g^i}\in B_{g^i}\cap \ker(U)$ for all $g^i\in C_6$,
\smallskip
\item $b_x\in B_{g}\cap U^{-1}(1)$,
\smallskip
\item $\alpha(b_x)=\xi b_x$,
\smallskip
\item $\alpha(b_{g^i})=b_{g^i}$ for all $g^i\in C_6$.
\end{enumerate}
By Theorem~\ref{algunas propiedades de las extensiones cleft} we know that
\begin{enumerate}
\smallskip
\item $B$ is a free left $C$-module with basis $\{b_{g^i}b_x^j : g^i\in C_6 \text{ and } 0\le j\le 2\}$.
\smallskip
\item The weak action of $H_{\mathcal{D}}$ on $C$ associated with $\gamma$ according to item~(5) of Theorem~\ref{equiv entre cleft, H-Galois con base normal e isomorfo a un producto cruzado}, is given by
\begin{align*}
&\quad g^i\rightharpoonup c = b_{g^i} c b^{-1}_{g^i},\\
& \quad g^i x \rightharpoonup c = b_{g^i} \bigl(b_x c - \alpha(c)b_x\bigr) b^{-1}_{g^{i+1}}
\shortintertext{and}
&\quad g^i x^2 \rightharpoonup c
= b_{g^i} \bigl( b^2_x c + \xi b_x\alpha(c)b_x + \alpha^2(c)b^2_x\bigr) b^{-1}_{g^{i+2}},
\end{align*}
for $c\in C$.
\smallskip
\item The two cocycle $\sigma\colon H_{\mathcal{D}}\otimes H_{\mathcal{D}} \to C$, associated with $\gamma$ according to item~(5) of Theorem~\ref{equiv entre cleft, H-Galois con base normal e isomorfo a un producto cruzado}, is given by
\begin{align*}
&\quad\qquad \sigma(g^i\otimes g^j) = b_{g^i}b_{g^j} b_{g^{i+j}}^{-1},\\
&\quad\qquad \sigma(g^i x\otimes g^j) = -\xi^j b_{g^i}b_{g^j} b_x b_{g^{i+j+1}}^{-1} + b_{g^i}b_x b_{g^j} b_{g^{i+j+1}}^{-1},\\
&\quad\qquad \sigma(g^i x^2\otimes g^j) = \xi^{2j+2} b_{g^i}b_{g^j} b^2_x b_{g^{i+j+2}}^{-1} + \xi^{j+1}b_{g^i}b_xb_{g^j} b_x b_{g^{i+j+2}}^{-1}+ b_{g^i}b^2_xb_{g^j} b_{g^{i+j+2}}^{-1},\\
&\quad\qquad \sigma(g^i \otimes g^j x) = 0,\\
&\quad\qquad \sigma(g^i x \otimes g^j x) = 0,\\
&\quad\qquad \sigma(g^i x^2\otimes g^j x) = -\xi^{2j} b_{g^i}b_{g^j} b_{g^{i+j+3}}^{-1} -\xi^{2j} b_{g^i}b_{g^j} b_{g^{i+j}}^{-1} + \xi^{2j} b_{g^i}b_{g^j} b_x^3 b_{g^{i+j+3}}^{-1},\\
&\quad\qquad \sigma(g^i \otimes g^j x^2) = 0,\\
&\quad\qquad \sigma(g^i x\otimes g^j x^2) = -\xi^{j} b_{g^i}b_{g^j}b_{g^{i+j+3}}^{-1} -\xi^{j} b_{g^i}b_{g^j} b_{g^{i+j}}^{-1} + \xi^{j} b_{g^i}b_{g^j}b_x^3 b_{g^{i+j+3}}^{-1}
\shortintertext{and}
&\quad\qquad \sigma(g^i x^2\otimes g^j x^2) = -\xi^{2j+1} b_{g^i}b_{g^j}b_x b_{g^{i+j+4}}^{-1} -\xi^{2j+1} b_{g^i}b_{g^j}b_x b_{g^{i+j+1}}^{-1} + \xi^{j+1} b_{g^i}b_xb_{g^j}b_{g^{i+j+4}}^{-1} \\
&\phantom{\quad\qquad \sigma(g^i x^2\otimes g^j x^2)} + \xi^{j+1} b_{g^i}b_xb_{g^j}b_{g^{i+j+1}}^{-1} + \xi^{2j+1} b_{g^i}b_{g^j}b^4_x b_{g^{i+j+4}}^{-1} - \xi^{j+1} b_{g^i}b_xb_{g^j}b^3_x b_{g^{i+j+4}}^{-1}.
\end{align*}
\end{enumerate}
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|
2,869,038,156,339 | arxiv | \section{Introduction}
Maximum likelihood method is widely used in high energy physics, such as the observation of the Higgs boson~\cite{higgs_observation_atlas, higgs_observation_cms} at the Large Hadron Collider (LHC). Typically in a measurement by ATLAS or CMS collaboration, we have to estimate tens of or even hundreds of systematic sources. For example, 214 nuisance parameters are involved in the measurement of higgs properties in the diphoton decay channel~\cite{higgs_diphoton}. They may affect the normalization or/and shape of the observable distribution differently. In practice, we introduce a nuisance parameter (NP) for each systematic uncertainty in the likelihood function. Due to many fitting parameters, it may take hours for one fit and even more time to understand the potential features presented in the fitting results. For example, the post-fit uncertainty for a systematic source may be much smaller than its initial estimation in a measurement. In other words, the corresponding nuisance parameter turns out to be over-constrained in the fit. Then the measurement may be aggressive as the fit does not consider the full uncertainty. On the other side, we care about which systematic sources have large contribution to the uncertainty of the parameter of interest (POI). This is important if the data statistics is not the main limiting factor. In this paper, we are trying to understand why a parameter could be over-constrained or have a large impact on POI uncertainty. Meanwhile, we also present simple formulae to estimate the constraint and impact directly. It should be noted that advanced numerical methods have been developed to estimate them precisely. Our formulae will never be a substitute, but help us understand the physics reasons behind the fitting features.
In Section~\ref{sec:simple_model}, we start with a simple model based on number-counting experiments and introduce the definition of constraint and impact for a nuissance parameter. It is extended to a more realistic model in Section~\ref{sec:realistic_model}. A toy experiment is performed for illustration in Section~\ref{sec:example}. The conclusion will be summarized in Section~\ref{sec:summary}.
\section{\label{sec:simple_model}A simple model}
Considering an experiment of counting number of events, let $n$ be the observed number of events, $b$ be the number of background events from Monte Carlo (MC) simulation and $s$ be the number of signal events to be determined. For the background prediction, let $\delta$ be the MC statistical uncertainty, and we introduce one systematic uncertainty $\Delta$. The likelihood function in this model is
\begin{equation}\label{eq:L_simple_model}
\mathcal{L}(s,\alpha,\gamma) = P(n|s+\gamma b + \alpha\Delta)\times P(m | \gamma m)\times G(\alpha|0,1) \:,
\end{equation}
where $s$ is the parameter of interest, which is the measurement target and can be used to discriminate the right theory model; $\alpha$ and $\gamma$ are two nuisance parameters as explained below; $m\equiv \frac{b^2}{\delta^2}$ is a constant; $P(n|\lambda) = \frac{\lambda^{n}}{n!}e^{-\lambda}$ is the Poisson distribution function with the expectation value $\lambda$, $G(x|\mu,\sigma) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ is the Gaussian distribution function with mean $\mu$ and standard deviation $\sigma$. The right-hand side of the equation is a
product of three factors. The first one is the Poisson probability of observing $n$ events with the expectation $s+\gamma b+\alpha\Delta$. We start with the expectation $s+b$. The second factor is to account for the background MC statistical uncertainty. It is considered by introducing a nuissance parameter $\gamma$ and an auxillary experiment with the observed number of events $m$ so that the original relative uncertainty $\frac{\delta}{b}$ is preserved. Thus the expectation $s+b$ is replaced by $s+\gamma b$. The third factor is
to include the estimation of the systematic uncertainty $\Delta$. This is done by introducing a nuisance parameter $\alpha$ which abides by the Gaussian distribution function with mean value 0 and variance 1 and replacing $s+\gamma b$ by $s+\gamma b + \alpha\Delta$. The Gaussian constraint is generally used across the measurements~\cite{atlas1,atlas2,cms1,cms2} by ATLAS and CMS collaborations. Ignoring the irrelevant constant terms, the log likelihood function is then
\begin{equation}
\ln \mathcal{L} = n \ln(s+\gamma b + \alpha \Delta) - (s+ \gamma b +\alpha\Delta) - \frac{\alpha^2}{2} + m \ln\gamma - \gamma m \:.
\end{equation}
Maximizing the likelihood function leads to the following estimation.
\begin{equation}
\hat{s} = n-b\:, \quad \hat{\alpha} = 0\:,\quad \hat{\gamma} = 1\: .
\end{equation}
Here a hat $\hat{}$ is added to represent the best-fit values.
Letting $\bold{V}$ denote the covariance matrix of the fitting parameters, the inverse of its estimation is related with the second-order derivatives of the log likelihood function evaluated at the best-fit values, as shown in the following equation~\cite{book:glen_cowan}.
\begin{equation}
(\bold{V}^{-1})_{ij} = - \frac{\partial^2 \ln \mathcal{L} (\hat{\theta}_i,\hat{\theta}_j)}{\partial \theta_i\partial\theta_j}
\end{equation}
where $\theta_i$s denote the parameters $(\theta_1,\theta_2,\theta_3)=(s,\alpha,\gamma)$. For this model, the inverse of $\bold{V}$ is
\begin{equation}
\bold{V}^{-1} = \frac{1}{n}
\begin{pmatrix}
1 & \Delta & b\\
\Delta & \Delta^2 + n & \Delta b \\
b & \Delta b& b^2+n\frac{b^2}{\delta^2}\\
\end{pmatrix} \: ,
\end{equation}
with the determinant $\det|\bold{V}^{-1}| = \frac{b^2}{n\delta^2}$. The diagonal elements of $\bold{V}$ give the uncertainty of the fitting parameters.
\begin{equation}~\label{eq:simple_model}
\hat{\sigma}_s =\sqrt{\bold{V}_{11}}= \sqrt{n + \Delta^2 + \delta^2 }\:, \quad \hat{\sigma}_{\alpha}=\sqrt{\bold{V}_{22}}= 1 \:, \quad \hat{\sigma}_{\gamma}=\sqrt{\bold{V}_{33}} = \frac{\delta}{b} \: .
\end{equation}
Based on the results above, let us introduce the definition of constraint and impact for a nuisance parameter studied in this paper. In this example, the nuisance parameter $\alpha$ corresponding to the systematic uncertainty $\pm \Delta$ is not over-constrained as $\hat{\sigma}_{\alpha}=1$ where 1 is our initial estimation. We define the constraint of a nuisance parameter as the ratio of the fitted variation to the input variation (unity by construction), namely, $\frac{\hat{\sigma}_{\alpha}}{1}=\hat{\sigma}_{\alpha}$. If $\hat{\sigma}_{\alpha}$ is much smaller than 1, we say $\alpha$ is over-constrained and we may worry because the uncertainty $\pm\Delta$ is not fully considered. On the other side, if $\hat{\sigma}_{\alpha}$ is higher than 1, it means larger uncertainty than initial estimation is considered. The latter case is usually not of our concern because the measurement is conservative. The POI uncertainty can be expressed as a quadrature sum, namely, $\hat{\sigma}_s^2 = \sqrt{n}^2 + \Delta^2 + \delta^2$. It has three parts, which represent the contribution from the data statistics, the systematic uncertainty ($\alpha$) and the MC statistical uncertainty ($\gamma$), respectively. Taking $\hat{\sigma}_s$ as a function of $\Delta$ and $\delta$, the impact on the POI ( $s$ in this model ) of the nuisance parameter $\alpha$, denoted by $\hat{\sigma}_s^{\alpha}$, can be defined as $\hat{\sigma}_s^{\alpha}\equiv \sqrt{\hat{\sigma}_s^2(\Delta,\delta)-\hat{\sigma}_s^2(0,\delta)}=\Delta$. Similarly, the impact of $\gamma$ can be defined as $\hat{\sigma}_s^{\gamma}\equiv\sqrt{\hat{\sigma}_s^2(\Delta,\delta) - \hat{\sigma}_s^2(\Delta,0)}=\delta$.
\section{\label{sec:realistic_model}A realistic model}
Extending the model above to a case that a measurement is performed in distributions of an observable, we resort to the binned likelihood estimation and the likelihood function becomes
\begin{equation}\label{eq:L_realistic_model}
\mathcal{L}(\mu,\alpha,\gamma) = \Pi_{i=1}^{N}P(n_i|\mu s_i+\gamma_i b_i + \alpha \Delta_i) P(m_i | \gamma_i m_i) \times G(\alpha|0,1) \:,
\end{equation}
where $N$ is the number of bins; $n_i$ is the observed number of events in the $i$-th bin while $s_i$ and $b_i$ are signal and background prediction; $\mu$ is the signal strength with respective to the prediction and is the parameter of interest; $\Delta_i$ is the systematical variation at the $i$-th bin with the corresponding nuisance parameter $\alpha$; $m_i\equiv \frac{b_i^2}{\delta_i^2}$ with $\delta_i$ being the MC statistical uncertainty at $i$-th bin and $\gamma_i$ being the
corresponding nuisance parameter.
Differently from previous model, here the signal strength $\mu$ is fitted to give the the signal magnitude while the signal shape is determined by the theory model or the well-known physics (for example, we use the Breit-Wigner formula convoluted with a Gaussian function to describe the shape of a resonance). This procedure is common in model-dependent measurements as well as many model-independent measurements. If the signal shape is determined by a model, we usually need to consider theoretical uncertainty due to this model.
Only one systematical uncertainty source is introduced in the current model. The uncertainty allows the background distribution to deviate from the prediction and the deviation at each bin should behave in a coherent way. Thus we introduce one nuisance parameter $\alpha$.
However, the MC statistical uncertainty should be considered differently. It is due to limited sample size in the MC simulation. It is usually true that the simulation is done randomly in all bins and there is no bin-by-bin correlation. Hence we introduce one nuisance parameter $\gamma_i$ ($i=1,2,\ldots,N$) for each bin.
Ignoring the constant terms, the log likelihood function is
\begin{equation}
\ln\mathcal{L} = -\frac{\alpha^2}{2} + \sum_{i=1}^N \left[ n_i\ln(\mu s_i+\gamma_i b_i +\alpha\Delta_i) - (\mu s_i +\gamma_i b_i +\alpha\Delta_i) + m_i\ln\gamma_i - \gamma_i m_i\right] \: .
\end{equation}
Unlike previous model considering a single number-counting experiment, the present model can be seen as a combination of $N$ number-counting measurements. It is nearly impossible to solve out the the best-fit $\mu$ analytically by maximizing the present log likelihood function. Let us use an Asimov dataset~\cite{asimov}, where $n_i = b_i + s_i$ for all bins. This option will not change the conclusion in this paper as we are studying the uncertainty of the fitting parameters. The best-fit values are then
\begin{equation}
\hat{\mu} = 1 \:, \hat{\alpha} = 0\:, \hat{\gamma} = 1 \: .
\end{equation}
Letting the parameters are arranged in the order of $\mu,\alpha,\gamma_1,\gamma_2,\cdots,\gamma_N$, the inverse of the covariance matrix $\bold{V}$ is
\begin{equation}
\bold{V}^{-1} =
\begin{pmatrix}
s\otimes s & s\otimes \Delta & s_1*b_{1} & s_2*b_{2} & \cdots & s_N*b_{N} \\
s\otimes \Delta & 1 + \Delta\otimes \Delta & \Delta_{1}*b_{1} & \Delta_{2}*b_{2} & \cdots & \Delta_{N}*b_{N} \\
s_1*b_{1} & \Delta_{1}*b_{1} & m_1 + \frac{b_{1}^2}{n_1} & 0 & \cdots & 0 \\
s_2*b_{2} & \Delta_{2}*b_{2} & 0 & m_2 + \frac{b_{2}^2}{n_2} & \cdots & 0 \\
\vdots & \vdots& \vdots & \vdots & \ddots & \vdots \\
s_N*b_{N} & \Delta_{N}*b_{N} & 0 & 0 & \cdots & m_N + \frac{b_{N}^2}{n_N} \\
\end{pmatrix} \: .
\end{equation}
To simplify the expression, the sign $*$ is introduced with the definition $x_i*y_i \equiv \frac{x_iy_i}{n_i}$ and the sign $\otimes$ is introduced with the definition $x\otimes y \equiv \sum_{i=1}^N \frac{x_iy_i}{\sqrt{n_i}^2}$ where the summation is over all bins. Here we keep the form $\sqrt{n_i}^2$ to remind us that it represents the Poisson fluctuation. Let us analyse some matrix elements in the first place.
\begin{itemize}
\item $s\otimes s = \sum_i \frac{s_i^2}{\sqrt{n_i}^2}$ represents the signal significance compared to the statistical fluctuation indicated by the denominator $\sqrt{n_i}$. We expect that this term determines the measurement precision of the signal strength $\mu$ if no systematic uncertainties or MC statistical uncertainty is present.
\item $\Delta \otimes \Delta = \sum_i \frac{\Delta_i^2}{\sqrt{n_i}^2}$ represents the significance of the systematic uncertainty compared to the Poisson statistical fluctuation. If this term is big, we expect that $\alpha$ could be over-constrained and may have a large impact on the $\mu$ uncertainty.
\item $s\otimes \Delta = \sum_i \frac{s_i\Delta_i}{\sqrt{n_i}^2}$ represents the correlation of the signal shape and the systematic variation. If this term is big, it means that the systematic variation is similar to the signal shape and we expect that $\alpha$ would have a large impact on the $\mu$ uncertainty.
\item $s_i*b_i$ ($\Delta_i* b_i$) describes the contribution to the shape correlation between signal (systematic uncertainty) and background from the $i$-th bin.
\end{itemize}
To obtain the covariance matrix itself, we decompose the inverse matrix into two parts, $\bold{A}$ and $\bold{B}$.
\begin{equation}
\bold{V}^{-1} = \bold{A}+ \bold{B} = \bold{A} (\mathbf{1} + \bold{A}^{-1}\bold{B})
\end{equation}
with $\mathbf{1}$ being the identity matrix.
The matrix $\bold{A}$ contains all the diagonal elements and the correlation term $s\otimes \Delta$ while the matrix $\bold{B}$ contains all other non-diagonal elements. It should be mentioned that it is easy to calculate the inverse matrix of $\bold{A}$.
\begin{equation}
\bold{A} =
\begin{pmatrix}
s\otimes s & s\otimes \Delta & 0 & 0 & \cdots & 0 \\
s\otimes \Delta & 1 + \Delta\otimes \Delta & 0& 0& \cdots & \\
0 & 0 & m_1 + \frac{b_{1}^2}{n_1} & 0 & \cdots & 0 \\
0 & 0 & 0 & m_2 + \frac{b_{2}^2}{n_2} & \cdots & 0 \\
\vdots & \vdots& \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & 0 & \cdots & m_N + \frac{b_{N}^2}{n_N} \\
\end{pmatrix}
\end{equation}
\begin{equation}
\bold{B} =
\begin{pmatrix}
0 & 0 & s_1*b_{1} & s_2*b_{2} & ... & s_N*b_{N} \\
0 & 0 & \Delta_{1}*b_{1} & \Delta_{2}*b_{2} & ... & \Delta_{N}*b_{N} \\
s_1*b_{1} & \Delta_{1}*b_{1} & 0 & 0 & ... & 0 \\
s_2*b_{2} & \Delta_{2}*b_{2} & 0 & 0 & ... & 0 \\
\vdots & \vdots& \vdots & \vdots & \ddots & \vdots \\
s_N*b_{N} & \Delta_{N}*b_{N} & 0 & 0 & ... & 0 \\
\end{pmatrix}
\end{equation}
Therefore the covariance matrix can be expressed in the following way.
\begin{equation}\label{eq:VBA}
\bold{V} = (\mathbf{1} + \bold{A}^{-1}\bold{B})^{-1}\bold{A}^{-1} = (\mathbf{1}+\sum_{i=1}^{\infty}(-\bold{A}^{-1}\bold{B})^i)\bold{A}^{-1}
\end{equation}
where the fact that $\mathbf{1}=(\mathbf{1}+\bold{x})(\mathbf{1}-\bold{x}+\bold{x}^2-\bold{x}^3+\cdots)$ is used. In the Appendix~\ref{app:Identity}, we show that it is valid to apply this identity in our case with the assumptions which will be introduced later. To further simplify the expression, we introduce the following sub-matrices according to the vanishing blocks in $\bold{V}$.
\begin{equation}
\bold{V}=
\begin{pmatrix}
\bold{V}_1 & \bold{V}_2 \\
\bold{V}_2^T & \bold{V}_3 \\
\end{pmatrix}
\:, \quad
\bold{A}^{-1} =
\begin{pmatrix}
\bold{a}_1 & 0 \\
0 & \bold{a}_2 \\
\end{pmatrix}
\:, \quad
\bold{B} =
\begin{pmatrix}
0 & \bold{b} \\
\bold{b}^{T} & 0 \\
\end{pmatrix}
\end{equation}
Here $\bold{a}_1$, $\bold{a}_2$ and $\bold{b}$ are
\begin{equation}\label{eq:a1}
\bold{a}_1 = \frac{1}{(1+\Delta\otimes \Delta)s\otimes s-(s\otimes \Delta)^2}
\begin{pmatrix}
1 + \Delta\otimes \Delta & - s\otimes \Delta \\
-s\otimes \Delta & s\otimes s\\
\end{pmatrix}\:,
\end{equation}
\begin{equation}
\bold{a}_2 =
\begin{pmatrix}
(m_1+\frac{b_1^2}{n_1})^{-1} & 0 & \cdots & 0 \\
0 & (m_2+\frac{b_2^2}{n_2})^{-1} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & (m_N+\frac{b_N^2}{n_N})^{-1} \\
\end{pmatrix}\:,
\end{equation}
and
\begin{equation}
\bold{b} =
\begin{pmatrix}
s_1 * b_1 & s_2 * b_2 & \cdots & s_N * b_N \\
\Delta_1 * b_1 & \Delta_2 * b_2 & \cdots & \Delta_N * b_N \\
\end{pmatrix}\:.
\end{equation}
Then $\bold{V}_1$ is the covariance matrix for $\mu$ and $\alpha$. From Eq.~\ref{eq:VBA} and letting $\bold{X}\equiv \bold{a}_1\bold{b}\bold{a}_2\bold{b}^T $, we can show that
\begin{equation}\label{eq:V1}
\bold{V}_1 = (\mathbf{1} + \bold{X} + \bold{X}^2 + \bold{X}^3 + \cdots) \bold{a}_1 \:.
\end{equation}
To make approximation, we write down the explict expression for $\bold{X}$.
\begin{equation}\label{eq:X}
\bold{X} = \bold{a}_1
\begin{pmatrix}
s\hat{\otimes}s & s\hat{\otimes} \Delta \\
s\hat{\otimes}\Delta & \Delta\hat{\otimes} \Delta \\
\end{pmatrix}
\approx
\begin{pmatrix}
\frac{s\hat{\otimes}s}{s\otimes s} & 0\\
0 & \frac{\Delta\hat{\otimes} \Delta}{1 + \Delta\otimes \Delta} \\
\end{pmatrix}
\end{equation}
Here the sign $\hat{\otimes}$ is defined as $x\hat{\otimes}y \equiv \sum_{i=1}^N \frac{x_iy_i}{n_i (1 + \frac{n_i}{\delta_i^2})}$. In Eq.~\ref{eq:X}, $\bold{X}$ is made diagonal if we only care about the diagonal elements of $\bold{V}_1$ and assume that the MC statistical uncertainty have the similar size relative to the Poisson fluctuation in all bins and that the non-diagonal elements are much smaller than the diagonal elements, namely, $(s\otimes \Delta)^2<<(s\otimes s)(\Delta\otimes\Delta)$. The details are presented in the Appendix~\ref{app:X}. These assumptions also allow us to express the covariance matrix as a power series as shown in the Appendix~\ref{app:Identity}. $s\otimes\Delta$ describes the correlation between signal shape and the systematical variation. Thus the approximation is reasonable because the signal shape is usually peaky while the systematic variation is random, which will be estimated quantitatively later. We emphasize that the approximation is only applied to $\bold{X}$, not to $\bold{a}_1$ in Eq.~\ref{eq:V1}. Therefore, this correlation is still partly considered. The diagnonal elements of $\bold{V}_1$ give the uncertainty of $\mu$ and $\alpha$. Defining $\check{\otimes}$ as $x\check{\otimes}y \equiv x\otimes y - x\hat\otimes y=\sum_i \frac{x_iy_i}{n_i + \delta_i^2}$, we have
\begin{equation}\label{eq:dmu}
\hat{\sigma}_{\mu} =\sqrt{(\bold{V}_1)_{11}} \approx \frac{1}{\sqrt{s\check{\otimes}s \left[1 - \frac{(s\otimes \Delta)^2}{s\otimes s(1+\Delta\otimes \Delta)}\right]}} \: ,
\end{equation}
and
\begin{equation}\label{eq:dalpha}
\hat{\sigma}_{\alpha} = \sqrt{(\bold{V}_1)_{22}} \approx \frac{1}{\sqrt{(1+\Delta\check{\otimes} \Delta)\left[1 -\frac{(s\otimes \Delta)^2}{s\otimes s(1+\Delta\otimes \Delta)}\right]}} \: .
\end{equation}
From Eq.~\ref{eq:dalpha}, it turns out that the nuisance parameter could be over-constrained if the systematic variation is significant compared to the combination of the Poisson statistical fluctuation and the MC statistical uncertainty as indicated by the term $\Delta \check{\otimes} \Delta=\sum_{i=1}^N\frac{\Delta_i^2}{\sqrt{n_i}^2 + \delta_i^2}$. We also see that the correlation term $s\otimes\Delta$ exists in both Eq.~\ref{eq:dmu} and Eq.~\ref{eq:dalpha}. This term is assumed to be much smaller than $(s\otimes s)(\Delta\otimes \Delta)$ in the derivation. Using the Cauchy-Schwarz inequality, it is true that $(s\otimes\Delta)^2\leq(s\otimes s)(\Delta\otimes\Delta)$. For a quantitative understanding, let us consider a simplified case where the signal events fall in a single bin, the background distribution is uniform and the systematic variation has the same absolute size in all bins. It is easy to show that
\begin{equation}\label{eq:cauchy}
\frac{(s\otimes\Delta)^2}{(s\otimes s)(\Delta\otimes\Delta)} = \frac{(\frac{s\Delta}{n/N})^2}{\frac{s^2}{n/N}\sum_{i=1}^N\frac{\Delta^2}{n/N}} = \frac{1}{N} \: .
\end{equation}
We can see that this assumption is valid as long as the signal shape is peaky enough (as indicated by the factor of $1/N$). The correlation term is of small contribution to the constraint of $\alpha$ in Eq.~\ref{eq:dalpha}. The impact on $\mu$ uncertainty, however, is determined by this correlation as shown in Eq.~\ref{eq:dmu}.
In the model above, only one systematic source is considered. It is not difficult to extend it to the case of multiple systematic sources. The log likelihood function is
\begin{equation}\label{eq:logL_Nsys}
\ln\mathcal{L} = -\sum_{j=1}^M\frac{\alpha_j^2}{2} + \sum_{i=1}^N\left[ n_i\ln(\mu s_i+\gamma_i b_i +\sum_{j=1}^{M}\alpha_j\Delta_i^{j}) - (\mu s_i +\gamma_i b_i +\sum_{j=1}^M\alpha_j\Delta_i^j) + m_i\ln\gamma_i - \gamma_i m_i\right] \: ,
\end{equation}
where $M$ is the number of systematic items with $M$ nuisance parameters $\alpha_1,\alpha_2,\cdots,\alpha_M$. With the parameters arranged in the order $(\mu,\alpha_1,\alpha_2,\cdots,\alpha_M,\gamma_1,\gamma_2,\cdots,\gamma_N)$, the inverse of the covariance matrix is
\begin{equation} \label{eq:invV_Nsys}
\bold{V}^{-1} =
\begin{pmatrix}
s\otimes s & s\otimes \Delta^1 & s\otimes \Delta^2 & \cdots & s\otimes\Delta^M & s_1*b_{1} & s_2*b_{2} & \cdots & s_N*b_{N} \\
s\otimes \Delta^1 & 1 + \Delta^1\otimes \Delta^1 & \Delta^1\otimes\Delta^2 & \cdots & \Delta^1\otimes\Delta^M & \Delta_{1}^1*b_{1} & \Delta_{2}^1*b_{2} & \cdots & \Delta_{N}^1*b_{N} \\
s\otimes\Delta^2 & \Delta^1\otimes\Delta^2 & 1 + \Delta^2\otimes\Delta^2 & \cdots &\Delta^2\otimes\Delta^M & \Delta_1^2*b_1 & \Delta_2^2*b_2&\cdots & \Delta_N^2 * b_N \\
\vdots & \vdots& \vdots & \cdots & \vdots & \vdots & \vdots &\cdots & \vdots\\
s\otimes\Delta^M & \Delta^1\otimes\Delta^M & \Delta^2\otimes\Delta^M & \cdots & 1+\Delta^M\otimes\Delta^M & \Delta_1^M*b_1 & \Delta_1^M*b_2 & \cdots & \Delta_N^M*b_N \\
s_1*b_{1} & \Delta_{1}^1*b_{1} & \Delta_1^2*b_1 & \cdots & \Delta_1^M*b_1 & m_1 + \frac{b_{1}^2}{n_1} & 0 & \cdots & 0 \\
s_2*b_{2} & \Delta_{2}^1*b_{2} & \Delta_2^2*b_2 & \cdots & \Delta_2^M*b_2 & 0 & m_2 + \frac{b_{2}^2}{n_2} & \cdots & 0 \\
\vdots & \vdots& \vdots & \cdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
s_N*b_{N} & \Delta_{N}^1*b_{N} &\Delta_N^2*b_N & \cdots & \Delta_N^M*b_N & 0 & 0 & \cdots & m_N + \frac{b_{N}^2}{n_N} \\
\end{pmatrix} \: .
\end{equation}
We can see that new elements $\Delta^i\otimes\Delta^j = \sum_{k=1}^N \frac{\Delta_k^i\Delta_k^j}{\sqrt{n_k}^2}$ appear. They represent the correlation between two different systematic uncertainties for $i\neq j$. We further assume that this kind of correlation is small ($(\Delta^i\otimes\Delta^j)^2<<(\Delta^i\otimes\Delta^i)(\Delta^j\otimes\Delta^j)$) and MC statistical uncertainty is also small. It is not difficult to derive (the calculation details and some discussions on the approximation precision can be found in the Appendix~\ref{app:main}) that
\begin{equation}\label{eq:dmu_Nsys}
\hat{\sigma}_{\mu} \approx \frac{1}{\sqrt{s\otimes s}} \sqrt{1 + \frac{s\hat{\otimes} s}{s\check{\otimes}s} + \sum_{j=1}^M \left[\frac{(s\otimes \Delta^j)^2}{s\otimes s(1+\Delta^j\otimes \Delta^j)} - \sum_{i\neq j} \frac{(s\otimes\Delta^i)(s\otimes\Delta^j)(\Delta^i\otimes\Delta^j)}{s\otimes s (1+\Delta^i\otimes\Delta^i)(1+\Delta^j\otimes\Delta^j)} \right]} \: ,
\end{equation}
and
\begin{equation}\label{eq:dalpha_Nsys}
\hat{\sigma}_{\alpha_i} \approx \frac{1}{\sqrt{1+\Delta^i \otimes \Delta^i}}\sqrt{1 + \frac{\Delta^i\hat{\otimes}\Delta^i}{1+\Delta^i\check{\otimes}\Delta^i} +\frac{(s\otimes\Delta^i)^2}{s\otimes s(1+\Delta^i\otimes\Delta^i)}+\sum_{j\neq i} \frac{(\Delta^i\otimes\Delta^j)^2}{(1+\Delta^i\otimes\Delta^i)(1+\Delta^j\otimes\Delta^j)}} \: .
\end{equation}
Eq.~\ref{eq:dmu_Nsys} can be decomposed into three terms
\begin{equation}\label{eq:dmu3}
\hat{\sigma}_{\mu}^2 = \hat{\sigma}_{\mu}^{02} + \sum_{j=1}^M\hat{\sigma}_{\mu}^{\alpha^j 2} + \hat{\sigma}_{\mu}^{\gamma2} \:,
\end{equation}
and the individual terms are
\begin{eqnarray}
&& \hat{\sigma}_{\mu}^{02} = \frac{1}{s\otimes s} \label{eq:impact0}\\
&& \hat{\sigma}_{\mu}^{\alpha^j2}=\hat{\sigma}_{\mu}^{02}\left[\frac{(s\otimes \Delta^j)^2}{s\otimes s(1+\Delta^j\otimes \Delta^j)}-\sum_{i\neq j} \frac{(s\otimes\Delta^i)(s\otimes\Delta^j)(\Delta^i\otimes\Delta^j)}{s\otimes s (1+\Delta^i\otimes\Delta^i)(1+\Delta^j\otimes\Delta^j)}\right] \label{eq:impact_alpha}\\
&& \hat{\sigma}_{\mu}^{\gamma2} = \hat{\sigma}_{\mu}^{02}\frac{s\hat{\otimes} s}{s\check{\otimes}s} \label{eq:impact_gamma}\: .
\end{eqnarray}
They represent the contribution to $\mu$ uncertainty from the data statistics, the systematic uncertainty ($\alpha^j$) and the MC statistical uncertainty ($\gamma$), respectively.
In many ATLS and CMS measurements involved with fits, a pruning algorithm is usually applied before performing the fit. It is to prune those systematic uncertainties that are not important for the measurement target and thus to reduce the fitting time. In practice, one would always perform two fits with or without using the pruning algorithm in case any important factors are missed. Part of the reason is that most of the pruning conditions originate from intuitive understanding and are not directly related with the signal sensitivity. The formulae above can be used to develop more reliable pruning criteria.
Here we propose three conditions.
\begin{eqnarray}
\frac{(s\otimes \Delta)^2}{s\otimes s(1+\Delta\otimes \Delta)} &<& 0.02 \label{eq:prune1}\\
\frac{(s\otimes \Delta)^2}{s\otimes s(1+\Delta\otimes \Delta)} &<& 0.1 \frac{s\hat{\otimes} s}{s\check{\otimes}s} \label{eq:prune2}\\
\frac{s_i^2/[n_i(1 + \frac{n_i}{\delta_i^2})]}{s\check{\otimes}s} &<& 0.02 \label{eq:prune3}
\end{eqnarray}
Basically, we can ignore a systematical uncertainty if its impact on the signal strength uncertainty is much smaller than the impact of data statistics (inequality~\ref{eq:prune1}) or MC statistical uncertainty (inequality~\ref{eq:prune2}). Numerically, we choose 0.02 as the threshold in the inequality~\ref{eq:prune1} because the change of $\hat{\sigma}_{\mu}$ due to omitting the systematical uncertainty is about 1~\% ($\sqrt{1+0.02}\approx 1 +0.01$) and 0.1 in the inequality~\ref{eq:prune1} so that the systematical effect is one order of magnitude smaller than that of the MC statistical uncertainty, but the thresholds can be re-optimized. The effect of the correlation between different systematical uncertainties is usually minor and thus not included in the inequalities (but we can always use the full expression above instead). Similarly, we can also ignore the nuisance parameter corresponding to the MC statistical uncertainty in the $i$-th bin according to the inequality~\ref{eq:prune3}.
In the end of this section, let us comment on the validity of the formulae. The main assumption is small correlation between signal shape and systematical variations and small correlation between different systematic uncertainties. The former part is usually true in the searches for resonance-like signals as shown in Eq.~\ref{eq:cauchy}, where the signal shape is peaky while the systematical variation is relatively smooth. However, the latter part is not always true. Taking the top-quark pair ($t\bar{t}$) background in any typical measurement at LHC as example, the $t\bar{t}$ production cross section uncertainty would be anti-correlated with the uncertainty of the tagging efficiency of jets originated from beauty hadrons. Both uncertainties affect the $t\bar{t}$ background normalization and this correlation may be not small inevitably. If both systematical uncertainties happen to be important to the signal sensitivity, we admit that it is not precise to calculate $\hat{\sigma}_\mu$ using Eq.~\ref{eq:dmu_Nsys}. On the other hand, we can always check the precision by looking at the omitted sub-leading terms. For $\hat{\sigma}_{\mu}$, these terms (see Appendix~\ref{app:main}) look like
\begin{eqnarray}
&& \sum_{i\neq j}\frac{(s\otimes\Delta^i)^2}{s\otimes s(1+\Delta^i\otimes\Delta^i)}\frac{(s\otimes\Delta^j)^2}{s\otimes s(1+\Delta^j\otimes\Delta^j)} \nonumber \\
+ && \sum_{i\neq j, i\neq k, j\neq k} \frac{(s\otimes\Delta^i) (\Delta^i\otimes\Delta^j) (\Delta^j\otimes\Delta^k) (s\otimes\Delta^k)}{s\otimes s(1+\Delta^i\otimes\Delta^i)(1+\Delta^j\otimes\Delta^j)(1+\Delta^k\otimes\Delta^k)}
\end{eqnarray}
They are important only when there are multiple systematical uncertainties that are mutually highly correlated or highly correlated with the signal shape. But we expect that this case is not often seen under normal circumstances. The other assumption is small MC statistical uncertainty. This is usually true. Otherwise, one would probably seek for data-driven methods or request to produce larger MC samples.
\section{\label{sec:example} A pseudo experiment for illustration}
In this section, we present a pseudo experiment of searching for a resonance on a mass spectrum. The signal is simulated with a Gaussian distribution $G(x|1000,50)$ with the resonance mass 1000~GeV and mass resolution 50~GeV. We assume the width of the resonance is small compared to the mass resolution. Two background components (denoted by ``bkg1'' and ``bkg2'') are introduced and simulated with exponential distributions with the decay parameter 1000~GeV and 5000~GeV. Each signal/background event is given a constant weight for simplicity. Table~\ref{tab:example_model} summarizes the information for the signal and background models. They are also shown in Fig.~\ref{fig:exam_nom}.
\begin{table}
\caption{\label{tab:example_model} Information for the signal and background models.
}
\begin{ruledtabular}
\begin{tabular}{l l l l}
& Weighted number of events & Model & Event wegiht\\
\hline
Signal & 100 & Gauss distribution $G(x|1000, 50)$ & 0.1 \\
bkg1 & 6000 & Exponential distribution $e^{-x/1000}$ & 0.2 \\
bkg2 & 4000 & Exponential distribution $e^{-x/5000}$ & 0.5 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}
\includegraphics[width=0.5\textwidth]{Cs_nominal.pdf}
\caption{\label{fig:exam_nom}
The signal and background distributions in the presudo experiment and the signal is scaled by a factor of 5 for illustration.
}
\end{figure}
We introduce 5 nuisance parameters, where 3 NPs represent 3 shape-only systematic uncertainties (applied to both signal and background components) and 2 NPs represent 2 norm-only systematic uncertainties (applied to bkg1 and bkg2 separately). Here ``shape-only'' means that the systematic item only affects the shape of the observable distribution while ``norm-only'' means it only affects the normalization. To implement the shape-only systematic uncertainties, we apply a random variation to the nominal distribution in each bin, where the random variation is generated with a Gaussian distribution with the mean 0 and different standard deviations.
As shown in Table~\ref{tab:example_systs}, ``ShapeBig/ShapeMedium/ShapeSmall'' label the size of shape-only the systematic effect. They correpsond to a standard deviation of $\Delta n$, $0.5\Delta n$, and $0.1\Delta n$ respectively. Here $\Delta n$ denotes the statistic uncertainty in that bin. A normalization uncertainty of 10~\% is applied to bkg1 and denoted by ``NormBig'' while that of 5~\% is applied to bkg2 and denoted by ``NormSmall''.
All the systematic uncertainties are summarized in Table~\ref{tab:example_systs}. The envelope plots for the systematic uncertainties are shown in Fig.~\ref{fig:exam_envelope1} and Fig.~\ref{fig:exam_envelope2}. In these plots, the blue/red histogram represents the ``high''/``low'' variation for a systematic item. They represent the initial estimation of this systematic uncertainty. In addition, MC statistical uncertainty is also considered.
\begin{table}
\caption{\label{tab:example_systs} Information for the systematic uncertainties. Here $\Delta n$ denotes the total statistical uncertainty.
}
\begin{ruledtabular}
\begin{tabular}{l l l}
Nuisance parameter name & Applied to sample & Variation Size \\
\hline
ShapeBig & signal, bkg1, bkg2 &$G(x|0,\Delta n)$ \\
ShapeMedium &signal, bkg1, bkg2 & $G(x|0, 0.5\Delta n)$ \\
ShapeSmall & signal, bkg1, bkg2 & $G(x|0, 0.1\Delta n)$ \\
NormBig & bkg1 & 10\% \\
NormSmall &bkg2 & 5\% \\
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}
\includegraphics[width=0.32\textwidth]{Cs_testSR_ShapeBig.pdf}
\includegraphics[width=0.32\textwidth]{Cs_testSR_ShapeMedium.pdf}
\includegraphics[width=0.32\textwidth]{Cs_testSR_ShapeSmall.pdf}
\caption{\label{fig:exam_envelope1}
Envelope plots for the shape-only systematic uncertainties, namely, ShapeBig (L), ShapeMedium (M) and ShapeSmall (R). The blue/red histograms represent the high/low variation of the systematic item. The signal is scaled by a factor of 5 for illustration. The lower pad shows the ratio of systematic variation and the nominal distribution. The black vertical error bars represent the MC statistical uncertainty and the hatch histogram represents the Poisson statistical fluctuation.
}
\end{figure}
\begin{figure}
\includegraphics[width=0.45\textwidth]{Cs_testSR_NormBig.pdf}
\includegraphics[width=0.45\textwidth]{Cs_testSR_NormSmall.pdf}
\caption{\label{fig:exam_envelope2}
Envelope plots for the norm-only systematic uncertainties, namely, NormBig (L) and NormSmall (R). The blue/red histograms represent the high/low variation of the systematic item. The signal is scaled by a factor of 5 for illustration. The lower pad shows the ratio of systematic variation and the nominal distribution. The black vertical error bars represent the MC statistical uncertainty and the hatch histogram represents the Poisson statistical fluctuation.
}
\end{figure}
For this toy measurement, the fit is performed using a tool based on the HistoFactory~\cite{histofactory}, where advanced numerical tools are used to determine the covariance matrix precisely. In Table~\ref{tab:example_results}, the fitting results and the approximate calculations using the Eqs.~\ref{eq:dalpha_Nsys}, \ref{eq:impact0}, \ref{eq:impact_alpha} and \ref{eq:impact_gamma} are summarized for comparison. To use the equations, we have (taking $s\otimes \Delta$ as example)
\begin{equation}\label{eq:example_SB}
s\otimes\Delta^{\text{high/low}} = \sum_{i=1}^N \frac{S_i(N_i^{\text{high/low}}-N_i^{\text{nom}})}{\sqrt{N_i^{\text{nom}}}^2} \: ,
\end{equation}
where $S_i$ is the number of signal events in $i$-th bin; $N_i^{\text{nom}}$ is the predicted total number of events; $N_i^{\text{high/low}}$ is the total number of events corresponding to the systematic ``high/low'' variation.
We can see that only the ShapeSmall NP is mildly constrained while all others are over-constrained. The calculated constraint is consistent with that from the fit for the shape-only systematic uncertainties while this consistence is not very good for the norm-only systematic uncertainties. One of the reasons is that we are using a linear interpolation strategy instead of the exponential interpolation strategy used in the fit~\cite{histofactory} when implementing the norm-only systematic uncertainty. Taking the model in Sec.~\ref{sec:simple_model} as example, the linear interpolation is $b\pm\alpha\Delta$ while the exponential interpolation is $b(1\pm\frac{\Delta}{b})^{\alpha^\prime}$ (a ``$\prime$'' is added to distinguish from that in linear interpolation). $\alpha\approx\alpha^\prime$ only if $\Delta/b$ is small and we can show that $\sigma_\alpha = \frac{\ln(1+\Delta/b)}{\Delta/b}\sigma_{\alpha^\prime}<\sigma_{\alpha^\prime}$. The calculated impact is also fairly consistent with that in the fit although the correlation between different systematic items and the correlation between signal shape and the systematic variation are not fully considered. Especially, it should be noted that the calculation is able to indicate which systematic items would be important. For example, ShapeMedium would have a larger impact than ShapeSmall as its size is bigger by our design. But from either the fit or the approximate calculation, its impact is smaller. There are two reasons behind. One is that the ShapeMedium variation is larger and thus the corresponding NP is more constrained. The other is that the correlation between the signal shape and the ShapeSmall variation turns out to be bigger than the ShapeMedium variation. The latter point can be also seen by comparing the middle and right plots in Fig.~\ref{fig:exam_envelope1}. Similarly, we find that the NormSmall systematic item has actually a larger impact than the NormBig systematic item.
\begin{table}
\caption{\label{tab:example_results}
Comparison of the fitting results and the approximate calculations.
}
\begin{ruledtabular}
\begin{tabular}{l | l l | l l }
& \multicolumn{2}{l|}{Constraint ($\hat{\sigma}_{\alpha}$ in Eq.~\ref{eq:dalpha_Nsys})} & \multicolumn{2}{l}{Impact ($\hat{\sigma}_{\mu}^0$,$\hat{\sigma}_{\mu}^{\alpha}$ and $\hat{\sigma}_{\mu}^{\gamma}$ in Eqs.~\ref{eq:impact0}-\ref{eq:impact_gamma})} \\
\hline
Nuisance parameter name & Fit & Calculation & Fit & Calculation\\
\hline
ShapeBig & $\pm0.26$ & $\pm0.24$ & $ _{-0.116}^{+0.112}$ & $_{-0.080}^{+0.082}$\\
ShapeMedium & $\pm0.48$ & $\pm0.48$ & $_{-0.045}^{+0.038}$ & $_{-0.015}^{+0.015}$\\
ShapeSmall & $\pm0.94$ & $\pm0.94$ & $_{-0.045}^{+0.045}$ & $_{-0.040}^{+0.040}$\\
NormBig & $\pm0.29$ & $\pm 0.23$ &$_{-0.015}^{+0.024}$& $_{-0.022}^{+0.022}$\\
NormSmall & $\pm0.78$ & $\pm 0.61$ &$_{-0.058}^{+ 0.051}$& $_{-0.044}^{+0.044}$\\
MC Stat. Unc. & & & $_{-0.19}^{+0.18}$ & $\pm0.17$ \\
Data statistics & & & $_{-0.28}^{+0.29}$ & $\pm0.29$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
In this pseudo experiment, the norm and shape components of a systematical item are considered separately. But it is trivial to apply the calculation if a systematic item affects both the normalization and shape of the observable distribution.
\section{\label{sec:summary}Summary}
In summary, the constraint and impact on the POI of nuisance parameters in maximum likelihood method are studied. Based on simplified models, we find that a nuisance parameter will be over-constrained if the corresponding variation is large compared to the total statistical uncertainty (the combination of the Poisson statistic fluctuation and the MC statistic uncertainty). It will have a large impact on the POI uncertainty if the corresponding variation has a strong correlation with the signal shape. Assuming small MC statistical uncertainty, small correlation between different systematic uncertainties and small correlation between signal shape and systematic variation, simple formulae (Eqs.~\ref{eq:dmu_Nsys},~\ref{eq:dalpha_Nsys},~\ref{eq:impact0},~\ref{eq:impact_alpha}, and \ref{eq:impact_gamma}) are derived to calculate the constraint and impact. A toy experiment is also performed and shows fair consistence between the calculation and that using the current fitting tool. In many measurements by ATLAS or CMS collaborations, complicated fits are involved and advanced numerical methods are developed to obtain the covariance matrix very precisely in the fitting tools. This study is not to provide a replacement of the numerical methods, but helps to cross-check the potential features in the fitting results in an easy and direct way. It can also help us to improve the pruning algorithms adopted in many fitting tools.
\section{Acknowledgement}
I admit that this work will not be finished without the close cooperation with Christopher John Mcnicol, Elisabetta Pianori and Paul Thompson. I would also like to thank Liang Zhong and Mengzhen Wang for discussions on some mathematics topics and Fang Dai for encouraging words.
\begin{appendix}
\section{Validation of the identity of $\mathbf{1} = (\mathbf{1}+\bold{x})(\mathbf{1}-\bold{x}+\bold{x}^2-\bold{x}^3+\cdots)$}\label{app:Identity}
In this section, let us explain it is valid to apply the identity of $\mathbf{1} = (\mathbf{1}+\bold{x})(\mathbf{1}-\bold{x}+\bold{x}^2-\bold{x}^3+\cdots)$ in our case. In the first place, it is not difficult to show that the sufficient and necessary condition for this identity is that the absolute value of every eigenvalue of $\bold{x}$ is less than 1. We then show this condition is satisfied in our case with $\bold{x} = \bold{A}^{-1}\bold{B}$. Using the sub-matrices $\bold{a}_1$, $\bold{a}_2$ and $\bold{b}$ defined in Sec.~\ref{sec:realistic_model}, we have
\begin{equation}
\bold{x} = \begin{pmatrix}
0 & \bold{a}_1\bold{b} \\
\bold{a}_2\bold{b}^T & 0 \\
\end{pmatrix}\: .
\end{equation}
The eigenvalues can be found by solving the equation $\det|\bold{x}-\lambda\mathbf{1}|=0$. We resort to Schur's determinant identity, namely,
\begin{equation}\label{eq:schur}
\det\left|\begin{matrix} \bold{A} & \bold{B} \\
\bold{C} & \bold{D} \\
\end{matrix}\right| = \det|\bold{D}|\det|\bold{A}-\bold{B}\bold{D}^{-1}\bold{C}|
\end{equation}
which holds if $\bold{D}$ is invertible. Applying it to $\det|\bold{x}-\lambda\mathbf{1}|$, we have
\begin{equation}
\det|\bold{x}-\lambda\mathbf{1}| = \det\left|\begin{matrix}
-\lambda \mathbf{1}_{2\times 2} & \bold{a}_1\bold{b} \\
\bold{a}_2\bold{b}^T & -\lambda \mathbf{1}_{N\times N} \\
\end{matrix}\right|
=\det|-\lambda\mathbf{1}_{N\times N}|\det|-\lambda\mathbf{1}_{2\times 2}+\frac{1}{\lambda}\bold{X}| \: ,
\end{equation}
where $\mathbf{1}_{2\times 2}$ and $\mathbf{1}_{N\times N}$ denote the $2\times 2$ and $N\times N$ identity matrices respectively and $\bold{X}\equiv \bold{a}_1\bold{b}\bold{a}_2\bold{b}^{T}$ as defined in Sec.~\ref{sec:realistic_model}. Assuming $(s\otimes s)(\Delta\otimes\Delta) >> (s\otimes\Delta)^2$, we find that $\bold{X}$ is approximately an upper triangular matrix as shown in Appendix~\ref{app:X}. Using Eq.~\ref{eq:appX_X}, it is easy to obtain
\begin{equation}
\det|\bold{x}-\lambda\mathbf{1}| \approx (-\lambda)^{N}(-\lambda + \frac{\bold{X}_{11}}{\lambda})(-\lambda + \frac{\bold{X}_{22}}{\lambda}) \:.
\end{equation}
Therefore, the only non-vanishing eigenvalues are $|\lambda|=\sqrt{\bold{X}_{11}}\approx \sqrt{\frac{s\hat{\otimes}s}{s\otimes s}}< 1$, and $ |\lambda|=\sqrt{\bold{X}_{22}}\approx \sqrt{\frac{\Delta\hat{\otimes}\Delta}{1+\Delta\otimes\Delta}}<1$. Both of them are less than 1, which guarantee that we can apply the identity $\mathbf{1} = (\mathbf{1}+\bold{x})(\mathbf{1}-\bold{x}+\bold{x}^2-\bold{x}^3+\cdots)$ with the assumption $(s\otimes s)(\Delta\otimes\Delta) >> (s\otimes\Delta)^2$.
\section{Approximation in the expression of $\bold{X}$}\label{app:X}
In this section, let us explain that the approximation in Eq.~\ref{eq:X} is from the main assumption $\epsilon \equiv \frac{(s\otimes\Delta)^2}{(s\otimes s)(\Delta\otimes\Delta)}<<1$. First of all, we can ignore the difference between $\hat{\otimes}$ and $\otimes$ in Eq.~\ref{eq:X} by assuming the MC statistical uncertainty has the same size relative to the Poisson fluctuation, namely, $\frac{\delta_i}{\sqrt{n_i}} = r$ for all bins. This assumption
is unnecessary because we only need $\frac{(s\otimes\Delta)^2}{(s\otimes s)(\Delta\otimes\Delta)}<<1$ which also leads to $\frac{(s\hat{\otimes}\Delta)^2}{(s\otimes s)(\Delta\otimes\Delta)}<<1$ with $0<s\hat{\otimes}\Delta<s\otimes\Delta$. But we keep using this assumption as it is usually true and brings convenience. Using Eq.~\ref{eq:a1}, Eq.~\ref{eq:X} becomes
\begin{eqnarray}\label{eq:appX_X}
\bold{X} &=& \bold{a}_1
\begin{pmatrix}
s\hat{\otimes}s & s\hat{\otimes}\Delta \\
s\hat{\otimes}\Delta & \Delta\hat{\otimes}\Delta\\
\end{pmatrix} \approx c\bold{a}_1
\begin{pmatrix}
s\otimes s & s \otimes\Delta \\
s\otimes\Delta & \Delta\otimes\Delta\\
\end{pmatrix}\\
\bold{X}_{11} &\approx& c \\
\bold{X}_{12} &\approx& c \frac{s\otimes\Delta}{(1+\Delta\otimes\Delta)s\otimes s - (s\otimes\Delta)^2} \approx c\frac{\epsilon_1}{s\otimes\Delta}\\
\bold{X}_{21} &\approx& 0 \\
\bold{X}_{22} &\approx& c\frac{(s\otimes s)(\Delta\otimes\Delta) - (s\otimes\Delta)^2}{(1+\Delta\otimes\Delta)s\otimes s - (s\otimes\Delta)^2} \approx c\frac{\Delta\otimes\Delta}{1+\Delta\otimes\Delta}(1-\frac{\epsilon}{1+\Delta\otimes\Delta}) \label{eq:X22}
\end{eqnarray}
where $c=\frac{1}{1+\frac{1}{r^2}} \approx \frac{s\hat{\otimes}s}{s\otimes s} \approx \frac{\Delta\hat{\otimes}\Delta}{\Delta\otimes \Delta}$ and $\epsilon_1 \equiv \frac{\Delta\otimes\Delta}{1+\Delta\otimes\Delta}\epsilon < \epsilon$.
We note that $\bold{X}$ is approximately an upper triangular matrix.
By the assumption $\epsilon<<1$, we can neglect the term proportional to $\epsilon$ for the element $\bold{X}_{22}$, but we cannot neglect $\bold{X}_{12}$ as it is possible that $s\otimes\Delta$ is small and $\frac{\epsilon_1}{s\otimes\Delta}$ is big.
The trick is that we can neglect $\bold{X}_{12}$ as long as we only care about the diagonal elements in the covariance matrix, $\bold{V}_1$. Noting that $\bold{V}_1 = (1+\bold{X} + \bold{X}^2+\ldots)\bold{a}_1$, we can show it order by order. The zeroth-order term is $\bold{V}^{(0)}=\bold{a}$ (in this section, we omit the subscript ``1'' in $\bold{V}_1$ and $\bold{a}_1$ for cleaness.) and does not contain terms about $\bold{X}_{12}$. Let us start with the first-order term, $\bold{V}^{(1)}=\bold{X}\bold{a}$ and focus on the diagonal element $\bold{V}_{11}^{(1)}$ (for the
other one $\bold{V}_{22}^{(1)}$, any term containing $\bold{X}_{12}$ will always contain $\bold{X}_{21}$ and thus be vanishing. This is also true for higher-order terms in $\bold{V}$.).
\begin{eqnarray}\label{eq:v1}
\bold{V}_{11}^{(1)} = \bold{X}_{11}\bold{a}_{11}(1 + \frac{\bold{X}_{12}\bold{a}_{21}}{\bold{X}_{11}\bold{a}_{11}}) \approx \bold{X}_{11}\bold{a}_{11}(1 - \frac{\epsilon_1}{1+\Delta\otimes\Delta} )
\end{eqnarray}
We can neglect the term proportional to $\epsilon$, which is equivalent to neglecting the element $\bold{X}_{12}$. Let us rewrite the equation above as $\frac{\bold{V}_{11}^{(1)}}{\bold{X}_{11}\bold{a}_{11}}=1+\mathcal{O}(\epsilon)$ where $\mathcal{O}(\epsilon)$ denotes any term of the order of $\epsilon$. For the second-order term $\bold{V}^{(2)}=\bold{X}^2\bold{a}$, the diagonal element can be expressed as (noting that $\bold{V}^{(2)}= \bold{X}\bold{V}^{(1)}$)
\begin{eqnarray}\label{eq:v2}
\bold{V}_{11}^{(2)}&=& \bold{X}_{11}\bold{V}_{11}^{(1)} ( 1 + \frac{\bold{X}_{12}\bold{V}_{21}^{(1)}}{\bold{X}_{11}\bold{V}_{11}^{(1)}}) \\
&=& \bold{X}_{11}\bold{V}_{11}^{(1)} ( 1 + \frac{\bold{X}_{12}\bold{X}_{22}\bold{a}_{21}}{\bold{X}_{11}\bold{X}_{11}\bold{a}_{11}}) \\
&=& \bold{X}_{11}\bold{V}_{11}^{(1)} \left[ 1 + \frac{\bold{X}_{22}}{\bold{X}_{11}}\left(\frac{\bold{V}_{11}^{(1)}}{\bold{X}_{11}\bold{a}_{11}}-1\right)\right] \: .
\end{eqnarray}
We can see that $\frac{\bold{V}_{11}^{(2)}}{\bold{X}_{11}\bold{V}_{11}^{(1)}} = 1 + \mathcal{O}(\epsilon)$ because of Eq.~\ref{eq:v1}. Similarly, the third-order term $\bold{V}_{11}^{(3)}$ is
\begin{equation}
\bold{V}_{11}^{(3)}= \bold{X}_{11}\bold{V}_{11}^{(2)} \left[ 1 + \frac{\bold{X}_{22}}{\bold{X}_{11}}\left(\frac{\bold{V}_{11}^{(2)}}{\bold{X}_{11}\bold{V}_{11}^{(1)}}-1\right)\right] \: ,
\end{equation}
and thus $\frac{\bold{V}_{11}^{(3)}}{\bold{X}_{11}\bold{V}_{11}^{(2)}}=1+\mathcal{O}(\epsilon)$.
Using the method of mathematical induction, we can show that $\frac{\bold{V}_{11}^{(n+1)}}{\bold{X}_{11}\bold{V}_{11}^{(n)}}=1+\mathcal{O}(\epsilon)$ for any $n\geq 1$. Therefore, if we only care about the diagonal elements of $\bold{V}$, neglecting the terms of the order of $\epsilon$ is equivalent to that we neglect $\bold{X}_{12}$ in the beginning and hence $\bold{X}$ is made diagonal in Eq.~\ref{eq:X}.
\section{Some calculation details}\label{app:main}
In this section, we present some calculation details to derive Eq.~\ref{eq:dmu_Nsys} and Eq.~\ref{eq:dalpha_Nsys} in the case of $M$ systematical uncertainty sources. The matrices $\bold{V}$, $\bold{A}^{-1}$ and $\bold{B}$ can be written in the following form.
\begin{equation}
\bold{V}=
\begin{pmatrix}
\bold{V}_1 & \bold{V}_2 \\
\bold{V}_2^T & \bold{V}_3 \\
\end{pmatrix}
\:, \quad
\bold{A}^{-1} = \begin{pmatrix} \bold{a}_1 & 0 \\ 0 & \bold{a}_2 \end{pmatrix} \:, \quad
\bold{B} = \begin{pmatrix} \bold{a}_3 & \bold{b} \\ \bold{b}^T & 0 \end{pmatrix} \: ,
\end{equation}
where the sub-matrices $\bold{a}_2$ and $\bold{b}$ are the same as in the main text while $\bold{a}_1$ and $\bold{a}_3$ are
\begin{eqnarray}
\bold{a}_1 &=& \begin{pmatrix}
\frac{1}{s\otimes s} & & & & \\
&\frac{1}{\Delta^1\otimes\Delta^1} & & & \\
&&\frac{1}{\Delta^2\otimes\Delta^2} & & \\
&&& \ddots & \\
&&&& \frac{1}{\Delta^M\otimes\Delta^M} \\
\end{pmatrix} \:,\\
\bold{a}_3 &=& \begin{pmatrix}
0 & s\otimes\Delta^1 & s\otimes\Delta^2 & \cdots & s\otimes\Delta^M\\
s\otimes\Delta^1&0 & \Delta^1\otimes\Delta^2 &\cdots & \Delta^1\otimes\Delta^M \\
s\otimes\Delta^2&\Delta^1\otimes\Delta^2&0&\cdots &\Delta^2\otimes\Delta^M \\
\vdots&\vdots&\vdots& \ddots & \vdots \\
s\otimes\Delta^M&\Delta^1\otimes\Delta^M&\Delta^2\otimes\Delta^M&\cdots &0 \\
\end{pmatrix} \: .
\end{eqnarray}
We assume that the covariance matrix $\bold{V}$ can be expressed as a series, namely, $\bold{V} = [1+\sum_{i=1}^{+\infty}(-\bold{A}^{-1}\bold{B})^i]\bold{A}^{-1}$. Let us investigate $\bold{x} \equiv \bold{A}^{-1}\bold{B}$ to check this assumption.
\begin{equation}
\bold{x} = \begin{pmatrix}
\bold{a}_1\bold{a}_3 & \bold{a}_1\bold{b} \\ \bold{a}_2\bold{b}^T & 0
\end{pmatrix}
\end{equation}
Using the same procedure presented in Appendix~\ref{app:Identity}, the eigenvalues of $\bold{x}$ can be found from the equation below.
\begin{eqnarray}
\det|\bold{x}| &=& \det|-\lambda\mathbf{1}_{N\times N}|\det|\bold{a}_1\bold{a}_3 - \lambda\mathbf{1}_{1+M,1+M} + \frac{1}{\lambda}\bold{X}| \label{eq:detx} \\
&\approx& (-\lambda)^N(-\lambda+\frac{1}{\lambda}\frac{s\hat{\otimes}s}{s\otimes s})\Pi_{i=1}^M\left(-\lambda+\frac{1}{\lambda}\frac{\Delta^i\hat{\otimes}\Delta^i}{1+\Delta^i\otimes\Delta^i}\right) \label{eq:detx1}
\end{eqnarray}
where $\bold{X}\equiv \bold{a}_1\bold{b}\bold{a}_2\bold{b}^T$, the same definition as in the main text. The terms containing $\bold{X}$ arise from the MC statistical uncertainty. It is difficult to solve Eq.~\ref{eq:detx}. But if the correlations are small, namely, $\frac{(s\otimes\Delta^i)^2}{(s\otimes s)(\Delta^i\otimes\Delta^i)}<<1$ and $\frac{(\Delta^i\otimes\Delta^j)^2}{(\Delta^i\otimes\Delta^i)(\Delta^j\otimes\Delta^j)}<<1$, and MC statistical uncertainty is small, we can expect the determinant in Eq.~\ref{eq:detx} is dominated by the contribution from the diagonal elements and hence we have Eq.~\ref{eq:detx1} in the limits $s\otimes\Delta^i \to 0$, $\Delta^i\otimes\Delta^j \to 0$ and $\delta^i \to 0$. Noting that $0<s\hat{\otimes}s<s\otimes s$ and $0<\Delta^i\hat{\otimes}\Delta^i < \Delta^i\otimes\Delta^i$, it is easy to see that the absolute value of all eigenvalues is less than 1 and thus it is valid for this series expansion.
Now let us present some calculation details for the main results Eq.~\ref{eq:dmu_Nsys} and~\ref{eq:dalpha_Nsys}. We are interested in the top left block of the $\bold{V}$, $\bold{V}_1$. The first few terms are
\begin{eqnarray}
\bold{V}_1^{(0)} &=& \bold{a}_1 \: ,\\
\bold{V}_1^{(1)} &=& -(\bold{a}_1\bold{a}_3)\bold{a}_1 \:, \\
\bold{V}_1^{(2)} &=& [(\bold{a}_1\bold{a}_3)^2 + \bold{X}] \bold{a}_1 \:, \\
\bold{V}_1^{(3)} &=&-[(\bold{a}_1\bold{a}_3)^3 + (\bold{a}_1\bold{a}_3)\bold{X} + \bold{X}(\bold{a}_1\bold{a}_3) ] \bold{a}_1 \:,\\
\bold{V}_1^{(4)} &=&[(\bold{a}_1\bold{a}_3)^4 + (\bold{a}_1\bold{a}_3)^2\bold{X} + (\bold{a}_1\bold{a}_3)\bold{X}(\bold{a}_1\bold{a}_3) + \bold{X}(\bold{a}_1\bold{a}_3)^2 + \bold{X}^2 ] \bold{a}_1 \:.
\end{eqnarray}
Let us focus on the diagonal elements of $\bold{V}_1$. We find that
\begin{eqnarray}
(\bold{a}_1\bold{a}_3\bold{X})_{ii} &=& \sum_{j\neq i} (\bold{a}_1)_{ii}(\bold{a}_3)_{ij}(\bold{a}_1)_{jj}(\bold{b}\bold{a}_2\bold{b}^T)_{ij} \:, \\
(\bold{X}\bold{a}_1\bold{a}_3)_{ii} &=& \sum_{j\neq i}(\bold{a}_1)_{ii}(\bold{b}\bold{a}_2\bold{b}^T)_{ij}(\bold{a}_1)_{jj}(\bold{a}_3)_{ji} \:,
\end{eqnarray}
which lead to $(\bold{a}_1\bold{a}_3\bold{X})_{ii} = (\bold{X}\bold{a}_1\bold{a}_3)_{ii}$ as $\bold{a}_1$, $\bold{a}_2$ and $\bold{a}_3$ are symmetrical matrices. Therefore we find that
\begin{eqnarray}
(\bold{V}_1)_{ii} =&& \left[ \sum_{n=0}^{+\infty}(-1)^n(\bold{a}_1\bold{a}_3)^n + \sum_{n=0}^{+\infty}(1+f_n(\bold{a}_1\bold{a}_3))\bold{X}^n \right]_{ii}(\bold{a}_1)_{ii}\:, \label{eq:V} \\
(\bold{V}_1)_{ii} \approx && \left[ \sum_{n=0}^{3}(-1)^n[(\bold{a}_1\bold{a}_3)^n]_{ii} + \sum_{n=0}^{+\infty}(\bold{X}_{ii})^n \right](\bold{a}_1)_{ii} \label{eq:V0} \\
= && \left[ \sum_{n=0}^{3}(-1)^n[(\bold{a}_1\bold{a}_3)^n]_{ii} + \frac{\bold{X}_{ii}}{1-\bold{X}_{ii}} \right](\bold{a}_1)_{ii} \:. \label{eq:V1}
\end{eqnarray}
Here $f_n(\bold{a}_1\bold{a}_3)$ is a power series about $\bold{a}_1\bold{a}_3$, for example, $f_1(\bold{a}_1\bold{a}_3) = \sum_{i=1}^{+\infty}(-1)^i(1+i)(\bold{a}_1\bold{a}_3)^i$. $f_n(\bold{a}_1\bold{a}_3)\bold{X}^n$ (and the non-diagonal elements of $\bold{X}$) represent the mixing contributions from MC statistical uncertainty and the correlation between signal shape and systematical variation or the correlation between different systematical uncertainties. To derive Eq.~\ref{eq:V}, we have to assume that both MC statistical uncertainty and these correlations are small so that it is valid to represent $\bold{V}$ as a series. Because MC statistical uncertainty and other systematical uncertainty are usually independent, it is difficult to keep terms in a consistent way if do not know their sizes. From Eq.~\ref{eq:V} to Eq.~\ref{eq:V0}, we keep the leading terms considering the correlation between different systematical uncertainties and the dominant terms considering MC statistical uncertainty and omit the mixing contributions, which seems reasonable.
The main results, namely, Eq.~\ref{eq:dmu_Nsys} and Eq.~\ref{eq:dalpha_Nsys}, are derived from the Eq.~\ref{eq:V1}. In practice, we care more about the systematic uncertainty source than the MC statistical uncertainty (if MC statistical uncertainty is dominant, we will usually resort to data-driving methods or increase the MC statistics). To estimate the precision of $\hat{\sigma}_{\mu}$ using Eq.~\ref{eq:dmu_Nsys}, we can investigate the term, $(\bold{a}_1\bold{a}_3)^4\bold{a}_1$ (In fact, it is not difficult to write down the expression for a general term $(\bold{a}_1\bold{a}_3)^n\bold{a}_1$ by induction).
\begin{eqnarray}
[(\bold{a}_1\bold{a}_3)^4]_{11}(\bold{a}_1)_{11} = && \sum_{i\neq j}\frac{(s\otimes\Delta^i)^2}{s\otimes s(1+\Delta^i\otimes\Delta^i)}\frac{(s\otimes\Delta^j)^2}{s\otimes s(1+\Delta^j\otimes\Delta^j)} \nonumber \\
+ && \sum_{i\neq j, i\neq k, j\neq k} \frac{(s\otimes\Delta^i) (\Delta^i\otimes\Delta^j) (\Delta^j\otimes\Delta^k) (s\otimes\Delta^k)}{s\otimes s(1+\Delta^i\otimes\Delta^i)(1+\Delta^j\otimes\Delta^j)(1+\Delta^k\otimes\Delta^k)}
\end{eqnarray}
Obviously, they are subleading contributions compared to Eq.~\ref{eq:dmu_Nsys}. They are important only when there are multiple systematical uncertainties which are highly correlated with each other or with the signal shape. For the precision of Eq.~\ref{eq:dalpha_Nsys} to calculate $\hat{\sigma}_{\alpha_i}$, we can look at the omitted sub-leading terms.
\begin{equation}
-2\sum_{j\neq i} \frac{(s\otimes\Delta^i)(s\otimes\Delta^j)(\Delta^i\otimes\Delta^j)}{s\otimes s(1+\Delta^i\otimes\Delta^i)(1+\Delta^j\otimes\Delta^j)} - \sum_{j\neq i, k\neq i, k\neq j} \frac{(\Delta^i\otimes\Delta^j)(\Delta^i\otimes\Delta^k)(\Delta^j\otimes\Delta^k)}{(1+\Delta^i\otimes\Delta^i)(1+\Delta^j\otimes\Delta^j)(1+\Delta^k\otimes\Delta^k)}
\end{equation}
They are minor contributions compared to Eq.~\ref{eq:dalpha_Nsys}. Using the pseudo experiment described in Sec.~\ref{sec:example}, we confirm that considering these terms will pull the calculated results closer to the results from the fitting tool. But the improvement is limited, and we do not want to include them to make the formulae too cumbersome.
\end{appendix}
|
2,869,038,156,340 | arxiv | \section{Introduction}
Astronomers are not sure how common the double nucleus phenomenon is. This happens because the centers of galaxies are obscured by dust and gas and, therefore, have yield few of their secrets. Fortunately this is beginning to change with the advent of space-based observations. Galaxies with reported double nuclei are M31 (Statler et al., 1999), the barred spiral galaxy M83 (Mast et al., 2006; Soria \& Wu, 2002), NGC 6240 (Komosca et al., 2003; Risaliti et al., 2006), NGC 3256 (Lira et al., 2007) and the dwarf elliptical galaxy VCC 128 (Debattista et al., 2006). A detailed catalog of disk galaxies with double nuclei was compiled by Cimeno et al. (2004).
On this basis, there is no doubt that the construction of a 3D dynamical model in order to study the motion in a galaxy with a double nucleus would be of interest. As far as the authors know there are few dynamical models for the study of motion in galaxies with double nuclei (see Jalali \& Rafie, 2001; Samlhus \& Sridhar, 2002).
This article can be considered as a continuation of the results presented in a recent paper (see Caranicolas \& Papadopoulos, 2009; hereafter CP). In Section 2 we present the dynamical model. In Section 3 the regular or chaotic character of orbits in the 2D model is explored. In Section 4 we study the nature of motion in the 3D model. A discussion and the conclusions of this research are presented in Section 5.
\section{The 3D-model}
The model is an extension of the model used in CP in the 3D space with an additional disk. Thus the potential of the first body is
\begin{equation}
V_1(x,y,z) = -\frac{M_d}{\left[x^2 + y^2 +\left(\alpha + \sqrt{h^2 + z^2}\right)^2\right]^{1/2}}
- \frac{M_{n1}}{\left(x^2 + y^2 + z^2 + c_{n1}^2\right)^{1/2}},
\end{equation}
where $M_d$, $M_{n1}$ is the mass of disk and nucleus 1 respectively, $\alpha$ is the disk's scale length, $h$ is the disk's scale height, while $c_{n1}$ is the scale length of nucleus 1. The second nucleus is described by the potential
\begin{equation}
V_2(x,y,z) = -\frac{M_{n2}}{\left(x^2 + y^2 + z^2 + c_{n2}^2\right)^{1/2}},
\end{equation}
where $M_{n2}$, $c_{n2}$ is the mass and the scale length of the nucleus 2 respectively. As in CP, the two bodies move in circular orbits in an inertial frame OXYZ with the origin at the center of mass of the system at a constant angular velocity $\Omega_p >0$.
Let $M_t=M_d+M_{n1}+M_{n2}$ be the total mass of the system and $R$ be the distance between the two bodies. A clockwise rotating frame Oxyz is used with axis Oz coinciding with the axis OZ and the axis Ox coinciding with the straight line joining the two bodies. In this frame, which rotates at an angular velocity $\Omega_p$, the two bodies have fixed positions $x_1,y_1=0$ and $x_2,y_2=0$. The total potential responsible for the motion of a test particle (star) with a unit mass is
\begin{equation}
\Phi(x,y,z) = \Phi_1(x,y,z) + \Phi_2(x,y,z),
\end{equation}
where
\begin{equation}
\Phi_1(x,y,z) = -\frac{M_d}{\sqrt{r_{a1}^2 + \left(\alpha + \sqrt{h^2 + z^2}\right)^2}}
- \frac{M_{n1}}{\sqrt{r_1^2 + c_{n1}^2}} - \frac{M_{n2}}{\sqrt{r_2^2 + c_{n2}^2}},
\end{equation}
\begin{equation}
\Phi_2(x,y,z) = -\frac{\Omega_p^2}{2}\left[\frac{M_{n2}}{M_t}r_{a2}^2 + \left(1 - \frac{M_{n2}}{M_t}\right)r_{a1}^2
- R^2\frac{M_{n2}}{M_t}\left(1 - \frac{M_{n2}}{M_t}\right)\right],
\end{equation}
and
\begin{equation}
r_{a1}^2 = (x - x_1)^2 + y^2, \ \ \ r_{a2}^2 = (x - x_2)^2 + y^2,
\end{equation}
\begin{equation}
r_1^2 = r_{a1}^2 + z^2, \ \ \ r_2^2 = r_{a2}^2 + z^2,
\end{equation}
with
\begin{equation}
x_1 = -\frac{M_{n2}}{M_t}R, \ \ \ x_2 = R - \frac{M_{n2}}{M_t}R.
\end{equation}
The angular frequency $\Omega_p$ is calculated as follows: The two bodies move about the center of mass of the system with angular velocities $\Omega_{p1}$, $\Omega_{p2}$ given by
\begin{equation}
\Omega_{p1} = \left[\frac{1}{x_1}\left(\frac{-dV_2(r)}{dr}\right)_{r=R}\right]^{1/2}, \ \ \
\Omega_{p2} = \left[\frac{1}{x_2}\left(\frac{dV_1(r)}{dr}\right)_{r=R}\right]^{1/2},
\end{equation}
where $r^2 = x^2 + y^2$. As the two bodies are not mass points the two angular frequencies are not equal. Nevertheless, we can make them equal by choosing properly the parameters $\alpha, h, c_{n1}, c_{n2}$ of the system. The authors must make clear that, after choosing properly the parameters the two frequencies may differ slightly, so that $\nu = (\Omega_{p1} - \Omega_{p2})/\Omega_{p1}$ is of order of $10^{-6}$ or smaller. Therefore, we consider the two angular frequencies practically equal, that is $\Omega_{p1} = \Omega_{p2} = \Omega_p$.
The equations of motion are written
\begin{equation}
\ddot{x} = - \frac{\partial \Phi}{\partial x} - 2 \Omega_p\dot{y}, \ \ \
\ddot{y} = - \frac{\partial \Phi}{\partial y} + 2 \Omega_p\dot{x}, \ \ \
\ddot{z} = - \frac{\partial \Phi}{\partial z},
\end{equation}
where the dot indicates derivative with respect to the time. The only integral of motion for the system of differential equations (10) is
\begin{equation}
J = \frac{1}{2} \left(p_x^2 + p_y^2 + p_z^2 \right) + \Phi(x,y,z) = E_J,
\end{equation}
where $p_x$, $p_y$ and $p_z$ are the momenta per unit mass conjugate to $x$, $y$ and $z$ respectively. This is the well known Jacobi integral and $E_J$ is its numerical value. In this work we use the same system of galactic units, as in CP. The unit of length is 1 kpc, the unit of mass is 2.325 $\times 10^7$ $M_{\odot}$ and the unit of time is 0.977 $\times 10^8$ yr. The velocity unit is equal to 10 km/s, while $G$ is equal to unity.
\section{Orbits in the 2D potential}
In this section we shall study the 2D potential. Thus we set $z=p_z=0$ in (11) obtaining
\begin{equation}
J_2 = \frac{1}{2}\left(p_x^2 + p_y^2 \right) + \Phi(x,y) = E_{J2},
\end{equation}
where $E_{J2}$ is the numerical value of $J_2$. As the system is now two-dimensional, we can use the classical method of the $(x,p_x)$, $y=0, p_y>0$, Poincar\'{e} phase plane in order to explore the regular or chaotic character of motion. The results obtained from the study of the 2D system will be used in order to determine the character of orbits in the 3D system.
Our results come from the numerical integration of the equations of motion, which was done using a Bulirsh-St\"{o}er method in double precision. The accuracy of the calculations was checked by the constancy of the Jacobi integral, which was conserved up to the twelfth significant figure.
Fig. 1 shows the $(x,p_x)$ phase plane when $M_d=1100, M_{n1}=100, M_{n2}=600, R=1.5, c_{n1}=0.10, c_{n2}=0.35, \Omega_p=22.1938, \alpha=0.3053, h=0.06$, while $E_{J2}=-2010$. As one can see there is a large unified chaotic sea surrounding both nuclei. One also observes two separate regular regions near each nucleus. It is of interest to note that the regular regions are around the retrograde periodic points in each of the two nuclei. Some smaller regular regions are also present near the smaller nucleus 1 on the left. One additional feature is the presence of several small islands near the heavy nucleus 2 on the right. These small islands are produced by secondary resonances.
Fig. 2 is similar to Fig. 1 but when $M_d=1100, M_{n1}=300, M_{n2}=400, R=1.5, c_{n1}=0.10, c_{n2}=0.25, \Omega_p=22.6244, \alpha=0.2181, h=0.06$, while $E_{J2}=-2100$. Here again we see a large unified chaotic sea surrounding both nuclei. The two regular areas around the retrograde periodic points of each nucleus are also present. On the other hand, a careful observer is able to see some significant differences between the two patterns. In Fig. 2 the regular area near the nucleus 1 on the left is now considerably larger than that observed in Fig. 1. Furthermore an additional small regular region has appeared near direct periodic point of the nucleus 1 on the left. Moreover the corresponding regular area near the nucleus 2 on the right has now become smaller.
\begin{figure}
\centering
\resizebox{0.6\hsize}{!}{\rotatebox{0}{\includegraphics*{Fig1.eps}}}
\caption{The $(x,p_x)$ phase plane when $M_d=1100, M_{n1}=100, M_{n2}=600, R=1.5,$
$c_{n1}=0.10, c_{n2}=0.35, \Omega_p=22.1938, \alpha=0.3053, h=0.06$. The value of $E_{J2}$ is equal to $-2010$.}
\end{figure}
\begin{figure}
\centering
\resizebox{0.6\hsize}{!}{\rotatebox{0}{\includegraphics*{Fig2.eps}}}
\caption{Similar to Fig. 1 but when $M_d=1100, M_{n1}=300, M_{n2}=400, R=1.5,$
$c_{n1}=0.10, c_{n2}=0.25, \Omega_p=22.6244, \alpha=0.2181, h=0.06$, while $E_{J2}=-2100$.}
\end{figure}
This can be explained, if we take into account that the observed chaotic motion near each nucleus is a result not only of the force coming from the nucleus itself but also of the force coming from the other nucleus. As the mass of nucleus 2 is smaller while the mass of the disk and the distance $R$ between the two nuclei are the same, the influence from this nucleus to the disk and the nucleus 1 system is smaller. This has as a result the reduction of chaotic region and a parallel increase of the regular regions associated with the nucleus 1 and the disk. On the other hand, for similar reasons, as the mass of nucleus 1 has increased, the regular area associated with nucleus 2 has become smaller.
The numerical results suggest that there are two kinds of chaotic orbits: (i) chaotic orbits approaching both nuclei and (ii) chaotic orbits that approach only one of the two nuclei. On the other hand, the regular orbits circulate around only one of the two nuclei.
\section{Orbits in the 3D potential}
In this section we shall study the properties of orbits in the 3D potential. In order to keep things simple we shall use our experience gained from the study of the 2D system in order to obtain a clear picture of the properties of orbits in the 3D dynamical model.
We are particularly interested to locate the initial conditions in the 3D model producing regular or chaotic orbits. A convenient way to obtain this is to start from the $(x,p_x)$ phase plane of the 2D system with the same value of the Jacobi integral used in the 2D system. Thus we take $E_J=E_{J2}$. For this purpose a large number of orbits were computed with initial conditions $(x_0,p_{x0},z_0)$, where $(x_0,p_{x0})$ is a point in the chaotic sea of Figs. 1 or 2 with all permissible values of $z_0$, and $p_{z0}=0$. Remember that, as we are on the phase plane, we have $y_0=0$, while in all cases the value of $p_{y0}$ was obtained from the Jacobi integral. All tested orbits were found to be chaotic. Therefore, one concludes that the majority of orbits in the 3D system are chaotic.
An interesting question one might ask is this. Are there any other chaotic orbits in the 3D system? In order to give an answer we have taken the sections of the 3D orbit with the plane $y=0$, when $p_y >0$. The set of the four-dimensional points $(x,p_x,z,p_z)$ was projected on the $(z,p_z)$ plane. If the projected points lie on an ``invariant curve" this suggests that the motion is regular, while, if not, this is an indication that the motion is chaotic. Fig. 3 shows such ``invariant curves" for orbits starting near the regular region on the right side of Fig. 1. In order to obtain the results shown in Fig. 3 we have taken the point $(x_0,p_{x0})=(-0.17,0.0)$ representing approximately the position of the periodic orbits in the $(x,p_x)$, $y=0, p_y>0$, phase plane and a set of values of $z_0=(0.02, 0.05, 0.08, 0.11, 0.15, 0.20, 0.25)$. Note that the numerical results indicate that, for small values of $z_0$ the motion is regular, while for larger values of $z_0$, the motion seems to be chaotic. Here we must emphasize, that the results of Fig. 3 are rather qualitative and can be considered as an indication that the transition from regularity to chaos occurs as the value of $z_0$ increases. Results not given here show a similar behavior near each regular region in Figs. 1 and 2.
\begin{figure}[!tH]
\centering
\resizebox{0.6\hsize}{!}{\rotatebox{0}{\includegraphics*{Fig3.eps}}}
\caption{Projection of the sections of the 3D orbit with the plane $y=0$, when $p_y >0$. The set of the four dimensional points $(x,p_x,z,p_z)$ is projected on the $(z,p_z)$ plane. }
\end{figure}
In order to estimate the degree of chaos in the 2D and the 3D system we have calculated the maximum LCE (Lyapunov Characteristic Exponent) (see Lichtenberg \& Lieberman, 1992) for a large number of chaotic orbits. Each LCE was calculated for a time period of $10^4$ time units. The LCE for the 2D orbits, starting in the chaotic sea of Fig. 1, was found in the range $5.0-5.2$, while in the chaotic sea of Fig. 2 it was found in the range $6.0-6.2$. The LCE for chaotic orbits in the 3D system with initial conditions $(x_0, p_{x0}, z_0)$ with $(x_0,p_{x0})$ in the chaotic sea of the Fig. 1 was found in the range $3.2-3.4$. The corresponding values of LCE for Fig.2 were found in the range $3.9-4.1$.
\begin{figure*}[!tH]
\centering
\resizebox{0.8\hsize}{!}{\rotatebox{0}{\includegraphics*{Fig4a.eps}}\hspace{1cm}
\rotatebox{0}{\includegraphics*{Fig4b.eps}}}
\vskip 0.1cm
\caption{(a-b): (a-left) A chaotic orbit approaching both nuclei. The initial conditions are $x_0=0.8, y_0=p_{x0}=0, z_0=0.1, p_{z0}=0$. The values of the other parameters are as in Fig. 1, while $E_J=-2010$. (b-right) A regular orbit circulating around nucleus 1 with initial conditions $x_0=1.4, y_0=p_{x0}=0, z_0=0.15, p_{z0}=0$. The values of the other parameters are as in Fig. 2 and $E_J=-2100$.}
\end{figure*}
\begin{figure*}[!tH]
\centering
\resizebox{0.8\hsize}{!}{\rotatebox{0}{\includegraphics*{Fig5a.eps}}\hspace{1cm}
\rotatebox{0}{\includegraphics*{Fig5b.eps}}}
\vskip 0.1cm
\caption{(a-b): (a-left) A regular orbit and (b-right) a chaotic orbit. The two orbits differ in initial conditions only in the value of $z_0$. See text for details.}
\end{figure*}
\begin{figure*}[!tH]
\centering
\resizebox{0.8\hsize}{!}{\rotatebox{0}{\includegraphics*{Fig6a.eps}}\hspace{1cm}
\rotatebox{0}{\includegraphics*{Fig6b.eps}}}
\vskip 0.1cm
\caption{(a-b): (a-left) Evolution of $L_z$ with the time for the regular orbit of Fig. 5a and (b-right) for the chaotic orbit of Fig. 5b.}
\end{figure*}
Fig. 4a shows a chaotic orbit with conditions $x_0=0.8, y_0=p_{x0}=0, z_0=0.1, p_{z0}=0$. The value of $p_{y0}$ is always found from the Jacobi integral. The values of the other parameters are as in Fig. 1, while $E_J=-2010$. Note that the orbit goes arbitrary close to both nuclei. It is interesting to observe that near the more massive nucleus the orbit is deflected to more higher values of $z$, while near the less massive nucleus the star stays close to the disk. Fig. 4b shows a regular orbit circulating around nucleus 1. The initial conditions are $x_0=1.4, y_0=p_{x0}=0, z_0=0.15, p_{z0}=0$. The values of the other parameters are as in Fig. 2, while $E_J=-2100$.
Fig. 5a shows a quasi-periodic orbit starting near the retrograde periodic point, which is close to nucleus 1. The initial conditions are $x_0=0, y_0=p_{x0}=0, z_0=0.1, p_{z0}=0$. The values of the other parameters are as in Fig. 2, while $E_J=-2100$. Fig. 5b shows an orbit with the same initial conditions, the same value of the Jacobi integral and the same values of the parameters as in Fig. 5a but when $z_0=0.5$. The orbit has now become chaotic and goes arbitrarily close to the nucleus 1. This orbit shows, from another point of view, that 3D orbits starting near the stable periodic points of the 2D system are regular only for small values of $z_0$.
Here the physical parameter playing an important role is the $L_z$ component of the angular momentum. From our previous experience we know that low angular momentum stars, on approaching a dense nucleus are scattered off the galactic plane displaying chaotic motion (Caranicolas \& Innanen, 1991; Caranicolas \& Papadopoulos, 2003). Of course here in 3D space things are more complicated than in an axially symmetric dynamical model, where $L_z$ is conserved (see Caranicolas \& Innanen, 1991). As $L_z$, is not conserved, we can compute numerically the mean value $< L_z >$ of $L_z$ using the formula
\begin{equation}
< L_z > = \frac{1}{n}\displaystyle\sum_{i}^{n}L_{zi}.
\end{equation}
Our numerical calculations suggest that the chaotic orbits have low values of $< L_z >$, while regular orbits have high values of $< L_z >$. Fig. 6a shows the evolution of $L_z$ with the time for the regular orbit of Fig. 5a. Here we find a value of $< L_z >=12.5$. Fig. 6b is similar to Fig. 6a but for the chaotic orbit of Fig. 5b. Here $< L_z >=-10.3$ The value of $n$ in both cases was $10^4$.
\section{Discussion and conclusions}
Observation data show that a small fraction of active galaxies have double nuclei (Eracleus \& Halpen, 2003; Xinwu \& Ting-Gui, 2006). It was this reason that motivated our construction of a 3D model in order to study the motion in a disk galaxy hosting a binary nucleus. It was found that the majority of orbits in the 2D system were chaotic. Two kinds of chaotic orbits were observed: (i) Chaotic orbits that approach both nuclei and (ii) Chaotic orbits that approach only one of the nuclei. The regular regions are confined mainly around the retrograde periodic points in both nuclei. All regular orbits go around nucleus 1 or nucleus 2 but not both. It was also found that the total velocity near each nucleus attains high values. The value of velocity depends on the mass of the nucleus and the value of its scale length. Regular motion corresponds to low velocities while chaotic motion is characterized by high velocities.
In order to understand the behavior of orbits in the 3D system we have used our knowledge from the 2D system. Of particular interest was the determination of the region of initial conditions in the $(x,p_x,z,p_y)=E_J$, $(y=p_z=0)$ phase space that produces regular or chaotic 3D orbits. As $p_{y0}$ was found always from the Jacobi integral we have used the same value of $E_J$, as in the 2D system and took initial condition $(x_0,p_{x0},z_0)$ such as $(x_0,p_{x0})$ lies in the chaotic region of the 2D system. It was found that the motion was chaotic for all permissible values of $z_0$. In the case when $(x_0,p_{x0})$ was inside a regular region, the corresponding 3D orbit was regular only for small values of $z_0$, while for larger values of $z_0$ the orbit was chaotic. The particular values of $z_0$ were different for each regular region of the 2D system. We did not feel that it was necessary to try to define the particular values of $z_0$ for each case.
An important role is played by the $L_z$ component of the test particle's angular momentum. It was found that the values of $< L_z >$ for regular orbits are larger, than those corresponding to chaotic orbits. Thus, the $L_z$ component of the angular momentum is a significant parameter connected with the regular or chaotic character of orbits in both 2D and 3D galactic models.
In order to estimate the degree of chaos in the 2D as well as in the 3D potentials, we have computed the maximum Lyapunov Characteristic Exponent (LCE) for a large number of orbits for a time period of $10^4$ time units. The numerical results indicate that the degree of chaos in 3D double nucleus systems is smaller than in similar 2D systems.
\section*{Acknowledgement}
\textit{The authors would like to thank an anonymous referee for his useful suggestions and comments.}
\section*{References}
|
2,869,038,156,341 | arxiv | \section{Introduction}
\label{sec:introduction}
The notion of \textit{self-contracted curve} captures, under a simple metric definition (see forthcoming Definition~\ref{def:self_contracted}), characteristic properties of the gradient flow of a convex function, relevant for convergence. The term appeared for the first time in~\cite{DLS2010}, where it was shown that \emph{planar} self-contracted curves are rectifiable. Later on, exploring an old geometrical idea of Manselli-Pucci (see~\cite{MP1991}), the previous result has been extended to any finite dimensional Euclidean space. In particular, in~\cite{DDDL2015} (and independently in \cite{LMV2015} assuming continuity of the curves) it was shown that the length of any self-contracted curve in $\mathbb{R}^d$ is controlled by a universal constant $C_d$ (depending only on the dimension of the space) times the diameter of the image of the curve.\smallskip
The aforementioned control of the length directly yields uniform estimates for the asymptotic behaviour of the bounded orbits of quasiconvex gradient systems, convex subgradient systems as well as of the bounded orbits of convex foliations --- all of them being typical instances of self-contracted curves. Self-contractedness is indeed strongly related to convexity. It was shown in~\cite{DL2018} that under mild assumptions every smooth self-contracted curve can be obtained as an orbit of some smooth convex function. \smallskip
Another important feature of the notion of self-contractedness is that it translates naturally to the discrete case, to include sequences $\{x_k\}_{k\in\mathbb{N}}$ generated by some algorithmic scheme. (The series $\sum_{k\geq 0}\Vert x_{k+1}-x_k\Vert$ corresponds to the \textit{length of the sequence}.) A typical example consists of the iterates generated by the proximal-point algorithm applied to a convex function. These iterations, being obtained as successive projections to the convex foliation given by the sublevel sets of the function, generate a self-contracted sequence (cf.~\cite{DDDL2015}). This provides an independent proof of the convergence of the proximal-point algorithm. \smallskip
The objective of this work is to show that other classical iterative schemes such as the gradient descent algorithm of a smooth convex function with Lipschitz gradient, the alternating projection algorithm for two closed convex sets and the average projection method for finitely many closed convex sets, also generate self-contracted sequences. Consequently, a prior universal estimate for the convergence of all of these methods can be deduced. This estimate neither depends on the specific function nor on the choice of proximal parameters, since all self-contracted sequences/curves lying in the given bounded set admit a universal bound for their length. \smallskip
Our approach relies strongly on interpreting the aforementioned algorithms as particular instances of the proximal-gradient method (Forward-Backward algorithm), see Algorithm~\ref{alg:proxgrad} and then establishes that the iterates of the latter give a self-contracted sequence, see Theorem~\ref{thm:proxgrad_selfcontracted}. \smallskip
The proof of this central result is surprisingly simple, making astute use of an additional quadratic decay stemming from the strong convexity that appears in the proximal operator. This being said, establishing directly self-contractedness for the alternating projection algorithm is not an easy task, and might be quite involved even in the particular case that one of the convex sets is in fact a convex cone. Indeed, the generated sequence of this algorithm (and in general of all of the aforementioned algorithms) cannot be obtained, in any obvious way, via successive projections to some convex foliation related to our data. The only exception is the fixed-step gradient descent algorithm of a $C^{1,1}$-convex function (which, being identified with the proximal-point algorithm of another convex function, it can indeed be obtained with successive projections to some convex foliation). Therefore, overall, this new simple approach gives a technique for establishing self-contractedness, without passing through a convex foliation, which up-to-now was the only known way to proceed. In particular, as a by-product, we obtain a new proof for establishing self-contractedness of the proximal-point algorithm (cf. Corollary~\ref{cor:prox}). \smallskip
Let us finally mention, for completeness, that self-contracted curves have also been considered in more general settings, emancipating from direct applications to asymptotic theory of dynamical systems or optimization algorithms. To this end, self-contracted curves have been studied in~\cite{DDDR2018} in Riemann manifolds, where rectifiability has been established via an involved proof that borrows heavily from the underlying Euclidean structure. Remarkably enough, recent works on the topic reveal that Euclidean structure is not a real restriction: generalizing the results of~\cite{L2016}, the authors in~\cite{ST2017} established that any self-contracted curve in any finite dimensional (potentianlly asymmetric) normed space is rectifiable. Futher extensions include CAT(0) spaces~\cite{O2020} and spaces with weak lower curvature bound~\cite{LOZ2019}. In view of these developments, it is possible that the notion of self-contracted curve will turn out to be relevant also for abstract dynamics in a metric setting (see \cite{AGS2008} e.g.)
\section{Preliminaries}%
\label{sec:preliminaries}
Throughout this paper, $\mathbb{R}^d$ will denote the $d$-dimensional Euclidean space and $\langle \cdot, \cdot\rangle$ its inner product which generates the distance $d(x,y):= \lVert x - y \rVert$. For a nonempty subset~$A$ we denote its \emph{diameter} by $\textup{diam}(A):= \sup\{ d(x,y) : x,y \in A \}$.
\begin{definition}[self-contracted curve]
\label{def:self_contracted}
Given a possibly unbounded interval $I\subset \mathbb{R}$, a map $\gamma: I \to \mathbb{R}^d$ is called \emph{self-contracted}, if for all $t_1,t_2,t_3 \in I$ such that $t_1 \le t_2 \le t_3$
\begin{equation*}
d(\gamma(t_3), \gamma(t_2)) \le d(\gamma(t_3), \gamma(t_1)).
\end{equation*}
\end{definition}
Note that although originally inspired by continuous curves, this definition does not require any form of continuity or smoothness for the curve $\gamma$. In particular, taking $\gamma$ to be constant on each interval $[n,n+1)$, for all $n\in\mathbb{N}$, the definition also covers the case of discrete sequences. Formalizing this, we call a sequence $\{x_{k}\}_{k\in\mathbb{N}}$ in $\mathbb{R}^d$ self-contracted if for all $k_1, k_2, k_3 \in \mathbb{N}$ such that $k_1 \le k_2 \le k_3$
\begin{equation*}
d(x_{k_3}, x_{k_2}) \le d(x_{k_3},x_{k_1}).
\end{equation*}
This seemingly innocent property of self-contractedness has remarkable consequences. It was proven in~\cite[Theorem~3.3]{DDDL2015} that every self-contracted curve in a finite dimensional Euclidean space is rectifiable and its length satisfies
$$\ell(\gamma) \le C_d\, \textup{diam}(\gamma(I)),$$
where $C_d$ denotes a constant only depending on the dimension of the space.
Therefore, any bounded self-contracted sequence $\{x_{k}\}_{k\in\mathbb{N}}$ converges to some $x_{\infty}$ and
\begin{equation}
\label{eq:bound-sc}
\sum_{k=1}^{\infty} d(x_{k+1}, x_{k} ) \le C_d \, d(x_0, x_{\infty}).
\end{equation}
The aim of this work is to establish the self-contractedness of several classical algorithms in convex optimization. Previous convergence proofs relied on specific Lyapunov functions, in particular, the characteristic property of Fejer monotonicity with respect to the solution set $\mathcal{S}$ (cf.~\cite[Definition~5.1]{bc}), meaning $d_{\mathcal{S}}(x_{k+1}) \le d_{\mathcal{S}}(x_{k})$. Making use of the additional information that the iterates form a self-contracted sequence, we obtain a \textit{data independent} bound given by~\eqref{eq:bound-sc}. This bound can be further improved, using Fejer monotonicity, to
\begin{equation*}
\sum_{k=1}^{\infty} d(x_{k+1}, x_{k} ) \le C_d \, d_{\mathcal{S}}(x_0),
\end{equation*}
whenever $x_{\infty}\in \mathcal{S}$ (which can always be ensured in the forthcoming algorithm).
\section{Proximal-gradient generates self-contracted iterates}%
\label{sec:proximal_gradient_is_self_contracted}
Consider the \emph{classical} problem
\begin{equation}
\label{eq:convex_splitting}
\min_{x \in \H{}}\, g(x) + f(x)
\end{equation}
for a proper, convex and lower semicontinuous function $g:\H\to\overline{\mathbb{R}}$ and a differentiable convex function $f:\H\to\mathbb{R}$ with $L$-Lipschitz continuous gradient. We associate with the above system the \emph{Forward-Backward} or \emph{Proximal-Gradient} (cf.~\cite[Section 27.3]{bc}) operator
\begin{equation}
\label{eq:prox-grad-op}
T_{\alpha}(x) := \prox{\alpha g}{x - \alpha \nabla f(x)}
\end{equation}
with stepsize $\alpha>0$.
\subsection{Stepsize bounded by the inverse of the Lipschitz constant}%
The most established method to solve the above problem is described below:
\begin{algo}[Proximal-Gradient-Method]%
\label{alg:proxgrad}
In the above setting, for $x_{0} \in \mathbb{R}^{d}$ and a sequence of stepsizes $\{\alpha_{k}\}_{k\in\mathbb{N}} \subseteq (0, 1/L)$, consider the following iterative scheme
\begin{equation*}
\quad x_{k+1} = T_{\alpha_{k}}(x_{k}), \quad \forall k \ge 0.
\end{equation*}
\end{algo}
\begin{theorem}[Main result]
\label{thm:proxgrad_selfcontracted}
The iterates generated by Algorithm~\ref{alg:proxgrad} (Proximal-Gradient-Method with variable stepsize) form a self-contracted sequence.
\end{theorem}
For the proof we shall make use of the following three lemmata. The first two are well known and will be quoted without proof. The third lemma is also quite standard for these problems. \smallskip
Before we proceed, let us first recall that the \emph{subdifferential $\partial \Phi$} of a convex function $\Phi:\mathbb{R}^{d} \to \overline{\mathbb{R}}$ at $x$ is defined as follows
\begin{equation*}
\partial \Phi(x) := \{p \in \mathbb{R}^{d}: \, \Phi(x) + \langle p, y -x \rangle \le \Phi(y),\, \forall y \in \mathbb{R}^{d}\}.
\end{equation*}
In particular, a point $x^{*}\in\mathbb{R}^d$ is a minimizer of $\Phi$ ($x^{*}\!\in\!\argmin{\Phi}$) if and only if $0\in \partial \Phi(x^{*})$. Furthermore, a function $\Phi$ is called \emph{$\sigma$-strongly convex} if for every $x,y\in \mathbb{R}^{d}$ and for every $p\in \partial \Phi(x)$
\begin{equation*}
\Phi(y) \ge \Phi(x) + \langle p, y - x \rangle + \frac{\sigma}{2}\Vert x - y \Vert^{2}.
\end{equation*}
The following result is straightforward. It will play an important role in the proof of Lemma~\ref{lem:decrease}.
\begin{lemma}[Quadratic decay]%
\label{lem:strongly-convex-function-values}
Let $\Phi:\H \to \overline{\mathbb{R}}$ be a $\sigma$-strongly convex function and let $x^{*}$ denote its global minimizer. Then, it holds
\begin{equation*}
\Phi(x) - \Phi(x^*) \ge \frac{\sigma}{2}\lVert x-x^* \rVert^2, \quad \forall x \in \H.
\end{equation*}
\end{lemma}
Whereas the previous statement gives a quadratic lower bound, the next one will give a quadratic upper bound. For a proof we refer to~\cite[Theorem~18.15]{bc} or~\cite{Bertsekas}.
\begin{lemma}[Descent Lemma]
\label{lem:descent-lemma}
Let $f:\mathbb{R}^{d}\!\to\!\mathbb{R}$ be a differentiable function with an $L$-Lipschitz gradient. Then for all $x,y \in \mathbb{R}^{d}$
\begin{equation*}
f(y) \le f(x) + \langle \nabla f(x), y-x \rangle + \frac{L}{2} \Vert y-x \Vert^2.
\end{equation*}
\end{lemma}
The following lemma is the core of our main result. It will give an estimation for the decrease of the objective function considered in~\eqref{eq:convex_splitting}, when applying the proximal-gradient operator $T_{\alpha}$ defined in~\eqref{eq:prox-grad-op}.
For the needs of the next lemma we denote
\begin{equation}
\label{eq:x-plus}
x^{+} = T_{\alpha}(x), \quad \forall x \in \mathbb{R}^{d}.
\end{equation}
\begin{lemma}%
\label{lem:decrease}
Fix $x \in \mathbb{R}^{d}$.
If the stepsize $\alpha>0$ is smaller than the inverse of the Lipschitz constant, i.e. $\alpha\le 1/L$, then for all $z\in \mathbb{R}^{d}$
\begin{equation*}
(g+f)(x^{+}) + \frac{1}{2 \alpha} \lVert x^{+}-z \rVert^2 \le
(g+f)(z) + \frac{1}{2 \alpha} \lVert x - z \rVert^2.
\end{equation*}
\end{lemma}
\begin{proof}
First note that~\eqref{eq:x-plus} is equivalent to
\begin{equation*}
x^{+} = \argmin_{z\in\H{}} \left\{ g(z) + f(x) + \langle \nabla f(x),z-x \rangle + \frac{1}{2 \alpha} \lVert z - x \rVert^2 \right\}.
\end{equation*}
We define for all $z\in \mathbb{R}^{d}$
\begin{equation*}
l_{x}(z) := f(x) + \langle \nabla f(x), z-x\rangle
\end{equation*}
and
\begin{equation*}
\Phi_{x}(z) = g(z) + l_{x}(z) + \frac{1}{2 \alpha} \Vert z - x \Vert^2.
\end{equation*}
Notice that $\Phi_{x}$ is $\frac1\alpha$-strongly convex.
Thus, by applying Lemma~\ref{lem:strongly-convex-function-values}, we have that for all $z\in\H$
\begin{equation*}
\Phi_{x}(x^{+}) + \frac{1}{2 \alpha} \lVert x^{+}-z \rVert^2 \le \Phi_{x}(z).
\end{equation*}
By the gradient inequality we know that
\begin{equation*}
l_{x}(z) = f(x) + \left\langle \nabla f(x), z - x \right\rangle \le f(z), \quad \forall z \in \H{}.
\end{equation*}
At the same time, by Lemma~\ref{lem:decrease} (Descent Lemma) and the fact that $1/\alpha\ge L$ we have that for all $z \in \mathbb{R}^d$
\begin{equation}
\label{eq:descent_lemma}
f(x^{+}) \le l_{x}(z) + \frac{1}{2 \alpha}\lVert x^{+} - x \rVert^2
\end{equation}
which in return shows the statement of the lemma.
\end{proof}
\bigskip
\begin{proof}[Proof of Theorem~\ref{thm:proxgrad_selfcontracted}]
Let $\{x_k\}_{k\in \mathbb{N}}$ be the sequence obtained by applying Lemma~\ref{lem:decrease} with $x:=x_{k}$, $x^{+}=x_{k+1}$ and $\alpha:=\alpha_{k}$ we get that
\begin{equation*}
(g+f)(x_{k+1}) + \frac{1}{2 \alpha_{k}} \lVert x_{k+1} - z \rVert^2 \le (g+f)(z) + \frac{1}{2 \alpha_{k}} \lVert x_{k} - z \rVert^2, \quad \forall z\in\H.
\end{equation*}
Setting $z=x_{k}$, the above yields that
\begin{equation*}
(g+f)(x_{k+1}) \le (g+f)(x_{k}), \quad \forall k \in \mathbb{N}.
\end{equation*}
Moreover, taking any $z\in\mathbb{R}^d$ such that $(g+f)(z)\le (g+f)(x_{k+1})$ we deduce
\begin{equation*}
\lVert x_{k+1} - z \rVert \le \lVert x_{k} - z \rVert.
\end{equation*}
In particular, for all $m > k+1$, we get
\begin{equation*}
\Vert x_{k+1} - x_{m} \Vert \le \Vert x_{k}- x_{m} \Vert.
\end{equation*}
Since $k$ is arbitrary in the above inequality, it can be replaced by $k+1$, yielding
\begin{equation*}
\Vert x_{k+2} - x_{m} \Vert \le \Vert x_{k+1}- x_{m} \Vert \le \Vert x_{k}- x_{m} \Vert.
\end{equation*}
Using this iterative argument for $l\in \{k+1, k+2, \dots, m\}$, we deduce
\begin{equation*}
\Vert x_{l} - x_{m} \Vert \le \Vert x_{k}- x_{m} \Vert.
\end{equation*}
This shows that the sequence is self-contracted, as asserted.
\end{proof}
\subsection{Stepsize determined via Backtracking}%
\label{sub:proximal_gradient_with_backtracking}
In practice, the Lipschitz constant of the gradient of $f$ is not always known and estimating it might lead to poor stepsizes and thus to slow convergence. In this case it is natural to use some kind of line search procedure to determine an appropriate stepsize.
We will describe one (reminiscent of Armijo test) as presented in~\cite{fista}.
The idea is the following: We want to apply Algorithm~\ref{alg:proxgrad} without the restriction $1/\alpha\ge L$ on the stepsize. In every iteration we start with an initial stepsize and decrease it until the statement of the Descent Lemma~\ref{lem:descent-lemma} is fulfilled, see below:
\begin{algo}[Proximal-Gradient with Backtracking Line search]%
\label{alg:prox_grad_backtracking}
For $x_{0} \in \H$, $\alpha>0$ and $0<q<1$ consider
\begin{equation*}
(\forall k \geq 0) \quad
\left\lfloor \begin{array}{l l}
\text{Set } \alpha_{k}=\alpha \smallskip \\
\text{while} \quad f(T_{\alpha}(x_{k})) > f(x_{k}) + \left\langle \nabla f(x_{k}), T_{\alpha}(x_{k})-x_{k} \right\rangle + \frac{1}{2 \alpha}\lVert T_{\alpha}(x_{k}) - x_{k} \rVert^2,\\ \text{do}
\qquad \alpha_{k} := q \alpha_{k}\smallskip\\
x_{k+1} = T_{\alpha_{k}}(x_{k}).
\end{array}\right.
\end{equation*}
\end{algo}
Note that the parameter $\alpha>0$ that will be finally chosen in each iteration might be larger than $1/L$. This means that, for the cost of some extra function evaluations, a precise knowledge of the Lipschitz constant of the gradient is no more required; in addition, the algorithm might produce larger steps than what would have been allowed in Algorithm~\ref{alg:alternating-projections}, yielding a faster convergence.
\begin{theorem}%
\label{thm:proxgrad_backtracking}
The iterates generated by the Proximal-Gradient-Method with Backtracking Algorithm~\ref{alg:prox_grad_backtracking} form a self-contracted sequence.
\end{theorem}
\begin{proof}
The proof follows the same lines as the one of Theorem~\ref{thm:proxgrad_selfcontracted}, the only difference being in~\eqref{eq:descent_lemma} of Lemma~\ref{lem:decrease}. This equation now holds true because of the way the iterations are chosen ensuring
\begin{equation*}
f(x_{k+1}) \le f(x_{k}) + \left\langle \nabla f(x_{k}), x_{k+1}-x_{k} \right\rangle + \frac{1}{2 \alpha}\lVert x_{k+1} - x_{k} \rVert^2.
\end{equation*}
(In the proof of Theorem~\ref{thm:proxgrad_selfcontracted} the above estimate was provided by Lemma~\ref{lem:descent-lemma}.)
\end{proof}
\subsection{Special cases: proximal-point algorithm, gradient descent}%
In problem~\eqref{eq:convex_splitting} we may consider separately the particular instances $f=0$ and $g=0$.
In the first case, the problem reduces to the minimization of a lower semicontinuous, convex function $g$ via the proximal-point algorithm. In particular, from Algorithm~\ref{alg:proxgrad} and the previous analysis, we deduce the following result, which first appeared (with a different proof) in~\cite[Theorem~4.17]{DDDL2015}
\begin{corollary}[Proximal-Point Algorithm]
\label{cor:prox}
Let $g: \mathbb{R}^{d} \to \overline{\mathbb{R}}$ be a convex, lower semicontinuous function and ${\{\alpha_k\}}_{k\in \mathbb{N}} \subseteq (0, +\infty)$. Then, for any $x_{0}\in \mathbb{R}^{d}$ the proximal sequence
$$
x_{k+1} = \prox{\alpha_{k}g}{x_{k}}, \quad \forall k \ge 0,
$$
is a self-contracted curve.
\end{corollary}
If $g=0$, the problem reduces to minimizing a smooth convex function with Lipschitz gradient via steepest descent. In particular, we obtain the following result, which is new. (While preparing the manuscript, the recent interesting preprint \cite{gupta2019path} came to our attention. The forthcoming result also appears there with a different proof (see~\cite[Lemma~3.1]{gupta2019path}).
\begin{corollary}[Steepest Descent]
\label{cor:gradient-descent}
Let $f:\mathbb{R}^{d} \to \mathbb{R}$ be a smooth convex function with $L$-Lipschitz gradient and $\{\alpha_k\}_{k\in \mathbb{N}}$ either be bounded from above by $1/L$ or produced by backtracking line search. Then, for any $x_{0} \in \mathbb{R}^{d}$, the sequence $\{x_{k}\}_{k\in \mathbb{N}}$ defined by
$$
x_{k+1} = x_{k} - \alpha_{k}\nabla f(x_{k}), \quad \forall k \ge 0,
$$
is self-contracted.
\end{corollary}
\subsection{A priory estimates for convergence}%
\label{sub:estimates}
An important consequence of Theorem~\ref{thm:proxgrad_selfcontracted}, Theorem~\ref{thm:proxgrad_backtracking} and Corollaries~\ref{cor:prox}--\ref{cor:gradient-descent}, is the following.
If the optimization problem~\eqref{eq:convex_splitting} has a solution, then the iterates form a bounded, self-contracted sequence. Therefore, by~\cite[Theorem~3.3]{DDDL2015} we deduce that the sequence of iterates $\{x_k\}_{k\in\mathbb{N}}$ has finite length, i.e.
\begin{equation*}
\sum_{k=0}^{\infty} \Vert x_{k+1} - x_{k} \Vert < + \infty\,,
\end{equation*}
and thus it converges to some point $x_{\infty}\in \mathbb{R}^d$. In addition, provided that the stepsize ${\{\alpha_k\}}_{k\in \mathbb{N}}$ does not go to zero too fast (for instance if it is not in $\ell^{1}(\mathbb{N})$), we deduce that $x_{\infty}$ will be a minimizer of our objective function. \smallskip
Moreover, the bound on the length of the sequence of the iterates
\begin{equation*}
\sum_{k=0}^{\infty} \Vert x_{k+1} - x_{k} \Vert < \, C_{d} \, d_{\mathcal{S}}(x_{0}), \quad \mathcal{S}=\argmin{(f+g)}
\end{equation*}
does not depend on the data $f,g$ but only on the dimension and the distance of the initial point to the set of minimizers.
\section{Projection Algorithms}%
\label{sec:projection_algorithms}
In this section we investigate the property of self-contractedness for various projection type algorithms. Although a direct approach for establishing this property would be quite involved, it turns out that the overall analysis simplifies significantly by utilizing the main result of the previous section.\smallskip
Let us introduce our main problem of finding the intersection of two closed convex sets $A,B\subset \H$
\begin{equation}\label{eq:vlad}
\text{Find} \quad x \in A \cap B.
\end{equation}
\subsection{Alternating Projections}%
\label{sub:alternating_projections}
The arguably best known algorithm for solving this problem is given by
\begin{algo}[Alternating Projections{\cite[page 186]{bauschke1993convergence}}]%
\label{alg:alternating-projections}
For $x_{0} \in \H$, consider the iterative scheme
\begin{equation*}
(\forall k \geq 0) \quad
\left\lfloor \begin{array}{l}
y_{k+1} = P_A(x_{k}) \smallskip \\
x_{k+1} = P_B(y_{k+1}).
\end{array}\right.
\end{equation*}
\end{algo}
We are going to prove that both sequences above are self-contracted. This will follow from the self-contractedness of the iterates of the Proximal-Gradient-Method.
\begin{theorem}%
\label{thm:AP-selfcontracted}
Let ${\{x_{k}\}}_{k\in\mathbb{N}}$ and ${\{y_{k}\}}_{k\in\mathbb{N}}$ be the two sequences of iterates generated by Algorithm~\ref{alg:alternating-projections}. Then, both sequences are self-contracted.
\end{theorem}
\begin{proof}
We interpret alternating projections as a proximal gradient scheme.
Define $f:= \frac12 d_A^{2}$ and note that a gradient step with respect to this function corresponds to a Projection onto $A$, i.e.\ $\textup{Id}-\nabla f= P_A$. Furthermore, we define $g:=\delta_B$ as the indicator function of the set $B$.
Thus,
\begin{equation*}
x_{k+1} = P_B(P_A(x_{k})) = P_B(x_{k} - \nabla f(x_{k})) = \prox{g}{x_{k} - \nabla f(x_{k})}.
\end{equation*}
Therefore, Theorem~\ref{thm:proxgrad_selfcontracted} shows that ${\{x_{k}\}}_{k\in\mathbb{N}}$ is self-contracted, whereas self-contractedness of ${\{y_{k}\}}_{k\in\mathbb{N}}$ follows from symmetry.
\end{proof}
\subsection{Averaged Projections}%
\label{sub:averaged_projections}
Consider the more general problem of finding the intersection of a finite number of closed convex sets ${\{C_{i}\}}_{i=1}^n$:
\begin{equation}
\label{eq:multiple-sets}
\text{Find}\quad x \in \bigcap_{i=1}^n C_i.
\end{equation}
One could clearly extend the method of alternating projections to this setting and end up with the cyclic projection method, which is given by
\begin{equation*}
x_{k+1} = P_n\circ P_{n-1}\circ \cdots \circ P_1 (x_{k}), \quad \forall k\ge0.
\end{equation*}
In practice, this method is often replaced by other schemes (see for instance~\cite{BCC2012} and references therein). We will focus on the following modification (cf. \cite[page~368]{bauschke1996projection}).
\begin{algo}[Averaged Projections]%
\label{alg:averaged-projection}
For $x_0\in\H$ consider the iterative scheme
\begin{equation}
\label{eq:averaged-projection}
x_{k+1} = \frac1n \sum_{i=1}^{n} P_{C_i}(x_{k}), \quad \forall k \ge 0.
\end{equation}
\end{algo}
\begin{proposition}%
\label{prop:averaged-projection}
The iterates generated by Algorithm~\ref{alg:averaged-projection} form a self-contracted sequence.
\end{proposition}
We will give to different proofs for the assertion above. The first one relies on reformulating the method of averaged projections as gradient descent (with fixed stepsize), whereas the second one is based on the interpretaton of this method as alternating projections over two closed convex sets in a product space.
\begin{proof}
\textit{(first proof)}
Let us define for $i \in \{1,2,\dots, n\}$
\begin{equation*}
f_i := \frac12 d_{C_i}^{2}
\end{equation*}
and notice that $f:= \sum_{i=1}^{n} f_{i}$ is convex and smooth with $L$-Lipschitz gradient, where $L\le n$.
Then, the sequence of~\eqref{eq:averaged-projection} can be equivalently defined by
\begin{equation*}
x_{k+1} = \left( \textup{Id} - \frac1n \nabla \bigg(\sum_{i=1}^{n} f_i \bigg) \right)(x_{k}),
\end{equation*}
with stepsize $\alpha_{k} = \alpha = 1/n \le 1/L$.
The result follows by applying Corollary~\ref{cor:gradient-descent}.
\bigskip
\noindent\textit{(second proof)}
Define the closed convex sets $\widehat{C}=\Pi_{i=1}^n C_i$ and $$\Delta=\{(y^{1}, y^{2}, \dots, y^{n}) \in \mathbb{R}^{d \times n} : y^{1}=y^{2}=\cdots=y^{n}\}.$$
Then,~\eqref{eq:averaged-projection} is equivalent to
\begin{equation*}
x_{k+1} = P_\Delta(P_{\widehat{C}}(x_{k})),
\end{equation*}
i.e. the alternating projections method applied to the sets $\widehat{C}$ and $\Delta$ in $\mathbb{R}^{d\times n}$.
Now can apply Theorem~\ref{thm:AP-selfcontracted}, and deduce the fact that ${(x_{k})}_{k\in\mathbb{N}}$ is self-contracted.
\end{proof}
To summarize, Algorithm~\ref{alg:alternating-projections} applied to problem~\eqref{eq:vlad} or Algorithm~\ref{alg:averaged-projection} applied to problem~\eqref{eq:multiple-sets} share the following common feature: whenever the problem is feasible or at least one of the involved sets is bounded, then the sequence of iterates is bounded. In both cases, the sequence $\{x_k\}_{k\in\mathbb{N}}$ is convergent and the estimates mentioned in Section~\ref{sec:preliminaries} hold true.
\medskip
\paragraph{General Conclusion.} The results of this work confirm the previous understanding~\cite{DLS2010,DDDL2015,DL2018,LMV2015} that self-contractedness relates to convexity, in both, the continuous and discrete setting. Let us mention that this concept has recently been relaxed~in~\cite{DDD2018} to so-called $\lambda$-curves (respectively $\lambda$-sequences). In that work, estimates, similar to the ones of self-contracted curves, have been obtained. However, the general asymptotic behavior of such curves remains unclear. At the same time this notion might be related to more complex settings such as accelerated convex methods or algorithms in nonconvex optimization.
\bigskip
\paragraph{Acknowledgments.} A major part of this work was done during a research
visit of the first author to the University of Chile (March to June 2019) and of
the second author to the University of Vienna (March 2020). These authors wish
to thank their hosts for hospitality.
|
2,869,038,156,342 | arxiv | \section{Introduction}
Offensive language can be defined as instances of profanity in communication, or any instances that disparage a person or a group based on some characteristic such as race, color, ethnicity, gender, sexual orientation, nationality, religion, \emph{etc.}~\cite{nockleby:2000}.
The ease of accessing social networking sites has resulted in an unprecedented rise of offensive content on social media.
With massive amounts of data being generated each minute, it is imperative to develop scalable systems that can automatically filter offensive content.
The first works in offensive language detection were primarily based on a lexical approach, utilizing surface-level features such as n-grams, bag-of-words, \emph{etc.}, drawn from the similarity of the task to another NLP task, {\em i.e.}, Sentiment Analysis (SA).
These systems perform well in the context of foul language but prove ineffective in detecting hate speech.
Consequently, the main challenge lies in discriminating profanity and hate speech from each other~\cite{zampieri:2019}.
On the other hand, recent deep neural network based approaches for offensive language detection fall prey to inherent biases in a dataset, leading to the systems being discriminative against the very classes they aim to protect.
Davidson et al., \shortcite{davidson:2019} presented the evidence of a systemic bias in classifiers, showing that such classifiers predicted tweets written in African-American English as abusive at substantially higher rates.
Table \ref{tab:table6} presents the scenarios where a tweet may be considered~hateful.
\begin{table*}[hbtp!]
\centering
\begin{tabular}{|c|l|}
\hline
\textbf{S.No.} & \textbf{Hateful Tweet Scenarios} \\
\hline
1 & uses sexist or racial slurs. \\
2 & attacks a minority. \\
3 & seeks to silence a minority. \\
4 & criticizes a minority (without a well-founded argument). \\
5 & promotes but does not directly use hate speech or violent crime. \\
6 & criticizes a minority and uses a straw man argument. \\
7 & blatantly misrepresents truth or seeks to distort views on a minority with unfounded claims.\\
8 & shows support of problematic hashtags. {\em E.g.} “\#BanIslam,” “\#whoriental,” “\#whitegenocide”\\
9 & negatively stereotypes a minority.\\
10 & defends xenophobia or sexism.\\
11 & contains an offensive screen name\\
\hline
\end{tabular}
\caption{Hateful Tweet Scenarios~\cite{waseem:2016}}
\label{tab:table6}
\vspace{-0.5cm}
\end{table*}
Syntactic features are essential for a model to detect latent offenses, {\em i.e.}, untargeted offenses, or where the user might mask the offense using the medium of sarcasm~\cite{schmidt:2017}.
Syntactic features prevent over-generalization on specific word classes, {\em e.g.}, profanities, racial terms, \emph{etc.}, instead examining the possible arrangements of the precise lexical internal features which factor in differences between words of the same class.
Hence, syntactic features can overcome the systemic bias, which may have arisen from the pejorative use of specific word classes.
A significant property of dependency parse trees is their ability to deal with morphologically rich languages with a relatively free word order~\cite{10.5555/1214993}.
Motivated by the nature of the modern Twitter vocabulary, which also follows a relatively free word order, we present an integration of syntactic features in the form of dependency grammar in a deep learning framework.
In this paper, we propose a novel architecture called \emph{Syntax-based LSTM} (\emph{SyLSTM}), which integrates latent features such as syntactic dependencies into a deep learning model.
Hence, improving the efficiency of identifying offenses and their targets while reducing the systemic bias caused by lexical features.
To incorporate the dependency grammar in a deep learning framework, we utilize the Graph Convolutional Network (GCN)~\cite{kipf:2016}.
We show that by subsuming only a few changes to the dependency parse trees, they can be transformed into compatible input graphs for the GCN.
The final model consists of two major components, a BiLSTM based Semantic Encoder and a GCN-based Syntactic Encoder in that order.
Further, a Multilayer Perceptron handles the classification task with a Softmax head.
The state-of-the-art BERT model requires the re-training of over $110M$ parameters when fine-tuning for a downstream task.
In comparison, the \emph{SyLSTM} requires only $\sim9.5M$ parameters and significantly surpasses BERT level performance.
Hence, our approach establishes a new state-of-the-art result for offensive language detection while being over ten times more parameter efficient than BERT.
We evaluate our model on two datasets; one treats the task of hate speech and offensive language detection separately~\cite{davidson:2017}.
The other uses a hierarchical classification system that identifies the types and targets of the offensive tweets as a separate task~\cite{zampieri:2019}.
\paragraph{Our Contribution:} The major contribution of this paper is to incorporate syntactic features in the form of dependency parse trees along with semantic features in the form of feature embeddings into a deep learning architecture.
By laying particular emphasis on sentence construction and dependency grammar, we improve the performance of automated systems in detecting hate speech and offensive language instances, differentiating between the two, and identifying the targets for the same.
Results (Section \ref{result}) show that our approach significantly outperforms all the baselines for the three tasks, {\em viz.}, identification of offensive language, the type of the offense, and the target of the offense.
The rest of the paper is organized as follows.
In Section \ref{related}, we discuss related work in this field. Section \ref{methodology} presents the design of \emph{SyLSTM}.
Section \ref{experimental} elaborates on the datasets and the experimental protocol.
Section \ref{result} presents the results and discussion, and Section \ref{conclusion} concludes the paper.
\section{Related Work}
\label{related}
Hate speech detection, as a topical research problem, has been around for over two decades.
One of the first systems to emerge from this research was called {\em Smokey}~\cite{spertus:1997}.
It is a decision-tree-based classifier that uses $47$ syntactic and semantically essential features to classify inputs in one of the three classes ($ flame $, $ okay $ or $ maybe $).
\emph{Smokey} paved the way for further research in using classical machine learning techniques to exploit the inherent features of Natural Language over a plethora of tasks such as junk filtering~\cite{sahami:1998}, opinion mining~\cite{wiebe:2005}~\emph{etc}.
Owing to the unprecedented rise of social networks such as Facebook and Twitter, most of the research on hate speech detection has migrated towards the social media domain.
To formalize this new task, a set of essential linguistic features was proposed~\cite{waseem:2016}.
Initial research in this direction focused more on detecting profanity, pursuing hate speech detection implicitly~\cite{nobata:2016,waseem:2017}.
Using these systems, trained for detecting profanities, to detect hate speech reveals that they fall prey to inherent biases in the datasets while also proving ineffective in classifying a plethora of instances of hate~speech~\cite{davidson:2019}.
Research has also shown the importance of syntactic features in detecting offensive posts and identifying the targets of such instances~\cite{chen:2012}.
On social media, it was found that hate speech is primarily directed towards specific groups, targeting their ethnicity, race, gender, caste, {\em etc.}~\cite{silva:2016}.
ElSherief et al. \shortcite{elsherief:2018} make use of linguistic features in deep learning models, which can be used to focus on these directed instances.
The problem with this approach is two-fold.
First, these linguistic features learn inherent biases within the datasets, thus discriminating against the classes they are designed to protect.
Second, the use of explicit linguistic features to detect hate speech leaves the model prone to the effects of domain shift.
Altogether, there is a need to develop more robust techniques for hate speech detection to address the above mentioned issues.
While the use of syntactic features for the task has proven useful, there has been little effort towards incorporating non-Euclidean syntactic linguistic structures such as dependency trees into the deep learning~sphere.
Graph Neural Networks (GNNs) provide a natural extension to deep learning methods in dealing with such graph structured data.
A special class of GNNs, known as Graph Convolutional Networks (GCNs), generalize Convolutional Neural Networks (CNNs) to non-Euclidean data.
The GCNs were first introduced by Bruna et al. \shortcite{bruna2013spectral}, following which, Kipf et al. \shortcite{kipf:2016} presented a scalable, first order approximation of the GCNs based on Chebyshev polynomials.
The GCNs have been extremely successful in several domains such as social networks~\cite{hamilton2017inductive}, natural language processing~\cite{marcheggiani2017encoding} and natural sciences~\cite{zitnik2018modeling}.
Marcheggiani and Titov \shortcite{marcheggiani2017encoding} were the first to show the effectiveness of GCNs for NLP by presenting an analysis over semantic role labelling.
Their experiments paved the way for researchers to utilize GCNs for feature extraction in NLP.
Since then, GCNs have been used to generate embedding spaces for words~\cite{vashishth:2018}, documents~\cite{peng:2018} and both words and documents together~\cite{yao:2019}.
Even though GCNs have been used in NLP, their inability to handle multirelational graphs has prevented researchers from incorporating the dependency parse tree in the deep feature space.
In this paper, we present a first approach towards transforming the dependency parse tree in a manner that allows the GCN to process it.
The final model is a combination of a BiLSTM based Semantic Encoder, which extracts semantic features and addresses long-range dependencies, and a GCN-based Syntactic Encoder, which extracts features from the dependency parse tree of the sentence.
Results show that the proposed approach improves the performance of automated systems in detecting hate speech and offensive language instances, differentiating between the two, and identifying the targets for the same.
\section{Methodology}
\label{methodology}
Traditionally, grammar is organized along two main dimensions: \emph{morphology} and \emph{syntax}.
While morphology helps linguists understand the structure of a word, the syntax looks at sentences and how each word performs in a sentence.
The meaning of a sentence in any language depends on the syntax and order of the words.
In this regard, a sentence that records the occurrence of relevant nouns and verbs ({\em e.g.}, Jews and kill) can prove helpful in learning the offensive posts and their targets~\cite{gitari:2015}.
Further, the syntactic structure I $\langle intensity \rangle$ $\langle user intent \rangle$ $\langle hate target \rangle$, {\em e.g.}, “I f*cking hate white people,” helps to learn more about offensive posts, their targets, and the intensity of the offense~\cite{silva:2016}.
Our approach incorporates both semantic features and the dependency grammar of a tweet into the deep feature space.
The following subsections present a detailed discussion on the proposed methodology.
\begin{table*}[hbt!]
\centering
\resizebox{1.8\columnwidth}{!}
\begin{tabular}{|c|c|}
\hline
\textbf{Preprocessing} & \textbf{Description} \\
\hline
{\em Replacing usernames} & replacing all usernames with ‘@user’. {\em Eg.} ‘@india’ to ‘@user’. \\
{\em Replacing URLs} & replacing URLs in a tweet with the word ‘url’. \\
{\em Hashtag Segmentation} & Eg. ‘\#banislam’ becomes ‘\# banislam’. \\
{\em Emoji Normalization} & normalizing emoji instances with text. {\em Eg.} ‘:)’ becomes ‘smiley face’. \\
{\em Compound Word Splitting} & split compound words. {\em E.g.} ‘putuporshutup’ to ‘put up or shut up’. \\
{\em Reducing Word Lengths} & reduce word lengths, exclamation marks, {\em E.g.} ‘waaaaayyyy’ to ‘waayy’. \\
\hline
\end{tabular}
}
\caption{Preprocessing Modules}
\label{tab:table5}
\vspace{-0.3cm}
\end{table*}
\begin{figure*}[hbt!]
\centering
\includegraphics[width=10cm,height=8cm]{figures/archi.png}
\caption{Model Architecture for \emph{SyLSTM}}
\label{fig:model}
\vspace{-0.5cm}
\end{figure*}
\subsection{Preprocessing}
Raw tweets usually have a high level of redundancy and noise associated with them, such as varying usernames, URLs, \emph{etc}.
In order to clean the data, we implement the preprocessing module described in Table \ref{tab:table5}.
\subsection{Model}
The proposed model \emph{SyLSTM} (Figure \ref{fig:model}) has the following six components:
\begin{enumerate}
\item Input Tokens: The tweet is passed through a word-based tokenizer after the preprocessing step. The tokenized tweet is then given as input to the model;
\item Embedding Layer: A mapping for each word to a low-dimensional feature vector;
\item BiLSTM Layer: used to extract a high-level feature space from the word embeddings;
\item GCN Layer: produces a weight vector according to the syntactic dependencies over the high-level features from step $3$.
Multiply with high-level feature space to produce new features with relevant syntactic information.
\item Feed Forward Network: reduces the dimensionality of the outputs of step $4$.
\item Output Layer: the last hidden states from step $3$ are concatenated with the output of step 5 as a residual connection and fed as input.
The feature space is finally used for hate speech detection.
\end{enumerate}
The detailed description of these components is given below.
\paragraph{Word Embeddings:} Given a sentence consisting of $ T $ words $S = \{x_1, x_2, ... , x_T\}$, every word $x_i$ is converted to a real valued feature vector $e_i$.
This is done by means of an embedding matrix which serves as a lookup table,
\begin{equation}
\label{eq:embedding_matrix}
\mathcal{E}^{(word)} \in \mathbb{R}^{|V| \times d^{(w)}},
\end{equation}
where, $|V|$ is the size of the vocabulary and $d^{(w)}$ is the dimensional size of the embeddings.
Each word in $ S $ is then mapped to a specific entry in this~matrix,
\begin{equation}
\label{eq:embedding}
e_i = \mathcal{E}^{(word)} . v_i,
\end{equation}
where, $v_i$ is a one hot vector of size $|V|$.
The entire sentence is fed into the proceeding layers as real-valued vectors $emb = \{e_1, e_2, ... , e_T\}$.
The embedding matrix can be initialized randomly and learned via backpropagation, or one can also use a set of pretrained embeddings.
Twitter posts generally use the modern \emph{internet lexicon} and hence have a unique vocabulary.
For our model, we use two different instances for the embedding space - first, a randomly initialized embedding space learned at the training time.
Second, a pretrained embedding space where we utilize the GloVe-Twitter Embeddings\footnotemark[1]\footnotetext[1]{\url{https://nlp.stanford.edu/projects/glove/}} $(d^{(w)} = 200)$.
These embeddings have been trained on 27B tokens parsed from a Twitter corpus~\cite{pennington:2014}.
Results indicate that models trained on the GloVe-Twitter Embeddings learn a stronger approximation of semantic relations in the twitter vocabulary, showcasing a more robust performance than their randomly initialized~counterparts.
\paragraph{Semantic Encoding with BiLSTM:} Most of the existing research on GCNs focuses on learning nodal representations in undirected graphs.
These are suited to single relational edges and can suffer from a severe semantic gap when operating on multirelational graphs.
To codify the relational edges' underlying semantics and resolve language on a temporal scale, we utilize the Bidirectional LSTM.
Using an adaptive gating mechanism, the LSTMs decide the degree of importance between features extracted at a previous time step to that at the current time step~\cite{hochreiter:1997}.
Consequently, they prove extremely useful in the context of hate speech detection, where hate speech can be distributed randomly at any part of the sentence.
Standard LSTMs process sequences in a temporal order hence ignoring future context.
Bidirectionality allows us access to both future and past contexts, which helps improve the cognition of hate speech in a tweet~\cite{8684825}.
We pass the sentence embedding vectors $emb = \{e_1, e_2, ... , e_T\}$ through a two-layered BiLSTM network with $32$ hidden units and a dropout of $0.4$.
As outputs, we extract the sequential vectors and the final hidden states for the forward and backward sequences.
The final hidden states for the forward and backward sequences are concatenated and used as a residual connection at a later stage, as shown in Figure \ref{fig:model}.
The sequential vectors are passed through a batch normalization layer with a momentum of $0.6$ and then fed into the GCN layer along with the dependency parse trees.
\begin{figure*}[hbt!]
\centering
\includegraphics[width=13cm,height=4cm]{figures/IMG_0061.png}
\caption{(a) Dependency Graph $ G $ with Nodal Embeddings (b) Adjacency Matrix $ A $ for the graph $ G $}
\label{fig:graph}
\vspace{-0.5cm}
\end{figure*}
\paragraph{Syntactic Encoding with GCN:} The dependency parse trees have specific characteristics which are rarely considered in general graphs.
On the one hand, they have multirelational edges.
And on the other hand, the definition of each type of edge is relatively broad, resulting in a huge difference in the semantics of edges with the same relationship.
For instance, an `amod' dependency may be presented in <Techniques, Computational> and <Techniques, Designed>, but their semantics are obviously different.
The GCN~\cite{kipf:2016} cannot handle such scenarios without introducing some changes to the structure of the input dependency parse tree.
First, inverse edges corresponding to each of the respective dependencies are introduced between all connected nodes.
Furthermore, to highlight the importance of specific words in the given context, we add self-loops over each node.
The dependency parse tree of a sentence is extracted using the NLP open-source package spaCy\footnotemark[2]\footnotetext[2]{\url{https://github.com/explosion/spaCy}}.
Hence, the extracted dependency parse tree is transformed into a graph $G = (V, E)$, where $V$ is the set of all vertices which represent the words in a tweet and $E$ is the set of all edges which highlight the dependency and their inverse relations.
The result is an undirected graph with self-loops (see Figure \ref{fig:graph}).
This comes as a natural extension to the dependency structure of the sentence, highlighting the importance of word positioning and combating possible confusions in identifying the direction of the dependency.
The graph is then fed into the GCN as a sparse adjacency matrix, with each dependency represented by a weight $\alpha$.
With the setup in place, the GCN performs a convolution operation over the graph $G$ represented by the adjacency matrix $A$.
Formally, the GCN performs the following~computation:
\begin{equation}
\label{eq:GCN}
H^{(l+1)} = \sigma(\tilde D^{-\frac{1}{2}} \tilde A \tilde D^{-\frac{1}{2}}H^{(l)}W^{(l)})
\end{equation}
where, $\tilde A = A + I_N$ is the adjacency matrix of the undirected graph $ G $ with added self-connections.
$I_N$ is the identity matrix, $\tilde D_{ii} = \Sigma_j \tilde A_{ij}$ and $W^{(l)}$ is a layer-specific trainable weight matrix.
$\sigma (\cdot)$ denotes an activation function, in our case the ReLU$(\cdot)$ = $\max(0, \cdot)$.
$H^{(l)} \in \mathbb{R}^{N \times D}$ is the matrix of activations in the $l^{th}$ layer; $H^{(0)} = L$.
The model learns hidden layer representations that encode both local graph structure (\emph{the dependencies}) and nodal features (\emph{the importance of the word in that context}).
Furthermore, the Semantic Encoder complements the Syntactic Encoder by addressing the long range spatial inabilities of the GCN~\cite{marcheggiani2017encoding}.
The sparse adjacency matrix leads to a problem with vanishing gradients.
We combat this by applying a batch normalization layer with a momentum of $0.6$ and applying a dropout of $0.5$.
We use the Xavier distribution to initialize the weight matrix and set the output dimension of the GCN as $32$.
\paragraph{Feed Forward Neural Network (FFNN):} The output of the GCN is then passed through a single layered FFNN to learn high-level features based on dependency structure.
The FFNN is activated using the non-linear ReLU activation function.
\paragraph{Output Layer:} The output from the FFNN is then concatenated with the last hidden states of the BiLSTM which is added as a residual connection.
The concatenated vector is then passed through a linear layer with a softmax head that produces a probability distribution over the required outputs.
\section{Experimental Setup}
\label{experimental}
This section describes the dataset and the experimental setup for the models reported in the paper.
\subsection{Datasets}
The primary motivation of this paper is the design of a methodology to integrate a neural network model with syntactic dependencies for improved performance over fine-grained offensive language detection.
Keeping in line with this ideology, we test our model on two separate datasets.
The following section describes these datasets at length.
\paragraph{Offensive Language Identification Dataset:}
This dataset was presented for a shared task on offensive language detection in the SemEval Challenge $2019$.
This was the first time that offensive language identification was presented as a hierarchical task.
Data quality was ensured by selecting only experienced annotators and using test questions to eliminate individuals below a minimum reliability threshold.
Tweets were retrieved using a keyword approach on the Twitter API.
The dataset forms a collection of $14,000$ English tweets annotated for three subtasks proceeding in a hierarchy~\cite{zampieri:2019}:
\begin{enumerate}
\item whether a tweet is offensive or not (A);
\item whether the offensive tweet is targeted (B);
\item whether the target of the offensive tweet is an individual, a group, or other (\emph{i.e.}, an organization, an event, an issue, a situation) (C).
\end{enumerate}
We choose this dataset because of the extended subtasks B and C.
An increase in performance over these will posit that our model has been successful in tackling its objectives.
We evaluate our model on all three subtasks.
\paragraph{Hate Speech and Offensive Language Dataset:} Motivated by the central problem surrounding the separation of hate speech from other instances of offensive language, Davidson et al. \shortcite{davidson:2017} curated a dataset annotating each tweet in one of three classes, hate speech (\textbf{HATE}), offensive language (\textbf{OFF}), and none (\textbf{NONE}).
They use a hate speech lexicon containing words and phrases identified by internet users as hate speech, compiled by \emph{Hatebase}.
These lexicons are used to extract English tweets from the Twitter API.
From this corpus, a random sample of $25k$ tweets containing terms from the lexicon was extracted.
The tweets were manually coded by CrowdFlower (CF) workers, with a final inter-annotator agreement of $92\%$.
\subsection{Baseline Models}
In the following section, we describe the design of all the baseline models used for comparison.
\paragraph{Linear-SVM:} SVMs have achieved state-of-the-art results for many text classification tasks and significantly outperform many neural networks over the OLID dataset~\cite{zampieri:2019}.
Hence, we use a Linear-SVM trained on word unigrams as a baseline.
We employ a Grid-search technique to identify the best hyperparameters.
\paragraph{Two-channel BiLSTM:} We design a two-channel BiLSTM as a second baseline, with the two input channels differentiated only by their embedding space.
One of the input channels learns the embedding space via backpropagation after a random initialization, while the other uses the pretrained BERT embeddings.
This choice is motivated by the contextual nature of the BERT embeddings.
This conforms with the ideation that certain words may be deemed offensive depending upon the context they are used in.
The BiLSTM itself is two layers deep and consists of $32$ hidden-units.
The final hidden states for the forward and backward sequences of each channel are concatenated and passed through an MLP with a softmax head for classification.
\paragraph{Fine-tuned BERT:} We also fine-tune a BERT model~\cite{devlin:2018} for this task.
We adapt the state-of-the-art BERT model which won the SemEval Challenge $2019$~\cite{liu:2019} and tune the hyperparameters of the model to get the best performance on our preprocessing strategy.
While fine-tuning this model, the choices over the loss function, optimizer, and learning rate schedule remain the same as those for the \emph{SyLSTM}.
\subsection{Training} We train our models using the standard cross-entropy loss.
The AdamW optimizer~\cite{loshchilov:2018} is chosen to learn the parameters.
To improve the training time and chances of reaching the optima, we adopt a cosine annealing~\cite{loshchilov:2017} learning rate scheduler.
The vocabulary of the models is fixed to the top $30,000$ words in the corpus. The initial learning rate is set to $0.001$, with a regularization parameter of $0.1$.
\subsection{Evaluation Metric}
The datasets exhibit large class imbalances over each task.
In order to address this problem, we use the Weighted F1-measure as the evaluation metric.
We also provide the precision and recall scores for a deeper insight into the model’s performance.
\section{Results}
\label{result}
We evaluate two instances of our model, (1) with a randomly initialized embedding matrix (referred to as \emph{SyLSTM}) and (2) utilizing the pretrained GloVe Twitter embeddings (referred to as \emph{SyLSTM*}).
A paired Student's t-test using the Weighted-F1 measure of the model's performance shows that our models significantly outperform each of the baselines across all the tasks (\emph{p $<$ 0.001}).
\subsection{Performance on Offensive Language Identification Dataset}
In this section, we present performance comparisons between the baselines and the \emph{SyLSTM} for the three subtasks.
We split the training data, using $10\%$ of the tweets to get a dev set.
The hyperparameters are tuned according to the performance on the dev set.
The results presented here demonstrate the performance over the predefined test set.
We also present the performance metrics for the trivial case, notably where the model predicts only a single label for each tweet.
By comparison, we show that the chosen baselines and our models perform significantly better than chance for each~task.
\paragraph{Offensive Language Detection:} The performance comparisons for discriminating between offensive (\textbf{OFF}) and non-offensive (\textbf{NOT}) tweets are reported in Table \ref{tab:table1}.
Neural network models perform substantially better than the Linear-SVM.
Our model (in gray) outperforms each of the baselines in this task.
\begin{table}[hbt!]
\centering
\resizebox{0.9\columnwidth}{!}
\begin{tabular}{l l l l}
\hline
\textbf{System} & \textbf{Precision} & \textbf{Recall} & \textbf{F1-score} \\
\hline
All OFF & 8.4 & 28.2 & 12.1 \\
All NOT & 52.4 & 72.7 & 60.4 \\
\hline
SVM & 77.7 & 80.2 & 78.6\\
BiLSTM & 81.7 & 82.8 & 82.0\\
BERT & \underline{87.3} & 85.8 & 85.7\\
\hline
\rowcolor[RGB]{230,230,230}
SyLSTM & 85.2 & \underline{88.1} & \underline{86.4}\\
\rowcolor[RGB]{230,230,230}
SyLSTM* & \textbf{87.6} & \textbf{88.1} & \textbf{87.4}\\
\hline
\end{tabular}
}
\caption{Offensive Language Detection}
\label{tab:table1}
\vspace{-0.5cm}
\end{table}
\paragraph{Categorization of Offensive Language:} This sub-task is designed to discriminate between targeted insults and threats (\textbf{TIN}) and untargeted (\textbf{UNT}) offenses, generally referring to profanity~\cite{zampieri:2019}.
Performance comparisons for the same are reported in Table 4.
Our model (in gray) shows a significant $4\%$ relative improvement in performance in comparison to the BERT model.
\begin{table}[hbt!]
\centering
\resizebox{0.9\columnwidth}{!}
\begin{tabular}{l l l l}
\hline
\textbf{System} & \textbf{Precision} & \textbf{Recall} & \textbf{F1-score} \\
\hline
All TIN & 78.7 & 88.6 & 83.4 \\
All UNT & 1.4 & 11.3 & 12.1 \\
\hline
SVM & 81.6 & 84.1 & 82.6\\
BiLSTM & 84.8 & 88.4 & 85.7\\
BERT & 88.4 & \underline{92.3} & 89.6\\
\hline
\rowcolor[RGB]{230,230,230}
SyLSTM & \underline{90.6} & 91.6 & \underline{91.4}\\
\rowcolor[RGB]{230,230,230}
SyLSTM* & \textbf{94.4} & \textbf{92.3} & \textbf{93.2}\\
\hline
\end{tabular}
}
\caption{Categorization of Offensive Language}
\label{tab:table2}
\vspace{-0.5cm}
\end{table}
\paragraph{Offensive Language Target Identification:} This sub-task is designed to discriminate between three possible targets: a group (\textbf{GRP}), an individual (\textbf{IND}), or others (\textbf{OTH}).
The results for the same are reported in Table \ref{tab:table3}.
Note that the three baselines produce almost identical results.
The low F1-scores for this task may be on account of the small size of the dataset and large class imbalances, factors that make it difficult to learn the best features for classification.
Our model (in gray) shows a $5.7\%$ relative improvement over the BERT model, hence showcasing its robustness when generalizing over smaller datasets.
\begin{table}[hbt!]
\centering
\resizebox{0.9\columnwidth}{!}
\begin{tabular}{l l l l}
\hline
\textbf{System} & \textbf{Precision} & \textbf{Recall} & \textbf{F1-score} \\
\hline
All GRP & 13.6 & 37.4 & 19.7 \\
All IND & 22.1 & 47.3 & 30.3 \\
ALL OTH & 3.4 & 16.2 & 5.4 \\
\hline
SVM & 56.1 & 62.4 & 58.3\\
BiLSTM & 56.1 & 65.8 & 60.4\\
BERT & 58.4 & 66.2 & 60.9\\
\hline
\rowcolor[RGB]{230,230,230}
SyLSTM & \underline{60.3} & \textbf{67.4} & \underline{63.4}\\
\rowcolor[RGB]{230,230,230}
SyLSTM* & \textbf{62.4} & \underline{66.3} & \textbf{64.4}\\
\hline
\end{tabular}
}
\caption{Offensive Language Target Identification}
\label{tab:table3}
\vspace{-0.5cm}
\end{table}
\subsection{Performance on Hate Speech and Offensive Language Dataset}
This section presents the performance comparisons between our model and the baselines for this multi-class classification problem.
The task presented by the dataset complies with our main objective of integrating syntactic dependencies in a neural network model to differentiate between offensive language and hate speech more efficiently.
The tweets are classified in one of three categories: hate speech (\textbf{HATE}), offensive language (\textbf{OFF}), and none (\textbf{NONE}).
The Linear-SVM and the neural network baselines produce very similar results, all of which are significantly better than chance (see Table \ref{tab:table4}).
The \emph{SyLSTM} (in gray) significantly outperforms all the~baselines.
\begin{table}[hbt!]
\centering
\resizebox{0.9\columnwidth}{!}
\begin{tabular}[width=\textwidth]{l l l l}
\hline
\textbf{System} & \textbf{Precision} & \textbf{Recall} & \textbf{F1-score} \\
\hline
All HATE & 0.2 & 6.1 & 0.4 \\
All OFF & 3.1 & 16.9 & 5.3 \\
All NONE & 58.8 & 77.2 & 66.7 \\
\hline
SVM & 84.9 & 90.1 & 88.2\\
BiLSTM & 90.3 & 90.2 & 90.3\\
BERT & \underline{91.2} & 90.4 & 91.0\\
\hline
\rowcolor[RGB]{230,230,230}
SyLSTM & 90.5 & \underline{91.4} & \underline{91.4}\\
\rowcolor[RGB]{230,230,230}
SyLSTM* & \textbf{92.3} & \textbf{92.8} & \textbf{92.7}\\
\hline
\end{tabular}
}
\caption{Hate Speech and Offensive Language Dataset}
\label{tab:table4}
\vspace{-0.5cm}
\end{table}
\subsection{Discussion}
The two-channel BiLSTM and the BERT model discussed in this paper act as strong syntax-agnostic baselines for this study.
The aforementioned results indicate the superiority of the \emph{SyLSTM} over such approaches.
The inability of existing dependency parsers to generate highly accurate dependency trees for a tweet may seem like a severe problem.
However,
since the dependency tree has been transformed to accommodate inverse dependency edges, we find that the resulting undirected graph acts as a single-relational graph where each edge represents a ``dependency".
The nature of the dependency is addressed by graph convolutions operating over the dynamic LSTM features.
Hence, the parser only needs to generate congruent copies of the actual dependency tree of the tweet.
\vspace{0.1cm}We tested the utility of enriching the features generated by a BERT encoder using a GCN.
Existing literature in this field integrates word embeddings learned using a GCN with the BERT model~\cite{lu2020vgcn}.
In contrast, our experiments dealt with a GCN mounted over a BERT encoder.
We note that this combination leads to over-parametrization and severe sparsity issues.
Since BERT models have been shown to learn fairly accurate dependency structures~\cite{clark:2019}, additional importance to dependency grammar over the same encoder network may be~unnecessary.
\section{Conclusion}
\label{conclusion}
In this paper, we present a novel approach called the \emph{SyLSTM} which demonstrates how GCNs can incorporate syntactic information in the deep feature space, leading to state-of-the-art results for fine-grained offensive language detection on Twitter Data.
Our analysis uncovers the Semantic and Syntactic Encoders' complementarity while revealing that the system's performance is largely unaffected for mislabeled dependencies over congruent dependency trees.
Leveraging the dependency grammar of a tweet provides a practical approach to simulating how humans read such texts.
Furthermore, the performance results of the \emph{SyLSTM} indicate the robustness of the architecture in generalizing over small datasets.
The added simplicity of the overall architecture promotes applicability over other NLP tasks.
The \emph{SyLSTM} can be used as an efficient and scalable solution towards accommodating graph-structured linguistic features into a neural network~model.
\vspace{0.2cm}\noindent {\bf Replication Package.} The replication package for this study is available at \url{https://github.com/dv-fenix/SyLSTM}.
|
2,869,038,156,343 | arxiv | \section{Introduction}
\end{center}
Logarithmic Sobolev inequalities are a powerful tool in Real
Analysis, Complex Analysis, Geometric Analysis, Convex Geometry
and Probability. The pioneer work by L. Gross \cite{Gr} puts
forward the equivalence between a class of Euclidean logarithmic Sobolev
inequalities and hypercontractivity of the associated heat
semigroup. Particularly, his logarithmic Sobolev inequality with respect to the Gaussian measure plays an important role in Ricci flow theory ({\it e.g.} \cite{Pe}), optimal transport theory ({\it e.g.} \cite{Tal}), probability theory ({\it e.g.} \cite{Le1}), among other applications. Later, Weissler \cite{W} introduced a log-Sobolev type inequality (also known as Euclidean $L^2$-entropy inequality) equivalent to the Gross's inequality with Gaussian measure. The Euclidean $L^2$-entropy inequality and its variants have been used in the study of optimal estimates for solutions of certain nonlinear diffusion equations, see for instance \cite{BGL}, \cite{BGL1}, \cite{Bro1}, \cite{DDG}, \cite{G0}, \cite{G} and references therein.
The Euclidean $L^p$-entropy inequality for $p \geq 1$ states that, for any function $u \in W^{1,p}(\R^n)$ with $\int_{\R^n} |u|^p dx = 1$,
\begin{equation}\label{dee}
Ent_{dx}(|u|^p):= \int_{\R^n} |u|^p \log(|u|^p)\; dx \leq \frac{n}{p} \log \left({\cal A}_0(p) \int_{\R^n} |\nabla u|^p\; dx\right)\; ,
\end{equation}
\n where $n \geq 2$, $p \geq 1$ and ${\cal A}_0(p)$ is the best possible constant in this inequality.
As mentioned above, the Euclidean $L^2$-entropy inequality was established by Weissler in \cite{W}. Thereafter, Carlen \cite{Carlen} showed that its extremal functions are precisely dilations and translations of the Gaussian function
\[
u_0(x) = \pi^{-\frac n2} e^{-|x|^2}\; .
\]
\n For $p = 1$, Ledoux \cite{Le} proved the inequality (\ref{dee}) and Beckner \cite{Be} classified its extremal
functions as normalized characteristic functions of balls. In \cite{Be}, Beckner also proved that (\ref{dee}) is valid for any $1 < p <
n$ and Del Pino and Dolbeault \cite{DPDo} characterized its extremal functions as dilations and translations of the function
\begin{equation} \label{ext}
u_0(x) = \pi^{-\frac n2} \frac{\Gamma(\frac{n}{2} +
1)}{\Gamma(\frac{n(p - 1)}{p} + 1)} e^{-|x|^{\frac{p}{p-1}}}\; .
\end{equation}
\n Finally, Gentil \cite{G} established the validity of (\ref{dee}) and that $u_0$ is an extremal function for any $p > 1$. Thanks to an uniqueness argument due to Del Pino and Dolbeault \cite{DPDo}, modulo dilations and translations, the classification also extends for any $p > 1$.
As a byproduct, they derived, for any $p > 1$,
\[
{\cal A}_0(p) = \frac{p}{n} \left( \frac{p - 1}{e} \right)^{p - 1}
\pi^{-\frac{p}{2}} \left( \frac{\Gamma(\frac{n}{2} +
1)}{\Gamma(\frac{n(p - 1)}{p} + 1)} \right)^{\frac{p}{n}}\; .
\]
In order to introduce sharp $L^p$-entropy inequalities within the Riemannian environment for $1 \leq p < n$, we first deduce an intermediate entropy inequality.
Let $(M,g)$ be a smooth closed Riemannian manifold of dimension $n \geq 2$ and $1 \leq p < n$. For any $p \leq s \leq p^*:=\frac{np}{n-p}$, H\"{o}lder's inequality gives
\[
||u||_{L^s(M)} \leq ||u||_{L^p(M)}^\alpha ||u||_{L^{p^*}(M)}^{1 -
\alpha}\; ,
\]
\n for all function $u \in L^{p^*}(M)$, where $\alpha = \frac{np - ns +
ps}{ps}$. Taking logarithm of both sides, one gets
\[
\log \left( \frac{||u||_{L^s(M)}}{||u||_{L^p(M)}} \right) +
(\alpha - 1) \log \left( \frac{||u||_{L^p(M)}}{||u||_{L^{p^*}(M)}}
\right) \leq 0\; .
\]
\n Since this inequality trivializes to an equality when $s = p$,
we may differentiate it with respect to $s$ at $s = p$. Then, a simple computation provides
\[
\int_M |u|^p \log |u|^p \; dv_g \leq \frac{n}{p} \log \left(
\int_M |u|^{p^*} \; dv_g\right)^{\frac{p}{p^*}}
\]
\n for all function $u \in L^{p^*}(M)$ with $||u||_{L^p(M)} = 1$.
Using the above inequality and the Sobolev embedding theorem for compact manifolds ({\it e.g.} \cite{Au4}), one gets constants ${\cal A}, {\cal B} \in \R$ such
that, for any $u \in H^{1,p}(M)$ with $\int_M |u|^p dv_g = 1$,
\begin{gather}\label{AB}
Ent_{dv_g}(|u|^p):= \int_{M} |u|^p \log |u|^p\; dv_g \leq \frac{n}{p} \log \left( {\cal A} \int_M |\nabla_g u|^p\; dv_g + {\cal B} \right)\; , \tag{$L({\cal A},{\cal B})$}
\end{gather}
\n where $dv_g$ denotes the Riemannian volume element, $\nabla_g$
is the gradient operator of $g$ and $H^{1,p}(M)$ is the Sobolev
space defined as the completion of $C^{\infty}(M)$ under the norm
\[
||u||_{H^{1,p}(M)} := \left( \int_{M} |\nabla_g u|^p\; dv_g + \int_{M} |u|^p\; dv_g \right)^{\frac{1}{p}}\; .
\]
The following definitions and notations related to (\ref{AB}) are
quite natural when one desires to introduce sharp $L^p$-entropy
inequalities in the Riemannian context.
The {\bf first best $L^p$-entropy constant} is defined by
\[
{\cal A}_0(p,g) := \inf \{ {\cal A} \in \R: \mbox{ there exists} \hspace{0,18cm} {\cal B} \in \R \hspace{0,18cm} \mbox{such that (\ref{AB})} \hspace{0,18cm} \mbox{holds for all} \hspace{0,18cm} u \in H^{1,p}(M) \hspace{0,18cm} \mbox{with} \hspace{0,18cm} \int_M |u|^p dv_g = 1\}\; .
\]
\n It follows directly that ${\cal A}_0(p,g)$ is well defined and moreover, by Jensen's inequality applied to (\ref{AB}), ${\cal A}_0(p,g)$ is positive for any $1 \leq p < n$.
The {\bf first sharp $L^p$-entropy inequality} states that there exists a constant ${\cal B} \in \R$ such that, for any $u \in H^{1,p}(M)$ with $\int_M |u|^p dv_g = 1$,
\[
\int_{M} |u|^p \log |u|^p\; dv_g \leq \frac{n}{p} \log \left( {\cal A}_0(p,g) \int_M |\nabla_g u|^p\; dv_g + {\cal B} \right)\; .
\]
If the preceding inequality is true, then we can define the {\bf second best $L^p$-entropy constant} as
\[
{\cal B}_0(p,g) := \inf \{ {\cal B} \in \R:\; (L({\cal A}_0(p,g),{\cal B})) \hspace{0,18cm} \mbox{holds for all} \hspace{0,18cm} u \in H^{1,p}(M) \hspace{0,18cm} \mbox{with} \hspace{0,18cm} \int_M |u|^p dv_g = 1\}
\]
and the {\bf second sharp $L^p$-entropy inequality} as the saturated version of ($L({\cal A}_0(p,g),{\cal B})$) on the functions $u \in H^{1,p}(M)$ with $\int_M |u|^p dv_g = 1$, that is
\[
\int_{M} |u|^p \log |u|^p\; dv_g \leq \frac{n}{p} \log \left( {\cal A}_0(p,g) \int_M |\nabla_g u|^p\; dv_g + {\cal B}_0(p,g) \right)\; .
\]
\n Note that ($L({\cal A}_0(p,g),{\cal B}_0(p,g))$) is sharp with respect to both the first and second best constants in the sense that none of them can be lowered.
In a natural way, one introduces the notion of extremal functions of ($L({\cal A}_0(p,g),{\cal B}_0(p,g))$). A function $u_0 \in H^{1,p}(M)$ satisfying $\int_M |u_0|^p dv_g = 1$ is said to be extremal, if
\[
\int_{M} |u_0|^p \log |u_0|^p\; dv_g = \frac{n}{p} \log \left( {\cal A}_0(p,g) \int_M |\nabla_g u_0|^p\; dv_g + {\cal B}_0(p,g) \right)\; .
\]
\n We denote by ${\cal E}_0(p,g)$ the set of all extremal functions of ($L({\cal A}_0(p,g),{\cal B}_0(p,g))$).
In \cite{Bro}, Brouttelande proved for dimensions $n \geq 3$ that ${\cal A}_0(2,g) = {\cal A}_0(2)$ and ($L({\cal A}_0(2,g),{\cal B})$) is valid for some constant ${\cal B} \in \R$. Subsequently, in \cite{Bro1}, he obtained the lower bound
\[
{\cal B}_0(2,g) \geq \frac{1}{2 n \pi e} \max_M R_g\; ,
\]
\n where $R_g$ stands for the scalar curvature of the metric $g$, and proved that if the inequality is strict, then the set ${\cal E}_0(2,g)$ is non-empty.
A simple lower bound for ${\cal B}_0(p,g)$ involving the volume $v_g(M)$ of $M$ also follows by taking a normalized constant function in ($L({\cal A}_0(p,g),{\cal B}_0(p,g))$), namely
\[
{\cal B}_0(p,g) \geq v_g(M)^{-p/n}\; ,
\]
\n provided that ${\cal B}_0(p,g)$ is well defined.
Our main contributions are gathered in the following result:
\begin{teor} \label{MT}
Let $(M,g)$ be a smooth closed Riemannian manifold of dimension $n \geq 2$. For any $1 < p \leq 2$ and $p < n$, we have:
\begin{itemize}
\item[{\bf (a)}] ${\cal A}_0(p,g) = {\cal A}_0(p)$;
\item[{\bf (b)}] there exists a constant ${\cal B} \in \R$ such that ($L({\cal A}_0(p,g),{\cal B})$) holds for all function $u \in H^{1,p}(M)$ with $\int_M |u|^p dv_g = 1$.
\end{itemize}
\end{teor}
Note that the above result extends to $1 < p \leq 2$ the corresponding one due to Brouttelande. However, his arguments do not apply to $p \neq 2$, so we here develop an alternative approach in order to prove Theorem \ref{MT}.
The idea of using subcritical interpolation inequalities for obtaining entropy inequalities was introduced by Bakry, Coulhon,
Ledoux and Sallof-Coste in \cite{Ba} within the Euclidean
environment. Namely, for $1 \leq p < n$, they showed how to produce
non-sharp $L^p$-entropy inequalities as a limit case of a class of
non-sharp Gagliardo-Nirenberg inequalities.
Later, this view point was explored in the sharp sense by Del Pino and Dolbeault \cite{DPDo} in order to establish the inequality
(\ref{dee}) for $1 < p < n$. Indeed, they considered a family of sharp Gagliardo-Nirenberg inequalities, interpolating
the $L^p$-Sobolev and $L^p$-entropy inequalities, whose extremal functions are explicitly known.
In trying to adapt the same idea of getting entropy inequalities as a limit case of subcritical interpolation inequalities to the
Riemannian context, the situation changes drastically mainly because extremal functions and second best constants are usually
unknown.
With the aim to make clear to the reader our program of proof and its main points of difficulty, a brief overview on related sharp Riemannian
Nash inequalities should be presented.
Let $1 < p \leq 2$ and $1 \leq q < p$. In \cite{CM2}, Ceccon and Montenegro established the existence of a constant $B \in \R$ such that the sharp $L^p$-Nash inequality
\begin{gather}\label{AB1}
\left( \int_M |u|^p\; dv_g \right)^{\frac{1}{\theta}} \leq \left( A_{opt} \int_M |\nabla_g u|^p\; dv_g + B \int_M |u|^p\; dv_g \right) \left(
\int_M |u|^q\; dv_g \right)^{\frac{p(1 - \theta)}{q \theta}} \tag{$I_{p,q}(A,B)$}
\end{gather}
\n holds for all function $u \in H^{1,p}(M)$, where $\theta = \frac{n(p - q)}{q(p - n) + np}$ and $A_{opt}$ is the first best $L^p$-Nash constant.
One also knows that
\begin{equation} \label{eq}
A_{opt} = N(p,q) \; ,
\end{equation}
\n where $N(p,q)$ stands for the best Euclidean $L^p$-Nash constant, see \cite{DHV} for $p = 2$ and \cite{CM2} for $1 < p < 2$. Namely, $N(p,q)$ is the best possible constant in the $L^p$-Nash inequality
\[
\left( \int_{\R^n} |u|^p\; dx \right)^{\frac{1}{\theta}} \leq A \left( \int_{\R^n} |\nabla u|^p\; dx \right) \left(
\int_{\R^n} |u|^q\; dx \right)^{\frac{p(1 - \theta)}{q \theta}}
\]
\n which holds for all function $C^\infty_0(\R^n)$, provided that $n \geq 2$ and $1 \leq q < p$. This last inequality was first proved by Nash in \cite{Na}. An alternative proof was given by Beckner in \cite{Be1}. For $p \neq 2$, we refer to the recent work \cite{Ce} by Ceccon.
One then defines the {\bf second best $L^p$-Nash constant} as
\[
B(p,q,g) := \inf \{ B \in \R:\; (I_{p,q}(N(p,q),B)) \hspace{0,18cm} \mbox{holds for all} \hspace{0,18cm} u \in H^{1,p}(M)\}\; .
\]
\n In this case, the {\bf second sharp $L^p$-Nash inequality} automatically holds on $H^{1,p}(M)$, namely
\[
\left( \int_M |u|^p\; dv_g \right)^{\frac{1}{\theta}} \leq \left( N(p,q) \int_M |\nabla_g u|^p\; dv_g + B(p,q,g) \int_M |u|^p\; dv_g \right) \left( \int_M |u|^q\; dv_g \right)^{\frac{p(1 - \theta)}{q \theta}}\; .
\]
Our strategy consists in rearranging the above inequality and applying logarithmic of both sides, so that
\begin{equation} \label{AE}
\frac{p}{\theta} \log{\frac{||u||_{L^p(M)}}{||u||_{L^q(M)}}} \leq \log{\frac{N(p,q) \int_M |\nabla_g u|^p\; dv_g + B(p,q,g) \int_M |u|^p\; dv_g} {\left( \int_M |u|^q\; dv_g \right)^{\frac{p}{q}}} }\, .
\end{equation}
\n and then letting the limit as $q \rightarrow p^-$.
The success of this plan will be result of the following three statements for $1 < p \leq 2$ and $p < n$:
\begin{itemize}
\item[{\bf (A)}] $N(p,q)$ converges to ${\cal A}_0(p)$ as $q \rightarrow p^-$
\item[{\bf (B)}] $B(p,q,g)$ is bounded for any $q < p$ close to $p$
\item[{\bf (C)}] ${\cal A}_0(p,g) \geq {\cal A}_0(p)$
\end{itemize}
\n In fact, assume for a moment that {\bf (A)}, {\bf (B)} and {\bf (C)} are true. Letting $q \rightarrow p^-$ in (\ref{AE}) and using {\bf (A)} and {\bf (B)}, after straightforward computations, one obtains the inequality ($L({\cal A}_0(p),{\cal B})$) for some constant ${\cal B}$. Indeed, as can easily be checked,
\[
\lim_{q \rightarrow p^-} \frac{p}{\theta} \log{\frac{||u||_{L^p(M)}}{||u||_{L^q(M)}}} =
\frac{p^3}{n} \lim_{q \rightarrow p^-} \frac{1}{p - q} \log{\frac{||u||_{L^p(M)}}{||u||_{L^q(M)}}} = \frac{p}{n} \int_M
|u|^p \log (|u|^p)\; dv_g
\]
\n for any $u \in H^{1,p}(M)$ satisfying $||u||_{L^p(M)} = 1$, see the proof of Proposition \ref{P.1} for a similar computation. Thus, one has ${\cal A}_0(p,g) \leq {\cal A}_0(p)$ and, thanks to {\bf (C)}, the conclusion of Theorem \ref{MT} follows.
We then describe some ideas involved in the proof of each one of the claims {\bf (A)}-{\bf (C)}. We begin by addressing {\bf (A)} and {\bf (C)} since their proofs are shortest.
\n {\bf On the proof of {\bf (A)}.} This claim is proved in Section 2 and uses Jensen's inequality and the fact to be proved that the best Nash constant $N(p,q)$ is increasing on $q$.
\n {\bf On the proof of {\bf (C)}.} This claim is proved in Section 3. Its proof is based on estimates of Gaussian bubbles. Precisely, we consider the following test function in ($L({\cal A}_0(p,g),{\cal B}_0(p,g))$):
\[
u_\varepsilon(exp_{x_0}(x)) = \eta(x) \varepsilon^{-\frac{n}{p}} u_0(\frac{x}{\varepsilon})
\]
\n defined locally around a point $x_0$ on $M$, where $\eta$ denotes a cutoff function supported in a ball centered at $0$. After using Cartan's expansion of the metric $g$ around $x_0$ and estimating each involved integral for $\varepsilon > 0$ small enough, the desired conclusion follows.
\n {\bf On the proof of {\bf (B)}.} This claim is proved in Section 4. Since its proof is rather long and technique, for a better understanding two important steps are presented under the form of lemma.
It suffices to prove the assertion for each sequence $(q_k) \subset (1, q)$ such that $q_k \rightarrow p$ as $k \rightarrow + \infty$. For such a sequence and each $k$, we consider the functional
\[
J_k(u) = \left( \int_M |\nabla_g u|^p\; dv_g + C_k \int_M |u|^p\; dv_g \right) \left( \int_M |u|^{q_k}\; dv_g \right)^{\frac{p(1 -
\theta_{q_k})}{q_k \theta_{q_k}}}
\]
\n on the set ${\cal H} = \{ u \in H^{1,p}(M):\; ||u||_{L^p(M)} = 1 \}$, where $C_k$ is defined as
\[
C_k = \frac{B(p,q_k,g) - (p-q_k)}{N(p,q_k)}\; .
\]
\n From its definition, we readily have $C_k < B(p,q_k,g) N(p,q_k)^{-1}$. Therefore, from the definition of $B(p,q_k,g)$, there exists a function $w_k \in {\cal H}$ satisfying $J_k(w_k) < N(p,q_k)^{-1}$, so that
\[
\inf_{u \in {\cal H}} J_k (u) < N(p,q_k)^{-1}\; .
\]
\n As usual, this last inequality leads to the existence of a $C^1$ minimizer $u_k \in {\cal H}$ for the functional $J_k$ on ${\cal H}$. The next steps consists in studying some fine properties satisfied by the sequence $(u_k)$ such as concentration and pointwise estimates. The main tools used here are blow-up method and Moser's iteration on elliptic PDEs.
We conclude the introduction by exposing some still open problems on sharp Riemannian entropy inequalities and related best constants.
Perhaps, contrary to what one might expect, it is not clear that the first best entropy constant ${\cal A}_0(p,g)$ is well defined and is equal to ${\cal A}_0(p)$ for all $p \geq n$. The great difficult in Riemannian $L^p$-entropy inequalities is that local-to-global type arguments do not usually work well. For example, the normalization condition $\int_M |u|^p dv_g = 1$ and the involved log functions in (\ref{AB}) do not allow a direct comparison to the corresponding flat case. In particular, it does not seem immediate that (\ref{AB}) is valid for some constants ${\cal A}$ and ${\cal B}$ and also that, for each $\varepsilon > 0$, there exists a constant ${\cal B}_\varepsilon \in \R$ such that
\[
\int_{M} |u|^p \log |u|^p\; dv_g \leq \frac{n}{p} \log \left( ({\cal A}_0(p) + \varepsilon) \int_M |\nabla_g u|^p dv_g + {\cal B}_\varepsilon \right)
\]
\n for all function $u \in H^{1,p}(M)$ with $\int_M |u|^p dv_g = 1$.
According to our contributions, the first best entropy constant ${\cal A}_0(p,g)$ is well defined for all $1 \leq p < n$ and equal to ${\cal A}_0(p)$ for all $1 < p \leq 2$ and $p < n$. In addition, based on developments over thirty years in the field of sharp entropy and Sobolev inequalities (see \cite{Au4}, \cite{DH} and references therein), one expects positive answers for the following questions:\\
\n {\bf Open problem 1.} Is ${\cal A}_0(p,g)$ well defined for all $p \geq n$?\\
\n {\bf Open problem 2.} Does ${\cal A}_0(p,g) = {\cal A}_0(p)$ hold in the three cases $p = 1, n \geq 2$, $p = 2, n = 2$ or $p > 2, n \geq 2$?\\
\n {\bf Open problem 3.} Is ($L({\cal A}_0(p,g),{\cal B})$) valid for some constant ${\cal B}$ in the two cases $p = 1, n \geq 2$ or $p = 2, n = 2$?\\
\n {\bf Open problem 4.} Is ($L({\cal A}_0(p,g),{\cal B})$) non-valid whenever $p > 2$, $n \geq 2$ and $(M,g)$ has positive scalar curvature somewhere?
\section{Proof of the assertion {\bf (A)}}
This section is devoted to the proof of the proposition.
\begin{propo} \label{P.1}
Let $n \geq 2$ and $1 \leq q < p$. We have
\begin{equation} \label{LN}
\lim_{q \rightarrow p^-} N(p,q) = {\cal A}_0(p) \; .
\end{equation}
\end{propo}
We first prove that
\begin{equation} \label{DN}
N(p,q) \leq {\cal A}_0(p)
\end{equation}
\n for all $1 < q < p$.
\n Let $u \in C^{\infty}_0(\R^n)$ such that $||u||_{L^p(\R^n)} = 1$. Using Jensen's inequality, we have
\[
-\ln \left(\int_{\R^n} |u|^q\; dx \right) = -\ln \left(\int_{\R^n} |u|^{q - p} |u|^p\; dx \right) \leq - \int_{\R^n} \ln(|u|^{q - p})
|u|^p\; dx = \frac{p - q}{p} \int_{\R^n} \ln(|u|^p) |u|^p\; dx\; .
\]
\n Joining this inequality with (\ref{dee}), one obtains
\[
\left(\int_{\R^n} |u|^q dx \right)^{- \frac{p^2}{n(p-q)}} \leq
{\cal A}_0(p) \int_{\R^n} |\nabla u|^p dx \; .
\]
\n Using the fact that $||u||_{L^p(\R^n)} = 1$, we then derive the Nash inequality
\[
\left( \int_{\R^n} |u|^p dx \right)^{\frac{1}{\theta}} \leq {\cal A}_0(p)
\left( \int_{\R^n} |\nabla u|^p dx \right) \left(\int_{\R^n} |u|^q dx
\right)^{\frac{p(1 - \theta)}{\theta q}}\; ,
\]
\n where
\[
\theta = \frac{n(p - q)}{qp - qn + np} \; .
\]
\n By homogeneity, the above inequality is valid for all function $u \in C^{\infty}_0(\R^n)$, so that the assertion (\ref{DN}) follows.
We now prove that $N(p,q)$ is monotonically increasing on $q$.
\n Let $1 < q_1 < q_2 < p$ fixed. An usual interpolation inequality yields
\[
\left( \int_{\R^n} |u|^{q_2}\; dx \right)^{\frac{1}{q_2}} \leq \left(\int_{\R^n} |u|^{q_1}\; dx \right)^{\frac{\mu}{q_1}}
\left(\int_{\R^n} |u|^{p}\; dx \right)^{\frac{1 - \mu}{p}}\; ,
\]
\n where $\frac{1}{q_2} = \frac{\mu}{q_1} + \frac{1 - \mu}{p}$.
\n Plugging this inequality in
\[
\left(\int_{\R^n} |u|^{p}\; dx \right)^{\frac{1}{\theta_2}} \leq N(p,q_2) \left( \int_{\R^n} |\nabla u|^p\; dx \right)
\left( \int_{\R^n} |u|^{q_2}\; dx \right)^{\frac{p(1 - \theta_2)}{q_2 \theta_2}} \; ,
\]
\n where $\theta_2 = \frac{n(p - q_2)}{q_2(p - n) + np}$, one gets
\[
\left(\int_{\R^n} |u|^p\; dx \right)^{1 + \mu\frac{1 - \theta_2}{\theta_2}} \leq N(p,q_2)
\left(\int_{\R^n} |\nabla u|^p\; dx \right) \left( \int_{\R^n} |u|^{q_1}\; dx\right)^{\frac{p}{q_1} (\mu
\frac{1 - \theta_2}{\theta_2})}\; .
\]
\n On the other hand, the definition of $\mu$ produces the relations
\[
1 + \mu \frac{1 -
\theta_2}{\theta_2} = \frac{1}{\theta_1}
\]
\n and
\[
\frac{p}{q_1}\left( \mu \frac{1 -
\theta_2}{\theta_2}\right) =
\frac{p}{q_1}\frac{1 - \theta_1}{\theta_1} \; ,
\]
\n where $\theta_1 = \frac{n(p - q_1)}{q_1(p - n) + np}$, so that $N(p,q_1) \leq N(p,q_2)$ and the monotonicity follows.
\n We then denote
\begin{equation}
A(p) = \lim_{q \rightarrow p^-} N(p,q)\; .
\end{equation}
\n It is clear by (\ref{DN}) that $A(p) \leq {\cal A}_0(p)$.
The remaining of the proof is devoted to show that $A(p) \geq {\cal A}_0(p)$.
\n Rearranging the sharp Nash inequality and taking logarithm of both sides, one has
\[
\frac{1}{\theta} \log \left(\frac{||u||_p}{||u||_q} \right) \leq \log \left(N(p,q) \frac{||\nabla u||_p^p}{||u||_q^p}\right)^{\frac{1}{p}}\; .
\]
\n Using the definition of $\theta$ and taking the limit on $q$, one gets
\[
\frac{p^2}{n} \lim_{q \rightarrow p^-} \frac{1}{p - q} \log \left(\frac{||u||_p}{||u||_q} \right)\leq \log \left(A(p) \frac{||\nabla u||_p^p}{||u||_p^p}\right)^{\frac{1}{p}}\; .
\]
\n We now compute the left-hand side limit. We first write
\[
\log \left(\frac{||u||_p}{||u||_q} \right) = \frac{1}{p} \log(||u||_p^p) - \frac{1}{q} \log(||u||_q^q) = \frac{q - p}{p}
\log(||u||_q) + \frac{1}{p} \left( \log(||u||_p^p) - \log(||u||_q^q) \right)\; .
\]
\n A straightforward computation then gives
\[
\lim_{q \rightarrow p^-} \frac{1}{p - q} \log \left(\frac{||u||_p}{||u||_q} \right) = \frac{1}{p} \int_{\R^n}
\frac{|u|^p}{||u||_p^p} \log \left(\frac{|u|}{||u||_p}\right)\; dx\; .
\]
\n Therefore,
\[
\int_{\R^n} \frac{|u|^p}{||u||_p^p} \log \left( \frac{|u|^p}{||u||_p^p} \right)\; dx \leq \frac{n}{p} \log \left( A(p) \frac{||\nabla
u||_p^p}{||u||_p^p}\right)\; ,
\]
\n or equivalently,
\[
\int_{\R^n}|u|^p \log (|u|^p)\; dx \leq \frac{n}{p} \log \left( A(p)
\int_{\R^n} |\nabla u|^p\; dx \right)
\]
\n for all function $u \in C^\infty_0(\R^n)$ with $||u||_{L^p(\R^n)} = 1$. So, $A(p) \geq {\cal A}_0(p)$ and the proof of Proposition \ref{P.1} follows. \bl
\section{Proof of the assertion {\bf (C)}}
In this section, we prove the following result:
\begin{propo}\label{P.2}
For each $n \geq 2$ and $1 < p < n$, we have
\[
{\cal A}_0(p,g) \geq {\cal A}_0(p)\; .
\]
\end{propo}
Using the assumption that $n \geq 2$ and $1 < p < n$, one gets constants ${\cal A}$ and ${\cal B}$ such that $L({\cal A},{\cal B})$ is valid. It suffices to show that ${\cal A} \geq {\cal A}_0(p)$.
One knows that $L({\cal A},{\cal B})$ is equivalent to
\begin{equation} \label{EF1}
\frac{1}{||u||_p^p} \int_M |u|^p \log(|u|^p)\; dv_g + (\frac{n}{p} - 1) \log (||u||_p^p) \leq \frac{n}{p} \log \left( {\cal A} \int_M |\nabla
u|_g^p\; dv_g + {\cal B} \int_M |u|^p\; dv_g \right)
\end{equation}
\n for all function $u \in C^{\infty}(M)$.
We first fix a point $x_0 \in M$ and an extremal function $u_0(x) = a e^{-b|x|^{\frac{p}{p-1}}} \in W^{1,p}(\R^n)$ for the sharp entropy inequality (\ref{dee}) (see \cite{DPDo} or \cite{G}), where $a$ and $b$ are positive constants chosen so that $||u_0||_{L^p(\R^n)} = 1$.
Consider now a geodesic ball $B(x_0,\delta) \subset M$ and a radial cutoff function $\eta \in C^\infty(B(0,\delta))$ satisfying $\eta = 1$ in $B(0,\frac{\delta}{2})$, $\eta = 0$ outside $B(0,\delta),\ 0 \leq \eta \leq 1$ in $B(0,\delta)$. For $\varepsilon > 0$ and $x \in B(0,\delta)$, set
\[
u_\varepsilon(exp_{x_0}(x)) = \eta(x) \varepsilon^{-\frac{n}{p}} u_0(\frac{x}{\varepsilon})\; .
\]
The asymptotic behavior of each integral of (\ref{EF1}) computed at $u_\varepsilon$ with $\varepsilon > 0$ small enough is now presented.
\n Denote
\[
I_1 = \int_{\R^n} u_0^p \log(u_0^p)\; dx, \ \ I_2 = \int_{\R^n} |\nabla u_0|^p\; dx\; ,
\]
\[
J_1 = \int_{\R^n} u_0^p |x|^2\; dx, \ \ J_2 = \int_{\R^n} |\nabla u_0|^p |x|^2\; dx,\ \ J_3 = \int_{\R^n} u_0^p \log(u_0^p) |x|^2\; dx\; .
\]
\n Using the expansion of volume element in geodesic coordinates
\[
\sqrt{det g} = 1 - \frac{1}{6} \sum_{i,j = 1}^{n} Ric_{ij}(x_0) x_i x_j + O(r^3)\; ,
\]
\n where $Ric_{ij}$ denotes the components of the Ricci tensor in these coordinates and $r=|x|$, one easily checks that
\begin{equation} \label{1}
\int_{M} u_\varepsilon^p\; dv_g = 1 - \frac{R_g(x_0)}{6n} J_1 \varepsilon^2 + O(\varepsilon^4)\; ,
\end{equation}
\begin{equation} \label{2}
\int_{M} u_\varepsilon^p \log(u_\varepsilon^p)\; dv_g = I_1 - n \log \varepsilon - \frac{R_g(x_0)}{6n} J_3 \varepsilon^2 + \frac{R_g(x_0)}{6} J_1 \varepsilon^2 \log \varepsilon + o(\varepsilon^4 \log \varepsilon)
\end{equation}
\n and
\begin{equation} \label{3}
\int_{M} |\nabla u_\varepsilon|^p\; dv_g = \varepsilon^{-p} \left(I_2 - \frac{R_g(x_0)}{6n} J_2 \varepsilon^2 + O(\varepsilon^4) \right)\; .
\end{equation}
\n Plugging $u_\varepsilon$ in (\ref{EF1}), from (\ref{1}), (\ref{2}) and (\ref{3}), one obtains
\[
\frac{I_1 - n \log \varepsilon - \frac{R_g(x_0)}{6n} J_3 \varepsilon^2 + \frac{R_g(x_0)}{6} J_1 \varepsilon^2 \log \varepsilon + o(\varepsilon^4 \log \varepsilon)}{1 - \frac{R_g(x_0)}{6n} J_1 \varepsilon^2 + O(\varepsilon^4)} + (\frac{n}{p} - 1) \log \left(1 - \frac{R_g(x_0)}{6n} J_1 \varepsilon^2 + O(\varepsilon^4)\right)
\]
\[
\leq -n \log \varepsilon + \frac{n}{p} \log\left( {\cal A} I_2 - \frac{R_g(x_0)}{6n} {\cal A} J_2 \varepsilon^2 + {\cal B} \varepsilon^p +
O(\varepsilon^q)\right)
\]
\n for $\varepsilon > 0$ small enough, where $q = \min\{4, p +2\}$.
\n Taylor`s expansion then guarantees that
\begin{equation}\label{taylor}
I_1 - n \log \varepsilon - \frac{R_g(x_0)}{6n} J_3 \varepsilon^2 +
\frac{R_g(x_0)}{6} J_1 \varepsilon^2 \log \varepsilon +
\frac{R_g(x_0)}{6n} I_1 J_1 \varepsilon^2 - \frac{R_g(x_0)}{6} J_1
\varepsilon^2 \log \varepsilon
\end{equation}
\[
- (\frac{n}{p} - 1) \frac{R_g(x_0)}{6n} J_1 \varepsilon^2 +
O(\varepsilon^4 \log \varepsilon) \leq - n \log \varepsilon +
\frac{n}{p} \log ({\cal A} \; I_2) - \frac{n}{p} \frac{R_g(x_0)}{6n}
\frac{J_2}{I_2} \varepsilon^2 + \frac{n}{p} \frac{{\cal B}}{{\cal A} \; I_2}
\varepsilon^p + O(\varepsilon^q)\; .
\]
\n After a suitable simplification, one arrives at
\[
I_1 - \frac{n}{p} \log({\cal A} \; I_2) \leq - \frac{R_g(x_0)}{6n} \left( I_1 J_1 + \frac{n}{p} \frac{J_2}{I_2} - (\frac{n}{p} - 1) J_1 -
J_3\right) \varepsilon^2 + O(\varepsilon^p)
\]
\n for $\varepsilon > 0$ small enough, so that
\[
I_1 \leq \frac{n}{p} \log({\cal A} \; I_2) \; .
\]
\n Thus, since $u_0$ is an extremal function of (\ref{dee}), one has
\[
I_1 = \frac{n}{p} \log({\cal A}_0(p) \; I_2)\; ,
\]
\n so that ${\cal A} \geq {\cal A}_0(p)$, and the conclusion of Proposition \ref{P.2} readily follows. \bl
\section{Proof of the assertion {\bf (B)}}
This section is devoted to the proof of the following theorem:
\begin{teor}\label{lB}
Let $(M,g)$ be a smooth closed Riemannian manifold of dimension $n \geq 2$. For each fixed $1 < p \leq 2$ and $p < n$, the best constant $B(p,q,g)$ is bounded for any $q < p$ close to $p$.
\end{teor}
Clearly, it suffices to prove this result for an arbitrary sequence $(q_k) \subset (1, p)$ converging to $p$ as $k \rightarrow + \infty$.
\n Define
\[
C_k = \frac{B(p,q_k,g) - (p-q_k)}{N(p,q_k)}\; .
\]
\n Since $C_k < B(p,q_k,g) N(p,q_k)^{-1}$, according to the definition of the best constant $B(p,q_k,g)$, we have
\begin{equation}\label{ldf}
\inf_{u \in {\cal H}} J_k (u) < N(p,q_k)^{-1}\; ,
\end{equation}
\n where ${\cal H} = \{ u \in H^{1,p}(M):\; ||u||_{L^p(M)} = 1 \}$ and
\[
J_k(u) = \left( \int_M |\nabla_g u|^p\; dv_g + C_k \int_M |u|^p\; dv_g \right) \left( \int_M |u|^{q_k}\; dv_g \right)^{\frac{p(1 -
\theta_k)}{q_k \theta_k}}\; .
\]
\n As can easily be checked, the functional $J_k$ is of class $C^1$.
\n The condition (\ref{ldf}) and an usual argument of direct minimization lead to a minimizer $u_k \in {\cal H}$ of $J_k$, so that
\begin{equation}\label{l3iha}
\nu_k := J_k(u_k) = \inf_{u \in {\cal H}} J_k(u)\; .
\end{equation}
\n Note that we can assume $u_k \geq 0$, since $|\nabla_g |u_k|| = |\nabla_g u_k|$ almost everywhere.
\n In addition, since $J_k$ is differentiable, $u_k$ satisfies the quasilinear elliptic equation
\begin{equation}\label{l3ep}
A_k \Delta_{p,g} u_k + A_k C_k u_k^{p - 1} + \frac{1 -
\theta_k}{\theta_k} B_k u_k^{q_k - 1} = \frac{\nu_k}{\theta_k} u_k^{p - 1}
\hspace{0,2cm} \mbox{on} \hspace{0,2cm} M\; ,
\end{equation}
\n where $\Delta_{p,g} = -{\rm div}_g(|\nabla_g|^{p-2} \nabla_g)$ is the $p$-Laplace operator of $g$,
\[
\theta_k = \frac{n(p - q_k)}{np + pq_k - nq_k}\; ,
\]
\[
A_k = \left(\int_M u_k^{q_k}\; dv_g \right)^{\frac{p( 1 - \theta_k)}{q_k \theta_k}}\; ,
\]
\[
B_k = \left(\int_M |\nabla_g u_k|^p\; dv_g + C_k \right) \left(\int_M u_k ^{q_k}\; dv_g \right)^{\frac{p(1 - \theta_k)}{q_k \theta_k} - 1}\; .
\]
\n So, by Serrin \cite{Se}, $u_k \in L^\infty(M)$ and, by Tolksdorf's \cite{To}, it follows that $u_k$ is of class $C^1$.
\n A relation that will be useful is
\begin{equation} \label{nu}
B_k \int_M u_k^{q_k}\; dv_g =\nu_k\; .
\end{equation}
\n Modulo a subsequence, we analyze two possible situations for the sequence $(A_k)$.
\begin{itemize}
\item[{\bf (i)}] $\lim_{k \rightarrow + \infty} A_k > 0$;
\item[{\bf (ii)}] $\lim_{k \rightarrow + \infty} A_k = 0$.
\end{itemize}
\n Assume that {\bf (i)} occurs. Taking $u_k$ as a test function in (\ref{l3ep}) and after using Proposition \ref{P.1}, one gets
\begin{equation} \label{lim1}
A_k C_k \leq \nu_k \leq c
\end{equation}
\n for $k$ large enough and some constant $c$ independent of $k$, so that $(C_k)$ is bounded. Thus, the conclusion of Theorem \ref{lB} follows directly from the definition of $C_k$.
The remaining of this section is dedicated to prove the boundedness of $(C_k)$ by assuming that the situation {\bf (ii)} occurs. In this case, the inequality
\begin{equation} \label{lim2}
||u_k||_{L^\infty(M)} A_k^{\frac{n}{p^2}} \geq 1
\end{equation}
\n follows from
\[
1 = ||u_k||_{L^p(M)}^p = \int_M u_k^p\; dv_g \leq ||u_k||_{L^\infty(M)}^{p - q_k} \int_M u_k^{q_k}\; dv_g
\]
\n and implies that $(u_k)$ blows up in $L^\infty(M)$. Part of the proof consists in performing a fine analysis of concentration of the sequence $(u_k)$.
\n At first, we have
\begin{equation} \label{asym}
\lim \limits_{k \rightarrow + \infty} \nu_k = {\cal A}_0(p)^{-1}\; .
\end{equation}
\n In fact, a combination between the Nash inequality
\[
\left( \int_M |u|^p\; dv_g \right)^{\frac{1}{\theta_k}} \leq \left( N(p,q_k) \int_M |\nabla_g u|^p\; dv_g + B(p,q_k,g) \int_M |u|^p\; dv_g \right) \left(
\int_M |u|^{q_k}\; dv_g \right)^{\frac{p(1 - \theta_k)}{q_k \theta_k}}
\]
\n and the definition of $C_k$ yields
\[
N(p,q_k)^{-1} \leq \left(\int_M |\nabla_g u_k|^p\; dv_g + C_k \right)\left(\int_M u_k^{q_k}\; dv_g \right)^{\frac{p(1 - \theta_k)}{q_k \theta_k}} + \frac{(p - q_k)}{N(p,q_k)} \; A_k = \nu_k + A_k \;\frac{p - q_k}{N(p,q_k)}\; .
\]
\n By Proposition \ref{P.1}, $N(p,q_k)$ remains away from zero for $k$ large enough. Therefore,
\[
\liminf \limits_{k \rightarrow + \infty} \nu_k \geq {\cal A}_0(p)^{-1}
\]
\n and so the limit (\ref{asym}) follows from (\ref{ldf}).
Let $x_k \in M$ be a maximum point of $u_k$, that is
\begin{equation}\label{l3ix}
u_k(x_k) = ||u_k||_{L^\infty(M)}\; .
\end{equation}
The following concentration property satisfied by the sequence $(u_k)$ plays an essential role in what follows. The main tools used in its proof are the blow-up method and Moser's iteration.
\begin{lema}\label{CA}
We have
\[
\lim \limits_{\sigma\rightarrow +\infty} \lim \limits_{k \rightarrow + \infty} \int_{B(x_k,\sigma A_k^{\frac{1}{p}})} u_k^p\; dv_g = 1\; .
\]
\end{lema}
\n {\bf Proof of Lemma \ref{CA}.} Let $\sigma > 0$. For each $x \in B(0, \sigma)$ and $k$ large, we define
\[
\begin{array}{l}
h_k(x) = g(\exp_{x_k} (A_k^{\frac{1}{p}} x))\; , \vspace{0,3cm}\\
\varphi_k(x) = A_k ^{\frac{n}{p^2}}
u_k(\exp_{x_k}(A_k^{\frac{1}{p}} x))\; .
\end{array}
\]
\n Note that the above expressions are well defined because we are assuming that $A_k \rightarrow 0$ as $k \rightarrow + \infty$ (situation {\bf (ii)}).
\n By (\ref{l3ep}) and (\ref{nu}), one easily deduces that
\[
\Delta_{p,h_k} \varphi_k + A_k C_k \varphi_k^{p - 1} + \frac{1 - \theta_k}{\theta_k} \nu_k \varphi_k^{q_k - 1} = \frac{\nu_k}{\theta_k} \varphi_k^{p - 1} \hspace{0,2cm} \mbox{on} \hspace{0,2cm}
B(0,\sigma) \; .
\]
\n Using the mean value theorem and the value of $\theta_k$, one gets
\begin{equation}\label{el1}
\Delta_{p,h_k} \varphi_k + A_k C_k \varphi_k^{p - 1} = \nu_k \left( \varphi_k^{q_k - 1} + \frac{pq_k - nq_k + np}{n} \varphi_k^{\rho_k} \log \varphi_k \right) \hspace{0,2cm} \mbox{on} \hspace{0,2cm} B(0,\sigma)
\end{equation}
\n for some $\rho_k \in (q_k - 1,p - 1)$.
\n Consider $\varepsilon > 0$ fixed such that $p + \varepsilon < \frac{np}{n - p}$. Since $\varphi_k^{\rho_k} \log (\varphi_k^\varepsilon) \leq \varphi_k^{p - 1 + \varepsilon}$, we then get
\[
\Delta_{p,h_k} \varphi_k + A_k C_k \varphi_k^{p - 1} \leq \nu_k \left( \varphi_k^{q_k - 1} + \frac{pq_k - nq_k + np}{n \varepsilon}
\varphi_k^{\varepsilon + p - 1} \right) \hspace{0,2cm} \mbox{on} \hspace{0,2cm} B(0,\sigma)\; .
\]
\n Once all coefficients of this equation are bounded, the Moser's iterative scheme (see \cite{Se}) produces
\[
(A_k^{\frac{n}{p^2}} ||u_k||_{L^{\infty}(M)})^p = \sup_{B(0,\sigma)} \varphi_k^p \leq c_\sigma \int_{B(0,2\sigma)} \varphi_k^p\; dv_{h_k} =
c_\sigma \int_{B(x_k,2 \sigma A_k^{\frac{1}{p}})} u_k^p\; dv_g \leq c_\sigma
\]
\n for all $k$ large enough, where $c_\sigma$ is a constant independent of $q$.
\n So, using (\ref{lim2}) in the above inequality, one readily obtains
\begin{equation}\label{lep}
1 \leq ||\varphi_k||_{L^{\infty}(B(0,\sigma))} \leq c_\sigma^{1/p}
\end{equation}
\n for all $k$ large enough.
\n By (\ref{lim1}), up to a subsequence, we have
\[
\lim_{k \rightarrow + \infty} A_k C_k = C \geq 0\; .
\]
\n Applying the Tolksdorf's elliptic theory to (\ref{el1}), thanks to (\ref{lep}), one easily checks that $\varphi_k \rightarrow \varphi$ in $C^1_{loc}(\R^n)$ and $\varphi \not \equiv 0$.
\n Letting now $k \rightarrow + \infty$ in (\ref{el1}) and using (\ref{asym}), one gets
\begin{equation} \label{ELim}
\Delta_{p,\xi} \varphi + C \varphi^{p - 1} = {\cal A}_0(p)^{-1} \left( \varphi^{p - 1} + \frac{p}{n} \varphi^{p - 1} \log (\varphi^p)\right) \hspace{0,2cm} \mbox{on} \hspace{0,2cm} \R^n \;,
\end{equation}
\n where $\Delta_{p,\xi}$ stands for the Euclidean $p$-Laplace operator.
\n For each $\sigma > 0$, we have
\begin{equation} \label{whole}
\int_{B(0,\sigma)} \varphi^p\; dx = \lim_{k \rightarrow + \infty} \int_{B(0,\sigma)} \varphi_k^p\; dv_{h_k} = \lim_{k \rightarrow + \infty}\int_{B(x_k,\sigma A_k^{\frac{1}{p}})} u_k^p\; dv_g \leq 1
\end{equation}
\n and, by $(\ref{ldf})$ and Proposition \ref{P.1},
\begin{equation} \label{grad}
\int_{B(0,\sigma)} |\nabla \varphi|^p\; dx = \lim_{k \rightarrow + \infty} \int_{B(0,\sigma)} |\nabla _{h_k} \varphi_k|^p\; dv_{h_k} = \lim_{k \rightarrow + \infty} \left( A_k \int_{B(x_k,\sigma A_k^{\frac{1}{p}})} |\nabla_g u_k|^p\; dv_g \right) \leq {\cal A}_0(p)^{-1}\; .
\end{equation}
\n In particular, one has $\varphi \in W^{1,p}(\R^n)$.
\n Consider then a sequence of nonnegative functions $(\varphi_k) \subset C^\infty_0(\R^n)$ converging to $\varphi$ in $W^{1,p}(\R^n)$. Taking $\varphi_k$ as a test function in (\ref{ELim}), we can write
\[
\frac{n}{p} {\cal A}_0(p) \int_{\R^n} |\nabla \varphi|^{p-2} \nabla \varphi \cdot \nabla \varphi_k\; dx + \frac{n}{p} {\cal A}_0(p) C \int_{\R^n} \varphi^{p-1} \varphi_k\; dx = \int_{\R^n} \varphi^{p-1} \varphi_k \log (\varphi^p)\; dx + \frac{n}{p} \int_{\R^n} \varphi^{p-1} \varphi_k \; dx
\]
\[
= \int_{[\varphi \leq 1]} \varphi^{p-1} \varphi_k \log (\varphi^p)\; dx + \int_{[\varphi \geq 1]} \varphi^{p-1} \varphi_k \log (\varphi^p)\; dx + \frac{n}{p} \int_{\R^n} \varphi^{p-1} \varphi_k \; dx
\]
\n Letting $k \rightarrow + \infty$ and applying Fatou's lemma and the dominated convergence theorem in the right-hand side, one gets
\[
\frac{n}{p} {\cal A}_0(p) \int_{\R^n} |\nabla \varphi|^p\; dx \leq \int_{[\varphi \leq 1]} \varphi^p \log (\varphi^p)\; dx + \int_{[\varphi \geq 1]} \varphi^p \log (\varphi^p)\; dx + \frac{n}{p} \int_{\R^n} \varphi^p\; dx
\]
\[
= \int_{\R^n} \varphi^p \log (\varphi^p)\; dx + \frac{n}{p} \int_{\R^n} \varphi^p\; dx\; .
\]
\n Rewriting this inequality in function of $\psi(x) = \frac{\varphi(x)}{||\varphi||_p}$, one has
\[
\frac{n}{p} {\cal A}_0(p) \int_{\R^n} |\nabla \psi|^p\; dx \leq \int_{\R^n} \psi^p \log (\psi^p)\; dx + \frac{n}{p} \int_{\R^n} \psi^p\; dx + \log ||\varphi||_p^p\; .
\]
\n Note that this inequality combined with
\[
\int_{\R^n} \psi^p \log (\psi^p)\; dx \leq \frac{n}{p} \log \left({\cal A}_0(p) \int_{\R^n} |\nabla \psi|^p\; dx \right)
\]
\n produces
\[
{\cal A}_0(p) \int_{\R^n} |\nabla \psi|^p\; dx \leq \log \left({\cal A}_0(p)\; \int_{\R^n} |\nabla \psi|^p\; dx\right) + 1 + \frac{p}{n} \log ||\varphi||_p^p\; .
\]
\n Since $\log x \leq x - 1$ for all $x > 0$, we find $\log ||\varphi||_{L^p(\R^n)}^p \geq 0$ or, equivalently, $||\varphi||_{L^p(\R^n)} \geq 1$.
\n Thus, by (\ref{whole}), we conclude that $||\varphi||_{L^p(\R^n)} = 1$, so that
\[
\lim_{\sigma \rightarrow \infty} \lim_{k \rightarrow + \infty} \int_{B(x_k,\sigma A_k^{\frac{1}{p}})} u_k^p\; dv_g = \int_{\R^n} \varphi^p\; dx = 1\; .
\]
\bl
By using the concentration property provided in Lemma \ref{CA}, we now establish a pointwise estimate for the sequence $(u_k)$. This result is the key ingredient in the final part of the proof of Theorem \ref{lB}. The necessary tools in its proof are the same ones of the previous proof.
\begin{lema}\label{UE}
For any $\lambda > 0$, there exists a constant $c_\lambda > 0$, independent of $q < p$, such that
\[
d_g(x,x_k)^\lambda u_k(x) \leq c_\lambda A_k^{\frac{\lambda}{p} - \frac{n}{p^2}}
\]
\n for all $x \in M$ and $k$ large enough, where $d_g$ stands for the distance with respect to the metric $g$.
\end{lema}
\n {\bf Proof of Lemma \ref{UE}.} Suppose by contradiction that the above statement fails. Then, there exist $\lambda_0 > 0$ and $y_k \in M$ for each $k$ such that $f_{k,\lambda_0}(y_k) \rightarrow + \infty$ as $k \rightarrow + \infty$, where
\[
f_{k,\lambda}(x) = d_g(x,x_k)^\lambda u_k(x) A_k^{\frac{n}{p^2} - \frac{\lambda}{p}} \; .
\]
\n Without loss of generality, we assume that $f_{k,\lambda_0}(y_k) = ||f_{k,\lambda_0}||_{L^\infty(M)}$.
\n From (\ref{lep}), we have
\[
f_{k,\lambda_0}(y_k) \leq c \frac{u_k(y_k)}{||u_k||_{\infty}} d_g(x_k,y_k)^{\lambda_0} A_k^{- \frac{\lambda_0}{p}} \leq c d_g(x_k,y_k)^{\lambda_0} A_k^{- \frac{\lambda_0}{p}}\; ,
\]
\n so that
\begin{equation}\label{inf}
d_g(x_k,y_k) A_k^{- \frac{1}{p}} \rightarrow + \infty\ \ {\rm as}\ \ k \rightarrow + \infty\; .
\end{equation}
\n For any fixed $\varepsilon \in (0,1)$ and $\sigma > 0$, we first claim that
\begin{equation}\label{3int}
B(y_k,\varepsilon d(x_k,y_k)) \cap B(x_k, \sigma A_k^{\frac{1}{p}}) = \emptyset
\end{equation}
\n for $k$ large enough.
\n Clearly, this assertion follows from
\[
d_g(x_k,y_k) \geq \sigma A_k^{\frac{1}{p}} + \varepsilon d(x_k,y_k)
\]
\n or equivalently,
\[
d_g(x_k, y_k)(1 - \varepsilon) A_k^{- \frac{1}{p}} \geq \sigma \; .
\]
\n But the above inequality is automatically satisfied, since $d_g(x_k,y_k) A_k^{- \frac{1}{p}} \rightarrow + \infty$ as $k \rightarrow + \infty$.
\n We assert that exists a constant $c > 0$, independent of $k$, such that
\begin{equation}\label{lim}
u_k(x) \leq c u_k(y_k)
\end{equation}
\n for all $x \in B(y_k, \varepsilon d_g(x_k,y_k))$ and $k$ large enough. Indeed,
\[
d_g(x,x_k) \geq d_g(x_k,y_k) - d_g(x,y_k) \geq (1 - \varepsilon) d_g(x_k,y_k)
\]
\n for all $x \in B(y_k, \varepsilon d_g(x_k,y_k))$. Thus,
\[
d_g(x_k,y_k)^{\lambda_0} u_k(y_k) A_k^{\frac{n}{p^2} - \frac{\lambda_0}{p}} = f_{k,\lambda_0}(y_k) \geq f_{k,\lambda_0}(x) = d_g(x,x_k)^{\lambda_0} u_k(x) A_k^{\frac{n}{p^2} -
\frac{\lambda_0}{p}}
\]
\[
\geq (1 - \varepsilon)^{\lambda_0} d_g(x_k,y_k)^{\lambda_0} u_k(x) \; A_k^{\frac{n}{p^2} - \frac{\lambda_0}{p}} \; ,
\]
\n so that
\[
u_k(x) \leq \left(\frac{1}{1 - \varepsilon}\right)^{\lambda_0} u_k(y_k)
\]
\n for all $x \in B(y_k, \varepsilon d_g(x_k,y_k))$ and $k$ large enough, as claimed.
\n We now define
\[
\begin{array}{l}
\tilde{h}_k(x) = g(\exp_{y_k} (A_k^{\frac{1}{p}} x))\; , \vspace{0,3cm}\\
\tilde{\varphi}_k(x) = A_k ^{\frac{n}{p^2}} u_k(\exp_{y_k}(A_k^{\frac{1}{p}} x)) \; .
\end{array}
\]
\n By (\ref{l3ep}) and (\ref{nu}), it readily follows that
\[
\Delta_{p,h_k} \tilde{\varphi}_k + A_k C_k \tilde{\varphi}_k^{p - 1} + \frac{1 - \theta_k}{\theta_k} \nu_k \; \tilde{\varphi}_k^{q_k - 1}
= \frac{\nu_k}{\theta_k} \tilde{\varphi}_k^{p - 1} \hspace{0,2cm} \mbox{on} \hspace{0,2cm} B(0,3) \; .
\]
\n Applying the mean value theorem, one obtains
\begin{equation}\label{leqgn}
\Delta_{p,h_k} \tilde{\varphi}_k + A_k C_k \tilde{\varphi}_k^{p - 1} = \nu_k \left( \tilde{\varphi}_k^{q_k - 1} + \frac{pq_k - nq_k +
np}{n} \tilde{\varphi}_k^{\rho_k} \log (\tilde{\varphi}_k)\right) \hspace{0,2cm} \mbox{on} \hspace{0,2cm} B(0,3) \; ,
\end{equation}
\n where $\rho_k \in (q_k - 1,p - 1)$.
\n For fixed $\varepsilon > 0$ such that $p + \varepsilon < \frac{np}{n - p}$, one has $\tilde{\varphi}_k^{\rho_k} \log (\tilde{\varphi}_k^\varepsilon)
\leq \tilde{\varphi}_k^{p - 1 + \varepsilon}$. So, the Moser's iterative scheme applied to (\ref{leqgn}) yields
\begin{equation}\label{pf}
\mu_k^{\frac{p}{p - q_k}} = (A_k^{\frac{n}{p^2}} u_k(y_k))^p \leq \sup_{B(0,1)} \tilde{\varphi}_k^p \leq c \int_{B(0,2)}
\tilde{\varphi}_k^p\; dv_{\tilde{h}_k} = c \int_{B(y_k,A_k^{\frac{1}{p}})} u_k^p\; dv_g
\end{equation}
\n for $k$ large enough, where
\begin{equation}\label{mu}
\mu_k = u_k(y_k)^{p - q_k} \int_M u_k^{q_k} dv_g\; .
\end{equation}
\n We next analyze two independent situations that can occur:
\begin{itemize}
\item[{\bf (I)}] $\mu_k \geq 1 - \theta_k$ for all $k$, up to a subsequence;
\item[{\bf (II)}] $\mu_k < 1 - \theta_k$ for $k$ large enough.
\end{itemize}
\n In each case, we derive a contradiction. If the assertion {\bf (I)} is satisfied, on the one hand, one has
\[
\liminf \limits_{k \rightarrow + \infty} \mu_k^{\frac{p}{p - q_k}} \geq e^{-\frac np}\; .
\]
\n On the other hand, by (\ref{inf}), one gets
\begin{equation}\label{sub}
B(y_{q_k}, A_{q_k}^{\frac{1}{p}}) \subset B(y_{q_k}, \varepsilon d(x_{q_k},y_{q_k}))
\end{equation}
\n for $k$ large enough. Thus, joining Lemma \ref{CA}, (\ref{3int}) and (\ref{pf}), one arrives at the contradiction
\[
0 < e^{-\frac np} \leq \lim_{k \rightarrow + \infty} \int_{B(y_{q_k},A_{q_k}^{\frac{1}{p}})} u_{q_k}^p\; dv_g = 0 \; .
\]
\n Assume then the assertion {\bf (II)}. In this case, for $k$ large, we set
\[
\begin{array}{l}
\tilde{h}_k(x) = g(\exp_{y_k}(A_k^{\frac{1}{p}} x)) \vspace{0,2cm}\\
\psi_k(x) = u_k(y_k)^{-1} u_k(\exp_{y_k}(A_k^{\frac{1}{p}} x))\; .
\end{array}
\]
\n Thanks to (\ref{l3ep}) and (\ref{nu}), we have
\[
\Delta_{p,h_k} \psi_k + A_k C_k \psi_k^{p - 1} + \frac{1 - \theta_k}{\theta_k} \frac{\nu_k}{\mu_k} \psi_k^{q_k - 1} =
\frac{\nu_k}{\theta_k} \psi_k^{p - 1} \hspace{0,2cm} \mbox{on} \hspace{0,2cm} B(0,3) \; .
\]
\n Rewriting this equation as
\[
\Delta_{p,h_k} \psi_k + A_k C_k \psi_k^{p - 1} + \frac{\nu_k}{\theta_k}\left(\frac{1 - \theta_k}{\mu_k} -1\right)
\psi_k^{q_k - 1} = \frac{\nu_k}{\theta_k}\left( \psi_k^{p - 1} - \psi_k^{q_k - 1}\right) \hspace{0,2cm} \mbox{on} \hspace{0,2cm} B(0,3) \; ,
\]
\n from the mean value theorem, one gets
\begin{equation} \label{norm}
\Delta_{p,h_k} \psi_k + A_k C_k \psi_k^{p - 1} + \frac{\nu_k}{\theta_k}\left(\frac{1 - \theta_k}{\mu_k} -1\right) \psi_k^{q_k - 1} = \frac{\nu_k (pq_k - nq_k + np)}{n}
\psi_k^{\rho_k} \log (\psi_k) \hspace{0,2cm} \mbox{on} \hspace{0,2cm} B(0,3)
\end{equation}
\n for some $\rho_k \in (q_k-1,p-1)$.
\n Using (\ref{inf}), (\ref{lim}) and the fact that $\frac{1 - \theta_k}{\mu_k} - 1 > 0$ for $k$ large, one easily deduces that $\psi_k \rightarrow \psi$ in
$W^{1,p}(B(0,2))$ and, by a Moser's iteration, one has $\psi \not \equiv 0$.
\n Let $h \in C^1_0(B(0,2))$ be a fixed cutoff function such that $h \equiv 1$ in $B(0,1)$ and $h \geq 0$. Taking $\psi h^p$ as a test function in (\ref{norm}), one obtains
\[
\limsup_{k \rightarrow + \infty} \frac{1}{\theta_k}\left(\frac{1 - \theta_k}{\mu_k} - 1\right) < c\; .
\]
\n Therefore, up to a subsequence, we can write
\[
\lim_{k \rightarrow + \infty} \frac{1}{\theta_k}\left(\frac{1 -
\theta_k}{\mu_k} - 1\right) = \gamma \geq 0\; ,
\]
\n so that $\mu_k \rightarrow 1$. Using the definition of $\theta_k$, we then derive
\[
\lim \limits_{k \rightarrow + \infty} \mu_k^{\frac{p}{p - q_k}} = e^{-(1 + \gamma)\frac{n}{p}}\; .
\]
\n At last, combining Lemma \ref{CA}, (\ref{3int}), (\ref{pf}) and the above limit, we obtain the contradiction
\[
0 < e^{-(1 + \gamma)\frac{n}{p}} \leq \lim_{k \rightarrow + \infty} \int_{B(y_k,A_k^{\frac{1}{p}})} u_k^p\; dv_g = 0\; .
\]
\bl\\
Finally, we turn our attention to the final argument of the proof of Theorem \ref{lB}. We recall that our goal is to prove that the sequence $(C_k)$ is bounded by assuming that $A_k \rightarrow 0$ as $k \rightarrow + \infty$ (situation {\bf (ii)}). This step consists of several integral estimates around the maximum point $x_k$ of $u_k$ and Lemma \ref{UE} plays a central role on some of them.
In what follows, several possibly different positive constants independent of $k$ will be denoted by $c$ or $c_1$.
Assume, without loss of generality, that the radius of injectivity of $M$ is greater than $2$. Let $\eta \in C^1_0(\R)$ be a cutoff function such that $\eta = 1$ on $[0,1)$, $\eta = 0$ on $[2, \infty)$ and $0 \leq \eta \leq 1$ and define $\eta_k(x) = \eta(d_g(x,x_k))$.
The sharp Euclidean $L^p$-Nash inequality asserts that
\[
\left( \int_{B(0,2)} (u_k \eta_k)^p\; dx \right)^{\frac{1}{\theta_k}} \leq N(p,q_k) \left( \int_{B(0,2)} |\nabla (u_k \eta_k)|^p\; dx\right) \left(\int_{B(0,2)} (u_k \eta_k)^{q_k}\; dx \right)^{\frac{p(1 - \theta_k)}{q_k \theta_k}}\; .
\]
\n Expanding the metric $g$ in normal coordinates around $x_k$, one locally gets
\[
(1 - c d_g(x,x_k)^2) dv_g \leq dx \leq (1 + c d_g(x,x_k)^2) dv_g
\]
\n and
\[
|\nabla(u_k \eta_k)|^p \leq |\nabla_g(u_k \eta_k)|^p (1 + c \; d_g(x,x_k)^2) \; .
\]
\n Thus,
\[
\left( \int_{B(0,2)} u_k^p \eta_k^p\; dx \right)^{\frac{1}{\theta_k}} \leq \left( N(p,q_k) A_k \int_{B(x_k,2)} |\nabla_g (u_k \eta_k)|^p\; dv_g + c A_k \int_{B(x_k,2)} |\nabla_g (u_k\eta_k)|^p d_g(x,x_k)^2\; dv_g \right)
\]
\[
\times \left( \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g} \right)^{\frac{p(1 - \theta_k)}{q_k \theta_k}} \; .
\]
\n Using now the inequality
\[
|\nabla_g (u_k \eta_k)|^p \leq |\nabla_g u_k|^p \eta_k^p + c |\eta_k \nabla_g u_k|^{p - 1} |u_k \nabla_g \eta_k| + c |u_k \nabla_g \eta_k|^p \; ,
\]
\n and denoting
\[
X_k = A_k \int_M \eta_k^p |\nabla_g u_k|^p d_g(x,x_k)^2\; dv_g
\]
\n and
\[
Y_k = A_k \int_M |\nabla_g u_k|^{p -1} |\nabla_g \eta_k| u_k\; dv_g \; ,
\]
\n we deduce that
\begin{equation}\label{af1}
\left( \int_{B(0,2)} u_k^p \eta_k^p\; dx \right)^{\frac{1}{\theta_k}} \leq \left( N(p,q_k) A_k \int_M |\nabla_g u_k|^p \eta_k^p\; dv_g + c X_k + c Y_k + c A_k \right)
\left( \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g} \right)^{\frac{p(1 - \theta_k)}{q_k \theta_k}}\; .
\end{equation}
\n On the other hand, choosing $u_k \eta_k^p$ as a test function in (\ref{l3ep}) and using (\ref{nu}), one gets
\[
N(p,q_k) A_k \int_M |\nabla_g u_k|^p \eta_k^p\; dv_g \leq 1 - N(p,q_k) A_k C_k + \frac{1}{\theta_k} \left( \int_M u_k^p \eta_k^p\; dv_g - \frac{\int_M u_k^{q_k} \eta_k^p\; dv_g}{\int_M u_k^{q_k}\;
dv_g} \right)
\]
\[
+ c \int_M |\nabla_g u_k|^{p -1} |\nabla_g \eta_k| u_k\; dv_g \; .
\]
\n Using (\ref{ldf}), Proposition \ref{P.1} and Lemma \ref{UE} with a suitable value of $\lambda$, the last integral can be estimated as
\[
\int_M |\nabla_g u_k|^{p -1} |\nabla_g \eta_k| u_k\; dv_g \leq c \left( \int_M |\nabla_g u_k|^p\; dv_g \right)^{\frac{p-1}{p}} \left( \int_{B(x_k,2) \setminus B(x_k,1)} u_k^p\; dv_g \right)^{\frac{1}{p}} \leq c A_k\; ,
\]
\n so that
\begin{equation}\label{af2}
N(p,q_k) A_k \int_M |\nabla_g u_k|^p \eta_k^p\; dv_g \leq 1 - c_1 A_k C_k + \frac{1}{\theta_k} \left( \int_M u_k^p \eta_k^p\; dv_g - \frac{\int_M u_k^{q_k} \eta_k^p\; dv_g}{\int_M u_k^{q_k}\;
dv_g} \right) + c A_k
\end{equation}
\n for $k$ large enough.
\n Let
\[
Z_k = \frac{1}{\theta_k} \left( \int_M u_k^p \eta_k^p\; dv_g - \frac{\int_M u_k^{q_k} \eta_k^{q_k}\; dv_g}{\int_M u_k^{q_k}\; dv_g} \right) \; .
\]
\n By the mean value theorem and Lemma \ref{UE}, there exists $\gamma_k \in (q_k,p)$ such that
\begin{equation}\label{af3}
\left| Z_k - \frac{1}{\theta_k}\left( \int_M u_k^p \eta_k^p\; dv_g - \frac{\int_M u_k^{q_k} \eta_k^p\; dv_g}{\int_M u_k^{q_k}\; dv_g} \right)
\right| \leq \frac{1}{\theta_k} \left| \frac{\int_M u_k^{q_k} (\eta_k^{q_k} - \eta_k^p)\; dv_g}{\int_M u_k^{q_k}\; dv_g} \right| \leq \frac{pq_k - nq_k + np}{n} \frac{\int_M \eta_k^{\gamma_k} | \log \eta_k |
u_k^{q_k}\; dv_g}{\int_M u_k^{q_k}\; dv_g}
\end{equation}
\[
\leq c \frac{\int_{B(x_k,2) \setminus B(x_k,1)} u_k^{q_k}\; dv_g}{\int_M u_k^{q_k}\; dv_g} \leq c A_k
\]
\n for $k$ large enough.
\n Plugging (\ref{af3}) into (\ref{af2}) and after (\ref{af2}) into (\ref{af1}), one arrives at
\begin{equation} \label{af4}
\left( \int_{B(0,2)} u_k^p \eta_k^p\; dx \right)^{\frac{1}{\theta_k}} \leq \left( 1 - c A_k C_k + Z_k + c X_k + c Y_k + c A_k\right) \left( \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g} \right)^{\frac{p(1 - \theta_k)}{q_k \theta_k}}
\end{equation}
\n for $k$ large enough.
\n In order to estimate $X_k$, we take $u_k d_g^2 \eta_k^p$ as a test function in (\ref{l3ep}). From this choice, we derive
\[
X_k \leq \frac{\nu_k}{\theta_k}\left( \int_M u_k^p \eta_k^p d_g(x,x_k)^2\; dv_g - \frac{\int_M u_k^{q_k} \eta_k^{q_k} d_g(x,x_k)^2\; dv_g}{\int_M u_k^{q_k}\;
dv_g}\right) + c A_k \int_M u_k \eta_k^p |\nabla_g u_k|^{p - 1} d_g(x,x_k)\; dv_g
\]
\[
+ c \frac{\int_M u_k^{q_k} \eta_k^{q_k} d_g(x,x_k)^2\; dv_g}{\int_M u_k^{q_k} dv_g} + c Y_k + c A_k\; .
\]
\n We now estimate the first two terms of the right-hand side above. Namely, after a change of variable, one has
\[
\frac{1}{\theta_k} \int_{M} \left| u_k^p \eta_k^p d_g(x,x_k)^2 - \frac{u_k^{q_k} \eta_k^{q_k} d_g(x,x_k)^2}{\int_M u_k^{q_k}\; dv_g}\right|\; dv_g \leq c A_k^{\frac{2}{p}} \frac{1}{\theta_k} \int_{B(0,2 A_k^{- \frac{1}{p}})} \left| \varphi_k^p \tilde{\eta}_k^p - \varphi_k^{q_k} \tilde{\eta}_k^{q_k} \right| |x|^2\; dx
\]
\[
= c A_k^{\frac{2}{p}} \int_{B(0,2 A_k^{- \frac{1}{p}})} (\varphi_k \tilde{\eta}_k)^{\rho_k} \left| \log(\varphi_k \tilde{\eta}_k) \right| |x|^2\; dx
\]
\n for some $\rho_k \in (q_k,p)$, where $\tilde{\eta}_k(x) = \eta_k(A_k^{\frac{1}{p}} x)$.
\n Thus, using Lemma \ref{UE} and the assumption $p \leq 2$, one obtains
\begin{equation} \label{est6}
\frac{1}{\theta_k} \int_{M} \left| u_k^p \eta_k^p d_g(x,x_k)^2 - \frac{u_k^{q_k} \eta_k^{q_k} d_g(x,x_k)^2}{\int_M u_k^{q_k}\; dv_g}\right|\; dv_g \leq c A_k\; .
\end{equation}
\n In particular, since
\[
\int_M u_k^p \eta_k^p d_g(x,x_k)^2\; dv_g - \frac{\int_M u_k^{q_k} \eta_k^{q_k} d_g(x,x_k)^2\; dv_g}{\int_M u_k^{q_k}\; dv_g} = \int_{M} \left( u_k^p \eta_k^p d_g(x,x_k)^2 - \frac{u_k^{q_k} \eta_k^{q_k} d_g(x,x_k)^2}{\int_M u_k^{q_k} \; dv_g}\; \right) dv_g\; ,
\]
\n we have
\[
\frac{1}{\theta_k}\left| \int_M u_k^p \eta_k^p d_g(x,x_k)^2\; dv_g - \frac{\int_M u_k^{q_k} \eta_k^{q_k} d_g(x,x_k)^2\; dv_g}{\int_M u_k^{q_k}\;
dv_g}\right| \leq c A_k
\]
\n for $k$ large enough.
\n Besides, thanks to (\ref{ldf}), Proposition \ref{P.1}, Lemma \ref{UE} and the fact that $p \leq 2$, we derive
\[
\int_M u_k \eta_k^p |\nabla_g u_k|^{p - 1} d_g(x,x_k)\; dv_g \leq c \left( \int_M |\nabla_g u_k|^p\; dv_g \right)^{\frac{p-1}{p}} \left( \int_{B(x_k,2)} u_k^p d_g(x,x_k)^p\; dv_g \right)^{\frac{1}{p}} \]
\[
\leq c A_k^{\frac{2-p}{p}} \left( \int_{B(0,2 A_k^{- \frac{1}{p}})} \varphi_k^p |x|^p\; dx \right)^{\frac{p-1}{p}} \leq c
\]
\n for $k$ large enough.
\n So, the above estimates guarantees that
\[
X_k \leq c \frac{\int_M u_k^{q_k} \eta_k^{q_k} d_g(x,x_k)^2\; dv_g}{\int_M u_k^{q_k}\; dv_g} + c Y_k + c A_k\; .
\]
\n Evoking again Lemma \ref{UE} and the condition $p \leq 2$, one has
\begin{equation}\label{f1}
\frac{\int_M u_k^{q_k} \eta_k^{q_k} d_g(x,x_k)^2\; dv_g}{\int_M u_k^{q_k}\; dv_g} \leq c A_k^{\frac{2}{p}} \int_{B(0,2A_k^{-1/p})} \varphi_k^{q_k} |x|^2\; dh_k \leq c A_k
\end{equation}
\n Also, it follows directly from (\ref{ldf}) and Proposition \ref{P.1} that
\begin{equation}\label{est3}
Y_k \leq c A_k^{\frac{1}{p}} \int_{B(x_k,2) \backslash B(x_k,1)} u_k^p\; dv_g \leq c A_k\; ,
\end{equation}
\n so that
\begin{equation}\label{f2}
X_k \leq c A_k\; .
\end{equation}
\n Thus, plugging (\ref{est3}) and (\ref{f2}) into (\ref{af4}), one gets
\[
\left(\int_{B(0,2)} u_k^p \eta_k^p\; dx \right)^{\frac{1}{\theta_k}} \leq \left(1 + Z_k - c_1 A_k C_k + c A_k \right) \left(\frac{\int_{B(0,2)} (u_k \eta_k)^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g} \right)^{\frac{p(1 - \theta_k)}{q_k \theta_k}}
\]
\n for $k$ large enough.
\n Taking logarithm of both sides and using the fact that $\frac{p(1 - \theta_k)}{q \theta_k} = \frac{1}{\theta_k} - \frac{n - p}{n}$, one has
\begin{equation}\label{d1}
\frac{1}{\theta_k} \left( \log \int_{B(0,2)} u_k^p \eta_k^p dx - \log \left( \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k} dx}{\int_M u_k^{q_k}\; dv_g}
\right) \right) \leq \log (1 + Z_k - c_1 A_k C_k + c A_k) - \frac{n - p}{n} \log \left( \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g}
\right)\; .
\end{equation}
\n By the mean value theorem,
\begin{equation}\label{d2}
\log \int_{B(0,2)} u_k^p \eta_k^p\; dx - \log \left( \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g} \right)
= \frac{1}{\tau_k} \left( \int_{B(0,2)} u_k^p \eta_k^p\; dx - \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g} \right)
\end{equation}
\n for some number $\tau_k$ between the expressions
\[
\int_{B(0,2)} u_k^p \eta_k^p\; dx\ \ {\rm and}\ \ \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g} \; .
\]
\n Using Cartan's expansion of $g$ in normal coordinates around $x_k$ and Lemma \ref{UE}, one obtains
\begin{equation} \label{est4}
\max\left\{ \left| \int_{B(0,2)} u_k^p \eta_k^p\; dx - \int_M u_k^p \eta_k^p\; dv_g \right|, \left| \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g} - \frac{\int_M u_k^{q_k} \eta_k^{q_k} dv_g}{\int_M u_k^{q_k} dv_g} \right|\right\} \leq c A_k
\end{equation}
\n for $k$ large enough. Indeed, since $p \leq 2$,
\[
\left| \int_{B(0,2)} u_k^p \eta_k^p\; dx - \int_M u_k^p \eta_k^p\; dv_g \right| \leq c \int_M u_k^p \eta_k^p d_g(x,x_k)^2 \; dv_g \leq c A_k^{\frac{2}{p}} \int_{B(0,2A_k^{-1/p})} \varphi_k^p |x|^2\; dv_{h_k} \leq c A_k
\]
\n and, by (\ref{f1}),
\[
\left| \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g} - \frac{\int_M u_k^{q_k} \eta_k^{q_k} dv_g}{\int_M u_k^{q_k} dv_g} \right| \leq c \frac{\int_M u_k^{q_k} \eta_k^{q_k} d_g(x,x_k)^2\; dv_g}{\int_M u_k^{q_k}\; dv_g} \leq c A_k\; .
\]
\n Moreover, we also have
\begin{equation} \label{est5}
\max\left\{ \left| \int_M (u_k \eta_k)^p\; dv_g - 1\right|, \left|\frac{\int_M u_k^{q_k} \eta_k^{q_k}\; dv_g}{\int_M u_k^{q_k}\; dv_g} - 1 \right|\right\} \leq c A_k
\end{equation}
\n for $k$ large enough. In fact, by Lemma \ref{UE},
\[
\left| \int_M u_k^p \eta_k^p\; dv_g - 1\right| = \left| \int_M u_k^p \eta_k^p\; dv_g - \int_M u_k^p \; dv_g\right| \leq c \int_{M \setminus B(x_k,1)} u_k^p \; dv_g \leq c A_k
\]
\n and
\[
\left|\frac{\int_M u_k^{q_k} \eta_k^{q_k}\; dv_g}{\int_M u_k^{q_k}\; dv_g} - 1 \right| \leq c \frac{\int_{M \setminus B(x_k,1)} u_k^{q_k}\; dv_g}{\int_M u_k^{q_k}\; dv_g} \leq c A_k\; .
\]
\n Thanks to (\ref{est4}) and (\ref{est5}), one easily deduces that $\tau_k^{-1} = 1 + O(A_k)$. Then, by (\ref{d2}),
\begin{equation}\label{d3}
\frac{1}{\theta_k} \left( \log \int_{B(0,2)} u_k^p \eta_k^p\; dx - \log \left( \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g} \right) \right)
= \frac{1}{\theta_k} \left( \int_{B(0,2)} u_k^p \eta_k^p\; dx - \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g} \right) (1 + O(A_k))\; .
\end{equation}
\n But, by Cartan's expansion and (\ref{est6}), we have
\[
\frac{1}{\theta_k} \left( \int_{B(0,2)} u_k^p \eta_k^p\; dx - \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g} \right) = \frac{1}{\theta_k} \left(\int_{B(0,2)} u_k^p \eta_k^p - \frac{u_k^{q_k} \eta_k^{q_k}}{\int_M u_k^{q_k}\; dv_g}\; dx \right)
\]
\[
= \frac{1}{\theta_k} \left(\int_M u_k^p \eta_k^p - \frac{u_k^{q_k} \eta_k^{q_k}}{\int_M u_k^{q_k}\; dv_g}\; dv_g \right) + \frac{1}{\theta_k} \left(\int_M \left( u_k^p \eta_k^p - \frac{u_k^{q_k} \eta_k^{q_k}}{\int_M u_k^{q_k}\; dv_g} \right) O(d_g(x, x_k)^2)\; dv_g \right)
\]
\[
= Z_k + O(A_k)
\]
\n for $k$ large enough.
\n Replacing this inequality in (\ref{d3}), one obtains
\[
\frac{1}{\theta_k} \left(\log \int_{B(0,2)} u_k^p \eta_k^p\; dx - \log \left( \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g} \right)\right) \geq Z_k - c A_k\; .
\]
\n In turn, plugging the above inequality in (\ref{d1}), one has
\[
Z_k - c A_k \leq \log (1 + Z_k - c_1 A_k C_k + c A_k) + \frac{n - p}{n} \left| \log \left( \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g} \right) \right|\; .
\]
\n Finally, using Cartan's expansion of $g$ in normal coordinates, Taylor's expansion of the function $\log$ and Lemma \ref{UE}, one gets
\[
\left| \log \left( \frac{\int_{B(0,2)} u_k^{q_k} \eta_k^{q_k}\; dx}{\int_M u_k^{q_k}\; dv_g} \right) \right| \leq c \frac{\int_{M \backslash B(x_k,1)} u_k^{q_k}\; dv_g}{\int_M u_k^{q_k}\; dv_g} + c \frac{\int_M u_k^{q_k} \eta_k^{q_k} d_g(x,x_k)^2\; dv_g}{\int_M u_k^{q_k}\; dv_g} \leq c A_k\; .
\]
\n In short, for a certain constant $c > 0$, we deduce that
\[
Z_k \leq \log (1 + Z_k - c_1 A_k C_k + c A_k) + c A_k
\]
\n for $k$ large enough.
\n Since $\log x \leq x - 1$ for all $x > 0$, we find
\[
Z_k \leq Z_k - c_1 A_k C_k + c A_k
\]
\n for $k$ large enough, so that the sequence $(C_k)$ is bounded and the proof of Theorem \ref{lB} follows.\bl \\
{\bf Acknowledgements.} The authors are indebted to the referee for his valuable suggestions and comments pointed out concerning this work. The first author was partially supported by CAPES through INCTmat and the second one was partially supported by CNPq and Fapemig.
|
2,869,038,156,344 | arxiv | \section{Introduction}
\subsection{Motivation}
A main focus of quantitative modeling in finance has been to
measure the risk of financial portfolios. In connection with the
widespread use of Value-at-Risk (VaR) and related risk measurement
methodologies, a considerable theoretical literature
\citep{acerbi2002smr,acerbi2007cmr,artzner1999cmr,contdeguestscandolo,FollmerHSchiedA:02convexriskmeasure,follmer2011sfi,frittelli2002por,heyde2006grm,mcneil2005qrm}
has focused on the design of appropriate risk measures for financial
portfolios. In this approach, a risk measure is represented as a
real-valued map assigning to each random variable
$X$---representing the payoff of a portfolio---a number which
measures the risk of this portfolio. A framework often used as a
starting point is the axiomatic setting of { \cite{artzner1999cmr},
which defines a {\it coherent risk measure} as a map
$\rho:L^{\infty}(\Omega,{\cal F},\mathbb{P})\to\mathbb{R}$ that is
\begin{enumerate}
\item \textit{monotone} (\emph{decreasing}): $\rho(X)\leq
\rho(Y)$ provided $X\geq Y$;
\item \textit{cash-additive (additive with respect to cash reserves)}:
$\rho(X+c)= \rho(X)-c$ for any $c\in\mathbb{R}$;
\item \textit{positive homogeneous}: $\rho(\lambda X)=\lambda\rho(X)$ for any $\lambda\geq 0$;
\item \textit{sub-additive}: $\rho(X+Y)\leq \rho(X)+\rho(Y)$.
\end{enumerate}
\cite{artzner1999cmr} argue that these axioms correspond to
desirable properties of a risk measure, such as the reduction of
risk under diversification. These axioms have provided an elegant
mathematical framework for the study of coherent risk measures, but
fail to take into account some key features encountered in the
practice of risk management, as illustrated by the following
(important) example.
Consider a central clearing facility or an exchange, in which
various market participants clear portfolios of financial
instruments. Any participant of the
clearing house must deposit a margin
requirement for the purpose of covering the potential cost of
liquidating the clearing participant's portfolio in case of
default. The risk measurement problem facing the exchange is
therefore to determine the margin requirement for each portfolio,
which is in this case the risk measure of the portfolio as seen by
the exchange. Unlike the situation of an investor evaluating his/her
own risk, the exchange is affected by the gains and losses of the
market participants in an asymmetric way. As long as the market
participant's positions results in a gain, the gain is kept by the
participant, but if the participant suffers a loss, the exchange may
have to step in and cover the loss in case the participant default.
It follows that, when measuring the risk posed {\it to the exchange}
by a participant's portfolio, it is only relevant to consider the
{\it losses} of this portfolio, not the gains. Indeed, the well
known Standard Portfolio ANalysis (SPAN) method introduced by the
Chicago Merchantile Exchange and used by many other exchanges,
computes the margin requirement of a financial portfolio with profit
and loss (P\&L) $X$ as the maximum {\em loss} of the portfolio
over a set of pre-selected stress scenarios
$\omega_1,\dots,\omega_n$:
\begin{align}\label{eq:SPAN}
\rho(X)= \max \{ - \min(X(\omega_1),0),..., - \min(X(\omega_n),0)\}.
\end{align}
As we can see in this example, the risk measure of a portfolio is
only based on the loss $\min(X,0)$ that is, the negative part of
$X$.
The argument that a risk measure should be based on losses, not
gains, is not restricted to the problem of computing margin
requirements for a central clearing facility. Indeed, a regulator
faces a similar issue when evaluating the cost of a bank failure:
these costs materialize only in scenarios when a bank undergoes
large losses resulting in its default, whereas the trading gains of
a bank do not positively affect the regulator's position. Thus, the
risk of a bank's portfolio, as viewed by the regulator, should also
be based on the magnitude of the bank's {\it loss}, not its
potential gains.
These examples show that, a risk measure used for determining
capital (or margin) requirements (called 'external' risk measure in
\cite{heyde2006grm}) should be solely based on the loss of a
portfolio. This property can be formulated by requiring the risk
measure $\rho(X)$ to depend only on the negative part of $X$,
representing the loss:
\begin{equation} \rho(X) = \rho(\ \min(X,0)\ ). \label{eq.loss}\end{equation}
This property is clearly not contained in the axioms of coherent
risk measures. In fact, the cash-additivity property implies that
coherent risk measures must depend on gains as well, which clearly
contradicts \eqref{eq.loss}. So, one may not simply add the
loss-dependence property \eqref{eq.loss} to the axioms of coherent
risk measures without reconsidering the other axioms. In fact, the
CME SPAN method does {\it not} verify the cash-additivity axiom and
therefore is not a coherent risk measure.\footnote{It is interesting
to note that the CME SPAN method was one of the motivations cited in
\cite{artzner1999cmr} for introducing the framework of 'coherent'
risk measures.}
Many revisions to the axioms of coherent risk measures have been
proposed and studied in the literature, replacing in particular
positive homogeneity and sub-additivity with the more general
convexity property
\citep{FollmerHSchiedA:02convexriskmeasure,follmer2011sfi,frittelli2002por},
co-monotonic sub-additivity \citep{heyde2006grm} or co-monotonic
convexity \citep{SongYan:2009riskmeasure}. But these alternative
frameworks still rely on cash-additivity and do not consider the
property of loss-dependence as formulated in \eqref{eq.loss}, so a
proper definition of loss-based risk measure calls for a systematic
revision of the axioms of \cite{artzner1999cmr} along new
directions.
Let us note here
that the property of loss-dependence is not exactly the same as
requiring the risk measure to depend on, say, the left tail of the
loss distribution (which is the case for Value at Risk or Expected
Shortfall, for instance). The example of a portfolio with random,
but positive payoffs shows the difference between these two.
We propose a new class of risk measures, {\em loss-based risk
measures}, which depend only on the loss of a financial portfolio,
and investigate the properties of such risk measures.
Since the cash-additivity property is incompatible with the
loss-based property, we remove the cash-additivity for risk
measures. However, it is worth mentioning that loss-based
risk measures, though not cash-additive,
do not necessarily violate the property
$\rho(X+\rho(X))=0$. Indeed, it is easy to verify that the
CME SPAN method satisfies this
property, without verifying the cash-additivity property.
\cite{ElKarouiRavanelli:2009CashSubadditive} challenge the axiom of
cash additivity from a different angle, showing that when
considering risk measures defined on {\em future} (instead of
discounted) value of portfolio gains in order to take into account
interest rate risk, cash additivity is inevitably violated and
needs to be replaced by cash subadditivity. Our approach provides an
independent motivation for relaxing cash-additivity as a property
for risk measures.
Another issue which is extremely important in practice but somewhat
neglected in the literature on risk measures is the issue of
statistical estimation and the design of robust estimators for
risk measures. Risk measures are usually defined in terms of a portfolio's
profit/loss (P\&L) or their distributions. Because these distributions are
not directly observed, one has to use historical data to {\it estimate } the risk of
each portfolio. For instance, one may use the historical P\&Ls of a
portfolio to estimate the distribution of the portfolio P\&L and
apply a risk measure on the estimated distribution to obtain an
estimate of the portfolio risk. Such a procedure, which starts
from historical data or simulated losses as input and obtains the estimated
risk of the portfolio as output, is called a risk estimator. Roughly
speaking, a risk estimator is robust if the resulting risk estimate
of a portfolio is not extremely sensitive to a small change in the
sample. A non-robust risk estimator may vary dramatically from day
to day which makes it difficult to use and even more difficult, if not impossible, to backtest. For
instance, if the clearing house computes margin requirements
according to a non-robust risk estimator, market participants may
find the resulting margin requirements to be extremely volatile and
thus unacceptable. Unfortunately, as shown in
\cite{contdeguestscandolo}, an unintended consequence of
subadditivity property is that it
requires dependence on extreme tail events,
which leads in turn to high sensitivity to outliers and lack of
robustness. We study these issues in the context of loss-based risk
measures, extending previous work of
\cite{contdeguestscandolo}, and characterize loss-based
risk measures which admit robust estimators.
Loss-based risk measures have in fact a long tradition in actuarial science: \cite{Hattendorff1868} is an early example. In the financial risk management context, this idea was explored by
\cite{jarrow02} and, in parallel with the present work, by \cite{Staum:2011ExcessInvarianceShortfall} in a discrete setting.
\cite{jarrow02} defines a risk measure that is the premium of the
put option on a portfolio's net value.
Our setting extends these examples, discussed in Section 2, and allows to
obtain a characterization of loss-based risk measures on a
general probability space and study the robustness of the associated
risk estimators.
\subsection{Main results}
We consider in this paper an alternative approach to defining risk
measures which addresses these concerns.
Starting from the requirement that risk measures of financial
portfolios should be based on their losses, not their gains, we
define the notion of {\it loss-based risk measure} and study the
properties of this class of risk measures.
We first provide a dual representation for convex loss-based risk
measures, which are loss-based risk measures satisfying a convexity
property. This representation is similar to that of convex risk
measures, and states that a convex loss-based risk measure is
worst-case expected loss (adjusted by some penalty). For {\it
statistical} convex loss-based risk measures i.e. which only depend
on the loss distribution, we provide another representation theorem
in terms of portfolio loss quantiles.
We provide abundant examples of convex loss-based risk measures,
many of which are obtained by simply replacing P\&Ls with their loss
parts in certain convex risk measures. However, we also provide an
example that cannot be constructed from convex risk measures in this
way. This example illustrates that convex loss-based risk measures
are not trivial extensions of convex risk measures. We further prove
that a convex loss-based risk measure can be constructed from a
convex risk measure by replacing P\&Ls with their loss parts if and
only if it satisfies a property which we call cash-loss additivity.
We then investigate the robustness of the risk estimators associated
with a family of statistical loss-based risk measures that include
both statistical convex loss-based risk measures and VaR on losses
as special cases. Using a notion of robustness for risk estimators
given in \cite{contdeguestscandolo}, we give a necessary and
sufficient condition for the risk estimators to be robust. Our
results imply that risk estimators associated with {\it convex } statistical
loss-based risk measures are {\em not} robust, whereas sample
loss quantiles are robust. These conclusions are further confirmed
by investigating the influence function of a large class of
statistical loss-based risk measures.
One of our main results is that the convexity property, which
leads to reduction of risk under diversification, cannot coexist
with robustness. Therefore, when choosing a risk measure, one has to
decide which property is more important, and the choice should be
dependent of the context in which the risk measure is used. For
example, if the risk measure is used to compute margin requirements
frequently, which is the case in a clearing house, robustness might
be more important than convexity. If the risk is used for asset
allocation, then robustness may not be the primary issue st stake
and convexity might be more relevant property in this case.
The paper is organized as follows. In Section \ref{sec:lossbased}
we define loss-based risk measures and provide the representation
theorems for convex loss-based risk measures before we provide
several examples of loss-based risk measures. Section
\ref{sec:robustness} is devoted to studying the qualitative
robustness of the risk estimators associated with a family of
statistical loss-based risk measures. In Section
\ref{sec:sensitivity} we perform sensitivity analysis on a set of
loss-based risk measures and investigate their influence functions.
Finally, Section \ref{se:Conclusions} concludes the paper.
\iffalse
Even more basic issues arise when considering the above axioms as
the sole characterization of a ``risk measure".
Intuitively, the risk of a portfolio is associated with the
magnitude of its losses, not its gains. In the case of a statistical
risk measure, this means that the risk measure should be defined in
terms of the {\it left} tail of the portfolio gain (distribution).
Indeed, the most popular risk measures---Value at Risk and expected
shortfall---are based on the left tail of the loss distribution.
However, this natural property is not implied by any of the axioms
of coherent risk measures as outlined in \cite{artzner1999cmr}.
If one further distinguishes, as in \cite{heyde2006grm}, `external'
risk measures used for determining capital requirements from
`internal' risk measures used for a firm's risk management
--determining position limits, allocation of capital, etc.---then
the seemingly innocent axiom of cash-additivity---that the risk of a
portfolio decreases by $c$ if combined with a cash position of size
$c$---also comes under question. This axiom makes sense if $\rho(X)$
is interpreted as a capital requirement for portfolio $X$, i.e., for
external risk measures. But if $\rho(X)$ is used to gauge the risk
of the portfolio $X$, this axiom implies that if one combines a
risky position $X$ with a cash position $c=\rho(X)$ then the
combined position $X+c$ has a ``zero risk": $\rho(X+c)=0$, which is
meaningless in a risk management context. Also, as noted already by
\cite{jarrow02}, the axiom of cash-additivity excludes some
natural examples of risk measures, such as the value of the put
option on a portfolio's net value \citep{jarrow02}.
Another issue, very important in practice but somewhat neglected in
the literature on risk measures, is the issue of statistical
estimation of risk, and the design of robust estimators for risk
measures: lack of robustness of a risk estimation procedure makes it
difficult, if not impossible, to use in practice. Unfortunately, as
shown in \cite{contdeguestscandolo}, another unintended consequence
of the axioms of coherent risk measures is that lead to non-robust
risk estimators: the subadditivity property requires dependence on
extreme tail events, which leads to high sensitivity to outliers
\cite{contdeguestscandolo}.
We claim that our theoretical framework is a more natural one for
the purpose of risk measurement and management of financial
portfolios, and englobes various natural examples of risk measures
used in practice.
Examples of
such loss-based risk measures have been considered by
\cite{jarrow02}, who studied the put option premium as a risk
measure, and \cite{Staum:2011ExcessInvarianceShortfall}, in the
context of a finite dimensional probability space. Our analysis
extends these results to a more general setting: we use a slightly
different set of axioms and obtain a characterization of these risk
measures on a general probability space.
provides a representation theorem in terms of a penalty function
(Theorem \ref{th:representationgeneral}), which adapts the results
of \cite{follmer2011sfi} to our setting. This representation result
is made more explicit in the case of distribution-based risk
measures (Theorem \ref{th:representationlawinvariant}) and examples
are given in Section \ref{sec:examples}. We then study the
statistical robustness, in the sense of \cite{contdeguestscandolo},
of estimators for loss-based risk measures. In the case of
statistical (i.e. distribution-based) risk measures we present a
general criterion for qualitative robustness of such risk estimators
(Section \ref{sec:robustness}). These results extend results of
\cite{contdeguestscandolo} to loss-based risk measures. In Section
\ref{sec:robustification} we provide statistically robust versions
of these risk estimators. The theoretical results on the qualitative
robustness of these estimators are confirmed by a quantitative
sensitivity analysis based on influence functions (Section
\ref{sec:sensitivity}).
\fi
\section{Loss-Based Risk Measures}\label{sec:lossbased}
Consider an atomless probability space $(\Omega, {\cal F},\mathbb{P})$
representing market scenarios. For a random variable $X$, denote by
$F_X(\cdot)$ its cumulative distribution function and $G_X(\cdot)$
its left-continuous quantile function. For any $p\in[1,\infty)$, let
$\mathbb{L}^p(\Omega, {\cal F},\mathbb{P})$ be the space of random variables $X$ with
the norm $\|X\|_p:=\left(\mathbb{E}[|X|^p]\right)^\frac{1}{p}$, and let
$\mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P})$ be the space of bounded random
variables.
Let
\begin{align*}
\qquad{\cal P}(\Omega,{\cal F},\mathbb{P}):=\{X\in \mathbb{L}^1(\Omega, {\cal F},\mathbb{P}):X\ge 0,||X||_1=1 \},
\end{align*}
\begin{align*}
{\rm and}\qquad {\cal M}(\Omega,{\cal F},\mathbb{P}):=\{X\in \mathbb{L}^1(\Omega, {\cal F},\mathbb{P}):X\ge 0,||X||_1\le1 \}.
\end{align*}
Then, ${\cal P}(\Omega,{\cal F},\mathbb{P})$ can also be regarded as the set of $\mathbb{P}$-absolutely continuous probability measures on $(\Omega,{\cal F},\mathbb{P})$, and ${\cal M}(\Omega,{\cal F},\mathbb{P})$ as the set of $\mathbb{P}$-absolutely continuous measures $\mu$ such that $\mu(\Omega)\leq1$.
Denote by ${\cal P}((0,1))$ the set of probability measures on the open
unit interval $(0,1)$, and ${\cal M}((0,1))$ the set of positive
measures $\mu$ on $(0,1)$ such that $\mu((0,1))\leq1$. Let
\begin{align}
\Psi((0,1)) :&=\left\{ \phi:(0,1)\mapsto\mathbb{R}_+ \mid \phi(\cdot)\text{ is decreasing on (0,1) and }\int_0^1\phi(z)dz\le 1 \right\},\\
\Phi((0,1)):&=\left\{ \phi:(0,1)\mapsto\mathbb{R}_+ \mid \phi(\cdot)\text{ is decreasing on (0,1) and }\int_0^1\phi(z)dz= 1 \right\},
\end{align}
both of which can be identified as subsets of ${\cal M}((0,1))$. Finally, for any random variables $X$, let $X\wedge 0:=\min(X,0)$.
\subsection{Definition}
A risk measure is a mapping which associates to a random variable
$X$, representing the future P\&L of a portfolio, a number $\rho(X)$
representing its risk. The set of random variables $X$ is often
taken to be $\mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P})$
\cite{artzner1999cmr,FollmerHSchiedA:02convexriskmeasure}, but one
may also allow for unbounded P\&Ls by defining risk measures as
maps on $\mathbb{L}^p(\Omega, {\cal F},\mathbb{P})$; see for instance,
\cite{FilipovicSvindland2008:CanonicalModelSpace} and the references
therein. In this section we will focus on risk measures defined on
$\mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P})$, following \cite{artzner1999cmr}. In
the study of robustness in Sections \ref{sec:robustness} and
\ref{sec:sensitivity}, we will revert back to the more general case
of unbounded losses.
\iffalse
\cite{artzner1999cmr} have argued that, in order for $\rho$ to qualify as a monetary risk measure, it has to satisfy at least the two following properties:
\begin{itemize}
\item[(i)] {\em Cash additivity:} adding cash to a portfolio reduces the risk by the same amount. $$ \forall X \in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P}), \forall \alpha \in\mathbb{R}, \rho(\alpha + X) = \rho(X) - \alpha$$
\item[(ii)] {\em Monotonicity:} if a portfolio $X$ has higher payoff than a portfolio $Y$ in all scenarios, then it has lower risk. $$\forall X,Y \in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P}), \qquad X \leq Y \Rightarrow \rho(X) \geq \rho(Y).$$
\end{itemize}
The cash-additivity property (i) is necessary to interpret $\rho(X)$
in terms of capital requirements for $X$: the capital requirement
for a portfolio $X$ may be satisfied by combining it with a position
in cash equal to $\rho(X)$ since $\rho(X + \rho(X)) =
\rho(X)-\rho(X)=0$. However if one uses $\rho(X)$ as a measure of
risk, one might be tempted to think that the position $X+c$ has ``no
residual risk", which is counterintuitive and, indeed, nonsensical if
taken literally. This observation suggests that the cash-invariance
property may not be a reasonable requirement for a good risk
measure.
Also, as noted by \cite{jarrow02}, the cash-additivity requirement
excludes a natural risk measure, the premium of a put on the loss:
$$ \rho(X)= \mathbb{E}[ \max(-X,0)]$$
Moreover, the initial motivation of risk measures is to quantify
the risk associated with portfolio's losses. Thus, a risk measure
should focus on the loss of a portfolio, and two portfolios which have
the same losses should lead to the same risk measure.
With these concerns in mind, we define a class of risk measures, which we call {\em loss-based risk measures}, as follows.
\fi
\begin{definition}[Loss-based risk measures]\label{de:riskmeasure}
A mapping $\rho:\mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P})\rightarrow \mathbb{R}_{+}$ is called a {\em loss-based risk measure} if it satisfies
\begin{itemize}
\item[\bf (a)] {\em Normalization for cash losses:} for any $\alpha \in\mathbb{R}_{+}$, $\rho(-\alpha) = \alpha$;
\item[\bf (b)] {\em Monotonicity:} for any $X,Y \in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P})$, if $ X \leq Y$, then $\rho(X) \geq
\rho(Y)$; and
\item[\bf (c)] {\em Loss-dependence:} for any $X\in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P})$, $ \rho(X) = \rho(X\wedge 0)$.
\end{itemize}
\end{definition}
The cash-loss property is a ``normalization" property stating that the risk of a (non-random) cash liability is its face value. This property ensures that the risk of a portfolio has the same unit as monetary payoffs. It is easy to observe that the cash-loss property is implied by the cash-additivity property in the set of axioms of coherent risk measures. However, the cash-additivity cannot be inferred from the cash-loss property. The monotonicity property says that a higher payoff leads to lower risk, which is a natural requirement for a meaningful risk measure and is also enforced in the definition of coherent risk measures. The loss-dependence property entails that the risk of a portfolio only depends on its losses, which is a desirable property when the risk measure is used to compute margin requirements or capital reserves. Loss-based risk measures can be seen as a special case of the more general notion of risk orders considered in \cite{DrapeauKupper2010:RiskPreferences}.
A loss-based risk measure $\rho$ is called {\em convex loss-based risk measure} if it satisfies
\begin{itemize} \item[\bf (d)] {\em convexity:} for any $X,Y \in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P})$ and $0<\alpha<1$, $\rho(\alpha X+(1-\alpha)Y) \leq \alpha
\rho(X)+(1-\alpha)\rho(Y)$.
\end{itemize}
As in the classical convex risk measures, the convexity property leads to a reduction of risk under diversification. Compared to convex risk measures, i.e., those risk measures satisfying the cash-additivity, monotonicity, and convexity properties, convex loss-based risk measures have an additional property---loss-dependence, and on the other hand, they replace the cash-additivity property with a weaker one---cash-loss property.
The following useful lemma shows that the monotonicity and convexity
properties together imply $\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$-continuity
for a risk measure. A similar result was proved in \cite[Proposition
3.1]{RuszczynskiShapiro2006:OptimizationConvexRiskFunctions}. We
provide a simpler proof, using elementary results.
\begin{lemma}\label{le:continuity} Any mapping $\rho:\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})\to\mathbb{R}$ that is monotone and convex is continuous on $\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$.\end{lemma}
\begin{proof}
Consider a sequence $(X_n)_{n\geq 1}$ that converges to
$X$ in $\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$. Then the sequence
$\varepsilon_n:=||X_n-X||_\infty$, $n\ge 1$, converges to zero
and we can assume without loss of generality that $\varepsilon_n\le
1,n\ge 1$. Let $\alpha_n:=\sqrt{\varepsilon_n}$, $\beta_n :=
\frac{\varepsilon_n}{\alpha_n} + ||X||_\infty + 1$, then
\begin{align*}
(1-\alpha_n)X_n - \alpha_n\beta_n = X_n - \varepsilon_n-\alpha_n(||X||_\infty + 1+X_n)\le X
\end{align*}
where the inequality is because $X_n-X\le ||X_n-X||_\infty$ and
$||X||_\infty + 1+X_n\ge 0$. By the monotonicity and convexity of
$\rho$, we derive
\begin{align*}
\rho(X)&\le \rho((1-\alpha_n)X_n-\alpha_n\beta_n)\\
&\le (1-\alpha_n)\rho(X_n) + \alpha_n\rho(-\beta_n)\\
&\le (1-\alpha_n)\rho(X_n) + \alpha_n\rho(-(||X||_\infty+2)),
\end{align*}
which leads to
\begin{align*}
\rho(X_n)\ge \frac{\rho(X) - \alpha_n\rho(-(||X||_\infty+2))}{1-\alpha_n}.
\end{align*}
Letting $n\to\infty$ and using the fact that $\alpha_n$ converges to zero, we
immediately derive $\liminf_{n\rightarrow \infty}\rho(X_n)\ge
\rho(X)$, i.e., $\rho$ is lower semi-continuous in
$\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$. A similar argument leads to the
upper semi-continuity of $\rho$ in $\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$.
\end{proof}
Convex loss-based risk measures are special cases of cash-subadditive risk measures defined in
\cite{ElKarouiRavanelli:2009CashSubadditive}. Indeed, consider any $X\in \mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$
and $\alpha\ge 0$. For any $\epsilon\in(0,1)$, we have
\begin{align*}
\rho((1-\epsilon)X - \alpha) &= \rho((1-\epsilon)X + \epsilon(-\frac{\alpha}{\epsilon}))\\
&\le (1-\epsilon)\rho(X) + \epsilon\rho(-\frac{\alpha}{\epsilon}))\\
& = (1-\epsilon)\rho(X) + \alpha,
\end{align*}
where the inequality is due to the convexity property and the
last equality is due to the cash-loss property. Letting
$\epsilon\downarrow 0$ and using the semi-continuity obtained in Lemma \ref{le:continuity}, we have
\begin{align*}
\rho(X-\alpha)\le \rho(X) + \alpha,\quad \forall X\in \mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P}),\alpha\ge 0,
\end{align*}
which also implies
\begin{align*}
\rho(X+\alpha)\ge \rho(X) - \alpha,\quad \forall X\in \mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P}),\alpha\ge 0.
\end{align*}
Thus, convex loss-based risk measures are cash-subadditive.
Although convex loss-based risk measures are special cases of cash-subadditive risk measures, they deserve to be highlighted and treated separately because the loss-dependence property is a desirable property in some of the risk management practices and thus needs to be enforced in the definition of risk measures. In addition, the cash-loss property, which is not implied by cash-subadditivity, is also a reasonable property to define risk measures. Therefore, it is of great interest to carefully investigate this particular class of cash-subadditive risk measures---convex loss-based risk measures. Even further, in Sections \ref{sec:robustness} and \ref{sec:sensitivity}, we study the robustness of the risk estimators associated with a family of loss-based risk measures, not necessarily convex and thus not necessarily cash-subadditive.
\iffalse
Property (a) is a ``normalization" property which states that the
risk of a (non-random) cash liability is its face value. Properties
(b) and (d) have well-known interpretations in terms of increasing
loss diversification, see e.g.
\cite{FollmerHSchiedA:02convexriskmeasure}. Property (c) entails
that {\it portfolios with the same losses have the same risk}. This
implies that in fact the monotonicity property (b) may be replaced
by the weaker
$$ X\wedge 0 \leq Y\wedge 0 \Rightarrow \quad\rho(X) \geq
\rho(Y). $$
Note that property (c) prevents the risk measure
from satisfying the cash-additivity property (i): a loss-based risk
measure is not ``coherent" in the sense of \cite{artzner1999cmr}.
Loss-based risk measures are related to the class of cash
subadditive risk measures defined by El Karoui and Ravanelli
\cite{ElKarouiRavanelli:2009CashSubadditive}. Indeed, properties
(a) and (c), together with a weak semi-continuity condition, imply
the cash subadditivity property defined in \cite[Def
3.1]{ElKarouiRavanelli:2009CashSubadditive}. To see this, let us
first assume the following semi-continuity condition: for any
$X_n\rightarrow X$ in $\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$, $\rho(X)\le
\text{liminf}_{n\rightarrow \infty}\rho(X_n)$. This condition is
weaker than the Fatou property commonly assumed in the risk measure
literature. Now, consider any $X\in \mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$
and $\alpha\ge 0$. For any $\epsilon\in(0,1)$, we have
\begin{align*}
\rho((1-\epsilon)X - \alpha) &= \rho((1-\epsilon)X + \epsilon(-\frac{\alpha}{\epsilon}))\\
&\le (1-\epsilon)\rho(X) + \epsilon\rho(-\frac{\alpha}{\epsilon}))\\
& = (1-\epsilon)\rho(X) + \alpha,
\end{align*}
where the inequality is due to the convexity property (d) and the
last equality is due to the Cash Loss property (a). Letting
$\epsilon\downarrow 0$ and using semi-continuity, we have
\begin{align*}
\rho(X-\alpha)\le \rho(X) + \alpha,\quad \forall X\in \mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P}),\alpha\ge 0,
\end{align*}
which also implies
\begin{align*}
\rho(X+\alpha)\ge \rho(X) - \alpha,\quad \forall X\in \mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P}),\alpha\ge 0.
\end{align*}
Thus, loss-based risk measures are cash--subadditive. On the other
hand, it is easy to see that cash subadditivity in the sense of
\cite{ElKarouiRavanelli:2009CashSubadditive} does not imply the Cash
Loss property (a).
However, our focus in this study is not only on cash-subadditivity:
the Loss-dependence property (c) is at least as important as
replacing cash additivity by cash subadditivity in a risk management
context. Thus, loss-based risk measures deserve to be highlighted
and treated separately although they are special cases of cash
subadditive risk measures
\cite{ElKarouiRavanelli:2009CashSubadditive}.
The following useful lemma shows that that convexity (d) and
monotonicity (b) together imply $\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$
continuity for a risk measure.
\begin{lemma}\label{le:continuity} Any map $\rho:\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})\to\mathbb{R}$ which is monotone and convex is continuous on $\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$.\end{lemma}
\begin{proof}
Consider a sequence $(X_n)_{n\geq 1}$ that converges to
$X$ in $\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$. Then the sequence
$\varepsilon_n:=||X_n-X||_\infty$, $n\ge 1$, converges to zero
and we can assume without loss of generality that $\varepsilon_n\le
1,n\ge 1$. Let $\alpha_n:=\sqrt{\varepsilon_n}$, $\beta_n :=
\frac{\varepsilon_n}{\alpha_n} + ||X||_\infty + 1$, then
\begin{align*}
(1-\alpha_n)X_n - \alpha_n\beta_n = X_n - \varepsilon_n-\alpha_n(||X||_\infty + 1+X_n)\le X
\end{align*}
where the inequality is because $X_n-X\le ||X_n-X||_\infty$ and
$||X||_\infty + 1+X_n\ge 0$. By the monotonicity and convexity of
$\rho$, we derive
\begin{align*}
\rho(X)&\le \rho((1-\alpha_n)X_n-\alpha_n\beta_n)\\
&\le (1-\alpha_n)\rho(X_n) + \alpha_n\rho(-\beta_n)\\
&\le (1-\alpha_n)\rho(X_n) + \alpha_n\rho(-(||X||_\infty+2)),
\end{align*}
which leads to
\begin{align*}
\rho(X_n)\ge \frac{\rho(X) - \alpha_n\rho(-(||X||_\infty+2))}{1-\alpha_n}.
\end{align*}
Letting $n\to\infty$ and using the fact that $\alpha_n$ converges to zero, we
immediately have $\liminf_{n\rightarrow \infty}\rho(X_n)\ge
\rho(X)$, i.e., $\rho$ is lower semi-continuous in
$\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$. A similar argument leads to the
upper semi-continuity of $\rho$ in $\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$.
\end{proof}
\fi
\subsection{Representation theorem for convex loss-based risk measures}
We now give a characterization of convex loss-based risk measures that
have a semi-continuity property, the Fatou property. A (loss-based)
risk measure $\rho$ satisfies the {\em Fatou property} if, for any
sequence $(X_n)_{n\geq 1}$ uniformly bounded in $\mathbb{L}^\infty(\Omega,
{\cal F},\mathbb{P})$ such that $X_n \rightarrow X$ almost surely, we have $\rho(X) \leq \liminf_{n\rightarrow \infty} \rho(X_n)$.
\begin{theorem}\label{th:representationgeneral}
The following are equivalent
\begin{enumerate}
\item $\rho$ is a convex loss-based risk measure satisfying the Fatou property.
\item There exists a convex function $V :{\cal M}(\Omega,{\cal F},\mathbb{P})\rightarrow [0,\infty]$ satisfying
\begin{align}\label{eq:representation_conditionV}
\inf_{\| Y \|_{1} \geq 1-\epsilon} V(Y) = 0 \quad \text{ for any } \quad \epsilon \in (0,1),
\end{align}
such that
\begin{align}\label{eq:representation}
\rho(X) = - \inf_{Y \in {\cal M}(\Omega,{\cal F},\mathbb{P})} \{\mathbb{E}[(X\wedge 0)Y] + V(Y)\},\quad \forall X\in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P}).
\end{align}
\end{enumerate}
\end{theorem}
\begin{proof}
Assume that $\rho$ is a convex loss-based risk measure satisfying the Fatou property. Following Theorem 2.1 in \cite{DelbaenFSchachermayerW:94na} or Theorem 3.2 in \cite{delbaen2002coherent}, $\rho$ is lower-semi-continuous under weak* topology if and only if it satisfies the Fatou property. By \cite[Theorem 6]{frittelli2002por} there exists
$V :\mathbb{L}^1(\Omega,{\cal F},\mathbb{P})\rightarrow (-\infty,\infty]$ such that
\begin{align*}
\rho(X) &= - \inf_{Y \in \mathbb{L}^1(\Omega,{\cal F},\mathbb{P})} \{\mathbb{E}[XY] + V(Y)\}\\
&=- \inf_{Y \in \mathbb{L}^1(\Omega,{\cal F},\mathbb{P})} \{\mathbb{E}[(X\wedge 0)Y] + V(Y)\},\quad \forall X\in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P}),
\end{align*}
where the second equality is due to the loss-dependence property. Furthermore, we have the dual relation
\begin{align*}
V(Y) = \sup_{X\in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P})} \{-\rho(X) - \mathbb{E}[XY]\},\quad \forall Y\in \mathbb{L}^1(\Omega, {\cal F},\mathbb{P}).
\end{align*}
For any $Y\in \mathbb{L}^1(\Omega, {\cal F},\mathbb{P})$, let $A:=\{Y<0\}$. If $\mathbb{P}(A)>0$,
\begin{align*}
V(Y) &\ge \sup_{n\ge 1} \{-\rho(n\mathbf{1}_A) - \mathbb{E}[n\mathbf{1}_AY]\} = \sup_{n\ge 1} \{- \mathbb{E}[n\mathbf{1}_AY]\} = +\infty,
\end{align*}
where the first equality holds because $n\mathbf{1}_A \geq 0$ and $\rho$ is monotone with $\rho(0)=0$. Now, if $Y\ge 0$ and $\|Y\|_{1}>1$, we have
\begin{align*}
V(Y) & \ge \sup_{\alpha\ge 0} \{-\rho(-\alpha) - \mathbb{E}[(-\alpha)Y]\}= \sup_{\alpha\ge 0} \{(\mathbb{E}[Y]-1)\alpha\}=+\infty.
\end{align*}
Thus, the domain of $V$ lies in ${\cal M}(\Omega,{\cal F},\mathbb{P})$. Next, it is easy to see that $V(Y)\ge 0$ for any $Y\in \mathbb{L}^1(\Omega, {\cal F},\mathbb{P})$ because $\rho(0)=0$. Finally, for any $\epsilon \in(0,1)$ and any $\alpha>0$,
\begin{align*}
\alpha &= \rho(-\alpha) = \sup_{Y \in {\cal M}(\Omega,{\cal F},\mathbb{P})} \{\alpha\mathbb{E}[ Y] - V(Y)\}\\
& = \max\left(\sup_{Y \in {\cal M}(\Omega,{\cal F},\mathbb{P}),\|Y\|_1< 1-\epsilon} \{\alpha\mathbb{E}[Y] - V(Y)\},\sup_{Y \in {\cal M}(\Omega,{\cal F},\mathbb{P}),\|Y\|_1\ge 1-\epsilon} \{\alpha\mathbb{E}[Y] - V(Y)\}\right)\\
&\le \max\left(\alpha(1-\epsilon),\sup_{Y \in {\cal M}(\Omega,{\cal F},\mathbb{P}),\|Y\|_1\ge 1-\epsilon} \{\alpha\mathbb{E}[Y] - V(Y)\}\right)\\
&\le \max\left( \alpha(1-\epsilon), \alpha -\inf_{Y \in {\cal M}(\Omega,{\cal F},\mathbb{P}),\|Y\|_1\ge 1-\epsilon}\{ V(Y)\}\right).
\end{align*}
Thus, we conclude that $V(\cdot)$ must satisfy \eqref{eq:representation_conditionV}.
On the other hand, one can check that $\rho$ represented in \eqref{eq:representation} is a convex loss-based risk measure satisfying the lower-semi-continuity under weak* topology and thus the Fatou property.
\end{proof}
We can see in the representation theorem that the domain of the
penalty function $V(\cdot)$ is a subset of ${\cal M}(\Omega,{\cal F},\mathbb{P})$,
the set of all positive measures with total mass less than one, which contrasts with the representation theorem for convex risk measures in which the domain of the penalty function is a subset of ${\cal P}(\Omega,{\cal F},\mathbb{P})$. This difference
is also observed in \cite[Theorem 4.3-(b)]{ElKarouiRavanelli:2009CashSubadditive}. Compared to the representation theorem in
\cite{ElKarouiRavanelli:2009CashSubadditive}, we have an additional
condition \eqref{eq:representation_conditionV}, which is due to the cash-loss
property. Moreover, the dual representation formula
\eqref{eq:representation} depends only on the negative part of $X$
due to the loss-dependence property.
If we compare the results in Theorem \ref{th:representationgeneral} with the representation of the general notion of risk orders in Theorem 2.6 of \cite{DrapeauKupper2010:RiskPreferences}, then we notice that Equation \eqref{eq:representation} represents a particular set of monotone quasi-convex risk functionals $R$ in \cite{DrapeauKupper2010:RiskPreferences} due to the additional features of convex loss-based risk measures such as the convexity and cash-loss properties.
\subsection{Statistical loss-based risk measures}
Most of the risk measures used in finance are statistical, or distribution-based risk measures, i.e. they depend on $X$ only through its distribution $F_X(\cdot)$:
$$ F_X(\cdot) = F_Y(\cdot) \quad \Rightarrow \quad \rho(X)=\rho(Y).$$
Following ideas from \cite{kusuoka2001lic},
\cite{FrittelliMEmanuelaRG:05lawinvariantriskmeasure}, \cite{Ruschendorf2006:LawInvariantConvexRiskMeasures},
\cite{jouini2006law}, and \cite{follmer2011sfi}, we derive a representation theorem for statistical convex
loss-based risk measures.
\begin{theorem}\label{th:representationlawinvariant}
Let $\rho$ be a statistical convex loss-based risk measure.
There exists a convex function $v :\Psi((0,1))\rightarrow [0,\infty]$ satisfying
\begin{align}\label{eq:penaltyfunctionlawinvariant}
\inf_{\int_0^1\phi(z)dz\ge 1-\epsilon} v(\phi) = 0\text{ for any }\epsilon \in (0,1),
\end{align}
such that
\begin{align}\label{eq:representationlawinvariant}
\rho(X) = - \inf_{\phi \in \Psi} \left\{\int_0^1(G_X(z)\wedge 0)\phi(z)dz + v(\phi)\right\},\quad \forall X\in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P}).
\end{align}
\end{theorem}
\begin{proof}
First, we remark that a statistical convex loss-based risk measure necessarily
satisfies the Fatou property. Indeed, as noted in Theorem 2.2,
\cite{jouini2006law}, distribution-based convex functionals in
$\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$ satisfy the Fatou property if and only if
they are lower-semi-continuous under the $\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$
norm. By Lemma \ref{le:continuity}, any convex loss-based risk measure is continuous under the $\mathbb{L}^\infty(\Omega,{\cal F},\mathbb{P})$ norm, so automatically satisfies the Fatou
property if it is distribution-based. We then need to build a
connection between \eqref{eq:representation} and
\eqref{eq:representationlawinvariant}
when the risk measure is distribution-based, which can be done following the lines of \cite[Theorem 2.1]{jouini2006law}.
\end{proof}
We can observe that in the representation theorem the domain of the penalty function $v(\cdot)$ is a subset of $\Psi$, which by definition is the set of positive measures on $(0,1)$ that have decreasing densities and have total mass less than or equal to one. By contrast, in the representation theorem for statistical convex risk measures, the domain of the penalty function is a subset of $\Phi$, which is a set of probability measures on $(0,1)$ with decreasing densities.
Motivated by the representation \eqref{eq:representationlawinvariant}, we sometimes abuse the notation by writing
\begin{align}
\rho(G(\cdot)) = - \inf_{\phi \in \Psi} \left\{\int_0^1(G(z)\wedge 0)\phi(z)dz + v(\phi)\right\}
\end{align}
for any bounded quantile functions, if we are considering a statistical convex loss-based risk measure.
\subsection{Loss-based versione of convex risk measures}
For any convex risk measure $\widetilde \rho$, we can define a new risk measure $\rho$ by applying $\rho$ to the loss part of each portfolio's P\&L, i.e., $\rho(X):=\widetilde \rho(X\wedge 0)$ for any $X\in\mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P})$. It is easy to verify that $\rho$ is a convex loss-based risk measure. We call $\rho$ the loss-based version of $\widetilde \rho$.
In the following, we show that a convex loss-based risk measure $\rho$ is the loss-based version of some convex risk measure if and only if it satisfies
\begin{itemize}
\item[\bf (e)] {\em cash-loss additivity:} for any $X\in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P})$, $X\le 0$, and $\alpha\in \mathbb{R}_+$, $\rho(X-\alpha)=\rho(X)+\alpha$.
\end{itemize}
The cash-loss additivity property says that for a portfolio that generates a pure loss, extracting certain amount of cash from the portfolio will increase its risk by the same amount.
On the one hand, if $\rho(X)= \widetilde \rho(X\wedge 0)$ for certain convex risk measure $\widetilde \rho$, then for any $X\in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P})$, $X\le 0$, and $\alpha\in \mathbb{R}_+$,
\begin{align*}
\rho(X-\alpha) = \widetilde \rho(X-\alpha) = \widetilde \rho(X)+\alpha = \rho(X)+\alpha,
\end{align*}
where the second equality is due to the cash-additivity property of $\widetilde \rho$.
On the other hand, suppose a convex loss-based risk measure $\rho$ satisfies the cash-loss additivity property. Define
\begin{align*}
\widetilde \rho(X) = \rho(X-\alpha_X)-\alpha_X,
\end{align*}
where $\alpha_X$ is any upper-bound of $X$. By the cash-loss additivity property for $\rho$, $\widetilde \rho$ is well-defined. Furthermore, it is easy to check that $\widetilde \rho$ is a convex risk measure and $\rho(X) = \rho(X\wedge 0) = \widetilde\rho(X\wedge 0)$.
However, even though $\rho$ satisfies the cash-loss additivity property, it is still not cash additive. Indeed, in general, we only have
\begin{align*}
\rho(X+\alpha) = \widetilde\rho((X+\alpha)\wedge 0)\ge \widetilde\rho(X\wedge 0 + \alpha) = \widetilde\rho(X\wedge 0) - \alpha = \rho(X) - \alpha,
\end{align*}
for any $X\in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P})$ and $\alpha \in \mathbb{R}$.
A natural question is whether any convex loss-based risk measure satisfies the cash-loss additivity, i.e., whether it is the loss-based version of certain convex risk measure. The answer is no, which is illustrated by a nontrivial and meaningful example presented in the following section. As a result, convex loss-based risk measures are nontrivial extensions of convex risk measures.
\subsection{Examples}\label{sec:examples}
\begin{example}[Put option premium]
\cite{jarrow02} argues that a natural measure of a firm's insolvency risk is the premium of a put option
on the firms equity, which is given by the positive part of its net value $X$ (assets minus liabilities), i.e. $\mathbb{E}^{\mathbb{Q}}[- \min(X,0)]$ where ${\mathbb{Q}}$ is an appropriately chosen
pricing model.
One can generalize this to any portfolio whose net value is represented by a random variable $X$: the downside risk of the
portfolio can be measured by
\begin{align}\label{eq:putpremium}
\rho(X):= \mathbb{E}[- \min(X,0)] \end{align}
This example satisfies all the properties in Definition
\ref{de:riskmeasure} and the convexity property: it is a convex loss-based risk measure. In particular, as noted by \cite{jarrow02}, it is not cash-additive. In the actuarial literature such risk measures have existed for more than 150 years, see \cite{Hattendorff1868}.
\end{example}
\begin{example}[Scenario-based margin requirements]
\iffalse
When determining margin requirements for derivative transactions,
the objective of a central clearing facility is to compute the
margin requirement of each participant in the clearinghouse in order
to cover losses incurred by the clearing participants portfolio over
a liquidation horizon $T$ (typically, a few days). A popular method
for computing such margin requirements --used for example by various
futures and options exchanges-- is to select a certain number of
``stress scenarios" for risk factors affecting the portfolio and
compute the margin requirement as the maximum loss over these
scenarios. If one denotes the portfolio P\&L over the horizon $T$ by
$X$, then the margin requirement $\rho(X)$ is given by
\begin{align}\label{eq:SPAN}
\rho(X)= \max \{ - \min(X(\omega_1),0),..., - \min(X(\omega_n),0)\} \end{align}
where $\omega_1,...,\omega_n$ are the stress scenarios. Naturally,
the clearinghouse only consider the losses of the portfolio when
computing the margin: as a result, such margin requirements may be
viewed as a loss-based risk measure.
This example satisfies all the properties of Definition
\ref{de:riskmeasure}: it is a loss-based risk measure.
This is the main idea behind the SPAN method used by the Chicago Merchantile
Exchange (CME).
Interestingly, the SPAN method was considered as an
initial motivation for the definition of coherent risk measures in
\cite{artzner1999cmr}. Yet it is easy to check that
\eqref{eq:SPAN} is loss-dependent and therefore not cash-additive,
so is not a coherent risk measure.
\fi
It is easy to verify that the margin requirement \eqref{eq:SPAN} that is used in the CME is a convex loss-based risk measure. This method of determining margin requirements is known as the SPAN method. Interestingly, this method was considered as an
initial motivation for the definition of coherent risk measures in
\cite{artzner1999cmr}. Yet it is easy to check that the margin requirement \eqref{eq:SPAN} is not cash-additive, thus not a coherent risk measure.
\end{example}
\begin{example}[Expected tail-loss]
This popular risk measure is defined as
\begin{align}\label{eq:esriskmeasure}
\rho(X):=-\frac{1}{\beta}\int_0^\beta(G_X(z)\wedge 0)dz,\quad X\in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P})
\end{align}
Note that, by construction, this risk measure focuses on the left tail of the loss distribution, since it only involves the quantile function on $(0,\beta)$. Nonetheless, the classical definition of expected shortfall $-\frac{1}{\beta}\int_0^\beta G_X(z) dz$ does not satisfy the loss-dependence property in Definition \ref{de:riskmeasure} since $G_X(\beta)$ might be greater than $0$ for some $X$. Therefore, we insert $G_X(z)\wedge 0$ in its definition to turn it into a loss-based risk measure. We notice that the put option premium is an expected tail-loss by taking $\beta =1$.
\end{example}
\begin{example}[Spectral loss measures]
A large class of statistical loss-based risk measures is obtained
by taking weighted averages of loss quantiles with various weight
functions $\phi\in\Phi$:
\begin{align}\label{eq:spectralriskmeasure}
\rho(X):=-\int_0^1(G_X(z)\wedge 0)\phi(z)dz,\quad X\in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P})
\end{align}
We call such a risk measures a {\em spectral loss measure}. By definition, this risk measure is the loss-based version of the spectral risk measures defined by \cite{acerbi2002smr}. As a result, it is a convex loss-based risk measure, and in addition it satisfies the positive homogeneity property:
for any $\lambda>0$, $\rho(\lambda X) = \lambda \rho(X)$. Notice
that the expected tail-loss is a spectral loss measure with
$\phi(z)=\frac{1}{\beta}1_{(0,\beta)}(z)$.
\end{example}
\begin{example}[Loss certainty equivalent]
Consider $u(\cdot)\in C^4(\mathbb{R}_+)$ which is strictly increasing and strictly convex. Assume $u'/u''$ is concave. Consider the following mapping:
\begin{align}\label{eq:CEriskmeasure}
\rho(X) := u^{-1}\left(\mathbb{E} u(|X\wedge 0|)\right),\quad X\in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P}).
\end{align}
It is clear that $\rho$ satisfies the cash-loss, monotonicity, and loss-dependence properties. Therefore, it is a loss-based risk measure. On the other hand, by \cite[Theorem 106]{HardyLittlewoodPolya:1959Inequalities}, $\rho$ is also convex. Thus, $\rho$ is a convex loss-based risk measure. We call $\rho$ a {\em loss certainty equivalent }. By definition, $\rho$ is distributional-based and
\begin{align}\label{eq:CEriskmeasurestatistical}
\rho(X) = u^{-1}\left(\int_0^1u(|G_X(z)\wedge 0|)dz\right),\quad X\in \mathbb{L}^\infty(\Omega, {\cal F},\mathbb{P}).
\end{align}
If $u(x) = x^p,x\ge 0$ for certain $p\ge 1$, we will speak of the {\em $L^p$ loss certainty equivalent }.
Here, when $p=1$, $u(\cdot)$ is not strictly convex. However, in this case, \eqref{eq:CEriskmeasure} is still well-defined and the risk measure is actually the put option premium, which is also a convex loss-based risk measure. Thus, we include the case $p=1$ here and identify the put option premium as a special case of $L^p$ loss certainty equivalent in the following.
One can show that the $L^p$ loss certainty equivalent is the only loss certainty equivalent that satisfies the positive homogeneity property. The $L^p$ loss certainty equivalent has the following dual representation
\begin{align}\label{eq:Lpriskmeasurerepresentation}
\rho(X) = -\inf_{Y\in {\cal M}^q(\Omega,{\cal F},\mathbb{P})}\mathbb{E}\left[(X\wedge 0) Y\right]
\end{align}
where $1<q\le \infty$ is the conjugate of $p$, i.e., $\frac{1}{p}+\frac{1}{q}=1$, and ${\cal M}^q(\Omega,{\cal F},\mathbb{P})$ is the set of all nonnegative random variables with $L^q$ norms less than or equal to one. Moreover, it has the distribution-based representation
\begin{align}\label{eq:Lpriskmeasurerepresentationlawinvariant}
\rho(X) = -\inf_{\phi\in \Psi^q((0,1))}\int_0^1(G_X(z)\wedge 0)\phi(z)dz.
\end{align}
where $\Psi^q((0,1))$ is the set of $\phi\in\Psi((0,1))$ such that $\int_0^1\phi(z)^qdz\le 1$.
The $L^p$ certainty equivalents are closely related to the {\em lower partial moments} in \cite{Fishburn1977:MeanRiskAnalysisBelowTargetReturns}. It is straightforward to show that $L^p$ loss certainty equivalents for $p>1$ do not satisfy the cash-loss additivity property, so are loss-based versions of certain convex risk measures. Therefore, $L^p$ loss certainty equivalents, which are an important family of risk measures and cannot be accommodated in the framework of convex risk measures, are convex loss-based risk measures.
Let $u(x) = e^{\beta x},x\ge 0$ for certain $\beta>0$, then the loss certainty equivalent becomes the {\em entropic loss certainty equivalent }, a loss-based version of the entropic risk measure studied in \cite{frittelli2002por}, \cite{FollmerHSchiedA:02convexriskmeasure}, and \cite{follmer2011sfi}. Actually, one can show that the entropic loss certainty equivalent and $L^1$ loss certainty equivalent are the only loss certainty equivalents that satisfy the cash-loss additivity property, see for instance Proposition 2.46 in \cite{follmer2011sfi}.
By recalling the representation theorems for the entropic risk measure,\footnote{For the representation theorems for the entropic risk measure, see for instance, \cite{frittelli2002por,FrittelliMEmanuelaRG:05lawinvariantriskmeasure}, \cite{FollmerHSchiedA:02convexriskmeasure}, and \cite{follmer2011sfi}.} we obtain the dual representation for the entropic loss certainty equivalent
\begin{align}\label{eq:entropyriskmeasurerepresentation}
\rho(X) = -\inf_{Y\in {\cal P}(\Omega,{\cal F},\mathbb{P})}\{\mathbb{E}[(X\wedge 0)Y] + V(Y)\},
\end{align}
where
\begin{align}
V(Y) = \mathbb{E}\left[Y\ln Y\right]- \inf_{Y\in {\cal P}(\Omega,{\cal F},\mathbb{P})}\mathbb{E}\left[Y\ln Y\right],
\end{align}
and the distributional-based representation
\begin{align}\label{eq:entropyriskmeasurerepresentationlawinvariant}
\rho(X) = -\inf_{\phi\in \Phi((0,1))}\int_0^1(G_X(z)\wedge 0)\phi(z)dz + v(\phi),
\end{align}
where
\begin{align}
v(\phi) = \int_0^1\phi(z)\ln \phi(z)dz-\inf_{\phi\in\Phi((0,1))} \int_0^1\phi(z)\ln \phi(z)dz.
\end{align}
\end{example}
\iffalse
\begin{example}
The {\em entropic risk measure}
\begin{align}\label{eq:entropyriskmeasure}
\rho(X) = \frac{1}{\beta}\log \left\{\mathbb{E}\left[e^{\beta|X\wedge 0|}\right]\right\}, \qquad \beta>0
\end{align}
is a loss-based statistical risk measure. See \cite{frittelli2002por,FollmerHSchiedA:02convexriskmeasure,follmer2011sfi}.
The entropic risk measure satisfies the cash-loss invariance property (e). It is well-known that the entropic risk measure
has the following representation
\begin{align}\label{eq:entropyriskmeasurerepresentation}
\rho(X) = -\inf_{Y\in {\cal P}(\Omega,{\cal F},\mathbb{P})}\{\mathbb{E}[(X\wedge 0)Y] + V(Y)\},
\end{align}
where
\begin{align}
V(Y) = \mathbb{E}\left[Y\ln Y\right]- \inf_{Y\in {\cal P}(\Omega,{\cal F},\mathbb{P})}\mathbb{E}\left[Y\ln Y\right],
\end{align}
and
\begin{align}\label{eq:entropyriskmeasurerepresentationlawinvariant}
\rho(X) = -\inf_{\phi\in \Phi((0,1))}\int_0^1(G_X(z)\wedge 0)\phi(z)dz + v(\phi),
\end{align}
where
\begin{align}
v(\phi) = \int_0^1\phi(z)\ln \phi(z)dz-\inf_{\phi\in\Phi((0,1))} \int_0^1\phi(z)\ln \phi(z)dz.
\end{align}
See e.g. \cite{frittelli2002por,FollmerHSchiedA:02convexriskmeasure,follmer2011sfi,FrittelliMEmanuelaRG:05lawinvariantriskmeasure}.
\end{example}
\begin{example}[One-sided moments] For $p\ge 1$, the p-th order
moment of the loss
\begin{align}\label{eq:Lpriskmeasure}
\rho(X) = \left\{\mathbb{E}\left[|X\wedge 0|^p\right]\right\}^{\frac{1}{p}}.
\end{align}
is a loss-based statistical risk measure. However, it does not satisfy cash-loss invariance property (e) except for $p=1$. On the other hand it satisfies the {\em positive homogeneity}, i.e., for any $\lambda>0$, $\rho(\lambda X) = \lambda \rho(X)$. It is also well-known in the literature that the $L^p$ risk measure has the following dual representation
\begin{align}\label{eq:Lpriskmeasurerepresentation}
\rho(X) = -\inf_{Y\in {\cal M}^q(\Omega,{\cal F},\mathbb{P})}\mathbb{E}\left[(X\wedge 0) Y\right]
\end{align}
where $1<q\le \infty$ is the conjugate of $p$, i.e., $\frac{1}{p}+\frac{1}{q}=1$, and ${\cal M}^q(\Omega,{\cal F},\mathbb{P})$ is the set of all nonnegative random variables with $L^q$ norms less or equal to one. It also has the distribution-based representation
\begin{align}\label{eq:Lpriskmeasurerepresentationlawinvariant}
\rho(X) = -\inf_{\phi\in \Psi^q((0,1))}\int_0^1(G_X(z)\wedge 0)\phi(z)dz.
\end{align}
where $\Psi^q((0,1))$ is the set of $\phi\in\Psi((0,1))$ such that $\int_0^1\phi(z)^qdz\le 1$.
\end{example}
\fi
\section{Robustness of Risk Estimators}
\label{sec:robustness} In practice, for measuring the risk of a
portfolio, aside from the theoretical choice of a risk measure, a
key issue is the estimation of the risk measure, which requires the
choice of a {\it risk estimator} \citep{contdeguestscandolo}.
In this
section we study the robustness property of empirical risk
estimators built from some statistical loss-based risk measure. We
follow the ideas in \citet{contdeguestscandolo} but we will view
risk measures as functionals on the set of quantile functions,
rather than the set of distribution functions. As we will see, this
makes the study of continuity properties of loss-based risk measures
easier. Moreover, from Theorem \ref{th:representationlawinvariant}, a statistical convex loss-based risk measure can be represented by \eqref{eq:representationlawinvariant}, which suggests that it is more natural to work directly on quantile functions. Note that the quantile representation of entropic risk measures and their different variants also appear in \cite{FollmerKnispel2012:ConvexCapitalRequirments}.
Denote by ${\cal Q}$ the set of all quantile functions and by ${\cal D}$ the set of all distribution functions. The L\'evy-Prokhorov metric between two distribution functions $F_1\in{\cal D}$ and $F_2\in{\cal D}$ is defined as
\begin{equation*}
d_P(F_1,F_2)\triangleq\inf\{\epsilon>0
\,:\,F_1(x-\epsilon)-\epsilon\leq F_2(x)\leq F_1(x+\epsilon)+\epsilon,\;\forall x\in\mathbb{R}\}.
\end{equation*}
This metric appears to be the most tractable one on ${\cal D}$ and it induces the same topology as the usual weak topology on ${\cal D}$.
The quantile set ${\cal Q}$ and the distribution set ${\cal D}$ are connected by the following one-on-one correspondence
\begin{align*}
\begin{array}{ccc}
{\cal D} &\rightarrow &{\cal Q}\\
F(\cdot) & \mapsto & F^{-1}(\cdot)
\end{array}
\end{align*}
where
\begin{align*}
F^{-1}(t) = \inf\left\{x\in\mathbb{R}\mid F(x)\ge t\right\},\quad t\in (0,1)
\end{align*}
is the left continuous inverse of $F(\cdot)$. Such a correspondence, together with the L\'evy-Prokhorov metric on ${\cal D}$ induces a metric on ${\cal Q}$ which we denote by $d$. The convergence under this metric can be characterized by the following: for any $G_n,G\in {\cal Q}$, $G_n\rightarrow G$ if and only if $G_n(z)\rightarrow G(z)$ at any continuity points of $G$. In the following, we only need the characterization of the convergence on ${\cal Q}$, so the choice of metric on ${\cal Q}$ is irrelevant once it leads to the same topology.
Most of the time, we work with quantile functions that are continuous on $(0,1)$ in order to avoid irregularities due to the presence of atoms. In practice, it is not restrictive to focus on continuous quantile functions. Indeed, people do assume the continuity of quantile functions in many applications, e.g., when computing the VaR. The study of discontinuous quantile functions is more technical and of little interest, so we choose not to pursue in this direction. In the following, we denote by ${\cal Q}_c$ the set of all continuous quantile functions. We also denote by ${\cal Q}^\infty$ the set of all bounded quantile functions and by ${\cal Q}^\infty_c$ the set of all bounded continuous quantile functions.
\subsection{A family of statistical loss-based risk measures}
Motivated by the representation \eqref{eq:representationlawinvariant}, we consider a family of statistical loss-based risk measures defined by
the following {\em Fenchel-Legendre transform}:
\begin{align}
\label{eq:rhofunction}
\rho(G(\cdot)):= -\inf_{m\in \text{dom}(v)}\left\{\int_{(0,1)} (G(z) \wedge 0) m(dz) + v(m)\right\},
\end{align}
where $v:{\cal M}((0,1))\rightarrow [0,\infty]$ is the {\em penalty function} satisfying
\begin{align}\label{eq:penaltyfunction}
\inf_{m((0,1))\ge 1-\epsilon} v(m) = 0\text{ for any }\epsilon \in (0,1),
\end{align}
and $\text{dom}(v)$ is the domain of $v$, i.e.,
\begin{align}
\text{dom}(v):=\{m\in {\cal M}((0,1))\mid v(m)<\infty\}.
\end{align}
It is easy to see that $\rho(G(\cdot))$ is well-defined for any
$G(\cdot)\in {\cal Q}$, and $\text{dom}(\rho)$, the domain of $\rho$,
contains ${\cal Q}^\infty$.
Straightforward calculation shows that the risk measure $\rho$ defined in \eqref{eq:rhofunction} satisfies the cash-loss, monotonicity, and loss-dependence properties, so it is a loss-based risk measure. It is clear to see from the representation \eqref{eq:representationlawinvariant} that statistical convex loss-based risk measures are special cases of the risk measures in \eqref{eq:rhofunction} by identifying each element in $\Psi$ as the density of a certain measure in ${\cal M}((0,1))$. The risk measures \eqref{eq:rhofunction}, however, do not necessarily satisfy the convexity property. For instance, if we choose $\text{dom}(v)=\{\delta_{\alpha}\}$ and $v(\delta_{\alpha})=0$, where $\delta_\alpha$ is the Dirac measure at $\alpha\in(0,1)$, then $\rho(G(\cdot)) = -G(\alpha)\wedge 0$, which is the $\alpha$-level VaR on losses. It is well-known that despite its wide use in practice, VaR does not satisfy the convexity property. To conclude, the risk measures defined by \eqref{eq:rhofunction} are a rich family that include both statistical convex loss-based risk measures that entail the convexity property and the VaR on losses that is popular in practice.
Finally, it is easy to observe that although $\rho$ in \eqref{eq:rhofunction} does not satisfy the convexity property, when viewed as a mapping on ${\cal Q}$, it satisfies the
following property.
\begin{enumerate}
\item[{\bf (d')}] {\em Quantile convexity:} for any $G_1(\cdot),G_2(\cdot)\in {\cal Q}$ and $0<\alpha<1$, $\rho(\alpha G_1(\cdot)+(1-\alpha)G_2(\cdot))\leq \alpha \rho(G_1(\cdot))+(1-\alpha)\rho(G_2(\cdot))$.
\end{enumerate}
Quantile convexity is related to co-monotonic subadditivity, discussed in \cite{heyde2006grm}, and co-monotonic convexity \citep{SongYan:2009riskmeasure}.
\subsection{Qualitative robustness}
In practice, in order to compute the risk measure $\rho(G(\cdot))$ of a portfolio whose P\&L has quantile $G(\cdot)$, one has first to estimate the distribution or quantile of the portfolio's P\&L from data, and then apply the risk measure $\rho$ to the estimated distribution or quantile. One of the popular ways is to apply the risk measure to the empirical distribution.
A {\em sequence of samples} of $G(\cdot)\in {\cal Q}$ is a sequence of random variables $X_1,X_2,\cdots$ which are i.i.d and follow the distribution $G^{-1}$. We denote by $\mathbf{X}$ this sequence of samples and $\mathbf{X}^n$ its first $n$ samples. For each sample size $n$, the {\em empirical distribution} is defined as
\begin{align}
F^{\text{emp}}_{\mathbf{X}^n}(x) = \frac{1}{n}\sum_{i=1}^n\mathbf{1}_{X_i\le x},\quad x\in\mathbb{R},
\end{align}
and the {\em empirical quantile} is defined as
\begin{align}
G^{\text{emp}}_{\mathbf{X}^n}(z) :=(F^{\text{emp}}_{\mathbf{X}^n})^{-1}(z) = X_{(\lfloor nz\rfloor +1)},\quad z\in (0,1),
\end{align}
where $\lfloor a\rfloor$ denotes the integer part of $a$ and $X_{(1)} \leq \cdots \leq X_{(n)}$. In practice, the quantity $\widehat{\rho}(\mathbf{X}^n) := \rho(G^{\text{emp}}_{\mathbf{X}^n}(\cdot))$ is computed as the estimated risk measure $\rho(G(\cdot))$ given the $n$ samples $\mathbf{X}^n$. The risk estimator $\widehat{\rho}$ is defined on $\cup_{n\geq1} \mathbb{R}^n$, the set of all possible sequences of samples $(\mathbf{X}^n)_{n\geq1}$, and has values in $\mathbb{R}^+$. Since the samples can be regarded as random variables, so is the risk estimator $\widehat{\rho}(\mathbf{X}^n)$. We denote by ${\cal L}_n(\widehat{\rho},G)$ the distribution function of $\widehat{\rho}(\mathbf{X}^n)$.
$\rho$ is said to be {\em consistent} at $G=F^{-1}\in\text{dom}(\rho)$ if
\begin{align}
\lim_{n\rightarrow \infty} \widehat{\rho}(\mathbf{X}^n) = \rho(F^{\text{emp}}_{\mathbf{X}^n})^{-1}) \qquad \text{almost surely.}
\end{align}
Because the true risk measure $\rho(G(\cdot))$ is estimated by $\widehat{\rho}(\mathbf{X}^n)$, the consistency is the minimal requirement for a meaningful risk measure. In the following we denote by ${\cal Q}^\rho$ the set of quantiles $G$ at which $\rho$ is well defined and consistent, and ${\cal Q}^\rho_c$ the continuous quantiles in ${\cal Q}^\rho$.
The following definition of {\it robust risk estimator} is
considered in \cite[Definition 4]{contdeguestscandolo}:
\begin{definition}[\citealt{contdeguestscandolo}]
Let $\rho$ be defined by \eqref{eq:rhofunction} and
$\mathcal{C}\subset{\cal Q}^\rho$ be a a set of plausible P\&L
quantiles. $\widehat{\rho}$ is {\em $\mathcal{C}$-robust} at $G\in \mathcal{C}$ if for any
$\varepsilon>0$ there exist $\delta>0$ and $n_0\ge 1$ such that, for
all $\widetilde G\in \mathcal{C}$,
\begin{align*}
d(\widetilde G,G)\le \delta \Longrightarrow d_P({\cal L}_n(\widehat{\rho},\widetilde G),{\cal L}_n(\widehat{\rho},G))<\varepsilon, \qquad \forall n \geq n_0.
\end{align*}
A risk estimator $\widehat{\rho}$ is called {\em $\mathcal{C}$-robust} if it is
$\mathcal{C}$-robust at any $G\in\mathcal{C}$.
\end{definition}
We can see that the definition does not rely on the choice of the
metric on the topological space ${\cal Q}$. The choice of the
L\'evy-Prokhorov distance on ${\cal D}$, however, is critical. As
pointed out by \cite{huber1981rs}, the use of a different metric,
even if it also metrizes the weak topology, may lead to a different
class of robust estimators. This metric is a natural choice in
robust statistics \citep{huber1981rs}. Alternative formulations are
proposed by \cite{kratschmer2011qualitative}.
The following proposition, taken from \cite{contdeguestscandolo},
shows that the robustness of the risk estimator $\widehat{\rho}$ is equivalent
to the continuity of the risk measure $\rho$ on ${\cal Q}$ under the
weak topology.
\begin{proposition}[\citealt{contdeguestscandolo}]\label{prop:robustness}
Let $\rho$ be a risk measure and $G\in \mathcal{C}\subset {\cal Q}^\rho$. The following are equivalent:
\begin{enumerate}
\item $\rho$, when restricted to $\mathcal{C}$, is continuous at $G$;
\item $\widehat{\rho}$ is $\mathcal{C}$-robust at $G$.
\end{enumerate}
\end{proposition}
In the following, we are going to investigate the continuity of $\rho$, which finally clarifies whether $\widehat{\rho}$ is robust or not. The following lemma is useful.
\begin{lemma}\label{le:uniformconvergence}
Let $-\infty<a<b<+\infty$, and $G_n,G$ be increasing functions on $[a,b]$. Suppose $G(z)$ is continuous on $[a,b]$ and $G_n(z)\rightarrow G(z)$ for each $z\in [a,b]$, then $G_n(z)\rightarrow G(z)$ uniformly on $[a,b]$.
\end{lemma}
\begin{proof}
Because $[a,b]$ is a closed interval and $G(z)$ is continuous on $[a,b]$, for each $\epsilon>0$, there exists $a=z_1<\cdots<z_m=b$ such that $\sup_{1\le i\le m-1}|G(z_{i+1})-G(z_i)|<\frac{\epsilon}{2}$. On the other hand, there exists $N$ such that when $n\ge N$, $|G_n(z_i)-G(z_i)|<\frac{\epsilon}{2}$ for every $z_i,i=1,\dots m$. Now, for any $z\in [a,b]$, there exist $z_i,z_{i+1}$ such that $z_i\le z\le z_{i+1}$. Thus,
\begin{align*}
G_n(z)-G(z)&\le G_n(z_{i+1})-G(z_i)\\
& = G_n(z_{i+1})-G(z_{i+1})+G(z_{i+1})-G(z_i)\\
&< \epsilon
\end{align*}
when $n\ge N$. Similarly, we have $G_n(z)-G(z)>-\epsilon$ when $n\ge N$. Therefore, $G_n(z)\rightarrow G(z)$ on $[a,b]$ uniformly.
\end{proof}
The following lemma shows that any risk measure $\rho$ in \eqref{eq:rhofunction} is consistent on ${\cal Q}_c^\infty$, i.e., ${\cal Q}_c^\infty\subset {\cal Q}^\rho_c$.
\begin{lemma}\label{le:consistency}
Let $\rho$ be given in \eqref{eq:rhofunction}. Then $\rho$ is consistent at any $G\in {\cal Q}_c$ that is bounded from below. In particular, ${\cal Q}_c^\infty \subset{\cal Q}^\rho_c$.
\end{lemma}
\begin{proof}
Let $G\in {\cal Q}_c$ that is bounded from below, and $X_1,X_2,\dots$ be its samples. By Glivenko-Cantelli theorem, $G^{\text{emp}}_{\mathbf{X}^n}(z)\rightarrow G(z),0<z<1$ almost surely. Furthermore, $\inf_{i=1,\dots,n}X_i\rightarrow \text{essinf} X_1$ almost surely, which shows that $G^{\text{emp}}_{\mathbf{X}^n}(0+)\rightarrow G(0+)$. Thus, if we extend $G^{\text{emp}}_{\mathbf{X}^n}$ and $G$ from $(0,1)$ to $[0,1)$ by setting $G^{\text{emp}}_{\mathbf{X}^n}(0):=G^{\text{emp}}_{\mathbf{X}^n}(0+)$ and $G(0):= G(0+)$, then $G(\cdot)$ is continuous on $[0,1)$ and $G^{\text{emp}}_{\mathbf{X}^n}(z)\rightarrow G(z),0\le z<1$ almost surely.
In the following, for each fixed $\omega$, let $G_n:=G^{\text{emp}}_{\mathbf{X}^n}$. For simplicity, we work with $U(\cdot) := -\rho(\cdot)$. We want to show that $U(G_n(\cdot))\rightarrow U(G(\cdot))$. On the one hand,
\begin{align*}
\limsup_{n\rightarrow \infty}U(G_n(\cdot))&\le \inf_{m\in \text{dom}(v)}\left[\limsup_{n\rightarrow \infty}\int_{(0,1)}(G_n\wedge0)(z)m(dz) + v(m)\right]\\
&\le\inf_{m\in \text{dom}(v)}\left[\int_{(0,1)}(G(z)\wedge0)m(dz) + v(m)\right]\\
& = U(G(\cdot)),
\end{align*}
where the second inequality is due to Fatou's lemma. On the other hand, for each $\eta<1$, by Lemma \ref{le:uniformconvergence}, $G_n(z)\rightarrow G(z)$ uniformly for $z\in(0,\eta]$. Thus, we have
\begin{align*}
\liminf_{n\rightarrow \infty}U(G_n(\cdot))& = \liminf_{n\rightarrow \infty}\inf_{m\in \text{dom}(v)}\left[\int_{(0,1)}(G_n(z)\wedge 0)m(dz) + v(m)\right]\\
& \geq \liminf_{n\rightarrow \infty}\inf_{m\in \text{dom}(v)}\left[\int_{(0,1)}(G_n(z\wedge \eta)\wedge 0)m(dz) + v(m)\right]\\
& = \inf_{m\in \text{dom}(v)}\left[\int_{(0,1)}(G(z\wedge \eta)\wedge 0)m(dz) + v(m)\right]\\
& = U(G(\cdot\wedge\eta)),
\end{align*}
where the second equality holds because $G_n(z\wedge \eta)\rightarrow G(z\wedge \eta)$ for $z\in(0,1)$ uniformly. Finally,
\begin{align*}
U(G(\cdot))\ge U(G(\cdot\wedge\eta)) &= \inf_{m\in \text{dom}(v)}\left[\int_{(0,1)}(G(z\wedge \eta )\wedge 0)m(dz) + v(m)\right]\\
&=\inf_{m\in \text{dom}(v)}\Big[\int_{(0,1)}(G(z )\wedge 0)m(dz)+ v(m)\\
&\qquad -\int_{(0,1)} (G(z)\wedge 0-(G(z\wedge \eta )\wedge 0))m(dz)\Big]\\
&\ge U(G(\cdot)) - \left[\lim_{z\uparrow 1}G(z)\wedge 0 - G(\eta)\wedge 0\right]\\
&\rightarrow U(G(\cdot))
\end{align*}
as $\eta \uparrow 1$. Therefore, we conclude $\liminf_{n\rightarrow \infty}U(G_n(\cdot))\ge U(G(\cdot))$.
\end{proof}
Lemma \ref{le:consistency} shows that any risk measure $\rho$ defined in \eqref{eq:rhofunction} is consistent at least at bounded continuous quantile functions. For any particular example of the risk measures in \eqref{eq:rhofunction}, it is possible to show that it is consistent at certain unbounded quantile functions.\footnote{For instance, \citet[Example 2.11]{contdeguestscandolo} show that spectral risk measures are consistent at any quantile functions at which the risk measures are well-defined.} In this paper, we consider the general risk measure in \eqref{eq:rhofunction} and mainly focus on investigating the robustness of risk estimators. For this reason, we do not explore the consistency for any particular risk measure in detail.
The following result provides a sufficient and necessary condition under which the risk measures defined in \eqref{eq:rhofunction} are
continuous on some subset of ${\cal Q}$.
\begin{theorem}
\label{th:weak_continuity}
Let $\rho$ be defined by \eqref{eq:rhofunction} and $\mathcal{C}$ be any subset of ${\cal Q}_c$ such that $\mathcal{C}\supseteq{\cal Q}_c^\infty$. The following are equivalent:
\begin{itemize}
\item[(i)] $\rho(\cdot)$, when restricted to $\mathcal{C}$, is continuous at any $G(\cdot) \in \mathcal{C}$.
\item[(ii)] There exists $0<\delta<1$ such that
\begin{eqnarray}
\sup_{m\in\text{dom}(v)} m((0,\delta))=0.
\end{eqnarray}
\end{itemize}
Furthermore, if (ii) holds, ${\cal Q}_c^\rho = {\cal Q}_c$.
\end{theorem}
\begin{proof}
\begin{itemize}
\item[$(ii)\Rightarrow(i)$] The proof is analogous to the proof of Lemma \ref{le:consistency}.
Let (ii) hold for some $0<\delta<1$. For simplicity, we work with $U(.) \triangleq -\rho(.)$ and first show that $U$ is lower-semi-continuous. For each $\eta<1$, we have
\begin{align*}
\liminf_{n\rightarrow \infty}U(G_n(\cdot))& = \liminf_{n\rightarrow \infty}\inf_{m\in \text{dom}(v)}\left[\int_{[\delta,1)}(G_n(z)\wedge 0)m(dz) + v(m)\right]\\
& \geq \liminf_{n\rightarrow \infty}\inf_{m\in \text{dom}(v)}\left[\int_{[\delta,1)}(G_n(z\wedge \eta)\wedge 0)m(dz) + v(m)\right]\\
& = \inf_{m\in \text{dom}(v)}\left[\int_{[\delta,1)}(G(z\wedge \eta)\wedge 0)m(dz) + v(m)\right]\\
& = U(G(\cdot\wedge\eta)),
\end{align*}
where the second equality holds because $G_n(z\wedge \eta)$ converges to $G(z\wedge\eta)$ uniformly for $z\in[\delta,1)$ where Lemma \ref{le:uniformconvergence} applies. Then, by monotonicity, we have $0\le U(G(\cdot))-U(G(\cdot\wedge \eta))$. Finally,
\begin{align*}
U(G(\cdot))\ge U(G(\cdot\wedge\eta)) &= \inf_{m\in \text{dom}(v)}\left[\int_{[\delta ,1)}(G(z\wedge \eta )\wedge 0)m(dz) + v(m)\right]\\
&=\inf_{m\in \text{dom}(v)}\Big[\int_{[\delta,1)}(G(z )\wedge 0)m(dz)+ v(m)\\
&\qquad -\int_{(0,1)} (G(z)\wedge 0-(G(z\wedge \eta )\wedge 0))m(dz)\Big]\\
&\le U(G(\cdot)) - \left[\lim_{z\uparrow 1}G(z)\wedge 0 - G(\eta)\wedge 0\right]\\
&\rightarrow U(G(\cdot))
\end{align*}
as $\eta \uparrow 1$.
Thus, we have $ \liminf_{n\rightarrow \infty}U(G_n(\cdot))\ge U(G(\cdot))$, and can conclude that $U$ is lower-semi-continuous.
Next, we show that $U$ is also upper-semi-continuous.
\begin{align*}
\limsup_{n\rightarrow \infty}U(G_n(\cdot))&\le \inf_{m\in \text{dom}(v)}\left[\limsup_{n\rightarrow \infty}\int_{(0,1)}(G_n\wedge0)(z)m(dz) + v(m)\right]\\
&\le\inf_{m\in \text{dom}(v)}\left[\int_{(0,1)}(G(z)\wedge0)m(dz) + v(m)\right]\\
& = U(G(\cdot)),
\end{align*}
where the second inequality is due to Fatou's lemma. Together with the lower-semi-continuity, $U(\cdot)$ is continuous at $G$, and so is $\rho$.
\item[$(i)\Rightarrow(ii)$] We prove it by contradiction. If (ii) is not true, there exists $\delta_n>0$, and $m_n\in\text{dom}(v)$, such that $\delta_n\rightarrow 0$ and $m_n((0,\delta_n))>0$. Define
\begin{align*}
G_n(z):=\begin{cases}
-\beta_n & 0<z\le\delta_n,\\
\frac{\beta_n}{\delta_n}(z-2\delta_n)& \delta_n< z\le 2\delta_n\\
0 & 2\delta_n<z<1,
\end{cases}
\quad n\ge 1,
\end{align*}
where
\begin{align*}
\beta_n:=\frac{v(m_n)+1}{m_n((0,\delta_n))},\quad n\ge 1.
\end{align*}
It is obvious that $G_n\in{\cal Q}_c^\infty\subset \mathcal{C}$, $n\ge 1$.
Because $\delta_n\downarrow 0$, $G_n(\cdot)\rightarrow 0$ in $\mathcal{C}$. On the other hand,
\begin{align*}
U(G_n(\cdot))\le \int_{(0,1)}G_n(z)m_n(dz) + v(m_n)\le -1,
\end{align*}
and $U(0)=0$. Thus, $U(\cdot)$ is not continuous at $0$ which is a contradiction.
\end{itemize}
Finally, if (ii) holds, for any $G(\cdot)\in {\cal Q}_c$, we have $\rho(G(\cdot)) = \rho(G(\cdot\vee \delta))$. Because $G(\cdot\vee \delta)$ is bounded from below, by Lemma \ref{le:consistency}, $\rho$ is consistent at $G(\cdot\vee \delta)$, and therefore consistent at $G(\cdot)$. In other words, ${\cal Q}_c^\rho = {\cal Q}_c$.
\end{proof}
Theorem \ref{th:weak_continuity} provides a sufficient and necessary condition for the risk measures in \eqref{eq:rhofunction} to be continuous under the weak topology on ${\cal Q}$. The reason for choosing the weak topology is because the resulting continuity result is directly connected to the robustness of the risk estimators according to Proposition \ref{prop:robustness}. In \cite{jouini2006law}, the continuity of statistical convex risk measures on ${\cal Q}$ under a different topology, named as {\em Lebesgue property}, is investigated. The authors find an equivalent characterization of the Lebesgue property. To compare the sufficient and necessary condition we derive in Theorem \ref{th:weak_continuity} and theirs, we extend the result in \cite{jouini2006law} to the case of the general risk measures in \eqref{eq:rhofunction} by mimicking the proof in that paper. Because the comparison is not the main theme of the paper, we place the details in Appendix \ref{se:lebesgue}.
Finally, we combine Proposition \ref{prop:robustness} and Theorem \ref{th:weak_continuity} to obtain a sufficient and necessary condition for the robustness of the risk estimators associated with the risk measures \eqref{eq:rhofunction}.
\begin{corollary}
\label{co:robustness}
Let $\rho$ be defined by \eqref{eq:rhofunction} and $\mathcal{C}$ be any subset of ${\cal Q}_c^\rho$ such that $\mathcal{C}\supseteq{\cal Q}_c^\infty$. Then the following are equivalent
\begin{enumerate}
\item $\widehat{\rho}$ is $\mathcal{C}$-robust
\item There exists $0<\delta<1$ such that
\begin{eqnarray}
\sup_{m\in\text{dom}(v)} m((0,\delta))=0.
\end{eqnarray}
\end{enumerate}
Furthermore, if $\widehat{\rho}$ is $\mathcal{C}$-robust, then ${\cal Q}_c^\rho = {\cal Q}_c$.
\end{corollary}
An immediate consequence of Corollary \ref{co:robustness} is that statistical convex loss-based risk measures do not lead to robust risk estimators.
\begin{corollary}
Let $\rho$ be a loss-based statistical risk measure and $\mathcal{C}$ be any subset of ${\cal Q}_c^\rho$ such that $\mathcal{C}\supseteq {\cal Q}_c^\infty$. Then, $\widehat{\rho}$ is not $\mathcal{C}$-robust.
\end{corollary}
\begin{proof}
By Theorem \ref{th:representationlawinvariant}, $\rho$ can be represented as \eqref{eq:representationlawinvariant} where the penalty function $v$ satisfies \eqref{eq:penaltyfunctionlawinvariant}. As a result, there exists a $\phi\in\Psi((0,1))\cap \text{dom} (v)$ such that $\int_0^1\phi(z)dz\ge \frac{1}{2}$. Because $\phi(\cdot)$ is decreasing on (0,1), we must have $\int_0^\delta \phi(z)dz>0$ for any $\delta>0$. By Corollary \ref{co:robustness}, $\widehat{\rho}$ is not $\mathcal{C}$-robust.
\end{proof}
This result reveals a dilemma: one has choose between convexity property, which leads to a reduction of risk under diversification, and the robustness of the risk estimator associated with this risk measure, which rendes it amenable to estimation and backtesting. In practice, when choosing a risk measure for a certain purpose, one has to decide which of the two properties is more important. For instance, if the risk measure is used to compute daily margin requirements in a clearing house, the robustness is the more important issue because a system generating unstable margin requirements may lead to large margin calls even in absence of any significant market event. However, the convexity property might be more relevant if the risk measure is used as an allocation tool, rather than a risk management tool.
On the other hand, recalling that the VaR on losses can be identified as a special case of the risk measures in \eqref{eq:rhofunction} by letting $\text{dom}(v)=\{\delta_{\alpha}\}$ and $v(\delta_{\alpha})=0$, Corollary \ref{co:robustness} shows that the 'histoical (loss-based) Value at Risk' i.e. the empirical loss quantile is a robust risk estimator.
These results are similar in spirit to previous results by \cite{contdeguestscandolo}.
\subsection{Robustification of risk estimators}
\label{sec:robustification}
Corollary \ref{co:robustness} provides a sufficient and necessary condition for a loss-based risk measure $\rho$ represented by equation \eqref{eq:rhofunction} to be continuous and therefore to lead to a robust empirical risk estimator. In particular, convex loss-based statistical risk measures lead to non-robust risk estimators. In the following, we provide one way to robustify risk estimator computed from statistical convex loss-based risk measures. Fixing some $\delta\in(0,1)$, for any statistical convex loss-based risk measure $\rho$,\footnote{The discussion in this subsection can also be applied to the risk measures given by representation \eqref{eq:rhofunction}. The restriction to statistical convex loss-based risk measures is only made to illustrate further the conflict between the convexity property and the robustness.} consider its {\em $\delta$-truncation}
\begin{eqnarray}
\label{eq:rhoc_phi_function}
\rho_\delta(G(\cdot)) := \rho(G(\cdot \vee \delta)),\quad G\in{\cal Q}.
\end{eqnarray}
The new risk measure $\rho_\delta$ leads to a new risk estimator, $\widehat{\rho}_\delta$, by plugging historical quantile functions. In the following, we are going to show that $\widehat{\rho}_\delta$ is robust. As a result, $\widehat{\rho}_\delta$ can be regarded as robustification of $\widehat{\rho}$, the risk estimator associated with $\rho$.
From the representation \eqref{eq:representationlawinvariant}, we can find the representation of $\rho_\delta$, which in turns shows that $\widehat{\rho}_\delta$ is robust.
Define the map $\pi: \Psi \mapsto {\cal M}((0,1))$ which associates to any density function $\phi$ in $\Psi$ a measure $m$ defined by
\begin{eqnarray*}
m(dz) :=
\left\{
\begin{array}{cc}
\phi(z)dz, & \delta<z<1, \\
\left(\int_{(0,\delta]} \phi(t)dt\right) \delta_{z}, & z=\delta, \\
0, & 0<z<\delta,
\end{array}
\right.
\end{eqnarray*}
where $\delta_{z}$ is the Dirac measure at $z$. The observation that $\pi$ is not a bijective map leads to an additional definition since the penalty function $v(m)$ for $m \in \pi(\Psi((0,1))):= \{m \mid m=\pi(\phi) \text{ for some } \phi \in \Psi((0,1)) \}$ cannot be derived uniquely from $v(\phi)$ where $m=\pi(\phi)$. Therefore, by denoting $\pi^{-1}(m) = \{\phi\in \Psi((0,1))\mid \pi(\phi)=m \}$, we define for $m \in \pi(\Psi)$,
\begin{eqnarray*}
v_\delta(m) := \inf_{\phi \in \Psi \cap \pi^{-1}(m)} v(\phi).
\end{eqnarray*}
From the representation \eqref{eq:representationlawinvariant}, we have
\begin{eqnarray}
\rho_\delta(G) &=& -\inf_{\phi \in \Psi}\left\{(G(\delta) \wedge 0) \int_{(0,\delta]} \phi(z)dz + \int_{(\delta,1)} (G(z) \wedge 0) \phi(z)dz + v(\phi)\right\} \nonumber \\
&=& -\inf_{m \in \pi(\Psi)} \left\{ \int_{(0,1)} (G(z) \wedge 0) m(dz) + v_\delta(m)\right\}. \label{eq:rhocfunction}
\end{eqnarray}
Finally, from \eqref{eq:penaltyfunctionlawinvariant} it is easy to see that $v_\delta$ satisfies \eqref{eq:penaltyfunction}.
From the representation \eqref{eq:rhocfunction}, it is immediate to see that the $\delta$-truncation $\rho_\delta$ is no longer convex since measures $m \in \pi(\Psi((0,1)))$ have a point mass at $\delta$ and therefore do not admit a density on $(0,1)$. On the other hand, it is also straightforward to see that $\widehat{\rho}_\delta$ is ${\cal Q}_c$-robust because each $m\in \pi(\Psi((0,1)))$ satisfies $m((0,\delta))=0$.
\begin{example}
The $\delta$-truncation of the spectral loss measure \eqref{eq:spectralriskmeasure} is given by
\begin{align}
\rho_\delta(G) = \int_\delta^1(G(z)\wedge 0)\phi(z)dz + G(\delta)\int_0^\delta \phi(z)dz,\quad G\in{\cal Q}.
\end{align}
\end{example}
\begin{example}
The $\delta$-truncation of the loss certainty equivalent is given by
\begin{align}\label{eq:CEriskmeasuretruncated}
\rho_\delta(G(\cdot)) = u^{-1}\left(\int_0^1u(|G(t\vee \delta)\wedge 0|)dt\right),\quad G\in{\cal Q}.
\end{align}
In particular, the $\delta$-truncation of the $L^p$ loss certainty equivalent is given by
\begin{align} \label{eq:Lpriskmeasuretruncated}
\rho_\delta(G) = \left[\int_0^1|G(z\vee \delta)\wedge 0|^pdz\right]^{\frac{1}{p}},\quad G\in{\cal Q},
\end{align}
and its representation is expressed as
\begin{align}
\rho_\delta(G) = -\inf_{m\in \pi(\Psi^q((0,1)))}\int_0^1(G(z)\wedge 0)m(dz),\quad G\in{\cal Q}.
\end{align}
The $\delta$-truncation of the entropic loss certainty equivalent is given by
\begin{align}\label{eq:entropyriskmeasuretruncated}
\rho_\delta(G) = \frac{1}{\beta} \log \left(\int_{(0,1)} e^{-\beta \,G(z\vee \delta) \wedge 0}\, dz \right),\quad G\in {\cal Q},
\end{align}
and its representation is expressed as
\begin{eqnarray}
\rho_\delta(G) = -\inf_{m \in \pi(\Phi((0,1)))} \left\{ \int_{(0,1)} (G(z) \wedge 0) m(dz) + v_\delta(m)\right\},\quad G\in {\cal Q}
\end{eqnarray}
where for each $m\in\pi(\Phi((0,1)))$
\begin{eqnarray*}
v_\delta(m) &=& \inf_{\phi \in \Phi \cap \pi^{-1}(m)} v(\phi)\\
&=& \frac{1}{\beta} \, \int_{(\delta,1)} m'(z) \log\left(m'(z)\right) \, dz + \frac{1}{\beta} \, \lim_{z \downarrow \delta} \left( m'(z) \log\left(m'(z)\right) \, z\right) - \inf_{\phi\in\Phi((0,1))}\int_0^1\phi(z)\ln\phi(z)dz.
\end{eqnarray*}
Here, $m'(z),\delta<z<1$ denotes the density of $m$ on $(\delta,1)$ w.r.t. to Lebesgue measure. This density is well-defined because $m \in\pi(\Phi((0,1)))$.
\end{example}
It is worth mentioning the paper \citet{contdeguestscandolo}, in which the authors also propose a way to robustify the risk estimator associated with the expected shortfall. The robustification suggested by those authors is different from ours. Indeed, \citet{contdeguestscandolo} propose to truncate the expected shortfall, denoted by $\textrm{ES}_\alpha$, in the following way
\begin{eqnarray*}
\text{ES}_{\delta,\alpha}(G) = \frac{1}{\alpha-\delta} \int_{(\delta,\alpha)} G(z) \, dz,
\end{eqnarray*}
and define the risk estimator associated with $\text{ES}_{\delta,\alpha}$ as the robustification. Applying their idea to the larger class of statistical convex loss-based risk measures would lead to defining the following truncation
\begin{eqnarray*}
\widetilde{\rho_\delta}(G) = -\inf_{\widetilde{\phi} \in \widetilde{\pi}(\Psi)} \, \left\{\int_{(0,1)} (G(z) \wedge 0) \widetilde{\phi}(z)dz + \widetilde{v_\delta}(\widetilde{\phi})\right\},
\end{eqnarray*}
where the map $\widetilde{\pi}: \Psi \rightarrow {\cal M}((0,1))$ associates to any function $\phi \in \Psi$ another function $\widetilde{\phi} \in {\cal M}((0,1))$ defined by
\begin{eqnarray*}
\widetilde{\phi}(z) := \frac{\phi(z) \mathbf{1}_{(\delta,1)}(z)}{\int_{(\delta,1)} \phi(z) dz} \not\in \Psi,
\end{eqnarray*}
and where the penalty function is given by
\begin{eqnarray*}
\widetilde{v}_\delta(\widetilde{\phi}) := \inf_{\phi \in \Psi \cap \widetilde{\pi}^{-1}(\widetilde{\phi})} v(\phi).
\end{eqnarray*}
Compared to the truncation $\rho_\delta$ considered in this paper, this new truncation $\widetilde{\rho_\delta}$ reassigns the probability weight attached to $z\in(0,\delta)$ evenly to $z\in(\delta,1)$. The new truncation $\widetilde{\rho_\delta}$ is less tractable than $\rho_\delta$ because it cannot be computed from the initial risk measure $\rho$ as in \eqref{eq:rhoc_phi_function}.
\section{Sensitivity Analysis of Risk Estimators}
\label{sec:sensitivity}
In the previous section, we have studied the robustness of risk estimators in a qualitative sense. One may argue that the above results rely on the choice of a topology (weak topology) together with a distance (L\'evy-Prokhorov distance) on the space of probability distributions. As illustrated in Appendix \ref{se:lebesgue}, the choice of a weaker topology could lead to different continuity properties for the risk measures and thus to different robustness properties for the corresponding risk estimators. Nonetheless, as noted in \cite{huber1981rs}, the choice of the weak continuity to study robustness is natural in statistics. To further illustrate this statement in this section, we study sensitivity properties of the risk estimators associated with loss certainty equivalents and their $\delta$-truncation versions without relying on any topology and show that the study leads to the same conclusions as before, i.e., loss certainty equivalents are not robust but their $\delta$-truncation versions are.
The study of the sensitivity properties may be done by quantifying the sensitivity of risk estimators using \textit{influence functions} \citep{contdeguestscandolo,hampel1974influence}. Fix a risk estimator $\widehat{\rho}$ for which the estimation is based on applying the risk measure $\rho$ to empirical quantile functions. Then, its sensitivity function at the quantile function $G$ of a distribution $F$, in the direction of the Dirac mass at $z$ is equal to
\begin{equation*}
\label{def:sensitivity}
S(z;G) \triangleq \lim_{\epsilon\to 0^+}\frac{\rho(\epsilon \delta_z + (1-\epsilon)F)-\rho(F)}{\epsilon},
\end{equation*}
for any $z\in\mathbb{R}$ such that the limit exists. Note that $S(z;G)$ is nothing but the directional derivative of the risk measure $\rho$ at $F$ in the direction $\delta_z\in\mathcal{D}$.
$S(z,G)$ measures the sensitivity of the risk estimator based on a large sample to the addition of a new observation \cite{contdeguestscandolo}. In the former work, the authors consider different estimation methods, using both empirical and parametric distributions. In that case, the definition of the sensitivity function should be considered with more attention since the risk measure $\rho$ would have to be replaced with an \textit{effective risk measure} incorporating both the choice of the risk measure and the estimation method as explained in \cite{contdeguestscandolo}.
\subsection{Unbounded sensitivity functions}
In this section, we compute the sensitivity function of the loss
certainty equivalent. We find that this risk measure has unbounded
sensitivity function which is consistent with our findings of
Section \ref{sec:robustness}. Note that, unlike the setting of
Section \ref{sec:robustness}, this result makes no reference to any
topology on the set of loss distributions.
\begin{proposition} \label{prop:sensitivity_CE}
The sensitivity function of the loss certainty equivalent
\eqref{eq:CEriskmeasure} is given by
\begin{align}\label{eq:sensitivity_CE}
S(z;G) = \frac{u(|z\wedge 0|)-u(\rho(G))}{u'(\rho(G))}.
\end{align}
\end{proposition}
\begin{proof}
By denoting $G_\epsilon(\cdot)$ the quantile function corresponding to the distribution function $F_\epsilon(\cdot)=(1-\epsilon)F(\cdot)+\epsilon \delta_z(\cdot)$, we have
\begin{eqnarray*}
\rho(G_\epsilon) &=& u^{-1}\left(\int_\mathbb{R} u(|x\wedge 0|)dF_\epsilon(x)\right)\\
&=& u^{-1}\left((1-\epsilon)\int_\mathbb{R} u(|x\wedge 0|)dF(x)+\epsilon u(|z\wedge 0|)\right).
\end{eqnarray*}
Now, simple calculus leads to \eqref{eq:sensitivity_CE}.
\end{proof}
From Proposition \ref{prop:sensitivity_CE}, if $\lim_{x\rightarrow \infty}u(x)=\infty$, $\lim_{z\downarrow -\infty}S(z;G)=+\infty$, showing that the sensitivity function is unbounded. In particular, for the $L^p$ and entropic risk measures, the sensitivity functions are unbounded.
\iffalse
In this section, we compute the sensitivity function of the entropic and $L^p$ risk measures. We find that these two risk measures have unbounded sensitivity function, which is consistent with our findings of Section \ref{sec:robustness}.
\begin{proposition} \label{prop:sensitivity_entropic}
The sensitivity function of the entropic risk measure is
\begin{eqnarray*}
S(z;G) = \frac{1}{\beta} \, \left[ \frac{e^{-\beta z\wedge 0}}{e^{\beta \, \rho(G)}} - 1\right].
\end{eqnarray*}
\end{proposition}
\begin{proof} By denoting $G_\epsilon(.)$ the quantile function corresponding to the distribution function $F_\epsilon(.)=(1-\epsilon)F(.)+\epsilon \delta_z(.)$, we have
\begin{eqnarray*}
\rho(G_\epsilon) &=& \frac{1}{\beta} \log\left( (1-\epsilon) \int_{(0,1)} e^{-\beta x\wedge 0} F(dx) + \epsilon \, e^{-\beta z\wedge 0}\right) \\
&=& \frac{1}{\beta} \log\left( e^{\beta \, \rho(G)} + \epsilon\, \left[e^{-\beta z\wedge 0} - e^{\beta \, \rho(G)} \right] \right) \\
&=& \frac{1}{\beta} \log\left( e^{\beta \, \rho(G)} \right) + \frac{1}{\beta} \log\left( 1 + \epsilon\, \frac{\left[e^{-\beta z\wedge 0} - e^{\beta \, \rho(G)} \right]}{e^{\beta \, \rho(G)}} \right) \\
&=& \rho(G) + \frac{1}{\beta} \log\left( 1 + \epsilon\, \left[\frac{e^{-\beta z\wedge 0}}{e^{\beta \, \rho(G)}} -1 \right] \right)
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
\frac{\rho(G_\epsilon) - \rho(G)}{\epsilon} &=& \frac{1}{\beta} \, \frac{\log\left( 1 + \epsilon\, \left[\frac{e^{-\beta z\wedge 0}}{e^{\beta \, \rho(G)}} -1 \right] \right)}{\epsilon} \\ &\stackrel{\epsilon \rightarrow 0}{\longrightarrow}& \frac{1}{\beta} \, \left[\frac{e^{-\beta z\wedge 0}}{e^{\beta \, \rho(G)}} -1 \right].
\end{eqnarray*}
\end{proof}
\begin{proposition} \label{prop:sensitivity_Lp}
The sensitivity function of the $L^p$ risk measure is
\begin{eqnarray*}
S(z;G) = \frac{\rho(G)}{p} \, \left[ \frac{(z\wedge 0)^p}{\rho(G)^p} -1\right].
\end{eqnarray*}
\end{proposition}
\begin{proof} By denoting $G_\epsilon(.)$ the quantile function corresponding to the distribution function $F_\epsilon(.)=(1-\epsilon)F(.)+\epsilon \delta_z(.)$, we have
\begin{eqnarray*}
\rho(G_\epsilon) &=& \left[ (1-\epsilon) \, \int_{(0,1)} (x\wedge 0)^p F(dx) + \epsilon \, (z\wedge 0)^p \right]^{\frac{1}{p}}\\
&=& \left[ \rho(G)^p + \epsilon\, ([z\wedge 0]^p-\rho(G)^p) \right]^{\frac{1}{p}}\\
&=& \rho(G) \, \left[ 1 + \epsilon\, \frac{[z\wedge 0]^p-\rho(G)^p}{\rho(G)^p} \right]^{\frac{1}{p}}
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
\frac{\rho(G_\epsilon) - \rho(G)}{\epsilon}
&=& \rho(G) \, \frac{\left\{ \left[ 1 + \epsilon\, \frac{[z\wedge 0]^p-\rho(G)^p}{\rho(G)^p} \right]^{\frac{1}{p}} -1 \right\}}{\epsilon}\\
&\stackrel{\epsilon \rightarrow 0}{\longrightarrow}& \frac{\rho(G)}{p} \, \left[ \frac{(z\wedge 0)^p}{\rho(G)^p} -1\right]. \end{eqnarray*}
\end{proof}
\fi
\subsection{Boundedness of sensitivity functions for robust risk estimators}
In this section, we compute the sensitivity functions of the
$\delta$-truncated versions of the loss certainty equivalents.
These truncated versions were introduced in Section
\ref{sec:robustification} in order to obtain robust risk
estimators. The conclusion of the following proposition is that by
truncating these risk measures, their sensitivity functions become
bounded. Therefore, the robustness properties of risk estimators
derived in Section \ref{sec:robustness} are consistent with the
sensitivity functions computed in this section.
\begin{proposition}\label{prop:sensitivity_robust_CE}
Consider the $\delta$-truncation of the loss certainty equivalent \eqref{eq:CEriskmeasuretruncated} with $0<\delta <1$. Assume $F(z)<1$ and $G(\cdot)$ is differentiable at $\delta$. Then, the sensitivity $S(z;G)$ can be computed as follows:
\begin{enumerate}
\item[\bf(i)] When $G(\delta)>0$,
\begin{align}\label{eq:SensitivityCEPositive}
S(z;G) = 0.
\end{align}
\item[\bf(ii)] When $G(\delta)=0$,
\begin{align}\label{eq:SensitivityCEZero}
S(z;G) =\frac{1}{u'(\rho_\delta(G))}\begin{cases}
0, & z\ge G(\delta),\\
\delta \, (1-\delta) \, G'(\delta), & z< G(\delta).
\end{cases}
\end{align}
\item[\bf(iii)] When $G(\delta)<0$,
\begin{align}\label{eq:SensitivityCENegative}
S(z;G) = \frac{1}{u'(\rho_\delta(G))}\left[-u(\rho_\delta(G)) + \begin{cases}
u(|z\wedge 0|) - \delta^2 u'(|G(\delta)|) G'(\delta) , & z>G(\delta),\\
u(|z\wedge 0|) , & z=G(\delta),\\
u(|G(\delta)|) + \delta(1-\delta)u'(|G(\delta)|)G'(\delta), & z<G(\delta).
\end{cases}\right]
\end{align}
\end{enumerate}
\end{proposition}
\begin{proof}
First let us recall some properties of left-continuous inverse functions. For any distribution function $\widetilde F$, denote by $\widetilde F^{-1}$ its left-continuous inverse. Then, for any $t\in[0,1],x\in\mathbb{R}$,
\begin{align}\label{eq:LeftInversePropertyOne}
\widetilde F(x)\ge t\Longleftrightarrow x\ge \widetilde F^{-1}(t),\quad \widetilde F(x)<t\Longleftrightarrow x< \widetilde F^{-1}(t),
\end{align}
and thus
\begin{align}\label{eq:LeftInversePropertyTwo}
\widetilde F(x-)<t\le \widetilde F(x)\Longleftrightarrow x = \widetilde F^{-1}(t).
\end{align}
\iffalse
and
\begin{align*}
\widetilde F(x-)> t\Longleftrightarrow x> \widetilde F^{-1}(t+),\quad \widetilde F(x-)\le t\Longleftrightarrow x\le \widetilde F^{-1}(t+).
\end{align*}\fi
In the following, we denote by $F(\cdot)$ the distribution function associated with $G(\cdot)$. By assumption, $G(\cdot)$ is differentiable and thus continuous at $\delta$. We claim that
\begin{align}\label{eq:LeftInversePropertyThree}
F(z-)\le \delta \le F(z) \Longleftrightarrow z = G(\delta),\quad F(z)>\delta \Longleftrightarrow z>G(\delta), \quad F(z)<\delta \Longleftrightarrow z<G(\delta).
\end{align}
Indeed, from \eqref{eq:LeftInversePropertyOne}, we deduce that $F(z)<\delta$ if only if $z<G(\delta)$. From \eqref{eq:LeftInversePropertyTwo}, we deduce that if $z = G(\delta)$, then $F(z-)\le \delta \le F(z)$. Thus, to prove \eqref{eq:LeftInversePropertyThree}, we only need to show that if $F(z-)\le \delta \le F(z)$, $z = G(\delta)$. Suppose $F(z-)\le \delta \le F(z)$. On the one hand, from \eqref{eq:LeftInversePropertyOne}, $z\ge G(\delta)$. On the other hand, for any $\varepsilon>0$ small enough, $F(z-\varepsilon)<\delta+\varepsilon$, which again by \eqref{eq:LeftInversePropertyOne} leads to $z-\epsilon<G(\delta+\varepsilon)$. Letting $\varepsilon\downarrow 0$, by the continuity of $G(\cdot)$ at $\delta$, we conclude that $z\le G(\delta)$.
Denote by $G_\epsilon(\cdot)$ the quantile function corresponding to the distribution function $F_\epsilon(\cdot)=(1-\epsilon)F(\cdot)+\epsilon \delta_z(\cdot)$. We only need to compute
\begin{align*}
A:=\lim_{\epsilon\downarrow 0}\frac{\int_0^1u(|G_\epsilon(t\vee \delta)\wedge 0|)dt - \int_0^1u(|G(t\vee \delta)\wedge 0|)dt}{\epsilon},
\end{align*}
and $S(z;G)$ follows from the chain rule.
Straightforward computation shows that
\begin{align*}
G_\epsilon(t) = \begin{cases}
G\left(\frac{t-\epsilon}{1-\epsilon}\right), & t>\epsilon + (1-\epsilon)F(z),\\
G\left(\frac{t}{1-\epsilon}\right), & t\le (1-\epsilon)F(z-),\\
z, & (1-\epsilon)F(z-)<t\le \epsilon + (1-\epsilon)F(z).
\end{cases}
\end{align*}
Thus,
\begin{align*}
&\int_\delta^1u(|G_\epsilon(t)\wedge 0|)dt\\
=& \int_\delta^1u(|G\left(\frac{t-\epsilon}{1-\epsilon}\right)\wedge 0|)\mathbf{1}_{\{t>\epsilon + (1-\epsilon)F(z)\}}dt + \int_\delta^1u(|G\left(\frac{t}{1-\epsilon}\right)\wedge 0|)\mathbf{1}_{\{t\le (1-\epsilon)F(z-)\}}dt \\
&\quad + \int_\delta^1u(|z\wedge 0|)\mathbf{1}_{\{(1-\epsilon)F(z-)<t\le \epsilon + (1-\epsilon)F(z)\}}dt\\
=& (1-\epsilon)\int_{\frac{\delta-\epsilon}{1-\epsilon}}^1u(|G\left(t\right)\wedge 0|)\mathbf{1}_{\{t>F(z)\}}dt + (1-\epsilon)\int_{\frac{\delta}{1-\epsilon}}^1u(|G\left(t\right)\wedge 0|)\mathbf{1}_{\{t\le F(z-)\}}dt \\
&\quad + \int_\delta^1u(|z\wedge 0|)\mathbf{1}_{\{(1-\epsilon)F(z-)<t\le \epsilon + (1-\epsilon)F(z)\}}dt\\
=&(1-\epsilon)\int_\delta^1u(|G\left(t\right)\wedge 0|)dt +(1-\epsilon)\int_{\frac{\delta-\epsilon}{1-\epsilon}}^{\delta}u(|G\left(t\right)\wedge 0|)\mathbf{1}_{\{t>F(z)\}}dt\\
&\quad - (1-\epsilon)\int_{\delta}^{\frac{\delta}{1-\epsilon}}u(|G\left(t\right)\wedge 0|)\mathbf{1}_{\{t\le F(z-)\}}dt - (1-\epsilon)\int_\delta^1u(|G\left(t\right)\wedge 0|)\mathbf{1}_{\{F(z-)<t\le F(z)\}}dt\\
&\quad + \int_\delta^1u(|z\wedge 0|)\mathbf{1}_{\{(1-\epsilon)F(z-)<t\le \epsilon + (1-\epsilon)F(z)\}}dt.
\end{align*}
It is then easy to show
\begin{align*}
\int_{\frac{\delta-\epsilon}{1-\epsilon}}^{\delta}u(|G\left(t\right)\wedge 0|)\mathbf{1}_{\{t>F(z)\}}dt &= \begin{cases}
0, & F(z)\ge \delta,\\
(1-\delta)u(|G(\delta)\wedge 0|)\epsilon + o(\epsilon), & F(z)<\delta,
\end{cases}\\
\int_{\delta}^{\frac{\delta}{1-\epsilon}}u(|G\left(t\right)\wedge 0|)\mathbf{1}_{\{t\le F(z-)\}}dt &= \begin{cases}
\delta u(|G(\delta)\wedge 0|)\epsilon + o(\epsilon), & F(z-)> \delta,\\
0, & F(z-)\le \delta.
\end{cases}
\end{align*}
Noticing \eqref{eq:LeftInversePropertyTwo}, we can show that
\begin{align*}
\int_\delta^1u(|G\left(t\right)\wedge 0|)\mathbf{1}_{\{F(z-)<t\le F(z)\}}dt = \begin{cases}
0, & F(z)< \delta,\\
(F(z)-\delta)u(|z\wedge 0|), & F(z-)\le\delta\le F(z),\\
(F(z)-F(z-))u(|z\wedge 0|), & \delta< F(z-).
\end{cases}
\end{align*}
Similarly, we can show that
{\footnotesize
\begin{align*}
\int_\delta^1u(|z\wedge 0|)\mathbf{1}_{\{(1-\epsilon)F(z-)<t\le \epsilon + (1-\epsilon)F(z)\}}dt = \begin{cases}
o(\epsilon), & F(z)< \delta,\\
\left[(1-\epsilon)(F(z)-\delta)+\epsilon (1-\delta)\right]u(|z\wedge 0|), & F(z-)\le \delta\le F(z),\\
\left[(1-\epsilon)(F(z)-F(z-))+\epsilon\right]u(|z\wedge 0|), & \delta<F(z-).
\end{cases}
\end{align*}}
Therefore,
{\footnotesize
\begin{align*}
&\int_\delta^1u(|G_\epsilon(t)\wedge 0|)dt=(1-\epsilon)\int_\delta^1u(|G\left(t\right)\wedge 0|)dt+ \begin{cases}
\left[u(|z\wedge 0|) - \delta u(|G(\delta)\wedge 0|)\right]\epsilon + o(\epsilon), & \delta<F(z-),\\
(1-\delta)u(|z\wedge 0|)\epsilon +o(\epsilon), & F(z-)\le\delta \le F(z),\\
(1-\delta)u(|G(\delta)\wedge 0|)\epsilon + o(\epsilon), & \delta>F(z).
\end{cases}
\end{align*}}
On the other hand
\begin{align*}
G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0 = \begin{cases}
G\left(\frac{\delta-\epsilon}{1-\epsilon}\right)\wedge 0 - G(\delta)\wedge 0, & \delta >\epsilon + (1-\epsilon)F(z),\\
G\left(\frac{\delta}{1-\epsilon}\right)\wedge 0- G(\delta)\wedge 0, & \delta\le (1-\epsilon)F(z-),\\
z\wedge 0- G(\delta)\wedge 0, & (1-\epsilon)F(z-)<\delta \le \epsilon + (1-\epsilon)F(z).
\end{cases}
\end{align*}
We discuss case by case.
\begin{enumerate}
\item $G(\delta)>0$. Because $G(\cdot)$ is continuous at $\delta$, it is easy to show that $G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0=0$ when $\epsilon$ is sufficiently small.
\item $G(\delta)=0$. In this case, we have
\begin{align*}
G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0 = \begin{cases}
G\left(\frac{\delta-\epsilon}{1-\epsilon}\right), & \delta >\epsilon + (1-\epsilon)F(z),\\
0, & \delta\le (1-\epsilon)F(z-),\\
z\wedge 0, & (1-\epsilon)F(z-)<\delta \le \epsilon + (1-\epsilon)F(z)
\end{cases}
\end{align*}
when $\epsilon$ is sufficiently small. Recalling \eqref{eq:LeftInversePropertyThree}, we conclude that
\begin{align*}
G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0 = \begin{cases}
0, & \delta <F(z-),\\
0, & F(z-)\le \delta\le F(z),\\
-(1-\delta)G'(\delta)\epsilon + o(\epsilon), & \delta>F(z).
\end{cases}
\end{align*}
\item $G(\delta)<0$. In this case, we have
\begin{align*}
G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0 = \begin{cases}
G\left(\frac{\delta-\epsilon}{1-\epsilon}\right) - G(\delta), & \delta >\epsilon + (1-\epsilon)F(z),\\
G\left(\frac{\delta}{1-\epsilon}\right)- G(\delta), & \delta\le (1-\epsilon)F(z-),\\
z- G(\delta), & (1-\epsilon)F(z-)<\delta \le \epsilon + (1-\epsilon)F(z).
\end{cases}
\end{align*}
when $\epsilon$ is small enough, leading to
\begin{align*}
G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0 = \begin{cases}
\delta G'(\delta)\epsilon + o(\epsilon), & \delta <F(z-),\\
0, & F(z-)\le \delta\le F(z),\\
-(1-\delta)G'(\delta)\epsilon + o(\epsilon), & \delta>F(z).
\end{cases}
\end{align*}
\end{enumerate}
Notice that
\begin{align*}
\int_0^1u(|G_\epsilon(t\vee \delta)\wedge 0|)dt - \int_0^1u(|G(t\vee \delta)\wedge 0|)dt = \int_\delta^1u(|G_\epsilon(t)\wedge 0|)dt - \int_\delta^1u(|G(t)\wedge 0|)dt\\
+ \delta \left(u(|G_\epsilon(\delta)\wedge 0|) - u(|G(\delta)\wedge 0|)\right).
\end{align*}
Then, when $G(\delta)>0$,
\begin{align*}
&\lim_{\epsilon\downarrow 0}\frac{\int_0^1u(|G_\epsilon(t\vee \delta)\wedge 0|)dt - \int_0^1u(|G(t\vee \delta)\wedge 0|)dt}{\epsilon}\\
=& \lim_{\epsilon\downarrow 0}\frac{\int_\delta^1u(|G_\epsilon(t)\wedge 0|)dt - \int_\delta^1u(|G(t)\wedge 0|)dt + \delta\left(u(|G_\epsilon(\delta)\wedge 0|) - u(|G(\delta)\wedge 0|)\right)}{\epsilon}\\
=& \lim_{\epsilon\downarrow 0}\frac{\int_\delta^1u(|G_\epsilon(t)\wedge 0|)dt - \int_\delta^1u(|G(t)\wedge 0|)dt - \delta u'(|G(\delta)\wedge 0|)\left(G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0\right)}{\epsilon}\\
=& 0.
\end{align*}
Similarly, when $G(\delta) = 0$,
{\footnotesize
\begin{align*}
\lim_{\epsilon\downarrow 0}\frac{\int_0^1u(|G_\epsilon(t\vee \delta)\wedge 0|)dt - \int_0^1u(|G(t\vee \delta)\wedge 0|)dt}{\epsilon}
= \begin{cases}
0, & \delta <F(z-),\\
0, & F(z-)\le \delta\le F(z),\\
u'(0)\delta (1-\delta) G'(\delta), & \delta>F(z).
\end{cases}
\end{align*}}
When $G(\delta)<0$,
\begin{align*}
&\lim_{\epsilon\downarrow 0}\frac{\int_0^1u(|G_\epsilon(t\vee \delta)\wedge 0|)dt - \int_0^1u(|G(t\vee \delta)\wedge 0|)dt}{\epsilon}\\
=& -\int_\delta^1u(|G(t)\wedge 0|)dt + \begin{cases}
u(|z\wedge 0|) - \delta u(|G(\delta)|) - \delta^2 u'(|G(\delta)|) G'(\delta) , & \delta <F(z-),\\
(1-\delta)u(|z\wedge 0|) , & F(z-)\le \delta\le F(z),\\
(1-\delta)u(|G(\delta)|) + \delta(1-\delta)u'(|G(\delta)|)G'(\delta), & \delta>F(z).
\end{cases}\\
=& -\int_0^1u(|G(t\vee \delta)\wedge 0|)dt + \begin{cases}
u(|z\wedge 0|) - \delta^2 u'(|G(\delta)|) G'(\delta) , & \delta <F(z-),\\
u(|z\wedge 0|) , & F(z-)\le \delta\le F(z),\\
u(|G(\delta)|) + \delta(1-\delta)u'(|G(\delta)|)G'(\delta), & \delta>F(z).
\end{cases}
\end{align*}
Finally, applying chain rule and \eqref{eq:LeftInversePropertyThree}, we immediately have \eqref{eq:SensitivityCEPositive}-\eqref{eq:SensitivityCENegative}.
\end{proof}
From Proposition \ref{prop:sensitivity_robust_CE}, it is easy to show that
\begin{align*}
\sup_{z\in \mathbb{R}}S(z;G) \le \frac{1}{u'(\rho_\delta(G))}\times \begin{cases}
0, & G(\delta)>0,\\
\delta(1-\delta)G'(\delta), &G(\delta)=0,\\
-u(\rho_\delta(G)) + u(|G(\delta)|) + \delta(1-\delta)u'(|G(\delta)|)G'(\delta), & G(\delta)<0.
\end{cases}
\end{align*}
Thus, the truncated loss certainty equivalent has bounded
sensitivity functions, which is consistent with the robustness
properties of risk estimators derived in Section
\ref{sec:robustness}.
\begin{remark}
If $F$ has a continuous positive density $f$ in the neighborhood of $G(\delta)$, then $G(\cdot)$ is differentiable at $\delta$ and
\begin{align*}
G'(\delta) = \frac{1}{f(G(\delta))}.
\end{align*}
Moreover, denoting by $\textrm{SVaR}_\delta(z;G)$ the sensitivity function of $\textrm{VaR}_\delta$ at $G$, which, from the proof of Proposition \ref{prop:sensitivity_robust_CE}, is
\begin{eqnarray*}
\textrm{SVaR}_\delta(z;G) = \begin{cases}
\frac{\delta}{f(G(\delta))} , & z> G(\delta),\\
0, & z = G(\delta),\\
-\frac{1-\delta}{f(G(\delta))}, & z< G(\delta),
\end{cases}
\end{eqnarray*} we can rewrite, for $G(\delta)<0$, the sensitivity of the
$\delta$-truncation of the loss certainty equivalent as: \begin{eqnarray*}
S(z;G) = \frac{1}{u'(\rho_\delta(G))}\left[-u(\rho_\delta(G)) +
u(|(z\vee G(\delta))\wedge 0|) - \delta
u'(|G(\delta)|)\textrm{SVaR}_\delta(z;G)\right]. \end{eqnarray*}
\end{remark}
\iffalse
In this section, we compute the sensitivity functions of the $\delta-$truncated versions of the entropic and $L^p$ risk measures. These truncated versions were introduced in Section \ref{sec:robustification}, in order to obtain robust risk estimators. The conclusion of the following propositions is that by truncating these two risk measures, their sensitivity functions become bounded. Therefore, the robustness properties of risk estimators derived in Section \ref{sec:robustness} are consistent with the sensitivity functions computed in this Section.
\begin{proposition}\label{prop:sensitivity_robust_entropic}
Consider the $\delta$-truncation of the entropic risk measure \eqref{eq:entropyriskmeasuretruncated} with $0<\delta <1$. Assume $F(z) = F(z-)<1$ and $G(\cdot)$ is differentiable at $\delta$. Then the sensitivity $S(z;G)$ can be computed as follows:
\begin{enumerate}
\item[\bf(i)] When $G(\delta)>0$,
\begin{align}\label{eq:senentrotruncase1}
S(z;G) = 0.
\end{align}
\item[\bf(ii)] When $G(\delta)=0$,
\begin{align}\label{eq:senentrotruncase2}
S(z;G) =\begin{cases}
0, & z\ge G(\delta),\\
\delta \, (1-\delta) \, G'(\delta), & z< G(\delta).
\end{cases}
\end{align}
\item[\bf(iii)] When $G(\delta)<0$,
\begin{align}\label{eq:senentrotruncase3}
S(z;G) = \frac{1}{\beta}\begin{cases}
\frac{e^{-\beta z\wedge 0} - \beta \delta^2 G'(\delta)e^{-\beta G(\delta)}}{e^{\beta \rho_\delta(G)}} -1, & z>G(\delta),\\
\frac{e^{-\beta G(\delta)}}{e^{\beta \rho_\delta(G)}}-1, & z=G(\delta),\\
\frac{e^{-\beta G(\delta)} + (1-\delta) \beta \delta G'(\delta)e^{-\beta G(\delta)}}{e^{\beta \rho_\delta(G)}}-1, & z<G(\delta).
\end{cases}
\end{align}
\end{enumerate}
\end{proposition}
\begin{proof}
The following obvious facts will be used in the proof: (i) for any distribution function $\widetilde F$, and $t\in[0,1],x\in\mathbb{R}$
\begin{align*}
\widetilde F(x)\ge t\Longleftrightarrow x\ge \widetilde F^{-1}(t),\quad \widetilde F(x)<t\Longleftrightarrow x< \widetilde F^{-1}(t);
\end{align*}
(ii) for any $x\in\mathbb{R}$, if $\widetilde F^{-1}(\cdot)$ is continuous at $\widetilde F(x)$, then $\widetilde F^{-1}(\widetilde F(x)) = x$; (iii) for any $t\in(0,1)$, if $\widetilde F(\cdot)$ is continuous at $\widetilde F^{-1}(t)$, then $\widetilde F(\widetilde F^{-1}(t))=t$. A direct consequence of these facts is
\begin{align}\label{eq:distquanequ}
F(z)>\delta \Longleftrightarrow z>G(\delta),\quad F(z)=\delta \Longleftrightarrow z=G(\delta), \quad F(z)<\delta \Longleftrightarrow z<G(\delta),
\end{align}
because $F(\cdot)$ is continuous at $z$ and $G(\cdot)$ is continuous at $\delta$.
\iffalse
Let $G_\epsilon(\cdot):=F_\epsilon^{-1}(\cdot)$, and we have
\begin{align*}
G_\epsilon(t) = \begin{cases}
G\left(\frac{t-\epsilon}{1-\epsilon}\right), & t>\epsilon + (1-\epsilon)F(z),\\
G\left(\frac{t}{1-\epsilon}\right), & t\le (1-\epsilon)F(z-),\\
z, & (1-\epsilon)F(z-)<t\le \epsilon + (1-\epsilon)F(z).
\end{cases}
\end{align*}
\fi
Because $F(z)=F(z-)$, direct computation shows that
\begin{align*}
G_\epsilon(t) = \begin{cases}
G\left(\frac{t-\epsilon}{1-\epsilon}\right), & t>\epsilon + (1-\epsilon)F(z),\\
G\left(\frac{t}{1-\epsilon}\right), & t\le (1-\epsilon)F(z),\\
z, & (1-\epsilon)F(z)<t\le \epsilon + (1-\epsilon)F(z).
\end{cases}
\end{align*}
Therefore, we have
\begin{align*}
\int_\delta^1 e^{-\beta G_\epsilon(t)\wedge 0}dt& = \int_\delta^1e^{-\beta G\left(\frac{t-\epsilon}{1-\epsilon}\right)\wedge 0}\mathbf{1}_{\{t>\epsilon + (1-\epsilon)F(z)\}}dt+\int_\delta^1e^{-\beta G\left(\frac{t}{1-\epsilon}\right)\wedge 0}\mathbf{1}_{\{t\le (1-\epsilon)F(z)\}}dt\\
& \quad +\int_\delta^1e^{-\beta z\wedge 0}\mathbf{1}_{\{(1-\epsilon)F(z)<t\le \epsilon + (1-\epsilon)F(z)\}}dt\\
& = {(1-\epsilon)} \left\{ \int_{\frac{\delta -\epsilon}{1-\epsilon}}^1e^{-\beta G\left(t\right)\wedge 0}\mathbf{1}_{\{t>F(z)\}}dt+\int_{\frac{\delta}{1-\epsilon}}^{\frac{1}{1-\epsilon}}e^{-\beta G\left(t\right)\wedge 0}\mathbf{1}_{\{t\le F(z)\}}dt \right\}\\
& \quad +\int_\delta^1e^{-\beta z\wedge 0}\mathbf{1}_{\{(1-\epsilon)F(z)<t\le \epsilon + (1-\epsilon)F(z)\}}dt\\
& = {(1-\epsilon)} \left\{\int_\delta^1e^{-\beta G\left(t\right)\wedge 0}dt + \int_{\frac{\delta -\epsilon}{1-\epsilon}}^\delta e^{-\beta G\left(t\right)\wedge 0}\mathbf{1}_{\{t>F(z)\}}dt-\int_\delta^{\frac{\delta}{1-\epsilon}}e^{-\beta G\left(t\right)\wedge 0}\mathbf{1}_{\{t\le F(z)\}}dt \right.\\
&\left. \quad - \int_{\frac{1}{1-\epsilon}}^1e^{-\beta G\left(t\right)\wedge 0}\mathbf{1}_{\{t\le F(z)\}}dt \right\}+\int_\delta^1e^{-\beta z\wedge 0}\mathbf{1}_{\{(1-\epsilon)F(z)<t\le \epsilon + (1-\epsilon)F(z)\}}dt,
\end{align*}
which, together with the continuity of $G(\cdot)$ at $\delta$, immediately yields
\begin{align}\label{eq:GeGdeltaint}
\int_\delta^1 e^{-\beta G_\epsilon(t)\wedge 0}dt = {(1-\epsilon)} \int_\delta^1e^{-\beta G\left(t\right)\wedge 0}dt + \begin{cases}
\left[e^{-\beta z\wedge 0}-\delta e^{-\beta G\left(\delta \right)\wedge 0}\right]\epsilon + o(\epsilon), & F(z)>\delta,\\
(1-\delta)e^{-\beta z\wedge 0} \epsilon + o(\epsilon), & F(z)=\delta.\\
(1-\delta)e^{-\beta G\left(\delta \right)\wedge 0}\epsilon + o(\epsilon), & F(z)<\delta,
\end{cases}
\end{align}
On the other hand, we have
\begin{align*}
G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0 = \begin{cases}
G\left(\frac{\delta-\epsilon}{1-\epsilon}\right)\wedge 0 - G(\delta)\wedge 0, & \delta >\epsilon + (1-\epsilon)F(z),\\
G\left(\frac{\delta}{1-\epsilon}\right)\wedge 0- G(\delta)\wedge 0, & \delta\le (1-\epsilon)F(z),\\
z\wedge 0- G(\delta)\wedge 0, & (1-\epsilon)F(z)<\delta \le \epsilon + (1-\epsilon)F(z).
\end{cases}
\end{align*}
In the following we discuss case by case.
We first consider the case when $G(\delta)>0$. In this case we immediately have $G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0=0$ when $\delta >\epsilon + (1-\epsilon)F(z)$ or $\delta\le (1-\epsilon)F(z)$ and $\epsilon$ small enough. When $(1-\epsilon)F(z)<\delta \le \epsilon + (1-\epsilon)F(z)$, we have $F(z)\ge \frac{\delta-\epsilon}{1-\epsilon}$ which is equivalent to $z\ge G\left(\frac{\delta-\epsilon}{1-\epsilon}\right)$. Thus $z>0$ when $\epsilon$ is small enough because $G(\cdot)$ is continuous at $\delta$. To summarize, we have
\begin{align}\label{eq:GeGdeltacase1}
G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0=0,\quad \text{when } G(\delta)>0\text{ and }\epsilon\text{ small enough}.
\end{align}
Next we consider the case when $G(\delta)<0$. We immediately have
\begin{align*}
G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0 = \begin{cases}
G\left(\frac{\delta-\epsilon}{1-\epsilon}\right)- G(\delta), & \delta >\epsilon + (1-\epsilon)F(z),\\
G\left(\frac{\delta}{1-\epsilon}\right)- G(\delta), & \delta\le (1-\epsilon)F(z),
\end{cases}
\end{align*}
when $\epsilon$ is small enough, because of the continuity of $G(\cdot)$ at $\delta$. When $(1-\epsilon)F(z)<\delta \le \epsilon + (1-\epsilon)F(z)$, we have $F(z)< \frac{\delta}{1-\epsilon}$ which is equivalent to $z< G\left(\frac{\delta}{1-\epsilon}\right)$. Thus $z<0$ when $\epsilon$ is small enough because $G(\cdot)$ is continuous at $\delta$. Therefore $G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0=z- G(\delta)$. To summarize, we have
\begin{align*}
G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0 = \begin{cases}
G\left(\frac{\delta-\epsilon}{1-\epsilon}\right)- G(\delta), & \delta >\epsilon + (1-\epsilon)F(z),\\
G\left(\frac{\delta}{1-\epsilon}\right)- G(\delta), & \delta\le (1-\epsilon)F(z),\\
z- G(\delta), & (1-\epsilon)F(z)<\delta \le \epsilon + (1-\epsilon)F(z).
\end{cases}
\end{align*}
Now we can see that, if $F(z)>\delta$, $G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0 = \delta G'(\delta)\epsilon + o(\epsilon)$, if $F(z)<\delta$, $G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0 = -(1-\delta)G'(\delta)\epsilon + o(\epsilon)$, and If $F(z)=\delta$ which implies $z=G(\delta)$ from \eqref{eq:distquanequ}, $G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0 = z-G(\delta)=0$. In conclusion, we have
\begin{align}\label{eq:GeGdeltacase2}
G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0 = \begin{cases}
\delta G'(\delta)\epsilon + o(\epsilon), & F(z)>\delta,\\
0, & F(z)=\delta,\\
-(1-\delta)G'(\delta)\epsilon + o(\epsilon), & F(z)<\delta,
\end{cases}
\quad \text{when }G(\delta)<0.
\end{align}
At last, when $G(\delta)=0$, we similarly have
\begin{align*}
G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0 = \begin{cases}
G\left(\frac{\delta-\epsilon}{1-\epsilon}\right), & \delta >\epsilon + (1-\epsilon)F(z),\\
0, & \delta\le (1-\epsilon)F(z),\\
z\wedge 0, & (1-\epsilon)F(z)<\delta \le \epsilon + (1-\epsilon)F(z),
\end{cases}
\end{align*}
when $\epsilon$ is small enough, which yields
\begin{align}\label{eq:GeGdeltacase3}
G_\epsilon(\delta)\wedge 0 - G(\delta)\wedge 0 = \begin{cases}
0 , & F(z)>\delta,\\
0, & F(z)=\delta,\\
-(1-\delta)G'(\delta)\epsilon +o(\epsilon),& F(z)<\delta,
\end{cases}\quad \text{when }G(\delta)=0.
\end{align}
Now, we define the following quantity:
\begin{align*}
A(\epsilon) := \int_0^1e^{-\beta G_\epsilon(t\vee \delta)\wedge 0}dt - \int_0^1e^{-\beta G(t\vee \delta)\wedge 0}dt &= \int_\delta^1e^{-\beta G_\epsilon(t)\wedge 0}dt - \int_\delta^1e^{-\beta G(z)\wedge 0}dt \\
&\quad + \delta e^{-\beta G(\delta)\wedge 0}\left[e^{-\beta(G_\epsilon(\delta)\wedge 0-G(\delta)\wedge 0)}-1\right].
\end{align*}
Combining \eqref{eq:GeGdeltaint},\eqref{eq:GeGdeltacase1},\eqref{eq:GeGdeltacase2}, and \eqref{eq:GeGdeltacase3}, we have
\begin{align*}
A(\epsilon) = { - (1-\delta)\epsilon }+ (1-\delta)\epsilon + o(\epsilon)=o(\epsilon),
\end{align*}
when $G(\delta)>0$,
\begin{align*}
A(\epsilon) &= { -(1-\delta)\epsilon }+
\begin{cases}
(1-\delta)\epsilon + o(\epsilon), & F(z)>\delta,\\
(1-\delta)\epsilon + o(\epsilon), & F(z)=\delta,\\
(1-\delta)(1+\delta\beta G'(\delta))\epsilon + o(\epsilon), & F(z)<\delta,
\end{cases}
\end{align*}
when $G(\delta)=0$, and
\begin{align*}
A(\epsilon)& = {- \epsilon \int_\delta^1e^{-\beta G\left(t\right)\wedge 0}dt} +
\begin{cases}
\left[e^{-\beta z\wedge 0}-\delta(1+\beta \delta G'(\delta))e^{-\beta G(\delta)}\right]\epsilon + o(\epsilon), & F(z)>\delta,\\
(1-\delta)e^{-\beta z\wedge 0}\epsilon + o(\epsilon), & F(z)=\delta.\\
(1-\delta)(1+\beta \delta G'(\delta))e^{-\beta G(\delta)}\epsilon + o(\epsilon), & F(z)<\delta,
\end{cases}
\end{align*}
when $G(\delta)<0$.
Finally, noticing
\begin{align*}
\rho_\delta(G_\epsilon(\cdot)) - \rho_\delta(G(\cdot)) = \frac{1}{\beta\int_0^1e^{-\beta G(t\vee \delta)\wedge 0 dt}}A(\epsilon) + o(\epsilon),
\end{align*}
and recalling \eqref{eq:distquanequ}, we immediately have \eqref{eq:senentrotruncase1}-\eqref{eq:senentrotruncase3}.
\end{proof}
\begin{proposition} \label{prop:sensitivity_robust_Lp}
Consider the $\delta$-truncation of the $L^p$ risk measure \eqref{eq:Lpriskmeasuretruncated} with $0<\delta <1$. Assume $F(z) = F(z-)<1$ and $G(\cdot)$ is differentiable at $\delta$. Then the sensitivity $S(z;G)$ can be computed as follows:
\begin{enumerate}
\item[\bf(i)] When $G(\delta)>0$,
\begin{align}\label{eq:senLptruncase1}
S(z;G) = 0.
\end{align}
\item[\bf(ii)] When $G(\delta)=0$, $S(z;G) = 0$ if $p>1$, and
\begin{align}\label{eq:senLptruncase2}
{\ S(z;G) =\begin{cases}
0, & z\ge G(\delta),\\
\delta (1-\delta) G'(\delta), & z< G(\delta).
\end{cases}}
\end{align}
if $p=1$.
\item[\bf(iii)] When $G(\delta)<0$,
\begin{align}\label{eq:senLptruncase3}
{\
S(z;G) = \frac{\rho_\delta(G)}{p}\begin{cases}
\frac{|z\wedge 0|^p - p \, |G(\delta)|^{p-1} \delta^2 G'(\delta)}{\rho_\delta(G)^p} -1, & z>G(\delta),\\
\frac{|G(\delta)|^p}{\rho_\delta(G)^p}-1, & z=G(\delta),\\
\frac{|G(\delta)|^p + p \, |G(\delta)|^{p-1} \, (1-\delta) \delta G'(\delta)}{\rho_\delta(G)^p}-1, & z<G(\delta).
\end{cases}}
\end{align}
\end{enumerate}
\end{proposition}
\begin{proof}
Following the same argument as in the proof of Proposition \ref{prop:sensitivity_robust_entropic}, we can easily show that
\begin{align*}
\int_\delta^1 \left|G_\epsilon(t)\wedge 0 \right|^p dt = (1-\epsilon) \int_\delta^1 \left|G(t)\wedge 0 \right|^p dt + \begin{cases}
\left( |z\wedge 0|^p - \delta |G(\delta)\wedge 0|^p \right)\epsilon + o(\epsilon), & F(z)>\delta,\\
(1-\delta)\left|z\wedge 0\right|^p \epsilon + o(\epsilon), & F(z)=\delta,\\
(1-\delta)\left|G(\delta)\wedge 0\right|^p \epsilon + o(\epsilon), & F(z)<\delta.
\end{cases}
\end{align*}
Now, we define the following quantity:
\begin{align*}
A(\epsilon) := \int_0^1 \left|G_\epsilon(t)\wedge 0 \right|^p dt - \int_0^1 \left|G(t)\wedge 0 \right|^p dt &= \int_\delta^1 \left|G_\epsilon(t)\wedge 0 \right|^p dt - \int_\delta^1 \left|G(t)\wedge 0 \right|^p dt \\
&\quad + \delta \left(\left|G_\epsilon(\delta)\wedge 0 \right|^p-\left|G(\delta)\wedge 0 \right|^p\right).
\end{align*}
When $G(\delta)>0$, we have immediately (see Proposition \ref{prop:sensitivity_robust_entropic})
\begin{align*}
\left|G_\epsilon(\delta)\wedge 0 \right|^p-\left|G(\delta)\wedge 0 \right|^p = 0,
\end{align*}
and therefore $A(\epsilon) = o(\epsilon)$.
When $G(\delta)=0$, we have $\left|G_\epsilon(\delta)\wedge 0 \right|^p-\left|G(\delta)\wedge 0 \right|^p = o(\epsilon)$ for $p>1$ and
\begin{align*}
\left|G_\epsilon(\delta)\wedge 0 \right|^p-\left|G(\delta)\wedge 0 \right|^p={\ -G_\epsilon(\delta)\wedge 0=
\begin{cases}
0 , & F(z)>\delta,\\
0, & F(z)=\delta,\\
(1-\delta)G'(\delta)\epsilon +o(\epsilon),& F(z)<\delta,
\end{cases}}
\end{align*}
for $p=1$, which gives $A(\epsilon)=o(\epsilon)$ for $p>1$ and
\begin{align*}
A(\epsilon) = \begin{cases}
0 , & F(z)>\delta,\\
0, & F(z)=\delta,\\
\delta(1-\delta)G'(\delta)\epsilon +o(\epsilon),& F(z)<\delta,
\end{cases}
\end{align*}
for $p=1$.
\iffalse
Therefore, we have
\begin{align*}
A(\epsilon) &=
\begin{cases}
o(\epsilon), & F(z)>\delta,\\
- \delta (1-\delta) G'(\delta) \epsilon \, \mathbf{1}_{p=1} + o(\epsilon), & F(z)<\delta,\\
o(\epsilon), & F(z)=\delta.
\end{cases}
\end{align*}
\fi
Finally, when $G(\delta)<0$, we have
\begin{align*}
\left|G_\epsilon(\delta)\wedge 0 \right|^p-\left|G(\delta)\wedge 0 \right|^p &= {\ p \, |G(\delta)|^{p-1}
\begin{cases}
-\delta G'(\delta) \epsilon +o(\epsilon), & F(z)>\delta,\\
0, & F(z)=\delta,\\
(1-\delta) G'(\delta)\epsilon +o(\epsilon),& F(z)<\delta,
\end{cases}}
\end{align*}
leading to
\begin{align*}
A(\epsilon)& = -\epsilon \int_\delta^1 \left|G(t)\wedge 0 \right|^p dt +{\
\begin{cases}
\left( |z\wedge 0|^p - \delta |G(\delta)|^p - p \, |G(\delta)|^{p-1} \delta^2 G'(\delta) \right)\epsilon + o(\epsilon), & F(z)>\delta,\\
(1-\delta)\left|z\wedge 0\right|^p \epsilon + o(\epsilon), & F(z)=\delta,\\
(1-\delta) \left( |G(\delta)|^p + p |G(\delta)|^{p-1}\,\delta G'(\delta) \right)\epsilon + o(\epsilon), & F(z)<\delta.
\end{cases}}
\end{align*}
Recalling \eqref{eq:distquanequ}, the sensitivity \eqref{eq:senLptruncase1}-\eqref{eq:senLptruncase3} follows easily.
\end{proof}
\begin{remark}
If $F$ has a continuous positive density $f$ in the neighborhood of $G(\delta)$, then $G(\cdot)$ is differentiable at $\delta$ and
\begin{align*}
G'(\delta) = \frac{1}{f(G(\delta))}.
\end{align*}
Moreover, denoting by $\textrm{SVaR}_\delta(z;G)$ the sensitivity function of $\textrm{VaR}_\delta$ at $G$
\begin{eqnarray*}
\textrm{SVaR}_\delta(z;G) = \begin{cases}
-\frac{\delta}{f(G(\delta))} , & z> G(\delta),\\
0, & z = G(\delta),\\
\frac{1-\delta}{f(G(\delta))}, & z< G(\delta),
\end{cases}
\end{eqnarray*}
we can rewrite, for $G(\delta)<0$, the sensitivity of the $\delta-$truncation of the entropic risk measure as:
\begin{eqnarray*}
S(z;G) = \frac{1}{\beta} \left[ \frac{e^{-\beta (G(\delta)\vee z) \wedge 0} + \delta \, \beta \, e^{-\beta G(\delta)} \, \textrm{SVaR}_\delta(z;G)}{e^{\beta \rho_\delta(G)}} -1\right], \qquad \text{for } G(\delta)<0,
\end{eqnarray*}
and the sensitivity of the $\delta-$truncation of the $L^p$ risk measure as:
\begin{eqnarray*}
S(z;G) = {\ \frac{\rho_\delta(G)}{p} \left[ \frac{| (G(\delta)\vee z) \wedge 0|^p + \delta \, p \, |G(\delta)|^{p-1} \textrm{SVaR}_\delta(z;G)}{\rho_\delta(G)^p} -1\right], \qquad \text{for } G(\delta)<0.}
\end{eqnarray*}
\end{remark}
\fi
\section{Conclusions}\label{se:Conclusions}
In this paper, we have proposed a new class of risk measures named as loss-based risk measures, provided two representation theorems for convex loss-based risk measures, and investigated the robustness of the risk estimators associated with a family of statistical loss-based risk measures that include both statistical convex loss-based risk measures and VaR on losses as special cases.
Motivated by the fact that the risk measures employed in some of the risk management practices only depend on portfolio losses, we propose the loss-dependence property, a characterizing property of loss-based risk measures, which is largely overlooked and actually cannot be accommodated in the existing risk measure frameworks. We have shown in the paper a dual representation theorem for convex loss-based risk measures and another representation theorem if the risk measures are furthermore distributional-based. In addition, we have provided abundant interesting examples of loss-based risk measures, some of which cannot be obtained in the existing risk measure frameworks by simple modification.
In order to address the issue that risk estimates that are extremely sensitive to single data points are useless in some of the risk management practices, we have investigated the robustness of the risk estimators associated with a family of statistical loss-based risk measures. We have found a sufficient and necessary condition for those risk estimators to be robust, and this result significantly improves the existing ones. From that condition, we have shown that statistical convex loss-based risk measures lead to non-robust risk estimators while the loss-based VaR leads to a robust risk estimator, and these results have been confirmed by performing further sensitivity analysis. The conflict between the convexity property of a risk measure and the robustness of the corresponding risk estimator suggests that we need to decide which of the properties is more relevant when choosing a risk measure for certain use.
|
2,869,038,156,345 | arxiv | \section*{Introduction}
The emergence of ordered structure as a result of self-assembly of building blocks is far from being well understood, and has received a great deal of attention \cite{Phillips2012a,Phillips2012b,Damasceno2012}. Understanding this underlying scheme (an example of `hidden order') may open up the doors for explaining phenomena that has thus far remained elusive (e.g., anomalies in water \cite{Brovchenko2008}). Studies have shown that order in structure is directly related to its fundamental building blocks \cite{Damasceno2012}. Recently, 2D quasicrystalline\ order has been achieved in a tetrahedral packing, when thermodynamic conditions are applied to an ensemble of tetrahedra \cite{Akbari2009}. This discovery has stimulated further investigations into quasicrystalline\ packings of tetrahedra. This paper presents an icosahedrally symmetric quasicrystal, as a packing of regular tetrahedra. We obtain this by using two approaches, with very different rationales, but ultimately obtain the same quasicrystalline\ packing. This structure might also help in the search for a perfect 4-coordinated quasicrystal\ \cite{Dmitrienko2001}: even though the tetrahedra in our quasicrystal\ are packed in such a way that faces of neighboring tetrahedra can be paired up (yielding a 4-connected network between tetrahedra centers), the structure cannot be considered physically as 4-coordinated, due to the excessively high ratio between the longest and shortest bonds stemming from each tetrahedron center (Figure~\ref{network}~{\bf b}).
\section*{First approach}
This is a guided decoration of the 3D slice of the Elser-Sloane quasicrystal\ \cite{Elser1987} (Figure~\ref{clusters}~{\bf a}), which contains four types of prototiles: icosidodecahedron (IDHD), dodecahedron (DHD), and icosahedron (IHD)---each as a section (boundary of a cap) of a 600-cell---and a golden tetrahedron. The way each of these polyhedra (Figure~\ref{clusters}~{\bf c}-IHD, {\bf e}-DHD, IDHD (not shown)) is decorated is guided by the arrangement of the tetrahedra in the 600-cells of the Elser-Sloan quasicrystal. For example, the IDHD is a slice through the equator of the 600-cell and is the boundary of a cap of 300 tetrahedra. Projecting these 300 tetrahedra into the IDHD hyperplane results in distorted tetrahedra, and gaps appear when tetrahedra are restored (while avoiding collisions) back to a regular shape. To maintain a higher packing density and a better tetrahedral coordination \cite{Dmitrienko2001}, extra tetrahedra can be introduced to fill in such gaps or, alternatively, some tetrahedra may be removed from each shell before regularization to avoid conflicts resulting from collisions. Both methods result in exactly the same quasicrystalline\ structure, as shown in Figure~\ref{clusters}~{\bf f}. This method can be thought as a decoration of the 3D slice of the Elser-Sloane quasicrystal, guided by the quasicrystal\ itself.
\begin{figure}[h!]
\centering
\includegraphics[width=\textwidth,clip]{clusters.png}
\caption{({\bf a}) 3D slice of the Elser-Sloane quasicrystal. ({\bf b}) 10-tetrahedron `ring'. ({\bf c}) 20-tetrahedron `ball' (icosahedral cluster), obtained by capping the ring with two 5-tetrahedron groups. ({\bf d}) 40-tetrahedron cluster, obtained by placing a tetrahedron on top of each face of the icosahedron in ({\bf c}). ({\bf e}) 70-tetrahedron dodecahedral cluster, obtained by adding 30 more tetrahedra in the crevices in ({\bf d}). ({\bf f}) Patch of the resulting quasicrystal, which contains the clusters shown in ({\bf c}), ({\bf d}) and ({\bf e}) around its center.}
\label{clusters}
\end{figure}
\section*{Second approach}
This is an direct decoration of the 3D Ammann tiling \cite{Peters1991}. A 3D Ammann tiling can be generated by cut-and-project from the $\mathbb{Z}^6$ lattice, and contains two prototiles: a prolate rhombohedron and an oblate rhombohedron.
The decoration consists in placing a 20-tetrahedron `ball' (Figure~\ref{clusters}~{\bf c}) at each vertex, and a 10-tetrahedron `ring' (Figure~\ref{clusters}~{\bf b}) around each edge of each of the rhombohedra, and then removing all those tetrahedra that do not intersect the rhombohedron. This process creates some clashes inside the oblate rhombohedron, and after excluding the appropriate tetrahedra, the packing remains face-to-face and therefore the resulting network of tetrahedron centers is 4-connected. (However, the bond-length distribution has a rather long tail, (Figure~\ref{network}~{\bf b}).) This process yields some pairs of tetrahedra with large overlaps shared between the balls and the rings, providing degrees of freedom to choose either tetrahedron of each pair, which in turn translates into the ability to flip 3 tetrahedra in each face of the prolate rhombohedra, suggesting a novel phason mechanism for this type of quasicrystal\ (Figure~\ref{prolate}~{\bf b,c}). The oblate rhombohedron, due to its flatness, does not have these degrees of freedom, there being only one way to choose the tetrahedra so that the resulting decoration will have a 3-fold axis of symmetry (Figure~\ref{oblate}).
\begin{figure}[h!]
\centering
\includegraphics[width=.54\textwidth,clip]{decorated_prolate_tile.pdf}
\caption{Decoration of the prolate rhombohedron ({\bf a}). ({\bf b}), ({\bf c}) The two conformations that 3 of tetrahedra associated to each face can have. There are 52 tetrahedra decorating the whole prolate rhombohedron: 16 of them lie completely inside it, while the remaining 36 are shared 50\% with corresponding tetrahedra in a face of an adjacent rhombohedron in the packing. The 3 tetrahedra (in each face) that can `flip' consist of 2 shared ones and one internal one.}
\label{prolate}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=.75\textwidth,clip]{decorated_oblate_tile-rot.pdf}
\caption{Decoration of the oblate rhombohedron ({\bf a}). ({\bf b}) The decoration consists of 36 tetrahedra, each shared 50\% with corresponding tetrahedra in an adjacent rhombohedron. ({\bf c}) Here only the 24 tetrahedra coming from the 20-tetrahedron `balls' (each centered at a vertex of the rhombohedron) are displayed, in order to get a better feel of the arrangement. The other 12 tetrahedra shown in ({\bf b}) come from the 10-tetrahedron `rings' (each centered at the midpoint of a rhombohedron edge).}
\label{oblate}
\end{figure}
\section*{Twisting and plane-class reduction}
The quasicrystal\ can also be obtained by placing a 40-tetrahedron cluster (Figure~\ref{twisted}~{\bf a}) at each vertex of the 3D Ammann tiling. Applying a `golden rotation' \cite{Fang2013} of $\arccos(\tau^2/2\sqrt{2}) \approx 22.2388^{\circ}$ (where $\tau=\frac{1}{2}(1+\sqrt{5})$ is the golden ratio) to each of the tetrahedra in the cluster, around an axis running through the tetrahedral center and the center of the cluster, yields a twisted quasicrystal\ (Figure~\ref{twisted}~{\bf b}-twisted 40-tetrahedron cluster, {\bf c}-the twisted quasicrystal). This golden twist reduces the total number of plane classes from 190 to 10. The resulting relative face rotation at each `face junction' between adjacent tetrahedra is $\arccos(\frac{1}{4}(3\tau-1)) = \frac{1}{3}\arccos(11/16) \approx 15.5225^{\circ}$.
\begin{figure}[h!]
\centering
\includegraphics[width=.75\textwidth,clip]{twisted.png}
\caption{({\bf a}) Non-twisted and ({\bf b}) twisted 40-tetrahedron clusters. ({\bf c}) Patch of the twisted quasicrystal.}
\label{twisted}
\end{figure}
\section*{Analysis}
Diffraction patterns of the non-twisted quasicrystal\ (Figure~\ref{clusters}~{\bf f}) reveal 2-, 3-, and 5-fold symmetry planes (Figure~\ref{diffract}), confirming the icosahedral symmetry of this quasicrystal. Its packing density is $\frac{65}{16464}(208800\sqrt{2}+64215\sqrt{5}-45499\sqrt{10}-294845) \approx 0.59783$. The derivation of this expression is too lengthy to be included here, and was done using the {\sl Mathematica} software. (A {\sl Mathematica} notebook containing code for this calculation is available upon request.) Basically, the calculation is done for each rhombohedron, by solving equations that minimize the rhombohedron's edge length relative to the tetrahedron's, in such a way that the various tetrahedra just touch each other. The degrees of freedom allowed in this process are shifts in the directions of the rhombohedron's edges and radially from them. Finally, the density values for both rhombohedra are combined using the fact that the relative frequencies of occurrence of the prolate and oblate rhombohedra in the Ammann tiling is the golden ratio.
The network of tetrahedral centers (Figure~\ref{network}~{\bf a}) is 4-connected, although it cannot be considered what the chemistry community would call ``4-coordinated,'' due to the rather wide range of bond lengths (Figure~\ref{network}~{\bf b}), with a ratio of 1.714 between the longest and shortest bonds.
\begin{figure}[h!]
\centering
\includegraphics[width=.9\textwidth,clip]{diffract.png}
\caption{Diffraction patterns on the 2-fold ({\bf a}), 3-fold ({\bf b}), and 5-fold ({\bf c}) planes of the quasicrystal\ defined by the tetrahedral centers.}
\label{diffract}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=.9\textwidth,clip]{network.png}
\caption{({\bf a}) Stereo pair of the network of tetrahedral centers. ({\bf b}) Distribution of bond lengths in the network.}
\label{network}
\end{figure}
\section*{Summary and outlook}
Using two seemingly unrelated approaches, we have surprisingly obtained the same quasicrystalline\ packing of regular tetrahedra with global icosahedral symmetry. After the fact, this convergence turned out not to be by chance. The reason is that a 3-dimensional slice (in the appropriate orientation) of the 4-dimensional Elser-Sloane quasicrystal\ can be obtained directly by cut-and-project from the $D_6$ lattice \cite{Moody1993}, $D_6$ being a sublattice of $\mathbb{Z}^6$ (and $\mathbb{Z}^6$ being a sublattice of $\frac{1}{2}D_6$, the 6-dimensional face-centered cubic lattice).
To our knowledge, an icosahedrally symmetric packing of tetrahedra with this relatively high density of almost 0.6 has not been shown before. Moreover, this packing provides a 4-connected network with bond lengths ranging from 0.437 and 0.749 (in units of the tetrahedron's edge length), a 1:1.714 ratio. These features are non-trivial among icosahedral arrangements of tetrahedra. For applications, an important step would be to shrink the range of bond lengths to what can be considered as a realistic 4-coordinated network.
We also considered the number of `plane classes' and ways to reduce it to the minimum possible. The above packing has 190 plane classes. By applying what we call the `golden twist' to each tetrahedron, the 190 plane classes of the original quasicrystal\ coalesce to only 10. (This is easily seen to be the minimum possible for an icosahedral arrangement of tetrahedra.)
This quasicrystal\ also suggests interesting alternatives to the classical phason flips, as shown in Figure~\ref{prolate}. We are investigating the dynamics of this and other types of phasons and their potential physical applications.
\bibliographystyle{abbrv}
|
2,869,038,156,346 | arxiv | \section{Introduction}
\label{sec:intro}
\input{tex/intro}
\section{Related Works}
\label{sec:related}
\input{tex/related}
\section{Parallel Triangle Counting Algorithms}
\label{sec:ptc}
\input{tex/ptc}
\section{Implementations}
\label{sec:impl}
\input{tex/impl}
\section{Experiments and Discussion}
\label{sec:exp}
\input{tex/exp}
\section{Conclusion}
\label{sec:conc}
\input{tex/conc}
\section*{Acknowledgements}
\label{sec:acks}
\input{tex/acks}
\bibliographystyle{abbrv}
\subsection[The Implementation of Subgraph-Matching-Based TC Algorithm]{The
Implementation of Subgraph-Match-\\ing-Based TC Algorithm}
We make several
optimizations to the algorithm proposed by Tran et
al.~\cite{Tran:2015:FSM} in both filtering and joining phases.
\subsubsection{Optimizations for Candidate Node Filtering}
The initial filtering phase takes nodes as processing units and uses the
massively parallel \emph{advance} operation in Gunrock to mark nodes with
same labels and larger degrees as candidates in a candidate\_set. We then use
another \emph{advance} to label candidate edges based on the candidate\_set
and use a \emph{filter} operation in Gunrock to prune out all non-candidate edges and
reconstruct the graph. Note that by reconstructing the data graph, we also update
node degree and neighbor list information. So we can run the above two steps for a few
iterations in order to prune out more edges.
\subsubsection{Optimizations for Candidate Edge Joining}
The most time-consuming part in subgraph matching is joining.
Unlike previous
\emph{backtracking}-based algorithms, our joining operation forms partial
solutions in the verification phase to save a substantial amount of
intermediate space.
First, we collect candidate edges for each query edge from the new graph.
This step can be achieved by first labeling each candidate edge with its
corresponding query edge id, then using the
select primitive from Merrill's \emph{CUB} library to assign edges into query edge bins. This approach is both simpler and more
load-balanced compared to the two-step (\emph{computing}-\emph{and-assigning}) output
strategy used by Tran et al.~\cite{Tran:2015:FSM}, which requires heavy use of
gather and scatter operations. When joining the candidate edges, we verify the
intersection rules between candidate edges specified by the query graph.
We do the actual
joining only when the candidate edges satisfy the intersection rules in order to reduce the number of
intermediate results and consequently, the amount of required memory.
\subsection{The Implementation of Set-Intersection-Based TC Algorithm}
The reason behind the high performance of our set-intersection-based
implementation lies in the specific optimizations we propose in the
two stages of the algorithm.
\subsubsection{Optimizations for Edge List Generation Phase} Previous GPU
implementations of intersection-based triangle counting compute intersections
for all edges in the edge list. We believe they make this decision because of
a lack of efficient edge-list-generating and -filtering primitives; instead, in
our implementation, we leverage Gunrock's operations to implement these
primitives and increase our performance. To implement the \emph{forward}
algorithm, we use Gunrock's advance V2E operator to generate the edge list that
contains all edges, and then use Gunrock's filter operator to get rid of
$e(u,v)$ where $d(u) < d(v)$. For edges with the same $d(u)$ and $d(v)$, we
keep the edge if the vertex ID of $u$ is smaller than $v$. This efficiently
removes half of the workload. We then use segmented reduction in Merrill's
\emph{CUB} library to generate a smaller induced subgraph contains only the
edges that have not been filtered to further reduce two thirds of the workload.
\subsubsection{Optimizations for Batch Set Intersection}
High-performance batch set intersection requires a similar focus as
high-performance graph traversal: effective load-balancing and GPU utilization.
In our implementation, we use the same dynamic grouping strategy proposed in
Merrill's BFS work~\cite{Merrill:2012:SGG}. We divide the edge lists into two
groups: (1)~small neighbor lists; and (2)~large neighbor lists. We implement two
kernels (TwoSmall and TwoLarge) that cooperatively compute intersections. Our
TwoSmall kernel uses one thread to compute the intersection of a node pair. Our
TwoLarge kernel partitions the edge list into chunks and assigns each chunk to
a separate thread block. Then each block uses the balanced path primitive from
the \emph{Modern GPU} library\footnote{Code is available at
\url{http://nvlabs.github.io/moderngpu}} to cooperatively compute
intersections. By using this 2-kernel strategy and carefully choosing
a threshold value to divide the edge list into two groups, we can process
intersections with same level of workload together to gain load balancing and
higher GPU resource utilization.
\subsection[The Implementation of Matrix-Multiplication-Based TC Algorithm]{The
Implementation of Matrix-Multiplic-\\ation-Based TC Algorithm}
\label{sec:schank-wagner-proof}
Our matrix-multiplication-based TC algorithm tests how well a highly
optimized matrix multiplication function from a standard GPU library (csrgemm
from cuSPARSE) performs compared to one implemented using a graph processing
library (subgraph matching and set intersection). The cuSPARSE SpGEMM function
allocates one CSR row in the left input matrix for each thread, which performs
the dot product in linear time. The $n$ rows need to be multiplied with $n$
columns as mentioned in Section~\ref{sec:ptc:mm}. This leads to a $O(\sum_{v
\in V}d(v)^{2})$ complexity, which is the same as the set-intersection-based TC
algorithm. For a detailed proof that shows this equivalence, see Schank and Wagner~\cite{Schank:2005:FCL}.
We compute the Hadamard product by assigning a thread to each row, which then
performs a multiplication by the corresponding value in adjacency matrix
$\textbf{A}$. We incorporate two optimizations: (1)~compute only the upper
triangular values, because the output of the Hadamard product is known to be
symmetric; (2)~use a counter to keep track of the number of triangles found by
each thread, allowing the sum of the number of triangles in each column with a single
parallel reduction.
Step 2 combines lines 5 and 6 in Algorithm~\ref{alg:tcmatrix} so that
the reduction can be performed without writing the nonzeros into global memory.
\subsection{TC using Subgraph Matching}
Subgraph matching is the task of finding all occurrences of a small query graph
in a large data graph. In our formulation, both graphs are undirected labeled
graphs. We can use subgraph matching to solve triangle counting problems by
first defining the query graph to be a triangle, then assigning a unified label
to both the triangle and the data graph. One advantage of using subgraph
matching to enumerate triangles in a graph is that we can get the listings of all
the triangles for free. Another advantage is that we can generalize the problem
to find the embeddings of triangles with certain label patterns.
Our method follows a filtering-and-joining procedure and is implemented using
the Gunrock~\cite{Wang:2016:GAH} programming model. Algorithm~\ref{alg:tcsm}
outlines our subgraph matching-based triangle counting implementation. In the
filtering phase, we focus on pruning away candidate vertices that cannot
contribute to the final solutions; we note that nodes with degree less than two
cannot be matched any query vertex, since every node in a triangle has a degree
of two. We use one step of Gunrock's \emph{Advance} operator to fill
a boolean candidate set (denoted as $c\_set$) for the data graph. If node $i$ in the
data graph is a candidate to node $j$ in the
query graph, then $c\_set[i][j]$ has a value of one. Otherwise, $c\_set[i][j]=0$. We then use our
$c\_set$ to label candidate edges corresponding to each query edge using another
\emph{Advance} and collect them using a \emph{Filter} operator. Then
we can get an edge list of the new data graph composed of only candidate
edges. After collecting candidate edges for each edge in the triangle, we test
if each combination of the edges satisfies the $intersection\_rule$ defined by
the input triangle. If so, we count them into the output edge list.
Figure~\ref{fig:sm} shows a running example of the graph sample from
Figure~\ref{fig:example}.
\begin{figure}[ht]
\includegraphics[height=0.26\textwidth]{fig/sm.pdf}
\centering
\caption{A simple graph example for SM-based TC illustration.\label{fig:sm}} \end{figure}
Since every query node has the same degree of
two and the degree of a node in a large connected graph is usually larger than
two, the filtering phase cannot prune away that many nodes simply based on node
degrees and candidate neighbor lists. The most important, and also the most
time-consuming, step is joining. Tran et
al.~\cite{Tran:2015:FSM} use a two-step scheme to collect candidate edges
into a hash table. First, they count the number of candidate edges for each
query edge. Then, they compute the address of each candidate edge and assign
them to the computed address in the hash table. In our implementation, we first
use a massively parallel advance operation to label each candidate edge with
its corresponding query edge id. Then we use the \emph{select} primitive from
Merrill's \emph{CUB}
library\footnote{\label{cub}\url{http://nvlabs.github.io/cub/}} to collect
candidate edges for each query edge.
\begin{algorithm}[!ht]\caption{TC using subgraph matching.}
\label{alg:tcsm}
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\begin{small}
\begin{algorithmic}[1]
\Require{Query Graph (Triangle) $T$, Data Graph $G$.}
\Ensure{Number of triangles $n$ and listings of all matches.}
\Procedure{initialize\_candidate\_set}{$T,G$}
\State
\Call{Advance}{$G$}\Comment{Fill c\_set based on node label and degree}
\EndProcedure
\State
\Procedure{collect\_candidate\_edges}{$G,T,c\_set$}
\State
\Call{Advance}{$G$}\Comment{Label candidate edges with query\_edge\_id}
\State
\Call{Filter}{$G$}\Comment{Collect candidate edges}
\State
\Return{$ELIST$}
\EndProcedure
\State
\Procedure{Join\_candidate\_edges}{$ELIST, intersection\_rule$}
\State \textbf{parallel for} each candidate edge combination $\{e_i,e_j,e_k\}$
\If {$\{e_i,e_j,e_k\}$ satisfy $intersection\_rules$}
\State Write $\{e_i,e_j,e_k\}$ to output list
\State Add 1 to count
\EndIf
\State
\Return{count, outputlist}
\EndProcedure
\end{algorithmic}
\end{small}
\end{algorithm}
\vfill\eject\subsection{TC using Set Intersection}
The extensive survey by Schank and Wagner~\cite{Schank:2005:FCL} shows several
sequential algorithms for counting and listing triangles in undirected graphs.
Two of the best performing algorithms, \emph{edge-iterator} and \emph{forward},
both use edge-based set intersection primitives. The optimal theoretical bound
of this operation coupled with its high potential for parallel implementation
make this method a good candidate for GPU implementation.
\begin{figure}[ht]
\includegraphics[width=0.5\textwidth]{fig/intersection-chart.pdf}
\caption{The workflow of intersection-based GPU TC algorithm.\label{fig:intersection-chart}}
\end{figure}
We can view the TC problem as a set intersection problem by the
following observation: An edge $e=(u,v)$, where $u,v$ are its two end nodes,
can form triangles with edges connected to both $u$ and $v$. Let the
intersections between the neighbor lists of $u$ and $v$ be $(w_1, w_2, \ldots,
w_N)$, where $N$ is the number of intersections. Then the number of triangles
formed with $e$ is $N$, where the three edges of each triangle are $(u, v),
(w_i, u), (w_i, v), i \in [1,N]$. In practice, computing intersections for
every edge in an undirected graph is redundant. We visit all the neighbor lists
using Gunrock's \emph{Advance} operator. Then we filter out half of the edges by degree
order. Thus, in general, set intersection-based TC algorithms have two stages:
(1)~forming edge lists; (2)~computing set intersections for two neighbor lists of
an edge. Different optimizations can be applied to either stage. Our GPU
implementation (shown in Algorithm~\ref{alg:tcintersection}) follows the
\emph{forward} algorithm and uses several operators of the high-performance
graph processing library Gunrock~\cite{Wang:2016:GAH}.
Figure~\ref{fig:intersection-chart} is the flow chart that shows how this
algorithm works on the running example (Figure~\ref{fig:example}).
\begin{algorithm}[!ht]\caption{TC using edge-based set intersection.} \label{alg:tcintersection}
\begin{small}
\begin{algorithmic}[1]
\Procedure{Form\_Filtered\_Edge\_List}{$G$}
\State
\Call{Advance}{G}
\State
\Call{Filter}{G}
\State
\Return{ELIST}
\EndProcedure
\State
\Procedure{Compute\_Intersection}{$G, ELIST$}
\State
\{SmallList,LargeList\} = \Call{Partition}{G, ELIST}
\State
\Call{LargeNeighborListIntersection}{LargeList}
\State
\Call{SmallNeighborListIntersection}{SmallList}
\EndProcedure
\State
\Procedure{TC}{$G$}
\State
ELIST=\Call{Form\_Filtered\_Edge\_List}{$G$}
\State
IntersectList=\Call{Compute\_Intersection}{$G, ELIST$}
\State
Count=\Call{Reduce}{IntersectList}
\State
\Return{Count}
\EndProcedure
\end{algorithmic}
\end{small}
\end{algorithm}
\subsection{TC using Matrix Multiplication}
\label{sec:ptc:mm}
The matrix multiplication formulation comes from Azad, Bulu\c{c}, and Gilbert's
algorithm for counting and enumerating triangles~\cite{Azad:2015:PTC}. Nodes in
the adjacency matrix $\mathbf{A}$ are arranged in some order. The lower
triangular matrix $\mathbf{L}$ represents all edges from row $i$ to column $k$
such that $k \leq i$. The upper triangular matrix $\mathbf{U}$ represents all
edges from row $k$ to column $j$ such that $k \leq j$. The dot product of row
$i$ of $\mathbf{L}$ with column $j$ of $\mathbf{U}$ is a count of all wedges
$i-k-j$ where $k \leq i$ and $k \leq j$. By performing a Hadamard product
(element-wise multiplication) with $A_{ij}$, we obtain an exact count of the
number of triangles $i-k-j$ for $k \leq i$ and $k \leq j$. Algorith~\ref{alg:tcmatrix}
shows the pseudocode of our GPU implementation and Figure~\ref{fig:mm}.
We illustrate this algorithm by giving a running example of the sample graph (Figure~\ref{fig:example}) in Figure~\ref{fig:mm}.
\begin{algorithm}[]
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\begin{small}
\begin{algorithmic}[1]
\Require{Adjacency matrix $\mathbf{A}$.}
\Ensure{Number of triangles $n$.}
\Procedure{Triangle\_Count\_Matrix}{}
\State{}Permute rows $\mathbf{A}$ so that it is ordered by
an increasing number of nonzeros.
\State{}Break matrix into a lower triangular piece
$\mathbf{L}$ and an upper triangular piece $\mathbf{U}$ such that $\mathbf{A = L + U}$.
\State{}Compute $\mathbf{B = LU}$.
\State{}Compute $\mathbf{C = A \circ B}$ where $\circ$ is the Hadamard product.
\State{}Compute $n = \frac{1}{2} \sum_{i} \sum_{j} A_{ij}$.
\EndProcedure{}
\end{algorithmic}
\end{small}
\caption{Counts how many triangles are in an undirected graph using matrix multiplication formulation. \label{alg:tcmatrix}}
\end{algorithm}
\begin{figure}
{\tiny \begin{eqnarray*}
\textbf{A} & = & \begin{pmatrix}
0 & 1 & 0 & 0 & 1 & 1 & 0 \\
1 & 0 & 1 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 & 1 & 1 & 0 \\
1 & 0 & 0 & 1 & 0 & 1 & 0 \\
1 & 1 & 0 & 1 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\textbf{L} & = & \begin{pmatrix}
0 & & & & & & \\
1 & 0 & & & & \makebox(0,0){\text{\huge0}} & \\
0 & 1 & 0 & & & & \\
0 & 0 & 1 & 0 & & & \\
1 & 0 & 0 & 1 & 0 & & \\
1 & 1 & 0 & 1 & 1 & 0 & \\
0 & 0 & 1 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\textbf{U} & = & \begin{pmatrix}
0 & 1 & 0 & 0 & 1 & 1 & 0 \\
& 0 & 1 & 0 & 0 & 1 & 0 \\
& & 0 & 1 & 0 & 0 & 1\\
& & & 0 & 1 & 1 & 0 \\
& & & & 0 & 1 & 0 \\
& \makebox(0,0){\text{\huge0}} & & & & 0 & 0 \\
& & & & & & 0
\end{pmatrix} \\
\textbf{B} & = & \textbf{LU} \\
& = & \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 2 & 2 & 0 \\
0 & 1 & 1 & 0 & 2 & 4 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 1
\end{pmatrix} \\
\textbf{C} & = & \textbf{A} \circ \textbf{B} \\
& = & \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 2 & 0 \\
0 & 1 & 0 & 0 & 2 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}
\end{eqnarray*}}
\caption{The number of triangles is yielded by summing the elements of
\textbf{C}, then dividing by two. In this example, the number of triangles is
$\frac{6}{2} = 3$. \label{fig:mm}}
\end{figure}
|
2,869,038,156,347 | arxiv | \section{Introduction}
The {\it Gaia}\xspace mission \citep{Gaia2016,Gaia2018} will give us a many-dimensional astrometric, photometric and spectroscopic perspective on the stars of the Milky Way. Already with the preliminary second data release (DR2), {\it Gaia}\xspace has provided astrometric positions and broad-band $G$ photometry for 1,692,919,135 sources. \edits{This begs the question, however: which stars are missing from {\it Gaia}\xspace DR2 and where do they lie on the sky?}
In this second work in our series investigating the completeness of the {\it Gaia}\xspace-verse, we quantify the completeness of the stars with positions $(\alpha,\delta)$ and $G$ magnitudes in {\it Gaia}\xspace DR2 (i.e. all 1,692,919,135 sources included in \edits{the DR2 source catalogue}) through a selection function. \edits{Such a selection function would be} immediately useful for those using star count overdensities to look for stellar structures in the Galaxy \edits{or for those looking to map the distribution of stars in the disk}. Furthermore, it is a necessary foundation for the selection functions of the more commonly used {\it Gaia}\xspace DR2 subsets that have proper motions and parallaxes, colour photometry, \edits{variable star classifications}, or radial velocities.
The simplest way to calculate a catalogue's selection function is to count the fraction of stars in another more complete catalogue that are missing; for example, \citet{Bovy2014} calculated the selection function of the spectroscopic APOGEE Red Clump Catalog in comparison to the photometric 2MASS survey. The APOGEE survey selected stars for observation from the 2MASS source catalogue \citep{Zasowski2013} and so 2MASS is guaranteed to be a superset of the stars observed by APOGEE. This approach is attractive because it is easy to understand and implement, and does not require any knowledge about the instrumentation or pipeline. However, it does require that the comparison catalogue is truly complete for the magnitude range of the catalogue of interest, and so cannot be used to compute the {\it Gaia}\xspace selection function, because there is no more complete catalogue for {\it Gaia}\xspace to be compared against. At the other extreme, if we have perfect knowledge of the instrumentation and pipeline that lead to a catalogue, then the selection function of that catalogue can be directly computed. We adopt a hybrid approach in this work: by using a small amount of knowledge about {\it Gaia}\xspace (the spinning-and-precessing scanning law) we are able to compute an empirically-driven selection function.
The key and entirely novel idea behind this paper is that the selection function of {\it Gaia}\xspace DR2 is approximately that the number of times that {\it Gaia}\xspace detected the source as it crossed the field-of-view was greater than four, as motivated in the preamble of Sec. \ref{sec:methodology}. We use the term `detections' to mean those occasions where the source transiting the {\it Gaia}\xspace field-of-view resulted in an astrometric measurement that contributed to the source's astrometric solution. \edits{There are many reasons why an observation of a source might not result in such a detection, as discussed later in the text}. \edits{The} number of detections is termed \textsc{astrometric\_matched\_observations} in the {\it Gaia}\xspace terminology and this number is given for every star in {\it Gaia}\xspace DR2.
In the first part of this paper we assume that the probability that a star is detected on each observation is solely a function of brightness, and thus by modelling the \textsc{astrometric\_matched\_observations} of all the stars in {\it Gaia}\xspace DR2 we deduce a first-order selection function for the entire catalogue. We describe our methodology in Sec. \ref{sec:methodology} and present our inferred selection function in Sec. \ref{sec:results}, including a map of the magnitude limit of 99\% completeness in the top panel of Fig. \ref{fig:completenessmaps}.
In the second part of this paper, we extend our selection function to account for crowding. {\it Gaia}\xspace can only \edits{simultaneously track} $1,050,000\;\mathrm{sources}\;\mathrm{deg}^{-2}$ \citep{Gaia2016} and will choose to track brighter stars ahead of fainter stars, and thus crowding acts as a second-order effect which limits the completeness of {\it Gaia}\xspace with respect to faint stars in dense regions of the sky. The effect of crowding is vitally important in the Galactic bulge and the Large and Small Magellanic Clouds. We describe the adaptation of our selection function to account for crowding in Sec. \ref{sec:crowding}, and show a revised map of the magnitude limit of 99\% completeness in the bottom panel of Fig. \ref{fig:completenessmaps}.
\edits{We put our results in the context of other attempts to map the completeness of {\it Gaia}\xspace DR2 in Sec. \ref{sec:discussion}, and also discuss how the methodology presented in this paper could be extended to the parallax and proper motion, colour photometry, variable star and radial velocity subsets of {\it Gaia}\xspace DR2. We end by presenting our new \textsc{Python} package \textsc{selectionfunctions} which will allow the reader to easily incorporate our selection functions in their own work.}
\section{Methodology}
\label{sec:methodology}
\begin{figure*}
\centering
\includegraphics[width=1.\linewidth,trim=0 0 0 0, clip]{./figs/cutjustification.pdf}
\caption{The distribution of all sources in {\it Gaia}\xspace DR2 (blue) with the cuts (red, dashed) used to select which sources were included in {\it Gaia}\xspace DR2 \citep{Lindegren2018}.}
\label{fig:cutjustification}
\end{figure*}
This work tackles the selection function of the {\it Gaia}\xspace DR2 source catalogue, which is the catalogue of 1,692,919,135 sources detected by {\it Gaia}\xspace that satisfied the basic astrometric quality cuts. Sources were included if their five parameter astrometric solution (position, parallax and proper motion) satisfied the quality cuts given by Eq.~11 of \citet{Lindegren2018} or -- failing that -- if their two parameter astrometric solution (position only) satisfied the quality cuts
\begin{equation}
\begin{cases}
\textsc{astrometric\_matched\_observations} &\geq 5, \\
\textsc{astrometric\_excess\_noise} &< 20\;\mathrm{mas}, \\
\sigma_{\mathrm{pos,max}} &< 100\;\mathrm{mas},
\end{cases}
\label{eq:lindegrentwo}
\end{equation}
where \textsc{astrometric\_matched\_observations} is the number of field-of-view transits that contributed measurements to the astrometric solution, \textsc{astrometric\_excess\_noise} quantifies the goodness-of-fit of the solution, and $\sigma_{\mathrm{pos,max}}$ is the semi-major axis of the position uncertainty ellipse \citep{Lindegren2018}. In practise, if a source has a valid five parameter astrometric solution then the two parameter astrometric solution would also have been valid, and so we can consider the cuts in Eq. \ref{eq:lindegrentwo} to define the selection of sources for the {\it Gaia}\xspace DR2 source catalogue. \edits{We show in Fig. \ref{fig:cutjustification} histograms of \textsc{astrometric\_matched\_observations}, \textsc{astrometric\_excess\_noise} and $\sigma_{\mathrm{pos,max}}$ for all sources in {\it Gaia}\xspace DR2.} While the cuts in the latter two of these occur far out in the tail of the distribution, the cut in \textsc{astrometric\_matched\_observations} occurs at the peak of the distribution. \citet{Lindegren2018} notes that most of the sources excluded by the cuts in Eq. \ref{eq:lindegrentwo} are spurious, and we therefore conclude that for genuine sources the only effective cut is $\textsc{astrometric\_matched\_observations}\geq5$. \edits{There is one additional requirement\footnote{\url{https://gea.esac.esa.int/archive/documentation/GDR2/Catalogue_consolidation/chap_cu9cva/sec_cu9cva_consolidation/ssec_cu9cva_consolidation_ingestion.html}} that there must have been at least ten $\textsc{phot\_g\_n\_obs}$ ($G$ CCD transits), however this cut is much weaker than that on \textsc{astrometric\_matched\_observations} because each focal plane transit can result in as many as nine $\textsc{phot\_g\_n\_obs}$.}
The approach we have developed to calculate the {\it Gaia}\xspace DR2 selection function is entirely novel, and so to aid clarity we give brief definitions of the key terminology here. A \textit{source} is any object which {\it Gaia}\xspace would report as a single entry in the DR2 catalogue (used interchangeably with \textit{star}). An \textit{observation} of a source is a single occasion on which a source could be seen by {\it Gaia}\xspace, because it is within {\it Gaia}\xspace's field-of-view. A \textit{detection} is an observation during which {\it Gaia}\xspace notices the presence of the source and obtains a measurement that subsequently contributes to the \edits{astrometric solution in} the {\it Gaia}\xspace DR2 catalogue. \edits{Observations only result in detections if the source is acquired by the SkyMapper CCD and selected to have a window tracked on-board, the source is confirmed by the AF1 CCD, the telemetry is successfully sent to the ground and processed by DPAC, and the measurements pass all the minimum astrometric and photometric quality criteria to make it to the Gaia DR2 catalogue.} The number of detections is given in {\it Gaia}\xspace DR2 as \textsc{astrometric\_matched\_observations} and is less than or equal to the number of observations. The \textit{detection probability} is the probability that {\it Gaia}\xspace will detect a source on a given observation. The \textit{selection function} is the probability that a source is detected on at least five observations of that source. The core assumption of this work is that a source having been detected at least five times is the sole requirement for inclusion in {\it Gaia}\xspace DR2.
\subsection{Flipping biased coins}
\label{sec:bias}
Our methodology is rooted in a simple problem that is often used to introduce Bayesian statistics: how do you determine the bias of a weighted coin? Suppose we flip a coin $n$ times and observe $k$ heads and $n-k$ tails. If the coin is fair, then the probability of observing $k$ heads is given by
\begin{equation}
\operatorname{P}(k|n) = \binom{n}{k}\left(\frac{1}{2}\right)^{n}, \label{eq:faircoin}
\end{equation}
because there are $\binom{n}{k}$ ways to pick which coin flips resulted in heads and on each flip there is a $\frac{1}{2}$ chance of either heads or tails. If the coin is biased, such that the probability of heads on any one flip is given by $\theta$, then the probability of $k$ heads is given by
\begin{equation}
\operatorname{P}(k|n,\theta) = \binom{n}{k}\theta^k(1-\theta)^{n-k}, \label{eq:binomial}
\end{equation}
which simplifies to Eq. \ref{eq:faircoin} in the case $\theta=\frac{1}{2}$. This is the probability mass function of a Binomial random variable.
If we do not know the precise bias of the coin, then we will want to quantify our knowledge in terms of a prior $\operatorname{P}(\theta)$, which gives our a priori belief that $\theta$ takes each possible value between 0 and 1. A commonly-used choice is the Beta distribution
\begin{equation}
\operatorname{P}(\theta|\alpha,\beta) = \frac{\theta^{\alpha-1}(1-\theta)^{\beta-1}}{\operatorname{B}(\alpha,\beta)},
\end{equation}
where the denominator is the beta function\footnote{The Beta distribution, the beta function and the $\beta$ variable are all separate entities.}
\begin{equation}
\operatorname{B}(\alpha,\beta) = \int_0^1 x^{\alpha-1}(1-x)^{\beta-1}\mathrm{d}x=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}.
\end{equation}
The Beta distribution is the conjugate prior\footnote{A distribution is a conjugate prior of a likelihood function if the posterior distribution belongs to the same probability distribution family as the prior distribution.} of the Binomial distribution, meaning that if your prior belief is that $\theta\sim\operatorname{Beta}(\alpha,\beta)$ and you subsequently observe $k$ heads from $n$ flips, then your posterior belief should be that $\theta\sim\operatorname{Beta}(\alpha+k,\beta+n-k)$. This pairing is known as the Beta-Binomial model. The probability of observing $k$ heads from $n$ flips is then, after marginalising over $\theta$,
\begin{equation}
\operatorname{P}(k|n,\alpha,\beta)=\binom{n}{k}\frac{\operatorname{B}(\alpha+k,\beta+n-k)}{\operatorname{B}(\alpha,\beta)}.
\end{equation}
If $\alpha=1$ and $\beta=1$ then the Beta prior is equivalent to the Uniform distribution over $(0,1)$, which corresponds to having no knowledge about the bias of the coin.
In Appendix \ref{sec:betabinomial} we have illustrated the range of possible shapes of the Binomial and Beta distributions, which may be of interest to readers unfamiliar with these distributions.
If we had multiple coins $i$, then we could either assume that they all have the same bias $\theta\sim\operatorname{Beta}(\alpha,\beta)$, or that they each have their own bias $\theta_i\sim\operatorname{Beta}(\alpha,\beta)$. In the former case the Beta distribution quantifies our uncertainty on the true bias of the coins, while in the latter case it quantifies the intrinsic spread in the biases.
The connection between this problem and the problem of the {\it Gaia}\xspace selection function is immediate: we are attempting to quantify the probability $\theta$ that {\it Gaia}\xspace detects a star $k$ times out of $n$ observations, which is the same statistical problem as quantifying the probability $\theta$ that a coin comes up heads $k$ times out of $n$ flips. However, {\it Gaia}\xspace DR2 only reports sources if $k\geq5$, which is analogous to flipping a number of coins multiple times but discarding any data from the coins where the number of heads was fewer than five. An additional complication is that the detection probability is likely to depend on the properties of the star, which requires us to stretch our analogy to breaking point in the form of the following fictitious story.
Suppose that we suspect that the Royal Mint has been producing biased pennies for the last \edits{fifty} years. Even worse, the bias of every coin is different, and the distribution of those biases appears to be changing from year to year. To investigate this troubling phenomenon, we establish the Penny Processing and Analysis Consortium and ask volunteers all over Britain to gather their pennies, flip them until they get bored, and send us a postcard with one line for each coin giving the number of flips, the number of heads and the year of minting. After the British public has dutifully flipped 1,692,919,135 pennies\footnote{A total of 22,820,373,346 pennies had been minted up until the 14$^{\mathrm{th}}$ October 2019 (\url{https://www.royalmint.com/currency/uk-currency/mintages/1-penny/}).} a reported total of 48,421,126,164 times, we are dismayed to discover that none of the coins in our dataset had come up heads less than five times. This seems unlikely to be true, given that 52,871,273 of the coins had shown heads exactly five times. It emerges that one of our researchers had carelessly stated during a TV interview that `we expect very few coins to have come up heads less than five times', and the public had taken this to heart, discarding the data about a coin if there were fewer than five heads. It now falls to us to infer the changing distribution of the penny biases from year-to-year, accounting for the missing data.
This story describes the exact same statistical problem that we have solved in this work. In place of the year of minting we binned the stars by their brightness (and later their neighbouring stellar density), and computed the distribution of detection probabilities in each of these bins, subject to the data truncation that none of the reported stars had fewer than five detections.
\subsection{Statistical framework}
\label{sec:framework}
Suppose that for each of the 1,692,919,135 sources in {\it Gaia}\xspace DR2 (labelled $i$) we have the number of detections $k_i$, the number of times that that source crossed the field-of-view $n_i$ and the $G_i$ magnitude. At the time of each observation there will be some probability that {\it Gaia}\xspace will successfully detect the star. Our first simplifying assumption is that the probability of detection is the same at every observation and we term this probability $\theta_i$. It is possible that the probability could vary with time (for instance, the amount of scattered light from the frayed edges of the solar shield will be a function of the phase of {\it Gaia}\xspace's orbit) and thus our $\theta_i$ will be an approximate average of the true probabilities. We discuss the difficulties of weakening this assumption in Sec. \ref{sec:poissonbinomial}.
We assumed that the only selection for a star to be included in {\it Gaia}\xspace DR2 was that $k_i\geq5$. If that selection had not been applied, then we could have modelled $k_i$ as a Binomial random variable with $n_i$ observations and probability of success $\theta_i$, and thus the probability mass function would be
\begin{equation}
P(k_i|n_i,\theta_i) = \begin{cases}
\binom{n_i}{k_i} \theta_i^{k_i} (1-\theta_i)^{n_i-k_i} &\mathrm{for}\;k_i \geq 0, \\
0 &\mathrm{otherwise}.\end{cases}
\end{equation}
The selection $k_i\geq5$ acts to adjust the probability mass function:
\begin{equation}
P(k_i|n_i,\theta_i,S) = \begin{cases}
\frac{1}{P(k\geq5|n_i,\theta_i)}\binom{n_i}{k_i} \theta_i^{k_i} (1-\theta_i)^{n_i-k_i} &\mathrm{for}\;k_i \geq 5, \\
0 &\mathrm{otherwise},\end{cases} \label{eq:pmf}
\end{equation}
where $P(k\geq5|n_i,\theta_i) = \operatorname{I}_{\theta_i}(5,n_i-4)$ is an \edits{incomplete beta function},
\begin{equation}
\operatorname{I}_{\theta}(\alpha,\beta) = \int_0^{\theta} x^{\alpha-1}(1-x)^{\beta-1}\mathrm{d}x,
\end{equation}
and is the survival function of the Binomial distribution.
We further assumed that the probability $\theta_i$ will depend primarily on the brightness of the star $G$ (although in Sec. \ref{sec:crowding} we additionally consider the neighbouring source density) \edits{and considered two different models of that dependency}. Our simple model (hereafter Model T) was to assume $\theta_i$ is directly a function of $G_i$, and thus that all stars with the same magnitude $G$ have the same detection probability $\theta = T(G)$. Our more realistic model (hereafter Model AB) was to assume that the $\theta$ for each star was drawn from a Beta distribution with parameters $A(G)$ and $B(G)$ that are each functions of the brightness. Model AB can account for stars of the same magnitude having a distribution of detection probabilities \edits{and allows that distribution to change with magnitude.}
We opted to model the functions $T(G)$, $A(G)$ and $B(G)$ as piecewise-constant over decimag bins in $G$ from $1.7$ to $23.5$, where the value of each function in each of the 218 bins $j$ is given by the hyper-parameters $T_j$, $A_j$ and $B_j$. We additionally require that $0<T(G)<1$, $A(G)>0$ and $B(G)>0$ due to the constraints on the parameters of the Binomial and Beta distributions. In practice, we further restricted the domains to $10^{-1}<A(G),B(G)<10^{4}$ to allow us to pre-compute the numerically expensive incomplete beta function appearing in Eq. \ref{eq:pmf}, but note that our posteriors only run up against these boundaries in bins with very few stars. We have illustrated the relationship between these parameters and the observables through the plate diagrams in Fig. \ref{fig:plate}.
\begin{figure}
\centering
\includegraphics[width=1.\linewidth,trim=0 10 0 10, clip]{./figs/plateT.pdf}
(a) Model T
\includegraphics[width=1.\linewidth,trim=0 10 0 0, clip]{./figs/plateAB.pdf}
(b) Model AB
\caption{Plate diagrams for our two models. The stars in {\it Gaia}\xspace DR2 are divided into bins by their $G$ magnitude and their number of detections $k$ is assumed to be Binomial-distributed with number of observations $n$ and probability of detection $\theta$. \textbf{Top:} In Model T the probability of detection is assumed to be the same $T_j$ for all the stars in each bin. \textbf{Bottom:} In Model AB the the probability of detection of each star in the bin is assumed to have been drawn from a Beta distribution with parameters $A_j$ and $B_j$.}
\label{fig:plate}
\end{figure}
\subsection{Computing the number of observations}
\label{sec:n}
\begin{figure*}
\centering
\includegraphics[width=1.\linewidth,trim=70 110 35 110, clip]{./figs/map_4096_lowres.pdf}
\caption{A \edits{Galactic} HEALPix map of the estimated number of times that {\it Gaia}\xspace looked at each location on the sky. The graticule marks out lines of longitude and latitude in the Ecliptic coordinate frame, which is the native frame of the {\it Gaia}\xspace scanning law. The colourmap has been histogram-equalised to increase the contrast. This figure is similar to Fig. 2 of \citet{Boubert2020}, but \edits{with the observations during periods that were excluded from the astrometric data-taking removed.}}
\label{fig:map}
\end{figure*}
Our methodology requires that we know the number of times $n_i$ that each of the stars in {\it Gaia}\xspace DR2 was observed, i.e. transited across either the preceding or following {\it Gaia}\xspace field-of-view (FoV). \edits{In Paper I of this series \citep{Boubert2020}, we precisely determined the {\it Gaia}\xspace scanning law over the period of DR2 using the recently published DR2 nominal scanning law\footnote{\url{https://www.cosmos.esa.int/web/gaia/scanning-law-pointings}} together with the detection times of each of the 550,737 variable sources with DR2 light-curves.} There are stretches of time during the 22 months covered by DR2 when {\it Gaia}\xspace was not able to obtain astrometric measurements or the astrometric measurements that were taken were erroneous, for example, due to the Ecliptic Pole Scanning Law, decontamination procedures, micro-meteroid impacts or station-keeping maneuvers \citep{Lindegren2018}, and it is vital to account for these gaps. \edits{We use the astrometric data gaps recently provided by the {\it Gaia}\xspace DPAC\footnote{\url{https://www.cosmos.esa.int/web/gaia/dr2-data-gaps}} and refer the interested reader to Paper I for more details.} We use the same methodology as detailed in Paper I to predict $n_i$ for all of the individual sources in {\it Gaia}\xspace DR2.
\edits{A crucial requirement of our methodology is that the $G$ magnitude reported in the {\it Gaia}\xspace catalogue is accurate, because the proportion of observations that result in detections will be much lower for fainter sources. If a truly bright source was reported to be faint, then the higher detection efficiency could greatly bias our posterior on the selection function. In Paper I we identified that almost all of the faintest stars in {\it Gaia}\xspace DR2 have spurious magnitudes resulting from a miscalibration due to a thunderstorm over Madrid, with most of the remainder being attributed to a similar second miscalibration event. We therefore do not consider the 72,785,162 sources with a predicted observation in the time periods $\mathrm{OBMT}=1388{-}1392\;\mathrm{rev}$ and $\mathrm{OBMT}=2211{-}2215\;\mathrm{rev}$ in the remainder of this work.}
When using the results of this work to map the selection function on the sky, we assume that the number of observations of a source is approximately equal to the number of observations of the centre of the nearest pixel of an \edits{\textsc{nside}=4096}, Equatorial, nested HEALPix map, which we show in Fig. \ref{fig:map}. This map reveals the intricate overlaps between the successive precessing scans of {\it Gaia}\xspace \edits{, which we will show in later sections result in equally intricate variations in the completeness of {\it Gaia}\xspace DR2 across the sky.}
\subsection{Implementation of Bayesian model}
In this section we detail the implementation of the models described in Sec. \ref{sec:framework}. We first binned the sources in {\it Gaia}\xspace DR2 by their $G$ magnitude, by their number of detections $k$ and by the number of observations $n$ we predicted in the previous section. Paradoxically, there were \edits{619,272} sources where their predicted $n$ was less than $k$\edits{, likely due to a combination of small uncertainties in our prediction of the numbers of observation and spurious duplicate detections of the bright stars (as discussed in Paper I). Rather than discard these sources, in Appendix \ref{sec:deconvolution} we deconvolve the distribution of all sources in $(n,k)$ for each slice in $G$ to account for these processes. We have verified that the results in the remainder of this paper are not substantially changed if we instead discard the sources with $n<k$ and leave the distribution of the other stars unchanged.} We note that the models shown in Fig. \ref{fig:plate} are entirely independent between the magnitude bins, and so we could calculate the posterior model in each bin separately. Within each magnitude bin the likelihood is only a function of $n$, $k$ and the hyper-parameters of the model, and thus we can calculate the likelihood once for each combination of $n$ and $k$ and then multiply the log-likelihood by the number of sources with that $n$ and $k$. For each of the models and each of the bins we use the affine invariant Markov chain Monte Carlo (MCMC) ensemble sampler \citep{Goodman2010} implemented in \textsc{emcee} \citep{Foreman2013,Foreman2019} to draw samples from the posterior over the hyper-parameters. In each case we used 32 walkers, drew samples until the chain was at least 50 times longer than the chain's autocorrelation time $\tau$, and discarded the first $5\lceil\tau\rceil$ samples as `burn-in'.
\subsubsection{Likelihood and priors for Model T}
The likelihood under Model T of $k$ \edits{reported detections} for a source in magnitude bin $j$ with $n$ \edits{predicted observations} is
\begin{equation}
P(k|n,T_j) = \begin{cases}
\frac{1}{\operatorname{I}_{T_j}(5,n-4)}\binom{n}{k} T_j^k (1-T_j)^{n-k} &\mathrm{for}\;k \geq 5, \\
0 &\mathrm{otherwise}.\end{cases} \label{eq:modelT}
\end{equation}
We assumed a uniform prior over $(0,1)$ for $T_j$ in each magnitude bin. The incomplete beta function is computationally-expensive to evaluate and hence we opted to pre-compute the value of $\operatorname{I}_x(5,n-4)$ at 10,000 equally spaced points $x\in(0,1)$ for each possible $n$. We then interpolated this grid with a cubic spline in place of explicitly evaluating the incomplete beta at each MCMC step.
If the selection $k\geq5$ is not applied then Model T would be identical to the Beta-Binomial model described in Sec. \ref{sec:bias}. Suppose we have a biased coin with a prior on the probability of success $p \sim \operatorname{Beta}(\alpha,\beta)$, where $\alpha$ and $\beta$ are fixed. We then conduct $N$ trials and in each trial $i$ conduct $n_i$ flips and observe $k_i$ successes with $k_i\sim\operatorname{Bin}(n_i,p)$. The posterior is simply $p\sim\operatorname{Beta}(\alpha+\Sigma k_i,\beta + \Sigma [n_i-k_i])$. We expect that {\it Gaia}\xspace will detect sources of intermediate brightness on almost every transit, and thus over this range in magnitude the Beta-Binomial model will give a close approximation to Model T because the incomplete beta function will be close to 1.
\subsubsection{Likelihood and priors for Model AB}
The Model AB is hierarchical as there is an additional parameter $\theta$ for each source that is drawn from a Beta distribution with hyper-parameters $A_j$ and $B_j$:
\begin{equation}
P(\theta|A_j,B_j) = \frac{\theta^{A_j-1}(1-\theta)^{B_j-1}}{\operatorname{B}(A_j,B_j)}.
\end{equation}
The likelihood under Model AB of $k$ detections for a source in magnitude bin $j$ with $n$ transits is then
\begin{equation}
P(k|n,\theta) = \begin{cases}
\frac{1}{\operatorname{I}_{\theta}(5,n-4)}\binom{n}{k} \theta^k (1-\theta)^{n-k} &\mathrm{for}\;k \geq 5, \\
0 &\mathrm{otherwise}.\end{cases}
\end{equation}
Obtaining the posterior of a Bayesian model with tens of millions of parameters is not \edits{generally} feasible and so we were forced to marginalise over the $\theta$ parameter of each source. For each possible combination of $n$ and $k$ we numerically evaluated the integral
\begin{equation}
J_{n,k}(A,B) = \int_0^1 \frac{\theta^{A+n-1}(1-\theta)^{B+n-k-1}}{\operatorname{I}_{\theta}(5,n-4)}\mathrm{d}\theta
\end{equation}
over a grid in $(A,B)$ with logarithmic spacings between $10^{-1}$ and $10^{4}$ in each direction. We then implemented a bivariate cubic spline in $(\log_{10}A,\log_{10}B)$ for each $n$ and $k$. The likelihood is thus simplified to
\begin{equation}
P(k|n,A_j,B_j) = \begin{cases}
\binom{n}{k}\frac{J_{n,k}(A_j,B_j)}{\operatorname{B}(A_j,B_j)} &\mathrm{for}\;k \geq 5, \\
0 &\mathrm{otherwise}.\end{cases} \label{eq:modelAB}
\end{equation}
We adopted \edits{log-}uniform priors for $A_j$ and $B_j$ over $(10^{-1},10^{4})$ to ensure that the MCMC walkers do not explore outside our pre-computed grid. The limits of this grid were determined through experimentation and we can confirm that the posteriors on $A_j$ and $B_j$ only extend to the limits in the case of bins with small numbers of sources (and thus broad unconstrained posteriors).
If the selection $k\geq5$ is not applied then the integral in the likelihood has the closed form $J_{n,k}(A,B)=\operatorname{B}(A+n,B+n-k)$. Unlike in the Beta-Binomial case discussed in the previous section this does not lead to an analytic posterior, but this simpler model does provide a valuable sanity check on Model AB.
\section{Results}
\label{sec:results}
\subsection{Posterior selection functions}
We computed the posteriors of both models with and without the $k\geq5$ selection, and show the median with \edits{$1\sigma$ error-bars} on $T$, $A$ and $B$ as a function of magnitude in Fig. \ref{fig:modelposterior}. The \edits{uncertainties} are large at both the bright ($G<8$) and faint ($G>22$) ends due to the small number of {\it Gaia}\xspace DR2 sources per decimag bin at these magnitudes. The difference between the posteriors with and without the $k\geq5$ selection is only apparent when the posterior indicates that the typical detection probability is less than 20\% ($T<0.2$ or $A/(A+B)<0.2$), because the odds of a star having $k<5$ are otherwise vanishingly small, given that that the average number of observations is \edits{28}.
\edits{As mentioned in Sec. \ref{sec:n}, we have not included any of the sources observed during the miscalibrated periods identified in Paper I and thus have removed almost all of the sources fainter than $G=22.1$, causing the large uncertainty in all three parameters at the faint end. There are likely to be some remaining miscalibrated sources which may bias our results, given that there are still sources reported to be as faint as $G=23.1$ in our sample. We note, for example, that $T(G)$ is smoothly declining over $20<G<21.7$, but then begins to rise again. We opt to exclude all datapoints for magnitudes fainter than $G=21.3$ from our analysis, motivated by empirical investigations of the run of each parameter with $G$ magnitude.}
\edits{We chose to model $T$, $A$ and $B$ as piece-wise to expedite the Bayesian inference by making it independent between each magnitude bin, but this is of course only an approximation. It would have been preferable to model the continuous run of $T$, $A$ and $B$ with $G$ across all magnitudes simultaneously. This would also result in a more precise estimation of $T$, $A$ and $B$, because their value at a particular $G$ would be informed by their value at neighbouring magnitudes. We opted to model the data-points shown in Fig. \ref{fig:modelposterior} as independent Gaussian Processes with a squared-exponential kernel. This required us to transform the $T$ data-points from $(0,1)$ to $(-\infty,+\infty)$ through a logit transform and the $A$ and $B$ data-points from $(0,+\infty)$ to $(-\infty,+\infty)$ through a log$_{10}$ transform, with the uncertainties being propagated by transforming the 16\% and 84\% percentiles and then averaging their distance from the transformed magnitudes. Through experimentation we fixed the length-scale of the Gaussian Processes at $0.3\;\mathrm{mag}$ and fixed the variances at $1.0$ for $T$ and $0.3$ for $A$ and $B$ (though note that these variances are in the logit and log$_{10}$ transformed spaces). An advantage of using Gaussian Processes is that we can use them to extrapolate at the bright and faint end. We chose the means of the Gaussian Processes to be -10, 0 and 4 for $T$, $A$ and $B$ respectively in order to ensure that the posteriors on the detection probability tended towards zero away from regions constrained by data. We evaluated the conditional distribution of these Gaussian Processes at 501 points uniformly spaced by $0.05\;\mathrm{mag}$ over $0\leq G \leq25$ and applied the reverse of the logit and log$_{10}$ transforms to obtain the median and $1$ and $2\sigma$ regions shown in Fig. \ref{fig:modelposterior}. We use interpolation of these points to obtain the results in the remainder of this work.}
The top panel of Fig. \ref{fig:completeness} shows the posterior distribution of the detection probability $\theta$ under Model AB\edits{, where we have taken the mean conditional value of the Gaussian Processes over $A(G)$ and $B(G)$ at each magnitude and computed the percentiles of the detection probability from the corresponding $\operatorname{Beta}(A,B)$ distribution. The contours shown in Fig. \ref{fig:completeness} thus only illustrate the spread of detection probabilities under our most likely model at each magnitude, and do not incorporate any uncertainty in the model parameters.}
\begin{figure}
\centering
\includegraphics[width=1.\linewidth,trim=0 0 0 0, clip]{./figs/model_t_posterior.pdf}
(a) Posterior on Model T
\includegraphics[width=1.\linewidth,trim=0 0 0 0, clip]{./figs/model_ab_posterior.pdf}
(b) Posterior on Model AB
\caption{The posteriors on our two models of the probability that {\it Gaia}\xspace detects a source each time the source crosses the field-of-view. In both panels we show the \edits{median value with the $1\sigma$ error-bars. We also show our post-hoc Gaussian Process fits which continuously interpolate these data points and allow us to extrapolate at the bright and faint end. The faint data points to the right of the dashed line at $G=21.3$ are those which are possibly biased due to sources with miscalibrated magnitudes and which we discarded before fitting the Gaussian Processes.} \textbf{Top:} In Model T, the detection probability is assumed to be a deterministic function of the magnitude of the source. \textbf{Bottom:} In Model AB, the detection probability of a source by {\it Gaia}\xspace is drawn from a Beta distribution whose parameters are deterministic functions of the magnitude of the star. To allow for comparison with Model T, we show the resulting posterior on the detection probability in the top panel of Fig. \ref{fig:completeness}.}
\label{fig:modelposterior}
\end{figure}
\subsection{Completeness as a function of magnitude}
\label{sec:completeness}
\begin{figure*}
\centering
\includegraphics[width=1.\linewidth,trim=0 0 0 0, clip]{./figs/completeness_map_lowres.pdf}
\caption{Our Model AB predicts the distribution of probabilities that a source is detected by {\it Gaia}\xspace as a function of magnitude, which we illustrate in the top panel with the median and $1$ and $2\sigma$ regions. These distributions can be used to compute the completeness of {\it Gaia}\xspace DR2 with respect to sources which are of magnitude $G$ and which have received $n$ observations. In the bottom panel, we show a grid of \edits{Galactic} completeness maps at the magnitudes labelled in the top panel. An \href{https://www.gaiaverse.space/publications/paper-ii}{interactive version} of this figure is available online (please note that the static HTML behind the webpage will take up to thirty seconds to download and decompress). \edits{We stress that the results shown here do not account for the deleterious effects of crowding on {\it Gaia}\xspace's completeness (as considered further in Sec. \ref{sec:crowding}) and so should be treated as a biased indication of how the completeness changes with source magnitude and the number of observations.}}
\label{fig:completeness}
\end{figure*}
Our inferred detection probabilities can be used to predict the completeness of {\it Gaia}\xspace DR2 as a function of magnitude and number of observations. The completeness is simply the fraction of stars with magnitude $G$ that were observed $n$ times and were detected on at least five occasions. We focused our attention on the more \edits{general} Model AB, where the probability of detection $\theta$ of each source is assumed to be Beta-distributed with parameters $A(G)$ and $B(G)$ which are each functions of magnitude. The completeness \edits{increases} with the number of observations because more observations equates to more occasions on which a source can be detected. We demonstrated in Fig. \ref{fig:map} that the number of observations that a source receives strongly varies with position on the sky, and thus the completeness of {\it Gaia}\xspace DR2 at a given magnitude can change from 0\% to 100\% depending \edits{upon} where the source is on the sky. Predicting the completeness from Model AB is not trivial and we outline the procedure we followed in Appendix \ref{sec:completenessAB}.
In the bottom panels of Fig. \ref{fig:completeness}, we show maps of the fraction of stars that would have been detected by {\it Gaia}\xspace, at \edits{selected} magnitudes where the distribution of detection probabilities appears to be changing. As expected, the high detection probability over $3<G<20$ implies that {\it Gaia}\xspace is essentially complete over the entire sky at these magnitudes. The dip between $10<G<12$ has a negligible effect on the completeness. The completeness does drop at the extreme bright end $G<3$ and at the faint end, falling from $100\%$ to $0\%$ over $20.0<G<21.5$.
The other features in the top panel of Fig. \ref{fig:completeness} can be explained by \edits{technical details of the {\it Gaia}\xspace instrumentation}. The decline at magnitudes brighter than (d) is due to saturation for $G<6$. \edits{The treatment of bright stars (particularly of very bright stars $G<3$) was preliminary in {\it Gaia}\xspace DR2 and will be improved in future data releases \citep{Gaia2018}, likely leading to a boost in completeness.} The more clearly resolved dip that occurs over the range $10<G<12$ is due to the ``different CCD gates'' \citep{Lindegren2018b}, which we discuss in more detail in Appendix \ref{sec:gating}. This happens well before the switch from $2\times2$ pixel binning to $4\times4$ pixel binning at $G=13$ \citep{Gaia2016} \edits{and so is likely to be unrelated}. The existence of features like these was always likely given the complexity of the {\it Gaia}\xspace~\edits{instrumentation}.
An alternative way to visualise the completeness of {\it Gaia}\xspace is to compute -- for each pixel on the sky -- the faintest magnitude at which {\it Gaia}\xspace will still see 99\% of the stars at that magnitude. We describe our procedure for computing that magnitude limit in Appendix \ref{sec:completenessAB}. The sky map of the result is shown in the top panel of Fig. \ref{fig:completenessmaps}. We can see that {\it Gaia}\xspace DR2 is complete down to $G=20$ over almost the entire sky, but in parts of the sky with many scans (the caustics at Ecliptic latitudes of $\pm45^{\circ}$ in particular) the magnitude limit is fainter than $G=21$.
\section{Effect of crowding on our results}
\label{sec:crowding}
The probability that a source is detected by {\it Gaia}\xspace is lower in highly crowded regions of the sky, because {\it Gaia}\xspace cannot assign windows to every source on every observation \edits{and some measurements are deleted on-board due a lack of bandwidth to downlink them.} This drop in detection probability decreases the number of sources that have at least five detections, thus causing the completeness of {\it Gaia}\xspace DR2 to drop in crowded regions. The effective crowding limit for the $G$ photometry and astrometry is $1,050,000\;\mathrm{sources}\;\mathrm{deg}^{-2}$ \citep{Gaia2016}. When {\it Gaia}\xspace has the option to assign a window to one of two stars, {\it Gaia}\xspace will always give the brighter star higher priority, and hence crowding acts to decrease the completeness of faint stars relative to bright stars. The windows required for BP/RP photometry and RVS spectroscopy are larger and so have lower source density crowding limits of $750,000\;\mathrm{sources}\;\mathrm{deg}^{-2}$ and $35,000\;\mathrm{sources}\;\mathrm{deg}^{-2}$ respectively, and therefore the effects of crowding will be even more vital to include in these subsets.
In the previous section we ignored the effect of crowding, with two likely consequences. First, the mean detection probability at faint magnitudes will be dragged to lower values by the stars which are missing observations due to crowding, and a portion of the detection probability spread in Model AB can be attributed to crowding changing the effective detection probabilities in regions with high source densities. Second, our selection functions in the previous section are averaged across the entire sky and so should not be applied specifically to regions with only a small range in source density, because the true selection function will deviate significantly. Given the often frustrating correlation between the stellar density of a region and how astrophysically interesting that region is to study, we decided to investigate the effect of crowding on our results.
We attempted to modify our method to fully account for crowding, but were unable to. The major difficulty we encountered is that the effect of crowding should depend on the true density of sources, but we only have access to the observed density of sources in {\it Gaia}\xspace DR2. A further complication is that crowding should not be a function solely of the density of sources, but rather of the distribution of those sources with magnitude. A proper accounting of crowding is thus beyond the scope of this paper. Nevertheless, in this section, we demonstrate the effect of crowding by fitting our Model T and AB selection functions to ten subsets of {\it Gaia}\xspace DR2 that are split by the source density in their vicinity.
We split the sky into an \textsc{nside} = 128 Equatorial HEALPix grid, computed the number of {\it Gaia}\xspace DR2 sources within each pixel, and divided those counts by the $0.21\;\mathrm{deg}^2$ pixel area to obtain the average source density in each pixel. We then grouped the pixels into regions by their source density such that each region contained an equal number of sources, and we show these regions in Fig. \ref{fig:skydensity}. Note that these regions are not contiguous on the sky and that the divisions in source density are irregularly spaced (see the colour bar of Fig. \ref{fig:skydensity} for the divisions). The maximum source density observed in one of these cells of $1,138,520\;\mathrm{sources}\;\mathrm{deg}^{-2}$ is slightly higher than the maximum crowding limit of {\it Gaia}\xspace, because the sources in a field which are assigned windows changes based on the scan angle.
\begin{figure*}
\centering
\includegraphics[width=1.\linewidth,trim=55 80 0 90, clip]{./figs/skydensitymap.pdf}
\caption{An Equatorial HEALPix map rotated to Galactic coordinates discretely coloured by the number of {\it Gaia}\xspace DR2 sources per square degree. Note that the edges of the bins in source density were chosen to ensure that an equal number of stars fall into each of the ten bins.}
\label{fig:skydensity}
\end{figure*}
We modelled the detection probability separately with Models T and AB for each of these pseudo-isodensity regions over the magnitude range \edits{$G < 16$. Below this range} we assumed that the selection function is independent of source density, and thus equal to the selection functions from Sec. \ref{sec:methodology}\edits{, because we should not expect bright stars $G<16$ to be strongly affected by crowding.} We verified the validity of this assumption by fitting Model T to the five least dense and the five most dense regions over the entire magnitude range and comparing the results. The practical motivation for restricting the magnitude range where we fit each region separately is that the dense regions have proportionally fewer bright stars and so the selection function at the bright end is \edits{more poorly} constrained.
\edits{The median of} our posterior on the detection probability for both models is illustrated in Fig. \ref{fig:crowding}\edits{, where we have discarded the data-points for $G>21.3$ and fitted Gaussian Processes as we did in Sec. \ref{sec:results}.} In Model T, the detection probability consistently decreases with increasing source density, and thus we are able to interpret much of the detection probability spread seen in Fig. \ref{fig:completeness} as due to the varying effect of crowding. We only plotted the Model AB posterior for the least and most dense regions to avoid congesting the figure, but these two distributions bracket the distributions of the other eight regions. The effect of crowding is hugely important, with the $1\sigma$ intervals of the distributions of detection probability in the least and most dense regions being entirely disjoint at $G=20$. The posterior for the least dense region is approximately flat until $G=20$ -- in contrast to the posterior shown in Fig. \ref{fig:modelposterior} -- and is thus representative of the true run of detection probability with magnitude when crowding is not an issue. The posterior for the most dense region exhibits a bump at around $G=18.5$ and we conjecture that this is due to the Red Clump stars in the LMC; this population contains a large number of stars at roughly the same apparent magnitude ($G\approx18.5$) at the same location on the sky, which thus all receive a similar number of observations and so potentially bias the inference at this magnitude. \edits{That the detection probability curves converge on the same behaviour at the faint end is due to the photon-limit becoming more dominant than crowding as a cause of missing detections, although we note that the detection probability of the curve in the densest regions should be further below that in the sparsest regions over the magnitude interval $21.0<G<21.5$. It is possible that there are either spurious sources or sources with miscalibrated magnitudes biasing our determination of the detection probability in this regime that have survived our cuts.}
\begin{figure*}
\centering
\includegraphics[width=1.\linewidth,trim=0 0 0 0, clip]{./figs/skydensitydetectionprobability.pdf}
\caption{\edits{Our posterior on the detection probability at the median parameters of our} Models T (top) and AB (bottom) when the effect of crowding is included. \edits{In Model T we assume the detection probability is simply a changing function of magnitude, whilst in Model AB we assume it is a changing distribution with magnitude.} In both Models, we assume that crowding only plays a role for \edits{magnitudes $G>16$}. The colour of the lines and regions indicates the pesudo-isodensity region which was used to fit that model. We only show the posteriors for the most and least dense regions in the bottom panel to avoid congesting the plot.}
\label{fig:crowding}
\end{figure*}
We show a map of the magnitude limit of 99\% completeness in the bottom panel of Fig. \ref{fig:completenessmaps}, which was computed following the methodology described in Appendix \ref{sec:completenessAB}. The difference in the magnitude limit of 99\% completeness with and without accounting for crowding is striking. In regions with few numbers of observations the magnitude limit has changed by as much as one magnitude. In some extremely dense regions which received few observations \edits{including a large portion of the Galactic bulge}, the magnitude limit is as bright as $G=18$, The competition between numbers of scans and crowding is most visible just West of the Galactic bulge, where the caustics of large numbers of observations push the magnitude limit fainter despite the effect of crowding. The completeness limit in crowded regions will be considerably fainter in later {\it Gaia}\xspace data releases as the scanning law fills in the gaps.
The explanation for the change in the selection function in regions of few observations between the two panels of Fig. \ref{fig:completenessmaps} is that our inference in Sec. \ref{sec:methodology} was biased by not accounting for crowding. There were many bright stars in crowded regions that had a low detection efficiency which biased Model AB to lower detection probabilities across the entire sky. Accounting for crowding is thus non-negotiable.
\edits{We end this section by reminding the reader that our prescription assumes that the effects of crowding are only conditional on the density of neighbouring sources in {\it Gaia}\xspace DR2, which is overly simplistic. We also note that we have not accounted for the incompleteness caused by the finite spatial resolution of {\it Gaia}\xspace, which means that stars in close pairs may not be resolved into separate sources. A further difficulty we have not considered is that bright stars can prevent {\it Gaia}\xspace from seeing faint neighbouring sources. We will return to these issues in later papers in this series, when we have identified a workable solution.}
\begin{figure*}
\centering
\includegraphics[width=1.\linewidth,trim=70 110 35 110, clip]{./figs/completeness_limit.pdf}
(a) {\it Gaia}\xspace DR2 completeness map based on our 1$^{\mathrm{st}}$-order selection function that only depends on the source magnitude and number of observations.\vspace{0.5cm}
\includegraphics[width=1.\linewidth,trim=70 110 35 110, clip]{./figs/skydensitycompletenessmap.pdf}
(b) {\it Gaia}\xspace DR2 completeness map based on our 2$^{\mathrm{nd}}$-order selection function that additionally depends on the nearby source density.
\caption{The magnitude limit down to which {\it Gaia}\xspace DR2 is 99\% complete varies across the sky due to the scanning law giving different places on the sky different numbers of observations. This limit can be directly calculated from the selection function and scanning law. Here we show HEALPix maps of this magnitude limit assuming either that the selection function only depends on source magnitude and number of observations (top) or that it additionally depends on the \edits{neighbouring} {\it Gaia}\xspace DR2 source density (bottom). \edits{The colour scale has been histogram-normalized separately in each panel to maximise the dynamic range of the plots in the appropriate region, with the end-points set by the 0.1\% and 99.9\% percentiles of the magnitude limit of the pixels. For the colour map we have used \textsc{cubehelix} \citep{cubehelix}}.}
\label{fig:completenessmaps}
\end{figure*}
\section{Discussion}
\label{sec:discussion}
\subsection{Previous attempts to quantify the completeness of {\it Gaia}\xspace DR2}
\label{sec:previous}
\citet{Arenou2018} investigated the completeness of {\it Gaia}\xspace DR2 as part of the DPAC validation of the catalogue. One approach they used was to calculate the 99$^{\mathrm{th}}$ percentile of $G$ magnitude in pixels on the sky, which they show in their Fig. 3. This definition is different from the magnitude limit we used in Fig. \ref{fig:completenessmaps}, which we defined to be the magnitude at which we predict that {\it Gaia}\xspace is no longer 99\% complete to stars of that magnitude. \citet{Arenou2018} find that in some parts of the sky their limiting magnitude is fainter than $G=21.5$ and in the Galactic bulge it is as bright as $G=19$, \edits{broadly consistent with our results}.
\edits{A further check carried out by \citet{Arenou2018} was to compare the completeness of {\it Gaia}\xspace DR2 with that of OGLE \citep{Udalski2008} across a series of fields in the disk, bulge and LMC chosen to have a range of source densities. A limitation of this comparison is the poorer spatial resolution of OGLE compared to {\it Gaia}\xspace, and so this comparison could only place upper limits on {\it Gaia}\xspace's completeness. Nevertheless, \citet{Arenou2018} were able to conclude that {\it Gaia}\xspace is almost complete at $G=18$ across the sky even in high source density regions.}
\edits{In order to improve the test of {\it Gaia}\xspace's completeness in the densest regions of the sky,} \citet{Arenou2018} \edits{compared} the completeness of {\it Gaia}\xspace DR2 relative to Hubble Space Telescope observations of 26 globular clusters. In their Fig. 7 they show the completeness as a function of both $G$ magnitude and the density of sources in the field, demonstrating that both of these factors strongly influence the completeness. We note that the completeness of {\it Gaia}\xspace to stars of the same magnitude in two globular clusters of the same density can be different due to the different number of observations each cluster will have received. This can \edits{likely} explain much of the scatter seen in their Fig. 7.
\citet{Rybizki2018b} estimated the completeness of {\it Gaia}\xspace DR2 in two ways and made them available through their \textsc{Python} package \textsc{gdr2\_completeness}\footnote{\url{https://github.com/jan-rybizki/gdr2_completeness}}. Their first method divided the sky into a \textsc{nside}=64 HEALPix map and computed the ratio of the number of stars in {\it Gaia}\xspace DR2 to the number of stars in 2MASS in each of the $G$ magnitude bins $(8,12)$, $(12,15)$ and $(15,18)$ in each of the pixels. This approach will poorly constrain the completeness in pixels with few stars and assumes that the completeness is not varying over large ranges of $G$. It is also limited to the relatively bright magnitude range of the 2MASS catalogue. Their second method assumed that the magnitude at which {\it Gaia}\xspace DR2 is no longer complete in a pixel on the sky is the mode of the observed magnitude distribution. This approach only returns the true completeness magnitude limit if the true magnitude number density is monotonically increasing, {\it Gaia}\xspace is 100\% complete up to the limit and 0\% complete beyond it. In general, this is a biased estimator that performs worse in pixels on the sky with few stars. However, their method demonstrates that {\it Gaia}\xspace drops in completeness in low extinction regions close to the Galactic center where there are high densities of bright sources.
\subsection{Applications in the {\it Gaia}\xspace-verse}
In the previous sections we demonstrated that {\it Gaia}\xspace is mostly complete to sources with magnitudes in the range $3<G<20$ (aside from in the densest regions of the sky), however there are further scanning-law-driven selections that can cause sources in the {\it Gaia}\xspace DR2 source catalogue to not have published parallaxes, proper motions, variability indicators or radial velocities.
\subsubsection{Stars with proper motions and parallaxes}
As mentioned in Sec. \ref{sec:methodology}, more stringent cuts were applied by \citet{Lindegren2018} when determining whether a source has a reported parallax and proper motion. In addition to cuts on a proxy for the size of the astrometric uncertainty and on $G<21$, \citet{Lindegren2018} cut to require that there were at least 6 clusters (\textsc{visibility\_periods\_used}) amongst the $k$ detections, as this ensured a reasonable spread of the measurements across the time period of {\it Gaia}\xspace observations. These additional cuts are challenging to interpret as a selection function and so we postpone their dissection to \edits{a later} paper of this series.
\subsubsection{Variable stars}
One of the major data products associated with {\it Gaia}\xspace DR2 were the catalogues of classified variable stars \citep{Mowlavi2018, Clementini2019}. The pipeline for these classifications discarded some fraction of the observations of each star as failing to meet their quality criteria, and thus defined $k_{\mathrm{good}}\leq k \leq n$ as the number of good observations of each star. Cuts were then made on $k_{\mathrm{good}}$ at different points in the variability pipeline (see Fig. 2 of \citealp{Holl2018}). The weakest of these cuts was that a source was only classified as variable or non-variable (\textsc{phot\_variable\_flag} in the {\it Gaia}\xspace DR2 main catalogue) if $k_{\mathrm{good}}\geq2$, while sources were only classified as RR Lyrae, Cepheids or Long Period Variables if $k_{\mathrm{good}}\geq12$ and as Short Period or Rotationally Modulated Variables if $k_{\mathrm{good}}\geq20$. For comparison, 826 million stars satisfy $k_{\mathrm{good}}\geq20$ compared to 833 million stars that satisfy $k\geq20$ and 1,124 million stars that satisfy $n\geq20$. These numbers suggest that the observation quality cuts applied by the DPAC are not removing many detection and thus any selection of the form $k_{\mathrm{good}}\geq K$ can be approximated by the selection $k\geq K$. This approximation is necessary because the quantity $k_{\mathrm{good}}$ was only published for certain subsets of the variability catalogues. The selection function of a specific variable star catalogue in {\it Gaia}\xspace DR2 (those stars that are classified as being a variable of a specific type) is likely to have at least four components, which we will illustrate for the Rotationally Modulated Variables (otherwise known as BY Dra-type stars):
\begin{enumerate}
\item Was the star observed at least $n\geq20$ times?
\item Was the star detected at least $k\geq20$ times?
\item Were at least $k_{\mathrm{good}}\geq20$ of those detections of good quality?
\item Does supervised classification based on the location of the star in the H-R diagram and the time series of photometric measurements from those good detections support the conclusion that this is a BY Dra-type star?
\end{enumerate}
The first three of these selections try to select only those stars where DPAC were confident in being able to reliably classify the variability, while the final selection attempts to select those stars that are actually BY Dra-type variables.
Curiously, the sky map of the number of {\it Gaia}\xspace-classified BY Dra-type variables (see Fig. 6 of \citealp{Holl2018}) suggests that the effective number of good detections required for the classification was much higher than 20, because the classified stars are only found along the caustics of many repeated scans and in regions covered by the Ecliptic Pole Scanning Law (EPSL). Strikingly -- apart from the EPSL -- the North and South Ecliptic polar cap regions, which typically have a quite high number of observations of $30<n<40$, are entirely devoid of BY Dra stars. \citet{Lanzafame2018} attempted to quantify the completeness of the {\it Gaia}\xspace BY Dra catalogue by extrapolating the $14\%$ completeness of their catalogue in the region around the Pleiades (in comparison to an existing catalogue) to the 38\% of the sky in which their catalogue contains BY Dra stars, and thus arriving at an upper limit of 5\% completeness. Comparing Fig. A.1. of \citet{Lanzafame2018} to our Fig. \ref{fig:map}, we can see that BY Dra stars appear to only be detected on the two caustics of between forty and fifty observations that cross through the Pleiades. It is clear that the reported 14\% completeness is being driven by the number of observations that part of the sky received, and thus that this number should not be extrapolated to the rest of the sky. Indeed, a large fraction of the 38\% of the sky containing {\it Gaia}\xspace BY Dra stars has more than fifty observations, and thus should be more than 14\% complete. In Fig. 6 of \citet{Holl2018}, the number of detected BY Dra variables increases towards the South Ecliptic Pole even as the true number density must be decreasing (due to increasing Galactic latitude), which affirms our conclusion that the completeness of the {\it Gaia}\xspace BY Dra variable catalogue is being strongly determined by the number of observations.
\edits{A further complication is that the ability of DPAC processing to classify a source as a variable will depend on the timings of the detections, not just their quantity. For instance, sources near the Ecliptic Poles will have received a large number of observations during the month-long EPSL, but those many observations will not be as constraining of the period of a Long Period variable star as a few observations evenly spaced across the 22 months of DR2.}
{\it Gaia}\xspace is revolutionary for the study of variable stars, because it is the first all-sky mission with large numbers of repeat visits with high photometric precision down to $G=20$ covering multi-year baselines. An accurate estimate of the completeness of the {\it Gaia}\xspace variable star catalogues will need to account for the unique spinning-and-precessing way that {\it Gaia}\xspace looks at the sky. While this is outside the scope of this paper, we plan to return to this topic in later papers in this series.
\subsubsection{Stars with radial velocities}
{\it Gaia}\xspace DR2 included the largest spectroscocopic radial velocity catalogue in history, with the on-board Radial Velocity Spectrometer (RVS) reporting radial velocities of 7,224,631 stars \citep{Cropper2018,Sartoretti2018,Katz2019}. The quantity \textsc{radial\_velocity} in {\it Gaia}\xspace DR2 is the median of the multiple $k_{\mathrm{RVS}}$ radial velocity measurements {\it Gaia}\xspace made of each star. DPAC required that a star had at least $k_{\mathrm{RVS}}\geq2$ measurements in order for the star to be published in {\it Gaia}\xspace DR2. This selection has the effect of making the {\it Gaia}\xspace DR2 RVS subset more complete in regions of the sky with more observations, as can be seen in Fig. 6 of \citet{Katz2019}. {\it Gaia}\xspace attempts to measure the radial velocity of a star if all of the following apply: 1) the star was detected by the SkyMapper CCDs, 2) the star is crossing the focal plane in one of the four of seven rows which have an RVS CCD, and 3) {\it Gaia}\xspace was able to assign a window to the star that does not partially overlap with a window already assigned to another star. The first requirement is not onerous because the {\it Gaia}\xspace DR2 RVS catalogue was limited to magnitudes $G_{\mathrm{RVS}}<12$ and {\it Gaia}\xspace detects \edits{most} stars this bright on almost every observation. The second requirement \edits{can be thought of as discarding} three out of every seven observations, and thus if we consider $k_{\mathrm{RVS}}\sim\operatorname{Bin}(n,\theta_{\mathrm{RVS}})$ then $\theta_{\mathrm{RVS}}\leq 4/7$. The third requirement greatly complicates the $k_{\mathrm{RVS}}\geq2$ selection, because the odds that there is already a star with an assigned window that would overlap are dramatically higher in denser regions of the sky. This explains the drop in completeness of {\it Gaia}\xspace RVS near the Galactic plane, and the particularly low 25\% completeness near the Galactic centre where the high density of sources combines with a region of small numbers of observations. Whether the windows of two sources will overlap depends on the angle that {\it Gaia}\xspace is scanning across that field, and thus changes between each scan. \edits{A further complication is that whether a radial velocity measurement can be extracted from the RVS spectrum will depend on the depth of the Calcium triplet which will vary with the surface gravity and effective temperature of the source.} We therefore conjecture that the probability that an observation of a star will result in a successful radial velocity measurement will be a \edits{complicated} function of the magnitude $G_{\mathrm{RVS}}$, the colour $G_{\mathrm{BP}}-G_{\mathrm{RP}}$ \edits{(acting as a proxy for the stellar type)} and the density of nearby bright sources.
The {\it Gaia}\xspace DR2 catalogue of radial velocities has been transformative for Galactic dynamics, but sophisticated applications have been hindered by the lack of a well-motivated selection function that accounts for the scanning law selection. The development of such a selection will be the subject of two later papers in this series.
\subsection{Poisson binomial distribution}
\label{sec:poissonbinomial}
One of the key assumptions of our methodology is that the detection probability $\theta$ of each source is the same every time that source is observed (see Sec. \ref{sec:framework}). There are several possible ways that this assumption could be violated in reality, for instance:
\begin{enumerate}
\item If the source is variable, then the detection probability will vary depending on the brightness of the star at the time of each observation.
\item If the source is faint and in a crowded region, then whether the source can be assigned a non-overlapping window can depend on the angle at which {\it Gaia}\xspace is scanning across the field, which changes between scans.
\item {\it Gaia}\xspace suffers from stray light being scattered by fibres at the edge of the sunshield \citep{Gaia2016}. The level of straylight contamination varies with the location of {\it Gaia}\xspace on its orbit, thus causing a time-dependence in the detection probability of faint sources.
\end{enumerate}
A possible improvement of our method would be to assume that the detection probability of a source is different on each observation, and thus that the number of detections is a Poisson-Binomial random variable. Suppose there were $n$ observations with detection probabilities $\{p_1,\dots p_n\}$. The probability of $k$ detections is then given by
\begin{equation}
\operatorname{P}(k|\{p_i\}) = \sum_{A\in F_k^n} \prod_{i\in A} p_i \prod_{j\in A^{\mathrm{c}}} (1-p_j),
\end{equation}
where $F_k^n$ is the set of all subsets of $k$ integers that can be picked from $\{1,\dots,n\}$. The Poisson-Binomial \edits{probability mass function} is computationally-challenging to evaluate for even moderately large $n\gtrsim20$, because the size of the set $F_k^n$ -- and thus the number of terms to be summed -- is $|F_k^n|=\binom{n}{k}$. However, if we assume that each $p_i\sim\operatorname{Beta}(\alpha,\beta)$, where we assume that the $\alpha$ and $\beta$ are the same across all observations of a single source, then we can marginalise away the $p_i$ by noting that
\begin{align}
\operatorname{P}(k&|\alpha,\beta)=\idotsint \operatorname{P}(k|\{p_i\})\prod_{j=1}^n\operatorname{P}(p_j|\alpha,\beta) \mathrm{d}p_j \nonumber\\
&= \sum_{A\in F_k^n} \prod_{i\in A} \int p_i \operatorname{B}_{\alpha,\beta}(p_i) \mathrm{d}p_i \prod_{j\in A^{\mathrm{c}}} \int (1-p_j) \operatorname{B}_{\alpha,\beta}(p_j) \mathrm{d}p_j \nonumber \\
&= \sum_{A\in F_k^n} \prod_{i\in A} \frac{\operatorname{B}(\alpha+1,\beta)}{\operatorname{B}(\alpha,\beta)} \prod_{j\in A^{\mathrm{c}}} \frac{\operatorname{B}(\alpha,\beta+1)}{\operatorname{B}(\alpha,\beta)} \nonumber\\
&= \sum_{A\in F_k^n} \prod_{i\in A} \frac{\alpha}{\alpha+\beta} \prod_{j\in A^{\mathrm{c}}} \frac{\beta}{\alpha+\beta} \nonumber\\
&= \sum_{A\in F_k^n} \left(\frac{\alpha}{\alpha+\beta} \right)^{|A|}\left(\frac{\beta}{\alpha+\beta} \right)^{|A^{\mathrm{c}}|} \nonumber\\
&= \binom{n}{k} \left(\frac{\alpha}{\alpha+\beta} \right)^{k} \left(1-\frac{\alpha}{\alpha+\beta} \right)^{n-k}.
\label{eq:pbin}
\end{align}
This is the form of the probability mass function of a Binomial random variable, and thus we can equivalently write that $k\sim\operatorname{Binomial}(n,\frac{\alpha}{\alpha+\beta})$. We note that the combination $\frac{\alpha}{\alpha+\beta}$ is the mean of the $\operatorname{Beta}(\alpha,\beta)$ distribution, which implies that -- when the $p_i\sim\operatorname{Beta}(\alpha,\beta)$ -- the number of detections $k$ is only sensitive to the ratio of $\alpha$ and $\beta$ and not their scale.
By marginalising the $\{p_i\}$ we have drastically simplified the evaluation of the Poisson-Binomial likelihood with Beta priors, and therefore enabled it for use with large $n$. However, this simplification can only be made if the data $k$ has not been truncated. If the data has been truncated $k\geq5$, then the likelihood must be re-normalised by the survival function,
\begin{equation}
\operatorname{S}(k\geq5|\{p_i\}) = 1- \sum_{l=0}^4\sum_{A\in F_l^n} \prod_{i\in A} p_i \prod_{j\in A^{\mathrm{c}}} (1-p_j),
\end{equation}
and thus this intimidating function would appear as a divisor inside the integral in the derivation above. It is possible that there exists a closed-form solution to this integral, but, if it exists, then it is likely to be in terms of obscure special functions. We plan to revisit the possibility of utilising a truncated Beta-Poisson-Binomial model \edits{in a later paper in this series}.
\subsection{Using our selection functions}
To aid the reader in using our selection functions, we have created a new \textsc{Python} module \textsc{selectionfunctions} (\url{https://github.com/gaiaverse/selectionfunctions}) based on the \textsc{dustmaps} package by \citet{Green2018}. This module allows the user to easily query our selection functions in any coordinate system. We have shown in Sec. \ref{sec:crowding} that when crowding is not accounted for the selection function is biased and we therefore do not provide the selection functions computed in Sec. \ref{sec:methodology} in \textsc{selectionfunctions}. However, we provide the ability to query both Model T and Model AB from Sec. \ref{sec:crowding}, and to ignore the effect of crowding by only returning the selection function computed for the least dense bin (the purple lines in Fig. \ref{fig:crowding}). We illustrate the simplicity of using our \textsc{selectionfunctions} package in the code snippet below, where we query our Model AB from Sec. \ref{sec:crowding} to calculate the selection function for the fastest main-sequence star in the Galaxy \citep[S5-HVS1,][]{Koposov2020}. \edits{In the future, we plan to include the selection functions of ground-based spectroscopic surveys \citep{Everall2020} and the ability to query the selection function of the intersection of multiple surveys, thus enabling astronomers to ask questions such as `what are the odds that a star in {\it Gaia}\xspace DR2 has an APOGEE radial velocity?'.}
\lstinputlisting[language=Python]{selectionfunctions.py}
\section{Conclusions}
\label{sec:conclusion}
The {\it Gaia}\xspace mission has broken astrometric, photometric and spectroscopic records with its second data release, but to \edits{fully exploit this remarkable dataset we need to know which stars are missing and where they lie on the sky. In this work we argued that the completeness of {\it Gaia}\xspace DR2 is driven by the spinning-and-precessing way that {\it Gaia}\xspace looks at the sky.} We computed the number of times that {\it Gaia}\xspace looked at each of the stars in {\it Gaia}\xspace DR2 and at each point on the sky. We then developed a statistical framework with which we answered the question: what are the odds that {\it Gaia}\xspace detects a star each time it observes it? The answer to this question is important, because to leading order {\it Gaia}\xspace DR2 contains all sources that were detected at least five times. We could answer this question because {\it Gaia}\xspace reports the number of times that each source was detected. We modelled the run of the detection probability with $G$-band magnitude, finding that {\it Gaia}\xspace DR2 is broadly complete over $7<G<20$ but that the completeness falls to 0\% over the range \edits{$20<G<21.3$}. We created an interactive visualisation to illustrate this result. We further calculated the magnitude up to which {\it Gaia}\xspace is complete at each location on the sky, finding that this magnitude limit varies from 18.0 to 21.3. We then extended our selection function to account for crowding, and found that crowding is a vitally important component of the {\it Gaia}\xspace selection function, particularly in the Galactic bulge \edits{and the Large and Small Magellanic Clouds.} We concluded by conjecturing on the necessity of accounting for the {\it Gaia}\xspace scanning law when modelling the selection functions of the {\it Gaia}\xspace DR2 parallax and proper motion, variable star and radial velocity sub-catalogues, topics that we will return to in later papers of this series. \edits{Finally, we presented a \textsc{Python} package that allows the reader to easily incorporate our selection functions in their work.} This work lays down the conceptual framework needed to grasp the selection functions of the {\it Gaia}\xspace catalogues, and thus is a fundamental stepping stone on the path to mapping our Galaxy with {\it Gaia}\xspace.
\section*{Acknowledgements}
The authors are grateful to the {\it Gaia}\xspace DPAC for making the scanning law publicly available and to Berry Holl for his help in interpreting it. DB thanks Magdalen College for his fellowship and the Rudolf Peierls Centre for Theoretical Physics for providing office space and travel funds. AE thanks the Science and Technology Facilities Council of
the United Kingdom for financial support. This work has made use of data from the European Space Agency (ESA) mission {\it Gaia}\xspace (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia}\xspace
Data Processing and Analysis Consortium (DPAC,
\url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC
has been provided by national institutions, in particular the institutions
participating in the {\it Gaia}\xspace Multilateral Agreement.
\bibliographystyle{mnras}
\section{Introduction}
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\section{Preparing and submitting a paper}
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\begin{equation}
a^2=b^2+c^2
\end{equation}
\end{verbatim}
\noindent which produces
\begin{equation}
a^2=b^2+c^2
\end{equation}
By default, the equations are numbered sequentially throughout the whole paper. If a paper has a large number of equations, it may be better to number them by section (2.1, 2.2 etc.). To do this, add the command \verb'\numberwithin{equation}{section}' to the preamble.
It is also possible to produce un-numbered equations by using the \LaTeX\ built-in \verb'\['\textellipsis\verb'\]' and \verb'$$'\textellipsis\verb'$$' commands; however MNRAS requires that all equations are numbered, so these commands should be avoided.
\subsection{Special symbols}
\begin{table}
\caption{Additional commands for special symbols commonly used in astronomy. These can be used anywhere.}
\label{tab:anysymbols}
\begin{tabular}{lll}
\hline
Command & Output & Meaning\\
\hline
\verb'\sun' & \sun & Sun, solar\\[2pt]
\verb'\earth' & \earth & Earth, terrestrial\\[2pt]
\verb'\micron' & \micron & microns\\[2pt]
\verb'\degr' & \degr & degrees\\[2pt]
\verb'\arcmin' & \arcmin & arcminutes\\[2pt]
\verb'\arcsec' & \arcsec & arcseconds\\[2pt]
\verb'\fdg' & \fdg & fraction of a degree\\[2pt]
\verb'\farcm' & \farcm & fraction of an arcminute\\[2pt]
\verb'\farcs' & \farcs & fraction of an arcsecond\\[2pt]
\verb'\fd' & \fd & fraction of a day\\[2pt]
\verb'\fh' & \fh & fraction of an hour\\[2pt]
\verb'\fm' & \fm & fraction of a minute\\[2pt]
\verb'\fs' & \fs & fraction of a second\\[2pt]
\verb'\fp' & \fp & fraction of a period\\[2pt]
\verb'\diameter' & \diameter & diameter\\[2pt]
\verb'\sq' & \sq & square, Q.E.D.\\[2pt]
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Additional commands for mathematical symbols. These can only be used in maths mode.}
\label{tab:mathssymbols}
\begin{tabular}{lll}
\hline
Command & Output & Meaning\\
\hline
\verb'\upi' & $\upi$ & upright pi\\[2pt]
\verb'\umu' & $\umu$ & upright mu\\[2pt]
\verb'\upartial' & $\upartial$ & upright partial derivative\\[2pt]
\verb'\lid' & $\lid$ & less than or equal to\\[2pt]
\verb'\gid' & $\gid$ & greater than or equal to\\[2pt]
\verb'\la' & $\la$ & less than of order\\[2pt]
\verb'\ga' & $\ga$ & greater than of order\\[2pt]
\verb'\loa' & $\loa$ & less than approximately\\[2pt]
\verb'\goa' & $\goa$ & greater than approximately\\[2pt]
\verb'\cor' & $\cor$ & corresponds to\\[2pt]
\verb'\sol' & $\sol$ & similar to or less than\\[2pt]
\verb'\sog' & $\sog$ & similar to or greater than\\[2pt]
\verb'\lse' & $\lse$ & less than or homotopic to \\[2pt]
\verb'\gse' & $\gse$ & greater than or homotopic to\\[2pt]
\verb'\getsto' & $\getsto$ & from over to\\[2pt]
\verb'\grole' & $\grole$ & greater over less\\[2pt]
\verb'\leogr' & $\leogr$ & less over greater\\
\hline
\end{tabular}
\end{table}
Some additional symbols of common use in astronomy have been added in the MNRAS class. These are shown in tables~\ref{tab:anysymbols}--\ref{tab:mathssymbols}. The command names are -- as far as possible -- the same as those used in other major astronomy journals.
Many other mathematical symbols are also available, either built into \LaTeX\ or via additional packages. If you want to insert a specific symbol but don't know the \LaTeX\ command, we recommend using the Detexify website\footnote{\url{http://detexify.kirelabs.org}}.
Sometimes font or coding limitations mean a symbol may not get smaller when used in sub- or superscripts, and will therefore be displayed at the wrong size. There is no need to worry about this as it will be corrected by the typesetter during production.
To produce bold symbols in mathematics, use \verb'\bmath' for simple variables, and the \verb'bm' package for more complex symbols (see section~\ref{sec:packages}). Vectors are set in bold italic, using \verb'\mathbfit{}'.
For matrices, use \verb'\mathbfss{}' to produce a bold sans-serif font e.g. \mathbfss{H}; this works even outside maths mode, but not all symbols are available (e.g. Greek). For $\nabla$ (del, used in gradients, divergence etc.) use \verb'$\nabla$'.
\subsection{Ions}
A new \verb'\ion{}{}' command has been added to the class file, for the correct typesetting of ionisation states.
For example, to typeset singly ionised calcium use \verb'\ion{Ca}{ii}', which produces \ion{Ca}{ii}.
\section{Figures and tables}
\label{sec:fig_table}
Figures and tables (collectively called `floats') are mostly the same as built into \LaTeX.
\subsection{Basic examples}
\begin{figure}
\includegraphics[width=\columnwidth]{example}
\caption{An example figure.}
\label{fig:example}
\end{figure}
Figures are inserted in the usual way using a \verb'figure' environment and \verb'\includegraphics'. The example Figure~\ref{fig:example} was generated using the code:
\begin{verbatim}
\begin{figure}
\includegraphics[width=\columnwidth]{example}
\caption{An example figure.}
\label{fig:example}
\end{figure}
\end{verbatim}
\begin{table}
\caption{An example table.}
\label{tab:example}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
Sun & 1.00 & 1.00\\
$\alpha$~Cen~A & 1.10 & 1.52\\
$\epsilon$~Eri & 0.82 & 0.34\\
\hline
\end{tabular}
\end{table}
The example Table~\ref{tab:example} was generated using the code:
\begin{verbatim}
\begin{table}
\caption{An example table.}
\label{tab:example}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
Sun & 1.00 & 1.00\\
$\alpha$~Cen~A & 1.10 & 1.52\\
$\epsilon$~Eri & 0.82 & 0.34\\
\hline
\end{tabular}
\end{table}
\end{verbatim}
\subsection{Captions and placement}
Captions go \emph{above} tables but \emph{below} figures, as in the examples above.
The \LaTeX\ float placement commands \verb'[htbp]' are intentionally disabled.
Layout of figures and tables will be adjusted by the publisher during the production process, so authors should not concern themselves with placement to avoid disappointment and wasted effort.
Simply place the \LaTeX\ code close to where the figure or table is first mentioned in the text and leave exact placement to the publishers.
By default a figure or table will occupy one column of the page.
To produce a wider version which covers both columns, use the \verb'figure*' or \verb'table*' environment.
If a figure or table is too long to fit on a single page it can be split it into several parts.
Create an additional figure or table which uses \verb'\contcaption{}' instead of \verb'\caption{}'.
This will automatically correct the numbering and add `\emph{continued}' at the start of the caption.
\begin{table}
\contcaption{A table continued from the previous one.}
\label{tab:continued}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
$\tau$~Cet & 0.78 & 0.52\\
$\delta$~Pav & 0.99 & 1.22\\
$\sigma$~Dra & 0.87 & 0.43\\
\hline
\end{tabular}
\end{table}
Table~\ref{tab:continued} was generated using the code:
\begin{verbatim}
\begin{table}
\contcaption{A table continued from the previous one.}
\label{tab:continued}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
$\tau$~Cet & 0.78 & 0.52\\
$\delta$~Pav & 0.99 & 1.22\\
$\sigma$~Dra & 0.87 & 0.43\\
\hline
\end{tabular}
\end{table}
\end{verbatim}
To produce a landscape figure or table, use the \verb'pdflscape' package and the \verb'landscape' environment.
The landscape Table~\ref{tab:landscape} was produced using the code:
\begin{verbatim}
\begin{landscape}
\begin{table}
\caption{An example landscape table.}
\label{tab:landscape}
\begin{tabular}{cccccccccc}
\hline
Header & Header & ...\\
Unit & Unit & ...\\
\hline
Data & Data & ...\\
Data & Data & ...\\
...\\
\hline
\end{tabular}
\end{table}
\end{landscape}
\end{verbatim}
Unfortunately this method will force a page break before the table appears.
More complicated solutions are possible, but authors shouldn't worry about this.
\begin{landscape}
\begin{table}
\caption{An example landscape table.}
\label{tab:landscape}
\begin{tabular}{cccccccccc}
\hline
Header & Header & Header & Header & Header & Header & Header & Header & Header & Header\\
Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit \\
\hline
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
\hline
\end{tabular}
\end{table}
\end{landscape}
\section{References and citations}
\subsection{Cross-referencing}
The usual \LaTeX\ commands \verb'\label{}' and \verb'\ref{}' can be used for cross-referencing within the same paper.
We recommend that you use these whenever relevant, rather than writing out the section or figure numbers explicitly.
This ensures that cross-references are updated whenever the numbering changes (e.g. during revision) and provides clickable links (if available in your compiler).
It is best to give each section, figure and table a logical label.
For example, Table~\ref{tab:mathssymbols} has the label \verb'tab:mathssymbols', whilst section~\ref{sec:packages} has the label \verb'sec:packages'.
Add the label \emph{after} the section or caption command, as in the examples in sections~\ref{sec:sections} and \ref{sec:fig_table}.
Enter the cross-reference with a non-breaking space between the type of object and the number, like this: \verb'see Figure~\ref{fig:example}'.
The \verb'\autoref{}' command can be used to automatically fill out the type of object, saving on typing.
It also causes the link to cover the whole phrase rather than just the number, but for that reason is only suitable for single cross-references rather than ranges.
For example, \verb'\autoref{tab:journal_abbr}' produces \autoref{tab:journal_abbr}.
\subsection{Citations}
\label{sec:cite}
MNRAS uses the Harvard -- author (year) -- citation style, e.g. \citet{author2013}.
This is implemented in \LaTeX\ via the \verb'natbib' package, which in turn is included via the \verb'usenatbib' package option (see section~\ref{sec:options}), which should be used in all papers.
Each entry in the reference list has a `key' (see section~\ref{sec:ref_list}) which is used to generate citations.
There are two basic \verb'natbib' commands:
\begin{description}
\item \verb'\citet{key}' produces an in-text citation: \citet{author2013}
\item \verb'\citep{key}' produces a bracketed (parenthetical) citation: \citep{author2013}
\end{description}
Citations will include clickable links to the relevant entry in the reference list, if supported by your \LaTeX\ compiler.
\defcitealias{smith2014}{Paper~I}
\begin{table*}
\caption{Common citation commands, provided by the \texttt{natbib} package.}
\label{tab:natbib}
\begin{tabular}{lll}
\hline
Command & Ouput & Note\\
\hline
\verb'\citet{key}' & \citet{smith2014} & \\
\verb'\citep{key}' & \citep{smith2014} & \\
\verb'\citep{key,key2}' & \citep{smith2014,jones2015} & Multiple papers\\
\verb'\citet[table 4]{key}' & \citet[table 4]{smith2014} & \\
\verb'\citep[see][figure 7]{key}' & \citep[see][figure 7]{smith2014} & \\
\verb'\citealt{key}' & \citealt{smith2014} & For use with manual brackets\\
\verb'\citeauthor{key}' & \citeauthor{smith2014} & If already cited in close proximity\\
\verb'\defcitealias{key}{Paper~I}' & & Define an alias (doesn't work in floats)\\
\verb'\citetalias{key}' & \citetalias{smith2014} & \\
\verb'\citepalias{key}' & \citepalias{smith2014} & \\
\hline
\end{tabular}
\end{table*}
There are a number of other \verb'natbib' commands which can be used for more complicated citations.
The most commonly used ones are listed in Table~\ref{tab:natbib}.
For full guidance on their use, consult the \verb'natbib' documentation\footnote{\url{http://www.ctan.org/pkg/natbib}}.
If a reference has several authors, \verb'natbib' will automatically use `et al.' if there are more than two authors. However, if a paper has exactly three authors, MNRAS style is to list all three on the first citation and use `et al.' thereafter. If you are using \bibtex\ (see section~\ref{sec:ref_list}) then this is handled automatically. If not, the \verb'\citet*{}' and \verb'\citep*{}' commands can be used at the first citation to include all of the authors.
\subsection{The list of references}
\label{sec:ref_list}
It is possible to enter references manually using the usual \LaTeX\ commands, but we strongly encourage authors to use \bibtex\ instead.
\bibtex\ ensures that the reference list is updated automatically as references are added or removed from the paper, puts them in the correct format, saves on typing, and the same reference file can be used for many different papers -- saving time hunting down reference details.
An MNRAS \bibtex\ style file, \verb'mnras.bst', is distributed as part of this package.
The rest of this section will assume you are using \bibtex.
References are entered into a separate \verb'.bib' file in standard \bibtex\ formatting.
This can be done manually, or there are several software packages which make editing the \verb'.bib' file much easier.
We particularly recommend \textsc{JabRef}\footnote{\url{http://jabref.sourceforge.net/}}, which works on all major operating systems.
\bibtex\ entries can be obtained from the NASA Astrophysics Data System\footnote{\label{foot:ads}\url{http://adsabs.harvard.edu}} (ADS) by clicking on `Bibtex entry for this abstract' on any entry.
Simply copy this into your \verb'.bib' file or into the `BibTeX source' tab in \textsc{JabRef}.
Each entry in the \verb'.bib' file must specify a unique `key' to identify the paper, the format of which is up to the author.
Simply cite it in the usual way, as described in section~\ref{sec:cite}, using the specified key.
Compile the paper as usual, but add an extra step to run the \texttt{bibtex} command.
Consult the documentation for your compiler or latex distribution.
Correct formatting of the reference list will be handled by \bibtex\ in almost all cases, provided that the correct information was entered into the \verb'.bib' file.
Note that ADS entries are not always correct, particularly for older papers and conference proceedings, so may need to be edited.
If in doubt, or if you are producing the reference list manually, see the MNRAS instructions to authors$^{\ref{foot:itas}}$ for the current guidelines on how to format the list of references.
\section{Appendices and online material}
To start an appendix, simply place the \verb' |
2,869,038,156,348 | arxiv | \section{Credits}
This document has been adapted
by Steven Bethard, Ryan Cotterell and Rui Yan
from the instructions for earlier ACL and NAACL proceedings, including those for
ACL 2019 by Douwe Kiela and Ivan Vuli\'{c},
NAACL 2019 by Stephanie Lukin and Alla Roskovskaya,
ACL 2018 by Shay Cohen, Kevin Gimpel, and Wei Lu,
NAACL 2018 by Margaret Michell and Stephanie Lukin,
2017/2018 (NA)ACL bibtex suggestions from Jason Eisner,
ACL 2017 by Dan Gildea and Min-Yen Kan,
NAACL 2017 by Margaret Mitchell,
ACL 2012 by Maggie Li and Michael White,
ACL 2010 by Jing-Shing Chang and Philipp Koehn,
ACL 2008 by Johanna D. Moore, Simone Teufel, James Allan, and Sadaoki Furui,
ACL 2005 by Hwee Tou Ng and Kemal Oflazer,
ACL 2002 by Eugene Charniak and Dekang Lin,
and earlier ACL and EACL formats written by several people, including
John Chen, Henry S. Thompson and Donald Walker.
Additional elements were taken from the formatting instructions of the \emph{International Joint Conference on Artificial Intelligence} and the \emph{Conference on Computer Vision and Pattern Recognition}.
\section{Introduction}
The following instructions are directed to authors of papers submitted to AACL-IJCNLP 2020 or accepted for publication in its proceedings.
All authors are required to adhere to these specifications.
Authors are required to provide a Portable Document Format (PDF) version of their papers.
\textbf{The proceedings are designed for printing on A4 paper.}
\section{Electronically-available resources}
AACL provides this description and accompanying style files at
\begin{quote}
\url{http://aacl2020.org//downloads/aacl-ijcnlp2020-templates.zip}
\end{quote}
We strongly recommend the use of these style files, which have been appropriately tailored for the AACL-IJCNLP 2020 proceedings.
\paragraph{\LaTeX-specific details:}
The templates include the \LaTeX2e{} source (\texttt{\small aacl-ijcnlp2020.tex}),
the \LaTeX2e{} style file used to format it (\texttt{\small aacl-ijcnlp2020.sty}),
an ACL bibliography style (\texttt{\small acl\_natbib.bst}),
an example bibliography (\texttt{\small aacl-ijcnlp2020.bib}),
and the bibliography for the ACL Anthology (\texttt{\small anthology.bib}).
\section{Length of Submission}
\label{sec:length}
The conference accepts submissions of long papers and short papers.
Long papers may consist of up to eight (8) pages of content plus unlimited pages for references.
Upon acceptance, final versions of long papers will be given one additional page -- up to nine (9) pages of content plus unlimited pages for references -- so that reviewers' comments can be taken into account.
Short papers may consist of up to four (4) pages of content, plus unlimited pages for references.
Upon acceptance, short papers will be given five (5) pages in the proceedings and unlimited pages for references.
For both long and short papers, all illustrations and tables that are part of the main text must be accommodated within these page limits, observing the formatting instructions given in the present document.
Papers that do not conform to the specified length and formatting requirements are subject to be rejected without review.
The conference encourages the submission of additional material that is relevant to the reviewers but not an integral part of the paper.
There are two such types of material: appendices, which can be read, and non-readable supplementary materials, often data or code.
Additional material must be submitted as separate files, and must adhere to the same anonymity guidelines as the main paper.
The paper must be self-contained: it is optional for reviewers to look at the supplementary material.
Papers should not refer, for further detail, to documents, code or data resources that are not available to the reviewers.
Refer to Appendices~\ref{sec:appendix} and \ref{sec:supplemental} for further information.
Workshop chairs may have different rules for allowed length and whether supplemental material is welcome.
As always, the respective call for papers is the authoritative source.
\section{Anonymity}
As reviewing will be double-blind, papers submitted for review should not include any author information (such as names or affiliations). Furthermore, self-references that reveal the author's identity, \emph{e.g.},
\begin{quote}
We previously showed \citep{Gusfield:97} \ldots
\end{quote}
should be avoided. Instead, use citations such as
\begin{quote}
\citet{Gusfield:97} previously showed\ldots
\end{quote}
Please do not use anonymous citations and do not include acknowledgements.
\textbf{Papers that do not conform to these requirements may be rejected without review.}
Any preliminary non-archival versions of submitted papers should be listed in the submission form but not in the review version of the paper.
Reviewers are generally aware that authors may present preliminary versions of their work in other venues, but will not be provided the list of previous presentations from the submission form.
Once a paper has been accepted to the conference, the camera-ready version of the paper should include the author's names and affiliations, and is allowed to use self-references.
\paragraph{\LaTeX-specific details:}
For an anonymized submission, ensure that {\small\verb|\aclfinalcopy|} at the top of this document is commented out, and that you have filled in the paper ID number (assigned during the submission process on softconf) where {\small\verb|***|} appears in the {\small\verb|\def\aclpaperid{***}|} definition at the top of this document.
For a camera-ready submission, ensure that {\small\verb|\aclfinalcopy|} at the top of this document is not commented out.
\section{Multiple Submission Policy}
Papers that have been or will be submitted to other meetings or publications must indicate this at submission time in the START submission form, and must be withdrawn from the other venues if accepted by AACL-IJCNLP 2020. Authors of papers accepted for presentation at AACL-IJCNLP 2020 must notify the program chairs by the camera-ready deadline as to whether the paper will be presented. We will not accept for publication or presentation the papers that overlap significantly in content or results with papers that will be (or have been) published elsewhere.
Authors submitting more than one paper to AACL-IJCNLP 2020 must ensure that submissions do not overlap significantly ($>$25\%) with each other in content or results.
\section{Formatting Instructions}
Manuscripts must be in two-column format.
Exceptions to the two-column format include the title, authors' names and complete addresses, which must be centered at the top of the first page, and any full-width figures or tables (see the guidelines in Section~\ref{ssec:title-authors}).
\textbf{Type single-spaced.}
Start all pages directly under the top margin.
The manuscript should be printed single-sided and its length should not exceed the maximum page limit described in Section~\ref{sec:length}.
Pages should be numbered in the version submitted for review, but \textbf{pages should not be numbered in the camera-ready version}.
\paragraph{\LaTeX-specific details:}
The style files will generate page numbers when {\small\verb|\aclfinalcopy|} is commented out, and remove them otherwise.
\subsection{File Format}
\label{sect:pdf}
For the production of the electronic manuscript you must use Adobe's Portable Document Format (PDF).
Please make sure that your PDF file includes all the necessary fonts (especially tree diagrams, symbols, and fonts with Asian characters).
When you print or create the PDF file, there is usually an option in your printer setup to include none, all or just non-standard fonts.
Please make sure that you select the option of including ALL the fonts.
\textbf{Before sending it, test your PDF by printing it from a computer different from the one where it was created.}
Moreover, some word processors may generate very large PDF files, where each page is rendered as an image.
Such images may reproduce poorly.
In this case, try alternative ways to obtain the PDF.
One way on some systems is to install a driver for a postscript printer, send your document to the printer specifying ``Output to a file'', then convert the file to PDF.
It is of utmost importance to specify the \textbf{A4 format} (21 cm x 29.7 cm) when formatting the paper.
Print-outs of the PDF file on A4 paper should be identical to the hardcopy version.
If you cannot meet the above requirements about the production of your electronic submission, please contact the publication chairs as soon as possible.
\paragraph{\LaTeX-specific details:}
PDF files are usually produced from \LaTeX{} using the \texttt{\small pdflatex} command.
If your version of \LaTeX{} produces Postscript files, \texttt{\small ps2pdf} or \texttt{\small dvipdf} can convert these to PDF.
To ensure A4 format in \LaTeX, use the command {\small\verb|\special{papersize=210mm,297mm}|}
in the \LaTeX{} preamble (below the {\small\verb|\usepackage|} commands) and use \texttt{\small dvipdf} and/or \texttt{\small pdflatex}; or specify \texttt{\small -t a4} when working with \texttt{\small dvips}.
\subsection{Layout}
\label{ssec:layout}
Format manuscripts two columns to a page, in the manner these
instructions are formatted.
The exact dimensions for a page on A4 paper are:
\begin{itemize}
\item Left and right margins: 2.5 cm
\item Top margin: 2.5 cm
\item Bottom margin: 2.5 cm
\item Column width: 7.7 cm
\item Column height: 24.7 cm
\item Gap between columns: 0.6 cm
\end{itemize}
\noindent Papers should not be submitted on any other paper size.
If you cannot meet the above requirements about the production of your electronic submission, please contact the publication chairs above as soon as possible.
\subsection{Fonts}
For reasons of uniformity, Adobe's \textbf{Times Roman} font should be used.
If Times Roman is unavailable, you may use Times New Roman or \textbf{Computer Modern Roman}.
Table~\ref{font-table} specifies what font sizes and styles must be used for each type of text in the manuscript.
\begin{table}
\centering
\begin{tabular}{lrl}
\hline \textbf{Type of Text} & \textbf{Font Size} & \textbf{Style} \\ \hline
paper title & 15 pt & bold \\
author names & 12 pt & bold \\
author affiliation & 12 pt & \\
the word ``Abstract'' & 12 pt & bold \\
section titles & 12 pt & bold \\
subsection titles & 11 pt & bold \\
document text & 11 pt &\\
captions & 10 pt & \\
abstract text & 10 pt & \\
bibliography & 10 pt & \\
footnotes & 9 pt & \\
\hline
\end{tabular}
\caption{\label{font-table} Font guide. }
\end{table}
\paragraph{\LaTeX-specific details:}
To use Times Roman in \LaTeX2e{}, put the following in the preamble:
\begin{quote}
\small
\begin{verbatim}
\usepackage{times}
\usepackage{latexsym}
\end{verbatim}
\end{quote}
\subsection{Ruler}
A printed ruler (line numbers in the left and right margins of the article) should be presented in the version submitted for review, so that reviewers may comment on particular lines in the paper without circumlocution.
The presence or absence of the ruler should not change the appearance of any other content on the page.
The camera ready copy should not contain a ruler.
\paragraph{Reviewers:}
note that the ruler measurements may not align well with lines in the paper -- this turns out to be very difficult to do well when the paper contains many figures and equations, and, when done, looks ugly.
In most cases one would expect that the approximate location will be adequate, although you can also use fractional references (\emph{e.g.}, this line ends at mark $295.5$).
\paragraph{\LaTeX-specific details:}
The style files will generate the ruler when {\small\verb|\aclfinalcopy|} is commented out, and remove it otherwise.
\subsection{Title and Authors}
\label{ssec:title-authors}
Center the title, author's name(s) and affiliation(s) across both columns.
Do not use footnotes for affiliations.
Place the title centered at the top of the first page, in a 15-point bold font.
Long titles should be typed on two lines without a blank line intervening.
Put the title 2.5 cm from the top of the page, followed by a blank line, then the author's names(s), and the affiliation on the following line.
Do not use only initials for given names (middle initials are allowed).
Do not format surnames in all capitals (\emph{e.g.}, use ``Mitchell'' not ``MITCHELL'').
Do not format title and section headings in all capitals except for proper names (such as ``BLEU'') that are
conventionally in all capitals.
The affiliation should contain the author's complete address, and if possible, an electronic mail address.
The title, author names and addresses should be completely identical to those entered to the electronical paper submission website in order to maintain the consistency of author information among all publications of the conference.
If they are different, the publication chairs may resolve the difference without consulting with you; so it is in your own interest to double-check that the information is consistent.
Start the body of the first page 7.5 cm from the top of the page.
\textbf{Even in the anonymous version of the paper, you should maintain space for names and addresses so that they will fit in the final (accepted) version.}
\subsection{Abstract}
Use two-column format when you begin the abstract.
Type the abstract at the beginning of the first column.
The width of the abstract text should be smaller than the
width of the columns for the text in the body of the paper by 0.6 cm on each side.
Center the word \textbf{Abstract} in a 12 point bold font above the body of the abstract.
The abstract should be a concise summary of the general thesis and conclusions of the paper.
It should be no longer than 200 words.
The abstract text should be in 10 point font.
\subsection{Text}
Begin typing the main body of the text immediately after the abstract, observing the two-column format as shown in the present document.
Indent 0.4 cm when starting a new paragraph.
\subsection{Sections}
Format section and subsection headings in the style shown on the present document.
Use numbered sections (Arabic numerals) to facilitate cross references.
Number subsections with the section number and the subsection number separated by a dot, in Arabic numerals.
\subsection{Footnotes}
Put footnotes at the bottom of the page and use 9 point font.
They may be numbered or referred to by asterisks or other symbols.\footnote{This is how a footnote should appear.}
Footnotes should be separated from the text by a line.\footnote{Note the line separating the footnotes from the text.}
\subsection{Graphics}
Place figures, tables, and photographs in the paper near where they are first discussed, rather than at the end, if possible.
Wide illustrations may run across both columns.
Color is allowed, but adhere to Section~\ref{ssec:accessibility}'s guidelines on accessibility.
\paragraph{Captions:}
Provide a caption for every illustration; number each one sequentially in the form:
``Figure 1. Caption of the Figure.''
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Type the captions of the figures and tables below the body, using 10 point text.
Captions should be placed below illustrations.
Captions that are one line are centered (see Table~\ref{font-table}).
Captions longer than one line are left-aligned (see Table~\ref{tab:accents}).
\begin{table}
\centering
\begin{tabular}{lc}
\hline
\textbf{Command} & \textbf{Output}\\
\hline
\verb|{\"a}| & {\"a} \\
\verb|{\^e}| & {\^e} \\
\verb|{\`i}| & {\`i} \\
\verb|{\.I}| & {\.I} \\
\verb|{\o}| & {\o} \\
\verb|{\'u}| & {\'u} \\
\verb|{\aa}| & {\aa} \\\hline
\end{tabular}
\begin{tabular}{lc}
\hline
\textbf{Command} & \textbf{Output}\\
\hline
\verb|{\c c}| & {\c c} \\
\verb|{\u g}| & {\u g} \\
\verb|{\l}| & {\l} \\
\verb|{\~n}| & {\~n} \\
\verb|{\H o}| & {\H o} \\
\verb|{\v r}| & {\v r} \\
\verb|{\ss}| & {\ss} \\
\hline
\end{tabular}
\caption{Example commands for accented characters, to be used in, \emph{e.g.}, \BibTeX\ names.}\label{tab:accents}
\end{table}
\paragraph{\LaTeX-specific details:}
The style files are compatible with the caption and subcaption packages; do not add optional arguments.
\textbf{Do not override the default caption sizes.}
\subsection{Hyperlinks}
Within-document and external hyperlinks are indicated with Dark Blue text, Color Hex \#000099.
\subsection{Citations}
Citations within the text appear in parentheses as~\citep{Gusfield:97} or, if the author's name appears in the text itself, as \citet{Gusfield:97}.
Append lowercase letters to the year in cases of ambiguities.
Treat double authors as in~\citep{Aho:72}, but write as in~\citep{Chandra:81} when more than two authors are involved. Collapse multiple citations as in~\citep{Gusfield:97,Aho:72}.
Refrain from using full citations as sentence constituents.
Instead of
\begin{quote}
``\citep{Gusfield:97} showed that ...''
\end{quote}
write
\begin{quote}
``\citet{Gusfield:97} showed that ...''
\end{quote}
\begin{table*}
\centering
\begin{tabular}{lll}
\hline
\textbf{Output} & \textbf{natbib command} & \textbf{Old ACL-style command}\\
\hline
\citep{Gusfield:97} & \small\verb|\citep| & \small\verb|\cite| \\
\citealp{Gusfield:97} & \small\verb|\citealp| & no equivalent \\
\citet{Gusfield:97} & \small\verb|\citet| & \small\verb|\newcite| \\
\citeyearpar{Gusfield:97} & \small\verb|\citeyearpar| & \small\verb|\shortcite| \\
\hline
\end{tabular}
\caption{\label{citation-guide}
Citation commands supported by the style file.
The style is based on the natbib package and supports all natbib citation commands.
It also supports commands defined in previous ACL style files for compatibility.
}
\end{table*}
\paragraph{\LaTeX-specific details:}
Table~\ref{citation-guide} shows the syntax supported by the style files.
We encourage you to use the natbib styles.
You can use the command {\small\verb|\citet|} (cite in text) to get ``author (year)'' citations as in \citet{Gusfield:97}.
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\subsection{References}
Gather the full set of references together under the heading \textbf{References}; place the section before any Appendices.
Arrange the references alphabetically by first author, rather than by order of occurrence in the text.
Provide as complete a citation as possible, using a consistent format, such as the one for \emph{Computational Linguistics\/} or the one in the \emph{Publication Manual of the American
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Authors should not rely on automated citation indices to provide accurate references for prior and related work.
The following text cites various types of articles so that the references section of the present document will include them.
\begin{itemize}
\item Example article in journal: \citep{Ando2005}.
\item Example article in proceedings, with location: \citep{borschinger-johnson-2011-particle}.
\item Example article in proceedings, without location: \citep{andrew2007scalable}.
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\end{itemize}
\paragraph{\LaTeX-specific details:}
The \LaTeX{} and Bib\TeX{} style files provided roughly follow the American Psychological Association format.
If your own bib file is named \texttt{\small aacl-ijcnlp2020.bib}, then placing the following before any appendices in your \LaTeX{} file will generate the references section for you:
\begin{quote}\small
\verb|\bibliographystyle{acl_natbib}|\\
\verb|
\section{Method}
\subsection{Model Architecture}
End-to-end ST models directly map a source speech utterance into a sequence of target tokens.
We use the S-Transformer architecture proposed by \citep{di-gangi-etal-2019-enhancing}, which achieves competitive performance on the MuST-C dataset~\cite{di-gangi-etal-2019-must}.
In the encoder, a two-dimensional attention is applied after the CNN layers and a distance penalty is introduced to bias the attention towards short-range dependencies.
We investigate two types of simultaneous translation mechanisms, flexible and fixed policy. In particular, we investigate monotonic multihead attention~\cite{ma2020monotonic}, which is an instance of flexible policy and the prefix-to-prefix model~\cite{ma-etal-2019-stacl}, an instance of fixed policy, designated by wait-$k$ from now on.
\begin{description}[style=unboxed,leftmargin=0cm]
\item[Monotonic Multihead Attention](MMA)~\cite{ma2020monotonic} extends monotonic attention
\cite{raffel2017online, arivazhagan-etal-2019-monotonic} to Transformer-based models.
Each head in each layer has an independent step probability
$p_{ij}$ for the $i$th target and $j$th source step, and then uses a closed form expected attention for training.
A weighted average and variance loss were proposed to control the behavior of the attention heads and thus the trade-offs between quality and latency.
\item[Wait-$k$]~\citep{ma-etal-2019-stacl} is a fixed policy that waits for $k$ source tokens,
and then reads and writes alternatively.
Wait-$k$ can be a special case of Monotonic Infinite-Lookback Attention (MILk) \cite{arivazhagan-etal-2019-monotonic} or MMA where the step-wise probability $p_{ij} = 0$ if $j - i < k$ else $p_{ij} = 1$.
\end{description}
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{figures/arch.png}
\caption{
Simul-ST architecture with pre-decision module.
Blue states in the figure indicate when the Simul-ST model triggers the simultaneous decision making process.}
\label{fig:arch}
\end{figure}
\subsection{Pre-Decision Module}
In SimulMT, READ or WRITE decisions are made at the token (word or BPE) level.
However, with speech input, it is unclear when to make such decisions.
For example, one could choose to read or write after each frame or after
generating each encoder state. Meanwhile, a frame typically only covers 10ms of the input while an encoder state generally covers 40ms of the input (assuming a subsampling factor of 4), while the average length of a word in our dataset is 270ms.
Intuitively, a policy like wait-$k$ will not have enough information
to write a token after reading a frame or generating an encoder state.
In principle, a flexible or model-based policy such as MMA should be able to handle granular input. Our analysis will show, however, that while MMA is more robust to the granularity of the input, it also performs poorly when the input is too fine-grained.
In order to overcome these issues, we introduce the notion of pre-decision module, which groups frames or encoder states, prior to making a decision.
A pre-decision module generates a series of trigger probabilities $p_{tr}$ on each encoder states to
indicate whether a simultaneous decision should be made.
If $p_{tr} > 0.5$, the model triggers the simultaneous decision making,
otherwise keeps reading new frames.
We propose two types of pre-decision module.
\begin{description}[style=unboxed,leftmargin=0cm]
\item[Fixed Pre-Decision]
A straightforward policy for a fixed pre-decision module is to trigger simultaneous decision making every fixed number of frames.
Let $\Delta t$ be the time corresponding to this fixed number of frames, with $\Delta t$ a multiple of $T_s$, and $r_e=\text{int}(|{\bm{X}}| / |{\bm{H}}|)$. $p_{tr}$ at encoder step $j$ is defined in \cref{eq:trigger-probablity-fixed}:
%
\begin{equation}
p_{tr}(j) =
\begin{cases}
1 & \text{if } \text{mod}(j \cdot r_e \cdot T_s, \Delta t) = 0, \\
0 & \text{Otherwise.}
\end{cases}
\label{eq:trigger-probablity-fixed}
\end{equation}
\item[Flexible Pre-Decision]
We use an oracle flexible pre-decision
module that uses the source boundaries either at the word or phoneme level. Let ${\bm{A}}$ be the alignment between encoder states and source labels (word or phoneme). ${\bm{A}}(h_i)$ represents the token that $h_i$ aligns to.
The trigger probability can then be defined in \cref{eq:trigger-probablity-flexible}:
\begin{equation}
p_{tr}(j) = \begin{cases}
0 & \text{if } {\bm{A}}(h_j) = {\bm{A}}(h_{j-1}) \\
1& \text{Otherwise.}
\end{cases}
\label{eq:trigger-probablity-flexible}
\end{equation}
\end{description}
\section{Task formalization}
\label{sec:task}
A SimulST model takes as input a sequence of acoustic features ${\bm{X}}=[{\bm{x}}_1,...{\bm{x}}_{|{\bm{X}}|}]$ extracted from speech samples every $T_{s}$ ms,
and generates a sequence of text tokens ${\bm{Y}}=[y_1,...,y_{|{\bm{Y}}|}]$ in a target language.
Additionally, it is able to generate $y_i$ with only partial input
${\bm{X}}_{1:n(y_i)}=[{\bm{x}}_1,...{\bm{x}}_{n(y_i)}]$,
where $n(y_i) \leq |{\bm{X}}|$ is the number of frames needed to generate the $i$-th target token $y_i$.
Note that $n$ is a monotonic function, i.e.\ $n(y_{i-1}) \leq n(y_i)$.
A SimulST model is evaluated with respect to quality, using BLEU~\cite{papineni2002bleu}, and latency. We introduce two latency evaluation methods for SimulST that are adapted from SimulMT.
We first define two types of delays to generate the word $y_i$, a
computation-aware (CA) and a non computation-aware (NCA) delay.
The CA delay of $y_i$, $d_{\text{CA}}(y_i)$,
is defined as the time that elapses (speech duration) from the beginning of the process to the prediction of $y_i$,
while the NCA delay for $y_i$ $d_{\text{CA}}(y_i)$ is defined by
$d_{\text{NCA}}(y_i) = T_{\text{s}} \cdot n(y_i)$.
Note that $d_\text{NCA}$ is an ideal case for $d_\text{CA}$ where the computational time for the model is ignored.
Both delays are measured in milliseconds.
Two types of latency measurement, $L_{CA}$ and $L_{NCA}$, are calculated accordingly: $L = \mathcal{C}({\bm{D}})$ where $\mathcal{C}$ is a latency metric and $\mathbf{D}=[d(y_1), ..., d(y_{|{\bm{Y}}|})]$.
To better evaluate the latency for SimulST, we introduce a modification to AL. We assume an oracle system that can perform perfect simultaneous translation
for both latency and quality, while in \citet{ma-etal-2019-stacl} the oracle is ideal only from the latency perspective. We evaluate the lagging based on time rather than steps. The modified AL metric is defined in \cref{eq:al}:
\begin{equation}
\text{AL} = \frac{1}{\tau(|{\bm{X}}|)} \sum^{\tau(|{\bm{X}}|)}_{i=1} d(y_i) - \frac{|{\bm{X}}|}{|{\bm{Y}}^*|} \cdot T_s \cdot (i - 1)
\label{eq:al}
\end{equation}
where $|{\bm{Y}}^*|$ is the length of the reference translation, $\tau(|{\bm{X}}|)$ is the index of the first target token generated when the model has read the full input. There are two benefits from this modification. The first is that latency is measured using time instead of steps, which makes it agnostic to preprocessing and segmentation. The second is that it is more robust and can prevent an extremely low and trivial value when the prediction is significantly shorter than the reference.
\section{Future works}
Given the results from this paper, the furture work lie in folloing three categories:
\begin{description}
\item[Incremental Encoder] From the \cref{sec:cal} we can find that encoder updates cause the major computation-aware latency when the step size is small.
Incremental encoder also allow a high-accuracy low-mismatch trainable pre desicion module \cite{ren-etal-2020-simulspeech}.
\item[Multi-Task Learning] Due to sparsity of speech data,
multi-task learning leverage the quality of the models.
Potentially we can also explore approaches for better latency by SimulST + SimulMT multi-task learning.
\item[Learned flexible pre-desicion module]
\end{description}
\section{Conclusion}
We investigated how to adapt SimulMT methods to end-to-end SimulST by introducing the concept of pre-decision module. We also adapted Average Lagging to be computation-aware. The effects of combining a fixed or flexible pre-decision module with a fixed or flexible policy were carefully analyzed. Future work includes building an incremental encoder to reduce the CA latency and design a learnable pre-decision module.
\subsection{Computation Aware Latency}
\label{sec:cal}
We also consider the computation-aware latency described in \cref{sec:task},
shown in \cref{fig:ca_curve}. The focus is on fixed pre-decision approaches in order to
understand the relation between the granularity of the pre-decision and the computation time.
\cref{fig:ca_curve} shows that as the step size increases, the difference between the NCA and the CA latency shrinks. This is because with larger step sizes, there is less overhead of recomputing the bidirectional encoder states \footnote{This is a common practice in SimulMT where the input length is significantly shorter than in SimulST~\cite{arivazhagan-etal-2019-monotonic, ma-etal-2019-stacl,arivazhagan-etal-2020-translation}}.
We recommend future work on SimulST to make use of CA latency as it reflects a more realistic evaluation, especially in low-latency regimes, and is able to distinguish streaming capable systems.
\section{Introduction}
Simultaneous speech translation (SimulST) generates a translation from an input speech utterance before the end of the utterance has been heard. SimulST systems aim at generating translations with maximum quality and minimum latency,
targeting applications such as video caption translations and real-time language interpretation.
While great progress has recently been achieved on both end-to-end speech translation~\cite{ansari-etal-2020-findings} and simultaneous text
translation (SimulMT)~\cite{grissom2014don, gu2017learning, luo2017learning, lawson2018learning, alinejad2018prediction, zheng-etal-2019-simultaneous, zheng-etal-2019-simpler,ma2020monotonic,arivazhagan-etal-2019-monotonic, arivazhagan-etal-2020-translation}, little work has combined the two tasks together~\cite{ren-etal-2020-simulspeech}.
End-to-end SimulST models feature a smaller model size, greater inference speed and fewer compounding errors compared to their cascade counterpart, which perform streaming speech recognition followed by simultaneous machine translation. In addition, it has been demonstrated that end-to-end SimulST systems can have lower latency than cascade systems~\cite{ren-etal-2020-simulspeech}.
In this paper, we study how to adapt methods developed for SimulMT to end-to-end SimulST.
To this end, we introduce the concept of pre-decision module. Such module guides how to group encoder states into meaningful units prior to making a READ/WRITE decision. A detailed analysis of the latency-quality trade-offs when combining a fixed or flexible pre-decision module with a fixed or flexible policy is provided. We also introduce a novel computation-aware latency metric, adapted from Average Lagging (AL)~\cite{ma-etal-2019-stacl}.
\section{Experiments}
We conduct experiments on the English-German portion of the MuST-C dataset~\cite{di-gangi-etal-2019-must}, where source audio, source transcript and target translation are available.
We train on 408 hours of speech and 234k sentences of text data.
We use Kaldi~\cite{povey2011kaldi} to extract 80 dimensional log-mel filter bank features,
computed with a 25$ms$ window size and a 10$ms$ window shift.
For text, we use SentencePiece~\cite{kudo2018sentencepiece} to generate a unigram vocabulary of size 10,000.
We use Gentle\footnote{https://lowerquality.com/gentle/} to generate the alignment between source text and speech
as the label to generate the oracle flexible pre-decision module. Translation quality is evaluated with case-sensitive detokenized BLEU with \textsc{SacreBLEU}~\cite{post-2018-call}.
The latency is evaluated with our proposed modification of AL~\cite{ma-etal-2019-stacl}.
All results are reported on the MuST-C dev set.
All speech translation models are first pre-trained on the ASR task where the target vocabulary is character-based, in order to initialize the encoder.
We follow the same hyperparameter settings from \cite{di-gangi-etal-2019-enhancing}.
We follow the latency regularization method introduced by \cite{ma2020monotonic, arivazhagan-etal-2019-monotonic}.
The objective function to optimize is
\begin{equation}
\begin{aligned}
L = - \text{log} \left( P({\bm{Y}}| {\bm{X}})\right) + \lambda \text{max}\left(\mathcal{C}({\bm{D}}), 0\right)
\end{aligned}
\label{eq:loss}
\end{equation}
Where $\mathcal{C}$ is a latency metric (AL in this case) and ${\bm{D}}$ is described in \cref{sec:task}.
Only samples with $\text{AL} > 0$ are regularized to avoid overfitting.
For the models with monotonic multihead attention,
we first train a model without latency with $\lambda_{\text{latency}}=0$.
After the model converges, $\lambda_{\text{latency}}$ is set to a desired value and we continue trining the model until convergence.
\section{Related Works}
The research of simultaneous machine translation lies in three categories.
The first one is the fixed policy, which uses a pre-designed rule to operate actions.
\cite{cho2016can, ma-etal-2019-stacl, dalvi-etal-2018-incremental} propose an incremental decoding method,
In the second category, a flexible policy is learnt from data with reinforcement learning or other methods.
\cite{grissom2014don, gu2017learning, luo2017learning, lawson2018learning, alinejad2018prediction, zheng-etal-2019-simultaneous, zheng-etal-2019-simpler}
The last one is monotonic attention which use an close form expected attention.
\cite{raffel2017online} first introduce hard monotonic attention for online linear time decoding.
\cite{chiu2018mocha} extended \citet{raffel2017online}'s work to let the model attend to a chunk of encoder state.
\cite{arivazhagan-etal-2019-monotonic} introduce the infinite lookback mechanism on top of the monotonic attention to improve the translation quality |
2,869,038,156,349 | arxiv | \section{Introduction}\label{secintro1}
Consider a linear model with response $\mathbf{y} \in\Re^{n}$, model
matrix $\mathbf{X}_{n \times p }$, regression coefficients
$\bolds\beta\in\Re^{p}$ and error ${\bolds\varepsilon} \in
\Re^{n}$:
\[
\mathbf{y} = \mathbf{X}\bolds\beta+ {\bolds\varepsilon}.
\]
We will assume that $\mathbf{X}$ contains a column of ones to account
for the intercept in the model.
Given data for the $i$th sample $(y_{i}, \mathbf{x}_{i})$, $i = 1,\ldots, n$ (where, $\mathbf{x}_{i} \in\Re^{p\times1}$) and
regression coefficients $\bolds\beta$, the
$i$th residual is given by the usual notation $r_{i} = y_{i} - \mathbf
{x}_{i}' \bolds\beta$ for $i = 1, \ldots, n$.
The traditional Least Squares (LS) estimator given by
\begin{equation}
\label{eqls} \hat{\bolds\beta}{}^{(\mathrm{LS})} \in\argmin _{\bolds\beta}
\sum_{i = 1}^{n} r_{i}^2
\end{equation}
is a popular and effective method for estimating the regression
coefficients when the error vector
${\bolds\varepsilon}$ has \emph{small} $\ell_2$-norm. However, in
the presence of outliers, the LS estimators do not work favorably---a
single outlier can have an arbitrarily large effect on the estimate.
The robustness of an estimator vis-a-vis outliers is often quantified
by the notion of
its finite sample breakdown point~[\citet{donoho1983notion,hampel1971general}].
The LS estimate~(\ref{eqls}) has a limiting (in the limit
$n\rightarrow\infty$ with $p$ fixed) breakdown point~[\citet
{hampel1971general}] of zero.
The Least Absolute Deviation (LAD) estimator given by
\begin{equation}
\label{eqlad} \hat{\bolds\beta}{}^{(\mathrm{LAD})} \in\argmin _{\bolds\beta}
\sum_{i = 1}^{n} \llvert r_{i}
\rrvert
\end{equation}
considers the
$\ell_{1}$-norm on the residuals, thereby implicitly assuming that the
error vector ${\bolds\varepsilon}$ has small
$\ell_{1}$-norm. The LAD estimator is not resistant to large
deviations in the covariates and, like the optimal LS solutions, has a
breakdown point of
zero (in the limit $n \rightarrow\infty$ with $p$ fixed).
M-estimators~[\citet{huber1973robust}] are obtained by minimizing
a loss function of the residuals of the form
$\sum_{i=1}^{n} \rho(r_{i})$, where $\rho(r)$ is a symmetric
function with a unique minimum at zero. Examples include the
Huber function and the Tukey function~[\citet{rousseeuw2005robust,huber2011robust}], among others. \mbox{M-}estimators often
simultaneously estimate the scale parameter along with the regression
coefficient.
M-estimators too are severely affected by the presence of outliers in
the covariate space.
A generalization of M-estimators are Generalized
\mbox{M-}estimators [\citet{rousseeuw2005robust,huber2011robust}],
which bound the influence of outliers in the covariate space by the
choice of a
weight function dampening the effect of outlying covariates. In some
cases, they have an improved
finite-sample breakdown point of $1/(p+1)$.
The repeated median estimator~[\citet{siegel1982robust}] with
breakdown point of approximately 50\%, was one of the earliest estimators
to achieve a very high breakdown point. The estimator however, is not
equivariant under linear transformations of the covariates.
\citet{rousseeuw1984least} introduced Least Median of Squares
(LMS)~[see also \citet{hampel1975beyond}] which
minimizes the median of the absolute residuals\footnote{Note that the
original definition of LMS~[\citet{rousseeuw1984least}] considers
the squared residuals instead of the absolute values. However, we will
work with the absolute values, since the problems are equivalent.}
\begin{equation}
\label{eqmedian1-lms}
\hat{\bolds\beta}{}^{(\mathrm{LMS})} \in \argmin
_{\bolds\beta} \Bigl( \operatorname{median}\limits
_{i
= 1, \ldots, n } \llvert r_{i}
\rrvert \Bigr).
\end{equation}
The LMS problem is equivariant and has a limiting breakdown point of
50\%---making
it the first equivariant estimator to achieve the maximal possible
breakdown point in the limit $n\rightarrow\infty$ with $p$ fixed.
Instead of considering the median, one may consider more generally, the
$q$th order statistic, which leads to the
Least Quantile of Squares (LQS) estimator:
\begin{equation}
\label{eqmedian-q} \hat{\bolds\beta}{}^{(\mathrm{LQS})} \in \argmin _{\bolds\beta}
\llvert r_{(q)}\rrvert,
\end{equation}
where $r_{(q)}$ denotes the residual, corresponding to the $q$th
ordered absolute residual:
\begin{equation}
\label{order-elts-1} \llvert r_{(1)} \rrvert \leq\llvert r_{(2)}
\rrvert \leq\cdots\leq\llvert r_{(n)} \rrvert.
\end{equation}
\citet{rousseeuw1984least} showed that if the sample points
$(y_{i}, \mathbf{x}_{i})$, $i = 1, \ldots, n$ are in general position,
that is,
for any subset of ${\mathcal I} \subset\{1, \ldots, n \}$ with $\llvert
{\mathcal I} \rrvert = p$, the $p\times p$ submatrix $X_{\mathcal I}$ has
rank $p$; an optimal LMS solution~(\ref{eqmedian1-lms}) exists and has
a finite sample breakdown point
of $(\lfloor n/2\rfloor- p + 2)/n$, where $\lfloor s\rfloor$ denotes
the largest integer smaller than or equal to $s$.
\citet{rousseeuw1984least} showed that the finite sample
breakdown point of the estimator~(\ref{eqmedian1-lms})
can be further improved to achieve the maximum possible finite sample
breakdown point
if one considers the estimator~(\ref{eqmedian-q}) with $q = \lfloor
n/2 \rfloor+ \lfloor(p +1)/2 \rfloor$.
The LMS estimator has low efficiency~[\citet{rousseeuw1984least}]. This can, however, be improved by using certain
post-processing methods on the LMS estimator---the one step M-estimator
of~\citet{bickel1975one} or
a reweighted least-squares estimator, where points with large values of
LMS residuals are given small weight are popular methods that are used
in this vein.
\subsection*{Related work}
It is a well recognized fact that the LMS problem is computationally
demanding due to the combinatorial nature of the problem.
\citet{bernholt-hard} showed that computing an optimal LMS
solution is NP-hard.
Many algorithms based on different approaches have been proposed for
the LMS problem over the past thirty years.
State of the art algorithms, however, fail to obtain a global minimum of
the LMS problem for problem sizes larger than $n = 50, p = 5$. This
severely limits the use of LMS for important real world multivariate
applications,
where $n$ can easily range in the order of a few thousands.
It goes without saying that a poor local minimum for the LMS problem
may be misleading from a statistical inference point of view [see also
\citet{Stromberg1993} and references therein for related
discussions on this matter].
The various algorithms presented in the literature for the LMS can be
placed into two very broad categories.
One approach computes an optimal solution to the LMS problem using
geometric characterizations of the fit---they typically rely on
complete enumeration and have complexity $O(n^p)$.
The other approach gives up on obtaining an
optimal solution and resorts to heuristics and/or randomized algorithms
to obtain approximate solutions to the LMS problem. These methods,
to the best of our knowledge, do not provide certificates about the
quality of the solution obtained. We describe below a brief overview of
existing algorithms for LMS.
Among the various algorithms proposed in the literature for the LMS
problem, the most popular seems to be
{\small\textsc{PROGRESS}} (Program for Robust Regression)~[\citet
{rousseeuw2005robust,Rousseeuw97recentdevelopments}]. The \mbox{algorithm}
does a complete enumeration of all $p$-subsets of the $n$ sample
points, computes the hyperplane passing through them and finds the
configuration leading to the smallest value of the objective. The
algorithm has a run-time complexity of $O(n^p)$ and assumes that
the data points are in general position.
For computational scalability, heuristics that randomly sample subsets
are often used.
See also~\citet{Barreto2006SCL16479631648203} for a recent work
on algorithms for the bivariate regression problem.
\citet{steele1986algorithms} proposed exact algorithms for LMS
for $p = 2$ with complexity $O(n^3)$ and some probabilistic speed-up methods
with complexity $O((n\log(n))^2)$.
\citet{Stromberg1993} proposed an exact algorithm for LMS with
run-time $O(n^{(p+2)}\log(n))$ using some
insightful geometric properties of the LMS fit. This method does a
brute force search
among ${n \choose p+1}$ different regression coefficient values and
scales up to problem sizes $n=50$ and $p=5$.
\citet{agullo-1997} proposed a finite branch and bound technique
with run-time complexity $O(n^{p+2})$ to obtain an optimal solution to
the LMS problem motivated
by the work of~\citet{Stromberg1993}. The algorithm showed
superior performance compared to methods preceding it
and can scale up to problem sizes $n \approx70, p \approx4$.
\citet{erickson2006least} give an exact algorithm with run-time
$O(n^p\log(n))$ for LMS
and also show that computing an optimal LMS solution requires $O(n^p)$ time.
For the two-dimensional case $p = 2$, \citet{guided-topo-jasa-87}
proposed an exact algorithm for LMS with complexity
$O(n^2)$ using the topological sweep-line technique.
\citet{Giloni2002LTS22581142258762} propose integer optimization
formulations for the LMS problem, however,
no computational experiments are reported---the practical performance
of the proposed method thus remains unclear.
\citet{Mount20072461} present an algorithm based on branch and
bound for $p=2$ for computing approximate solutions to the LMS problem.
\citet{mount-quantile} present a quantile approximation algorithm
with approximation factor $\varepsilon$ with complexity
$O(n\log(n) + (1/\varepsilon)^{O(p)})$.
\citet{chakraborty2008optimization} present probabilistic search
algorithms for a class of problems in robust statistics.
\citet{Nunkesser20103242} describe computational procedures based
on heuristic search strategies using evolutionary algorithms for some
robust statistical estimation problems including LMS.
\citet{Hawkins199381} proposes a probabilistic algorithm for LMS
known as the ``Feasible Set Algorithm'' capable of
solving problems up to sizes $n = 100, p = 3$.
\citet{Bernholt05computingthe} describes a randomized algorithm
for computing the LMS running in
$O(n^p)$ time and $O(n)$ space, for fixed $p$.
\citet{Olson97anapproximation} describes an approximation
algorithm to compute an optimal LMS solution within an approximation
factor of two using randomized
sampling methods---the method has (expected) run-time complexity of
$O(n^{p-1}\log(n))$.
\subsection*{Related approaches in robust regression}
Other estimation procedures that achieve a high breakdown point and
good statistical efficiency
include the least trimmed squares estimator~[\citet
{rousseeuw1984least,rousseeuw2005robust}], which minimizes the sum of
squares of the $q$ smallest squared residuals. Another popular approach
is based on S-estimators~[\citet
{rousseeuw1984least,rousseeuw2005robust}], which are
a type of M-estimators of scale on the residuals.
These estimation procedures like the LMS estimator are
NP-hard~[\citet{bernholt-hard}].
We refer the interested reader to~\citet{hubert2008high} for a
nice review of various robust statistical methods and
their applications~[\citet
{meer1991robust,comp-vision-siam-review-99,rousseeuw2006robustness}].
\subsection*{What this paper is about}
In this paper, we propose a computationally tractable framework to
compute a globally optimal solution to the LQS problem~(\ref
{eqmedian-q}), and in particular the LMS problem
via modern optimization methods: first-order methods from continuous
optimization and mixed integer optimization (MIO), see~\citet
{bertsimas2005optimization}.
Our view of computational tractability is not polynomial time solution
times as these do not exist for the LQS problem unless P${}={}$NP. Rather it
is the ability of a method to solve problems of practical interest in
times that are appropriate for the application addressed.
An important advantage of our framework is that it easily adapts to
obtain solutions to more general variants of~(\ref{eqmedian-q}) under
polyhedral constraints, that is,
\begin{equation}
\label{eqmedian-q-abs-gen} \mini_{\bolds\beta}\qquad \llvert r_{(q)} \rrvert,\qquad
\mbox{subject to}\qquad \mathbf{A}\bolds\beta\leq\mathbf{b},
\end{equation}
where $\mathbf{A}_{m \times p}, \mathbf{b}_{m \times1}$ are given
parameters in the problem representing
side constraints on the variable $\bolds\beta$ and ``$\leq$''
denotes component wise inequality.
This is useful if one would like to incorporate some form of
regularization on the $\bolds\beta$ coefficients, for example,
$\ell_{1}$ regularization~[\citet{Ti96}] or generalizations thereof.
\subsection*{Contributions}
Our contributions in this paper may be summarized as follows:
\begin{longlist}[(3)]
\item[(1)] We use MIO to find a provably optimal solution to the LQS
problem. Our
framework has the appealing characteristic that if we terminate the
algorithm early, we obtain a solution with a guarantee
on its suboptimality. We further show that the MIO approach leads to an
optimal solution for \emph{any}
dataset where the data-points $(y_{i}, \mathbf{x}_{i})$'s are not
necessarily in general position.
Our framework enables us to provide a simple proof of the breakdown
point of the LQS objective value, generalizing the existing results for
the problem.
Furthermore, our approach is readily generalizable to
problems of the type~(\ref{eqmedian-q-abs-gen}).
\item[(2)] We introduce a variety of solution methods based on modern
continuous optimization---first-order
subdifferential based minimization, sequential linear optimization and
a hybrid version of these two methods that provide near optimal
solutions for
the LQS problem. The MIO algorithm is found to significantly benefit
from solutions obtained by the continuous optimization methods.
\item[(3)] We report computational results with both synthetic and
real-world datasets that show that
the MIO algorithm with warm starts from the continuous optimization
methods solve small ($n=100$) and medium ($n=500$) size LQS problems to
provable optimality in under two hours, and outperform all publicly
available methods for large-scale ($n={}$10,000) LQS problems, but
without showing provable optimality in under two hours of computation time.
\end{longlist}
\subsection*{Structure of the paper}
The paper is organized as follows. Section~\ref{secmio-method1}
describes MIO approaches for the LQS problem.
Section~\ref{secconts-opt-methods1} describes continuous optimization
based methods for obtaining
local minimizers of the LQS problem.
Section~\ref{secprops-lqs} describes properties of an optimal LQS
solution. Section~\ref{seccomps-1} describes computational
results and experiments. The last section contains our key conclusions.
\section{Mixed integer optimization formulation}\label{secmio-method1}
In this section, we present an exact MIO formulation for the LQS problem.
For the sake of completeness, we will first introduce the definition of
a linear MIO problem.
The generic MIO framework concerns the following optimization problem:
\begin{eqnarray}\label{mio-gen-setup}
\mini &\qquad& \mathbf{c}'\bolds{\alpha} + \mathbf{d}' \bolds{\theta},\nonumber
\\
&&A\bolds{\alpha} + B\bolds\theta\bolds\geq \mathbf{b},
\nonumber\\[-8pt]\\[-8pt]\nonumber
&&\bolds\alpha\in\Re^n_{+},
\\
&&\bolds\theta\in\{0, 1\}^m,\nonumber
\end{eqnarray}
where $\mathbf{c}\in\Re^{n}, \mathbf{d}\in\Re^{m}, A \in\Re^{k
\times n}, B \in\Re^{k \times m}, \mathbf{b} \in\Re^{k}$ are the
given parameters of the problem;
$\Re^{n}_{+}$ denotes the nonnegative $n$-dimensional orthant, the
symbol $\bolds{\geq}$ denotes element-wise inequalities and
we optimize over both continuous ($\bolds{\alpha}$) and discrete
($\bolds\theta$) variables. For background on MIO,
see~\citet{bertsimas2005optimization}.
Consider a list of $n$ numbers $\llvert r_{1}\rrvert, \ldots, \llvert r_{n}\rrvert $, with the
ordering described in~(\ref{order-elts-1}).
To model the sorted $q$th residual, that is, $\llvert r_{(q)}\rrvert $, we need to
express the fact that $r_{i} \leq\llvert r_{(q)}\rrvert $
for $q$ many residuals $\llvert r_{i}\rrvert $'s from $\llvert r_{1}\rrvert, \ldots, \llvert r_{n}\rrvert $. To
do so, we introduce the binary variables $z_{i}$, $i = 1, \ldots, n$
with the interpretation
\begin{equation}
\label{zi-def-1} z_{i} = \cases{ 1, &\quad if $\llvert r_{i}
\rrvert \leq\llvert r_{(q)}\rrvert $,
\cr
0, &\quad otherwise.}
\end{equation}
We further introduce auxiliary continuous variables $\mu_{i},
\bar{\mu}_{i} \geq0$, such that
\begin{equation}
\label{order-resid-1} \llvert r_{i}\rrvert - \mu_{i} \leq\llvert
r_{(q)} \rrvert \leq\llvert r_{i}\rrvert + \bar{\mu
}_{i},\qquad i = 1, \ldots, n,
\end{equation}
with the conditions
\begin{eqnarray}
\label{eqnmus-1}
\mbox{if } \llvert r_{i}\rrvert &\geq& \llvert
r_{(q)}\rrvert,\qquad\mbox{then }\bar{\mu}_{i}=0,
\mu_{i} \geq0\quad\mbox{and}
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
\mbox{if } \llvert r_{i}\rrvert &\leq& \llvert
r_{(q)}\rrvert,\qquad\mbox{then } \mu_{i}=0, \bar{\mu}_{i} \geq0.
\end{eqnarray}
We thus propose the following MIO formulation:
\begin{eqnarray}\label{lqs-obs-1}
&& \mini\qquad \gamma,\nonumber
\\
&&\qquad\mbox{subject to}\qquad \llvert r_{i}\rrvert + \bar{
\mu}_{i} \geq\gamma,\qquad i = 1,\ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad}\gamma\geq\llvert r_{i}\rrvert - \mu_{i},\qquad i = 1,\ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad}M_{u}z_{i} \geq\bar{\mu}_{i},\qquad i = 1,\ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad}M_{\ell} ( 1- z_{i}) \geq\mu_{i},\qquad i = 1,\ldots, n,
\\
&&\phantom{\qquad\mbox{subject to}\qquad}\sum_{i=1}^{n} z_{i} = q,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad}\mu_{i} \geq0,\qquad i = 1, \ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad}\bar{\mu}_{i} \geq0,\qquad i = 1, \ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad}z_{i} \in\{0, 1\},\qquad i = 1, \ldots, n,\nonumber
\end{eqnarray}
where, $\gamma, z_{i},\mu_{i}, \bar{\mu}_{i}$, $i = 1, \ldots,n$ are the optimization variables,
$M_{u}, M_{\ell}$ are the so-called \emph{Big-M} constants.
Let us denote the optimal solution of problem~(\ref{lqs-obs-1}), which
depends on $M_{\ell}, M_{u}$,
by $\gamma^*$.
Suppose we consider $M_{u}, M_{\ell} \geq\max_{i} \llvert r_{(i)}\rrvert $---it
follows from formulation~(\ref{lqs-obs-1}) that
$q$ of the $\mu_{i}$'s are zero. Thus, $\gamma^*$ has to be larger
than at least $q$ of the $\llvert r_{i}\rrvert $ values.
By arguments similar to the above, we see that,
since $(n - q)$ of the $z_{i}$'s are zero, at least $(n - q)$ many
$\bar{\mu}_{i}$'s are zero. Thus,
$\gamma^{*}$ is less than or equal to at least $(n - q)$ many of the
$\llvert r_{i}\rrvert, i=1, \ldots, n$ values. This shows that
$\gamma^{*}$ is indeed equal to $\llvert r_{(q)}\rrvert $, for $M_{u}, M_{\ell}$
sufficiently large.
We found in our experiments that, in formulation~(\ref{lqs-obs-1}), if
$z_{i}=1$, then $\bar{\mu}_{i}=M_{u}$ and if $z_{i}= 0$ then
$\mu_{i}=M_{\ell}$.
Though this does not interfere with the definition of~$\llvert r_{(q)}\rrvert $, it
creates a difference in the strength of the MIO formulation.
We describe below how to circumvent this shortcoming.
From~(\ref{eqnmus-1}), it is clear that $\bar{\mu}_{i}\mu_{i}
=0$, $\forall i = 1, \ldots, n$.
The constraint $\bar{\mu}_{i}\mu_{i} =0$ can be modeled via
integer optimization using Specially Ordered Sets of type~1 [\citet{bertsimas2005optimization}], that is, SOS-1 constraints
as follows:
\begin{equation}
\label{sos-def-1} \mu_{i}\bar{\mu}_{i}=0\quad\iff\quad (\mu_{i}, \bar{\mu}_{i} ) \dvtx \mbox{SOS-1},
\end{equation}
for every $i = 1, \ldots, n$.
In addition, observe that, for $M_{\ell}$ sufficiently large and every
$i\in\{1, \ldots, n\}$
the constraint $M_{\ell} ( 1- z_{i}) \geq\mu_{i} \geq0$ can be
modeled\footnote{To see why this is true, observe that
$(\mu_{i}, z_{i}) \dvtx \mbox{SOS-1}$ is equivalent to $\mu_{i}z_{i}
=0$. Now, since \mbox{$z_{i} \in\{ 0, 1\}$}, we have the following possibilities:
$z_{i}= 0$, in which case $\mu_{i}$ is free; if $z_{i} = 1$, then $\mu
_{i} = 0$. }
by a SOS-1 constraint---$(\mu_{i}, z_{i}) \dvtx \mbox{SOS-1}$.
In light of this discussion, we see that
\begin{equation}
\label{eqn-sos-split1} \llvert r_{i}\rrvert - \llvert r_{(q)}\rrvert
= {\mu}_{i} - \bar{\mu}_{i},\qquad ( \mu _{i},
\bar{\mu}_{i} ) \dvtx \mbox{SOS-1}.
\end{equation}
We next show that $\llvert r_{(q)}\rrvert \geq\bar{\mu}_{i}$ and $\mu_{i}
\leq\llvert r_{i}\rrvert $ for all $i = 1, \ldots, p$.
When $\llvert r_{i}\rrvert \leq\llvert r_{(q)}\rrvert $, it follows from the above
representation that
\[
{\mu}_{i} =0 \quad\mbox{and}\quad\bar{\mu}_{i} =
\llvert r_{(q)}\rrvert - \llvert r_{i}\rrvert \leq\llvert
r_{(q)}\rrvert.
\]
When $\llvert r_{i}\rrvert > \llvert r_{(q)}\rrvert $, it follows that $\bar{\mu}_{i} =0$.
Thus, it follows that $0 \leq\bar{\mu}_{i} \leq\llvert r_{(q)}\rrvert $ for
all $i = 1, \ldots, n$.
It also follows by a similar argument that
$0 \leq\mu_{i} \leq\llvert r_{i}\rrvert $ for all $i$.
Thus, by using $\mbox{SOS-1}$ type of constraints, we can avoid the
use of \emph{Big-M}'s appearing in formulation~(\ref{lqs-obs-1}), as follows:
\begin{eqnarray}\label{lqs-obs-1-mod}
&&\mini\qquad \gamma,\nonumber
\\
&&\qquad\mbox{subject to}\qquad \llvert r_{i}\rrvert - \gamma=\mu_{i} - \bar{\mu }_{i},\qquad i = 1,\ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad}\sum_{i=1}^{n} z_{i} = q,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad}\gamma\geq\bar{\mu}_{i},\qquad i = 1,\ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad}\bar{\mu}_{i} \geq0, \qquad i = 1,\ldots, n,
\\
&&\phantom{\qquad\mbox{subject to}\qquad}\mu_{i}\geq0, \qquad i = 1, \ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad}(\bar{\mu}_{i}, \mu_{i}) \dvtx \mbox{SOS-1}, \qquad i = 1, \ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad}( z_{i}, {\mu}_{i} ) \dvtx \mbox{SOS-1},\qquad i = 1,\ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad}z_{i} \in\{0, 1\},\qquad i = 1, \ldots, n.\nonumber
\end{eqnarray}
Note, however, that the constraints
\begin{equation}
\label{const-1} \llvert r_{i}\rrvert - \gamma= \mu_{i} -
\bar{\mu}_{i},\qquad i = 1, \ldots, n
\end{equation}
are not convex in $r_{1}, \ldots, r_{n}$.
We thus introduce the following variables ${r}_{i}^{+}, {r}_{i}^{-}$, $i
= 1, \ldots, n$ such that
\begin{eqnarray}\label{resids-split-1}
r_{i}^{+} + r_{i}^{-}= \llvert r_{i}\rrvert,\qquad
y_{i} - \mathbf{x}_{i}'\bolds\beta= r_{i}^{+} -r_{i}^{-},
\nonumber\\[-8pt]\\[-8pt]
\eqntext{r^+_{i} \geq0, r^-_{i}\geq0, r^+_{i}r^-_{i}=0, i =1, \ldots, n.}
\end{eqnarray}
The constraint $r^+_{i}r^-_{i}=0$ can be modeled via SOS-1 constraints
\[
\bigl( r^+_{i}, r^-_{i} \bigr) \dvtx \mbox{SOS-1}\qquad
\mbox{for every } i = 1, \ldots, n.
\]
This leads to the following MIO for the LQS problem that we use in this paper:
\begin{eqnarray}\label{lqs-reg-form-1}
&&\mini\qquad \gamma,\nonumber
\\
&&\qquad\mbox{subject to}\qquad r_{i}^{+} + r_{i}^{-}- \gamma= \bar{\mu }_{i} - \mu_{i}, \qquad i = 1,\ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad} r_{i}^{+} - r_{i}^{-} =y_{i} - \mathbf{x}_{i}'\bolds\beta, \qquad i =1,\ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad} \sum_{i=1}^{n} z_{i} = q,
\\
&&\phantom{\qquad\mbox{subject to}\qquad}\gamma\geq\mu_{i} \geq0, \qquad i = 1,\ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad}\mu_{i} \geq0, \qquad i = 1,\ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad} \bar{\mu}_{i}\geq0, \qquad i = 1, \ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad} r_{i}^{+} \geq0, r_{i}^{-} \geq0,\qquad i = 1,\ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad} (\bar{\mu}_{i}, \mu_{i}) \dvtx \mbox{SOS-1},\qquad i =1, \ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad} \bigl( r^+_{i}, r^-_{i} \bigr) \dvtx \mbox{SOS-1},\qquad i= 1,\ldots, n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad} ( z_{i}, {\mu}_{i} ) \dvtx \mbox{SOS-1},\qquad i = 1, \ldots,n,\nonumber
\\
&&\phantom{\qquad\mbox{subject to}\qquad} z_{i} \in\{0, 1\}, \qquad i = 1, \ldots, n.\nonumber
\end{eqnarray}
To motivate the reader, we show in Figure~\ref{fig-alco-data1} an
example that illustrates that the MIO formulation~(\ref
{lqs-reg-form-1}) leads to a provably optimal solution
for the LQS problem. We give more details in Section~\ref{seccomps-1}.
\begin{figure
\includegraphics{1223f01.eps}
\caption{Figure showing the typical evolution of the MIO
formulation~(\protect\ref{lqs-reg-form-1}) for the ``Alcohol''
dataset with $n = 44, q = 31$ with $p=5$ (left panel) and
$p = 7$ (right panel). Global solutions for both the problems are found
quite quickly in both examples, but it takes longer to certify global
optimality via the lower bounds.
As expected, the time taken for the MIO to certify convergence to the
global optimum increases with increasing $p$.}\label{fig-alco-data1}
\end{figure}
\section{Continuous optimization based methods} \label{secconts-opt-methods1}
We describe two main approaches based on continuous optimization for
the LQS problem. Section~\ref{secseq-LP1} presents a method based on
sequential linear optimization and Section~\ref{secsubgrad1} describes a
first-order subdifferential based method for the LQS problem.
Section~\ref{sechybrid-1} describes hybrid combinations of the
aforementioned approaches,
which we have found, empirically, to provide high quality solutions.
Section~\ref{secinit-strategies} describes initialization strategies
for the algorithms.
\subsection{Sequential linear optimization}\label{secseq-LP1}
We describe a sequential linear optimization approach to obtain a local
minimum of problem~(\ref{eqmedian-q}).
We first describe the algorithm, present its convergence analysis and
describe its iteration complexity.
\subsubsection*{Main description of the algorithm}\label{secset-up-1}
We decompose the $q$th ordered absolute residual as follows:
\begin{equation}
\label{eqmedian2} \llvert r_{(q)}\rrvert = \bigl\llvert y_{(q)} -
\mathbf{x}'_{(q)}\bolds\beta\bigr\rrvert = \underbrace{\sum
_{i= q }^{n} \bigl\llvert y_{(i)} -
\mathbf{x}'_{(i)}\bolds\beta \bigr\rrvert }_{H_{q}(\bolds\beta)} -
\underbrace{\sum_{i=q+1}^{n} \bigl\llvert
y_{(i)} - \mathbf{x}'_{(i)}\bolds\beta\bigr\rrvert
}_{H_{q+1}(\bolds\beta)},
\end{equation}
where, we use the notation $H_{m}(\bolds\beta) = \sum_{i=
m}^{n} \llvert y_{(i)} - \mathbf{x}'_{(i)}\bolds\beta\rrvert $ to denote the
sum of the largest $m$ ordered
residuals $\llvert r_{(i)}\rrvert:=\llvert y_{(i)} - \mathbf{x}'_{(i)}\bolds\beta\rrvert$,
$i = 1, \ldots, n$ in absolute value.
The function $H_{m}(\bolds\beta)$
can be written as
\begin{eqnarray}\label{top-r-convex-1}
&& H_{m}(\bolds\beta):= \max_{\mathbf{w}}\qquad
\sum_{i=1}^{n} w_{i} \bigl\llvert
y_{i} - \mathbf{x}'_{i}\bolds\beta\bigr\rrvert\nonumber
\\
&&\qquad \mbox{subject to}\qquad \sum_{i=1}^{n} w_{i} = n-m +1,
\\
&&\phantom{\qquad \mbox{subject to}\qquad} 0 \leq w_{i} \leq1,\qquad i = 1, \ldots, n.\nonumber
\end{eqnarray}
Let us denote the feasible set in problem~(\ref{top-r-convex-1}) by
\[
{\mathcal W}_{m}:= \Biggl\{ \mathbf{w} \dvtx \sum
_{i=1}^{n} w_{i} = n-m +1,
w_{i} \in[0, 1], i = 1, \ldots, n \Biggr\}.
\]
Observe that for every $\mathbf{w}\in{\mathcal W}_{m}$ the function
$\sum_{i=1}^{n} w_{i} \llvert y_{i} - \mathbf{x}'_{i}\bolds\beta\rrvert $ is
convex in $\bolds\beta$. Furthermore, since
$H_{m}(\bolds\beta)$ is the point-wise supremum
with respect to $\mathbf{w}$ over~${\mathcal W}_{m}$, the function
$H_{m}(\bolds\beta)$ is convex in $\bolds\beta$
[see~\citet{BV2004}].
Equation~(\ref{eqmedian2}) thus shows that $\llvert r_{(q)}\rrvert $ can be written as the
difference of two convex functions, namely,
$H_{q}(\bolds\beta)$ and $H_{q+1}(\bolds\beta)$.
By taking the dual of problem~(\ref{top-r-convex-1}) and invoking
strong duality, we have
\begin{eqnarray}\label{lp-duality-1}
&& H_{m}(\bolds\beta) = \min_{\theta,\bolds\nu}\qquad \theta (n-m+1) + \sum_{i=1}^{n}\nu_{i}\nonumber
\\
&&\qquad\mbox{subject to}\qquad \theta+ \nu_{i} \geq\bigl\llvert y_{i} - \mathbf {x}'_{i}\bolds\beta\bigr\rrvert,\qquad i = 1, \ldots,n,
\\
&&\phantom{\qquad\mbox{subject to}\qquad}\nu_{i} \geq0,\qquad i = 1, \ldots, n.\nonumber
\end{eqnarray}
Representation~(\ref{top-r-convex-1}) also provides
a characterization of the set of subgradients of~$H_{m}(\bolds\beta)$:
\begin{eqnarray}
\label{subdiff-top-r-conv} && {\large{\bolds\partial} H_{m}(\bolds\beta)}
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
&&\qquad =
\operatorname{conv} \Biggl\{ \sum_{i=1}^{n}-
w^*_{i} \sgn\bigl(y_{i} - \mathbf{x}'_{i}
\bolds\beta\bigr)\mathbf{x}_{i} \dvtx \mathbf{w}^* \in\argmax_{\mathbf{w} \in{\mathcal W}_{m} }
{\mathcal L}(\bolds\beta, \mathbf{w}) \Biggr\},
\end{eqnarray}
where ${\mathcal L}(\bolds\beta, \mathbf{w}) = \sum_{i=1}^{n}
w_{i} \llvert y_{i} - \mathbf{x}'_{i}\bolds\beta\rrvert $ and ``conv($S$)''
denotes the convex hull of set~$S$.
An element of the set of subgradients (\ref{subdiff-top-r-conv}) will
be denoted by $\partial H_{m} (\bolds\beta)$.
Recall that~(\ref{eqmedian2}) expresses the $q$th ordered absolute
residual as a difference of two convex functions. Now,
having expressed $H_{q}(\bolds\beta)$ as the value of a Linear Optimization (LO)
problem~(\ref{lp-duality-1}) (with $m=q$) we linearize the function
$H_{q+1}(\bolds\beta)$.
If $\bolds\beta_{k}$ denotes the value of the estimate at
iteration $k$,
we linearize $H_{q+1}(\bolds\beta)$ at $\bolds\beta_{k}$
as follows:
\begin{equation}
\label{eqapprox-1aa} H_{q+1}(\bolds\beta) \approx H_{q+1}(\bolds
\beta_{k} ) + \bigl\langle\partial H_{q+1}(\bolds
\beta_{k} ), \bolds\beta- \bolds\beta_{k} \bigr\rangle,
\end{equation}
where $\partial H_{q+1}(\bolds\beta_{k} )$ is a subgradient of
$H_{q+1}(\bolds\beta_{k} )$ as defined in~(\ref{subdiff-top-r-conv}), with \mbox{$m = (q+1)$}.
Combining (\ref{lp-duality-1}) and (\ref{eqapprox-1aa}), we obtain
that, the minimum of problem~(\ref{eqmedian2}) with respect to
$\bolds\beta$ can be approximated by
solving the following LO problem:
\begin{eqnarray}\label{lqs-lp-almost-20}
&& \min_{\bolds{\nu}, \theta, \bolds\beta}\qquad \theta( n - q +1 ) + \sum
_{i=1}^{n} \nu_{i} - \bigl\langle\partial H_{q+1}(\bolds \beta_{k}), \bolds\beta\bigr\rangle\nonumber
\\
&&\qquad\mbox{subject to}\qquad \theta+ \nu_{i} \geq\bigl\llvert
y_{i} - \mathbf {x}'_{i}\bolds\beta\bigr\rrvert, \qquad i = 1, \ldots,n,
\\
&&\phantom{\qquad\mbox{subject to}\qquad}\nu_{i} \geq0,\qquad i = 1, \ldots, n.\nonumber
\end{eqnarray}
Let $\bolds\beta_{k+1}$ denote a minimizer of problem~(\ref
{lqs-lp-almost-20}). This leads to an iterative optimization procedure
as described in Algorithm~\ref{algoseq-LO1}.
\begin{algorithm}[t]
\caption{Sequential linear optimization algorithm for the LQS problem}\label{algoseq-LO1}
\begin{longlist}[1.]
\item[1.] Initialize with $\bolds\beta_{1}$, and for $k \geq1$
perform the following steps 2--3 for a predefined tolerance parameter ``Tol.''
\item[2.] Solve the linear optimization problem~(\ref{lqs-lp-almost-20})
and let $(\bolds\nu_{k+1}, \theta_{k+1}, \bolds\beta_{k
+1})$ denote a minimizer.
\item[3.] If
$ ( \llvert y_{(q)} - \mathbf{x}'_{(q)}\bolds\beta_{k}\rrvert -
\llvert y_{(q)} - \mathbf{x}'_{(q)}\bolds\beta_{k+1}\rrvert ) \leq
\mathrm{Tol}\cdot\llvert y_{(q)} - \mathbf{x}'_{(q)}\bolds\beta
_{k}\rrvert $
exit; else go to step 2.
\end{longlist}
\end{algorithm}
We next study the convergence properties of Algorithm~\ref{algoseq-LO1}.
\subsubsection*{Convergence analysis of Algorithm~\protect\ref{algoseq-LO1}}\label{secset-up-3}
In representation~(\ref{eqmedian2}), we replace $H_{q}(\bolds
\beta)$ by its dual representation~(\ref{lp-duality-1}) to obtain
\begin{eqnarray}\label{lqs-lp-almost-1}
&& f_{q}(\bolds\beta):= \min_{\bolds{\nu}, \theta}\qquad F(\bolds\nu, \theta, \bolds\beta):= \theta (n - q + 1 ) + \sum
_{i=1}^{n} \nu_{i} - H_{q+1}(\bolds\beta)\nonumber
\\
&&\qquad\mbox{subject to}\qquad \theta+ \nu_{i} \geq\bigl\llvert
y_{i} - \mathbf {x}'_{i}\bolds\beta\bigr\rrvert, \qquad i = 1, \ldots,n,
\\
&&\phantom{\qquad\mbox{subject to}\qquad} \nu_{i} \geq0, \qquad i = 1, \ldots, n.\nonumber
\end{eqnarray}
Note that the minimum of problem~(\ref{lqs-lp-almost-1})
\begin{eqnarray}
\label{lqs-lp-almost-1-equi}
&& \min_{\bolds{\nu}, \theta, \bolds\beta}\qquad F(\bolds \nu, \theta, \bolds\beta)\nonumber
\\
&&\qquad\mbox{subject to}\qquad \theta+ \nu_{i} \geq\bigl\llvert
y_{i} - \mathbf {x}'_{i}\bolds\beta\bigr\rrvert,\qquad i = 1, \ldots,n,
\\
&&\phantom{\qquad\mbox{subject to}\qquad}\nu_{i} \geq0,\qquad i = 1, \ldots, n\nonumber
\end{eqnarray}
equals to $\min_{\bolds\beta} f_{q}(\bolds\beta)$, which
is also the minimum of~(\ref{eqmedian-q}), that is,
$\min_{\bolds\beta} f_{q}(\bolds\beta) = \min_{\bolds\beta} \llvert r_{(q)}\rrvert $.
The objective function $F(\bolds\nu, \theta, \bolds\beta
)$ appearing in~(\ref{lqs-lp-almost-1}) is the sum of a linear
function in $(\bolds\nu, \theta)$ and a concave function in
$\bolds\beta$ and the constraints are convex.
Note that the function
\begin{eqnarray}\label{major-1-Hfn}
&& Q\bigl((\bolds\nu, \theta, \bolds
\beta); \bar{\bolds\beta}\bigr)
\nonumber\\[-8pt]\\[-8pt]\nonumber
&&\qquad = \theta( n-q+1 ) + \sum
_{i=1}^{n} \nu_{i} - \bigl\langle\partial
H_{q+1}(\bar{\bolds\beta}), \bolds \beta- \bar{\bolds\beta}
\bigr\rangle- H_{q+1}(\bar{\bolds\beta}),\nonumber
\end{eqnarray}
which is linear in the variables $(\bolds\nu, \theta,
\bolds\beta)$ is a linearization of $F(\bolds\nu, \theta, \bolds\beta)$ at the
point~$\bar{\bolds\beta}$. Since $H_{q+1}(\bolds
\beta)$ is convex in $\bolds\beta$, the function
$Q((\bolds\nu, \theta, \bolds\beta); \bar{\bolds\beta})$
is a majorizer of $F(\bolds\nu, \theta, \bolds\beta)$
for \emph{any} fixed $\bar{\bolds\beta}$ with equality
holding at $\bar{\bolds\beta}= \bolds\beta$, that is,
\[
Q\bigl((\bolds\nu, \theta, \bolds\beta); \bar{\bolds\beta}\bigr) \geq F(
\bolds\nu, \theta, \bolds \beta) \qquad \forall\bolds\beta\quad\mbox{and}\quad Q\bigl((
\bolds\nu, \theta, \bar{\bolds\beta}); \bar{\bolds\beta}\bigr) = F(
\bolds\nu, \theta, \bar{\bolds\beta}).
\]
Observe that problem~(\ref{lqs-lp-almost-20}) minimizes the function
$Q((\bolds\nu, \theta, \bolds\beta); \bolds\beta_{k})$.
It follows that for every fixed $\bar{\bolds\beta}$, an
optimal solution of the following linear optimization problem:
\begin{eqnarray}\label{lqs-lp-almost-2}
&& \min_{\bolds{\nu}, \theta, \bolds\beta}\qquad Q\bigl((\bolds\nu, \theta, \bolds
\beta); \bar{\bolds\beta}\bigr)\nonumber
\\
&&\qquad\mbox{subject to}\qquad \theta+ \nu_{i} \geq\bigl\llvert
y_{i} - \mathbf {x}'_{i}\bolds\beta\bigr
\rrvert, \qquad i = 1, \ldots,n,
\\
&&\phantom{\qquad\mbox{subject to}\qquad} \nu_{i} \geq0, \qquad i = 1, \ldots, n,\nonumber
\end{eqnarray}
provides an upper bound to the minimum of problem~(\ref
{lqs-lp-almost-1-equi}), and hence the global minimum of the LQS
objective function.
We now define the first-order optimality conditions of problem~(\ref
{lqs-lp-almost-1}).
\begin{mydef}\label{defloc-min-1}
A point $(\bolds\nu_*, \theta_*, \bolds\beta_*)$
satisfies the first-order optimality conditions for the minimization
problem~(\ref{lqs-lp-almost-1-equi}) if (a) $(\bolds\nu_*,
\theta_*, \bolds\beta_*)$ is feasible for problem~(\ref
{lqs-lp-almost-1}) and (b) $(\bolds\nu_*, \theta_*, \bolds
\beta_*)$ is a minimizer of the following LO problem:
\begin{eqnarray}\label{stat-F-fn-1}
&& \Delta_{*}:= \min_{\bolds\nu, \theta, \bolds\beta}\qquad \left\langle\nabla F (\bolds\nu_*, \theta_*, \bolds \beta_*), \left(\matrix{ \bolds\nu-
\bolds\nu_{*}
\vspace*{-1.5pt}\cr
\theta- \theta_{*}
\vspace*{-1.5pt}\cr
\bolds\beta- \bolds
\beta_{*}} \right)\right\rangle\nonumber
\\
&&\qquad\mbox{subject to}\qquad \theta+ \nu_{i} \geq\bigl\llvert
y_{i} - \mathbf {x}'_{i}\bolds\beta\bigr
\rrvert, \qquad i = 1, \ldots,n,
\\
&&\phantom{\qquad\mbox{subject to}\qquad} \nu_{i} \geq0, \qquad i = 1, \ldots, n,\nonumber
\end{eqnarray}
where, $ \nabla F (\bolds\nu_*, \theta_*, \bolds\beta
_*)$ is a subgradient of the function $F (\bolds\nu_*, \theta
_*, \bolds\beta_*)$.
\end{mydef}
It is easy to see that, if $(\bolds\nu_*, \theta_*, \bolds
\beta_*)$ satisfies the first-order optimality conditions as in
Definition~\ref{defloc-min-1}, then
$\Delta_{*} = 0 $.
\begin{rem}\label{remloc-min-1}
Note that if $(\bolds\nu_*, \theta_*, \bolds\beta_*)$
satisfies the first-order optimality conditions for the minimization
problem~(\ref{lqs-lp-almost-1-equi}), then $\bolds\beta_*$
satisfies the first-order stationarity conditions for the LQS
minimization problem~(\ref{eqmedian-q}).
\end{rem}
Let us define $\Delta_{k}$ as a measure of suboptimality of the tuple
$(\bolds\nu_{k}, \theta_{k}, \bolds\beta_{k})$
from first-order stationary conditions, given in Definition~\ref{defloc-min-1}
\begin{equation}
\label{eqdefdelta-k} \Delta_{k}:= \left\langle\nabla F(\bolds
\nu_{k}, \theta_{k}, \bolds\beta_{k }), \pmatrix{
\bolds\nu_{k+1} - \bolds\nu_{k}
\vspace*{-1.5pt}\cr
\theta_{k+1} -
\theta_{k}
\vspace*{-1.5pt}\cr
\bolds\beta_{k +1} - \bolds\beta_{k}}
\right\rangle,
\end{equation}
where $ \{(\bolds\nu_{k}, \theta_{k}, \bolds\beta
_{k }) \}_{k \geq1}$ are as defined in Algorithm~\ref{algoseq-LO1}.
Note that $\Delta_{k} \leq0$. If $\Delta_{k}=0$, then the point
$(\bolds\nu_{k}, \theta_{k}, \bolds\beta_{k})$ satisfies
the first-order stationary conditions.
If $\Delta_{k}<0$, then we can improve the solution further.
The following theorem presents the rate at which $\Delta_{k}
\rightarrow0$.
\begin{teo}\label{labthm1}
\textup{(a)} The sequence $(\bolds\nu_{k}, \theta_{k}, \bolds
\beta_{k })$ generated by Algorithm~\ref{algoseq-LO1} leads~to a
decreasing sequence of objective values $F(\bolds\nu_{k+1},
\theta_{k+1}, \bolds\beta_{k+1}) \leq F(\bolds\nu_{k},\break
\theta_{k}, \bolds\beta_{k}), k \geq1$
that converge to a value $F_*$.
\textup{(b)} The measure of suboptimality $\{\Delta_{k}\}_{K \geq k \geq
1}$ admits a $O(1/K)$ convergence rate, that is,
\[
\frac{ F(\bolds\nu_{1}, \theta_{1}, \bolds\beta_{1}) -
F_* }{K} \geq\min_{k = 1,\ldots,K} ( -\Delta_{k} ),
\]
where $F(\bolds\nu_{k}, \theta_{k}, \bolds\beta_{k})
\downarrow F_*$.
\textup{(c)} As $K \rightarrow\infty$ the sequence satisfies the
first-order stationary conditions as in Definition~\ref{defloc-min-1}
for problem~(\ref{lqs-lp-almost-1-equi}).
\end{teo}
\begin{pf}
Part (a). Since the objective function in~(\ref
{lqs-lp-almost-2}) is a linearization of the concave function~(\ref
{lqs-lp-almost-1}), Algorithm~\ref{algoseq-LO1} leads to a decreasing
sequence of objective values:
\[
f_{q}(\bolds\beta_{k+1})= F(\bolds\nu_{k+1}, \theta
_{k+1}, \bolds\beta_{k +1}) \leq F(\bolds\nu_{k},
\theta _{k}, \bolds\beta_{k})= f_{q}(\bolds
\beta_{k}).
\]
Thus, the sequence $F(\bolds\nu_{k}, \theta_{k}, \bolds
\beta_{k})$ is decreasing and bounded below, hence it converges---we
denote the limit as $F_*$.
Part (b).
We make use of the concavity of $ F(\bolds\nu, \theta,
\bolds\beta)$ which follows since it can be written as the sum
of a linear function in $(\bolds\nu, \theta)$ and $-H_{q+1}
(\bolds\beta)$, which is a
concave function in $\bolds\beta$.
This gives rise to the following inequality:
\begin{eqnarray}\label{bound-1}
&& F(\bolds\nu_{k+1}, \theta_{k+1}, \bolds
\beta_{k +1}) - F(\bolds\nu_{k}, \theta_{k}, \bolds
\beta_{k })
\nonumber\\[-8pt]\\[-8pt]\nonumber
&&\qquad \leq \left\langle\nabla F(\bolds\nu_{k},
\theta_{k}, \bolds \beta_{k }), \pmatrix{ \bolds
\nu_{k+1} - \bolds\nu_{k}
\vspace*{-1.5pt}\cr
\theta_{k+1} -
\theta_{k}
\vspace*{-1.5pt}\cr
\bolds\beta_{k +1} - \bolds\beta_{k}}
\right\rangle.
\end{eqnarray}
Considering inequality~(\ref{bound-1}) for $k = 1, \ldots, K$, the
notation~(\ref{eqdefdelta-k}) and adding up the terms we have
\begin{eqnarray}
&& \sum_{k=1}^{K} \bigl( F(\bolds
\nu_{k}, \theta_{k}, \bolds\beta_{k}) - F(\bolds
\nu_{k+1}, \theta_{k+1}, \bolds\beta_{k+1}) \bigr)
\geq \sum_{k = 1}^{K} ( -
\Delta_{k} ),
\end{eqnarray}
that is,
\begin{equation}
F(\bolds\nu_{1}, \theta_{1}, \bolds\beta
_{1}) - F(\bolds\nu_{K+1}, \theta_{K+1}, \bolds\beta
_{K+1}) \geq K \Bigl( \min_{k = 1,\ldots, K} ( -
\Delta_{k} ) \Bigr),\label{bound-2-1}
\end{equation}
that is,
\begin{equation}
\frac{ F(\bolds\nu_{1}, \theta_{1},
\bolds\beta_{1}) - F_* }{K} \geq \Bigl( \min_{k = 1,\ldots, K} (
-\Delta_{k} ) \Bigr).\label{bound-2-2}
\end{equation}
In the above, while moving from line~(\ref{bound-2-1}) to~(\ref
{bound-2-2}) we made use of the fact that
$F(\bolds\nu_{K+1}, \theta_{K+1}, \bolds\beta_{K+1})
\geq F_*$, where the decreasing sequence $F(\bolds\nu_{k},
\theta_{k}, \bolds\beta_{k})$
converges to $F_*$. Equation~(\ref{bound-2-2}) provides a convergence
rate for the algorithm.
Part (c).
As $K \rightarrow\infty$, we see that
$\Delta_{k} \rightarrow0$---corresponding to the first-order
stationarity condition~(\ref{stat-F-fn-1}). This also
corresponds to a local minimum of~(\ref{eqmedian-q}).
This completes the proof of the theorem.
\end{pf}
\subsection{A first-order subdifferential based algorithm for the LQS problem}\label{secsubgrad1}
Subgradient descent methods have a long history in nonsmooth convex
optimization~[\citet{shor1985minimization,nesterov2004introductory}].
If computation of the subgradients turns out to be inexpensive, then
subgradient based methods are quite effective in
obtaining a moderate accuracy solution with relatively low
computational cost. For nonconvex and nonsmooth functions,
a subgradient need not exist, so the notion of a subgradient needs to
be generalized.
For nonconvex, nonsmooth functions having certain regularity properties
(e.g., Lipschitz functions) subdifferentials exist and form a natural
generalization of
subgradients~[\citet{clarke1990optimization}].
Algorithms based on subdifferential information oracles [see,
e.g.,~\citet{shor1985minimization}]
are thus used as natural generalizations of subgradient methods for nonsmooth,
nonconvex optimization problems.
While general subdifferential-based methods can become quite
complicated based on appropriate choices
of the subdifferential and step-size sequences, we
propose a simple subdifferential based method for approximately
minimizing $f_{q}(\bolds\beta)$ as we describe below.
Recall that $f_{q}(\bolds\beta)$ admits a representation as the
difference of two simple convex functions of the form~(\ref{eqmedian2}).
It follows that $f_{q}(\bolds\beta)$ is Lipschitz~[\citet{rock-conv-96}],
almost everywhere differentiable and any element belonging to the set difference
\[
\partial f_{q}(\bolds\beta) \in{\large{\bolds\partial }}
{H}_{q}(\bolds\beta) - {\large{\bolds\partial}} {H}_{q+1}(
\bolds\beta),
\]
where, ${\large{\bolds\partial}} {H}_{m}(\bolds\beta)$
(for $m = q, q+1$) is the set of subgradients
defined in~(\ref{subdiff-top-r-conv});
is a \emph{subdifferential}~[\citet{shor1985minimization}] of
$f_{q}(\bolds\beta)$.
In particular, the quantity
\[
\partial f_{q}(\bolds\beta) = -\sgn\bigl(y_{(q)} - \mathbf
{x}'_{(q)}\bolds\beta\bigr)\mathbf{x}_{(q)}
\]
is a subdifferential of the function $f_{q}(\bolds\beta)$ at
$\bolds\beta$.
Using the definitions above, we propose a first-order subdifferential based
method for the LQS problem as described in Algorithm~\ref{algoGD}, below.
\begin{algorithm}[t]
\caption{Subdifferential based algorithm for the LQS problem}\label{algoGD}
\begin{longlist}[1.]
\item[1.] Initialize $\bolds\beta_{1}$, for $\operatorname{MaxIter}
\geq k\geq1$ do the following:
\item[2.] $\bolds\beta_{k+1} = \bolds\beta_{k} - \alpha_{k}
\partial f_{q}(\bolds\beta_{k})$
where $\alpha_{k}$ is a step-size.
\item[3.] Return $\min_{1 \leq k \leq\operatorname{MaxIter}}
f_{q}(\bolds\beta_{k})$ and
$\bolds\beta_{k^*}$ at which the minimum is attained, where $k^*
= \argmin_{1 \leq k \leq\operatorname{MaxIter}} f_{q}(\bolds
\beta_{k})$.
\end{longlist}
\end{algorithm}
While various step-size choices are possible, we found the following
simple fixed step-size sequence to be quite useful in our experiments:
\[
\alpha_{k} = \frac{1}{\max_{i = 1, \ldots, n} \llVert \mathbf{x}_{i}\rrVert _2},
\]
where, the quantity $\max_{i = 1, \ldots, n} \llVert \mathbf{x}_{i}\rrVert _2$
may be interpreted as an upper bound to the subdifferentials
of $f_{q}(\bolds\beta)$. Similar constant step-size based rules
are often used in subgradient descent methods for convex optimization.
\subsection{A hybrid algorithm}\label{sechybrid-1}
Let $\hat{\bolds\beta}_{\mathrm{GD}}$ denote the estimate
produced by Algorithm~\ref{algoGD}.
Since Algorithm~\ref{algoGD}
runs with a fixed step-size, the estimate $\hat{\bolds\beta
}_{\mathrm{GD}}$ need not be a local minimum of the LQS problem.
Algorithm~\ref{algoseq-LO1}, on the other hand, delivers an estimate
$\hat{\bolds\beta}_{\mathrm{LO}}$, say,
which\vspace*{2pt} is a \emph{local} minimum of the LQS objective function. We
found that if
$\hat{\bolds\beta}_{\mathrm{GD}}$ obtained from the
subdifferential method is used as a warm-start for the sequential
linear optimization algorithm,
the estimator obtained improves upon $\hat{\bolds\beta
}_{\mathrm{GD}}$
in terms of the LQS objective value. This leads to the proposal of a
hybrid version of Algorithms~\ref{algoseq-LO1}~and~\ref{algoGD}, as
presented in Algorithm~\ref{algohybrid} below.
\begin{algorithm}[b]
\caption{A hybrid algorithm for the LQS problem}\label{algohybrid}
\begin{longlist}[1.]
\item[1.] Run Algorithm~\ref{algoGD} initialized with $\bolds\beta
_{1}$ for $\operatorname{MaxIter}$ iterations.
Let $\hat{\bolds\beta}_{\mathrm{GD}}$ be the solution.
\item[2.] Run Algorithm~\ref{algoseq-LO1} with $\hat{\bolds
\beta}_{\mathrm{GD}}$ as the initial solution and Tolerance parameter
``Tol'' to obtain
$\hat{\bolds\beta}_{\mathrm{LO}}$.
\item[3.] Return $\hat{\bolds\beta}_{\mathrm{LO}}$ as the
solution to Algorithm~\ref{algohybrid}.
\end{longlist}
\end{algorithm}
\subsection{Initialization strategies for the algorithms}\label{secinit-strategies}
Both Algorithms~\ref{algoseq-LO1}~and~\ref{algoGD} are
sensitive to initializations $\bolds\beta_{1}$.
We run each algorithm for a prescribed number of runs ``RUNS'' (say),
and consider the solution that gives the best objective value among them.
For the initializations, we found two strategies to be quite useful.
\subsubsection*{Initialization around LAD solutions}
One\vspace*{1pt} method is based on the LAD solution, that is, $\hat{\bolds\beta}{}^{(\mathrm{LAD})}$ and random initializations around $\hat {\bolds\beta}{}^{(\mathrm{LAD})}$
given by $ [\hat{\beta}^{(\mathrm{LAD})}_{i} - \eta\llvert \hat{\beta}^{(\mathrm{LAD})}_{i}\rrvert, \hat{\beta}^{(\mathrm{LAD})}_{i} +
\eta\llvert \hat{\beta}^{(\mathrm{LAD})}_{i}\rrvert ]$, for
$i = 1, \ldots, p$, where $\eta$ is a predefined \mbox{number} say $\eta\in
\{ 2, 4\}$. This initialization strategy leads to $\bolds\beta
^{1}$, which we denote by the ``LAD'' initialization.
\subsubsection*{Initialization around Chebyshev fits}
Another initialization strategy is inspired by a geometric
characterization of the LQS solution [see~\citet{Stromberg1993}
and also Section~\ref{secprops-lqs}].
Consider a subsample $\mathcal J \subset\{1, \ldots, n\}$ of size of
size $(p+1)$ and the associated
$\ell_\infty$ regression fit (also known as the Chebyshev fit) on the
subsample $(y_{i}, \mathbf{x}_{i}), i \in\mathcal J$ given by
\[
\hat{\bolds\beta}_{\mathcal J} \in\argmin_{\bolds\beta
} \Bigl( \max
_{i \in\mathcal J} \bigl\llvert y_{i} - \mathbf
{x}_{i}'\bolds\beta\bigr\rrvert \Bigr).
\]
Consider a number of random subsamples $\mathcal J$ and the associated
coefficient-vector $\hat{\bolds\beta}_{\mathcal J}$ for every
$\mathcal J$.
The estimate $\hat{\bolds\beta}_{\mathcal J^*}$ that produces
the minimum value of the LQS objective function is taken as
$\bolds\beta_{1}$.
We denote $\bolds\beta_{1}$ chosen in this fashion as the best
Chebyshev fit or ``Cheb'' in short.
Algorithm~\ref{algohybrid}, in our experience, was found to be less
sensitive to initializations.
Experiments demonstrating the different strategies described above are
discussed in Section~\ref{seccomps-1}.
\section{Properties of the LQS solutions for arbitrary datasets}\label{secprops-lqs}
In this section, we prove that key properties
of optimal LQS solutions hold without assuming that the data
$(\mathbf{y}, \mathbf{X})$ are in general position as it is done in
the literature to date~[\citet{rousseeuw1984least,rousseeuw2005robust,Stromberg1993}].
For this purpose, we utilize the MIO characterization of the
LQS problem.
Specifically:
\begin{longlist}[(2)]
\item[(1)]
We show in Theorem~\ref{teoexist} that an optimal solution
to the LQS problem (and in particular the LMS problem)
always exists, for \emph{any} $(\mathbf{y}, \mathbf{X})$ and $q$.
The theorem also shows that an optimal LQS solution is given by
the $\ell_{\infty}$ or Chebyshev regression fit to a subsample of
size $q$ from the sample $(y_{i}, \mathbf{x}_{i})$, $i = 1, \ldots, n$,
thereby generalizing the results of \citet{Stromberg1993}, which
require $(\mathbf{y}, \mathbf{X})$ to be in general position.
\item[(2)] We show in Theorem \ref{propkey2} that the absolute values of
some of the residuals are equal to the optimal solution value of the
LQS problem,
without assuming that the data is in general position.
\item[(3)] We show in Theorem~\ref{teo-break-1} a new result that the
breakdown point of the optimal value of the LQS objective is $(n-q+1)/n$
without assuming that the data is in general position. For the LMS
problem $q=n-\lfloor n/2 \rfloor$, which leads to the sample breakdown
point of\vadjust{\goodbreak}
LQS objective of $(\lfloor n/2 \rfloor+1)/n$, independent of the
number of covariates $p$.
In contrast, it is known that LMS solutions have a sample breakdown
point of $(\lfloor n/2 \rfloor-p+2)/n$ (when the data is in general position).
\end{longlist}
\begin{teo}\label{teoexist}
The LQS problem is equivalent to the following:
\begin{equation}
\label{max-l2-norm1} \min_{\bolds\beta} \llvert r_{(q)}\rrvert =
\min_{ {\mathcal I} \in
\Omega_{q} } \Bigl( \min_{\bolds\beta} \llVert
\mathbf{y}_{I} - \mathbf{X}_{I}\bolds\beta\rrVert
_\infty \Bigr),
\end{equation}
where, $ \Omega_{q}:= \{ {\mathcal I} \dvtx {\mathcal I} \subset \{
1, \ldots, n \}, \llvert {\mathcal I} \rrvert = q \} $
and $(\mathbf{y}_{I}, \mathbf{X}_{I})$ denotes the subsample $(y_{i},
\mathbf{x}_{i}), i \in{\mathcal I}$.
\end{teo}
\begin{pf}
Consider the MIO formulation~(\ref{lqs-reg-form-1}) for the LQS problem.
Let us take a vector of binary variables $\bar{z}_{i}\in\{ 0, 1\}$, $i
= 1, \ldots, n$ with $\sum_{i} \bar{z}_{i} = q$,
feasible for problem~(\ref{lqs-reg-form-1}). This vector $\bar{\mathbf{z}}:= (\bar{z}_{1}, \ldots, \bar{z}_{n})$ gives rise to a
subset $\mathcal I \in\Omega_{q}$ given by
\[
\mathcal I = \bigl\{ i | \bar{z}_{i} = 1, i
\in\{1, \ldots, n \} \bigr\}.
\]
Corresponding to this subset $\mathcal I$ consider the subsample
$(y_{\mathcal I}, \mathbf{X}_{\mathcal I})$ and the associated
optimization problem
\begin{equation}
\label{max-norm-reg-1} T_{\mathcal I} = \min_{\bolds\beta} \llVert
\mathbf{y}_{I} - \mathbf{X}_{I}\bolds\beta\rrVert
_\infty,
\end{equation}
and let $\bolds\beta_{\mathcal I}$ be a minimizer of~(\ref
{max-norm-reg-1}).
Observe that $\bar{\mathbf{z}}, \bolds\beta_{\mathcal I}$ and
$\bar{r}_{i} = y_{i} - \mathbf{x}_{i}'\bolds\beta_{\mathcal
I}$, $i = 1, \ldots, n$
is feasible for
problem~(\ref{lqs-reg-form-1}). Furthermore, it is easy to see that,
if $\mathbf{z}$ is taken to be equal to $\bar{\mathbf{z}}$,
then the minimum value of
problem~(\ref{lqs-reg-form-1}) with respect to the variables
$\bolds\beta$ and $r^+_{i}, r^{-}_{i}, \mu_{i}, \bar{\mu
}_{i}$ for $i = 1, \ldots, n$
is given by $\llvert \bar{r}_{(q)}\rrvert = T_{\mathcal I}$. Since every choice of
$\mathbf{z} \in\{0,1\}^n$ with $\sum_i z_{i} = q$ corresponds to a subset
$\mathcal I \in\Omega_{q}$, it follows that the minimum value of
problem~(\ref{lqs-reg-form-1}) is given by the minimum value of
$T_{\mathcal I}$ as $\mathcal I$ varies over
$\Omega_{q}$.
Note that the minimum in problem~(\ref{max-l2-norm1}) is
attained since it is a minimum over finitely many subsets $\mathcal I
\in\Omega_{q}$.
This shows that an optimal solution to the LQS problem always exists,
without any assumption
on the geometry or orientation of the sample points $(\mathbf{y},
\mathbf{X})$.
This completes the proof of the equivalence~(\ref{max-l2-norm1}).
\end{pf}
\begin{cor}\label{cor-1}
Theorem~\ref{teoexist} shows that an optimal LQS solution for \emph{any} sample $(\mathbf{y}, \mathbf{X})$
is given by the Chebyshev or $\ell_{\infty}$ regression fit to a
subsample of size
$q$ from the $n$ sample points. In particular, for every optimal LQS solution
there is a
${\mathcal I}_* \in\Omega_{q}$ such that
\begin{equation}
\hat{\bolds\beta}{}^{(\mathrm{LQS})} \in\argmin _{\bolds\beta} \llVert
\mathbf{y}_{{\mathcal I}_*} - \mathbf {X}_{ {\mathcal I}_*}\bolds\beta\rrVert
_{\infty}.
\end{equation}
\end{cor}
We next show that, at an optimal solution
of the LQS problem, some of the absolute values of the residuals are
all equal to the minimum objective value of the LQS problem,
generalizing earlier work by \citet{Stromberg1993}.
Note that problem~(\ref{max-norm-reg-1}) can be written as the
following linear optimization problem:
\begin{eqnarray}\label{max-norm-reg-1-eq}
&& \mini_{t, \bolds\beta}\qquad t,
\nonumber\\[-8pt]\\[-8pt]\nonumber
&&\qquad \mbox{subject to}\qquad - t
\leq{y}_{i} - \mathbf{x}'_{i}\bolds\beta\leq t,\qquad
i \in {\mathcal I}_*.
\end{eqnarray}
The Karush Kuhn Tucker (KKT) [\citet{BV2004}] optimality
conditions of problem~(\ref{max-norm-reg-1-eq}) are given by
\begin{eqnarray} \label{line-kkt-1}
\sum_{i \in{\mathcal I}_* } \bigl( \nu^{-}_{i}
+ \nu^{+}_{i} \bigr) &=& 1,\nonumber
\\
\sum_{i \in{\mathcal I}_*} \bigl( \nu^{-}_{i}
- \nu^{+}_{i} \bigr) \mathbf{x}_{i} &=&
\mathbf{0},\nonumber
\\
\nu^{+}_{i} \bigl( {y}_{i} -
\mathbf{x}'_{i}\hat{\bolds\beta} - t^* \bigr) &=&0\qquad \forall i \in{\mathcal I}_*,
\\
\nu^{-}_{i} \bigl( {y}_{i} -
\mathbf{x}'_{i}\hat{\bolds\beta} + t^* \bigr) &=& 0\qquad \forall i \in{\mathcal I}_*,\nonumber
\\
\nu^{+}_{i}, \nu^{-}_{i} &\geq& 0\qquad\forall i \in{\mathcal I}_*,\nonumber
\end{eqnarray}
where $\hat{\bolds\beta}, t^*$ are optimal
solutions\footnote{We use the shorthand $\hat{\bolds\beta
}$ in place of $\hat{\bolds\beta}{}^{(\mathrm{LQS})}$.}
to~(\ref{max-norm-reg-1-eq}).
Let us denote
\begin{equation}
\label{eqistar}
{\mathcal I}^{+}:= \bigl\{ i | i
\in{\mathcal I}_*, \bigl\llvert \nu^+_{i} - \nu^{-}_{i}
\bigr\rrvert >0 \bigr\},
\end{equation}
clearly, on this set of indices at least one of $\nu^+_{i}$ or $\nu
^-_{i}$ is nonzero, which implies
$\llvert {y}_{i} - \mathbf{x}'_{i}\hat{\bolds\beta}\rrvert = t^*$.
This gives the following bound:
\[
\bigl\llvert {\mathcal I}^{+} \bigr\rrvert \leq \bigl\llvert \bigl\{
i \in { \mathcal I}_* \dvtx \bigl\llvert {y}_{i} - \mathbf{x}'_{i}
\hat {\bolds\beta} \bigr\rrvert = t^* \bigr\} \bigr\rrvert.
\]
It follows from~(\ref{line-kkt-1}) that $\llvert {\mathcal I}^{+}
\rrvert > \rnk ([\mathbf{x}_{i}, i \in{\mathcal I}^{+}]
)$. We thus have
\[
\bigl\llvert \bigl\{ i \in{ \mathcal I}_* \dvtx \bigl\llvert {y}_{i} -
\mathbf {x}'_{i}\hat{\bolds\beta}\bigr\rrvert = t^*
\bigr\} \bigr\rrvert \geq \bigl\llvert {\mathcal I}^{+} \bigr\rrvert >
\rnk \bigl(\bigl[\mathbf{x}_{i}, i \in{\mathcal I}^{+}\bigr]
\bigr).
\]
In particular, if the $\mathbf{x}_{i}$'s come from a continuous distribution
then with probability one:
\[
\rnk \bigl(\bigl[\mathbf{x}_{i}, i \in{\mathcal I}^{+}
\bigr] \bigr) = p\quad\mbox{and}\quad \bigl\llvert \bigl\{ i \in{ \mathcal I}_* \dvtx
\bigl\llvert {y}_{i} - \mathbf {x}'_{i}
\hat{\bolds\beta}\bigr\rrvert = t^* \bigr\} \bigr\rrvert \geq(p +1).
\]
This leads to the following theorem.
\begin{teo} \label{propkey2}
Let ${\mathcal I}_* \in\Omega_{q}$ denote a subset of size $q$ which
corresponds to an optimal LQS solution (see Corollary~\ref{cor-1}).
Consider the KKT optimality conditions of the Chebyshev fit to this
subsample $(\mathbf{y}_{{\mathcal I}_{*}}, \mathbf{X}_{{\mathcal
I}_*})$ as given
by~(\ref{line-kkt-1}). Then
\[
\bigl\llvert \bigl\{ i \in{ \mathcal I}_* \dvtx \bigl\llvert {y}_{i} -
\mathbf {x}'_{i}\hat{\bolds\beta}\bigr\rrvert = t^*
\bigr\} \bigr\rrvert \geq \bigl\llvert {\mathcal I}^{+} \bigr\rrvert >
\rnk \bigl(\bigl[\mathbf{x}_{i}, i \in{\mathcal I}^{+}\bigr]
\bigr),
\]
where $\hat{\bolds\beta},{\mathcal I}^{+}$ are as defined
in~(\ref{line-kkt-1}) and~(\ref{eqistar}).
\end{teo}
\subsection{Breakdown point and stability of solutions}
In this section, we revisit the notion of a breakdown point of
estimators and derive sharper results for the problem without the assumption
that the data is in general position.
Let $\Theta(\mathbf{y}, \mathbf{X})$ denote an estimator based on a
sample $(\mathbf{y}, \mathbf{X})$.
Suppose the original sample is $(\mathbf{y}, \mathbf{X})$ and $m$ of
the sample points have been replaced arbitrarily---let
$(\mathbf{y} + \Delta_{\mathbf{y}}, \mathbf{X} + \Delta_{\mathbf
{X}})$ denote the perturbed sample. Let
\begin{equation}
\label{eqbreakdown-est1} \alpha\bigl(m; \Theta; (\mathbf{y}, \mathbf{X})\bigr)=\sup
_{(\Delta
_{\mathbf{y}}, \Delta_{\mathbf{X}} )} \bigl\llVert \Theta(\mathbf{y}, \mathbf{X}) -\Theta(
\mathbf{y} + \Delta_{\mathbf{y}}, \mathbf{X} + \Delta_{\mathbf{X}}) \bigr
\rrVert,
\end{equation}
denote the maximal change in the estimator under this perturbation,
where $\llVert \cdot\rrVert $ denotes the standard Euclidean norm.
The finite sample breakdown point of the estimator $\Theta$ is defined
as follows:
\begin{equation}
\label{eqbreakdown-est2} \eta\bigl( \Theta; (\mathbf{y}, \mathbf{X}) \bigr):=\min
_{m} \biggl\{ \frac{m}{n} \bigg| \alpha\bigl(m; \Theta; (
\mathbf{y}, \mathbf {X})\bigr) = \infty \biggr\}.
\end{equation}
We will derive the breakdown point of the minimum value of the LQS
objective function, that is,
$\llvert r_{(q)}\rrvert = \llvert y_{(q)} - \mathbf{x}'_{(q)} \hat{\bolds\beta}{}^{(\mathrm{LQS})}\rrvert $, as defined in~(\ref{eqmedian2}).
\begin{teo}\label{teo-break-1}
Let $ \hat{\bolds\beta}{}^{(\mathrm{LQS})}$ denote an
optimal solution and
$\Theta:=\Theta(\mathbf{y}, \mathbf{X})$ denote the optimum
objective value to the LQS problem
for a given dataset $(\mathbf{y}, \mathbf{X})$,
where the $(y_{i}, \mathbf{x}_{i})$'s are not necessarily in general position.
Then the finite sample breakdown point of $\Theta$ is $(n - q + 1) / n$.
\end{teo}
\begin{pf}
We will first show that the breakdown point of $\Theta$ is strictly
greater than $(n - q) /n$.
Suppose we have a corrupted sample $(\mathbf{y} + \Delta_{\mathbf
{y}}, \mathbf{X} + \Delta_{\mathbf{X}})$, with $m = n - q $ replacements
in the original sample. Consider the equivalent LQS formulation~(\ref{max-l2-norm1}) and let ${\mathcal I}_{0}$ denote the unchanged
sample indices.
Consider the inner convex optimization
problem appearing in~(\ref{max-l2-norm1}), corresponding to the index
set ${\mathcal I}_{0}$:
\begin{equation}
\label{eqn-maxmin-1} T_{{\mathcal I}_{0} } (\mathbf{y} + \Delta_{\mathbf{y}}, \mathbf{X}
+ \Delta_{\mathbf{X}}) = \min_{\bolds\beta} \llVert \mathbf
{y}_{{\mathcal I}_{0}} - \mathbf{X}_{{\mathcal I}_{0}} \bolds \beta\rrVert
_\infty,
\end{equation}
with\vspace*{1pt} $\bolds\beta_{ {\mathcal I}_{0} } (\mathbf{y} + \Delta
_{\mathbf{y}}, \mathbf{X} + \Delta_{\mathbf{X}})$ denoting a minimizer
of the convex optimization problem~(\ref{eqn-maxmin-1}).
Clearly, both a minimizer and the minimum objective value are finite
and neither depends upon $(\Delta_{\mathbf{y}}, \Delta_{\mathbf{X}})$.
Suppose
\[
T_{{\mathcal I}_*}(\mathbf{y} + \Delta_{\mathbf{y}}, \mathbf{X} +
\Delta_{\mathbf{X}}) = \min_{{\mathcal I} \in\Omega_{q}} T_{\mathcal I}(\mathbf{y}
+ \Delta_{\mathbf{y}}, \mathbf{X} + \Delta_{\mathbf{X}})
\]
denotes the minimum value of the LQS objective function corresponding
to the perturbed sample, for some ${\mathcal I}_* \in\Omega_{q}$,
then it follows that: $T_{{\mathcal I}_*}(\mathbf{y} + \Delta
_{\mathbf{y}}, \mathbf{X} + \Delta_{\mathbf{X}}) \leq T_{{\mathcal
I}_{0} } (\mathbf{y} + \Delta_{\mathbf{y}}, \mathbf{X} + \Delta
_{\mathbf{X}})$---which clearly implies that
the quantity $\llVert T_{{\mathcal I}_{0} } (\mathbf{y} + \Delta_{\mathbf
{y}}, \mathbf{X} + \Delta_{\mathbf{X}}) - \Theta\rrVert $ is bounded above
and the bound does not depend upon $ (\Delta_{\mathbf{y}}, \Delta
_{\mathbf{X}} )$. This shows that the breakdown point of $\Theta$ is
strictly larger than $\frac{(n - q)}{n}$.\vadjust{\goodbreak}
We will now show that the breakdown point of the estimator is less than
or equal to $(n - q+1)/n$.
If the number of replacements is given by $m = n - q + 1$, then
it is easy to see that every ${\mathcal I} \in\Omega_{q}$ includes a
sample $i_{0}$ (say) from the replaced sample units.
If $ (\delta_{y_{i_0}}, \delta'_{\mathbf{x}_{i_{0}}})$ denotes\vspace*{1pt} the
perturbation corresponding to the $i_0$th sample, then it is easy to
see that
\[
T_{\mathcal I}(\mathbf{y} + \Delta_{\mathbf{y}}, \mathbf{X} +
\Delta_{\mathbf{X}}) \geq\bigl\llvert (y_{i} - \mathbf{x}_{i_{0}}
\bolds \beta_{\mathcal I}) + \bigl(\delta_{y_{i_0}} -
\delta'_{\mathbf
{x}_{i_{0}}}\bolds\beta_{\mathcal I}\bigr) \bigr\rrvert,
\]
where $ \bolds\beta_{\mathcal I}$ is a minimizer for the
corresponding problem~(\ref{eqn-maxmin-1}) (with ${\mathcal I}_{0} =
{\mathcal I}$).
It is possible to choose $\delta_{y_{i_0}}$
such that the r.h.s. of the above inequality becomes arbitrarily
large. Thus, the finite-sample breakdown point of the estimator $\Theta
$ is $\frac{(n - q)+1}{n}$.
\end{pf}
For the LMS problem $q=n-\lfloor n/2 \rfloor$, which leads to the
sample breakdown point of $\Theta$ of $(\lfloor n/2 \rfloor+1)/n$,
independent of the number of covariates $p$.
In contrast, LMS solutions have a sample breakdown point of $(\lfloor
n/2 \rfloor-p+2)/n$. In other words, the optimal solution value
is more robust than optimal solutions to the LMS problem.
\section{Computational experiments}\label{seccomps-1}
In this section, we perform computational experiments demonstrating the
effectiveness of our algorithms in terms of quality of solutions
obtained, scalability and
speed.
All computations were done in MATLAB version {\texttt R2011a} on a
64-bit linux machine, with 8 cores and 32 GB RAM.
For the MIO formulations we used {\textsc{Gurobi}} [\citet{gurobi}] via its MATLAB interface.
We consider a series of examples including synthetic and real-world
datasets showing that our proposed methodology
consistently finds high quality solutions of problems of sizes up to $n
={}$10,000 and $p=20$.
For moderate-large sized examples, we observed that
global optimum solutions are obtained usually within a few minutes (or
even faster), but it takes longer to deliver a certificate of global optimality.
Our continuous optimization based methods
enhance the performance of the MIO formulation, the margin of
improvement becomes more significant with increasing problem sizes.
In all the examples, there is an appealing common theme---if the MIO
algorithm is terminated early, the procedure provides a bound on its
suboptimality.
In Section~\ref{synthetic-egs}, we describe the synthetic datasets
used in our experiments.
Section~\ref{secdeeper-1} presents a deeper understanding of
Algorithms~\ref{algoseq-LO1},~\ref{algoGD} and~\ref{algohybrid}.
Section~\ref{seccompare-methods} presents comparisons of
Algorithms~\ref{algoseq-LO1},~\ref{algoGD} and~\ref{algohybrid} as
well as the MIO algorithm
with state of the art algorithms for the LQS. In Section~\ref{secreal-data}, we illustrate the performance of our algorithms on
real-world data sets.
Section~\ref{secglobal-opt} discusses the evolution of lower bounds
and global convergence certificates for the problem.
Section~\ref{seclarge-scale-method1} describes scalability
considerations for larger problems.
\subsection{Synthetic examples}\label{synthetic-egs}
We considered a set of synthetic examples,
following~\citet{Rousseeuw2006CLR11170811117088}.
We generated the model matrix $\mathbf{X}_{n \times p}$ with i.i.d.
Gaussian entries $N(0,100)$ and took $\bolds\beta\in\Re^{p}$
to be a vector of all ones.
Subsequently, the response is generated as $\mathbf{y} = \mathbf
{X}\bolds{\beta} + \bolds\varepsilon$, where $\varepsilon
_{i}\sim\mathrm{N}(0,10)$, $i = 1, \ldots, n$.
Once $(\mathbf{y}, \mathbf{X})$ have been generated, we corrupt a
certain proportion $\pi$ of the sample in two different ways:
\begin{longlist}[(A)]
\item[(A)] $\lfloor\pi n \rfloor$ of the samples are chosen at
random and the first coordinate of the data matrix $\mathbf{X}$, that is,
$x_{1j}$'s are replaced by $x_{1j}\leftarrow x_{1j} + 1000$.
\item[(B)] $\lfloor\pi n \rfloor$ of the samples are chosen at
random out of which the covariates of half of the points are changed as
in item~(A);
for the remaining half of the points the responses are corrupted as
$y_{j} \leftarrow y_{j} + 1000$. In this set-up, outliers are added in
\emph{both}
the covariate and response spaces.
\end{longlist}
We considered seven different examples for different values of $(n, p,
\pi)$:
\begin{description}
\item[Moderate-scale:] We consider four moderate-scale examples Ex-1--Ex-4:
\begin{enumerate}
\item[Ex-1:] Data is generated as per (B) with $(n, p, \pi)= (201, 5, 0.4)$.
\item[Ex-2:] Data is generated as per (B) with $(n, p, \pi)= (201,
10, 0.5)$.
\item[Ex-3:] Data is generated as per (A) with $(n, p, \pi)= (501, 5, 0.4)$.
\item[Ex-4:] Data is generated as per (A) with $(n, p, \pi)= (501,
10, 0.4)$.
\end{enumerate}
\item[Large-scale:] We consider three large-scale examples, Ex-5--Ex-7:
\begin{enumerate}
\item[Ex-5:] Data is generated as per (B) with $(n, p, \pi)= (2001,
10, 0.4)$.
\item[Ex-6:] Data is generated as per (B) with $(n, p, \pi)= (5001,
10, 0.4)$.
\item[Ex-7:] Data is generated as per (B) with $(n, p, \pi)= (10,001, 20, 0.4)$.
\end{enumerate}
\end{description}
\subsection{A deeper understanding of
Algorithms~\texorpdfstring{\protect\ref{algoseq-LO1}}{1},
\texorpdfstring{\protect\ref{algoGD}{2}}
and~\texorpdfstring{\protect\ref{algohybrid}}{3}}\label{secdeeper-1}
For each of the synthetic examples Ex-1--Ex-4, we compared the
performances of the different continuous optimization based algorithms
proposed in this paper---Algorithms~\ref{algoseq-LO1}, \ref{algoGD} and~\ref{algohybrid}. For each of the
Algorithms~\ref{algoseq-LO1},~\ref{algoGD}, we considered two
different initializations, following the strategy described in
Section~\ref{secinit-strategies}:
\begin{longlist}[(Cheb)]
\item[(LAD)] This is the initialization from the LAD solution, with
$\eta= 2$ and number of random initializations taken to be 100. This is
denoted in Table~\ref{table-gran-1-small} by the moniker ``LAD.''
\begin{sidewaystable
\tablewidth=\textwidth
\tabcolsep=0pt
\caption{Table showing performances of different continuous
optimization based methods proposed in this paper, for examples, Ex-1--Ex-4.
For every example, the top row ``Accuracy'' is Relative Accuracy
[see~(\protect\ref{rel-accu})] and
the numbers inside parenthesis denotes standard errors (across the
random runs); the lower row denotes the time taken (in cpu seconds).
Results are averaged over 20 different random instances of the problem.
Algorithm~\protect\ref{algohybrid} seems to be the clear winner among
the different examples, in terms of the quality
of solutions obtained.
Among all the algorithms considered, Algorithm~\protect\ref{algohybrid} seems to be least sensitive to initializations}\label{table-gran-1-small}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}@{}lccccccc@{}}
\hline
\multirow{2}{71pt}{\break \break \textbf{Example} $\bolds{(n,p,\pi)}$\break \textbf{q}} & & \multicolumn{6}{c@{}}{\textbf{Algorithm used}}\\[-6pt]
& & \multicolumn{6}{c@{}}{\hrulefill}\\
& & \multicolumn{2}{c}{\textbf{Algorithm~\ref{algoseq-LO1}}} & \multicolumn{2}{c}{\textbf{Algorithm~\ref{algoGD}}} & \multicolumn{2}{c@{}}{\textbf{Algorithm~\ref{algohybrid}}}\\[-6pt]
& & \multicolumn{2}{c}{\hrulefill} & \multicolumn{2}{c}{\hrulefill} & \multicolumn{2}{c@{}}{\hrulefill}\\
& & \textbf{(LAD)} & \textbf{(Cheb)} & \textbf{(LAD)} & \textbf{(Cheb)} & \textbf{(LAD)} & \textbf{(Cheb)}\\
\hline
Ex-1 (201, 5, 0.4) & Accuracy & 49.399 (2.43) & 0.0 (0.0) & 0.233 (0.03)& 0.240 (0.02) & 0.0 (0.0) & 0.0 (0.0)\\
$q=121$ & Time (s) & \phantom{0}24.05 & \phantom{0}83.44 & 3.29 & 83.06 & \phantom{0}36.13 & 118.43
\\[3pt]
Ex-2 (201, 10, 0.5)& Accuracy &43.705 (2.39) & 5.236 (1.73) & 1.438 (0.07) & 1.481 (0.10) & 0.0 (0.0) & 0.0 (0.0) \\
$q=101$ & Time (s) & \phantom{0}54.39 & 133.79 & 3.22 & 73.14 & \phantom{0}51.89 & 125.55
\\[3pt]
Ex-3 (501, 5, 0.4) & Accuracy & 2.897 (0.77) & 0.0 (0.0) & 0.249 (0.05) & 0.274 (0.06) & 0.0 (0.0) & 0.0 (0.0)\\
$q=301$ & Time (s) & \phantom{0}83.01 & 158.41 & 3.75 & 62.36 & 120.90 & 179.34
\\[3pt]
Ex-4 (501, 10, 0.4) & Accuracy & 8.353 (2.22) & 11.926 (2.31) & 1.158 (0.06) & 1.083 (0.06) & 0.0 & 0.0 \\
$q=301$ & Time (s) &192.02 & 240.99 & 3.76 & 71.45 & 155.36 & 225.09 \\
\hline
\end{tabular*}
\end{sidewaystable}
\item[(Cheb)] This is the initialization from the Chebyshev fit. For
every initialization, forty different subsamples were taken to estimate
$\bolds\beta_{1}$, 100 different initializations were
considered. This method is denoted by the moniker ``Cheb'' in
Table~\ref{table-gran-1-small}.
\end{longlist}
Algorithm~\ref{algoseq-LO1}, initialized at the ``LAD'' method
(described above) is denoted by Algorithm~\ref{algoseq-LO1} (LAD), the
same notation
carries over to the other remaining combinations of Algorithms~\ref
{algoseq-LO1} and \ref{algoGD} with initializations ``LAD'' and ``Cheb.''
Each of the methods
Algorithms~\ref{algoGD} (LAD) and~\ref{algoGD} (Cheb), lead
to an initialization for Algorithm~\ref{algohybrid}---denoted by
Algorithms~\ref{algohybrid} (LAD) and~\ref{algohybrid}
(Cheb), respectively.
In all the examples, we set the MaxIter counter for Algorithm~\ref
{algoGD} at 500 and took the step-size sequence as described in
Section~\ref{secsubgrad1}.
The tolerance criterion ``Tol'' used in Algorithm~\ref{algoseq-LO1}
(and consequently Algorithm~\ref{algohybrid}), was set to $10^{-4}$.
Results comparing these methods are
summarized in Table~\ref{table-gran-1-small}. To compare the different
algorithms in terms of the quality of solutions obtained, we do the following.
For every instance, we run all the algorithms and obtain the best
solution among them, say, $f_*$. If
$f_{\mathrm{alg}}$ denotes the value of the LQS objective function for
algorithm ``alg,'' then we define the relative accuracy of the solution
obtained by ``alg'' as
\begin{equation}
\label{rel-accu} \mbox{Relative Accuracy} = (f_{\mathrm{alg}} -
f_{*})/f_{*} \times 100.
\end{equation}
To obtain the entries in Table~\ref{table-gran-1-small}, the relative accuracy
is computed for every algorithm (six in all: Algorithms~\ref{algoseq-LO1}---\ref{algohybrid}, two types for each ``LAD'' and
``Cheb'') for every random problem instance corresponding to a
particular example type; and the results are averaged (over 20 runs).
The times reported for Algorithms~\ref{algoseq-LO1} (LAD) and~\ref{algoseq-LO1} (Cheb) includes the times taken to perform
the LAD and Chebyshev fits, respectively.
The same thing applies to Algorithms~\ref{algoGD} (LAD) and~\ref{algoGD} (Cheb).
For Algorithm~\ref{algohybrid} (Cheb) [resp., Algorithm~\ref{algohybrid} (LAD)],
the time taken equals the time taken by Algorithm~\ref{algoGD} (Cheb)
[resp., Algorithm~\ref{algoGD} (LAD)]
and the time taken to perform the Chebyshev (resp., LAD) fits.
In Table~\ref{table-gran-1-small}, we see that Algorithm~\ref{algoGD}
(LAD) converges quite quickly in all the examples. The quality of the
solution, however, depends upon
the choice of \mbox{$p$---}for $p=10$ the algorithm converges to a lower
quality solution when compared to $p=5$. The time till convergence for
Algorithm~\ref{algoGD} is less sensitive to the problem
dimensions---this is in contrast to the other algorithms, where
computation times show a monotone trend depending upon the sizes of $(n,p)$.
Algorithm~\ref{algoGD} (Cheb) takes more time than Algorithm~\ref{algoGD} (LAD), since it spends a
considerable amount of time in performing multiple Chebyshev fits (to
obtain a good initialization). Algorithm~\ref{algoseq-LO1} (LAD) seems
to be sensitive to the
type of initialization used; Algorithm~\ref{algoseq-LO1} (Cheb) is
more stable and it appears that the multiple Chebyshev initialization
guides Algorithm~\ref{algoseq-LO1} (Cheb) to higher quality solutions.
Algorithm~\ref{algohybrid} (both variants) seem to be the clear winner
among the various algorithms---this does not come as a surprise since,
intuitively it aims
at combining the \emph{best features} of its constituent algorithms.
Based on computation times,
Algorithm~\ref{algohybrid} (LAD) outperforms Algorithm~\ref
{algohybrid} (Cheb),\vadjust{\goodbreak} since it avoids the computational overhead of
computing several Chebyshev fits.
\begin{table
\tabcolsep=0pt
\caption{Table showing performances of various algorithms for the LQS
problem for different moderate-scale examples as described in the text.
For each example, ``Accuracy'' is Relative Accuracy [see~(\protect\ref{rel-accu})],
the numbers within brackets denote the standard errors;
the lower row denotes the averaged cpu time (in secs) taken by the
algorithm. All results are averaged over 20 random examples.
The {MIO formulation}~(\protect\ref{lqs-reg-form-1}) warm-started
with {Algorithm}~\protect\ref{algohybrid} seems to be the best
performer in terms of obtaining the best solution.
The combined time taken by {MIO formulation}~(\protect\ref{lqs-reg-form-1}) (warm-start) and {Algorithm}~\protect\ref{algohybrid}
(which is used as a warm-start) equals the run-time
of {MIO formulation}~(\protect\ref{lqs-reg-form-1}) (cold-start)}\label{table-notgran-1-small}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}@{}lc cccc@{}}
\hline
\multirow{2}{71pt}{\break \break \textbf{Example} $\bolds{(n,p,\pi)}$\break \textbf{q}} & & \multicolumn{4}{c@{}}{\textbf{Algorithm used}}\\[-6pt]
& & \multicolumn{4}{c@{}}{\hrulefill}\\
& & \multirow{3}{32pt}{\centering{\textbf{LQS}\break \textbf{(MASS)}}} & & \multicolumn{2}{c@{}}{\textbf{MIO formulation~(\ref{lqs-reg-form-1})}}\\[-6pt]
& & & & \multicolumn{2}{c@{}}{\hrulefill}\\
& & & \textbf{Algorithm \ref{algohybrid}} & \textbf{(Cold-start)} & \textbf{(Warm-start)}\\
\hline
Ex-1 (201, 5, 0.4) &Accuracy &24.163 (1.31) & 0.0 (0.0) & 60.880 (5.60) & 0.0 (0.0) \\
$q=121$ & Time (s) & 0.02 & \phantom{0}36.13 & \phantom{0}71.46 & \phantom{0}35.32 \\[3pt]
Ex-2 (201, 10, 0.5) &Accuracy & 105.387 (5.26) & 0.263 (0.26) & 56.0141 (3.99) & 0.0 (0.0) \\
$q=101$ & Time (s) & 0.05 & \phantom{0}51.89 & 193.00 & 141.10\\[3pt]
Ex-3 (501, 5, 0.4) & Accuracy & 9.677 (0.99) & 0.618 (0.27) & 11.325 (1.97) & 0.127 (0.11) \\
$q=301$ & Time (s) & 0.05 &120.90 & 280.66 & 159.76\\[3pt]
Ex-4 (501, 5, 0.4) &Accuracy &29.756 (1.99) & 0.341 (0.33) & 27.239 (2.66) & 0.0 (0.0) \\
$q=301$ & Time (s) & 0.08 & 155.36 & 330.88 & 175.52 \\
\hline
\end{tabular*}
\end{table}
\subsection{Comparisons: Quality of the solutions obtained}\label{seccompare-methods}
In this section, we shift our focus from studying the detailed dynamics
of Algorithms~\ref{algoseq-LO1}---\ref{algohybrid}; and compare
the performances of Algorithm~\ref{algohybrid} (which seems to be the
best among the algorithms we propose in the paper),
the MIO formulation~(\ref{lqs-reg-form-1}) and state-of-the art
implementations of the LQS problem as
implemented in the popular \texttt{R}-package \texttt{MASS} (available from
\texttt{CRAN}).
For the MIO formulation~(\ref{lqs-reg-form-1}), we considered two
variations: MIO formulation~(\ref{lqs-reg-form-1}) (cold-start), where
the MIO algorithm is not provided
with any advanced warm-start and MIO formulation~(\ref
{lqs-reg-form-1}) (warm-start), where the MIO algorithm is provided with
an advanced warm-start obtained by Algorithm~\ref{algohybrid}.
The times taken by MIO formulation~(\ref{lqs-reg-form-1}) (warm-start)
do not include the times taken by Algorithm~\ref{algohybrid}, the
combined times are similar to the times
taken by MIO formulation~(\ref{lqs-reg-form-1}) (cold-start).
The focus here is on comparing the quality of upper bounds to the LQS problem.
We consider the same datasets used in Section~\ref{secdeeper-1} for
our experiments.
The results are shown in Table~\ref{table-notgran-1-small}. We see
that MIO formulation~(\ref{lqs-reg-form-1}) (warm-start) is the clear
winner among all the examples,
Algorithm~\ref{algohybrid} comes a close second. MIO formulation~(\ref
{lqs-reg-form-1}) (cold-start) in the absence of advanced warm-starts
as provided by Algorithm~\ref{algohybrid} requires more time to obtain
high quality upper bounds. The state-of-the art algorithm for LQS
(obtained from the \texttt{R}-package \texttt{MASS}) delivers a solution very quickly,
but the solutions obtained are quite far from the global minimum.
\subsection{Performance on some real-world datasets}\label{secreal-data}
We considered a few real-world datasets popularly used in the context
of robust statistical estimation, as available from the {\texttt R}-package \texttt{robustbase} [\citet
{robust-base-manual,robust-base-paper}].
We used the ``Alcohol'' dataset~(available from the same package),
which is aimed at studying
the solubility of alcohols in water to understand alcohol transport in
living organisms. This dataset
contains physicochemical characteristics of $n = 44$
aliphatic alcohols and measurements on seven numeric variables:
SAG solvent accessible surface-bounded molecular volume ($x_{1}$),
logarithm of the octanol-water partitions coefficient ($x_{2}$), polarizability
($x_{3}$), molar refractivity ($x_{4}$), mass ($x_{5}$), volume ($x_{6}$)
and the response ($y$) is taken to be the logarithm of the solubility.
We consider two cases from the Alcohol dataset---the first one has $n =
44$, $p = 5$ where the five covariates were $x_{1},x_{2}, x_{4}, x_{5}, x_{6}$;
the~second example has all the six covariates and an intercept term,
which leads to $p = 7$. We used the MIO formulation~(\ref
{lqs-reg-form-1}) (cold-start) for both the cases.
The evolution of the MIO (with upper and lower bounds) for the two
cases are shown in
Figure~\ref{fig-alco-data1}. As expected, the time taken for the
algorithm to converge is larger for $p=7$ than for $p=5$.\vadjust{\goodbreak}
We considered a second dataset created by~\citet
{hawkins1984location} and available from the \texttt{R}-package
{\texttt{robustbase}}. The dataset consists of 75
observations in four dimensions (one response and three explanatory
variables), that is, $n = 75, p = 3$.
We computed the LQS estimate for this example for $q \in\{60, 45\}$.
We used both
the MIO formulation~(\ref{lqs-reg-form-1}) (cold-start) and MIO
formulation~(\ref{lqs-reg-form-1}) (warm-start) and observed that the
latter showed superior convergence speed to global optimality (see
Figure~\ref{fig-hbk-data1}). As expected, the time taken for
convergence was found to increase with decreasing
$q$-values. The results are shown in Figure~\ref{fig-hbk-data1}.
\begin{figure
\includegraphics{1223f02.eps}
\caption{Figure showing the evolution of the MIO formulation~(\protect\ref{lqs-reg-form-1}) for the HBK dataset with different values of
$q$ with and without warm-starts.
(Top row) MIO formulation warm-started with the least squares solution
(which we denote by ``cold-start''), for $q=60$ (left panel) and $q=45$
(right panel).
(Bottom row) MIO formulation warm-started with Algorithm~\protect\ref{algohybrid} for $q=60$ (left panel) and $q=45$ (right panel).}\label{fig-hbk-data1}
\end{figure}
\subsection{Certificate of lower bounds and global optimality}\label{secglobal-opt}
The MIO formulation~(\ref{lqs-reg-form-1}) for the LQS problem
converges to the global solution. With the aid of advanced MIO
warm-starts as
provided by Algorithm~\ref{algohybrid} the MIO obtains a very high
quality solution very quickly---in most of the examples the solution
thus obtained, indeed turns out to be the global minimum. However, the
certificate of global optimality comes later as the lower bounds of the
problem ``evolve'' slowly; see, for example, Figures~\ref{fig-alco-data1} and~\ref{fig-hbk-data1}.
We will now describe a regularized version of the MIO formulation,
which we found to be quite useful in speeding up the convergence of the
MIO algorithm without any loss in the
accuracy of the solution.
The LQS problem formulation does not contain any explicit
regularization on $\bolds\beta$, it is rather implicit (since
$\hat{\bolds\beta}{}^{(\mathrm{LQS})}$ will be generally bounded).
We thus consider the following modified version of the LQS
problem~(\ref{eqmedian-q}):
\begin{eqnarray}\label{eqmedian-q-bound}
&& \mini_{\bolds\beta}\qquad \llvert r_{(q)}\rrvert,
\nonumber\\[-8pt]\\[-8pt]\nonumber
&&\qquad\mbox{subject to}\qquad \llVert \bolds\beta- \bolds\beta_{0}\rrVert _\infty\leq M
\end{eqnarray}
for some predefined $\bolds\beta_{0}$ and $M\geq0$.
If $\hat{\bolds\beta}_{M}$ solves problem~(\ref
{eqmedian-q-bound}), then it is the global minimum
of the LQS problem in the $\ell_{\infty}$-ball $\{ \bolds\beta
\dvtx -M\mathbf{1} \leq\bolds\beta- \bolds\beta_{0} \leq
M\mathbf{1} \}$.
In particular, if $\hat{\bolds\beta}{}^{\mathrm{LQS}}$ is
the solution to problem~(\ref{eqmedian-q}), then by
choosing $\bolds\beta_{0}= \mathbf{0}$ and $M \geq\llVert \hat {\bolds\beta}{}^{\mathrm{LQS}} \rrVert _\infty$ in~(\ref
{eqmedian-q-bound}); both problems~(\ref{eqmedian-q}) and~(\ref
{eqmedian-q-bound})
will have the same solution. The MIO formulation of problem~(\ref
{eqmedian-q-bound}) is a very simple modification of~(\ref
{lqs-reg-form-1}) with additional box-constraints on
$\bolds\beta$ of the form $ \{ \bolds\beta\dvtx -M\mathbf{1}
\leq\bolds\beta- \bolds\beta_{0} \leq M\mathbf{1} \}$.
Our empirical investigation suggests that the MIO formulation~(\ref{lqs-reg-form-1}) in presence of box-constraints\footnote{Of course, a
very large value of $M$ will render the box-constraints to be ineffective.}
produces tighter lower bounds than the unconstrained MIO
formulation~(\ref{lqs-reg-form-1}), for a given time limit.
As an illustration of formulation~(\ref{eqmedian-q-bound}), see
Figure~\ref{fig-hbk-data-fraction}, where we use the MIO
formulation~(\ref{lqs-reg-form-1}) with box constraints.
We consider two cases corresponding to $M \in\{ 3, 40\}$; in both the
cases we took $\bolds\beta_{0} = \hat{\bolds\beta}{}^{(\mathrm{LS})}=(0.08,-0.36,0.43)$.
Both these boxes (which are in fact, quite large, given that $\llVert
\hat{\bolds\beta}{}^{\mathrm{(LS)}}\rrVert _\infty= 0.43$)
contains the (unconstrained)
global solution for the problem. As the figure shows, the evolution of
the lower bounds of the
MIO algorithm toward the global optimum depends upon the radius of the box.
\begin{figure
\includegraphics{1223f03.eps}
\caption{Figure showing the effect of the bounding box for the
evolution of the MIO formulation~(\protect\ref{lqs-reg-form-1}) for
the HBK dataset, with
$(n,p,q) = (75, 3, 45)$. The left panel considers a bounding box of
diameter 6 and the right panel considers a bounding box of diameter 80
centered around the least squares solution.}\label{fig-hbk-data-fraction}
\end{figure}
We argue that formulation~(\ref{eqmedian-q-bound}) is a more desirable
formulation---the constraint may behave as a regularizer to shrink
coefficients or
if one seeks an unconstrained LQS solution, there are effective ways to
choose to $\bolds\beta_{0}$ and $M$. For example,
if $\bolds\beta_{0}$ denotes the solution obtained by
Algorithm~\ref{algohybrid}, then for $M=\eta\llVert \bolds\beta_{0}
\rrVert _{\infty}$, for $\eta\in[1, 2]$ (say),
the solution to~(\ref{eqmedian-q-bound}) corresponds to a global
optimum of the LQS problem inside a box of diameter $2M$ centered at
$\bolds\beta_{0}$.
For moderate sized problems with $n \in\{ 201, 501\}$, we found this
strategy to be useful in certifying global optimality within a
reasonable amount of time.
Figure~\ref{figmio-moderate-p} shows some examples.
\begin{figure
\includegraphics{1223f04.eps}
\caption{Figure showing evolution of MIO in terms of upper/lower
bounds (left panel) and Optimality gaps (in \%) (right panel).
Top and middle rows display an instance of Ex-1 with $(n, p, q) = (201,
5, 121)$ with different initializations, that is, MIO~(\protect\ref{lqs-reg-form-1}) (cold-start) and
MIO~(\protect\ref{lqs-reg-form-1}) (warm-start), respectively.
Bottom row considers an instance of Ex-3 with $(n, p, q) = (501, 5,
301)$.}\label{figmio-moderate-p}
\end{figure}
\subsection{Scalability to large problems}\label{seclarge-scale-method1}
We present our findings for large-scale experiments performed on
synthetic and real data-set, below.
\subsubsection*{Synthetic large scale examples}
For large-scale problems
with $n \geq5000$ with $p \geq10$, we found that Algorithm~\ref
{algoseq-LO1} becomes computationally demanding due to the
associated LO problems~(\ref{lqs-lp-almost-20}) appearing in step~2 of
Algorithm~\ref{algoseq-LO1}.
On the other hand,
Algorithm~\ref{algoGD} remains computationally inexpensive. So for
larger problems, we propose using a modification of Algorithm~\ref
{algoseq-LO1}---we\vspace*{1pt} run
Algorithm~\ref{algoGD} for several random initializations around the
$\hat{\bolds\beta}{}^{(\mathrm{LAD})}$ solution and find the
best solution among them.
The regression coefficient thus obtained is used
as an initialization for Algorithm~\ref{algoseq-LO1}---we call this
Algorithm~\ref{algohybrid} (large-scale).
Note that this procedure deviates from the vanilla Algorithm~\ref
{algohybrid} (described in Section~\ref{sechybrid-1}), where, we do
\emph{both} steps~1~and~2 for every initialization $\bolds
\beta_{1}$.
For each of the examples Ex-5--Ex-7, Algorithm~\ref{algoGD}
was run for $\operatorname{MaxIter} = 500$, for 100 different
initializations around the LAD solution, the best solution was used as
an initialization for
Algorithm~\ref{algoseq-LO1}. Table~\ref{table-notgran-1-large}
presents the results obtained with
Algorithm~\ref{algohybrid} (large-scale). In addition, the
aforementioned algorithms, Table~\ref{table-notgran-1-large} also presents
MIO (warm-start), that is, MIO formulation~(\ref{lqs-reg-form-1})
warm-started with
Algorithm~\ref{algohybrid} (large-scale) and the LQS algorithm from
the \texttt{R}-package \texttt{MASS}.
\subsubsection*{Large scale examples with real datasets}
In addition to the
above, we considered a large environmental dataset from the \texttt
{R}-package {\texttt{robustbase}}
with hourly measurements of NOx pollution content in the ambient air.
The dataset has $n = 8088$ samples with $p = 4$ covariates (including
the intercept).
The covariates are
square-root of the windspeed $(x_{1})$,
day number ($x_{2}$),
log of hourly sum of NOx emission of cars ($x_{3}$) and intercept,
with response being log of hourly mean of NOx concentration in ambient
air $(y)$.
We considered three different values of $q \in\{ 7279, 6470, 4852 \}$
corresponding to the $90$th, $80$th and $60$th quantile, respectively.
We added a small amount of contamination by changing $\lfloor0.01n
\rfloor$ sample points according to item~(B) in Section~\ref{synthetic-egs}. On the modified dataset,
we ran three different algorithms: Algorithm~\ref{algohybrid}
(large-scale),\footnote{In this example, we initialized Algorithm~\ref
{algoGD} with the best Chebyshev fit from forty different subsamples.
Algorithm~\ref{algoGD} was run for $\operatorname{MaxIter}=500$, with five hundred
random initializations. The best solution was taken as the starting
point of Algorithm~\ref{algohybrid}.} MIO (warm-start), that is, MIO
formulation~(\ref{lqs-reg-form-1}) warm-started with
Algorithm~\ref{algohybrid} (large-scale) and
the LQS algorithm from the \texttt{R}-package \texttt{MASS}. In all the
following cases, the MIO algorithm was run for a maximum of two hours.
We summarize our key findings below:
\begin{table
\tabcolsep=0pt
\caption{Table showing performances of various Algorithms for the LQS
problem for different moderate/large-scale examples as described in the text.
For each example, ``Accuracy'' is Relative Accuracy [see~(\protect\ref{rel-accu})]
the numbers within brackets denote the standard errors;
the lower row denotes the averaged cpu time (in secs) taken for the
algorithm. All results are averaged over 20 random examples}\label{table-notgran-1-large}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}@{}lc cccc@{}}
\hline
\multirow{2}{71pt}{\break \break \textbf{Example} $\bolds{(n,p,\pi)}$\break \textbf{q}} & & \multicolumn{4}{c@{}}{\textbf{Algorithm used}}\\[-6pt]
& & \multicolumn{4}{c@{}}{\hrulefill}\\
& & \multirow{3}{32pt}{\centering{\textbf{LQS}\break \textbf{(MASS)}}} & \multirow{3}{47pt}{\centering{\textbf{Algorithm~\ref{algohybrid} (large-scale)}}} & \multicolumn{2}{c@{}}{\textbf{MIO formulation~(\ref{lqs-reg-form-1})}}\\[-6pt]
& & & & \multicolumn{2}{c@{}}{\hrulefill}\\
& & & & \textbf{(Cold-start)} & \textbf{(Warm-start)}\\
\hline
Ex-5 (2001, 10, 0.4) & Accuracy & 65.125 (2.77) &0.0 (0.0) & 273.543 (16.16) & 0.0 (0.0) \\
$q=1201$ & Time (s) & 0.30 &13.75 & 200& 100\\[3pt]
Ex-6 (5001, 10, 0.4)& Accuracy & 52.092 (1.33) & 0.0 & 232.531 (17.62) & 0.0 (0.0) \\
$q=3001$ & Time (s) & 0.69 & 205.76 & 902 & 450.35 \\[3pt]
Ex-7 (10,001, 20, 0.4)& Accuracy &146.581 (3.77) & 0.0 (0.0) & 417.591 (4.18) & 0.0 (0.0) \\
$q=6001$ &Time (s) & 1.80& 545.88 & 1100 & 550 \\
\hline
\end{tabular*}
\end{table}
\begin{longlist}[(3)]
\item[(1)] For $q = 7279$, the best solution was obtained by MIO
(warm-start) in about 1.6 hours.
Algorithm~\ref{algohybrid} (large-scale) delivered a solution with
relative accuracy~[see~(\ref{rel-accu})] $0.39$\%
in approximately six minutes.
The LQS algorithm from \texttt{R}-package \texttt{MASS}, delivered a solution
with relative accuracy $2.8$\%.
\item[(2)] For $q = 6470$, the best solution was found by MIO (warm-start)
in 1.8 hours.
Algorithm~\ref{algohybrid} (large-scale) delivered a solution with
relative accuracy [see~(\ref{rel-accu})] $0.19$\%
in approximately six minutes.
The LQS algorithm from \texttt{R}-package \texttt{MASS}, delivered a solution
with relative accuracy $2.5$\%.
\item[(3)] For $ q = 4852$, the best solution was found by MIO (warm-start)
in about 1.5 hours.
Algorithm~\ref{algohybrid} (large-scale) delivered a solution with
relative accuracy~[see~(\ref{rel-accu})] $0.14$\%
in approximately seven minutes.
The LQS algorithm from \texttt{R}-package \texttt{MASS}, delivered a solution
with relative accuracy $1.8$\%.
\end{longlist}
Thus, in all the examples above, MIO warm-started with Algorithm~\ref
{algohybrid} (large-scale) obtained the best upper bounds.
Algorithm~\ref{algohybrid} (large-scale) obtained very high quality
solutions, too, but the solutions were all improved by MIO.
\section{Conclusions}
In this paper, we proposed algorithms for LQS problems based on a
combination of first-order methods from continuous optimization and
mixed integer optimization. Our key conclusions are:
\begin{longlist}[(3)]
\item[(1)] The MIO algorithm with warm start from the continuous
optimization algorithms solves to provable optimality problems of small
($n=100$)
and medium ($n=500$) size problems in under two hours.\vadjust{\goodbreak}
\item[(2)] The MIO algorithm with warm starts finds high quality solutions
for large ($n={}$10,000) scale problems in under two hours outperforming
all state of the art algorithms
that are publicly available for the LQS problem. For problems of this
size, the MIO algorithm does not provide a certificate of optimality in
a reasonable amount of time.
\item[(3)] Our framework enables us to show the existence of an optimal
solution for the LQS problem for \emph{any}
dataset, where the data-points $(y_{i}, \mathbf{x}_{i})$'s are not
necessarily in general position.
Our MIO formulation leads to a simple proof of the breakdown point of
the LQS optimum objective value that holds for general datasets
and our framework can easily incorporate extensions of the LQS
formulation with
polyhedral constraints on the regression coefficient vector.
\end{longlist}
|
2,869,038,156,350 | arxiv | \section{Introduction}
Multi-component plasmas comprise different species that, in the presence of waves, may be in the state of relative
macroscopic motion. In such a situation, friction between the species may lead to wave damping (though not always, as
we are going to show in the forthcoming text). For example, neutrals in a weakly ionized plasma represent a barrier for
electron and ion motion in a wave field. A similar friction appears in a fully ionized plasma when the electron and
ion components do not share the same momentum. The interaction is described by a friction force $\vec F_j= m_j n_j
\nu_{jl} (\vec v_j-\vec v_l)$ in the momentum equation for the species $j$. Momentum conservation implies that for its
counterpart $l$, $\vec F_l= m_l n_l \nu_{lj} (\vec v_l-\vec v_j)$, where $m_j n_j \nu_{jl}= m_l n_l \nu_{lj}$. If the
two species $j$ and $l$ have a large mass difference, the friction response of the heavier component is
typically omitted as negligible in the literature. However, this may yield completely wrong results as we shall demonstrate in the
forthcoming text using the ion acoustic (IA) mode as an example.
In the presence of high frequency waves $\omega \gg \Omega_i=e B_0/m_i$ in a plasma placed in an external magnetic
field $\vec B_0= B_0 \vec e_z$, ions will follow nearly straight lines regardless of the direction of the wave-number
vector $\vec k$ and the magnetic field vector. For electrons, in view of the mass difference, the opposite may hold,
$\omega \ll \Omega_e=e B_0/m_e$, hence they will behave as magnetized and their perpendicular and parallel dynamics
will be essentially different \cite{k1}. Ions can behave as un-magnetized in the perturbed state also in case of
collisions provided that $\nu_i>\Omega_i$ even if at the same time $\Omega_i > \omega$, or for short wavelengths
$\lambda<\rho_i$, $\rho_i=v_{{\scriptscriptstyle T} i}/\Omega_i$, $v_{{\scriptscriptstyle T} i}^2=\kappa T_i/m_i$. In the case of an inhomogeneous
equilibrium, with a density gradient perpendicular to the magnetic field vector, in the unperturbed state the ions may
behave as un-magnetized in case of a low temperature, when their diamagnetic drift velocity becomes negligible as
compared to electrons [for singly charged ions $v_{*i}/v_{*e}=T_i/T_e$, where $v_{*j}= \kappa T_j n_{j0}'/(q_j B_0
n_{j0})$, and $n_j'=d n_j/dx$ denotes the equilibrium density gradient]. The same holds in the presence of numerous
collisions as above, $\nu_i>\Omega_i$, when their diamagnetic effects are absent too.
In all these situations, and neglecting the electron polarization drift (inertia-less limit), the wave will still
have the basic properties of the IA mode. Within the two-fluid theory such a mode in an inhomogeneous plasma [that may be
called ion-acoustic-drift (IAD) mode] may in fact become growing \cite{k1}-\cite{v1} in the simultaneous presence of
collisions and the mentioned equilibrium density gradient perpendicular to $\vec B_0$.
Within the kinetic theory the mode is also growing in the presence of the same density gradient and this even without
collisions (due to purely kinetic effects), and the physics of the growth rate is similar to the standard drift wave
instability \cite{k2}. It requires that the wave frequency is below the electron diamagnetic drift frequency
$\omega_{*e}=v_{*e} k_\bot$.
On the other hand, keeping the electron inertia results in the instability of the lower-hybrid-drift (LHD) type
\cite{k3}-\cite{h1}. In some other limits the effects of the same density gradient yield growing ion plasma (Langmuir)
oscillations \cite{k3}, or growing electron-acoustic oscillations \cite{moh}.
In the present manuscript the friction force effects on the IA wave are discussed, both for fully and partially ionized
un-magnetized plasmas, and for inhomogeneous plasmas with magnetized electrons. The latter implies growing modes within
both the fluid and kinetic descriptions, and in the manuscript these two instabilities are compared.
\section{IA wave in fully and partially ionized collisional plasmas}
The equations used further in this section are the momentum equations for the ions, the electrons and the neutral particles, respectively:
\be
m_i n_i \left(\frac{\partial}{\partial t} + \vec v_i\cdot \nabla\right)\vec v_i= - e n_i \nabla \phi- \kappa T_i \nabla
n_i - m_i n_i \nu_{ie} (\vec v_i - \vec v_e) - m_i n_i \nu_{in} (\vec v_i - \vec v_n),\label{s1} \ee
\be
m_e n_e \left(\frac{\partial}{\partial t} + \vec v_e\cdot \nabla\right)\vec v_e= e n_e \nabla \phi- \kappa T_e \nabla
n_e - m_e n_e \nu_{ei} (\vec v_e - \vec v_i) - m_e n_e \nu_{en} (\vec v_e - \vec v_n),\label{s2} \ee
and
\be
m_n n_n \left(\frac{\partial}{\partial t} + \vec v_n\cdot \nabla\right)\vec v_n= - \kappa T_n \nabla n_n - m_n n_n
\nu_{ni} (\vec v_n - \vec v_i) - m_n n_n \nu_{ne} (\vec v_n - \vec v_e),\label{s3} \ee
and the continuity equation
\be
\frac{\partial n_j}{\partial t}+ \nabla\cdot(n_j \vec v_j)=0, \quad {j=e, i, n}. \label{s4} \ee
This set of equations is closed either by using the quasi-neutrality or the Poisson equation. The differences between
the two cases are discussed below.
\subsection{Friction in electron-ion plasma}
The continuity equation (\ref{s4}) yields
\be
v_{i1}=\omega n_{i1}/(k n_0), \quad v_{e1}=\omega n_{e1}/(k n_0),
\label{c} \ee
so that the velocity difference in the friction term $v_e - v_i\equiv 0$ if the quasi-neutrality is used. The IA mode propagates without any damping. Hence, the friction force in a fully ionized plasma in this limit
cancels out exactly even without using the momentum balance. The physical reason for this is the assumed exact balance of
the perturbed densities: what one plasma component loses the other component receives, this is valid at every
position in the wave and no momentum is lost.
A typical mistake seen in the literature is to take the friction force term for electrons only, in the form $m_e n_e
\nu_{ei} \vec v_e$. This comes with the excuse of the large mass difference, so that the displacement of the much
heavier ion fluid, caused by the electron friction is neglected. In the case of a fully ionized electron-ion plasma
this yields a false damping of the IA mode within the quasi-neutrality limit:
\be
\omega=\pm k(c_s^2 + v_{{\sss T}i}^2)^{1/2} - \nu_{ei}/2.\label{ins} \ee
On the other hand, if the Poisson equation is used instead of the quasi-neutrality, one obtains \cite{v1a}
%
\be
\omega=\pm k v_s \left(1-r_{de}^2k^2 \frac{\nu_{ie}^2r_{de}^2}{v_s^2}\right)^{1/2} - i \nu_{ie}r_{de}^2k^2.\label{puas}
\ee
Here, we have used the momentum conservation $\nu_{ie}=m_e\nu_{ei}/m_i$ and $v_s^2= c_s^2+ v_{{\sss T}i}^2$,
$r_{de}=v_{{\sss T}e}/\omega_{pe}$. The physical reason for damping in the present case is the fact that the detailed balance
$n_{i1}=n_{e1}$ does not hold, because of the electric field which takes part for small enough wave-lengths. It can
easily be seen that for any realistic parameters the second term in the real part of the frequency in Eq.~(\ref{puas}) is
much below unity and the mode is never evanescent. However, in partially ionized plasmas (see below) this may be
completely different.
\subsection{Friction and collisions in partially ionized plasma}
Keeping the quasi-neutrality limit, we now discuss the IA wave damping in plasmas comprising neutrals as well. In view
of the results presented above, the electron-ion friction terms in Eqs.~(\ref{s1}) and (\ref{s2}) will cancel each other
out and in a few steps one derives the following dispersion equation containing the collisions of plasma species with neutrals
and vice versa:
\[
\omega^3+ i \omega^2 \left(\nu_{en} \frac{m_e}{m_i} + \nu_{in}\right) \left(1 + \frac{m_i}{m_n}\frac{n_0}{n_{n0}}
\right) - k^2 c_s^2 \,\omega
\]
\be
- i k^2 c_s^2 \,\frac{m_e}{m_n} \frac{n_0}{n_{n0}} \left(\nu_{en} + \frac{m_i}{m_e} \nu_{in}\right)=0. \label{dn} \ee
In the derivation, the ion and neutral thermal terms are neglected. The ion thermal terms would give the modified
mode frequency $\omega^2=k^2 c_s^2 (1+ T_i/T_e)$. Hence, even if $T_e=T_i$ the wave frequency is only modified by a
factor $2^{1/2}$. The neutral thermal terms are discussed further in the text.
\begin{figure}
\includegraphics[height=6cm, bb=15 15 270 220, clip=,width=.5\columnwidth]{h22.eps}\\
\caption{\label{figs} Frequency $\omega_r$ and absolute value of the IA mode damping $|\gamma|$ in terms of the number
density of neutrals. Details of the mode behavior in the region A are better seen in the linear scale (small figure
inside). }
\end{figure}
Note that in deriving Eq.~(\ref{dn}), the momentum conservation condition $\nu_{ie}=m_e\nu_{ei}/m_i$ is nowhere used: the
e-i and i-e friction terms exactly vanish in view of Eq.~(\ref{c}).
Equation~(\ref{dn}) is solved numerically for a plasma containing electrons, protons, and neutral hydrogen atoms using the
following set of parameters: $T_e= 4\;$eV, $n_0=10^{18}\;$m$^{-3}$, $k=10\;$m$^{-1}$, with \cite{bk} $\sigma_{en}=1.14
\cdot 10^{-19}\;$m$^{-2}$. The neutral density is varying in the interval $10^{16} - 10^{23}\;$m$^{-3}$. The ion and hydrogen
temperatures are taken $T_i=T_n=T_e/20$, satisfying the condition of their small thermal effects. This also gives
\cite{kr}, $\sigma_{in}=2.24 \cdot 10^{-18}\;$m$^{-2}$. The results are presented in Fig.~1. The IA mode propagates in
two distinct regions A and B.
Only a limited left part of the region A would correspond to the 'standard' IA wave behavior in a collisional plasma: the
mode is damped and the damping is proportional to the neutral number density. Hence, in this region it may be more or
less appropriate to use the approximate expressions for the friction force, like (in the case of electrons) $F_e\simeq
m_e n_0\nu_{en} v_{e1}$. However, this domain is very limited because in the rest of the domain the frequency drops
and the mode becomes non-propagating for $n_{n0}\geq 3.8 \cdot 10^{19}\;$m$^{-3}$ (this is the lower limit of the region C
in Fig.~1).
Increasing the neutral number density, after some critical value (in the present case this is around $n_{n0} \simeq
10^{20}\;$m$^{-3}$) the IA mode reappears again in the region B, with a frequency starting from zero. For even larger
neutrals number densities, the mode damping in fact vanishes completely and the wave propagates freely but with a
frequency that is many orders of magnitude below the ideal case $k c_s \simeq 196\;$kHz. This behavior can be explained
in the following manner. For a relatively small number of collisions the IA mode is weakly damped because initially
neutrals do not participate in the wave motion and do not share the same momentum. Increasing the number of neutrals,
the damping may become so strong that the wave becomes evanescent. However, for much larger collision frequencies
(i.e., for a lower ionization ratio), the tiny population of electrons and ions is still capable of dragging neutrals
along and all three components move together. The plasma and the neutrals become so strongly coupled that the two
essentially different fluids participate in the electrostatic wave together. In this regime, the stronger the
collisions are, the less wave damping there is! Yet, this a bit counter-intuitive behavior comes with a price: the wave frequency
and the wave energy flux becomes reduced by several orders of magnitude.
\begin{figure}
\includegraphics[height=6cm, bb=15 15 270 220, clip=,width=.5\columnwidth]{k3b.eps}\\
\caption{\label{figk} Frequency $\omega_r$ and absolute value of the IA mode damping $|\gamma|$ in terms of the wave
number. The line $k c_s$ shows part of the graph of the ideal mode. Details of the domain $a$ are better seen in the
linear scale (small figure inside). }
\end{figure}
Similar effects may be expected by varying the wave-length. The previous role of the varying density of neutrals is
now replaced by the the ratio of the mean free path of a species $\lambda_{fj}=v_{{\scriptscriptstyle T} j}/\nu_j$ (with respect to
their collision with neutrals) and the wavelength. This ratio now determines the coupling between the plasma and
the neutrals. The mode behavior is directly numerically checked by fixing $n_{n0}=10^{20}\;$m$^{-3}$,
$n_0=10^{18}\;$m$^{-3}$, and for other parameters same as above. For these parameters we have $\lambda_{fe}=v_{{\scriptscriptstyle T}
e}/\nu_{en}=0.09$ m, and $\lambda_{fi}=v_{{\scriptscriptstyle T} i}/\nu_{in}=0.004$. The numerical results are presented in Fig.~2
for $k$ varying in the interval $0.2 - 80\;$m$^{-1}$. The mode vanishes in the interval $c$, between
$k\simeq 10\;$m$^{-1}$ and $k\simeq 25.6\;$m$^{-1}$. The explanation is similar as before. Note that for $k=0.2\;$m$^{-1}$ (in the
region $a$) we have $\omega_r\simeq 390\;$Hz, and this is about one order below $k c_s$. Compared to the mode behavior
in Fig.~1, this implies that the mode in the present domain $a$ is in the regime equivalent to the domain B from
Fig.~1; here, in Fig.~2, these large wave-lengths imply well coupled plasma-neutrals, where the frequency is reduced and
the damping is small. The region $a$ is also given separately in linear scale together with the dotted line describing the
ideal mode $k c_s$. Clearly, in general the realistic behavior of the wave is beyond recognition and completely
different as compared to the ideal case.
\begin{figure}
\includegraphics[height=6cm, bb=25 10 655 530, clip=,width=.5\columnwidth]{limits.eps}
\caption{\label{figl} The two lines give the lower and upper values of the neutrals' density $n_{n0}$ between which,
for the given plasma density $n_0$, the IA mode does not propagate. }
\end{figure}
After checking for various sets of plasma densities, it appears that the evanescence region reduces and vanishes for
larger plasma densities $n_0$. This is presented in Fig.~3 for the same parameters as above, by taking $k=10\;$m$^{-1}$,
but for a varying plasma density $n_0$. The two lines represent boundary values of the number densities of
neutrals, for the given plasma density, at which the IA mode vanishes; for the neutrals densities between the two
lines the IA mode does not propagate. The symbols $*$ on the two lines denote the boundaries of the region $C$ from
Fig.~1. It is seen that for the given case the IA mode propagates without evanescence for the plasma densities above
$n_0=3.8 \cdot 10^{18}\;$m$^{-3}$. Physical reason for a larger non-propagating domain for low plasma density is
obvious, namely the tiny plasma population is less efficient in inducing a synchronous motion of neutrals. In the other
limit, the opposite happens and the forbidden region eventually vanishes.
\begin{figure}
\includegraphics[height=6cm, bb=40 14 655 525, clip=,width=.5\columnwidth]{limits-k.eps}
\caption{\label{fig11} Values of the wave-number, in terms of the plasma density, for which the IA wave becomes
evanescent. In the region between the lines the mode does not propagate. }
\end{figure}
A similar check is done by varying the wave-number and the plasma density, and the result is presented in Fig.~4 for a
fixed $n_{n0}=10^{20}\;$m$^{-3}$. The lines represent the values $(n_0, k)$ at which the IA wave becomes evanescent.
There can be no wave in the region between the lines. On the other hand, there is no evanescence for the plasma density
above $n_0=1.2 \cdot 10^{19}\;$m$^{-3}$. Here $*$ denote the boundaries of the region $c$ from Fig.~2.
All these results clearly indicate that in practical measurements in laboratory and space plasmas, the IA mode
can hardly be detected and recognized as the IA mode unless collisions are correctly taken into account (using full
friction terms), and the mode is sought in the corresponding domain which follows from our Eq.~(\ref{dn}).
\subsection{Thermal effects of neutrals}
Keeping the pressure terms for ions and neutrals yields the following dispersion equation
\[
\omega^4+ i \omega^3 \left(\nu_{in} + \nu_{en} \frac{m_e}{m_i}\right) \left(1+ \frac{m_i}{m_n}
\frac{n_0}{n_{n0}}\right) - k^2 \left(v_s^2 + v_{{\sss T}n}^2\right) \omega^2
\]
\be
- i \omega k^2 \left[\frac{n_0}{n_{n0}} \frac{m_i}{m_n}v_s^2 \left(\nu_{in} + \nu_{en} \frac{m_e}{m_i}\right) +
\nu_{in} v_{{\sss T}n}^2\right] + k^4 v_{{\sss T}n}^2 v_s^2=0. \label{g} \ee
Here, $v_s^2=c_s^2 + v_{{\sss T}i}^2$. Without collisions, this yields two independent modes, viz.\ the ion-acoustic mode and the
gas thermal (GT) mode, $(\omega^2- k^2 v_{{\sss T}n}^2)(\omega^2- k^2 v_s^2)=0$. The collisions couple the two modes, and in order to compare with
the previous cases we solve Eq.~(\ref{g}) for $k=10\;$m$^{-1}$, $n_0=10^{18}\;$m$^{-3}$, $T_e=4\;$eV, $T_i=T_e/20$, and in
terms of the density and temperature of neutrals. For a low thermal contribution of neutrals (i.e., a low neutral
temperature, or/and heavy neutral atoms) the previous results remain valid. Larger values of $v_{{\sss T}n}$ introduce new
effects, this is checked by varying the temperature $T_n$. The ion thermal terms do not make much difference, as
explained earlier. The real part of the frequency $\omega_g$ of the gas thermal mode is presented in Fig.~\ref{gg},
and this only in a limited region that includes the evanescence area $C$ from Fig.~\ref{figs}. The damping is not
presented but the mode is in fact heavily damped.
\begin{figure}
\includegraphics[height=6cm, bb=15 15 280 225, clip=,width=.5\columnwidth]{og.eps}
\caption{\label{gg} The real part of the frequency of damped gas thermal mode in terms of the number density of
neutrals and for several temperatures of the neutrals gas. }
\end{figure}
The explanation of the figure is as follows. The starting solution for $T_n=0$ is in fact the line $\omega_g=0$, and
this case would correspond to the the IA mode from Fig.~\ref{figs}. For some finite $T_n$ there appears the GT mode.
For a low gas temperature the mode becomes evanescent for a higher density of neutrals (the dot and dash-dot lines in
Fig.~\ref{gg}). This evanescence is accompanied with the previously discussed evanescence and re-appearance of the IA
mode (described earlier and no need to be presented here again). However, for still larger $T_n$, the IA and GT modes
become indistinguishable and propagate as one single mode. This is presented by the two upper (the full and
dashed) lines in Fig.~\ref{gg}, that go up for large enough $n_{n0}$. Also given are the corresponding ideal values $k
v_{{\sss T}n}$ that appear to be much above the actual wave frequency $\omega_g$ in such a collisional plasma, but this
remains so only until the neutral density $n_{n0}$ exceeds some critical value. After that the wave in fact behaves
as less and less collisional and the wave frequency is increased.
\section{IA wave instability in inhomogeneous partially ionized plasma}
\subsection{Fluid description in collisional plasma}
In the previous text, collisions were shown to yield damping of the IA mode. However, if the plasma is inhomogeneous,
implying the presence of source of free energy in the system, a drift-type instability of the IA wave may develop if there is
a magnetic field
$\vec B_0 = B_0 \vec e_z$ present, and the electrons (ions) are magnetized (un-magnetized). The magnetic field introduces a
difference in the parallel and perpendicular dynamics of the magnetized species so that the continuity condition in this
case can be written as
\be
\frac{\partial n_{j1}}{\partial t} + n_{j0} \nabla\cdot \vec v_{j1} + \vec v_{j1}\cdot \nabla n_{j0}=0. \label{e2} \ee
Here, $\nabla\equiv \nabla_\bot + \nabla_z$. For the {\em un-magnetized} species the direction of the wave plays no role
so that $\nabla\rightarrow i \vec k$, $k^2=k_y^2+ k_z^2$. On the other hand, for the equilibrium gradient along the
$x$-axis and for perturbations of the form $\sim f(x)\exp(-i \omega t + i k_y y + i k_z z)$, where $|(d f/dx)/f|, |(d
n_{j0}/dx)/n_0|\ll k_y$, we apply a local approximation, and for ions the last term in Eq.~(\ref{e2}) vanishes because of the
assumed geometry. The ions' dynamics is basically the same as in the previous sections.
The electron momentum equation (\ref{s2}) will now include the Lorentz force term $- e n_e \vec v_e\times \vec B$.
Repeating the derivation from Ref.~\cite{v1}, the total perpendicular electron velocity can be written as
\be
v_{e\bot}= \frac{1}{1+ \nu_{en}^2 \alpha^2/\Omega_e^2}\left[\frac{1}{B_0}\vec
e_z\times \nabla_\bot \phi +\frac{\nu_{en} \alpha}{\Omega_e} \frac{\nabla_\bot \phi}{B_0} - \frac{v_{\scriptscriptstyle{T}
e}^2\nu_{en} \alpha}{\Omega_e^2} \frac{\nabla_\bot n_e}{n_e} - \frac{v_{\scriptscriptstyle{T} e}^2}{\Omega_e} \vec e_z\times
\frac{\nabla_\bot n_e}{n_e}\right]. \label{e5} \ee
In the direction along the magnetic field vector, the perturbed electron velocity is
\be
v_{ez1}=\frac{i k_z v_{{\sss T}e}^2}{\nu_{en}} \frac{\omega^2 + \nu_{ne}^2}{\omega^2 - i \nu_{ne} \omega} \left(\frac{e
\phi_1}{\kappa T_e} - \frac{n_{e1}}{n_0} \right). \label{e6} \ee
Here, $\alpha=\omega/(\omega + i \nu_{ne})$, and for magnetized electrons, $|\nu_{en}^2 \alpha^2/\Omega_e^2|\ll 1$ in
the denominator in Eq.~(\ref{e5}). Using these equations in the continuity condition (\ref{e2}) for electrons one
obtains
\be
\frac{n_{e1}}{n_0}= \frac{\omega_{*e} + i D_p + i D_z (\omega^2 + \nu_{ne}^2)/(\omega^2- i \nu_{ne} \omega)}{ \omega +
i D_p + i D_z (\omega^2 + \nu_{ne}^2)/(\omega^2- i \nu_{ne} \omega)} \frac{e \phi_1}{\kappa T_e}, \label{e7} \ee
\[
D_p= \nu_{en}\alpha k_y^2 \rho_e^2, \quad D_z= k_z^2 v_{{\sss T}e}^2/\nu_{en}, \quad \rho_e=v_{{\sss T}e}/\Omega_e.
\]
The term $D_p$ describes the effects of collisions on the electron perpendicular dynamics and is usually omitted in the
literature. However, as shown in a recent study \cite{v1}, it can strongly modify the growth rate of the drift and
IA-drift wave instability in the limit of small parallel wave-number $k_z$.
Neglecting the neutral dynamics is equivalent to setting $\nu_{ne}=0$. This yields $\alpha=1$, and Eq.~(\ref{e7}) becomes
identical to the corresponding expression in Refs.~\cite{mih,v5}. For a negligible $D_p$, Eq.~(\ref{e7}) becomes the
same as the corresponding equation from Ref.~\cite{v3}.
For negligible ion thermal effects, the final dispersion equation reads
%
\be
\frac{k^2 c_s^2 }{\omega^2} = \frac{\omega_{*e} + i D_p + i D_z (\omega^2 + \nu_{ne}^2)/(\omega^2- i \nu_{ne} \omega)}{
\omega + i D_p + i D_z (\omega^2 + \nu_{ne}^2)/(\omega^2- i \nu_{ne} \omega)}. \label{e9a} \ee
Equation~(\ref{e9a}) can be solved numerically keeping in mind a number of conditions used in their derivations, like smallness of
the plasma beta to remain in electrostatic limit, smallness of the parallel phase velocity as compared to the electron thermal
speed because of the massless electrons limit, also the ratio $D_p/D_z$ should be kept not too big or too small in
order to have the assumed effects of electron collisions in perpendicular direction. We plan to compare this
collisional instability with the kinetic instability due to the presence of the density gradient. Therefore, the wave
frequency should be below the electron diamagnetic frequency etc.
\begin{figure}
\includegraphics[height=7cm, bb=5 5 280 220, clip=,width=.6\columnwidth]{argon-3b.eps}
\caption{\label{fig1a} Real part of the frequency from Eq.~(\ref{e9a}) (full lines) and the corresponding growth rates
(dashed lines), both normalized to the electron diamagnetic drift frequency, for three values of neutral number
density. The lines I, II, III correspond (respectively) to $n_{n0}=10^{19}, \, 10^{18}, \,10^{17}\;$m$^{-3}$. The line
$\gamma_k$ is the kinetic growth-rate from Eq.~(\ref{kg}) (for the same parameters as line II). }
\end{figure}
\begin{figure}
\includegraphics[height=7cm, bb=5 5 280 220, clip=,width=.6\columnwidth]{ln.eps}
\caption{\label{fig1b} The real and imaginary parts of the frequency for the line II from Fig.~\ref{fig1a}, in terms
of the characteristic density inhomogeneity scale length $L_n=(dn_0/dx)^{-1}$, and for the angle $\theta$ at the
maximum on Fig.~\ref{fig1a}.}
\end{figure}
We solve Eq.~(\ref{e9a}) for an electron-argon plasma in the presence of parental argon atoms. As an example we take
$T_e= 4\;$eV, $T_i=T_n=T_e/30$, $n_0=10^{15}\;$m$^{-3}$, $B_0=1.2\cdot 10^{-2}\;$T, $k=500\;$m$^{-1}$, $L_n =0.05\;$m, and take
several values for the density of neutrals. The result in terms of the angle of the propagation
$\theta=\arctan(k_z/k_y)$ is presented in Fig.~5. The three lines (full for the real part of the frequency, and dashed
for the growth rates) are for $n_{n0}=10^{19}, \, 10^{18}, \,10^{17}\;$m$^{-3}$. It is seen that i) the instability is
angle dependent and there exists an angle of preference and an instability window in terms of $\theta$ within which
the mode is most easily excited, ii) this angle of preference is shifted towards smaller values for lower values of the
neutral density, and iii) in the same time the instability window becomes considerably reduced. This shows an
interesting possibility of launching the IA-drift wave in a certain direction by simply varying the pressure of the
neutral gas.
Varying the density scale length $L_n=(dn_0/dx)^{-1}$ the wave frequency may become above $\omega_{*e}$ and in this
case the instability vanishes. As an example, this is demonstrated in Fig.~\ref{fig1b} for the parameters corresponding
to the line II from Fig.~\ref{fig1a} and for the angle $\theta$ at the maximum growth rate. The growth rate changes
the sign for $\omega\simeq \omega_{*e}$.
\subsection{Comparison with collision-less kinetic gradient-driven IA wave instability}
Keeping the same model of magnetized (un-magnetized) electrons (ions), within the kinetic theory the perturbed number
density for electrons can be written as \cite{w}
\be
\frac{n_{e1}}{n_0}= \frac{e \phi_1}{\kappa T_e} \left\{1+ i \left(\frac{\pi}{2}\right)^{1/2}
\frac{\omega-\omega_{*e}}{k_z v_{{\sss T}e}} \exp\left[-\omega^2/(2 k_z^2 v_{{\sss T}e}^2)\right]\right\}. \label{ew} \ee
In the derivation of Eq.~(\ref{ew}) the electron Larmor radius corrections are neglected in terms of the type $I_n(b)
\exp(-b)$, $b=k_\bot^2 \rho_e^2$, where $I_n$ denotes the modified Bessel function of the first kind, order $n$, and
only $n=0$ terms are kept for the present case of frequencies much below the gyro-frequency.
The ion number density can be calculated using the kinetic description for un-magnetized species, the derivation is
straight-forward and it yields \cite{v6}
\be
\frac{n_{i1}}{n_{i0}}= - \frac{e \phi_1}{m_iv_{{\scriptscriptstyle T} i}^2} \left[ 1- J_+ \left(\frac{\omega_i}{k v_{{\scriptscriptstyle
T}i}}\right)\right]. \label{e11} \ee
Here, $J(\eta)= [\eta/(2 \pi)^{1/2}] \int_c d \zeta\exp(- \zeta^2/2)/(\eta-\zeta)$ is the plasma dispersion function,
and $\zeta=v/v_{{\scriptscriptstyle T} i}$. In the case $|\eta|\gg 1$, and assuming $|Re(\eta)|\gg Im(\eta)$, an expansion is used
for $J(\eta)$. This together with the quasi-neutrality yields the kinetic dispersion equation for the IA-drift wave:
\[
\Delta(\omega, k) \equiv 1-\frac{k^2 c_s^2}{\omega^2} - \frac{3 k^4 v_{{\sss T}i}^2 c_s^2}{\omega^4}
\]
\be
+ i (\pi/2)^{1/2} \left\{\frac{\omega-\omega_{*e}}{k_z v_{{\sss T}e}} \exp\left[-\omega^2/(2 k_z^2 v_{{\sss T}e}^2)\right]
+\frac{T_e}{T_i} \frac{\omega}{k v_{{\sss T}i}} \exp\left[-\omega^2/(2 k^2 v_{{\sss T}i}^2)\right] \right\}. \label{ek} \ee
The real part of Eq.~(\ref{ek}) yields the spectrum
\be
\omega_k^2=\frac{k^2 c_s^2}{2} \left[1+ \left(1+ 12T_i/T_e\right)^{1/2}\right]. \label{s} \ee
The kinetic growth rate is given by
\[
\gamma_k \simeq -Im \Delta/(\partial Re \Delta/\partial \omega)=-\frac{(\pi/2)^{1/2} \omega_k^3}{ 2 k^2 c_s^2} \times
\]
\be
\times \left\{\frac{\omega_k-\omega_{*e}}{k_z v_{{\sss T}e}} \exp\left[-\omega_k^2/(2 k_z^2 v_{{\sss T}e}^2)\right] +\frac{T_e}{T_i}
\frac{\omega_k}{k v_{{\sss T}i}} \exp\left[-\omega_k^2/(2 k^2 v_{{\sss T}i}^2)\right] \right\}.\label{kg} \ee
Here, the index $k$ is used to denote kinetic expressions. The electron contribution in Eq.~(\ref{kg}) yields a kinetic
instability provided that $\omega_k< \omega_{*e}$.
Equation~(\ref{kg}) is solved numerically and compared with the growth rate obtained from the collisional IA-drift mode
(\ref{dn}). For a fixed $k=500\;$m$^{-1}$ as in Figs.~\ref{fig1a} and \ref{fig1b}, the normalized frequency
$\omega_k/\omega_{*e}=0.485$, and the result for the growth rate is presented by the line $\gamma_k$ in
Fig.~\ref{fig1a} for the parameters corresponding to the line II from the fluid analysis (i.e., for $n_{n0}= 10^{18}\;$m$^{-3}$).
The larger kinetic growth rate appears also to be angle dependent, yet with a much wider instability window
as compared to the collisional gradient driven instability obtained from the fluid theory.
\section{Summary}
The analysis of the ion acoustic wave presented here shows the importance of collisions in describing the wave
behavior. Without a proper analytical description, the identification of the mode in the laboratory and space
observations may be rather difficult because one might fruitlessly search for the wave in a very inappropriate domain,
as can be concluded from the graphs presented here, and in particular from Fig.~2. Not only the wave frequency may
become orders of magnitude below an expected ideal value, but also the mode may completely vanish. A similar analysis
of the effects of collisions may be performed for other plasma modes as well, like the Alfv\'{e}n wave etc, as
predicted long ago in classic Ref.~\cite{tan}. The impression is that these effects are frequently overlooked in the
literature, hence the necessity for the quantitative analysis given in the present work that can be used as a good
starting point for an eventual experimental check of the wave behavior in collisional plasmas. Particularly interesting
for experimental investigations may be the angle dependent mode behavior given in Sec.~3, where it is shown that the
strongly growing mode may be expected within a given narrow instability window in terms of the angle of propagation.
Comparison with the kinetic theory shows a less pronounced angle dependent peak, yet this kinetic effect can
effectively be smeared out in the presence of numerous collisions, that are known to reduce kinetic effects in any
case, and the sharp angle dependence that follow from pure fluid effects should become experimentally detectable.
\vspace{1cm}
\paragraph{Acknowledgements:}
The results presented here are obtained in the framework of the projects G.0304.07 (FWO-Vlaanderen), C~90347
(Prodex), GOA/2009-009 (K.U.Leuven). Financial support by the European Commission through the SOLAIRE Network
(MTRN-CT-2006-035484) is gratefully acknowledged.
\pagebreak
|
2,869,038,156,351 | arxiv | \section{Introduction}
\label{section1}
Timely information updates from wireless sensors to the destination are critical in real-time monitoring and control systems.
In order to describe the timeliness of information updates, the metric Age of Information(AoI) is proposed\cite{kaul2012real}. Different from general performance metrics such as delay and throughput, AoI refers to the time elapsed since the generation of the latest received information. A lower AoI usually reflects the more timely information received by the destination.
Therefore, the AoI-minimal status updating policies in sensor networks have been widely studied\cite{sun2017update,kadota2018scheduling,tang2020minimizing}.
In sensor-based information updating systems, energy is consumed in the process of sensing and transmitting updates. If the sensor's energy comes from the grid, it pays the electricity bill. If the sensor's energy comes from its own non-rechargeable battery, the price of sensing and transmitting updates is the cost of frequent battery replacement. We call these sources \textit{reliable energy} since they enable sensors reliable to operate until the power grid is cut off or sensors' batteries are exhausted. There is clearly a price to be paid for using reliable energy to update.
In order to reduce the reliable energy consumption, a reasonable idea is to introduce energy harvesting technology\cite{ma2019sensing}. Energy harvesting can continuously replenish energy for the sensor by extracting energy from solar power, ambient RF and thermal energy. The harvested energy is stored in the sensor’s rechargeable battery. Since the harvested energy is renewable, it can be used for free. Hence, reliable energy can serve as backup energy. The design of coexistence of reliable backup energy and harvested energy has been researched and promoted in academia and industry\cite{jackson2019capacity,instruments2019bq25505}. However, because the harvested energy arrives sporadically and irregularly, and the capacity of rechargeable batteries is limited, we still need to schedule the usage of energy properly to reduce the cost of using reliable backup energy while maintaining the timeliness of information updates(i.e. the average AoI).
Intuitively, the average AoI and the cost of using reliable energy cannot be minimized simultaneously. On the one hand, a lower average AoI means that the sensor senses and transmits updates more frequently, which will increase the consumption of reliable backup energy since the harvested energy is limited. On the other hand, to reduce the cost of reliable backup energy, the sensor will only exploit the harvested energy. Due to the uncertainty of the energy harvesting behavior, the average AoI of the system will inevitably increase.
Therefore, in this paper, we focus on achieving the best trade-off between the average AoI and the cost of reliable backup energy in a sensor-based information update system where an energy harvesting sensor with reliable backup energy sends timely updates to the destination through an erasure channel. Related work includes\cite{wu2017optimal,bacinoglu2018achieving,arafa2019age,wu2020optimal,wu2020delay,draskovic2021optimal}. \cite{wu2017optimal,bacinoglu2018achieving,arafa2019age} investigate AoI-minimal status updating policies for sensor networks that rely solely on harvested energy. In \cite{wu2020optimal,wu2020delay,draskovic2021optimal}, although the sensors can use both harvested energy and reliable energy, the authors only optimize for delay or throughput and ignore the timeliness of the system. Based on our settings, we will minimize the long-term average weighted sum of the AoI and the paid reliable energy cost to find the optimal information updating policy. The structure of the optimal policy will be analyzed theoretically, and its performance will be demonstrated through simulation.
\section{SYSTEM MODEL and Problem Formulation}
\label{section2}
\subsection{System Model Overview}
In this paper, we consider a point-to-point information update system where a wireless sensor and a destination are connected by an erasure channel, as shown in Fig.~\ref{system}. Wireless sensors can use the free harvest energy stored in the rechargeable battery and the reliable backup energy that needs to be paid to generate and send real-time environmental status information. The destination keeps track of the environment status through the received updates. We apply the metric Age of Information to measure the freshness of the status information available at the destination.
Without loss of generality, time is slotted with equal length and indexed by $t\in\mathbb N$.
At the beginning of each time slot, the sensor decides whether to generate and transmit an update to the destination or stay idle.
The decision action at slot $t$, denoted by $a[t]$, takes value from action set $\mathcal{A}=\left\{0,1\right\}$, where $a[t] =1$ means that the sensor decides to generate and transmit an update to the destination while $a[t]=0$ means the sensor is idle. The channel between the sensor and the destination is assumed to be noisy and time-invariant, and each update will be corrupted with probability $p$ during transmission (Note $p \in (0,1)$). The destination will feed back an instantaneous ACK to the sensor through an error-free channel when it has successfully received an update and a NACK otherwise. We assume the above processes can be completed in one time slot.
\begin{figure}[tbp]
\centerline{\includegraphics[width=0.5\textwidth]{fig/system_model_version5_v2.eps}}
\caption{System model.}
\label{system}
\end{figure}
\subsection{Age of Information}
Age of Information (AoI) is defined as the elapsed time since the generation of the latest successfully received update in this paper.
Let $U[t]$ be the time slot when the most recently received update is generated before time slot $t$, and $\Delta[t]$ denote the AoI of destination in time slot $t$. Then, the AoI is given by
\begin{equation}
{\Delta[t]} = t - U[t].
\label{AoI}
\end{equation}
In particular, the AoI will decrease to one if a new update is successfully received. Otherwise it will increase by one. To summarize, the evolution of AoI can be expressed as follows:
\begin{equation}
\label{equ_evolving_AoI}
\Delta[t+1]=
\begin{cases}
1, &\text{ successful transmission},\\
\Delta[t]+1, &\text{ otherwise}.
\end{cases}
\end{equation}
A sample path of AoI is depicted in Fig.~\ref{AoI_evolution}.
\begin{figure}[tbp]
\centerline{\includegraphics[width=0.5\textwidth]{fig/AoI_evolution_update.eps}}
\caption{A sample path of AoI with initial age 1.}
\label{AoI_evolution}
\end{figure}
\subsection{Description of Energy Supply}
We assume that only the sensor's measurement and transmission process will consume energy, and other energy consumption is ignored.
The energy unit is normalized, so the generation and transmission for each update will consume one energy unit. As previously described, the energy sources of the sensor include energy harvested from nature and reliable backup energy. The sensor can store the harvested energy in a rechargeable battery for later use. The maximum capacity of the rechargeable battery is $B$ units ($B > 1$). Let $b(t)$ be the accumulated harvested energy in time slot $t$. Since the energy to be harvested is relatively limited, sometimes $b(t)$ does not reach an energy unit. So we consider using the Bernoulli process with the parameter $\lambda$ to approximately capture the arrival process of harvested energy, which is also adopted in \cite{valentini2016optimal,dong2020energy,gindullina2021age}. That is, we have $\Pr\left\{b(t)=1\right\}= \lambda$ and $\Pr\left\{b(t)=0\right\}= 1-\lambda$ in each time slot $t$.
For reliable backup energy, we assume that it contains much more energy units than the rechargeable battery can store, so the energy it contains is infinite. However, it needs to be used for a fee. Therefore, when the power of the rechargeable battery is not 0, the sensor will prioritize using the energy in the rechargeable battery for status update, otherwise, it will automatically switch to the reliable backup energy until the sensor has harvested energy. Defining the power of the rechargeable battery at the beginning of time slot $t$ as the battery state $q[t]$, then the evolution of battery state between time slot $t$ and $t+1$ can be summarized as follows:
\begin{equation}
\label{q_evolution}
q[t+1] = \min \{ q[t] + b[t] - a[t]u(q[t]),B\} ,
\end{equation}
where $u(\cdot)$ is unit step function, which is defined as
\begin{equation}
\label{indicator}
u(x)=
\begin{cases}
1,&\text{if $x>0$},\\
0,&\text{otherwise}.
\end{cases}
\end{equation}
Suppose that under paid reliable energy supply, the cost of generating and transmitting an update is a non-negative value $C_r$. Define $E[t]$ as the paid reliable energy costs at the time slot $t$, then we have
\begin{equation}
\label{E_cost}
E[t] = {C_r}a[t](1-u(q[t])).
\end{equation}
\subsection{Problem Formulation}
Let $\Pi$ denote the set of non-anticipated policies in which scheduling decision $a[t]$ are made based on the action history $\left\{a[k]\right\}_{k=0}^{t-1}$, the AoI evolution $\left\{\Delta[k]\right\}_{k=0}^{t-1}$, the evolution of battery state $\left\{q[k]\right\}_{k=0}^{t-1}$ as well as the system parameters(i.e. $p$, $\lambda$, etc.).
In order to keep the information freshness at the destination, the sensor needs to send updates.
However, due to the randomness of harvested energy arrivals, the battery energy may sometimes be insufficient to support updates, and the sensor has to take energy from reliable backup energy. To balance the information freshness and the paid reliable backup energy costs, we aim to find the optimal information updating policy $\pi \in \Pi$ that achieves the minimum of the time-average weighted sum of the AoI and the paid reliable backup energy costs.
The problem is formulated as follows:
\begin{equation}
\begin{aligned}
\label{problem}
&\mathop {\min }\limits_{\pi \in \Pi } \mathop {\lim\sup }\limits_{T \to \infty } \frac{1}{T}{\mathbb E}\left\{\sum\limits_{t = 0}^{T-1} [{\Delta [t]} + \omega E[t]]\right\}, \\
&\text{s}.\text{t}.\qquad (2),(3),(5), \\
\end{aligned}
\end{equation}
where $\omega$ is the positive weighting factor.
\section{Optimal policy analysis}
\label{section3}
In this section, we aim to solve the problem (\ref{problem}) and obtain the optimal policy. It is difficult to solve the original problem directly due to the random erasures and the temporal dependency in both AoI evolution and battery state evolution. So we reformulate the original problem as a time-average cost MDP with infinite state space and analyze the structure of the optimal policy.
\subsection{Markov Decision Process Formulation}
According to the system description mentioned in the previous section, the MDP is formulated as follows:
\begin{itemize}
\item \textbf{State Space}. The state of a sensor $\textbf{x}[t]$ in slot $t$ is a couple of the current destination-AoI and the battery state, i.e., $(\Delta[t], q[t])$. Define $\mathcal{B}=\left\{0,1,...,B \right\}$. The state space $\mathcal{S}= \mathbb Z^+ \times \mathcal{B}$ is thus infinite countable.
\item \textbf{Action Space}. The sensor's action $a[t]$ in time slot $t$ only takes value from the action set $\mathcal{A}=\left\{0,1\right\}$.
\item \textbf{Transition Probabilities}. Denote $\Pr ({\textbf{x}}[t + 1]|{\textbf{x}}[t],a[t])$ as the transition probability that current state $\textbf{x}[t]$ transits to next state $\textbf{x}[t+1]$ after taking action $a[t]$. Suppose the current state ${\textbf{x}}[t] = (\Delta, q)$ and action $a[t] = a$, then the transition probability is divided into two following cases conditioned on different values of action.
\textbf{\emph{Case 1}}. $a=0$,
\begin{equation}
\label{transition_case1_v2}
\begin{cases}
\Pr \{ (\Delta+1, q+1)|(\Delta, q),0\}=\lambda, &\text{ if }q < B, \\
\Pr \{(\Delta+1, B)|(\Delta, B),0\}= 1, &\text{ if } q = B, \\
\Pr \{(\Delta+1, q)|(\Delta, q),0\}= 1-\lambda, &\text{ if }q < B. \\
\end{cases}
\end{equation}
\textbf{\emph{Case 2}}. $a=1$,
\begin{equation}
\label{transition_case2_v2}
\begin{cases}
\Pr \{ (\Delta+1, q)|(\Delta, q),1\}=p\lambda, &\text{ if }q>0, \\
\Pr \{ (1, q)|(\Delta, q),1\}=(1-p)\lambda, &\text{ if }q>0, \\
\Pr \{\Delta+1, q-1)|(\Delta, q),1\}=p(1-\lambda), &\text{ if }q>0, \\
\Pr \{ (1, q-1)|(\Delta, q),1\}=(1-p)(1-\lambda), &\text{ if }q >0, \\
\Pr \{ (\Delta+1, 1)|(\Delta, 0),1\}=p\lambda, &\text{ if }q=0, \\
\Pr \{ (1, 1)|(\Delta, 0),1\}=(1-p)\lambda, &\text{ if }q=0, \\
\Pr \{(\Delta+1, 0)|(\Delta, 0),0\}= p(1-\lambda), &\text{ if } q = 0, \\
\Pr \{(1, 0)|(\Delta, 0),0\}= (1-p)(1-\lambda), &\text{ if } q = 0. \\
\end{cases}
\end{equation}
In both cases, the evolution of AoI still follows equation \eqref{equ_evolving_AoI} and the evolution of battery state follows \eqref{q_evolution}.
\item \textbf{One-step Cost}. For the current state $\textbf{x}=(\Delta, q)$, the one-step cost $C(\textbf{x},a)$ of taking action $a$ is expressed by
\begin{equation}
\label{onestepcost}
C(\textbf{x},a) = \Delta + \omega {C_r}a(1-u(q)).
\end{equation}
\end{itemize}
After the above modeling, the original problem \eqref{problem} is transformed into obtaining the optimal policy for the MDP to minimize the average cost in an infinite horizon:
\begin{equation}
\label{trans_problem}
\mathop {\lim\sup }\limits_{T \to \infty } \frac{1}{T}{\mathbb E_\pi}\left\{ \sum\limits_{t = 0}^{T-1} C(\textbf{x}[t],a[t])\right\}.
\end{equation}
Denote $\Pi_{SD}$ as the set of stationary deterministic policies. Given observation$(\Delta[t],q[t])=(\Delta,q)$, the policy $\pi \in \Pi_{SD}$ selects action $a[t]=\pi(\Delta,q)$, where $\pi(\cdot):(\Delta,q)\to\left\{0,1\right\}$ is a deterministic function from state space $\mathcal{S}$ to action space $\mathcal{A}$.
According to \cite{altman1999constrained} , there exists a stationary deterministic policy to minimize the above unconstrained MDP with infinite countable state and action space under certain verifiable conditions. In the next section, the structural properties of the optimal policy are investigated.
\subsection{Structure Analysis of Optimal Policy}
According to \cite{sennott1989average}, there exits a value function $V(\textbf{x})$ which satisfies the following Bellman equation for the infinite horizon average cost MDP:
\begin{equation}
\lambda + V(\textbf{x}) = \min_{a \in \mathcal{A}} \left\{ C(\textbf{x},a) + \sum_{\textbf{x}^\prime \in \mathcal{S}} \Pr (\textbf{x}^\prime|\textbf{x},a)V(\textbf{x}^\prime) \right\},
\label{bel_equation}
\end{equation}
where $\lambda$ is the average cost by following the optimal policy. Denote $Q(\textbf{x},a)$ as the state-action value function which means the value of taking action $a$ in state $\textbf{x}$. We have:
\begin{equation}
Q(\textbf{x},a)=C(\textbf{x},a) + \sum_{\textbf{x}^\prime \in \mathcal{S}} \Pr (\textbf{x}^\prime|\textbf{x},a)V(\textbf{x}^\prime).
\label{Qfunction}
\end{equation}
So the optimal policy $\pi ^\star \in \Pi_{SD}$ in state $\textbf{x}$ can be expressed as follows:
\begin{equation}
{\pi ^\star}(\textbf{x}) = \arg \mathop {\min }\limits_{a \in \mathcal{A}} Q(\textbf{x},a).
\label{piandQequation}
\end{equation}
Next, we first prove the monotonicity of the value function on different dimensions, which is summarized in the following lemma.
\begin{lemma}
For a fixed channel erasure probability $p$, given the battery state $q$ and for any $1\leq \Delta_1\leq\Delta_2$, we have
\begin{equation}
\label{lemma1_part1}
V(\Delta_1,q)\le V(\Delta_2,q),
\end{equation}
and, given AoI $\Delta\geq 1$,
\begin{equation}
\label{lemma1_part2}
V(\Delta,q)\ge V(\Delta,q+1)
\end{equation}
holds for any $q\in \left\{0,1,...,B-1\right\}$.
\label{lemma1}
\end{lemma}
\begin{IEEEproof}
See Appendix \ref{app_proof_lemma_monitonic} in Supplementary Material \cite{tsinghua.edu}.
\end{IEEEproof}
Based on Lemma \ref{lemma1}, we then establish the incremental property of the value function, which is shown in the following lemma.
\begin{lemma}
For a fixed channel erasure probability $p$, for any $\Delta_1 \le \Delta_2$ and given $q \in \mathcal{B}$, we have:
\begin{equation}
\label{lemm2_formula1}
V(\Delta_2,q)-V(\Delta_1,q)\ge \Delta_2-\Delta_1.
\end{equation}
And, for any $q \in \left\{0,1,...,B-1\right\}$ and $\Delta \in \mathbb Z^+ $, we have:
\begin{equation}
\label{lemm2_formula2}
V(\Delta+1,q+1)-V(\Delta,q+1)\ge p[V(\Delta+1,q)-V(\Delta,q)].
\end{equation}
\label{lemma2}
\end{lemma}
\begin{IEEEproof}
See Appendix \ref{app_proof_lemma_creasement} in Supplementary Material \cite{tsinghua.edu}.
\end{IEEEproof}
With Lemma \ref{lemma1} and Lemma \ref{lemma2}, we directly provide our main result in the following Theorem.
\begin{theorem}
Assuming that the channel erasure probability $p$ is fixed. For given battery state $q$, there exists a threshold $\Delta_q$ , such that when $\Delta\ < \Delta_q$, the optimal action $\pi^\star (\Delta,q)=0$, i.e., the sensor keeps idle; when $\Delta \ge \Delta_q$, the optimal action $\pi^\star(\Delta,q)=1$, i.e., the sensor chooses to generate and transmit a new update.
\label{theorem1}
\end{theorem}
\begin{IEEEproof}
The optimal policy is of a threshold structure if $Q(\textbf{x},a)$ has a sub-modular structure, that is,
\begin{equation}
Q(\Delta,q,0)- Q(\Delta,q,1) \leq Q(\Delta+1,q,0)- Q(\Delta+1,q,1).
\end{equation}
We will divide the whole proof into the following three cases:
\textbf{Case 1}. When $q=0$, for any $\Delta \in \mathbb Z^+$ we have:
\begin{align}
&Q(\Delta,q,0)-Q(\Delta,q,1)\nonumber\\
=&\Delta+ \lambda V(\Delta+1,q+1)+(1-\lambda)V(\Delta+1,q)\nonumber\\
&-\Delta-\omega{C_r}-p\lambda V(\Delta+1,q+1)+p(1- \lambda)V(\Delta+1,q)\nonumber\\
&-(1-p)\lambda V(1,q+1)-(1-p)(1-\lambda) V(1,q)\nonumber\\
=&(1-p)\lambda(V(\Delta+1,q+1)-V(1,q+1))\nonumber\\
&+(1-p)(1-\lambda)(V(\Delta+1,q)-V(1,q))-\omega{C_r}.
\end{align}
Therefore, we have
\begin{align}
&Q(\Delta+1,q,0)-Q(\Delta+1,q,1) - [Q(\Delta,q,0)-Q(\Delta,q,1)]\nonumber\\
=&(1-p)\lambda(V(\Delta+2,q+1)-V(\Delta+1,q+1)) \nonumber\\
&+(1-p)(1-\lambda)(V(\Delta+2,q)-V(\Delta,q))\nonumber\\
\overset{(a)}{\geq}& 0,
\end{align}
where the last inequality $(a)$ is due to the monotonicity property revealed by \eqref{lemma1_part1} in Lemma \ref{lemma1}.
\textbf{Case 2}. When $q \in \left\{1,...,B-1\right\}$,for any $\Delta \in \mathbb Z^+$ we have:
\begin{align}
&Q(\Delta+1,q,0)-Q(\Delta+1,q,1)-[Q(\Delta,q,0)-Q(\Delta,q,1)]\nonumber\\
=&Q(\Delta+1,q,0)-Q(\Delta,q,0)-[Q(\Delta+1,q,1)-Q(\Delta,q,1)]\nonumber\\
=&\lambda[V(\Delta+2,q+1)-V(\Delta+1,q+1)]\nonumber\\
&-p\lambda[V(\Delta+2,q)-V(\Delta+1,q)]\nonumber\\
&+(1-\lambda)[V(\Delta+2,q)-V(\Delta+1,q)]\nonumber\\
&-p(1-\lambda)[V(\Delta+2,q-1)-V(\Delta+1,q-1)]\nonumber\\
\overset{(a)}{\geq}& 0,
\label{submodular}
\end{align}
where the last inequality $(a)$ is due to the incremental property revealed by \eqref{lemm2_formula2} in Lemma \ref{lemma2}.
\textbf{Case 3}. When $q=B$,for any $\Delta \in \mathbb Z^+$ we have:
\begin{align}
&Q(\Delta+1,q,0)-Q(\Delta+1,q,1)-[Q(\Delta,q,0)-Q(\Delta,q,1)]\nonumber\\
=&Q(\Delta+1,q,0)-Q(\Delta,q,0)-[Q(\Delta+1,q,1)-Q(\Delta,q,1)]\nonumber\\
=&(1-\lambda)[V(\Delta+2,q)-V(\Delta+1,q)]\nonumber\\
&-p(1-\lambda)[V(\Delta+2,q-1)-V(\Delta+1,q-1)]\nonumber\\
\overset{(a)}{\geq}& 0,
\label{B_submodular}
\end{align}
where the last inequality (a) is also due to the incremental property revealed by \eqref{lemm2_formula2} in Lemma \ref{lemma2}.
Therefore, we have completed the whole proof.
\end{IEEEproof}
Theorem \ref{theorem1} reveals the threshold structure of the optimal policy: if the optimal action in a certain state is to generate and transmit an update, then in the state with the same battery state and larger AoI, the optimal action must be the same.
Based on this unique threshold structure, we propose a modified value iteration algorithm to solve the optimal policy, as shown in Algorithm \ref{mvia}. Specifically, We first iterate the Bellman equation \eqref{bel_equation} to obtain the value function. Then based on the threshold structure, \added{the optimal policy can be obtained without calculating the equation \eqref{piandQequation} in each state}\deleted{we can find the optimal policy by equation \eqref{piandQequation} without traversing all the states}, which reduces the computational complexity.
\begin{algorithm}[tb]
\caption{Modified Value Iteration Algorithm}
\label{mvia}
\begin{algorithmic}[1]
\REQUIRE ~~\\
Iteration number $K$ and iteration threshold $\epsilon$.
\ENSURE ~~\\
Optimal policy $\pi^\star(\textbf{x})$ for all state $\textbf{x}$.
\STATE \textbf{Initialization: }$V_0(\textbf{x})= 0.$
\FOR{episodes $k = 0,1,2,...,K$}
\FOR{state $\textbf{x}\in \mathcal{S}$}
\FOR{action $a\in \mathcal{A}$}
\STATE $Q_k(\textbf{x},a)\leftarrow C(\textbf{x},a) +\underset{\textbf{x}^\prime \in \mathcal{S}}{\sum} \Pr (\textbf{x}^\prime|\textbf{x},a)V_k(\textbf{x}^\prime)$
\ENDFOR
\STATE ${V_{k + 1}}(\textbf{x}) \leftarrow \mathop {\min }\limits_{a \in \mathcal{A}} Q_k(\textbf{x},a)$
\ENDFOR
\IF{$\|V_{k+1}(\textbf{x})-V_{k}(\textbf{x})\|\leq \epsilon$}
\FOR{$\textbf{x}=(\Delta,q) \in \mathcal{S}$}
\IF{$\pi^\star(\Delta-1,q)=1$}
\STATE $\pi^\star(\textbf{x})\leftarrow 1$,
\ELSE
\STATE ${\pi^\star}(\textbf{x}) \leftarrow \arg \mathop {\min }\limits_{a \in \mathcal{A}} Q_k(\textbf{x},a)$
\ENDIF
\ENDFOR
\ENDIF
\ENDFOR
\end{algorithmic}
\end{algorithm}
\section{Numerical Result}
\label{section4}
In this section, we first show the threshold structure of optimal policy by the simulation results. Then we compare the performance of the optimal policy with the zero-wait policy, the periodic policy, the randomized policy, the energy first policy under different system parameters such as weighting factor $\omega$, energy harvesting probability $\lambda$ and erasure probability $p$. Note that the zero-wait policy means the sensor generates and transmits an update in every time slot\cite{sun2017update}, while the periodic policy means the sensor periodically generates and sends updates to the destination. The randomized policy refers to that the sensor chooses to send an update or stay idle in each time slot with the same probability. The energy first policy means that the sensor only uses the harvested energy, that is, as long as the battery state is not 0, it will choose to sense and send updates, otherwise it will remain idle. Obviously, the energy first policy will not incur the cost of reliable energy. In our simulation, we assume that the cost of reliable energy $C_r$ for one update equals to $2$ and the maximum battery capacity $B$ equals to $20$.
Fig.~\ref{fig:thresold_blue} shows the optimal policy under different \replaced{system parameters}{channel erasure probability and energy harvesting probability}. All the subfigures in Fig.~\ref{fig:thresold_blue} exhibits the threshold structure described in Theorem \ref{theorem1}. Note that the weighting factor $\omega$ is set to be $10$ , which is neither too small nor too large. Intuitively, when $\omega$ is too small, the optimal action for every state should be 0, and when $\omega$ is too large, the optimal action for every state should be 1.
Fig.~\ref{fig:thresold_blue} shows that when the AoI is small, even if the battery state is not 0, the optimal action in the corresponding state is to keep idle. When the AoI is large or the battery state is large, the optimal action is to measure and send updates.
\deleted{Then, we show the average cost performance of optimal policy in Fig.~\ref{costwithb_big} under different weighting factor $\omega$.} \deleted{Fig.~\ref{costwithb_big} shows the system performance of the optimal policy under different setting. }
\added{Fig.~\ref{costwithb_big} shows the time average cost under different policies, i.e., the zero-wait policy, the periodic policy, the randomized policy, the energy first policy and the proposed optimal policy.}
\deleted{The optimal policy is compared with the zero-wait policy and the periodic policy \replaced{conditioned on the }{under the same} channel erasure probability $p=0.2$ and the energy harvesting probability $\lambda=0.5$}\deleted{ in this simulation}
Here we set the period \added{of the periodic policy }to 5 and 10 for comparison \added{without loss of generality}. It can be found that under different weighting factor $\omega$, the optimal policy proposed in this paper can obtain the minimum long-term average cost compared with the other policies, which indicates the best trade-off between the average AoI and the cost of reliable energy. When $\omega$ tends to $0$, the zero-wait policy tends to be optimal. \replaced{Since}{When} there is no need to consider the update cost brought by paid reliable backup energy, the optimal policy \replaced{should maximize the utilization of the updating opportunities.}{is to update information in every time slot.}
\deleted{In }Fig.~\ref{costwithlambda_big}\replaced{ reveals}{, we present} the impact of \deleted{different }energy harvesting probabilities $\lambda$\deleted{ on different policies}. \deleted{In this simulation, we also set }The channel erasure probability $p$ is set to be $0.2$ and weighting factor $\omega$ is $10$\deleted{in this simulation}. It \added{also} can be found \deleted{from Fig.~\ref{costwithlambda_big} }that the proposed optimal update policy outperforms all other policies under different energy harvesting probabilities\deleted{the zero-wait policy and the periodic policy (period = 5)}. The interesting point is that when the probability of energy harvesting tends to 1, \replaced{i.e.}{that is}, energy arrives in each time slot, the performance of the zero-wait policy and the energy first policy is \replaced{equal}{close} to the optimal policy, while there is still a performance gap between the optimal policy and the other two polices. This is intuitive because when the free harvested energy is sufficient, the optimal policy must be to generate and transmit updates in every time slot. However, the periodic policy and the randomized policy still keep idle in many time slots\deleted{send updates in many time slots}, which will \added{lead to a higher average AoI and thus increase the average cost}\deleted{increase the average cost of AoI term and finally increases the whole system cost}\deleted{leads to an increase in the average cost of the system}.
In Fig.~\ref{costwithp_big}, we compare the above five policies under different erasure probabilities $p$. \replaced{The simulation settings are }{In this simulation, we set }the energy harvesting probability $\lambda=0.5$ and weighting factor $\omega=10$. It can be found \deleted{from Fig.~\ref{costwithp_big} }that when erasure probability increases from 0 to 0.9, the proposed optimal update policy always performs better than the other baseline policies. Note when $p = 1$, all the updates are erased by the noisy channel. So it is meaningless to discuss this case.
\begin{figure}[tbp]
\centerline{\includegraphics[width=0.5\textwidth]{fig/threshold_bigfonts_blue_version3.eps}}
\caption{Optimal policy conditioned on different parameters.}
\label{fig:thresold_blue}
\end{figure}
\begin{figure}[tbp]
\centerline{\includegraphics[width=0.4\textwidth]{fig/cost_with_omega_version1_six_alg.eps}}
\caption{Performance comparison of the zero-wait policy, periodic policy (period = 5), periodic policy (period = 10), randomized policy, energy first policy and proposed policy versus the weighting factor $\omega$ with simulation conditions $p=0.2$, $\lambda=0.5$ and $B=20$.}
\label{costwithb_big}
\end{figure}
\begin{figure}[tbp]
\centerline{\includegraphics[width=0.4\textwidth]{fig/cost_with_lambda_bigfonts_version1_six_alg.eps}}
\caption{Comparison of the zero-wait policy, periodic policy (period = 5), periodic policy (period = 10), randomized policy, energy first policy and proposed policy versus the energy harvesting probability with simulation conditions $p=0.2$, $\omega=10$ and $B=20$.}
\label{costwithlambda_big}
\end{figure}
\begin{figure}[tbp]
\centerline{\includegraphics[width=0.4\textwidth]{fig/cost_with_p_bigfonts_version2_six_alg.eps}}
\caption{Comparison of the zero-wait policy, periodic policy (period = 5), periodic policy (period = 10), randomized policy, energy first policy and proposed policy versus the erasure probability with simulation conditions $\lambda=0.5$, $\omega=10$ and $B=20$.}
\label{costwithp_big}
\end{figure}
\section{Conclusion}
\label{section5}
In this paper, we have studied the optimal updating policy for an information update system where a wireless sensor sends updates over an erasure channel using both harvested energy and reliable backup energy. Theoretical analysis indicates the threshold structure of the optimal policy and simulation results verify its performance.
\bibliographystyle{./IEEEtran}
|
2,869,038,156,352 | arxiv | \section{Introduction
\label{sec:intro}}
Since the discovery of starlight polarization over 70 years ago
\citep{Hiltner_1949a,Hall_1949},
polarization has become a valuable tool for study of both the physical
properties of interstellar dust and the structure of the interstellar
magnetic field. Starlight polarization arises because
initially unpolarized starlight becomes linearly polarized as a result
of linear dichroism produced by aligned dust grains
in the interstellar medium (ISM).
While the physics of dust grain alignment is not yet fully understood,
early investigations \citep{Davis+Greenstein_1951} showed how
spinning dust grains
could become aligned with their shortest axis
parallel to the magnetic field direction. Subsequent studies
have identified a number of important physical processes that were
initially overlooked
\citep[see the review by][]{Andersson+Lazarian+Vaillancourt_2015},
but it remains clear that in the
diffuse ISM the magnetic field establishes the direction
of grain aligment, with the dust grains tending to align
with their short axes parallel to the local magnetic field.
\citet{van_de_Hulst_1957} noted
that if the magnetic field direction was not uniform,
starlight propagating through the dusty
ISM would become circularly
polarized.
This was further discussed by \citet{Serkowski_1962} and
\citet{Martin_1972c}.
The birefringence of the dusty ISM is responsible for
converting linear polarization to circular polarization
\citep{Serkowski_1962,Martin_1972c}.
The strength of the resulting circular polarization depends on the
changes in the magnetic field direction and also on the optical properties
of the dust.
Circular polarization of optical light from the Crab Nebula
was observed by \citet{Martin+Illing+Angel_1972}.
Circular polarization
of starlight was subsequently observed by
\citet{Kemp_1972} and \citet{Kemp+Wolstencroft_1972};
the observed degree of circular polarization, $|V|/I \ltsim 0.04\%$,
was small but measurable.
As had been predicted, the circular polarization $V$ changed sign
as the wavelength varied from blue to red,
passing through zero near the wavelength $\sim$$0.55\micron$
where the linear polarization peaked \citep{Martin+Angel_1976}.
Because the circular polarization depends on the change in magnetic field \added{direction}
along the line of sight, it can in principle
be used to study the structure of the Galactic magnetic field.
Data for 36 stars near the Galactic
Plane suggested a systematic bending of the field for Galactic longitudes
$80^\circ \ltsim \ell < 100^\circ$
\citep{Martin+Campbell_1976}.
However,
these studies do not appear to have been pursued,
presumably because
sufficiently bright and reddened stars are sparse.
In the infrared, circular polarization has been measured for bright
sources in molecular clouds
\citep{Serkowski+Rieke_1973,
Lonsdale+Dyck+Capps+Wolstencroft_1980,Dyck+Lonsdale_1981}.
Measurements of linear and
circular polarization were used
to constrain the magnetic field structure in the
Orion molecular cloud OMC-1
\citep{Lee+Draine_1985,Aitken+Hough+Chrysostomou_2006}.
Circular polarization has also been observed in the infrared
(K$_{\rm s}$ band) in
reflection nebulae
\citep{Kwon+Tamura+Hough+etal_2014,
Kwon+Tamura+Hough+etal_2016,
Kwon+Nakagawa+Tamura+etal_2018},
but in this case
scattering is \deleted{thought to be} important
\citep{Fukushima+Yajima+Umemura_2020}.
\added{Scattering can convert linear to circular polarization},
making interpretation dependent
on the uncertain scattering geometry.
It was long understood that the nonspherical and aligned grains responsible
for starlight polarization must emit far-infrared radiation which would
be linearly polarized.
Observations of this polarized emission now allow the
magnetic field direction projected on the sky to be mapped in the general ISM
\citep[see, e.g.,][]{Planck_int_results_xix_2015,Planck_int_results_xxi_2015,
Fissel+Ade+Angile+etal_2016}.
Ground-based observations have provided polarization
maps for high surface-brightness regions at submm frequencies
\citep[e.g.,][]{Dotson+Vaillancourt+Kirby+etal_2010},
and the Statospheric Observatory for Infrared Astronomy (SOFIA) is
providing polarization maps of bright regions in the far-infrared
\citep[e.g., OMC-1:][]{Chuss+Andersson+Bally+etal_2019}.
ALMA observations of mm and submm emission from protoplanetary disks
find that the radiation is often linearly polarized.
Scattering may contribute to the polarization
\citep{Kataoka+Muto+Momose+etal_2015},
but the observed polarization directions and wavelength dependence
appear to
indicate that a substantial fraction of
the polarized radiation arises from thermal
emission from aligned dust grains
\citep{Lee+Li+Yang+etal_2021}.
Previous theoretical discussions of circular polarization
were mainly concerned with
infrared and optical wavelengths where
initially unpolarized starlight
becomes polarized as a result of linear dichroism.
In a medium with changing polarization direction, the resulting circular
polarization is small because the linear polarization itself is
typically
only a few \%, and the optical ``phase shift''
\added{(between the two linear polarization modes)}
produced by the aligned medium
is likewise small. At far-infrared wavelengths,
however, the radiation is already
substantially polarized when it is emitted, with linear polarizations
of 20\% or more under favorable conditions
\citep{Planck_2018_XII}.
While absorption optical depths tend to be small at long wavelengths,
the optical properties of the dust are
such that phase shift cross sections at submillimeter wavelengths
can be much larger than
absorption cross sections, raising the possibility that a medium with changing
alignment direction might exhibit measurable levels of circular polarization
at far-infrared or submm wavelengths.
The present paper discusses polarized radiative transfer in a medium with
partially aligned nonspherical grains, including both absorption and
thermal emission.
We estimate the expected degree of circular polarization
for emission from molecular clouds and protoplanetary disks.
For nearby molecular clouds, the far-infrared circular polarization is
very small, and probably unobservable.
The circular
polarization is predicted to be larger for so-called infrared dark
clouds (IRDCs), although it is still small.
For protoplanetary disks the circular polarization may be measurable,
but will depend on how the
direction of grain alignment changes in the disk.
The paper is organized as follows. The equations describing propagation of
partially-polarized radiation are presented in Section
\ref{sec:radiative transfer}, and
the optics of partially-aligned dust mixtures
are summarized in Section \ref{sec:dust}.
Section \ref{sec:clouds} estimates the circularly polarized emission from
molecular clouds, including IRDCs.
Section \ref{sec:disks} discusses the alignment of solid particles in
stratified protoplanetary disks resembling HL Tau.
If the grain alignment is due to dust-gas streaming, the emission may
be circularly-polarized.
The results are discussed in Section \ref{sec:discussion},
and summarized in Section \ref{sec:summary}.
\section{\label{sec:radiative transfer}
Polarized Radiative Transfer}
\subsection{Refractive Index of a Dusty Medium}
Aligned dust grains result in linear dichroism -- the attenuation
coefficient depends on the linear polarization of the radiation.
Linear dichroism is
responsible for the polarization of starlight -- initially unpolarized
light from a star becomes linearly polarized as the result of
polarization-dependent attenuation by aligned dust grains.
We adopt the convention that the electric field
$E \propto {\rm Re}[e^{imkz-i\omega t}]$ for a wave
propagating in the $+\bzhat$ direction, where
$k\equiv\omega/c = 2\pi/\lambda$ is the wave vector
{\it in vacuo}, and $m(\omega)$
is the complex refractive index of the dusty medium.
For radiation polarized with $\bE\parallel\behat_j$,
the complex refractive index is
\beq
m_j \equiv 1 + m_j^\prime + i m_j^{\prime\prime}
~~~.
\eeq
The real part $m_j^\prime$ describes retardation of the wave,
relative to propagation {\it in vacuo}. The phase delay $\phi$ varies as
\beq
\frac{d\phi_j}{dz} =
\frac{2\pi}{\lambda} m_j^\prime = n_d C_{{\rm pha},j}
~~~,
\eeq
where $n_d$ is the number density of dust grains, and
$C_{{\rm pha},j}$ is the ``phase shift'' cross section of a grain.
The imaginary part $m_j^{\prime\prime}$ describes attenuation of the energy
flux $F$:
\beq
\frac{d\ln F}{dz} =
- \frac{4\pi}{\lambda} m_j^{\prime\prime} = -n_d C_{{\rm ext},j}
~~~,
\eeq
where $C_{{\rm ext},j}$ is the extinction cross section.
\subsection{Transfer Equations for the Stokes Parameters}
Consider a beam of radiation characterized by the usual
Stokes vector ${\bf S}\equiv(I,Q,U,V)$.
The equations describing transfer of radiation through a dichroic
and birefringent medium with
changing magnetic field direction have
been discussed by \citet{Serkowski_1962} and
\citet{Martin_1974}.\footnote{
Our axes $\bxhat$ and $\byhat$ correspond, respectively,
to axes 2 and 1 in \citet{Martin_1974}.}
The discussions have asssumed that the aligned grains
polarize the light by preferential attenuation of one of the polarization
modes, with circular polarization then arising from differences in propagation
speed of the linearly polarized modes.
For submicron particles, scattering is negligible
at far-infrared wavelengths, because the grain is small compared to the
wavelength.
However, the grains are themselves able to radiate,
and aligned grains will emit polarized radiation.
Let the direction of the static magnetic field $\bB_0$ be
\beq
\bbhat \equiv \frac{\bB_0}{|\bB_0|} =
(\hat{\bf n}\cos\Psi + \hat{\bf e}\sin\Psi )\sin\gamma
+ \bzhat \cos\gamma
\eeq
where $\hat{\bf n}$ and $\hat{\bf e}$ are unit vectors in the North and East
directions, $\bzhat = \hat{\bf n}\times\hat{\bf e}$
is the direction of propagation, and $\sin\gamma=1$ if $\bbhat$ is in the
plane of the sky.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=6.0cm,
clip=true,trim=0.5cm 5.0cm 0.5cm 2.5cm
{f1.pdf}
\caption{\footnotesize\label{fig:coords}
Angle $\Psi$, and directions $\bxhat$, $\byhat$.
\btdnote{f1.pdf}}
\end{center}
\end{figure}
Let $\bxhat$ and $\byhat$ be orthonormal vectors in the plane of the sky, with
$\bxhat$ parallel to the projection of $\bB_0$ on the plane of the sky
(see Figure \ref{fig:coords}):
\beqa
\bxhat &\,=\,& \hat{\bf n} \cos\Psi + \hat{\bf e} \sin\Psi
\\
\byhat &=& -\hat{\bf n} \sin\Psi + \hat{\bf e} \cos\Psi
~.
\eeqa
If the dust grains are partially
aligned with their short axes tending to be parallel to $\bB_0$, we expect
$C_{{\rm ext},y} > C_{{\rm ext},x}$.
At long wavelengths ($\lambda \gg 10\micron$) we also expect
$C_{{\rm pha},y} > C_{{\rm pha},x}$.
We assume that the dust grains themselves have no overall chirality, hence
circular dichroism and circular birefringence can be neglected so long
as the response of the magnetized plasma is negligible, which is
generally the case for $\nu \gtsim 30\GHz$.
Following the notation of \citet{Martin_1974}, define
\beqa
\delta &\,\equiv\,&
n_d ~
\frac{(C_{{\rm ext},y}+C_{{\rm ext},x})}{2}
= \frac{2\pi}{\lambda} \left(m_x^{\prime\prime} + m_y^{\prime\prime}\right)
\\
\Delta\sigma
&\equiv&
n_d ~
\frac{(C_{{\rm ext},y}-C_{{\rm ext},x})}{2}
=
\frac{2\pi}{\lambda}\left(m_y^{\prime\prime} - m_x^{\prime\prime}\right)
\\
\Delta\epsilon
&\equiv&
n_d ~
\frac{(C_{{\rm pha},y}-C_{{\rm pha},x})}{2}
=
\frac{2\pi}{\lambda}\frac{\left(m_y^\prime - m_x^\prime\right)}{2}
~.
\eeqa
If scattering is neglected,
the propagation of the Stokes parameters is given
by\footnote{Eq.\ (\ref{eq:propagation}) conforms to the IEEE and
IAU conventions for the Stokes parameters
\citep{Hamaker+Bregman_1996}: $Q>0$ for $\bE$ along the N-S direction,
$U>0$ for $\bE$ along the NE-SW direction, $V>0$ for ``right-handed''
circular polarization
($\bE$ rotating in the counterclockwise direction as viewed on the sky).}
\beq \label{eq:propagation}
\frac{d}{dz}
\left(
\begin{array}{c}
I \\ Q \\ U \\ V \\
\end{array}
\right)
=
\left(
\begin{array}{c c c c}
-\delta & \Delta\sigma\cos2\Psi & \Delta\sigma\sin2\Psi & 0 \\
\Delta\sigma\cos2\Psi & -\delta & 0 & \Delta\epsilon\sin2\Psi \\
\Delta\sigma\sin2\Psi & 0 & -\delta & -\Delta\epsilon\cos2\Psi \\
0 & -\Delta\epsilon\sin2\Psi & \Delta\epsilon\cos2\Psi & -\delta\\
\end{array}
\right)
\left(
\begin{array}{c}
I-B(T_d)\\ Q \\ U \\ V \\
\end{array}
\right)
~,
\eeq
where
$B(T_d)$ is the intensity of blackbody radiation
for dust temperature $T_d$.
Eq.\ (\ref{eq:propagation})
differs from \citet{Martin_1974} only by
replacement of $I$ by $(I-B)$ on the
right-hand side to allow for thermal emission
\citep[see also][]{Reissl+Wolf+Brauer_2016}.
It is apparent that Eq.\ (\ref{eq:propagation})
is consistent with thermal equilibrium blackbody radiation, with $d{\bf S}/dz=0$
for
${\bf S}=(B,0,0,0)$.
\section{\label{sec:dust}
Optical Properties of the Dust}
We now assume that the grains can be
approximated by spheroids.
\citet{Draine+Hensley_2021c} found that observations of
starlight polarization and
far-infrared polarization appear to be
consistent with dust with oblate spheroidal shapes, with
axial ratio $b/a\approx 1.6$ providing a good fit to observations.
\added{Observations of the diffuse ISM are consistent with
\beq
\label{eq:opacity}
\frac{\delta}{\nH} = \frac{\tau}{\NH} \approx 6.5\times10^{-27}
\left(\frac{\lambda}{\mm}\right)^{-1.8} \cm^2\Ha^{-1}
\eeq
for $100\micron \ltsim \lambda \ltsim 1\cm$
\citep{Hensley+Draine_2021a,Draine+Hensley_2021a}.}
Let $\bbhat$ be a ``special'' direction in space for grain alignment: the short
axis $\bahat_1$ of the grain
may be preferentially aligned either parallel or perpendicular
to $\bbhat$. For grains in the diffuse ISM, $\bbhat$ is the
magnetic field direction, and the short axis $\bahat_1$ tends to be
parallel to $\bbhat$. In protostellar disks, however, other alignment
mechanisms may operate, and $\bbhat$ may not be parallel to the magnetic
field.
We \replaced{will assume}{approximate}
the grains \replaced{to be}{by} oblate spheroids, spinning with short axis
$\bahat_1$ parallel to the angular momentum $\bJ$.
For oblate spheroids,
the fractional alignment is defined to be
\beq \label{eq:falign}
\falign\equiv
\frac{3}{2}
\langle (\bahat_1\cdot\bbhat)^2\rangle -
\frac{1}{2}
~~~,
\eeq
where $\langle ...\rangle$ denotes averaging over the grain population.
If $\bJ\parallel\bbhat$, then $\falign\rightarrow1$;
if $\bJ$ is randomly-oriented, then $\falign=0$;
if $\bJ\perp\bbhat$, then $\falign\rightarrow-\frac{1}{2}$.
The ``modified picket fence approximation'' \citep{Draine+Hensley_2021c}
relates $\delta$, $\Delta\sigma$,
and $\Delta\epsilon$ to $\falign$ and the angle $\gamma$:
\beqa \label{eq:delta from MPFA}
\delta &~=~& n_d
\left[ \frac{C_{{\rm abs},a}+2C_{{\rm abs},b}}{3}
+ \falign\left(\cos^2\gamma-\frac{1}{3}\right)
\frac{\left(C_{{\rm abs},b}-C_{{\rm abs},a}\right)}{2}
\right]
\\
\Delta \sigma &=& n_d \falign\sin^2\gamma ~
\frac{\left(C_{{\rm abs},b} - C_{{\rm abs},a}\right)}{2}
\\
\Delta\epsilon &=& n_d \falign\sin^2\gamma ~
\frac{\left(C_{{\rm pha},b} - C_{{\rm pha},a}\right)}{2}
~~~.
\eeqa
In the Rayleigh limit (grain radius $a\ll\lambda$) we have
\citep{Draine+Lee_1984}
\beqa
C_{{\rm abs},j} &~=~&
\frac{2\pi V}{\lambda}
\frac{\epsilon_2}
{|1+(\epsilon-1)L_j|^2}
\\
C_{{\rm pha},j} &~=~&
\frac{\pi V}{\lambda}
\frac{\left\{
(\epsilon_1-1)\left[1+L_j(\epsilon_1-1)\right]+\epsilon_2^2L_j
\right\}
}
{|1+(\epsilon-1)L_j|^2}
~~~,
\eeqa
where $\epsilon(\lambda)\equiv\epsilon_1+i\epsilon_2$ is the complex dielectric
function of the grain material, and
$L_a$ and $L_b=(1-L_a)/2$ are dimensionless ``shape factors''
\citep{van_de_Hulst_1957,Bohren+Huffman_1983} that depend on the
axial ratio of the spheroid.
\citet{Draine+Hensley_2021a} have estimated
$\epsilon(\lambda)$ of astrodust for
different assumed axial ratios.
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=10.0cm,
clip=true,trim=0.5cm 5.0cm 0.5cm 2.5cm
{f2.pdf}
\caption{\label{fig:Cpha/Cabs}
The ratios $\Delta\sigma/\delta$ and $\Delta \epsilon/\delta$
for oblate astrodust
spheroids with porosity $\calP=0.2$, axial ratio $b/a=1.6$,
alignment fraction
$\falign=0.5$, and $\sin^2\gamma=1$ (magnetic field in the plane of the
sky).
The power-law approximation (\ref{eq:powerlaw}) for $\Delta\epsilon/\delta$
is also shown.
\btdnote{f2.pdf}
}
\end{center}
\end{figure}
Figure \ref{fig:Cpha/Cabs} shows the
dimensionless ratios $\Delta\sigma/\delta$ and $\Delta\epsilon/\delta$
for oblate astrodust spheroids with \added{porosity $\calP=0.2$,} $b/a=1.6$
($L_a=0.464$, $L_b=0.268$)
and
$\falign=0.5$, for the case where the magnetic field is in the plane of
the sky ($\sin\gamma=1$).
The relatively high opacity that enables ``astrodust'' to reproduce the
observed far-infrared emission and polarization also implies that $\epsilon_1$
has to be fairly large at long wavelengths
\citep{Draine+Hensley_2021a}.
This causes $\Delta\epsilon/\delta$ to be relatively large, as seen in
Figure \ref{fig:Cpha/Cabs}.
For $\lambda \gtsim 70\micron$, oblate astrodust grains with $b/a=1.6$ have
\beqa \label{eq:Delta sigma/delta in FIR}
\frac{\Delta\sigma}{\delta}
&\,\approx\,&
0.38 \falign \sin^2\gamma
\\ \label{eq:powerlaw}
\frac{\Delta\epsilon}{\delta}
&\approx&
9.0\left(\frac{\lambda}{\mm}\right)^{0.7}
\falign \sin^2\gamma
~~~.
\eeqa
Eq.\ (\ref{eq:Delta sigma/delta in FIR}) and (\ref{eq:powerlaw})
neglect the weak dependence of $\delta$ on $\falign$ and $\gamma$
(see Eq.\ \ref{eq:delta from MPFA}).
Eqs.\ (\ref{eq:Delta sigma/delta in FIR}) and
((\ref{eq:powerlaw}) are shown in Figure \ref{fig:Cpha/Cabs} for
$\falign\sin^2\gamma=0.5$.
\section{\label{sec:clouds}
Circular Polarization from Interstellar Clouds}
\subsection{Grain Alignment}
A spinning grain develops a magnetic moment from the Barnett effect
(if it has unpaired electrons)
and the Rowland effect (if it has a net charge).
For submicron grains, the resulting net magnetic
moment is large enough that
the Larmor precession period in the local interstellar magnetic field is
short compared to the timescales for other mechanisms
to change the direction of
the grain's angular momentum $\bJ$. The rapid
precession of $\bJ$ around
the local magnetic field $\bB_0$ and the resulting averaging of grain
optical properties
establishes $\bB_0$ as the special direction for grain alignment --
grains will
be aligned with their short axis preferentially oriented either parallel or
perpendicular to $\bB_0$.
Paramagnetic dissipation,
radiative torques, or systematic streaming of the grains relative to the
gas will determine whether the grains
align with their short axes preferentially
parallel or perpendicular to $\bB_0$. Although the details of the physics of
grain alignment are not yet fully understood, it is now
clear
that grains in diffuse and translucent clouds tend to align with short axes
$\bahat_1$ tending to be parallel to $\bB_0$, i.e., with
$\falign>0$ (see Eq.\ \ref{eq:falign}).
If the dust
grains are modeled by oblate spheroids with axial ratio $b/a=1.6$,
a mass-weighted alignment fraction $\falign\approx 0.5$
can reproduce the
highest observed levels of polarization of both starlight and far-infrared
emission from dust in diffuse clouds (including
diffuse molecular clouds) \citep{Draine+Hensley_2021c}.
In dark clouds, the fractional polarization of the thermal emission is
generally lower than in diffuse clouds.
The lower fractional polarization may indicate
lower values of $\falign$ within dark clouds,
but it could also
result from a nonuniform
magnetic field in the cloud, with the overall linear polarization fraction
reduced by beam-averaging over regions with different
polarization directions.
If the reduced values of linear polarization are
due to \added{systematic}
changes in magnetic field direction along the line-of-sight,
the emission from the cloud could
become partly circularly-polarized. We now estimate
what levels of circular polarization might be present.
\subsection{\label{sec:nearby MCs}
Nearby Molecular Clouds}
Planck has observed linearly polarized emission from many molecular
clouds.
To estimate the levels of circular polarization that might be present,
we consider one
illustrative example, in the ``RCrA-Tail'' region in the R Corona Australis
molecular cloud
\citep[see Fig.\ 11 in][]{Planck_int_results_xix_2015}.
The polarized emission in this region has a number of local maxima.
One of the polarized flux maxima coincides with a total emission peak
near
$(\ell,b)\approx (-0.9^\circ,-18.7^\circ)$,
with total intensity $I(353\GHz)\approx 4\MJy\sr^{-1}$ and
linear polarization fraction $p\approx 2.5\%$.
For an assumed dust temperature
$T_d\approx 15\K$, the observed intensity
$I(353\GHz)=4\MJy\sr^{-1}$ implies
$\tau(353\GHz)\approx 1.3\times10^{-4}$.
For diffuse ISM dust \citep[see, e.g.][]{Hensley+Draine_2021a}, this would
correspond to $A_V\approx 5\,$mag.
For simple assumptions about the angle $\Psi$ characterizing the
projection of the magnetic field on the sky, we can obtain approximate
analytic solutions to the radiative transfer equations
(\ref{eq:propagation}), valid for $\tau \ll 1$
(see Appendix \ref{app:uniform twist}).
Define $d\tau^\prime \equiv \delta dz$.
Suppose that $T_d$, $(\Delta\sigma/\delta)$,
and $(\Delta\epsilon/\delta)$ are constant, and
assume that the
magnetic field direction has a smooth twist along the line of sight,
with $\Psi$ varying linearly with $\tau^\prime$ as $\tau^\prime$ varies
from $0$ to $\tau$:
\beq \label{eq:linear Psi}
\Psi = \Psi_0 + \alpha\tau^\prime ~~~,~~~~\alpha\equiv \frac{\Delta\Psi}{\tau}
~~~.
\eeq
For $\tau\ll 1$,
the linear and circular polarization fractions are then
(see Appendix \ref{app:uniform twist})
\beqa \label{eq:p approx}
p &~\approx~&
\left(\frac{\Delta\sigma}{\delta}\right)
\frac{\left[1-\cos(2\Delta\Psi)\right]^{1/2}}{\Delta\Psi}
\\ \label{eq:V/I approx}
\frac{V}{I} &\approx&
\left(\frac{\Delta\sigma}{\delta}\right)
\left(\frac{\Delta\epsilon}{\delta}\right)
\frac{\tau}{2\Delta\Psi}
\left[1 - \frac{\sin(2\Delta\Psi)}{2\Delta\Psi}\right]
~~~.
\eeqa
Eq.\ (\ref{eq:p approx}--\ref{eq:V/I approx}) are for the special case of
an isothermal medium with a uniform twist in the alignment direction.
If we assume diffuse cloud dust properties (Eq.\ \ref{eq:Delta sigma/delta in FIR}, \ref{eq:powerlaw}) but with
$\falign\sin^2\gamma=0.075$
and a twist angle $\Delta\Psi=90^\circ$,
we can reproduce the observed polarization $p\approx 2.5\%$
in the RCrA-Tail region.
With these parameters, Eq. (\ref{eq:V/I approx}) predicts
circular polarization
\replaced{$V/I \approx 7\times10^{-7}$}
{$V/I \approx 7\times10^{-7}
(\lambda/850\micron)^{-1.1}$},
far below
current sensitivity limits.
It is clear that
measurable levels of circular polarization in the far-infrared
will require much larger optical
depths $\tau$.
\subsection{Infrared Dark Clouds}
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=8.0cm,
clip=true,trim=0.5cm 5.0cm 0.5cm 2.5cm
{f3.pdf}
\caption{\label{fig:stokes params}\footnotesize
Stokes parameters for radiation from a dust slab resembling
the Brick IRDC, as a function of
the optical depth $\tau^\prime$
along the path.
The properties of astrodust at $850\GHz$ ($350\micron$)
are assumed, with
\replaced{$\falign\sin^2\gamma=0.12$}{$\falign\sin^2\gamma=0.075$},
and a field rotation $\Delta\Psi=90^\circ$.
\added{Black curves: numerical results. Red curves: analytic approximations
(\ref{eq:I(tau)}--\ref{eq:p}).}
\btdnote{f3.pdf}
}
\end{center}
\end{figure}
Typical giant molecular clouds \added{(GMCs)},
such as the Orion Molecular Cloud,
have mass surface densities resulting in
$A_V \approx 10{\rm\,mag}$ of extinction, and are therefore
referred to as ``dark clouds''.
However, in the inner Galaxy, a number of clouds have
been observed that appear to be ``dark'' (i.e., opaque) \added{even} in the
mid-infrared.
These ``infrared dark clouds'' (IRDCs) have
dust masses per area an order of magnitude larger
than ``typical'' giant molecular clouds.
Because of the much larger extinction in IRDCs, the circular
polarization may be much larger than in normal GMCs.
The ``Brick'' (G0.253+0.016) is a well-studied IRDC
\citep{Carey+Clark+Egan+etal_1998,
Longmore+Rathborne+Bastian+etal_2012}.
With an estimated mass $M>10^5\Msol$ and high estimated density
($\nH>10^4\cm^{-3}$), the Brick appears to be forming stars
\citep{Marsh+Ragan+Whitworth+Clark_2016,
Walker+Longmore+Bally+etal_2021}, although with
no signs of high-mass star formation.
It has been mapped at $70-500\micron$ by Herschel Space Observatory
\citep{Molinari+Schisano+Elia+etal_2016} and at
$220\GHz$ by ACT
\citep{Guan+Clark+Hensley+etal_2021}.
Polarimetric maps have been made at $220\GHz$ by ACT, and at $850\GHz$
by the CSO \citep{Dotson+Vaillancourt+Kirby+etal_2010}.
The Northeastern region at $(\ell,b)=(16^\prime,2^\prime)$
has $I(600\GHz)\approx 5000\MJy\sr^{-1}$
\citep{Molinari+Schisano+Elia+etal_2016}
and $I(220\GHz)\approx 90\MJy\sr^{-1}$
\citep{Guan+Clark+Hensley+etal_2021}.
For an assumed
dust temperature $T_d\approx 20\K$, this
indicates optical depths
$\tau(600\GHz)\approx 0.05$, $\tau(220\GHz)\approx0.005$.
Astrodust would then have
$\tau(850\GHz)\approx 0.09$, and $\tau(353\GHz) \approx 0.014$ --
about 100 times
larger than in the R CrA molecular cloud.
The fractional polarization is expected to be approximately independent
of frequency in the submm.
At $220\GHz$,
\citep{Guan+Clark+Hensley+etal_2021} report
a linear polarization
of \replaced{$3.6\%$}{$1.8\%$}
at 220 GHz for the Northeastern end of the cloud,
\added{$(\ell,b)\approx(16^\prime,2.5^\prime)$
(Yilun Guan 2021, private communication).}
The CSO polarimetry suggests a \replaced{smaller}{similar}
fractional polarization \added{at 850\,GHz}.
\deleted{We take the fractional polarization at
$(\ell,b)=(16^\prime,2^\prime)$
to be $\sim$3\%.}
While this fractional polarization is relatively small compared to the
highest values ($\sim 20\%$) observed by Planck in diffuse clouds, it is
still appreciable, requiring significant grain alignment in a substantial
fraction of the cloud volume
(i.e., not just in the surface layers of the IRDC).
The inferred average magnetic field direction $\Psi\approx 20^\circ$
\citep{Guan+Clark+Hensley+etal_2021}
differs by
$\sim$$60^\circ$ from the $\Psi\approx80^\circ$ field direction
indicated by the $220\GHz$ polarization
outside the cloud, demonstrating that the magnetic field in this region
is far from uniform.
As a simple example,
we suppose, as we did for the RCrA-Tail region above, that
the projected field rotates
by $\Delta\Psi=90^\circ$ from the far side of the ``Brick'' to the near side.
We calculate the
circular polarization
at $850\GHz$ ($350\micron$)
for the estimated total optical depth $\tau(850\GHz)=0.09$ of the Brick.
We use the estimated properties of astrodust in the diffuse ISM,
with $\falign\sin^2\gamma=$\replaced{$0.12$}{$0.075$}
to approximately reproduce the $\sim$\replaced{$3\%$}{$1.8\%$}
polarization observed for the Brick.
Figure 2 shows the polarization state of the radiation as it propagates
through the cloud from $\tau^\prime=0$ to $\tau^\prime=\tau$. The fractional
polarization $p$ starts off at $\sim$\replaced{4.7}{2.9}\%,
dropping to $\sim$\replaced{3\%}{1.8\%} at
$\tau^\prime=\tau$ as the result of the assumed magnetic field twist of
$\Delta\Psi=90^\circ$.
The resulting $850\GHz$
circular polarization $V/I$ is small, only
$\sim$\replaced{$0.065\%$}{$0.025\%$}.
Measuring such low levels of circular polarization will be challenging.
For $\Delta\epsilon/\delta \propto \lambda^{0.7}$
(see Figure \ref{fig:Cpha/Cabs}) and the absorption coefficient
$\delta \propto \lambda^{-1.8}$
\added{(see Eq.\ \ref{eq:opacity})}, the circular polarization from an IRDC
is expected to vary as $V/I \propto \lambda^{-1.1}$.
For the adopted parameters ($\Delta\Psi=90^\circ$,
$\tau(850\GHz)=0.09$,
$\falign\sin^2\gamma=$\replaced{$0.12$}{$0.075$}), the Brick would have
\beq
\frac{V}{I} \approx \replaced{0.065}{0.025}\%\left(\frac{350\micron}{\lambda}\right)^{1.1}
~~~
\eeq
\added{for $70\micron \ltsim \lambda \ltsim 1\cm$.}
While much larger than for normal GMCs, this estimate for the
circularly-polarized emission from the Brick is \deleted{still} small, and
measuring it will be challenging.
\added{\subsection{PDR Seen Through a Molecular Cloud}}
\begin{figure}
\begin{center}
\includegraphics[angle=0,width=10.0cm,
clip=true, trim=0.5cm 0.5cm 0.5cm 0.5cm]
{f4.pdf}
\caption{\footnotesize\label{fig:pdrspec}
\added{
Polarization of dust continuum from a molecular cloud with a
PDR on the far side.
The magnetic field in the cold cloud is assumed to have a systematic
twist along the line of sight, with a twist angle
$\Delta\Psi=60^\circ$.
The dust is assumed to be partially aligned, with
$\falign\sin^2\gamma=0.1$ for the warm dust ($T_1=80\K$) in the PDR,
and $\falign\sin^2\gamma=0.05$ for the cold dust ($T_2=15\K$)
in the rest of the cloud.
}}
\end{center}
\end{figure}
\added{The warm dust surrounding
an embedded HII region may allow measurement of circular polarization
at wavelengths as short as $\sim$$20\micron$.
We consider a cloud with optical depth $\tau(353\GHz)=4\times10^{-4}$,
somewhat greater than the R Corona Australis cloud example considered in Section
\ref{sec:nearby MCs}, but small compared to the Brick.
The far edge of the cloud is assumed to contain
warm dust in a photodissociation region (PDR)
with optical depth
$\tau_1(\lambda)$.
The PDR is assumed to contribute
10\% of the total column density through the
the molecular cloud, with
dust heated to $T_1=80\K$.
The dust
in the rest of the molecular cloud is cold, $T_2=15\K$.
We assume the dust in the PDR to be moderately aligned, with
$\falign\sin^2\gamma=0.1$, whereas for the dust in the rest of the molecular
cloud we take $\falign\sin^2\gamma=0.05$.
Using the analytic approximation for the ``two-zone'' model
in Appendix \ref{app:two zone}, we find the
the fractional linear polarization $p$ and circular polarization $V/I$ shown in
Figure \ref{fig:pdrspec}.
At $\lambda \ltsim 100\micron$, the
polarization is the combination of polarized emission
from the warm dust in the PDR and dichroic absorption by the cool dust.
At longer wavelengths, $\lambda > 300\micron$, dichroic absorption is minimal,
and we see the sum of the polarized emission from the warm and cool regions.
The polarization angle rotates as the ratio of warm emission
to cool emission drops with increasing wavelength.
The features at $10< \lambda < 30\micron$ arise from the strong silicate
absorption bands at $10$ and $18\micron$.
The circular polarization reaches
$V/I=0.02\%$ at $\lambda=20\micron$
but declines as $\sim\!\lambda^{-1.1}$ at longer wavelengths.
}
\bigskip
\section{\label{sec:disks}
Circular Polarization from Protoplanetary Disks}
Protoplanetary disks can have dust surface densities well in excess of
IRDCs, raising the possibility that $\tau$ may be large enough to generate
measurable circular polarization if the grains are locally aligned
{\it and} the
alignment direction varies along the optical path.
\subsection{Grain Alignment in Protoplanetary Disks}
Gas densities in protoplanetary disks exceed interstellar gas densities by
many orders of magnitude.
The observed thermal emission spectra from young protoplanetary
disks appear to require that
most of the solid material be in particles
with sizes that may be as large as $\sim$mm
\citep{Beckwith+Sargent_1991,Natta+Testi_2004,Draine_2006a},
orders of magnitude larger than \added{the submicron grains}
in the diffuse ISM.
The physics of grain alignment in protoplanetary
disks differs substantially
from the processes in the diffuse ISM.
One important difference from interstellar clouds
is that in protoplanetary disks the Larmor precession period
for the grain sizes of interest is {\it long}
compared to the time for the grain
to undergo collisions with a mass of gas atoms equal to the grain mass
\citep{Yang_2021}.
With Larmor precession no longer important,
the magnetic field no longer determines the preferred direction
for grain alignment. Instead, the ``special'' direction
may be either the local direction of
gas-grain streaming -- in which case, $\bbhat\parallel\bvdrift$ -- or
perhaps the direction of anisotropy in the radiation field
-- in which case, $\bbhat\parallel\br$. Whether grains will tend to align
with short axes $\bahat_1$ parallel or perpendicular to $\bbhat$
\added{(i.e., $\falign>0$ or $\falign<0$)} is a separate
question.
\subsubsection{Alignment by Radiative Torques?}
Radiative torques resulting from outward-directed radiation provide
\replaced{another}{one}
possible mechanism for grain alignment.
Starlight torques have been found to be very important for both
spinup and alignment of interstellar grains
\citep{Draine+Weingartner_1996,
Draine+Weingartner_1997,
Weingartner+Draine_2003,
Lazarian+Hoang_2007a}.
With mm-sized grains, both stellar radiation and infrared emission from the
disk may be capable of exerting systematic torques large enough to
affect the spin of the grain. However, the radiation pressure
$\sim L_\star/4\pi R^2c \approx 5\times10^{-9}(L_\star/\Lsol)(100\AU/R)^2\erg\cm^{-3}$ is small compared to the gas pressure
$\sim 8\times10^{-5}(\nH/10^{10}\cm^{-3})(T/100\K)\erg\cm^{-3}$.
If the grain streaming velocity exceeds $\sim 10^{-4} c_s$, where $c_s$
is the sound speed, systematic torques exerted by gas atoms may dominate
radiative torques.
Studies of realistic grain geometries are needed to clarify the relative
importance of gaseous and radiative torques.
\subsubsection{Alignment by Grain Drift?}
The differential motion of dust and gas in three-dimensional disks
has been discussed by
\citet{Takeuchi+Lin_2002}.
Grains well above or below
the midplane will
sediment toward the midplane, with
$\bvdrift \parallel \bz_{\rm disk}$, where
$z_{\rm disk}$ is height above the midplane.
Dust grains close to the midplane will be in near-Keplerian orbits, but will
experience a ``headwind'', with $\bvdrift \parallel \hat{\phi}$.
Vertical and azimuthal drift velocities will in general differ, with different
dependences on grain size and radial distance from the protostar.
\citet{Gold_1952} proposed
grain drift relative to the gas as an alignment mechanism.
For hypersonic motion,
Gold concluded that needle-shaped particles would tend to align
with their short axes perpendicular to
$\bvdrift$.
\citet{Purcell_1969a} analyzed spheroidal shapes, finding that significant
alignment requires hypersonic
gas-grain velocities
if the grains are treated as rigid bodies.
The degree of grain alignment \added{of spheroidal grains}
is increased when dissipative processes
within the grain are included \citep{Lazarian_1994b}, but the degree of
alignment is small unless the streaming is supersonic.
\citet{Lazarian+Hoang_2007b} discussed mechanical alignment of
subsonically-drifting grains with ``helicity'', arguing that helical
grains would preferentially acquire angular momentum parallel or antiparallel
to $\bvdrift$; internal dissipation would then cause the short
axis to tend to be {\it parallel} to $\bvdrift$.
\citet{Lazarian+Hoang_2007b} based their analysis on a simple geometric
model of a spheroidal grain with a single projecting panel.
More realistic irregular geometries have been considered by
\citet{Das+Weingartner_2016} and \citet{Hoang+Cho+Lazarian_2018}.
However, these studies all assumed Larmor
precession to be rapid compared to the gas-drag time, and are therefore
not directly
applicable to protoplanetary disks.
It appears possible that, averaged over
the ensemble of irregular grain shapes,
the net effect of gas-grain streaming in protoplanetary disks may
be (1) suprathermal angular momenta tending to be perpendicular to
$\bvdrift$, and (2) tendency of grains to align with short axes
perpendicular to $\bvdrift$.
Below, we consider the
consequences of this conjecture.
\subsection{The HL Tau Disk as an Example}
ALMA has observed a number of protoplanetary disks
\citep[e.g.,][]{Andrews+Huang+Perez+etal_2018}. HL Tau remains one of the
best-observed cases: it is nearby ($\sim$$140\pc$), bright, and
\deleted{only} moderately inclined ($i \approx 45^\circ$).
The optical depth in the disk is large, with
beam-averaged $\tau(3.1\mm)\approx 0.13$ at $R\approx 100\AU$.\footnote{
At $R\approx 100\AU$, $I_\nu(3.1\mm)\approx 1.1\times10^3\MJy\sr^{-1}$
\citep{Kataoka+Tsukagoshi+Pohl+etal_2017,Stephens+Yang+Li+etal_2017},
implying $\tau\approx 0.13$ if
the dust temperature $T_d\approx30\K$ \citep{Okuzumi+Tazaki_2019}.}
Given that the dust is visibly concentrated in rings, and the
possibility that there may be
additional unresolved substructure, the actual optical depth of the
emitting regions at $100\AU$ is likely to be larger.
The polarization in HL Tau has been mapped by ALMA at $870\micron$,
$1.3\mm$, and $3.1\mm$
\citep{Kataoka+Tsukagoshi+Pohl+etal_2017,
Stephens+Yang+Li+etal_2017}.
The observed polarization patterns show
considerable variation from one frequency to another,
complicating interpretation.
Both intrinsic polarization from
aligned grains and polarization resulting from scattering appear
to be contributing to the overall polarization.
\citet{Mori+Kataoka_2021} argue that
polarized
emission makes a significant contribution to the polarization,
at least at $3.1\mm$.
The $3.1\mm$ polarization pattern is generally azimuthal
\citep{Stephens+Yang+Li+etal_2017}. If due to polarized emission,
this would require that the radiating dust
grains have
short axes preferentially oriented in the radial direction.
The alignment mechanism is unclear.
\citet{Kataoka+Okuzumi+Tazaki_2019} favor radiative torques, with the
grain's short axis assumed to be {\it parallel} to the radiative flux, in the
radial direction.
This would be consistent with the observation that the linear
polarization tends to be in the azimuthal direction.
If radiative torques are responsible for grain alignment in protoplanetary
disks, then we do not expect the thermal emission from the disk to be
circularly polarized, because the grains in the upper and lower layers
of the disk will tend to have the same alignment direction as the grains near
the midplane.
If there is no change in the direction of
the grain alignment along a ray, there will be no circular polarization.
Here we instead
suppose that grain alignment is dominated by gas-grain streaming due
to systematic motion of the dust grains relative to the local gas.
If we define $\bbhat \parallel \bvdrift$ we can apply the
discussion above.
As discussed above,
we conjecture that the irregular grains align with short axes tending
to be \emph{perpendicular} to $\bvdrift$, thus $\falign<0$.
\begin{table}[t]
\footnotesize
\begin{center}
\caption{\label{tab:disk pars}A Stratified Disk Example$^{\rm a}$}
\begin{tabular}{c c c c}
& lower layer & midplane & upper layer \\
Parameter & $j= 1$ & $j= 2$ & $j= 3$ \\
\hline
$\tau_j$ & $0.05$ & $0.2$ & $0.05$ \\
$B(T_{d,j})/B(T_{d,2})$ & $2$ & $1$ & $2$ \\
$\sin\gamma$ & $\cos\theta_i$ & $1\rightarrow\sin\theta_i$
& $\cos\theta_i$ \\
$\Psi_j$ & $0$ & $0\rightarrow180^\circ$ & $180^\circ$ \\
$\falign$ & $-0.2$ & $-0.2$ & $-0.2$ \\
$(\Delta\sigma/\delta)_j$& $0.38\falign\sin^2\gamma$
& $0.38\falign\sin^2\gamma$
& $0.38\falign\sin^2\gamma$ \\
$(\Delta\epsilon/\delta)_j$ & $19\falign\sin^2\gamma$
& $19\falign\sin^2\gamma$
& $19\falign\sin^2\gamma$ \\
\hline
\multicolumn{4}{l}{For $\lambda=3.1\mm$.}
\end{tabular}
\end{center}
\end{table}
\begin{figure}[b]
\begin{center}
\includegraphics[angle=0,width=7.0cm,
clip=true,trim=0.5cm 0.5cm 0.5cm 0.5cm
{f5ab.pdf}
\includegraphics[angle=0,width=7.0cm,
clip=true,trim=0.5cm 0.5cm 0.5cm 0.5cm
{f5cd.pdf}
\caption{\label{fig:disk}\footnotesize
(a) Grain drift directions in a stratified disk (see text).
(b) Upper figure: the direction of linear polarization if the
grains are aligned by $\bv_{\rm drift}$,
with parameters in Table \ref{tab:disk pars}.
The length of the line segment is proportional to the fractional
polarization, with the scale bar showing $2\%$ fractional polarization.
Lower figure:
quadrupolar pattern for circular polarization $V$ for the
example discussed in the text.
(c) Fractional linear polarization $p=(Q^2+U^2)^{1/2}/I$
for stratified disk model (see Table \ref{tab:disk pars}),
viewed at inclination $\theta_i=45^\circ$
(see text),
as a function of azimuthal angle in the disk plane. $\phi=0$ is along
the minor axis.
(d) Circular polarization $V/I$ for this
model.
\btdnote{f4ab.pdf, f4cd.pdf}
}
\end{center}
\end{figure}
As before, let $\gamma$ be the angle between the line-of-sight and
$\bbhat$,
and let $\Psi$ be
the angle (relative to north) of the projection of $\bbhat$ on the
plane of the sky.
For illustration, we take the disk to have the major axis in the E-W direction
(see Figure \ref{fig:disk}), with inclination $i$.
Thus vertical drifts correspond to $\Psi=0$ and $180^\circ$.
The treatment of radiative transfer developed above for magnetized
clouds can be reapplied to protoplanetary disks --
the only difference is that if the grains align with their short
axis tending to be perpendicular to $\bvdrift$
then $\falign<0$, implying
$\Delta\sigma<0$ and $\Delta\epsilon<0$.
The direction and magnitude of $\bvdrift$ will vary with height in the disk.
$\bvdrift$ may be approximately normal to the disk plane
for grains that are falling toward the midplane, whereas $\bvdrift$
will be azimuthal
for grains near the midplane, with Keplerian rotation causing them to move
faster than the pressure-supported gas disk.
Thus, grain orientations may vary both vertically
and azimuthally. With $\Psi$ varying along a ray, the emerging radiation may
be partially circularly-polarized.
The observed linear polarization of a few percent suggests that
$|\Delta\sigma/\delta| \approx $ a few \%.
We do not expect $\Psi$ to vary linearly with $\tau$ as in
Eq.\ (\ref{eq:linear Psi}):
the variation of $\Psi$ along the ray will depend on
the varying grain dynamics along the ray.
To investigate what levels of circular polarization might be present,
we consider an idealized model with three dust layers:
layer 2 is the dust near the midplane,
and layers 1 and 3 contain the dust below and above the midplane.
Conditions in layers 1 and 3 are assumed to be identical.
Let $\tau_j$ be the optical depth
through layer $j$.
Assume that $\bbhat$ is normal to the disk in layers 1 and 3, and
azimuthal in layer 2 (see Figure \ref{fig:disk}).
Thus $\Psi_1=\Psi_3$.
For small values of $\tau_1$, $\tau_2$, and $\tau_3$ we can approximate
the radiative transfer (see Appendix \ref{app:three zone model}):
\beqa
I_1 &=& B_1\tau_1 \left(1-\frac{1}{2}\tau_1\right) e^{-\tau_2-\tau_3}
\\
I_2 &=& B_2\tau_2 \left(1-\frac{1}{2}\tau_2\right) e^{-\tau_3}
\\
I_3 &=& B_3\tau_3 \left(1-\frac{1}{2}\tau_3\right)
\\
I &\,\approx\,&
I_1+I_2+I_3
\\ \label{eq:Q}
Q &\approx&
-\left(\frac{\Delta\sigma}{\delta}\right)_{\!3}
\cos(2\Psi_3)
\left[I_1+I_3-\tau_3(I_1+I_2)\right]
-\left(\frac{\Delta\sigma}{\delta}\right)_{\!2}
\cos(2\Psi_2)
\left(I_2-\tau_2I_1\right)
\\ \label{eq:U}
U &\approx&
-\left(\frac{\Delta\sigma}{\delta}\right)_{\!3}
\sin(2\Psi_3)
\left[I_1+I_3-\tau_3(I_1+I_2)\right]
-\left(\frac{\Delta\sigma}{\delta}\right)_{\!2}
\sin(2\Psi_2)
\left(I_2-\tau_2I_1\right)
\\ \label{eq:V}
V &\approx&
\sin(2\Psi_2-2\Psi_1)
\left[
\left(\frac{\Delta\epsilon}{\delta}\right)_{\!2}
\left(\frac{\Delta\sigma}{\delta}\right)_{\!1}
\tau_2 I_1
+
\left(\frac{\Delta\epsilon}{\delta}\right)_{\!3}
\left(\frac{\Delta\sigma}{\delta}\right)_{\!2}
\tau_3 I_2
\right]
~.
\eeqa
The direction and magnitude
of linear polarization at selected positions are shown
in Figure \ref{fig:disk} for a stratified disk model with parameters
given in Table \ref{tab:disk pars}, viewed at inclination
$\theta_i=45^\circ$.
Figure \ref{fig:disk}{c,d} show the linear and circular polarization
as a function of azimuthal angle (in the disk plane) for this model.
In addition to accurate results from numerical integration, the results
from the analytic approximation (Eqs. \ref{eq:Q}--\ref{eq:V})
are also plotted. The analytic
approximation is seen to provide
fair accuracy, even though $\tau_2=0.2$ is not small.
The circular polarization $V/I$ is quite accurate, but
in Figure \ref{fig:disk}(c),
the analytic approximation slightly
overestimates the linear polarization fraction.
However, the analytic approximations were developed for $\tau\ll 1$,
and here the total optical depth
$\tau_1+\tau_2+\tau_3=0.3$ is not small.
For this model, the linear polarization varies from $1.4\%$ to $3.2\%$ around
the disk, with average value $\sim$$2.5\%$.
The linear polarization tends to be close to
the azimuthal direction, with largest
values on the major axis, and smallest values along the minor axis of
inclined disk (see Figure \ref{fig:disk}).
The predicted circular polarization $|V|/I$
is small but perhaps detectable, with
$V/I$ varying from positive to negative from one
quadrant to another (see Figure \ref{fig:disk}),
with maxima $|V|/I \approx 0.2\%$ (see Figure \ref{fig:disk}(d)).
\citet{Stephens+Yang+Li+etal_2017} mapped $V$ over the HL Tau disk
at $3.3\mm$, $1.3\mm$, and $870\micron$.
The $3.3\mm$ $V$ map does not appear to show any statistically significant
detection, with upper limits $|V/I| \ltsim 1\%$.
At $1.3\mm$ and $870\micron$ the NW side of the major axis may have
$V/I \approx -1\%$, but whether this is real rather than an instrumental
artifact remains unclear. In any event, the likely importance of
scattering at these shorter wavelengths will complicate interpretation.
\section{\label{sec:discussion}
Discussion}
For typical molecular clouds we conclude that
the circular polarization will
be undetectably
small at the far-infrared and submm wavelengths where the clouds
radiate strongly. Probing the magnetic field structure in such clouds
using circular polarization is feasible only at shorter infrared wavelengths
where the extinction is appreciable, using
embedded infrared sources (stars, protostars, \added{or PDRs}).
The thermal dust emission from
so-called IR dark clouds (IRDCs) in the inner Galaxy -- such
as the ``Brick'' -- can show appreciable levels of linear polarization,
demonstrating both that there is appreciable grain alignment
\emph{and} that the magnetic field structure in the cloud, while showing
evidence of rotation, is relatively coherent.
IRDCs
have large enough column densities that the resulting circular polarization
may reach detectable levels. For one position on the Brick and
plausible assumptions concerning the field,
we estimate a circular polarization
$|V/I| \approx \replaced{0.06}{0.025}\%$ at $850\GHz$.
If the circular polarization can be detected and mapped in IRDCs,
it would provide constraints on the 3-dimensional magnetic field structure.
Unfortunately, the predicted $V/I$ is small, especially
at longer wavelengths (we expect
$V/I \propto \lambda^{-1.1}$), and detection will be
challenging.
Protoplanetary disks may offer the best opportunity to measure
circular polarization at submm wavelengths.
If there are significant
changes in the direction of grain alignment between the dust near the
midplane and dust well above and below
the midplane, linear dichroism and birefringence will
produce circular polarization.
Alignment processes in protoplanetary disks remain uncertain, but we
suggest that grain drift may cause the grains near the midplane to
be aligned with long axes
preferentially in the azimuthal direction, while grains above and below
the midplane may
be aligned with long axes
tending to be in the vertical direction (normal to the disk).
If the grains are small enough that scattering can be neglected, we calculate
the linear and circular polarization that would be expected for such
a model.
A characteristic quadrupole pattern of circular polarization is predicted
for this kind of grain alignment (see Figure \ref{fig:disk}).
Eq.\ (\ref{eq:V}) can be used to estimate the circular polarization
at wavelengths $\lambda \gtsim 100\micron$ where thermal emission is
strong and the grains may be approximated by the Rayleigh limit.
We present a simple example to show the linear and circular polarization that
might be present in protoplanetary disks, such as the disk around HL Tau.
This example is not being put forward as a realistic model for HL Tau, but
simply to illustrate the possible circular polarization from dust aligned by
streaming in a stratified disk.
If observed,
this would help clarify the physical processes responsible for grain
alignment in protoplanetary disks. Absence of this circular polarization
would indicate that the preferred direction for grain
alignment in high-altitude regions is the same as the preferred direction
near the midplane, or else that grain alignment occurs
only in the midplane, or only in the upper layers.
\added{If circular polarization is detected and mapped in a protoplanetary
disk,
interpretation will require
radiative transfer
models that include the birefringence and dichroism discussed here as well
as the circular polarization produced by scattering of linearly polarized
radiation.
Models will be sensitive to the spatial distribution of the
dust, and also to the sizes and scattering properties of the solid particles.
Maps of $V/I$ at multiple frequencies would strongly constrain
protoplanetary disk models.}
\section{\label{sec:summary}
Summary}
\begin{enumerate}
\item
We present the transfer equations for the Stokes parameters, including
the effects of thermal emission.
Once the properties of the medium are specified, these equations
can easily be integrated numerically.
For small optical depths, analytic solutions are given for
clouds with a uniform twist to the magnetic field, and for stratified
clouds with uniform alignment within individual strata.
\item
Using the ``astrodust'' grain model \citep{Draine+Hensley_2021a}
we calculate the relevant optical properties of dust grains for producing
linear and circular polarization in the far-infrared and submm.
By adjusting the assumed degree of dust alignment $\falign$, these dust
properties may approximate the properties of dust in protoplanetary disks,
at wavelengths where scattering can be neglected.
\item
At submm wavelengths, the ``phase shift'' cross section $C_{\rm pha}$
tends to be
much larger than the absorption cross section
$C_{\rm abs}$.
We estimate $C_{\rm pha}/C_{\rm abs} \approx 24(\lambda/\mm)^{0.7}$.
\item
The far-IR emission from dust in diffuse clouds, and in normal
molecular clouds, will have very low levels of circular polarization,
below current and foreseen sensitivities.
\item
If the magnetic field in IRDCs has a significant
systematic twist,
the emission from IRDCs \added{ -- such as the ``Brick'' --} may have
$V/I \approx \replaced{0.06}{0.025}\% (\lambda/350\micron)^{-1.1}$
\item
If dust grains in protoplanetary disks are aligned in different
directions in different strata, the resulting submm emission may
be circularly polarized with peak
$V/I\approx 0.2\% (\lambda/350\micron)^{-1.1}$
for one simple example
\added{with parameters suggested by HL Tau}.
Measuring the circular polarization can constrain
the mechanisms responsible for grain alignment in protoplanetary disks.
\end{enumerate}
This work was supported in part by NSF grant AST-1908123.
I thank \added{Yilun Guan,}
Chat Hull and Joseph Weingartner for helpful discussions,
Robert Lupton for availability of the SM package. \added{
I thank the anonymous referee for helpful suggestions that improved
this paper.}
|
2,869,038,156,353 | arxiv | \section*{Introduction}
\begin{tabsection}
It is a common pattern in mathematics that things that are easy to
define are hard to compute and things that are hard to define come with
lots of machinery to compute them\footnote{Quote taken from a lecture
by Janko Latschev.}. On the other hand, mathematics can be very
enjoyable if these different definitions can be shown to yield
isomorphic objects. In the present article we want to promote such a
perspective towards topological group cohomology, along with its
specialization to Lie group cohomology.
It has become clear in the last decade that concretely accessible
cocycle models for cohomology theories (understood in a broader sense)
are as important as abstract constructions. Examples for this are
differential cohomology theories (cocycle models come for instance from
(bundle) gerbes, an important concept in topological and conformal
field theory), elliptic cohomology (where cocycle models are yet
conjectural but have nevertheless already been quite influential) and
Chas-Sullivan's string topology operations (which are subject to
certain very well behaved representing cocycles). This article describes an
easily accessible cocycle model for the more complicated to define
cohomology theories of topological and Lie groups
\cite{Segal70Cohomology-of-topological-groups,Wigner73Algebraic-cohomology-of-topological-groups,Deligne74Theorie-de-Hodge.-III,Brylinski00Differentiable-Cohomology-of-Gauge-Groups}.
The cocycle model is a seemingly obscure mixture of (abstract) group
cohomology, added in a continuity condition only around the identity.
Its smooth analogue has been used in the context of Lie group
cohomology and its relation to Lie algebra cohomology
\cite{TuynmanWiegerinck87Central-extensions-and-physics,WeinsteinXu91Extensions-of-symplectic-groupoids-and-quantization,Neeb02Central-extensions-of-infinite-dimensional-Lie-groups,Neeb04Abelian-extensions-of-infinite-dimensional-Lie-groups,Neeb06Towards-a-Lie-theory-of-locally-convex-groups,Neeb07Non-abelian-extensions-of-infinite-dimensional-Lie-groups},
which is where our original motivation stems from. The basic message
will be that all the above concepts of topological and Lie group
cohomology coincide for finite-dimensional Lie groups and coefficients
modeled on quasi-complete locally convex spaces. Beyond
finite-dimensional Lie groups the smooth and the continuous concepts
begin to diverge, but still all continuous concepts agree.
There is a na{\"i}ve notion of topological group cohomology for a
topological group $G$ and a continuous $G$-module $A$. It is the
cohomology of the complex of continuous cochains with respect to the
usual group differential. This is what we call ``globally continuous''
group cohomology and denote it by $H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)$. It cannot
encode the topology of $G$ appropriately, for instance
$H^{2}_{\ensuremath{\op{glob},\cont}}(G,A)$ can only describe abelian extensions which are
topologically trivial bundles. However, in case $G$ is contractible it
will turn out that the more elaborate cohomology groups from above
coincide with $H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)$. In this sense, the deviation from
the above cohomology groups from being the globally continuous ones
measures the non-triviality of the topology of $G$. On the other hand,
the comparison between $H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)$ and the other cohomology
groups for topologically trivial \emph{coefficients} $A$ will give rise
to a comparison theorem between the other cohomology groups. It is this
circle of ideas that the present article is about.
The paper is organized as follows. In the first section we review the
construction and provide the basic facts of what we call locally
continuous group cohomology $H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$ (respectively the
locally smooth cohomology $H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A)$ for $G$ a Lie group and
$A$ a smooth $G$-module). Since it will become important in the sequel
we highlight in particular that for loop contractible coefficients
these cohomology groups coincide with the globally continuous
(respectively smooth) cohomology groups $H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)$
(respectively $H^{n}_{\ensuremath{\op{glob},\sm}}(G,A)$). In the second section we then
introduce what we call simplicial continuous cohomology
$H^{n}_{\ensuremath{\op{simp},\cont}}(G,A)$ and construct a comparison morphism
$H^{n}_{\ensuremath{\op{simp},\cont}}(G,A)\to H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$. The third section explains
how simplicial cohomology may be computed in a way similar to computing
sheaf cohomology via \v{C}ech cohomology (the fact that this gives
indeed $H^{n}_{\ensuremath{\op{simp},\cont}}(G,A)$ will have to wait until the next section).
The first main point of this paper comes in Section
\ref{sect:Comparison_Theorem}, where we give the following axiomatic
characterization of what we call a cohomology theory for topological
groups.
\begin{nntheorem}[Comparison Theorem]
Let $G$ be a compactly generated topological group and let
$\cat{G-Mod}$ be the category of locally contractible $G$-modules.
Then there exists, up to isomorphism, exactly one sequence of functors
$(H^{n}\ensuremath{\nobreak\colon\nobreak}\cat{G-Mod}\to\cat{Ab})_{n\in\ensuremath{\mathbb{N}}_{0}}$ admitting natural
long exact sequences for short exact sequences in $\cat{G-Mod}$ such
that
\begin{enumerate}
\item $H^{0}(A)=A^{G}$ is the invariants functor
\item $H^{n}(A)=H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)$ for contractible $A$.
\end{enumerate}
\end{nntheorem}
There is one other way of defining cohomology groups $H^{n}_{\ensuremath{\op{SM}}}(G,A)$
which it due to Segal and Mitchison
\cite{Segal70Cohomology-of-topological-groups}. This construction will
turn out to be the one which is best suited for establishing the
Comparison Theorem. However, we then show that under some mild
assumptions (guaranteed for instance by the metrizability of $G$) all
cohomology theories that we had so far (except the globally continuous)
obey these axiomatics. The rest of the section in then devoted to
showing that almost all other concepts of cohomology theories for
topological groups also fit into this scheme. This includes the ones
considered by Flach in
\cite{Flach08Cohomology-of-topological-groups-with-applications-to-the-Weil-group},
the measurable cohomology of Moore from
\cite{Moore76Group-extensions-and-cohomology-for-locally-compact-groups.-III}
and the mixture of measurable an locally continuous cohomology of
Khedekar and Rajan from
\cite{KhedekarRajan10On-Cohomology-theory-for-topological-groups}. The
only exception that we know not fitting into this scheme is the
continuous bounded cohomology (see
\cite{Monod01Continuous-bounded-cohomology-of-locally-compact-groups,Monod06An-invitation-to-bounded-cohomology}),
which differs from the above concepts by design.
The second main point comes with Section \ref{sect:examples}, where we
exploit the interplay between the different constructions. For
instance, we construct cohomology classes that deserve to be named
string classes, and we construct topological crossed modules associated
to third cohomology classes. Moreover, we show how to extract the
purely topological information contained in an element in
$H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$ by relating an explicit formula for this with a
structure map for the spectral sequence associated to
$H^{n}_{\ensuremath{\op{simp},\cont}}(G,A)$. Furthermore, $H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A)$ maps naturally
to Lie algebra cohomology and we use the previous result to identify
situations where this map becomes an isomorphism. Almost none of the
consequences mentioned here could be drawn from one model on its own,
so this demonstrates the strength of the unified framework.
In the last two sections, which are independent from the rest of the
paper, we provide some details on the constructions that we use.
\end{tabsection}
\section*{Acknowledgements}
\begin{tabsection}
Major parts of the research on this article were done during research
visits of CW in Nantes (supported by the program ``Math\'ematiques des
Pays de Loire'') and of FW in Hamburg (supported by the University of
Hamburg). Both authors want to thank Chris Schommer-Pries for various
enlightening discussions and conversations, which influenced the paper
manifestly (see \cite{Pries09Smooth-group-cohomology} for a bunch of
ideas pointing towards the correct direction). Thanks go also to Rolf Farnsteiner for pointing
out the usefulness of the language of $\delta$-functors.
\end{tabsection}
\section*{Conventions}
\begin{tabsection}
Since we will be working in the two different regimes of compactly
generated Hausdorff spaces and infinite-dimensional Lie groups we have
to choose the setting with some care.
Unless further specified, $G$ will throughout be a group in the
category $\cat{CGHaus}$ of compactly generated Hausdorff spaces (cf.\
\cite{Whitehead78Elements-of-homotopy-theory,Mac-Lane98Categories-for-the-working-mathematician}
or \cite{Hovey99Model-categories}) and $A$ will be a locally
contractible $G$-module in this category\footnote{From the beginning of
Section \ref{sect:Comparison_Theorem} we will also assume that $A$ is
locally contractible.}. This means that the multiplication
(respectively action) map is continuous with respect to the
\emph{compactly generated topology} on the product. Note that the
topology on the product may be finer than the product topology, so this
may not be a topological group (respectively module) as defined below.
To avoid confusion, we denote the compactly generated product by
$X\ktimes Y$ ($X^{\ktimes[n]}$ for the $n$-fold product) and the
compactly generated topology on $C(X,Y)$ by $\ensuremath{C_{\boldsymbol{k}}}(X,Y)$ for $X,Y$ in
$\cat{CGHaus}$.
If $X$ and $Y$ are arbitrary topological spaces, then we refer to the
product topology by $X\ptimes Y$. With a topological group
(respectively topological module) we shall mean a group (respectively
module) in this category, i.e., the multiplication (respectively
action) is continuous for the product topology.
Frequently we will assume, in addition, that $G$ is a (possibly
infinite dimensional) Lie group and that $A$ is a smooth
$G$-module\footnote{This assumption seems to be quite restrictive for
either side, but it is the natural playground on which homotopy theory
and (infinite-dimensional) Lie theory interacts.}. With this we mean
that $G$ is a group in the category $\cat{Man}$ of manifolds, modeled
modeled on locally convex vector spaces (see
\cite{Hamilton82The-inverse-function-theorem-of-Nash-and-Moser,Milnor84Remarks-on-infinite-dimensional-Lie-groups,Neeb06Towards-a-Lie-theory-of-locally-convex-groups}
or \cite{GlocknerNeeb11Infinite-dimensional-Lie-groups-I} for the
precise setting) and $A$ is a $G$-modules in this category. This means
in particular that the multiplication (respectively action) map is
smooth for the product smooth structure. To avoid confusion we refer to
the product in $\cat{Man}$ by $X\mtimes Y$ (and $X^{\mtimes[n]}$).
Note that we set things up in such a way that the smooth setting is a
specialization of the topological one, which is in turn a
specialization of the compactly generated one. This is true since
smooth maps are in particular continuous and since the product topology
is coarser than the compactly generated one. Note also that all
topological properties on $G$ (except the existence of good covers)
that we will assume are satisfied for metrizable $G$ and all smoothness
properties are satisfied for metrizable and smoothly paracompact $G$.
The existence of good cover (as well as metrizability and smooth
paracompactness) is in turn satisfied for large classes of
infinite-dimensional Lie groups like mapping groups or diffeomorphism
groups
\cite{KrieglMichor97The-Convenient-Setting-of-Global-Analysis,SperaWurzbacher10Good-coverings-for-section-spaces-of-fibre-bundles}.
We shall sometimes have to impose topological conditions on the
topological spaces underlying $G$ and $A$. We will do so by leisurely
adding the corresponding adjective. For instance, a contractible
$G$-module $A$ is a $G$-module such that $A$ is contractible as a
topological space.
\end{tabsection}
\section{Locally continuous and locally smooth cohomology}
\label{sect:locally_smooth_cohomology}
\begin{tabsection}
One of our main objectives will be the relation of locally continuous
and locally smooth cohomology for topological or Lie groups to other
concepts of topological group cohomology. In this section, we recall
the basic notions and properties of locally continuous and locally
smooth cohomology. These concepts already appear in the work of
Tuynman-Wiegerinck
\cite{TuynmanWiegerinck87Central-extensions-and-physics}, of
Weinstein-Xu
\cite{WeinsteinXu91Extensions-of-symplectic-groupoids-and-quantization}
and have been popularized recently by Neeb
\cite{Neeb02Central-extensions-of-infinite-dimensional-Lie-groups,Neeb04Abelian-extensions-of-infinite-dimensional-Lie-groups,Neeb06Towards-a-Lie-theory-of-locally-convex-groups,Neeb07Non-abelian-extensions-of-infinite-dimensional-Lie-groups}.
There has also appeared a slight variation of this by measurable locally smooth
cohomology in
\cite{KhedekarRajan10On-Cohomology-theory-for-topological-groups}.
\end{tabsection}
\begin{definition}\label{def:locsm}
For any pointed topological space $(X,x)$ and abelian topological
group $A$ we set
\begin{align*}
C_{\ensuremath{ \op{loc}}}(X,A):=\{f\ensuremath{\nobreak\colon\nobreak} X\to A\mid f\text{ is continuous on some neighborhood of }x\}.
\end{align*}
If, moreover, $X$ is a smooth manifold and $A$ a Lie group, then we set
\begin{align*}
C_{\ensuremath{ \op{loc}}}^{\infty}(X,A):=\{f\ensuremath{\nobreak\colon\nobreak} X\to A\mid f\text{ is smooth on some neighborhood of }x\}.
\end{align*}
With this we set $C^{n}_{\ensuremath{ \op{loc},\cont}}(G,A):=C_{\ensuremath{ \op{loc}}}(G^{\ktimes[n]},A)$,
where we choose the identity in $G^{n}$ as base-point. We call these
functions (by some abuse of language) \emph{locally continuous group
cochains}. The ordinary group differential
\begin{align}
(\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} f)(g_0,\ldots,g_n)&=g_0.f(g_1,\ldots,g_n)\,+\notag\\
&+\sum_{j=1}^n(-1)^jf(g_0,\ldots,g_{j-1}g_j,\ldots,g_n)\,+\,(-1)^{n+1}
f(g_0,\ldots,g_{n-1})\label{eqn:group_differential}
\end{align}
turns $(C_{\ensuremath{ \op{loc},\cont}}^{n}(G,A),\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}})$ into a cochain complex. Its
cohomology will be denoted by $H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$ and be called the
\emph{locally continuous group cohomology}.
If $G$ is a Lie group and $A$ a smooth $G$-module, then we also
consider the sub complex
$C^{n}_{\ensuremath{ \op{loc},\sm}}(G,A):=C_{\ensuremath{ \op{loc}}}^{\infty}(G^{\mtimes[n]},A)$ and call its
cohomology $H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A)$ the \emph{locally smooth group
cohomology}.
\end{definition}
\begin{tabsection}
These two concepts should not be confused with the continuous
\emph{local cohomology} (respectively the smooth \emph{local
cohomology}) of $G$, which is given by the complex of germs of
continuous (respectively smooth) $A$-valued functions at the identity
(which is isomorphic to the Lie algebra cohomology for a
finite-dimensional Lie group $G$, see Remark
\ref{rem:connection_to_Lie_algebra_cohomology}). It is crucial that the
cocycles in the locally continuous cohomology actually are extensions
of locally defined cocycles and this extension is extra information
they come along with. Note for instance, that not all locally defined
homomorphisms of a topological groups extend to global homomorphisms
and that not all locally defined 2-cocycles extend to globally defined
cocycles
\cite{Smith51The-complex-of-a-group-relative-to-a-set-of-generators.-I,Smith51The-complex-of-a-group-relative-to-a-set-of-generators.-II,Est62Local-and-global-groups.-I,Est62Local-and-global-groups.-II}.
\end{tabsection}
\begin{remark}\label{rem:long_exact_coefficient_sequence}
(cf.\ \cite[App.\
E]{Neeb04Abelian-extensions-of-infinite-dimensional-Lie-groups}) Let
\begin{equation}
A\xrightarrow{\alpha}B\xrightarrow{\beta}C \label{eqn:shot_exact_coefficient_sequence}
\end{equation}
be a \emph{short exact sequence of $G$-modules in $\cat{CGHaus}$},
i.e., the underlying sequence of abstract abelian groups is exact and
$\beta$ (or equivalently $\alpha$) has a continuous local section. The
latter is equivalent to demanding that
\eqref{eqn:shot_exact_coefficient_sequence} is a locally trivial
principal $A$-bundle. Then composition with $\alpha$ and $\beta$
induces a sequence
\begin{equation}\label{eqn:short_exact_sequence_of_complexes}
C^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)\xrightarrow{\alpha_{*}} C^{n}_{\ensuremath{ \op{loc},\cont}}(G,B)\xrightarrow{\beta _{*}}
C^{n}_{\ensuremath{ \op{loc},\cont}}(G,C),
\end{equation}
which we claim to be a short exact sequence of chain complexes.
Injectivity of $\alpha_{*}$ and $\ensuremath{\operatorname{im}}(\alpha_{*})\ensuremath{\nobreak\subseteq\nobreak}\ker(\beta_{*})$ is
clear. Since a local trivialization of the bundle induces a continuous
left inverse to $\alpha$ on some neighborhood of $\ker(\beta)$, we also
have $\ker(\beta_{*})\ensuremath{\nobreak\subseteq\nobreak} \ensuremath{\operatorname{im}}(\alpha_{*})$. To see that $\beta_{*}$ is
surjective, we choose a local continuous section $\sigma\ensuremath{\nobreak\colon\nobreak} U\to B$
which we extend to a global (but not necessarily continuous) section
$\sigma\ensuremath{\nobreak\colon\nobreak} C\to B$. Thus if $f\in C^{n}_{\ensuremath{ \op{loc},\cont}}(G,C)$, then
$ \sigma \circ f \in C^{n}_{\ensuremath{ \op{loc},\cont}}(G,B)$ with
$\beta_{*}(\sigma \circ f)=\beta \op{\circ} \sigma \op{\circ} f=f$ and
$\beta_{*}$ is surjective.
Since \eqref{eqn:short_exact_sequence_of_complexes} is exact, it induces
a long exact sequence
\begin{equation}\label{eqn:long_exact_coefficient_sequence}
\cdots\to H^{n-1}_{\ensuremath{ \op{loc},\cont}}(G,C) \to H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)\to H^{n}_{\ensuremath{ \op{loc},\cont}}(G,B) \to H^{n}_{\ensuremath{ \op{loc},\cont}}(G,C) \to H^{n+1}_{\ensuremath{ \op{loc},\cont}}(G,A) \to\cdots
\end{equation}
in the locally continuous cohomology.
If, in addition, $G$ is a Lie group and
\eqref{eqn:shot_exact_coefficient_sequence} is a \emph{short exact
sequence of smooth $G$-modules}, i.e., a smooth locally trivial
principal $A$-bundle, then the same argument shows that $\alpha_{*}$
and $\beta_{*}$ induce a long exact sequence
\begin{equation*}
\cdots\to H^{n-1}_{\ensuremath{ \op{loc},\sm}}(G,C) \to H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A)\to H^{n}_{\ensuremath{ \op{loc},\sm}}(G,B) \to H^{n}_{\ensuremath{ \op{loc},\sm}}(G,C) \to H^{n+1}_{\ensuremath{ \op{loc},\sm}}(G,A) \to\cdots
\end{equation*}
in the locally smooth cohomology.
\end{remark}
\begin{remark}
The low-dimensional cohomology groups $H^{0}_{\ensuremath{ \op{loc},\cont}}(G,A)$,
$H^{1}_{\ensuremath{ \op{loc},\cont}}(G,A)$ and $H^{2}_{\ensuremath{ \op{loc},\cont}}(G,A)$ have the usual
interpretations. $H^{0}_{\ensuremath{ \op{loc},\cont}}(G,A)=A^{G}$ are the $G$-invariants of
$A$, $H^1_{\ensuremath{ \op{loc},\cont}}(G,A)$ (respectively $H^{1}_{\ensuremath{ \op{loc},\sm}}(G,A)$) is the
group of equivalence classes of continuous (respectively smooth)
crossed homomorphisms modulo principal crossed homomorphisms. If $G$ is
connected, then $H^{2}_{\ensuremath{ \op{loc},\cont}}(G,A)$ (respectively
$H^{2}_{\ensuremath{ \op{loc},\sm}}(G,A)$) is the group of equivalence classes of abelian
extensions
\begin{equation}\label{eqn:locally_trivial_abelian_extension}
A\to \wh{G}\to G
\end{equation}
which are continuous (respectively smooth) locally trivial principal
$A$-bundles over $G$ \cite[Sect.\
2]{Neeb04Abelian-extensions-of-infinite-dimensional-Lie-groups}.
\end{remark}
\begin{remark}\label{rem:globally_continuous_cohomology}
The cohomology groups $H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$ and $H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A)$ are
variations of the more na{\"i}ve \emph{globally continuous} cohomology
groups $H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)$ and \emph{globally smooth} cohomology
groups $H^{n}_{\ensuremath{\op{glob},\sm}}(G,A)$, which are the cohomology groups of the chain
complexes
\begin{equation*}
C^{n}_{\ensuremath{\op{glob},\cont}}(G,A):=C(G^{\ktimes[n]},A)
\quad\text{ and }\quad
C^{n}_{\ensuremath{\op{glob},\sm}}(G,A):=C^{\infty}(G^{\mtimes[n]},A),
\end{equation*}
endowed with the differential \eqref{eqn:group_differential}. We
obviously have
\begin{equation*}
H^{0}_{\ensuremath{ \op{loc},\cont}}(G,A)=H^{0}_{\ensuremath{\op{glob},\cont}}(G,A)\quad\text{ and }\quad
H^{0}_{\ensuremath{ \op{loc},\sm}}(G,A)=H^{0}_{\ensuremath{\op{glob},\sm}}(G,A).
\end{equation*}
Since crossed homomorphisms are continuous (respectively smooth) if and
only if they are so on some identity neighborhood (see for example
\cite[Lemma III.1]
{Neeb04Abelian-extensions-of-infinite-dimensional-Lie-groups}), we also have
\begin{equation*}
H^{1}_{\ensuremath{ \op{loc},\cont}}(G,A)=H^{1}_{\ensuremath{\op{glob},\cont}}(G,A)\quad\text{ and }\quad
H^{1}_{\ensuremath{ \op{loc},\sm}}(G,A)=H^{1}_{\ensuremath{\op{glob},\sm}}(G,A).
\end{equation*}
Moreover, the argument from Remark
\ref{rem:long_exact_coefficient_sequence} also shows that we have a
long exact sequence
\begin{equation*}
\cdots\to H^{n-1}_{\ensuremath{\op{glob},\cont}}(G,C) \to H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)\to H^{n}_{\ensuremath{\op{glob},\cont}}(G,B) \to H^{n}_{\ensuremath{\op{glob},\cont}}(G,C) \to H^{n+1}_{\ensuremath{\op{glob},\cont}}(G,A) \to\cdots
\end{equation*}
if the exact sequence $A\xrightarrow{\alpha} B\xrightarrow{\beta} C$
has a global continuous section (and respectively for the globally
smooth cohomology if $A\xrightarrow{\alpha} B\xrightarrow{\beta} C$ has
a global smooth section).
Now assume that $A$ is contractible (respectively smoothly
contractible) and that $G$ is connected and paracompact (respectively smoothly
paracompact). In this case, the bundle
\eqref{eqn:locally_trivial_abelian_extension} has a global continuous
(respectively smooth) section and thus the extension
\eqref{eqn:locally_trivial_abelian_extension} has a representative in
$H^{2}_{\ensuremath{\op{glob},\cont}}(G,A)$ (respectively $H^{2}_{\ensuremath{\op{glob},\sm}}(G,A)$), cf.\
\cite[Prop.\
6.2]{Neeb04Abelian-extensions-of-infinite-dimensional-Lie-groups}.
Moreover, the argument in \cite[Prop.\
6.2]{Neeb04Abelian-extensions-of-infinite-dimensional-Lie-groups} also
shows that two extensions of the form
\eqref{eqn:locally_trivial_abelian_extension} are in this case
equivalent if and only if the representing globally continuous
(respectively smooth) cocycles differ by a globally continuous
(respectively smooth) coboundary, and thus the canonical homomorphisms
\begin{equation*}
H^{2}_{\ensuremath{\op{glob},\cont}}(G,A)\to H^{2}_{\ensuremath{ \op{loc},\cont}}(G,A) \quad\text{ and }\quad
H^{2}_{\ensuremath{\op{glob},\sm}}(G,A)\to H^{2}_{\ensuremath{ \op{loc},\sm}}(G,A)
\end{equation*}
are isomorphisms in this case.
\end{remark}
\begin{tabsection}
It will be crucial in the following that the latter observation
also holds for a large
class of contractible coefficients in arbitrary dimension (and in the
topological case also for not necessarily paracompact $G$). For this,
recall that $A$ is called \emph{loop-contractible} if there exists a
contracting homotopy $\rho\ensuremath{\nobreak\colon\nobreak} [0,1]\times A\to A$ such that
$\rho_{t}\ensuremath{\nobreak\colon\nobreak} A\to A$ is a group homomorphism for each $t\in [0,1]$.
If $A$ is a Lie group, then it is called \emph{smoothly
loop-contractible} if $\rho$ is, in addition, smooth. In particular,
vector spaces are smoothly loop-contractible, but in the topological
case there exist more elaborate and important examples (see Section
\ref{sect:Comparison_Theorem}).
\end{tabsection}
\begin{proposition}\label{prop:loc=glob_for_contractible_coefficients}
If $A$ is loop-contractible, and the product topology on all $G^{n}$ is
compactly generated, then the inclusion
$C^{n}_{\ensuremath{\op{glob},\cont}}(G,A)\hookrightarrow C^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$ induces an
isomorphism $H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)\cong H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$.
If $G$ is a Lie group such that all $G^{\mtimes[n]}$ are smoothly
paracompact and $A$ is a smooth $G$-module which is smoothly
loop-contractible, then
$C^{n}_{\ensuremath{\op{glob},\sm}}(G,A)\hookrightarrow C^{n}_{\ensuremath{ \op{loc},\sm}}(G,A)$ induces an
isomorphism $H^{n}_{\ensuremath{\op{glob},\sm}}(G,A)\cong H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A)$.
\end{proposition}
\begin{proof}
This is \cite[Prop.\ III.6, Prop.\
IV.6]{FuchssteinerWockel11Topological-Group-Cohomology-with-Loop-Contractible-Coefficients}.
\end{proof}
In the
case of discrete $A$ we note that there is no difference
between the locally continuous and locally smooth cohomology groups.
This is immediate since continuous and smooth maps
into discrete spaces are both the same thing as constant maps on connected
components.
\begin{lemma}
If $G$ is a Lie group and $A$ is a discrete $G$-module, then the
inclusion $C_{\ensuremath{ \op{loc},\sm}}^{n}(G,A)\hookrightarrow C_{\ensuremath{ \op{loc},\cont}}^{n}(G,A)$
induces an isomorphism in cohomology
$H_{\ensuremath{ \op{loc},\sm}}^{n}(G,A)\to H_{\ensuremath{ \op{loc},\cont}}^{n}(G,A)$.
\end{lemma}
In the finite-dimensional case, we also note that there is no difference
between the locally continuous and locally smooth cohomology groups.
\begin{proposition}\label{prop:locc=locs_in_finite_dimensions}
Let $G$ be a finite-dimensional Lie group, $\mathfrak{a}$ be a
quasi-complete locally convex space\footnote{A locally convex space is
said to be quasi-complete if each bounded Cauchy net converges.} on
which $G$ acts smoothly, $\Gamma\ensuremath{\nobreak\subseteq\nobreak}\mathfrak{a}$ be a discrete
submodule and set $A=\mathfrak{a}/\Gamma$. Then the inclusion
$C_{\ensuremath{ \op{loc},\sm}}^{n}(G,A)\hookrightarrow C_{\ensuremath{ \op{loc},\cont}}^{n}(G,A)$ induces an
isomorphism $H_{\ensuremath{ \op{loc},\sm}}^{n}(G,A)\cong H_{\ensuremath{ \op{loc},\cont}}^{n}(G,A)$.
\end{proposition}
\begin{proof}
(cf.\ \cite[Cor.\
V.3]{FuchssteinerWockel11Topological-Group-Cohomology-with-Loop-Contractible-Coefficients})
If $\Gamma=\{0\}$, then this is implied by Proposition
\ref{prop:loc=glob_for_contractible_coefficients} and \cite[Thm.\
5.1]{HochschildMostow62Cohomology-of-Lie-groups}. The general case then
follows from the previous lemma, the short exact sequence for the
coefficient sequence $\Gamma\to \mathfrak{a}\to A$ and the Five Lemma.
\end{proof}
\begin{remark}
Proposition \ref{prop:locc=locs_in_finite_dimensions} does not remain
valid in infinite dimensions. In fact, the group of continuous free
loops $G=C(\ensuremath{\mathbb{S}}^{1},K)$ into a simple compact 1-connected Lie group $K$
provides a counterexample. The group $G$ is a Banach Lie group in the
compact-open topology modeled on its Lie algebra $\ensuremath{\mathfrak{g}}=C(\ensuremath{\mathbb{S}}^{1},\ensuremath{\mathfrak{k}})$
where $\ensuremath{\mathfrak{k}}$ is the Lie algebra of $K$.
On the one hand, $H^{2}_{\ensuremath{ \op{loc},\sm}}(G,\op{U}(1))$ vanishes. This is true,
for instance, since its Lie algebra $\ensuremath{\mathfrak{g}}$ has trivial second continuous
Lie algebra cohomology $H^{2}_{\op{Lie},c}(\ensuremath{\mathfrak{g}},\ensuremath{\mathbb{R}})$ \cite[Cor.\ 13,
Thm.\ 16]{Maier02Central-extensions-of-topological-current-algebras}
and the sequence
\begin{equation*}
\ensuremath{\operatorname{Hom}}(\pi_{1}(G),U(1))\to H^{2}_{\ensuremath{ \op{loc},\sm}}(G,\op{U(1)})\to
H^{2}_{\op{Lie},c}(\ensuremath{\mathfrak{g}},\ensuremath{\mathbb{R}})
\end{equation*}
is exact \cite[Thm.\
7.12]{Neeb02Central-extensions-of-infinite-dimensional-Lie-groups}.
Since $\pi_{1}(G)\cong \pi_{2}(K)\oplus \pi_{1}(K)=0$ this shows that
$H^{2}_{\ensuremath{ \op{loc},\sm}}(G,\op{U(1)})$ vanishes.
On the other hand, the 2-connected space $G\langle 2\rangle$ (second
step in the Whitehead tower) gives rise to a non-trivial element in
$H^{2}_{\ensuremath{ \op{loc},\cont}}(G,\op{U}(1))$, since it may be realized as a central
extension
\begin{equation}\label{eqn:whitehead-tower}
U(1)\to G\langle 2\rangle \to G
\end{equation}
of topological groups which is a locally trivial principal
$U(1)$-bundle. In order to verify this, we note that a bundle
$U(1)\to Q\to G$ admits on $Q$ the structure of a topological group
turning it into a central extension if and only if it is
multiplicative, i.e., if
$\ensuremath{\operatorname{pr}}_{2}^{*}(Q)\otimes m^{*}(Q)\otimes \ensuremath{\operatorname{pr}}_{1}^{*}(Q)$ is trivial as a
bundle on $G\times G$ \cite[Prop.\ VII.1.3.5]{72SGA-7}. The inclusion
$C^{\infty}(\ensuremath{\mathbb{S}}^{1},K)\hookrightarrow G$ is a homotopy equivalence
\cite[Remark
A.3.8]{Neeb02Central-extensions-of-infinite-dimensional-Lie-groups} and
thus the universal central extension of $C^{\infty}(\ensuremath{\mathbb{S}}^{1},K)$ gives
rise to a multiplicative bundle $U(1)\to P\to G$, which carries the
structure of central extension of topological groups. Since
$\pi_{2}(G)\to \pi_{1}(U(1))$ is an isomorphism, this cannot not be
trivial, and thus $H^{2}_{\ensuremath{ \op{loc},\cont}}(G,U(1))$ does not vanish.
This phenomenon might be related to the missing
smooth paracompactness of $G$ or to the lack of ``automatic
smoothness'' in infinite dimensions (see for example
\cite{Glockner05Holder-continuous-homomorphisms-between-infinite-dimensional-Lie-groups-are-smooth}).
\end{remark}
\begin{remark}\label{rem:alternative-locally-continuous-cohomology}
For a topological group $G$ and a topological $G$-module $A$ there also
exists a variation of the locally continuous group cohomology, which
are the cohomology groups of the cochain complex
$(C_{\ensuremath{ \op{loc},\cont}}(G^{\ptimes[n]},A),\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}})$ (note the difference in the
topology that we put on $G^{n}$). We denote this by
$H^{n}_{\ensuremath{ \op{loc},\op{top}}}(G,A)$. The same argument as above yields long exact
sequences from short exact sequences of topological $G$-modules that
are locally trivial principal bundles. Moreover, they coincide with the
corresponding globally continuous cohomology groups
$H^{n}_{\ensuremath{ \op{glob},\op{top}}}(G,A)$ of $(C(G^{\ptimes[n]},A),\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}})$ if $A$ is loop
contractible \cite[Cor.\
II.8]{FuchssteinerWockel11Topological-Group-Cohomology-with-Loop-Contractible-Coefficients}.
We will use these cohomology groups very seldom.
\end{remark}
\section{Simplicial group cohomology}
\label{sect:simplicial_group_cohomology}
\begin{tabsection}
The cohomology groups that we introduce in this section date back to
\cite[Sect.\ 3]{Wigner73Algebraic-cohomology-of-topological-groups} and
have also been worked with for instance in
\cite{Deligne74Theorie-de-Hodge.-III,Friedlander82Etale-homotopy-of-simplicial-schemes,Brylinski00Differentiable-Cohomology-of-Gauge-Groups,Conrad03Cohomological-Descent}.
Since the simplicial cohomology groups are defined in terms of sheaves
on simplicial space, we first recall some facts about it. The material
is largely taken from
\cite{Deligne74Theorie-de-Hodge.-III,Friedlander82Etale-homotopy-of-simplicial-schemes}
and \cite{Conrad03Cohomological-Descent}.
\end{tabsection}
\begin{definition}
Let $X_{\bullet}\ensuremath{\nobreak\colon\nobreak} \Delta^{\op{op}}\to \cat{Top}$ be a simplicial
space, i.e., a collection of topological spaces $(X_{k})_{k\in\ensuremath{\mathbb{N}}_{0}}$,
together with continuous face maps maps
$d_{k}^{i}\ensuremath{\nobreak\colon\nobreak} X_{k}\to X_{k-1}$ for $i=0,\ldots,k$ and continuous
degeneracy maps $s_{k}^{i}\ensuremath{\nobreak\colon\nobreak} X_{k}\to X_{k+1}$ for $i=0,\ldots,k$
satisfying the simplicial identities
\begin{equation*}
\begin{cases}
d_{k}^{i}\op{\circ}d_{k+1}^{j}=d_{k}^{j-1}\op{\circ}d_{k+1}^{i}& \text{if }i<j\\
d_{k+1}^{i}\op{\circ}s_{k}^{j}=s^{j-1}_{k-1}\op{\circ}d^{i}_{k} & \text{if }i<j\\
d_{k+1}^{i}\op{\circ}s_{k}^{j}= \ensuremath{\operatorname{id}}_{X_{k}} &\text{if }i=j\tx{ or }i=j+1\\
d_{k+1}^{i}\op{\circ}s_{k}^{j}= s^{{j}}_{k-1}\op{\circ}d^{i-1}_{k} & \text{if }i>j+1\\
s^{i}_{k+1}\op{\circ}s^{j}_{k}=s^{j+1}_{k+1}\op{\circ}s^{i}_{k} & \text{if }i\leq j.
\end{cases}
\end{equation*}
(cf.\ \cite{GoerssJardine99Simplicial-homotopy-theory}). Then a sheaf
$E^{\bullet}$ on $X_{\bullet}$ consists of sheaves $E^{k}$ of abelian
groups on each space $X_{k}$ and a collection of morphisms
$D^{k}_{i}\ensuremath{\nobreak\colon\nobreak} {d_{k}^{i}}^{*}E^{k-1}\to E^{k}$ (for $k\geq 1$) and
$S^{k}_{i}\ensuremath{\nobreak\colon\nobreak} {s_{k}^{i}}^{*}E^{k+1}\to E^{k}$, obeying the
simplicial identities
\begin{equation*}
\left\{ \begin{array}{r@{\,}c@{\,}ll}
D^{k}_{j}\op{\circ}{d_{k}^{j}}^{*}D^{k+1}_{i}&=&D^{k}_{i}\op{\circ}{d_{k}^{i}}^{*}D^{k+1}_{j-1} &\text{ if }i<j\\
S^{k+1}_{j}\op{\circ}{s_{k+1}^{j}}^{*}D^{k}_{i}&=&D^{i}_{k+1}\op{\circ}{d_{k+1}^{i}}^{*}S^{k+2}_{j-1} &\text{ if }i<j\\
S^{k+1}_{j}\op{\circ}{s_{k+1}^{j}}^{*}D^{k}_{i}&=& \ensuremath{\operatorname{id}}_{E^{k}} &\text{ if }i=j\tx{ or }i=j+1\\
S^{k+1}_{j}\op{\circ}{s_{k+1}^{j}}^{*}D^{k}_{i}&=&D^{i-1}_{k+1}\op{\circ}{d_{k+1}^{i-1}}^{*}S^{{j}}_{k} &\text{ if }i>j+1\\
S^{k+1}_{j}\op{\circ}{s_{k}^{j}}^{*}S^{k}_{i}&=&S^{k+1}_{i}\op{\circ}{s_{k+1}^{i}}^{*}S^{k}_{j+1} &\text{ if }i\leq j.
\end{array}\right.
\end{equation*}
A morphism of sheaves $u\ensuremath{\nobreak\colon\nobreak} E^{\bullet}\to F^{\bullet}$ consists of
morphisms $u^{k}\ensuremath{\nobreak\colon\nobreak} E^{k}\to F^{k}$ compatible with $D^{k}_{i}$ and
$S^{k}_{i}$ (cf.\ \cite[5.1]{Deligne74Theorie-de-Hodge.-III}).
\end{definition}
\begin{tabsection}
Note that $E^{\bullet}$ is \emph{not} what one usually would call a
simplicial sheaf since the latter usually refers to a sheaf (on some
arbitrary site) with values in simplicial sets or, equivalently, to a
simplicial object in the category of sheaves (again, on some arbitrary
site). However, one can interpret sheaves on $X_{\bullet}$ as sheaves
on a certain site \cite[5.1.8]{Deligne74Theorie-de-Hodge.-III},
\cite[Def.\ 6.1]{Conrad03Cohomological-Descent}.
\end{tabsection}
\begin{remark}
Sheaves on $X_{\bullet}$ and their morphisms constitute a category
$\ensuremath{\op{Sh}}(X_{\bullet})$. Since morphisms in $\ensuremath{\op{Sh}}(X_{\bullet})$ consist of
morphisms of sheaves on each $X_{k}$, $\ensuremath{\op{Sh}}(X_{\bullet})$ has naturally
the structure of an abelian category (sums of morphisms, kernels and
cokernels are simply taken space-wise). Moreover, $\ensuremath{\op{Sh}}(X_{\bullet})$
has enough injectives, since simplicial sheaves on sites do so
\cite[Prop.\ II.1.1, 2\superscript{nd} proof]{Milne80Etale-cohomology},
\cite[p.\ 36]{Conrad03Cohomological-Descent}.
\end{remark}
\begin{definition}
(\cite[5.1.13.1]{Deligne74Theorie-de-Hodge.-III}) The
\emph{section functor} is the functor
\begin{equation*}
\Gamma\ensuremath{\nobreak\colon\nobreak} \ensuremath{\op{Sh}}(X_{\bullet})\to \cat{Ab},\quad
F^{\bullet}\mapsto \ker(D^{1}_{0}-D^{1}_{1}),
\end{equation*}
where $D^{1}_{i}$ denotes the homomorphism of the groups of global
sections $\Gamma(E^{0})\to \Gamma(E^{1})$, induced from the morphisms
of sheaves $D^{1}_{i}\ensuremath{\nobreak\colon\nobreak} {d_{1}^{i}}^{*}E^{0}\to E^{1}$.
\end{definition}
\begin{lemma}
The functor $\Gamma$ is left exact.
\end{lemma}
\begin{definition}
(\cite[5.2.2]{Deligne74Theorie-de-Hodge.-III}) The cohomology groups
$H^{n}(X_{\bullet},E^{\bullet})$ are the right derived functors of
the section functor $\Gamma$.
\end{definition}
\begin{tabsection}
Since injective (or acyclic) resolutions in $\ensuremath{\op{Sh}}(X_{\bullet})$ are not
easily dealt with (cf.\ \cite[p.\ 36]{Conrad03Cohomological-Descent} or
the explicit construction in \cite[Prop.\
2.2]{Friedlander82Etale-homotopy-of-simplicial-schemes}), the groups
$H^{n}(X_{\bullet},E^{\bullet})$ are notoriously hard to access.
However, the following proposition provides an important link to
cohomology groups of the sheaves on each single space of $X_{\bullet}$.
\end{tabsection}
\begin{proposition}
(\cite[5.2.3.2]{Deligne74Theorie-de-Hodge.-III}, \cite[Prop.\
2.4]{Friedlander82Etale-homotopy-of-simplicial-schemes}) If
$E^{\bullet}$ is a sheaf on $X_{\bullet}$, then there is a first
quadrant spectral sequence with
\begin{equation}\label{eqn:spectral_sequence}
E_{1}^{p,q}=H^{q}_{\ensuremath{\op{Sh}}}(X_{p},E^{p})\Rightarrow
H^{p+q}(X_{\bullet},E^{\bullet}).
\end{equation}
\end{proposition}
\begin{remark}\label{rem:double_complex_for_spectral_sequence}
We will need the crucial step from the proof of this proposition, so we repeat it here. It
is the fact that the spectral sequence arises from a double complex
\begin{equation*}
\xymatrix@R=2em@C=2em{ &&&&\\
F_{2}^{\bullet}\ar@{}[u]|(.75){\displaystyle\vdots} & \Gamma(F^{0}_{2}) \ar[r]^{d^{0}_{2}}\ar@{}[u]|(.75){\displaystyle\vdots}
& \Gamma(F^{1}_{2})\ar[r]^{d^{1}_{2}}\ar@{}[u]|(.75){\displaystyle\vdots} & \Gamma(F^{2}_{2}) \ar@{}[r]|(0.71){\displaystyle\cdots}\ar@{}[u]|(.75){\displaystyle\vdots}&\\%
F_{1}^{\bullet} & \Gamma(F^{0}_{1})\ar[u] \ar[r]^{d^{0}_{1}}& \Gamma(F^{1}_{1})\ar[u]\ar[r]^{d^{1}_{1}}
& \Gamma(F^{2}_{1})\ar@{}[r]|(0.71){\displaystyle\cdots}\ar[u] &\\%
E^{\bullet} & \Gamma(E^{0})\ar[u]\ar[r]^{d^{0}} & \Gamma(E^{1})\ar[u]\ar[r]^{d^{1}} & \Gamma(E^{2})\ar[u]\ar@{}[r]|(0.71){\displaystyle\cdots}& \\
& X_{0} & X_{1} & X_{2}\ar@{}[r]|(0.71){\displaystyle\cdots}&
%
%
\ar "3,1" -/d 1em/; "2,1" -/u 1em/
\ar "4,1" -/d 1em/; "3,1" -/u 1em/
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
\ar@<.45em> "5,4" -/r 1.25em/; "5,3" -/l 1.25em/
\ar "5,4" -/r 1.25em/; "5,3" -/l 1.25em/
\ar@<-.45em> "5,4" -/r 1.25em/; "5,3" -/l 1.25em/
%
\ar@<.225em> "5,3" -/r 1.25em/; "5,2" -/l 1.25em/
\ar@<-.225em> "5,3" -/r 1.25em/; "5,2" -/l 1.25em/
%
%
\ar "5,1" -/d:a(-50) 2em/; "5,5" -/d:a(-50) 2em/
\ar "5,1" -/d:a(-50) 2em/ ; "1,1" -/d:a(-50) 2em/
}
\end{equation*}
where each $F^{\bullet}_{{q}}$ is a sheaf on $X_{\bullet}$,
$E^{q}\to F^{q}_{\bullet}$ is an injective resolution in $\ensuremath{\op{Sh}}(X_{q})$
\cite[Lemma 6.4]{Conrad03Cohomological-Descent} and $d^{p}_{q}$ is the
alternating sum of morphisms induced from the $D_{i}^{p}$, respectively
for each sheaf $F^{\bullet}_{q}$. Now taking the vertical differential
first gives the above form of the $E_{1}$-term of the spectral sequence.
\end{remark}
\begin{corollary}\label{cor:acyclic_sheaves_give_augmentation_row_cohomology}
If $E^{\bullet}$ is a sheaf on $X_{\bullet}$ such that each $E^{k}$ is
acyclic on $X_{k}$, then $H^{n}(X_{\bullet},E^{\bullet})$ is the
cohomology of the Moore complex of the cosimplicial group of sections
of $E^{\bullet}$. More precisely, it is the cohomology of the complex
$(\Gamma(X_{k},E^{k}),d)$ with differential given by
\begin{equation*}
d^{k}\gamma=\sum_{i=0}^{k}(-1)^{i}D^{k}_{i}{d_{k}^{i}}^{*}(\gamma)
\quad\text{ for }\quad \gamma\in \Gamma(X_{k},E^{k}).
\end{equation*}
\end{corollary}
\begin{proof}
The $E_{1}$-term of the spectral sequence from the previous proposition
is concentrated in the first column due to the acyclicity of $E^{k}$
and yields the described cochain complex.
\end{proof}
\begin{remark}\label{rem:simplicail_sheaf_of_continuous_functions}
The simplicial space that we will work with is the classifying
space\footnote{The geometric realization of $BG_{\bullet}$ yields a
model for the (topological) classifying space of $G$
\cite{Segal68Classifying-spaces-and-spectral-sequences}, whence the
name.} $BG_{\bullet}$, associated to $G$. It is given by setting
$BG_{n}:=G^{\ktimes[n]}$ for $n\geq 1$ and $BG_{0}=\op{pt}$ and the
standard simplicial maps given by multiplying adjacent elements
(respectively dropping the outermost off) and inserting identities.
On $BG_{\bullet}$ we consider the sheaf $A_{\ensuremath{\op{glob},\cont}}^{\bullet}$, given
on $BG_{n}=G^{n}$ as the sheaf of continuous $A$-valued functions
$\underline{A}^{\ensuremath{c}}_{G^{n}}$. We turn this into a sheaf on
$BG_{\bullet}$ by introducing the following morphisms $D_{i}^{n}$ and
$S_{i}^{n}$. The structure maps on $BG_{\bullet}$ are in this case
given by inclusions and projections. Indeed, the face maps factor
through projections
\begin{equation*}
\xymatrix{G^{n}\ar@/_2em/[rr]^{d_{n}^{i}}\ar[r]^(.35){\cong}&G^{n-1}
\ktimes G\ar[r]^(.55){\ensuremath{\operatorname{pr}}}&G^{n-1}}.
\end{equation*}
Thus ${d_{n}^{i}}^{*}\underline{A}^{\ensuremath{c}}_{G^{n}}(U)=C(d_{n}^{i}(U),A)$ and we may
set
\begin{equation*}
(D_{i}^{n}f)(g_{0},\ldots,g_{n})=\begin{cases}
f(d_{n}^{i}(g_{0},\ldots,g_{n}))&\text{if }i>0\\
g_{0}.f(g_{1},\ldots,g_{n})&\text{if }i=0
\end{cases}.
\end{equation*}
Similarly,
\begin{equation*}
{s_{n}^{i}}^{*}\underline{A}^{\ensuremath{c}}_{G^{n+1}}(U)=\lim_{\overrightarrow{~V~}}C(V,A),
\end{equation*}
where $V$ ranges through all open neighborhoods of $s_{n}^{i}(U)$, has
a natural homomorphism $S_{i}^{n}$ to $\underline{A}^{\ensuremath{c}}_{G^{n}}(U)= C(U,A)$,
given by precomposition with $s_{n}^{i}$.
If, in addition, $G$ is a Lie group, then we also consider the slightly
different simplicial space $BG^{\infty}_{\bullet}$ with
$BG^{\infty}_{n}=G^{\mtimes[n]}$ and the same maps. If $A$ is a smooth
$G$-module, we obtain in the same way the sheaf $A^{\bullet}_{\ensuremath{\op{glob},\sm}}$
on $BG^{\infty}_{\bullet}$ by considering on each $BG^{\infty}_{n}$ the
sheaf $\underline{A}^{\ensuremath{s}}_{G^{n}}$ of smooth $A$-valued functions (in
order to make sense out of the latter we have to consider
$BG^{\infty}_{\bullet}$ instead of $BG_{\bullet}$).
\end{remark}
\begin{definition}
The \emph{continuous simplicial group cohomology} of $G$ with
coefficients in $A$ is defined to be
$H^{n}_{\ensuremath{\op{simp},\cont}}(G,A):=H^{n}(BG_{\bullet},A^{\bullet}_{\ensuremath{\op{glob},\cont}})$. If $G$
is a Lie group and $A$ a smooth $G$-module, then the \emph{smooth
simplicial group cohomology} of $G$ with coefficients in $A$ is defined
to be
$H^{n}_{\ensuremath{\op{simp},\sm}}(G,A):=H^{n}(BG^{\infty}_{\bullet},A^{\bullet}_{\ensuremath{\op{glob},\sm}})$.
\end{definition}
\begin{lemma}\label{eqn:long_exact_coefficient_sequence_in_simplicial_cohom}
If $A\xrightarrow{\alpha}B\xrightarrow{\beta}C$ is a short exact
sequence of $G$-modules in $\cat{CGHaus}$, then composition with
$\alpha$ and $\beta$ induces a long exact sequence
\begin{equation*}
\cdots\to H^{n-1}_{\ensuremath{\op{simp},\cont}}(G,C) \to H^{n}_{\ensuremath{\op{simp},\cont}}(G,A)\to H^{n}_{\ensuremath{\op{simp},\cont}}(G,B) \to H^{n}_{\ensuremath{\op{simp},\cont}}(G,C) \to H^{n+1}_{\ensuremath{\op{simp},\cont}}(G,A) \to\cdots.
\end{equation*}
If, moreover $G$ is a Lie group and
$A\xrightarrow{\alpha}B\xrightarrow{\beta}C$ is a short exact sequence
of smooth $G$-modules, then $\alpha$ and $\beta$
induce a long exact sequence
\begin{equation*}
\cdots\to H^{n-1}_{\ensuremath{\op{simp},\sm}}(G,C) \to H^{n}_{\ensuremath{\op{simp},\sm}}(G,A)\to H^{n}_{\ensuremath{\op{simp},\sm}}(G,B) \to H^{n}_{\ensuremath{\op{simp},\sm}}(G,C) \to H^{n+1}_{\ensuremath{\op{simp},\sm}}(G,A) \to\cdots
\end{equation*}
\end{lemma}
\begin{proof}
Since kernels and cokernels of a sheaf $E^{\bullet}$ are simply the
kernels and cokernels of $E^{k}$, this follows from the exactness
of the sequences of sheaves of continuous functions
$\underline{A}^{\ensuremath{c}}\to \underline{B}^{\ensuremath{c}}\to \underline{C}^{\ensuremath{c}}$ (and
similarly for the smooth case).
\end{proof}
\begin{proposition}\label{prop:simp=glob_for_contractible_coefficients}
If $G^{\ktimes[n]}$ is paracompact for each
$n\geq 1$ and $A$ is contractible, then
\begin{equation*}
H^{n}_{\ensuremath{\op{simp},\cont}}(G,A)\cong H^{n}_{\ensuremath{\op{glob},\cont}}(G,A).
\end{equation*}
If, moreover, $G$ is a Lie group, $A$ is a smoothly
contractible\footnote{By this we mean that there exists a contraction
of $A$ which is smooth as a map $[0,1]\mtimes A\to A$} smooth
$G$-module and if $G^{\mtimes[n]}$ is smoothly
paracompact for each $n\geq 1$, then
\begin{equation*}
H^{n}_{\ensuremath{\op{simp},\sm}}(G,A)\cong H^{n}_{\ensuremath{\op{glob},\sm}}(G,A).
\end{equation*}
\end{proposition}
\begin{proof}
In the case of contractible $A$ the sheaves $\underline{A}$ are soft
and thus acyclic on paracompact spaces \cite[Thm.\
II.9.11]{Bredon97Sheaf-theory}. The first claim thus follows from
Corollary \ref{cor:acyclic_sheaves_give_augmentation_row_cohomology}.
In the smooth case, the requirements are necessary to have the softness
of the sheaf of smooth $A$-valued functions on each $G^{k}$ as well,
since we then can extend sections from closed subsets (cf.\
\eqref{eqn:sections_on_closed_subsets}) by making use of smooth
partitions of unity.
\end{proof}
\begin{remark}
The requirement on $G^{\ktimes[n]}$ to be paracompact for each
$n\geq 1$ is for instance fulfilled if $G$ is metrizable, since then
$G^{\ktimes[n]}=G^{\ptimes[n]}$ is so and metrizable spaces are
paracompact. If $G$ is, in addition, a smoothly paracompact Lie group,
then \cite[Cor.\
16.17]{KrieglMichor97The-Convenient-Setting-of-Global-Analysis} shows
that $G^{\mtimes[n]}$ is also smoothly paracompact.
However, metrizable topological groups are not the most general
compactly generated topological groups that can be of interest. Any
$G$ that is a CW-complex has the property that $G^{\ktimes[n]}$
is a CW-complex and thus is in particular paracompact.
\end{remark}
We now introduce a second important sheaf on $BG_{\bullet}$.
\begin{remark}
For an arbitrary pointed topological space $(X,x)$ and an abelian
topological group $A$, we denote by $A^{\ensuremath{ \op{loc},\cont}}_{X}$ the sheaf
\begin{equation*}
U\mapsto \begin{cases}
C_{\ensuremath{ \op{loc}}}(U,A)
&\text{if }x\in U\\
\ensuremath{\operatorname{Map}}(U,A) &\text{if }x\notin U
\end{cases}
\end{equation*}
and call it the \emph{locally continuous sheaf} on $X$. If $X$ is a
manifold and $A$ an abelian Lie group, then we similarly set
\begin{equation*}
A^{\ensuremath{ \op{loc},\sm}}_{X}(U)= \begin{cases}
C_{\ensuremath{ \op{loc}}}^{\infty}(U,A)
&\text{if }x\in U\\
\ensuremath{\operatorname{Map}}(U,A) &\text{if }x\notin U.
\end{cases}
\end{equation*}
Obviously, these sheaves have the sheaves of continuous functions
$\underline{A}$ and of smooth functions $\underline{A}^{\ensuremath{s}}$ as sub sheaves.
As in Remark \ref{rem:simplicail_sheaf_of_continuous_functions}, the
sheaves $A^{\ensuremath{ \op{loc},\cont}}_{G^{k}}$ assemble into a sheaf $A^{\bullet}_{\ensuremath{ \op{loc},\cont}}$
on $BG_{\bullet}$. Likewise, if $G$ is a Lie group and $A$ is smooth,
the sheaves $A^{\ensuremath{ \op{loc},\sm}}_{G^{k}}$ assemble into a sheaf
$A^{\bullet}_{\ensuremath{ \op{loc},\sm}}$ on $BG^{\infty}_{\bullet}$.
\end{remark}
We learned the importance of the following fact from
\cite{Pries09Smooth-group-cohomology}.
\begin{proposition}
If $X$ is regular, then $A^{\ensuremath{ \op{loc},\cont}}_{X}$ and $A^{\ensuremath{ \op{loc},\sm}}_{X}$ are soft
sheaves. In particular, these both sheaves are acyclic if $X$ is paracompact.
\end{proposition}
\begin{proof}
In order to show that $A^{\ensuremath{ \op{loc},\cont}}_{X}$ is soft we have to show that
sections extend from closed subsets. Let $C\ensuremath{\nobreak\subseteq\nobreak} X$ be closed and
\begin{equation}\label{eqn:sections_on_closed_subsets}
[f]\in A^{\ensuremath{ \op{loc},\cont}}_{X}(C)=\lim_{\overrightarrow{~U~}}A^{\ensuremath{ \op{loc},\cont}}_{X}(U)
\end{equation}
be a section over $C$, where the limit runs over all open
neighborhoods of $C$ (cf.\ \cite[Th.\ II.9.5]{Bredon97Sheaf-theory}).
Thus $[f]$ is represented by some $f\ensuremath{\nobreak\colon\nobreak} U\to A$ for $U$ an open
neighborhood of $C$. The argument now distinguishes the relative position
of the base point $x$ which enters the definition of $A^{\ensuremath{ \op{loc},\cont}}_{X}$ with
respect to $U$.
If $x\in U$, then we may extend $f$ arbitrarily
to obtain a section on $X$ which restricts to $[f]$. If $x\notin U$,
then we choose $V\ensuremath{\nobreak\subseteq\nobreak} X$ open with $C\ensuremath{\nobreak\subseteq\nobreak} V$ and $x\notin \ol{V}$ and
define $\wt{f}$ to coincide with $f$ on $U\cap V$ and to vanish
elsewhere. This defines a section on $X$ restricting to $[f]$. This
argument works for $A^{\ensuremath{ \op{loc},\sm}}_{X}$ as well. Since soft sheaves on
paracompact spaces are acyclic \cite[Thm.\
II.9.11]{Bredon97Sheaf-theory}, this finishes the proof.
\end{proof}
Together with Corollary
\ref{cor:acyclic_sheaves_give_augmentation_row_cohomology}, this now implies
\begin{corollary}
If $G^{\ktimes[n]}$ is paracompact for all $n\geq 1$, then
\begin{equation}\label{eqn:locally_continuous_cohomology_as_simplicial_cohomology}
H^{n}(BG_{\bullet},A^{\bullet}_{\ensuremath{ \op{loc},\cont}})\cong H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A).
\end{equation}
If $G$ is a Lie group and $G^{\ptimes[n]}$ is paracompact for all $n\geq 1$, then
\begin{equation*}
H^{n}(BG_{\bullet},A^{\bullet}_{\ensuremath{ \op{loc},\sm}})\cong H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A).
\end{equation*}
\end{corollary}
Note that the second of the previous assertions does not require each $G^{\mtimes[n]}$ to be \emph{smoothly} paracompact, plain paracompactness of the underlying topological space suffices.
\begin{remark}\label{rem:comparison_homomorphisms_from_simpc_to_locc}
From the isomorphisms
\eqref{eqn:locally_continuous_cohomology_as_simplicial_cohomology} we
also obtain natural morphisms
\begin{equation*}
H^{n}_{\ensuremath{\op{simp},\cont}}(G,A)\to H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)\quad\text{ and }\quad
H^{n}_{\ensuremath{\op{simp},\sm}}(G,A)\to H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A),
\end{equation*}
induced from the morphisms of sheaves
$A^{\bullet}_{\ensuremath{\op{glob},\cont}}\to A^{\bullet}_{\ensuremath{ \op{loc},\cont}}$ and
$A^{\bullet}_{\ensuremath{\op{glob},\sm}}\to A^{\bullet}_{\ensuremath{ \op{loc},\sm}}$ on $BG_{\bullet}$
and $BG_{\bullet}^{\infty}$.
\end{remark}
\section{\v{C}ech cohomology}
\begin{tabsection}
In this section, we will explain how to compute the cohomology groups
introduced in the previous section in terms of \v{C}ech cocycles. This
will also serve as a first touching point to the locally continuous
(respectively smooth) cohomology from the first section in degree 2.
The proof that all these cohomology theories are isomorphic in all
degrees (under some technical conditions) will have to wait until
Section \ref{sect:Comparison_Theorem}.
\end{tabsection}
\begin{definition}
Let $X_{\bullet}$ be a \emph{semi-simplicial space}, i.e., a collection
of topological spaces $(X_{k})_{k\in\ensuremath{\mathbb{N}}_{0}}$, together with continuous
face maps maps $d_{k}^{i}\ensuremath{\nobreak\colon\nobreak} X_{k}\to X_{k-1}$ for $i=0,\ldots,k$ such
that $d_{k-1}^{i}\op{\circ}d_{k}^{j}=d_{k-1}^{j-1}\op{\circ}d_{k}^{i}$
if $i<j$. Then a \emph{semi-simplicial cover} (or simply a
\emph{cover}) of $X_{\bullet}$ is a semi-simplicial space $\mc{U}_{\bullet}$,
together with a morphism
$f_{\bullet}\ensuremath{\nobreak\colon\nobreak} \mc{U}_{\bullet}\to X_{\bullet}$ of semi-simplicial
spaces such that
\begin{equation*}
\mc{U}_{k}=\coprod _{j\in J_{k}}U^{j}_{k}
\end{equation*}
for $(U_{k}^{j})_{j\in J_{k}}$ an open cover of $X_{k}$ and
$\left.f_{k}\right|_{U_{k}^{j}}$ is the inclusion
$U_{k}^{j}\hookrightarrow X_{k}$. The cover is called \emph{good} if
each $(U_{k}^{j})_{j\in J_{k}}$ is a good cover, i.e., all
intersections $U_{k}^{j_{0}}\cap\ldots\cap U_{k}^{j_{l}}$ are contractible.
\end{definition}
\begin{remark}\label{rem:simplicial_covers}
It is easy to construct semi-simplicial covers from covers of the $X_k$.
In particular, we can construct good
covers in the case that each $X_{k}$ admits good cover, i.e., each
cover has a refinement which is a good cover. Indeed, given an arbitrary
cover $(U^{i})_{i\in I}$ of $X_{0}$, denote $I$ by $J_{0}$ and the cover
by $(U_{0}^{j})_{j\in J_{0}}$.
We then obtain a cover of $X_{1}$ by
pulling the cover $(U_{0}^{j})_{j\in J_{0}}$ back along $d_{1}^{0}$,
$d_{1}^{1}$, $d_{1}^{2}$ and take a common refinement
$(U_{1}^{j})_{j\in J_{1}}$ of the three covers. By definition, $J_{1}$
comes equipped with maps $\varepsilon _{1}^{1,2,3}\ensuremath{\nobreak\colon\nobreak} J_{1}\to J_{0}$
such that $d_{1}^{i}(U_{1}^{j})\ensuremath{\nobreak\subseteq\nobreak} U_{0}^{\varepsilon _{1}^{i}(j)}$. We
may thus define the face maps of
\begin{equation*}
\mc{U}_{1}:=\coprod _{j\in J_{1}}U^{j}_{1}
\end{equation*}
to coincide with $d_{1}^{i}$. In this way we then proceed to arbitrary
$k$. In the case that each $X_{k}$ admits good covers, we may refine the
cover on each $X_{k}$ before constructing the cover on $X_{k+1}$ and
thus obtain a good cover of $X_{\bullet}$.
The previous construction can be made more canonical in the case
that $X_{\bullet}=BG_{\bullet}$ for a compact Lie group $G$. In this
case, there exists a bi-invariant metric on $G$, and we let $r$ be the
injectivity radius of the exponential map at the identity (and thus at
each $g\in G$). Then $(U^{g,r})_{g\in G}$ is a good open cover of $G$,
where $U^{g,r}$ denotes the open ball of radius $r$ around $g$. Now the
triangle inequality shows that
$U^{g_{1},r/2}\cdot U^{g_{2},r/2}=U^{g_{1}g_{2},r}$, which is obviously
true for $g_{1}=g_{2}=e$ and thus for arbitrary $g_{1}$ and $g_{2}$ by
the bi-invariance of the metric. Thus
$(U^{g_{1},r/2}\times U^{g_{2},r/2})_{(g_{1},g_{2}) \in G^{2}}$ gives a
cover of $G^{2}$ compatible with the face maps
$d_{1}^{i}\ensuremath{\nobreak\colon\nobreak} G^{2}\to G$. Likewise,
\begin{equation*}
(U^{g_{1},r/2^{k}}\times \ldots\times U^{g_{k},r/2^{k}})_{(g_{1},\ldots,g_{k})\in G^{k}}
\end{equation*}
gives a cover of $G^{k}$ compatible with the face maps
$d_{k}^{i}\ensuremath{\nobreak\colon\nobreak} G^{k}\to G^{k-1}$. Since each cover of $G^{k}$ consists
of geodesically convex open balls in the product metric, this
consequently comprises a canonical good open cover of $BG_{\bullet}$.
\end{remark}
\begin{definition}\label{def:cech_cohomology}
Let $\mc{U}_{\bullet}$ be a cover of the semi-simplicial space
$X_{\bullet}$ and ${E}^{\bullet}$ be a sheaf on
$X_{\bullet}$\footnote{Sheaves on semi-simplicial spaces are defined
likewise by omitting the degeneracy morphisms.}. Then the
\emph{\v{C}ech complex} associated to $\mc{U}_{\bullet}$ and
$E^{\bullet}$ is the double complex
\begin{equation*}
\check{C}^{p,q}(\mc{U}_{\bullet},E^{\bullet}):=\prod_{i_{0},\ldots,i_{q}\in I_{p}}
E^{p}(U_{i_{0},\ldots,i_{q}}),
\end{equation*}
where we set, as usual,
$U_{i_{0},\ldots,i_{q}}:=U_{i_{0}}\cap\ldots\cap U_{i_{q}}$. The two
differentials
\begin{equation*}
d_{h}:=\sum_{i=0}^{p}(-1)^{i+q}D^{p}_{i} \op{\circ} {d_{p}^{i}}^{*} \ensuremath{\nobreak\colon\nobreak} \check{C}^{p,q}(\mc{U}_{\bullet},E^{\bullet})\to \check{C}^{p+1,q}(\mc{U}_{\bullet},E^{\bullet})\quad\text{ and }\quad
d_{v}:=\check{\delta}\ensuremath{\nobreak\colon\nobreak} \check{C}^{p,q}(\mc{U}_{\bullet},E^{\bullet})\to \check{C}^{p,q+1}(\mc{U}_{\bullet},E^{\bullet})
\end{equation*}
turn $\check{C}^{p,q}(\mc{U}_{\bullet},E^{\bullet})$ into a double
complex. We denote by
$\check{H}^{n}(\mc{U}_{\bullet},E^{\bullet})$ the cohomology of the
associated total complex and call it the \emph{\v{C}ech Cohomology} of
$E^{\bullet}$ with respect to $\mc{U}_{\bullet}$.
\end{definition}
\begin{proposition}
Suppose $G^{\ktimes[n]}$ is paracompact for each $n\geq 1$ and that
$\mc{U}_{\bullet}$ is a good cover of $BG_{\bullet}$\footnote{We may
also interpret $BG_{\bullet}$ as a semi-simplicial space by forgetting
the degeneracy maps.}. If $A\xrightarrow{\alpha}B\xrightarrow{\beta}C$
is a short exact sequence of $G$-modules in $\cat{CGHaus}$, then
composition with $\alpha$ and $\beta$ induces a long exact sequence
\begin{equation*}
\cdots\to \check{H}^{n-1}(\mc{U}_{\bullet},C^{\bullet}_{\ensuremath{\op{glob},\cont}}) \to
\check{H}^{n}(\mc{U}_{\bullet},A^{\bullet}_{\ensuremath{\op{glob},\cont}}) \to
\check{H}^{n}(\mc{U}_{\bullet},B^{\bullet}_{\ensuremath{\op{glob},\cont}}) \to
\check{H}^{n}(\mc{U}_{\bullet},C^{\bullet}_{\ensuremath{\op{glob},\cont}}) \to
\check{H}^{n+1}(\mc{U}_{\bullet},A^{\bullet}_{\ensuremath{\op{glob},\cont}}) \to \cdots.
\end{equation*}
Moreover, for each sheaf $E^{\bullet}$ on $BG_{\bullet}$ there is a
first quadrant spectral sequence with
\begin{equation*}
E_{1}^{p,q}\cong \check{H}^{q}(G^{\ktimes[p]},E^{p})\Rightarrow \check{H}^{p+q}(\mc{U}_{\bullet},E^{\bullet}).
\end{equation*}
In particular, if $A$ is contractible, then
\begin{equation*}
\check{H}^{n}(\mc{U}_{\bullet},A^{\bullet}_{\ensuremath{\op{glob},\cont}})\cong H^{n}_{\ensuremath{\op{glob},\cont}}(G,A).
\end{equation*}
\end{proposition}
\begin{proof}
Each short exact sequence $A\to B\to C$ induces a short exact sequence
of the associated double complexes and thus a long exact sequence
between the cohomologies of the total complexes. The columns of the
double complex $\check{C}^{p,q}(\mc{U}_{\bullet},E^{\bullet})$ are just
the \v{C}ech complexes of the sheaf $E^{p}$ on $G^{p}$ for the open
cover $\mc{U}_{p}$. Since the latter is good by assumption, the
cohomology of the columns is isomorphic to the \v{C}ech cohomology of
$G^{p}$ with coefficients in the sheaf $\underline{A}$.
If $A$ is contractible, then the sheaf $\underline{A}$ is soft on each
$G^{\ktimes[n]}$ and thus acyclic. Hence the $E_{1}$-term of the
spectral sequence is concentrated in the first column. Since
$E_{1}^{0,q}=C(G^{q},A)$ and the horizontal differential is just the
standard group differential, this shows the claim.
\end{proof}
\begin{remark}\label{rem:morphism_from_locally_continuous_to_Cech_cohomology}
For a connected topological group $G$ and a topological $G$-module $A$
we will now explain how to construct an isomorphism
$H^{2}_{\ensuremath{ \op{loc},\op{top}}}(G,A)\cong \check{H}^{2}(\mc{U}_{\bullet},A_{\ensuremath{\op{glob},\cont}}^{\bullet})$
in quite explicit terms (where $\mc{U}_{\bullet}$ now is a good cover
of the semi simplicial space $(G^{\ptimes[n]})_{n\in\ensuremath{\mathbb{N}}_{0}}$). To a
cocycle $f\in C_{\ensuremath{ \op{loc},\cont}}(G\ptimes G,A)$ with $\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} f=0$ we associate the
group $A\times_{f}G$ with underlying set $A\times G$ and multiplication
$(a,g)\cdot (b,h)=(a+g.b+f(g,h),gh)$. Assuming that $U\ensuremath{\nobreak\subseteq\nobreak} G$ is such
that $\left.f\right|_{U\times U}$ is continuous and $V\ensuremath{\nobreak\subseteq\nobreak} U$ is an open
identity neighborhood with $V=V^{-1}$ and $V^{2}\ensuremath{\nobreak\subseteq\nobreak} U$, there exists a
unique topology on $A\times_{f}G$ such that
$A\times V\hookrightarrow A\times _{f}G$ is an open embedding. In
particular, $\ensuremath{\operatorname{pr}}_{2}\ensuremath{\nobreak\colon\nobreak} A\times_{f}G\to G$ is a continuous
homomorphism and $x\mapsto (0,x)$ defines a continuous section thereof
on $V$. Consequently, $A\times_{f}G\to G$ is a continuous principal
$A$-bundle.
The topological type of this principal bundle is classified by a
\v{C}ech cocycle $\tau(f)$, which can be obtained from the system of
continuous sections
\begin{equation*}
\sigma_{g}\ensuremath{\nobreak\colon\nobreak} gV\to A\times_{f}G,\quad x\mapsto (0,g)\cdot
\sigma(g^{-1}x)=(f(g,g^{-1}x),x),
\end{equation*}
the associated trivializations
$A\times gV\ni (a,x)\mapsto \sigma_{g}(x)\cdot (a,e)=(f(g,g^{-1}x)+x.a,x)\in\ensuremath{\operatorname{pr}}_{2}^{-1}(gV)$
and is thus given on the cover $(gV)_{g\in G}$ by
\begin{equation*}
\tau(f)_{g_{1},g_{2}}\ensuremath{\nobreak\colon\nobreak} g_{1}^{~}V\cap g_{2}^{~}V\to A,
\quad x\mapsto
f(g_{2}^{~},g_{2}^{-1}x)-f(g_{1}^{~},g_{1}^{-1}x)=g_{1}^{~}.f(g_{1}^{-1}g_{2}^{~},g_{2}^{-1}x)-f(g_{1}^{~},g_{1}^{-1}g_{2}^{~}).
\end{equation*}
The multiplication
$\mu\ensuremath{\nobreak\colon\nobreak} (A\times_{f}G)\times(A\times_{f}G)\to A\times_{f}G$ may be
expressed in terms of these local trivializations (although it might
not be a bundle map in the case of non-trivial coefficients). For this,
we pull back the cover $(gV)_{g\in G}$ via the multiplication to
$G\times G$ and take a common refinement of this with the cover
$(gV\times hV)_{(g,h)\in G\times G}$, over which the bundle
$(A\times_{f}G)\times(A\times_{f}G)\to G\times G$ trivializes. A direct
verification shows that $(V_{g,h})_{(g,h)\in g\times G}$ with
\begin{equation*}
V_{g,h}:=\{(x,y)\in G\times G:x\in gV, y\in hV,xy\in ghV\}
\end{equation*}
and the obvious maps does the job. Expressing $\mu$ in terms of these
local trivializations, we obtain the representation
\begin{equation*}
((a,x),(b,y))\mapsto
\big((xy)^{-1}.\big[f(g,g^{-1}x)+a.x+x.f(h,h^{-1}y)+xy.b+f(x,y)-f(gh,(gh)^{-1}xy)\big],xy\big)
\end{equation*}
for $(x,y)\in V_{(g,h)}$. Since this is a continuous map
$A^{2}\times V_{(g,h)}\to A\times V_{gh}$ and since $G$ acts
continuously on $A$ it follows that
\begin{equation*}
\mu(f)_{g,h}\ensuremath{\nobreak\colon\nobreak} V_{g,h}\to A,\quad (x,y)\mapsto
f(g,g^{-1}x)+x.f(h,h^{-1}y)+f(x,y)-f(gh,(gh)^{-1}xy)
\end{equation*}
is indeed a continuous map.
A straight forward computation with the definitions of $d_{v}$, $d_{h}$
from Definition \ref{def:cech_cohomology} and the definitions of
$D^{k}_{i}$ from Remark
\ref{rem:simplicail_sheaf_of_continuous_functions} shows that
$d_{h}(\tau(f))=d_{v}(\mu(f))$ in this case. Moreover, the cocycle
identity for $f$ shows that $d_{h}(\mu(f))=0$. Thus $(\mu(f),\tau(v))$
comprise a cocycle in the total complex of
$\check{C}^{p,q}(\mc{U}_{\bullet},E^{\bullet})$ if we extend
$(gV)_{g\in G}$ and $(V_{g,h})_{(g,h\in G\times G)}$ to a cover of
$BG_{\bullet}$ as described in Remark \ref{rem:simplicial_covers}.
The reverse direction is more elementary. One associates to a cocycle
$(\Phi,\tau)$ in the total complex of
$\check{C}^{p,q}(\mc{U}_{\bullet},E^{\bullet})$ a principal bundle
$A\to P_{\tau}\to G$ clutched from the \v{C}ech cocycle $\tau$. Then
$\Phi$ defines a map $P_{\tau}\times P_{\tau}\to P_{\tau}$ (not
necessarily a bundle map, if $G$ acts non-trivially on $A$) whose
continuity and associativity may be checked directly in local
coordinates. Thus $P_{\tau}\to G$ is an abelian extension given by an
element in $H^{2}_{\ensuremath{ \op{loc},\cont}}(G,A)$. By making the appropriate choices, one
sees that these constructions are inverse to each other on the nose.
\end{remark}
\section{The Comparison Theorem via soft modules}
\label{sect:Comparison_Theorem}
\begin{tabsection}
We now describe a method for deciding whether certain cohomology groups
are the same. The usual, and frequently used technique for this is to
invoke Buchsbaum's criterion
\cite{Buchsbaum55Exact-categories-and-duality}, which also runs under
the name universal $\delta$-functor or ``satellites''
\cite{CartanEilenberg56Homological-algebra,Grothendieck57Sur-quelques-points-dalgebre-homologique,Weibel94An-introduction-to-homological-algebra}.
The point of this section is that a more natural requirement on the
various cohomology groups, which can often be checked right away for
different definitions, implies this criterion. The reader who is
unfamiliar with these techniques might wish to consult the independent
Section \ref{sect:universal_delta_functors} before continuing.
In order to make the comparison accessible, we have to introduce yet
another definition of cohomology groups $H^{n}_{\ensuremath{\op{SM}}}(G,A)$ for a
$G$-module $A$ in $\cat{CGHaus}$ due to Segal and Mitchison
\cite{Segal70Cohomology-of-topological-groups}. We give some detail on
this in Section
\ref{sect:some_information_on_moore_s_and_segal_s_cohomology_groups};
for the moment it is only important to recall that
$A\mapsto H^{n}_{\ensuremath{\op{SM}}}(G,A)$ is a $\delta$-functor for exact sequences
of locally contractible $G$-modules that are principal bundles
\cite[Prop.\ 2.3]{Segal70Cohomology-of-topological-groups} and that for
contractible $A$, one has natural isomorphisms
$H^{n}_{\ensuremath{\op{SM}}}(G,A)\cong H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)$ \cite[Prop.\
3.1]{Segal70Cohomology-of-topological-groups}.
\end{tabsection}
\begin{remark}
In what follows, we will consider a special kind of classifying space
functor, introduced by Segal in
\cite{Segal68Classifying-spaces-and-spectral-sequences}. The
classifying space $BG$ and the universal bundle $EG$ are constructed by
taking $BG=|BG_{\bullet}|$ (where $|\mathinner{\cdot}|$ denotes
geometric realization), and $EG=|EG_{\bullet}|$, where $EG_{\bullet}$
denotes the simplicial space obtained from the nerve of the pair
groupoid of $G$. The resulting $EG$ is contractible. The nice thing
about this construction of $BG$ is that it is functorial and that the
natural map $E(G\ktimes G)\to EG\ktimes EG$ is a homeomorphism. In
particular, $EG$ and $BG$ are again abelian groups in $\cat{CGHaus}$
provided that $G$ is so.
\end{remark}
\begin{definition}\label{def:segalsCohomology}
(cf.\ \cite{Segal70Cohomology-of-topological-groups}) On $\ensuremath{C_{\boldsymbol{k}}}(G,A)$, we
consider the $G$-action $(g.f)(x):=g.(f(g^{-1}\cdot x))$\footnote{This
really is the action one wants to consider, as one sees in \cite[Prop.\
3.1]{Segal70Cohomology-of-topological-groups}. Some calculations in
\cite[Ex.\ 2.4]{Segal70Cohomology-of-topological-groups} seem to use
the action $(g.f)(x)=f(g^{-1}\cdot x)$, we clarify this in Section
\ref{sect:some_information_on_moore_s_and_segal_s_cohomology_groups}.},
which obviously turns $\ensuremath{C_{\boldsymbol{k}}}(G,A)$ into a $G$-module in $\cat{CGHaus}$.
If $A$ is contractible, then we call the module $\ensuremath{C_{\boldsymbol{k}}}(G,A)$ a \emph{soft
module}. Moreover, for arbitrary $A$ we set $E_{G}(A):=\ensuremath{C_{\boldsymbol{k}}}(G,EA)$ and
$B_{G}(A):=E_{G}(A)/i_{A}(A)$, where
$i_{A}\ensuremath{\nobreak\colon\nobreak} A\hookrightarrow \ensuremath{C_{\boldsymbol{k}}}(G,EA)$ is the closed embedding
$A\hookrightarrow EA$, composed with the closed embedding
$EA\hookrightarrow \ensuremath{C_{\boldsymbol{k}}}(G,EA)$ of constant functions.
\end{definition}
\begin{lemma}\label{lem:global_section_for_BGA}
The sequence $A\to E_{G}(A)\to B_{G}(A)$ has a local continuous
section. If $A$ is contractible, then it has a global continuous
section.
\end{lemma}
\begin{proof}
The first claim is contained in \cite[Prop.\
2.1]{Segal70Cohomology-of-topological-groups}, the second follows from
\cite[App.\ (B)]{Segal70Cohomology-of-topological-groups}.
\end{proof}
\begin{proposition}\label{prop:soft_modules_are_acyclic_for_globcont}
Soft modules are acyclic for the globally continuous group
cohomology, i.e., $H^{n}_{\ensuremath{\op{glob},\cont}}(G,\ensuremath{C_{\boldsymbol{k}}}(G,A))$ vanishes for
contractible $A$ and $n\geq 1$.
\end{proposition}
\begin{proof}
This is already implicitly contained in \cite[Prop.\
2.2]{Segal70Cohomology-of-topological-groups}. See also \cite[Prop.\
17]{Pries09Smooth-group-cohomology} and Section
\ref{sect:some_information_on_moore_s_and_segal_s_cohomology_groups}.
\end{proof}
\begin{tabsection}
The following theorem now shows that all cohomology
theories considered so far are in fact isomorphic, at least if the
topology of $G$ is sufficiently well-behaved.
\end{tabsection}
\begin{theorem}[Comparison Theorem]\label{thm:comparison_theorem}
Let $\cat{G-Mod}$ be the category of locally contractible $G$-modules
in $\cat{CGHaus}$. We call a sequence
$A\xrightarrow{\alpha} B\xrightarrow{\beta} C$ in $\cat{G-Mod}$ short
exact if the underlying exact sequence of abelian groups is short exact
and $\alpha$ (or equivalently $\beta$) has a local continuous section.
If $(H^{n}\ensuremath{\nobreak\colon\nobreak}\cat{G-Mod}\to\cat{Ab})_{n\in\ensuremath{\mathbb{N}}_{0}}$ is a
$\delta$-functor such that
\begin{enumerate}
\renewcommand{\labelenumi}{\theenumi}
\renewcommand{\theenumi}{\arabic{enumi}.}
\item \label{eqn:comparison_theorem1} $H^{0}(A)=A^{G}$ is the
invariants functor
\item \label{eqn:comparison_theorem3} $H^{n}(A)=H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)$
for contractible $A$,
\end{enumerate}
then $(H^{n})_{n\in\ensuremath{\mathbb{N}}_{0}}$ is equivalent to
$(H^{n}_{\ensuremath{\op{SM}}}(G,\mathinner{\:\cdot\:}))_{n\in\ensuremath{\mathbb{N}}_{0}}$ as
$\delta$-functor. Moreover, each morphism between $\delta$-functors
with properties \ref{eqn:comparison_theorem1} and
\ref{eqn:comparison_theorem3} that is an isomorphism for $n=0$ is
automatically an isomorphism of $\delta$-functors.
\end{theorem}
\begin{proof}
The functors $I(A):=E_{G}(A)$ and $U(A):=B_{G}(A)$ make Theorem
\ref{thm:moores_comparison_theorem} applicable. In order to check the
requirements of the first part, we have to show that
$H^{n\geq 1}(E_{G}(A))$ vanishes, which in turn follows from property
\ref{eqn:comparison_theorem3} and Proposition
\ref{prop:soft_modules_are_acyclic_for_globcont}.
To check the requirements of the second part of Theorem
\ref{thm:moores_comparison_theorem} we observe that if $f\ensuremath{\nobreak\colon\nobreak} A\to B$
is a closed embedding with a local continuous section, then $f(A)$ is
also closed in $E_{G}(B)$ and thus we may set $Q_{f}:=E_{G}(B)/f(A)$.
The local sections of $f\ensuremath{\nobreak\colon\nobreak} A\to B$ and $B\to E_{G}(B)$ then also
provide a section of the composition $A\to E_{G}(B)$, and
$A\to E_{G}(B)\to Q_{f}$ is short exact. The morphism
$B_{G}(A)\to Q_{f}$ can now be taken to be induced by
$f_{*}\ensuremath{\nobreak\colon\nobreak} E_{G}(A)\to E_{G}(B)$, since it maps $A$ to $f(A)$ by
definition. Likewise, $\iota_{B}$ maps $f(A)\ensuremath{\nobreak\subseteq\nobreak} B$ into
$f(A)\ensuremath{\nobreak\subseteq\nobreak} E_{G}(B)$, so induces a morphism
$\gamma_{f}\ensuremath{\nobreak\colon\nobreak} B/f(A)\cong C\to Q_{f}=E_{G}(B)/f(A)$. The diagrams
\eqref{eqn:morphism_of_delta_functors} thus commute by construction.
\end{proof}
\begin{tabsection}
The property of $G$-modules to be locally contractible is essential
in order to provide a local section of the embedding $A\to E_{G}(A)$
\cite[Prop.\ A.1]{Segal70Cohomology-of-topological-groups}. We will
assume this from now on without any further reference.
\end{tabsection}
\begin{remark}
Property \ref{eqn:comparison_theorem3} of the Comparison Theorem may be
weakened to
\begin{equation*}
H^{n}(A)=H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)
\text{ for \emph{loop} contractible }A,
\end{equation*}
where loop contractible means that there exists a contracting homotopy
$\rho \ensuremath{\nobreak\colon\nobreak} [0,1]\times A\to A$ such that each $\rho _{t}$ is a group
homomorphisms.
If this is the case, then one may still apply Theorem
\ref{thm:moores_comparison_theorem}: Since $EA$ is loop contractible
(\cite[Ex.\ 5.5.]{Fuchssteiner11Cohomology-of-local-cochains} and
\cite[Rem.\ on p.\
217]{BrownMorris77Embeddings-in-contractible-or-compact-objects}) so is
$E_{G}=\ensuremath{C_{\boldsymbol{k}}}(G,EA)$ and thus $H^{n\geq 1}(E_{G}(A))$ still vanishes. In
this case, it is then a consequence of Theorem
\ref{thm:moores_comparison_theorem} that $H^{n}(A)=H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)$
for \emph{all} contractible modules $A$.
\end{remark}
\begin{corollary}
If $G^{\ktimes[n]}$ is paracompact for each $n\geq 1$, then
$H^{n}_{\ensuremath{\op{SM}}}(G,A)\cong H^{n}_{\ensuremath{\op{simp},\cont}}(G,A)$.
\end{corollary}
\begin{corollary}
If $G^{\ptimes[n]}$ is compactly generated for each $n\geq 1$, then we
have $H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)\cong H^{n}_{\ensuremath{\op{SM}}}(G,A)$\footnote{This is also
the main theorem in \cite{Pries09Smooth-group-cohomology}, whose proof
remains unfortunately incomplete.}. If, moreover each $G^{\ptimes[n]}$
is paracompact, then the morphisms
\begin{equation*}
H^{n}_{\ensuremath{\op{simp},\cont}}(G,A)\to H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A),
\end{equation*}
from Remark \ref{rem:comparison_homomorphisms_from_simpc_to_locc} are
isomorphisms.
\end{corollary}
\begin{corollary}
Let $G$ be a finite-dimensional Lie group, $\mathfrak{a}$ be a
quasi-complete locally convex space on which $G$ acts smoothly,
$\Gamma\ensuremath{\nobreak\subseteq\nobreak}\mathfrak{a}$ be a discrete submodule and set
$A=\mathfrak{a}/\Gamma$. Then the natural morphisms
\begin{equation}\label{eqn:finite-dimensional_G}
H^{n}_{\ensuremath{\op{simp},\sm}}(G,A)\to H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A)\to H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)
\leftarrow H^{n}_{\ensuremath{\op{simp},\cont}}(G,A)
\end{equation}
are all isomorphisms.
\end{corollary}
\begin{proof}
The second is an isomorphism by Proposition
\ref{prop:locc=locs_in_finite_dimensions} and the third by the
preceding corollary.
Since $H^{n}_{\ensuremath{\op{simp},\sm}}(G,\Gamma)\to H^{n}_{\ensuremath{\op{simp},\cont}}(G,\Gamma)$ is an
isomorphism by definition and
$H^{n}_{\ensuremath{\op{simp},\sm}}(G,\mf{a})\to H^{n}_{\ensuremath{\op{simp},\cont}}(G,\mf{a})$ is an
isomorphism by Proposition
\ref{prop:simp=glob_for_contractible_coefficients} and
\cite{HochschildMostow62Cohomology-of-Lie-groups}, the fist one in
\eqref{eqn:finite-dimensional_G} is an isomorphism by the Five Lemma.
\end{proof}
\begin{corollary}
If $G^{\ktimes[n]}$ is paracompact for each $n\geq 1$, and
$\mc{U}_{\bullet}$ is a good cover of $BG_{\bullet}$, then
$H^{n}_{\ensuremath{\op{SM}}}(G,A)\cong\check{H}^{n}(\mc{U}_{\bullet},A^{\bullet}_{\ensuremath{\op{glob},\cont}})$.
\end{corollary}
\begin{remark}
Analogous to Corollary
\ref{cor:acyclic_sheaves_give_augmentation_row_cohomology} one sees
that if each $G^{\ktimes[n]}$ is paracompact, $\mc{U}_{\bullet}$ is a
good cover of $BG_{\bullet}$ and $E^{\bullet}$ is a sheaf on
$BG_{\bullet}$ with each $E^{n}$ is acyclic, then
$\check{H}^{n}(\mc{U}_{\bullet},E^{\bullet})$ is the cohomology of the
first column of the $E_{1}$-term. This shows in particular that
$\check{H}^{n}(\mc{U}_{\bullet},A^{\bullet}_{\ensuremath{ \op{loc},\cont}})\cong H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$.
Moreover, the morphism of sheaves
$A^{\bullet}_{\ensuremath{\op{glob},\cont}}\to A^{\bullet}_{\ensuremath{ \op{loc},\cont}}$ induces a morphism
\begin{equation}
\label{eqn:morphism_from_cech_to_locally_continuous_cohomology}
\check{H}^{n}(\mc{U}_{\bullet},A^{\bullet}_{\ensuremath{\op{glob},\cont}})\to \check{H}^{n}(\mc{U}_{\bullet},A^{\bullet}_{\ensuremath{ \op{loc},\cont}})
\xrightarrow{\cong} H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A).
\end{equation}
This morphism can be constructed in (more or less) explicit terms by
the standard staircase argument for double complexes with acyclic rows
(note that by the acyclicity of $A^{n}_{\ensuremath{ \op{loc},\cont}}$ we may choose for each
locally smooth \v{C}ech $q$-cocycle
$\gamma_{i_{0},\ldots,i_{q}}\ensuremath{\nobreak\colon\nobreak} U_{i_{0}}\cap\ldots\cap U_{i_{q}}\to A$
on $G^{p}$ a locally smooth \v{C}ech cochain
$\eta_{i_{0},\ldots,i_{q-1}}$ such that $\check{\delta}(\eta)=\gamma$).
It is obvious that
\eqref{eqn:morphism_from_cech_to_locally_continuous_cohomology} defines
a morphism of $\delta$-functors. From the previous results and the
uniqueness assertion of Theorem \ref{thm:moores_comparison_theorem} it
now follows that
\eqref{eqn:morphism_from_cech_to_locally_continuous_cohomology} is in
fact an isomorphism provided $G^{\ptimes[n]}$ is compactly generated
and paracompact for each $n\geq 1$.
\end{remark}
\begin{remark}
In \cite[Prop.\
5.1]{Flach08Cohomology-of-topological-groups-with-applications-to-the-Weil-group}
it is shown that for $G$ a topological group and $A$ a $G$-module, such
that the sheaf of continuous functions has no cohomology, the
cohomology group of
\cite{Flach08Cohomology-of-topological-groups-with-applications-to-the-Weil-group}
coincide with $H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)$. By \cite[Lem.\
6]{Flach08Cohomology-of-topological-groups-with-applications-to-the-Weil-group}
we also have long exact sequences, so the cohomology groups from
\cite[Sect.\
3]{Flach08Cohomology-of-topological-groups-with-applications-to-the-Weil-group}
(which are anyway very similar to $H^{n}_{\ensuremath{\op{simp},\cont}}(G,A)$, see also
\cite{Lichtenbaum09The-Weil-etale-topology-for-number-rings}) also
agree with $H^{n}_{\ensuremath{\op{SM}}}(G,A)$.
There is a slight variation of the latter cohomology groups by
Schreiber
\cite{Schreiber11Differential-Cohomology-in-a-Cohesive-infty-Topos} in
the smooth setting and over the big topos of all cartesian smooth
spaces. The advantage of this approach is that it is embedded in a
general setting of differential cohomology. In the case that $G$ is
compact and $A$ is discrete or $A=\mf{a}/\Gamma$ for $\mf{a}$
finite-dimensional, $\Gamma\ensuremath{\nobreak\subseteq\nobreak}\mf{a}$ discrete and $G$ acts trivially
on $A$ it has been shown in \cite[Prop.\
3.3.12]{Schreiber11Differential-Cohomology-in-a-Cohesive-infty-Topos}
that the cohomology groups
$H^{n}_{\op{Smooth\infty{}Grpd}}({{\mathbf{B}}}G,A)$ from
\cite{Schreiber11Differential-Cohomology-in-a-Cohesive-infty-Topos} are
isomorphic to\footnote{This assertion is not stated explicitly but
follows from \cite[Prop.\
3.3.12]{Schreiber11Differential-Cohomology-in-a-Cohesive-infty-Topos}
by the vanishing of $H^{n}_{\ensuremath{\op{glob},\sm}}(G,\mf{a})$ \cite[Thm.\
1]{Est55On-the-algebraic-cohomology-concepts-in-Lie-groups.-I-II} and
the long exact coefficient sequence.}
$\check{H}^{n}(\mc{U}_{\bullet},A^{\bullet}_{\ensuremath{\op{glob},\sm}})$ (where
$\mc{U}_{\bullet}$ is a good cover of $BG^{\infty}_{\bullet}$).
\end{remark}
\begin{remark}
We now compare $H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$ with the cohomology groups from
\cite{Moore76Group-extensions-and-cohomology-for-locally-compact-groups.-III}.
For this we assume that $G$ is a second countable locally compact group
of finite covering dimension. A Polish $G$-module is a separable
complete metrizable\footnote{We will throughout assume that the metric
is bounded. This is no lose of generality since we may replace each
invariant metric $d(x,y)$ with the topologically equivalent bounded
invariant metric $\frac{d(x,y)}{1+d(x,y)}$.} abelian topological group
$A$ together with a jointly continuous action $G\times A\to A$.
Morphisms of Polish $G$-modules are continuous group homomorphisms
intertwining the $G$-action. If $G$ is a locally compact group and $A$
is a Polish $G$-module, then $H^{n}_{\ensuremath{\op{Moore}}}(G,A)$ denotes the
cohomology of the cochain complex
\begin{equation*}
C^{n}_{\ensuremath{\op{\mu}}}(G,A):=\{f\ensuremath{\nobreak\colon\nobreak} G^{n}\to A:f\text{ is Borel measurable}\}
\end{equation*}
with the group differential $\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}}$ from \eqref{eqn:group_differential}.
It has already been remarked in
\cite{Wigner73Algebraic-cohomology-of-topological-groups} that these
are isomorphic to $H^{n}_{\ensuremath{\op{simp},\cont}}(G,A)$, we give here a detailed proof
of this and extend the result slightly.
On the category of Polish $G$-modules we consider as short exact
sequences those sequences
$A\xrightarrow{\alpha} B\xrightarrow{\beta} C$ for which the underlying
sequence of abstract abelian groups is exact, $\alpha$ is an
(automatically closed) embedding and $\beta$ is open. From \cite[Prop.\
11]{Moore76Group-extensions-and-cohomology-for-locally-compact-groups.-III}
it follows that from this we obtain natural long exact sequences, i.e.,
$H^{n}_{\ensuremath{\op{Moore}}}(G,\:\mathinner{\cdot}\:)$ is a $\delta$-functor.
Moreover, it follows from \cite[Prop.\
3]{Wigner73Algebraic-cohomology-of-topological-groups} and from the
remarks before \cite[Thm.\
2]{Wigner73Algebraic-cohomology-of-topological-groups} that each
locally continuous cochain $f\ensuremath{\nobreak\colon\nobreak} G^{\ptimes[n]}\to C$ can be lifted
to a locally continuous cochain $\wt{f}\ensuremath{\nobreak\colon\nobreak} G^{\ptimes[n]}\to B$. This
is due to the assumption on $G$ to be finite-dimensional. From this it
follows as in Remark \ref{rem:long_exact_coefficient_sequence} that
$A\xrightarrow{\alpha} B\xrightarrow{\beta} C$ also induces a long
exact sequence for $H^{n}_{\ensuremath{ \op{loc},\cont}}(G,\:\mathinner{\cdot}\:)$ (this is
the reason why we chose $H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$ for this comparison).
On the category of Polish $G$-modules we now consider the functors
\begin{equation*}
A \mapsto \ol{E}_{G}(A):=C(G,U(I,A)),
\end{equation*}
where $U(I,A)$ is the group of Borel functions from the unit interval
$I$ to $A$. Moreover, $U(I,A)$ is a Polish $G$-module \cite[Sect.\
2]{Moore76Group-extensions-and-cohomology-for-locally-compact-groups.-III}
and coincides with the completion of the metric abelian topological
group $A$-valued step-functions on the right-open unit interval
$[0,1)$, endowed with the metric
\begin{equation*}
d(f,g ):=\int_{0}^{1}d_{A}(f(t)g(t))\;dt,
\end{equation*}
see also
\cite{BrownMorris77Embeddings-in-contractible-or-compact-objects,Keesling73Topological-groups-whose-underlying-spaces-are-separable-Frechet-manifolds,HartmanMycielski58On-the-imbedding-of-topological-groups-into-connected-topological-groups}.
In particular, $U(I,A)$ inherits the structure of a $G$-module and so
does $\ol{E}_{G}(A)$. Moreover, it is contractible and thus
$\ol{E}_{G}(A)$ is soft. Since $G$ is $\sigma$-compact we also have
that $C(G,U(I,A))$ is completely metrizable.
Now $A$ embeds as a closed submodule into $\ol{E}_{G}(A)$ and we set
$\ol{B}_{G}(A):=\ol{E}_{G}(A)/A$. Thus
\begin{equation*}
A\to \ol{E}_{G}(A)\to \ol{B}_{G}(A)
\end{equation*}
becomes short exact since orbit projection of continuous group actions
are automatically open. In virtue of Theorem
\ref{thm:moores_comparison_theorem} and the fact that the locally
continuous cohomology vanishes for soft modules this furnishes a
morphism of $\delta$-functors from
$H^{n}_{\ensuremath{ \op{loc},\cont}}(G,\:\mathinner{\cdot}\:)$ to
$H^{n}_{\ensuremath{\op{Moore}}}(G,\:\mathinner{\cdot}\:)$ (the constructions of
$Q_{f}, \beta_{f}$ and $\gamma_{f}$ from Theorem
\ref{thm:comparison_theorem} carry over to the present setting).
Moreover, the functors $A\mapsto I(A)$ and $A\mapsto U(A)$ that Moore
constructs in \cite[Sect.\
2]{Moore76Group-extensions-and-cohomology-for-locally-compact-groups.-III}
satisfy $H^{n}_{\ensuremath{\op{Moore}}}(I(A))=0$ \cite[Thm.\
4]{Moore76Group-extensions-and-cohomology-for-locally-compact-groups.-III}.
Thus Remark \ref{rem:weaker_comparison_theorem} shows that
$H^{n}_{\ensuremath{ \op{loc},\cont}}(G,\:\mathinner{\cdot}\:)$ and
$H^{n}_{\ensuremath{\op{Moore}}}(G,\:\mathinner{\cdot}\:)$ are isomorphic (even as
$\delta$-functors) on the category of Polish $G$-modules. This also
extends \cite[Thm.\
C]{Austin10Continuity-properties-of-Moore-cohomology} to arbitrary
locally contractible coefficients.
In addition, this shows that the mixture of measurable and locally
continuous cohomology groups $H^{n}_{lcm}(G,A)$ from
\cite{KhedekarRajan10On-Cohomology-theory-for-topological-groups} does
also coincide with $H^{n}_{\ensuremath{\op{Moore}}}(G,A)$. Indeed, the morphism
$H^{n}_{lcm}(G,A)\to H^{n}_{\ensuremath{\op{Moore}}}(G,A)$ of $\delta$-functors
\cite[Cor.\
1]{KhedekarRajan10On-Cohomology-theory-for-topological-groups} is
surjective for each $n$ and contractible $A$ (since then
$H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)\to H^{n}_{\ensuremath{\op{Moore}}}(G,A)$ is surjective) and also
injective (since
$H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)\to H^{n}_{lcm}(G,A)\to H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$ is so).
Thus
$H^{n}_{lcm}(G,A)\cong H^{n}_{\ensuremath{\op{Moore}}}(G,A)\cong H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)$
for each $n$ and contractible $A$ and the Comparison Theorem shows that
$H^{n}_{lcm}(G,\:\mathinner{\cdot}\:)$ is isomorphic to
$H^{n}_{\ensuremath{ \op{loc},\cont}}(G,\:\mathinner{\cdot}\:)$, also as $\delta$-functor.
\end{remark}
\begin{remark}\label{rem:bounded_cohomology}
Whereas all preceding cohomology theories fit into the framework of the
Comparison Theorem, bounded continuous cohomology
\cite{Monod01Continuous-bounded-cohomology-of-locally-compact-groups,Monod06An-invitation-to-bounded-cohomology}
does not. First of all, this concept considers locally compact $G$ and
Banach space coefficients $A$, whence all of the above cohomology
theories agree to give $H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)$. The bounded continuous
cohomology $H^{n}_{\ensuremath{ bc}}(G,A)$ is the cohomology of the sub complex
of bounded continuous functions $(C_{\ensuremath{ bc}}(G^{n},A),\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}})$. Thus there
is a natural comparison map
\begin{equation*}
H^{n}_{\ensuremath{ bc}}(G,A)\to H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)
\end{equation*}
which is obviously an isomorphism for compact $G$. However, bounded
cohomology unfolds its strength not before considering non-compact
groups, where the above map is in general not an isomorphism \cite[Ch.\
9]{Monod01Continuous-bounded-cohomology-of-locally-compact-groups},
even not for Lie groups \cite[Ex.\
9.3.11]{Monod01Continuous-bounded-cohomology-of-locally-compact-groups}.
In fact, bounded cohomology is \emph{designed} to make the above map
\emph{not} into an isomorphism in order to measure the deviation of $G$
from being compact.
\end{remark}
\begin{tabsection}
Despite the last example, the properties of the Comparison Theorem seem
to be the essential ones for a large class of important concepts of
cohomology groups for topological groups. We thus give it the following
name.
\end{tabsection}
\begin{definition}
A \emph{cohomology theory for $G$} is a $\delta$-functor
$(F^{n}\ensuremath{\nobreak\colon\nobreak} \cat{G-Mod}\to \cat{Ab})_{n\in\ensuremath{\mathbb{N}}}$ satisfying conditions
\ref{eqn:comparison_theorem1} and \ref{eqn:comparison_theorem3} of the
Comparison Theorem.
\end{definition}
\begin{remark}\label{rem:properties}
We end this section with listing properties that any cohomology theory
for $G$ has. We will always indicate the concrete model that we are
using, the isomorphisms of the models are then due to the corollaries
of this section. Parts of these facts have already been established for
the various models in the respective references.
\begin{enumerate}
\item If $A$ is discrete and each $G^{\ktimes[n]}$ is paracompact,
then
$H^{n}_{\ensuremath{\op{SM}}}(G,A)\cong H^{n}_{\pi_{1}(BG)}(BG,\underline{A})$ is
the cohomology of the topological classifying space twisted by
the $\pi_{1}(BG)\cong \pi_{0}(G)$-action on $A$ (note that
$G_{0}$ acts trivially since $A$ is discrete). This follows from
\cite[Prop.\ 3.3]{Segal70Cohomology-of-topological-groups}, cf.\
also \cite[6.1.4.2]{Deligne74Theorie-de-Hodge.-III}. If,
moreover, $G$ is $(n-1)$-connected, then
$H^{n+1}_{\ensuremath{\op{SM}}}(G,A)\cong \ensuremath{\operatorname{Hom}}(\pi_{n}(G),A)$.
\item If $G$ is contractible and each $G^{\ptimes[n]}$ is compactly
generated, then
$H^{n}_{\ensuremath{\op{SM}}}(G,A)\cong H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)\cong H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)$.
This follows from \cite[Thm.\
5.16]{Fuchssteiner11A-spectral-sequence-connecting-continuous-with-locally-continuous-group-cohomology}.
\item If $G$ is compact and $A=\mf{a}/\Gamma$ for $\mf{a}$ a
quasi-complete locally convex space which is a continuous
$G$-module and $\Gamma$ a discrete submodule, then
$H^{n}_{\ensuremath{\op{SM}}}(G,A)\cong H^{n+1}_{\pi_{1}(BG)}(BG,\Gamma)$. This
follows from the vanishing of
$H^{n}_{\ensuremath{\op{SM}}}(G,\mf{a})\cong H^{n}_{\ensuremath{\op{glob},\cont}}(G,\frak{a})$ (cf.\
\cite[Thm.\ 2.8]{Hu52Cohomology-theory-in-topological-groups} or
\cite[Lem.\
IX.1.10]{BorelWallach00Continuous-cohomology-discrete-subgroups-and-representations-of-reductive-groups})
and the long exact sequence induced from the short exact
sequence $\Gamma\to \mf{a}\to A$. In particular, if $G$ is a
compact Lie group and $A$ is finite-dimensional, then
\begin{equation*}
H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)\cong H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A)\cong
H^{n+1}_{\pi_{1}(BG)}(BG,\Gamma).
\end{equation*}
\end{enumerate}
\end{remark}
\section{Examples and applications} \label{sect:examples}
\begin{tabsection}
The main motivation for this paper is that locally continuous and
locally smooth cohomology are somewhat easy to handle, but lacked so
far a conceptual framework. On the other hand, the simplicial
cohomology groups or the ones introduced by Segal and Mitchison are
hard to handle in degrees $\geq 3$. We will give some results that one
can derive from the interaction of these different concepts.
\end{tabsection}
\begin{example}\label{ex:string_cocycles}
There is a bunch of cocycles which (or, more precisely, whose
cohomology classes) deserve to be named ``String Cocycle'' (or, more
precisely, ``String Class''). For this example, we assume that $G$ is a
compact simple and 1-connected Lie group (which is thus automatically
2-connected). There exists for each $g\in G$ a path
$\alpha_{g}\in C^{\infty}([0,1],G)$ with $\alpha_{g}(0)=e$,
$\alpha_{g}(1)=g$ and for each $g,h\in G$ a filler\footnote{From the
2-connectedness of $G$ is only follows that there exist continuous
fillers, that these can chosen to be smooth follows from the density of
$C^{\infty}(\Delta^{n},G)$ in $C(\Delta^{n},G)$ \cite[Cor.\
14]{Wockel06A-Generalisation-of-Steenrods-Approximation-Theorem}.}
$\beta_{g,h}\in C^{\infty}(\Delta^{2},G)$ for the triangle
$(\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \alpha)(g,h,k)= g.\alpha_{h}-\alpha_{gh}+\alpha_{g}$ (Figure
\ref{fig:pic_triangle}).
\begin{figure}[htbp]
\centering \includegraphics[width=0.5\textwidth]{triangle.pdf}
\caption{$\beta_{g,h}$ fills $(\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \alpha)(g,h)$}
\label{fig:pic_triangle}
\end{figure}
\noindent Moreover,
$(\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \beta)(g,h,k)=g.\beta_{h,k}-\beta_{gh,k}+\beta_{g,hk}-\beta_{g,h}$
bounds a tetrahedron which can be filled with
$\gamma_{g,h,k}\in C^{\infty}(\Delta^{3},G)$ (Figure
\ref{fig:pic_cocycle}).
\begin{figure}[htbp]
\centering \includegraphics[width=0.65\textwidth]{cocycle.pdf}
\caption{$\gamma_{g,h,k}$ fills $(\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \beta)(g,h,k)$}
\label{fig:pic_cocycle}
\end{figure}
\noindent In addition, $\alpha$, $\beta$ and $\gamma$, interpreted as
maps $G^{n}\to C^{\infty}(\Delta^{n},G)$ for $n=1,2,3$, can be chosen
to be smooth on some identity neighborhood. From these choices we can
now construct the following cohomology classes (which in turn are
independent of the above choices as a straight-forward check shows,
cf.\ \cite[Rem.\
1.12]{Wockel08Categorified-central-extensions-etale-Lie-2-groups-and-Lies-Third-Theorem-for-locally-exponential-Lie-algebras}).
\begin{enumerate}
\item Since
$\partial \ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}}(\gamma)=\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}}(\partial \gamma)=\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}}^{2}\beta =0$,
the map
\begin{equation*}
(g,h,k,l)\mapsto(\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \gamma)(g,h,k,l)
\end{equation*}
takes values in the singular 3-cycles on $G$ and thus gives rise
to map
$\theta_{3}\ensuremath{\nobreak\colon\nobreak} G^{4}\to H_{3}(G)\cong \pi_{3}(G)\cong \ensuremath{\mathbb{Z}}$ (see
also Example \ref{ex:path-space_construction}). This map is
locally smooth since $\gamma$ was assumed to be so and it is a
cocycle since $\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} (\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}}(\gamma))=0$ (note that it is not a
coboundary since $\gamma$ does not take values in the singular
cycles but only in the singular chains).
\item The cocycle $\sigma_{3}\ensuremath{\nobreak\colon\nobreak} G^{3}\to U(1)$ from \cite[Ex.\
4.10]{Wockel08Categorified-central-extensions-etale-Lie-2-groups-and-Lies-Third-Theorem-for-locally-exponential-Lie-algebras}
obtained by setting
\begin{equation*}
\sigma_{3}(g,h,k):=\exp\left(\int_{\gamma_{g,h,k}}\omega\right),
\end{equation*}
where $\omega$ is the left-invariant 3-from on $G$ with
$\omega(e)=\langle [\mathinner{\cdot},\mathinner{\cdot}],\mathinner{\cdot}\rangle$
normalized such that $[\omega]\in H^{3}_{\op{dR}}(G)$ gives a
generator of $H^{3}_{\op{dR}}(G,\ensuremath{\mathbb{Z}})\cong \ensuremath{\mathbb{Z}}$ and
$\exp\ensuremath{\nobreak\colon\nobreak} \ensuremath{\mathbb{R}}\to U(1)$ is the exponential function of $U(1)$
with kernel $\ensuremath{\mathbb{Z}}$. Since $\omega$ is in particular an integral
3-form, this implies that $\sigma_{3}$ is a cocycle because
$\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}}(\gamma)(g,h,k,l)$ is a piece-wise smooth singular cycle
and thus
\begin{equation*}
\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \sigma_{3}(g,h,k,l)=\exp\left(\int _{\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \gamma(g,h,k,l)}\omega \right)=1.
\end{equation*}
Since $\gamma$ is smooth on some identity neighborhood,
$\sigma_{3}$ is so as well. Now
\begin{equation*}
\wt{\sigma}_{3}(g,h,k):=\int_{\gamma(g,h,k)}\omega
\end{equation*}
provides a locally smooth lift of $\sigma_{3}$ to $\ensuremath{\mathbb{R}}$. Thus the
homomorphism
$\delta\ensuremath{\nobreak\colon\nobreak} H^{3}_{\ensuremath{ \op{loc},\sm}}(G,U(1))\to H^{4}_{\ensuremath{ \op{loc},\sm}}(G,\ensuremath{\mathbb{Z}})$ maps
$[\sigma_{3}]$ to $[\theta_{3}]$ since
\begin{equation*}
\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \wt{\sigma}_{3}=\int_{\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}}\gamma}\omega
\end{equation*}
and integration of piece-wise smooth representatives along
$\omega$ provides the isomorphism $\pi_{3}(G)\cong \ensuremath{\mathbb{Z}}$. We will
justify calling $\sigma_{3}$ a sting cocycle in Remark
\ref{rem:sting_cocycle_2}.
\item The locally smooth cocycles arising as characteristic cocycles
\cite[Lem.\
3.6.]{Neeb07Non-abelian-extensions-of-infinite-dimensional-Lie-groups}
from the strict models
\cite{BaezCransStevensonSchreiber07From-loop-groups-to-2-groups,NikolausSachseWockel11A-Smooth-Model-for-the-String-Group}
of the string 2-group. In the case of the model from
\cite{BaezCransStevensonSchreiber07From-loop-groups-to-2-groups}
this gives precisely $\sigma_{3}$.
\end{enumerate}
Suppose $\mc{U}_{\bullet}$ is a good cover of $BG_{\bullet}$. The model
from
\cite{Schommer-Pries10Central-Extensions-of-Smooth-2-Groups-and-a-Finite-Dimensional-String-2-Group}
is constructed by showing that
$\check{H}^{3}(\mc{U}_{\bullet},U(1)^{\bullet}_{\ensuremath{\op{glob},\sm}})$ classifies
central extensions of finite-dimensional group stacks
\begin{equation*}
[{*}/U(1)]\to [\Gamma] \to G
\end{equation*}
and then taking the isomorphism
\begin{equation*}
\check{H}^{3}(\mc{U}_{\bullet},U(1)^{\bullet}_{\ensuremath{\op{glob},\sm}})\cong H^{3}_{\ensuremath{\op{SM}}}(G,U(1))\cong H^{4}(BG,\ensuremath{\mathbb{Z}})\cong \ensuremath{\mathbb{Z}}
\end{equation*}
(cf.\ Remark \ref{rem:properties}), yielding for each generator a model
for the string group. We will see below that the classes from above are
also generators in the respective cohomology groups and thus represent
the various properties of the string group. For instance, we expect
that the class $[\sigma_{3}]$ will be the characteristic class for
representations of the string group.
\end{example}
The previous construction can be generalized as follows.
\begin{example}\label{ex:path-space_construction}
Let $G$ be a $(n-1)$-connected Lie group and denote by
$C^{\infty}_{*}(\Delta^{k},G)$ the group of based smooth $k$-simplices
in $G$ (the same construction works for locally contractible
topological groups and the continuous $k$-simplices). Then we may
choose for each $1\leq k\leq n$ maps
\begin{equation*}
\alpha_{k}\ensuremath{\nobreak\colon\nobreak} G^{k}\to C^{\infty}_{*}(\Delta^{k},G),
\end{equation*}
such that each $\alpha_{k}$ is smooth on some identity neighborhood and
that
\begin{equation*}
\partial\alpha_{k}(g_{1},\ldots,g_{k})=\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}}(\alpha_{k-1})(g_{1},\ldots,g_{k}).
\end{equation*}
In the latter formula, we interpret $C^{\infty}_{*}(\Delta^{k},G)$ as a
subset of the group $\langle C(\Delta^{k},G)\rangle_{\ensuremath{\mathbb{Z}}}$ of singular
$k$-chains in $G$, which becomes a $G$-module if we let $G$ act by left
multiplication. Since $G$ is $(n-1)$-connected, we can inductively
choose $\alpha_{k}$, starting with $\alpha_{0}\equiv e$.
Now consider the map
\begin{equation*}
\theta_{n}:=\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}}(\alpha_{n})\ensuremath{\nobreak\colon\nobreak} G^{n+1}\to \langle C(\Delta^{n},G)\rangle_{\ensuremath{\mathbb{Z}}}.
\end{equation*}
Since
\begin{equation}
\partial \theta_{n}=\partial \ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}}(\alpha_{n})=\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}}(\partial \alpha_{n})=\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}}^{2}(\alpha_{n-1})=0,
\end{equation}
$\theta_{n}$ takes values in the singular $n$-cycles on $G$ and thus
gives rise to a map
$\theta_{n}\ensuremath{\nobreak\colon\nobreak} G^{n+1}\to H_{n}(G)\cong \pi_{n}(G)$.
Moreover, $\theta_{n}$ is a group cocycle %
and it is locally smooth since $\alpha_{n}$ is so. Of course, this
means here that $\theta_{n}$ even vanishes on some identity
neighborhood (in the product topology). It is straight forward to show
that different choices for $\alpha_{k}$ yield equivalent cocycles.
These are the characteristic cocycles for the $n$-fold extension
\begin{equation}\label{eqn:n-fold_extension}
\pi_{n}(G)\to \wt{\Omega ^{n}G}\to P_{e}\Omega^{n-1}G\to \cdots\to P_{e}\Omega G\to P_{e}G \to G,
\end{equation}
($P_{e}$ denoted pointed paths and $\Omega $ pointed loops) of
topological groups spliced together from the short exact sequences
\begin{equation*}
\pi_{n}(G)\to \wt{\Omega^{n}G}\to \Omega^{n}G
\quad\text{ and }\quad
\Omega^{n}G\to P_{e}\Omega^{n-1}G \to \Omega^{n-1}G\text{ for }n\geq 0.
\end{equation*}
Moreover, the exact sequence
\begin{equation*}
\wt{\Omega ^{n}G}\to \Omega^{n-1}G\to \cdots\to \Omega G\to P_{e}G
\end{equation*}
gives rise to a simplicial topological group $\Pi_{n}(G)$ and we have
canonical morphisms
\begin{equation*}
B^{n}\pi_{n}(G)\to \Pi_{n}(G)\to \ul{G}.
\end{equation*}
Here, $B^{n}\pi_{n}(G)$ is the nerve of the $(n-1)$-groupoid with only
trivial morphisms up to $(n-2)$ and $\pi_{n}(G)$ as $(n-1)$-morphisms
and $\ul{G}$ is the nerve of the groupoid with objects $G$ and only
identity morphisms. Taking the geometric realization
$|\mathinner{\cdot}|$ gives (at least for metrizable $G$) now an
extension of groups in \cat{CGHaus}
\begin{equation*}
K(n,\pi_{n}(G))\simeq |B^{n}\pi_{n}(G)|\to |\Pi_{n}(G)|\to |G|=G,
\end{equation*}
which can be shown to be an $n$-connected cover
$G\langle n\rangle\to G$ with the same methods as in
\cite{BaezCransStevensonSchreiber07From-loop-groups-to-2-groups}.
\end{example}
\begin{remark}\label{rem:X-Mod}
Recall that a crossed module $\mu:M\to N$ is a group homomorphism
together with an action by automorphisms of $N$ on $M$ such that $\mu$
is equivariant and such that for all $m,m'\in M$, the Peiffer identity
\begin{equation*}
\mu(m). m'=mm'm^{-1}
\end{equation*}
holds. Taking into account topology, we suppose that $M$ and $N$ are
groups in \cat{CGHaus}, $\mu$ continuous and $(n,m)\mapsto n.m$ is
continuous. We call a closed subgroup $H$ of a group in \cat{CGHaus}
{\it split} if the multiplication map $G\ktimes H\to G$ defines a
topological $H$-principal bundle (see \cite[Def.\
2.1]{Neeb07Non-abelian-extensions-of-infinite-dimensional-Lie-groups}).
We will throughout use the constructions in the smooth setting from
\cite{Neeb07Non-abelian-extensions-of-infinite-dimensional-Lie-groups},
which carry over to the present topological setting. In this case, we
have in particular that $G\to G/H$ has a continuous local section. In
order to avoid extensions coming from non-complemented topological
vector spaces, we suppose that all our crossed modules are {\it
topologically split}, i.e., we suppose that $\ker(\mu)$ is a split
topological subgroup of $M$, that $\ensuremath{\operatorname{im}}(\mu)$ is a split topological
subgroup of $N$, and that $\mu$ induces a homeomorphism
$M/\ker(\mu)\cong\ensuremath{\operatorname{im}}(\mu)$. In case $M$ and $N$ are (possibly infinite
dimensional) Lie groups, this implies that the corresponding sequence
of Lie algebras is topologically split as a sequence of topological
vector spaces.
\end{remark}
Using the above methods, we can now show the following:
\begin{theorem}
If each $G^{\ptimes[n]}$ is compactly generated, then the set of
equivalence classes of crossed modules with cokernel $G$ and kernel $A$
is in bijection with the cohomology space $H^3_{\ensuremath{ \op{loc},\cont}}(G,A)$.
\end{theorem}
\begin{proof}
It is standard to associate to a (topologically split) crossed module a
locally continuous $3$-cocycle (see \cite[Lem.\
3.6]{Neeb07Non-abelian-extensions-of-infinite-dimensional-Lie-groups}).
To show that this defines an injection of the set of equivalence
classes into $H^3_{\ensuremath{ \op{loc},\cont}}(G,A)$, we use the continuous version of
\cite[Th.\
3.8]{Neeb07Non-abelian-extensions-of-infinite-dimensional-Lie-groups}.
Namely, the existence of an extension $A\to\hat{N}\xrightarrow{q}N$
such that $M$ is $N$-equivariantly isomorphic to $q^{-1}(\ensuremath{\operatorname{im}}(\mu))$
provides two morphisms of four term exact sequences linking
$A\to M\to N\to G$ to the trivial crossed module
$A\to A \xrightarrow{0}G\to G$.
Therefore we focus here on surjectivity, i.e. we reconstruct a crossed
module from a given locally continuous $3$-cocycle. For this, embed the
$G$-module $A$ in a soft $G$-module:
\begin{equation*}
0\to A\to E_G(A)\to B_G(A)\to 0
\end{equation*}
. Observe that $H_{\ensuremath{\op{SM}}}^n(G,E_G(A))\cong H^{n}_{\ensuremath{\op{glob},\cont}}(G,E_{G}(A))$
vanishes for $n\geq 1$ (Proposition
\ref{prop:soft_modules_are_acyclic_for_globcont}). The vanishing shows
now that the connecting homomorphism of the associated long exact
sequence induces an isomorphism
\begin{equation*}
\delta:H_{\ensuremath{ \op{loc},\cont}}^2(G,B_G(A))\cong H_{\ensuremath{ \op{loc},\cont}}^3(G,A),
\end{equation*}
where we have used the isomorphism of $H_{\ensuremath{\op{SM}}}^{n}$ and
$H_{\ensuremath{ \op{loc},\cont}}^{n}$. Thus for the given $3$-cocycle $\gamma$, there exists
a locally continuous $2$-cocycle $\alpha$ with values in $B_G(A)$ such
that $\delta[\alpha]=[\gamma]$. Using $\alpha$, we can form an abelian
extension
\begin{equation*}
0\to B_G(A)\to B_G(A)\times_{\alpha} G\to G\to 1.
\end{equation*}
Now splicing together this abelian extension with the short exact
coefficient sequence
\begin{equation*}
0\to A\to E_G(A)\to B_G(A)\to 0
\end{equation*}
gives rise to a crossed module $\mu:E_G(A)\to B_G(A)\times_{\alpha} G$
which is topologically split in the above sense. Indeed, the
coefficient sequence is topologically split by assumption, and the
abelian extension has a continuous local section by construction.
Finally, the fact that the $3$-class associated to this crossed module
is $[\gamma]$ follows from $\delta[\alpha]=[\gamma]$. Some details for
this kind of construction can also be found in
\cite{Wagemann06On-Lie-algebra-crossed-modules}.
\end{proof}
\begin{remark}
In the case of locally compact second countable $G$ and metrizable $A$
the module $EA$ is metrizable
\cite{BrownMorris77Embeddings-in-contractible-or-compact-objects} and
since $G$ is in particular $\sigma$-compact $C(G,EA)=E_{G}(A)$ is also
metrizable. Thus the above crossed module is a crossed module of
metrizable topological groups. In particular, if we take a generator
$[\alpha]\in H^{3}_{SM}(G,U(1))\cong H^{4}(G,\ensuremath{\mathbb{Z}})\cong \ensuremath{\mathbb{Z}}$ for $G$ a
simple compact 1-connected Lie group, then the crossed module
\begin{equation*}
U(1) \to E_{G}(U(1)) \to B_{G}(U(1))\times_{\alpha}G \to G
\end{equation*}
gives yet another (topological) model for the string 2-group.
\end{remark}
\begin{remark}
(cf.\ \cite[Def.\ 19]{Pries09Smooth-group-cohomology}) The locally
continuous cohomology can be topologized as follows. For an open
identity neighborhood $U\ensuremath{\nobreak\subseteq\nobreak} G^{\ktimes[n]}$ we have the bijection
\begin{equation*}
C_{U}^{n}(G,A):=\{f\ensuremath{\nobreak\colon\nobreak} G^{n}\to A:\left.f\right|_{U}\text{ is continuous}\}\cong
C(U,A)\times A^{G^{n}\setminus U}.
\end{equation*}
This carries a natural topology coming from
$\ensuremath{C_{\boldsymbol{k}}}(U,A)\ktimes A^{G^{n}\setminus U}$, when first endowing
$A^{G^{n}\setminus U}$ with the product topology and then taking the
induced compactly generated topology. If $U\ensuremath{\nobreak\subseteq\nobreak} V$, then the inclusion
$C_{U}^{n}(G,A)\hookrightarrow C^{n}_{V}(G,A)$ is continuous so that
the direct limit
\begin{equation*}
\lim_{\overrightarrow{U\in \mf{U}}} C_{U}^{n}(G,A) \cong C_{\ensuremath{ \op{loc},\cont}}^{n}(G,A)
\end{equation*}
carries a natural topology. The differential $\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}}$ is continuous and
the cohomology groups $H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$ inherits the corresponding
quotient topology.
\end{remark}
\begin{remark}\label{rem:products}
There is a classical way of constructing {\it products} for some of the
cohomology theories which we have considered here. Let us recall these
definitions. The easiest product is the usual {\it cup product} for the
locally continuous (respectively the locally smooth) group cohomology
$H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$ (respectively $H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A)$) \cite[Ch.\
VIII.9]{MacLane63Homology}. In the following, we will stick to
$H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$, noting that all constructions carry over word by
word to $H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A)$ for a Lie group $G$ and a smooth
$G$-module $A$.
Suppose that the two $G$-modules $A$ and $A'$ have a tensor product in
\cat{CGHaus}. The simplicial cup product (see \cite{MacLane63Homology}
equation (9.7) p. 246) in group cohomology yields a homomorphism
\begin{equation*}
\cup:H^p_{\ensuremath{ \op{loc},\cont}}(G,A)\otimes H^q_{\ensuremath{ \op{loc},\cont}}(G,A')\to H^{p+q}_{\ensuremath{ \op{loc},\cont}}(G,A\otimes A'),
\end{equation*}
where the $G$-module $A\otimes A'$ is given the diagonal action.
In case the $G$-module $A$ has its tensor product $A\otimes A$ in
\cat{CGHaus} and has a product, i.e. a homomorphism of $G$-modules
$\alpha:A\otimes A\to A$, we obtain an {\it internal cup product}
\begin{equation*}
\cup:H^p_{\ensuremath{ \op{loc},\cont}}(G,A)\otimes H^q_{\ensuremath{ \op{loc},\cont}}(G,A)\to H^{p+q}_{\ensuremath{ \op{loc},\cont}}(G,A)
\end{equation*}
by post composing with $\alpha$. The product reads then explicitly for
cochains $c\in C^p_{\ensuremath{ \op{loc},\cont}}(G,A)$ and $c'\in C^q_{\ensuremath{ \op{loc},\cont}}(G,A)$
\begin{equation*}
c\cup c'(g_0,\ldots,g_{p+q})=\alpha(c(g_0,\ldots,g_{p}),c'(g_{p},\ldots, g_{p+q})).
\end{equation*}
On the other hand, Segal-Mitchison cohomology $H^{n}_{\ensuremath{\op{SM}}}(G,A)$ is a
(relative) derived functor, and therefore the setting of \cite[Sect.\
XII.10]{MacLane63Homology} is adapted. Observe that our choice of exact
sequences does not satisfy all the demands of a {\it proper class} of
exact sequences \cite[Sect.\ XII.4]{MacLane63Homology} (it does not
satisfy the last two demands) and we neither have automatically enough
proper injectives or projectives. Nevertheless, we have explicit
acyclic resolutions for each module in \cat{CGHaus} which are exact
sequences in our sense. We have the universality property for the
functor $H^{n}_{\ensuremath{\op{SM}}}(G,A)$ \cite[Sect.\ XII.8]{MacLane63Homology} by
Theorem \ref{thm:moores_comparison_theorem}. Therefore we obtain
products for Segal-Mitchison cohomology by universality as in
\cite[Th.\ XII.10.4]{MacLane63Homology} for two $G$-modules $A$ and
$A'$ which have a tensor product in \cat{CGHaus}.
By the uniqueness statement in \cite[Th.\ XII.10.4]{MacLane63Homology},
the isomorphism $H^{n}_{\ensuremath{\op{SM}}}(G,A)\cong H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$ respects
products. Note also that the differentiation homomorphism
$D_{n}\ensuremath{\nobreak\colon\nobreak} H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A)\to H^{n}_{\op{Lie},c}$ that we will turn
to in Remark \ref{rem:connection_to_Lie_algebra_cohomology} is
compatible with products.
\end{remark}
\begin{tabsection}
We now give an explicit description of the purely topological
information contained in a locally continuous cohomology class. If $G$
is a connected topological group and $A$ is a topological $G$-module,
then there is an exact sequence
\begin{equation}\label{eqn:tau2}
0 \to H^{2}_{\ensuremath{ \op{glob},\op{top}}}(G,A)\to H^{2}_{\ensuremath{ \op{loc},\op{top}}}(G,A)\xrightarrow{\tau_{2}} \check{H}^{1}(G,\underline{A})
\end{equation}
\cite[Sect.\
2]{Wockel09Non-integral-central-extensions-of-loop-groups}, where
$\tau_2$ assigns to an abelian extension $A\to \hat{G}\to G$ the
characteristic class of the underlying principal $A$-bundle. By
definition, we have that $\ensuremath{\operatorname{im}}(\tau_2)$ are those classes in
$\check{H}^{1}(G,\underline{A})$ whose associated principal $A$-bundles
admit a compatible group structure. These are precisely the bundles
$A\to P\to G$ for which the bundle
$\ensuremath{\operatorname{pr}}_{1}^{*}(P)\otimes \mu^{*}(P)\otimes \ensuremath{\operatorname{pr}}_{2}^{*}(P)$ on $G\times G$
is trivial \cite[Prop.\ VII.1.3.5]{72SGA-7}.
We will now establish a similar behavior of the map $\tau_{n}$ for
arbitrary $n$.
\end{tabsection}
\begin{proposition}\label{prop:tau_is_delta_functor}
Let $G$ be a connected topological group and $A$ be a topological
$G$-module. Suppose that the cocycle $f\in C^{n}_{\ensuremath{ \op{loc},\op{top}}}(G,A)$ is
continuous on the identity neighborhood $U\ensuremath{\nobreak\subseteq\nobreak} G^{n}$ and let $V\ensuremath{\nobreak\subseteq\nobreak} G$ be open
such that $e\in V$ and $V^{2}\times \ldots\times V^{2}\ensuremath{\nobreak\subseteq\nobreak} U$. Then the
map
\begin{multline*}
\tau(f)_{g_{1},\ldots,g_{n}}\ensuremath{\nobreak\colon\nobreak} g_{1}V\cap \ldots\cap g_{n}V\to A,\quad
x\mapsto
g_{1}^{~}. f(g_{1}^{-1}g_{2}^{~},\ldots,g_{n-1}^{-1}g_{n}^{~},g_{n}^{-1}x)
-(-1)^{n} f(g_{1}^{~},g_{1}^{-1}g_{2}^{~},\ldots,g_{n-1}^{-1}g_{n}^{~})
\end{multline*}
defines a continuous \v{C}ech $(n-1)$-cocycle on the open cover
$(gV)_{g\in G}$. Moreover, this induces a well-defined map
\begin{equation*}
\tau_{n}\ensuremath{\nobreak\colon\nobreak} H^{n}_{\ensuremath{ \op{loc},\op{top}}}(G,A)\to \check{H}^{n-1}(G,\underline{A}),\quad
[f]\mapsto [\tau(f)]
\end{equation*}
which is a morphism of $\delta$-functors.
\end{proposition}
\begin{proof}
We first note that $\tau(f)_{g_{1},\ldots,g_{n}}$ depends continuously
on $x$. Indeed, the first term depends continuously on $x$ since
$g_{1}V\cap \ldots\cap g_{n}V\neq \emptyset$ implies that
$g_{k-1}^{-1}g_{k}^{~}\in V^{2}$ and $f$ is continuous on
$V^{2}\times\ldots\times V^{2}$ by assumption. Since the second term
does not depend on $x$, this shows continuity. Now the cocycle identity
for $f$, evaluated on
$(g_{1}^{~},g_{1}^{-1}g_{2}^{~},\ldots,g_{n-1}^{-1}g_{n}^{~},g_{n}^{-1}x)$,
shows that $\tau(f)_{g_{1},\ldots,g_{n}}(x)$ may also be written as
$(\check \delta (\kappa(f)))_{g_{1},\ldots,g_{n}}(x) $, where
\begin{equation*}
\kappa(f)_{g_{2},\ldots,g_{n}}(x):=f(g_{2}^{~},g_{2}^{-1}g_{3}^{~},\ldots,g_{n}^{-1}x).
\end{equation*}
Note that $\kappa(f)_{g_{2},\ldots,g_{n}}$ does not depend continuously
on $x$ and thus the above assertion does not imply that $\tau(f)$ is a
coboundary. However, $\check{\delta}^{2}=0$ now implies that $\tau(f)$
is a cocycle.
It is clear that the class $[\tau(f)]$ in
$\check{H}^{n-1}(G,\underline{A})$ does not depend on the choice of $V$
since another such choice $V'$ yields a cocycle given by the same
formula on the refined cover $(g(V\cap V'))_{g\in G}$. Moreover, if $f$
is a coboundary, i.e., $f=\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} b$ for $b\in C^{n-1}_{\ensuremath{ \op{loc},\cont}}(G,A)$
(where we assume w.l.o.g. that $b$ is also continuous on
$V^{2}\times \ldots\times V^{2}$), then we set
\begin{equation*}
\rho(b)_{g_{1},\ldots,g_{n-1}}(x):=
g_{1}. b(g_{1}^{-1}g_{2}^{~},\ldots,g_{n-1}^{-1}x)+(-1)^{n}
b(g_{1}^{~},g_{1}^{-1}g_{2},\ldots,g_{n-2}^{-1}g_{n-1}^{~}).
\end{equation*}
As above, this defines a continuous function on
$g_{1}V\cap \ldots\cap g_{n-1}V\neq \emptyset$ and thus a \v{C}ech
cochain. A direct calculation shows that
$\check{\delta}(\rho(f))=\tau(f)$ and thus that the class $[\tau(f)]$
only depends on the class of $f$.
We now turn to the second claim, for which we have to check that for
each exact sequence $A\hookrightarrow B\xrightarrow{q} C$ of
topological $G$-modules the diagram
\begin{equation*}
\xymatrix{
H^{n}_{\ensuremath{ \op{loc},\cont}}(G,C)
\ar[r]^{\delta_{n}}\ar[d]^{\tau_{n}}&
H^{n+1}_{\ensuremath{ \op{loc},\cont}}(G,A)
\ar[d]^{\tau_{n+1}}\\
\check{H}^{n-1}(G,\underline{C})
\ar[r]^{{\delta}_{n-1}}&
\check{H}^{n}(G,\underline{A})
}
\end{equation*}
commutes. For this, we recall that $\delta_{n}$ is constructed by
choosing for $[f]\in H^{n}_{\ensuremath{ \op{loc},\cont}}(G,C)$ a lift
$\wt{f}\ensuremath{\nobreak\colon\nobreak} G^{n}\to B$ and then setting
$\delta_{n}([f])=[\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \wt{f}]$. After possibly shrinking $V$, we can
assume that $f$ is continuous on $V^{2}\times \ldots\times V^{2}$ ($n$
factors) and that $\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \wt{f}$ is continuous on
$V^{2}\times \ldots\times V^{2}$ ($n+1$ factors).
Since $q$ is a homomorphism, $\wt{f}$ also gives rise to lifts
\begin{equation*}
\wt{\tau(f)}_{g_{1},\ldots,g_{n}}(x):=
g_{1}^{~}. \wt{f}(g_{1}^{-1}g_{2}^{~},\ldots,g_{n-1}^{-1}g_{n}^{~},g_{n}^{-1}x)
-(-1)^{n} \wt{f}(g_{1}^{~},g_{1}^{-1}g_{2}^{~},\ldots,g_{n-1}^{-1}g_{n}^{~})
\end{equation*}
of $\tau(f)_{g_{1},\ldots,g_{n}}$, which obviously depends continuously
on $x$ on $g_{1}V\cap \ldots \cap g_{n}V$. Thus we have that
${\delta}_{n-1}(\tau_{n}([f]))$ is represented by the \v{C}ech cocycle
\begin{equation*}
\check{\delta}(\wt{\tau(f)})_{g_{0},\ldots,g_{n}}.
\end{equation*}
On the other hand, $\tau_{n+1}(\delta_{n}([f]))$ is represented by
$\tau(\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \widetilde{f})_{g_{0},\ldots,g_{n}}$, whose value on $x$ is
given by
\begin{gather*}
g_{0}^{~}. \ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \widetilde{f}(g_{0}^{-1}g_{1}^{~},\ldots,g_{n-1}^{-1}g_{n}^{~},g_{n}^{-1}x)
-(-1)^{n+1} \ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \widetilde{f}(g_{0}^{~},g_{0}^{-1}g_{1}^{~},\ldots,g_{n-1}^{-1}g_{n}^{~})=\\
g_{0}^{~}.\Big[g_{0}^{-1}g_{1}^{~}.\widetilde{f}(g_{1}^{-1}g_{2}^{~},\ldots,g_{n}^{-1}x)\pm\ldots
\dashuline{+(-1)^{k}\widetilde{f}(g_{0}^{-1}g_{1}^{~},\ldots,g_{k-1}^{-1}g_{k+1}^{~},\ldots,g_{n}^{-1}x)}\pm\ldots\\
+\underline{(-1)^{n+1} \widetilde{f}(g_{0}^{-1}g_{1}^{~},\ldots,g_{n-1}^{-1}g_{n}^{~})}\Big]
-(-1)^{n+1}\Big[
g_{0}^{~}.\underline{\widetilde{f}(g_{0}^{-1}g_{1}^{~},\ldots,g_{n-1}^{-1}g_{n}^{~})}
\pm\ldots\\ \dashuline{-(-1)^{k}\widetilde{f}(g_{0}^{~},g_{0}^{-1}g_{1}^{~},\ldots,g_{k-1}^{-1}g_{k+1}^{~},\ldots,g_{n-1}^{-1}g_{n}^{~})}\pm\ldots
+(-1)^{n+1}\widetilde{f}(g_{0}^{~},g_{0}^{-1}g_{1}^{~},\ldots,g_{n-2}^{-1}g_{n-1}^{~})
\Big]
\end{gather*}
The underlined terms cancel and the sum of the dashed terms gives
$(-1)^{k}\wt{\tau(f)}_{g_{0},\ldots,\widehat{g_{k}},\ldots,g_{n}}(x)$.
This shows that
\begin{equation*}
\check{\delta}(\wt{\tau(f)})_{g_{1},\ldots,g_{n}}(x) = \tau(\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \widetilde{f})_{g_{1},\ldots,g_{n}}(x).
\end{equation*}
\end{proof}
\begin{tabsection}
We will now identify the map $\tau$ with one of the edge homomorphisms
in the spectral sequence associated to $H^{n}_{\ensuremath{\op{simp},\cont}}(G,A)$.
\end{tabsection}
\begin{proposition}\label{prop:tau_is_edge}
For $n\geq 1$ the edge homomorphism of the spectral sequence
\eqref{eqn:spectral_sequence} induces a homomorphism
\begin{equation*}
\op{edge}_{n+1}\ensuremath{\nobreak\colon\nobreak} H^{1+n}_{\ensuremath{\op{simp},\cont}}(G,A)\to H^{1+n}_{\ensuremath{\op{simp},\cont}}(G,A)/\mathcal{F}^{2}H^{2+n}_{\ensuremath{\op{simp},\cont}}(G,A)\cong E_{\infty}^{1,n}\to E_{1}^{1,n}\cong H^{n}_{\ensuremath{\op{Sh}}}(G,\underline{A}),
\end{equation*}
where $\mathcal{F}$ denotes the standard column filtration (cf.\ Remark
\ref{rem:double_complex_for_spectral_sequence}). If, moreover,
$G^{\ptimes[n]}$ is compactly generated, paracompact and admits good
covers for all $n\geq 1$ and $A$ is a topological $G$-module, then the
diagram
\begin{equation}\label{eqn:edge_morphism_commuting_square}
\vcenter{\xymatrix{
H^{n+1}_{\ensuremath{\op{simp},\cont}}(G,A)
\ar[d]^{\cong}\ar[rr]^(.54){\op{edge}_{n+1}}&&
H^{n}_{\ensuremath{\op{Sh}}}(G,\underline{A})
\ar[d]^{\cong}\\
H^{n+1}_{\ensuremath{ \op{loc},\cont}}(G,A)
\ar[rr]^{\tau_{n+1}}&&
\check{H}^{n}(G,\underline{A})
}}
\end{equation}
commutes.
\end{proposition}
\begin{proof}
We first note that $H^{n}_{\ensuremath{ \op{loc},\op{top}}}(G,A)=H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$ under the
above assumptions. Since $BG_{0}=\op{pt}$, we have
$E_{1}^{0,q}=H^{q}_{\ensuremath{\op{Sh}}}(\op{pt},\underline{A})=0$ for all $q\geq 1$
and thus the edge homomorphism $E_{\infty}^{1,p}\to E_{1}^{1,p}$. Since
we have $\mathcal{F}^{p}H^{p+q}_{\ensuremath{\op{simp},\cont}}(G,A)=H^{p+q}_{\ensuremath{\op{simp},\cont}}(G,A)$
for $p=0,1$, $q\geq 1$ this gives the desired form of
$\op{edge}_{q+1}$. Since this construction commutes with the connecting
homomorphisms, it is a morphism of $\delta$-functors. Moreover, the
isomorphism
$H^{n}_{\ensuremath{\op{Sh}}}(G,\underline{\mathinner{\:\cdot\:}})\cong \check{H}^{n}(G,\underline{\mathinner{\:\cdot\:}})$
is even an isomorphism of $\delta$-functors. By virtue of the
uniqueness assertion for morphisms of $\delta$-functors from Theorem
\ref{thm:moores_comparison_theorem}, it thus remains to verify that
that \eqref{eqn:edge_morphism_commuting_square} commutes for $n=1$.
The construction from Remark
\ref{rem:morphism_from_locally_continuous_to_Cech_cohomology} gives an
isomorphism
$H^{2}_{\ensuremath{ \op{loc},\op{top}}}(G,A)\cong \check{H}^{2}(\mc{U}_{\bullet},A_{\ensuremath{ \op{loc},\cont}}^{\bullet})$,
where $\mc{U}_{\bullet}$ is a good cover of $BG_{\bullet}$ chosen such
that $\mc{U}_{k}$ refines the covers of $G^{k}$ constructed there.
Since this construction commutes with the connecting homomorphisms, the
isomorphism
$H^{2}_{\ensuremath{ \op{loc},\op{top}}}(G,A)\cong \check{H}^{2}(\mc{U}_{\bullet},A_{\ensuremath{ \op{loc},\cont}}^{\bullet})$
is indeed the one from the unique isomorphism of the corresponding
$\delta$-functors. Now $\tau_{2}$ coincides with the morphism
$H^{2}_{\ensuremath{ \op{loc},\op{top}}}(G,A)\cong \check{H}^{2}(\mc{U}_{\bullet},A_{\ensuremath{ \op{loc},\cont}}^{\bullet})\to \check{H}^{1}(G,A)$,
given by projecting the cocycle $(\mu,\tau)$ in the total complex of
$\check{C}^{p,q}(\mc{U}_{\bullet},E^{\bullet})$ to the \v{C}ech cocycle
$\tau$. Since this is just the corresponding edge homomorphism, the
diagram \eqref{eqn:edge_morphism_commuting_square} commutes for $n=1$.
\end{proof}
\begin{remark}
In case the action of $G$ on $A$ is trivial, Proposition
\ref{prop:tau_is_edge} also holds for $n=0$. Indeed, then the
differential $A\cong E_{1}^{0,0}\to E_{1}^{1,0}\cong C^{\infty}(G,A)$,
which is given by assigning the principal crossed homomorphism to an
element of $A$, vanishes. This shows commutativity of
\eqref{eqn:edge_morphism_commuting_square} also for $n=0$.
\end{remark}
\begin{remark}
The other edge homomorphism is induced from the identification
$ C^{n}_{\ensuremath{\op{glob},\cont}}(G,A)\cong H^{0}_{\ensuremath{\op{Sh}}}(G^{n},A)\cong E_{1}^{n,0}$,
which shows $E_{2}^{n,0}\cong H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)$. It coincides with
the morphism $H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)\to H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$ induced by the
inclusion $C^{n}_{\ensuremath{\op{glob},\cont}}(G,A)\hookrightarrow C^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$ (cf.\
also \cite[Remarks in \S3]{Segal70Cohomology-of-topological-groups}).
\end{remark}
The following is a generalization of \eqref{eqn:tau2} in case $A$ is
discrete.
\begin{corollary}\label{cor:injectivity_of_tau}
If $n\geq 1$, $G$ is $(n-1)$-connected and $A$ is a discrete
$G$-module, then
$\tau_{n+1}\ensuremath{\nobreak\colon\nobreak} H_{\ensuremath{ \op{loc},\cont}}^{n+1}(G,A)\to \check{H}^{n}(G,\underline{A})$
is injective if $G^{\ptimes[n]}$ is compactly generated, paracompact
and admits good covers for all $n\geq 1$.
\end{corollary}
\begin{proof}
If $G$ is $(n-1)$-connected, and $A$ is discrete, then $E_{1}^{p,q}$ of
the spectral sequence \eqref{eqn:spectral_sequence} vanishes if
$q\leq n-1$. Thus
$E_{\infty}^{1,n-1}=\ker(d_{1}^{1,n-1})\ensuremath{\nobreak\subseteq\nobreak} E_{1}^{1,n-1}\cong \check{H}^{n-1}(G,\underline{A})$
and $\op{edge}_{n}$ coincides with the embedding
\begin{equation*}
H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)\cong H^{n}_{\ensuremath{\op{simp},\cont}}(G,A)\cong E_{\infty}^{1,n-1}\hookrightarrow E_{1}^{1,n-1}\cong \check{H}^{n-1}(G,A).
\end{equation*}
\end{proof}
\begin{remark}\label{rem:sting_cocycle_2}
An explicit analysis of the differentials of the spectral sequence
\eqref{eqn:spectral_sequence} shows that for discrete $A$ with trivial
$G$-action and $(n-1)$-connected $G$ the image of
$\tau_{n+1}\ensuremath{\nobreak\colon\nobreak} H^{n+1}_{\ensuremath{ \op{loc},\cont}}(G,A)\to \check{H}^{n}(G,\underline{A})$
consists of those cohomology classes
$c\in\check{H}^{n}(G,\underline{A})$ which are \emph{primitive}, i.e.,
for which
\begin{equation*}
\ensuremath{\operatorname{pr}}_{1}^{*}c\otimes \mu^{*}c \otimes \ensuremath{\operatorname{pr}}_{2}^{*}c=0.
\end{equation*}
Since the primitive elements generate the rational cohomology of a
compact Lie group $G$ \cite[p.\ 167, Thm.\
IV]{GreubHalperinVanstone73Connections-curvature-and-cohomology.-Vol.-II:-Lie-groups-principal-bundles-and-characteristic-classes},
it follows that all non-torsion elements in the lowest cohomology
degree are primitive in this case.
In particular, if $G$ is a compact, simple and $1$-connected (thus
automatically $2$-connected), the generator of
$\check{H}^{2}(G,\underline{U(1)})\cong \check{H}^{3}(G,\underline{\ensuremath{\mathbb{Z}}})\cong \ensuremath{\mathbb{Z}}$
is primitive and thus
$\tau_{4}\ensuremath{\nobreak\colon\nobreak} H^{4}_{\ensuremath{ \op{loc},\cont}}(G,\ensuremath{\mathbb{Z}})\to \check{H}^{3}(G,\underline{\ensuremath{\mathbb{Z}}})$
is an isomorphism. Since the diagram
\begin{equation*}
\xymatrix{
H^{4}_{\ensuremath{ \op{loc},\cont}}(G,\ensuremath{\mathbb{Z}})\ar[r]^{\tau_{4}^{\ensuremath{\mathbb{Z}}}}\ar[d]^{\cong}&\check{H}^{3}(G,\underline{\ensuremath{\mathbb{Z}}})\ar[d]^{\cong}\\
H^{3}_{\ensuremath{ \op{loc},\cont}}(G,U(1)) \ar[r]^{\tau_{3}^{U(1)}}&\check{H}^{2}(G,\underline{U(1)})
}
\end{equation*}
commutes by Proposition \ref{prop:tau_is_delta_functor}, this shows
that $\tau_{3}^{U(1)}$ is also an isomorphism. Since the string class
$[\sigma_{3}]$ from Example \ref{ex:string_cocycles} maps under
$\tau_{3}$ to a generator
\cite{BrylinskiMcLaughlin93A-geometric-construction-of-the-first-Pontryagin-class},
this shows that $[\sigma_{3}]$ gives indeed a generator of
$H^{3}_{\ensuremath{ \op{loc},\cont}}(G,U(1))$, and $[\theta_{3}]$ gives a generator of
$H^{4}_{\ensuremath{ \op{loc},\cont}}(G,\ensuremath{\mathbb{Z}})$.
\end{remark}
\begin{remark}\label{rem:connection_to_Lie_algebra_cohomology}
One reason for the importance of \emph{locally smooth} cohomology is
that it allows for a direct connection to Lie algebra cohomology and
thus may be computable in algebraic terms. This relation is induced by
the differentiation homomorphism
\begin{equation*}
H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A)\xrightarrow{D_{n}} H^{n}_{\op{Lie},c}(\ensuremath{\mathfrak{g}},\mf{a}),
\end{equation*}
where $H^{n}_{\op{Lie},c}$ denotes the continuous Lie algebra
cohomology, $\ensuremath{\mathfrak{g}}$ is the Lie algebra of $G$ and $\mf{a}$ the induced
infinitesimal topological $\ensuremath{\mathfrak{g}}$-module (cf.\ \cite[Thm.\
V.2.6]{Neeb06Towards-a-Lie-theory-of-locally-convex-groups}).
Suppose $G$ is finite-dimensional. Then the kernel of $D_{n}$ consists
of those cohomology classes $[f]$ that are represented by cocycles
vanishing on some neighborhood of the identity. For $\Gamma=\{0\}$ this
follows directly from
\cite{Swierczkowski71Cohomology-of-group-germs-and-Lie-algebras}, where
it is shown that the differentiation homomorphism from the cohomology
of \emph{locally defined} smooth group cochains to Lie algebra
cohomology is an isomorphism. Thus if $[f]\in\ker(D_{n})$, then there
exists a locally defined smooth map $b$ with $\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} b-f=0$ wherever
defined. Since we can extend $b$ arbitrarily to a locally smooth
cochain this shows the claim. In the case of non-trivial $\Gamma$ one
may deduce the claim from the case of trivial $\Gamma$ since $\mf{a}$
and $A=\mf{a}/\Gamma$ are isomorphic as local Lie groups so that
$A$-valued local cochains can always be lifted to $\mf{a}$-valued local
cochains. If $A^{\delta}$ denotes $A$ with the discrete topology, then
the isomorphism
$H^{n}_{\pi_{1}(BG)}(BG,\underline{A^{\delta}})\cong H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A^{\delta})$
from Remark \ref{rem:properties} induces an exact sequence
\begin{equation*}
H^{n}_{\pi_{1}(BG)}(BG,\underline{A^{\delta}})\to H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A)\xrightarrow{D_{n}} H^{n}_{\op{Lie},c}(\ensuremath{\mathfrak{g}},\mf{a})
\end{equation*}
(see also
\cite{Neeb02Central-extensions-of-infinite-dimensional-Lie-groups,Neeb04Abelian-extensions-of-infinite-dimensional-Lie-groups}
for an exhaustive treatment of $D_{2}$ for general infinite-dimensional
$G$). From the van Est spectral sequence
\cite{Est58A-generalization-of-the-Cartan-Leray-spectral-sequence.-I-II}
it follows that if $G$ ist $n$-connected (more general $G$ may be
infinite-dimensional with split de Rham complex
\cite{Beggs87The-de-Rham-complex-on-infinite-dimensional-manifolds}),
then differentiation induces an isomorphism
\begin{equation*}
H^{n}_{\ensuremath{\op{glob},\sm}}(G,\mf{a})\to H^{n}_{\op{Lie},c}(\ensuremath{\mathfrak{g}},\mf{a}).
\end{equation*}
For $G$ an $(n-1)$-connected Lie group this is not true any more, for
instance the Lie algebra $3$-cocycle
$\langle [\mathinner{\cdot},\mathinner{\cdot}],\mathinner{\cdot}\rangle$
from Example \ref{ex:string_cocycles} is non-trivial but
$H^{3}_{\ensuremath{\op{glob},\sm}}(G,\ensuremath{\mathbb{R}})$ vanishes by \cite[Thm.\
1]{Est55On-the-algebraic-cohomology-concepts-in-Lie-groups.-I-II} for
compact and connected $G$.
However, there exists integrating cocycles when considering
\emph{locally smooth} cohomology: If $G$ is an $(n-1)$-connected
finite-dimensional Lie group and $A\cong{\mathfrak a}/\Gamma$ is a
finite-dimensional smooth module for $\mf{a}$ a finite-dimensional
$G$-module and $\Gamma$ a discrete submodule, then
$D_{n}\ensuremath{\nobreak\colon\nobreak} H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A)\to H^{n}_{\op{Lie},c}(\mathfrak{g},\mathfrak{a})$
is injective and its image consists of those cohomology classes
$[\omega]$ whose associated period homomorphism $\ensuremath{\operatorname{per}}_{[\omega]}$
\cite[Def.\
V.2.12]{Neeb06Towards-a-Lie-theory-of-locally-convex-groups} has image
in $\Gamma$. In fact, $H^{n}_{\ensuremath{ \op{loc},\sm}}(G,A^{\delta})$ vanishes (by Corollary
\ref{cor:injectivity_of_tau}), and thus $D_{n}$ is injective. Surjectivity of
$D_{n}$ may be seen from the following standard integration argument.
If $\omega$ is a Lie algebra $n$-cocycle, then the associated
left-invariant $n$-form $\omega^{l}$ is closed \cite[Lem.\
3.10]{Neeb02Central-extensions-of-infinite-dimensional-Lie-groups}. If
we make the choices of $\alpha_{k}$ for $1\leq k\leq n$ as in Example
\ref{ex:path-space_construction}, then
\begin{equation*}
\Omega(g_{1},\ldots,g_{n}):= \int_{\alpha_{n}(g_{1},\ldots,g_{n})}\omega^{l}
\end{equation*}
defines
\begin{itemize}
\item a locally smooth group cochain on $G$, since $\alpha_{n}$
depends smoothly on $(g_{1},\ldots,g_{n})$ on an identity
neighborhood and the integral depends smoothly
$\alpha_{n}(g_{1},\ldots,g_{n})$.
\item a group cocycle, since
\begin{equation*}
\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \Omega (g_{0},\ldots,g_{n})=\int_{\ensuremath{\op{\mathtt{d}_{\mathrm{gp}}}} \alpha (g_{0},\ldots,g_{n})}\omega^{l}\in \ensuremath{\operatorname{per}}_{\omega}(\pi_{n}(G))\ensuremath{\nobreak\subseteq\nobreak} \Gamma.
\end{equation*}
\end{itemize}
A straight forward calculation, similar to the ones in
\cite{Neeb02Central-extensions-of-infinite-dimensional-Lie-groups} or
\cite{Neeb04Abelian-extensions-of-infinite-dimensional-Lie-groups} now
shows that $D_{n}([\Omega])=[\omega]$. We expect that large parts of
this remark can be generalized to arbitrary infinite-dimensional $G$
with techniques similar to those of
\cite{Neeb02Central-extensions-of-infinite-dimensional-Lie-groups,Neeb02Central-extensions-of-infinite-dimensional-Lie-groups}.
\end{remark}
\section{\texorpdfstring{$\delta$}{δ}-Functors}
\label{sect:universal_delta_functors}
\begin{tabsection}
In this section we recall the basic setting of (cohomological)
$\delta$-functors (sometimes also called ``satellites''), as for
instance exposed in \cite[Chap.\
3]{CartanEilenberg56Homological-algebra}, \cite[Sect.\
III.5]{Buchsbaum55Exact-categories-and-duality}, \cite[Sect.\
2]{Grothendieck57Sur-quelques-points-dalgebre-homologique} or
\cite[Sect.\
4]{Moore76Group-extensions-and-cohomology-for-locally-compact-groups.-III}.
It will be important that the arguments work in more general categories
than abelian ones, the only thing one needs is a notion of short exact
sequence.
\end{tabsection}
\begin{definition}
A \emph{category with short exact sequences} is a category $\cat{C}$,
together with a distinguished class of composable morphisms
$A\to B\to C$. The latter are called a short exact sequence. A
morphisms between $A\to B\to C$ and $A'\to B'\to C'$ consists of
morphisms $A\to A'$, $B\to B'$ and $C\to C'$ such that the diagram
\begin{equation*}
\xymatrix{A\ar[r]\ar[d]&B\ar[r]\ar[d]&C\ar[d]\\A'\ar[r]&B'\ar[r]&C'}
\end{equation*}
commutes.
A (cohomological) $\delta$-functor on a category with short exact
sequences is a sequence of functors
\begin{equation*}
(H^{n}\ensuremath{\nobreak\colon\nobreak} \cat{C}\to\cat{Ab})_{n\in\ensuremath{\mathbb{N}}_{0}}
\end{equation*}
such that for each exact $A\to B\to C$ there exist morphisms
$\delta_{n}\ensuremath{\nobreak\colon\nobreak} H^{n}(C)\to H^{n+1}(A)$ turning
\begin{equation*}
H^{0}(A)\to H^{0}(B)\to H^{0}(C)\xrightarrow{\delta_{0}} \cdots \xrightarrow{\delta_{n-1}}H^{n}(A)\to H^{n}(B)
\to H^{n}(C)\xrightarrow{\delta_{n}} \cdots
\end{equation*}
into an exact sequence\footnote{Note that we do not require $H^{0}$ to
be left exact.} and that for each morphism of exact sequences the
diagram
\begin{equation}\label{eqn:delta_functor}
\xymatrix{
H^{n}(C)\ar[r]^{\delta_{n}}\ar[d]&H^{n+1}(A)\ar[d]\\
H^{n}(C')\ar[r]^{\delta_{n}}&H^{n+1}(A')}
\end{equation}
commutes. A morphisms of $\delta$-functors from $(H^{n})_{n\in\ensuremath{\mathbb{N}}_{0}}$
to $(G^{n})_{n\in\ensuremath{\mathbb{N}}_{0}}$ is a sequence of natural transformations
$(\varphi^{n}\ensuremath{\nobreak\colon\nobreak} H^{n}\Rightarrow G^{n})_{n\in\ensuremath{\mathbb{N}}_{0}}$ such that for
each short exact $A\to B\to C$ the diagram
\begin{equation}\label{eqn:morphism_of_delta_functors}
\xymatrix{
H^{n}(C)\ar[r]^{\delta_{n}}\ar[d]^{\varphi^{n}_{C}}&
H^{n+1}(A)\ar[d]^{\varphi_{A}^{n+1}}\\
G^{n}(C)\ar[r]^{\delta_{n}}&G^{n+1}(A)}
\end{equation}
commutes. An isomorphism of $\delta$-functors is then a morphism for
which all $\varphi^{n}$ are natural isomorphisms of functors.
\end{definition}
\begin{theorem}\label{thm:moores_comparison_theorem}
Let $\cat{C}$ be a category with short exact sequences. Let
$F\ensuremath{\nobreak\colon\nobreak} \cat{C}\to\cat{Ab}$, $I\ensuremath{\nobreak\colon\nobreak} \cat{C}\to\cat{C}$ and
$U\ensuremath{\nobreak\colon\nobreak} \cat{C}\to\cat{C}$ be functors, $\iota_{A}\ensuremath{\nobreak\colon\nobreak} A\to I(A)$ and
$\zeta_{A}\ensuremath{\nobreak\colon\nobreak} I(A)\to U(A)$ be natural such that
$A\xrightarrow{\iota_{A}} I(A)\xrightarrow{\zeta_{A}} U(A)$ is short
sequence and let $(H_{n})_{n\in\ensuremath{\mathbb{N}}_{0}}$ and
$(G_{n})_{n\in\ensuremath{\mathbb{N}}_{0}}$ be two $\delta$-functors.
\begin{enumerate}
\renewcommand{\labelenumi}{\theenumi}
\renewcommand{\theenumi}{\arabic{enumi}.}
\item \label{item:moores_comparison_theorem_1} If $\alpha\ensuremath{\nobreak\colon\nobreak} H^{0}\Rightarrow G^{0}$ is a natural
transformation and $H^{n}(I(A))=0$ for all $A$ and all
$1\leq n\leq m$, then there exist natural transformations
$\varphi^{n}\ensuremath{\nobreak\colon\nobreak} H^{n}\Rightarrow G^{n}$, uniquely determined
by requiring that $\varphi^{0}=\alpha$ and that
\begin{equation*}
\xymatrix{
H^{n}(U(A)) \ar[r]^{\delta_{n}}\ar[d]_{\varphi^{n}_{U(A)}} & H^{n+1}(A) \ar[d]^{\varphi^{n+1}_{A}}\\
G^{n}(U(A)) \ar[r]^{\overline{\delta}_{n}} & G^{n+1}(A)
}
\end{equation*}
commutes for $0\leq n< m$. In particular, if
$H^{n}(I(A))=0=G^{n}(I(A))$ for all $n\geq 0$, then
$\varphi^{n}$ is an isomorphism of functors for all $n\in\ensuremath{\mathbb{N}}$ if
and only if it is so for $n=0$.
\item \label{item:moores_comparison_theorem_2} Assume, moreover, that for any short exact sequence
$A\xrightarrow{f} B\to C$ the morphism $A\to I(B)$ can be
completed to a short exact sequence $A\to I(B)\to Q_{f}$ such
that there exist morphisms $U(A)\xrightarrow{\beta_{f}}Q_{f}$
and $C\xrightarrow{\gamma_{f}} Q_{f}$ making
\begin{equation}\label{eqn:moores_comparison_theorem}
\vcenter{ \xymatrix{
A\ar[r]^{\iota_{A}}\ar@{=}[d] & I(A) \ar[r]^{\zeta_{A}}\ar[d]^{I(f)} & U(A) \ar[d]^{\beta_{f}}\\
A\ar[r] & I(B)\ar[r] & Q_{f}
} }\quad\text{ and }\quad
\vcenter{ \xymatrix{
A\ar[r]^{f}\ar@{=}[d] & B \ar[r]\ar[d]^{\iota_{B}} & C \ar[d]^{\gamma_{f}}\\
A\ar[r] & I(B)\ar[r] & Q_{f}
} }
\end{equation}
commute. Then the diagram
\begin{equation*}
\xymatrix{
H^{n}(C) \ar[r]^{\delta_{n}}\ar[d]_{\varphi^{n}_{C}} & H^{n+1}(A) \ar[d]^{\varphi^{k}_{A}}\\
G^{n}(C) \ar[r]^{\overline{\delta}_{n}} & G^{n+1}(A)
}
\end{equation*}
also commutes for $0\leq m<m$. In particular, if $H^{n}(I(A))=0$
for all $A$ and all $n\geq 1$, then $(\varphi^{n})_{n\in\ensuremath{\mathbb{N}}_{0}}$
is a morphism of $\delta$-functors.
\end{enumerate}
\end{theorem}
\begin{proof}
The proof of \cite[Thm.\
II.5.1]{Buchsbaum55Exact-categories-and-duality} (cf.\ also \cite[Thm.\
2]{Moore76Group-extensions-and-cohomology-for-locally-compact-groups.-III})
carries over to this more general setting. The claims are shown by
induction, so we assume that $\varphi ^{n}$ is constructed up to
$n\geq 0$. Then we consider for arbitrary $A$ the diagram (recall that
$H^{n+1}(I(A))=0$)
\begin{equation*}
\vcenter{\xymatrix{
H^{n}(I(A))
\ar[r] \ar[d]^{\varphi^{n}_{I(A)}} &
H^{n}(U(A))
\ar[r]^{\delta_{n}} \ar[d]^{\varphi^{n}_{U(A)}} &
H^{n+1}(A)
\ar[r] &
0\\
G^{n}(I(A))
\ar[r] &
G^{n}(U(A))
\ar[r]^{\overline{\delta}_{n}} &
G^{n+1}(A)
}},
\end{equation*}
which shows that there is a unique
$\varphi ^{n+1}_{A}\ensuremath{\nobreak\colon\nobreak} H^{n+1}(A)\to G^{n+1}(A)$ making this diagram
commute. To check naturality take $f\ensuremath{\nobreak\colon\nobreak} A\to B$. By the construction
of $\varphi_{A}^{n+1}$, the induction hypothesis and the construction
of $\varphi_{B}^{n+1}$ the diagrams
\begin{equation*}
\vcenter{ \xymatrix{
H^{n}(U(A))
\ar[d]^{\varphi_{U(A)}^{n}}\ar[r]^{\delta_{n}^{U(A)}} &
H^{n+1}(A)
\ar[d]^{\varphi_{A}^{n+1}}\\
G^{n}(U(A))
\ar[r]^{\overline{\delta}_{n}^{U(A)}} &
G^{n+1}(A)
}}
\quad\text{ and }\quad
\vcenter{\xymatrix{
H^{n}(U(A))
\ar[d]^{\varphi_{U(A)}^{n}}\ar[rr]^{H^{n}(U(f))} &&
H^{n}(U(B))
\ar[d]^{\varphi_{U(B)}^{n}}\ar[r]^{\delta_{n}^{U(B)}} &
H^{n+1}(B)
\ar[d]^{\varphi_{B}^{n+1}}\\
G^{n}(U(A))
\ar[rr]^{{G^{n}(U(f))}} &&
G^{n}(U(B))
\ar[r]^{\overline{\delta}_{n}^{U(B)}} &
G^{n+1}(B)
}}
\end{equation*}
commute. Since $(H_{n})_{n\in\ensuremath{\mathbb{N}}_{0}}$ and $(G_{n})_{n\in\ensuremath{\mathbb{N}}_{0}}$ are
$\delta$-functors we know that
$H^{n+1}(f)\op{\circ} \delta_{n}^{U(A)}= \delta_{n}^{U(B)} \op{\circ} H^{n}(U(f)) $
and that
$G^{n+1}(f)\op{\circ} \overline{\delta}_{n}^{U(A)}= \overline{\delta}_{n}^{U(B)} \op{\circ} G^{n}(U(f)) $.
We thus conclude that
\begin{equation*}
\varphi_{B}^{n+1}\op{\circ} H^{n+1}(f) \op{\circ} \delta_{n}^{U(A)}=
G^{n+1}(f) \op{\circ} \varphi_{A}^{n+1} \op{\circ} \delta_{n}^{U(A)}
\end{equation*}
holds. Since $\delta_{n}^{U(A)}$ is an epimorphism this shows
naturality of $\varphi^{n+1}$ and finishes the proof of the first
claim.
To show the second claim we note that the first diagram of
\eqref{eqn:moores_comparison_theorem} gives rise to a diagram
\begin{equation*}
\vcenter{\xymatrix@=1.5em{
H^{n}(U(A))
\ar[ddd]_{\varphi_{U(A)}^{n}}
\ar[dr]^(0.6){H^{n}(\beta_{f})}
\ar@/^15pt/[drr]^{\delta_{n}^{U(A)}}\\
& H^{n}(Q_{f})
\ar[r]^{\delta_{n}^{Q_{f}}}
\ar[d]_{\varphi_{Q_{f}}^{n}}&
H^{n+1}(A)
\ar[r]
\ar[d]^{\varphi_{A}^{n+1}}&
0\\
& G^{n}(Q_{f})
\ar[r]^{\overline{\delta}_{n}^{Q_{f}}} &
G^{n+1}(A)\\
G^{n}(U(A))
\ar[ur]_(0.6){G^{n}(\beta_{f})}
\ar@/_15pt/[urr]_{{\overline{\delta}_{n}^{U(A)}}}
}}.
\end{equation*}
The outer diagram commutes by construction of $\varphi_{A}^{n+1}$ (see above),
the already shown naturality of $\varphi^{n}$
shows that the trapezoid on the left
commutes and the two triangles are commutative because $H$ and $G$ are
$\delta$-functors. This implies that the whole diagram commutes. In particular,
we have
$\varphi_{A}^{n+1}\op{\circ}\delta_{n}^{Q_{f}}=\overline{\delta}_{n}^{Q_{f}}\op{\circ} \varphi_{Q_{f}}^{n}$.
The latter now implies that
\begin{equation*}
\xymatrix{
H^{n}(C)
\ar[d]^{\delta_{n}^{C}}
\ar[rr]^{H^{n}(\gamma_{f})} &&
H^{n}(Q_{f})
\ar[d]^{\delta_{n}^{Q_{f}}}
\ar[r]^{\varphi^{n}_{Q_{f}}} &
G^{n}(Q_{f})
\ar[d]^{\overline{\delta}_{n}^{Q_{f}}}
&&
G^{n}(C)
\ar[d]^{\overline{\delta}_{n}^{C}}
\ar[ll]_{G^{n}(\gamma_{f})}\\
H^{n+1}(A)
\ar@{=}[rr] &&
H^{n+1}(A)
\ar[r]^{\varphi_{A}^{n+1}} &
G^{n+1}(A)
&&
G^{n}(A)
\ar@{=}[ll]
}
\end{equation*}
commutes and since
$G^{n}(\gamma_{f})\op{\circ}\varphi_{C}^{n}=\varphi_{Q_{f}}^{n}\op{\circ} H^{n}(\gamma_{f})$
we eventually conclude that
\begin{equation*}
\overline{\delta}_{n}^{C}\op{\circ}\varphi_{C}^{n}=
\overline{\delta}_{n}^{Q_{f}}\op{\circ}G^{n}(\gamma_{f})\op{\circ} \varphi_{C}^{n}=
\overline{\delta}_{n}^{Q_{f}}\op{\circ} \varphi_{Q_{f}}^{n}\op{\circ} H^{n}(\gamma_{f})=
\varphi_{A}^{n+1}\op{\circ}\delta_{n}^{Q_{f}}\op{\circ}H^{n}(\gamma_{f})=
\varphi_{A}^{n+1}\op{\circ}\delta_{n}^{C}.
\end{equation*}
\end{proof}
\begin{remark}\label{rem:weaker_comparison_theorem}
The preceding theorem also shows the following slightly stronger
statement. Assume that we have for each $\delta$-functor
$H=(H^{n})_{n\in\ensuremath{\mathbb{N}}_{0}}$ and $G=(G^{n})_{n\in\ensuremath{\mathbb{N}}_{0}}$ (defined on the
\emph{same} category with short exact sequences) \emph{different}
functors $I$, $U$ and $I'$, $U'$ such that $H^{n}(I(A))=0=G^{n}(I'(A))$
for all $n\geq 1$ and all $A$. Suppose that the assumptions of Theorem
\ref{thm:moores_comparison_theorem}
(\ref{item:moores_comparison_theorem_2}) are fulfilled for one of the
functors $I$ or $I'$.
If $\alpha\ensuremath{\nobreak\colon\nobreak} H^{0}\rightarrow G^{0}$ is an isomorphism, then the
natural transformations $\varphi^{n}\ensuremath{\nobreak\colon\nobreak} H^{n}\Rightarrow G^{n}$
(resulting from extending $\alpha$) and
$\psi^{n}\ensuremath{\nobreak\colon\nobreak} G^{n}\Rightarrow H^{n}$ (resulting from extending
$\alpha^{-1}$) are in fact isomorphisms of $\delta$-functors. This
follows immediately from the uniqueness assertion since the diagrams
\begin{equation*}
\vcenter{\xymatrix{
H^{n}(U(A)) \ar[r]^{\delta_{n}}\ar[d]_{\varphi^{n}_{U(A)}} & H^{n+1}(A) \ar[d]^{\varphi^{k}_{A}}\\
G^{n}(U(A)) \ar[r]^{\overline{\delta}_{n}} & G^{n+1}(A)
}}
\quad\quad\quad
\vcenter{\xymatrix{
G^{n}(U(A)) \ar[r]^{\overline{\delta}_{n}}\ar[d]_{\psi^{n}_{U(A)}} & G^{n+1}(A) \ar[d]^{\psi^{k}_{A}}\\
H^{n}(U(A)) \ar[r]^{{\delta}_{n}} & H^{n+1}(A)
}}
\end{equation*}
(and likewise for $U'$) commute for arbitrary $A$ due to the property
of being a $\delta$-functor.
\end{remark}
\begin{remark}
Usually, one would impose some additional conditions on a category with
short exact sequences, for instance that it is additive (with zero
object), that for a short exact sequences $A\to B\to C$ the square
\begin{equation*}
\xymatrix{A\ar[r]\ar[d]&0\ar[d]\\ensuremath{\mathbb{B}}\ar[r]&C }
\end{equation*}
is a pull-back and a push-out, that short exact sequences are closed
under isomorphisms and that certain pull-backs and push-outs exist
\cite{Buehler09Exact-Categories}. These assumptions will then help in
constructing $\delta$-functors. However, the above setting does not
require this, all the assumptions are put into the requirements on the
$\delta$-functor.
\end{remark}
\begin{example}
Suppose $G$ is paracompact. On the category of $G$-modules in $\cat{CGHaus}$, we consider the short
exact sequences $A\xrightarrow{\alpha}B\xrightarrow{\beta}C$ such that
$\beta$ (or equivalently $\alpha$) has a continuous local section and
the functor $A\mapsto \check{H}^{n}(G,\underline{A})$ (or equivalently
$A\mapsto H^{n}_{\ensuremath{\op{Sh}}}(G,\underline{A})$). Then the functors
$A\mapsto E_{G}(A)$ and $A\mapsto B_{G}(A)$ from Definition
\ref{def:segalsCohomology} satisfy $\check{H}^{n}(G,E_{G}(A))=0$ since
$E_{G}(A)$ is contractible.
\end{example}
\begin{remark}\label{rem:counterexample_to_tus_comparison_theorem}
The argument given in the proof of \cite[Prop.\
6.1(b)]{Tu06Groupoid-cohomology-and-extensions} in order to draw the
conclusion of the first part of Theorem
\ref{thm:moores_comparison_theorem} from weaker assumptions is false as
one can see as follows. First note that the proof only uses
$I(U(A))\cong U(I(A))$, the more restrictive assumptions on the
categories to be abelian and on the natural inclusion
$A\hookrightarrow I(A)$ to satisfy $I(i_{A})=i_{I(A)}$ may be replaced
by this.
The requirements of \cite[Prop.\
6.1(b)]{Tu06Groupoid-cohomology-and-extensions} are satisfied if we set
$I(A)=E_{G}(A)$, $U(A)=B_{G}(A)$ and $i_{A}$ as in Definition
\ref{def:segalsCohomology}. In fact, the exactness of the functor $E$
shows that
\begin{equation*}
0\to EA\to E \ensuremath{C_{\boldsymbol{k}}}(G,EA) \to E B_{G}(A)\to 0
\end{equation*}
is exact and since this sequence has a continuous section by
\cite[Thm.\ B.2]{Segal70Cohomology-of-topological-groups}, we also have
that
\begin{equation*}
0\to \ensuremath{C_{\boldsymbol{k}}}(G,EA)\to \ensuremath{C_{\boldsymbol{k}}}(G,E \ensuremath{C_{\boldsymbol{k}}}(G,EA)) \to \ensuremath{C_{\boldsymbol{k}}}(G,E B_{G}(A))\to 0
\end{equation*}
is exact. Consequently, we have
\begin{equation*}
E_{G}(B_{G}(A))=\ensuremath{C_{\boldsymbol{k}}}(G,E B_{G}(A))\cong \ensuremath{C_{\boldsymbol{k}}}(G,E \ensuremath{C_{\boldsymbol{k}}}(G,EA))/\ensuremath{C_{\boldsymbol{k}}}(G,EA) =B_{G}(E_{G}(A)).
\end{equation*}
However, the two sequences of functors
$A\mapsto H^{n}_{\ensuremath{\op{SM}}}(G,A)\cong H^{n}_{\ensuremath{ \op{loc},\cont}}(G,A)$ and
$A\mapsto H^{n}_{\ensuremath{\op{glob},\cont}}(G,A)$ vanish on $E_{G}(A)$ for $n=1$, but are
different:
\begin{itemize}
\item $H^{2}_{\ensuremath{\op{glob},\cont}}(G,A)$ is not isomorphic to $H^{2}_{\ensuremath{ \op{loc},\cont}}(G,A)$,
for instance for $G=C^{\infty}(\ensuremath{\mathbb{S}}^{1},K)$ ($K$ compact, simple
and 1-connected) and $A=\op{U}(1)$.
\item For non-simply connected $G$, the universal cover gives rise to
a an element in $H^{2}_{\ensuremath{ \op{loc},\cont}}(G,\pi_{1}(G))$, not contained in
the image of $H^{2}_{\ensuremath{\op{glob},\cont}}(G,\pi_{1}(G))$.
\item The string classes from Example \ref{ex:string_cocycles} gives
an element in $H^{3}_{\ensuremath{ \op{loc},\cont}}(K,\op{U}(1))$, not contained in the
image of $H^{3}_{\ensuremath{\op{glob},\cont}}(K,\op{U}(1))$.
\end{itemize}
\end{remark}
\section{Supplements on Segal-Mitchison cohomology}
\label{sect:some_information_on_moore_s_and_segal_s_cohomology_groups}
\begin{tabsection}
We shortly recall the definition of the cohomology groups due to Segal
and Mitchison from \cite{Segal70Cohomology-of-topological-groups}.
Moreover, we also establish the acyclicity of the soft modules from
above for the globally continuous group cohomology and show
$H^{n}_{\ensuremath{\op{SM}}}(G,A')\cong H^{n}_{\ensuremath{\op{glob},\cont}}(G,A')$ for contractible $A'$.
Consider the long exact sequence
\begin{equation}\label{eqn:resolution_for_Segal_cohomology}
A\to E_{G}A\to E_{G}(B_{G}A)\to E_{G}(B_{G}^{2}A)\to E_{G}(B_{G}^{3}A) \cdots.
\end{equation}
This serves as a resolution of $A$ for the invariants functor
$A\mapsto A^{G}$ and the cohomology groups $H^{n}_{\ensuremath{\op{SM}}}(G,A)$ are the
cohomology groups of the complex
\begin{equation}\label{eqn:complex_for_Segal_cohomology}
(E_{G}A)^{G}\to (E_{G}(B_{G}A))^{G}\to(E_{G}(B_{G}^{2}A))^{G}\to
(E_{G}(B_{G}^{3}A))^{G}\cdots .
\end{equation}
We now make the following observations:
\begin{enumerate}
\renewcommand{\labelenumi}{\theenumi}
\renewcommand{\theenumi}{\arabic{enumi}.}
\item \label{item:soft_explanation1} \cite[Ex.\
2.4]{Segal70Cohomology-of-topological-groups} For an arbitrary
short exact sequence $\ensuremath{C_{\boldsymbol{k}}}(G,A)\to B\to C$, the sequence
\begin{equation*}
\ensuremath{C_{\boldsymbol{k}}}(G,A)^{G}\to B^{G}\to C^{G}
\end{equation*}
is exact, i.e., $B^{G}\to C^{G}$ is surjective. Indeed, for
$y\in C^{G}$ choose an inverse image $x\in B$ and observe that
$g.x-x$ may be interpreted as an element of $\ensuremath{C_{\boldsymbol{k}}}(G,A)$ for each
$g\in G$. If we define
\begin{equation*}
\psi(g,h):=(g.x-x)(h)\quad\text{ and }\quad
\xi(h):=h.\psi(h^{-1},e)\footnote{Note that the leading $h$ is missing in \cite[Ex.\
2.4]{Segal70Cohomology-of-topological-groups}.},
\end{equation*}
then we have $g.\xi-\xi=g.x-x$ since
\begin{align*}
(g.\xi-\xi)(h)=&g.(\xi(g^{-1}h))-\xi(h)=
h.(\psi(h^{-1}g,e))-h.\psi(h^{-1},e)\\
=&h.((h^{-1}.g.x-x)(e)-(h^{-1}.x-x)(e))\\
=&h.((h^{-1}.(g.x-x))(e))=(g.x-x)(h).
\end{align*}
Thus $x-\xi$ is $G$-invariant and maps to $y$.
\item It is not necessary to work with the resolution
\eqref{eqn:resolution_for_Segal_cohomology}, any resolution
\begin{equation}\label{eqn:alternative_resolution_for_Segal_cohomology}
A\to A_{0}\to A_{1}\to A_{2}\to \cdots
\end{equation}
(i.e., a long exact sequence of abelian groups such that the
constituting short exact sequences have local continuous
sections) with $A_{i}$ of the form $\ensuremath{C_{\boldsymbol{k}}}(G,A'_{i})$ for some
contractible $A'_{i}$ would do the job. Indeed, then the
double complex
\begin{equation*}
\xymatrix@=0.75em{
\vdots & \vdots & \vdots &
\vdots \\
E_{G}(B_{G}^{2}A) \ar[r]\ar[u] & E_{G}(B_{G}^{2} (\ensuremath{C_{\boldsymbol{k}}}(G,A'_{0})))\ar[r] \ar[u]& E_{G}(B_{G} ^{2}(\ensuremath{C_{\boldsymbol{k}}}(G^{2},A'_{1})))\ar[r]\ar[u] &
E_{G}(B_{G}^{2}( \ensuremath{C_{\boldsymbol{k}}}(G^{3},A'_{2})))\ar[r]\ar[u]&\cdots\\
E_{G}(B_{G}A) \ar[r]\ar[u] & E_{G}(B_{G} (\ensuremath{C_{\boldsymbol{k}}}(G,A'_{0})))\ar[r] \ar[u]& E_{G}(B_{G} (\ensuremath{C_{\boldsymbol{k}}}(G^{2},A'_{1})))\ar[r]\ar[u] &
E_{G}(B_{G}( \ensuremath{C_{\boldsymbol{k}}}(G^{3},A'_{2})))\ar[r]\ar[u]&\cdots\\
E_{G}(A) \ar[r]\ar[u] & E_{G}(\ensuremath{C_{\boldsymbol{k}}}(G,A'_{0}))\ar[r] \ar[u]& E_{G}(\ensuremath{C_{\boldsymbol{k}}}(G^{2},A'_{1}))\ar[r]\ar[u] &
E_{G}(\ensuremath{C_{\boldsymbol{k}}}(G^{3},A'_{2}))\ar[r]\ar[u]&\cdots\\
A \ar[r]\ar[u] & \ensuremath{C_{\boldsymbol{k}}}(G,A'_{0})\ar[r] \ar[u]& \ensuremath{C_{\boldsymbol{k}}}(G^{2},A'_{1})\ar[r]\ar[u] &
\ensuremath{C_{\boldsymbol{k}}}(G^{3},A'_{2})\ar[r]\ar[u]&\cdots
}
\end{equation*}
has exact rows and columns (cf.\ \cite[Prop.\
2.2]{Segal70Cohomology-of-topological-groups}), which remain
exact after applying the invariants functor to it by the
observation from \ref{item:soft_explanation1} Thus the
cohomology of the first row is that of the first column, showing
that the cohomology of \ref{eqn:complex_for_Segal_cohomology} is
the same as the cohomology of
$A_{0}^{G}\to A_{1}^{G}\to A_{2}^{G}\to\cdots$.
In particular, for contractible $A'$ we may replace
\eqref{eqn:resolution_for_Segal_cohomology} in the definition of
$H_{\ensuremath{\op{SM}}}^{n}(G,A')$ by
\begin{equation*}
A'\to E'_{G}A\to {E'}_{G}(B'_{G}A')\to E'_{G}({B'}_{G}^{2}A)\to E'_{G}({B'}_{G}^{3}A') \cdots
\end{equation*}
with $E'_{G}(A'):=\ensuremath{C_{\boldsymbol{k}}}(G,A')$ and $B'_{G}(A):=E'_{G}(A)/A$ (the
occurrence of $E$ in the definition $E_{G}(A):=\ensuremath{C_{\boldsymbol{k}}}(G,EA)$ only
serves the purpose of making the target contractible).
\item \label{item:soft_explanation2} Since $A'$ is assumed to be
contractible, the short exact sequence
$A'\to E'_{G}(A')\to B'_{G}(A')$ has a global continuous section
\cite[App.\ B]{Segal70Cohomology-of-topological-groups}, and
thus the sequence
\begin{equation*}
\ensuremath{C_{\boldsymbol{k}}}(G,A')\to \ensuremath{C_{\boldsymbol{k}}}(G,E'_{G}(A'))\to \ensuremath{C_{\boldsymbol{k}}}(G,B'_{G}(A'))
\end{equation*}
is exact. In particular, the isomorphism
$\ensuremath{C_{\boldsymbol{k}}}(G,E'_{G}(A'))\cong E'_{G}(\ensuremath{C_{\boldsymbol{k}}}(G,A'))$ shows that
\begin{equation*}
B'_{G}(\ensuremath{C_{\boldsymbol{k}}}(G,A')):=E'_{G}(\ensuremath{C_{\boldsymbol{k}}}(G,A'))/\ensuremath{C_{\boldsymbol{k}}}(G,A')\cong \ensuremath{C_{\boldsymbol{k}}}(G,E'_{G}(A'))/\ensuremath{C_{\boldsymbol{k}}}(G,A')\cong \ensuremath{C_{\boldsymbol{k}}}(G,B'_{G}(A'))
\end{equation*}
is again of the form $\ensuremath{C_{\boldsymbol{k}}}(G,A'')$ with $A''$ contractible.
\end{enumerate}
These observations, together with an inductive argument, imply that the
sequence
\begin{equation*}
A^{G}\to (E'_{G}A)^{G}\to (E'_{G}(B_{G}A))^{G}\to(E_{G}({B'}_{G}^{2}A))^{G}\to
(E'_{G}({B'}_{G}^{3}A))^{G}\cdots
\end{equation*}
is exact for $A=\ensuremath{C_{\boldsymbol{k}}}(G,A')$ and contractible $A'$, and finally that
$H^{n}_{\ensuremath{\op{SM}}}(G,A)$ vanishes for $n\geq 1$. What also follows is that
for contractible $A'$, we have
$H^{n}_{\ensuremath{\op{SM}}}(G,A')\cong H^{n}_{\ensuremath{\op{glob},\cont}}(G,A')$ (cf.\ \cite[Prop.\
3.1]{Segal70Cohomology-of-topological-groups}). Indeed,
$\ensuremath{C_{\boldsymbol{k}}}(G^{k},A')\cong \ensuremath{C_{\boldsymbol{k}}}(G,\ensuremath{C_{\boldsymbol{k}}}(G^{k-1},A'))$ and thus
\begin{equation*}
A' \to \ensuremath{C_{\boldsymbol{k}}}(G,A')\to \ensuremath{C_{\boldsymbol{k}}}(G^{2},A')\to \ensuremath{C_{\boldsymbol{k}}}(G^{3},A')\to \cdots
\end{equation*}
serves as a resolution of the form
\eqref{eqn:alternative_resolution_for_Segal_cohomology}. Dropping $A'$
and applying the invariants functor to it then gives the (homogeneous
version of) the complex $C^{n}_{\ensuremath{\op{glob},\cont}}(G,A')$.
\end{tabsection}
\bibliographystyle{new} \def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}}
|
2,869,038,156,354 | arxiv | \section{Introduction}
When discussing the influence of fermions on the dynamics of some
theory, e.g. of the Standard Particle Model (SM) in a cosmology
setting, it is mandatory to integrate out the fermions. This procedure
in one-loop order already becomes quite involved if arbitrary chiral
couplings of space-time dependent outer bosonic fields are
considered. The extensive work of Salcedo \cite{Salcedo1,Salcedo2}
resulted in an effective action to leading order in covariant
derivatives of such fields. Basis is a refined calculation in momentum
space, handling of anomalies using the Wess-Zumino-Witten model (WZW),
and last but not least, a very practical shorthand notation.
Such effective actions are quite important in evaluating some models.
For example, in ref.~\cite{Smit} Smit discussed a form of 'cold'
electroweak baryogenesis at the end of electroweak scale
inflation~\cite{Tranberg} which could very well work if the rephasing
invariant
\be
J=s_1^2s_2s_3c_1c_2c_3 \sin(\delta)=(3.0\pm0.3)\times10^{-5}
\ee
of the Jarlskog determinant is not accompanied by further suppressions
through mass ratios. It was proposed \cite{Smit} that derivative terms
in the effective action that are analytic in the time-dependent masses
considered non-perturbatively could be very important in
non-equilibrium. Such effects were also observed in
ref.~\cite{Konstandin2}. In the work~\cite{Smit} the fourth order
derivative result of ref.~\cite{Salcedo2} turned out not to contain
CP violation, but the claim was made that higher orders of the
imaginary part of the effective action will do.
Worldline methods in first quantized quantum field theory are ideally
adapted for calculating effective actions: One considers the
propagation of a particle in some space-time dependent background
\cite{Schwinger}, but in x-space path integral formulation
\cite{Feynman}. This method~\cite{Strassler, Schmidt4}, also related to the infinite
tension limit of String theory \cite{Bern1,Bern2}, was used heavily
for the discussion of various effective actions in one-loop
\cite{Schmidt4,Schmidt5,Schmidt6,Schmidt1} and two-loop
\cite{Schmidt2loop2,Schmidt2loop,Schmidt3,TwoQCD} order. For example, the high
order in the inverse mass calculation of ref.~\cite{Schmidt3} could
hardly be done with other methods.
The present paper provides a formalism to determine higher order
contributions to the imaginary part of the effective action using the
worldline formalism. We are concerned with the effective action of a
multiplet of N Dirac fermions coupled to an arbitrary matrix-valued
set of fields, including a scalar $\Phi$, a pseudoscalar $\Pi$, a
vector \textit{A}, a pseudovector \textit{B}, and an antisymmetric
tensor \textit{$K_{\mu \nu}$}. One peculiar feature of the imaginary
part of the effective action is that it cannot be written in a
manifest chiral covariant way, due to the presence of the chiral
anomaly. One possibility to arrive at a closed expression for the
effective action is to abandon manifest chiral covariance as it was
done in ref.~\cite{Gagne2}. The resulting expression is rather
complicated and not well suited for higher order
calculations. Alternatively, it was proposed in ref.~\cite{Salcedo2}
to determine the covariant current for which a manifestly chiral
covariant expression exists and to take account of the anomaly when
integrating the current to yield the effective action. Following this
idea, we present a worldline path integral formulation of the
covariant current.
Before we do so, we review the worldline formalism by discussing the
derivation of the real part of the effective action. A single Dirac
fermion in the presence of both a scalar and pseudoscalar fields in
the context of the worldline formalism was first treated in
ref.~\cite{Axial1}, the inclusion of a pseudovector in
ref.~\cite{Axial2}. In our discussion of the real part of the
effective action we will follow the elegant subsequent work of
refs.~\cite{Gagne1,Gagne2}.
Section \ref{sec_act} contains the derivation of the real part of the
effective action, the derivation of the covariant current and the
matching procedure to obtain the imaginary part of the effective
action. In section \ref{sec_low}, we briefly reproduce the results in
lowest order from ref.~\cite{Salcedo2}. As a novel result we present
the imaginary part of the effective action in two dimensions in next
to leading order in section~\ref{sec_next}.
\section{Effective Action\label{sec_act}}
We are concerned with the effective action
\begin{equation}
\label{W}
i\, W [ \Phi,\Pi,A,B,K ]=\log {\mathrm{Det\,}} i\, [ \, i\, {\slashsym5\partial} -\Phi +
\, i\, \, \gamma^5 \Pi + {\slashsym9A} + \gamma^5 {\slashsym6B} +
\, i \, \gamma^{\mu} \gamma^{\nu} K_{\mu \nu} ],
\end{equation}
and its continuation to Euclidean space. The $\gamma$ matrices remain
unaffected by the continuation, but it is useful to introduce the
following notation, $(\gamma_E)_j
\equiv i \gamma_j$, $(\gamma_E)_4 \equiv
\gamma_0$, and $(\gamma_E)_5 \equiv \gamma_5$.
After Wick-rotation, $t\rightarrow -it$, one obtains with this new
notation
\begin{equation}
\label{wickrot}
{\slashsym5\partial} \rightarrow i {\slashsym5\partial}_E, \, {\slashsym9A} \rightarrow
i {\slashsym9A}_E, \, {\slashsym6B} \rightarrow i {\slashsym6B}_E,
\, \gamma^{\mu} \gamma^{\nu} K_{\mu \nu} \rightarrow
- (\gamma_E)_{\mu} (\gamma_E)_{\nu} K_{E \mu \nu}.
\end{equation}
From now on, the $E$ subscript will be suppressed.
The effective action of \eq~(\ref{W}) now reads
\begin{equation}
\label{WE}
-W [ \Phi,\Pi,A,B,K ] = \log {\mathrm{Det\,}} [ {\mathcal{O}} ],
\end{equation}
with the operator ${\mathcal{O}}$ in momentum space defined by
\begin{equation}
\label{O}
{\mathcal{O}} \equiv {\slashsym8p} - i \Phi(x) - \gamma_5 \Pi(x) - {\slashsym9A}(x)
- \gamma_5 {\slashsym6B}(x) + \gamma_{\mu} \gamma_{\nu} K_{\mu \nu}.
\end{equation}
As in ref.~\cite{Gagne1, Salcedo1}, the real and imaginary parts of
the effective action are analyzed separately
\begin{equation}
\label{Sep}
-W^{+}-i\,W^{-}=\log \left(\vert {\mathrm{Det\,}}[{\mathcal{O}}]\vert\right)+i\,\arg\left({\mathrm{Det\,}}[{\mathcal{O}}]\right).
\end{equation}
A perturbative expansion in weak fields \cite{Gagne1} shows that
graphs with an even number of $\gamma_5$ vertices are real, and graphs
with an odd number of $\gamma_5$ vertices are imaginary. This will
prove useful when the behavior of the effective action under
complex conjugation is explored later on.
\subsection{Real Part of the Effective Action}
Our intention is to obtain a worldline representation for the
effective action with manifest chiral and gauge invariance. This is
unproblematic for the real part, but it causes certain difficulties
for the imaginary part due to the chiral anomaly. In order to
familiarize the reader with the worldline method we review the
derivation for the real part in four dimensions as it was presented in
refs.~\cite{Gagne1,Gagne2}.
\subsubsection{Construction of a Positive Operator for the Real Part of the Effective Action}
In order to use the worldline formalism, one has to rewrite the
effective action in terms of a positive operator, thus obtaining
\begin{equation}
W^{+}=-\frac{1}{2}\log {\mathrm{Det\,}} [ {\mathcal{O}}^{\dagger} {\mathcal{O}} ].
\end{equation}
The problem with this operator is that it contains terms linear in the
$\gamma$ matrices, what makes the transition to a path integral of
Grassman fields problematic. One way to avoid this problem is by
doubling the fermion system and exchanging the operator ${\mathcal{O}}$ for a
Hermitian operator $\Sigma$ yielding
\begin{equation}
\label{WR}
W^{+}=-\frac{1}{2}\log {\mathrm{Det\,}} [ {\mathcal{O}}^{\dagger} {\mathcal{O}} ] = -\frac{1}{4} \log {\mathrm{Det\,}} [ \Sigma^2 ],
\qquad \quad
\Sigma \equiv
\begin{pmatrix}
0 & {\mathcal{O}} \\
{\mathcal{O}}^{\dagger} & 0 \\
\end{pmatrix}.
\end{equation}
Since $\Sigma$ is Hermitian, one can use the Schwinger integral
representation of the logarithm without any restrictions. One obtains
\begin{equation}
\label{WRint}
W^{+}=\frac{1}{4}\int_0^{\infty}\frac{dT}{T} {\mathrm{Tr\,}} \exp(-T \Sigma^2).
\end{equation}
At this point, it is natural to introduce six $8\times8$ Hermitian
$\Gamma_A$ matrices. These matrices satisfy
$\lbrace\Gamma_A,\Gamma_B\rbrace=2\delta_{A B}$, with $A,B=1..6$ and
are defined as
\begin{equation}
\label{Gdef}
\Gamma_{\mu}=
\begin{pmatrix}
0 & \gamma_{\mu} \\
\gamma_{\mu} & 0 \\
\end{pmatrix},\quad
\Gamma_5=
\begin{pmatrix}
0 & \gamma_5 \\
\gamma_5 & 0 \\
\end{pmatrix}, \quad
\Gamma_6=
\begin{pmatrix}
0 & i\, \mathbbm{l}_4 \\
-i\, \mathbbm{l}_4 & 0 \\
\end{pmatrix}.
\end{equation}
For later use we also introduce the equivalent of $\gamma_5$,
\begin{equation}
\label{G7def}
\Gamma_7 = -i\,\prod_{A=1}^6 \Gamma_A=
\begin{pmatrix}
\mathbbm{l}_4 & 0 \\
0 & -\mathbbm{l}_4 \\
\end{pmatrix},
\end{equation}
and $\Gamma_7$ anticommutes with all other $\Gamma$ matrices.
Expressing $\Sigma$ in terms of these new matrices yields~\cite{Gagne2},
\begin{equation}
\label{Sigmanew}
\Sigma=\Gamma_{\mu}(p_{\mu}-A_{\mu})-\Gamma_6 \Phi-\Gamma_5 \Pi
-i\, \Gamma_{\mu} \Gamma_5 \Gamma_6 B_{\mu}
-i\, \Gamma_{\mu} \Gamma_{\nu} \Gamma_6 K_{\mu \nu}.
\end{equation}
The aim is to turn \eq~(\ref{Sigmanew}) into an expression which is
manifestly chiral covariant. This can be achieved by changing to a
basis in which $i\, \Gamma_5 \Gamma_6$ is diagonal~\cite{Gagne1} using
the following transformation
\begin{equation}
\label{Change}
M^{-1}i\Gamma_5\Gamma_6 M=
\begin{pmatrix}
\mathbbm{l}_4 & 0 \cr 0 & -\mathbbm{l}_4 \\
\end{pmatrix}, \qquad M=
\begin{pmatrix}
\mathbbm{l}_2 & 0 & 0 & 0 \\
0 & 0 & 0 & \mathbbm{l}_2 \\
0 & 0 & \mathbbm{l}_2 & 0 \\
0 & \mathbbm{l}_2 & 0 & 0 \\
\end{pmatrix}.
\end{equation}
In this basis, $\Sigma$ takes the form
\begin{equation}
\label{newSigma}
\tilde{\Sigma} = M^{-1} \Sigma M =
\begin{pmatrix}
\gamma_{\mu}(p_{\mu}-A_{\mu}^L) & \gamma_5(-i\, H+\frac{1}{2}
\gamma_{\mu} \gamma_{\nu} K_{\mu \nu}^s) \\
-\gamma_5(-i\, H^{\dagger}+\frac{1}{2} \gamma_{\mu} \gamma_{\nu}
K_{\mu \nu}^{s \dagger}) & \gamma_{\mu}(p_{\mu}-A_{\mu}^R) \\
\end{pmatrix},
\end{equation}
which is manifestly chiral covariant. Here $A^L=A+B$, $A^R=A-B$,
$H=\Phi-i\,\Pi$, $K^s=K-i\,\tilde{K}$ and $\tilde K_{\mu\nu}=\frac12
\epsilon_{\mu\nu\rho\sigma} K^{\rho\sigma}$ have been defined.
The square of $\tilde{\Sigma}$ constitutes a positive operator which
is suitable for the worldline formalism. However, even though this
expression contains only even combinations of $\gamma$ matrices, the
coherent state formalism cannot yet be used to transform this
expression into a fermionic path integral. In the coherent state
formalism, the $\gamma_5$ matrices have to be rewritten as a product
of the other $\gamma$ matrices, what would result again in odd
combinations. One possible solution of this problem is to enlarge the
Clifford space, replacing the $\gamma$ matrices by $\Gamma$ matrices
\be
\gamma_A \to \Gamma_A = \gamma_A \otimes
\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix}, \quad A \in [1\dots5].
\ee
The matrix $\Gamma_5$ is then independent from the other $\Gamma$
matrices and the coherent state formalism with six (instead of four)
operators can be used. The doubling of the Clifford space inside the
trace has to be compensated by a factor $\frac12$, such that
\eq~(\ref{WRint}) reads
\begin{equation}
\label{newWR}
W^{+}=\frac{1}{8}\int_0^{\infty}\frac{dT}{T} {\mathrm{Tr\,}} \exp(-T \hat{\Sigma}^2),
\end{equation}
and the operator $\hat{\Sigma}^2$ is given by
\begin{eqnarray}
\label{Sigma2}
\hat{\Sigma}^2=(p-{\mathcal{A}})^2+{\mathcal{H}}^2+
\frac{1}{2}{\mathcal{K}}_{\mu \nu} {\mathcal{K}}_{\mu \nu}+\frac{i}{2} \Gamma_{\mu} \Gamma_{\nu}({\mathcal{F}}_{\mu \nu}+
\lbrace {\mathcal{H}},{\mathcal{K}}_{\mu \nu}\rbrace+i\, [{\mathcal{K}}_{\mu \rho},{\mathcal{K}}_{\rho \nu}])\nonumber \\
+i\, \Gamma_{\mu} \Gamma_5({\mathcal{D}}_{\mu}{\mathcal{H}}+\{ p_{\nu}-{\mathcal{A}}_{\nu},{\mathcal{K}}_{\mu \nu}\})-
\frac12 \Gamma_{\mu \rho \sigma} \Gamma_5 {\mathcal{D}}_{\mu} {\mathcal{K}}_{\rho \sigma}-
\frac{1}{4}\Gamma_{\mu \nu \rho \sigma} {\mathcal{K}}_{\mu \nu} {\mathcal{K}}_{\rho \sigma},
\end{eqnarray}
with enlarged background fields defined by
\begin{equation}
{\mathcal{A}}_{\mu}=
\begin{pmatrix}
A_{\mu}^L & 0 \cr 0 & A_{\mu}^R \\
\end{pmatrix}, \;
{\mathcal{H}} =
\begin{pmatrix}
0 & i\, H \\
-i\, H^{\dagger} & 0 \\
\end{pmatrix},
\; {\mathcal{K}}_{\mu \nu} =
\begin{pmatrix}
0 & i\,K_{\mu \nu}^s \\
-i\,K_{\mu \nu}^{s\dagger} \\
\end{pmatrix}.
\end{equation}
$\Gamma_{A_1 ... A_k} \equiv \Gamma_{[A_1}...\Gamma_{A_k]}$ denotes
the anti-symmetrized product of $k$ $\Gamma$ matrices, and the
field-strength and the covariant derivative have been defined as
\begin{equation}
\label{defF}
{\mathcal{F}}_{\mu \nu}=\partial_{\mu} {\mathcal{A}}_{\nu}-\partial_{\nu} {\mathcal{A}}_{\mu}-i\,
[{\mathcal{A}}_{\mu},{\mathcal{A}}_{\nu}], \qquad {\mathcal{D}}_{\mu} {\mathcal{\chi}} = \partial_{\mu} {\mathcal{\chi}} -i\, [{\mathcal{A}}_{\mu},{\mathcal{\chi}}].
\end{equation}
The $\hat{\Sigma}^2$ operator is seen to be manifestly gauge and
chiral covariant. It also contains $\Gamma$ matrices to even powers
only, and is well suited for the worldline path integral
representation.
\subsubsection{Worldline Path Integral}
With the use of the coherent state formalism~\cite{coherent,Gagne1},
one can perform the transition from $\Gamma$ matrices to a path
integral over Grassman fields $\psi$, with the correspondence
$\Gamma_A\Gamma_B\rightarrow2\psi_A\psi_B$ and
$\Gamma_A\Gamma_B\Gamma_C\Gamma_D\rightarrow4\psi_A\psi_B\psi_C\psi_D$,
as long as $A$, $B$, $C$, and $D$ are all different. The final form
for the real part of the effective action is
\begin{equation}
\label{WRfinal}
W^{+}=\frac{1}{8}\int_0^{\infty}\frac{dT}{T}{\mathcal{N}}\int {\mathcal{D}} x \int_{AP} {\mathcal{D}} \psi\,
\mathrm{tr}\,{\mathcal{P}} e^{-\int_0^Td\tau{\mathcal{L}}(\tau)}.
\end{equation}
Here ${\mathcal{N}}$ denotes a normalization constant coming from a momentum
integration and AP stands for antiperiodic boundary conditions, which
must be fulfilled by the Grassman variables $\psi(T)=-\psi(0)$. The
Lagrangian is given by
\begin{eqnarray}
\label{LagrangianR}
{\mathcal{L}}(\tau)& = &\frac{\dot{x}^2}{4}+
\frac{1}{2}\psi_A\dot{\psi}_A-i\,\dot{x}_{\mu}{\mathcal{A}}_{\mu} +
{\mathcal{H}}^2 + \frac{1}{2}{\mathcal{K}}_{\mu\nu}{\mathcal{K}}_{\mu\nu}+
2i\,\psi_{\mu}\psi_5\left({\mathcal{D}}_{\mu}{\mathcal{H}}+
i\,\dot{x}_{\nu}{\mathcal{K}}_{\mu\nu}\right) \nonumber \\
&& + \, i\, \psi_{\mu}\psi_{\nu}\left({\mathcal{F}}_{\mu\nu}+
\lbrace{\mathcal{H}},{\mathcal{K}}_{\mu\nu}\rbrace +i\, [{\mathcal{K}}_{\mu \rho},{\mathcal{K}}_{\rho \nu}] \right) \nn \\
&& - \, \psi_{\mu}\psi_{\nu}\psi_{\rho}\left( 2 \psi_5{\mathcal{D}}_{\mu}{\mathcal{K}}_{\mu\nu}+
\psi_{\sigma}{\mathcal{K}}_{\mu\nu}{\mathcal{K}}_{\rho\sigma}\right).
\end{eqnarray}
The periodic boundary conditions for the field $x(\tau)$ suggest to
separate the zero modes of the free field operator
$\frac{d^2}{d\tau^2}$. The fields $x(\tau)$ are split into a constant
part and a $\tau$ dependent part according to $x(\tau)=x_0+y(\tau)$,
with $\partial_{\tau}x_0=0$ and $\int_0^Td\tau\,y(\tau)=0$, and the
measure in the integral is changed into ${\mathcal{D}} x={\mathcal{D}} y\,d^Dx_0$. The
Green function is defined on a subspace orthogonal to the zero
modes. The $\psi_A$ fields contain no zero modes, so that the propagators
for the $y(\tau)$ and $\psi_A(\tau)$ fields read
\begin{eqnarray}
\label{PropR}
\langle y(\tau_1) y(\tau_2) \rangle & = & \frac{(\tau_1-\tau_2)^2}{T}-
\vert \tau_1-\tau_2 \vert, \nonumber \\
\langle \psi_{A}(\tau_1) \psi_{B}(\tau_2) \rangle & = &
\frac{1}{2} \delta_{AB} \mathrm{sign}\left(\tau_1 - \tau_2 \right).
\end{eqnarray}
This formalism can then be used to determine the real part of the
effective action as discussed in ref.~\cite{Gagne2}.
\subsection{Imaginary Part of the Effective Action}
As in the case of the real part of the effective action, one requires
a positive operator in order to use the Schwinger trick. Even though
this is still possible for the imaginary part, gauge and chiral
invariance cannot be manifestly conserved due to the chiral
anomaly. For example, in ref.~\cite{Gagne1,Gagne2} a parameter
$\alpha$ is introduced, which breaks the chiral invariance, but leads
to a positive operator. However the resulting expression is not
appropriate for higher order calculations since the breaking of
manifest chiral invariance leads to a large number of contributions in
the perturbative expansion of the path integral.
The aim of the present work is to present a worldline representation
of the effective current for which a manifestly chiral covariant
expression exists. This current can then be integrated to obtain the
effective action~\cite{Hoker,Salcedo2,Konstandin}. This integration
rather proceeds by matching: First, a general effective action is
proposed, which has the expected chiral and covariant properties. The
functional variation of this action is then matched to the covariant
current that is obtained using the worldline formalism. This method
has the advantage that it is both gauge and chiral invariant at each
stage of the calculation. The anomaly only leads to additional
complications in the matching procedure of the lowest order
contributions as will be discussed in detail in the next section.
Starting point of our analysis is the functional derivative of the
imaginary part of the effective action in \eq~(\ref{Sep})
\begin{equation}
\label{WI}
\delta W^{-}=\frac{1}{2}\delta\left(\log{\mathrm{Det\,}} {\mathcal{O}}-
\log{\mathrm{Det\,}} {\mathcal{O}}^{\dagger}\right)=\frac{1}{2}{\mathrm{Tr\,}}\left(\delta{\mathcal{O}}\frac{1}{{\mathcal{O}}}-
\delta{\mathcal{O}}^{\dagger}\frac{1}{{\mathcal{O}}^{\dagger}}\right).
\end{equation}
This expression can be rewritten in terms of a positive operator which
can be used to employ the worldline representation in combination with
the heat kernel formula. Incidentally, it can also be expressed in a
manifestly chiral covariant form, what simplifies higher order
calculations tremendously as compared to the formalism presented in
ref.~\cite{Gagne2}.
\subsubsection{Construction of a Positive Operator for
the Imaginary Part of the Effective Action}
The expression in \eq~(\ref{WI}) can be transformed using the
operator $\Sigma$ defined in \eq~(\ref{WR})
\begin{equation}
\delta W^{-}=\frac{1}{2}{\mathrm{Tr\,}}
\begin{pmatrix}
0 & \delta{\mathcal{O}} \\
-\delta{\mathcal{O}^{\dagger}} & 0 \\
\end{pmatrix}
\begin{pmatrix}
0 & 1/{\mathcal{O}^{\dagger}} \\
1/{\mathcal{O}} & 0 \\
\end{pmatrix},
\end{equation}
which, with the introduction of a new matrix $\chi$, can be rewritten
as
\begin{equation}
\label{dWI}
\delta W^{-}=\frac{1}{2}{\mathrm{Tr\,}} \chi \delta\Sigma \Sigma^{-1},
\end{equation}
with
\begin{equation}
\Sigma=
\begin{pmatrix}
0 & {\mathcal{O}} \\
{\mathcal{O}}^{\dagger} & 0 \\
\end{pmatrix}, \qquad
\chi =
\begin{pmatrix}
\mathbbm{l}_4 & 0 \\
0 & -\mathbbm{l}_4 \\
\end{pmatrix}.
\end{equation}
To produce the positive definite operator $\Sigma^2$ in
\eq~(\ref{dWI}), we multiply and divide by $\Sigma$, using the cyclic
property of the trace and the fact that $\Sigma$ anticommutes with
$\chi$, to obtain
\begin{eqnarray}
\label{dWIplusb}
\delta W^{-}&=&\frac{1}{4}{\mathrm{Tr\,}} \left( \chi\delta\Sigma\Sigma +
\Sigma\chi\delta\Sigma\right) \Sigma^{-2}
\nonumber\\
&=&\frac{1}{4}{\mathrm{Tr\,}}\chi\left[\delta\Sigma,\Sigma\right]\Sigma^{-2}.
\end{eqnarray}
Since the last factor is a positive operator, it can be reexpressed as
an integral, similar to the expression of the real part of the
effective action in \eq~(\ref{Sigma2}), namely
\begin{eqnarray}
\label{dWIplus}
\delta W^{-}&=&\frac{1}{4}{\mathrm{Tr\,}}\int_0^{\infty}dT
\chi\left[\delta\Sigma,\Sigma\right]e^{-T\Sigma^2}.
\end{eqnarray}
As in the case for the real part, the chiral covariance can be made
manifest by changing to an appropriate basis. With the help of the
matrix $M$ in \eq~(\ref{Change}), one obtains again
\begin{equation}
\label{newSigma2}
\tilde{\Sigma}= \gamma_{\mu}(p_{\mu}-{\mathcal{A}}_{\mu})-\gamma_5 {\mathcal{H}}-
\frac{i}{2} \gamma_{\mu} \gamma_{\nu} \gamma_5 {\mathcal{K}}_{\mu \nu}.
\end{equation}
The additional factors $\chi \left[\delta\Sigma,\Sigma\right]$ read
\begin{equation}
\label{chitilde}
M^{-1} \chi M = \tilde{\chi} =
\begin{pmatrix}
\gamma_5 & 0 \\
0 & -\gamma_5 \\
\end{pmatrix}
= \chi \,\gamma_5,
\end{equation}
and for the case $\delta\tilde\Sigma=-\gamma_{\mu}\delta{\mathcal{A}}_{\mu}$
\begin{eqnarray}
\label{commutator}
\left[ \delta\tilde\Sigma ,\tilde\Sigma \right]&=&-
\gamma_{\mu\nu}\left\lbrace \delta{\mathcal{A}}_{\mu},p_{\nu}-
{\mathcal{A}}_{\nu}\right\rbrace-i\,{\mathcal{D}}_{\mu}\delta{\mathcal{A}}_{\mu}-
\gamma_5\gamma_{\mu}\left\lbrace\delta{\mathcal{A}}_{\mu},{\mathcal{H}}\right\rbrace\nonumber\\
&&+i\,\gamma_5\gamma_{\mu}\left[\delta{\mathcal{A}}_{\nu},{\mathcal{K}}_{\mu\nu}\right]-
\frac{i}2\,\gamma_5\gamma_{\mu\lambda\sigma}
\left\lbrace\delta{\mathcal{A}}_{\mu},{\mathcal{K}}_{\lambda\sigma}\right\rbrace.
\end{eqnarray}
To use the coherent state formalism, it is again necessary to enlarge
the Clifford algebra and to replace the $\gamma$ matrices by $\Gamma$
matrices. However, taking into account the factor $\gamma_5$ in
\eq~(\ref{chitilde}) the imaginary part of the effective action contains only
odd combinations of $\gamma$ matrices. Thus, the replacement
\be
\gamma_A \to \Gamma_A = \gamma_A \otimes
\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix}, \quad A \in [1\dots5]
\ee
has to be compensated by a factor
\begin{equation}
\label{compensator}
-\frac{i}2 \Gamma_7\Gamma_6= \mathbbm{l}_4 \otimes
\begin{pmatrix}
0 & \frac12 \\
\frac12 & 0 \\
\end{pmatrix}.
\end{equation}
The overall factor $\Gamma_7$ changes the boundary condition of
the fermionic sector from antiperiodic to periodic as explained in
ref.~\cite{Gagne1}. This means that the fermionic sector contains
zero modes, which have to be separated in the same way as was done
for the bosonic sector.
Including the factor in \eq~(\ref{compensator}) to compensate for the
doubling of the Clifford space, one obtains
\begin{equation}
\label{WIm2}
\delta W^{-}=\frac{i}{8}{\mathrm{Tr\,}}\int_0^{\infty}dT\Gamma_7\Gamma_6\chi w(T)e^{-T\hat\Sigma^2},
\end{equation}
where $\hat\Sigma^2$ is given in \eq~(\ref{Sigma2}), and the insertion
due to the commutator yields
\begin{eqnarray}
w(T)&=&-\frac{1}{2}\Gamma_5\Gamma_{\mu\nu}\left\lbrace \delta{\mathcal{A}}_{\mu},p_{\nu}-
{\mathcal{A}}_{\nu}\right\rbrace-i\,\Gamma_5{\mathcal{D}}_{\mu}\delta{\mathcal{A}}_{\mu}-
\Gamma_{\mu}\left\lbrace\delta{\mathcal{A}}_{\mu},{\mathcal{H}}\right\rbrace \nonumber\\
&&+i\,\Gamma_{\mu}\left[\delta{\mathcal{A}}_{\nu},{\mathcal{K}}_{\mu\nu}\right]-
\frac{i}{2} \Gamma_{\mu\lambda\sigma}
\left\lbrace\delta{\mathcal{A}}_{\mu},{\mathcal{K}}_{\lambda\sigma}\right\rbrace.
\end{eqnarray}
To transform this expression into a worldline path integral, a similar
procedure as for the real part of the effective action can be
followed. Products of $\Gamma$ matrices can be replaced by Grassman
fields, however in this case the Jacobian of the transformation
contains additional contributions from the zero modes
\begin{eqnarray}
{\mathcal{D}}\theta{\mathcal{D}}\bar\theta &\equiv&
d\theta_3d\theta_2d\theta_1d\bar\theta_1d\bar\theta_2d\bar\theta_3
{\mathcal{D}}\theta'{\mathcal{D}}\bar\theta' \nonumber\\
&=&\frac{1}{J}d\psi_1^0d\psi_2^0d\psi_3^0d\psi_4^0d\psi_5^0d\psi_6^0{\mathcal{D}}\psi'.
\end{eqnarray}
The factor $J$ only includes the Jacobian for the zero modes, while
the Jacobian for the orthogonal modes is absorbed in the normalization
of the correlation functions of the $\psi_A'$. $J$ can be calculated
from the definition of the Grassman fields $\psi$ in the coherent
state formalism~\cite{Gagne1} and yields in $D$ dimension
\begin{equation}
J=\det\left(\frac{\partial\theta,\bar\theta}{\partial\psi}\right)=(-i)^{(D+2)/2}.
\end{equation}
The final result can be expressed as
\begin{equation}
\label{current}
\delta W^{-}=\frac{1}{8} \, \tr \int_0^{\infty}dT {\mathcal{N}}\int{\mathcal{D}} x
\int_{P}{\mathcal{D}}\psi \,\chi w(T){\mathcal{P}} e^{-\int_0^Td\tau{\mathcal{L}}(\tau)}.
\end{equation}
The Lagrangian is of the same form as in the real
part, \eq~(\ref{LagrangianR}),
\begin{eqnarray}
\label{LagrangianI}
{\mathcal{L}}(\tau)& = &\frac{\dot{x}^2}{4}+\frac{1}{2}\psi_A\dot{\psi}_A-
i\,\dot{x}_{\mu}{\mathcal{A}}_{\mu} +
{\mathcal{H}}^2-\frac{1}{2}{\mathcal{K}}_{\mu\nu}{\mathcal{K}}_{\mu\nu}+2i\,\psi_{\mu}\psi_5\left({\mathcal{D}}_{\mu}{\mathcal{H}}+
i\,\dot{x}_{\nu}{\mathcal{K}}_{\mu\nu}\right) \nonumber \\
& + & i\, \psi_{\mu}\psi_{\nu}\left({\mathcal{F}}_{\mu\nu}+
\lbrace{\mathcal{H}},{\mathcal{K}}_{\mu\nu}\rbrace\right) -
2\psi_{\mu}\psi_{\nu}\psi_{\rho}\left(\psi_5{\mathcal{D}}_{\mu}{\mathcal{K}}_{\mu\nu}+
\frac{1}{2}\psi_{\sigma}{\mathcal{K}}_{\mu\nu}{\mathcal{K}}_{\rho\sigma}\right).
\end{eqnarray}
and the trivial integration over $\psi_6$ can been carried
out, so that the insertion yields
\begin{eqnarray}
\label{InsertionI}
w(T)&=&-4i\,\psi_5\psi_{\mu}\psi_{\nu}\delta{\mathcal{A}}_{\mu}\dot{x}_{\nu}-
2i\,\psi_5{\mathcal{D}}_{\mu}\delta{\mathcal{A}}_{\mu}-
2\psi_{\mu}\left\lbrace\delta{\mathcal{A}}_{\mu},{\mathcal{H}}\right\rbrace\nonumber\\
&&+2i\,\psi_{\mu}\left[\delta{\mathcal{A}}_{\nu},{\mathcal{K}}_{\mu\nu}\right]-
2i\,\psi_{\mu}\psi_{\lambda}\psi_{\sigma}
\left\lbrace\delta{\mathcal{A}}_{\mu},{\mathcal{K}}_{\lambda\sigma}\right\rbrace.
\end{eqnarray}
The normalization ${\mathcal{N}}$ coming from the momentum integration, satisfies
\begin{equation}
{\mathcal{N}}\int{\mathcal{D}} x e^{-\int_0^Td\tau\frac{\dot{x}}{4}}=(4\pi T)^{-D/2}\int d^D x.
\end{equation}
The Green function for the bosonic field $x$ is the same as for the
real part of the effective action, \eq~(\ref{PropR}), while the Green
function of the Grassman fields $\psi_A$ differs due to the presence
of the zero modes. The fermionic fields are split according to
$\psi_A(\tau)=\psi_A^0+\psi_A^{'}(\tau)$, with
$\partial_{\tau}\psi_A^0=0$ and $\int_0^Td\tau\psi_A^{'}(\tau)=0$ and
the measure turns into
${\mathcal{D}}\psi=d\psi_1d\psi_2d\psi_3d\psi_4d\psi_5 {\mathcal{D}}\psi'$. The Green
function for the $\psi_A^{'}$ fields, defined on a space orthogonal to
the zero modes, reads
\begin{equation}
\left\langle \psi_A^{'}(\tau_1)\psi_B^{'}(\tau_2) \right\rangle =
\delta_{A B}\left( \frac{1}{2} \textrm{sign}(\tau_1-\tau_2)-\frac{(\tau_1-\tau_2)}{T}\right).
\end{equation}
These results can be easily generalized to different dimensions. In
two dimension, one obtains an additional overall factor $-i$ from the
Jacobian of the zero modes and the fermionic measure reads
${\mathcal{D}}\psi=d\psi_1d\psi_2d\psi_5 {\mathcal{D}}\psi'$.
\subsubsection{The Effective Density}
The effective density is obtained by varying with respect to the ${\mathcal{H}}$
field, so that $\delta\tilde\Sigma=-\gamma_5\delta{\mathcal{H}}$. In comparison
to the worldline representation of the covariant current only the
insertion changes into
\begin{equation}
\label{commDensity}
\left[ \delta\tilde\Sigma ,\tilde\Sigma \right]=
-\gamma_5\gamma_{\mu}\left\lbrace \delta{\mathcal{H}},p_{\mu}-
{\mathcal{A}}_{\mu}\right\rbrace+\left[\delta{\mathcal{H}},{\mathcal{H}}\right]+
\frac{i}{2}\gamma_{\mu}\gamma_{\nu}\left[\delta{\mathcal{H}},{\mathcal{K}}_{\mu\nu}\right].
\end{equation}
The corresponding insertion $w(T)$ in the path integral reads then
\begin{equation}
\label{insertionDensity}
w(T)=-2i\,\psi_{\mu}\dot{x}_{\mu}\delta{\mathcal{H}}+
2\psi_5\left[\delta{\mathcal{H}},{\mathcal{H}}\right]+
2i\,\psi_{\mu}\psi_{\nu}\left[\delta{\mathcal{H}},{\mathcal{K}}_{\mu\nu}\right].
\end{equation}
Since $\delta{\mathcal{A}}$ carries an index, the effective current is of one
order lower than the effective density and usually results in less
terms to calculate. The advantage of the effective density lies in
the matching process, since the factors in the effective density
consist of the same type as found in the effective action. They both
combine the same type of object, ${\mathcal{D}} {\mathcal{H}}$ and ${\mathcal{F}}$, to the same kind of
order, while the effective current combines the terms to a lower
order. Besides, there is no distinction between a consistent effective
density and a covariant effective density, as there is for the
effective current, as will be explained in the next section.
\subsubsection{Distinction between the Consistent and the Covariant Current}
With \eq~(\ref{current}) an expression for the covariant current which
is chiral and gauge covariant was derived. This current cannot be
the variation of the effective action, since the effective action
contains the chiral anomaly, and in fact the covariant current is not
a variation of any action. The reason for this is that performing the
variation does not commute with the regularization procedure we used,
namely the Schwinger trick. On the other hand, knowing the chiral
anomaly, one can reproduce the so-called consistent current that
denotes the true variation of the effective action.
To explain the relation between the two currents, we define a general
variation
\begin{equation}
\delta_Y = \int dx \, Y_{\mu}^a(x) \frac{\delta}{\delta {\mathcal{A}}_{\mu}(x)},
\end{equation}
so that a gauge variation $\delta_{\xi}$ is given by
\begin{equation}
\label{GaugeVar}
\delta_{\xi}=\int dx \, \left({\mathcal{D}}_{\mu}\xi\right)(x)\frac{\delta}{\delta {\mathcal{A}}_{\mu}(x)}.
\end{equation}
Two subsequent variations have then the commutator
$[\delta_{Y},\delta_{\xi}]=\delta_{[Y,\xi]}$ and in order to find the
transformation properties of the consistent current, one can apply
this commutator to the effective action
\begin{equation}
\label{GaugeComm}
[\delta_Y,\delta_{\xi}]W^{-}[{\mathcal{A}}_{\mu}]=
\delta_{[Y,\xi]}W^{-}[{\mathcal{A}}_{\mu}].
\end{equation}
Using the anomalous Ward identity~\cite{Anomaly}
\begin{equation}
\label{AnomWard}
\delta_{\xi}W^{-}[{\mathcal{A}}_{\mu}]=\int dx \, \xi(x)G[{\mathcal{A}}_{\mu}](x),
\end{equation}
with $G[{\mathcal{A}}_{\mu}]$ denoting the consistent anomaly, one can
evaluate both sides of \eq~(\ref{GaugeComm}) to obtain
\bea
\int dx \, [Y_{\mu},\xi](x)\frac{\delta}{\delta {\mathcal{A}}_{\mu}(x)}W^{-}[{\mathcal{A}}_{\mu}] &=&
\delta_Y\int dx \, \xi(x)G[{\mathcal{A}}_{\mu}](x) \nn \\
&& -\delta_{\xi}\int dx \, Y_{\mu}(x)\frac{\delta}{\delta {\mathcal{A}}_{\mu}(x)}W^{-}[{\mathcal{A}}_{\mu}].
\eea
Defining the consistent current as the variation of the effective
action
\begin{equation}
\langle j^{\mu}(x)\rangle=\frac{\delta}{\delta {\mathcal{A}}_{\mu}(x)}W^{-}[{\mathcal{A}}_{\mu}],
\end{equation}
one finds
\begin{eqnarray}
\int dx \, Y_{\mu}(x)\delta_{\xi}\langle j^{\mu}(x)\rangle =
\int dx \, Y_{\mu}[\langle j^{\mu}(x)\rangle,\xi](x)+
\int dx \, \xi(x)\delta_YG[{\mathcal{A}}_{\mu}](x).
\end{eqnarray}
Since $Y$ was a general variation this leads to
\begin{equation}
\delta_{\xi}\langle j^{\mu}(x)\rangle=[\langle j^{\mu}(x)\rangle,\xi]+
\int dy \, \xi^b(y)\frac{\delta}{\delta {\mathcal{A}}_{\mu}(x)}G[{\mathcal{A}}_{\mu}](y).
\end{equation}
This shows that only if the anomaly vanishes, the current transforms
covariantly. This relation can be used to determine the connection
between the consistent current, i.e. the true variation of the action,
and the covariant current. The latter is obtained by adding an object
$P^{\mu}[{\mathcal{A}}_{\mu}]$, called the Bardeen-Zumino polynomial
\cite{BardeenZumino}, to the consistent current so that the sum transforms
covariantly
\begin{equation}
\label{defCovCurrent}
\langle\bar{j^{\mu}}\rangle=\langle j^{\mu}\rangle+\langle P^{\mu}\rangle.
\end{equation}
This implies the following gauge transformation property for the BZ polynomial
\begin{equation}
\label{condP}
\delta_{\xi}P^{\mu}[{\mathcal{A}}_{\mu}](x)=
\left[P^{\mu}[{\mathcal{A}}_{\mu}],\xi\right](x)-
\int dy \, \xi(y)\frac{\delta}{\delta {\mathcal{A}}_{\mu}(x)}G[{\mathcal{A}}_{\mu}](y).
\end{equation}
It is not obvious that such an object exists, but using
\begin{equation}
\label{defP}
P^{\mu}[{\mathcal{A}}_{\mu}]=\frac{1}{48\pi^2}\epsilon^{\mu\nu\lambda\sigma}
\tr \chi\left({\mathcal{A}}_{\nu}{\mathcal{F}}_{\lambda\sigma}+
{\mathcal{F}}_{\lambda\sigma}{\mathcal{A}}_{\nu} + i\,{\mathcal{A}}_{\nu}{\mathcal{A}}_{\lambda}{\mathcal{A}}_{\sigma}\right),
\end{equation}
and the consistent anomaly~\cite{Anomaly}
\begin{equation}
G[{\mathcal{A}}_{\mu}]=\frac{1}{24\pi^2}\epsilon^{\mu\nu\lambda\sigma}
\tr \, \chi \, \partial_{\mu}\left({\mathcal{A}}_{\nu}\partial_{\lambda}{\mathcal{A}}_{\sigma} -
\frac{i}{2}{\mathcal{A}}_{\nu}{\mathcal{A}}_{\lambda}{\mathcal{A}}_{\sigma}\right),
\end{equation}
it can be shown that the definition of $P^{\mu}$ in \eq~(\ref{defP})
provides a unique polynomial in ${\mathcal{A}}_{\mu}$ that satisfies
\eq~(\ref{condP}). The corresponding functions in two dimensions are
given by
\be
P^{\mu}= \frac{1}{4\pi}\epsilon^{\mu\nu}\tr {\mathcal{A}}_{\nu}, \quad
G[{\mathcal{A}}_{\mu}]=\frac{1}{4\pi}\epsilon^{\mu\nu}
\tr \, \chi \, \partial_{\mu} \, {\mathcal{A}}_{\nu}.
\ee
As stated above, the path integral in \eq~(\ref{current}) constitutes
a worldline representation of the covariant current. To obtain the
imaginary part of the effective action from the covariant current one
can use the following ansatz
\begin{equation}
\label{ansatz_action}
W^{-}=\Gamma_{gWZW}+W_c^{-}.
\end{equation}
Here, $\Gamma_{gWZW}$ is an extended gauged Wess-Zumino-Witten
action~\cite{WZ,Witten,Hoker}, which is chosen to reproduce
the correct chiral anomaly, and $W_c^{-}$ denotes a chiral invariant
part. The variation of the functional $\Gamma_{gWZW}$, consists of
a part that saturates the anomaly, namely the BZ polynomial, and a
covariant remainder which has to be added to the variation of $W_c^{-}$
to yield the covariant current.
\subsubsection{The Wess-Zumino-Witten action}
When the effective action is separated into two parts, it is required
by the non-covariant part that it reproduces the anomaly. It is well
known that the WZW action has this property.
The ungauged WZW action in four dimension is e.g. of the form
\begin{equation}
\label{ungWZW}
\Gamma({\mathbf{U}})=\frac{i}{48\pi^2}\int_Qd^5x \, \epsilon^{abcde}
\tr \left[\frac{1}{5}{\mathbf{U}}^{-1}\partial_a{\mathbf{U}}\Ub^{-1}\partial_b{\mathbf{U}}
{\mathbf{U}}^{-1}\partial_c{\mathbf{U}}\Ub^{-1}\partial_d{\mathbf{U}}\Ub^{-1}\partial_e{\mathbf{U}}\right],
\end{equation}
where Q is a five-dimensional space with boundary $\partial Q$ equal
to the $R^4$ flat Euclidean space. The matrix ${\mathbf{U}}$ is a unitary
matrix, and is usually related to the case where the mass can be
expressed as a constant times that unitary matrix. We are interested
in the more general case when the mass matrix is not of this form
which is called extended WZW action. In addition, the presence of the
background gauge fields makes a gauging of the action mandatory. The
gauged extended WZW action can be generally expressed as the integral
in five dimensions~\cite{Hoker}. Unlike the action itself, the
resulting current turns out to be a total derivative in five
dimensions, such that it can be represented by an integral over the
physical four-dimensional space
\begin{eqnarray}
\delta\Gamma_{gWZW}&=&
\frac{1}{96\pi^2}\int d^4x \, \epsilon^{\mu\nu\lambda\sigma}\tr \,\chi
\left[\delta{\mathcal{A}}_{\mu}\left(
-{\mathcal{H}}^{-1}{\mathcal{D}}_{\nu}{\mathcal{H}}\Hc^{-1}{\mathcal{D}}_{\lambda}{\mathcal{H}}\Hc^{-1}{\mathcal{D}}_{\sigma}{\mathcal{H}}\right.\right.\nonumber\\
&&\quad+{\mathcal{D}}_{\nu}{\mathcal{H}}\Hc^{-1}{\mathcal{D}}_{\lambda}{\mathcal{H}}\Hc^{-1}{\mathcal{D}}_{\sigma}{\mathcal{H}}\Hc^{-1}-
i\,\left\lbrace{\mathcal{H}}^{-1}{\mathcal{D}}_{\nu}{\mathcal{H}}-{\mathcal{D}}_{\nu}{\mathcal{H}}\Hc^{-1},{\mathcal{F}}_{\lambda\sigma}\right\rbrace\nonumber\\
&&\quad+\frac{i}{2}{\mathcal{H}}\left\lbrace{\mathcal{H}}^{-1}{\mathcal{D}}_{\nu}{\mathcal{H}},{\mathcal{F}}_{\lambda\sigma}\right\rbrace{\mathcal{H}}^{-1}
-\frac{i}{2}{\mathcal{H}}^{-1}\left\lbrace{\mathcal{D}}_{\nu}{\mathcal{H}}\Hc^{-1},{\mathcal{F}}_{\lambda\sigma}\right\rbrace{\mathcal{H}}\nonumber\\
&&\quad\left.\left.-2\left\lbrace{\mathcal{A}}_{\nu},{\mathcal{F}}_{\lambda\sigma}\right\rbrace-
2i\,{\mathcal{A}}_{\nu}{\mathcal{A}}_{\lambda}{\mathcal{A}}_{\sigma}\right)\right],
\end{eqnarray}
or in two dimensions
\begin{eqnarray}
\label{Contr2d}
\delta\Gamma_{gWZW}&=&
\frac{1}{8\pi}\int d^2x \,\epsilon^{\mu\nu} \tr \, \chi\left[\delta{\mathcal{A}}_{\mu}
\left(-i\,{\mathcal{H}}^{-1}{\mathcal{D}}_{\nu}{\mathcal{H}}+i\,{\mathcal{D}}_{\nu}{\mathcal{H}}\Hc^{-1}-2{\mathcal{A}}_{\nu}\right)\right].
\end{eqnarray}
Notice that in both cases the last term in the current denotes the BZ
polynomial. The remaining chiral covariant terms have to be subtracted
from the covariant current before it is matched to the effective
action according to the ansatz made in \eq~(\ref{ansatz_action}).
\section{Lowest Order Effective Action\label{sec_low}}
\subsection{Effective covariant current}
In order to reproduce the results from ref.~\cite{Salcedo2}, we
neglect in this section the antisymmetric field $K_{\mu\nu}$. The
fields ${\mathcal{A}}$ and ${\mathcal{H}}$ are matrices of some internal group, and we
only assume that ${\mathcal{H}}(x_0)$ is nowhere singular. With this in mind, we
restate our result \eq~(\ref{current}) from the last section in $D$
dimensions
\begin{equation}
\label{dWIm2}
\delta W^{-}=-\frac{i^{D/2}}{8}\, \mathrm{tr}\int_0^{\infty}dT {\mathcal{N}}
\int{\mathcal{D}} x \int_{P}{\mathcal{D}}\psi \,\chi w(T){\mathcal{P}} e^{-\int_0^Td\tau{\mathcal{L}}(\tau)},
\end{equation}
with
\begin{eqnarray}
\label{ohneK}
{\mathcal{L}}(\tau) &=&\frac{\dot{x}^2}{4}+\frac{1}{2}\psi_A\dot{\psi}_A-
i\,\dot{x}_{\mu}{\mathcal{A}}_{\mu} + {\mathcal{H}}^2+2i\,\psi_{\mu}\psi_5{\mathcal{D}}_{\mu}{\mathcal{H}}+
i\,\psi_{\mu}\psi_{\nu}{\mathcal{F}}_{\mu\nu},\nonumber\\
w(T)&=&-4i\,\psi_5\psi_{\mu}\psi_{\nu}\delta{\mathcal{A}}_{\mu}\dot{x}_{\nu}-
2i\,\psi_5{\mathcal{D}}_{\mu}\delta{\mathcal{A}}_{\mu}-2\psi_{\mu}\left\lbrace\delta{\mathcal{A}}_{\mu},{\mathcal{H}}\right\rbrace.
\end{eqnarray}
Next, the derivative expansion of the heat kernel is used. In the
derivative expansion terms are classified by the number of covariant
indices that they carry, so that ${\mathcal{D}}_{\mu}{\mathcal{H}}$ is of first order,
while ${\mathcal{F}}_{\mu\nu}$ is of second order. The worldline formalism is
well suited for this expansion, and there are two major advantages
compared to the more traditional methods used e.g. in
ref.~\cite{Salcedo2}. First, the tedious manipulations using the
$\gamma$ algebra are avoided. Secondly, the momentum integration is
omitted and replaced by the rather trivial integration in $\tau$
space.
The coordinate is split as $x(\tau)=x_0+y(\tau)$, and we work in the
Fock-Schwinger gauge~\cite{Fockgauge2}, in which ${\mathcal{A}}(x)\cdot y=0$. In
this gauge, expressions remain gauge covariant and the field ${\mathcal{A}}$ can
be expressed in terms of the field strength tensor ${\mathcal{F}}_{\mu\nu}$ by
\begin{equation}
\label{expA}
{\mathcal{A}}_{\mu}(x)=\int_0^1d\alpha \, \alpha \, {\mathcal{F}}_{\rho\mu}(x_0+\alpha y)y_{\rho}.
\end{equation}
All background fields can then be expanded around the point $x_0$ in terms
of covariant derivatives
\begin{equation}
X(x_0+y(\tau))=\exp\left(y(\tau)\cdot{\mathcal{D}}_{x_0}\right)X(x_0),
\end{equation}
where ${\mathcal{D}}_{x_0}$ refers to the covariant derivative in
\eq~(\ref{defF}) with respect to $x_0$. With the expansion of the
field strength tensor in terms of covariant derivatives and
\eq~(\ref{expA}), one can rewrite the field ${\mathcal{A}}$ as
\begin{eqnarray}
{\mathcal{A}}_{\mu}(x)= \frac{1}{2}y_{\rho}{\mathcal{F}}_{\rho\mu}(x_0)+
\frac{1}{3}y_{\alpha}y_{\rho}{\mathcal{D}}_{\alpha}{\mathcal{F}}_{\rho\mu}(x_0)
+\frac{1}{4\cdot2!}y_{\alpha}y_{\beta}y_{\rho}
{\mathcal{D}}_{\alpha}{\mathcal{D}}_{\beta}{\mathcal{F}}_{\rho\mu}(x_0)+\ldots \,\,.
\end{eqnarray}
Since we will not carry out the integration with respect to $x_0$ we
use the following notation in $D$ dimensions
\begin{equation}
\label{norm}
\left\langle X\right\rangle_D= - \left( \frac{i}{4\pi} \right)^{D/2}
\mathrm{tr}\chi\int d^D x_0 X.
\end{equation}
It is important to remember that $\chi$ and ${\mathcal{H}}$ anticommute; hence, when
the cyclic property of the trace is used, a minus sign is
generated, for example
\bea
\label{norm_ex}
\left\langle \epsilon^{\mu\nu\lambda\sigma}
{\mathcal{H}} {\mathcal{F}}_{\mu\nu} {\mathcal{H}}^3 {\mathcal{F}}_{\lambda\sigma}\right\rangle&=&
-\left\langle \epsilon^{\mu\nu\lambda\sigma}
{\mathcal{F}}_{\mu\nu} {\mathcal{H}}^3 {\mathcal{F}}_{\lambda\sigma}{\mathcal{H}}\right\rangle=
-\left\langle \epsilon^{\mu\nu\lambda\sigma}
{\mathcal{H}}^3 {\mathcal{F}}_{\lambda\sigma}{\mathcal{H}} {\mathcal{F}}_{\mu\nu} \right\rangle \nn \\
&=&-\left\langle \epsilon^{\mu\nu\lambda\sigma}
{\mathcal{H}}^3 {\mathcal{F}}_{\mu\nu}{\mathcal{H}} {\mathcal{F}}_{\lambda\sigma} \right\rangle.
\eea
After expanding the mass field
${\mathcal{H}}(x)^2={\mathcal{H}}^2(x_0)+y_{\mu}{\mathcal{D}}_{\mu}{\mathcal{H}}^2(x_0)+\ldots$, the field
${\mathcal{H}}(x_0)$ is treated non-perturbatively. Since all the fields can be
matrices of some internal space the resulting expressions normally
cannot be expressed in closed form. For this case we use the labeled
operator notation laid down in ref.~\cite{Feynman2,Salcedo1}. The
notation works as follows: In an expression
$f(A_1,B_2,\ldots)XY\ldots$, the labels of the operators $A$, $B$,
$\ldots$ denote the position of that operator with respect to the
remaining operators $XY\ldots$. For instance, for the function
$f(A,B)=\alpha(A)\beta(B)$, the expression $f(A_1,B_2)XY$ represents
$\alpha(A)X\beta(B)Y$. In the case at hand, the operator appearing in
the functions is always $m:={\mathcal{H}}(x_0)$, such that general functions $f$
can be easily interpreted in the basis where $m$ is diagonal. Using
this notation, \eq~(\ref{norm_ex}) can be recast as
\bea
\left\langle \epsilon^{\mu\nu\lambda\sigma}
{\mathcal{H}} {\mathcal{F}}_{\mu\nu} {\mathcal{H}}^3 {\mathcal{F}}_{\lambda\sigma}\right\rangle&=&
\left\langle \epsilon^{\mu\nu\lambda\sigma}
m_1 m_2^3 {\mathcal{F}}_{\mu\nu}{\mathcal{F}}_{\lambda\sigma}\right\rangle=
-\left\langle \epsilon^{\mu\nu\lambda\sigma}
m_3 m_2^3 {\mathcal{F}}_{\mu\nu}{\mathcal{F}}_{\lambda\sigma}\right\rangle \nn \\
&=& -\left\langle \epsilon^{\mu\nu\lambda\sigma}
m_2 m_1^3 {\mathcal{F}}_{\mu\nu}{\mathcal{F}}_{\lambda\sigma}\right\rangle
=-\left\langle \epsilon^{\mu\nu\lambda\sigma}
{\mathcal{H}}^3 {\mathcal{F}}_{\mu\nu}{\mathcal{H}} {\mathcal{F}}_{\lambda\sigma} \right\rangle.
\eea
This notation can also be used to simplify the matrix valued
derivative. Using the definition
\begin{equation}
(\nabla f )(m_1,m_2) := \frac{f(m_1)-f(m_2)}{m_1-m_2},
\end{equation}
it is possible to prove that
\be
{\mathcal{D}}_{\mu}f(m)=(\nabla f)(m_1,m_2) \, {\mathcal{D}}_{\mu} {\mathcal{H}}.
\ee
For example, in the polynomial case $f(m)=m^3$ one obtains
\bea
{\mathcal{D}}_{\mu}f(m) &=& {\mathcal{D}}_{\mu} ({\mathcal{H}}^3)
= {\mathcal{D}}_{\mu} {\mathcal{H}} \, {\mathcal{H}}^2 + {\mathcal{H}} {\mathcal{D}}_{\mu} {\mathcal{H}} \, {\mathcal{H}} + {\mathcal{H}}^2 {\mathcal{D}}_{\mu} {\mathcal{H}} \nn \\
&=& (m_2^2 + m_1 m_2 + m_1^2 ) \, {\mathcal{D}}_{\mu} {\mathcal{H}}
= \frac{m_1^3 - m_2^3}{m_1 - m_2} \, {\mathcal{D}}_{\mu} {\mathcal{H}} \nn \\
&=& (\nabla f)(m_1,m_2) \, {\mathcal{D}}_{\mu} {\mathcal{H}}.
\eea
As mentioned earlier, non-polynomial expressions are hereby
interpreted in a basis where $m$ is diagonal, so that for
$m=\textrm{diag}(d_1, \dots, d_n)$
\be
\frac{f(m_1)-f(m_2)}{m_1-m_2} \, X
= \frac{f(d_i)-f(d_j)}{d_i-d_j} \, X_{ij}.
\ee
More general, this suggests the following definition for the case with several
variables:
\begin{equation}
\nabla_k f(m_1,\ldots,m_n) =
\frac{f\left(m_1,\ldots,{\hat m}_{k+1},\ldots,m_n\right)
-f\left(m_1,\ldots,\hat{m_{k}},\ldots,m_n\right)}{m_k-m_{k+1}},
\end{equation}
where $\hat{m_k}$ indicates that the corresponding argument is left
out.
If all arguments of the functions are of the same type one can further
simplify the notation and use subscripts to refer to the argument of
the function, e.g. $f(m_1,m_2)=:f_{12}$ and we employ this notation
in the following. Additionally, negative arguments will be denoted by
underlining the corresponding index,
$f(-m_1,m_2)=:f_{\underline{1}2}$. More applications of the labeled
operator notation can be found in refs.~\cite{Salcedo1,Salcedo2}.
The path ordering in \eq~(\ref{dWIm2}) is defined by
\begin{equation}
\label{defPath}
{\mathcal{P}}\prod_{i=1}^N\int_0^Td\tau_i\equiv N!
\int_0^Td\tau_1\int_0^{\tau_1}d\tau_2\cdots\int_0^{\tau_{N-1}}d\tau_N=
N!\int_0^Td\tau_1\cdots\int_0^Td\tau_N\prod_{i-1}^{N-1}\theta(\tau_i-\tau_{i+1}).
\end{equation}
Separating the Lagrangian \eq~(\ref{ohneK}) into
${\mathcal{L}}(\tau)={\mathcal{L}}_0(\tau)+{\mathcal{H}}^2(x_0)+{\mathcal{L}}_1(\tau)$, with
\begin{eqnarray}
{\mathcal{L}}_0(\tau)&=&\frac{\dot{x}^2}{4}+\frac{1}{2}\psi_A\dot{\psi}_A, \nonumber\\
{\mathcal{L}}_1(\tau)&=&-i\,\dot{x}_{\mu}{\mathcal{A}}_{\mu}(x)+2i\,\psi_{\mu}\psi_5{\mathcal{D}}_{\mu}{\mathcal{H}}(x)+
i\,\psi_{\mu}\psi_{\nu}{\mathcal{F}}_{\mu\nu}(x)+y_{\mu}{\mathcal{D}}_{\mu}{\mathcal{H}}^2(x_0)+\ldots,
\end{eqnarray}
the terms of the expansion of ${\mathcal{H}}^2(x)$, except the leading term
${\mathcal{H}}^2(x_0)$, are attributed to ${\mathcal{L}}_1(\tau)$, and treated
perturbatively. Notice that ${\mathcal{L}}_0$ commutes with the rest of the
Lagrangian, so that the expansion of the path ordered exponential in
\eq~(\ref{dWIm2}) takes the form
\begin{eqnarray}
\label{expansionPath}
{\mathcal{P}} e^{-\int_0^T d\tau {\mathcal{L}}(\tau)}=e^{-\int_0^T d\tau {\mathcal{L}}_0(\tau)}
\left(e^{-T{\mathcal{H}}^2(x_0)}+
\int_0^Td\tau_1e^{-(T-\tau_1){\mathcal{H}}^2(x_0)}\left(-{\mathcal{L}}_1(\tau_1)\right)
e^{-\tau_1{\mathcal{H}}^2(x_0)}\right.\nonumber\\
\quad\left.
+\int_0^Td\tau_1\int_0^{\tau_1}d\tau_2e^{-(T-\tau_1){\mathcal{H}}^2(x_0)}
\left(-{\mathcal{L}}_1(\tau_1)\right)e^{-(\tau_1-\tau_2){\mathcal{H}}^2(x_0)}\left(-{\mathcal{L}}_1(\tau_2)\right)
e^{-\tau_2{\mathcal{H}}^2(x_0)}+\ldots\right).
\end{eqnarray}
When performing the $\psi$ integrals, the zero modes have to be
saturated and at least a factor $\psi_1^0 \dots\psi_D^0\psi_5^0$ is
required from the Grassman fields in order to contribute. The first
term in
\eq~(\ref{expansionPath}) lacks the appropriate $\psi$ factor except
in two dimensions, where the first term of the insertion
\eq~(\ref{ohneK}) already has the appropriate factor. However it
contains a factor $\dot{x}_{\mu}$ which must be contracted with a
similar factor to form a Green function, hence it does not contribute
and can be left out. The rest of \eq~(\ref{expansionPath}) can be
simplified using the labeled operator notation. Using the expression
$m_n^2$ to denote ${\mathcal{H}}^2(x_0)$ in $n$th position, one obtains
\begin{eqnarray}
\label{expansionPath2}
{\mathcal{P}} e^{-\int_0^T d\tau {\mathcal{L}}(\tau)}&=&e^{-\int_0^T d\tau {\mathcal{L}}_0(\tau)}\left(
-\int_0^Td\tau_1e^{-T m_1^2-\tau_1\left(m_2^2-m_1^2\right)}{\mathcal{L}}_1(\tau_1)
\right.\nonumber\\
&&\hskip -1 cm \left.+\int_0^Td\tau_1\int_0^{\tau_1}d\tau_2e^{-T m_1^2-
\tau_1\left(m_2^2-m_1^2\right)-
\tau_2\left(m_3^2-m_2^2\right)}{\mathcal{L}}_1(\tau_1){\mathcal{L}}_1(\tau_2)+\ldots\right).
\end{eqnarray}
The evaluation of the worldline path integral can be summarized as
follows: First, all fields in \eq~(\ref{expansionPath2}) and the
insertion are expanded around $x_0$. Next, the functional integration
over the $y$ fields is carried out, generating bosonic Green
functions. Then, the $\psi$ integrations are performed saturating the
zero modes and generating fermionic Green functions. Finally, the $T$
and $\tau$ integrations are performed.
Before presenting the actual calculation, we comment on the behavior
of the effective action under complex conjugation. As noted earlier,
in any contribution to the imaginary part of the action the field
$\psi_5$ appears an odd number of times. If one attributes a factor
$i$ to the operators ${\mathcal{F}}$ and $\delta{\mathcal{A}}$ one observes that the
remaining expressions in the current in \eq~(\ref{ohneK}) are
real. Accordingly, all expressions in $W^-$ are real as long as a
factor $i$ is attributed to the operator ${\mathcal{F}}$. In addition, notice
that the effective action has to be an even function in the masses due
to chiral invariance.
In order to showcase the method, we present the lowest order
calculation in two dimensions. The lowest order contribution coming
from the first term in the insertion is given by
\begin{equation}
-4i\,\psi_5\psi_{\mu}\psi_{\nu}\dot{y}_{\nu}(T)
\int_0^Td\tau_1e^{-T m_1^2-\tau_1\left(m_2^2-m_1^2\right)}
y_{\alpha}(\tau){\mathcal{D}}_{\alpha}{\mathcal{H}}^2 \, \delta{\mathcal{A}}_{\mu}(T).
\end{equation}
Performing the $y$ and $\psi$ integrals one obtains
\begin{eqnarray}
&& \hskip -2 cm
\frac{i}{2}\left\langle \, \epsilon^{\mu\nu}
(m_1+m_2) \int_0^{\infty}\frac{dT}{T}\int_0^Td\tau_1e^{-T m_1^2-\tau_1\left(m_2^2-m_1^2\right)}
\dot{g}_B(T,\tau_1) \, {\mathcal{D}}_{\mu}{\mathcal{H}} \,\delta{\mathcal{A}}_{\nu}
\right\rangle \nonumber\\
&=&
\frac{i}{2}\left\langle \epsilon^{\mu\nu}
\,J^2_{12}(m_1+m_2) \, {\mathcal{D}}_{\mu}{\mathcal{H}} \, \delta{\mathcal{A}}_{\nu}
\right\rangle.
\end{eqnarray}
The second term of the insertion does not contribute at lowest order
since it is already of second order in derivatives but lacks the
appropriate fermionic factor to saturate the zero modes. The third
term of the insertion leads only to one contribution of the form
\begin{equation}
\label{C2dlo2c}
-2\psi_{\mu}\left\lbrace\delta{\mathcal{A}}_{\mu},{\mathcal{H}}\right\rbrace
\int_0^Td\tau_1e^{-T m_1^2-\tau_1\left(m_2^2-m_1^2\right)}
\left(-2i\,\psi_{\nu}\psi_5{\mathcal{D}}_{\nu}{\mathcal{H}}\right).
\end{equation}
yielding
\begin{eqnarray}
&& \hskip -2 cm
-\frac{i}{2}\left\langle \,\epsilon^{\mu\nu}
(m_1-m_2)\int_0^{\infty}\frac{dT}{T}
\int_0^Td\tau_1e^{-T m_1^2-\tau_1\left(m_2^2-m_1^2\right)}\, {\mathcal{D}}_{\mu}{\mathcal{H}} \,\delta{\mathcal{A}}_{\nu}
\right\rangle\nonumber\\
&=& -\frac{i}{2}\left\langle \epsilon^{\mu\nu} \,J^1_{12}(m_1-m_2){\mathcal{D}}_{\mu}{\mathcal{H}} \, \delta{\mathcal{A}}_{\nu}
\right\rangle.
\end{eqnarray}
The factor $(m_1-m_2)$ results from the anticommutator in
\eq~(\ref{C2dlo2c}), and the sign change in the cyclic property of the
trace as explained in \eq~(\ref{norm_ex}). The integrals $J$ are given
in Appendix \ref{app:Integrals}. The total current is hence given by
\be
\label{2dcovcont}
\delta W^{-}
= - i \left\langle \,\epsilon^{\mu\nu}
A^1_{12}{\mathcal{D}}_{\mu}{\mathcal{H}} \,\delta{\mathcal{A}}_{\nu} \right\rangle,
\ee
\be
A^1_{12} =
\frac{1}{m_1-m_2} - \frac{m_1m_2\log\left(m_1^2/m_2^2\right)}{(m_1-m_2)(m_1^2-m_2^2)},
\ee
where the function $A^1_{12}$ has been defined.
This agrees with the results obtained in ref.~\cite{Salcedo2}.
\subsection{Effective Density}
The effective density can be obtained similar as the covariant
current, utilizing the insertion in
\eq~(\ref{insertionDensity}). Neglecting the antisymmetric tensor
${\mathcal{K}}$, the insertion is
\begin{equation}
\label{insertionDensity2}
w(T)=-2i\,\psi_{\mu}\dot{x}_{\mu}\delta{\mathcal{H}}+2\psi_5\left[\delta{\mathcal{H}},{\mathcal{H}}\right].
\end{equation}
The contributions to the effective density are
\begin{eqnarray}
\label{cF2dlo}
\delta W^{-}&=&\left\langle\epsilon^{\mu\nu} \left(
\frac{i}{4}\left(J^1_{12}(m_1+m_2) + J^2_{12}(m_1-m_2)\right) {\mathcal{F}}_{\mu\nu} \right.\right.\nonumber\\
&&\left.\left.
- \frac{1}{2}\left(J^5_{123}(m_1+m_3) + J^6_{123}(m_1+m_2) - J^7_{123}(m_2+m_3)\right)
{\mathcal{D}}_{\mu}{\mathcal{H}}{\mathcal{D}}_{\nu}{\mathcal{H}}
\right) \delta{\mathcal{H}} \right\rangle\nonumber\\
&=& \left\langle \epsilon^{\mu\nu}
\left( -\frac{i}2 B^1_{12}{\mathcal{F}}_{\mu\nu}
+ B^2_{123}{\mathcal{D}}_{\mu}{\mathcal{H}}{\mathcal{D}}_{\nu}{\mathcal{H}}\right) \delta{\mathcal{H}} \right\rangle.
\end{eqnarray}
where the functions $B_{12}$ and $B_{123}$ are given by
\begin{eqnarray}
B^1_{12}&=& - \frac{1}{m_1+m_2} -\frac{m_1m_2}
{(m_1+m_2)(m_1^2-m_2^2)}\log\left(\frac{m_1^2}{m_2^2}\right),\\
B^2_{123}&=&B_{123}^R+B_{123}^L\log(m_1^2)+
B_{\underline{23}1}^L\log(m_2^2)+B_{\underline{3}12}^L\log(m_2^2),
\end{eqnarray}
with
\begin{equation}
B_{123}^R=\frac{1}{(m_1-m_2)(m_2-m_3)(m_1+m_3)},
\end{equation}
\begin{equation}
B_{123}^L=\frac{(m_1^3+m_1m_2m_3)}{(m_1-m_2)(m_1+m_3)(m_1^2-m_2^2)(m_1^2+m_3^2)},
\end{equation}
in accordance with ref.~\cite{Salcedo2}.
\subsection{Effective Action}
We proceed and briefly present the derivation of the imaginary part of
the effective action following ref.~\cite{Salcedo2}. Using the ansatz
in \eq~(\ref{ansatz_action}), the most general functional for $W_c^-$
consistent with chiral and gauge invariance in two dimensions reads
\begin{equation}
W_c^{-}=\left\langle \epsilon^{\mu\nu}N_{12}{\mathcal{D}}_{\mu}{\mathcal{H}}{\mathcal{D}}_{\nu}{\mathcal{H}} \right\rangle.
\end{equation}
An additional term proportional to ${\mathcal{F}}$ could be added but it can be
removed by partial integration. Notice that $N_{12}$ is a real
function according to the comments made in the last section.
The function $N_{12}$ has some nontrivial restrictions. First of all,
the function $N_{12}$ is even in $m$ such that
\be
N(-m_1, -m_2) := N_{\underline{1}\underline{2}}= N_{12}.
\ee
Because of the cyclic property of the trace one obtains
\begin{equation}
N_{12}=N_{\underline{3}2}=N_{\underline{2}1}=N_{2\underline{1}},
\end{equation}
and due to the Hermiticity of $W^{-}$
\begin{equation}
N_{12}=-N_{32}=-N_{\underline{1}2}=-N_{21}.
\end{equation}
Varying $W_{c}^{-}[{\mathcal{A}},{\mathcal{H}}]$ with respect to ${\mathcal{A}}$, one obtains
\begin{equation}
\delta W_c^{-}=
-i \left\langle\epsilon^{\mu\nu} \left(-2\,(m_1+m_2)N_{12}\right)
{\mathcal{D}}_{\mu} {\mathcal{H}} \delta{\mathcal{A}}_{\nu} \right\rangle.
\end{equation}
Comparing this to \eq~(\ref{2dcovcont}) and adding the covariant
contribution in \eq~(\ref{Contr2d}) coming from $\Gamma_{gWZW}$ one
has
\begin{equation}
\frac{1}{m_1-m_2} - \frac{m_1m_2\log\left(m_1^2/m_2^2\right)}
{(m_1-m_2)(m_1^2-m_2^2)}=\frac{1}{2m_1}-\frac{1}{2m_2}-2\,(m_1+m_2)N_{12},
\end{equation}
which finally leads to
\begin{equation}
\label{N12exp}
N_{12}=\frac12 \frac{m_1m_2}{m_1^2-m_2^2}
\left(\frac{\log(m_1^2/m_2^2)}{m_1^2-m_2^2}-\frac{1}{2}\left(\frac{1}{m_1^2}+
\frac{1}{m_2^2}\right)\right).
\end{equation}
At higher order, the matching of the effective potential to the
current potentially becomes more intricate. On the other hand, the
anomaly only contributes to the leading order, such that the knowledge
of the covariant current (that in higher order coincides with the
consistent current) suffices to determine the effective action.
\subsection{Four Dimensions\label{4dim}}
For completeness, we also present the results for the effective action
and the effective current in four dimensions. The matching procedure
proceeds the same way as in ref.~\cite{Salcedo2}, and we do not repeat
it here.
The effective current in four dimensions consists out of three terms
and reads
\begin{equation}
\label{covcur4d}
\delta W_{d=4}^{-}= - i \left\langle\epsilon^{\mu\nu\lambda\sigma}
\left( - \frac{i}{2}A^2_{123}{\mathcal{F}}_{\nu\lambda}{\mathcal{D}}_{\mu}{\mathcal{H}}
- \frac{i}{2}A^3_{123}{\mathcal{D}}_{\mu}{\mathcal{H}}{\mathcal{F}}_{\nu\lambda}
- \,A^4_{1234}{\mathcal{D}}_{\mu}{\mathcal{H}}{\mathcal{D}}_{\nu}{\mathcal{H}}{\mathcal{D}}_{\lambda}{\mathcal{H}}
\right)\delta{\mathcal{A}}_{\sigma} \right\rangle,
\end{equation}
while the effective density can be written as
\begin{eqnarray}
\label{den4d}
\delta W_{d=4}^{-}\lefteqn{ = \left\langle \epsilon^{\mu\nu\lambda\sigma}\left(
\frac{1}{4}B^3_{123}{\mathcal{F}}_{\mu\nu}{\mathcal{F}}_{\lambda\sigma}
+\frac{i}{2}B^4_{1234}{\mathcal{F}}_{\lambda\sigma}{\mathcal{D}}_{\mu}{\mathcal{H}}{\mathcal{D}}_{\nu}{\mathcal{H}}\right.\right.}\nonumber\\
& & \quad +\frac{i}{2}B^5_{1234}{\mathcal{D}}_{\mu}{\mathcal{H}}{\mathcal{F}}_{\lambda\sigma}{\mathcal{D}}_{\nu}{\mathcal{H}}
+\frac{i}{2}B^6_{1234}{\mathcal{D}}_{\mu}{\mathcal{H}}{\mathcal{D}}_{\nu}{\mathcal{H}}{\mathcal{F}}_{\lambda\sigma} \nn \\
&& \left. \left. \phantom{\frac12} - B^7_{12345}{\mathcal{D}}_{\mu}{\mathcal{H}}{\mathcal{D}}_{\nu}{\mathcal{H}}{\mathcal{D}}_{\lambda}
{\mathcal{H}}{\mathcal{D}}_{\sigma} {\mathcal{H}} \right) \delta{\mathcal{H}} \right\rangle.
\end{eqnarray}
The functions $A^2_{123}$, $A^3_{123}$, $A^4_{1234}$, $B^3_{123}$,
$B^4_{1234}$, $B^5_{1234}$, $B^6_{1234}$, and $B^7_{12345}$ are given in
Appendix~\ref{app:ResEff}.
\section{Next to Leading Order Effective Action in Two Dimensions\label{sec_next}}
In this section we present as a novel result the imaginary part of the
effective action in next to leading order and two dimensions. Even
though the results are rather lengthy, the evaluation of the worldline
path integral involves only very basic integrals such that it can be
easily implemented using a computer algebra system.
In two dimensions and in next to leading order, the imaginary part of
the effective action takes the form
\begin{eqnarray}
\label{Wc}
W^c&=&\left\langle \epsilon^{\mu\nu}\left(
Q_{12} {\mathcal{D}}_{\mu} {\mathcal{D}}_{\alpha}{\mathcal{H}}{\mathcal{D}}_{\alpha} {\mathcal{D}}_{\nu}{\mathcal{H}}
+\frac{i}{2}P_{12} {\mathcal{F}}_{\mu\nu}{\mathcal{D}}_{\alpha}{\mathcal{D}}_{\alpha}{\mathcal{H}} \right. \right. \nn \\
&& +\widetilde{R}_{123}{\mathcal{D}}_{\alpha}{\mathcal{D}}_{\alpha}{\mathcal{H}}{\mathcal{D}}_{\mu}{\mathcal{H}}{\mathcal{D}}_{\nu}{\mathcal{H}}
+\frac{i}{2}\widehat{R}_{123}{\mathcal{F}}_{\mu\nu}{\mathcal{D}}_{\alpha}{\mathcal{H}}{\mathcal{D}}_{\alpha}{\mathcal{H}} \nn \\
&& \left.\left.
+ M_{1234}{\mathcal{D}}_{\mu}{\mathcal{H}}{\mathcal{D}}_{\alpha}{\mathcal{H}}{\mathcal{D}}_{\nu}{\mathcal{H}}{\mathcal{D}}_{\alpha}{\mathcal{H}}\right)\right\rangle.
\end{eqnarray}
At next to leading order the action is chiral invariant and the
effective action can hence be immediately obtained by matching with
the covariant current that in this order coincides with the consistent
current. These functions must have the following properties
\be
\label{eq:PropP}
P_{12}=-P_{\underline{12}}=P_{21}, \quad
Q_{12}=Q_{\underline{12}}=-Q_{21},
\ee
\be
\widetilde{R}_{123}=-\widetilde{R}_{\underline{123}}=\widetilde{R}_{\underline{21}3}, \quad
\widehat{R}_{123}=\widehat{R}_{\underline{123}}=-\widehat{R}_{\underline{21}3},
\ee
\be
M_{1234}=M_{\underline{1234}}=
-M_{\underline{34}12}=M_{4321}.
\ee
We have chosen a rather general imaginary effective action at the
required order which preserves gauge and chiral invariance, but we
have included a larger number of terms than necessary to perform the
matching process with the effective current. In fact, the matching
process could be done with solely the functions $Q_{12}$,
$\widetilde{R}_{123}$, $\widehat{R}_{123}$, and $M_{1234}$. Instead,
we have decided to include the additional term $P_{12}$, in order to
have the option of simplifying the action by a judicious choice of
this extra function. For example, the extra function can be used to
ensure that all functions remain finite at the coincidence limit,
as will be explained later on.
The calculation from the worldline formalism leads to the following
contributions to the covariant current
\begin{eqnarray}
\label{eq:WLcont}
\delta W^c&=&-i \, \epsilon^{\mu\nu} \left\langle
I^1_{12}{\mathcal{D}}_{\mu}{\mathcal{D}}_{\alpha}{\mathcal{D}}_{\alpha}{\mathcal{H}}\delta{\mathcal{A}}_{\nu}
+ i\, I^2_{12}{\mathcal{D}}_{\alpha}{\mathcal{F}}_{\mu\alpha}\delta{\mathcal{A}}_{\nu}
+I^3_{123}{\mathcal{D}}_{\alpha}{\mathcal{D}}_{\alpha}{\mathcal{H}}{\mathcal{D}}_{\mu}{\mathcal{H}}\delta{\mathcal{A}}_{\nu}\right.\nonumber\\
&&\!\!+I^4_{123}{\mathcal{D}}_{\mu}{\mathcal{H}}{\mathcal{D}}_{\alpha}{\mathcal{D}}_{\alpha}{\mathcal{H}}\delta{\mathcal{A}}_{\nu}
+I^5_{123}{\mathcal{D}}_{\mu}{\mathcal{D}}_{\alpha}{\mathcal{H}}{\mathcal{D}}_{\alpha}{\mathcal{H}}\delta{\mathcal{A}}_{\nu}
+I^6_{123}{\mathcal{D}}_{\alpha}{\mathcal{H}}{\mathcal{D}}_{\mu}{\mathcal{D}}_{\alpha}{\mathcal{H}}\delta{\mathcal{A}}_{\nu}\nonumber\\
&&\!\!+ i\, I^7_{123}{\mathcal{F}}_{\mu\alpha}{\mathcal{D}}_{\alpha}{\mathcal{H}}\delta{\mathcal{A}}_{\nu}
+ i \, I^8_{123}{\mathcal{D}}_{\alpha}{\mathcal{H}}{\mathcal{F}}_{\mu\alpha}\delta{\mathcal{A}}_{\nu}
+I^9_{1234}{\mathcal{D}}_{\mu}{\mathcal{H}}{\mathcal{D}}_{\alpha}{\mathcal{H}}{\mathcal{D}}_{\alpha}{\mathcal{H}}\delta{\mathcal{A}}_{\nu}\nonumber\\
&&\!\!\left.
+I^{10}_{1234}{\mathcal{D}}_{\alpha}{\mathcal{H}}{\mathcal{D}}_{\mu}{\mathcal{H}}{\mathcal{D}}_{\alpha}{\mathcal{H}}\delta{\mathcal{A}}_{\nu}
+I^{11}_{1234}{\mathcal{D}}_{\alpha}{\mathcal{H}}{\mathcal{D}}_{\alpha}{\mathcal{H}}{\mathcal{D}}_{\mu}{\mathcal{H}}\delta{\mathcal{A}}_{\nu}\right\rangle.
\end{eqnarray}
The coefficient functions are given in Appendix \ref{curr_nlo}. In
order to express the current in this form, partial integration has
been used to remove terms of the form ${\mathcal{D}}\delta{\mathcal{A}}$. In addition,
indices that are contracted with the $\epsilon$ tensor have been moved
to the left, such that a term of the form ${\mathcal{D}}_{\alpha}{\mathcal{D}}_{\mu}$
yields a sum of terms of the type ${\mathcal{D}}_{\mu}{\mathcal{D}}_{\alpha}$ and
${\mathcal{F}}_{\mu\alpha}$.
The contributions from the variation of \eq~(\ref{Wc}) can be grouped
in three levels, with the first level having only contributions from
$Q$ and $P$; the second level from the previous ones and
$\widetilde{R}$ and $\widehat{R}$; the last level with all
functions. Adding the contributions from the worldline method and the
variation of \eq~(\ref{Wc}) one obtains for the first level the
following two equations
\begin{eqnarray}
\label{eq:1stLevel}
\,P_{\underline{2}1} + (m_1+m_2)Q_{12} &=&I^1_{12},\nonumber\\
(m_1 + m_2)P_{12}-(m_1^2-m_2^2)Q_{\underline{2}1} &=&I^2_{12},
\end{eqnarray}
which have the solution
\begin{eqnarray}
\label{Q}
Q_{12}&=&\frac{I^{2}_{\underline{2}1}}{m_1^2-m_2^2}-
\frac{P_{\underline{2}1}}{m_1+m_2}.
\end{eqnarray}
Besides, there arises the following restriction which is satisfied and
can serve as a check for the corresponding terms in the effective
current
\be
(m_1+m_2)I^1_{\underline{2}1} = - I^2_{12}, \quad
I^1_{12}= - I^1_{\underline{12}}, \quad
I^2_{12}=I^2_{\underline{12}}.
\ee
The matching equations for the next level are
\begin{eqnarray}
\label{eq:2ndLevela}
- \nabla_{2}\left((m_1 + m_2) Q_{\underline{2}1} -P_{\underline{2}1}\right)
+ Q_{12} + Q_{\underline{2}1}
+(m_1+m_3)( -\widetilde{R}_{123} +\widetilde{R}_{\underline{3}12})&=&I^3_{123},\\
\label{eq:2ndLevelb}
\nabla_{2}\left((m_1 + m_2)(Q_{12}+Q_{\underline{2}1}) \right)
-(m_1+m_3)\widetilde{R}_{\underline{3}12}
-2 Q_{12} -2 Q_{\underline{2}1}
+ \widehat{R}_{\underline{3}12}&=&I^5_{123},\\
\label{eq:2ndLevelc}
\nabla_{2}\left((m_1 + m_2)P_{12} \right)
- (m_1 - m_2)\nabla_{2}\left((m_1+ m_2) Q_{\underline{2}1} \right) && \nn \\
-2 P_{12} + 2 (m_1 - m_2) Q_{\underline{2}1}
+ (m_1 + m_3)(Q_{13} + 2 Q_{\underline{3}1})&&\nonumber\\
- \widehat{R}_{123} (m_1 + m_3)
+(m_1 - m_2) (m_1 + m_3) \widetilde{R}_{\underline{3}12} &=&I^7_{123},
\end{eqnarray}
and their complex conjugates.
The first \eq~(\ref{eq:2ndLevela}) is of the form
\be
\widetilde{R}_{123} - \widetilde{R}_{\underline{3}12}
=- \frac{ \tilde I^3_{123}}{m_1+m_3},
\ee
and a set of solutions to \eqs~(\ref{eq:2ndLevela}) and
(\ref{eq:2ndLevelb}) is hence given by
\begin{eqnarray}
\label{eq:SolRRR}
\widetilde{R}_{123}&=&-\frac{1}{2}
\left(\frac{\widetilde{I}_{123}}{m_1+m_3}\right)_{123}
-\frac{1}{2}\left(\frac{\widetilde{I}_{123}}{m_1+m_3}\right)_{\underline{3}12}
-\frac{1}{2}\left(\frac{\widetilde{I}_{123}}{m_1+m_3}\right)_{\underline{23}1}\\
\label{eq:SolRRRR}
\widehat{R}_{123}&=&\widehat{I}_{\underline{23}1}-(m_1-m_2)\widetilde{R}_{123}.
\end{eqnarray}
The functions $\widetilde{I}$ and $\widehat{I}$ are hereby defined as
\begin{eqnarray}
\label{eq:widetildeI}
\widetilde{I}_{123}&=&I^3_{123}
+ \nabla_{2}\left((m_1 + m_2) Q_{\underline{2}1} - P_{\underline{2}1}\right)
- Q_{12}- Q_{\underline{2}1}, \\
\label{eq:widehatI}
\widehat{I}_{123}&=&I^5_{123}
-\nabla_{2}\left((m_1 + m_2)(Q_{12}+Q_{\underline{2}1})\right)
+ 2 Q_{12}+ 2 Q_{\underline{2}1}.
\end{eqnarray}
The last \eq~(\ref{eq:2ndLevelc}) leads to a constraint on the $I$
functions that is given in Appendix~\ref{curr_nlo}.
The function $\widetilde{R}$ possesses the required symmetries, and it
reproduces the effective current correctly, but it is not necessarily
finite in the coincidence limit, $m_1 \to -m_3$. One way of solving
this problem is to choose the function $P$ appropriately which up to
this point remained undetermined. Such a choice is e.g. given by
\be
P_{12} = \frac{I^2_{12}}{m_1+m_2}, \quad Q_{12} = 0,
\ee
which leaves $\widetilde{I}$ as
\bea
\label{eq:widetildeIb}
\widetilde{I}_{123} &=&I^3_{123}
-\frac{I^2_{\underline{2}1}}{(m_1-m_2)(m_2-m_3)}
+\frac{I^2_{\underline{3}1}}{(m_1-m_3)(m_2-m_3)}.
\eea
With this choice, $\widetilde{R}$ is finite in the coincidence limit,
as can be checked explicitly, and since $\widehat{I}$ is also finite,
so is $\widehat{R}$.
For the last level, the following three equations hold
\begin{eqnarray}
\label{eqM1}
\nabla_{1} \widehat{R}_{\underline{3}12}
- (\nabla_{2}+\nabla_{3})
\left((m_1 + m_3) \widetilde{R}_{\underline{3}12} \right)
+ 2 \widetilde{R}_{\underline{3}12} &&\nonumber\\
- 2 (m_1 + m_4) M_{1234} &=&I^{9}_{1234},\\
\label{eqM2}
( \nabla_{3} - \nabla_1 )\left( \widetilde{R}_{\underline{3}12} (m_1 + m_3)\right)
+ \nabla_{2} \widehat{R}_{\underline{3}12}
- 2 \widetilde{R}_{\underline{3}12} + 2 \widetilde{R}_{\underline{4}23}
&&\nonumber\\
-2 (m_1 + m_4) M_{\underline{4}123} &=&I^{10}_{1234},\\
\label{eqM3}
(\nabla_{1}+\nabla_{2}) \left((m_1 + m_3) \widetilde{R}_{\underline{3}12} \right)
+ \nabla_{3} \widehat{R}_{\underline{3}12}
- 2 \widetilde{R}_{\underline{4}23} &&\nonumber\\
+2 (m_1 + m_4) M_{1234})&=&I^{11}_{1234}.
\end{eqnarray}
One of these equations can be used to determine $M$, while the other
two lead again to constraints on the $I$ functions. the sum of the
three equations has the especially simple form
\begin{eqnarray}
-2 (m_1 + m_4 ) M_{\underline{4}123}&=&
- (\nabla_{1} + \nabla_{2} + \nabla_{3} ) \widehat{R}_{\underline{3}12}
+ I^{9}_{1234} + I^{10}_{1234} + I^{11}_{1234}.
\end{eqnarray}
Since all previous functions in the effective action have been chosen
finite in the coincidence limit, so is $M_{1234}$.
\eqs~(\ref{eqM1}) and (\ref{eqM3}) show that $M_{1234}$ is finite in
the limit $m_1\to m_2$, while \eq~(\ref{eqM2}) shows that $M_{1234}$
is finite in the limit $m_1\to -m_4$. This concludes the discussion of
the next to leading order contributions in two dimensions.
\section{Conclusions}
We presented a worldline formalism to calculate the imaginary part of
the covariant current of a general chiral model in the derivative
expansion and our results are best summarized by \eqs~(\ref{dWIm2})
and (\ref{ohneK}).
The resulting covariant current can be used to reproduce the imaginary
part of the effective action by integration or matching. The advantage
of this approach, compared to explicit formulas of the effective
action, as given e.g. in ref.~\cite{Gagne2}, consists in the manifest
chiral covariance. Chiral covariance reduces the number of possible
contributions to the current enormously and makes even next to leading
order calculations manageable as demonstrated in section
\ref{sec_next} in the case of two dimensions. Besides the chiral
covariance, the use of the worldline formalism is essential in our
approach. The evaluation of the worldline path integral requires
neither performing Dirac algebra nor integrating over momentum space,
in contrast to the more traditional methods used in
ref.~\cite{Salcedo2}.
In principle, it is possible to use the presented formalism to
determine the effective CP violation resulting from integrating out
the fermions of the Standard model. For example, in next to leading
order in four dimensions, an operator of the form ${\mathcal{D}}{\mathcal{H}} {\mathcal{D}}{\mathcal{H}} {\mathcal{F}}
{\mathcal{F}}$ could arise from the CP violation in the CKM matrix. Since the
mass terms of the fermions are treated non-perturbatively in the
derivative expansion, the resulting effective CP-violating operator is
not necessarily proportional to the Jarlskog determinant. The
discussion of this question is postponed to a forthcoming publication.
\section*{Acknowledgments}
T.K. is supported by the Swedish Research Council
(Vetenskapsr{\aa}det), Contract No.~621-2001-1611. A.H. is supported
by CONACYT/DAAD, Contract No.~A/05/12566.
|
2,869,038,156,355 | arxiv | \section{Introduction\label{intro}}
In this article, we consider flat surfaces with a finite number of conical singularities, that is, surfaces provided with a flat structure with conical singular points. A conical singular point has a total angle different from $2\pi$.
Finding good parameters for these surfaces is still an open question (one should mention here that there are well-known good parameters for the subset of flat surfaces defined by quadratic differentials on Riemann surfaces). We treat here the question of classifying flat pairs of pants with one singularity. The decomposition of surfaces into pairs of pants is a common practice to study various structures on surfaces (see for instance \cite{pre05560297}). The idea, which is usually attributed to Grothendieck, and which is developed by Feng Luo in \cite{pre05560297}, is to provide building blocks (generators) which permit to reconstruct any surface endowed with the studied structure following some specific rules (relations). It should be noted however that there exist flat surfaces which are not decomposable into flat pairs of pants by disjoint simple closed geodesics (for an example of flat surface of genus $3$ with one singularity, see Example \ref{nondecomp}). But, it is possible to decompose such a surface into pairs of pants if we sacrifice some rules of decomposition, such as the simplicity of geodesics in the decomposition.
In Section \ref{prelim}, we give a definition of flat surface, we recall the formula of Gauss-Bonnet and we give some examples. In Section \ref{class}, we classify flat pairs of pants with one singularity and we study their space of parameters. In Section \ref{teich}, we present the Teichm\"uller space of flat pairs of pants with one singularity and we study its topology. In Section \ref{exmps}, we discuss the decomposability of a surface into pairs of pants.
This work uses ideas from \cite{1149.57029,1020.57003} in which the authors consider the similar question for hyperbolic surfaces with singular points.
\section{Preliminaries \label{prelim}}
\begin{defin}
Let $M_{g, k}$ (denoted by $M$ if there is no confusion) be a surface of genus $g$ with $k$ boundary components, provided with a flat metric $d (., .)$ with finitely many conical singularities $\varSigma= \{s_1,\ldots, s_n\}$. We assume all the boundary components of $M$ are closed curves (homeomorphic to circles). We denote by $\partial M$ the boundary of $M$ and by $\text{Int}(M)=M\backslash\partial M$ the interior of $M$. More precisely:
\begin{itemize}
\item For each point $x\in \text{Int}(M) \backslash\varSigma$ there is a neighborhood $U(x)$ isometric to a disc in the Euclidean plane.
\item For each point $s\in\varSigma \cap \text{Int}(M)$ there is a neighborhood isometric to a Euclidean cone of total angle $0<\theta<+\infty$, with $\theta\neq2\pi$.
\item For each point $x\in\partial M\backslash\varSigma$ there is a neighborhood isometric to a Euclidean sector with angle measure $\pi$ at the image of $x$.
\item For each point $s\in \varSigma\cap\partial(M)$ there is a neighborhood isometric to a Euclidean sector of angle $0<\theta<+\infty$, with $\theta\neq\pi$.
\end{itemize}
The metric $d(.,.)$ will be called a \emph{flat structure} on $M$, and $M$ will be called a \emph{flat surface} (\emph{with boundary} if it exists).
\end{defin}
\begin{defin}
The curvature $\kappa$ at a conical singularity $s$ of total angle $\theta$ is $\kappa=2\pi-\theta$. The curvature at a singular point on the boundary of angle $\theta$ is $\kappa=\pi-\theta$.
\end{defin}
This definition of curvature is motivated by the following formula:
\begin{prop} [Gauss-Bonnet formula for closed surfaces {\cite[p. 113]{0669.53001}}, {\cite[p. 190]{0146.44103}} or {\cite[p. 85-86]{0611.53035}} for a proof]
Let $M$ be a closed flat surface of genus $g$ with $n$ conical singularities. Let $\theta_i,\,i=1,\ldots,n$ be the total angles at the singularities. The formula of Gauss-Bonnet which connects the number of singularities with their total angles and the genus is:
\begin{equation}
\sum_{i=1}^{n} (2\pi - \theta_i) = (4-4g)\pi
\label{gb}
\end{equation}
\end{prop}
The Euler characteristic of this surface is given by $\chi(M)=2-2g$, and the formula of Gauss-Bonnet in terms of the Euler characteristic is:
\[\sum_{i=1}^{n} (2\pi - \theta_i) = 2\pi\chi(M) \,.\]
\begin{cor}[Gauss-Bonnet formula for a disc with $b$ holes]
Let $M$ be a flat surface of genus $0$ with boundary, with $n$ singularities in the interior and $m$ singularities on the boundary, and let $b$ be the number of boundary components. Let $\theta_i,\,i=1,\ldots,n$ be the total angles of conical singularities in the interior, and let $\tau_j,\,j=1,\ldots,m$ be the total angles of singularities on the boundary. Then,
\begin{equation}
\sum_{i=1}^{n} (2\pi - \theta_i) + \sum_{j=1}^{m} (\pi - \tau_j)= (4-2b)\pi \, .
\label{gbc}
\end{equation}
\end{cor}
\begin{proof}
By taking the double of $M$ (in this operation, each singularity on the boundary is glued to its copy) we obtain a closed flat surface of genus $g=b-1$ with $2n+m$ singularities. By the formula of Gauss-Bonnet (\ref{gb}) we get the result.
\end{proof}
More generally, the Euler characteristic of a surface $M$ of genus $g$ with $b$ boundary components is given by $\chi(M)=2-2g-b$, and the formula of Gauss-Bonnet in terms of the Euler characteristic is:
\[\sum_{i=1}^{n} (2\pi - \theta_i) + \sum_{j=1}^{m} (\pi - \tau_j)= 2\pi\chi(M) \,.\]
\begin{exmp}
Let $M$ be a pair of pants (a surface homeomorphic to a disc with two holes), equipped with a flat structure with one conical singularity $s$ in its interior and such that each component of the boundary $\partial M=c_1\cup c_2 \cup c_3$ is geodesic without singularities.
From $s$ we take three geodesic segments $d_i, i=1, 2,3$ which realize the distances between $s$ and $c_i$ respectively, Figure \ref{pantc}. We denote by $l_i, r_i$ the lengths of $c_i, d_i$ respectively. We cut and open $M$ along the geodesic segments $d_i$. Then we obtain a connected surface $P$ which can be decomposed into three rectangles $R_i,\,i=1,2,3$ whose lengths of sides are $l_i, r_i$ respectively, and one triangle whose sides have lengths $l_i$, as shown in Figure \ref{pantc1}.
Following the inverse procedure, we consider a triangle $T$ of side lengths $l_i$, and three rectangles $R_i$ each of which shares a side with $T$ and the other side being of length $r_i$ as shown in Figure \ref{pantc1}.
Now, by identifying the sides of equal length $r_i$, we obtain a flat pair of pants with one conical singularity. Note that the total angle of the singularity is $\theta=4\pi$.
This example will be studied later in Section \ref{class}.
\end{exmp}
\begin{exmp}
We cannot build a flat pair of pants with no singularities at all. Indeed, if we assume that such a pair of pants exists, by the formula (\ref{gbc}) we find $0=-2\pi$ which is impossible.
\end{exmp}
Finally, by gluing along their boundaries $2g-2$ flat pair of pants, each one with a single singularity, we obtain a closed flat surface of genus $g$ with $2g-2$ conical singularities.
\section{The geometry of a flat pair of pants with a conical singularity \label{class}}
In this section we assume that $M$ is a flat (singular) surface with boundary homeomorphic to a disc with two holes. We assume further that $M$ has only one conical singularity $s$. We call such a surface a \emph{flat pair of pants with one singularity} (or just \emph{flat pair of pants}). An example of such structure is obtained by taking the metric associated to two transverse measured foliations with one singularity on $M$ (see Figure \ref{teichf}). It is clear from this picture that the parameter space for the transverse measures of these two transverse foliations has dimension four, and it produces a space of flat structures of dimension four on the pair of pants, with one singular point. It will follow from the discussion below (see the remark after Definition \ref{teichm}) that there are examples of flat structures on $M$ which do not arise from pairs of transverse measured foliations. The aim of this section is to find a set of real parameters which determine the geometry of $M$.
\begin{figure}
\centering
\includegraphics[width=0.35\textwidth,bb=0 0 457 457]{Diagrammef.eps}
\caption{The two transverse measured foliations on the pair of pants induce a flat structure with one singular point\label{teichf}}
\end{figure}
The first natural guess is that the lengths of boundary geodesics might be good parameters, in analogy with the case of hyperbolic pairs of pants (described for instance in \cite{0731.57001}). This is easily seen to be false, as one can glue Euclidean cylinders to boundaries of flat pairs of pants, changing the isometry type, without changing the boundary component length (see Figure \ref{cylind}). Thus, the parameter space for flat structures on pair of pants is more complicated than the parameter space for hyperbolic structure on such surfaces.
\begin{figure}
\centering
\includegraphics[width=0.25\textwidth,bb=0 0 430 495]{Diagramme18.eps}
\caption{We can change the isometry type of a flat pair of pants without changing the boundary curve lengths\label{cylind}}
\end{figure}
Let $\partial M = c_1 \cup c_2 \cup c_3$ denote the boundary and denote by $d(.,.)$ the distance function on $M$. Let $d_i$ be a geodesic segment which realizes the distance $d (s, c_i)$ between the singularity $s$ and the boundary component $c_i$. This segment intersects the boundary component with a right angle. We denote by $l_i$ the length of $c_i$ and by $r_i$ the length of $d_i$. Obviously, the parameters $r_i,\,i=1,2,3$ depend on the position of $s$. We have the following proposition.
\begin{prop}
The real parameters $l_i, r_i, i=1, 2,3$ determine a unique flat pair of pants $M$ with one singularity up to isometry, where the $l_i$ are the lengths of the boundary components $c_i$, and the $r_i$ are the lengths of the geodesic segments between $s$ and the $c_i$.
\label{params}
\end{prop}
\begin{proof}
Every boundary component $c_i,\, i\in\{1,2,3\}$ is geodesic, even if the singularity is on a boundary component, since by the formula of Gauss-Bonnet, the angle at $s$ will be $3\pi$. By our assumption on the uniqueness of the singularity, $d_i$ does not meet another singularity on the component $c_i$ of the boundary, and so, since $c_i$ is geodesic, $d_i$ is orthogonal to $c_i$. For the same reasons, $d_i$ does not share any of its interior points with boundary components.
\textit{First case:} We assume that $s\in Int(M)$. This implies that the lengths of $d_i$ are different from zero, $r_i\neq 0, i=1,2,3$. We cut $M$ along $d_1\cup d_2 \cup d_3$, see Figure \ref{pantc}, to obtain a polygonal flat surface $P$ homeomorphic to a disc without singularities in the interior, see Figure \ref{pantc1}.
\begin{figure}
\centering
\includegraphics[width=0.35\textwidth, bb=0 0 430 259]{Diagramme1.eps}
\caption{\label{pantc}Flat pair of pants with one singularity in the interior}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth,bb=0 0 517 385]{Diagramme2.eps}
\caption{\label{pantc1}After cut. $l_i,\,r_i$ are the lengths of $c_i,\,d_i$ respectively}
\end{figure}
None of the parameters $l_i, i=1, 2, 3$ can be zero, because if one of them is zero, the surface $M$ looses one of its boundary components. This fact implies that we always have three copies of $s$ after the cut. Let us denote by $s_i, i=1,2,3$ the copy which lies between $d_{(i+1)\mod 3}$ and $d_{(i+2)\mod 3}$, Figure \ref{pantc1}. We can draw in $P$ the geodesics between these copies. These geodesic segments bound a triangle $T=(s_1 s_2 s_3)$ , Figure \ref{pantat}, which can be degenerate, with none of its angles greater than $\pi$. For instance, in Figure \ref{pantap} the angle $\measuredangle(s_2 s_1 s_3)= \pi$. This angle cannot be greater than $\pi$, because this implies that $d_1$, along which the cut was made, was not a geodesic segment which represents the distance between $s$ and $c_1$. This contradicts the assumption.
\begin{figure}
\centering
\includegraphics[width=0.35\textwidth,bb=0 0 430 401]{Diagramme8.eps}
\caption{\label{pantat}Here we see the triangle on the pair of pants}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth,bb=0 0 448 380]{Diagramme3.eps}
\caption{\label{pantap}The triangle is degenerate}
\end{figure}
We recognize easily, Figure \ref{pantc1}, in addition to $T$, three rectangles $R_i,\, i=1,2,3$. A rectangle $R_i,\,i\in\{1,2,3\}$ is bounded by a side of $T$, $c_i$ and the two copies of $d_i$. $P$ is composed of $T$ and $R_i,\, i=1,2,3$.
A rectangle is uniquely determined up to isometry by its length and height, and a triangle is uniquely determined up to isometry by its lengths of sides. Then, given the parameters $l_i,r_i,\,i=1,2,3$, where the $l_i,\,i=1,2,3$ satisfy the triangle inequalities, we can construct a unique pair of pants with one singularity by gluing together the three rectangles $R_i$ and the triangle $T$ given by the parameters $l_i, r_i,\, i=1, 2,3$.
The triangle could be degenerate, and this case lies on the boundary of the space of triangles. This happens only when
\begin{equation}
l_i=l_{(i+1)\mod 3} + l_{(i+2)\mod 3}
\label{degen}
\end{equation}
for some $i \in \{1,2,3\}$. Figure \ref{pantap}.
\textit{Second case:} We assume that $s\in \partial M$. Then at least one of $r_i,\, i=1,2,3$ equals zero.
More precisely, $s$ belongs to only one $c_i,\,i\in\{1,2,3\}$, and so only one $r_i$ equals zero, because if $s$ belongs to more than one boundary component, it does not have a neighborhood homeomorphic to a disc. In such a case we call this a \emph{degenerate pair of pants}. We will outline these cases later.
\begin{figure}
\centering
\includegraphics[width=0.35\textwidth,bb=0 0 430 401]{Diagramme4.eps}
\caption{\label{pantabord}Singularity on the boundary}
\end{figure}
Without loss of generality, assume $s\in c_1$, as in Figure \ref{pantabord}. Cut along $d_2 \cup d_3$ ($d_1$ being reduced to $s$) to obtain a polygonal flat surface $P$ homeomorphic to a disc without singularities in the interior. For the same reason as before, we have three copies of $s$. The geodesic segment between $s_2,s_3$ is $c_1$ itself since $c_1$ is geodesic.
The geodesic segments between the copies of $s$ bound a triangle $T=(s_1 s_2 s_3)$ which can be degenerate, with none of its angles greater than $\pi$ (for the same reason as in the first case). It is easy to recognize, in addition to $T$, two rectangles $R_i,\, i=2,3$. Figure \ref{hepta}.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth,bb=0 0 517 274]{Diagramme5.eps}
\caption{\label{hepta}The resulting heptagon after cut}
\end{figure}
Knowing the parameters $l_i,r_i\,,i=1,2,3$ we can determine if the singularity is on the boundary and on which component. The rectangles $R_i$ and the triangle $T$ are uniquely determined by the parameters and so is the flat structure.
If Relation (\ref{degen}) is satisfied, the triangle is degenerate. In this case we notice that only one $r_i,\, i\in\{1,2,3\}$ that correspond to the boundaries on the right hand side of this identity can take the value zero. The other cases are degenerate.
We proved that to each flat structure on $M$ with one singular point we may associate unique set of values $l_i, r_i\,, i=1,2,3$. Conversely, to each choice of values, we may associate a unique flat structure on $M$.
We deduce that each flat pair of pants $M$ is uniquely determined, up to isometry, by the parameters $l_i$ end $r_i$. And vice versa.
\end{proof}
Let us give the limits of parameters in view of the last proof. The length parameters can take any values satisfying these conditions:
\begin{itemize}
\item $0<l_i<\infty,\,i=1,2,3$,
\item $l_i\leq l_{(i+1)\mod 3}+l_{(i+2)\mod 3} \, ,\, i=1,2,3$,
\item $0\leq r_i < \infty,\,i=1,2,3$,
\item $0<r_i+r_{(i+1)\mod 3},\,i=1,2,3$
\item If $l_i=l_{(i+1)\mod 3}+l_{(i+2)\mod 3} \, , \text{for some }\, i\in\{1,2,3\}$ then $0<r_i<\infty$
\end{itemize}
\begin{rem}
Let $M$ be a flat surface with one singularity $s$, which is homeomorphic to a disc with $n$ holes, $n\geq 3$. We denote by $c_i$ the boundary components of $M$ and by $d_i$ the geodesic segments from $s$ to $c_i$. If we set $l_i$ the lengths of $c_i$ and $r_i$ the lengths of $d_i$, then the parameters $l_i, r_i$ do not characterize $M$ in the sense of Proposition \ref{params}. This follows because if we cut $M$ along $d_1 \cup d_2 \cup \dots \cup d_n$ then instead of a triangle $(s_1 s_2 s_3)$ we get a $(n+1)$-gon which, of course, is not uniquely determined by the lengths of its edges.
\end{rem}
We return to the case where $M$ is a pair of pants and $\partial M = c_1 \cup c_2 \cup c_3$. By the previous discussion, where we saw how a flat pair of pants with one singularity can be cut into pieces, three rectangles and one triangle, we find it convenient to introduce these terms: We say that \emph{the triangle is degenerate} if the condition:
\[l_i=l_{(i+1)\mod 3}+l_{(i+2)\mod 3} \, , \text{for some }\, i\in\{1,2,3\}\]
is satisfied. If not, we say that \emph{the triangle is non-degenerate}. We say that \emph{the rectangle} $R_i\,, i\in\{1,2,3\}$ \emph{is degenerate} if $r_i=0$ for some $i\in\{1,2,3\}$. If not, we say that \emph{the rectangle} $R_i$ is \emph{non-degenerate}.
\begin{rem}[Degenerate cases]
A degenerate pair of pants appears when the singularity does not have a neighborhood homeomorphic to a disc in the Euclidean plane. From the previous proof, we can conclude the following degenerate cases:
\begin{itemize}
\item In the case when the triangle is non-degenerate, if two or three rectangles $R_i,\, i\in\{1,2,3\}$ are degenerate we have a degenerate pair of pants. A neighborhood of the singularity is similar to one of the forms in Figure \ref{neibh}.
\begin{figure}
\centering
\begin{tabular}{c c c c c}
\includegraphics[width=0.25\textwidth,bb=0 0 343 258]{Diagramme20.eps}
& & & &
\includegraphics[width=0.20\textwidth,bb=0 0 318 316]{Diagramme21.eps}
\\ \\
Two rectangles are degenerate
& & & &
Three rectangles are degenerate
\\
\end{tabular}
\caption{\label{neibh}Neighborhood of singularity}
\end{figure}
\item In the case when the triangle is degenerate, that is, for some $i\in\{1,2,3\}$ identity (\ref{degen}) is satisfied, if both rectangles $R_{(i+1)\mod 3},R_{(i+2)\mod 3}$ are degenerate we have a degenerate pair of pants. A neighborhood of the singularity is similar to Figure \ref{neibh2}.
\begin{figure}
\centering
\includegraphics[width=0.20\textwidth,bb=0 0 258 173]{Diagramme22.eps}
\caption{\label{neibh2}The triangle and two rectangles are degenerate}
\end{figure}
\end{itemize}
We should finally pay attention that in this case when $R_i$ is degenerate, the resulting surface is not a pair of pants since it has four holes. Figure \ref{imposib}.
\begin{figure}
\centering
\includegraphics[width=0.25\textwidth,bb=0 0 288 227]{Diagramme23.eps}
\caption{\label{imposib}Impossible case}
\end{figure}
\end{rem}
After the previous discussion, we will introduce new parameters for flat pairs of pants. These parameters seem to be more convenient. For this, let now $k_i$ be a geodesic segment which realizes the distance between $c_{(i+1)\mod 3}$ and $c_{(i+2)\mod 3},\, i=1, 2, 3$, see Figure \ref{pantapa}. Denote by $l_i$ the length of $c_i$ and by $a_i$ the length of $k_i$.
\begin{figure}
\centering
\includegraphics[width=0.35\textwidth,bb=0 0 430 401]{Diagramme6.eps}
\caption{\label{pantapa}The geodesic $k_i$ represents the distance $a_i$ between $c_{(i+1)\mod 3}$ and $c_{(i+2)\mod 3}$}
\end{figure}
\begin{defin}
The six non-negative parameters $l_i, a_i, i=1,2,3$ will be called the \emph{distance parameters} of $M$.
\end{defin}
We have the following theorem.
\begin{prop}
\label{params2}
The distance parameters $l_i, a_i, i=1,2,3$ determine a unique flat pair of pants $M$ with one singularity.
\end{prop}
\begin{proof}
By the proof of Proposition \ref{params} we see that flat pairs of pants with one singularity are uniquely determined by the parameters $l_i, r_i, i=1,2,3$, and correspond to one of the following cases:
\begin{enumerate}
\item If the triangle is non-degenerate, we distinguish two cases.
\begin{enumerate}
\item If all the rectangles $R_i$ are non-degenerate, it is easy, after the cut, to see that geodesic segments between boundaries pass all by the singularity. Figure \ref{nondegen}.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth,bb=0 0 536 382]{Diagramme7.eps}
\caption{\label{nondegen}After cut, $l_i,\,a_i$ are the lengths of $c_i,\,k_i$ respectively}
\end{figure}
So we have:
\begin{equation}
r_i+r_{(i+1) \mod 3}=a_{(i+2)\mod 3}, \, i=1,2,3 \,.
\label{rig}
\end{equation}
These relations prove that knowing $l_i, a_i,\, i=1,2,3$ we can determine $ r_i,\, i=1, 2, 3 $, and so, all the parameters needed to determine the flat pair of pants in this case, by Proposition \ref{params}.
\item If only one rectangle is degenerate then the pair of pants can be cut into two rectangles and one non-degenerate triangle as in Figure \ref{hepta}. By the same relations (\ref{rig}) we can determine the flat pair of pants. The singularity, in this case, is on the boundary component which corresponds to the degenerate rectangle.
\end{enumerate}
\item If the triangle is degenerate, we also distinguish two cases.
\begin{enumerate}
\item If all the rectangles $R_i,\,i=1,2,3$ are non-degenerate, we see that the relations (\ref{rig}) also hold here and so the flat pair of pants is well determined, in this case, by the distance parameters. Figure \ref{pantap}.
\item If one of $R_{(i+1)\mod 3}, R_{(i+2)\mod 3}$ is degenerate then the pair of pants can be cut into two rectangles. Again, it is determined by relations (\ref{rig}). Figure \ref{2rect}.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth,bb=0 0 456 313]{Diagramme9.eps}
\caption{\label{2rect}Permitted gluing of two rectangles}
\end{figure}
\end{enumerate}
\end{enumerate}
\end{proof}
Here we give the limits of the new parameters in view of the last proof. The distance parameters can take any values in the limit of these conditions:
\begin{enumerate}[1) ]
\item $0<l_i<\infty,\,i=1,2,3$,
\item $l_i\leq l_{(i+1)\mod 3} + l_{(i+2)\mod 3} \, ,\, i=1,2,3$,
\item $0< a_i < \infty,\,i=1,2,3$,
\item $a_i\leq a_{(i+1)\mod 3} + a_{(i+2)\mod 3} \, ,\, i=1,2,3$, (Using Relations (\ref{rig})),
\item If $l_i=l_{(i+1)\mod 3} + l_{(i+2)\mod 3}$ , for some $i\in\{1,2,3\}$, then \\ $a_i<a_{(i+1)\mod3}+a_{(i+2)\mod3}$ . (Using Relations (\ref{rig})).
\end{enumerate}
It is easy to see that these conditions are equivalent to preceding ones. In addition, it is interesting to see that these conditions are equivalent to the fact that we have two Euclidean triangles (first four conditions) for which a degenerate case of one triangle prevents a degenerate case of the other (fifth condition).
We denote by $\mathcal{C}$ the set of all flat structures defined by the parameters $l_i,\, a_i,\, i=1,2,3$ under the previous conditions, and by $\mathcal{B}$ the set of all 6-tuples $(l_1,l_2,l_3,a_1,a_2,a_3)\in \mathbb{R}^6$ which satisfy these conditions. We can define a one-to-one mapping $\Phi: \mathcal{C}\rightarrow\mathcal{B}$ such that to a flat structure $\mathfrak{f}\in\mathcal{C}$ corresponds the unique 6-tuple $(l_1,l_2,l_3,a_1,a_2,a_3)$ of $\mathcal{B}$ which determines $\mathfrak{f}$. Obviously we have $\Phi(\mathcal{C})=\mathcal{B}$.
\section{The Teichm\"uller space of flat pairs of pants with one singularity\label{teich}}
Let us denote by $\mathcal{F}(M)$ the space of all flat structures with one singularity on a pair of pants $M$. We fix an orientation on $M$ and let $\mathtt{Homeo}^+(M,\partial)$ be the set of homeomorphisms of $M$ which preserve the orientation and each boundary component of $M$ (setwise). It is well known that each element of $\mathtt{Homeo}^+(M,\partial)$ is isotopic to the identity (see Expos\'e 2 in \cite{0731.57001}). The space $\mathtt{Homeo}^+(M,\partial)$ acts on $\mathcal{F}(M)$ as follows: If $h\in \mathtt{Homeo}^+(M,\partial)$ and $\mathfrak{f}\in \mathcal{F}(M)$ then $(h,\mathfrak{f})\mapsto h*\mathfrak{f}$ where $h*\mathfrak{f}(x,y):=\mathfrak{f}(h(x),h(y))$.
\begin{defin}
\label{teichm}
We define the Teichm\"uller space $\mathcal{T}(M)$ of $M$ as the quotient $\mathcal{F}(M)\slash \mathtt{Homeo}^+(M,\partial)$.
\end{defin}
Obviously, $\mathcal{T}(M)$ consists of all flat structures which belong to $\mathcal{C}$, the set of all flat structures defined by the parameters $l_i,\, a_i,\, i=1,2,3$. To each flat structure with one singularity on $M$ we may associate a unique configuration, which consists of a triangle $T$ and three rectangles $R_i,\,i=1,2,3$, glued as in Figure \ref{pantc1}. Conversely, to each configuration we may associate a unique flat structure with one singularity on $M$. This defines a mapping $\Phi:\mathcal{T}(M)\rightarrow \mathcal{B}$ which is one-to-one, and $\Phi(\mathcal{T}(M))=\mathcal{B}$.
\begin{rem}[Due to Athanase Papadopoulos]
Let $\mathcal{T}_\mathcal{M}(M)$ be the Teichm\"uller space of flat structures with one singularity on a pair of pants defined by a pair of transverse measured foliations. This space is of dimension 4. Since $\mathcal{T}(M)$, the Teichm\"uller space of all flat structures with one singularity on a pair of pants, is of dimension 6, as we showed here, then $\mathcal{T}_\mathcal{M}(M)\subsetneq \mathcal{T}(M)$. Thus, the space of flat structures we counter here is larger than the space of flat structures induced by quadratic differentials with one zero.
\end{rem}
\begin{prop}
The mapping $\Phi:\mathcal{T}(M)\rightarrow \mathcal{B}$ is continuous and open. Thus, $\Phi$ is homeomorphism.
\end{prop}
\begin{proof}
As explained in Proposition \ref{params2}, $\Phi$ is a bijection. If we consider the Euclidean distance between tuples of parameters $(l_1,l_2,l_3,a_1,a_2,a_3)$ and the following distance between two flat structures $\mathfrak{f}_1,\mathfrak{f}_2\in \mathcal{T}(M)$:
\[d(\mathfrak{f}_1,\mathfrak{f}_2)=\sup_{x,y\in M} |\mathfrak{f}_1(x,y)-\mathfrak{f}_2(x,y)|\]
we can see easily that both $\Phi$ and $\Phi^{-1}$ are continuous.
\end{proof}
Obviously, $\mathcal{B}$ is a non bounded convex subset of $\mathbb{R}^6$. In fact, the set of parameters $l_i,\,i=1,2,3$ can be seen as the space of all triangles which could be degenerate, but without those which have a side of length zero. This is, a non bounded pyramid of three faces with its sides and apex deleted. This space is convex in $\mathbb{R}^3$. The same can be said about the set of parameters $a_i,\,i=1,2,3$. Thus, $\mathcal{B}$ is convex in $\mathbb{R}^6$.
The boundary $\partial\mathcal{B}$ has six components which are non bounded convex subsets of $\mathbb{R}^5$. They correspond to cases of equality in the triangle inequality. Let us denote by $\partial_i \mathcal{B}_l\, ,\, i\in\{1,2,3\}$ the boundary component defined by
\[l_i = l_{(i+1)\mod 3} + l_{(i+2)\mod 3} \, .\]
The sets $\partial_i \mathcal{B}_l,\,i=1,2,3$ are pairwise disjoint subsets of $\partial \mathcal{B}$. Similarly, we denote by $\partial_j \mathcal{B}_a\, ,\, j\in\{1,2,3\}$ the boundary component defined by
\[a_j= a_{(j+1)\mod 3} + a_{(j+2)\mod 3} \, . \]
The sets $\partial_j \mathcal{B}_a,\,j=1,2,3$ are also pairwise disjoint subsets of $\partial \mathcal{B}$.
The intersection $\partial_{i,j}(\partial \mathcal{B)}=\partial_i\mathcal{B}_l \cap\partial_j\mathcal{B}_a$, is homeomorphic to $\mathbb{R}^4$ if $i\neq j$, and empty if $i=j$. This means that each boundary component of $\mathcal{B}$ has two convex connected boundary components homeomorphic to $\mathbb{R}^4$.
Then, by the homeomorphism $\Phi$, we have the following description of the Teichm\"uller space:
\begin{cor}
$\mathcal{T}(M)$ is homeomorphic to a non-compact submanifold of $\mathbb{R}^6$ of dimension $6$, with a natural cell-structure, having six cells of codimension one on its boundary.
\end{cor}
We know that any convex subset of Euclidean n-space $\mathbb{E}^n$ is contractible. This gives us the following result:
\begin{thm}
$\mathcal{T}(M)$ is a contractible space.
\end{thm}
In the next section we start a discussion of the decomposition into pairs of pants of closed surfaces with one singularity. Details will be given in subsequent work.
\section{Closed flat surfaces with one singularity \label{exmps}}
In general, a flat surface with one singularity is not decomposable by disjoint simple closed geodesics into pairs of pants. For this, we have the following example:
\begin{exmp}
\label{nondecomp}
Take a closed flat surface $M_3$ of genus $3$ with one singularity $s$. Suppose that we are able to decompose it by disjoint simple closed geodesics into pairs of pants. By the fact that there is no flat pair of pants without any singularity, we conclude that every pair of pants resulting from the decomposition has the singularity $s$ on its boundary. We know that a decomposition of $M_3$ by disjoint simple closed geodesics gives rise to four pairs of pants. This means that four boundary components should share the singularity. This is impossible under the present rules of identification (one boundary component is identified to one boundary component). Then, the decomposition is impossible.
\end{exmp}
The decomposition becomes possible if we change the rules of composition. For example, admitting geodesics to be non-simple or non-disjoint. That is, admitting the identification of parts of boundary components rather than boundary components entirely. I will not discuss here these rules, their study can be dealt with a separate work later.
The result in Example \ref{nondecomp} can be generalized as follows:
Let $M$ be a closed flat surface of genus $g\geq 3$ with a single conical
singularity $s$. Then $M$ cannot be decomposed into pairs of pants by disjoint simple closed geodesics.
\section*{Acknowledgements}
I would like to thank Athanase Papadopoulos for all the corrections and discussions during the preparation of this article. I also thank Daniele Alesandrini for his time.
\bibliographystyle{abbrv}
|
2,869,038,156,356 | arxiv | \section{Introduction}
Historically, calorimetry experiments at ambient pressure have led to an abundance of new physics, from Einstein's quantum theory of solids \cite{Pais1979} and Debye's vibrational model,\cite{Debye1912} to the discovery of superfluidity \cite{Donnelly1995} and a recent test of renormalization-group theory.\cite{Lipa1996} In Earth science, calorimetric measurements of silicates have proven essential for understanding magmatic processes.\cite{Lange1990}
Some research has aimed to make the same type of measurements at high pressures ($>$ 1 GPa). A few groups have demonstrated the ability to detect phase transitions at low temperatures (tens of mK to tens of K) in heavy fermion compounds up to $\sim 10$ GPa. \cite{Demuer2000,Fernandez-Panella2011,Wilhelm1999,Bouquet2000,Sidorov2010} In making these measurements, they have shown that qualitative calorimetry inside diamond cells is possible, but it is unclear whether such measurements are quantitatively accurate, even in relative values of specific heat as temperature is varied. In some dynamic high-pressure experiments, pyrometry-based temperature measurements allow calculation of specific heats as a shock wave decays.\cite{hicks2006,Eggert2009} However, the short timescales (nanoseconds) of the experiments can result in large uncertainties in the measurement, including the degree and nature of equilibrium achieved, and the large strain rates may generate a high density of atomic defects. Overall, although some pioneering experiments have been carried out, little is known about the specific heat of materials at pressures above a few GPa, or even about the potential for making calorimetric measurements at high pressures.
If successfully developed, high-pressure calorimetry techniques could have far-reaching applications. They could aid in the detection and characterization of high-pressure entropy-driven transformations, such as melting and order-disorder transitions.\cite{Hazen1996} The energetics of high-pressure phase transitions could reveal new information relevant to Earth science and to materials physics, such as the latent heat of melting in the Earth's core or the pressure-dependence of pre-melting phenomena. At low temperatures, calorimetry could be used to map out first- and second-order phase boundaries in pressure-temperature space, possibly helping in the search for quantum phase transitions (e.g. Ref. \onlinecite{Petrova2012}).
We focus here on calorimetry in the diamond-anvil cell because such cells currently achieve the highest pressures under static compression, and they allow samples to be probed by numerous other techniques (spectroscopy, diffraction, etc.). Static methods offer the widest range of time periods over which the sample can be probed, and may be essential for reaching thermodynamic equilibrium.
Before presenting the main model and results, we explain the need for new modeling of modulation calorimetry, a technique that is over 100 years old. Also, to show the potential for realization of the technique modeled here, we outline available technologies.
\section{Barriers to the use of traditional calorimetry models and methods}
Adaptation of typical calorimetric measurements to high pressures is a challenge because the volume at pressure - thermal insulation as well as sample - is small and therefore difficult to insulate thermally; relevant dimensions are of order $\mu$m to tens of $\mu$m so it is difficult to maintain adiabatic conditions. Moreover, insulating materials are liquids and solids with negligible porosity, hence large thermal conductivity and heat capacity, causing many equations of modulation calorimetry to be invalid.
Most modulation calorimetry experiments are analyzed using an approximate solution to the heat equation, with a single timescale of heat loss. Specifically, the sample is assumed to be thermally connected to a heater via a link with thermal conductance $K_h$, to a thermometer via $K_t$ and to a thermal bath via $K_b$.\cite{Sullivan1968, Kraftmakher2004} In this sense, the sample and its surroundings are modeled as if they are disjoint pieces. Heat flow to the thermal bath is assumed to be small and temperature gradients between the sample, heater and thermometer are assumed to be negligible.
The result, which is not true in diamond cells, but which provides a baseline to compare against, is that the sample's temperature varies according to
\begin{equation}
T_\omega = \frac{p_\omega}{\sqrt{C^2\omega^2 + K_b^2}}
\label{eqn:basic}
\end{equation}
where $T_\omega$ is the amplitude of temperature oscillation at the given frequency, $\omega$, $p_\omega$ is the amplitude of power oscillation, and $C$ is the heat capacity of sample and addenda. \footnote{This is Eq. (2.3c) of Ref. \onlinecite{Kraftmakher2004} with $Q'$ instead of $K_b$, Eq. (2) of Ref. \onlinecite{Baloga1977} with $\Gamma$ instead of $K_b$, Eq. (1) of Ref. \onlinecite{Bouquet2000}, or Eq. (11) of Ref. \onlinecite{Sullivan1968} with $\tau_1 = C/K_b$, $\tau_2 \to 0$, and $K_b/K_s \to 0$.}
\begin{mdframed}
Definition: \hspace{5 mm}
The \textit{addenda} are the materials close to the sample that heat diffuses into and away from during a heating cycle, effectively adding an apparent thermal mass to the sample. They include the heater and thermometer if they are separate from the sample, as well as any material that is within $\sim 1$ thermal diffusion length of the sample.
\end{mdframed}
To solve for both heat capacity and thermal conductance to the bath, Eq. (\ref{eqn:basic}) can be fitted to data at variable frequency, or a second equation can be used:
\begin{equation}
\tan\phi = C\omega/K_b
\label{eqn:phase}
\end{equation}
where $\phi$ is the phase shift between heat-source and temperature oscillations at a given frequency.
In any high-pressure system, including diamond-anvil cells, a model of disjoint sample, heater, thermometer, and thermal bath is not reasonable since the sample is contiguous with other liquids or solids on all sides, resulting in at least two problems: the addenda contribution to measured heat capacity can be large, and the extent of the addenda can depend on frequency; at lower frequencies thermal diffusion extends further into the surrounding material. Indeed, Ref. \onlinecite{Baloga1977} has shown that a 500 $\mu$m-thick Invar sample heated at 0 to 2 Hz in a pressure cell with $\leq 170$ MPa argon gas surrounding the sample is not well described by Eq. (\ref{eqn:basic}). Rather, the addenda contribution increases from approximately zero at ambient pressure (their Fig. 2) to $\sim 100\%$ at 150 to 170 MPa (their Fig. 3). \footnote{Ref. \onlinecite{Baloga1977} uses an analytic expression for an addenda contribution to heat capacity at variable frequency and shows that it fits their data.}
But at high enough frequency, it is possible that nanogram samples inside diamond cells could be heated in a manner that is close enough to adiabatic so that the error in measured value of a sample's heat capacity is small. For comparison, our previous work shows that heating timescales of 1 ns to 1 $\mu$s are required for pulsed heating experiments to reveal the latent heat expected during melting or other first order phase change of 1 $\mu$m-thick metal samples.\cite{Geballe2012}
\section{Available technologies for calorimetry in diamond-cells}
Several existing technologies can be exploited in the design of modulation calorimetry in diamond cells. Examples of specific Joule- and laser-heating designs are described in Appendices A,B.
Joule-heating, which allows heat to be deposited inside metallic samples, can be accomplished by lithographically fabricating wires and metallic samples onto the diamonds or onto a thin layer of thermal insulation,\cite{Weir2009a} or by positioning thin foils between insulating layers and through an electrically-insulating gasket.\cite{Zha2004,Komabayashi2009} Tapered electrical leads can connect the $\mu$m sized sample to electrical connectors that connect to commercial AC power supplies capable of outputting waveforms at kHz, MHz or GHz frequencies. Voltage measuring leads can be connected to lock-in amplifiers or analog-to-digital converters that monitor power oscillations, and perhaps also temperature oscillations via the third harmonic technique described in Appendix C. The background temperature can be measured using a thermocouple if the entire diamond-cell is heated, or by spectroradiometry if the sample's temperature is at least 1000 K. A more detailed Joule-heating design is presented in Part II of this publication.
Laser-heating, which deposits heat on the surface of metallic samples, can be accomplished with 100 MHz frequency oscillations with commercial diode laser modules (e.g. Newport LQD series), or with several GHz using more-specialized electrical modulation of a diode laser source (e.g. Ref. \onlinecite{Melentiev2001}). Pyrometric or spectroradiometric temperature measurements can be made with fast light-collecting technologies, such as intensified CCD cameras and photodiodes with nanosecond resolution. Incident laser power can be measured at the laser source by using a power meter, while changes in laser absorption can be monitored by measuring reflectivity from the sample area using a photodiode.
\begin{table}
\begin{center}
\begin{tabular}{ {c} p{1.3 cm} {c} {c}}
& & \multicolumn{2}{c}{Temperature measurement} \\
\cline{3-4}
& \multicolumn{1}{c|}{} & Internal & \multicolumn{1}{c|}{Surface}\\
\cline{2-4}
Heat& \multicolumn{1}{|c|}{Internal} & I/I & \multicolumn{1}{c|}{I/S} \\
Source & \multicolumn{1}{|c|}{Surface} & - & \multicolumn{1}{c|}{S/S} \\
\cline{2-4}
\end{tabular}
\caption{Summary of possible calorimetry designs with two-letter labels for the combinations studied here.}
\label{table:summary}
\end{center}
\end{table}
\section{Scope of analysis}
Our analysis discriminates between three types of temperature measurement and heating scheme, summarized in Table \ref{table:summary}: internal heating experiments with internal temperature measurement (``I/I'', such as Joule heating with the third harmonic temperature measurement technique), internal heating experiments with surface temperature measurement (``I/S'', such as Joule heating with spectroradiometry), and surface heating experiments with surface temperature measurement (``S/S'', such as laser heating of metals with pyrometry or spectroradiometry). We do not consider ``S/I'' because most experimental designs that allow for internal temperature measurement also allow for an internal heating source (I/I), which is likely to give more accurate calorimetry results because no heat transport is required to equilibrate the heated region with the region of temperature measurement. For example, if the amplitude of temperature oscillations is determined by measuring electrical resistance, then resistive (Joule) heating could also be used to deposit heat; if temperature is determined from electromagnetic radiation that is interior to the sample, it is likely that electromagnetic radiation (e.g., time-modulated laser heating) could also be deposited in the sample's interior. More nearly adiabatic conditions result from using the internal heating source in both cases.
We do not consider heating sources or temperature measurements that are far away from the sample because this increases the difficulty in extracting calorimetric information about the sample itself. To give a sense of the difficulty, typical diamond anvils are $\sim 2$ mm in each linear dimension, giving a thermal mass that is $\sim 10^5$ times the thermal mass of a typical diamond cell sample (10 $\mu$m thick, 100 $\mu$m in diameter). The gasket and epoxy that border the diamonds are also large compared to the sample, and their heat capacities would be difficult to calibrate since their dimensions typically vary from experiment to experiment.
We do not explicitly consider heating of samples by the ``hot plate'' method, in which one material (usually a thin metal foil) absorbs heat, which then diffuses into the sample of interest, despite the fact that this method can be useful for measuring a variety of properties (e.g. inteface conductance\cite{Cahill2004} and thermal diffusivity \cite{Cahill1990,Beck2007,Imada2014}). In fact, it is the diversity of uses that makes this heating method difficult to analyze comprehensively; temperature evolution is affected by both transport and thermodynamic properties of both the heat absorber and the sample. Nonetheless, we expect that thermal measurements using hot plate heating in diamond cells will be useful in the future, and therefore discuss the topic briefly in the discussion sections of this paper and the companion paper.
For simplicity, we assume local thermodynamic equilibrium is reached within the heated area, ignoring the possibility of kinetic barriers to phase transitions even though kinetics are known to affect many phase transformations. \cite{Birge1997} One way to account for, or at least estimate the influence of, kinetic effects is to reverse transformations by slowly raising and then slowly lowering temperature.
In principle, measurement of phase shifts could be used in conjunction with the amplitude of oscillations and two equations, (\ref{eqn:basic}) and (\ref{eqn:phase}), to solve for the heat capacity and thermal conductance of the link to the temperature bath. Unfortunately, Eq. (\ref{eqn:phase}) is not a good approximation for diamond-cell samples, resulting in little improvement at frequencies greater than 1 MHz (Appendix D).
\begin{table*}
\begin{center}
\begin{tabular}{ p{3 in} p{1.5 in} c }
& Sample (Fe) & Insulator (KBr)\\
\hline
$d$: Layer thickness ($\mu$m) & 5 & 10 \\
$\rho$: Density (g cm$^{-3}$) & 7.9 & 2.75 \\
$c$: Specific heat (J g$^{-1}$ K$^{-1}$) & 0.45 & 0.45 \\
$k$: Thermal conductivity (W m$^{-1}$ K$^{-1}$) & 80 & 4.8 \\
$D$: Thermal diffusivity ($\mu$m$^2$ $\mu$s$^{-1}$) & 22 & 3.9 \\
$r$: Resistivity ($\Omega$ m) & $9.7 \times 10^{-8}$ & - \\
$d\textrm{log}r/dT$: Temperature coefficient of resistance (K$^{-1}$) & 0.0064 & - \\
$\delta_{\textrm{skin}}$: Skin depth ($\mu$m)&
1000 if internal & - \\
& 0.1 if surface & \\
\hline
\end{tabular}
\caption{Properties of the sample and insulator used in our reference simulations.}
\label{table:mat_props}
\end{center}
\end{table*}
\section{Modeling scheme}
We model heat flow during modulated heating of metal samples pressed between symmetric layers of thermal insulation in a diamond anvil cell, as depicted schematically in Figs. \ref{fig:RH_schematic}, \ref{fig:LH_schematic}. Metals are chosen in this study since they are easier to heat via Joule-heating or laser-heating, and because they are likely choices for use as standard heaters in the future. We typically assume a total thickness from diamond to diamond of 30 $\mu$m, with sample thickness ranging from 1.7 to 15 $\mu$m, the remaining space consisting of thermal insulation between diamond and sample. We approximate the heat flow in the central part of the heated area as occurring solely in the axial direction of the diamond anvil cell (``$z$'' in Fig. \ref{fig:LH_schematic}), allowing reduction of the heat equation to one dimension along the axis:
\begin{equation}
\frac{\partial T(z,t)}{\partial t} = \frac{1}{\rho C} \frac{\partial (k\frac{\partial T}{\partial z})}{\partial z} + Q(z,t)
\label{eqn:T_diff}
\end{equation}
where $T$ is temperature at time $t$ and position $z$, $Q$ is a heating source, and $\rho$, $C$, and $k$ are material properties defined in Table \ref{table:mat_props} (assumed to be temperature-independent). In a previous publication, we used examples of two-dimensional axial simulations to show that the assumption of purely axial heat flow in diamond cells is valid at high frequencies ($f > D_{sam}/\textrm{width}$).\cite{Geballe2012}
We assume two heating sources of equal power are distributed through a skin depth, $\delta_{skin}$, from each side and that they vary sinusoidally in time with frequency $f$:
\begin{equation}
Q(z,t) = p_0\left( e^{-\frac{z-d_{\textrm{sam}}/2}{\delta_{\textrm{skin}}}} +e^{-\frac{z+d_{\textrm{sam}}/2}{\delta_{\textrm{skin}}}}\right)\sin(2\pi f t)
\label{eqn:Q}
\end{equation}
for $-d_{\textrm{sam}}/2 < z < d_{\textrm{sam}}/2$ and $Q(z,t) = 0$ otherwise. In Appendix E, a different heating source is used to simulate one-sided heating. We also tested a more realistic heating source that is always positive, $Q \propto 1+\sin(2\pi f t)$, and found no change in the resulting temperature variations.\footnote{The only change due to addition of a constant background heating source to a sinusoid (e.g. $Q \propto 1+\sin(2\pi f t)$ or $Q \propto 100 + \sin(2\pi f t)$) is in time-averaged temperature profile. The dynamic temperature response is identical.}
Layer thicknesses and material properties used in our reference simulations are listed in Table \ref{table:mat_props}. The material properties match ambient pressure-temperature values of an iron sample (except with a relative magnetic permeability of 1) and a single crystal potassium bromide insulator.
As in Refs. \onlinecite{Geballe2012,kiefer2005}, we assume that thermal conduction through the diamond anvils is so efficient that the temperature at the culet surface (diamond tip) is a constant (e.g. 300 K). In the main text, symmetric heating allows us to simulate one half of the sample chamber, from $z =0$ to $z= d_\textrm{sam} + d_\textrm{ins}$, as long as we enforce the boundary condition that no heat flows across the mid-plane of the sample, $\frac{\partial T}{\partial z}|_{z=0} = 0$.
The initial condition is that $T$ is constant in space, which is the average temperature distribution assuming the heating source described in Eq. (\ref{eqn:Q}).
We solve Eq. (\ref{eqn:T_diff}) by implementing the Crank-Nicholson numerical method described in Appendix F, with a typical time step of $0.01/f$ for frequency $f$, and a 10 to 50 nm mesh (i.e. 100-times smaller than the smallest sample dimension). We typically simulate 10 heating cycles and fit the final 5 cycles to a sinusoidal function in order to extract an amplitude of temperature oscillation. For laser-heating, we assume that the measured temperature is a weighted average of temperature to the fourth power in order to approximate the effect of the Stefan-Boltmann law:
\[T_{\textrm{meas}} = \left(A \int_0^{d_{\textrm{sam}}}{T^4(z) e^{-\left(\frac{z-d_{\textrm{sam}}}{\delta_{\textrm{skin}}}\right)}dz}\right) ^{1/4}
\]
where A is a normalization factor. In the case of Joule heating with third harmonic temperature measurement described below, we average $T$ rather than $T^4$, and since the skin depth is large compared to sample thickness, the exponential term approaches $e^0$. Hence,
\[T_{\textrm{meas}} = A \int_0^{d_{\textrm{sam}}}{T(z)dz} \]
The total heat capacity of sample plus addenda that would be inferred, is
\begin{equation}
C_\textrm{total} = \frac{p_\omega}{\omega T_\omega}
\label{eqn:C_meas_simple}
\end{equation}
where $p_\omega$ is the amplitude of power oscillations that is absorbed through the full thickness of the sample ($p = \int_{-d_{sam}}^{d_{sam}}{Q(z,T)dz}$), and $T_\omega$ is the amplitude of oscillation of $T_{\textrm{meas}}$, and $\omega$ is the angular frequency of both oscillations. \footnote{In the case of Joule-heating, we will redefine $\omega$ as the angular frequency of current or voltage oscillations, which are 2-fold smaller than the frequency of power oscillations, meaning ``$\omega$''s in Eq. (\ref{eqn:C_meas_simple}) will be replaced by ``$2\omega$''s in this paper and in Part II.}
\begin{mdframed}
Note: \hspace{5 mm} The variable $c$ or $C$ is used in several contexts in this study. Lower-case $c_{\textrm{sam}}$ and $c_{\textrm{ins}}$ are specific heats of sample and insulation material (units: Jg$^{-1}$K$^{-1}$, i.e. the material property).
Upper-case $C$ is the apparent heat capacity given by the ratio of energy input divided by temperature change, whether or not the conditions are adiabatic: in gneray they are not (units: J/K, i.e., the product of specific heat and mass). A subscript or superscript ``total'' implies the total heat capacity of sample plus addenda that would be inferred from power deposited and temperature oscillation measured, as opposed to a true heat capacity of sample or insulator. Superscript ``ref'' refers to the reference properties of Table \ref{table:mat_props}. For example, $C_\textrm{total}^\textrm{ref}$ is the total heat capacity that would be measured according to the simulation results using the reference properties.
\end{mdframed}
\begin{figure}
\begin{center}
\includegraphics[width=3.5in]{Internal_surf_comparison_v4-eps-converted-to.pdf}
\caption{Total heat heat capacities per unit area ($\frac{p_\omega}{\omega T_\omega}$) divided by sample heat capacity per unit area ($\rho_{\textrm{sam}}c_{\textrm{sam}}d_{\textrm{sam}}$) as a function of the frequency of heating modulation. Black circles indicate that heat is deposited internally and temperature is measured internally in our reference experiment, red diamonds indicate that heat is deposited internally and temperature is measured at the surface ($\delta_{\textrm{skin}} = 100$ nm), and blue triangles indicate that both temperature measurement and heat deposition take place in the surface ($\delta_{\textrm{skin}} = 100$ nm). The yellow band marks $< 10 \%$ error in heat capacity measurement.}
\label{fig:int_surf_comp}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=3.5in]{AC_calorim_errors_v4.png}
\caption{Error in heat capacity of sample due to addendum contribution vs. heating frequency. Black circles represent the reference experiment with internal heating and temperature measurement, while colors indicate that a single parameter has been changed by a factor of 10 from the reference. Yellow highlights the region with $< 10\%$ error.}
\label{fig:cal_errs_var_props}
\end{center}
\end{figure}
\begin{figure*}
\begin{center}
\includegraphics[width=5in]{K_f_limits_v9.png
\caption{Contours of error in measurement of heat capacity due to addendum contribution in a parameter space that describes two key properties of the proposed Joule-heating experiments: thermal conductivity of the insulation and the frequency of heating. Yellow shading marks $< 10\%$ error. We assume internal heating, internal temperature measurement, reference material properties (except for thermal conductivity, which varies), and reference geometries. The y-axes to the right of the figure show alternative changes from the reference experiment (KBr insulation, 5 $\mu$m-thick sample) that would result in approximately the same addenda contribution as the thermal conductivity of insulation plotted on the left-hand-side. In particular, we have mapped the effected of changes in thermal conductivity of insulation, $k_\textrm{ins}$, onto changes in thermal effusivity, $\sqrt{\rho_\textrm{ins} c_\textrm{ins} k_\textrm{ins}}$ or changes in sample thickness.}
\label{fig:K_f_limits}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=6.5in]{C-f_all_v2.png}
\caption{Total heat capacity of sample plus addenda (top row) and ratios of total heat capacity to total heat capacity in the reference model (bottom row) upon variation of the sample's specific heat (red curves), the sample's thermal conductivity (pink curves), the insulation's thermal conductivity (blue curves), the insulation's specific heat (cyan curves), and the skin depth of the heating source and temperature measurement probe (green curves). All four horizontal axes show frequency. The vertical axes in the top row are total heat capacities normalized by the true value of heat capacity of the sample. The vertical axes in the bottom row are the same total heat capacities, but normalized by the total heat capacity of sample plus addenda in the reference state ($C_\textrm{total}^{\textrm{ref}}$). The first column assumes internal heating and temperature measurement (I/I), while the second column assumes surface heating and temperature measurement (S/S, for which $\delta_{\textrm{skin}} = 100$ nm). Shades of red indicate the magnitude of specific heat changes (see legend), while yellow highlights accurate measurements (within 10$\%$) for 1-fold, 3-fold, 10-fold and 30-fold increases in sample heat capacity.}
\label{fig:C_changes}
\end{center}
\end{figure*}
\section{Results}
First, we assume the material properties and sample-chamber dimensions listed in \ref{table:mat_props}. We simulate heating experiments at 4 kHz to 100 MHz using the three geometries of heat source/temperature measurement described above, (I/I), (I/S) and (S/S). The total heat capacity inferred from power and temperature oscillations is shown in Fig. \ref{fig:int_surf_comp}.
At frequencies low compared to the timescale of heat conduction out of the sample, total heat capacity includes a large addenda contribution. For example, at 9 kHz, the addenda contribution is as large as the sample contribution to heat capacity (i.e. $C_\textrm{total}/C = 2$).
Near 1 MHz frequency, the total heat capacity drops to within $\sim 10\%$ of the sample's heat capacity in all cases (i.e. $1.1 > C_\textrm{total}/C > 0.9$) . At higher frequencies, the proposed measurement becomes even more accurate in the case of (I/I): total heat capacity asymptotically approaches the sample's value, while the addenda contribution becomes negligible. Examples showing this approach to adiabatic heating are shown in Appendix A, along with an idea of how it could be experimentally realized.
In the other cases, we also expect addenda contribution to heat capacity to become small at high frequency, but other details cause complications. In the case of I/S, the temperature measurement is near the sample surface, which is highly susceptible to thermal diffusion into the insulation. In fact, if temperature were measured at the true interface between sample and insulation, we expect insulation to contribute a measurable addendum at all frequencies. Here, we have assumed a 100 nm skin-depth, causing the slow decrease of total heat capacity at 10 to 100 MHz, when the lengthscale of diffusion starts to approach the skin-depth.
High-frequency experiments using (S/S) involve further complication. In this case, heat diffuses to all material near the sample-insulation interface (i.e. into both sample and addenda), but not to the sample's interior (see Appendix B for an example). Hence, at high frequency, total heat capacity is far smaller than the full sample's heat capacity. There is one lucky frequency at which addenda additions exactly cancel the reductions in heated sample, but its value depends on several material properties, including two that cannot be measured easily in a diamond cell: the skin depths of the heating source and of the temperature measurement probe.
To gain more insight into the total heat capacities that would be measured in Joule-heating with internal temperature measurement (I/I), we simulate a range of other experiments using the (I/I) scheme. A 3-fold increase in the sample's total heat capacity (i.e. a denser, greater specific-heat, or thicker sample) causes a $\sim 10$-fold reduction in the frequency requirement for low-addenda heat capacity measurements (Fig. \ref{fig:cal_errs_var_props}) The same effect results from 10-fold decreases in density, specific heat, or thermal conductivity of insulation. Thermal conductivity of the sample has a relatively small effect.
To state our findings more succinctly, we introduce the term ``thermal effusivity'', a material property that describes the ability of a material to absorb heat from its surface via conduction, which is defined by $\sqrt{\rho c k}$, with $\rho$, $c$ and $k$ being density, heat capacity and thermal conductivity. We can summarize the previous two observations by the following: a 3-fold decrease in the ratio of thermal effusivity of insulation to total heat capacity of sample causes a 10-fold decrease in frequency needed to achieve a fixed value of addenda contribution to total heat capacity. The inverse is also true: a 3-fold decrease in insulation thermal effusivity to sample heat capacity causes a 10-fold increase in frequency requirement.
Motivated by the large effect of insulation effusivity, a property that can be tuned over a wide range, we extend our calculations to more extreme values. In particular, we vary the thermal conductivity of insulation by two orders of magnitude in each direction, and present the results as contours of heat capacity measurement error in Fig. \ref{fig:K_f_limits} . Also, we project these simulated errors onto two other axes, the effusivity of the insulation and the thickness of metal sample, where we assume the addenda contribution is a function of frequency and the single variable $\sqrt{\rho_\textrm{ins} c_\textrm{ins} k_\textrm{ins}}/(\rho_\textrm{sam} c_\textrm{sam} d_\textrm{sam})$, as suggested by Fig. \ref{fig:cal_errs_var_props}. The result, Fig. \ref{fig:K_f_limits}, is meant as a guide for design of experiments. For example, use of polycrystalline KCl \cite{El-Sharkawy1984} or silica glass instead of single crystal KBr improves accuracy of heat capacity measured, whereas use of single crystal alumina decreases accuracy. Motivated by this observation, we chose to use silica glass instead of KCl in Part II of this two-part publication.
Variations in sample thickness also cause a large effect in the accuracy of measured heat capacity, with 20 $\mu$m-thick samples allowing frequencies less than 100 kHz to result in heat capacity measurements with 10$\%$ accuracy. We also note that a possible trick to reduce error in heat capacity measurement is to measure heat capacities at two (or more) sample thicknesses. The thinner sample is more sensitive to addenda than the thicker one, allowing deconvolution of the two contributions to measured heat capacity: the sample's contribution and the addenda's contribution. Detailed model results are presented in Appendix G.
Finally, in order to guide experiments seeking to detect changes in material properties (rather than absolute values), we study the frequency-dependence of changes in total heat capacity as various material properties change. Fig. \ref{fig:C_changes} shows the results for I/I and S/S schemes. The I/S scheme has been omitted as the results are similar to the S/S results. We normalize the results in two ways, via the heat capacity of the sample alone before changing the material property, $C^\textrm{ref}$, and via the heat capacity of sample plus addenda before changing the material property, $C^\textrm{ref}_\textrm{total}$. The former normalization may be more intuitive, but only the latter normalization would be experimentally feasible, so we discuss it here. At $\geq 300$ kHz and $\geq 1$ GHz, the ratio of total heat capacities, $C_\textrm{total}/C^\textrm{ref}_\textrm{total}$, approaches the correct values for (I/I) and (S/S), respectively. At lower frequencies (10 kHz and 10 MHz), measured heat capacities are more sensitive to increases in heat capacity than to all other changes in material properties considered here ($k_{\textrm{ins}}$, $c_{\textrm{ins}}$, $k_{\textrm{sam}}$), except for a change in skin depth, $\delta_{\textrm{skin}}$. Therefore, by scanning frequency by a few orders of magnitude around 10 kHz (for Joule-heating, I/I) or 10 MHz (for laser-heating, S/S), increases in heat-capacity may be identified with little ambiguity.
\section{Discussion}
Among the possible experiments considered here, two types of calorimetry measurements are shown to be feasible: absolute measurements of the heat capacities of Joule-heated metals with internal temperature measurement (I/I) and relative measurements of Joule- or laser-heated metals with surface temperature measurement (I/S) or (S/S). In both cases, high heating frequencies must be used ($\sim 100$ kHz to 10 MHz), and to allow for the most robust interpretations, frequency should be varied over such a range.
The requirement of high heating frequency can be understood by analyzing heat flow in the small volume of a diamond cell sample chamber; at frequencies lower than the frequency at which heat leaves the sample, the addenda contribution to heat capacity is large. Specifically, the characteristic timescale for heat conduction through the insulation to the diamonds is $\frac{d_\textrm{ins}^2}{D_\textrm{ins}}$, which is 25 $\mu$s for the 10 $\mu$m-thick KBr insulation assumed here. But there is a second source of heat loss: diffusion of heat into the insulation itself. The timescale of heat loss to the insulation is,
\begin{eqnarray}
\tau_{\textrm{into ins.}}
&=& \left( \frac{\rho_\textrm{sam} c_\textrm{sam} d_\textrm{sam}}{\textrm{eff}_\textrm{ins}}\right)^2
\label{eqn:into_ins_timescale}
\end{eqnarray}
where $\textrm{eff}_{ins} = \sqrt{\rho_{ins}c_{ins}k_{ins}}$ is the effusivity of the insulation. For the iron sample and KBr insulation assumed here, this timescale is 50 $\mu$s, meaning the insulation becomes a significant part of the addenda at frequencies less than a few tens of kHz.
In fact, the insulation contribution to total heat capacity is more difficult to correct for than heat flow to the diamonds anvils. Appendix D shows that total heat capacity can be corrected for contributions from the heat bath (i.e. the diamonds) by measuring the phase shift of the modulated temperature relative to the modulated heating source. Unfortunately, this correction does not seem to account for heat lost to insulation, leaving a large addenda contribution to total heat capacity.
Despite these challenges, we have shown that absolute calorimetry measurements of metals can be made with better than $10\%$ error using Joule heating, and relative heat capacities can be measured using either Joule- or laser-heating. In all cases, careful experimental design (e.g. frequency scanning) is required to discriminate changes in sample specific heat from changes in other properties of the sample or insulation. The simulation results presented here provide a roadmap for such design.
Joule heating affords a controlled way to deposit heat, and the third-harmonic temperature measurement described in Appendix C provides a way to measure temperature oscillations in the exact place that heat is deposited. Key experimental challenges are (1) to build a circuit capable of delivering high-frequency electrical power to the tips of diamonds with little electrical distortion and (2) to press a metal foil of uniform cross-sectional area between well-insulated diamond tips. If such engineering challenges are met, realization of the internal heating/internal temperature measurement experiment modeled here would provide the first absolute measurements of specific heat at $> 10$ GPa of pressure.
Laser heating allows simpler sample preparation and avoids the danger that a broken electrical lead ends an experiment prematurely. Assuming reliable temperature and power measurements, relative heat capacity can be measured at a range of frequencies from $\sim 1$ MHz to 1 GHz, and changes in specific heat can be discriminated from changes in other material properties by referring to Fig. \ref{fig:C_changes} (lower right panel). Indeed, temperature oscillations can be measured via thermal emissions, but estimation of the absorbed fraction of incident laser power is complicated by the potential for changes in reflectivity and opacity of sample and insulation, for example at a phase transition of interest. That is, the possibility of transition-induced changes in material properties would complicate interpretation of experimental measurements, especially if time dependencies (e.g., kinetics) play a role in the frequency range being investigated. Total reflectance and absorbance could be monitored, but even so, there is a possibility that the location of absorbed laser power remains unknown. If the insulation begins absorbing laser power at high temperatures, the model results presented here are of little use. On the other hand, there is little chance (in the absence of an insulator-metal transition) that the electrical conductivity of an insulator increases so much that it significantly alters the location at which Joule-heating power is deposited.
In order to quantify specific heat, the geometry of the sample must be known. The starting material in the proposed Joule-heating experiment should therefore be a strip of metal with uniform cross sectional area, and a soft pressure medium should be used on at least one side of the metal sample so that deformation is limited. Still, shearing is inevitable in diamond-cell experiments, and it typically results in thinning of a sample upon compression. Therefore, an accurate measurement of thickness is crucial to quantify specific heat. At least three options exist: (1) the metal's thickness can be inferred via white-light interference measurements from (i) diamond-to-diamond and (ii) diamond-to-metal on each side, (2) the metal's surface area can be measured \textit{in situ} via optical imaging, or (3) purely elastic strain following a known equation of state can be assumed upon decompression, and the metal's geometry can be measured (or inferred) after decompression to ambient pressure (i.e. \textit{ex situ}). One way to infer sample geometry precisely after decompression is to use the resistance or heat capacity measurement itself and the known ambient pressure value of resistivity or specific heat, as in the high-pressure resistivity measurements of Ref. \onlinecite{Gomi2013}.
A variety of potential benefits of the proposed measurements exist. First, high-pressure phase transitions of metals could be mapped in pressure-temperature space using either Joule- or laser-heating, and phase boundaries that are already documented using structural probes can be confirmed using an entropy-sensitive probe. Little accuracy is required to identify first-order phase transitions; relative heat capacity can be relied upon, and in principle, measured values of heat capacity can differ from true values by a factor of up to $\frac{L}{c_{sam}\sigma_T}$, where $L$ is the latent heat of phase transition and $\sigma_T$ is the precision of the temperature measurement. In fact, several studies of heavy-fermion compounds at high pressures have already documented phase transitions via \textit{relative} specific heat measurements up to $\sim 10$ GPa at $\leq 20$ K.\cite{Demuer2000,Wilhelm1999,Bouquet2000}
Second, the energetics of single phases can be studied and compared against models such as the Debye model. Deviations from models would be especially interesting for the metals that exhibit large increases in specific heat at temperatures 100s of K below melting at ambient pressure (i.e. ``pre-melting'').\cite{Ubbelohde1978}
Once a metal such as tungsten has been well-characterized using high-pressure calorimetry, it can be used as a standard ``hot plate'' heater. The insulator can be replaced with a non-metallic sample of interest, and relative calorimetry experiments can be performed to determine changes in their effusivities (referring to the curves labeled $c_{\textrm{ins}}$ and $k_{\textrm{ins}}$ in Fig. \ref{fig:C_changes}). This would complement current techniques that infer thermal diffusivities from pulsed heating experiments, but which require assumptions of heat capacities to infer thermal conductivities. \cite{Cahill2004,Cahill1990,Beck2007,Imada2014} A ``hot-plate'' technique could allow characterization of the energetics of transitions such as substitutional disordering in multi-valent minerals, or the dissociation of molecular fluids. As in the case of mapping pressures and temperatures of temperature-driven transitions in Joule-heated metals, rough calorimetric measurements are likely sufficient to identify the latent heat of many first-order phase transitions, to detect large increases in specific heat near second-order phase transitions, or to discriminate between the two.
We have also described the limitations of calorimetric measurements in diamond cells. First, absolute calorimetry is not possible in surface-heated samples. This precludes the possibility of using laser heating to study the heat capacity of metallic samples (without use of a heat capacity standard). Second, even when using internal heating, the addenda from relatively good thermal insulators such as KBr cause large biases in measured heat capacities at frequencies below $\sim 100$ kHz.
\section{Conclusions}
Many opportunities exist for novel calorimetry experiments at high-pressures using high-frequencies heating sources. Since ``insulators'' in diamond cells are thin layers of dense solids or liquids, significant addenda contributions to measured heat capacities are unavoidable at low frequencies. But those contributions can be limited to $< 10 \%$ of the sample's heat capacity by use of high-frequencies ($\sim 100$ kHz to 10 MHz), internal heating (e.g. Joule heating), relatively low thermal conductivity insulators (e.g. alkali-halides or glasses), and relatively thick samples ($\geq 5$ to 20 $\mu$m).
|
2,869,038,156,357 | arxiv | \section{Introduction}
This paper investigates the irreducibility of tensor modules for the Lie algebra of divergence zero vector fields on a torus. The Lie algebra of derivations of $A=\mathbb{C}[t_1^{\pm1},\dots,t_{N}^{\pm1}]$, denoted by $\mathcal{D}$, may be interpreted as the Lie algebra of polynomial vector fields on an $N$-dimensional torus (see Section 2). In \cite{R} Rao determines the conditions for irreducibility of tensor modules for $\mathcal{D}$. This originates from analogous results established earlier by Rudakov in \cite{Ru} for the Lie algebra of vector fields on an affine space. Tensor modules were also studies by Shen in \cite{S}.
Let $\mathfrak{gl}_N$ be the Lie algebra of $N\times N$ matrices with complex entries, and $\mathfrak{sl}_N$ its subalgebra of trace zero matrices. Then $\mathfrak{gl}_N=\mathfrak{sl}_N\oplus\mathbb{C} I$ where $I$ is the identity matrix. An irreducible $\mathfrak{gl}_N$-module can be constructed by taking $V(\lambda)$, the unique finite dimensional irreducible $\mathfrak{sl}_N$-module of highest weight $\lambda$, and specifying that $I$, since it is central, acts by complex number $b$. Denote this $\mathfrak{gl}_N$-module by $V(\lambda,b)$. The tensor product $V(\lambda,b)\otimes A$ becomes a module for a specified action of $\mathcal{D}$. Rao shows that these modules are irreducible unless $(\lambda,b)=(\omega_k,k)$ for $k\in\{1,\dots,N-1\}$, where $\omega_k$ is a fundamental dominant weight (see Section 3), or if $\lambda$ is zero. In these special cases irreducible quotients and submodules are determined.
The current paper is motivated by the results of \cite{R}, and considers the restriction of these tensor modules to the subalgebra of divergence zero vector fields, denoted $\mathcal{D}_{\text{div}}$. It will be shown in section 2 that the restriction to divergence zero vector fields results in limiting the action to only the $\mathfrak{sl}_N$ part of the modules. Thus the $\mathcal{D}_{\text{div}}$-modules considered here are of the form $V(\lambda)\otimes A$. A streamlined proof of Rao's result was given in \cite{GZ} and the current paper has drawn inspiration from these ideas and applied them here. The result, given in Theorem \ref{results}, mirrors that of Rao's and it is shown that these $\mathcal{D}_{\text{div}}$-modules are irreducible if $\lambda\neq\omega_k,0$.
To understand the cases when $\lambda=\omega_k,0$, the corresponding tensor modules can be realized as the modules of differential $k$-forms on a torus. The tensor modules $V(\omega_k)\otimes A$ for $k\in\{1,\dots,N-1\}$ are isomorphic to $\Omega^k$, the module (over Laurent polynomials) of differential $k$-forms on a torus. The modules of functions $\Omega^0$, and $N$-forms $\Omega^N$, correspond to tensor modules $V(0)\otimes A$, where $V(0)$ is a one-dimensional $\mathfrak{sl}_N$-module. It is useful to consider the de Rham complex of differential forms,
\[\Omega^0\xrightarrow{d}\Omega^1\xrightarrow{d}\dots\xrightarrow{d}\Omega^N\]
(where $d$ is the exterior derivative) since $d$ yields a homomorphism between tensor modules. Thus kernels and images of $d$ are submodules of the $\Omega^k$ and these will be used to determine submodules of the $\mathcal{D}_{\text{div}}$-modules.
In Section 2 the Lie algebra $\mathcal{D}_{\text{div}}$ will be presented, as well as the notation for its homogeneous elements, and a spanning set. The above-mentioned $\mathcal{D}_{\text{div}}$-modules are defined in Section 3 and classified as being minuscule or non minuscule. Two important properties of $\mathcal{D}_{\text{div}}$-modules are given in this section pertaining to the size of these modules and the action of $\mathcal{D}_{\text{div}}$. In Section 4 it is seen that highest weight vectors may always be obtained in modules of non minuscule type. The final section contains the main result, and here it is proven that non minuscule $\mathcal{D}_{\text{div}}$-modules are irreducible, while minuscule modules have irreducible submodules and quotients.
\section{Preliminaries}
Fix $N\in\mathbb{N}$ and consider the column vector space $\mathbb{C}^{N+1}$ with standard basis $\{e_1,\dots,e_{N+1}\}$. Let $(\cdot|\cdot)$ be symmetric bilinear form $(u|v)=u^Tv\in\mathbb{C}$, where $u,v\in\mathbb{C}^{N+1}$ and $u^T$ denotes matrix transpose.
Let $A=\mathbb{C}[t_1^{\pm1},\dots,t_{N+1}^{\pm1}]$ be the Laurent polynomials over $\mathbb{C}$. Elements of $A$ will be presented with the multi-index notation $t^r=t_1^{r_1}\dots t_{N+1}^{r_{N+1}}$ for $r=(r_1,\dots,r_{N+1})\in\mathbb{Z}^{N+1}$. Also, for $i\in\{1,\dots,N+1\}$, let $d_i=t_i\frac{\partial}{\partial t_i}$. The vector space of all of derivations of $A$, $\text{Der}(A)$, forms a Lie algebra called the Witt algebra denoted here by $\mathcal{D}$. Note that $\text{Der}(A)=\text{Span}_{\mathbb{C}}\left\{t^rd_i|i\in\{1,\dots,N+1\}, r\in\mathbb{Z}^{N+1}\right\}$. Homogeneous (in the power of $t$) elements of $\mathcal{D}$ will be denoted $D(u,r)=\sum_{i=1}^{N+1}u_it^rd_i$ for any $u\in\mathbb{C}^{N+1},r\in\mathbb{Z}^{N+1}$. Its Lie bracket is given by
\[[D(u,r),D(v,s)]=D\left((u|s)v-(v|r)u,r+s\right)\]
for $u,v\in\mathbb{C}^{N+1},r,s\in\mathbb{Z}^{N+1}$.
Geometrically, $\mathcal{D}$ may be interpreted as the Lie algebra of (complex-valued) polynomial vector fields on an $N+1$-dimensional torus, via the mapping $t_j=e^{ix_j}$ for all $j\in\{1,\dots,N+1\}$, where $x_j$ represents the $j$th angular coordinate. This has an interesting subalgebra, the Lie algebra of divergence-zero vector fields, and the corresponding subalgebra of $\mathcal{D}$ is denoted by $\mathcal{D}_{\text{div}}$.
\begin{prop}
Let $u\in\mathbb{C}^{N+1},r\in\mathbb{Z}^{N+1}$. Then $D(u,r)\in\mathcal{D}_{\text{div}}$ if and only if $(u|r)=0$.
\end{prop}
\begin{proof}
Note that under the mapping $t_j=e^{ix_j}$, $\frac{\partial}{\partial x_j}=\frac{\partial t_j}{\partial x_j}\cdot\frac{\partial}{\partial t_j}=it_j\frac{\partial}{\partial t_j}=id_j$, so that $v=\sum_{j=1}^{N+1}f_j(t)d_j\in\mathcal{D}$ can be written $v=-i\sum_{j=1}^{N+1}f_j(t)\frac{\partial}{\partial x_j}$. The divergence of $v$ with respect to the natural volume form in angular coordinates is then $-i\sum_{j=1}^{N+1}\frac{\partial f_j}{\partial x_j}=\sum_{j=1}^{N+1}t_j\frac{\partial f_j}{\partial t_j}$. Thus $\text{div} D(u,r)=\sum_{j=1}^{N+1}u_jr_jt^r=0$ if and only if $(u|r)=0$.
\end{proof}
This proposition uses elements of $\mathcal{D}$ which are homogeneous in $t$, and by linear independence of the powers of $t$ it may be applied to the general case. An element of $\mathcal{D}$ is in $\mathcal{D}_{\text{div}}$ if and only if its homogeneous components are in $\mathcal{D}_{\text{div}}$. It follows that $\mathcal{D}_{\text{div}}=\text{Span}_{\mathbb{C}}\left\{d_a,r_{b}t^rd_a-r_at^rd_{b}|a,b\in\{1,\dots,N+1\}, r\in\mathbb{Z}^{N+1}\right\}$.
Note that for $D(u,r),D(v,s)\in\mathcal{D}_{\text{div}}$, $\left((u|s)v-(v|r)u|r+s\right)=0$, demonstrating that $\mathcal{D}_{\text{div}}$ is closed under the bracket of $\mathcal{D}$. $\mathcal{D}_{\text{div}}$ has Cartan subalgebra $\mathcal{H}=\text{Span}_{\mathbb{C}}\{d_j|j\in\{1,\dots,N+1\}\}=\{D(u,0)|u\in\mathbb{C}^{N+1}\}$.
\section{$\mathcal{D}_{\text{div}}$-Modules}
Let $\mathfrak{sl}_{N+1}$ be the Lie algebra of all $(N+1)\times (N+1)$ traceless matrices with entries in $\mathbb{C}$. Fix a root system $\Phi$, of type $A_N$, with positive roots $\Phi^+$ and simple roots $\Delta=\{\alpha_1,\dots,\alpha_N\}$.
Let $V(\lambda)$ be a finite dimensional irreducible $\mathfrak{sl}_{N+1}$-module of highest weight $\lambda=C_1\omega_1+\dots+C_N\omega_N$, where $\langle\omega_i,\alpha_j\rangle\equiv\frac{2(\omega_i|\alpha_j)}{(\alpha_j|\alpha_j)}=\delta_{ij}$ and $C_i\in\mathbb{Z}_{\geq0}$ for all $i,j\in\{1,\dots,N\}$.
Any weight $\gamma$ of $V(\lambda)$ can be written $\gamma=\lambda-\sum_{i=1}^N\gamma_i\alpha_i$ where the $\gamma_i$'s are non-negative integers. For convenience assign to each weight a \emph{label}, defined as the $N$-tuple $(\langle\gamma,\alpha_1\rangle,\dots,\langle\gamma,\alpha_N\rangle)$. Thus $\lambda$ has label $(C_1,\dots,C_N)$. Note that here a weight of $V(\lambda)$, means only those assigned to a nonzero weight space.
Let $\gamma$ be a weight in $V(\lambda)$ and $\alpha\in\Phi$ (not necessarily simple). Then the set of weights of the form $\gamma+i\alpha$, for $i\in\mathbb{Z}$, is called the \emph{$\alpha$-string through $\gamma$}. let $r,q\in\mathbb{Z}_{\geq0}$ be the largest integers such that $\gamma-r\alpha$ and $\gamma+q\alpha$ are weights of $V(\lambda)$. Then $\gamma+i\alpha$ is a weight for all $-r\leq i\leq q$; i.e. the $\alpha$-string through $\gamma$ is unbroken from $\gamma-r\alpha$ to $\gamma+q\alpha$. Furthermore it can be seen that $\langle\gamma,\alpha\rangle=r-q$. These unbroken sets of weights will be referred to as \emph{weight strings}, and the number of weights in this set its \emph{length}. In this particular case the weight string is in the $\alpha$ direction.
The Weyl group for $A_N$, is isomorphic to $S_{N+1}$, the symmetric group on $N+1$ symbols, and so elements of the Weyl group will be denoted with cycle notation. The reflection through simple root $\alpha_i$ is the transposition $(i,i+1)$, and the reflection through a root of the form $\alpha_i+\dots+\alpha_{i+j}$ is the transposition $(i,i+j+1)$, for $1\leq i<i+j\leq N$.
\begin{lem}\label{size of diagram}
Let $\lambda_{\ell}$ denote the lowest weight of $V(\lambda)$. Write $\lambda_{\ell}=\lambda-\sum_{i=1}^N\kappa_i\alpha_i$, where $\kappa_i\in\mathbb{Z}_{\geq0}$. Then for $1\leq j\leq\lfloor \frac{N+1}{2}\rfloor$,
\[\kappa_j=\kappa_{N+1-j}=\sum_{i=1}^NC_i +\sum_{i=2}^{N-1}C_i +\dots+\sum_{i=j}^{N+1-j}C_i\]
\end{lem}
\begin{proof}\label{}
Consider the Weyl group element
\[w=\prod_{p=0}^{\lfloor\frac{N}{2}\rfloor}(1+p, N+1-p).\]
Then $w(\alpha_i)=-\alpha_{N+1-i}$ for $1\leq i\leq N$ and it follows that $w$ maps the highest weight of the $\mathfrak{sl}_{N+1}$-module to its lowest weight. The above transpositions are disjoint and hence commute. Let
\[\lambda_j=\left(\prod_{p=0}^{j-1}(1+p,N+1-p)\right)(\lambda)\]
for $j=1,\dots,\lfloor\frac{N}{2}\rfloor+1$ and $\lambda_0=\lambda$. Then
\begin{align*}
\lambda_1&=(1,N+1)(\lambda)\\
&=\lambda_0-\langle\lambda_0,\alpha_1+\dots+\alpha_N\rangle(\alpha_1+\dots+\alpha_N)\\
&=\lambda_0-\left(\sum_{i=1}^NC_i\right)(\alpha_1+\dots+\alpha_N).
\end{align*}
Assume $\lambda_j=\lambda_{j-1}-\sum_{i=j}^{N+1-j}C_i(\alpha_j+\dots+\alpha_{N+1-j})$ for some $j\geq1$. Then
\begin{align*}
\lambda_{j+1}&=(1+j,N+1-j)(\lambda_j)\\
&= \lambda_j-\langle\lambda_j,\alpha_{j+1}+\dots+\alpha_{N-j}\rangle(\alpha_{j+1}+\dots+\alpha_{N-j})\\
&= \lambda_j-\left\langle\lambda-\left(\sum_{i=1}^NC_i\right)(\alpha_1+\dots+\alpha_N)-\left(\sum_{i=2}^{N-1}C_i\right)(\alpha_2+\dots+\alpha_{N-1})\right.\\
&\quad-\dots\\
&\quad- \left.\left(\sum_{i=j}^{N-j}C_i\right)(\alpha_j+\dots+\alpha_{N-j}),\alpha_{j+1}+\dots+\alpha_{N-j}\right\rangle(\alpha_{j+1}+\dots+\alpha_{N-j})\\
&= \lambda_j-\langle\lambda,\alpha_{j+1}+\dots+\alpha_{N-j}\rangle(\alpha_{j+1}+\dots+\alpha_{N-j})\\
&\quad- \left(\sum_{i=1}^NC_i\right)\left\langle\alpha_1+\dots+\alpha_N,\alpha_{j+1}+\dots+\alpha_{N-j}\right\rangle(\alpha_{j+1}+\dots+\alpha_{N-j})\\
&\quad- \left(\sum_{i=2}^{N-1}C_i\right)\left\langle\alpha_2+\dots+\alpha_{N-1},\alpha_{j+1}+\dots+\alpha_{N-j}\right\rangle(\alpha_{j+1}+\dots+\alpha_{N-j})\\
&\quad-\dots\\
&\quad- \left(\sum_{i=j}^{N-j}C_i\right)\left\langle\alpha_j+\dots+\alpha_{N-j},\alpha_{j+1}+\dots+\alpha_{N-j}\right\rangle(\alpha_{j+1}+\dots+\alpha_{N-j})\\
&= \lambda_j-\left(\sum_{i=j+1}^{N-j}C_i\right)(\alpha_{j+1}+\dots+\alpha_{N-j}),
\end{align*}
since the angled bracket evaluates to zero in all but the second term. Then by induction
\[w(\lambda)=\lambda_{\lfloor\frac{N}{2}\rfloor+1}=\lambda-\sum_{i=1}^{\lfloor \frac{N}{2}\rfloor+1}\sum_{j=i}^{N+1-i}C_j\sum_{k=i}^{N+1-i}\alpha_k\]
with the final term being zero if $N$ even. Since $w(\lambda)=\lambda_{\ell}$, extracting coefficients of the $\alpha_i$ yields the result.
\end{proof}
Denote $F^{\sigma}(\lambda)= V(\lambda)\otimes\mathbb{C}[q_1^{\pm1},\dots,q_{N+1}^{\pm1}]$ for $\sigma\in\mathbb{C}^{N+1}$. Homogeneous elements of $F^{\sigma}(\lambda)$ will be denoted $v(n)=v\otimes q^n$ for $v\in V(\lambda),n\in\mathbb{Z}^{N+1}$. Then $F^{\sigma}(\lambda)$ becomes a module for $\mathcal{D}_{\text{div}}$ via the action
\[D(u,r).v(n)=(u|n+\sigma)v(n+r)+(ru^T)v(n+r),\]
for $D(u,r)\in\mathcal{D}_{\text{div}}$. Note that $ru^T$ is an $(N+1)\times (N+1)$ matrix with trace zero, as $(u|r)=0$ by assumption, and so the second term involves the $\mathfrak{sl}_{N+1}$ module action. Elements of $\mathfrak{sl}_{N+1}$ are denoted in the usual way, where $E_{ij}$ is the matrix with a 1 in the $(i,j)$ position and zeros elsewhere.
Modules $V(\lambda)$ and $F^{\sigma}(\lambda)$ will be called \emph{minuscule} if $\lambda=\omega_i$ or $\lambda=0$ for $i\in\{1,\dots,N\}$ (see \cite{BL} 2.11.15). Write $F^{\sigma}(\lambda)=\bigoplus_{n\in\mathbb{Z}^{N+1}}V(\lambda)\otimes q^n$ and let $v(n)\in V(\lambda)\otimes q^n$. Then
$D(e_i,0).v(n)=(n_i+\sigma_i)v(n)$ for all $i\in\{1,\dots,N+1\}$, so that the subspace $V(\lambda)\otimes q^n$ is homogeneous in $q$ and thus $F^{\sigma}(\lambda)$ is a graded module with respect to the action of $\mathcal{H}$. Any submodule $M$ of $F^{\sigma}(\lambda)$ will inherit this gradation and so $M=\bigoplus_{n\in\mathbb{Z}^{N+1}}(V(\lambda)\otimes q^n)\cap M$. It follows that $M$ contains elements of the form $v(n)$ (homogeneous elements).
\begin{prop}\label{actions}
Let $M$ be a submodule of $F^{\sigma}(\lambda)$. If $v(n)\in M$ then $(E_{ij}^kv)(n)\in M$ for any integer $k\geq2$ and any $i\neq j\in\{1,\dots,N+1\}$.
\end{prop}
\begin{proof}\label{}
Note that $D(e_j,re_i)\in\mathcal{D}_{\text{div}}$ for any $r\in\mathbb{Z}$, as $(e_j|e_i)=0$ for $i\neq j$. Fix an integer $k\geq2$, and choose $r_1,\dots,r_k\in\mathbb{Z}\setminus\{0\}$ such that $\sum_{p=1}^kr_p=0$. From the module action given above, $D(e_j,re_i).v(n)=\left((e_j|n+\sigma)+rE_{ij}\right)v(n+re_i)$. Thus
\begin{align*}
D(e_i,r_1e_j)\dots D(e_i,r_ke_j).v(n) &= (K_0+r_1E_{ij})\dots(K_0+r_kE_{ij})v\left(n+\sum_{p=1}^kr_p\right)\\
&= \left(K_0^k+K_1E_{ij}+K_2E_{ij}^2+\dots+K_kE_{ij}^k\right)v(n),
\end{align*}
where $K_0=(e_j|n+\sigma)$, $K_1=K_0^{k-1}\sum_{p=1}^kr_p=0$, and $K_k$ is nonzero. For the case $k=2$, take $r_1=1,r_2=-1$. Then $D(e_j,e_i).D(e_j,-e_i).v(n)=K_0^2v(n)-E_{ij}^2v(n)\in M$, and hence $E_{ij}^2v(n)\in M$. The result follows by induction on $k$.
\end{proof}
\section{Existence of Weight Vectors}
Recall that the root system of type $\mathfrak{sl}_{N+1}$ has highest root $\theta=\alpha_1+\dots+\alpha_N$. A weight string of maximal length in the $\theta$ direction will be called a \emph{maximal $\theta$-string}.
\begin{prop}\label{max theta string}
A $\theta$-string through $\lambda$ is a maximal $\theta$-string, and the highest weight of any maximal $\theta$-string has the form $\lambda-\sum_{i=2}^{N-1}\gamma_i\alpha_i$ for some non-negative integers $\gamma_i$.
\end{prop}
\begin{proof}\label{}
Let $\mathcal{S}_{\lambda}$ be the $\theta$-string through $\lambda$. Since $\lambda$ is the highest weight of $\mathcal{S}_{\lambda}$ the lowest weight of $\mathcal{S}_{\lambda}$ is $\lambda-\langle\lambda,\theta\rangle\theta=\lambda-\left(\sum_{i=1}^NC_i\right)\theta$, and thus $\mathcal{S}_{\lambda}$ has length $1+\sum_{i=1}^NC_i$. Let $\mathcal{S}_{\gamma}$ be the $\theta$-string through some other weight $\gamma$, where $\gamma$ is the highest weight of $\mathcal{S}_{\gamma}$ and $\gamma=\lambda-\sum_{j=1}^N\gamma_j\alpha_j$ for some non-negative integers $\gamma_j$.\\
Suppose $\mathcal{S}_{\gamma}$ has length greater than that of $\mathcal{S}_{\lambda}$, say $r+1+\sum_{i=1}^NC_i$ for some positive integer $r$. Then the lowest weight of $\mathcal{S}_{\gamma}$ is $\gamma-\left(r+\sum_{i=1}^NC_i\right)\theta=\lambda-\left(\sum_{j=1}^N\gamma_j\alpha_j\right)-\left(r+\sum_{i=1}^NC_i\right)\theta=\lambda-\sum_{j=1}^N\left(\gamma_j+r+\left(\sum_{i=1}^NC_i\right)\right)\alpha_j$. This contradicts Lemma \ref{size of diagram} since a multiple of $\alpha_1$ (or $\alpha_N$) greater than $\sum_{i=1}^NC_i$ cannot be subtracted from $\lambda$; i.e. no such weight exists. Thus $\mathcal{S}_{\gamma}$ cannot have a length greater than that of $\mathcal{S}_{\lambda}$, so $\mathcal{S}_{\lambda}$ has maximal length.\\
Suppose now that $\mathcal{S}_{\gamma}$ has the same (maximal) length as $\mathcal{S}_{\lambda}$. Then the lowest weight of $\mathcal{S}_{\gamma}$ is $\lambda-\sum_{j=1}^N\left(\gamma_j+\left(\sum_{i=1}^NC_i\right)\right)\alpha_j$. This contradicts Lemma \ref{size of diagram} in the same way as above unless both $\gamma_1$ and $\gamma_N$ are zero.
\end{proof}
\begin{cor}\label{sl3 theta string}
Irreducible $\mathfrak{sl}_3$-modules have exactly one maximal $\theta$-string.
\end{cor}
\begin{proof}\label{}
For the $N=2$ case, $\lambda$ is the only weight meeting the criteria of Proposition \ref{max theta string} forcing the only maximal $\theta$-string to be the one through $\lambda$.
\end{proof}
\begin{cor}\label{length at least 3}
In a non minuscule module the length of any maximal $\theta$-string is at least 3.
\end{cor}
\begin{proof}\label{}
As seen in the proof of Proposition \ref{max theta string}, the length of the $\theta$-string through $\lambda$ is $1+\langle\lambda,\theta\rangle=1+\sum_{i=1}^NC_i$. For a non minuscule module $\sum_{i=1}^NC_i\geq2$.
\end{proof}
\begin{remark}\label{labels}
Let $V(\lambda)$ be a non minuscule module and $N\geq 2$. The label assigned to $\lambda$, $(C_1,\dots,C_N)$, falls into at least one of the following three cases:
\begin{enumerate}
\item[(i)]
$(C_1,\dots,C_{N-1})$ is a highest weight label for a non minuscule $\mathfrak{sl}_N$-module,
\item[(ii)]
$(C_2,\dots,C_N)$ is a highest weight label for a non minuscule $\mathfrak{sl}_N$-module, or
\item[(iii)]
$C_1=C_N=1$ and $C_2=\dots=C_{N-1}=0$.
\end{enumerate}
\end{remark}
Applying Lemma \ref{size of diagram} to case (iii) shows that $\lambda_{\ell}=\lambda-2\theta$ and hence it is the lowest weight for the maximal $\theta$-string through $\lambda$. Suppose $\mathcal{S}_{\gamma}$ is a maximal $\theta$-string through the weight $\gamma=\lambda-\sum_{i=1}^n\gamma_i\alpha_i$ for $\gamma_i\in\mathbb{Z}_{\geq0}$. Then $\mathcal{S}_{\gamma}$ has lowest weight $\gamma-2\theta=\lambda-\sum_{i=1}^n\gamma_i\alpha_i-2\theta$. This forces each $\gamma_i$ to be zero, and thus $\mathcal{S}_{\lambda}$ is the only maximal $\theta$-string.
Let $M$ be a nonzero submodule of $F^{\sigma}(\lambda)$. The goal for the remainder of this section is to show that if $F^{\sigma}(\lambda)$ is non minuscule then $v_{\lambda}(n)\in M$ for all $n\in\mathbb{Z}^{N+1}$ where $v_{\lambda}$ is a highest weight vector in $V(\lambda)$. Once $v_{\lambda}(n)$ is obtained it will be used in the next section to generate $V(\lambda)\otimes q^n$ for each $n$, thus generating all of $F^{\sigma}(\lambda)$.
\begin{prop}\label{highest weight component}
$M$ contains a vector $v(n)$, for some $n$, such that $v$ has a nonzero component in $V_{\lambda}$, the highest weight space of $V(\lambda)$.
\end{prop}
\begin{proof}\label{}
Let $v(m)\in M$ for some $m\in\mathbb{Z}^{N+1}$, and $v\in V(\lambda)$. Let $v=\sum_{\gamma\in\Lambda}v_{\gamma}$ be the decomposition of $v$ into weight vectors where $\Lambda$ is the set of weights of $V(\lambda)$. If $v_{\lambda}\neq0$ there is nothing to prove, so suppose $v_{\lambda}=0$. From the definition of the highest weight it follows that there is some $\mu$ in $\Lambda$ such that $E_{ij}v_{\mu}\in V_{\mu'}\neq0$ and $v_{\mu'}=0$ in the above decomposition, for some $i<j\in\{1,\dots,N+1\}$. Applying $D(e_j,e_i)$ to $v(m)$ yields $(e_j|m+\sigma)v(m+e_i)+E_{ij}v(m+e_i)=v'(m+e_i)$, where $v'$ has a nonzero component in $V_{\mu'}$. If $\mu'\neq\lambda$ this process may be repeated by choosing another $(i,j)$-pair in the same way, until the highest weight space has been reached.
\end{proof}
\begin{prop}\label{highest weight vector}
Let $V(\lambda)$ be non minuscule and suppose for some $n\in\mathbb{Z}^{N+1}$ that $v(n)\in M$ has nonzero component in $V_{\lambda}$. Then $v_{\lambda}(n)\in M$, where $v_{\lambda}$ is a highest weight vector of $V(\lambda)$.
\end{prop}
\begin{proof}\label{}
Suppose for some $v(n)\in M$ that $v$ has a nonzero component $v_{\lambda}$ which lies in the highest weight space of $V(\lambda)$. Such a vector exists by Proposition \ref{highest weight component}. Then $v=\sum_{\gamma\in\Lambda}v_{\gamma}$, where $\Lambda$ is a set of weights of $V(\lambda)$ and $v_{\lambda}\neq0$. By Proposition \ref{max theta string} there is a maximal $\theta$-string of length $\langle\lambda,\theta\rangle+1\geq3$ which has highest weight $\lambda$. Note that $E_{N+1,1}$ maps $V_{\mu}$ into $V_{\mu-\theta}$ for any weight $\mu$ of $V(\lambda)$. The goal in what follows is annihilate those components different from $v_{\lambda}$. This will be accomplished by moving $v$ back and forth along the $\theta$-strings.
In the case $N=1$, an irreducible $\mathfrak{sl}_2$-module is a single $\theta$-string. For $N=2$, Corollary \ref{sl3 theta string} says that irreducible $\mathfrak{sl}_3$-modules have only one maximal $\theta$-string. In both cases these strings necessarily have highest weight $v_{\lambda}$. By Proposition \ref{actions}, $E_{N+1,1}^{\langle\lambda,\theta\rangle}v(n)\in M$ where $E_{N+1,1}^{\langle\lambda,\theta\rangle}v\in V_{\lambda-\langle\lambda,\theta\rangle\theta}$. All $v_{\gamma}\neq v_{\lambda}$ are annihilated because their weights either do not lie in the unique maximal $\theta$-string, or if they do, they are of the form $\lambda-i\theta$ for $i\geq1$. In either case $E_{N+1,1}^{\langle\lambda,\theta\rangle}\left(v_{\gamma}\right)=0$, for $\gamma\neq\lambda$, due to maximal length. Applying $E_{1,N+1}^{\langle\lambda,\theta\rangle}$ to $E_{N+1,1}^{\langle\lambda,\theta\rangle}v(n)$ then yields a scalar multiple of $v_{\lambda}(n)$. Now proceed by induction on $N$ with base case $N=2$.
Assume now $N\geq3$ and $v(n)\in M$ where $v=\sum_{\gamma\in\Lambda}v_{\gamma}$. Applying Proposition \ref{actions} yields $E_{N+1,1}^{\langle\lambda,\theta\rangle}v(n)=\sum_{\mu\in\Lambda_*}w_{\mu}(n)\in M$ where each $\mu\in\Lambda_*$ is given by $\mu=\gamma-\langle\lambda,\theta\rangle\theta$ for some $\gamma\in\Lambda$. By maximality of the length of a maximal $\theta$-string, any $\mu\in\Lambda_*$ is a lowest weight in the $\theta$-string which it belongs; i.e. the only $\gamma\in\Lambda$ for which $E_{N+1,1}^{\langle\lambda,\theta\rangle}v_{\gamma}\neq0$ are those which are the highest weight of their maximal $\theta$-string. In particular $\lambda-\langle\lambda,\theta\rangle\theta\in\Lambda_*$. Then $v'(n)=E_{1,N+1}^{\langle\lambda,\theta\rangle}\left(\sum_{\mu\in\Lambda_*}w_{\mu}(n)\right)=\sum_{\gamma\in\Lambda^*}v'_{\gamma}(n)\in M$ where $\lambda\in\Lambda^*$ with $v'_{\lambda}\neq0$ and $\Lambda^*\subset\Lambda$ are weights which occur as the highest weight in their respective maximal $\theta$-string.
Let $\Lambda_1$ be the set of all weights in $V(\lambda)$ of the form $\lambda-\sum_{i=1}^{N-1}a_i\alpha_i$, and $\Lambda_N$ be the set of all weights in $V(\lambda)$ of the form $\lambda-\sum_{i=2}^{N}b_i\alpha_i$, where $a_i,b_i\in\mathbb{Z}_{\geq 0}$. Let $V(\lambda)_1$ and $V(\lambda)_N$ be the span of all weight vectors with weights in $\Lambda_1$ and $\Lambda_N$ respectively. Then $V(\lambda)_1$ is an $\mathfrak{sl}_N$-module, for the copy of $\mathfrak{sl}_N$ generated by $\{E_{ij}|1\leq i,j\leq N,i\neq j\}\subset\mathfrak{sl}_{N+1}$, with highest weight label $(C_1,\dots,C_{N-1})$. Similarly $V(\lambda)_N$ is an $\mathfrak{sl}_N$-module, for the copy of $\mathfrak{sl}_N$ generated by $\{E_{ij}|2\leq i,j\leq N+1,i\neq j\}\subset\mathfrak{sl}_{N+1}$, and has highest weight label $(C_2,\dots,C_{N})$. Modules $V(\lambda)_1$ and $V(\lambda)_N$ are irreducible since any critical vector they contain is also a critical vector for $\mathfrak{sl}_{N+1}$ in $V(\lambda)$ which is irreducible by assumption.
By Proposition \ref{max theta string} $\Lambda^*$ is a subset of both $\Lambda_1$ and $\Lambda_N$. By remark \ref{labels}, either $V(\Lambda)_1$ or $V(\lambda)_N$ is non minuscule, or there is only one maximal $\theta$-string. In the latter $\Lambda^*$ is empty and so $v'(n)$ is a scalar multiple of $v_{\lambda}(n)$. Otherwise $v'$ lies in a non minuscule $\mathfrak{sl}_N$-module $W$ (either $V(\Lambda)_1$ or $V(\lambda)_N$) and has nonzero component in $V_{\lambda}$. By induction $v_{\lambda}(n)\in W$ and thus $v_{\lambda}(n)\in M$.
\end{proof}
\begin{lem}\label{first component}
If $v_{\lambda}(n)\in M$ then $v_{\lambda}(n+re_1)\in M$ for any $r\in\mathbb{Z}$.
\end{lem}
\begin{proof}
Let $v_{\lambda}(n)\in M$ for some $n\in\mathbb{Z}^{N+1}$. This result is clearly true for $r=0$, so suppose $r\neq 0$. Then
\[D(e_{N+1},re_1).v_{\lambda}(n)=(e_{N+1}|n+\sigma)v_{\lambda}(n+re_1)+rE_{1,N+1}v_{\lambda}(n+re_1),\]
where the second term is zero as $v_{\lambda}$ is a highest weight vector, and if $(e_{N+1}|n+\sigma)\neq0$ then $v_{\lambda}(n+re_1)\in M$. Suppose $(e_{N+1}|n+\sigma)=0$. By proposition \ref{actions} $E_{N+1,1}^2v_{\lambda}(n)\in M$. This is a weight vector whose weight lies on the maximal $\theta$-string through $\lambda$ and is nonzero since the string has length at least 3. So
\begin{multline*}
D(e_{N+1},-re_1).E_{N+1,1}^2v_{\lambda}(n)\\
=(e_{N+1}|n+\sigma)E_{N+1,1}^2v_{\lambda}(n-re_1)-rE_{1,N+1}E_{N+1,1}^2v_{\lambda}(n-re_1).
\end{multline*}
The first term has coefficient zero and the second is a scalar multiple of $E_{N+1,1}v_{\lambda}(n-re_1)$. Then
\begin{multline*}
D(e_{N+1},2re_1).E_{N+1,1}v_{\lambda}(n-re_1)\\
=(e_{N+1}|n-re_1+\sigma)E_{N+1,1}v_{\lambda}(n+re_1)+2rE_{1,N+1}E_{N+1,1}v_{\lambda}(n+re_1)
\end{multline*}
Again the first term is zero and the second term is a nonzero scalar multiple of $v_{\lambda}(n+re_1)$. Thus $v_{\lambda}(n+re_1)\in M$ for all $r\in\mathbb{Z}$.
\end{proof}
\begin{prop}\label{all highest weight vectors}
$v_{\lambda}(m)\in M$ for any $m\in\mathbb{Z}^{N+1}$.
\end{prop}
\begin{proof}
By Propositions \ref{highest weight component} and \ref{highest weight vector} there exists of $v_{\lambda}(n)\in M$ for some $n\in\mathbb{Z}^{N+1}$. Using Lemma \ref{first component} assume that $(e_{1}|n+\sigma)\neq0$. Then for any $r_{N+1}\in\mathbb{Z}$
\begin{multline*}
D(e_1,r_{N+1}e_{N+1}).v_{\lambda}(n)\\
= (e_{1}|n+\sigma)v_{\lambda}(n+r_{N+1}e_{N+1})+r_{N+1}E_{N+1,1}v_{\lambda}(n+r_{N+1}e_{N+1}).
\end{multline*}
Since the first term above is nonzero it follows from Proposition \ref{highest weight vector} that $v_{\lambda}(n+r_{N+1}e_{N+1})\in M$. Thus $v_{\lambda}(n+r_{N+1}e_{N+1})\in M$ for all $r_{N+1}\in\mathbb{Z}$. Let $r=\sum_{i=1}^Nr_ie_i\in\mathbb{Z}^{N+1}$. Then
\begin{multline*}
D(e_{N+1},r).v_{\lambda}(n+r_{N+1}e_{N+1})\\=(e_{N+1}|n+r_{N+1}e_{N+1}+\sigma)v_{\lambda}(n+r+r_{N+1}e_{N+1})
+ \sum_{i=1}^Nr_iE_{i,N+1}v_{\lambda}(n+r+r_{N+1}e_{N+1}),
\end{multline*}
with the second term being zero by the highest weight property of $v_{\lambda}$. When $(e_{N+1}|n+r_{N+1}e_{N+1}+\sigma)\neq0$ it follows by the arbitrary choice of $r$ that $v_{\lambda}(n+r+r_{N+1}e_{N+1})\in M$ for all $r\in\mathbb{Z}^{N}\times\{0\}$.
Suppose there exists $r_{N+1}\in\mathbb{Z}$ such that $(e_{N+1}|n+r_{N+1}e_{N+1}+\sigma)=0$, say when $r_{N+1}=r_{N+1}^*\in\mathbb{Z}$. For any $r=\sum_{i=1}^Nr_ie_i\in\mathbb{Z}^{N+1}$ and $\bar{r}_{N+1}\neq r_{N+1}^*$, the above shows that $v_{\lambda}(n+r+\bar{r}_{N+1})\in M$. By Lemma \ref{first component} $v_{\lambda}(n+ae_1+\sum_{i=2}^Nr_ie_i+\bar{r}_{N+1}e_{N+1})\in M$ where $(e_1|n+ae_1+\sum_{i=2}^Nr_ie_i+\bar{r}_{N+1}e_{N+1}+\sigma)\neq0$. Then
\begin{multline*}
D(e_1,(r_{N+1}^*-\bar{r}_{N+1})e_{N+1}).v_{\lambda}(n+ae_1+\sum_{i=2}^Nr_ie_i+\bar{r}_{N+1}e_{N+1})
\\= (e_1|n+ae_1+\sum_{i=2}^Nr_ie_i+\bar{r}_{N+1}e_{N+1}+\sigma)v_{\lambda}(n+ae_1+\sum_{i=2}^Nr_ie_i+r_{N+1}^*e_{N+1})
\end{multline*}
and so $v_{\lambda}(n+ae_1+\sum_{i=2}^Nr_ie_i+r_{N+1}^*e_{N+1})\in M$. Apply Lemma \ref{first component} again to get that $v_{\lambda}(n+r+r_{N+1}^*e_{N+1})\in M$. This along with the previous paragraph shows that $v_{\lambda}(m)\in M$ for any $m\in\mathbb{Z}^{N+1}$.
\end{proof}
\section{Generating the Module}
\begin{thm}\label{non minuscule lemma}
If $F^{\sigma}(\lambda)$ is non minuscule then it is irreducible as a $\mathcal{D}_{\text{div}}$-module.
\end{thm}
\begin{proof}
From the theory of highest weight modules for semisimple Lie algebras, $V(\lambda)$ is spanned by $v_{\lambda}$ and the vectors $y_1\dots y_kv_{\lambda}$, where, for each $p\in\{1,\dots,k\}$, $y_p=E_{ij}$ for some $(i,j)$-pair with $i>j$. Let $M$ be a nonzero submodule of $F^{\sigma}(\lambda)$. Proceed by induction on $k$ to show that $y_1\dots y_kv_{\lambda}(m)\in M$ for any $m\in\mathbb{Z}^{N+1}$. By Proposition \ref{all highest weight vectors}, $v_{\lambda}(m)\in M$ for any $m\in\mathbb{Z}^{N+1}$, which is the basis for the induction.
Suppose for some $k\geq0$ that $y_1\dots y_kv_{\lambda}(m)\in M$ for any $m\in\mathbb{Z}^{N+1}$ and any $y_1,\dots,y_k$ as above. Then for any $i,j\in\{1,\dots,N+1\}$ with $i>j$
\[D(e_j,e_i).y_1\dots y_kv_{\lambda}(m)
=(e_j|m+\sigma)y_1\dots y_kv_{\lambda}(m+e_i)+E_{ij}y_1\dots y_kv_{\lambda}(m+e_i).\]
Since $y_1\dots y_kv_{\lambda}(m+e_i)\in M$ by the induction hypothesis, $E_{ij}y_1\dots y_kv_{\lambda}(m+e_i)\in M$. Because $m$ is arbitrary, it follows that $E_{ij}y_1\dots y_kv_{\lambda}(n)\in M$ for any $n\in\mathbb{Z}^{N+1}$. Thus $V(\lambda)(m)\subseteq M$ for all $m\in\mathbb{Z}^{N+1}$ and hence $M=F^{\sigma}(\lambda)$.
\end{proof}
The $k$-fold wedge product $\bigwedge^k(\mathbb{C}^{N+1})$ is a highest weight $\mathfrak{sl}_{N+1}$-module via the action
\[X(v_1\wedge\dots\wedge v_k)=\sum_{p=1}^kv_1\wedge\dots\wedge Xv_p\wedge\dots\wedge v_k\]
for any $X\in\mathfrak{sl}_{N+1} $ and $v_1,\dots,v_k\in\mathbb{C}^{N+1}$. $\bigwedge^k(\mathbb{C}^{N+1})$ has highest weight vector $e_1\wedge\dots\wedge e_k$. It follows from $(E_{ii}-E_{i+1,i+1})(e_1\wedge\dots\wedge e_k)=\delta_{ik}$ that $\bigwedge^k(\mathbb{C}^{N+1})$ has highest weight $\omega_k$ and so $\bigwedge^k(\mathbb{C}^{N+1})\cong V(\omega_k)$ for $1\leq k\leq N$, and $\bigwedge^0(\mathbb{C}^{N+1})\cong\bigwedge^{N+1}(\mathbb{C}^{N+1})\cong V(0)$. For convenience set $\omega_0=\omega_{N+1}=0$ so that $V(0)=V(\omega_0)=V(\omega_{N+1})$.
In \cite{R} some $\mathcal{D}$-submodules of $\bigwedge^k(\mathbb{C}^{N+1})\otimes \mathbb{C}[q_1^{\pm1},\dots,q_{N+1}^{\pm1}]$ are identified. As $\mathcal{D}_{\text{div}}$-modules they are
\[W_k^{\sigma}=\bigoplus_{n\in\mathbb{Z}^{N+1}}\left(\mathbb{C}(n+\sigma)\wedge\mathbb{C}^{N+1}\wedge\dots\wedge\mathbb{C}^{N+1}\right)\otimes q^n,\]
and if $\sigma\in\mathbb{Z}^{N+1}$ there is a larger submodule
\[\tilde{W}_k^{\sigma}=W_k^{\sigma}\oplus\left(\mathbb{C}^{N+1}\wedge\dots\wedge\mathbb{C}^{N+1}\right)\otimes q^{-\sigma}.\]
Let $\tilde{W}_k^{\sigma}=W_k^{\sigma}$ in the case $\sigma\not\in\mathbb{Z}^{N+1}$. For $k=0$ this $\mathcal{D}_{\text{div}}$-module is simply $\mathbb{C}[q_1^{\pm1},\dots,q_{N+1}^{\pm1}]$, and will be denoted $F^{\sigma}(\omega_0)$. The module action reduces to
\[D(u,r).q^n=(u|n+\sigma)q^{n+r}.\]
If $\sigma\in\mathbb{Z}^{N+1}$ then $F^{\sigma}(\omega_0)$ has submodule $\tilde{W}_0^{\sigma}=\mathbb{C} q^{-\sigma}$.
\begin{lem}\label{wedge weight vector}
Let $N\geq 2$ and $1\leq k\leq N-1$. If $M$ is a submodule of $\bigwedge^k(\mathbb{C}^{N+1})\otimes \mathbb{C}[q_1^{\pm1},\dots,q_{N+1}^{\pm1}]$ which properly contains $\tilde{W}_k^{\sigma}$, then $M$ contains an element of the form $u(n)$, where $u$ is a weight vector in $\bigwedge^k(\mathbb{C}^{N+1})$, and $n+\sigma\neq 0$.
\end{lem}
\begin{proof}
Since $M$ properly contains $\tilde{W}_k^{\sigma}$, it contains a nonzero vector $v(n)\in F^{\sigma}(\omega_k)\setminus\tilde{W}_k^{\sigma}$ where $n+\sigma\neq0$. Thus $(e_x|n+\sigma)\neq 0$ for some $x\in\{1,\dots,N+1\}$ and so $\{n+\sigma,e_1,\dots,e_{x-1},e_{x+1},\dots,e_{N+1}\}$ is a basis for $\mathbb{C}^{N+1}$. Write
\[v=\sum_{\substack{p_1,\dots,p_k=1\\p_1,\dots,p_k\neq x}}^{N+1}\gamma_{p_1,\dots,p_k}e_{p_1}\wedge\dots\wedge e_{p_k}\]
(modulo $\tilde{W}_k^{\sigma}$) and assume that the coefficients satisfy $\gamma_{p_1,\dots,p_k}=\text{sgn}(\pi)\gamma_{\pi(p_1,\dots,p_k)}$ for any permutation $\pi$. Since $v\neq0$, at least one of the above coefficients is nonzero, say $\gamma_{i_1,\dots,i_k}$. Assume indices $i_1,\dots,i_k,x,y$ are distinct. Note that in the right hand side of
\begin{multline*}
D(e_x,-e_{y}).D(e_{i_1},e_{y}).v(n)\\=(e_x|n+\sigma)(e_{i_1}|n+\sigma)v(n)
-(e_{i_1}|n+\sigma)E_{yx}v(n)+(e_x|n+\sigma)E_{yi_1}v(n)-E_{yx}E_{yi_1}v(n),
\end{multline*}
the first term is in $M$, while the second and fourth terms are both zero since $e_x$ does not appear in $v$. Thus the third term, call it $v^{(1)}(n)$, is in $M$ and recall $(e_x|n+\sigma)\neq0$. Then,
\begin{align*}
v^{(1)}(n) &= (e_x|n+\sigma)E_{yi_1}v(n)\\
&= (e_x|n+\sigma)\sum_{\substack{p_2,\dots,p_k=1\\p_2,\dots,p_k\neq x}}^{N+1}\gamma_{i_1,p_2,\dots,p_k}e_{y}\wedge e_{p_2}\wedge\dots\wedge e_{p_k}(n)\\
&\quad+ (e_x|n+\sigma)\sum_{\substack{p_1,p_3,\dots,p_k=1\\p_1,p_3,\dots,p_k\neq x}}^{N+1}\gamma_{p_1,i_1,p_3,\dots,p_k}e_{p_1}\wedge e_{y}\wedge e_{p_3}\wedge\dots\wedge e_{p_k}(n)\\
&\quad+ \dots\\
&\quad+ (e_x|n+\sigma)\sum_{\substack{p_1,\dots,p_{k-1}=1\\p_1,\dots,p_{k-1}\neq x}}^{N+1}\gamma_{p_1,\dots,p_{k-1},i_1}e_{p_1}\wedge\dots\wedge e_{p_{k-1}}\wedge e_{y}(n)
\end{align*}
Relabelling indices, the above can be written
\[(e_x|n+\sigma)\sum_{\substack{p_2,\dots,p_k=1\\p_2,\dots,p_k\neq x}}^{N+1}\Gamma e_{y}\wedge e_{p_2}\wedge\dots\wedge e_{p_k}(n),\]
with $\Gamma=\gamma_{i_1,p_2,\dots,p_k}+(-1)\gamma_{p_2,i_1,p_3,\dots,p_k}+\dots+(-1)^{k-1}\gamma_{p_2,\dots,p_{k},i_1}$ which follows from the alternating property of the wedge product. By the assumption on the coefficients, $\gamma_{p_2,\dots,p_s,i_1,p_{s+1},\dots,p_k}=(-1)^{s}\gamma_{i_1,p_2,\dots,p_k}$ for $2\leq s\leq k$, and so
\[v^{(1)}(n)=k(e_x|n+\sigma)\sum_{\substack{p_2,\dots,p_k=1\\p_2,\dots,p_k\neq x}}^{N+1}\gamma_{i_1,p_2,\dots,p_k} e_{y}\wedge e_{p_2}\wedge\dots\wedge e_{p_k}(n).\]
Note that the coefficients in this summation are the same as those from $v$ which have first index equal to $i_1$. Doing this process again, by applying $D(e_x,-e_{i_1})D(e_{i_2},e_{i_1})$ to $v^{(1)}(n)$ in the first step, will yield
\[v^{(2)}(n)=k^2(e_x|n+\sigma)^2\sum_{\substack{p_{3},\dots,p_k=1\\p_{3},\dots,p_k\neq x}}^{N+1}\gamma_{i_1,i_2,p_{3},\dots,p_k} e_{y}\wedge e_{i_1}\wedge e_{p_3}\wedge\dots\wedge e_{p_k}(n).\]
Repeating (with suitable choices of indices) another $k-2$ times yields
\[k!(e_x|n+\sigma)^k\gamma_{i_1,\dots,i_k}e_y\wedge e_{i_1}\wedge\dots\wedge e_{i_{k-1}}(n)\]
which is a nonzero because both $(e_x|n+\sigma)\neq 0$ and $\gamma_{i_1,\dots,i_k}\neq0$. The wedge product $e_y\wedge e_{i_1}\wedge\dots\wedge e_{i_{k-1}}$ is a weight vector in $\bigwedge^k(\mathbb{C}^{N+1})$ and so the proof is complete.
\end{proof}
\begin{thm}\label{results}
Let $N\geq 1$ and $0\leq k\leq N$.
\begin{enumerate}
\item[(a)]
If $\lambda\neq0,\omega_k$ then $F^{\sigma}(\lambda)$ is irreducible.
\item[(b)]
$W_k^{\sigma}$ and $F^{\sigma}(\omega_k)/\tilde{W}_k^{\sigma}$ are irreducible and $F^{\sigma}(\omega_k)/\tilde{W}_k^{\sigma}\cong W_{k+1}^{\sigma}$
\end{enumerate}
\end{thm}
\begin{proof}
Part (a) follows from Theorem \ref{non minuscule lemma}.
For $0\leq k\leq N$, the linear map $\psi_k:F^{\sigma}(\omega_k) \rightarrow F^{\sigma}(\omega_{k+1})$ defined by
\[v_1\wedge\dots\wedge v_k(n) \mapsto (n+\sigma)\wedge v_1\wedge\dots\wedge v_k(n)\]
is a $\mathcal{D}_{\text{div}}$-module homomorphism, where ker$(\psi_k)=\tilde{W}_k^{\sigma}$ and Im$(\psi_k)=W_{k+1}^{\sigma}$. Thus
\[F^{\sigma}(\omega_k)/\tilde{W}_k^{\sigma}\cong W_{k+1}^{\sigma}.\]
Lemma \ref{wedge weight vector} will be used to show that $F^{\sigma}(\omega_k)/\tilde{W}_k^{\sigma}$ is irreducible (and hence that $W_{k+1}^{\sigma}$ is irreducible) for $1\leq k\leq N-1$ with $N\neq2$. The remaining cases are $F^{\sigma}(\omega_0)/\tilde{W}_0^{\sigma}$ and $F^{\sigma}(\omega_N)/\tilde{W}_N^{\sigma}$ for any $N\geq 1$. The $F^{\sigma}(\omega_0)/\tilde{W}_0^{\sigma}$ case is handled first by showing directly that $W_1^{\sigma}$ is irreducible.
Any nonzero submodule $M_1$ of $W_1^{\sigma}=\bigoplus_{n\in\mathbb{Z}^{N+1}}\mathbb{C}(n+\sigma)\otimes q^n$, contains a homogeneous vector $(n+\sigma)\otimes q^n$ for some $n\in\mathbb{Z}^{N+1}$ where $(n+\sigma)\neq0$. Choose $e_x$ so that $(e_x|n+\sigma)\neq0$ and $r\in\mathbb{Z}^{N+1}$ with $(e_x|r)=0$. Then $D(e_x,r).(n+\sigma)\otimes q^n=(e_x|n+\sigma)(n+r+\sigma)\otimes q^{n+r}$ and so $(n+r+\sigma)\otimes q^{n+r}\in M_1$ for any $r\in\{e_x\}^{\perp}\cap\mathbb{Z}^{N+1}$. Let $e_j\neq e_x$, and $r_x\in\mathbb{Z}$. Then
\begin{multline*}
D(e_j-e_x,r_x(e_x+e_j)).(n+r-r_xe_j+\sigma)\otimes q^{n+r-r_xe_j}\\
=(e_j-e_x|n+r-r_xe_j+\sigma)(n+r+r_xe_x+\sigma)\otimes q^{n+r+r_xe_x}.
\end{multline*}
If $(e_j-e_x|n+r-r_xe_j+\sigma)\neq0$ this implies $(m+\sigma)\otimes q^m\in M_1$ for any $m\in\mathbb{Z}^{N+1}$. Otherwise if $(e_j-e_x|n+r-r_xe_j+\sigma)=0$ then consider instead
\begin{multline*}
D(e_j+e_x,r_x(e_x-e_j).(n+r+r_xe_j+\sigma)\otimes q^{n+r+r_xe_j}\\
=(e_j+e_x|n+r+r_xe_j+\sigma)(n+r+r_xe_x+\sigma)\otimes q^{n+r+r_xe_x}
\end{multline*}
so that $(e_j+e_x|n+r+r_xe_j+\sigma)\neq 0$ and again this yields that $(m+\sigma)\otimes q^m\in M_1$ for any $m\in\mathbb{Z}^{N+1}$. Hence $W_1^{\sigma}$ is irreducible.
It remains to show that $F^{\sigma}(\omega_k)/\tilde{W}_k^{\sigma}$ is irreducible for $1\leq k\leq N$. Let $M$ be a submodule of $F^{\sigma}(\omega_k)$ which properly contains $\tilde{W}_k^{\sigma}$.
For $N\geq 2$ with $1\leq k\leq N-1$, Lemma \ref{wedge weight vector} shows that $M$ contains a vector of the form $e_y\wedge e_{i_1}\wedge\dots\wedge e_{i_{k-1}}(n)$, with $n$ such that $(e_x|n+\sigma)\neq0$ for some $x\in\{1,\dots,N+1\}$. For any $r\in\{e_x\}^{\perp}$,
\[D(e_x,r).e_y\wedge e_{i_1}\wedge\dots\wedge e_{i_{k-1}}(n)=(e_x|n+\sigma)e_y\wedge e_{i_1}\wedge\dots\wedge e_{i_{k-1}}(n+r).\]
Thus $e_y\wedge e_{i_1}\wedge\dots\wedge e_{i_{k-1}}(n+r)\in M$ for any $r\in\{e_x\}^{\perp}$. So for $a\not\in\{x,y,i_1,\dots,i_{k-1}\}$,
\begin{multline*}
D(e_x-e_a,r_x(e_x+e_a)).e_y\wedge e_{i_1}\wedge\dots\wedge e_{i_{k-1}}(n+r-r_xe_a)\\
=(e_x-e_a|n+r-r_xe_a+\sigma)e_y\wedge e_{i_1}\wedge\dots\wedge e_{i_{k-1}}(n+r+r_xe_x)
\end{multline*}
If $(e_x-e_a|n+r-r_xe_a+\sigma)\neq0$ this yields that $e_y\wedge e_{i_1}\wedge\dots\wedge e_{i_{k-1}}(m)\in M$ for all $m\in\mathbb{Z}^{N+1}$. If $(e_x-e_a|n+r-r_xe_a+\sigma)=0$ apply instead $D(e_x+e_a,r_x(e_x-e_a))$ to $e_y\wedge e_{i_1}\wedge\dots\wedge e_{i_{k-1}}(n+r+r_xe_a)$ and obtain the same result.
Using the fact that $e_y\wedge e_{i_1}\wedge\dots\wedge e_{i_{k-1}}(m)\in M$ for all $m\in\mathbb{Z}^{N+1}$, any vector of the form $e_{b_1}\wedge e_{b_2}\wedge\dots\wedge e_{b_k}(m)$ can be obtained by doing the following. To replace $e_j\in\{e_y,e_{i_1},\dots,e_{i_{k-1}}\}$ with $e_a$, $a\not\in\{y,i_1,\dots,i_{k-1}\}$ in $e_y\wedge e_{i_1}\wedge\dots\wedge e_{i_{k-1}}(m)$, note that
\begin{multline*}
D(e_j,e_a).e_y\wedge e_{i_1}\wedge\dots\wedge e_j\wedge\dots\wedge e_{i_{k-1}}(m-e_a)\\
=(e_j|m-e_a+\sigma)e_y\wedge e_{i_1}\wedge\dots\wedge e_j\wedge\dots\wedge e_{i_{k-1}}(m)+e_y\wedge e_{i_1}\wedge\dots\wedge e_a\wedge\dots\wedge e_{i_{k-1}}(m).\end{multline*}
The first term on the right hand side is already in $M$ which implies $e_y\wedge e_{i_1}\wedge\dots\wedge e_a\wedge\dots\wedge e_{i_{k-1}}(m)\in M$. This process of swapping out $e_j$'s can be repeated until the desired vector is obtained. Vectors of the form $e_{b_1}\wedge e_{b_2}\wedge\dots\wedge e_{b_k}(m)$ form a basis for $V(\omega_k)(m)$, and thus $M$ must be all of $F^{\sigma}(\omega_k)$. It follows that $F^{\sigma}(\omega_k)/\tilde{W}_k^{\sigma}$ is irreducible.
Consider the case $k=N$ for $N\geq 1$. $M$ contains a nonzero vector $v(n)$, where $v(n)\not\in\tilde{W}_N^{\sigma}$ such that $n+\sigma\neq0$. Suppose $(e_x|n+\sigma)\neq0$ for some $x\in\{1,\dots,N+1\}$, and so $\{n+\sigma,e_1,\dots,e_{x-1},e_{x+1},\dots,e_{N+1}\}$ is a basis for $\mathbb{C}^{N+1}$. From the definition of $\tilde{W}_N^{\sigma}$ it follows that $v(n)$ is a nonzero scalar multiple of $e_1\wedge\dots\wedge e_{x-1}\wedge e_{x+1}\wedge\dots\wedge e_{N+1}(n)$ plus some vector in $\tilde{W}_N^{\sigma}$. Assume without loss of generality that $v(n)=e_1\wedge\dots\wedge e_{x-1}\wedge e_{x+1}\wedge\dots\wedge e_{N+1}(n)$. Note that
\[V(\omega_N)(n)= \mathbb{C} v(n)\oplus(n+\sigma)\wedge\left(\bigwedge^{N-1}\mathbb{C}^{N+1}\right)(n)\]
and so $V(\omega_N)(n)\subset M$. For any $r\in\{e_x\}^{\perp}\cap\mathbb{Z}^{N+1}$, $D(e_x,r).v(n)=(e_x|n+\sigma)v(n+r)$, thus $v(n+r)\in M$. Since $(e_x|n+r+\sigma)=(e_x|n+\sigma)\neq0$ it follows that
\[V(\omega_N)(n+r)= \mathbb{C} v(n+r)\oplus(n+r+\sigma)\wedge\left(\bigwedge^{N-1}\mathbb{C}^{N+1}\right)(n+r)\]
and so $V(\omega_N)(n+r)\subset M$ for any $r\in\{e_x\}^{\perp}\cap\mathbb{Z}^{N+1}$. It remains to show that $V(\omega_N)(n+r+r_xe_x)\subset M$ for any $r_x\in\mathbb{Z}$.
Let $j>x$, $r_x\in\mathbb{Z}\setminus\{0\}$, and $r\in\{e_x\}^{\perp}\cap\mathbb{Z}^{N+1}$. Then
\[D(e_j,r_xe_x).v_{\omega_N}(n+r)=(e_j|n+r+\sigma)v_{\omega_N}(n+r+r_xe_x),\]
where $v_{\omega_N}$ is a highest weight vector in $V(\omega_N)$. If $(e_j|n+r+\sigma)\neq0$ then $v_{\omega_N}(n+r+r_xe_x)\in M$, and by the proof of Lemma \ref{non minuscule lemma} $V(\omega_N)(n+r+r_xe_x)\subset M$. Let $v_{\ell}$ be a lowest weight vector of $V(\omega_N)$, and $j<x$, then
\[D(e_j,r_xe_x).v_{\ell}(n+r)=(e_i|n+r+\sigma)v_{\ell}(n+r+r_xe_x).\]
If $(e_i|n+r+\sigma)\neq0$ then $v_{\ell}(n+r+r_xe_x)\in M$. Again the proof of Lemma \ref{non minuscule lemma} shows how to generate all of $V(\omega_N)(n+r+r_xe_x)$, only this time the $y_p=E_{ij}$ where $i<j$ are applied to lowest weight vector $v_{\ell}(n+r+r_xe_x)$. Suppose for some $r\in\{e_x\}^{\perp}\cap\mathbb{Z}^{N+1}$, say $r=r_0$, that $(e_j|n+r_0+\sigma)=0$ for all $j\neq x$; i.e. $(n+r_0+\sigma)=Ke_x\neq0$. Then for $j\neq x$
\begin{multline*}
D(e_j-e_x,r_x(e_j+e_x)).v(n+r_0-r_xe_j)=K v(n+r_0+r_xe_x)\\
+r_xe_1\wedge\dots\wedge e_{j-1}\wedge e_x\wedge e_{j+1}\wedge\dots\wedge e_{x-1}\wedge e_{x+1}\wedge\dots\wedge e_{N+1}(n+r_0+r_xe_x).
\end{multline*}
Since $(n+r_0+r_xe_x+\sigma)$ is a scalar multiple of $e_x$, the second term above is in $\tilde{W}_N^{\sigma}$, and thus $v(n+r_0+r_xe_x)\in M$. If $(e_x|n+r_0+r_xe_x+\sigma)\neq0$ then
\begin{multline*}
V(\omega_N)(n+r_0+r_xe_x)\\
=\mathbb{C} v(n+r_0+r_xe_x)\oplus(n+r_0+r_xe_x+\sigma)\wedge\left(\bigwedge^{N-1}\mathbb{C}^{N+1}\right)(n+r_0+r_xe_x)
\end{multline*}
and so $V(\omega_N)(n+r_0+r_xe_x)\subset M$. If $(e_x|n+r_0+r_xe_x+\sigma)=0$, then $(n+r_0+r_xe_x+\sigma)=0$, in which case $V(\omega_N)(n+r_0+r_xe_x)\subset\tilde{W}_N^{\sigma}\subset M$. Thus $V(\omega_N)(m)\subset M$ for all $m\in\mathbb{Z}^{N+1}$ and so $M=F^{\sigma}(\omega_N)$.
\end{proof}
\section{Acknowledgements}
Many thanks to Professor Yuly Billig for his expert guidance on this project.
|
2,869,038,156,358 | arxiv | \section{Comment on ``Parametric amplification in Josephson junction embedded transmission lines''}
\title{Comment on ``Parametric amplification in Josephson junction embedded transmission lines''}
\begin{abstract}
Recently Yaakobi and co-workers [Phys. Rev. B \textbf{87}, 144301 (2013)] theoretically studied four-wave mixing and parametric amplification in a nonlinear transmission line consisting of capacitively shunted Josephson junctions. By deriving and solving the coupled-mode equations, they have arrived at the conclusion that in a wide frequency range around the pump frequency exponential parametric gain (in which the signal grows exponentially with distance) can be achieved. However, we have found a mathematical error in their derivation of the coupled-mode equations (Equation (A13)), which leads to the wrong expression of the gain factor and invalidates their conclusions on the gain and bandwidth. In this comment, we present the correct expression for the parametric gain. We show that for a transmission line with weak dispersion or positive dispersion ($\Delta k>0 $), as is the case discussed by Yaakobi et al, while quadratic (power) gain can occur around the pump frequency, exponential gain is impossible. Furthermore, for a transmission line with proper intrinsic or engineered dispersion, exponential gain occurs at frequencies where the phase matching condition is met, while around the pump frequency the gain is still quadratic.
\end{abstract}
\author{S. Chaudhuri}
\affiliation{Stanford University, Department of Physics, Stanford, CA 94305}
\author{J. Gao}
\affiliation{National Institute of Standards and Technology, Boulder, CO 80305}
\date{\today}
\maketitle
Superconducting parametric amplifiers are currently of great interest because of their promise in achieving quantum-limited noise, which has important applications in the readout of superconducting quantum bits \cite{siddiqi} and astronomical low-temperature detectors. \cite{day} In a recent paper \cite{yaakobi} (referred to as ``the Paper'' hereafter), Yaakobi and co-workers theoretically studied four-wave mixing and parametric amplification in a nonlinear transmission line comprised of capacitively shunted Josephson junctions. By deriving and solving the coupled mode equations, they have derived an exponential gain factor g (signal grows exponentially with the distance, $y_s(x) \sim e^{gx}$).
The final expression for g, implied by the Paper, is
\renewcommand \theequation{\Roman{equation}}
\begin{equation}
g = \sqrt{-\Delta_p}=\sqrt{\left(\frac{3\gamma \tilde{k}_{p} ( \tilde{k}_{s} \tilde{k}_{i})^{1/2}}{4} B_{p,0}^{2} \right)^{2} - \left(\frac{3\gamma \tilde{k}_{p}^{2}}{8} B_{p,0}^{2} \left( 1+ \frac{\Delta \tilde{k}}{\tilde{k}_{p}} \right) + \frac{\Delta k}{2} \right)^{2} }
\label{eqn:g_wrong}
\end{equation}
which reaches a maximum positive value of $g_\mathrm{max}$ at the pump frequency (Eqn. (57)) \footnote{Throughout this comment, equations labeled with arabic numbers and roman numerals refer to equations defined in the original paper and this comment, respectively. } and remains positive in a bandwidth $\Delta \omega_B$ (Eqn.~(56)). Yaakobi et al have drawn the conclusion that this traveling wave parametric amplifier architecture can generate exponential gain in a frequency range around the pump frequency. In this comment, we show that this conclusion is invalid, due to an error in their derivation of the gain factor.
In careful examination of their appendix, we have found a mathematical error in the derivation of the coupled-mode equations. In the cubic expansion of the mixing products, the coefficients in front of the terms $k_p^2k_iA_p^2A_i^*e^{-i\Psi}$ and $k_p^2k_sA_p^2A_s^*e^{-i\Psi}$ in Eqn. (A13) should be 1 instead of 2. This leads to a factor of 2 reduction in $\mu$ in Eqn.~(31) and a different value of $g$. The discussions on the gain and bandwidth, based on the wrong expression of $g$, Eqn.~\ref{eqn:g_wrong}, are therefore invalid. Here we present the correct expression for $g$,
\begin{equation}
g = \sqrt{-\Delta_p}=\sqrt{\left(\frac{3\gamma \tilde{k}_{p} ( \tilde{k}_{s} \tilde{k}_{i})^{1/2}}{8} B_{p,0}^{2} \right)^{2} - \left(\frac{3\gamma \tilde{k}_{p}^{2}}{8} B_{p,0}^{2} \left( 1+ \frac{\Delta \tilde{k}}{\tilde{k}_{p}} \right) + \frac{\Delta k}{2} \right)^{2} }
\label{eqn:g_correct}
\end{equation}
To make a connection with established fiber parametric amplifier theory \cite{agrawal}, we write the exponential gain factor $g$ and the signal power gain $G_{s}$ in a more general form,
\begin{equation}
g = \sqrt{-\Delta_{p}}=\sqrt{\frac{\omega_{s}\omega_{i}}{\omega_{p}^{2}} \left( \Delta \Phi_4 \right)^{2}- \left( \frac{\kappa_{4}}{2} \right)^{2}}
\label{eqn:g_sc}
\end{equation}
\begin{equation}
G_s = \left| \mathrm{cosh}(gL) + \frac{i\kappa_{4}}{2g} \mathrm{sinh}(gL) \right|^{2},\\
\label{eqn:Gs_sc}
\end{equation}
where $\Delta \Phi_4$ is the pump self-phase modulation per unit length, $\kappa_{4}=\Delta k + 2\Delta\Phi_4$ is a measure of the phase mismatch, and $L$ is the length of the traveling-wave structure. For the Josephson junction transmission line discussed in the Paper, $\Delta \Phi_4 = \frac{3\gamma \tilde{k}_{p}^{2}}{8} B_{p,0}^{2}$. The expression Eqn.~\ref{eqn:g_sc} matches the corrected $g$ value in Eqn.~\ref{eqn:g_correct} in the long-wavelength limit ($\omega_{m} \propto \tilde{k}_{m}$ and $\Delta \tilde{k}/\tilde{k}_p \ll 1$), and Eqn.~\ref{eqn:Gs_sc} matches Eqn. (51) in the Paper. The expressions for $g$ and $G_s$ also apply to other four-wave mixing traveling wave parametric amplifiers whose phase velocity exhibits quadratic nonlinearity with respect to the wave amplitude, such as the dispersion-engineered traveling-wave kinetic inductance (DTWKI) amplifier \cite{eom}.
The correct gain expressions show significantly different features from those discussed in the Paper. A direct implication from Eqn.~\ref{eqn:g_sc} is that $\Delta_p$ has to be negative in order for exponential gain to occur, which in turn requires $\Delta k < 0$. From Eqn.~(A32) we see that $\Delta k \geq 0$ for the transmission line discussed in the Paper. This can also be seen from a non-negative $\Delta_p$ in Fig.~\ref{fig1}(a), where $\Delta_p$ is plotted for the set parameters given on page 5 of the Paper. Therefore, exponential gain is impossible at any frequency for this transmission line. Also, the gain coefficient g is always imaginary.
Furthermore, the transmission line discussed in the Paper has dispersion coming from the ladder network and the Josephson capcitance ($C_J$). The amount of dispersion is small near the pump frequency where $\Delta k \ll 2\Delta \Phi_4$ holds.
For a transmission line with no dispersion ($\Delta k =0$) or with weak dispersion ($|\Delta k| \ll 2\Delta \Phi_4$), Eqn.~\ref{eqn:g_sc} and Eqn.~\ref{eqn:Gs_sc} further reduce to
\begin{equation}
\Delta_p = \left( 1-\frac{\omega_s}{\omega_p} \right)^2\Delta \Phi_4^2,~~ g =\sqrt{-\Delta_p}= i \left|1-\frac{\omega_{s}}{\omega_{p}} \right| \Delta\Phi_{4}
\label{eqn:g_linear}
\end{equation}
\begin{equation}
G_s = 1+ \left( \frac{1}{\left| 1- \frac{\omega_{s}}{\omega_{p}} \right|^{2}}-1 \right) \mathrm{sin}^{2} \left( \left| 1-\frac{\omega_{s}}{\omega_{p}} \right| L \Delta\Phi_{4} \right)
\label{eqn:Gs_linear}
\end{equation}
As $\omega_{s} \rightarrow \omega_{p}$, $g\rightarrow 0$ and the signal power gain $G_s$ approaches a maximum of $G_{s,max}=1+(L\Delta\Phi_{4})^{2}$. The power gain around the pump frequency becomes quadratic (signal amplitude grows linearly with distance). Therefore we conclude that, \textbf{for the case considered by Yaakobi et al, while quadratic gain can occur around the pump frequency, exponential gain is impossible at all frequencies.}
\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{Fig1}
\caption{$\Delta_p$ and $G_s$ as a function of signal frequency from Eqn.~\ref{eqn:g_sc} and \ref{eqn:Gs_sc}. The parameters used to generate these curves are from page 5 of the Paper. The total self-phase shift is set to $L\Delta\Phi_{4}=3$ radians, resulting in a maximum gain of $G_{s,max}=1+(L\Delta\Phi_{4})^{2}=10$ dB (from Eqn.~\ref{eqn:Gs_linear}).}
\label{fig1}
\end{figure}
For a transmission line with proper dispersion, $|\Delta k| \sim 2\Delta\Phi_{4}$ and $\Delta k < 0$ at some frequency, exponential gain can occur. When the phase-matching condition $\kappa_4 \simeq 0$ is met or nearly met, $g$ can become real and positive. However, since $\Delta k \simeq 0$ for signal frequencies near the pump, as long as $k$ is a continous function of $\omega$, the gain around the pump will still be quadratic. These features have been verified in the measured gain profiles from both the optical fiber parametric amplifier (with intrinsic dispersion) and the DTWKI amplifier (with engineered dispersion), where one can identify exponential gain regions in two broader frequency bands detuned from the pump frequency and a quadratic gain region in a narrower band around the pump \cite{agrawal,eom}.
We want to point out that the Josephson junction transmission line considered by Yaakobi et al can still operate as a parametric amplifer. However, the gain and bandwidth are much reduced from those discussed in the Paper. Plotted in Fig.~\ref{fig1}(b) is the gain profile for a total self-phase shift of $L\Delta\Phi_{4}=3$ radians for the parameters given on pg. 5 of the Paper. Now the 3-dB bandwidth reduces to $\sim 30\%$ and the maximum gain is 10~dB. Because the maximum gain is limited to be quadratic, in order to achieve a certain gain, $L\Delta\Phi_{4}$ has to be larger than the exponential case discussed in the Paper. This means a longer line (more junctions) or driving the junctions closer to their critical current $I_c$. The amplifier presented in Fig.~\ref{fig1}(b) could potentially give a maximum exponential gain of $G_s = 20$~dB, according to Eqn.~\ref{eqn:Gs_sc}, and a broader bandwidth, if proper dispersion is introduced and the phase-matching condition is met. Therefore, it is very interesting to consider applying the dispersion engineering concept used in the DTWKI amplifer to the Josephson junction traveling wave parametric amplifier.
|
2,869,038,156,359 | arxiv | \section{Introduction}
Humans understand the world by decomposing scenes into objects that can interact
with each other. Analogously, autonomous systems’ reasoning and scene understanding capabilities could benefit from decomposing scenes into objects and modeling each of these independently.
This approach has been proven beneficial to perform a wide variety of computer vision tasks without explicit supervision, including unsupervised object detection~\cite{Eslami_AttendInferRepeatSceneUnderstanding_2016}, future frame prediction~\cite{Weis_UnmaskingInductiveBiasesOfUnsupervisedObjectRepresentationsForVideoSequences_2020,Greff_IodineMultiObjectRepresentationLearningWithIterativeVariationalInference_2019}, and object tracking~\cite{He_TrackingByAnnimation_2019,Veerapaneni_EntityAbstractioninVisualModelBasedReinforcementLearning_2020}.
Recent works propose extracting object-centric representations without the need for explicit supervision through the use of deep variational auto-encoders~\cite{Kingma_AutoEncodingVariationalBayes_2013} (VAEs) with spatial attention mechanisms~\cite{Burgess_MonetUnsupervisedSceneDecompositionRepresentation_2019,crawford2019spatially}.
However, training these models often presents several difficulties, such as long training times, requiring a large number of trainable parameters, or the need for large curated datasets. Furthermore, these methods suffer from the inherent lack of interpretability which is characteristic of deep neural networks (DNNs).
\begin{figure*}[t]
\centering
\includegraphics[width=0.92\linewidth]{imgs/PCDNet.png}
\caption{PCDNet decomposition framework. First, the Phase Correlation (PC) Cell estimates the $N$ translation parameters that best align each learned prototype to the objects in the image, and uses them to obtain $(P \times N)$ object and mask candidates. Second, the color module assigns a color to each of the transformed prototypes. Finally, a greedy selection algorithm reconstructs the input image by iteratively combining the colorized object candidates that minimize the reconstruction error.}
\label{fig:phacord}
\end{figure*}
To address the aforementioned issues, we propose a novel image decomposition framework: the \emph{\textbf{P}hase-\textbf{C}orrelation \textbf{D}ecomposition \textbf{Net}work } (PCDNet). Our method assumes that an image is formed as an arrangement of multiple objects, each belonging to one of a finite number of different classes.
Following this assumption, PCDNet decomposes an image into its object components, which are represented as transformed versions of a set of learned object prototypes.
The core building block of the PCDNet framework is the \emph{Phase Correlation Cell} (PC Cell). This is a differentiable module that exploits the frequency-domain representations of an image and a prototype to estimate the transformation parameters that best align a prototype to its corresponding object in the image.
The PC Cell localizes the object prototype in the image by applying the phase-correlation method~\cite{Alba_PahseCorrelationImageAlignment_2012}, i.e., finding the peaks in the cross-correlation matrix between the input image and the prototype. Then, the PC Cell aligns the prototype to its corresponding object in the image by performing the estimated phase shift in the frequency domain.
PCDNet is trained by first decomposing an image into its object components, and then reconstructing the input by recombining the estimated object components following the ``dead leaves'' model approach, i.e., as a superposition of different objects.
The strong inductive biases introduced by the network structure allow our method to learn fully interpretable prototypical object-centric representations without any external supervision while keeping the number of learnable parameters small.
Furthermore, our method also disentangles the position and color of each object in a human-interpretable manner.
In summary, the contributions of our work are as follows:
\begin{itemize}
\item We propose the PCDNet model, which decomposes an image into its object components, which are represented as transformed versions of a set of learned object prototypes.
\item Our proposed model exploits the frequency-domain representation of images so as to disentangle object appearance, position, and color without the need for any external supervision.
\item Our experimental results show that our proposed framework outperforms recent methods for joined unsupervised object discovery, image decomposition, and segmentation on benchmark datasets, while significantly reducing the number of learnable parameters, allowing for high throughput, and maintaining interpretability.
\end{itemize}
\section{Related Work}
\subsection{Object-Centric Representation Learning}
The field of representation learning~\cite{Bengio_RepresentationLearningReview_2013} has seen much attention in the last decade, giving rise to great advances in learning hierarchical representations~\cite{Paschalidou_LearningUnsupervisedPartDecompositionOf3DObjects_2020,Stanic_HierarchicalRelationalInference_2020} or in disentangling the underlying factors of variation in the data~\cite{Locatello_ChallengingAssumptionsInLearningOfDissentangledRepresentations_2019,Burgess_UnderstandingDisentanglingInBetaVAE_2018}.
Despite these successes, these methods often rely on learning representations at a scene level, rather than learning in an object-centric manner, i.e., simultaneously learning representations that address multiple, possibly repeating, objects.
In the last few years, several methods have been proposed to perform object-centric image decomposition in an unsupervised manner.
A first approach to object-centric decomposition combines VAEs with attention mechanisms to decompose a scene into object-centric representations. The object representations are then decoded to reconstruct the input image.
These methods can be further divided into two different groups depending on the class of latent representations used.
On the one hand, some methods~\cite{Eslami_AttendInferRepeatSceneUnderstanding_2016,Kosiorek_SequentialAttendInferRepeat_2018,Stanic_HierarchicalRelationalInference_2020,He_TrackingByAnnimation_2019} explicitly encode the input into factored latent variables, which represent specific properties such as appearance, position, and presence.
On the other hand, other models~\cite{Burgess_MonetUnsupervisedSceneDecompositionRepresentation_2019,Weis_UnmaskingInductiveBiasesOfUnsupervisedObjectRepresentationsForVideoSequences_2020,Locatello_ObjectCentricLearningWithSlotAttention_2020} decompose the image into unconstrained per-object latent representations.
Recently, several proposed methods~\cite{Greff_IodineMultiObjectRepresentationLearningWithIterativeVariationalInference_2019,Engelcke_GenesisGeneraticeSceneInferenceWithObjectCentricRepresentations_2019,Engelcke_GenesisV2InferringObjectRepresentationsWithoutIterativeRefinement_2021,Veerapaneni_EntityAbstractioninVisualModelBasedReinforcementLearning_2020,Lin_UnsupervisedObjectOrientedSceneRepresentationViaSpatialAttentionAndDecomposition_2020} use parameterized spatial mixture models with variational inference to decode object-centric latent variables.
Despite these recent advances in unsupervised object-centric learning, most existing methods rely on DNNs and attention mechanisms to encode the input images into latent representations, hence requiring a large number of learnable parameters and high computational costs. Furthermore, these approaches suffer from the inherent lack of interpretability characteristic of DNNs.
Our proposed method exploits the strong inductive biases introduced by our scene composition model in order to decompose an image into object-centric components without the need for deep encoders, using only a small number of learnable parameters, and being fully interpretable.
\subsection{Layered Models}
The idea of representing an image as a superposition of different layers
has been studied since the introduction of the ``dead leaves'' model by~\cite{matheron1968schema}.
This model has been extended to handle natural images and scale-invariant representations~\cite{Lee_OcclusionModelsForNaturalImages_2001}, as well as video sequences~\cite{Jojic_LearningFlexibleSpritesInVideoLayers_2001}.
More recently, several works~\cite{Yang_LRGan_2017,lin2018st,Zhang_CopyPasteGAN_2020,aksoy2017unmixing,Arandjelovic_ObjectDiscoveryWithCopyPasteGAN_2019,Sbai_UnuspervisedImageDecompositioninVectorLayers_2020} combine deep neural networks and ideas from layered image formation for different generative tasks, such as editing or image composition.
However, the aforementioned approaches are limited to foreground/background layered decomposition, or to represent the images with a small number of layers.
The work most similar to ours was recently presented by~\cite{Monnier_UnsupervisedLayeredImageDecompositionIntoObjectPrototypes_2021}.
The authors propose a model to decompose an image into overlapping layers, each containing an object from a predefined set of categories. The object layers are obtained with a cascade of spatial transformer networks, which learn transformations that align certain object sprites to the input image.
While we also follow a layered image formation, our PCDNet model is not limited to a small number of layers, hence being able to represent scenes with multiple objects. PCDNet represents each object in its own layer, and uses learned alpha masks to model occlusions and superposition between layers.
\subsection{Frequency-Domain Neural Networks}
Signal analysis and manipulation in the frequency domain is one of the most widely used tools in the field of signal processing~\cite{Proakis_DigitalSignalProcessing_2004}. However, frequency-domain methods are not so developed for solving computer vision tasks with neural networks. They mostly focus on specific applications such as compression~\cite{Xu_LearningInTheFrequencyDomain_2020,Gueguen_FasterNeuralNetworkStraightFromJPEG_2018}, image super-resolution and denoising~\cite{Fritsche_FrequencySeparationForRealWorldSuperResolution_2019,Villar_DeepLearningDesignsForSuperResolutionNoisyImages_2021,Kumar_ConvNeuralNetworksForWaveletDomainSuperResolution_2017}, or accelerating the calculation of convolutions~\cite{Mathieu_FastTrainingOfCNNsThroughFFTs_2013,Ko_EnergyEfficientTrainingCNNFrequencyDomain_2017}.
In recent years, a particular family of frequency-domain neural networks---the \emph{phase-correlation networks}---has received interest from the research community and has shown promise for tasks such as future frame prediction~\cite{Farazi_LocalFrequencyDomainTransformerNwtworksForVideoPrediction_2021,Wolter_ObjectCenteredFourierMotionEstimation_2020} and motion segmentation~\cite{Farazi_MotionSegmentationUsingFrequencyDomainTransformerNetworks_2020}.
Phase-correlation networks compute normalized cross-correlations in the frequency domain and operate with the phase of complex signals in order to estimate motion and transformation parameters between consecutive frames, which can be used to obtain accurate future frame predictions requiring few learnable parameters.
Despite these recent successes, phase-correlation networks remain unexplored beyond the tasks of video prediction and motion estimation. Our proposed method presents a first attempt at applying phase correlation networks for the tasks of scene decomposition and unsupervised object-centric representation learning.
\begin{figure*}[t]
\begin{subfigure}{0.55\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/PCCell.png}
\caption{PC Cell}
\label{fig: pccell}
\end{subfigure}
\hspace{0.0\textwidth}
\begin{subfigure}{0.45\textwidth}
\centering
\vspace{0.65cm}
\includegraphics[width=1.0\linewidth]{imgs/ColorModule.png}
\vspace{0.2cm}
\caption{Color Module}
\label{fig:color module}
\end{subfigure}
\caption{\textbf{(a)}: Inner structure of the PC Cell. First, the translation parameters are estimated by finding the correlation peaks between the object prototype and the input image. Second, the prototype is shifted by phase shifting in the frequency domain. \textbf{(b)}: The \emph{Color Module} estimates color parameters from the input and aligns the color channels of a translated object prototype.}
\label{a}
\end{figure*}
\section{Phase-Correlation Decomposition Network}
In this section, we present our image decomposition model: PCDNet. Given an input image $\textbf{I}$, PCDNet aims at its decomposition into $N$ independent objects ${\mathcal{O}} = \{\Object{1}, \Object{2}, ..., \Object{N}\}$.
In this work, we assume that these objects belong to one out of a finite number $P$ of classes, and that there is a known upper bound to the total number of objects present in an image ($N_{\max}$).
Inspired by recent works in prototypical learning and clustering~\cite{Li_PrototypicalContrastiveLearningOfUnsupervisedRepresentations_2020,Monnier_DeepTransformationInvariantClustering_2020}, we design our model such that the objects in the image can be represented as transformed versions of a finite set of object prototypes $\mathcal{P} = \{\Prototype{1}, \Prototype{2}, ..., \Prototype{P}\}$.
Each object prototype $\Prototype{i} \in {\mathbb R}^{H,W}$ is learned along with a corresponding alpha mask $\Mask{i} \in {\mathbb R}^{H,W}$, which is used to model occlusions and superposition of objects.
Throughout this work, we consider object prototypes to be in gray-scale and of smaller size than the input image.
PCDNet simultaneously learns suitable object prototypes, alpha masks and transformation parameters in order to accurately decompose an image into object-centric components.
An overview of the PCDNet framework is displayed in \Figure{fig:phacord}. First, the \emph{PC Cell} (\Section{subsec: pc-cell}) estimates the candidate transformation parameters that best align the object prototypes to the objects in the image, and generates object candidates based on the estimated parameters. Second, a \emph{Color Module} (\Section{subsec: color module}) transforms the object candidates by applying a learned color transformation. Finally, a \emph{greedy selection algorithm} (\Section{subsec: greedy selection}) reconstructs the input image by iteratively selecting the object candidates that minimize the reconstruction error.
\subsection{Phase-Correlation Cell}
\label{subsec: pc-cell}
The first module of our image decomposition framework is the PC Cell, as depicted in \Figure{fig:phacord}. This module first estimates the regions of an image where a particular object might be located, and then shifts the prototype to the estimated object location.
Inspired by traditional image registration methods~\cite{Reddy_FFTBasedImageRegistration_1996,Alba_PahseCorrelationImageAlignment_2012}, we adopt an approach based on phase correlation.
This method estimates the relative displacement between two images by computing the normalized cross-correlation in the frequency domain.
Given an image $\textbf{I}$ and an object prototype $\Prototype{}$, the PC Cell first transforms both inputs into the frequency domain using the \emph{Fast Fourier Transform} (FFT, $\mathcal{F}$).
Second, it computes the phase differences between the frequency representations of image and prototype, which can be efficiently computed as an element-wise division in the frequency domain. Then, a localization matrix $\mathbf{L}$ is found by applying the inverse FFT ($\mathcal{F}^{-1}$) on the normalized phase differences:
\vspace{-0.2cm}
\begin{align}
& \mathbf{L} = \mathcal{F}^{-1} \Big( \frac{\mathcal{F}(\textbf{I}) \odot \overline{\mathcal{F}(\Prototype{})}} {|| \mathcal{F}(\textbf{I}) \odot \overline{\mathcal{F}(\Prototype{})} || + \epsilon} \Big),
\label{eq:phase corr}
\end{align}
where $\overline{\mathcal{F}(\Prototype{})}$ denotes the complex conjugate of $\mathcal{F}(\Prototype{})$, $\odot$ is the Hadamard product, $||\cdot||$ is the modulus operator, and $\epsilon$ is a small constant to avoid division by zero.
Finally, the estimated relative pixel displacement ($\boldsymbol{\delta}_{x,y} = (\delta_x, \delta_y)$) can then be found by locating the correlation peak in $\mathbf{L}$:
\vspace{-0.4cm}
\begin{align}
& \boldsymbol{\delta}_{x,y} = \arg \max (\mathbf{L}) \; .
\end{align}
In practical scenarios, we do not know in advance which objects are present in the image or whether there are more than one objects from the same class. To account for this uncertainty, we pick the largest $N_{\max}$ correlation values from $\mathbf{L}$ and consider them as candidate locations for an object.
Finally, given the estimated translation parameters, the PC Cell relies on the Fourier shift theorem to align the object prototypes and the corresponding alpha masks to the objects in the image. Given the translation parameters $\delta_x$ and $\delta_y$, an object prototype is shifted using
\begin{align}
& \Template{} = \mathcal{F}^{-1}( \mathcal{F}(\Prototype{}) \exp(-i 2 \pi (\delta_x \mathbf{f}_x + \delta_y \mathbf{f}_y )),
\end{align}
where $\mathbf{f}_x$ and $\mathbf{f}_y$ denote the frequencies along the horizontal and vertical directions, respectively.
\Figure{fig: pccell} depicts the inner structure of the PC Cell, illustrating each of the phase correlation steps and displaying some intermediate representations, including the magnitude and phase components of each input, the normalized cross-correlation matrix, and the localization matrix $\mathbf{L}$.
\subsection{Color Module}
\label{subsec: color module}
The PC Cell module outputs translated versions of the object prototypes and their corresponding alpha masks. However, these translated templates need not match the color of the object represented in the image. This issue is solved by the \emph{Color Module}, which is illustrated in \Figure{fig:color module}. It learns color parameters from the input image, and transforms the translated prototypes according to the estimated color parameters.
Given the input image and the translated object prototype and mask, the color module first obtains a masked version of the image containing only the relevant object. This is achieved through an element-wise product of the image with the translated alpha mask.
The masked object is fed to a neural network, which learns the color parameters (one for gray-scale and three for RGB images). Finally, these learned parameters are applied to the translated object prototypes with a channel-wise affine transform.
Further details about the color module are given in \Appendix{app: color module}.
\subsection{Greedy Selection}
\label{subsec: greedy selection}
The PC Cell and color modules produce $T = N_{\max} \times P$ translated and colorized candidate objects ($\mathcal{T}=\{\Template{1}, ..., \Template{T}\}$) and their corresponding translated alpha masks ($\{\Mask{1}, ..., \Mask{T}\}$).
The final module of the PCDNet framework selects, among all candidates, the objects that minimize the reconstruction error with respect to the input image.
The number of possible object combinations grows exponentially with the maximum number of objects and the number of object candidates ($T^{N_{\max}}$), which quickly makes it infeasible to evaluate all possible combinations.
Therefore, similarly to \cite{Monnier_UnsupervisedLayeredImageDecompositionIntoObjectPrototypes_2021}, we propose a greedy algorithm that selects in a sequential manner the objects that minimize the reconstruction loss.
The greedy nature of the algorithm reduces the number of possible object combinations to $T \times N_{\max}$, hence scaling to images with a large number of objects and prototypes.
The greedy object selection algorithm operates as follows.
At the first iteration, we select the object that minimizes the reconstruction loss with respect to the input, and add it to the list of selected objects. Then, for each subsequent iteration, we greedily select the object that, combined with the previously selected ones, minimizes the reconstruction error.
This error is computed using \Equation{eq: error}, which corresponds to the mean squared error between the input image ($\textbf{I}$) and a combination of the selected candidates ($\mathcal{G}(\mathcal{T})$).
The objects are combined recursively in an overlapping manner, as shown in \Equation{eq: proto combination}, so that the first selected object ($\Template{1}$) corresponds to the one closest to the viewer, whereas the last selected object ($\Template{N}$) is located the furthest from the viewer:
\begin{align}
\textbf{E}(\textbf{I}, \mathcal{T}) = \; & ||\textbf{I} - \mathcal{G}(\mathcal{T})||_2^2
\label{eq: error} \\
%
\mathcal{G}(\mathcal{T}) = \; & \Template{i+1} \odot (1 - \Mask{i}) + \Template{i} \odot \Mask{i} \nonumber \\
&\forall \, i \, \in \, \{N-1, ..., 1\} \label{eq: proto combination} .
\end{align}
An example of this image composition is displayed in \Figure{fig:phacord}.
This reconstruction approach inherently models relative depths, allowing for a simple, yet effective, modeling of occlusions between objects.
\subsection{Training and Implementation Details}
\label{subsec: training}
We train PCDNet in an end-to-end manner to reconstruct an image as a combination of transformed object prototypes. The training is performed by minimizing the reconstruction error ($\mathcal{L}_{MSE}$), while regularizing the prototypes to with respect to the $\ell_1$ norm ($\mathcal{L}_{L1}$), and enforcing smooth alpha masks with a total variation regularizer~\cite{Rudin_TotalVariationBasedImageRestoration_1994} ($\mathcal{L}_{TV}$).
Specifically, we minimize the following loss function:
\vspace{-0.3cm}
\begin{align}
\mathcal{L}_{} &= \mathcal{L}_{MSE} + \lambda_{L1} \; \mathcal{L}_{L1} + \lambda_{TV} \; \mathcal{L}_{TV}
\label{eq: loss function} \\
\mathcal{L}_{MSE} &= ||\textbf{I} - \mathcal{G}(\mathcal{T}')||_2^2 \\
\mathcal{L}_{L1} &= \frac{1}{P} \sum_{\Prototype{} \in \mathcal{P}} ||\Prototype{}||_1 \\
\mathcal{L}_{TV} &= \frac{1}{P} \sum_{\Mask{} \in \mathcal{M}} \sum_{i,j} |\Mask{i+1,j} - \Mask{i,j}| + |\Mask{i,j+1} - \Mask{i,j}|
\end{align}
where $\mathcal{T}'$ corresponds to the object candidates selected by the greedy selection algorithm,
$\mathcal{P}$ are the learned object prototypes, and $\mathcal{M}$ to the corresponding alpha masks.
Namely, minimizing \Equation{eq: loss function} decreases the reconstruction error between the combination of selected object candidates ($\mathcal{G}(\mathcal{T}')$) and the input image, while keeping the object prototypes compact, and the alpha masks smooth.
\begin{table*}[t!]
\centering
\captionof{table}{Object discovery evaluation results on the Tetrominoes dataset. PCDNet outperforms SOTA methods, while using a small number of learned parameters. Moreover, our PCDNet has the highest throughput out of all evaluated methods. For each metric, the best result is highlighted in boldface, whereas the second best is underlined.}
\label{table: ari eval}
\vspace{-0.0cm}
\normalsize
\begin{tabular}{ p{5.3cm} | P{2.cm} | P{2.3cm} P{2.3cm}}
\midrule[2pt]
\textbf{Model} & \textbf{ARI (\%) $\uparrow$} & \textbf{ Params $\downarrow$} & \textbf{Imgs/s $\uparrow$} \\
\midrule[1.4pt]
Slot MLP~\cite{Locatello_ObjectCentricLearningWithSlotAttention_2020} & 35.1 & -- & -- \\
Slot Attention~\cite{Locatello_ObjectCentricLearningWithSlotAttention_2020} & 99.5 & \underline{229,188} & 18.36\\
ULID~\cite{Monnier_UnsupervisedLayeredImageDecompositionIntoObjectPrototypes_2021} & \underline{99.6} & 659,755 & \underline{52.31} \\
IODINE~\cite{Greff_IodineMultiObjectRepresentationLearningWithIterativeVariationalInference_2019} & 99.2 & 408,036 & 16.64\\
PCDNet (ours) & \textbf{99.7} & \textbf{28,130} & \textbf{58.96} \\
\midrule[2pt]
\end{tabular}
\end{table*}
\begin{figure}[t]
\centering
\includegraphics[width=0.99\linewidth]{imgs/tetris.png}
\caption{Object prototypes (top) and segmentation alpha masks (bottom) learned on the Tetrominoes dataset. Our model is able to discover in an unsupervised manner all 19 pieces.}
\label{fig: tetris}
\end{figure}
In our experiments, we noticed that the initialization and update strategy of the object prototypes is of paramount importance for the correct performance of the PCDNet model.
The prototypes are initialized with a small constant value (e.g., 0.2), whereas the center pixel is assigned an initial value of one, enforcing the prototypes to emerge centered in the frame.
During the first training iterations, we notice that the greedy algorithm selects some prototypes with a higher frequency that others, hence learning much faster.
In practice, this prevents other prototypes from learning relevant object representations, since they are not updated often enough.
To reduce the impact of uneven prototype discovery, we add, with a certain probability, some uniform random noise to the prototypes during the first training iterations. This prevents the greedy algorithm from always selecting, and hence updating, the same object prototypes and masks.
In datasets with a background, we add a special prototype to model a static background.
In these cases, the input images are reconstructed by overlapping the objects selected by the greedy algorithm on top of the background prototype.
This background prototype is initialized by averaging all training images, and its values are refined during training.
\section{Experimental Results}
In this section, we quantitatively and qualitatively evaluate our PCDNet framework for the tasks of unsupervised object discovery and segmentation.
PCDNet is implemented in Python using the PyTorch framework~\cite{Paszke_AutomaticDifferneciationInPytorch_2017}.
A detailed report of the hyper-parameters used is given in \Appendix{subsec: Model and Hyper-Parameter Details}.
\begin{figure*}[t!]
\hspace{0.02\textwidth}
\begin{subfigure}{0.11\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/a.png}
\caption{}
\label{fig: orig}
\end{subfigure}
\hspace{0.01\textwidth}
\begin{subfigure}{0.11\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/b.png}
\caption{}
\label{fig: recons}
\end{subfigure}
\hspace{0.01\textwidth}
\begin{subfigure}{0.11\textwidth}
\centering
\includegraphics[width=1.01\linewidth]{imgs/c1.png}
\caption{}
\label{fig: obj1}
\end{subfigure}
\hspace{0.01\textwidth}
\begin{subfigure}{0.11\textwidth}
\centering
\includegraphics[width=1.01\linewidth]{imgs/c2.png}
\caption{}
\label{fig: obj2}
\end{subfigure}
\hspace{0.01\textwidth}
\begin{subfigure}{0.11\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/c3.png}
\caption{}
\label{fig: obj3}
\end{subfigure}
\hspace{0.01\textwidth}
\begin{subfigure}{0.11\textwidth}
\centering
\includegraphics[width=.99\linewidth]{imgs/d.png}
\caption{}
\label{fig: segmentation}
\end{subfigure}
\hspace{0.01\textwidth}
\begin{subfigure}{0.11\textwidth}
\centering
\includegraphics[width=.99\linewidth]{imgs/e.png}
\caption{}
\label{fig: instance}
\end{subfigure}
\hspace{0.02\textwidth}
\vspace{-0.0cm}
\caption{Qualitative decomposition and segmentation results on the Tetrominoes dataset. Last row shows a failure case. \textbf{(a)}: Original image. \textbf{(b)}: PCDNet Reconstruction. \textbf{(c)-(e)}: Colorized transformed object prototypes. \textbf{(f)}: Semantic segmentation masks. Colors correspond to the prototype frames in \Figure{fig: tetris}. \textbf{(g)}: Instance segmentation masks.}
\label{fig: qualitative tetris}
\end{figure*}
\subsection{Tetrominoes Dataset}
\label{subsec: Image Decomposition}
We evaluate PCDNet for image decomposition and object discovery on the Tetrominoes dataset~\cite{Greff_IodineMultiObjectRepresentationLearningWithIterativeVariationalInference_2019}. This dataset contains 60.000 training images and 320 test images of size $35 \times 35$, each composed of three non-overlapping Tetris-like sprites over a black background. The sprites belong to one out of 19 configurations and have one of six random colors.
\Figure{fig: tetris} displays the 19 learned object prototypes and their corresponding alpha masks from the Tetrominoes dataset. We clearly observe how PCDNet accurately discovers the shape of the different pieces and their tiled texture.
\Figure{fig: qualitative tetris} depicts qualitative results for unsupervised object detection and segmentation.
In the first three rows, PCDNet successfully decomposes the images into their object components and precisely segments the objects into semantic and instance masks. The bottom row shows an example in which the greedy selection algorithm leads to a failure case.
For a fair quantitative comparison with previous works, we evaluate our PCDNet model for object segmentation using the Adjusted Rand Index~\cite{Hubert_ComparingPartitionsARI_1985} (ARI) on the ground truth foreground pixels. ARI is a clustering metric that measures the similarity between two set assignments, ignoring label permutations, and ranges from 0 (random assignment) to 1 (perfect clustering).
We compare the performance of our approach with several existing methods: Slot MLP and Slot Attention~\cite{Locatello_ObjectCentricLearningWithSlotAttention_2020}, IODINE~\cite{Greff_IodineMultiObjectRepresentationLearningWithIterativeVariationalInference_2019} and Unsupervised Layered Image Decomposition~\cite{Monnier_UnsupervisedLayeredImageDecompositionIntoObjectPrototypes_2021} (ULID).
\Table{table: ari eval} summarizes the evaluation results for object discovery on the Tetrominoes dataset. From \Table{table: ari eval}, we see how PCDNet outperforms SOTA models, achieving 99.7\% ARI on the Tetrominoes dataset. PCDNet uses only a small percentage of learnable parameters compared
to other methods (e.g., only 6\% of the parameters from IODINE), and has the highest inference
throughput (images/s). Additionally, unlike other approaches, PCDNet obtains disentangled representations for the object appearance, position, and color in a human-interpretable manner.
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth]{imgs/prototypes_atari.png}
\caption{Object prototypes learned on the Space Invaders dataset. PCDNet discovers prototypes corresponding to the different elements from the game, e.g., aliens and ships.}
\label{fig:prototypes atari}
\end{figure}
\subsection{Space Invaders Dataset}
\label{subsec: Space Invaders Dataset}
In this experiment, we use replays from humans playing the Atari game \emph{Space Invaders}, extracted from the Atari Grand Challenge dataset~\cite{Kurin_TheAtariGranChallengeDataset_2017}.
PCDNet is trained to decompose the Space Invaders images into 50 objects, belonging to one of 14 learned prototypes of size $20 \times 20$.
\Figure{fig:comp atari} depicts a qualitative comparison between our PCDNet model with SPACE~\cite{Lin_UnsupervisedObjectOrientedSceneRepresentationViaSpatialAttentionAndDecomposition_2020} and Slot Attention~\cite{Locatello_ObjectCentricLearningWithSlotAttention_2020}.
Slot Attention achieves an almost perfect reconstruction of the input image. However, it fails to decompose the image into its object components, uniformly scattering the object representations across different slots. In \Figure{fig:comp atari} (subplot labeled as \emph{Slot}) we show how one of the slots simultaneously encodes information from several different objects.
SPACE successfully decomposes the image into different object components, which are recognized as foreground objects. Nevertheless, the reconstructions appear blurred and several objects are not correct.
PCDNet achieves the best results among all compared methods. Our model successfully decomposes the input image into accurate object-centric representations. Additionally, PCDNet learns semantic understanding of the objects.
\Figure{fig:comp atari} depicts a segmentation of an image from the Space Invaders dataset.
Further qualitative results on the Space Invaders dataset are reported in \Appendix{section: further qualitative results}.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.91\linewidth]{imgs/comp_atari.png}
\caption{Comparison of different object-centric models on the Space Invaders dataset. PCDNet is the only one among the compared methods which successfully decomposes the image into accurate object components, and that has semantic knowledge of the objects. The color of each object corresponds to the frame of the corresponding prototype in \Figure{fig:prototypes atari}.}
\label{fig:comp atari}
\end{figure*}
\begin{figure*}[h!]
\centering
\includegraphics[width=0.91\linewidth]{imgs/CarExps.png}
\caption{Object discovery on the NGSIM dataset. PCDNet learns different vehicle prototypes in an unsupervised manner.}
\label{fig:cars}
\end{figure*}
\subsection{NGSIM Dataset}
\label{subsec: NGSIM Dataset}
In this third experiment, we apply our PCDNet model to discover vehicle prototypes from real traffic camera footage from the Next Generation Simulation (NGSIM) dataset~\cite{NGSIM_Dataset}.
We decompose each frame into up to 33 different objects, belonging to one of 30 learned vehicle prototypes.
\Figure{fig:cars} depicts qualitative results on the NGSIM dataset.
We see how PCDNet is applicable to real-world data, accurately reconstructing the input image, while learning prototypes for different types of vehicles.
Interestingly, we notice how PCDNet learns the car shade as part of the prototype. This is a reasonable observation, since the shades are projected towards the bottom of the image throughout the whole video.
\section{Conclusion}
We proposed PCDNet, a novel image composition model that decomposes an image, in a fully unsupervised manner, into its object components, which are represented as transformed versions of a set of learned object prototypes.
PCDNet exploits the frequency-domain representation of images to estimate the translation parameters that best align the prototypes to the objects in the image.
The structured network used by PCDNet allows for an interpretable image decomposition, which disentangles object appearance, position and color without any external supervision.
In our experiments, we show how our proposed model outperforms existing methods for unsupervised object discovery and segmentation on a benchmark synthetic dataset, while significantly reducing the number of learnable parameters, having a superior throughput, and being fully interpretable. Furthermore, we also show that the PCDNet model can also be applied for unsupervised prototypical object discovery on more challenging synthetic and real datasets.
We hope that our work paves the way towards further research on phase correlation networks for unsupervised object-centric representation learning.
\section*{ACKNOWLEDGMENTS}
This work was funded by grant BE 2556/18-2 (Research Unit FOR 2535
Anticipating Human Behavior) of the German Research Foundation (DFG).
\bibliographystyle{apalike}
{\small
\section{Introduction}
Humans understand the world by decomposing scenes into objects that can interact
with each other. Analogously, autonomous systems’ reasoning and scene understanding capabilities could benefit from decomposing scenes into objects and modeling each of these independently.
This approach has been proven beneficial to perform a wide variety of computer vision tasks without explicit supervision, including unsupervised object detection~\cite{Eslami_AttendInferRepeatSceneUnderstanding_2016}, future frame prediction~\cite{Weis_UnmaskingInductiveBiasesOfUnsupervisedObjectRepresentationsForVideoSequences_2020,Greff_IodineMultiObjectRepresentationLearningWithIterativeVariationalInference_2019}, and object tracking~\cite{He_TrackingByAnnimation_2019,Veerapaneni_EntityAbstractioninVisualModelBasedReinforcementLearning_2020}.
Recent works propose extracting object-centric representations without the need for explicit supervision through the use of deep variational auto-encoders~\cite{Kingma_AutoEncodingVariationalBayes_2013} (VAEs) with spatial attention mechanisms~\cite{Burgess_MonetUnsupervisedSceneDecompositionRepresentation_2019,crawford2019spatially}.
However, training these models often presents several difficulties, such as long training times, requiring a large number of trainable parameters, or the need for large curated datasets. Furthermore, these methods suffer from the inherent lack of interpretability which is characteristic of deep neural networks (DNNs).
\begin{figure*}[t]
\centering
\includegraphics[width=0.92\linewidth]{imgs/PCDNet.png}
\caption{PCDNet decomposition framework. First, the Phase Correlation (PC) Cell estimates the $N$ translation parameters that best align each learned prototype to the objects in the image, and uses them to obtain $(P \times N)$ object and mask candidates. Second, the color module assigns a color to each of the transformed prototypes. Finally, a greedy selection algorithm reconstructs the input image by iteratively combining the colorized object candidates that minimize the reconstruction error.}
\label{fig:phacord}
\end{figure*}
To address the aforementioned issues, we propose a novel image decomposition framework: the \emph{\textbf{P}hase-\textbf{C}orrelation \textbf{D}ecomposition \textbf{Net}work } (PCDNet). Our method assumes that an image is formed as an arrangement of multiple objects, each belonging to one of a finite number of different classes.
Following this assumption, PCDNet decomposes an image into its object components, which are represented as transformed versions of a set of learned object prototypes.
The core building block of the PCDNet framework is the \emph{Phase Correlation Cell} (PC Cell). This is a differentiable module that exploits the frequency-domain representations of an image and a prototype to estimate the transformation parameters that best align a prototype to its corresponding object in the image.
The PC Cell localizes the object prototype in the image by applying the phase-correlation method~\cite{Alba_PahseCorrelationImageAlignment_2012}, i.e., finding the peaks in the cross-correlation matrix between the input image and the prototype. Then, the PC Cell aligns the prototype to its corresponding object in the image by performing the estimated phase shift in the frequency domain.
PCDNet is trained by first decomposing an image into its object components, and then reconstructing the input by recombining the estimated object components following the ``dead leaves'' model approach, i.e., as a superposition of different objects.
The strong inductive biases introduced by the network structure allow our method to learn fully interpretable prototypical object-centric representations without any external supervision while keeping the number of learnable parameters small.
Furthermore, our method also disentangles the position and color of each object in a human-interpretable manner.
In summary, the contributions of our work are as follows:
\begin{itemize}
\item We propose the PCDNet model, which decomposes an image into its object components, which are represented as transformed versions of a set of learned object prototypes.
\item Our proposed model exploits the frequency-domain representation of images so as to disentangle object appearance, position, and color without the need for any external supervision.
\item Our experimental results show that our proposed framework outperforms recent methods for joined unsupervised object discovery, image decomposition, and segmentation on benchmark datasets, while significantly reducing the number of learnable parameters, allowing for high throughput, and maintaining interpretability.
\end{itemize}
\section{Related Work}
\subsection{Object-Centric Representation Learning}
The field of representation learning~\cite{Bengio_RepresentationLearningReview_2013} has seen much attention in the last decade, giving rise to great advances in learning hierarchical representations~\cite{Paschalidou_LearningUnsupervisedPartDecompositionOf3DObjects_2020,Stanic_HierarchicalRelationalInference_2020} or in disentangling the underlying factors of variation in the data~\cite{Locatello_ChallengingAssumptionsInLearningOfDissentangledRepresentations_2019,Burgess_UnderstandingDisentanglingInBetaVAE_2018}.
Despite these successes, these methods often rely on learning representations at a scene level, rather than learning in an object-centric manner, i.e., simultaneously learning representations that address multiple, possibly repeating, objects.
In the last few years, several methods have been proposed to perform object-centric image decomposition in an unsupervised manner.
A first approach to object-centric decomposition combines VAEs with attention mechanisms to decompose a scene into object-centric representations. The object representations are then decoded to reconstruct the input image.
These methods can be further divided into two different groups depending on the class of latent representations used.
On the one hand, some methods~\cite{Eslami_AttendInferRepeatSceneUnderstanding_2016,Kosiorek_SequentialAttendInferRepeat_2018,Stanic_HierarchicalRelationalInference_2020,He_TrackingByAnnimation_2019} explicitly encode the input into factored latent variables, which represent specific properties such as appearance, position, and presence.
On the other hand, other models~\cite{Burgess_MonetUnsupervisedSceneDecompositionRepresentation_2019,Weis_UnmaskingInductiveBiasesOfUnsupervisedObjectRepresentationsForVideoSequences_2020,Locatello_ObjectCentricLearningWithSlotAttention_2020} decompose the image into unconstrained per-object latent representations.
Recently, several proposed methods~\cite{Greff_IodineMultiObjectRepresentationLearningWithIterativeVariationalInference_2019,Engelcke_GenesisGeneraticeSceneInferenceWithObjectCentricRepresentations_2019,Engelcke_GenesisV2InferringObjectRepresentationsWithoutIterativeRefinement_2021,Veerapaneni_EntityAbstractioninVisualModelBasedReinforcementLearning_2020,Lin_UnsupervisedObjectOrientedSceneRepresentationViaSpatialAttentionAndDecomposition_2020} use parameterized spatial mixture models with variational inference to decode object-centric latent variables.
Despite these recent advances in unsupervised object-centric learning, most existing methods rely on DNNs and attention mechanisms to encode the input images into latent representations, hence requiring a large number of learnable parameters and high computational costs. Furthermore, these approaches suffer from the inherent lack of interpretability characteristic of DNNs.
Our proposed method exploits the strong inductive biases introduced by our scene composition model in order to decompose an image into object-centric components without the need for deep encoders, using only a small number of learnable parameters, and being fully interpretable.
\subsection{Layered Models}
The idea of representing an image as a superposition of different layers
has been studied since the introduction of the ``dead leaves'' model by~\cite{matheron1968schema}.
This model has been extended to handle natural images and scale-invariant representations~\cite{Lee_OcclusionModelsForNaturalImages_2001}, as well as video sequences~\cite{Jojic_LearningFlexibleSpritesInVideoLayers_2001}.
More recently, several works~\cite{Yang_LRGan_2017,lin2018st,Zhang_CopyPasteGAN_2020,aksoy2017unmixing,Arandjelovic_ObjectDiscoveryWithCopyPasteGAN_2019,Sbai_UnuspervisedImageDecompositioninVectorLayers_2020} combine deep neural networks and ideas from layered image formation for different generative tasks, such as editing or image composition.
However, the aforementioned approaches are limited to foreground/background layered decomposition, or to represent the images with a small number of layers.
The work most similar to ours was recently presented by~\cite{Monnier_UnsupervisedLayeredImageDecompositionIntoObjectPrototypes_2021}.
The authors propose a model to decompose an image into overlapping layers, each containing an object from a predefined set of categories. The object layers are obtained with a cascade of spatial transformer networks, which learn transformations that align certain object sprites to the input image.
While we also follow a layered image formation, our PCDNet model is not limited to a small number of layers, hence being able to represent scenes with multiple objects. PCDNet represents each object in its own layer, and uses learned alpha masks to model occlusions and superposition between layers.
\subsection{Frequency-Domain Neural Networks}
Signal analysis and manipulation in the frequency domain is one of the most widely used tools in the field of signal processing~\cite{Proakis_DigitalSignalProcessing_2004}. However, frequency-domain methods are not so developed for solving computer vision tasks with neural networks. They mostly focus on specific applications such as compression~\cite{Xu_LearningInTheFrequencyDomain_2020,Gueguen_FasterNeuralNetworkStraightFromJPEG_2018}, image super-resolution and denoising~\cite{Fritsche_FrequencySeparationForRealWorldSuperResolution_2019,Villar_DeepLearningDesignsForSuperResolutionNoisyImages_2021,Kumar_ConvNeuralNetworksForWaveletDomainSuperResolution_2017}, or accelerating the calculation of convolutions~\cite{Mathieu_FastTrainingOfCNNsThroughFFTs_2013,Ko_EnergyEfficientTrainingCNNFrequencyDomain_2017}.
In recent years, a particular family of frequency-domain neural networks---the \emph{phase-correlation networks}---has received interest from the research community and has shown promise for tasks such as future frame prediction~\cite{Farazi_LocalFrequencyDomainTransformerNwtworksForVideoPrediction_2021,Wolter_ObjectCenteredFourierMotionEstimation_2020} and motion segmentation~\cite{Farazi_MotionSegmentationUsingFrequencyDomainTransformerNetworks_2020}.
Phase-correlation networks compute normalized cross-correlations in the frequency domain and operate with the phase of complex signals in order to estimate motion and transformation parameters between consecutive frames, which can be used to obtain accurate future frame predictions requiring few learnable parameters.
Despite these recent successes, phase-correlation networks remain unexplored beyond the tasks of video prediction and motion estimation. Our proposed method presents a first attempt at applying phase correlation networks for the tasks of scene decomposition and unsupervised object-centric representation learning.
\begin{figure*}[t]
\begin{subfigure}{0.55\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/PCCell.png}
\caption{PC Cell}
\label{fig: pccell}
\end{subfigure}
\hspace{0.0\textwidth}
\begin{subfigure}{0.45\textwidth}
\centering
\vspace{0.65cm}
\includegraphics[width=1.0\linewidth]{imgs/ColorModule.png}
\vspace{0.2cm}
\caption{Color Module}
\label{fig:color module}
\end{subfigure}
\caption{\textbf{(a)}: Inner structure of the PC Cell. First, the translation parameters are estimated by finding the correlation peaks between the object prototype and the input image. Second, the prototype is shifted by phase shifting in the frequency domain. \textbf{(b)}: The \emph{Color Module} estimates color parameters from the input and aligns the color channels of a translated object prototype.}
\label{a}
\end{figure*}
\section{Phase-Correlation Decomposition Network}
In this section, we present our image decomposition model: PCDNet. Given an input image $\textbf{I}$, PCDNet aims at its decomposition into $N$ independent objects ${\mathcal{O}} = \{\Object{1}, \Object{2}, ..., \Object{N}\}$.
In this work, we assume that these objects belong to one out of a finite number $P$ of classes, and that there is a known upper bound to the total number of objects present in an image ($N_{\max}$).
Inspired by recent works in prototypical learning and clustering~\cite{Li_PrototypicalContrastiveLearningOfUnsupervisedRepresentations_2020,Monnier_DeepTransformationInvariantClustering_2020}, we design our model such that the objects in the image can be represented as transformed versions of a finite set of object prototypes $\mathcal{P} = \{\Prototype{1}, \Prototype{2}, ..., \Prototype{P}\}$.
Each object prototype $\Prototype{i} \in {\mathbb R}^{H,W}$ is learned along with a corresponding alpha mask $\Mask{i} \in {\mathbb R}^{H,W}$, which is used to model occlusions and superposition of objects.
Throughout this work, we consider object prototypes to be in gray-scale and of smaller size than the input image.
PCDNet simultaneously learns suitable object prototypes, alpha masks and transformation parameters in order to accurately decompose an image into object-centric components.
An overview of the PCDNet framework is displayed in \Figure{fig:phacord}. First, the \emph{PC Cell} (\Section{subsec: pc-cell}) estimates the candidate transformation parameters that best align the object prototypes to the objects in the image, and generates object candidates based on the estimated parameters. Second, a \emph{Color Module} (\Section{subsec: color module}) transforms the object candidates by applying a learned color transformation. Finally, a \emph{greedy selection algorithm} (\Section{subsec: greedy selection}) reconstructs the input image by iteratively selecting the object candidates that minimize the reconstruction error.
\subsection{Phase-Correlation Cell}
\label{subsec: pc-cell}
The first module of our image decomposition framework is the PC Cell, as depicted in \Figure{fig:phacord}. This module first estimates the regions of an image where a particular object might be located, and then shifts the prototype to the estimated object location.
Inspired by traditional image registration methods~\cite{Reddy_FFTBasedImageRegistration_1996,Alba_PahseCorrelationImageAlignment_2012}, we adopt an approach based on phase correlation.
This method estimates the relative displacement between two images by computing the normalized cross-correlation in the frequency domain.
Given an image $\textbf{I}$ and an object prototype $\Prototype{}$, the PC Cell first transforms both inputs into the frequency domain using the \emph{Fast Fourier Transform} (FFT, $\mathcal{F}$).
Second, it computes the phase differences between the frequency representations of image and prototype, which can be efficiently computed as an element-wise division in the frequency domain. Then, a localization matrix $\mathbf{L}$ is found by applying the inverse FFT ($\mathcal{F}^{-1}$) on the normalized phase differences:
\vspace{-0.2cm}
\begin{align}
& \mathbf{L} = \mathcal{F}^{-1} \Big( \frac{\mathcal{F}(\textbf{I}) \odot \overline{\mathcal{F}(\Prototype{})}} {|| \mathcal{F}(\textbf{I}) \odot \overline{\mathcal{F}(\Prototype{})} || + \epsilon} \Big),
\label{eq:phase corr}
\end{align}
where $\overline{\mathcal{F}(\Prototype{})}$ denotes the complex conjugate of $\mathcal{F}(\Prototype{})$, $\odot$ is the Hadamard product, $||\cdot||$ is the modulus operator, and $\epsilon$ is a small constant to avoid division by zero.
Finally, the estimated relative pixel displacement ($\boldsymbol{\delta}_{x,y} = (\delta_x, \delta_y)$) can then be found by locating the correlation peak in $\mathbf{L}$:
\vspace{-0.4cm}
\begin{align}
& \boldsymbol{\delta}_{x,y} = \arg \max (\mathbf{L}) \; .
\end{align}
In practical scenarios, we do not know in advance which objects are present in the image or whether there are more than one objects from the same class. To account for this uncertainty, we pick the largest $N_{\max}$ correlation values from $\mathbf{L}$ and consider them as candidate locations for an object.
Finally, given the estimated translation parameters, the PC Cell relies on the Fourier shift theorem to align the object prototypes and the corresponding alpha masks to the objects in the image. Given the translation parameters $\delta_x$ and $\delta_y$, an object prototype is shifted using
\begin{align}
& \Template{} = \mathcal{F}^{-1}( \mathcal{F}(\Prototype{}) \exp(-i 2 \pi (\delta_x \mathbf{f}_x + \delta_y \mathbf{f}_y )),
\end{align}
where $\mathbf{f}_x$ and $\mathbf{f}_y$ denote the frequencies along the horizontal and vertical directions, respectively.
\Figure{fig: pccell} depicts the inner structure of the PC Cell, illustrating each of the phase correlation steps and displaying some intermediate representations, including the magnitude and phase components of each input, the normalized cross-correlation matrix, and the localization matrix $\mathbf{L}$.
\subsection{Color Module}
\label{subsec: color module}
The PC Cell module outputs translated versions of the object prototypes and their corresponding alpha masks. However, these translated templates need not match the color of the object represented in the image. This issue is solved by the \emph{Color Module}, which is illustrated in \Figure{fig:color module}. It learns color parameters from the input image, and transforms the translated prototypes according to the estimated color parameters.
Given the input image and the translated object prototype and mask, the color module first obtains a masked version of the image containing only the relevant object. This is achieved through an element-wise product of the image with the translated alpha mask.
The masked object is fed to a neural network, which learns the color parameters (one for gray-scale and three for RGB images). Finally, these learned parameters are applied to the translated object prototypes with a channel-wise affine transform.
Further details about the color module are given in \Appendix{app: color module}.
\subsection{Greedy Selection}
\label{subsec: greedy selection}
The PC Cell and color modules produce $T = N_{\max} \times P$ translated and colorized candidate objects ($\mathcal{T}=\{\Template{1}, ..., \Template{T}\}$) and their corresponding translated alpha masks ($\{\Mask{1}, ..., \Mask{T}\}$).
The final module of the PCDNet framework selects, among all candidates, the objects that minimize the reconstruction error with respect to the input image.
The number of possible object combinations grows exponentially with the maximum number of objects and the number of object candidates ($T^{N_{\max}}$), which quickly makes it infeasible to evaluate all possible combinations.
Therefore, similarly to \cite{Monnier_UnsupervisedLayeredImageDecompositionIntoObjectPrototypes_2021}, we propose a greedy algorithm that selects in a sequential manner the objects that minimize the reconstruction loss.
The greedy nature of the algorithm reduces the number of possible object combinations to $T \times N_{\max}$, hence scaling to images with a large number of objects and prototypes.
The greedy object selection algorithm operates as follows.
At the first iteration, we select the object that minimizes the reconstruction loss with respect to the input, and add it to the list of selected objects. Then, for each subsequent iteration, we greedily select the object that, combined with the previously selected ones, minimizes the reconstruction error.
This error is computed using \Equation{eq: error}, which corresponds to the mean squared error between the input image ($\textbf{I}$) and a combination of the selected candidates ($\mathcal{G}(\mathcal{T})$).
The objects are combined recursively in an overlapping manner, as shown in \Equation{eq: proto combination}, so that the first selected object ($\Template{1}$) corresponds to the one closest to the viewer, whereas the last selected object ($\Template{N}$) is located the furthest from the viewer:
\begin{align}
\textbf{E}(\textbf{I}, \mathcal{T}) = \; & ||\textbf{I} - \mathcal{G}(\mathcal{T})||_2^2
\label{eq: error} \\
%
\mathcal{G}(\mathcal{T}) = \; & \Template{i+1} \odot (1 - \Mask{i}) + \Template{i} \odot \Mask{i} \nonumber \\
&\forall \, i \, \in \, \{N-1, ..., 1\} \label{eq: proto combination} .
\end{align}
An example of this image composition is displayed in \Figure{fig:phacord}.
This reconstruction approach inherently models relative depths, allowing for a simple, yet effective, modeling of occlusions between objects.
\subsection{Training and Implementation Details}
\label{subsec: training}
We train PCDNet in an end-to-end manner to reconstruct an image as a combination of transformed object prototypes. The training is performed by minimizing the reconstruction error ($\mathcal{L}_{MSE}$), while regularizing the prototypes to with respect to the $\ell_1$ norm ($\mathcal{L}_{L1}$), and enforcing smooth alpha masks with a total variation regularizer~\cite{Rudin_TotalVariationBasedImageRestoration_1994} ($\mathcal{L}_{TV}$).
Specifically, we minimize the following loss function:
\vspace{-0.3cm}
\begin{align}
\mathcal{L}_{} &= \mathcal{L}_{MSE} + \lambda_{L1} \; \mathcal{L}_{L1} + \lambda_{TV} \; \mathcal{L}_{TV}
\label{eq: loss function} \\
\mathcal{L}_{MSE} &= ||\textbf{I} - \mathcal{G}(\mathcal{T}')||_2^2 \\
\mathcal{L}_{L1} &= \frac{1}{P} \sum_{\Prototype{} \in \mathcal{P}} ||\Prototype{}||_1 \\
\mathcal{L}_{TV} &= \frac{1}{P} \sum_{\Mask{} \in \mathcal{M}} \sum_{i,j} |\Mask{i+1,j} - \Mask{i,j}| + |\Mask{i,j+1} - \Mask{i,j}|
\end{align}
where $\mathcal{T}'$ corresponds to the object candidates selected by the greedy selection algorithm,
$\mathcal{P}$ are the learned object prototypes, and $\mathcal{M}$ to the corresponding alpha masks.
Namely, minimizing \Equation{eq: loss function} decreases the reconstruction error between the combination of selected object candidates ($\mathcal{G}(\mathcal{T}')$) and the input image, while keeping the object prototypes compact, and the alpha masks smooth.
\begin{table*}[t!]
\centering
\captionof{table}{Object discovery evaluation results on the Tetrominoes dataset. PCDNet outperforms SOTA methods, while using a small number of learned parameters. Moreover, our PCDNet has the highest throughput out of all evaluated methods. For each metric, the best result is highlighted in boldface, whereas the second best is underlined.}
\label{table: ari eval}
\vspace{-0.0cm}
\normalsize
\begin{tabular}{ p{5.3cm} | P{2.cm} | P{2.3cm} P{2.3cm}}
\midrule[2pt]
\textbf{Model} & \textbf{ARI (\%) $\uparrow$} & \textbf{ Params $\downarrow$} & \textbf{Imgs/s $\uparrow$} \\
\midrule[1.4pt]
Slot MLP~\cite{Locatello_ObjectCentricLearningWithSlotAttention_2020} & 35.1 & -- & -- \\
Slot Attention~\cite{Locatello_ObjectCentricLearningWithSlotAttention_2020} & 99.5 & \underline{229,188} & 18.36\\
ULID~\cite{Monnier_UnsupervisedLayeredImageDecompositionIntoObjectPrototypes_2021} & \underline{99.6} & 659,755 & \underline{52.31} \\
IODINE~\cite{Greff_IodineMultiObjectRepresentationLearningWithIterativeVariationalInference_2019} & 99.2 & 408,036 & 16.64\\
PCDNet (ours) & \textbf{99.7} & \textbf{28,130} & \textbf{58.96} \\
\midrule[2pt]
\end{tabular}
\end{table*}
\begin{figure}[t]
\centering
\includegraphics[width=0.99\linewidth]{imgs/tetris.png}
\caption{Object prototypes (top) and segmentation alpha masks (bottom) learned on the Tetrominoes dataset. Our model is able to discover in an unsupervised manner all 19 pieces.}
\label{fig: tetris}
\end{figure}
In our experiments, we noticed that the initialization and update strategy of the object prototypes is of paramount importance for the correct performance of the PCDNet model.
The prototypes are initialized with a small constant value (e.g., 0.2), whereas the center pixel is assigned an initial value of one, enforcing the prototypes to emerge centered in the frame.
During the first training iterations, we notice that the greedy algorithm selects some prototypes with a higher frequency that others, hence learning much faster.
In practice, this prevents other prototypes from learning relevant object representations, since they are not updated often enough.
To reduce the impact of uneven prototype discovery, we add, with a certain probability, some uniform random noise to the prototypes during the first training iterations. This prevents the greedy algorithm from always selecting, and hence updating, the same object prototypes and masks.
In datasets with a background, we add a special prototype to model a static background.
In these cases, the input images are reconstructed by overlapping the objects selected by the greedy algorithm on top of the background prototype.
This background prototype is initialized by averaging all training images, and its values are refined during training.
\section{Experimental Results}
In this section, we quantitatively and qualitatively evaluate our PCDNet framework for the tasks of unsupervised object discovery and segmentation.
PCDNet is implemented in Python using the PyTorch framework~\cite{Paszke_AutomaticDifferneciationInPytorch_2017}.
A detailed report of the hyper-parameters used is given in \Appendix{subsec: Model and Hyper-Parameter Details}.
\begin{figure*}[t!]
\hspace{0.02\textwidth}
\begin{subfigure}{0.11\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/a.png}
\caption{}
\label{fig: orig}
\end{subfigure}
\hspace{0.01\textwidth}
\begin{subfigure}{0.11\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/b.png}
\caption{}
\label{fig: recons}
\end{subfigure}
\hspace{0.01\textwidth}
\begin{subfigure}{0.11\textwidth}
\centering
\includegraphics[width=1.01\linewidth]{imgs/c1.png}
\caption{}
\label{fig: obj1}
\end{subfigure}
\hspace{0.01\textwidth}
\begin{subfigure}{0.11\textwidth}
\centering
\includegraphics[width=1.01\linewidth]{imgs/c2.png}
\caption{}
\label{fig: obj2}
\end{subfigure}
\hspace{0.01\textwidth}
\begin{subfigure}{0.11\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/c3.png}
\caption{}
\label{fig: obj3}
\end{subfigure}
\hspace{0.01\textwidth}
\begin{subfigure}{0.11\textwidth}
\centering
\includegraphics[width=.99\linewidth]{imgs/d.png}
\caption{}
\label{fig: segmentation}
\end{subfigure}
\hspace{0.01\textwidth}
\begin{subfigure}{0.11\textwidth}
\centering
\includegraphics[width=.99\linewidth]{imgs/e.png}
\caption{}
\label{fig: instance}
\end{subfigure}
\hspace{0.02\textwidth}
\vspace{-0.0cm}
\caption{Qualitative decomposition and segmentation results on the Tetrominoes dataset. Last row shows a failure case. \textbf{(a)}: Original image. \textbf{(b)}: PCDNet Reconstruction. \textbf{(c)-(e)}: Colorized transformed object prototypes. \textbf{(f)}: Semantic segmentation masks. Colors correspond to the prototype frames in \Figure{fig: tetris}. \textbf{(g)}: Instance segmentation masks.}
\label{fig: qualitative tetris}
\end{figure*}
\subsection{Tetrominoes Dataset}
\label{subsec: Image Decomposition}
We evaluate PCDNet for image decomposition and object discovery on the Tetrominoes dataset~\cite{Greff_IodineMultiObjectRepresentationLearningWithIterativeVariationalInference_2019}. This dataset contains 60.000 training images and 320 test images of size $35 \times 35$, each composed of three non-overlapping Tetris-like sprites over a black background. The sprites belong to one out of 19 configurations and have one of six random colors.
\Figure{fig: tetris} displays the 19 learned object prototypes and their corresponding alpha masks from the Tetrominoes dataset. We clearly observe how PCDNet accurately discovers the shape of the different pieces and their tiled texture.
\Figure{fig: qualitative tetris} depicts qualitative results for unsupervised object detection and segmentation.
In the first three rows, PCDNet successfully decomposes the images into their object components and precisely segments the objects into semantic and instance masks. The bottom row shows an example in which the greedy selection algorithm leads to a failure case.
For a fair quantitative comparison with previous works, we evaluate our PCDNet model for object segmentation using the Adjusted Rand Index~\cite{Hubert_ComparingPartitionsARI_1985} (ARI) on the ground truth foreground pixels. ARI is a clustering metric that measures the similarity between two set assignments, ignoring label permutations, and ranges from 0 (random assignment) to 1 (perfect clustering).
We compare the performance of our approach with several existing methods: Slot MLP and Slot Attention~\cite{Locatello_ObjectCentricLearningWithSlotAttention_2020}, IODINE~\cite{Greff_IodineMultiObjectRepresentationLearningWithIterativeVariationalInference_2019} and Unsupervised Layered Image Decomposition~\cite{Monnier_UnsupervisedLayeredImageDecompositionIntoObjectPrototypes_2021} (ULID).
\Table{table: ari eval} summarizes the evaluation results for object discovery on the Tetrominoes dataset. From \Table{table: ari eval}, we see how PCDNet outperforms SOTA models, achieving 99.7\% ARI on the Tetrominoes dataset. PCDNet uses only a small percentage of learnable parameters compared
to other methods (e.g., only 6\% of the parameters from IODINE), and has the highest inference
throughput (images/s). Additionally, unlike other approaches, PCDNet obtains disentangled representations for the object appearance, position, and color in a human-interpretable manner.
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth]{imgs/prototypes_atari.png}
\caption{Object prototypes learned on the Space Invaders dataset. PCDNet discovers prototypes corresponding to the different elements from the game, e.g., aliens and ships.}
\label{fig:prototypes atari}
\end{figure}
\subsection{Space Invaders Dataset}
\label{subsec: Space Invaders Dataset}
In this experiment, we use replays from humans playing the Atari game \emph{Space Invaders}, extracted from the Atari Grand Challenge dataset~\cite{Kurin_TheAtariGranChallengeDataset_2017}.
PCDNet is trained to decompose the Space Invaders images into 50 objects, belonging to one of 14 learned prototypes of size $20 \times 20$.
\Figure{fig:comp atari} depicts a qualitative comparison between our PCDNet model with SPACE~\cite{Lin_UnsupervisedObjectOrientedSceneRepresentationViaSpatialAttentionAndDecomposition_2020} and Slot Attention~\cite{Locatello_ObjectCentricLearningWithSlotAttention_2020}.
Slot Attention achieves an almost perfect reconstruction of the input image. However, it fails to decompose the image into its object components, uniformly scattering the object representations across different slots. In \Figure{fig:comp atari} (subplot labeled as \emph{Slot}) we show how one of the slots simultaneously encodes information from several different objects.
SPACE successfully decomposes the image into different object components, which are recognized as foreground objects. Nevertheless, the reconstructions appear blurred and several objects are not correct.
PCDNet achieves the best results among all compared methods. Our model successfully decomposes the input image into accurate object-centric representations. Additionally, PCDNet learns semantic understanding of the objects.
\Figure{fig:comp atari} depicts a segmentation of an image from the Space Invaders dataset.
Further qualitative results on the Space Invaders dataset are reported in \Appendix{section: further qualitative results}.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.91\linewidth]{imgs/comp_atari.png}
\caption{Comparison of different object-centric models on the Space Invaders dataset. PCDNet is the only one among the compared methods which successfully decomposes the image into accurate object components, and that has semantic knowledge of the objects. The color of each object corresponds to the frame of the corresponding prototype in \Figure{fig:prototypes atari}.}
\label{fig:comp atari}
\end{figure*}
\begin{figure*}[h!]
\centering
\includegraphics[width=0.91\linewidth]{imgs/CarExps.png}
\caption{Object discovery on the NGSIM dataset. PCDNet learns different vehicle prototypes in an unsupervised manner.}
\label{fig:cars}
\end{figure*}
\subsection{NGSIM Dataset}
\label{subsec: NGSIM Dataset}
In this third experiment, we apply our PCDNet model to discover vehicle prototypes from real traffic camera footage from the Next Generation Simulation (NGSIM) dataset~\cite{NGSIM_Dataset}.
We decompose each frame into up to 33 different objects, belonging to one of 30 learned vehicle prototypes.
\Figure{fig:cars} depicts qualitative results on the NGSIM dataset.
We see how PCDNet is applicable to real-world data, accurately reconstructing the input image, while learning prototypes for different types of vehicles.
Interestingly, we notice how PCDNet learns the car shade as part of the prototype. This is a reasonable observation, since the shades are projected towards the bottom of the image throughout the whole video.
\section{Conclusion}
We proposed PCDNet, a novel image composition model that decomposes an image, in a fully unsupervised manner, into its object components, which are represented as transformed versions of a set of learned object prototypes.
PCDNet exploits the frequency-domain representation of images to estimate the translation parameters that best align the prototypes to the objects in the image.
The structured network used by PCDNet allows for an interpretable image decomposition, which disentangles object appearance, position and color without any external supervision.
In our experiments, we show how our proposed model outperforms existing methods for unsupervised object discovery and segmentation on a benchmark synthetic dataset, while significantly reducing the number of learnable parameters, having a superior throughput, and being fully interpretable. Furthermore, we also show that the PCDNet model can also be applied for unsupervised prototypical object discovery on more challenging synthetic and real datasets.
We hope that our work paves the way towards further research on phase correlation networks for unsupervised object-centric representation learning.
\section*{ACKNOWLEDGMENTS}
This work was funded by grant BE 2556/18-2 (Research Unit FOR 2535
Anticipating Human Behavior) of the German Research Foundation (DFG).
\bibliographystyle{apalike}
{\small
|
2,869,038,156,360 | arxiv | \section{Introduction}\label{sec::introduction}
The new generation of wireless technology has proliferated a large amount of connected devices in \textit{Internet of Thing} (IoT) networks~\cite{Liu2018a}.
\change{Modern IoT networks allow only sporadic communication, where} a small unknown subset of devices \change{is allowed} to be active at any given instant.
\change{Because} it is infeasible to assign orthogonal signature sequences to all \change{devices and because} the channel coherence time is limited in large-scale IoT networks, detecting active devices and estimating their \textit{Channel State Information} (CSI) \change{is the key} to \change{improving} communication efficiency.
Recently, \change{sparse} signal processing techniques have been proposed to support massive connectivity in IoT networks by exploiting the sparsity pattern over the devices~\cite{Chen2018,Liu2018b,Liu2018c}.
\change{This} sparsity pattern \change{can be exploited by using} a high dimensional group \textit{Least absolute shrinkage and selection operator} (Lasso) formulation~\cite{He2018,Jiang2019}.
In~\cite{Chen2018}, approximate message passing based approaches have been developed for joint channel estimation and user activity detection with non-orthogonal signature sequences. \change{By conducting a rigorous performance analysis},~\cite{Liu2018b,Liu2018c} showed that the probabilities of the missed device detection and the false alarm \change{is close} to zero \change{under mild assumptions}. \change{Moreover, a} joint user detection and channel estimation method \change{for cloud radio access networks} was developed in~\cite{He2018} \change{by using} various optimization methods. The trade-off between the computational cost and estimation accuracy was further \change{analyzed} in~\cite{Jiang2019} \change{by using methods from the field of} conic geometry.
\change{High} dimensional group Lasso \change{problems pose} significant computational challenges\change{, because} a fast and tailored numerical algorithm is essential to meet \change{real-time requirements}. \change{Since} first-order methods \change{have a low complexity per iteration, these} methods \boris{have been} investigated exhaustively \change{for solving} group Lasso problems. \change{For instance,} in~\cite{Jiang2019}, a primal-dual gradient method \change{has been used} to solve \change{this} problem \change{achieving an} improved convergence rate based on smoothing techniques. An \change{alternative to this is to use the}~\textit{Fast Iterative Shrinkage-Thresholding Algorithm} (FISTA)~\cite{Beck2009}, which \change{does}, however, \change{not fully exploit the distributed structure}.
Another option is \change{to apply} \textit{Proximal Gradient} (ProxGradient) method\change{s}~\cite{Parikh2014}, which can distribute the main iteration but \change{require} a centralized line search procedure. \boris{Moreover},~\cite{He2018} applied the \textit{Alternating Direction Method of Multipliers} (ADMM)~\cite{Boyd2011} to solve the \textit{Joint Activity Detection and Channel Estimation} (JADCE) problem in cloud radio access network. By exploiting the distributed structure, the numerical simulations \change{show} that ADMM can reduce the computational cost significantly. However, because \change{ADMM is not invariant under scaling,} it is advisable to apply a pre-conditioner, as the iterates may converge slowly otherwise.
In this letter, we \change{focus} on reducing the computational costs for solving the sparse signal processing problem \change{for} massive device connectivity. \change{Our goal is} to develop a simple and efficient decomposition method that converges fast and reliably to a minimizer \change{of} the group Lasso problems. \boris{In detail, we analyze a tailored version of the recently proposed \textit{Augmented Lagrangian based Alternating Direction Inexact Newton} (ALADIN) method}, which has originally been developed for solving distributed non-convex optimization problems~\cite{Houska2016}. \boris{Extensive} numerical results demonstrate that the proposed method \boris{outperforms ADMM, ProxGradient, and FISTA} in terms of converge speed and running time.
\yuning{
\textbf{Notation} For a given symmetric and positive definite matrix, $\Sigma\succeq0$, the notation $\|x\|_{\Sigma}^2 = x^\top \Sigma x$
is used. The Kronecker product of two matrices $A \in \mathbb{C}^{k \times l}$ and $B \in \mathbb{C}^{m \times n}$ is denoted by $A \otimes B$ and $\mathrm{vec}(A)$ denotes the vector that is obtained by stacking all columns of $A$ into one long vector. The reverse operation is denoted by $\mathrm{mat}$, such that $\mathrm{mat}(\mathrm{vec}(A)) = A$. The $(n\times n)$-unit matrix is denoted by~$\mathbb I_n$. Moreover, the notation $A^\herm = \Real(A)^\top - \mathrm{i}\, \Imag(A)^\top$ with $\mathrm{i} = \sqrt{-1}$ is used to denote the Hermitian transpose of $A$.}
\section{System Model and Problem Formulation}
\label{sec::model}
This section concerns an IoT network with single \textit{Base Station} (BS) supporting $N$ devices. It is assumed that the BS is equipped with $M$ antennas and each device has only one antenna. \yuning{The BS receives the signal
\begin{equation}\label{eq::linear_model_1}
Y \;=\; QSH + \Omega
\end{equation}
during the uplink transmission in $L$ coherence blocks.
Here, $Y\in\mathbb{C}^{L\times M}$ denotes the received signal matrix, $H\in\mathbb{C}^{N\times M}$ the associated channel matrix, and $Q\in\mathbb{C}^{L\times N}$ a given signature matrix. The rows of the additive noise $\Omega\in\mathbb{C}^{L\times M}$ have i.i.d.~Gaussian distributions with zero means. Moreover, the activity matrix $S\in\mathbb{S}_+^{N}$ is a diagonal matrix with $S_{i,i} = 1$ if the $i$-th device is active, but $S_{i,i}=0$ if the $i$-th device is inactive.}
The goal of JADCE is to estimate the channel matrix $H$ and detect the activity matrix $S$.
Let us introduce the vectorized optimization variable,
\[
x = \mathrm{vec} \left( \, \left[ \Real(SH) \; , \; \Imag(SH) \right]^\top \, \right) \in \mathbb R^{2MN} \; ,
\]
which stacks the real and imaginary parts of the matrix $SH$ row-wise into a vector $x$. This has the advantage that the JADCE problem can be written in the form of a Group Lasso problem,
\begin{equation}\label{eq::groupLasso}
\min_x\;\;\frac{1}{2}\Vert Ax - b \Vert_2^2 + \gamma\|x\|_{2,1}\;\;\text{with}\;\;\|x\|_{2,1} = \sum_{i=1}^{N}\|x_i\|_2\; .
\end{equation}
Here, $x_i \in \mathbb R^{2M}$ denotes the $i$-th subblock of $x$, such that we can write
$$x = [x_1^\top, \ldots, x_N^\top ]^\top .$$
Consequently, the matrix $A\in\mathbb{R}^{2LM\times 2MN}$ and the vector $b\in\mathbb{R}^{2LM}$ are given by
\begin{align}
A=\;& \Real(Q) \otimes
\begin{bmatrix}
\mathbb I_M & 0\\
0 & \mathbb I_M
\end{bmatrix}
+ \Imag(Q) \otimes
\begin{bmatrix}
0 & -\mathbb I_M \\
\mathbb I_M & 0
\end{bmatrix}
\label{eq::A} \\[0.1cm]
\text{and }\; b =\;& \mathrm{vec} \left( \, \left[ \Real(Y) \; , \; \Imag(Y) \right]^\top \, \right) \; . \label{eq::b}
\end{align}
Problem~\eqref{eq::groupLasso} is \change{a} non-differentiable \change{optimization problem for which} classical \change{sub-gradient methods often exhibit a rather} slow convergence. As discussed in Section~\ref{sec::introduction}, several first-order methods have been applied to solve~\eqref{eq::groupLasso} such as FISTA~\cite{Beck2009}, ProxGradient~\cite{Parikh2014} and ADMM~\cite{Boyd2011,He2018}. However, in order to achieve \change{fast} convergence, these methods require either a centralized step such as \change{the} line search \change{routine} in \change{the} ProxGradient method or a \change{pre-conditioner that scales all} variables in advance.
\change{In order to mitigate these issues, the following section develops a tailored ALADIN algorithm for solving~\eqref{eq::groupLasso}.}
\section{Algorithm}\label{sec::algorithm}
\change{This section develops a tailored variant of ALADIN for
solving the group Lasso problem~\eqref{eq::groupLasso}.}
\subsection{Augmented Lagrangian based Alternating Direction Inexact Newton Method}
\boris{The goal of} ALADIN \boris{is} to solve distributed optimization \boris{problems} \change{of the form}
\begin{equation}\label{eq::distProb}
\min_{z}\;\;\sum_{i=0}^{N}f_i(z_i)\quad \text{s.t.}\;\;\sum_{i=0}^{N}C_iz_i = d\qquad \mid\lambda\;,
\end{equation}
where the objectives, $f_i$, are convex functions with closed \boris{epigraphs.} \change{The matrices $C_i$ and the vector $d$ can be used to model the coupling constraint}. \yuning{Here, the notation $|\,\lambda$ behind the affine constraint is used to say that the multiplier of this constraint is denoted by $\lambda$.}
\begin{algorithm}[H]
\caption{\change{ALADIN}}
\textbf{Input:}
\begin{itemize}
\item Initial guess $(\boris{z^0},\lambda^0)$, tolerance $\epsilon>0$ and \yuning{symmetric} scaling matrices $\Sigma_i\succ 0$ for all $\change{i \in \{ 0,1,\ldots,N \}}$.
\end{itemize}
\textbf{Initialization:}
\begin{itemize}
\item Set $k=0$.
\end{itemize}
\textbf{Repeat:}
\begin{enumerate}
\item \textit{Parallelizable Step:} Solve
\[
\xi_i^k =\text{arg}\min_{\xi_i}\;\;f_i(\xi_i) + (C_i^\top\lambda^{k})^\top\xi_i + \frac{1}{2}\Vert\xi_i -z_i^k \Vert_{\Sigma_i}^2
\]
and evaluate
\begin{equation}\label{eq::gradient}
g_i = \Sigma_i(z_i^k-\xi_i^k) - C_i^\top\lambda^{k}
\end{equation}
for all $i \in \{ 0,1,...,N \}$ in parallel.
\item Terminate if $\max_{i}\;\|\xi_i^k-z_i^k\|\leq \epsilon$.
\item \textit{Consensus Step:} Solve
\begin{equation}\label{eq::cQP}
\begin{split}
z^{k+1}=\text{arg}\min_{z}&\;\;\sum_{i=0}^{N}\left(\frac{1}{2}\Vert z_i-\xi_i^k\Vert_{\Sigma_i}^2 + g_i^\top z_i\right)\\
\text{s.t.}&\;\;\sum_{i=0}^NC_iz_i = d \qquad \mid \change{\lambda^{k+1}}\; ,
\end{split}
\end{equation}
and set $k \leftarrow k+1$.
\end{enumerate}
\label{alg::aladin}
\end{algorithm}
\yuning{
Algorithm~\ref{alg::aladin} outlines a tailored version of ALADIN~\cite{Houska2016} for solving~\eqref{eq::distProb}. The algorithm also has two main steps, a parallelizable step and a consensus step.}
The parallelizable Step~1) solves $N+1$ \boris{small-scale unconstrained optimization} problems and \boris{computes the vectors $g_i$} in parallel. \change{Here, $g_i$ is, by construction, an} element of the subdifferential of $f_i$ at $\xi_i^k$\change{,}
\[
0 \in \partial f_i(\xi_i^k) + C_i^\top\lambda^{k} + \Sigma_i(\xi_i^k -z_i^k)\; \Rightarrow\; g_i\in\partial f_i(\xi_i^k)\;.
\]
If the termination criterion in Step~2) is satisfied, we have
\[
-C_i^\top \lambda^k \in \partial f_i(\xi_i^k) + \mathcal{O}(\epsilon)\;,\;i=0,1,...,N\;.
\]
Moreover, the \change{particular construction of the consensus QP~\eqref{eq::cQP} ensures that the} iterates $z^k$ are feasible \change{and}
\[
\sum_{i=0}^N C_i z^k_i = b \;\Rightarrow\;\sum_{i=0}^N C_i \xi^k_i - b = \mathcal{O}(\epsilon)
\]
upon termination\change{. This implies that} $\xi^k$ satisfies the stationarity and primal feasibility condition of~\eqref{eq::distProb} up to a small error of order $\mathcal{O}(\epsilon)$.
\boris{Notice that both the primal and the dual iterates, $(z^k,\lambda^k)$, are updated in Step~3) before the iteration continuous.}
\change{\boris{Because~\eqref{eq::distProb} is a convex optimization problem, the set of primal solutions~\cite{Boyd2004}} is} given by
\[
\mathbb{X}^* = \left\{
z\left|
\exists \, \lambda^*\in\mathbb{R}^{n_c}\,:
\;
\begin{split}
-\sum_{i=0}^{N}C_i^\top\lambda^*\in&\partial\left(\sum_{i=0}^{N}f_i(z_i) \right)\\
\sum_{i=0}^{N}C_iz_i^*=& d
\end{split}
\right.
\right\}\; \boris{,}
\]
\boris{where $n_c$ denotes the number of coupled equality constraints in~\eqref{eq::distProb}.}
\change{Theorem~\ref{thm:convex} summarizes an important convergence guarantee for Algorithm~\ref{alg::aladin}.}
\begin{theorem}
\label{thm:convex}
\change{If Problem~\eqref{eq::distProb} is feasible and if strong duality holds for~\eqref{eq::distProb},
then the iterates of Algorithm 1 converge globally to $\mathbb{X}^*$,}
\[
\lim_{k\to\infty}\underset{z\in\mathbb{X}^*}{\min}\;\;\|\xi^k-z\| \;=\; 0 \;.
\]
\end{theorem}
\noindent
A complete proof of Theorem~\ref{thm:convex} can be found in~\cite{Houska2017}. \boris{Notice that Algorithm~\ref{alg::aladin} is not invariant with respect to scaling, but the statement of the above theorem holds for any choice of the positive definite matrices $\Sigma_i \succ 0$.}
\subsection{\yuning{ALADIN for Group Lasso}}
In order to apply Algorithm~\ref{alg::aladin} for solving~\eqref{eq::groupLasso}, we split $A$ into $N$ \yuning{column} blocks, $A =[A_1,\ldots,A_N]$, \final{where each block $A_i$ contains the coefficients with respect to $x_i$. Problem~\eqref{eq::groupLasso} can be rewritten in the group distributed form}
\yuning{
\begin{equation}\label{eq::reGroupLasso}
\min_{z} \; \frac{1}{2} \left\| z_0 - b \right\|_2^2 + \gamma\sum_{i=1}^N \| z_i \|_2\quad \text{s.t.} \;z_0 -\sum_{i=1}^{N}A_i z_i = 0\, \final{,}
\end{equation}
\final{where the auxiliary variable $z_0$ is used to reformulate the affine consensus constraints.}
Now,~\eqref{eq::groupLasso} can be written in the form of~\eqref{eq::distProb} by setting\yuning{
\[
\begin{array}{rclrclrcl}
f_0(z_0) &=& \frac{1}{2}\|z_0-b\|_2^2 \; , & C_0 &=& \mathbb I_{2LM}\;, & d &=& 0\;,\\[0.12cm]
f_i(z_i) &=& \gamma\|z_i\|_2 \; , & C_i &=& -A_i
\end{array}
\]
for all $i\in\{ 1,...,N \}$.} Because this optimization problem is feasible and because strong duality trivially holds for this problem, Algorithm~\ref{alg::aladin} can be applied and Theorem~\ref{thm:convex} guarantees convergence. \yuning{
In this implementation we set
\[
\Sigma_0 = \mathbb I_{2LM} \quad \text{and} \quad \Sigma_i= \rho\, \mathbb I_{2M}
\]
for all $i \in \{ 1, \ldots, N \}$. Here, $\rho > 0$ denotes a tuning parameter.} Step~1 can be implemented by using a soft-thresholding operator $\mathcal{S}_\kappa: \mathbb{R}^{2M}\rightarrow\mathbb{R}^{2M}$,
\[
\mathcal{S}_\kappa(a) = \max (1-\kappa/\|a\|_2, 0)\cdot a \; ,
\]
which allows us to write Step~1 in the form
\yuning{
\begin{subequations}
\begin{align}\label{eq::localX0}
\xi_0^{k} = \;& \frac{1}{2}(z_0^k+b-\lambda^k)\;,\\\label{eq::localXi}
\xi_{i}^{k}=\;& \mathcal{S}_{\gamma/\rho}(z_i^k + A_{i}^\top \lambda^k/\rho)\;,\;i=1,...,N.
\end{align}
\end{subequations}
As elaborated in Algorithm~\ref{alg::aladin}, the coupled QP in Step~3 has only affine equality constraints. This means that its parametric solution can be worked out explicitly. To this end, we write out the KKT conditions of~\eqref{eq::cQP} as\yuning{
\begin{subequations}
\begin{align}\label{eq::KKTcQP1}
z_0^{k+1}\;=\;& b-\lambda^{k+1}\;,\\\label{eq::KKTcQP2}
z_i^{k+1}\;=\;& 2\xi_i^k - z_i^k +A_i^\top\Delta\lambda^{k+1}/\rho\;,\;i\in\{1,...,N\}\\\label{eq::KKTcQP3}
z_0^{k+1}\;=\;& \sum_{i=1}^{N}A_iz_i^{k+1}
\end{align}
\end{subequations}
with $\Delta \lambda^{k+1}=\lambda^{k+1}-\lambda^k$. Here, we have substituted the explicit expression~\eqref{eq::gradient} for $g_i$.} Combining~\eqref{eq::KKTcQP1} with~\eqref{eq::localX0} yields
$\xi_0^k = z_0^k$ for all $k\in\mathbb{N}_{\geq 1}$, which implies that the update of $\xi_0^k$ can be omitted from the iterations in Algorithm~\ref{alg::aladin}.
\yuning{Next,~\eqref{eq::KKTcQP2} and~\eqref{eq::KKTcQP3} can be resorted, which yields
\begin{equation}\label{eq::solcQP}
\Delta \lambda^{k+1} \;=\;2 \Lambda^{-1}\left(\sum_{i=1}^{N}A_i(z_i^k-\xi_i^{k})\right)\;.
\end{equation}
Here, the inverse of matrix $\Lambda = \mathbb I_{2ML} + AA^\top/\rho$ can be worked out explicitly by substituting~\eqref{eq::A}. For this aim, we introduce the shorthands
\begin{align}
\Lambda_1 =\;& \left( \rho\, \mathbb I_L + \Real(Q Q^\herm) \right) \otimes
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix} \\[0.1cm]
\text{and} \quad \Lambda_2 =\;& \Imag(Q Q^\herm) \otimes
\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}
\end{align}
such that the inverse can be written in the form
\begin{eqnarray}
\label{eq::LambdaInv}
\Lambda^{-1} = \rho \left[ \Lambda_1 + \Lambda_2 \right]^{-1} \otimes \mathbb I_M \; .
\end{eqnarray}
This implies that the matrix $\Lambda^{-1}$ does not have to be computed directly, but it is sufficient to pre-compute the inverse of the $(2L \times 2L)$-matrix $\Lambda_1 + \Lambda_2$. This is possible with reasonable computational effort, because modern IoT networks often consist of a large number of devices but have a limited number of channel coherence blocks, i.e., $L\ll N$.
An assoiated tailored version of ALADIN for solving~\eqref{eq::groupLasso} can be written in the following form:
\begin{equation}
\begin{split}\label{eq::aladin}
\begin{array}{c}
\textsc{\small parallel}\\
\textsc{\small step}
\end{array}\;&\left\{\begin{aligned}
z_i^{k+1} & = \xi_{i}^{k} + A_i^\top \Delta \lambda^{k}/\rho +\final{(\xi_{i}^{k}-z^k_i)} \\[0.12cm]
\lambda^{k+1} &= \lambda^k + \Delta \lambda^k \\[0.12cm]
k &\leftarrow k+1 \\[0.12cm]
\xi_{i}^{k}&= S_{\gamma/\rho}(z_i^{k}+ A_{i}^\top \lambda^{k}/\rho)\\[0.12cm]
w_i^{k}&=A_i(z_i^{k}-\xi_{i}^{k} )
\end{aligned} \right.\\
\begin{array}{c}
\textsc{\small consensus}\\
\textsc{\small step}
\end{array}& \;\;
\Delta \lambda^{k} = 2\Lambda^{-1}\left(\sum_{i=1}^{N}w_i^{k} \right)\;.
\end{split}
\end{equation}
An implementation of the consensus step in~\eqref{eq::aladin} requires each agent to send vectors $w_i^k$ to the \textit{Fusion Center} (FC)\boris{, which computes the vector $\Delta \lambda^k$ and sends it back it to the agents. The solution $(z_i^{k+1},\lambda^{k+1})$ of the QP can then be computed in parallel by using the result for $\Delta \lambda$. Notice that this means that the running index $k$ must be updated after $(z_i^{k+1},\lambda^{k+1})$ are computed. Notice that the consensus step can be implemented by substituting~\eqref{eq::LambdaInv}, which yields
\[
2 \Lambda^{-1} \left( \sum_{i=1}^{N} w_i^{k} \right) =
2 \rho \, \mathrm{vec} \left(
\mathrm{mat} \left( \sum_{i=1}^{N} w_i^{k} \right) \left[ \Lambda_1^\top + \Lambda_2^\top \right]^{-1} \right) .
\]
Thus, the consensus step has the computational complexity $\mathcal{O}( L^2 M + LMN)$ assuming that the matrix $\left[ \Lambda_1^\top + \Lambda_2^\top \right]^{-1}$ has already been precomputed. The complexity of all other steps is $\mathcal O(LMN)$, as one can exploit the sparsity pattern of the matrix $A$, as given in~\eqref{eq::A}.
}
The above tailored ALADIN algorithm can be compared to an associated tailored version of ADMM~\cite{He2018,Boyd2011} for solving~\eqref{eq::reGroupLasso}, \final{given by}
\begin{equation}\label{eq::admm}
\begin{split}
\begin{array}{c}
\textsc{\small parallel}\\
\textsc{\small step}
\end{array}\;&\left\{\begin{aligned}
z_i^{k+1} & = \xi_{i}^{k} + A_i^\top \Delta \lambda^{k}/\rho \\[0.12cm]
\lambda^{k+1} &= \lambda^k + \Delta \lambda^k \\[0.12cm]
k &\leftarrow k+1 \\[0.12cm]
\xi_{i}^{k}&= S_{\gamma/\rho}(z_i^{k}+ A_{i}^\top \lambda^{k}/\rho)\\[0.12cm]
w_i^{k}&=A_i(z_i^{k}-\xi_{i}^{k} )
\end{aligned} \right.\\
\begin{array}{c}
\textsc{\small consensus}\\
\textsc{\small step}
\end{array}& \;\;
\Delta \lambda^{k} = \Lambda^{-1}\left(\sum_{i=1}^{N}w_i^{k} \right)\;.
\end{split}
\end{equation}
Notice that the ADMM iteration~\eqref{eq::admm} and the ALADIN iteration~\eqref{eq::aladin} coincide except for the update of the variable $z_i^{k+1}$, \final{where ALADIN introduces the additional term $\xi_i^k-z_i^k$. Intuitively, one could interpret this terms as a momentum term---similar to Nesterov's momentum term used in traditional gradient methods~\cite{Nesterov2018}, which can help to speed up convergence.}
Consequently, both methods have the same computational complexity per iteration. However, in the following, we will show---by a numerical comparison of these two methods---that ALADIN converges, on average, much faster than ADMM.
\section{Numerical Results}\label{sec::results}
This section illustrates the numerical performance of the proposed algorithm. We randomly \boris{generate} \change{problem instances \boris{in the form of~\eqref{eq::groupLasso}} by analyzing a scenario for which} the BS in \change{the} IoT network is equipped with $M=100$ antennas\change{. The} number of devices is set to \yuning{$N=2000$}. \change{Additionally}, we fix the number of active device \change{to} \yuning{$50$}\change{. The} signature sequence length \change{is set to} $L=10$. The signature matrix $Q$ and \boris{additive} noise matrix $\Omega$ are dense with entries drawn from \change{normal distributions} with covariance matrices $I$ and $0.01I$, respectively. \yuning{Similar to~\cite{Boyd2011}, we set $\gamma=0.5\gamma_{\max}$ with
\[
\gamma_{\max} = \max_{i} \; \|A_i^\top b\|_2 > 0 \; .
\]
\yuning{We set $\rho = 0.8\gamma$ for both ALADIN and ADMM}. All implementations use \texttt{MATLAB 2018b} with Intel Core i7-8550U [email protected], 4 Cores.
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.89\linewidth]{Jiang_WCL2020-0304_R1_fig1}
\caption{Comparison of \change{the} convergence behavior \change{of ADMM, ALADIN, ProxGradient, and FISTA.}}
\label{fig::Convergence1}
\end{figure}
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.89\linewidth]{Jiang_WCL2020-0304_R1_fig2}
\caption{Comparison of \change{the} scalability \change{of ADMM, ALADIN, ProxGradient, and FISTA.}}
\label{fig::Convergence2}
\end{figure}
We compare ALADIN with \change{three existing} methods: ADMM~\cite{Boyd2011}, FISTA~\cite{Beck2009} and ProxGradient~\cite{Parikh2014}.
Figure~\ref{fig::Convergence1} shows the convergence performance comparison for a randomly generated problem, which indicates the superior performance of the proposed method.
\yuning{Additionally, Figure~\ref{fig::Convergence2} shows the average number of iterations all four methods versus $N$, all for a large number of randomly generated problems (over $1000$). Here, the termination tolerance has been set to $10^{-5}$. Moreover, the average run-times of ALADIN and ADMM per iteration are listed in Table~\ref{tab::runtime} (for $N=2000$). In summary, one may state that, if the termination tolerance is set to $10^{-5}$ and all parameters are set as stated above, ALADIN converges approximately five times faster than ADMM, six time faster than FISTA and eight times faster than ProxGradient taking into account that all of these methods have the same computational complexity per iteration, $\mathcal O(LMN)$, as long as $L \leq N$.}
\begin{table}[htbp!]
\centering
\setlength{\tabcolsep}{4pt
\renewcommand{\arraystretch}{1.2
\caption{\label{tab::runtime} Run-time of ADMM and ALADIN per iteration.}
\begin{tabular}{lcccc}
\toprule
&\multicolumn{2}{c}{ADMM} & \multicolumn{2}{c}{ALADIN} \\
\midrule
One iteration & 0.077 [s] & 100\% & 0.083 [s] & 100\% \\[0.1cm]
Parallel step & 0.043 [s] & 55.8\% & 0.053 [s] & 63.4\% \\ [0.1cm]
Consensus step & 0.034 [s] & 44.2\% & 0.030 [s] & 36.6 \% \\
\bottomrule
\end{tabular}
\end{table}
\section{Conclusion}\label{sec::conclusion}
In this letter, we proposed a \boris{tailored version
of the Augmented Lagrangian based Alternating Direction Inexact Newton \mbox{(ALADIN)} method} for \change{enabling massive connectivity} in \change{an IoT network}, \change{by solving} group Lasso \change{problems in a distributed manner}. \change{Theorem~\ref{thm:convex}} summarized \change{a general} global convergence guarantee \change{for ALADIN in the context of solving group Lasso problems}. \change{The highlight of this paper, however, is the illustration of} performance of the proposed method\change{. Here,} we compared \change{ALADIN} with three \change{state-of-the-art} algorithms. \change{Our} numerical results \change{indicate} that ALADIN outperforms \change{all other tested methods} in terms of \change{overall run-time by about a factor five}.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
2,869,038,156,361 | arxiv | \section{Introduction}
Research on the context of the Hawking temperature has gained momentum during last two decades.
It has been shown that the Hawking temperature \cite{haw} is modified in the presence of dark energy in an emergent gravity scenario for Schwarzschild, Reissner-Nordstrom and Kerr background in \cite{gm1,gm2}. As seen in \cite{gm1,gm2}, for an emergent gravity metric $\tilde G_{\mu\nu}$ is conformally transformed into $\bar G_{\mu\nu}$ where $\bar G_{\mu\nu}= g_{\mu\nu} - \partial _{\mu}\phi\partial_{\nu}\phi$ ($g_{\mu\nu}$ is the gravitational metric) for Dirac-Born-Infeld(DBI) \cite{born} type Lagrangian having $\phi$ as $k-$essence scalar field. The Lagrangian for $k-$essence scalar fields contains non-canonical kinetic terms.
The general form of the Lagrangian for $k-$essence model is: $L=-V(\phi)F(X)$ where $X=\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$ and it does not
depend explicitly on $\phi$ to start with \cite{gm1,gm2,babi,scherrer}.
Relativistic field theories with canonical kinetic terms have the distinction from those with non-canonical kinetic terms associated with $k-$essence, since the nontrivial dynamical solutions of the k-essence equation of motion not only spontaneously break Lorentz invariance but also change the metric for the perturbations around these solutions. Thus the perturbations propagate in the so called {\it emergent} or analogue curved spacetime \cite{babi} with the metric different from the gravitational one. Relevant literatures \cite{gorini} for such fields discuss about cosmology, inflation, dark matter, dark energy and strings.
The motivation of this work is to calculate the Hawking temperature in the presence of dark energy for an emergent gravity metric which is also a blackhole metric. We consider two cases, (a) when the gravitational metric is a Kerr-Newman and (b) when the gravitational metric Kerr-Newman-AdS.
In \cite{umetsu}-\cite{aliev}, discuss about Hawking radiation for Kerr, Kerr-Newman, Kerr-Newman-AdS etc. black holes using different techniques. Here we calculate the Hawking temperature for emergent gravity metric for Kerr-Newman and Kerr-Newman-AdS backgrounds using tunneling mechanism. These temperatures are different from usual temperatures of Kerr-Newman and Kerr-Newman-AdS black holes.
In section 2, we have described $k-$essence and emergent gravity where the metric $\tilde G_{\mu\nu}$ contains the
dark energy field $\phi$ and this field should satisfy the emergent gravity equations of motion.
Again, for $\tilde G_{\mu\nu}$ is to be a blackhole metric, it has to satisfy the Einstein field equations. The formalism for $k-$essence and emergent gravity used is as described in \cite{babi}.
In section 3 and 5, we have shown that for Kerr-Newman and Kerr-Newman-AdS both cases, the emergent gravity metrics are mapped on to the Kerr-Newman and Kerr-Newman-AdS type metrics in the presence of dark energy. The emergent metric satisfies Einstein equations for large $r$ and the dark energy field $\phi$ satisfies the emergent gravity equations of motion for along $\theta=0$ at $r\rightarrow\infty$.
We have calculated the Hawking temperature for emergent gravity metrics for Kerr-Newman and Kerr-Newman-AdS backgrounds in section 4 and 6 respectively. We clarify that the Hawking temperature is spherically symmetric from very general conditions and taking $\theta=0$ does not therefore affect this property of the Hawking temperature. It has been shown elaborately in \cite{mann}, how the Hawking temperature is independent of $\theta$, although the metric functions depend on $\theta$. Also Hawking temperature is purely horizon phenomenon of the spacetime where the temperature is not depending on $\theta$. So we can say that the Hawking temperature is spherically symmetric.
\section{$k-$essence and Emergent Gravity}
The $k-$essence scalar field $\phi$ minimally coupled to the gravitational field $g_{\mu\nu}$ has action \cite{babi}
\begin{eqnarray}
S_{k}[\phi,g_{\mu\nu}]= \int d^{4}x {\sqrt -g} L(X,\phi)
\label{eq:1}
\end{eqnarray}
where $X={1\over 2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$.
The energy momentum tensor is
\begin{eqnarray}
T_{\mu\nu}\equiv {2\over \sqrt {-g}}{\delta S_{k}\over \delta g^{\mu\nu}}= L_{X}\nabla_{\mu}\phi\nabla_{\nu}\phi - g_{\mu\nu}L
\label{eq:2}
\end{eqnarray}
$L_{\mathrm X}= {dL\over dX},~~ L_{\mathrm XX}= {d^{2}L\over dX^{2}},
~~L_{\mathrm\phi}={dL\over d\phi}$ and
$\nabla_{\mu}$ is the covariant derivative defined with respect to the gravitational metric $g_{\mu\nu}$.
The equation of motion is
\begin{eqnarray}
-{1\over \sqrt {-g}}{\delta S_{k}\over \delta \phi}= \tilde G^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi +2XL_{X\phi}-L_{\phi}=0
\label{eq:3}
\end{eqnarray}
where
\begin{eqnarray}
\tilde G^{\mu\nu}\equiv L_{X} g^{\mu\nu} + L_{XX} \nabla ^{\mu}\phi\nabla^{\nu}\phi
\label{eq:4}
\end{eqnarray}
and $1+ {2X L_{XX}\over L_{X}} > 0$.
Carrying out the conformal transformation
$G^{\mu\nu}\equiv {c_{s}\over L_{x}^{2}}\tilde G^{\mu\nu}$, with
$c_s^{2}(X,\phi)\equiv{(1+2X{L_{XX}\over L_{X}})^{-1}}\equiv sound ~ speed $.
Then the inverse metric of $G^{\mu\nu}$ is
\begin{eqnarray} G_{\mu\nu}={L_{X}\over c_{s}}[g_{\mu\nu}-{c_{s}^{2}}{L_{XX}\over L_{X}}\nabla_{\mu}\phi\nabla_{\nu}\phi]
\label{eq:5}
\end{eqnarray}
A further conformal transformation \cite{gm1,gm2} $\bar G_{\mu\nu}\equiv {c_{s}\over L_{X}}G_{\mu\nu}$ gives
\begin{eqnarray} \bar G_{\mu\nu}
={g_{\mu\nu}-{{L_{XX}}\over {L_{X}+2XL_{XX}}}\nabla_{\mu}\phi\nabla_{\nu}\phi}
\label{eq:6}
\end{eqnarray}
Here one must always have $L_{X}\neq 0$ for the sound speed
$c_{s}^{2}$ to be positive definite and only
then equations $(1)-(4)$ will be physically meaningful,
since $L_{X}=0$ implies $L$ is independent of
$X$, then from equation (\ref{eq:1}), $L(X,\phi)\equiv L(\phi)$ i.e., $L$ becomes a function of pure potential and
the very definition of $k-$essence fields becomes meaningless because such fields correspond to lagrangians where the kinetic energy dominates over the potential energy. Also the very concept of minimal coupling of $\phi$ to $g_{\mu\nu}$ becomes redundant, so the equation (\ref{eq:1}) meaningless and equations (\ref{eq:4}-\ref{eq:6}) ambiguous.
For the non-trivial configurations of the $k-$ essence field $\phi$, $\partial_{\mu}\phi\neq 0$ (for a scalar field,$\nabla_{\mu}\phi\equiv \partial_{\mu}\phi$ ) and $\bar G_{\mu\nu}$ is not conformally
equivalent to $g_{\mu\nu}$. So this $k-$ essence field $\phi$ field has the properties different from canonical scalar fields defined with $g_{\mu\nu}$ and the local causal structure is also different from those defined with $g_{\mu\nu}$.
Further, if $L$ is not an explicit function of $\phi$
then the equation of motion $(3)$ is reduces to;
\begin{eqnarray}
-{1\over \sqrt {-g}}{\delta S_{k}\over \delta \phi}
= \bar G^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi=0
\label{eq:7}
\end{eqnarray}
We shall take the Lagrangian as $L=L(X)=1-V\sqrt{1-2X}$ with $V$ is a constant.
This is a particular case of the DBI lagrangian \cite{gm1,gm2,born}
\begin{eqnarray}
L(X,\phi)= 1-V(\phi)\sqrt{1-2X}
\label{eq:8}
\end{eqnarray}
for $V(\phi)=V=constant$~~and~~$kinetic ~ energy ~ of~\phi>>V$ i.e.$(\dot\phi)^{2}>>V$. This is typical for the $k-$essence field where the kinetic energy dominates over the potential energy.
Then $c_{s}^{2}(X,\phi)=1-2X$.
For scalar fields $\nabla_{\mu}\phi=\partial_{\mu}\phi$. Thus (\ref{eq:6}) becomes
\begin{eqnarray}
\bar G_{\mu\nu}= g_{\mu\nu} - \partial _{\mu}\phi\partial_{\nu}\phi
\label{eq:9}
\end{eqnarray}
Note the rationale of using two conformal transformations:
the first is used to identify the inverse metric $G_{\mu\nu}$, while
the second realises the mapping onto the
metric given in $(9)$ for the lagrangian $L(X)=1 -V\sqrt{1-2X}$.
\section{Kerr-Newman metric and emergent gravity}
We consider the gravitational metric $g_{\mu\nu}$ is Kerr-Newman (KN) and denote
$\partial_{0}\phi\equiv\dot\phi$, $\partial_{r}\phi\equiv\phi '$. We consider
that the $k-$ essence scalar field $\phi\equiv\phi (r,t)$. The line element
of Kerr-Newman metric is \cite{kn}
\begin{eqnarray}
ds^2_{KN}=f(r,\theta)dt^2-\frac{dr^2}{g(r,\theta)}+2H(r,\theta)d\phi dt
\nonumber\\-K(r,\theta)d\phi^2-\Sigma(r,\theta) d\theta^2
\label{eq:10}
\end{eqnarray}
where,
$f(r,\theta)=\frac{\Delta(r)-\alpha^{2}sin^{2}\theta}{\Sigma(r,\theta)}$;\\
$g(r,\theta)=\frac{\Delta(r)}{\Sigma(r,\theta)}$;\\
$H(r,\theta)=\frac{\alpha sin^{2}\theta(r^{2}+\alpha^{2}-\Delta(r))}{\Sigma(r,\theta)}$;\\
$K(r,\theta)=\frac{(r^{2}+\alpha^{2})^{2}-\Delta(r)\alpha^{2} sin^{2}\theta}{\Sigma(r,\theta)}sin^{2}\theta$;\\
$\Sigma(r,\theta)=r^{2}+\alpha^{2}cos^{2}\theta$;\\
$\Delta(r)=r^{2}+\alpha^{2}+Q^2-2GMr$.
\vspace{0.2in}
It is to be noted that the above metric (\ref{eq:10}) also rediscovered in \cite{umetsu}. In \cite{mann}, elaborately shown how
the Hawking temperature is not depending on $\theta$ although the metric functions depend on $\theta$. In our case the emergent gravity metric (\ref{eq:9}) $\bar G_{\mu\nu}$ contains extra terms (first derivative of $k-$essence scalar fields) {\it but these extra terms are still not depended on $\theta$. Therefore, the modified Hawking temperature will still be independent of $\theta$.
For this reason, we will choose our evaluation for some fixed $\theta$, i.e., $\theta=0$ only.}
Assuming the Kerr-Newman metric along $\theta=0$. Then the above line element (\ref{eq:10}) becomes
\begin{eqnarray}
ds^2_{KN,\theta=0}=F(r)dt^2-\frac{1}{F(r)}dr^2
\label{eq:11}
\end{eqnarray}
with $\it{F(r)=\frac{\Delta(r)}{\Sigma}}$ and $\Sigma=r^2+\alpha^2$.
Also in \cite{umetsu1} was shown that the four dimensional spherically non-symmetric Kerr-Newman metric (\ref{eq:10}) transformed into a two dimensional spherically symmetric metric (\ref{eq:11}) in the region near the horizon by the method of dimensional reduction.
The emergent gravity metric (\ref{eq:9}) components are
\begin{eqnarray}
\bar G_{00}=g_{00}-(\partial _{0}\phi)^{2}={\Delta \over{\Sigma}}- \dot\phi ^{2}\nonumber\\
\bar G_{11}= g_{11} - (\partial _{r}\phi)^{2}= -{\Sigma\over{\Delta}} - (\phi ') ^{2}\nonumber\\
\bar G_{01}=\bar G_{10}=-\dot\phi\phi '.
\label{eq:12}
\end{eqnarray}
Then the emergent gravity line element (\ref{eq:12}) along $\theta=0$ becomes
\begin{eqnarray}
ds^{2,emer}_{KN}=({\Delta \over{\Sigma}}- \dot\phi ^{2})dt^{2}
-({\Sigma\over{\Delta}} + (\phi ') ^{2})dr^{2}-2\dot\phi\phi 'dtdr
\nonumber\\
\label{eq:13}
\end{eqnarray}
Now transform the coordinates \cite{gm1,gm2} from ($t,r$) to ($\omega,r$) such that
\begin{eqnarray}
d\omega=dt-({\dot\phi \phi ' \over{{\Delta \over{\Sigma}}- \dot\phi^{2}}})dr
\label{eq:14}
\end{eqnarray}
and considering
\begin{eqnarray}
\dot\phi^{2}={\Delta^{2}\over{\Sigma^{2}}}(\phi ')^{2}
\label{eq:15}
\end{eqnarray}
we get the line element (\ref{eq:13}):
\begin{eqnarray}
ds^{2,emer}_{KN}=({\Delta \over{\Sigma}}-\dot \phi^{2})d\omega^{2}
-\frac{dr^2}{({\Delta \over{\Sigma}}-\dot \phi^{2})}
\label{eq:16}
\end{eqnarray}
We consider the solution of equation (\ref{eq:15}) as $\phi(r,t)=\phi_{1}(r)+\phi_{2}(t)$.
Then the equation (\ref{eq:15}) reduces to
\begin{eqnarray}
\dot\phi_{2}^{2}={\Delta^{2}\over{\Sigma^2}}(\phi_{1} ')^{2}=K
\label{eq:17}
\end{eqnarray}
where $K$ is a constant and $K\neq 0$ since $k-$essence scalar field will have {\it non-zero} kinetic energy.
Now from (\ref{eq:17}) we get
$\dot \phi_{2}=\sqrt{K}$ and
$\phi_{1} '=\sqrt{K}[\frac{(r^{2}+\alpha^{2})}{r^{2}-2GMr+\alpha^{2}+Q^2}]$
Therefore the solution of (\ref{eq:15}) is
\begin{eqnarray}
\phi(r,t)=\phi_{1}(r)+\phi_{2}(t)\nonumber\\
=\sqrt{K}[(r-GM)+GM~ln[(r-GM)^2\nonumber\\+\alpha^2+Q^2-(GM)^2]+\frac{2G^2 M^2-Q^2}{\sqrt{\alpha^2+Q^2-(GM)^2}}\nonumber\\tan^{-1}(\frac{r-GM}{\sqrt{\alpha^2+Q^2-(GM)^2}})] + \sqrt{k}t\nonumber\\
\label{eq:18}
\end{eqnarray}
where
$\phi_{1}(r)=\sqrt{K}[(r-GM)+GM~ln[(r-GM)^2+\alpha^2+Q^2-(GM)^2]+\frac{2G^2 M^2-Q^2}{\sqrt{\alpha^2+Q^2-(GM)^2}}tan^{-1}(\frac{r-GM}{\sqrt{\alpha^2+Q^2-(GM)^2}})] $
and $\phi_{2}(t)=\sqrt{k}t$ and choosing integration constant to be zero.
Therefore the line element (\ref{eq:16}) becomes
\begin{eqnarray}
ds^{2,emer}_{KN}=({\Delta \over{\Sigma}}-K)d\omega^{2}-{1\over{({\Delta \over{\Sigma}}-K)}}dr^{2}\nonumber\\
=\frac{\beta\Delta'}{\Sigma}d\omega^2-\frac{\Sigma}{\beta\Delta'}dr^2
\label{eq:19}
\end{eqnarray}
where $\beta=1-K$, $M'=\frac{M}{1-K}$, $\Delta'=r^2-2GM'r+Q'^2+\alpha^2$ and $Q'=\frac{Q}{\sqrt{1-K}}$.
This new metric (\ref{eq:19}) is also Kerr-Newman (KN) type along $\theta=0$ in the presence of dark energy.
{\it Note that $K\neq 1$ since $\beta$ cannot be zero, as then the metric (\ref{eq:19}) becomes singular. Also we have the total energy density is unity ($\Omega_{matter} +\Omega_{radiation} +\Omega_{dark energy}= 1$) \cite{gm2,wein}. So we can say that the dark energy density i.e., kinetic energy ($\dot\phi_{2}^{2}=K$) of $k-$essence scalar field (in unit of critical density) cannot be greater than unity.
Again $K$ cannot be greater than $1$ because the metric (\ref{eq:19}) will lead to wrong signature. The possibility of non-zero $K$ appears because that would imply the absence of dark energy.
Therefore, the only allowed values of $K$ are $0 < K < 1$. So there is no question of $K$ approaching towards unity and confusions regarding this limit is avoided}.
It can be shown that, for $r\rightarrow\infty$, this metric (\ref{eq:19}) is an approximate solution of Einstein's equation.
Also mention that the mass and charge of this type black hole are modified as $M'=\frac{M}{1-K}$, $Q'=\frac{Q}{\sqrt{1-K}}$ respectively in the presence of dark energy density term $K=\dot\phi_{2}^2$.
Now we can show that the $k-$essence scalar field $\phi(r,t)$ given by equation (\ref{eq:18}) to satisfy the emergent equation of motion (\ref{eq:7}) along the symmetry axis $\theta=0$ at $r\rightarrow\infty$. For $\theta=0$, the emergent equation of motion (\ref{eq:7}) takes the form
\begin{eqnarray}
\bar G^{00}\partial_{0}^{2}\phi_{\mathrm 2}
+ \bar G^{11}\partial_{1}^{2}\phi_{\mathrm 1}
-\bar G^{11}\Gamma_{11}^{1}\partial_{1}\phi_{\mathrm 1}\nonumber\\
+\bar G^{01}\nabla_{0}\nabla_{1}\phi
+\bar G^{10}\nabla_{1}\nabla_{0}\phi= 0.
\label{eq:20}
\end{eqnarray}
The first term vanishes since $\phi_{2}(t)$ is linear in $t$
and the last two terms vanish because $\bar G^{01}=\bar G^{10}=0$.
Using the expression for $$\Gamma_{11}^{1}= \frac{GM (\alpha^2-r^2)+Q^2 r}{(r^2+\alpha^2)(r^2-2GMr +\alpha^2+Q^2)}$$
the second and third terms for $r\rightarrow\infty$ goes as $\frac{(1-K)\sqrt{K}}{r^{2}}$. From the Planck collaboration results \cite{planck1,planck2}, we have the value of dark energy density (in unit of critical density) $K$ is about $0.696$. Therefore, the second and third terms of (\ref{eq:20}) is negligible as the denominator goes to infinity. Therefore, in this limit the emergent equation of motion is satisfied.
\section{The Hawking temperature for KN type metric in the presence of dark energy}
We use the tortoise coordinate defined by \cite{wheeler,umetsu1}
\begin{eqnarray}
dr^{*}=\frac{dr}{f(r)}
\label{eq:21}
\end{eqnarray}
with $f(r)=\frac{\beta\Delta'}{\Sigma}$ then the emergent line element (\ref{eq:19}) can be written as
\begin{eqnarray}
ds^{2,emer}_{KN}=f(r)(d\omega-dr^{*})(d\omega+dr^{*})
\label{eq:22}
\end{eqnarray}
At near the horizon the equation (\ref{eq:21}) can be written as
\begin{eqnarray}
dr^{*}=\frac{(r^2+\alpha^2)dr}{\beta(r-r_{+})(r-r_{-})}
\label{eq:23}
\end{eqnarray}
with $r_{+}=GM'+\sqrt{(GM')^2-Q'^2-\alpha^2}$ and $r_{-}=GM'-\sqrt{(GM')^2-Q'^2-\alpha^2}$.
Integrating equation (\ref{eq:23}) we get
\begin{eqnarray}
r^{*}= \frac{1}{\beta}[r+({r_{+}^{2}+\alpha^{2}\over{r_{+}-r_{-}}})ln~|r-r_{+}|\nonumber\\
+({r_{-}^{2}+\alpha^{2}\over{r_{-}-r_{+}}})ln~|r-r_{-}|]+C
\label{eq:24}
\end{eqnarray}
where $C$ is an integration constant.
The above equation (\ref{eq:24}) can be written in terms of surface gravity when $r>r_{+}$ as \cite{umetsu1}
\begin{eqnarray}
r^{*}=\frac{r}{\beta}+\frac{1}{2\chi_{+}}ln~(\frac{r-r_{+}}{r_{+}})+\frac{1}{2\chi_{-}}ln~(\frac{r-r_{-}}{r_{-}})
\label{eq:25}
\end{eqnarray}
with surface gravity ($+$ sign for outer horizon and $-$ sign for inner horizon)
\begin{eqnarray}
\chi_{\pm}\equiv\frac{1}{2}f'(r)\mid_{r=r_{\pm}}=\frac{\beta}{2}[\frac{r_{\pm}-r_{\mp}}{r_{\pm}^{2}+\alpha^2}].
\label{eq:26}
\end{eqnarray}
Also we calculate the Hawking temperature \cite{haw} for (\ref{eq:19}) using {\it tunneling formalism} \cite{mann,mitra,jiang,murata,ma} for the two horizons as follows.
We going over to the Eddington-Finkelstein coordinates $(v,r)$ or
$(u,r)$ along $\theta=0$ i.e., {\it introducing advanced and retarded null coordinates} \cite{gm2} $$v=\omega+r^{*}~~;~~u=\omega-r^{*}$$. Using this coordinate the line element
(\ref{eq:19}) becomes
\begin{eqnarray}
ds^{2,emer}_{KN}=({\beta\Delta' \over{r^{2}+\alpha^{2}}})dv^{2}-2dv dr\nonumber\\
={\beta(r-r_{+})(r-r_{-}) \over{r^{2}+\alpha^{2}}}dv^{2}-2dv dr.
\label{eq:27}
\end{eqnarray}
Also we calculate the Hawking temperature \cite{haw} for (\ref{eq:27}) using {\it tunneling formalism} \cite{mann,mitra,jiang,murata,ma} for the two horizons as follows.
A massless particle in a black hole background is described by the Klein-Gordon equation
\begin{eqnarray}
\hbar^2(-\bar G)^{-1/2}\partial_\mu( \bar G ^{\mu\nu}(-\bar G)^{1/2}\partial_\nu\Psi)=0.
\label{eq:28}
\end{eqnarray}
We can expands $\Psi$ as
\begin{eqnarray}
\Psi=exp({i\over{\hbar}}S+...)
\label{eq:29}
\end{eqnarray}
to obtain the leading order in $\hbar$ the Hamilton-Jacobi equation is
\begin{eqnarray}
\bar G^{\mu\nu}\partial_\mu S \partial_\nu S=0.
\label{eq:30}
\end{eqnarray}
We consider $S$ is independent of $\theta$ and $\phi$. Then the above equation (\ref{eq:30})
\begin{eqnarray}
2{\partial S\over{\partial v}}{\partial S\over{\partial r}}+
(\frac{\beta(r^{2}-2GM^{'}r+\alpha^{2}+Q'^{2})}{r^{2}+\alpha^{2}})({\partial S\over{\partial r}})^{2}=0\nonumber\\
\label{eq:31}
\end{eqnarray}
The action $S$ is assumed to be of the form
\begin{eqnarray}
S=-Ev+W(r)+J(x^{i})
\label{eq:32}
\end{eqnarray}
Then
\begin{eqnarray}
\partial_{v}S=-E~;~\partial_{r}S=W^{'}~;~\partial_{i}S=J_{i}
\label{eq:33}
\end{eqnarray}
$J_{i}$ are constants chosen to be zero.
Now putting the values of equation (\ref{eq:33}) in equation (\ref{eq:31}) we get
\begin{eqnarray}
-2EW^{'}(r)+(\frac{\beta(r^{2}-2GM^{'}r+\alpha^{2}+Q'^{2})}{r^{2}+\alpha^{2}})
(W^{'}(r))^{2}=0.\nonumber\\
\label{eq:34}
\end{eqnarray}
Then
\begin{eqnarray}
W(r)=\int \frac{[E(r^{2}+\alpha^{2})+E(r^{2}+\alpha^{2})]dr}{\beta(r-r_{+})(r-r_{-})}\nonumber\\
=2\pi i ({E\over{\beta}})({r_{+}^{2}+\alpha^{2}\over{r_{+}-r_{-}}})
+2\pi i ({E\over{\beta}})({r_{-}^{2}+\alpha^{2}\over{r_{-}-r_{+}}})\nonumber\\
=W(r_{+})+W(r_{-})
\label{eq:35}
\end{eqnarray}
The two values of $W(r)$ correspond to the outer and inner horizons respectively.
Therefore the equation (\ref{eq:32}) becomes
\begin{eqnarray}
S=-Ev+2\pi i ({E\over{\beta}})({r_{+}^{2}+\alpha^{2}\over{r_{+}-r_{-}}})
+2\pi i ({E\over{\beta}})({r_{-}^{2}+\alpha^{2}\over{r_{-}-r_{+}}})+J(x^{i})\nonumber\\
\label{eq:36}
\end{eqnarray}
So the tunneling rates are
\begin{eqnarray}
\Gamma^{KN}_{+emergent} \sim e^{-2~Im S{+}} \sim e^{-2~Im W(r_{+})}\nonumber\\
=e^{-4\pi ({E\over{\beta}})({r_{+}^{2}+\alpha^{2}\over{r_{+}-r_{-}}})}=e^{-{E\over{K_{B}T_{+}}}}
\label{eq:37}
\end{eqnarray}
and
\begin{eqnarray}
\Gamma^{KN}_{-emergent} \sim e^{-2~Im S{-}} \sim e^{-2~Im W(r_{-})}\nonumber\\
=e^{-4\pi ({E\over{\beta}})({r_{-}^{2}+\alpha^{2}\over{r_{-}-r_{+}}})}=e^{-{E\over{K_{B}T_{-}}}}
\label{eq:38}
\end{eqnarray}
where $K_{B}$ is Boltzman constant.
From these above two expressions (\ref{eq:37}) and (\ref{eq:38}) the corresponding Hawking temperatures of the two horizons are
\begin{eqnarray}
T_{+emergent}^{KN}=\frac{\hbar c^{3}\beta}{4\pi k_{B}}(\frac{r_{+}-r_{-}}{r_{+}^{2}+\alpha^{2}})~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
=\frac{\hbar c^{3}\beta}{2\pi k_{B}}[\frac{\sqrt{(GM')^2-\alpha^2-Q'^2}}{2GM'(GM'+\sqrt{(GM')^2-\alpha^2-Q'^2})-Q'^2}]\nonumber\\
\label{eq:39}
\end{eqnarray}
and
\begin{eqnarray}
T_{-emergent}^{KN}=\frac{\hbar c^{3}\beta}{4\pi k_{B}}(\frac{r_{-}-r_{+}}{r_{-}^{2}+\alpha^{2}})~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
=-\frac{\hbar c^{3}\beta}{2\pi k_{B}}[\frac{\sqrt{(GM')^2-\alpha^2-Q'^2}}{2GM'(GM'-\sqrt{(GM')^2-\alpha^2-Q'^2})-Q'^2}]\nonumber\\
\label{eq:40}
\end{eqnarray}
with $\beta=1-K$.
The usual Hawking temperature for Kerr-Newman black hole is \cite{mann}
\begin{eqnarray}
T_{\pm}^{KN}=\frac{\hbar c^{3}}{4\pi k_{B}}(\frac{r_{\pm}-r_{\mp}}{r_{\pm}^{2}+\alpha^{2}})~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
=\frac{\hbar c^{3}}{2\pi k_{B}}[\frac{\sqrt{(GM)^2-\alpha^2-Q^2}}{2GM(GM\pm\sqrt{(GM)^2-\alpha^2-Q^2})-Q^2}]\nonumber\\
\label{eq:41}
\end{eqnarray}
{\it The above temperatures (\ref{eq:39},\ref{eq:40}) are modified in the presence of dark energy. These temperatures are different from usual Hawking temperature (\ref{eq:41}) as the presence of terms $\beta=1-K$, $M'=\frac{M}{1-K}$ and $Q'=\frac{Q}{\sqrt{1-K}}$ where $K$ is the dark energy density (in unit of critical density).}
\section{Kerr-Newman-AdS background}
We consider the gravitational metric $g_{\mu\nu}$ is Kerr-Newman-AdS (KNAdS).
The line element of KNAdS metric \cite{jiang,murata,ma,cald,aliev} is
\begin{eqnarray}
ds^2_{KNAdS}=\frac{1}{\Sigma}[\Delta_{r}-\Delta_{\theta}\alpha^2 sin^{2}\theta]dt^2
-\frac{\Sigma}{\Delta_{r}}dr^2-\frac{\Sigma}{\Delta_{\theta}}d\theta^2\nonumber\\-\frac{1}{\Sigma(\Xi)^2}[\Delta_{\theta}(r^2+\alpha^2)^2-\Delta_{r}\alpha^2 sin^{2}\theta]sin^{2}\theta~d\phi^2\nonumber\\+\frac{2\alpha}{\Sigma\Xi}[\Delta_{\theta}(r^2+\alpha^2)-\Delta_{r}]sin^{2}\theta~dtd\phi ~~~~~~~~~
\label{eq:42}
\end{eqnarray}
where
\begin{eqnarray}
\Sigma=r^2+\alpha^2 cos^{2}\theta ;~~~\Xi=1-\frac{\alpha^2}{l^2}
\label{eq:43}
\end{eqnarray}
\begin{eqnarray}
\Delta_{\theta}=1-\frac{\alpha^2}{l^2}cos^{2}\theta;~
\Delta_{r}=(r^2+\alpha^2)(1+\frac{r^2}{l^2})-2GMr+Q^2.\nonumber\\
\label{eq:44}
\end{eqnarray}
The parameters $M$ and $\alpha$ are related to the mass and
angular momentum of the black hole, $G$ is the gravitational constant and $l$ is the curvature radius determined by the negative cosmological constant ($\Lambda<0$) $\Lambda=-\frac{3}{l^2}$.
Again we choose symmetric axis along $\theta=0$ as before since in \cite{mann} elaborately shown that the Hawking temperature is independent of $\theta$.
Then the line element (\ref{eq:42}) reduces to
\begin{eqnarray}
ds^2_{KNAdS,\theta=0}=F(r)dt^2-\frac{1}{F(r)}dr^2
\label{eq:45}
\end{eqnarray}
with $F(r)=\frac{\Delta_{r}}{\Sigma}$ and $\Sigma=r^2+\alpha^2$.
Using this (\ref{eq:45}) the emergent gravity metric (\ref{eq:9}) components are
\begin{eqnarray}
\bar G_{00}=g_{00}-(\partial _{0}\phi)^{2}={\Delta_{r} \over{\Sigma}}- \dot\phi ^{2}\nonumber\\
\bar G_{11}= g_{11} - (\partial _{r}\phi)^{2}= -{\Sigma\over{\Delta_{r}}} - (\phi ') ^{2}\nonumber\\
\bar G_{01}=\bar G_{10}=-\dot\phi\phi '.
\label{eq:46}
\end{eqnarray}
Again we consider the $k-$essence scalar field $\phi(r,t)$ is spherically symmetric. So the emergent gravity line element for KNAdS background along $\theta=0$ is
\begin{eqnarray}
ds^{2,emer}_{KNAdS}=({\Delta_{r} \over{\Sigma}}- \dot\phi ^{2})dt^{2}
-({\Sigma\over{\Delta_{r}}} + (\phi ') ^{2})dr^{2}-2\dot\phi\phi 'dtdr.
\nonumber\\
\label{eq:47}
\end{eqnarray}
Transform the coordinates $(t,r)$ to $(\omega,r)$ as
\begin{eqnarray}
d\omega=dt-({\dot\phi \phi ' \over{{\Delta_{r} \over{\Sigma}}- \dot\phi^{2}}})dr
\label{eq:48}
\end{eqnarray}
and we choose
\begin{eqnarray}
\dot\phi^{2}={\Delta_{r}^{2}\over{\Sigma^{2}}}(\phi ')^{2}.
\label{eq:49}
\end{eqnarray}
Then the line element (\ref{eq:47}) becomes
\begin{eqnarray}
ds^{2,emer}_{KNAdS}=({\Delta_{r} \over{\Sigma}}-\dot \phi^{2})d\omega^{2}
-\frac{dr^2}{({\Delta_{r} \over{\Sigma}}-\dot \phi^{2})}
\label{eq:50}
\end{eqnarray}
We consider again the solution of equation (\ref{eq:49}) as $\phi(r,t)=\phi_{1}(r)+\phi_{2}(t)$.
Then the equation (\ref{eq:49}) is
\begin{eqnarray}
\dot\phi_{2}^{2}={\Delta_{r}^{2}\over{\Sigma^2}}(\phi_{1} ')^{2}=K
\label{eq:51}
\end{eqnarray}
where $K$ is a constant and $K\neq 0$.
From (\ref{eq:51}) we get
$\dot \phi_{2}=\sqrt{K}$ and
$\phi_{1} '=\sqrt{K}[\frac{(r^{2}+\alpha^{2})}{(r^{2}+\alpha^2)(1+\frac{r^2}{l^2})-2GMr+Q^2}]$.
So the solution of equation (\ref{eq:49}) is
\begin{eqnarray}
\phi(r,t)=\phi_{1}(r)+\phi_{2}(t)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
=\frac{C\sqrt{K}}{2}~ln|r^2+\lambda r+m|+\frac{D\sqrt{K}}{2}~ln|r^2-\lambda r+n|\nonumber\\+\frac{\sqrt{K}(2A-\lambda C)}{2\sqrt{m-\frac{\lambda^2}{4}}}~tan^{-1}(\frac{r+\frac{\lambda}{2}}{\sqrt{m-\frac{\lambda^2}{4}}})~~~~~~~~~~~~~~~~\nonumber\\
+\frac{\sqrt{K}(2B+\lambda D)}{2\sqrt{n-\frac{\lambda^2}{4}}}~tan^{-1}(\frac{r-\frac{\lambda}{2}}{\sqrt{n-\frac{\lambda^2}{4}}})
+\sqrt{K}t\nonumber\\
\label{eq:52}
\end{eqnarray}
where
\begin{eqnarray}
\phi_{1}(r)=\sqrt{K}\int{\frac{(r^{2}+\alpha^{2})}{(r^{2}+\alpha^2)(1+\frac{r^2}{l^2})-2GMr+Q^2}dr}\nonumber\\
=\sqrt{K}\int{\frac{(r^{2}+\alpha^{2})}{(r^2+\lambda r+m)(r^2-\lambda r+n)}dr}\nonumber\\
=\frac{C\sqrt{K}}{2}~ln|r^2+\lambda r+m|+\frac{D\sqrt{K}}{2}~ln|r^2-\lambda r+n|\nonumber\\+\frac{\sqrt{K}(2A-\lambda C)}{2\sqrt{m-\frac{\lambda^2}{4}}}~tan^{-1}(\frac{r+\frac{\lambda}{2}}{\sqrt{m-\frac{\lambda^2}{4}}})~~~~~\nonumber\\
+\frac{\sqrt{K}(2B+\lambda D)}{2\sqrt{n-\frac{\lambda^2}{4}}}~tan^{-1}(\frac{r-\frac{\lambda}{2}}{\sqrt{n-\frac{\lambda^2}{4}}})~~~~~~
\label{eq:53}
\end{eqnarray}
and
\begin{eqnarray}
\phi_{2}(t)=\sqrt{K}t.
\label{eq:54}
\end{eqnarray}
Now we clarify the parameters of the above equation (\ref{eq:52}):
$C=\frac{-1}{2\lambda^2+m-n}~,~D=\frac{1}{2\lambda^2+m-n}~,~$
$A=\frac{1}{m+n}[\alpha^2-\frac{n(m-n)}{2\lambda^2+m-n}]~,~$
$B=\frac{1}{m+n}[\alpha^2+\frac{m(m-n)}{2\lambda^2+m-n}]~,~$
$\lambda=[(\frac{1}{2}(-T+\sqrt{T^2+4H^3}))^{1/3}+(\frac{1}{2}(-T-\sqrt{T^2+4H^3}))^{1/3}-\frac{2(l^2+\alpha^2)}{3}]^{1/2}~,~$
$m=\frac{1}{2}[(l^2+\alpha^2)+\lambda^2+\frac{2GMl^2}{\lambda}]~,~$
$n=\frac{1}{2}[(l^2+\alpha^2)+\lambda^2-\frac{2GMl^2}{\lambda}]~,~$
$H=-\frac{1}{9}[l^4+2(3\alpha^2+2Q^2)l^2+\alpha^4]~,~$
$T=-\frac{1}{27}[2l^6-6(12Q^2+11\alpha^2-18G^{2}M^2)l^4-6\alpha^2(12Q^2+11\alpha^2)l^2+2\alpha^6]$.
For this type of $k-$essence scalar field $\phi$ (\ref{eq:52}), the line element (\ref{eq:50}) reduces to
\begin{eqnarray}
ds^{2,emer}_{KNAdS}=({\Delta_{r} \over{\Sigma}}-K)d\omega^{2}-{1\over{({\Delta_{r} \over{\Sigma}}-K)}}dr^{2}\nonumber\\
=\frac{\beta\Delta_{r}'}{\Sigma}d\omega^2-\frac{\Sigma}{\beta\Delta_{r}'}dr^2
\label{eq:55}
\end{eqnarray}
where $\beta=1-K$, $M'=\frac{M}{1-K}$, $\Delta_{r}'=(r^2+\alpha^2)(1+\frac{r^2}{l'^2})-2GM'r+Q'^2$, $Q'=\frac{Q}{\sqrt{1-K}}$ and $l'^2=(1-K)l^2$.
Similar reasons as before here also the only allowed values of $K$ are $0 < K < 1$. Also it can be shown that this metric (\ref{eq:55}) is an approximate solution of Einstein's equations at $r\rightarrow\infty$ along $\theta=0$. Note that the parameters $M, Q, l$ are also modified in the presence of dark energy density ($K$).
We can show that the $k-$essence scalar field (\ref{eq:52}) is satisfied emergent gravity equation of motions (\ref{eq:7}) along $\theta=0$ at $r\rightarrow\infty$. For $\theta=0$, the emergent equation of motion (\ref{eq:7}) takes the form
$\bar G^{00}\partial_{0}^{2}\phi_{\mathrm 2}
+ [\bar G^{11}\partial_{1}^{2}\phi_{\mathrm 1}
-\bar G^{11}\Gamma_{11}^{1}\partial_{1}\phi_{\mathrm 1}]
+\bar G^{01}\nabla_{0}\nabla_{1}\phi
+\bar G^{10}\nabla_{1}\nabla_{0}\phi= 0.$
The first term vanishes since $\phi_{2}(t)$ is linear in $t$
and the last two terms vanish because $\bar G^{01}=\bar G^{10}=0$.
Using the value of $$\Gamma_{11}^{1}=\frac{1}{\Sigma\Delta}[\frac{-2r^5}{l^2}+\frac{2\pi \alpha^2 r^3}{3}-GM(r^2-\alpha^2)+(Q^2-\frac{\alpha^4}{l^2})r]$$ we get
the terms within third bracket are vanished at $r\rightarrow\infty$.
\section{The Hawking temperature for KNAdS type metric in the presence of dark energy}
We calculate the Hawking temperature using tunneling formalism \cite{mitra,ma,cald,aliev}. The horizons of the metric (\ref{eq:55}) in the presence of dark energy are determined by
\begin{eqnarray}
\Delta_{r}'=(r^2+\alpha^2)(1+\frac{r^2}{l'^2})-2GM'r+Q'^2\nonumber\\
=\frac{\beta}{l'^{2}}[r^4+r^2(\alpha^2+l'^2)-2GM'l'^{2}r+l'^{2}(\alpha^2+Q'^2)]\nonumber\\
=\frac{\beta}{l'^{2}}(r-r_{++}^{d})(r-r_{--}^{d})(r-r_{+}^{d})(r-r_{-}^{d})=0~~~~~~
\label{eq:56}
\end{eqnarray}
The equation $\Delta_{r}'=0$ has four roots, two real positive roots and two complex roots. We denote $r_{++}^{d}$ and $r_{--}^{d}$ are complex roots and $r_{+}^{d}$ and $r_{-}^{d}$ are positive real roots in the presence of dark energy $(K)$. Here we consider $r_{+}^{d}>r_{-}^{d}$ so that $r_{+}^{d}$ is the black hole event horizon and $r_{-}^{d}$ is the Cauchy horizon of the KNAdS type black hole (\ref{eq:55}).
Now we use the Eddington-Finkelstein coordinates $(v,r)$ or
$(u,r)$ along $\theta=0$ i.e., {\it advanced and retarded null coordinates} \cite{gm2} $$v=\omega+r^{*}~~;~~u=\omega-r^{*}$$ with
\begin{eqnarray}
dr^{*}=\frac{(r^2+\alpha^2)dr}{\frac{\beta}{l'^{2}}(r-r_{++}^{d})(r-r_{--}^{d})(r-r_{+}^{d})(r-r_{-}^{d})}
\label{eq:57}
\end{eqnarray}
we get the emergent gravity line element (\ref{eq:55}) becomes
\begin{eqnarray}
ds^{2,emer}_{KNAdS}=\frac{\beta\Delta_{r}'}{\Sigma}dv^{2}-2dvdr\nonumber\\
=[\frac{\frac{\beta}{l'^{2}}(r-r_{++}^{d})(r-r_{--}^{d})(r-r_{+}^{d})(r-r_{-}^{d})}{r^2+\alpha^2}]dv^2-2dvdr.\nonumber\\
\label{eq:58}
\end{eqnarray}
Proceedings exactly same as KN type case we can calculate the Hawking temperatures for KNAdS type black hole (\ref{eq:58}) as:
\begin{eqnarray}
T^{KNAdS}_{+emergent}=\frac{\hbar c^{3}\beta}{4\pi k_{B}l^2}[\frac{(r_{+}^{d}-r_{++}^{d})(r_{+}^{d}-r_{--}^{d})(r_{+}^{d}-r_{-}^{d})}{(r_{+}^{d})^2+\alpha^2}]\nonumber\\
=-\frac{\hbar c^{3}(1-K)\Lambda}{12\pi k_{B}}[\frac{(r_{+}^{d}-r_{++}^{d})(r_{+}^{d}-r_{--}^{d})(r_{+}^{d}-r_{-}^{d})}{(r_{+}^{d})^2+\alpha^2}]~~~~~~~~
\label{eq:59}
\end{eqnarray}
and
\begin{eqnarray}
T^{KNAdS}_{-emergent}=\frac{\hbar c^{3}\beta}{4\pi k_{B}l^2}[\frac{(r_{-}^{d}-r_{++}^{d})(r_{-}^{d}-r_{--}^{d})(r_{-}^{d}-r_{+}^{d})}{(r_{-}^{d})^2+\alpha^2}]\nonumber\\
=-\frac{\hbar c^{3}(1-K)\Lambda}{12\pi k_{B}}[\frac{(r_{-}^{d}-r_{++}^{d})(r_{-}^{d}-r_{--}^{d})(r_{-}^{d}-r_{+}^{d})}{(r_{-}^{d})^2+\alpha^2}]~~~~~~~~
\label{eq:60}
\end{eqnarray}
where $k_{B}$ is the Boltzman constant. These temperatures $T^{KNAdS}_{+emergent}$ and $T^{KNAdS}_{-emergent}$ are different from usual Hawking temperature for KNAdS black hole as reported on \cite{jiang}-\cite{aliev}.
Here $\Lambda<0$, $r_{+}^{d}$ and $r_{-}^{d}$ are positive and $r_{+}^{d}>r_{-}^{d}$; $r_{++}^{d}$ and $r_{--}^{d}$ are complex conjugate, these make sure that the temperature of event horizon is positive.
Note that an Anti-de Sitter (AdS) space has negative cosmological constant in a vacuum, where empty space itself has negative energy density but positive pressure, unlike our accelerated Universe where observations of distant supernovae indicate a positive cosmological constant corresponding to the de-Sitter space \cite{wein} which has positive energy density but negative pressure. Dark energy is one of the candidate being regarded as the origin of this accelerated expansion where pressure is negative. So for AdS space, the cosmological constant is negative which cannot be associated with dark energy. Therefore, here the dark energy comes only from the $k-$essence scalar field. Also note that the Hawking temperature for KNAdS black hole is already eavaluated in \cite{ma}.
\section{Conclusion}
In this work we have determined the Hawking temperatures in the presence of dark energy for emergent gravity metrics having
Kerr-Newman and Kerr-Newman-AdS backgrounds.
We have shown that in the presence of dark energy the Hawking temperatures are modified. We did the calculation
for Kerr-Newman and Kerr-Newman-AdS background metrics along $\theta=0$ since the Hawking temperature is independent of $\theta$ and show that the modified metrics i.e., emergent gravity black hole metrics for both cases are satisfies Einstein's equations for large $r$ and the emergent black hole always radiates. These new emergent gravity metrics are mapped on to the Kerr-Newman and Kerr-Newman-AdS type metrics.
Throughout the paper our analysis is done in the
context of dark energy in an emergent gravity scenario having $k-$essence scalar fields $\phi$ with a Dirac-Born-Infeld type
lagrangian. In both cases the
scalar field $\phi(r,t)=\phi_{1}(r)+\phi_{2}(t)$ also satisfies the emergent gravity equations of motion at
$r\rightarrow\infty$ for $\theta=0$.
\vspace{0.3in}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~***~~~~~~~~~~~~~~~~~~~~~~
The authors would like to thank the referees for illuminating suggestions to improve the manuscript.
|
2,869,038,156,362 | arxiv | \section{Conclusion}\label{sec:conclusion}
We proposed a task-independent privacy-respecting data crowdsourcing framework TIPRDC.
A feature extractor is learned to hide privacy information features and maximally retain original information from the raw data. By applying TIPRDC, a user can locally extract features from the raw data using the learned feature extractor, and the data collector will acquire the extracted features only to train a DNN model for the primary learning tasks. Evaluations on three benchmark datasets show that TIPRDC attains a better privacy-utility tradeoff than existing solutions. The cross-dataset evaluations on CelebA and LFW shows the transferability of TIPRDC, indicating the practicability of proposed framework.
\section{Discussion and Limitations}\label{sec:discussion}
\section{Evaluation}\label{sec:evaluation}
In this section, we evaluate TIPRDC's performance on three real-world datasets, with a focus on the utility-privacy tradeoff. We also compare TIPRDC with existing solutions proposed in the literature and visualize the results.
\subsection{Experiment Setup}
We evaluate TIPRDC, especially the learned feature extractor, on two image datasets and one text dataset. We implement TIPRDC with PyTorch, and train it on a server with 4$\times$NVIDIA TITAN RTX GPUs. We apply mini-batch technique in training with a batch size of 64, and adopt the AdamOptimizer \cite{adam} with an adaptive learning rate in the hybrid learning procedure. The architecture configurations of each module are presented in Table \ref{tb:model_arch_img} and \ref{tb:model_arch_text}. For evaluating the performance, given a primary learning task, a simulated data collector trains a normal classifier using features processed by the learned feature extractor, and such normal classifier has the same architecture of the classifier presented in Table \ref{tb:model_arch_img} and \ref{tb:model_arch_text}. The utility and privacy of the extracted features $E_\theta(x)$ are evaluated by the classification accuracy of primary learning tasks and specified private attribute, respectively.
We adopt CelebA \cite{liu2015faceattributes}, LFW \cite{kumar2009attribute} and the dialectal tweets dataset (DIAL) \cite{blodgett2016demographic} for the training and testing of TIPRDC. CelebA consists of more than 200K face images. Each face image is labeled with 40 binary facial attributes. The dataset is split into 160K images for training and 40K images for testing. LFW consists of more than 13K face images, and each face image is labeled with 16 binary facial attributes. We split LFW into 10K images for training and 3K images for testing. DIAL consists of 59.2 million tweets collected from 2.8 million users, and each tweet is annotated with three binary attributes. DIAL is split into 48 million tweets for training and 11.2 million tweets for testing.
\begin{table}[h]
\caption{The architecture configurations of each module for CelebA and LFW.}\vspace{-0.15in}
\begin{tabular}{l|l|l}
\hline
\textbf{Feature Extractor} & \textbf{Classifier} & \textbf{MI Estimator} \\ \hline
2$\times$conv3-64 & 3$\times$conv3-256 & 3$\times$conv3-64 \\
maxpool & maxpool & maxpool \\ \hline
2$\times$conv3-128 & 3$\times$conv3-512 & 2$\times$conv3-128 \\
maxpool & maxpool & maxpool \\ \hline
\textbf{} & 3$\times$conv3-512 & 3$\times$conv3-256 \\
& maxpool & maxpool \\ \cline{2-3}
& 2$\times$fc-4096 & 3$\times$conv3-512 \\
& fc-label length & maxpool \\ \cline{2-3}
& & 3$\times$conv3-512 \\
& & maxpool \\ \cline{3-3}
& & \begin{tabular}[c]{@{}l@{}}fc-4096\\ fc-512\\ fc-1\end{tabular} \\ \hline
\end{tabular}\label{tb:model_arch_img}
\end{table}\vspace{-0.2in}
\begin{table}[h]
\caption{The architecture configurations of each module for DIAL.}\vspace{-0.15in}
\begin{tabular}{l|l|l}
\hline
\textbf{Feature Extractor} & \textbf{Classifier} & \textbf{MI Estimator} \\ \hline
embedding-300 & 2$\times$lstm-300 & embedding-300 \\ \hline
lstm-300 & fc-150 & lstm-300 \\ \cline{1-1} \cline{3-3}
& fc-label length & 2$\times$lstm-300 \\ \cline{2-3}
& & fc-150 \\
\textbf{} & & fc-1 \\ \hline
\end{tabular}\label{tb:model_arch_text}
\end{table}\vspace{-0.2in}
\subsection{Comparison Baselines}
We select four types of data privacy-preserving baselines \cite{liu2019privacy}, which have been widely applied in the literature, and compare them with TIPRDC. The details settings of the baseline solutions are presented as below.
\begin{itemize}
\item \textbf{Noisy} method perturbs the raw data $x$ by adding Gaussian noise $\mathcal{N}(0,\sigma^2)$, where $\sigma$ is set to 40 according to \cite{liu2019privacy}. The noisy data $\Bar{x}$ will be delivered to the data collector. The Gaussian noise injected to the raw data can provide strong guarantees of differential privacy using less local noise. This scheme has been widely applied in federated learning \cite{papernot2018scalable,truex2019hybrid}.
\item \textbf{DP} approach injects Laplace noise the raw data $x$ with diverse privacy budgets \{0.1, 0.2, 0.5, 0.9\}, which is a typical differential privacy method. The noisy data $\Bar{x}$ will be submitted to the data collector.
\item \textbf{Encoder} learns the latent representation of the raw data $x$ using a DNN-based encoder. The extracted features $z$ will be uploaded to the data collector.
\item \textbf{Hybrid} method \cite{osia2020hybrid} further perturbs the above encoded features by performing principle components analysis (PCA) and adding Laplace noise with varying noise factors privacy budgets \{0.1, 0.2, 0.5, 0.9\}.
\end{itemize}
\subsection{Evaluations on CelebA and LFW}
\textbf{Comparison of utility-privacy tradeoff:} We compare the utility-privacy tradeoff offered by TIPRDC with four privacy-preserving baselines.
In our experiments, we set `young' and `gender' as the private labels to protect in CelebA, and consider detecting `gray hair' and `smiling' as the primary learning tasks to evaluate the utility.
With regard to LFW, we set `gender' and `Asian' as the private labels, and choose recognizing `black hair' and `eyeglass' as the classification tasks.
Figure \ref{fig:comparsion} summarizes the utility-privacy tradeoff offered by four baselines and TIPRDC. Here we evaluate TIPRDC with four discrete choices of $\lambda\in\{1, 0.9, 0.5, 0\}$.
As Figure \ref{fig:comparsion} shows, although TIPRDC cannot always outperform the baselines in both utility and privacy, it still achieve the best utility-privacy tradeoff under most experiment settings.
For example, in Figure \ref{fig:comparsion} (h), TIPRDC achieves the best tradeoff by setting $\lambda=0.9$. Specifically, the classification accuracy of `Asian' on LFW is 55.31\%, and the accuracy of `eyeglass' is 86.88\%. This demonstrates that TIPRDC can efficiently protect privacy while maintaining high utility of extracted features.
In other four baselines, Encoder method can maintain a good utility of the extracted features, but it fails to protect privacy due to the high accuracy of private labels achieved by the adversary. Noisy, DP and Hybrid methods offer strong privacy protection with sacrificing the utility.
\begin{figure*}[t]
\centering
\subfigure[]{\includegraphics[scale=0.26]{up_1.eps}}
\subfigure[]{\includegraphics[scale=0.26]{up_2.eps}}
\subfigure[]{\includegraphics[scale=0.26]{up_3.eps}}
\subfigure[]{\includegraphics[scale=0.26]{up_4.eps}}
\subfigure[]{\includegraphics[scale=0.26]{up_5.eps}}
\subfigure[]{\includegraphics[scale=0.26]{up_6.eps}}
\subfigure[]{\includegraphics[scale=0.26]{up_7.eps}}
\subfigure[]{\includegraphics[scale=0.26]{up_8.eps}}\vspace{-0.2in}
\caption{Utility-privacy tradeoff comparison of TIPRDC with four baselines on CelebA and LFW.}
\label{fig:comparsion}
\end{figure*}
\textbf{Impact of the utility-privacy budget $\lambda$:} An important step in the hybrid learning procedure (see Equation \ref{eq:hybrid}) is to determine the utility-privacy budget $\lambda$.
To determine the optimal $\lambda$, we evaluate the utility-privacy tradeoff on CelebA and LFW by setting different $\lambda$. Specifically, we evaluate the impact of $\lambda$ with four discrete choices of $\lambda\in\{1, 0.9, 0.5, 0\}$. The private labels and primary learning tasks in CelebA and LFW are set as same as the above experiments.
As Figure \ref{fig:budget} illustrates, the classification accuracy of primary learning tasks will increase with a smaller $\lambda$, but the privacy protection will be weakened. Such phenomenon is reasonable, since the smaller $\lambda$ means hiding less privacy information in features but retaining more original information from the raw data according to Equation \ref{eq:hybrid}. For example, in Figure \ref{fig:budget} (a), the classification accuracy of `gray hair' on CelebA is 84.36\% with $\lambda=1$ and increases to 91.52\% by setting $\lambda=0$; the classification accuracy of `young' is 65.63\% and 81.85\% with decreasing $\lambda$ from 1 to 0, respectively. Overall, $\lambda=0.9$ is an optimal utility-privacy budget for experiment settings in both CelebA and LFW.
\begin{figure}[t]
\centering
\subfigure[raw image]{\includegraphics[scale=0.3]{raw.eps}}
\subfigure[$\lambda=1$]{\includegraphics[scale=0.3]{gender_lambda0.eps}}
\subfigure[$\lambda=0.9$]{\includegraphics[scale=0.3]{gender_lambda01.eps}}
\subfigure[$\lambda=0.5$]{\includegraphics[scale=0.3]{gender_lambda05.eps}}
\subfigure[$\lambda=0$]{\includegraphics[scale=0.3]{gender_lambda1.eps}}\vspace{-0.2in}
\caption{Visualize the impact of the utility-privacy budget $\lambda$ when protecting `gender' in CelebA.}
\label{fig:visualize}
\end{figure}
\begin{figure*}[t]
\centering
\subfigure[CelebA]{\includegraphics[scale=0.215]{celeb_1.eps}}
\subfigure[CelebA]{\includegraphics[scale=0.215]{celeb_2.eps}}
\subfigure[CelebA]{\includegraphics[scale=0.215]{celeb_3.eps}}
\subfigure[LFW]{\includegraphics[scale=0.215]{lfw_1.eps}}
\subfigure[LFW]{\includegraphics[scale=0.215]{lfw_2.eps}}
\subfigure[LFW]{\includegraphics[scale=0.215]{lfw_3.eps}}\vspace{-0.2in}
\caption{The impact of the utility-privacy budget $\lambda$ on CelebA and LFW.}
\label{fig:budget}
\end{figure*}
We further visualize how different options of $\lambda$ influence the utility maintained by the learned feature extractor. This is done by training a decoder with the reversed architecture of the feature extractor, and then the decoder aims to reconstruct the raw data by taking the extracted feature as input.
Here we adopt the setting in Figure \ref{fig:budget}(b) as an example, where `gender' is protected when training the feature extractor.
As Figure \ref{fig:visualize} shows, decreasing $\lambda$ allows a more informative image to be reconstructed.
This means more information is retained in the extracted feature with a smaller $\lambda$, which is consist with the results shown in Figure \ref{fig:budget}(c).
Additionally, if we compare Figure \ref{fig:budget}(c) vs. Figure \ref{fig:budget}(a-b) and Figure \ref{fig:budget}(f) vs. Figure \ref{fig:budget}(d-e), it can be observed that protecting more private attributes leads to slight degradation in utility with slightly enhanced privacy protection under a particular $\lambda$.
For example, given $\lambda=0.9$, the accuracy of `smiling' slightly decreases from 90.13\% in Figure \ref{fig:budget}(a) and 89.33\% in Figure \ref{fig:budget}(c) to 88.16\%.
The accuracy of `young' slightly decreases from 65.77\% in Figure \ref{fig:budget}(a) to 64.85\% in Figure \ref{fig:budget}(c). The reason is that the feature related to the private attributes has some intrinsic correlations to the feature related to the primary learning tasks.
Therefore, more correlated features may be hidden if more private attributes need to be protected.
\textbf{Effectiveness of privacy protection:} We quantitatively evaluate the effectiveness of privacy protection offered by TIPRDC by simulating an adversary to infer the private attribute through training a classifier.
As presented in Table \ref{tb:model_arch_img}, we adopt a default architecture for simulating the adversary's classifier. However, an adversary may train the classifier with different architectures. We implement three additional classifiers as an adversary in our experiments.
The architectural configurations of those classifiers are presented in Table \ref{tb:ablation}.
We train those classifiers on CelebA by considering recognizing `smiling' as the primary learning task, and `gender' as the private attribute that needs to be protected.
Table \ref{tab:bruteforce} presents the average classification accuracy for adversary classifiers on testing data. The results show that although we apply a default architecture for simulating the adversary classifier when training the feature extractor, the trained feature extractor can effectively defend against privacy leakage no matter what kinds of architecture are adopted by an adversary in the classifier design.
\begin{table}[t]
\caption{The ablation study with different architecture configurations of the adversary classifier.}\vspace{-0.15in}
\centering
\begin{tabular}{ccc}
\hline
\multicolumn{1}{c|}{V-CL\#16} & \multicolumn{1}{c|}{V-CL\#19} & Res-CL \\ \hline
\multicolumn{3}{c}{Input Feature Maps ($54\times44\times128$)} \\ \hline
\multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}3$\times$conv3-256\\ maxpool\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}4$\times$conv3-256\\ maxpool\end{tabular}} & \begin{tabular}[c]{@{}c@{}}$\begin{bmatrix}3 \times 3(2), & 128 \\ 3 \times 3,& 128 \end{bmatrix}$\\ $\begin{bmatrix}3 \times 3, & 128 \\ 3 \times 3,& 128 \end{bmatrix}$\end{tabular} \\ \hline
\multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}3$\times$conv3-512\\ maxpool\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}4$\times$conv3-512\\ maxpool\end{tabular}} & $\begin{bmatrix}3 \times 3, & 256 \\ 3 \times 3,& 256 \end{bmatrix} \times 2$ \\ \hline
\multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}3$\times$conv3-512\\ maxpool\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}4$\times$conv3-512\\ maxpool\end{tabular}} & $\begin{bmatrix}3 \times 3, & 512 \\ 3 \times 3,& 512 \end{bmatrix} \times 2$ \\ \hline
\multicolumn{2}{c|}{\begin{tabular}[c]{@{}c@{}}2$\times$fc-4096\\ fc-label length\end{tabular}} & \begin{tabular}[c]{@{}c@{}}avgpool\\ fc-label length\end{tabular} \\ \hline
\end{tabular}\vspace{-2mm}
\label{tb:ablation}
\end{table}
\begin{table}[t]
\centering
\caption{The classification accuracy of `gender' on CelebA with different adversary classifiers ($\lambda=0.9$).}\vspace{-0.15in}
\resizebox{.48\textwidth}{!}{
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{2}{*}{\textbf{Training Classifier}} & \multicolumn{4}{|c|}{\textbf{Adversary Classifier}}\\
\cline{2-5}
& Default architecture in Table 1 & V-CL\#16 & V-CL\#19 & Res-CL \\
\hline
Default architecture in Table 1 & 59.02\% & 59.83\% & 60.77\% & 61.05\% \\
\hline
\end{tabular}}
\label{tab:bruteforce}
\end{table}
\textbf{Evaluate the transferability of TIPRDC:} The data collector usually trains the feature extractor of TIPRDC before collecting the data from users. Hence, the transferability of the feature extractor determines the usability of TIPRDC. We evaluate the transferability of TIPRDC by performing cross-dataset evaluations. Specifically, we train the feature extractor of TIPRDC using either CelebA or LFW dataset and test the utility-privacy tradeoff on the other dataset. In this experiment, we choose recognizing `black hair' as the primary learning task, and `gender' as the private attribute that needs to be protected. As Table \ref{tab:cross} illustrates, the feature extractor that is trained using one dataset can still effectively defend against private attribute leakage on the other dataset, while maintaining the classification accuracy of the primary learning task. For example, if we train the feature extractor using CelebA and then test it on LFW, the accuracy of `gender' decreases to 56.87\% compared with 57.31\% by directly training the feature extractor using LFW. The accuracy of `black hair' marginally increases to 89.27\% from 88.12\%. The reason is that CelebA offers a larger number of training data so that the feature extractor can be trained for a better performance. Although there is a marginal performance drop, the feature extractor that is trained using LFW still works well on CelebA. The cross-dataset evaluations demonstrate good transferability of TIPRDC. \vspace{-0.1in}
\begin{table}[t]
\centering
\caption{Evaluate the transferability of TIPRDC with cross-dataset experiments ($\lambda=0.9$).}\vspace{-0.15in}
\begin{tabular}{c|c|c|c}
\hline
\textbf{Training Dataset} & \textbf{Test Dataset} & \textbf{`gender'} & \textbf{`black hair'} \\
\hline
LFW & CelebA & 59.73\% & 87.15\% \\
LFW & LFW & 57.31\% & 88.12\% \\
\hline
CelebA & LFW & 56.87\% & 89.27\% \\
CelebA & CelebA & 58.82\% & 88.98\% \\
\hline
\end{tabular}
\label{tab:cross}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[scale=0.3]{text_1.eps}\vspace{-0.15in}
\caption{The impact of the utility-privacy budget $\lambda$ on DIAL.}
\label{fig:dial}
\end{figure}
\subsection{Evaluation on DIAL}
To quantitatively evaluate the utility-privacy tradeoff of TIPRDC on DIAl, we choose `race' as the private attribute that needs to be protected and predicting `mentioned' as the primary learning task. The binary mention task is to determine if a tweet mentions another user, i.e., classifying conversational vs. non-conversational tweets. Similar to the experiment settings in CelebA and LFW, we evaluate the utility-privacy tradeoff on DIAL by setting different $\lambda$ with four discrete choices of $\lambda\in\{1, 0.9, 0.5, 0\}$.
As Figure \ref{fig:dial} shows, the classification accuracy of primary learning tasks will increase with a smaller $\lambda$, but the privacy protection will be weakened, showing the same phenomenon as the evaluations on CelebA and LFW. For example, the classification accuracy of `mentioned' is 67.71\% with $\lambda=1$ and increases to 76.87\% by setting $\lambda=0$, and the classification accuracy of `race' increases by 21.57\% after changing $\lambda$ from 1 to 0.
\section{Design of TIPRDC}\label{sec:framework_design}
\subsection{Overview}
The critical module of TIPRDC is the feature extractor. As presented in Section \ref{sec:problem_formulation}, there are two goals for learning the feature extractor, such that it can hide private attribute from features while retaining as much information of the raw data as possible to maintain the utility for primary learning task. To this end, we design a hybrid learning method to train the feature extractor, including the privacy adversarial training (PAT) algorithm and the MaxMI algorithm. In particular, we design the PAT algorithm, which simulates the game between an adversary who makes efforts to infer private attributes from the extracted features and a defender who aims to protect user privacy. By applying PAT to optimize the feature extractor, we enforce the feature extractor to hide private attribute $u$ from extracted features $z$, which is goal 1 introduced in Section \ref{sec:problem_formulation}. Additionally, we propose the MaxMI algorithm to achieve goal 2 presented in Section \ref{sec:problem_formulation}. By performing MaxMI to train the feature extractor, we can enable the feature extractor to maximize the mutual information between the information of the raw data $x$ and the joint information of the private attribute $u$ and the extracted feature $z$.
As Figure \ref{fig:framework_design} shows, there are three neural network modules in the hybrid learning method: \textit{feature extractor}, \textit{adversarial classifier} and \textit{mutual information estimator}. The feature extractor is the one we aim to learn by performing the proposed hybrid learning algorithm. The adversarial classifier simulates an adversary in the PAT algorithm, aiming to infer private attribute $u$ from the eavesdropped features. The mutual information estimator is adopted in MaxMI algorithm to measure the mutual information between the raw data $x$ and the joint distribution of the private attribute $u$ and the extracted feature $z$. All three modules are end-to-end trained using our proposed hybrid learning method.
\begin{figure}[t]
\centering
\captionsetup{width=1.0\linewidth}
\includegraphics[scale=0.35]{framework.pdf}
\caption{The hybrid learning method for training the feature extractor.}
\label{fig:framework_design}
\end{figure}
Before presenting the details of each algorithm, we give the following notations. As same as presented in Section \ref{sec:problem_formulation}, we adopt $x$, $u$ and $z$ as the raw data, the private attribute and the extracted feature, respectively. We denote $E_\theta$ as the feature extractor that is parameterized with $\theta$, and then $z$ can be expressed as $E_\theta(x)$. Let $E_\Psi$ represent the classifier, where $\Psi$ indicates the parameter set of the classifier. We adopt $y=E_\Psi(E_\theta(x))$ to denote the prediction generated by $E_\Psi$. Let $E_\omega$ denotes the mutual information estimator, where $\omega$ represents the parameter set of the mutual information estimator.
\subsection{Privacy Adversarial Training Algorithm}
We design the PAT algorithm to achieve goal 1, enabling the feature extractor to hide $u$ from $E_\theta(x)$. The PAT algorithm is designed by simulating the game between an adversary who makes efforts to infer private attributes from the extracted features and a defender who aims to protect user privacy. We can apply any architecture to both the feature extractor and the classifier based on the requirement of data format and the primary learning task. The performance of the classifier ($C$) is measured using the cross-entropy loss function as:
\begin{equation}
\mathcal{L}(C)=CE(y,u)=CE(E_\Psi(E_\theta(x),u),
\end{equation}
where $CE\left[\cdot\right]$ stands for the cross entropy loss function. When we simulate an adversary who tries to enhance the accuracy of the adversary classifier as high as possible, the classifier needs to be optimized by minimizing the above loss function as:
\begin{equation}
\label{eq:adv_lc}
\Psi=\underset{\Psi}{\arg\min} \mathcal{L}(C).
\end{equation}
On the contrary, when defending against private attribute leakage, we train the feature extractor in PAT that aims to degrade the performance of the classifier. Consequently, the feature extractor can be trained using Equation \ref{eq:lc} when simulating a defender:
\begin{equation}
\label{eq:lc}
\theta=\underset{\theta}{\arg\max} \mathcal{L}(C).
\end{equation}
Based on Equation \ref{eq:adv_lc} and \ref{eq:lc}, the feature extractor and the classifier can be jointly optimized using Equation \ref{eq:pat}:
\begin{equation}
\label{eq:pat}
\theta,\Psi=\arg\max_\theta \min_\Psi \mathcal{L}(C) ,
\end{equation}
which is consistent with Equation \ref{equ:altgoal1}.
\subsection{MaxMI Algorithm}
For goal 2, we propose MaxMI algorithm to make the feature extractor retain as much as information from the raw data as possible, in order to maintain the high utility of the extracted features. Specifically, the Jensen-Shannon mutual information estimator~\cite{nowozin2016f,hjelm2018learning} is adopted to measure the lower bound of the mutual information $I(x; z,u)$. Here we adopt $E_\omega$ as the the Jensen-Shannon mutual information estimator, and we can rewrite Equation \ref{eq:jsd} as:
\begin{align}
&\mathcal{I}(x;z,u)\ge \mathcal{I}_{\theta,\omega}^{(JSD)}(x;z,u) \nonumber\\
&:=\mathbb{E}_{x}\left[-sp(-E_\omega(x,E_\theta(x),u))\right]-\mathbb{E}_{x,x'}\left[sp(E_\omega(x',E_\theta(x),u))\right],
\end{align}
where $x'$ is an random input data sampled independently from the same distribution of $x$, $sp(z)=log(1+e^z)$ is the softplus function. Hence, to maximally retain the original information, the feature extractor and the mutual information estimator can be optimized using Equation~\ref{eq:new_jsd}:
\begin{align}
\theta,\omega&=\arg\max_\theta \max_\omega \mathcal{I}_{\theta,\omega}^{(JSD)}(x;z,u), \label{eq:new_jsd}
\end{align}
which is consistent with Equation \ref{eq:max_jsd}. Considering the difference of $x$, $z$ and $u$ in dimensionality, we feed them into the $E_\omega$ from different layers as illustrated in Figure \ref{fig:framework_design}. For example, the private attribute $u$ may be a binary label, which is represented by one bit. However, $x$ may be high dimensional
data (e.g., image), and hence it is not reasonable feed both $x$ and $u$ from the first layer of $E_\omega$.
\subsection{Hybrid Learning Method}
Finally, the feature extractor is trained by alternatively performing PAT algorithm and MaxMI algorithm. As aforementioned, we also introduce an utility-privacy budget $\lambda$ to balance the tradeoff between protecting privacy and retaining the original information. Therefore, combining Equation~\ref{eq:pat} and \ref{eq:new_jsd}, the objective function of training the feature extractor can be formulated as:
\begin{equation}
\theta,\Psi,\omega=\arg\max_\theta(\lambda \min_\Psi \mathcal{L}(C) + (1-\lambda) \max_\omega\mathcal{I}_{\theta,\omega}^{(JSD)}(x;z,u)). \label{eq:hybrid}
\end{equation}
Algorithm~\ref{Algo:TIPRDC} summarizes the hybrid learning method of TIPRDC. Before performing the hybrid learning, we first jointly pretrain the feature extractor and the adversarial classifier normally without adversarial objective to obtain the best performance on classifying a specific private attribute. Then within each training batch, we first perform PAT algorithm and MaxMI algorithm to update $\Psi$ and $\omega$, respectively. Then, the feature extractor will be updated according to Equation \ref{eq:hybrid}.
\begin{algorithm}
\caption{Hybrid Learning Method}
\label{Algo:TIPRDC}
\begin{algorithmic}[1]
\Require
Dataset $\mathcal{D}$
\Ensure
$\theta$
\For{every epoch}
\For{every batch}
\State $\mathcal{L}(C)\rightarrow$ update $\psi$ (performing PAT)
\State $-\mathcal{I}_{\theta, \omega}^{JSD}(x;z,u)\rightarrow$ update $\omega$ (performing MaxMI)
\State $-\lambda\mathcal{L}(C)-(1-\lambda)\mathcal{I}_{\theta, \omega}^{JSD}(x;z,u)\rightarrow$ update $\theta$
\EndFor
\EndFor
\end{algorithmic}
\end{algorithm}
\section{Introduction}\label{sec:introduction}
Deep learning has demonstrated an impressive performance in many applications, such as computer vision \cite{krizhevsky2012imagenet,he2016deep} and natural language processing \cite{bahdanau2014neural,wu2016google,oord2016wavenet}. Such success of deep learning partially benefits from various large-scale datasets (e.g., ImageNet \cite{deng2009imagenet}, MS-COCO \cite{lin2014microsoft}, etc.), which can be used to train powerful deep neural networks (DNN). The datasets are often crowdsourced from individual users to train DNN models. For example, companies or research institutes that want to implement face recognition systems may collect the facial images from employees or volunteers. However, those data that are crowdsourced from individual users for deep learning applications often contain private information such as gender, age, etc. Unfortunately, the data crowdsourcing process can be exposed to serious privacy risks as the data may be misused by the data collector or acquired by the adversary.
It is recently reported that many large companies face data security and user privacy challenges. The data breach of Facebook, for example, raises users' severe concerns on sharing their personal data. These emerging privacy concerns hinder generation or use of large-scale crowdsourcing datasets and lead to hunger of training data of many new deep learning applications.
A number of countries are also establishing laws to protect data security and privacy.
As a famous example, the new European Union‘s General Data Protection Regulation (GDPR) requires companies to not store personal data for a long time, and allows users to delete or withdraw their personal data within 30 days. It is critical to design a data crowdsourcing framework to protect the privacy of the shared data while maintaining the utility for training DNN models.
\begin{figure*}[t]
\centering
\includegraphics[scale=0.3]{kdd.eps}\vspace{-0.5cm}
\caption{The overview of TIPRDC.}
\label{fig:overview}
\end{figure*}
Existing solutions to protect privacy are struggling to balance the tradeoff between privacy and utility.
An obvious and widely adopted solution is to transform the raw data into task-oriented features, and users only upload the extracted features to corresponding service providers, such as Google Now \cite{googlenow} and Google Cloud \cite{googlecloud}.
Even though transmitting only features are generally more secure than uploading raw data, recent developments in model inversion attacks \cite{mahendran2015understanding,dosovitskiy2016inverting,dosovitskiy2016generating} have demonstrated that adversaries can exploit the acquired features to reconstruct the raw image, and hence the person on the raw image can be re-identified from the reconstructed image.
In addition, the extracted features can also be exploited by an adversary to infer private attributes, such as gender, age, etc. Ossia \textit{et al.} \cite{osia2020hybrid} move forward by applying dimentionality reduction and noise injection to the features before uploading them to the service provider. However, such approach leads to unignorable utility loss. Inspired by Generative Adversarial Networks (GAN), several adversarial learning approaches \cite{liu2019privacy,li2019deepobfuscator,oh2017adversarial,kim2019training} have been proposed to learn obfuscated features from raw images. Unfortunately, those solutions are designed for known primary learning tasks, which limits their applicability in the data crowdsourcing where the primary learning task may be unknown or changed when training a DNN model.
The need of collecting large-scale crowdsourcing dataset under strict requirement of data privacy and limited applicability of existing solutions motivates us to design a privacy-respecting data crowdsourcing framework: the raw data from the users are locally transformed into an intermediate representation that can remove the private information while retaining the discriminative features for primary learning tasks.
In this work, we propose TIPRDC -- a task-independent privacy-respecting data crowdsourcing framework with anonymized intermediate representation. The ultimate goal of this framework is to learn a feature extractor that can remove the privacy information from the extracted intermediate features while maximally retaining the original information embedded in the raw data for primary learning tasks.
As Figure \ref{fig:overview} shows, participants can locally run the feature extractor and submit only those intermediate representations to the data collector instead of submitting the raw data.
The data collector then trains DNN models using these collected intermediate representations, but both the data collector and the adversary cannot accurately infer any protected private information.
Compared with existing adversarial learning methods \cite{liu2019privacy,li2019deepobfuscator,oh2017adversarial,kim2019training}, TIPRDC does not require the knowledge of the primary learning task and hence, directly applying existing adversarial training methods becomes impractical.
It is challenging to remove all concerned private information that needs to be protected while retaining everything else for unknown primary learning tasks.
To address this issue, we design a hybrid learning method to learn the anonymized intermediate representation. The learning purpose is two-folded: (1) hiding private information from features; (2) maximally retaining original information. Specifically, we hide private information from features by performing our proposed privacy adversarial training (PAT) algorithm, which simulates the game between an adversary who makes efforts to infer private attributes from the extracted features and a defender who aims to protect user privacy. The original information are retained by applying our proposed MaxMI algorithm, which aims to maximize the mutual information between the feature of the raw data and the union of the private information and the retained feature.
In summary, our key contributions are the follows:
\begin{itemize}
\item To the best of our knowledge, TIPRDC is the first privacy-respecting data crowdsourcing framework for deep learning without the knowledge of any specific primary learning task. By applying TIPRDC, the learned feature extractor can hide private information from features while maximally retaining the information of the raw data.
\item We propose a privacy adversarial training algorithm to enable the feature extractor to hide privacy information from features. In addition, we also design the MaxMI algorithm to maximize the mutual information between the raw data and the union of the private information and the retained feature, so that the original information from the raw data can be maximally retained in the feature.
\item We quantitatively evaluate the utility-privacy tradeoff with applying TIPRDC on three real-world datasets, including both image and text data. We also compare the performance of TIPRDC with existing solutions.
\end{itemize}
The rest of this paper is organized as follows. Section \ref{sec:related_work} reviews the related work. Section \ref{sec:problem_formulation} presents the problem formulation. Section \ref{sec:framework_design} describes the framework overview and details of core modules. Section \ref{sec:evaluation} evaluates the framework. Section \ref{sec:conclusion} concludes this paper.
\section{Problem Formulation}\label{sec:problem_formulation}
There are three parties involved in the crowdsourcing process: \textit{user}, \textit{adversary}, and \textit{data collector}.
Under the strict requirement of data privacy, a data collector offers options to a user to specify any private attribute that needs to be protected. Here we denote the private attribute specified by a user as $u$.
According to the requirement of protecting $u$, the data collector will learn a feature extractor $f_\theta(z | x,u)$ that is parameterized by weight $\theta$, which is the core of TIPRDC.
The data collector distributes the data collecting request associated with the feature extractor to users. Given the raw data $x$ provided by a user, the feature extractor can locally extract feature $z$ from $x$ while hiding private attribute $u$.
Then, only extracted feature $z$ will be shared with the data collector, which can training DNN models for primary learning tasks using collected $z$. An adversary, who may be an authorized internal staff of the data collector or an external hacker, has access to the extracted feature $z$ and aims to infer private attribute $u$ based on $z$. We assume an adversary can train a DNN model via collecting $z$, and then the trained model takes a user's extracted feature $z$ as input and infers the user's private attribute $u$.
The critical challenge of TIPRDC is to learn the feature extractor, which can hide private attribute from features while maximally retaining original information from the raw data.
Note that it is very likely that the data and private attribute are somehow correlated. Therefore, we cannot guarantee no information about $u$ will be contained in $z$ unless we enforce it in the objective function when training the feature extractor.
The ultimate goal of the feature extractor $f$ is two-folded:
\begin{itemize}
\item \textbf{Goal 1:} make sure the extracted features conveys no private attribute;
\item \textbf{Goal 2:} retain as much information of the raw data as possible to maintain the utility for primary learning tasks.
\end{itemize}
\begin{figure}[tb]
\centering
\captionsetup{width=1.0\linewidth}
\includegraphics[width=0.6\linewidth]{set.png}
\caption{Optimal outcome of feature extractor $f_\theta(z|x,u)$.}
\label{fig:set}
\end{figure}
Formally, Goal 1 can be formulated as:
\begin{equation}
\label{equ:min}
\min_\theta I(z; u),
\end{equation}
where $I(z; u)$ represents the mutual information between $z$ and $u$. On the other hand, Goal 2 can be formulated as:
\begin{equation}
\label{equ:max}
\max_\theta I(x; z | u).
\end{equation}
In order to avoid any potential conflict with the objective of Goal 1, we need to mitigate counting in the information where $u$ and $x$ are correlated. Therefore, the mutual information in Equation~\ref{equ:max} is evaluated under the condition of private attribute $u$. Note that
\begin{equation}
I(x; z | u) = I(x; z,u)-I(x;u).
\end{equation}
Since both $x$ and $u$ are considered fixed under our setting, $I(x;u)$ will stay as a constant during the optimization process of feature extractor $f_\theta$. Therefore, we can safely rewrite the objective of Goal 2 as:
\begin{equation}
\label{equ:altmax}
\max_\theta I(x; z,u),
\end{equation}
which is to maximize the mutual information between $x$ and joint distribution of $z$ and $u$.
As Figure~\ref{fig:set} illustrates, we provide an intuitive demonstration of the optimal outcome of the feature extractor using a Venn diagram. Goal 1 is fulfilled as no overlap exist between $z$ (green area) and $u$ (orange area); and Goal 2 is achieved as $z$ and $u$ jointly (all colored regions) cover all the information of $x$ (blue circle).
It is widely accepted in the previous works that precisely calculating the mutual information between two arbitrary distributions are likely to be infeasible \cite{peng2018variational}. As a result, we replace the mutual information objectives in Equation~\ref{equ:min} and \ref{equ:altmax} with their upper and lower bounds for effective optimization. For Goal 1, we utilize the mutual information upper bound derived in~\cite{song2018learning} as:
\begin{equation}
\label{equ:upper}
I(z;u) \leq \mathbb{E}_{q_\theta(z)}D_{KL}(q_\theta(u|z)||p(u)),
\end{equation}
for any distribution $p(u)$.
Note that the term $q_\theta(u|z)$ in Equation~(\ref{equ:upper}) is hard to estimate and hence we instead replace the KL divergence term with its lower bound by introducing a conditional distribution $p_{\Psi}(u | z)$ parameterized with $\Psi$. It was shown in~\cite{song2018learning} that:
\begin{align}
&\mathbb{E}_{q_\theta(z)}\left[\log p_{\Psi}(u | z) - \log p(u)\right] \nonumber\\ &\leq \mathbb{E}_{q_\theta(z)}D_{KL}(q_\theta(u|z)||p(u))
\end{align}
Hence, the Equation~\ref{equ:min} can be rewritten as an adversarial training objective function:
\begin{equation}
\label{equ:goal1}
\min_\theta \max_\Psi \mathbb{E}_{q_\theta(z)}\left[\log p_{\Psi}(u | z) - \log p(u)\right],
\end{equation}
As $\log p(u)$ is a constant number that is independent of $z$, Equation~\ref{equ:goal1} can be further simplified to:
\begin{equation}
\label{equ:altgoal1}
\max_\theta \min_\Psi -\mathbb{E}_{q_\theta(z)}\left[\log p_{\Psi}(u | z) \right],
\end{equation}
which is the cross entropy loss of predicting $u$ with $p_{\Psi}(u | z)$, i.e., $CE\left[p_{\Psi}(u | z)\right]$.
This objective function can be interpreted as an adversarial game between an adversary $p_{\Psi}$ who tries to infer $u$ from $z$ and a defender $q_{\theta}$ who aims to protect the user privacy.
For Goal 2, we adopt the previously proposed Jensen-Shannon mutual information estimator~\cite{nowozin2016f,hjelm2018learning} to estimate the lower bound of the mutual information I(x; z,u). The lower bound is formulated as follows:
\begin{align}
\label{eq:jsd}
&\mathcal{I}(x;z,u)\ge \mathcal{I}_{\theta,\omega}^{(JSD)}(x;z,u) \nonumber\\
&:=\mathbb{E}_{x}\left[-sp(-E_\omega(x;f_\theta(x),u))\right]-\mathbb{E}_{x,x'}\left[sp(E_\omega(x';f_\theta(x),u))\right],
\end{align}
where $x'$ is an random input data sampled independently from the same distribution of $x$, $sp(z)=log(1+e^z)$ is the softplus function and $E_\omega$ is a discriminator modeled by a neural network with parameters $\omega$. Hence, to maximally retain the original information, the feature extractor and the mutual information estimator can be optimized using Equation~\ref{eq:max_jsd}:
\begin{equation}
\max_\theta \max_\omega \mathcal{I}_{\theta,\omega}^{(JSD)}(x;z,u)\label{eq:max_jsd}.
\end{equation}
Finally, combining Equation~\ref{equ:altgoal1} and \ref{eq:max_jsd}, the objective function of training the feature extractor can be formulated as:
\begin{equation}
\max_\theta(\lambda \min_\Psi CE\left[p_{\Psi}(u | z)\right] + (1-\lambda) \max_\omega\mathcal{I}_{\theta,\omega}^{(JSD)}(x;z,u)),
\end{equation}
where $\lambda \in [0,1]$ serves as a utility-privacy budget. A larger $\lambda$ indicates a stronger privacy protection, while a smaller $\lambda$ allowing more original information to be retained in the extracted features.
\section{Related Work}\label{sec:related_work}
\textbf{Data Privacy Protection}: Many techniques have been proposed to protect data privacy, most of which are based on various anonymization methods including \textit{k}-anonymity \cite{sweeney2002k}, \textit{l}-diversity \cite{mahendran2015understanding} and \textit{t}-closeness \cite{li2007t}.
However, these approaches are designed for protecting sensitive attributes in a static database and hence, are not suitable to our addressed problem -- data privacy protection in the online data crowdsourcing for training DNN models.
Differential privacy \cite{duchi2013local,erlingsson2014rappor,bassily2015local,qin2016heavy,smith2017interaction,avent2017blender,wang2017locally} is another widely applied technique to protect privacy of an individual's data record, which provides a strong privacy guarantee. However, the privacy guarantee provided by differential privacy is different from the privacy protection offered by TIPRDC in data crowdsourcing. The goal of differential privacy is to add random noise to a user’s true data record such that two arbitrary true data records have close probabilities to generate the same noisy data record.
Compared with differential privacy, our goal is to hide private information from the features such that an adversary cannot accurately infer the protected private information through training DNN models. Osia \textit{et al.} \cite{osia2020hybrid} leverage a combination of dimensionality reduction, noise addition, and Siamese fine-tuning to protect sensitive information from features, but it does not offer the tradeoff between privacy and utility in a systematic way.
\textbf{Visual Privacy Protection:} Some works have been done to specifically preserve privacy in images and videos. De-identification is a typical privacy-preserving visual recognition approach to alter the raw image such that the identity cannot be visually recognized.
There are various techniques to achieve de-identification, such as Gaussian blur \cite{oh2016faceless}, identity obfuscation \cite{oh2016faceless}, mean shift filtering \cite{winkler2014trusteye} and adversarial image perturbation \cite{oh2017adversarial}.
Although those approaches are effective in protecting visual privacy, they all limit the utility of the data for training DNN models.
In addition, encryption-based approaches \cite{gilad2016cryptonets,yonetani2017privacy} have been proposed to guarantee the privacy of the data, but they require specialized DNN models to directly train on the encrypted data. Unfortunately, such encryption-based solutions prevent general dataset release and introduce substantial computational overhead. All the above practices only consider protecting privacy in specific data format, i.e., image and video, which limit their applicability across diverse data modalities in the real world.
\textbf{Tradeoff between Privacy and Utility using Adversarial Networks}: With recent advances in deep learning, several approaches have been proposed to protect data privacy using adversarial networks and simulate the game between the attacker and the defender who defend each other with conflicting utility-privacy goals.
Pittaluga \textit{et al.} \cite{pittaluga2019learning} design an adversarial learning method for learning an encoding function to defend against performing inference for specific attributes from the encoded features.
Seong \textit{et al.} \cite{oh2017adversarial} introduce an adversarial network to obfuscate the raw image so that the attacker cannot successfully perform image recognition. Wu \textit{et al.} \cite{wu2018towards} design an adversarial framework to explicitly learn a degradation transform for the original video inputs, aiming to balance between target task performance and the associated privacy budgets on the degraded video.
Li \textit{et al.} \cite{li2019deepobfuscator} and Liu \textit{et al.} \cite{liu2019privacy} propose approaches to learn obfuscated features using adversarial networks, and only obfuscated features will be submitted to the service provider for performing inference. Attacker cannot train an adversary classifier using collected obfuscated features to accurately infer a user's private attributes or reconstruct the raw data.
The same idea behind the above solutions is that using adversarial networks to obfuscate the raw data or features, in order to defending against privacy leakage. However, those solutions are designed to protect privacy information while targeting some specified learning tasks, such as face recognition, activity recognition, etc.
Our proposed TIPRDC provides a more general privacy protection, which does not require the knowledge of the primary learning task.
\textbf{Differences between TIPRDC and existing methods:} Compared with prior arts, our proposed TIPRDC has two distinguished features: (1) a more general approach that be applied on different data formats instead of handling only image data or static database; (2) require no knowledge of primary learning tasks. |
2,869,038,156,363 | arxiv | \section{Introduction}
Coloured quarks and gluons are the main objects in the fundamental theory of
the strong interactions, quantum chromodynamics (QCD), which are not directly
observed in experiment. At intermediate and low energies the laws
governing interactions of these fundamental particles bring about complex
rearrangements inside a quark-gluon system which eventually result in
two-quark and three-quark colourless formations---hadrons. A second transition
appears at a comparable scale,
namely the spontaneous chiral symmetry breakdown,
leading to the pseudo-Goldstone boson octet. These phenomena cannot be
consistently described within perturbation theory in the strong coupling
constant $\alpha_s$ which is no longer small at the typical hadronic energy
scale. Therefore, it is mandatory to develop nonperturbative methods to solve
these problems.
There are few nonperturbative methods employed in particle physics.
Noteworthy are lattice field theory,
quantization on the light cone, and QCD sum rules. Despite all their
advantages, these approaches have not yet lead to a unified description of
confinement
and spontaneous chiral symmetry breakdown $(\chi SB)$.
Both phenomena are supposed to
arise from colour gauge interactions. Yet, the complex process of hadron
formation remains obscure. This stimulates numerous attempts to construct
models aimed at describing individual major features of the process.
One can find explanation of spontaneous symmetry breakdown in the
Nambu--Jona-Lasinio model, chiral quark model, approaches with stochastic
vacuum fields, dual QCD, etc. A large class of such models involve path
integral
methods for calculation of Green functions that approximate QCD in the
infrared region. The guiding principle is iterative integration first with
respect to rapidly varying fields and then with respect to slowly varying
ones, an idea borrowed from the general theory of phase transitions.
According to this idea, slow modes mainly determine the
asymptotic behaviour of the system at small energies and momenta (infrared
asymptotics) and require thorough study in terms of collective
variables. The contribution from ``fast'' fields is less substantial and
can be taken into account by perturbation theory. The
Nambu--Jona-Lasinio model (NJL) \cite{1}, on which we concentrate below, is
one of these schemes. The boundary between ``slow'' and ``fast'' fields is
largely conventional being dictated by physical considerations. The chiral
$SU(3)_L\otimes SU(3)_R$ symmetry of the QCD lagrangian is broken at
energies of the order of $\Lambda_{\chi SB}\sim 1\,$GeV if small quark
masses are ignored. This is what allows one to relate $\Lambda_{\chi SB}$
to the boundary between ``large'' and ``small'' momenta when looking into
infrared QCD asymptotics from the symmetry point of view.
Chiral symmetry breakdown is a fundamental issue because it is related
to the problem of elementary particle masses and chiral symmetry lends
itself to a controlable expansion (in energy) at
energies below $\Lambda_{\chi SB}$. At these energies
the QCD vacuum
becomes highly non-perturbative due to the condensation of quark-antiquark
pairs as a consequence of the
spontaneous symmetry violation. It is believed that any serious model
of the strong interactions has to account for these facts. This is the
case for the celebrated NJL model which was originally formulated not
in terms of quark but rather nucleonic degrees of freedom. In what follows
we will be concerned with a modern extension based on the quark description.
Assuming that the NJL model in the form that involves all essential QCD
symmetries may be a reasonable approximation of QCD in the intermediate
energy range, we develop a new theoretical method. It allows arbitrary
N-point functions describing strong and electroweak interactions of basic
meson states to be calculated through the effective four-quark interaction of
the NJL model \cite{2}. Now there are two approaches to the problem. One
is based on the classical treatment of four-fermion Lagrangians (the
Hartree--Fock approximation plus Bethe -- Salpeter type equations for bound
states) \cite{3}-\cite{7}. The other involves explicit separation of
collective degrees of freedom in the functional integral of the theory
(bosonization) \cite{8}-\cite{14}. In this framework one usually derives
the Lagrangian based on a derivative expansion
instead of directly calculating the Green functions.
For technical reasons, such a derivative expansion can only be performed
to the first few orders.
{\it We have constructed such a formal scheme that at the same
time incorporates bosonization and has all advantages of the pure fermionic
approach.} It allows to combine the advantages of both of the abovementioned
methods. The results of our more general method can be directly compared
with the ones based on standard bosonization by performing a chiral expansion
based on the nonlinear Bethe -- Salpeter equations.
Originally formulated in terms of boson variables linearly transforming
with respect to the chiral group action, the method can be extended to
nonlinear realization of symmetry. This is one of the significant advantages
of the approach as compared with Hartree and Fock's classical technique,
which is linear is this sense. Now chiral symmetry consequences are often
investigated via nonlinear chiral Lagrangians. They are a series in energy
$(E)$, which is equivalent to normal expansion with an increasing number
of derivatives. As a rule, investigations are mostly confined to the
${\cal O}(E^4)$ approximation. Recently a program like this was used for
the extended NJL model \cite{14}. In the present paper we show how
one can generalize these results without any reference to an expansion
in $E^2$.
Apart from pure theoretical developments to generalize the results of
\cite{14}, we carry out explicit calculations. Our primary concern
is the $\pi\pi$ scattering amplitude. We shall calculate it both in the
linear and nonlinear approaches using the standard Lagrangian of the
extended NJL model with scalar, pseudoscalar, vector, and axial-vector
four-quark interactions. We show that there is no difference between these
two approaches. The expressions for the total amplitude $A(s,t,u)$ are
the same in both cases. It has to be so because the same Lagrangian is
used. We consider this point in detail to show the
self-consistency of our method.
The paper is organized as follows. In the next section we introduce the
Lagrangian and notation as well as review the main steps of
the momentum-space bosonization technique.
We then consider the nonlinear realization of
chiral symmetry. We shall show that in this case equations for masses of
bound states coincide with analogous equations in the linear approach. The
expression for the weak pion decay constant $f_\pi$ does not change
either. All this allows us to get common rules for constructing chiral
expansions in the linear and nonlinear approaches.
In section 3 we derive the $\pi\pi$ scattering amplitudes and the main
low-energy characteristics of the process in question. Theoretical
derivations of the previous section will be the starting point. At first
we shall use the model with linear realization of chiral symmetry. In this
section our calculations are an extension of earlier investigations [2,
15]. The corresponding amplitude includes the exchange of scalar and
vector mesons as well as quark box diagrams. We give explicit expressions
for this amplitude in terms of a few standard integrals. We obtain the
low-energy expansion for the scattering amplitude and recover Weinberg's
\cite{16} formula (to lowest order in
the energy expansion). Examining the model arising from classification
of collective excitations based on nonlinear representations of the chiral
group, we arrive at the same expression for the $\pi\pi$ scattering
amplitude. We demonstrate this point by direct calculations and comparing
the different contributions. In this part of the paper we not only generalize
the known result of \cite{14}, we also prove the equivalence of this approach
with the other known results. Our method looks like a bridge between
different
NJL models. Finally we present our results, comparing different variants of
the NJL model with experiment and chiral perturbation results.
In section 4 we calculate the scalar form factor of the pion.
We pay particular
attention to its relation to the value of the constituent quark mass.
We conclude with a summary of our results.
\section{The theoretical background}
In Ref.\cite{17} the extended NJL model was investigated and it
was shown how to calculate
Fourier transforms of $N$-point Green functions by a simple
method which was proposed in \cite{2}. This calculation
was done in the linear realization of
chiral symmetry. In this section we are developing the method to apply it
to the extended NJL model with nonlinear realization of chiral symmetry.
To describe collective excitations we shall use induced representations of
the chiral group. Calculations of quark loop diagrams will involve
renormalization of collective variables,
as is typical of our approach. As a result, we
are able to calculate amplitudes of physical processes to
any accuracy in $E^2$ in models with nonlinear realization of chiral
symmetry.
\subsection{The linear approach}
In this section we summarize previous results (see also paper \cite{17}).
The reader who is familiar with the momentum-space bosonization method can
use this section as a collection of our notation and main formulae to be
found in this paper. The main idea of this method consists in the
construction of special bosonic variables to be used for the description
of the observable mesonic states. As a result, it extends the usual
treatment of bosonized NJL models, which was formulated in \cite{8}
and developed in [9,10,12]. The standard approach is essentially
linked to the derivative expansion of the effective meson Lagrangian.
The momentum-space bosonization method does not use this approximation.
It involves the bosonization procedure and has all the advantages of the pure
fermionic approach (Hartree-Fock plus Bethe-Salpeter approximation).
Instead of step-by-step construction of the effective Lagrangian
describing the dynamics of collective excitations we develop a method that
allows direct calculation of amplitudes for particular physical processes.
The amplitudes derived accumulate the entire information on the process under
investigation, just as if we had a total effective meson Lagrangian of the
bosonized NJL model (in the one-loop approximation).
Consider the extended $SU(2)_L\otimes SU(2)_R$ NJL Lagrangian with the
local four-quark interactions
\begin{eqnarray}
\label{la:1}
L(q)=\bar{q}(i\gamma^\mu\partial_\mu -\widehat{m})q
\!&+&\!\frac{G_S}{2}
\left[(\bar{q}\tau_aq)^2+(\bar{q}i\gamma_5\tau_aq)^2
\right]\nonumber \\
\!&-&\!\frac{G_V}{2}
\left[(\bar{q}\gamma^{\mu}\tau_aq)^2+
(\bar{q}\gamma_{5}\gamma^{\mu}\tau_aq)^2
\right],
\end{eqnarray}
where $\bar{q}=(\bar{u}, \bar{d})$ are coloured $(N_c=3)$ current quark
fields with current mass $\widehat{m}=\mbox{diag}(\widehat{m}_u,
\widehat{m}_d)$, $\tau_a=(\tau_0, \tau_i),\, \tau_0=I,\, \tau_i\, (i=1, 2,
3)$ are the Pauli matrices of the flavour group $SU(2)_f$. The constants of
the four-quark interactions are $G_S$ for the scalar and pseudoscalar cases,
$G_V$ for the vector and the axial-vector cases. The current mass term
explicitly breaks the $SU(2)_L\otimes SU(2)_R$ chiral symmetry of the
Lagrangian (\ref{la:1}). In what follows, we shall only consider the
isospin symmetrical case $\widehat{m}_u= \widehat{m}_d=\widehat{m}$. With
boson fields introduced in the standard way, the Lagrangian takes the form
\begin{eqnarray}
\label{la:2}
L(q, \bar{\sigma}, \tilde{\pi}, \tilde{v}, \tilde{a})\!
&=&\!\bar{q}\left(i\gamma^\mu\partial_\mu -\widehat{m}
+\bar{\sigma}
+i\gamma_5\tilde{\pi}
+\gamma^\mu\tilde{v}_\mu
+\gamma_5\gamma^\mu\tilde{a}_\mu
\right)q\nonumber \\
\!&-&\!\frac{\bar{\sigma}^2_a+\tilde{\pi}^2_a}{2G_S}
+\frac{\tilde{v}^2_{\mu a}+\tilde{a}^2_{\mu a}}{2G_V}.
\end{eqnarray}
Here $\bar{\sigma}=\bar{\sigma}_a\tau_a,\ \tilde{\pi}=\tilde{\pi}_a\tau_a,
\ \tilde{v}_{\mu}=\tilde{v}_{\mu a}\tau_a,\ \tilde{a}_{\mu}=\tilde{a}_{\mu a}
\tau_a$. The vacuum expectation value of the scalar field $\bar{\sigma}_0$
turns out to be different from zero $(<\bar{\sigma}_0>\neq 0)$. To obtain
the physical field $\tilde{\sigma}_0$ with $<\tilde{\sigma}_0>=0$ one
performs a field shift leading to a new quark mass $m$ to be identified with
the mass of the constituent quarks
\begin{equation}
\label{la:3}
\bar{\sigma}_0-\widehat{m}=\tilde{\sigma}_0-m,\qquad
\bar{\sigma}_i=\tilde{\sigma}_i,
\end{equation}
where $m$ is determined from the gap equation (see (\ref{la:9}) below).
Let us integrate out the quark fields in the generating functional
associated with the Lagrangian (\ref{la:2}). Evaluating the resulting quark
determinant by a loop expansion one obtains
\begin{eqnarray}
\label{la:4}
L(\tilde{\sigma}, \tilde{\pi}, \tilde{v}, \tilde{a})\!
&=\!&-i\mbox{Tr}\ln
\left[1+
(i\gamma^\mu\partial_\mu -m)^{-1}
(\tilde{\sigma}+i\gamma_5\tilde{\pi}+\gamma^\mu\tilde{v}_\mu
+\gamma_5\gamma^\mu\tilde{a}_\mu )
\right]_{\Lambda}\nonumber \\
\!&-&\!\frac{\bar{\sigma}^2_a+\tilde{\pi}^2_a}{2G_S}
+\frac{\tilde{v}^2_{\mu a}+\tilde{a}^2_{\mu a}}{2G_V}.
\end{eqnarray}
The NJL model belongs to the set of nonrenormalizable theories. Hence, to
define it completely as an effective model, a regularization scheme must
be specified to deal with the quark-loop integrals in harmony with general
symmetry requirements. As a result, an additional parameter $\Lambda$
appears, which characterizes the scale of the quark-antiquark forces
responsible for the dynamic chiral symmetry breaking. From the meson mass
spectrum it is known that $\Lambda\sim 1\,\hbox{GeV}$. Here, we shall make
use of the a modified
Pauli--Villars \cite{18} regularization, which preserves gauge
invariance and chiral symmetry.
In this form it was used in \cite{19}-\cite{20}. The
Pauli--Villars cut-off $\Lambda$ is introduced in the following way,
\begin{equation}
\label{la:5}
e^{-im^2z}
\rightarrow R(z)=e^{-im^2z}\left[1-(1+iz\Lambda^2)
e^{-iz\Lambda^2}\right],
\end{equation}
\begin{equation}
\label{la:6}
m^2e^{-im^2z}
\rightarrow iR'(z)
=m^2R(z)-iz\Lambda^4e^{-iz(\Lambda^2+m^2)},
\end{equation}
where only one Pauli--Villars regulator has been introduced. In this case
the expressions for the basic loop integrals $I_i$ coincide with those
obtained by the usual covariant cut-off scheme. Let us give our
definitions for these integrals. To simplify the formulae we introduce the
following notation
\begin{equation}
\label{la:7}
\Delta (p)=\frac{1}{p^2-m^2},\qquad
\tilde{d}^4q=\frac{d^4q}{(2\pi )^4}.
\end{equation}
Then we have
\begin{eqnarray}
\label{la:8}
&&I_1=iN_c\int\tilde{d}^4q\Delta (q)
=\frac{N_c}{(4\pi )^2}\left[\Lambda^2-
m^2\ln\left(1+\frac{\Lambda^2}{m^2}\right)\right],\\
&&I_2(p^2)=-iN_c\int\tilde{d}^4q\Delta (q)\Delta (q+p) \nonumber \\
&&\ \ \ \ \ \ \ \ \
=\frac{N_c}{16\pi^2}\int_0^1dy\int_0^{\infty}\frac{dz}{z}R(z)
e^{\frac{i}{4}zp^2(1-y^2)}, \\
&&J_2(p^2)=\frac{3N_c}{32\pi^2}\int_0^1dy(1-y^2)\int_0^{\infty}\frac{dz}{z}
R(z)e^{\frac{i}{4}zp^2(1-y^2)},\\
&&I_3(p_1, p_2)=-iN_c\int\tilde{d}^4q\Delta (q)\Delta (q+p_1)\Delta
(q+p_2),\\
&&I_4(p_1,p_2,p_3)=-iN_c\int\tilde{d}^4q\Delta (q)\Delta (q+p_1)
\Delta (q+p_2)\Delta (q+p_3).
\end{eqnarray}
Consider the first terms of the logarithm expansion in (\ref{la:4}). From
the requirement for the terms linear in $\tilde{\sigma}$ to vanish we get
a modified gap equation
\begin{equation}
\label{la:9}
m-\widehat{m}=8mG_SI_1.
\end{equation}
The terms quadratic in the boson fields lead to the amplitudes
\begin{eqnarray}
\label{la:10}
\Pi^{PP}(p^2)
&=&\left[8I_1-G^{-1}_S+p^2g^{-2}(p^2)\right]
\varphi^+_P\varphi^-_P,\\
\Pi^{SS}(p^2)
&=&\left[8I_1-G^{-1}_S+(p^2-4m^2)g^{-2}(p^2)\right]
\varphi^+_S\varphi^-_S,\\
\Pi^{VV}(p^2)
&=&\left[g^{\mu\nu}G_V^{-1}+4(p^{\mu}p^{\nu}-g^{\mu\nu}p^2)
g_V^{-2}(p^2)\right]\varepsilon^{*V}_{\mu}(p)
\varepsilon^V_{\nu}(p),\\
\Pi^{AA}(p^2)
&=&\left[g^{\mu\nu}\left( G_V^{-1}+4m^2g^{-2}(p^2)\right)
\right.\nonumber \\
& &\quad +\left. 4(p^{\mu}p^{\nu}-g^{\mu\nu}p^2)
g_V^{-2}(p^2)\right]\varepsilon^{*A}_{\mu}(p)
\varepsilon^A_{\nu}(p),\\
\Pi^{PA}(p^2)
&=&2img^{-2}(p^2)p^{\mu}\varepsilon^{*A}_{\mu}(p)
\varphi^-_P,\\
\Pi^{AP}(p^2)
&=&-2img^{-2}(p^2)p^{\mu}\varepsilon^A_{\mu}(p)
\varphi^+_P.
\end{eqnarray}
Here $\varepsilon^V_{\mu}(p), \varepsilon^A_{\mu}(p)$
are the polarization vectors of the vector and axial-vector fields.
We have introduced the symbols $\varphi^-_P=1$ and $\varphi^-_S=1$ to
explicitly show the pseudoscalar and scalar field contents of the
pertinent two-point functions. The functions $g(p^2)$ and $g_V(p^2)$
are determined by the following integrals
\begin{equation}
\label{la:11}
g^{-2}(p^2)= 4I_2 (p^2),
\end{equation}
\begin{equation}
\label{la:12}
g^{-2}_V(p^2)=\frac{2}{3}J_2 (p^2).
\end{equation}
Let us diagonalize the quadratic form $(14)+(17)+(18)+(19)$
by redefining the axial fields
\begin{eqnarray}
\label{la:13}
\varepsilon^{A}_{\mu}(p)
&\rightarrow &
\varepsilon^{A}_{\mu}(p)-i\beta (p^2)p_{\mu}
\varphi^{-}_{P}, \\
\varepsilon^{*A}_{\mu}(p)
&\rightarrow &
\varepsilon^{*A}_{\mu}(p)+i\beta (p^2)p_{\mu}
\varphi^{+}_{P}.
\end{eqnarray}
This determines the function $\beta (p^2)$,
\begin{equation}
\label{la:14}
\beta (p^2)=\frac{8mI_2(p^2)}{G_V^{-1}+16m^2I_2(p^2)}.
\end{equation}
Consequently, one has no more mixing
between pseudoscalar and axial-vector fields.
The self-energy of the pseudoscalar field takes the form
\begin{equation}
\label{la:15}
\Pi^{PP}(p^2)=
\left[8I_1-G^{-1}_S+p^2g^{-2}(p^2)
\left(1-2m\beta(p^2)\right)\right]
\varphi^+_P\varphi^-_P.
\end{equation}
Now we can construct special boson variables that will describe
the observed mesons. These field functions $\phi$\footnote{Here and in the
following we will use the common symbol $\phi$ for the all set of
meson fields: $\pi _a, \sigma _a, v_a, a_a$.}
correspond to bound quark -- antiquark states and are derived via the
following transformations
\begin{eqnarray}
\label{la:16}
\tilde{\pi}^a(p)&=&Z^{-1/2}_{\pi}g_{\pi}(p^2)\pi^a(p),\\
\tilde{\sigma}^a(p)&=&Z^{-1/2}_{\sigma}g(p^2)\sigma^a(p),\\
\tilde{v}^a(p)&=&\frac{1}{2}Z^{-1/2}_{v}g_V(p^2)v^a(p),\\
\tilde{a}^a(p)&=&\frac{1}{2}Z^{-1/2}_{a}g_V(p^2)a^a(p),
\end{eqnarray}
where
\begin{equation}
\label{la:17}
g_{\pi}(p^2)=\frac{g(p^2)}{\sqrt{1-2m\beta (p^2)}}
=g(p^2)\sqrt{1+16m^2G_VI_2(p^2)}.
\end{equation}
The new bosonic fields have the self-energies
\begin{eqnarray}
\label{la:18}
\Pi ^{\pi , \sigma}_{ab}(p^2)\!&=&\!\delta _{ab}Z^{-1}_{\pi , \sigma}
\left[p^2-m^2_{\pi , \sigma}(p^2)\right], \nonumber \\
\Pi ^{v, a}_{\mu\nu , ab}(p^2)\!&=&\!\delta _{ab}Z^{-1}_{v, a}
\left\{p_{\mu}p_{\nu}-g_{\mu\nu}\left[ p^2-m^2_{v, a}(p^2)\right]
\right\}.
\end{eqnarray}
The $p^2$-dependent masses are equal to
\begin{eqnarray}
\label{la:19}
m^2_{\pi}(p^2)\!&=&\! (G^{-1}_S-8I_1)g^2_{\pi}(p^2)
=\widehat{m}(mG_S)^{-1}g^2_\pi (p^2),\\
m^2_{\sigma}(p^2)\!
&=&\!\left[1-2m\beta (p^2)\right]m^2_{\pi}(p^2)+4m^2,\\
m^2_v(p^2)\!&=&\!\frac{g^2_V(p^2)}{4G_V}=\frac{3}{8G_VJ_2(p^2)},\\
m^2_a(p^2)\!&=&\! m^2_v(p^2)+6m^2\frac{I_2(p^2)}{J_2(p^2)}.
\end{eqnarray}
These equations coincide with the conditions for appearance of
quark -- antiquark bound state as deduced in the pure fermion approach from
analysis of the Bethe -- Salpeter equations.
The constants $Z_\phi$ are determined by the requirement
that the inverse meson field propagators $\Pi^\phi$
satisfy the normalization conditions
\begin{eqnarray}
\label{la:20}
\Pi ^{\pi , \sigma}(p^2)\!&=&\!
p^2-m^2_{\pi , \sigma}+{\cal O}\left(
(p^2-m^2_{\pi , \sigma})^2\right), \nonumber \\
\Pi ^{v, a}_{\mu\nu}(p^2)\!&=&\! -g_{\mu\nu}\left[
p^2-m^2_{v, a}+{\cal O}\left(
(p^2-m^2_{v, a})^2\right)\right],
\end{eqnarray}
around the physical mass points $p^2=m^2_\phi$,
respectively. The conditions (\ref{la:20}) lead to the values
\begin{eqnarray}
\label{la:21}
Z_{\pi}\!&=&\! 1+\frac{m^2_{\pi}[1-2m\beta (m^2_\pi )]}{I_2(m^2_\pi )}
\frac{\partial I_2(p^2)}{\partial p^2}
\bigg\vert_{p^2=m^2_{\pi}},\\
Z_{\sigma}\!&=&\! 1+\frac{
m^2_{\sigma}-4m^2}{I_2(m^2_{\sigma})}
\frac{\partial I_2(p^2)}{\partial p^2} \bigg\vert_{p^2=
m^2_{\sigma}},\\
Z_v\!&=&\! 1+\frac{m^2_v}{J_2(m^2_v)}
\frac{\partial J_2(p^2)}{\partial p^2} \bigg\vert_{p^2=
m^2_v},\\
Z_a\!&=&\! 1+\frac{m^2_a}{J_2(m^2_a)}
\frac{\partial J_2(p^2)}{\partial p^2} \bigg\vert_{p^2=
m^2_a}-\frac{6m^2}{J_2(m^2_a)}
\frac{\partial I_2(p^2)}{\partial p^2} \bigg\vert_{p^2=
m^2_a}.
\end{eqnarray}
In the following, when omitting an argument of a running coupling constant
or a running mass, we always assume that its value is taken on the
mass-shell of the corresponding particle. The symbol of this particle
will be used for that. For example, on the pion mass-shell
$m^2_\pi (p^2\! =\! m^2_\pi )=m^2_\pi ,\ \beta(m^2_\pi )=\beta_\pi$ and
so on.
Using the expressions (\ref{la:18}), one can obtain the two-point meson
Green functions $\Delta^\phi (p)$. For example, in the scalar
and vector field case the relations
\begin{equation}
\label{la:22}
\Pi^{\sigma}_{ab}(p^2)\Delta^{\sigma}_{bc}(p^2)=\delta_{ac}, \quad
\Pi^v_{\mu\nu ,ab}(p^2)\Delta^{v, \nu\sigma}_{bc}(p)=\delta_{ac}
\delta^{\sigma}_\mu
\end{equation}
give
\begin{equation}
\label{la:23}
\Delta^{\sigma}_{ab}(p^2)=\frac{\delta_{ab}Z_\sigma}{p^2-m^2_\sigma (p^2)},
\quad
\Delta^{v, \mu\nu}_{ab}(p)=\frac{\delta_{ab}Z_v}{m^2_v(p^2)}
\frac{p^\mu p^\nu -g^{\mu\nu}m^2_v(p^2)}{p^2-m^2_v(p^2)}.
\end{equation}
This picture corresponds to the calculations in the framework of the
pure fermionic NJL model where the Bethe--Salpeter equation sums an
infinite class of fermion bubble diagrams.
\subsection{The non-linear approach}
Phenomenological meson fields with appropriate transformation properties
under a nonlinear action of the chiral
group can be introduced as follows \cite{wein}, \cite{cwz}. Let $G$ be a
continuous symmetry group of the initial Lagrangian and $H$ a maximum
subgroup of group $G$ which leaves the vacuum invariant. Then an arbitrary
transformation of the group $G$ can be represented as $G=K(\zeta )H(\eta )$,
where $\zeta , \eta$ are the parameters determining the parametrization of
the $G$ group space. Acting from the left on the $G$ group element by an
arbitrary transformation of the same group
$G(g)K(\zeta )H(\eta )=K(\zeta ')H(\eta ')$,
one can find out how the parameters $\zeta$ and $\eta$ are transformed
under transformations of the group. It is essential that in this case the
transformation for $\zeta$ does not involve the parameters $\eta$:
$\zeta '=\zeta '(\zeta , g)$. Each parameter $\zeta^i$ is associated with
a local Goldstone field $\pi^i(x)$ so that the local fields $\pi^i(x)$
obey the transformation rule
\begin{equation}
\label{tb:1}
\pi^{i'}(x)=\zeta^{i'}(\pi^i(x), g).
\end{equation}
The non-linear transformation of the group $G$ on the matter fields is
constructed in the following way,
\begin{equation}
Q \rightarrow Q' = h(\pi ,g) Q ,
\end{equation}
\begin{equation}
R \rightarrow R' = h(\pi ,g) R h^\dagger (\pi ,g) ,
\end{equation}
where $Q$, are the quark fields, $R$, the vector, axial-vector or scalar H
multiplets, and $h(\pi ,g) \!\in\! H$.
Let us consider the Lagrangian (\ref{la:2}). In this case $G=SU(2)_L\otimes
SU(2)_R$. The quark field $q(x)$ can be represented as
$q(x)=q_L(x)+q_R(x)$, where $q_L(x)=P_Lq(x),\ q_R(x)=P_Rq(x)$. The
projection operators $P_{L,R}$ are $P_{R,L}=(1\pm\gamma_5)/2$. The fields
$q_{L,R}(x)$ transform linearly under action of chiral subgroups
$SU(2)_{L,R}$:
\begin{equation}
\label{tb:2}
q_{L}(x)\rightarrow g_{L}(x)q_{L}(x), \qquad
q_{R}(x)\rightarrow g_{R}(x)q_{R}(x).
\end{equation}
Let us introduce the notation $\bar{\sigma}_a+i\gamma_5\tilde{\pi}_a=M_a$.
Then we can write $\bar{\sigma}+i\gamma_5\tilde{\pi}=MP_R+M^\dagger P_L$.
Now let us represent the complex $2\times 2$ matrix $M=M_a\tau_a$ as a
product of the unitary matrix $\xi$ and the Hermitian matrix $S$
\begin{equation}
\label{tb:3}
M=\xi S\xi .
\end{equation}
The matrix $\xi$ is parametrized by Goldstone fields, and its
transformation law corresponds to the nonlinear transformation
(\ref{tb:1})
\begin{equation}
\label{tb:4}
\xi\rightarrow g_L(x)\xi h^\dagger (\pi , g_{L,R})=
h(\pi , g_{L,R})\xi g^\dagger_R(x).
\end{equation}
The map $\xi (\pi )\! : G/H\rightarrow G$ is thus the local section of the
principal $H$-bundle $G\rightarrow G/H$, where $h(\pi , g)\!\in\! H$ is a
compensating $H$ transformation which brings us back to our canonical
choice for coset representative in the new coset specified by $\pi '$.
The exponential parametrization $\xi (\pi )=\exp [i\pi /(2F)]$ corresponds to
the choice of a normal coordinate system in the coset space $G/H$.
New quark variables
\begin{equation}
\label{tb:5}
Q_R=\xi q_R,\qquad Q_L=\xi^\dagger q_L,\qquad Q=Q_R+Q_L
\end{equation}
are transformed by the nonlinear representation of the group
$G$, eq.(44), and can be used to describe the
constituent quark fields in the approach under consideration.
Let us rewrite the Lagrangian (\ref{la:2}) as
\begin{eqnarray}
\label{tb:6}
L&=&{\bar Q}\left[i\gamma^\mu\nabla_\mu +S-\frac{1}{2}(\Sigma
+\gamma_5\Delta )+\gamma^\mu\left(W_\mu^{(+)}
-\gamma_5W_\mu^{(-)}\right)\right]Q
\nonumber \\
&-&\frac{1}{4G_S}\mbox{\rm Tr}S^2+\frac{1}{4G_V}\mbox{\rm Tr}
\left(W_\mu^{(+)}W_\mu^{(+)}+W_\mu^{(-)}W_\mu^{(-)}\right).
\end{eqnarray}
Here we use the following notations:
\begin{eqnarray}
\label{tb:7}
&&\Sigma =\xi^\dagger\widehat{m}\xi^\dagger +\xi\widehat{m}\xi , \qquad
\Delta =\xi^\dagger\widehat{m}\xi^\dagger -\xi\widehat{m}\xi , \\
&&\nabla_\mu =\partial_\mu +\Gamma_\mu -\frac{i}{2}\gamma_5\xi_\mu , \\
&&\Gamma_\mu =\frac{1}{2}\left(\xi\partial_\mu\xi^\dagger +
\xi^\dagger\partial_\mu\xi\right), \quad
\xi_\mu =i(\xi\partial_\mu\xi^\dagger -
\xi^\dagger\partial_\mu\xi ).
\end{eqnarray}
To describe the vector $W_\mu^{(+)}$ and axial-vector $W_\mu^{(-)}$ mesons we
use new variables
\begin{equation}
\label{tb:8}
W_\mu^{(\pm )}=\frac{1}{2}\left[\xi^\dagger
\left(\tilde{v}_\mu +\tilde{a}_\mu\right)\xi\pm
\xi\left(\tilde{v}_\mu -\tilde{a}_\mu\right)\xi^\dagger\right].
\end{equation}
We take the matrix $S$ in the form $S=\widehat{m}-m+s(x)$. Then the
condition for the tadpole not to appear (after integration over quark
variables) in the case of the scalar field $s(x)$ will be the familiar gap
equation (\ref{la:9}).
Among the terms quadratic in meson fields only the amplitudes
\begin{eqnarray}
\label{tb:9}
\Pi^{PP}(p^2)&=&\frac{4}{F^2}\left[2\widehat{m}(\widehat{m}-m)I_1+
(m-\widehat{m})^2p^2I_2(p^2)\right]\varphi^i\varphi^j, \\
\Pi^{PA}(p^2)&=&-\frac{8im}{F}(m-\widehat{m})p^\mu\epsilon^{j*}_\mu
(p)I_2(p^2)\varphi^i, \\
\Pi^{AP}(p^2)&=&\frac{8im}{F}(m-\widehat{m})p^\mu\epsilon^{i}_\mu
(p)I_2(p^2)\varphi^j
\end{eqnarray}
will be different than in the linear case (see (\ref{la:10})-(19)).
Let us consider a standard replacement in a case like this
\begin{equation}
\label{tb:10}
\epsilon^i_\mu (p)\rightarrow\epsilon^i_\mu (p)+
ip_\mu\tilde{\beta}\varphi^i, \quad
\epsilon^{j*}_\mu (p)\rightarrow\epsilon^{j*}_\mu (p)-
ip_\mu\tilde{\beta}\varphi^j.
\end{equation}
The condition for cancellation of nondiagonal terms (absence of the
pseudoscalar -- axial-vector transition) fixes the form of the function
$\tilde{\beta}(p^2)$:
\begin{equation}
\label{tb:11}
\tilde{\beta}(p^2)=\frac{8m(m-\widehat{m})I_2(p^2)}{F[G^{-1}_V
+16m^2I_2(p^2)]}.
\end{equation}
The self-energy of the pseudoscalar mode takes the form
\begin{equation}
\label{tb:12}
\Pi^{PP}(p^2)=\frac{4(m-\widehat{m})^2}{F^2}\left[
\frac{2\widehat{m}}{\widehat{m}-m}I_1+p^2I_2(p^2)
\left(1-\frac{2m\tilde{\beta}(p^2)F}{m-\widehat{m}}\right)
\right]\varphi^i\varphi^j.
\end{equation}
Thus, in the nonlinear case only the pseudoscalar field $\pi (p)$ will have a
renormalization other than in the linear approach. The physical field
$\pi^{ph}(p)$ should be introduced by the following replacement
\begin{equation}
\label{tb:13}
\pi (p)=Z^{-1/2}_\pi\tilde{g}_\pi (p^2)\pi^{ph}(p).
\end{equation}
For other fields the transformations remain unchanged (see (27)-(29)). The
function $\tilde{g}_\pi (p^2)$ is determined from (\ref{tb:12})
\begin{equation}
\label{tb:14}
\tilde{g}^2_\pi (p^2)=\frac{F^2[1+16m^2G_VI_2(p^2)]}
{4(m-\widehat{m})^2I_2(p^2)}
=\left(\frac{Fg_\pi}{m-\widehat{m}}\right)^2.
\end{equation}
Hence it follows in particular that all equations for the masses of collective
modes coincide with the ones in the linear case (\ref{la:19})-(35).
This also applies to the equation of the pion mass, as seen from the chain of
transformations
\begin{equation}
\label{tb:15}
m^2_\pi (p^2)=\frac{8\widehat{m}}{F^2}(m-\widehat{m})\tilde{g}^2_\pi (p^2)I_1
=\frac{\widehat{m}[1+16m^2G_VI_2(p^2)]}{4mG_SI_2(p^2)},
\end{equation}
where we employed the gap equation (\ref{la:9}). The form factor $f_\pi
(p^2)$ appearing in the vertex of the weak pion decay
$\pi\rightarrow\ell\nu_\ell$ is easily found to be
\begin{equation}
\label{tb:16}
f_\pi (p^2)=\frac{4Z^{-1/2}_\pi m(m-\widehat{m})\tilde{g}_\pi (p^2)I_2(p^2)}
{F[1+16m^2G_VI_2(p^2)]}=\frac{m}{\sqrt{Z_\pi}g_\pi (p^2)}.
\end{equation}
The latter equality reveals that it coincides with a similar expression
derived in the linear model \cite{17}. Since the Bethe--Salpeter equation
for the masses of collective states and the expression for the constant $f_\pi$
coincide in the two approaches under consideration, there exists a unified
approach to construct chiral expansions.
\subsection{Chiral expansion}
At low energies the behaviour of scattering amplitudes or matrix elements
for currents can be described in terms of Taylor expansions in powers of
momenta. Yet, singularities, arising from the presence of light pseudoscalar
particles in the theory, restrict the applicability of Taylor expansions.
One has to take into account all these singularities to extend the
divergence region for the series in momenta. This is possible because it
is quite clear why the pion mass is small. The pion is a Goldstone boson
and its mass is expressed in terms of current quarks which make it
different from zero. Current quark masses are small and can be taken into
account through perturbation theory. New combined Taylor expansion in
momenta and current quark masses arising in this case form the basis of
chiral perturbation theory \cite{21}.
Let $\stackrel{\circ}{m}$ and $f$ be the values of the constituent quark mass $m$ and the
pion decay constant $f_\pi$ in the chiral limit where $\widehat{m}=0$. In
this case the pion mass is zero. The chiral series for this quantity
begins with a term linear in $\widehat{m}$:
\begin{equation}
\label{ce:1}
\stackrel{\circ}{m}^2_\pi=\frac{\widehat{m}\stackrel{\circ}{m}}{G_Sf^2}.
\end{equation}
The weak pion decay constant $f_\pi$ is (see eq.(64))
\begin{equation}
\label{ce:2}
f_\pi =\frac{m}{\sqrt{Z_\pi}g_\pi}.
\end{equation}
Using (\ref{la:17}) and (\ref{la:11}), we arrive at
\begin{equation}
\label{ce:3}
f_\pi^2=4Z_\pi^{-1}\delta m^2I_2(m^2_\pi ).
\end{equation}
Hence, at $\widehat{m}\rightarrow 0$ we get
\begin{equation}
\label{ce:4}
f^2=4\!\stackrel{\circ}{\delta}\stackrel{\circ}{m}^2\stackrel{\circ}{I}_2,
\end{equation}
which is non--vanishing in the chiral limit.
Here and below we use the following notation
\begin{equation}
\label{ce:5}
I_2=I_2(0),\quad\stackrel{\circ}{I}_2=\lim_{\widehat{m}\rightarrow 0}I_2.
\end{equation}
\begin{equation}
\label{ce:6}
\quad\delta =1-2m\beta_\pi ,\quad\stackrel{\circ}{\delta}=\lim_{\widehat{m}
\rightarrow 0}\delta .
\end{equation}
All these quantities are convenient to simplify the formulae. Note that
the equality
\begin{equation}
\label{ce:7}
1-4G_Vf^2=\stackrel{\circ}{\delta}
\end{equation}
is valid. We shall employ the gap equation (\ref{la:9}) to expand the
constituent quark mass in a series in powers of the current
quark mass,\footnote{We remark that only analytic terms in the current
quark masses can appear within the Hartree approximation employed here.}
\begin{equation}
\label{ce:8}
m=\sum_{i=0}^{\infty}c^{(m)}_i\widehat{m}^i .
\end{equation}
Here
\begin{equation}
\label{ce:9}
c_0^{(m)}=\stackrel{\circ}{m} =\lim_{\widehat{m}\rightarrow 0} m,\qquad
c_i^{(m)}=\lim_{\widehat{m}\rightarrow 0} \frac{1}{i!}
\frac{\partial^i m}{\partial\widehat{m}^i}.
\end{equation}
The mass $\stackrel{\circ}{m}$ is a solution of the gap equation at $\widehat{m}=0$. To
find other coefficients of the series we differentiate equation
(\ref{la:9}) with respect to the current quark mass $\widehat{m}$. As a
result, we get the following expression
\begin{equation}
\label{ce:10}
4G_{S}M^2(m)=\frac{1}{m'}-\frac{\widehat{m}}{m}.
\end{equation}
We introduced the abbreviation $M^2(m)=4m^2I_2$. In the case of exact chiral
symmetry with $\widehat{m}=0$ one can derive the value of $M^2(\stackrel{\circ}{m} )$ from
equation (68). Bearing in mind these remarks, we get the following
relation from the above equation:
\begin{equation}
\label{ce:11}
c_1^{(m)}=\frac{\stackrel{\circ}{\delta}}{4G_{S}f^2}.
\end{equation}
Other coefficients of the series are calculated through successive
differentiation of the gap equation. For example, differentiating equation
(74) we get
\begin{equation}
\label{ce:12}
8G_SMM'=\frac{\widehat{m}m'}{m^2}-\frac{1}{m}-\frac{m''}{(m')^2}.
\end{equation}
On the other hand, relying on the definition of $M$ one has
\begin{equation}
\label{ce:13}
\frac{MM'}{mm'}=\frac{M^2}{m^2}-\frac{3h_1}{4\pi^2},
\qquad h_1=\left(\frac{\Lambda^2}{\Lambda^2+m^2}\right)^2.
\end{equation}
In the chiral limit it follows from these two equations that
\begin{equation}
\label{ce:14}
c_2^{(m)}=\frac{3(c_1^{(m)})^2}{2\!\stackrel{\circ}{m}}\left(
\frac{\stackrel{\circ}{m}^2\stackrel{\circ}{h}_1\stackrel{\circ}{\delta}}
{2\pi^2f^2}-1\right).
\end{equation}
For simplicity we use the symbol $\stackrel{\circ}{h}_1$ to denote the chiral limit
\begin{equation}
\label{ce:15}
\stackrel{\circ}{h}_1 =\lim_{\widehat{m}\rightarrow 0}h_1=\left(\frac{\Lambda^2}{\Lambda^2+\mch^2}\right)^2 .
\end{equation}
Thus we obtain
\begin{equation}
\label{ce:16}
m=\stackrel{\circ}{m}\left[1+\frac{\stackrel{\circ}{m}_\pi^2\stackrel{\circ}{\delta}}{4\stackrel{\circ}{m}^2}
-\frac{3\stackrel{\circ}{m}_\pi^4\stackrel{\circ}{\delta}^2}{32\stackrel{\circ}{m}^4}
\left(1-\frac{\stackrel{\circ}{m}^2\stackrel{\circ}{h}_1\stackrel{\circ}{\delta}}{2\pi^2f^2}
\right)+{\cal O}(\stackrel{\circ}{m}_\pi^6 )\right].
\end{equation}
Another important example is the chiral expansion for the pion mass. While in
the case of the quark mass we employed the gap equation, here we need a
pion mass equation (\ref{la:19}), which can be conveniently represented as
\begin{equation}
\label{ce:17}
\left(m^2_\pi -4m\widehat{m}\frac{G_V}{G_S}\right)I_2(m^2_\pi )=
\frac{\widehat{m}}{4mG_S}.
\end{equation}
It follows from the equation that this kind of expansion begins with a
term proportional to $\widehat{m}$. We have already called it
$\stackrel{\circ}{m}^2_\pi$ (see (65)). Let us find the first few
coefficients of the series
\begin{equation}
\label{ce:18}
m^2_\pi =\sum_{i=1}^{\infty}c^{(m_\pi )}_i\widehat{m}^i.
\end{equation}
Obviously, $c^{(m_\pi )}_1=\stackrel{\circ}{m}\! (G_Sf^2)^{-1}$. To find other coefficients
of the series we represent equation (81) as
\begin{equation}
\label{ce:19}
4mG_S\left[\sum_{i=0}^{\infty}c^{(m_\pi )}_{i+1}\widehat{m}^i
-4m\frac{G_V}{G_S}\right]I_2(m^2_{\pi})=1.
\end{equation}
Using the expansion (80) we isolate combinations to the same
powers of $\widehat{m}$ on the left-hand side of the equation and set them
equal to zero. Thus we can calculate coefficients of the expansion
(82). For example
\begin{equation}
\label{ce:20}
c^{(m_\pi )}_2=\frac{\stackrel{\circ}{\delta}}{(2f^2G_S)^2}\left[\frac{\stackrel{\circ}{m}^2\stackrel{\circ}{h}_1}{2\pi^2f^2}
\stackrel{\circ}{\delta}(3\stackrel{\circ}{\delta}-1)+
(1-2\stackrel{\circ}{\delta})\right].
\end{equation}
\begin{eqnarray}
\label{ce:21}
&&c^{(m_\pi )}_3=\frac{\stackrel{\circ}{\delta}^2}{4\!\stackrel{\circ}{m}\! (2f^2G_S)^3}\left\{
8\!\stackrel{\circ}{\delta}\! -3+\frac{\stackrel{\circ}{m}^2\stackrel{\circ}{h}_1}{2\pi^2f^2}
\left[\frac{\stackrel{\circ}{\delta}\! (2-10\!\stackrel{\circ}{\delta}
+21\!\stackrel{\circ}{\delta}^2)\!\stackrel{\circ}{m}^2\stackrel{\circ}{h}_1}{2\pi^2f^2}\right.\right.
\nonumber \\
&&\qquad\ \ \ \ \ \ \left.\left.
+7\!\stackrel{\circ}{\delta}\!-20\!\stackrel{\circ}{\delta}^2\!\!
-\frac{4}{5}-\frac{4(15\!\stackrel{\circ}{\delta}^2\!\!
-10\!\stackrel{\circ}{\delta}\! +2)\!\stackrel{\circ}{m}^2}
{5(\Lambda^2+\stackrel{\circ}{m}^2)}\right]\right\}.
\end{eqnarray}
Below we give the results of similar calculations for the pion decay
constant $f_\pi$ on the basis of the expression (\ref{ce:2}).
\begin{equation}
\label{ce:22}
f_\pi =\sum_{i=0}^{\infty}c^{(f_\pi )}_i\widehat{m}^i.
\end{equation}
In this case the corresponding coefficients of the chiral series are
\begin{equation}
\label{ce:23}
c^{(f_\pi )}_0=f,
\end{equation}
\begin{equation}
\label{ce:24}
c^{(f_\pi )}_1=\frac{\stackrel{\circ}{\delta}^2}{4\!\stackrel{\circ}{m}\! fG_S}\left[1-
\frac{3\stackrel{\circ}{m}^2\stackrel{\circ}{h}_1\stackrel{\circ}{\delta}}{(2\pi f)^2}\right],
\end{equation}
\begin{eqnarray}
\label{ce:25}
& &c^{(f_\pi )}_2=-\frac{3\!\stackrel{\circ}{\delta}^3\!\!
(2-\!\stackrel{\circ}{\delta})}{2f(4\!\stackrel{\circ}{m}\! fG_S)^2}+
\frac{\stackrel{\circ}{\delta}^2\stackrel{\circ}{h}_1}{2f(4\pi f^2G_S)^2}
\left\{3\!\stackrel{\circ}{\delta}^2\!\left[\frac{\Lambda^2+2\!\stackrel{\circ}{m}^2}
{\Lambda^2+\!\stackrel{\circ}{m}^2}\right.\right.\nonumber \\
& &\left.\left.+\frac{3}{2}(1-\!\stackrel{\circ}{\delta} )
-\frac{9\!\stackrel{\circ}{m}^2\stackrel{\circ}{h}_1\stackrel{\circ}{\delta}}{(4\pi f)^2}(2-\!\stackrel{\circ}{\delta} )\right]
+\frac{\stackrel{\circ}{m}^2\stackrel{\circ}{h}_1\stackrel{\circ}{\delta}}{(2\pi f)^2}-
\frac{\Lambda^2+3\!\stackrel{\circ}{m}^2}{5(\Lambda^2+\!\stackrel{\circ}{m}^2)}\right\}.
\end{eqnarray}
Concluding the section we point out that chiral expansions derived here
for the main meson characteristics prove to be helpful in establishing
correspondence between the results obtained here and the known low-energy
theorems of current algebra. We shall use them to analyze $\pi\pi$
scattering and to consider the scalar radius of the pion.
\section{$\pi\pi$-scattering}
The formal scheme described in the previous section gives the
possibility of evaluating any mesonic N-point function through the
parameters of the model: $\Lambda , \widehat{m}, G_S, G_V$. Let us
use it for the evaluation of the $\pi\pi$-scattering amplitude. We
have done that in the same approach earlier \cite{2}, but without
considering the spin-one mesons. Now we would like to explore the role
of vector and axial-vector particles in this process.
The model with linearly realized chiral symmetry was already employed for
this purpose \cite{15}. However, the results of \cite{15} should be
considered only as a first approximation to this problem. The $\pi -a_1$
mixing was neglected there. This breaks chiral symmetry, which is
recovered only if the constituent quark mass $m$ goes to infinity. In our
case we exactly reproduce Weinberg's result at the level $E^2$ and we also
evaluate all higher order corrections in $E^2$.
The extended NJL model with nonlinearly realized chiral symmetry has not
been used to study low-energy $\pi\pi$ scattering so far, though some
general conclusions can be drawn from the results of \cite{14}. Chiral
expansion parameters $L_i$ were calculated in the model, the standard
heat kernel method being used to get the first terms of the effective
meson Lagrangian (up-to-and-including the ${\cal O}(E^4)$ terms). We shall
advance farther and find a general (to any accuracy in $E^2$) form of the
$\pi\pi$ scattering amplitude in the principal approximation in $1/N_c$.
In our calculations we use the conventional Mandelstam variables:
\begin{equation}
\label{a:1}
s=(q_1+q_2)^2,\quad t=(q_1-q_3)^2,\quad u=(q_1-q_4)^2
\end{equation}
for the scattering process $\pi^a(q_1)+\pi^b(q_2)\rightarrow
\pi^c(q_3)+\pi^d(q_4)$. The $\pi\pi$ scattering amplitude $T_{ab;cd}$
has the following isotopic structure
\begin{equation}
\label{a:2}
T_{ab;cd}(s,t,u)=\delta_{ab}\delta_{cd}A(s,t,u)+
\delta_{ac}\delta_{bd}A(t,s,u)+
\delta_{ad}\delta_{cb}A(u,t,s).
\end{equation}
It follows that the amplitudes with definite isospin are
\begin{eqnarray}
\label{a:3}
&&T^0(s,t,u)=A(t,s,u)+A(u,t,s)+3A(s,t,u), \nonumber \\
&&T^1(s,t,u)=A(t,s,u)-A(u,t,s), \nonumber \\
&&T^2(s,t,u)=A(t,s,u)+A(u,t,s).
\end{eqnarray}
Meson tree diagrams corresponding to the amplitude $A(s,t,u)$ are of the
same form both in the linear and nonlinear approach (see Fig.1). Only the
internal structure of quark-loop-based meson vertices will be different.
\subsection{The amplitude (linear approach)}
Let us calculate the amplitude $A(s,t,u)$ in the linear approach. The
vector meson and scalar meson exchange diagrams of Fig.1 include the
triangular vertices $\rho\rightarrow\pi\pi$ and $\sigma\rightarrow\pi\pi$.
We calculate these vertices in accordance with the diagrams of Fig.2,
taking into account the $\pi a_1$ mixing. The $\rho^a_\mu
(p)\rightarrow\pi^b(p_1) \pi^c(p_2)$ amplitude is equal to
\begin{equation}
\label{a:4}
M_{\rho\pi\pi}=\frac{1}{4}\mbox{Tr}(\tau_a[\tau_b,
\tau_c]_-)(p_1-p_2)^\mu\varepsilon_\mu (p)
f_{\rho\pi\pi}(p_1,p_2),
\end{equation}
where
\begin{equation}
\label{a:5}
f_{\rho\pi\pi}(p_1,p_2)=\frac{g_\rho (p^2)}{\sqrt{Z_{\rho}}}F(p_1,p_2).
\end{equation}
The function $F(p_1,p_2)$ has the form (for on-shell pions)
\begin{eqnarray}
\label{a:6}
&&F(p^2)=\frac{1}{Z_\pi}\biggl\{1-\frac{m\beta_\pi p^2}{
m^2_\rho (p^2)}+\frac{\delta}{p^2-4m^2_\pi}
\biggl[(p^2-2m^2_\pi )\left(\frac{I_2(p^2)}{I_2(m^2_\pi )}-1
\right) \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+2m^4_\pi\frac{I_3(-p_1, p_2)}{I_2(m^2_\pi )}
\biggr]\biggr\} .
\end{eqnarray}
Here $p=p_1+p_2$, with $p_1, p_2$ being the pion momenta. This function
obeys the requirement of universality of electromagnetic interactions,
$F(0)=1$. To see this one has to use the equality
\begin{equation}
\label{a:7}
I_2(0)-I_2(m^2_\pi )-m^2_\pi I_3(-p_1, p_2)\big\vert_{p^2=0}=
2m^2_\pi\frac{\partial I_2(p^2)}{\partial p^2}
\bigg\vert_{p^2=m^2_{\pi}}.
\end{equation}
The $\sigma^{a}(p)\rightarrow\pi^{b}(p_1)\pi^{c}(p_2)$
amplitude has the form
\begin{equation}
\label{a:8}
M_{\sigma\pi\pi}(p_1,p_2)=\frac{1}{4}\mbox{Tr}\left(\tau_{a}
[\tau_{b},\tau_{c}]_+\right)f_{\sigma\pi\pi}(p_1,p_2).
\end{equation}
The vertex function $f_{\sigma\pi\pi}(p_1,p_2)$ on the pion mass shell is
\begin{eqnarray}
\label{a:9}
&&f_{\sigma\pi\pi}(p^2)=\frac{16mg^2_{\pi}g(p^2)}
{Z_{\pi}\sqrt{Z_{\sigma}}}\left\{\left[1\! -\!\frac{p^2}{4m^2}(1-\delta^2)
\right]\!
I_2(p^2)+\frac{m^2_\pi\delta}{2m^2}(1-\delta )I_2(m^2_\pi )
\right.\nonumber \\
&&\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\frac{\delta^2}{2}(p^2-2m^2_\pi )I_3(-p_1,p_2)\right\}.
\end{eqnarray}
Using these expressions we obtain the contributions of the meson-exchange
diagrams in Fig.1 to the amplitude $A(s,t,u)$,
\begin{equation}
\label{a:10}
A_{\rho}(s,t,u)=4G_V\left[\frac{(s-u)F^2(t)}{1-\frac{8}{3}G_VtJ_2(t)}+
\frac{(s-t)F^2(u)}{1-\frac{8}{3}G_VuJ_2(u)}\right].
\end{equation}
\begin{equation}
\label{a:11}
A_{\sigma}(s,t,u)=\frac{Z_\sigma f^2_{\sigma\pi\pi}(s)}
{m^2_\sigma (s)-s}.
\end{equation}
Now let us consider the remaining diagrams, the boxes. Their total
number is 16. In Fig. 3 we display only structurally different diagrams.
Their calculation is quite cumbersome. Here we collect only the final result,
\begin{eqnarray}
\label{a:12}
&&A_{rest}(s,t,u)=4g^4_\pi Z^{-2}_\pi
\left\{4[s\beta^2_\pi I_2(m^2_\pi )-I_2(s)]
+(1-\delta^2)^2\frac{s}{m^2}
\right.\nonumber \\
&&\times [I_2(s)-I_2(m^2_\pi )]
+\frac{2}{3}\beta^4_\pi [(s-u)tJ_2(t)+(s-t)uJ_2(u)]
\nonumber \\
&&-\left. \delta^2[4(s-2m^2_\pi )I_3(q_1,-q_2)
-4\beta^2_\pi C(s,t,u)+\delta^2B(s,t,u)]\right\},
\end{eqnarray}
where
\begin{eqnarray}
\label{a:13}
&&\!\!\!\!\!
B(s,t,u)=(2m^4_\pi -us)I_4(q_1,q_1+q_2,q_4)
+(2m^4_\pi -ts)I_4(q_1,q_1+q_2,q_3)\nonumber \\
&&\ \ \ \ \ \
-(2m^4_\pi -ut)I_4(q_1,q_1-q_3,q_4)].
\end{eqnarray}
\begin{equation}
\label{a:14}
C(s,t,u)=\left(\frac{s-u}{s+u}\right)\!\left\{(t-2m^2_\pi )
[I_2(t)-I_2(m^2_\pi )]+2m^4_\pi I_3(q_1,q_3)\right\}
+(t\leftrightarrow u).
\end{equation}
Let us consider the low-energy expansion of the total $\pi\pi$-scattering
amplitude
\begin{equation}
\label{a:15}
A(s,t,u)=A_{\rho}(s,t,u)+A_{\sigma}(s,t,u)+A_{rest}(s,t,u).
\end{equation}
The self-consistent procedure to do that is chiral
perturbation theory \cite{21}. The chiral expansion in powers of
external momenta and current quark masses in the case of the
$\pi\pi$ scattering amplitude has to lead to Weinberg's celebrated
theorem \cite{16} at the $E^2$ level,
\begin{equation}
\label{lee:1}
A(s,t,u)=\frac{s-\stackrel{\circ}{m}^2_\pi}{f^2}+{\cal O}(E^4),
\end{equation}
where ${\cal O}(E^4)$ is the short form for the terms of order $q^4,\
\stackrel{\circ}{m}^2_{\pi}\! q^2,\ \stackrel{\circ}{m}^4_\pi$
and higher. In the case at hand the simplest way to get this result is
to consider first of all the momentum expansion and only partly use the
quark mass expansion. We shall do a chiral expansion for the pion mass,
$m_\pi$, and the constituent quark mass, $m$, only at the last stage. In
accordance with these remarks one obtains
\begin{equation}
\label{lee:2}
A_{\rho}=\frac{4g^4_\pi}{Z^2_\pi}
\left(\frac{3s-4m^2_\pi}{m^2}
\right)\!\delta (1-\delta )I_2+{\cal O}(E^4).
\end{equation}
\begin{eqnarray}
\label{lee:3}
&&A_{\sigma}=\frac{4g^4_\pi}{Z^2_\pi}
\left\{4I_2 +\frac{I_2}{m^2}
\left[s\left(2\delta^2\! -1\right)-m^2_\pi\delta (4\delta\! -3)
\right]\right.\nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \
\left. +\frac{h_1}{8\pi^2 m^2}\left[s-3\delta^2(s-2m^2_\pi )\right]
\right\}+{\cal O}(E^4).
\end{eqnarray}
\begin{equation}
\label{lee:4}
A_{rest}=-\frac{4g^4_\pi}{Z^2_\pi}
\left\{4I_2(1-\beta^2_\pi s)
+\frac{h_1}{8\pi^2 m^2}\left[s-3\delta^2(s-2m^2_\pi )\right]
\right\}+{\cal O}(E^4).
\end{equation}
Some remarks are useful here. We use momentum expansions of
integrals $I_2(s)=I_2+(sh_1)/(32\pi^2 m^2)+\ldots ,\
I_3(q_1,q_2)=-3h_1/(32\pi^2 m^2)+\ldots$. One can see that not only
divergent terms $(\sim I_2)$ contribute at this level. There are finite
contributions which are proportional to $h_1$. Full cancellation of
these terms in the full amplitude
\begin{equation}
\label{lee:5}
A(s,t,u)=\frac{4g^4_\pi\delta I_2}{Z^2_\pi m^2}(s-m^2_\pi )+{\cal O}(E^4)
\end{equation}
is a good test of self-consistency of our approach. In order to preserve
chiral symmetry they must start only at the next-to-leading order.
Noting that
\begin{equation}
\label{lee:6}
\frac{4g^4_\pi\delta I_2}{Z^2_\pi m^2}=\frac{I_2}{Z^3_\pi f^2_\pi
I_2(m^2_\pi )}=\frac{1}{f^2}+{\cal O}(E^2),
\end{equation}
we have full agreement of our amplitude with the low-energy theorem
(105).
Now let us consider the $E^4$ correction to Weinberg's result. After some
lengthy calculations one obtains
\begin{equation}
\label{lee:7}
A^{(4)}(s,t,u)=\frac{1}{96\pi^2f^4}\left\{2\bar{\it l}_1(s-2\!\stackrel{\circ}{m}_\pi^2)^2
+\bar{\it l}_2\left[s^2+(t-u)^2\right]\right\}.
\end{equation}
Here
\begin{equation}
\label{lee:8}
\bar{{\it l}}_1=\frac{12\pi^2f^2}{\stackrel{\circ}{m}^2}\stackrel{\circ}{\delta}^2\!
\left[\stackrel{\circ}{\delta}-\frac{\stackrel{\circ}{m}^2\stackrel{\circ}{h}_1}{\pi^2f^2}+
\frac{9\!\stackrel{\circ}{\delta}^3\stackrel{\circ}{m}^4\stackrel{\circ}{h}_1^2}{(2\pi f)^4}\right]
-9\!\stackrel{\circ}{\delta}^4\stackrel{\circ}{h}_2-
\frac{4\pi^2f^2}{\stackrel{\circ}{m}^2\stackrel{\circ}{\delta}}(1-\stackrel{\circ}
{\delta}^2)^2,
\end{equation}
\begin{equation}
\label{lee:9}
\!\!\!\!\!\!\!\!\!
\bar{\it l}_2=3\!\stackrel{\circ}{\delta}^4\stackrel{\circ}{h}_2
+\frac{2\pi^2f^2}{\stackrel{\circ}{m}^2\stackrel{\circ}{\delta}}
(1-\!\stackrel{\circ}{\delta}^2)^2+6\!\stackrel{\circ}{\delta}^2\!
(1-\!\stackrel{\circ}{\delta}^2)\stackrel{\circ}{h}_1 ,
\end{equation}
and
\begin{equation}
\label{lee:10}
\stackrel{\circ}{h}_2=\left(\frac{\Lambda ^2
+3\stackrel{\circ}{m}^2}{\Lambda ^2 +\stackrel{\circ}{m}^2}\right)\!\stackrel{\circ}{h}_1.
\end{equation}
Let us compare our result (111) with the known estimations of the
${\cal O}(E^4)$ terms. One can see that the structure of the term
$A^{(4)}(s,t,u)$ fully agree with the early result of \cite{22}.
There parameters $\bar{\it l}_1$ and $\bar{\it l}_2$ in the limiting case,
when $G_V\rightarrow 0$, i.e. $\stackrel{\circ}{\delta}\rightarrow 1$,
go over into the known result \cite{20}, \cite{2}.
In the case of $SU(3)$ symmetry this set of parameters are known as $L_i$.
They have been obtained in the paper \cite{14} on the basis of $SU(3)\otimes
SU(3)$ NJL Lagrangian with spin-one mesons. The constants $L_i$ specify
the general effective meson Lagrangian at order of $E^4$. One should,
however, not compare them directly to those analyzed by Gasser and
Leutwyler since their analysis includes the effect of meson loops\footnote{
And thus leads to a scale-dependence not present in the NJL model.}
If one considers the
$SU(2)$ limit of the effective meson Lagrangian from \cite{14} and compare it
with the known $SU(2)$ effective meson Lagrangian from \cite{20} one can
relate the $SU(3)$ low-energy parameters $L_i$ with the $SU(2)$ constants
$\bar{\it l}_i$. In particular we have
\begin{equation}
\label{n:4}
L_2=2L_1=\frac{\bar{\it l}_2}{192\pi^2},\quad
L_3=\frac{\bar{\it l}_1-\bar{\it l}_2}{192\pi^2}.
\end{equation}
Or
\begin{equation}
\label{n:5}
L_2=2L_1=\frac{1}{64\pi^2}\left[\stackrel{\circ}{\delta}^4
\stackrel{\circ}{h}_2\! +\frac{2\pi^2f^2}{3\!\stackrel{\circ}{\delta}
\stackrel{\circ}{m}^2}(1-\stackrel{\circ}{\delta}^2)^2
+2\!\stackrel{\circ}{\delta}^2\!
(1-\stackrel{\circ}{\delta}^2)\stackrel{\circ}{h}_1\right].
\end{equation}
\begin{eqnarray}
\label{n:6}
&&L_3=\frac{1}{64\pi^2}\left[-4\!\stackrel{\circ}{\delta}^4
\stackrel{\circ}{h}_2\! -\frac{2\pi^2f^2}{\stackrel{\circ}{\delta}
\stackrel{\circ}{m}^2}(1-\stackrel{\circ}{\delta}^2)^2+2\!\stackrel{\circ}{\delta}^2
\!(4\!\stackrel{\circ}{\delta}^2-3)\stackrel{\circ}{h}_1
\right. \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left.
+\frac{4\pi^2f^2\!\stackrel{\circ}{\delta}^3}{\stackrel{\circ}{m}^2}\left(1-
\frac{3\!\stackrel{\circ}{\delta}\stackrel{\circ}{m}^2\stackrel{\circ}{h}_1}{4\pi^2f^2}\right)^2\right].
\end{eqnarray}
To compare our formulae with similar expressions derived in \cite{14}, we
point out the following rules of correspondence:
\begin{equation}
\label{n:7}
\Gamma (0,x)\sim\frac{4\pi^2f^2}{3\!\stackrel{\circ}{\delta}\stackrel{\circ}{m}^2},
\quad\Gamma (1,x)\sim\stackrel{\circ}{h}_1 ,\quad\Gamma (2,x)\sim\stackrel{\circ}{h}_2,
\quad g_A\sim\stackrel{\circ}{\delta}.
\end{equation}
Here, we give the notation of \cite{14} on the left side and our equivalents
on the right side. Now it is easy to make sure that this part of our
calculations yields results which agree fully with earlier estimates.
Let us calculate scattering lengths $a^I_l$ and effective range
parameters $b^I_l$ up to and including ${\cal O}(E^4)$ terms. The result
is given in the Appendix since it is not new. The subsequent numerical
estimates will reveal that the $E^4$ approximation of the extended NJL
model is a very good approximation to the Hartree solution.
\subsection{The amplitude (non-linear approach)}
Let us calculate the $\pi\pi$ scattering amplitude in the nonlinear
approach. Though we used the same initial Lagrangian, the scheme developed
differs from the approach with linearly realized chiral symmetry. The
difference arises from variation in the structure of vertices describing the
interaction of Goldstone particles with quarks. Note that the structure of
similar vertices for scalar, vector, and axial-vector fields remains
unchanged. As before, we begin with the amplitude $\rho^a_\mu
(p)\rightarrow\pi^b (p_1)\pi^c(p_2)$ (see (93)). To distinguish
linear results from their nonlinear counterparts, we shall use symbols
with a tilde above them. For example, after calculation of the diagrams in
Fig. 4, instead of the function $F(p^2)$ (see (95)) we get a new
expression $\tilde{F}(p^2)$:
\begin{eqnarray}
\label{npp:1}
&&\tilde{F}(p^2)=\frac{4g^2_\pi}{Z_\pi (m-\widehat{m})^2}
\left\{\frac{p^2}{12}(1-g_A^2)J_2(p^2)-\widehat{m}(mg_A-\widehat{m})
I_2(m_\pi^2)
\right. \nonumber \\
&&\!\!\!\!\!\!\!\!\!\! \left. +\frac{(mg_A-\widehat{m})^2}{p^2-4m^2_\pi}
\left[(p^2-2m^2_\pi )\left(I_2(p^2)-I_2(m^2_\pi )\right)
+2m^4_\pi I_3(-p_1,p_2)\right]\right\}.
\end{eqnarray}
The constant $g_A$ is equal to $1-2\tilde{\beta}_\pi F$. It can be related
to the known constant $\delta$: $(mg_A-\widehat{m})=(m-\widehat{m})
\delta$. If $\widehat{m}, p^2\rightarrow 0$, then
$\tilde{F}(p^2)\rightarrow 0$. This off-shell behaviour of the form factor
essentially differs from the behaviour of $F(p^2)$, where $F(0)=1$. The
contribution to the $\pi\pi$ scattering amplitude from the $\rho$ meson
exchange diagram will be described by formula (99), where one
should make a change $F(p^2)\rightarrow\tilde{F}(p^2)$. We call this
contribution $\tilde{A}_\rho (s,t,u)$. For the given amplitude,
owing to the above property of the function $\tilde{F}(p^2)$, the chiral
expansion begins only with the terms ${\cal O}(E^6)$.
The vertex function for the two-pion decay of a scalar
particle\footnote{Similarly to the linear case, we shall use the symbol
$\sigma$ for this particle, though in the Lagrangian the symbol $s$ was
used.} will also differ from the linear one. It
is described by the second group of diagrams in Fig. 4.
\begin{eqnarray}
\label{npp:2}
&&\tilde{f}_{\sigma\pi\pi}(p^2)=\frac{4g(p^2)g^2_\pi}{Z_\pi
\sqrt{Z_\sigma}(m-\widehat{m})^2 }
\left\{(1-2g_A)(mg_A-\widehat{m})m^2_\pi I_2(m^2_\pi )
\right. \nonumber \\
&&\ \ \ \ \ \ \ \ \ \
+\left[mg^2_A+\widehat{m}(1-2g_A)\right]p^2I_2(p^2)
-4m\widehat{m}(m-\widehat{m})I_2(p^2) \nonumber \\
&&\ \ \ \ \ \ \ \ \ \
\left. +2m(mg_A-\widehat{m})^2(p^2-2m^2_\pi )I_3(-p_1, p_2)
\right\}.
\end{eqnarray}
The form factor $\tilde{f}_{\sigma\pi\pi}(p^2)$ also tends to zero at
$\widehat{m}, p^2\rightarrow 0$. Changing $f_{\sigma\pi\pi}$ for
$\tilde{f}_{\sigma\pi\pi}$ in (100) we get the contribution to the
$\pi\pi$ scattering amplitude from the exchange of the given scalar. We
denote it by $\tilde{A}_\sigma (s,t,u)$. The chiral expansion of
$\tilde{A}_\sigma (s,t,u)$ begins with terms of order
${\cal O}(E^4)$.
Now let us consider the remaining group of diagrams displayed in Fig. 5.
Here we do not show diagrams resulting from elimination of $\pi a_1$
mixing. These diagrams are easily derived from the given ones by the
corresponding inserts in vertices $\xi$ with one pion at the end. Without
going into detail, we shall give the total contribution to the $\pi\pi$
scattering amplitude from all the diagrams taken together,
\begin{eqnarray}
\label{npp:3}
&&\tilde{A}_{rest}(s,t,u)=\frac{4g^4_\pi}{Z_\pi^2(m-\widehat{m})^4}
\left\{(mg_A-\widehat{m})\left[(m-3\widehat{m})(s-m^2_\pi )
\right.\right. \nonumber \\
&&\left. +2\widehat{m}(1-2g_A)m^2_\pi\right]\!I_2(m^2_\pi )
+\frac{(1-g_A^2)^2}{24}\left[(s-u)tJ_2(t)+(s-t)uJ_2(u)\right]
\nonumber \\
&&+\widehat{m}\left[\widehat{m}(1-g_A)^2s+2g_A(mg_A-\widehat{m})s
-4\widehat{m}(m-\widehat{m})^2\right]I_2(s)
\nonumber \\
&&+g^2_A(mg_A-\widehat{m})(mg_A-3\widehat{m})s
\left[I_2(s)-I_2(m^2_\pi )\right]
\nonumber \\
&&+(mg_A-\widehat{m})^2
\left[4\widehat{m}(m-\widehat{m})(s-2m^2_\pi )I_3(q_1, -q_2)
\right. \nonumber \\
&& \left.\left. -(1-g^2_A)C(s,t,u)-(mg_A-\widehat{m})^2B(s,t,u)
\right]\right\}.
\end{eqnarray}
Here we use the notation (102) and (103) adopted earlier.
The expressions for the separate amplitudes $A_{\rho ,\sigma , rest}(s,t,u)$
in the linear and nonlinear approaches are different. Before we will consider
the total amplitude $A(s,t,u)=\tilde{A}_\rho +\tilde{A}_\sigma +
\tilde{A}_{rest}$ let us show that its chiral expansion coincide up to and
including the terms ${\cal O}(E^4)$ with the corresponding result in the
linear approach. Indeed, in the main ${\cal O}(E^2)$ approximation we get
Weinberg's result which follows from the term
\begin{equation}
\label{n:1}
\frac{4g^4_\pi}{Z_\pi^2(m-\widehat{m})^4}
(mg_A-\widehat{m})(m-3\widehat{m})(s-m^2_\pi )I_2(m^2_\pi )=
\frac{s-\stackrel{\circ}{m}_\pi^2}{f^2}+\mbox{\cal O}(E^4).
\end{equation}
In the ${\cal O}(E^4)$ approximation the contribution comes from only two
amplitudes, $\tilde{A}_{\sigma}(s,t,u)$ and $\tilde{A}_{rest}(s,t,u)$.
As was established, $\rho$ meson exchange diagrams lead to an amplitude
beginning with the term of the order ${\cal O}(E^6)$. Here we get
\begin{eqnarray}
\label{n:2}
&&\tilde{A}^{(4)}_{\sigma}=\frac{8\!\stackrel{\circ}{m}\!\widehat{m}}{f^4}
\left\{\left[2\!\stackrel{\circ}{m}\!\widehat{m}-s\!\stackrel{\circ}{\delta}^2\!\! -\!\stackrel{\circ}{\delta} (1-2\!\stackrel{\circ}{\delta} )\!\stackrel{\circ}{m}_\pi^2
\right]\!\stackrel{\circ}{I}_2 +\frac{3\!\stackrel{\circ}{\delta}^2\stackrel{\circ}{h}_1}{16\pi^2}(s-2\!\stackrel{\circ}{m}_\pi^2)\right\}
\nonumber \\
&&\ \ \ \ \ \ \ \
+\frac{\stackrel{\circ}{\delta}}{4\!\stackrel{\circ}{m}^2\!\! f^2}\left[\stackrel{\circ}{\delta}\! (s-2\!\stackrel{\circ}{m}_\pi^2)
\left(1-\frac{3\stackrel{\circ}{m}^2\stackrel{\circ}{\delta}\stackrel{\circ}{h}_1}{4\pi^2f^2}\right)+\stackrel{\circ}{m}_\pi^2\right]^2,
\end{eqnarray}
\begin{eqnarray}
\label{n:3}
&&\tilde{A}^{(4)}_{rest}=-\frac{8\!\stackrel{\circ}{m}\!\widehat{m}}{f^4}
\left\{\left[2\!\stackrel{\circ}{m}\!\widehat{m}-s\!\stackrel{\circ}{\delta}^2\!\! -\!\stackrel{\circ}{\delta} (1-2\!\stackrel{\circ}{\delta} )\!\stackrel{\circ}{m}_\pi^2
\right]\!\stackrel{\circ}{I}_2 +\frac{3\!\stackrel{\circ}{\delta}^2\stackrel{\circ}{h}_1}{16\pi^2}(s-2\!\stackrel{\circ}{m}_\pi^2)\right\}
\nonumber \\
&&\ \ \ \ \ \ \ \ \ \
+\frac{(1-\!\stackrel{\circ}{\delta}^2)^2}{6f^4}[(s-u)t+(s-t)u]\stackrel{\circ}{I}_2
-\frac{\stackrel{\circ}{m}_\pi^2\stackrel{\circ}{\delta}}{4\!\stackrel{\circ}{m}^2\!\! f^2}\left[2s\!\stackrel{\circ}{\delta}
+(1-4\!\stackrel{\circ}{\delta} )\!\stackrel{\circ}{m}_\pi^2\right]
\nonumber \\
&&\ \ \ \ \ \ \ \ \ \
+\frac{\stackrel{\circ}{\delta}^2\stackrel{\circ}{h}_1}{8\pi^2f^4}\left\{s(s-\!\stackrel{\circ}{m}_\pi^2)\stackrel{\circ}{\delta}^2
-\stackrel{\circ}{m}_\pi^2\left[3s(1-\!\stackrel{\circ}{\delta} )+2\!\stackrel{\circ}{m}_\pi^2(3\!\stackrel{\circ}{\delta} -2)\right]
\right. \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left. +(1-\!\stackrel{\circ}{\delta}^2)[(s-u)(t+\!\stackrel{\circ}{m}_\pi^2)+(s-t)(u+\!\stackrel{\circ}{m}_\pi^2)]\right\}
\nonumber \\
&&\ \ \ \ \ \ \ \ \ \
-\frac{\stackrel{\circ}{\delta}^4\stackrel{\circ}{h}_2}{8\pi^2f^4}\left[2\!\stackrel{\circ}{m}_\pi^4\! +ut-s(t+u)\right].
\end{eqnarray}
Summing the expressions, we arrive at a result coinciding excately with
the result of similar linear calculations (111). Noteworthy is
that contributions proportional to $\widehat{m}$ are fully cancelled by
the summation. Thus, despite all the differences in the two approaches,
they are equivalent up-to-and-including
the terms of the order ${\cal O}(E^4)$ in the chiral expansion.
\subsection{Equivalence of the linear and the non-linear approach}
Haag's theorem \cite{23} in axiomatic field theory
states the independence of the $S$-matrix elements on mass-shell from
the choice of interpolating fields. In the framework of the Lagrangian
approach the same result exists \cite{24}. In our case it means in
particular that if we did all correctly the total amplitude $A(s,t,u)$
should be the same in both cases. To demonstrate it let us first consider
the $\rho\pi\pi$ form factors $F(p^2)$ and $\tilde{F}(p^2)$
(see (95) and (119)).
One then obtains
\begin{equation}
\label{h:1}
\tilde{F}(p^2)-F(p^2)=\frac{1}{Z_\pi}\left(\frac{m}{m-\widehat{m}}\right)
\left[\frac{p^2}{m^2_{\rho}(p^2)}-1\right].
\end{equation}
It is clear now that on the $\rho$ mass shell these form factors coincide
with each other. We have a similar behaviour for $f_{\sigma\pi\pi}$
and $\tilde{f}_{\sigma\pi\pi}$
(see expressions (98) and (120)),
\begin{equation}
\label{h:2}
\tilde{f}_{\sigma\pi\pi}(p^2)-f_{\sigma\pi\pi}(p^2)=
\frac{4g^2_\pi g(p^2)I_2(p^2)}{Z_\pi\sqrt{Z_\sigma}(m-\widehat{m})}
\left[p^2-m^2_\sigma (p^2)\right].
\end{equation}
Again at $p^2=m^2_\sigma$ one gets zero for this difference.
We are ready now to compare the amplitudes
\begin{equation}
\label{h:3}
\Delta_\alpha (s,t,u)=\tilde{A}_\alpha (s,t,u)-A_\alpha (s,t,u),
\end{equation}
where $\alpha =\rho , \sigma , rest$. The result can be written as
\begin{eqnarray}
\label{h:4}
&&\!\!\!\!
\Delta_\rho (s,t,u)=\frac{4g^4_\pi\delta (1-\delta )}{Z^2_\pi
m(m-\widehat{m})}\left[\left(\frac{2\widehat{m}-m}
{m-\widehat{m}}\right)(3s-4m^2_\pi )I_2(m^2_\pi )
\right. \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left. -D(s,t,u)+2\delta C(s,t,u)\right].
\end{eqnarray}
\begin{eqnarray}
\label{h:5}
&&\!\!\!\!
\Delta_\sigma (s,t,u)=\frac{4g^4_\pi}{Z^2_\pi (m-\widehat{m})^2}
\left\{m^2_\pi\delta I_2(m^2_\pi )+(4m^2-s)I_2(s)
\right. \nonumber \\
&&-4m(m-\widehat{m})\left[2\left(1-\frac{s(1-\delta^2)}{4m^2}\right)I_2(s)
+\frac{m^2_\pi}{m^2}\delta (1-\delta )I_2(m^2_\pi )
\right. \nonumber \\
&&\left.\left. +\delta^2(s-2m^2_\pi )I_3(q_1,-q_2)\right]\right\}.
\end{eqnarray}
\begin{eqnarray}
\label{h:6}
&&\Delta_{rest}(s,t,u)=\frac{4g^4_\pi}{Z^2_\pi}\left\{\left[
\frac{s+4m(m-2\widehat{m})}{(m-\widehat{m})^2}-
\frac{2s(1-\delta^2)}{m(m-\widehat{m})}\right]I_2(s)
\right. \nonumber \\
&&+\frac{m^2_\pi\delta}{m(m-\widehat{m})^2}
\left[4\widehat{m}(1-\!\delta\! )-m+\frac{3s}{m^2_\pi}(1-\!\delta\! )
(m-2\widehat{m})\right]I_2(m^2_\pi )
\nonumber \\
&&+\frac{4m\delta^2}{(m-\widehat{m})}(s-2m^2_\pi )I_3(q_1,-q_2)
\nonumber \\
&&\left.
+\frac{\delta(1-\delta )}{m(m-\widehat{m})}\left[D(s,t,u)-
2\delta C(s,t,u)\right]\right\}.
\end{eqnarray}
Here the notation
\begin{equation}
\label{h:7}
D(s,t,u)=\frac{(1-\delta )}{6\delta m^2}\left(\delta +
\frac{\widehat{m}}{m-\widehat{m}}\right)
[(s-u)tJ_2(t)+(s-t)uJ_2(u)]
\end{equation}
has been used. Summing (128)-(130) it is easy to see that
\begin{equation}
\label{h:8}
\Delta_\rho (s,t,u)+\Delta_\sigma (s,t,u)+\Delta_{rest}(s,t,u)=0.
\end{equation}
This is what we wanted to show.
\subsection{The full result}
Now we can go on and estimate the role of higher order contributions
in the amplitude obtained. For direct comparison with the empirical
numbers we fix parameters in order to obtain the pion decay constant,
$f_\pi$, and the pion mass, $m_\pi$, close to their physical values,
$f_\pi\simeq 93$ MeV and $m_\pi\simeq 139$ MeV, respectively. We take
two sets of parameters, set $I$ with a
constituent quark mass $m \simeq m_\rho /2$, set
$II$ with a rather low quark mass. In the first case we get a bound state
for the $\rho$ - meson, in the other case the $\rho$ - meson mass lies above
the threshold for production of a pair of 'free' quarks, a problem related
to the absence of confinement of the model. Nevertheless, as will be discussed
in the next section, a better description of the scalar form factor
of the pion is only achieved for rather low constituent quark masses, so that,
with due care, it is of interest to study pion observables for this case too.
The model parameter $G_V$ is obtained by fixing the mass of the
$\rho$ meson, $m_\rho\simeq 770$ MeV, as long as $m_\rho < 2 m$. For the
low constituent mass case, the $\rho$ is embedded deeply in the $\bar q q$
continuum. To avoid the complications of defining this isovector-vector state
under such circumstances \cite{fred}, we choose to fit the scattering length
$a_1^1$ to fix $G_V$. This is an unambiguous and rather simple procedure.
As a result, the four parameters of
the model for set $I$ have the values: $G_S=9.41\:\mbox{GeV}^{-2},\ G_V=11.29
\:\mbox{GeV}^{-2},\ m=390\:\mbox{MeV}\ (\widehat{m}=3.9\:\mbox{MeV})$ and
$\Lambda =1$ GeV. With these parameters, we find $f_\pi =92\:\mbox{MeV},
\ m_\pi =139\:\mbox{MeV},\ m_\rho =770\:\mbox{MeV},\ \delta=0.62.$
For set $II$ we obtain $G_S=1.083\:\mbox{GeV}^{-2},\ G_V=8.8\:
\mbox{GeV}^{-2},\ m=200\:\mbox{MeV}\ (\widehat{m}=1.0\:\mbox{MeV})$,
$\Lambda =2.5$ GeV, $f_\pi =92.7\:\mbox{MeV},
\ m_\pi =139\:\mbox{MeV}, \ \delta=0.69.$ For obvious reasons,
phase shifts are always presented for $\sqrt {s} < 2m$.
We also calculate the set of parameters in the limiting case
$G_V\rightarrow 0$ or $\delta \rightarrow 1$, already determined in \cite{2}.
Here we use $G_S=7.74\:\mbox{GeV}^{-2},\quad \Lambda =1\:\mbox{GeV}$
and $m=242\:\mbox{MeV}\quad (\widehat{m}=5.5\:\mbox{MeV})$. Therefore one can
always compare the predictions of the NJL and extended NJL models.
Apart from that, we need values of the main physical quantities in
the chiral limit $\widehat{m}\rightarrow 0$. In this
case we obtain for set $I$, $\stackrel{\circ}{m} =382\:\mbox{MeV},
\ f=90.9\:\mbox{MeV},\ \stackrel{\circ}{\delta} =0.628$ and for set $II$,
$\stackrel{\circ}{m} =183\:\mbox{MeV},
\ f=88.4\:\mbox{MeV},\ \stackrel{\circ}{\delta} =0.73$
The leading term of the chiral expansion for the pion mass is $\stackrel{\circ}{m}_\pi =138.42$
MeV in $I$ and $\stackrel{\circ}{m}_\pi =141$ MeV in set $II.$
First, we give the values for the constants $L_2$ and $L_3$: we have
for set $I$, $L_2=1.2\cdot 10^{-3}$, $L_3=-3.2\cdot 10^{-3}$ and for set
$II$, $L_2=2.0\cdot 10^{-3}$, $L_3=-2.2\cdot 10^{-3}$.
For comparison, we give
the scale--dependent empirical values $L_2 (m_\rho)
=1.2\cdot 10^{-3}$, $L_3 (m_\rho )=-3.6\cdot 10^{-3}$. However such a
comparison has to be taken cum grano salis since the $L_i$ calculated
within the Hartree approximation are simple c-numbers.
Whereas set $I$ lead to values closer to the experimental ones, set $II$ is
compatible with the results of \cite{14}.
Second, we consider the $\pi\pi$ threshold parameters. The results of our
calculations are presented in the Table. The first and second columns list
the numbers based on the chiral expansion ($E^4$ approximation), for sets $I$
and $II$. The formulae used are given in the Appendix. The third and fourth
columns show the results of the exact calculations on the basis of the full
amplitudes $A(s,t,u)$, for $I$ and $II$. We compare them with the results
from the standard NJL approach without vector mesons \cite{2}, within
the current algebra \cite{16}, and with the experimental data \cite{25}.
\begin{table}
\begin{tabular}{||l||l|l|l|l|l|l|l||} \hline
$a^I_l$ & ${\cal O}(E^4) [I]$ & ${\cal O}(E^4) [II]$ & total [I] & total [II]
& NJL & SMT & exp. \\ \hline
$a^0_0$ & 0.16 & 0.19 & 0.17 & 0.19 & 0.19 &0.16 &$0.26\pm 0.05$ \\
$b^0_0$ &0.18&0.25&0.19&0.25&0.27&0.18&$0.25\pm 0.03$ \\
$a^2_0$ &-0.046&-0.043&-0.047&-0.045&-0.044&-0.045&$-0.028\pm 0.012$ \\
$b^2_0$ &-0.088&-0.078&-0.090&-0.079&-0.079&-0.089&$-0.082\pm 0.008$ \\
$a^1_1$ &0.034&0.037&0.038&0.039$^*$&0.034&0.030&$0.038\pm 0.002$ \\
$a^0_2\times 10^{4}$&5.9&21.0&6.9&18.5&16.7&&$17\pm 3$ \\
$a^2_2\times 10^{4}$&-2.1&4.7&-2.5&0.0 &3.2&&$1.3\pm 3$ \\ \hline
\end{tabular}\\[0.5cm]
{\small {\bf Table:} The $\pi\pi$ scattering lengths and effective ranges
in the extended NJL model in comparison with the same calculations in the
original NJL model, without spin-one mesons \cite{2}, and the soft meson
theorems (SMT) \cite{16}. The experimental data are taken from \cite{25}.}
The '$^*$' denotes an input quantity.
\end{table}
{}From this comparison one infers the following.
All scattering lengths and range parameters
are mainly determined by the ${\cal O}(E^4)$ approximation. The effect of
higher-order terms is noticeable only in the D-waves.
If we compare the results of numerical calculations with our
previous estimations of low-energy $\pi\pi$ scattering parameters within
the NJL model without vector mesons, we notice that the
agreement with the experimental data is of comparable quality (for set II)
but poorer for set I. This confirms the observation of Ref.\cite{27}
that within the Hartree approximation one needs to work with a small
constituent quark mass. Similar findings were obtained in Ref.\cite{14}.
Alternatively, one could think of going beyond the Hartree approximation
and, in particular, include pion loops. This is a difficult problem
which deserves further studies \cite{loops}.
In Fig.6a,b we show the S and P-wave phase shifts $\delta_0^0$, $\delta_0^2$
and $\delta_1^1$ for set I in comparison with the available data ($\sqrt s
\le 700$~MeV). The calculated phases are in reasonable agreement with the data.
For set II, we are confined to $\sqrt s \le 400$~MeV and therefore only
show the S-wave phases in Fig.6c. New data in this region will come once
DA$\Phi$NE is operating and $K_{\ell 4}$ decays have been analyzed.
\section{The scalar form factor of the pion}
Another quantity closely related to $\pi\pi$ scattering is the scalar
form factor of the pion, defined by
\begin{equation}
<\pi^a (p')\mid\widehat{m} (u\bar{u}+d\bar{d})\mid \pi^b (p)>=
\delta^{ab}\Gamma_\pi (t)
\label{e:1}
\end{equation}
where $t=(p-p')^2$ is the square of the invariant four-momentum transfer.
In the framework of the original NJL model this quantity was previously
determined \cite{26} and agreed quite well with the empirical scalar form
factor of the pion for low momentum transfers at a fairly small constituent
quark mass of $241.8$ MeV. In particular, the scalar pion radius was
demonstrated to impose powerful constraints on the parameters of the NJL
model. We anticipate that with the large constituent mass as in set I, the
scalar radius will be largely underestimated.
The relevant Feynman diagrams to calculate this quantity
are depicted in Fig.7. There is a sum of
two contributions to the scalar pion form factor in the NJL model: one is
the direct coupling of the operator $\widehat{m}(u\bar{u}+d\bar{d})$
(double-dashed line) to the two pions via a quark triangle (we call it
bare coupling), and the other corresponds to rescattering of quarks into
the scalar meson $\sigma$, which in turn couples to the pions. In Fig.7a
we show these contributions, with the pion legs amputated. After solving the
Bethe--Salpeter equation for this vertex, $\Gamma$, one obtains the scalar
pion form factor by attaching the pion legs, Fig.7b. With eqs.
(20) and (98), the result for the scalar form factor of the
pion $\Gamma_\pi (t)$ can be cast in the form
\begin{equation}
\Gamma_\pi (t)=\frac{\widehat m (4g_\pi)^2}{Z_\pi}
\left[1+\frac{{\cal J}_s(t)g^2(t)}{m^2_\sigma (t)-t}\right]{\cal K}(t)
\label{e:2}
\end{equation}
where
\begin{equation}
{\cal K}(t)=\left[m-\frac{t\beta_\pi}{2}(1+\delta )\right]\! I_2(t)
+m_\pi^2 \beta_\pi\delta I_2(m_\pi^2)
+\frac{m}{2}\delta^2(t-2 m_\pi^2)I_3(p,p'),
\label{e:3}
\end{equation}
and ${\cal J}_s(t)=8I_1+4(t-4m^2)I_2(t)$ is the fundamental quark bubble in
the scalar channel. For the same parameter sets used in the evaluation of the
scattering parameters, we obtain $\Gamma_\pi (0)=1.007m_\pi ^2$, (set $I$),
and $\Gamma_\pi (0)=0.946m_\pi ^2$, (set $II$), consistent
with the $\chi PT$ prediction \cite{21}. By expanding in powers of
$t$ we extract the scalar mean square radius of the pion
\begin{equation}
\frac {\Gamma_\pi(t)}{\Gamma_\pi(0)} =1+\frac{1}{6}<r^2>^s_\pi t+
{\cal O}(t^2).
\label{e:4}
\end{equation}
Its numerical value $<r^2>^s_\pi =0.043\:\mbox{fm}^2$ calculated with the
parameter set $I$ is one order of magnitude smaller than the empirical
value $<r^2>^s_\pi =(0.55 \pm 0.15)\:\mbox{fm}^2$ \cite{27} extracted
from phase shift analyses \cite{28}. For set $II$ we increase the mean scalar
radius $<r^2>^s_\pi =0.53\:\mbox{fm}^2$ in good agreement with the
empirical number. To understand these numbers in more
detail, we investigate the chiral expansion of the scalar form factor
up-to-and-including terms of order $E^4$,
\begin{equation}
\label{e:5}
\Gamma_\pi (0)=\stackrel{\circ}{m}_\pi^2\left\{1+\frac{\stackrel{\circ}{m}_\pi^2\stackrel{\circ}{\delta}}
{2\!\stackrel{\circ}{m}^2}\left[1-2\!\stackrel{\circ}{\delta}\!
-\frac{\stackrel{\circ}{m}^2\stackrel{\circ}{h}_1\stackrel{\circ}{\delta}\!
(1-3\!\stackrel{\circ}{\delta})}{2\pi^2f^2}
\right]\right\}+{\cal O}(E^6).
\end{equation}
In the leading order of the chiral expansion we get exactly the current
algebra result $\Gamma_\pi (0)=\stackrel{\circ}{m}_\pi^2$ \cite{29}, as expected, since all
symmetries are respected in the evaluation of the form factor. The $E^4$
contribution leads to $\Gamma_\pi (0)=1.005\stackrel{\circ}{m}_\pi^2$ for set $I$ and
$\Gamma_\pi (0)=0.943\stackrel{\circ}{m}_\pi^2$ in set $II.$
For the scalar radius of the pion we have
\begin{equation}
\label{e:6}
<r^2>^s_\pi =\frac{3\!\stackrel{\circ}{\delta}^2}{2\!\stackrel{\circ}{m}^2}
\left[1-\frac{3\!\stackrel{\circ}{m}^2\stackrel{\circ}{\delta}\stackrel{\circ}{h}_1}
{(2\pi f)^2}\right]
\end{equation}
\noindent which is $\approx 0.057\:\mbox{fm}^2$ in set $I$ and
$\approx 0.70\:\mbox{fm}^2$ in set $II$,
i.e. in this quantity one observes large
higher order effects. This is not unexpected from previous calculations
in chiral perturbation
theory beyond one loop \cite{28}. These higher order effects are essentially
accounted for by the low constituent quark mass in agreement with the
findings of Ref.\cite{27}. We stress again that alternatively one might want
to consider pion loop effects, which, however, goes beyond the scope of the
present manuscript.
\section{Summary and conclusions}
In the present paper we have developped a method which allows for the
calculation
of $N$-point functions within the extended Nambu--Jona-Lasinio model with
nonlinear realization of chiral symmetry \cite{14}. This extends
the ideas described in \cite{2}. Using the path
integral technique and the Hartree approximation, we work in
momentum space to circumvent the standard way of employing the heat kernel
expansion for constructing an effective meson Lagrangian. With the heat kernel
method, one manages to get only the first few terms of the Lagrangian
(in general the
${\cal O}(E^4)$ approximation) and faces growing difficulties when
calculating terms of yet higher orders. Examining the one-loop
approximation for the effective action in momentum space, we find
transformations of collective variables which permit to calculate these higher
order terms in a straightforward manner. The two-point functions describing
propagation of collective excitations lead to results coinciding with those
obtained from analysis of Bethe--Salpeter equations on bound
quark-antiquark states for two-particle Green functions of quark fields.
Using equations for the mass of constituent quarks (gap equation) and the
pion mass, we construct chiral expansions for these quantities and the
constant $f_\pi$. We have also calculated the $\pi\pi$ scattering
amplitude. This
program is applied both to the linear and nonlinear description of
transformation properties of physical fields. We have demonstrated explicitely
that the $\pi\pi$ scattering
amplitude $A(s,t,u)$ obtained in both approaches is completely equivalent.
We have carried out the chiral expansion for
this amplitude and calculated the pertinent
scattering lengths and range parameters. In addition we have obtained the
corresponding low-energy $\pi \pi$ phase shifts. All these calculations
have been performed for two sets of parameters, the first and the second
one having a large ($\sim 400$ MeV) and a small ($\sim 200$ MeV)
constituent mass, respectively.
In general, the conclusion is
that the corrections derived by considering terms of the effective meson
Lagrangian with higher derivatives are small for S--
and P--waves but are significant in the D--waves. Agreement with the
experimental data is of comparable quality than in the model ignoring vector
modes for the low constituent quark mass case. This conclusion has been
sharpened further by studying
the scalar pion radius. Only for low constituent masses one can describe this
quantity within the Hartree approximation of the (extended) NJL model
\cite{27}. Such low constituent quark masses are also obtained in the currently
popular estimates of the chiral perturbation theory low-energy constants from
extended NJL models \cite{14}. Further research in this direction should be
concerned with a consistent implementation of pion loop effects (for some first
attempts see e.g.
\cite{loops}).
\vspace{1cm}
|
2,869,038,156,364 | arxiv | \section{Introduction}\label{sec:intro}
In Merriam Webster, sarcasm is defined as \emph{“a mode of satirical wit depending for its effect on bitter, caustic, and often ironic language that is usually directed against an individual”\footnote{https://www.merriam-webster.com/dictionary/sarcasm}.}
The use of sarcasm is found to be beneficial for increasing creativity and humor on both the speakers and the addressees in conversations \cite{bowman2015large,burgers2012verbal}. Therefore, researches on sarcasm have an influence on downstream application tasks such as dialogue system, content creation, and recommendation.
Machines with sarcasm are often seen as intelligent, imaginative, and witty, which fits the key goal of \emph{Artificial general intelligence}\footnote{https://en.wikipedia.org/wiki/Artificial\_general\_intelligence}.
Over the years, studies have investigated sarcasm detection and textual sarcasm generation. Sarcasm detection aims to detect whether the input data is sarcastic, which has been explored in some research work \cite{davidov2010semi,gonzalez2011identifying,riloff2013sarcasm,joshi2015harnessing,ghosh2015sarcastic,muresan2016identification,ghosh2017magnets,ghosh2017role}. However, research on sarcasm generation stays in textual (text-to-text) sarcasm generation \cite{joshi2015sarcasmbot,peled2017sarcasm,zhu2019neural,mishra2019modular,chakrabarty2020,oprea2021chandler}, that is, outputting sarcastic text for the input text. Till now, there is no work attempting to generate sarcastic texts for images, while enabling machines to perceive visual information and generate sarcastic text will increase the richness and funniness of content or conversation, and serve downstream tasks such as content creation and dialogue systems.
In this study, we for the first time formulate and investigate a new problem of cross-modal sarcasm generation (CMSG).
Cross-modal sarcasm generation is a challenging problem as it should not only retain the characteristics of sarcasm but also make the information generated in a different modality related to the original modality.
In addition, there should be some inconsistency between the semantic information of the two modalities, which requires imagination. For example, the literal and intended meaning is reversed.
The information of the two modalities should have the effect of enhancing or producing sarcasm.
Sarcasm factors are defined as follows: 1) be evaluative, 2) be based on the inconsistency of the ironic utterance with the context, 3) be based on a reversal of valence between the literal and intended meaning, 4) be aimed at some target, and 5) be relevant to the communicative situation in some way \cite{burgers2012verbal,burgers2011finding}.
Moreover, there is insufficient high-quality cross-modal sarcasm training data, which makes cross-modal sarcasm generation more difficult. Experiment results on one of our baseline BLIP \cite{li2022blip} demonstrate that the existing cross-modal sarcasm dataset \cite{cai2019multi} is unable to solve problems in a supervised way.
To address the above problems, we focus on generating sarcastic texts from images and propose an Extraction-Generation-Ranking based Modular method (EGRM) for unsupervised cross-modal sarcasm generation (shown in Figure \ref{fig:framework}).
We introduce to extract and obtain diverse image information at different levels through image tagging and sentimental descriptive captioning for generating sarcastic texts. A sarcastic texts generation module is proposed to generate a set of candidate sarcastic texts.
In the sarcastic texts generation module, we first reverse the valence (RTV) of the sentimental descriptive caption and use it as the first sentence. Then the cause relation of commonsense reasoning is adopted to deduce the consequence of the image information, and the consequence and image tags are used to generate a set of candidate sarcastic texts.
As the cross-modal sarcasm generation task involves the evaluation from multiple perspectives, we propose a comprehensive ranking method that considers image-text relation, sarcasticness, and grammaticality to rank the candidate texts.
Two examples of the generated image-text pairs are shown in Figure \ref{fig:intro}.
The main contributions of our work are as follows: 1) For the first time, we formulate the problem of cross-modal sarcasm generation and analyze its challenges. 2) We propose a novel and non-trivial extraction-generation-ranking based modular method (EGRM) to address the challenging cross-modal sarcasm generation task. EGRM uses commonsense-based consequence and image tags to generate imaginative sarcastic texts, which makes the two modalities relevant and inconsistent to produce sarcasm.
Moreover, we consider the performance of candidate sarcastic texts from multiple perspectives, including image-text relation, semantic inconsistency, and grammar, and propose a comprehensive ranking method that simultaneously considers the performance of candidate texts from multiple perspectives to select the best-generated text.
Our method does not rely on cross-modal sarcasm training data. 3) Human evaluation results show the superiority of our proposed method in terms of sarcasticness, humor, and overall performance.
\begin{figure*}[t]
\centering
\includegraphics[width=2\columnwidth]{framework.png}
\caption{The overall framework of EGRM. EGRM consists of three modules: image information extraction, sarcastic texts generation, and comprehensive ranking. In the sarcastic texts generation module, RTV reverses the valence of the SD Caption. COMET is a commonsense reasoning method used to infer the consequence of the SD Caption. }
\label{fig:framework}
\end{figure*}
\section{Related Work}
\subsection{Textual Sarcasm Generation}
Research on Textual Sarcasm Generation is relatively preliminary. The limited amount of research on textual sarcasm generation is mainly divided into two categories, one is to generate a \emph{sarcasm response} based on the input utterance \cite{joshi2015sarcasmbot,oprea2021chandler}, and the other is to generate a \emph{sarcasm paraphrase} based on the input utterance \cite{peled2017sarcasm,mishra2019modular,chakrabarty2020}.
Joshi et al. \shortcite{joshi2015sarcasmbot} introduced a rule-based sarcasm generation module named SarcasmBot. SarcasmBot implements eight rule-based sarcasm generators, each of which generates a kind of sarcasm expression.
Peled and Reichart \shortcite{peled2017sarcasm} proposed a novel task of sarcasm interpretation which generate a non-sarcastic utterance conveying the same message as the original sarcastic utterance. They also proposed a supervised sarcasm interpretation algorithm based on machine translation.
However, it is impractical to train supervised generative models with deep neural networks due to the lack of large amounts of high-quality cross-modal sarcasm data. Therefore, we turn to unsupervised approaches.
Mishra et al. \shortcite{mishra2019modular} introduced a retrieval-based framework that employs reinforced neural sequence-to-sequence learning and information retrieval and is trained only using unlabeled non-sarcastic and sarcastic opinions.
Chakrabarty et al. \shortcite{chakrabarty2020} presented a retrieve-and-edit-based framework to instantiate two major characteristics of sarcasm: reversal of valence and semantic incongruity with the context, which could include shared commonsense or world knowledge between the speaker and the listener.
Oprea1 et al. \shortcite{oprea2021chandler} proposed Chandler, a system not only generates sarcastic responses but also explanations for why each response is sarcastic.
However, these works are mainly generating sarcasm text based on input utterance, and there is no existing research on cross-modal sarcasm generation. Enabling machines to perceive visual information and generate sarcasm information for communication will increase the richness and humor of communication and serve downstream tasks such as content creation. Therefore, we focus on cross-modal sarcasm generation.
\subsection{Image Captioning}
Image Captioning is the task of describing the content of an image in words.
Recent works on image captioning have concentrated on using the deep neural network to solve the MS-COCO Image Captioning Challenge\footnote{\url{http://mscoco.org/dataset/\#captions-challenge2015}}.
CNN family is often used as the image encoder and the RNN family is used as the decoder to generate sentences \cite{vinyals2015show,karpathy2015deep,donahue2015long,yang2016review,wang2021high}. Many methods have been proposed to improve the performance of image captioning. Previous work used reinforcement learning methods \cite{ranzato2015sequence,rennie2017self,liu2018context}, high-level attributes detection \cite{wu2016value,you2016image,yao2017boosting}, visual attention mechanism \cite{xu2015show,lu2017knowing,pedersoli2017areas,anderson2018bottom,pan2020x}, contrastive or adversarial learning \cite{dai2017contrastive,dai2017towards}, scene graph detection \cite{yao2018exploring,yang2019auto,shi2020improving} , and transformer \cite{cornia2020meshed,li2019entangled,luo2021dual,ji2021improving,xian2022dual,mao2022rethinking,wang2022geometry,kumar2022dual}.
A slightly related branch of our research in image captioning is sentimental image captioning which generates captions with emotions.
Mathews et al. \shortcite{mathews2016senticap} proposed SentiCap, a switching architecture with factual and sentimental caption paths, to generate sentimental descriptive captions. You et al. \shortcite{you2018image} introduced Direct Injection and Sentiment Flow to better solve the sentimental image captioning problem. Nezami et al. \shortcite{nezami2018senti} proposed an attention-based model namely SENTI-ATTEND to better add sentiments to image captions.
Li et al. \shortcite{li2021image} introduce an Inherent Sentiment Image Captioning (InSenti-Cap) method via an attention mechanism.
However, cross-modal sarcasm generation involves creativity as well as correlations and inconsistencies among different modalities, existing image captioning methods cannot meet the requirement.
\section{Methodology}
Due to the low quality and insufficient quantity of existing cross-modal sarcasm training data, which is confirmed in the experiment results of our pre-trained supervised baseline BLIP, we focus on unsupervised cross-modal sarcasm generation.
However, retrieval-based methods for generating sarcasm sentences are limited by the quality of the retrieval corpus and the ability of multi-keyword retrieval. The sentences generated by rule-based methods are easily limited by the proposed rules and have worse performance on tasks requiring creativity and imagination like sarcastic texts generation. Therefore, we propose a modular cross-modal sarcasm generation method, which has a key component of constrained text generation and is able to generate more imaginative and creative sarcastic texts.
The overall framework of our proposed Extraction-Generation-Ranking based Modular method (EGRM) is shown in Figure \ref{fig:framework}. Given an image, EGRM generates a sarcastic text related to the input image. EGRM consists of three modules: image information extraction, sarcastic texts generation, and comprehensive ranking, as shown in Figure \ref{fig:framework}. The image information extraction module extracts and obtains diverse image information at different levels, including image tags and sentimental descriptive caption (SD Caption). In the sarcastic texts generation module, we first reverse the valence (RTV) of the sentimental descriptive caption and use it as the first sentence. Then the cause relation of commonsense reasoning is adopted to deduce the consequence of the image information, and the consequence and image tags are used to generate a set of rest texts via constrained text generation techniques. The first sentence and each rest text are concatenated to form a candidate sarcastic text set.
At last, we propose a comprehensive ranking module with multiple metrics to measure various aspects of the generated texts and the highly ranked one is selected. As shown in Figure \ref{fig:framework}, the candidate image-text pairs are ranked and selected by using multiple metrics.
\subsection{Image Information Extraction}
As a cross-modal sarcasm generation task, it is crucial to extract and obtain important and diverse information from the input image that is useful for generating sarcastic texts.
We obtain image tags $x_t$ and sentimental descriptive caption $x_c$ from the image. Particularly, a popular object detection method YOLOv5 \cite{glenn_jocher_2022_6222936} is adopted to detect objects in the image and record image tags. SentiCap \cite{mathews2016senticap}, a switching recurrent neural network with word-level regularization, is used to generate sentimental descriptive image caption.
\subsection{Sarcastic Texts Generation}
As shown in the upper-left part of Figure \ref{fig:framework}, there are two branches in the sarcastic texts generation module.
The top branch generates the first sentence $y_f$ from the sentimental descriptive caption (SD Caption) $x_c$. The bottom branch generates a set of rest texts $(y_{r_1},y_{r_2},...y_{r_k})$ from the given sentimental descriptive caption $x_c$ and image tags $x_t$. $k$ represents the total number of generated texts.
Concretely, we generate multiple rest texts by using different pre-trained models with different image tags and consequence collocations as input.
The first sentence is then concatenated with each generated rest text to produce a set of candidate sarcastic texts $Y$, where each candidate text $y_i \in Y$.
The sarcastic texts generation method needs to satisfy the correlation between image and text and also the inconsistency of the two modalities.
This means that the content of the generated text should be related to the image. At the same time, there is some inconsistency in the semantic information of the generated text with regard to the image, such as forming inversion or obtaining some contrast content, which is related to the image but not directly reflected by the image, through certain imagination and reasoning.
Firstly, we obtain our first sentence $y_f$ based on the sentimental descriptive caption generated from the input image to achieve image-text relevance. We \textbf{r}everse \textbf{t}he \textbf{v}alence (RTV) of the caption to make the text and image inconsistent.
Considering that sarcasm usually occurs in positive sentiment towards a negative situation (i.e., sarcastic criticism) \cite{chakrabarty2020,kreuz2002asymmetries}, we invert the negative sentiment expressed by the caption, so that the first sentence contains context with positive sentiment.
Specifically, we obtain the negative score of the evaluative word from SentiWordNet \cite{esuli2006sentiwordnet} and use WordNet \cite{miller1995wordnet} to replace the evaluative words with its antonyms similar to the $R^3$ method \cite{chakrabarty2020}. We do nothing if there is no negative sentiment in the caption.
To sum up, the first sentence is obtained as $y_f = \text{RTV}(x_c)$. For example, for a raining image, we may reverse the first sentence ``a \textbf{bad} rainy day'' to ``a \textbf{good} rainy day'', which produces sarcasm and humor and may enhance sarcasm by the rest generated text.
The key to producing sarcasm is the reversal of valence between the literal and intended meaning as well as the relevance of the communicative situation. In the CMSG task, we should make some semantic inconsistency between the connotation expressed by the text and the real information shown by the image in the specific situation of the image.
To achieve this goal, we propose to use the commonsense-based consequence inferred by information from the image modality and the image tags to generate the rest texts, which will be concatenated after the first sentence.
The reason we use the image information to deduce the consequence $c$ is that commonsense reasoning can infer the cause relation and the possible consequence in the scene shown in the image, making the intention of the sarcasm clearer and the effect of the sarcasm more intense.
Taking the first example in Figure \ref{fig:intro} as an instance, commonsense reasoning result shows that information in the image may cause a \textbf{crash}. We may not feel sarcastic when we read the first sentence ``a man on a surfboard riding a wave in the ocean''. However, we feel sarcastic and funny when we imagine a man riding a wave and suddenly falls down from the surfboard which causes a crash. By using the commonsense-based consequence, the model is able to capture the deeper information contained in the image and imagine possible situations based on the commonsense-based consequence to generate more realistic sarcastic texts.
For inferring commonsense-based consequence, we extract verbs, nouns, adverbs, and adjectives, which denote as $\mathbf{w}$, from the sentimental descriptive caption $x_c$ and feed them to COMET to infer the consequence. Detailed information can be seen in these papers \cite{bosselut2019comet,chakrabarty2020,speer2017conceptnet}. Therefore, the commonsense-based consequence $c$ is obtained by $c = \text{COMET}(\mathbf{w})$.
Using image tags makes the image and text more relevant and makes it clearer who caused the consequence. In this way, we can generate sarcastic texts that are related to the image and inconsistent with the real semantic content.
For instance, both SC-$R^3$ and our method infer the consequence ``crash'' of the first example in Figure \ref{fig:intro}. SC-$R^3$ retrieves sentences from the corpus according to the commonsense-based consequence and gets a sentence ``The ceiling came down with a terrific crash.'', which is irrelevant to the image. The result is not only non-ironic but also confusing. Our method considers image tags and the commonsense-based consequence, and the generated texts have image-text correlation and inconsistency, which makes this image-text pair produce sarcasm.
To implement the cross-modal sarcastic texts generation module, we generate the rest texts based on a recently proposed constrained text generation method CBART \cite{he2021parallel}.
For instance, given image tag ``bananas'' and consequence ``fall down'' as input, the model may generate ``The adults are convinced their bananas will fall down the tree'', which can be seen in Figure \ref{fig:framework}.
As shown in the upper-left part of Figure \ref{fig:framework}, by using different numbers of tags, changing different pre-trained models, and using commonsense-based consequence inferred by information from the image modality, the sarcastic texts generation module can generate a variety of different sarcastic texts for selection. We use four pre-trained models to generate texts which are the base model initialized with BART-base model training on One-Billion-Word \cite{chelba2013one} dataset (base-One-Billion-Word), the base model initialized with BART-base model training on Yelp\footnote{https://www.yelp.com/dataset} dataset (base-Yelp), the large model initialized with BART-large model training on One-Billion-Word dataset (large-One-Billion-Word), and the large model initialized with BART-large model training on Yelp dataset (large-Yelp). Different pre-trained models can generate
diverse rest texts, making our candidate sarcastic texts more abundant.
For more details about CBART, please read the paper of CBART \cite{he2021parallel}.
\subsection{Comprehensive Ranking}
In the CMSG task, we need to convert the image to the text of the target sarcasm style $s_t$.
Given an input image $x$, the conditional likelihood of the generated sarcastic text $y$ is divided into three terms:
\begin{equation}
\begin{aligned}
p(y \mid x, s_{t})
&=\frac{p(y,x, s_{t})}{p(x, s_{t})}
\propto p(x,[y, s_{t}])\\
&=\ p(x \mid[y, s_{t}]) \ \ p([y, s_{t}])\\
&=\underbrace{p(x \mid [y, s_{t}])}_{\text {Image-Text Relation }}
\underbrace{p(s_{t} \mid y)}_{\text {Sarcasticness}} \underbrace{p\ ({\ y \ })}_{\text {Grammaticality}},
\end{aligned}\label{eq:rank}
\end{equation}
where $[\cdot]$ groups related terms (e.g., $[y, s_{t}]$) together. In the CMSG task, the first term of Equation \ref{eq:rank}, $p(x\mid[y, s_{t}])$ measures the \emph{Image-Text Relation} between the input image $x$ and the output target text $y$. It
calculates the correlation between the image and the generated text. The second term, $p(s_{t} \mid y)$, can be seen as a measure of \emph{Sarcasticness}. The third term, $p(y)$, measures the overall \emph{Grammaticality} of the output text $y$, which also shows the fluency of the generated text.
Finally, we rank our $k$ candidate sarcastic texts generated in the cross-modal sarcastic texts generation module according to the decomposition in Equation \ref{eq:rank}. For the $i$-th candidate text $y_i$, the ranking score is computed as:
\begin{equation}
\begin{aligned}
p_{crank}(y_i \mid x, s_{t}) \propto p(x \mid[y_i, s_{t}]) \ p\ (s_{t} \mid y_i) \ p\ (y_i),
\end{aligned}\label{eq:crank}
\end{equation}
where $p_{crank}$ represents the comprehensive ranking probability for $y_i$.
We choose the size of candidate sarcastic texts $k$ by conducting experiments on the validation data. Finally, the average $k$ of our method is 36.
All that remains is to describe how to calculate each term in Equation \ref{eq:crank}. To calculate the first term, image-text relation, we adopt a reference-free metric CLIPScore \cite{hessel2021clipscore} which measures the cosine similarity between the visual CLIP \cite{radford2021learning} embedding $v$ of the image $x$ and the textual CLIP embedding $e$ of a candidate text $y_i$. We presume $p(x\mid[y_i, s_{t}]) = \text{CLIPScore}(x,y_i) = w \cdot max(cos(e, v), 0)$ and $w$ follows the settings of CLIPScore \cite{hessel2021clipscore}, which is set as 2.5.
For calculating the second term, sarcasticness, we use semantic incongruity ranking \cite{chakrabarty2020} which fine-tunes RoBERTa-large \cite{liu2019roberta} on the Multi-NLI \cite{skalicky2018linguistic} dataset to calculate the contradictory score between the first sentence of the image description after reversing the valence and the rest text.
For the third term, we use perplexity (PPL) to calculate the existing probability of the texts, and we use the pre-trained model BERT \cite{kenton2019bert} to calculate the probability.
\section{Experimental Setup}
\subsection{Dataset}
As we do not need parallel cross-modal sarcasm data for training, we conduct the experiment on a testing subset of 503 images in the SentiCap \cite{mathews2016senticap} dataset, which uses images from the MSCOCO \cite{lin2014microsoft} validation partition and adds sentiment captions to those images.
Automatic metrics for each method are calculated on these 503 images. Considering the time and economic cost of human evaluation, we randomly selected 150 images as the test set for human evaluation. Since there are eight systems, the human evaluation is conducted on a total of 1200 image-text pairs.
Datasets for training pre-trained models for the sarcastic texts generation module are the One-Billion-Word \cite{chelba2013one} dataset and the Yelp\footnote{https://www.yelp.com/dataset} dataset.
One-Billion-Word is a public dataset for language modeling produced from the WMT 2011 News Crawl data. The Yelp dataset contains business reviews on Yelp.
\begin{table*}[]
\centering
\resizebox{\textwidth}{2.1cm}{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\textbf{Method} & \textbf{TL} & \textbf{CLIPScore} & \textbf{Sarcasticness} & \textbf{Image-Text Relation} & \textbf{Humor} & \textbf{Grammaticality} & \textbf{Overall} \\ \hline
\textbf{SC-MTS}\shortcite{mishra2019modular} & 9.43 & 19.70 & 0.65 & 0.98 & 0.71 & 0.88 & 0.73 \\ \hline
\textbf{BLIP}\shortcite{li2022blip} & 9.87 & \textbf{27.23} & 1.31 & \textbf{3.29} & 1.91 & 3.31* & 1.95 \\ \hline
\textbf{SC-$R^3$}\shortcite{chakrabarty2020} & 19.11* & 25.15 & 2.22* & 2.86 & 2.21* & 3.30 & 2.29* \\ \hline
\textbf{EGRM} (Ours) & \textbf{25.65} & 25.31* & \textbf{2.85} & \textbf{3.29} & \textbf{2.78} & \textbf{3.41} & \textbf{2.90} \\ \hline \hline
{\textbf{EGRM-woCS}} & {24.99} & {25.14} & {2.24} & {2.97} & {2.27} & {3.37} & {2.38} \\ \hline
\textbf{EGRM-woTag} & 25.99 & 24.78 & 2.26 & 2.91 & 2.28 & 3.32 & 2.37 \\ \hline
\textbf{EGRM-woS} & 30.99 & 24.12 & 2.39 & 2.91 & 2.33 & 3.16 & 2.42 \\ \hline
\textbf{EGRM-woGI} & 26.24 & 25.25 & 2.34 & 2.90 & 2.28 & 3.18 & 2.39 \\ \hline
\end{tabular}}
\caption{Evaluation results of all methods. The scores in columns 4$\sim$8 are human evaluation results, and the scale ranges from 0 (not at all) to 5 (very). The upper part of the table shows the comparison of our method and three baseline methods, and the lower part shows the results of our ablation study. As shown in the upper part of the table, our proposed EGRM has the best performance among all methods on all metrics except CLIPScore, on which EGRM is ranked 2nd (denoted by *).}
\label{tab:res}
\end{table*}
\subsection{Compared Methods}
As CMSG is a new task, we design the following three comparison methods, and the first two methods do not need parallel cross-modal sarcasm training data while the third one relies on such data for training.
\begin{itemize}
\item \textbf{SC-$R^3$}: We use the $R^3$ model released by Chakrabarty et al. \cite{chakrabarty2020} as it is the state-of-the-art textual sarcasm generation system to transform input texts into sarcastic paraphrases. We input the captions generated by SentiCap \cite{mathews2016senticap} to $R^3$ to generate sarcastic texts.
\item \textbf{SC-MTS}: We input the captions generated by SentiCap to MTS \cite{mishra2019modular} to generate sarcastic texts.
\item \textbf{BLIP}: This is a pre-trained image captioning model \cite{li2022blip}, and we fine-tune it on the parallel cross-modal sarcasm dataset proposed by Cai et al. \cite{cai2019multi}. It is considered a representative of the supervised methods.
\end{itemize}
To explore the effectiveness of various parts of our proposed model EGRM, we ablate some components of EGRM and evaluate their performance. These are termed as:
\begin{itemize}
\item \textbf{EGRM-woCS}: the EGRM method without using the commonsense-base consequence to generate sarcastic texts. The goal of EGRM-woCS is to analyze the effect of commonsense reasoning consequence in the sarcastic texts generation module.
\item \textbf{EGRM-woTag}: the EGRM method without using image tags to generate sarcastic texts. The goal of EGRM-woTag is to analyze the effect of image tags in the sarcastic texts generation module.
\item \textbf{EGRM-woS}: the EGRM method without using sarcasticness ranking during comprehensive ranking. The goal of EGRM-woS is to analyze the effect of sarcasticness ranking in the comprehensive ranking module.
\item \textbf{EGRM-woGI}: the EGRM method without using grammaticality ranking and image-text relation ranking during comprehensive ranking. The goal of EGRM-woGI is to analyze the effect of grammaticality ranking and image-text relation ranking in the comprehensive ranking module.
\item \textbf{EGRM}: the complete method with all components.
\end{itemize}
\subsection{Evaluation Criteria}
The difficulty of evaluating the CMSG task is that it is a creative and imaginative study, and there is no standard sarcastic text for reference. In addition, the difference in the average text length generated by different methods may cause problems in traditional generation evaluation metrics. These reasons make traditional generation evaluation metrics like BLEU \cite{papineni2002bleu}, one of the most popular evaluation metrics in text generation tasks, unsuitable in the CMSG task involving creativity and imagination. This problem also exists in textual sarcasm generation task \cite{mishra2019modular,chakrabarty2020}. Therefore, human evaluation is mainly used for evaluation, and we use ClipScore \cite{hessel2021clipscore}, a popular reference-free image captioning metric, to evaluate the image-text relevance.
Referring to the textual sarcasm generation metric WL \cite{mishra2019modular} for calculating the percentage of length increment, the notion behind which is that sarcasm typically requires more context than its literal version and requires to have more words present at the target side, we calculate the length of the generated text to assist in evaluating the performance of the model, and we name this metric total length (TL).
For human evaluation, we evaluate a total of 1200 generated image-text pairs since there are eight different systems with 150 image-text pairs each in our research.
Inspired by the evaluation method in previous work \cite{chakrabarty2020}, we propose five criteria to evaluate the performance of the cross-modal sarcasm generation methods: 1) \textbf{Sarcasticness} (How sarcastic is the image-text pair?),
2) \textbf{Image-Text Relation} (How relevant are the image and text?),
3) \textbf{Humor} (How funny is the image-text pair?) \cite{skalicky2018linguistic},
4) \textbf{Grammaticality} (How grammatical are the texts?) \cite{chakrabarty2020},
5) \textbf{Overall} (What is the overall quality of the image-text pair on the cross-modal sarcasm generation task?).
We design an MTurk CMSG task where each Turker was asked to score the image-text pairs from all the eight methods. Each Turker was given the image together with a set of sarcastic texts generated by all eight systems. Each criterion is rated on a scale from 0 (not at all) to 5 (very much). The Turker can grade with decimals like 4.3. As CMSG is a difficult task requiring imagination, each image-text pair was scored by three individual Turkers. Each Turker is paid \$281.08 for the whole evaluation of 1,200 image-text pairs, which is roughly \$0.23 per image-text pair. Figure \ref{fig:instruction} shows the instructions released to the Turkers.
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{instructions.png}
\caption{Instructions for human evaluation.}
\label{fig:instruction}
\end{figure}
\section{Experimental Results}
\begin{figure*}[t]
\includegraphics[width=2.1\columnwidth]{example.png}
\caption{Examples of generated outputs from different systems.}
\label{fig:res-example}
\end{figure*}
\subsection{Quantitative Results}
Table \ref{tab:res} shows the scores on automatic metrics and human evaluation metrics of different methods. As shown in the upper part of the table, our proposed EGRM has the best performance among all comparison methods on all metrics except CLIPScore, on which EGRM ranks second.
The ablation study in Table \ref{tab:res} demonstrates that our full model EGRM is superior to ablation methods in all criteria except the total length.
In terms of sarcasticness, our full model attains the highest average score, which shows our model meets the most important requirement of the CMSG task. According to the scores, EGRM gets the highest score on the humor criteria, which shows the potential contribution of our method for improving the interestingness and humor in content creation and communication. Moreover, the grammaticality of EGRM is good and the overall score of EGRM is the highest among all the methods.
The total length of the generated paragraph of EGRM is longer than SC-MTS, BLIP, and SC-$R^3$. This can be seen as an auxiliary basis for sarcasm as sarcasm typically requires more context than its literal version and requires to have more words present on the target side.
On the CLIPScore metric, we observe that EGRM does not have better performance than the pre-trained image captioning method BLIP, which is designed for generating textual descriptions of images. However, the CMSG task requires imagination and the method should imagine and generate text that is inconsistent with the image as well as relevant to the image, which leads to the CLIPScore of our method designed for the CMSG task being no better than the pre-trained image captioning method BLIP. Moreover, we can observe that EGRM and BLIP have the best performance among all the four methods on the image-text relation criteria in human evaluation. This is because when human judges consider whether the text is related to the image, they may allow reasonable imagination. Although BLIP has a higher CLIPScore, it cannot solve the CMSG problem due to the poor performance on sarcasticness.
This also shows the existing parallel cross-modal sarcasm data is unable to train a good supervised model for the CMSG task, due to the limitations in scale and quality.
\subsection{Ablation Study}
We concentrate our ablation study on the criteria of sarcasticness and overall performance, as we consider these metrics as the main criteria for the success of cross-modal sarcasm generation.
As shown in Table \ref{tab:res}, the full model (EGRM) outperforms the other four ablation methods.
EGRM-woCS has the worst performance in terms of sarcasticness among the ablation methods. This indicates that the commonsense-based consequence used in the sarcastic texts generation module, which is the inferring result of the image information based on commonsense reasoning, is important for sarcasticness. This is because the inconsistency between the commonsense reasoning consequence and the information of the image modality is the key to generating sarcasticness.
EGRM-woTag has the worst overall performance among the ablation methods. Because the combination of image information and inferring consequence can generate sarcastic image-text pairs where the two modalities are relevant, a text that is not related to the image may be regarded as incomprehensible in the generated text.
The experimental results of EGRM-woCS and EGRM-woTag show that the use of image tags and commonsense-based consequences in the generation module is crucial to generating image-text related and imaginary sarcastic texts.
EGRM-woS ranks first among the four ablation methods in terms of sarcasticness and overall performance while EGRM-woGI is slightly worse than EGRM-woS. However, both EGRM-woS and EGRM-woGI are worse than EGRM with a large margin, which demonstrates the importance of the three ranking criteria.
Moreover, image-text relation criteria are significant for sarcasticness because sarcasm is based on the correlation between the text and the image. If the text is not related to the image, the sarcasm is more likely to be poor, and sometimes it will be incomprehensible.
\subsection{Qualitative Analysis}
Figure \ref{fig:res-example} demonstrates several examples generated from different methods.
Taking the text generated by EGRM from the first image in Figure \ref{fig:res-example} as an example, the image shows a kite flying in the sun. The person flying the kite is more likely to be full of joy. However, they may be kite flyers who may suffer from sunburn from overexposure to the sun and headaches from heat stroke. The pleasure of the image modality and the pain of the sunburn and the headache in the text modality are inconsistent, which produces sarcasm.
Moreover, the kite-flyers may think that the kite can help them block the sun and reduce sunburn and headaches, which is sarcastic about the stupidity of the kite-flyers.
However, the results of SC-MTS and BLIP seem not to be sarcastic and the result of SC-$R^3$ seems to be confusing.
The second example shows that our approach is imaginative and humorous. EGRM imagines many people wearing umbrellas as traffic jams, and it satirizes road congestion caused by lots of umbrellas. The sarcasticness and humor score of EGRM in the second example is 4.33 and 3.83.
The third image shows a plate of food that does not look delicious. However, EGRM says that the veggies are perfect and the carrots are fresh, which makes the deliciousness displayed in the text and the bad taste displayed in the image reversed and inconsistent, making the image-text pair sarcastic. The other three comparison methods do not seem to produce sarcasm.
\section{Conclusion and Future Work}
We are the first to formulate the problem of cross-modal sarcasm generation and analyze the challenges of this task. We focus on generating sarcastic texts from images and proposed an extraction-generation-ranking based modular method with three modules to solve the problem without relying on any cross-modal sarcasm training data. Quantitative results and qualitative analysis reveal the superiority of our method.
In future work, we will explore generating sarcasm of different styles or categories. We will also try to build a large-scale high-quality parallel cross-modal sarcasm dataset for future researches in this field.
|
2,869,038,156,365 | arxiv | \section{Introduction}\label{Section1}
Consider the time-dependent Ginzburg-Landau system of equations
\begin{subequations}
\label{eq:1}
\begin{alignat}{2}
\frac{\partial\psi}{\partial t}- \nabla_{\kappa A}^2\psi +i\kappa\phi \psi=\kappa^2(1-|\psi|^2)\psi & \quad \text{ in }(0,+\infty)\times \Omega \,,\\
\frac{1}{c}\Big(\frac{\partial A}{\partial t}+\nabla\phi\Big)+ \text{\rm curl\,}^2A
=\frac{1}{\kappa}\Im(\bar\psi\nabla_{\kappa A}\psi) &
\quad \text{ in } (0,+\infty)\times \Omega\,, \\
\psi=0 &\quad \text{ on } (0,+\infty)\times \partial\Omega_c\,, \\
\nabla_{\kappa A}\psi\cdot\nu=0 & \quad \text{ on } (0,+\infty)\times \partial\Omega_i\,, \\
\frac{\partial\phi}{\partial\nu} = - c\kappa J(x) &\quad \text{ on } (0,+\infty)\times \partial\Omega_c\,, \\
\frac{\partial\phi}{\partial\nu}=0 &\quad \text{ on } (0,+\infty)\times \partial\Omega_i \,, \\[1.2ex]
\Xint-_{\partial\Omega}\text{\rm curl\,} A(t,x) \, ds = \kappa h_{ex} \,,&\quad {\text{ on } (0,+\infty)}\\
\psi(0,x)=\psi_0(x) & \quad \text{ in } \Omega\,, \\
A(0,x)=A_0(x) & \quad \text{ in } \Omega \,.
\end{alignat}
\end{subequations}
In the above $\psi$ denotes the order parameter, $A$ the
magnetic potential, $\phi$ the electric potential, $\kappa$
the Ginzburg-Landau parameter, which is a material property, and
$$c=\kappa^2/\sigma$$ where $\sigma$ is the normal conductivity of the
sample. Finally, $h_{ex}$, or the average magnetic field on $\partial\Omega$
divided by $\kappa$, is constant in time. Length has been scaled with
respect to the penetration depth (see \cite{alhe14}). \\
Unless otherwise stated we
shall assume in the sequel that $$\kappa\geq1\,.$$ We further assume that
$(\psi_0,A_0)\in H^1(\Omega,{\mathbb{C}})\times H^1(\Omega,{\mathbb{R}}^2)$ and that
\begin{equation}\label{condinit}
\|\psi_0\|_\infty\leq 1 \,.
\end{equation}
We use the notation $\nabla_A=\nabla-iA$, $\Delta_A=\nabla_A^2$ and $ds$ for the induced
measure on $\pa \Omega$. We have also used above the standard notation
\begin{displaymath}
\Xint-_{\partial \Omega} = \frac{1}{\ell(\partial \Omega)}\int \,.
\end{displaymath}
The domain $\Omega\subset\subset{\mathbb{R}}^2$ has the same characteristics as in \cite{alhe14},
in particular its boundary $\pa \Omega$ contains a smooth interface,
denoted by $\partial\Omega_c\,$, with a conducting metal which is at the normal
state. Thus, we require that $\psi$ vanishes on $\partial\Omega_c$ in (\ref{eq:1}c).
We make the following assumptions on the current $J$
\begin{equation}
\label{eq:2}
(J1)\quad J\in C^2(\overline{\partial\Omega_c},{\mathbb{R}})\,,
\end{equation}
\begin{equation}
\label{eq:3}
(J2) \quad \int_{\partial\Omega_c}J \,ds=0 \,,
\end{equation}
and
\begin{equation}\label{eq:75aa}
(J3)\quad \mbox{the sign of } J \mbox{ is constant on each
connected component of } \partial\Omega_c\,.
\end{equation}
We allow for $J\neq0$ at the corners despite the fact that no current is
allowed to enter the sample through the insulator. The rest of
$\partial\Omega$, denoted by $\partial\Omega_i$ is adjacent to an insulator. By
convention, we extend $J$ as equal to $0$ on $\partial \Omega_i$.
To simplify some of our regularity arguments we introduce the
following geometrical assumption (for further discussion we refer the
reader to Appendix A in \cite{alhe14}) on $\partial\Omega$:
\begin{equation}\label{propertyR}
(R1)\,\left\{
\begin{array}{l}
(a) \; \pa \Omega_i \mbox{ and } \pa \Omega_c \mbox{ are of class } C^3\,;\\
(b) \mbox{ Near each edge, } \pa
\Omega_i \mbox{ and } \pa \Omega_c
\mbox{ are} \\ \quad \mbox{ flat and meet with an angle of } \frac \pi 2\,.
\end{array}\right.
\end{equation}
We also require that
\begin{equation}\label{hyptopolo}
(R2) \quad\quad \mbox{Both } \partial\Omega_c \mbox{ and } \partial\Omega_i \mbox{ have two components}.
\end{equation}
\begin{figure}
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\caption{Typical superconducting sample. The arrows denote the direction of the current flow ($J_{in}$ for the inlet, and $J_{out}$ for the outlet).}
\label{fig:1}
\end{figure}
\nopagebreak
Figure 1 presents a typical sample with properties (R1) and
(R2), where the current flows into the sample from one connected
component of $\partial\Omega_c$, and exits from another part, disconnected from
the first one. Most wires would fall into the above class of domains.
The system \eqref{eq:1} is invariant to the gauge transformation
\begin{equation}
\label{eq:4}
A'= A + \nabla\omega\,, \; \phi'= \phi-
\frac{\partial\omega}{\partial t}\,, \; \psi' = \psi e^{i\omega} \,.
\end{equation}
We thus choose, as in \cite{alhe14}, the Coulomb gauge, i.e., we assume
\begin{equation}
\label{eq:5}
\begin{cases}
\text{\rm div\,} A =0 & \text{ in } (0,+\infty)\times \Omega\,, \\
A\cdot\nu =0 & \text{ on }(0,+\infty)\times \partial\Omega\,,
\end{cases}
\end{equation}
where the divergence is computed with respect to the spatial coordinates only.
From (\ref{eq:25}b), (\ref{eq:25}d), and (\ref{eq:25}f), we know that $\text{\rm curl\,}
A$ is constant on each connected component of $\partial \Omega_i$. Let then
$\{\partial\Omega_{i,j}\}_{j=1}^{2}$ denote the set of connected components of
$\partial\Omega_i$. We can write, for $j=1,2\,$,
\begin{equation}
\label{eq:6}
\text{\rm curl\,} A|_{\partial\Omega_{i,j}} =
h_j \, \kappa\,,
\end{equation}
where $h_1$ and $h_2$ are constants.
Note that $h_1$ and $h_2$ can be determined from
$J$ and $h_{ex}$ via the formula \cite{alhe14}
\begin{equation}
\label{eq:7}
h_j= h_{ex} - \Xint-_{\pa \Omega} \, |\Gamma(\tilde x, x_j)| \,
J(\tilde x) ds (\tilde x)\, \mbox{ for any } x_j \in \pa
\Omega_{i,j}\,, \quad j=1,2\,,
\end{equation}
where $\Gamma(x,x_0)$ is the portion of $\partial\Omega$ connecting $x_0$ and $x$ in
the positive trigonometric direction. We assume that $h_{ex}$ and $J$ are such
that
\begin{equation}
\label{eq:8}
h_1h_2 <0 \,,
\end{equation}
and without any loss of generality we can assume $h_2>0$. Let
\begin{equation}
\label{eq:9}
h=\text{\rm max}(|h_1|,|h_2|)\,.
\end{equation}
We assume that
\begin{equation}
\label{hyph}
h>1
\end{equation}
and distinguish below between two
different cases
\begin{subequations}
\label{eq:10}
\begin{equation}
\label{eq:5a}
1<h\leq \frac{1}{\Theta_0} \quad \mbox{or} \quad \frac{1}{\Theta_0}<h\,, \tag{\ref{eq:10}a,b}
\end{equation}
\end{subequations}
where $\Theta_0$ is given by \eqref{eq:20} ($\Theta_0\sim 0.59$).
In \cite{alhe14} we have established the global existence of solutions for
\eqref{eq:1}, such that
\begin{align*} \label{prop1coul}
&\psi_c\in C([0,+\infty);W^{1+\alpha,2}(\Omega,{\mathbb{C}}))\cap
H^1_{\text{\rm loc}}([0,+\infty);L^2(\Omega,{\mathbb{C}}))\,, \, \forall \alpha <1\,, \\
&A_c\in C([0,+\infty); W^{1,p}(\Omega,{\mathbb{R}}^2))\cap
H^1_{\text{\rm loc}}([0,+\infty);L^2(\Omega,{\mathbb{R}}^2))\,, \forall p\geq 1\,, \\
&\phi_c\in L^2_{\text{\rm loc}}([0,+\infty); H^1(\Omega))\,.
\end{align*}
We next define, as in \cite[Subsection 2.2]{alhe14}, (in a slightly
different manner as the definition here is $c$-independent) the
normal fields. They are defined as the weak solution -- $(\phi_n,A_n)\in
H^1(\Omega)\times H^1(\Omega,{\mathbb{R}}^2)$ -- of \eqref{eq:5} and
\begin{subequations}
\label{eq:11}
\begin{empheq}[left={\empheqlbrace}]{alignat=2}
&\, \text{\rm curl\,}^2A_n + \nabla\phi_n = 0 \qquad &\text{in } &\Omega\,, \\
& - \frac{\partial\phi_n}{\partial\nu} = J \qquad &\text{on } &\partial\Omega\,, \\
& \Xint-_{\partial\Omega}\text{\rm curl\,} A_n\,ds = h_{ex}\,, && \\
&\int_\Omega \phi_n \,dx =0\, . &&
\end{empheq}
\end{subequations}
Note that $(0,\kappa A_n,c\kappa\phi_n)$ is a steady-state solution of \eqref{eq:1}.
By \eqref{eq:11} we have (cf. \cite{alhe14})
\begin{equation}
\label{eq:12}
\begin{cases}
\Delta B_n = 0 & \text{in } \Omega\,, \\
\frac{\partial B_n}{\partial\tau} = J & \text{on } \partial\Omega\,, \\
\Xint-_{\partial\Omega}B_n(x)\,ds = h_{ex} \,. &
\end{cases}
\end{equation}
By taking the
divergence of (\ref{eq:11}a) we also obtain
\begin{subequations}
\label{eq:11b}
\begin{empheq}[left={\empheqlbrace}]{alignat=2}
& \; \Delta\phi_n=0 & \qquad \text{in } \Omega\,, \\
&- \frac{\partial\phi_n}{\partial\nu} = J & \qquad\text{on } \partial\Omega\,, \\
&\;\int_\Omega \phi_n \,dx =0\, . &
\end{empheq}
\end{subequations}
We recall from \cite[(2.16) and (2.17)] {alhe14} that
\begin{equation}\label{reguphi}
B_n\in W^{2,p}(\Omega)\,,\, \phi_n \in W^{2,p}(\Omega) \,,\, \forall p>1\,.
\end{equation}
We focus attention in this work on the exponential decay of $\psi$ in
regions where $|B_n|>1$. For steady-state solutions of \eqref{eq:1}
in the absence of electric current ($J=0$) we may set $\phi\equiv0$ and the
magnetic field is then constant on the boundary. The exponential
decay of $\psi$ away from the boundary has been termed ``surface
superconductivity'' and has extensively been studied (cf.
\cite{fohe09} and the references within). More recently, the case of
a non-constant magnetic field has been studied as well
\cite{attar2014ground,heka15}. In these works $\phi$ still identically
vanishes but nevertheless $\nabla B_n\neq0$ in view of the presence of a
current source term $\text{\rm curl\,} h_{ex}$ in (\ref{eq:1}b). In particular
in \cite{heka15} it has been established, in the large $\kappa$ limit for
the case $1\ll h\ll\kappa$ that $\psi$ is exponentially small away from
$B_n^{-1}(0)$.
In the absence of electric current the time-dependent case is of
lesser interest, since every solution of \eqref{eq:1} converges to a
steady-state solution \cite{lid97, feta01}. This result has been obtained in \cite{feta01} by
using the fact that the Ginzburg-Landau energy functional is a
Lyapunov function in this case. In contrast, when $J\neq0$ this
property of the energy functional is lost, and convergence to a
steady-state is no-longer guaranteed. In \cite{alhe14} the global
stability of the normal state has been established for $h=\mathcal O(\kappa)$
in the large $\kappa$ limit. In the present contribution, for the same
limit, we explore the behavior of the solution for $1<h\ll\kappa$ , and
establish exponential decay of $\psi$ in every subdomain of $\Omega$ where
$|B_n|>1$. We do that for both steady-state solutions (whose
existence we need to assume) and time-dependent ones. We also study
the large-domain limit, where we obtain weaker results for
steady-state solutions only.
Let, for $j=1,2$,
\begin{equation}
\label{eq:13}
\omega_{j} = \{x\in \Omega \, : \, (-1)^jB_n(x)>1 \} \,.
\end{equation}
Our first theorem concerns steady state solutions and their
exponential decay, in certain subdomains of $\Omega$, in the large $\kappa$ limit.
\begin{theorem}
\label{mainsteadystate}
Let for $\kappa \geq 1$, $(\psi_\kappa,A_\kappa, \phi_\kappa)$ be a time-independent solution of \eqref{eq:1}.
Suppose that for some $j\in\{1,2\}$ we have
\begin{equation}
\label{eq:14}
1<|h_j| \,.
\end{equation}
Then, for any compact set $K\subset \omega_j \cup \partial \Omega_c $, there
exist $C>0$, $\alpha >0$, and
$\kappa_0\geq 1 $, such that for any $\kappa\geq \kappa_0$ we have
\begin{equation}
\label{eq:15}
\int_{K} |\psi_\kappa(x)|^2 \,dx \leq C e^{- \alpha \kappa} \,.
\end{equation}
If, in addition,
\begin{equation}
\frac{1}{\Theta_0}<|h_j| \,,
\end{equation}
then (\ref{eq:15}) is satisfied for any compact subset $K\subset\overline{\omega_j}$.
\end{theorem}
In addition to the above exponential decay, which is limited to the
region where the normal magnetic field is large, we establish a weaker
decay of $\psi_\kappa$ in the entire domain.
\begin{proposition}
\label{entire}
Under the assumptions \eqref{eq:2}-\eqref{eq:10} there exists
$C(J,\Omega)>0$ such that, for $\kappa \geq 1$,
\begin{equation}
\label{eq:16}
\| \psi_\kappa \|_2 \leq C(J,\Omega) \, (1+c^{-1/2})^{1/3}\kappa^{-1/6} \,.
\end{equation}
\end{proposition}
Theorem \ref{mainsteadystate} is extended to the time dependent case
in the following way.
\begin{theorem}
\label{main}
Let $(\psi_\kappa,A_\kappa, \phi_\kappa)$ denote a time-dependent solution of
\eqref{eq:1}. Assuming $c=1$, under the conditions of Theorem
\ref{mainsteadystate} and \eqref{condinit}, for any compact set $K\subset
\omega_j\cup \partial \Omega_c$ there exist $C>0$ , $\alpha >0$, and $\kappa_0\geq 1$, such
that for any $\kappa\geq \kappa_0$ we have
\begin{equation}
\label{eq:17}
\limsup_{t\to\infty}\int_{K} |\psi_\kappa(t,x)|^2 \,dx \leq C e^{- \alpha \kappa} \,.
\end{equation}
\end{theorem}
Finally, we consider steady-state solutions of (\ref{eq:1}) in the
large domain limit, i.e., we set $\kappa=c=1$ and stretch $\Omega$ by a factor
of $R \gg 1 $. Let $\Omega^R$ be the image of $\Omega$ under the map $x\to Rx$.
We consider again steady-state solutions of \eqref{eq:1}.
\begin{subequations}
\label{eq:18}
\begin{alignat}{2}
& \Delta_A\psi + \psi \left( 1 - |\psi|^{2} \right)-i\phi\psi =0 & \quad \text{ in } \Omega^R\, ,\\
& \text{\rm curl\,}^2A + \nabla\phi = \Im(\bar\psi\, \nabla_A\psi) & \quad \text{ in } \Omega^R\,,\\
&\psi=0 &\quad \text{ on } \partial\Omega_c^R\,, \\
&\nabla_A\psi\cdot\nu=0 & \quad \text{ on } \partial\Omega_i^R\,,\\
& \frac{\partial\phi}{\partial\nu}= \frac{F(R)}{R}J &\quad \text{ on } \partial\Omega_c^R\,,\\
& \frac{\partial\phi}{\partial\nu}= 0 &\quad \text{ on } \partial\Omega_i^R\,,\\[1.2ex]
&\Xint-_{\partial\Omega^R}\text{\rm curl\,} A(x)\,ds = F(R)h_{ex}\,.
\end{alignat}
\end{subequations}
In the above $F(R)=R^{\gamma}$ for some $0<\gamma<1$. We study the above
problem in the limit $R\to\infty$. Assuming again (\ref{eq:2})-(\ref{eq:10})
we establish the following result.
\begin{proposition}
\label{largedomain}
Let $(\psi,A,\phi)$ denote a solution of \eqref{eq:18}.
Then, there exists a compact set $K\subset\Omega$, $C>0$, $R_0>0$, and $\alpha >0$,
such that for any $R>R_0\,$ we have
\begin{equation}
\label{eq:126a}
\int_{K_R} |\psi (x)|^2 \,dx \leq C e^{-\alpha R} \,,
\end{equation}
where $K_R$ is the image of $K$ under the map $x\to Rx$.
\end{proposition}
Note that $h_{ex}$ must be of $\mathcal O(J)$, otherwise $(0,A_n,\phi_n)$ would
be the unique solution.
Physically \eqref{eq:126a} demonstrates that there is a
significant portion of the superconducting sample which remains,
practically, at the normal state, for current densities which may be
very small. This result stands in
contrast with what one finds in standard physics handbooks
\cite{poetal99} where the critical current density, for which the
fully superconducting state looses its stability, is tabulated a
material property. However, our results suggest that the critical
current depends also on the geometry of the superconducting sample. In
fact, according to Proposition \ref{largedomain}, this current density
must decay in the large domain limit. In two-dimensions, our result
suggests that one should search for a critical current (and not
current density), whereas in three-dimensions a density with respect
to cross-section circumference (instead of area) should be obtained.
We note that Proposition \ref{largedomain} is certainly not
optimal. In fact, we expect the following conjecture to be true.
\begin{conjecture}
\label{conj}
Under the conditions of Proposition \ref{largedomain}, for any
compact set $K\subset\Omega\setminus B_n^{-1}(0)$, there exist $R_0>0$, $C>0$, and $\alpha >0$,
such that for any $R>R_0$ \eqref{eq:126a} is satisfied.
\end{conjecture}
The rest of this contribution is organized as follows. In the next
section we establish some preliminary results related to the
eigenvalues of the magnetic Laplacian in the presence of
Dirichlet-Neumann corners. We use these results in Section 3 where we
establish Theorem \ref{mainsteadystate} and Proposition~\ref{entire}.
In Section 4 we consider the time-dependent problem and establish, in
particular, Theorem~\ref{main}. Finally, in the last section, we
obtain some weaker results for steady-state solutions of (\ref{eq:1})
in the large domain limit.
\section{Magnetic Laplacian Ground States}
\label{sec:2}
In this section, we analyze the spectral properties of the Schr\"odinger
operator with constant magnetic field in a sector. The Neumann problem
has been addressed by V. Bonnaillie-No\"el in \cite{Bon}. In the sequel
we shall need, however, a lower bound for the ground state energy of
the above operator on a Dirichlet-Neumann sector, i.e., a Dirichlet
condition is prescribed on one side of the sector and the magnetic
Neumann condition on the other side. We begin by the following
auxiliary lemma whose main idea has been introduced to us by M.~Dauge
\cite{da14}. Hereafter the norms in the Lebesgue spaces $L^p(\Omega)$,
$L^p(\Omega,\mathbb R^2)$ and $L^p(\Omega,\mathbb C)$ will be denoted by
$\|\cdot\|_{L^p(\Omega)}$ or $\|\cdot\|_p$, and the norms in the Sobolev spaces
$W^{k,p}(\Omega,\mathbb R)$, $W^{k,p}(\Omega,\mathbb R^2)$ and $W^{k,p}(\Omega,\mathbb C)$
will be denoted by $\|\cdot\|_{W^{k,p}(\Omega)}$ or $\|\cdot\|_{k,p}$.
\begin{lemma}
\label{lem:Dirichlet-Neumann}
Let $S_\alpha$ denote an infinite sector of angle $\alpha \in (0,\pi]$, i.e.,
\begin{displaymath}
S_\alpha = \{ (x,y)\in{\mathbb{R}}^2 \,: \, 0<\arg(x+iy)<\alpha \}\,.
\end{displaymath}
Let further
\begin{displaymath}
{\mathcal H}_\alpha = \{ u\in H^1(S_\alpha) \,: \, u(r\cos\alpha,r\sin\alpha)=0\,, \; \forall r>0\}\,,
\end{displaymath}
and
\begin{displaymath}
\Theta^{DN}_\alpha = \inf_{\begin{subarray}{c}
u\in {\mathcal H}_\alpha \\
\|u\|_2 =1
\end{subarray}}
\int_{S_\alpha} |(\nabla-iF) u|^2 \,dx \,,
\end{displaymath}
where $F$ is a magnetic potential satisfying $\text{\rm curl\,} F = 1$ in
$S_\alpha$. Then,
\begin{equation}
\label{eq:19}
\Theta^{DN}_\alpha =\Theta_0\,,
\end{equation}
where
\begin{equation}
\label{eq:20}
\Theta_0 = \inf_{\begin{subarray}{c}
u\in H^1(S_\pi) \\
\|u\|_2 =1
\end{subarray}}
\int_{S_\pi} |(\nabla-iF)u|^2 \,dx \,.
\end{equation}
\end{lemma}
\begin{proof}
Let $0<\alpha_1<\alpha_2\leq\pi$ and $u\in{\mathcal H}_{\alpha_1}$. Let further
$\tilde{u}\equiv u$ in $S_{\alpha_1}$ and $\tilde{u}\equiv0$ in
$S_{\alpha_2}\setminus S_{\alpha_1}$. Clearly $\tilde{u}\in{\mathcal H}_{\alpha_2}$, and hence it
follows that $\Theta^{DN}_{\alpha_1}\geq\Theta^{DN}_{\alpha_2}$. Consequently,
\begin{displaymath}
\Theta^{DN}_\alpha \geq \Theta^{DN}_\pi\,, \quad \forall\alpha\leq\pi.
\end{displaymath}
From the definition of $\Theta_0$ it follows, however, that
$\Theta^{DN}_\pi\geq\Theta_0$, and hence, $ \Theta^{DN}_\alpha \geq \Theta_0$ for all
$\alpha\in(0,\pi]$. The proof of \eqref{eq:19} now easily follows
from the proof of Persson's Theorem \cite[Appendix B]{fohe09}
providing the upper bound $\Theta_0$ for the essential spectrum of the
magnetic Dirichlet-Neumann Laplacian in $S_\alpha$.
\end{proof}
Let ${\mathcal D}\subset\Omega$ have a smooth boundary, except at the corners of $\partial\Omega$. As
in \cite{alhe14} we let
\begin{equation}
\label{eq:21}
\mu_\epsilon(\mathcal A,{\mathcal D}) = \inf_{
\begin{subarray}{c}
u\in {\mathcal H}({\mathcal D}) \\
\|u\|_2 =1
\end{subarray}} \int_{\mathcal D} |\epsilon\nabla-i\mathcal A u|^2 \,dx \,,
\end{equation}
wherein
\begin{displaymath}
{\mathcal H}({\mathcal D}) = \{ u\in H^1({\mathcal D})\,:\, u=0 \text{ on } \partial{\mathcal D}\setminus (\partial{\mathcal D}\cap \partial\Omega_i)\,\} \,.
\end{displaymath}
Let
\begin{displaymath}
{\mathcal S} = \overline{\partial\Omega_c}\cap\overline{\partial\Omega_i}\cap\overline{\partial{\mathcal D}}
\end{displaymath}
denote the corners of $\Omega$ belonging to $\partial{\mathcal D}$. Following \cite{boda06}
we set for a given magnetic potential $\mathcal A \in C^1(\overline{\Omega},\mathbb R^2)$,
\begin{displaymath}
b=\inf_{x\in{\mathcal D}}|\text{\rm curl\,} \mathcal A| \quad ; \quad b^\prime = \inf_{x\in\partial{\mathcal D}\cap\partial\Omega_i}|\text{\rm curl\,} \mathcal A|\,.
\end{displaymath}
The following proposition is similar to a result in \cite{boda06}
obtained for a Neumann boundary condition. Here we treat a
Dirichlet-Neumann boundary condition and allow, in addition, some
dependence of the magnetic potential on the semi-classical parameter.
\begin{proposition}
Let $a\in W^{1,\infty}({\mathcal D},\mathbb R^2)$. There exist $C>0$ and $\epsilon_0>0$ such that for all
$0<\epsilon<\epsilon_0$ we have
\begin{equation}
\label{eq:22}
\mu_\epsilon(\mathcal A+\epsilon^{1/2}a,{\mathcal D}) \geq \epsilon \min(b,\Theta_0b^\prime) [1-C(1+\|\nabla a\|_\infty^2)\epsilon^{1/3}]\,.
\end{equation}
\end{proposition}
\begin{proof}
The case $b=0$ being trivial, we assume that $b>0$. We begin by
introducing for any $\epsilon >0$ a partition of unity (cf. also
\cite{hemo96}), i.e., families $\{\eta_i\}_{i=1}^K\subset C^\infty(\Omega)$, and
$\{x_i\}_{i=1}^K\subset{\mathcal D}$ satisfying
\begin{displaymath}
\sum_{i=1}^K\eta_i^2=1 \quad ; \quad {\rm supp}\,\eta_i \subset B(x_i,\epsilon^{1/3}) \quad
; \quad \sum_{i=1}^K|\nabla\eta_i|^2 \leq \frac{C}{\epsilon^{2/3}} \,,
\end{displaymath}
where $C>0$ is independent of $\epsilon$.
It can be easily verified that for any $u\in H^1(\Omega,{\mathbb{C}})$
\begin{equation}
\label{eq:23}
\|(\epsilon\nabla-i[\mathcal A+\epsilon^{1/2}a])u\|_2^2=\sum_{i=1}^K\big[
\|(\epsilon\nabla-i[\mathcal A+\epsilon^{1/2}a])(\eta_iu)\|_2^2 - \epsilon^2\|u\nabla\eta_i\|_2^2 \big]\,.
\end{equation}
We now set
\begin{displaymath}
v_i=\eta_iu\exp(-i\epsilon^{-1/2}a(x_i)\cdot x) \,,
\end{displaymath}
to obtain that
$$\aligned
& \|(\epsilon\nabla-i[\mathcal A+\epsilon^{1/2}a])(\eta_iu)\|_2^2\\
=&\|(\epsilon\nabla-i[\mathcal A+\epsilon^{1/2}(a-a(x_i))])v_i\|_2^2\\
\geq&
(1-\epsilon^{1/3})\|(\epsilon\nabla-i\mathcal A)v_i\|_2^2 - \epsilon^{2/3}\|a-a(x_i))v_i\|_2^2\\
\geq&
(1-\epsilon^{1/3})\mu_\epsilon(\mathcal A,{\mathcal D}) \|v_i\|_2^2 - C\epsilon^{4/3}\|v_i\|_2^2\,.
\endaligned
$$
Substituting the above into \eqref{eq:23} yields
\begin{equation}
\label{eq:24}
\|(\epsilon\nabla-i[\mathcal A+\epsilon^{1/2}a])u\|_2^2 \geq (1-\epsilon^{1/3})\mu_\epsilon(\mathcal A,{\mathcal D}) \|u\|_2^2 - C\epsilon^{4/3}(1+\|\nabla a\|_\infty^2)\|u\|_2^2\,.
\end{equation}
By following the same steps of the proof of Theorem 7.1 in \cite{boda06}
we can establish that
\begin{equation}\label{eq:74a}
\mu_\epsilon(\mathcal A,{\mathcal D}) \geq \epsilon \min(b,\Theta_0b^\prime,\Theta^{DN}_\alpha\inf_{x\in{\mathcal S}}|\text{\rm curl\,} \mathcal A| )
(1-C\epsilon^{1/2})\,.
\end{equation}
The lemma now follows from \eqref{eq:19}, \eqref{eq:24}, and
\eqref{eq:74a}.
\end{proof}
\section{Steady-State Solutions}\label{sec:3}
We begin by considering steady-state solutions of \eqref{eq:1}
$(\psi_k,A_\kappa,\phi_\kappa)\in H^1(\Omega,{\mathbb{C}}\times{\mathbb{R}}^2\times{\mathbb{R}})$ in
the limit $\kappa\to + \infty$. Hence, we look at the
system \cite[Section 5]{alhe14}
\begin{subequations}
\label{eq:25}
\begin{alignat}{2}
- & \nabla_{\kappa A_\kappa}^2\psi_\kappa+i\kappa\phi_\kappa \psi_\kappa =\kappa^2(1-|\psi_\kappa|^2)\psi_\kappa & \quad \text{ in } \Omega \,,\\
- &\text{\rm curl\,}^2A_\kappa+ \frac{1}{c}\nabla\phi_\kappa =\frac{1}{\kappa}\Im(\bar\psi_\kappa\nabla_{\kappa A_\kappa}\psi_\kappa) &
\quad \text{ in } \Omega\,, \\
&\psi_\kappa =0 &\quad \text{ on } \partial\Omega_c\,, \\
&\nabla_{\kappa A_\kappa}\psi_\kappa\cdot\nu=0 & \quad \text{ on } \partial\Omega_i\,, \\
& \frac{\partial\phi_\kappa}{\partial\nu} = - c\kappa J(x) &\quad \text{ on } \partial\Omega_c\,, \\
&\frac{\partial\phi_\kappa}{\partial\nu}=0 &\quad \text{ on } \partial\Omega_i \,, \\[1.2ex]
&\Xint-_{\partial\Omega}\text{\rm curl\,} A_\kappa \, ds = \kappa h_{ex} \,,&
\end{alignat}
\end{subequations}
with the additional gauge restriction \eqref{eq:5}. In the above
$(\psi_k,A_\kappa,\phi_\kappa)$ is the same as $(\psi,A,\phi)$ in \eqref{eq:1}. The
subscript $\kappa$ has been added to emphasize the limit we consider
here. We assume in addition (\ref{eq:2})-(\ref{eq:10}). By the strong
maximum principle we easily obtain that
\begin{equation}
\label{eq:26}
\|\psi_\kappa\|_\infty <1 \,.
\end{equation}
Let $h$ be given by
\eqref{eq:9}. It has been demonstrated in \cite{alhe14} that for
some $h_c>0$, when $h<h_c\kappa$, the normal state looses its stability.
Since we consider cases for which $1<h\ll\kappa$ it is reasonable to expect
that other steady-state solutions would exist. We note, however, that
in contrast with the case $J=0$, where the existence of steady-state
solutions can be proved using variational arguments (inapplicable in
our case), existence of steady-state solutions to \eqref{eq:25} is yet an open problem
when an electric current is applied. We shall address time-dependent
solutions in the next section.
Next we set
\begin{subequations}
\label{eq:27}
\begin{equation}
\label{eq:28}
A_{1,\kappa}=A_\kappa-\kappa A_n\,,\quad
\phi_{1,\kappa}=\phi_\kappa-c\kappa\phi_n\,.\tag{\ref{eq:27}a,b}
\end{equation}
\end{subequations}
Set further
\begin{subequations}
\label{eq:29}
\begin{equation}
\label{eq:30}
B_\kappa =\text{\rm curl\,} A_\kappa\,,\quad
B_{1,\kappa}=\text{\rm curl\,} A_{1,\kappa} \,. \tag{\ref{eq:29}a,b}
\end{equation}
\end{subequations}
By
(\ref{eq:25}b) we then have
\begin{subequations}
\label{eq:31}
\begin{empheq}[left={\empheqlbrace}]{alignat=2}
&\text{\rm curl\,} B_{1,\kappa} + \frac{1}{c}\nabla\phi_{1,\kappa} = \frac{1}{\kappa}\Im(\bar\psi_\kappa\, \nabla_{\kappa A_\kappa}\psi_\kappa) & \text{ in }
\Omega\,,\\
&\frac{\partial\phi_{1,\kappa}}{\partial\nu}= 0 & \text{ on } \partial\Omega\,, \\
&\Xint-_{\partial\Omega}B_{1,\kappa}(x)\,ds = 0 \,. &\label{eq:42c}
\end{empheq}
\end{subequations}
Note that since $\partial B_{1,\kappa}/\partial\tau=\partial\phi_{1,\kappa}/\partial\nu=0$ on $\partial\Omega$
we must have by (\ref{eq:31}c) that
\begin{equation}
\label{eq:32}
B_{1,\kappa}|_{\partial\Omega}\equiv 0\,.
\end{equation}
Taking the divergence of (\ref{eq:25}b) yields, with the aid of the
imaginary part of (\ref{eq:25}a), that $\phi_\kappa $ is a weak
solution of
\begin{equation}
\label{eq:33}
\begin{cases}
-\Delta\phi_\kappa + c \,|\psi_\kappa|^2 \phi_\kappa =0 & \text{ in } \Omega\,, \\
\frac{\partial\phi_\kappa}{\partial\nu}= c\kappa J & \text{ on } \partial\Omega_c\,,\\
\frac{\partial\phi_\kappa}{\partial\nu}= 0 & \text{ on } \partial\Omega_i\,.
\end{cases}
\end{equation}
By assumption $\phi_\kappa\in H^1(\Omega)$ and hence, by \cite[Proposition
A.2]{alhe14} we obtain that $\phi_\kappa\in W^{2,p}(\Omega)$ for all $p\geq2$, hence $\phi_\kappa \in C^1(\overline{\Omega})$. By
(\ref{eq:27}b) we then have
\begin{displaymath}
\begin{cases}
-\Delta\phi_{1,\kappa} + c\, |\psi_\kappa|^2 \phi_{1,\kappa} =- \kappa c^2|\psi_k|^2 \phi_n & \text{ in } \Omega\,, \\
\frac{\partial\phi_{1,\kappa}}{\partial\nu}= 0 & \text{ on } \partial\Omega \,.
\end{cases}
\end{displaymath}
Let $K=\|\phi_n\|_\infty$ and $w=\phi_{1,\kappa} +K\kappa c$. Clearly,
\begin{displaymath}
\begin{cases}
-\Delta w+ c\, |\psi_\kappa|^2 w =- \kappa c^2|\psi_k|^2(\phi_n-K)\geq 0 & \text{ in } \Omega\,, \\
\frac{\partial w}{\partial\nu}= 0 & \text{ on } \partial\Omega \,.
\end{cases}
\end{displaymath}
It can be easily verified that $w$ is the minimizer in $H^1(\Omega)$ of
\begin{displaymath}
{\mathcal J}(v) = \|\nabla v\|_2^2 + c\|\psi_\kappa v\|_2^2+\kappa c^2\langle|\psi_k|^2(\phi_n-K),v\rangle \,.
\end{displaymath}
As
\begin{displaymath}
{\mathcal J}(v_+) \leq {\mathcal J}(v)\,,
\end{displaymath}
it easily follows that $ w\geq 0$, that is
\begin{displaymath}
\phi_{1,\kappa}+K\kappa c \geq 0 \,.
\end{displaymath}
In a similar manner we obtain
\begin{displaymath}
\phi_{1,\kappa}-K\kappa c \leq 0 \,,
\end{displaymath}
which together with (\ref{eq:27}b) yields
\begin{equation}
\label{eq:34}
\|\phi_\kappa\|_\infty \leq C(\Omega,J)\, c\, \kappa \,.
\end{equation}
We now apply again Proposition A.2 in \cite{alhe14} to obtain that for
any $p\geq2$
\begin{equation}\label{estlp}
\|\phi_\kappa\|_{2,p} \leq C(\Omega,J)\, c\, \kappa \,.
\end{equation}
We note that all elliptic estimates must be taken with special care
since $\Omega$ possesses corners. The necessary details (with references
therein) can be found in Appendices A and B of \cite{alhe14} .
Next we set for $\delta >0$ and $\kappa\geq 1$,
\begin{displaymath}
D_\delta(\kappa) = \{ x\in\Omega \,: \,|B_\kappa(x)|>(1+\delta)\kappa \} \,,
\end{displaymath}
and
\begin{equation}
\label{eq:35}
S_\delta = \{ x\in\Omega \,: \,|B_n(x)|>(1+\delta)\}\,.
\end{equation}
By either (\ref{eq:10}a) or (\ref{eq:10}b), it follows that for $0<\delta <
h-1$, $S_\delta\neq\emptyset\,$. Below we show that the same is true for $ D_\delta(\kappa)$.
Note that \eqref{eq:8} implies that $S_\delta$ consists of two disjoint
sets:
\begin{equation}\label{sdj}
S_{\delta} = S_{\delta,1} \cup S_{\delta,2}\,,
\end{equation}
one near $\partial\Omega_{i,1}$ denoted by $S_{\delta,1}$, and one
near $\partial\Omega_{i,2}$ denoted by $S_{\delta,2}\,$. \\
We then let
\begin{equation}
\label{eq:36}
\mathcal C_{\delta,j}=\partial S_{\delta,j}\setminus (\partial\Omega\cap\partial S_{\delta,j})\,,\quad j=1,2\,,
\end{equation}
and
\begin{equation}\label{Cdi}
\mathcal C_\delta=\mathcal C_{\delta,1}\cup \mathcal C_{\delta,2} \,.
\end{equation}
We can now state and prove
\begin{lemma}\label{lemma3.1}
For any $0<\alpha<1$ there exists $ \kappa_0=\kappa_0(\Omega,J,\alpha)>0$ such that for
all $\kappa \geq \kappa_0$ and $0<\delta<h-1$, we have
\begin{equation}
\label{eq:37}
S_{\delta+\kappa^{-\alpha}} \subset D_\delta(\kappa) \,.
\end{equation}
\end{lemma}
\begin{proof} {\it Step 1: Prove that
for some $C(\Omega,J)>0$
\begin{equation}
\label{eq:43}
\|A_\kappa\|_\infty \leq C\, \kappa \,.
\end{equation}}
Taking the divergence of (\ref{eq:31}a) yields, with the aid of (\ref{eq:31}b),
\begin{equation}
\label{eq:38}
\begin{cases}
- \Delta\phi_{1,\kappa} = - \frac{c}{\kappa}\, \text{\rm div\,} \Im(\bar\psi_\kappa\, \nabla_{\kappa A_\kappa
}\psi_\kappa) & \text{ in } \Omega\,, \\
\frac{\partial\phi_{1,\kappa}}{\partial\nu}= 0 & \text{ on } \partial\Omega \,.
\end{cases}
\end{equation}
Multiplying the above equation by $\phi_{1,\kappa }$ and integrating by parts
then yields, with the aid of \eqref{eq:26} and (\ref{eq:25}c,d),
\begin{equation}
\label{eq:39}
\|\nabla \phi_{1,\kappa} \|_2 \leq \frac{c}{\kappa}\, \|\nabla_{\kappa A_\kappa}\psi_\kappa\|_2 \,.
\end{equation}
Taking the inner product of (\ref{eq:25}a) with $\psi_\kappa$ yields, after
integration by parts
\begin{equation}
\label{eq:40}
\|\nabla_{\kappa A_\kappa}\psi_\kappa\|_2^2 = \kappa^2\, \|\psi_\kappa\|_2^2 \,.
\end{equation}
By \eqref{eq:39} we then obtain that
\begin{displaymath}
\|\nabla\phi_{1,\kappa}\|_2 \leq Cc\,.
\end{displaymath}
Since $\text{\rm curl\,} B_{1,\kappa}=\nabla_\perp B_{1,\kappa}\,$, the boundedness of $ \|\nabla
B_{1,\kappa}\|_2$ then easily follows from the above and
\eqref{eq:31}. Consequently,
\begin{equation}
\label{eq:41}
\frac{1}{c}\|\nabla \phi_{1,\kappa} \|_2+ \|\nabla B_{1,\kappa}\|_2 \leq C\,.
\end{equation}
Note that $\nabla\phi_{1,\kappa}$ and $\nabla_\perp B_{1,\kappa}$ are respectively the $L^2$ projections
of $\Im(\bar\psi_\kappa\, \nabla_{\kappa A_\kappa}\psi_\kappa)$ on
\begin{displaymath}
H^0_0(\text{\rm curl\,}, \Omega)=\{\widehat V \in L^2(\Omega,\mathbb R^2)\,:\, \text{\rm curl\,} \widehat V=0\}\,,
\end{displaymath}
and
\begin{displaymath}
\mathcal H^0_d:= \{\widehat W \in L^2(\Omega,\mathbb R^2)\,:\, \text{\rm div\,} \widehat
W=0\mbox{ and } \widehat W \cdot \nu =0\mbox{ on } \pa \Omega \}\,.
\end{displaymath}
Next, we attempt to estimate $\|\nabla \phi_{1,\kappa} \|_p$ and $\|\nabla B_{1,\kappa}\|_p$ for
any \linebreak $p>2$. Since $\Omega$ is simply-connected, we may conclude
from \eqref{eq:5}, (\ref{eq:29}a) and Remark B.2 in \cite{alhe14} that
there exists for any $p>2$ a constant \linebreak $C(p,\Omega)>0$ such that
\begin{displaymath}
\|A_\kappa\|_{1,p} \leq C\, \|B_\kappa\|_p \,,
\end{displaymath}
for all $\kappa\geq 1$. Sobolev embeddings then imply
\begin{equation}
\label{eq:42}
\|A_\kappa\|_\infty \leq C\, \|B_\kappa\|_p\,.
\end{equation}
Since $ \|\nabla B_{1,\kappa}\|_2$ is uniformly bounded for all $\kappa \geq 1$, we
obtain from \eqref{eq:32}, the Poincar\'e inequality, and Sobolev embeddings
that, for any $p>2$ there exists a constant $C(p,\Omega)>0$ such that we have
\begin{equation}\label{estB1p}
\|B_{1,\kappa}\|_p \leq C(p,\Omega)\,.
\end{equation}
Hence, recalling from \eqref{reguphi} that $B_n\in L^p$ and independent
of $c$ and $\kappa$, as $J$ is independent of $\kappa$, there
exists a constant $C>0$ such that
\begin{equation}\label{estBp}
\|B_\kappa\|_p=\|B_{1,\kappa}+\kappa B_n\|_p\leq C\, \kappa\,.
\end{equation}
Combining the above computations with \eqref{eq:42} then yields
\eqref{eq:43}.
\vspace{1ex}
{\it Step 2: Prove \eqref{eq:37}.}
\vspace{1ex}
We first rewrite (\ref{eq:25}a,c,d) in the following form
\begin{displaymath}
\begin{cases}
\Delta\psi_\kappa = 2i\kappa A_\kappa\cdot\nabla_{\kappa A_\kappa}\psi_\kappa + |\kappa A_\kappa|^2\psi_\kappa- \kappa^2\psi_\kappa\big( 1 - |\psi_\kappa|^{2}
\big)+i\kappa\phi_\kappa\psi_\kappa & \text{ in } \Omega\, ,\\
\psi_\kappa =0 & \text{on } \partial\Omega_c \,, \\
\frac{\partial\psi_\kappa}{\partial\nu}=i\kappa (A_\kappa \cdot\nu) \psi_\kappa=0\, & \text{ on } \partial\Omega_i \,,\\
\end{cases}
\end{displaymath}
where the last equality follows from (\ref{eq:5}). By \eqref{eq:43},
\eqref{eq:34}, the fact $\|\psi_\kappa\|_\infty\leq 1$, Proposition~A.3 and
Remark~A.4 in \cite{alhe14} (note that $\psi_\kappa$ vanishes at the
corners) we obtain that for some $C(\Omega,p,J)$
\begin{displaymath}
\|\psi_\kappa\|_{2,p} \leq C\big[\kappa^4 + \kappa^2\|\nabla_{\kappa A_\kappa}\psi_\kappa\|_p\big]\,, \quad \forall p>2\,.
\end{displaymath}
Sobolev embedding and \eqref{eq:43} then yield
\begin{equation}
\label{eq:44}
\|\nabla_{\kappa A_\kappa}\psi_\kappa\|_\infty \leq C\big[\kappa^4 +
\kappa^2\|\nabla_{\kappa A_\kappa}\psi_\kappa\|_p\big] \,.
\end{equation}
We now use a standard interpolation theorem to obtain that
\begin{displaymath}
\|\nabla_{\kappa A_\kappa}\psi_\kappa\|_p\leq \|\nabla_{\kappa A_\kappa}\psi_\kappa\|_2^{2/p} \|\nabla_{\kappa A_\kappa}\psi_\kappa\|_\infty^{1-2/p}\,.
\end{displaymath}
Substituting \eqref{eq:44} in conjunction with \eqref{eq:40} into the
above inequality then yields
\begin{equation}
\label{eq:45}
\|\nabla_{\kappa A_\kappa}\psi_\kappa\|_p\leq C\Big[\kappa^4 + \kappa^2\|\nabla_{\kappa A_\kappa}\psi_\kappa\|_p\Big]^{1-2/p}\kappa^{2/p}\,.
\end{equation}
Suppose first that
\begin{displaymath}
\kappa^2< \|\nabla_{\kappa A_\kappa}\psi_\kappa\|_p \,.
\end{displaymath}
Then, we have
\begin{displaymath}
\|\nabla_{\kappa A_\kappa}\psi_\kappa\|_p \leq C\|\nabla_{\kappa A_\kappa}\psi_\kappa\|_p^{1-2/p}\kappa^{2(1-1/p)}\,.
\end{displaymath}
Hence,
\begin{equation}
\label{eq:46}
\|\nabla_{\kappa A_\kappa}\psi_\kappa\|_p \leq C\kappa^{p-1} \,.
\end{equation}
Next, assume that
\begin{displaymath}
\|\nabla_{\kappa A_\kappa}\psi_\kappa\|_p\leq \kappa^2 \,,
\end{displaymath}
to obtain that
\begin{displaymath}
\|\nabla_{\kappa A_\kappa}\psi_\kappa\|_p\leq C\kappa^{4-6/p} \,.
\end{displaymath}
From the above, together with \eqref{eq:46} we easily conclude that,
for any $2<p\leq3$, there exists a constant $C$ such that
\begin{equation}
\label{eq:47}
\frac{1}{\kappa}\|\nabla_{\kappa A_\kappa}\psi_\kappa\|_p\leq C\kappa^{3(1-2/p)}\,.
\end{equation}
To continue, we need a $W^{1,p}$ estimates for the solution of $\Delta
u=f$ where $f\in W^{-1,p}$. We thus apply \cite[Theorem 7.1]{gima12},
which is valid for any domain which is bilipschitz equivalent to the
unit cube, to \eqref{eq:38}. This yields that, for some $C(p,\Omega)>0$,
we have
\begin{equation}\label{reglp}
\|\nabla\phi_{1,\kappa}\|_p \leq C \frac{c}{\kappa} \|\nabla_{\kappa A_\kappa}\psi_\kappa\|_p\,, \quad \forall\kappa \geq 1\,,\; 2<p\leq3\,.
\end{equation}
From \eqref{eq:47} and \eqref{eq:31} we then obtain that
\begin{equation}
\label{eq:48}
\frac{1}{c}\|\nabla \phi_{1,\kappa} \|_p + \|\nabla B_{1,\kappa} \|_p \leq C\, \kappa^{3(p-2)/p}\,.
\end{equation}
Upon \eqref{eq:48} and \eqref{eq:32} we use the Poincar\'e inequality
together with Sobolev embeddings to conclude that
\begin{equation}
\label{B1k}
\|B_{1,\kappa}\|_\infty \leq C \kappa^{3(p-2)/p}\,,\quad \forall\kappa>1\,,\; 2<p\leq3\,.
\end{equation}
Let $x\in S_{\delta+\kappa^{-\alpha}}\,$, namely
\begin{displaymath}
|B_n(x)|>(1+\delta+\kappa^{-\alpha})\,.
\end{displaymath}
From \eqref{B1k}, for some $C(p,\Omega,J)>0$ we have that
\begin{displaymath}
\aligned
|B_\kappa(x)|>&\kappa|B_n(x)| - |B_{1,\kappa}(x)| \geq(1+\delta+\kappa^{-\alpha})\kappa - C \kappa^{3(p-2)/p}\\
=&(1+\delta)\kappa+[\kappa^{1-\alpha}- C \kappa^{3(p-2)/p}].
\endaligned
\end{displaymath}
By choosing
\begin{displaymath}
2<p<\min\Big(3,\frac{6}{2+\alpha}\Big)\,,
\end{displaymath}
we have $\kappa^{1-\alpha}- C \kappa^{3(p-2)/p}>0$ for sufficiently large $\kappa\,$. Thus
$$|B_\kappa(x)|>(1+\delta)\kappa\,,
$$
and hence $x\in D_\delta(\kappa)$. Consequently, $S_{\delta+\kappa^{-\alpha}}\subset D_\delta(\kappa)\,$.
\end{proof}
As a byproduct of the proof, we also obtain
\begin{proposition}
For any $2<p\leq 3 $, there exists $\kappa_0\geq 1$ and $C>0$ such that
\begin{equation}
\label{eq:49}
\|A_{1,\kappa}\|_{2,p} \leq C \kappa^{3(p-2)/p}\,,\quad \forall \kappa \geq \kappa_0\,.
\end{equation}
\end{proposition}
\begin{proof}
The proof follows immediately from \eqref{eq:31}, \eqref{eq:48}, and
Proposition B.3 in \cite{alhe14}.
\end{proof}
We can now prove the following semi-classical Agmon estimate for
$\psi_\kappa$, establishing that it must be exponentially small in $S_\delta$.
\begin{proposition}
Suppose that $h$ satisfies (\ref{eq:10}b). Let then $j\in\{1,2\}$ be such that
$h_j>1/\Theta_0$. There exist $C>0$ and
$\delta_0>0$, such that, for any $0<\delta\leq \delta_0\,$, some $\kappa_0(\delta)$ can be found,
for which
\begin{equation}
\label{eq:50}
\kappa \geq \kappa_0(\delta) \Rightarrow\int_{S_{\delta,j}} \exp\Big(\delta^{1/2}\kappa \,
d(x,\mathcal C_{\delta,j})\Big) |\psi_\kappa|^2 \,dx \leq \frac{C}{\delta^{3/2}} \,,
\end{equation}
where $S_{\delta,j}$ is introduced in \eqref{sdj} and $\mathcal C_{\delta,j}$ in \eqref{eq:36}.
\end{proposition}
\begin{proof}
For $\delta >0$, let $\eta\in C^\infty(\Omega,[0,1])$ satisfy
\begin{equation}
\label{eq:51}
\eta(x)=
\begin{cases}
1 & x\in S_{\delta,j}\,, \\
0 & x\in\Omega\setminus S_{\delta/2,j} \,.
\end{cases}
\end{equation}
By \eqref{eq:12} and \eqref{reguphi}, it follows that $\nabla B_n$ is
bounded and independent of both $\delta$ and $\kappa$. Consequently,
there exists a constant $C_1 >0$ such that
\begin{displaymath}
d(\mathcal C_{\delta,j},\mathcal C_{\delta/2,j}) \geq \frac{\delta}{C_1}\,,
\end{displaymath}
and hence, for some $C(\Omega,J)$ and all $0<\delta<\delta_0$ we can choose $\eta$ such that
\begin{displaymath}
|\nabla\eta| \leq \frac{C}{\delta} \,.
\end{displaymath}
Let further $$\zeta=\chi\, \eta$$ where
\begin{displaymath}
\chi=
\begin{cases}
\exp(\alpha_\delta\kappa d(x,\mathcal C_{\delta,j})) &\text{if } x\in S_{\delta,j}\,, \\
1 &\text{if } x\in\Omega\setminus S_{\delta,j}\,.
\end{cases}
\end{displaymath}
We leave the determination of $\alpha_\delta$ to a later stage. We further
define, for any $r\in (0,r_0)$, $\eta_r\in C^\infty(\Omega,[0,1])$ and $\tilde \eta_r\in C^\infty(\Omega,[0,1])$ such that
\begin{equation}
\label{eq:52}
\eta_r(x)=
\begin{cases}
1 & d(x,\partial\Omega_i)>r \\
0 & d(x,\partial\Omega_i)<r/2 \,,
\end{cases}
\qquad\text{and}\qquad
|\nabla\eta_r|^2 + |\nabla \tilde \eta_r|^2 \leq \frac{C}{r^2} \,,
\end{equation}
and
$$
\eta_r^2 + \tilde{\eta}_r ^2 = 1\,.
$$
Fix $0<\alpha<1$. Multiplying (\ref{eq:25}a) by $\zeta^2\bar{\psi}$,
integrating by parts yields for the real part
$$\aligned
& \|\nabla_{\kappa A_\kappa}(\zeta\tilde{\eta}_{\kappa^{-1/2}}\psi_\kappa)\|_2^2+
\|\nabla_{\kappa A_\kappa}(\zeta\eta_{\kappa^{-1/2}}\psi_\kappa)\|_2^2 \\
\leq& \kappa^2\|\zeta\psi_\kappa\|_2^2
+\|\zeta\psi_\kappa\nabla\eta_{\kappa^{-1/2}}\|_2^2+\|\zeta\psi_\kappa\nabla\tilde{\eta}_{\kappa^{-1/2}}\|_2^2 + \|\psi_\kappa\nabla\zeta\|_2^2 \,.
\endaligned
$$
Observing that $\langle\psi_\kappa\nabla\chi,\psi_\kappa\nabla\eta\rangle=0$, we obtain
\begin{displaymath}
\|\psi_\kappa\nabla\zeta\|_2^2 \leq \alpha_\delta^2\kappa^2\|\psi_\kappa\zeta\|_2^2 + \|\psi_\kappa\nabla\eta\|_2^2 \,.
\end{displaymath}
Hence,
\begin{equation}\label{eq:53}
\aligned
& \|\nabla_{\kappa A_\kappa}(\zeta\tilde{\eta}_{\kappa^{-1/2}}\psi_\kappa)\|_2^2+
\|\nabla_{\kappa A_\kappa}(\zeta\eta_{\kappa^{-1/2}}\psi_\kappa)\|_2^2 \\
\leq& \kappa^2\big(1+ \alpha_\delta^2+C\kappa^{-1}\big)\|\zeta\psi_\kappa\|_2^2 + \|\psi_\kappa\nabla\eta\|_2^2\,.
\endaligned
\end{equation}
We now use \eqref{eq:22} and \eqref{eq:49} to obtain, for sufficiently
small $\delta$,
\begin{equation}
\label{eq:54}
\aligned
\|\nabla_{\kappa A_\kappa}(\zeta\tilde{\eta}_{\kappa^{-1/2}}\psi_\kappa)\|_2^2
\geq&
\kappa^4\mu_{\kappa^{-2}}(A_n+\kappa^{-1}A_{1,\kappa},S_{\delta/2})\|\zeta\tilde{\eta}_{\kappa^{-1/2}}\psi_\kappa\|_2^2\\
\geq & \kappa^2\min(\Theta_0h_j,1+\delta/2)
[1-C\kappa^{-2/3}]\|\zeta\tilde{\eta}_{\kappa^{-1/2}}\psi_\kappa\|_2^2 \\
\geq & (1+\frac \delta 2 )\, \kappa^2[1-C\kappa^{-2/3}] \|\zeta\tilde{\eta}_{\kappa^{-1/2}}\psi_\kappa\|_2^2 \,.
\endaligned
\end{equation}
By \cite[Theorem 2.9]{AHS} we have, since $\zeta\eta_{\kappa^{-1/2}}\psi_\kappa$ vanishes on $\partial \Omega$,
\begin{equation}
\label{eq:55}
\|\nabla_{\kappa A_\kappa}(\zeta\eta_{\kappa^{-1/2}}\psi_\kappa)\|_2^2 \geq
(1+\delta/2-\kappa^{-1/2})\kappa^2\|\zeta\eta_{\kappa^{-1/2}}\psi_\kappa\|_2^2\,.
\end{equation}
Consequently, by \eqref{eq:53}, \eqref{eq:54}, and
\eqref{eq:55}, and by choosing
$$
\alpha_\delta^2 = \frac{\delta}{4}\,$$ we obtain, that for $\kappa\geq \kappa(\delta)$, with
$\kappa(\delta)$ sufficiently large:
\begin{displaymath}
\kappa^2 \frac{\delta}{8} \|\zeta\psi_\kappa\|_2^2 \leq \kappa^2 \left( \frac{\delta}{4} - \widehat C \kappa^{-\frac 12}\right)\, \|\zeta\psi_\kappa\|_2^2 \leq \|\psi_\kappa\nabla\eta\|_2^2\,,
\end{displaymath}
from which \eqref{eq:50} easily follows.
\end{proof}
Next we consider currents satisfying only (\ref{eq:10}a). Let, for $j=1,2$,
\begin{equation} \label{eq:56}
\omega_{\delta,j} = \{x\in \Omega \,: \, (-1)^jB_n(x)>1+\delta \;;\;d(x,\partial\Omega_i) > \delta\,\} \,,\end{equation}
and
\begin{equation} \label{Gamma}
\Gamma_{\delta,j}= \partial\omega_{\delta,j} \setminus\partial\Omega_c\cap\partial\omega_{\delta,j} \,.
\end{equation}
We can now state
\begin{proposition}
Suppose that for some $j\in\{1,2\}$ we have
\begin{equation}
\label{eq:57}
1<|h_j| \,.
\end{equation}
Then, there exist $C>0$, $\delta_0>0$, such that for any $0<\delta<\delta_0$,
some $\kappa_0 (\delta)>0$ can be found, for which
\begin{equation} \label{eq:58}
\kappa\geq \kappa_0(\delta)\Rightarrow \int_{\omega_{\delta,j}} \exp\Big(\delta^{1/2}\kappa d(x,
\Gamma_{\delta,j})\Big) |\psi_\kappa|^2 \,dx \leq \frac{C}{\delta^{3/2}} \,.
\end{equation}
\end{proposition}
\begin{proof}
Without loss of generality we may assume $h_j>0$; otherwise we apply
to \eqref{eq:25} the transformation
$(\psi_\kappa,A_\kappa,\phi_\kappa)\to(\bar{\psi_\kappa},-A_\kappa,-\phi_\kappa)$. Let
\begin{equation}\label{check1}
{ \check{\chi}=}
\begin{cases}
\exp\Big(\frac 12 \delta^{1/2}\kappa d(x,\Gamma_{\delta,j})\Big) &\text{if } x\in \omega_{\delta,j}\,, \\
1 &\text{if } x\in\Omega\setminus \omega_{\delta,j}\,.
\end{cases}
\end{equation}
Let further $\eta$ and $\eta_r$ be given by \eqref{eq:51} and
\eqref{eq:52} respectively. Then set
\begin{equation}\label{check2}
\check{\zeta}= \eta_\delta\, \eta\, \check \chi \,.
\end{equation}
The proof proceeds in the same manner as in the previous proposition
with $\zeta$ replaced by $\check{\zeta}$ with the difference that now
$\check{\zeta} \psi_\kappa(x)$ vanishes for all $ x\in\partial \Omega_i$. Consequently,
(\ref{eq:10}a) is no longer necessary (see \eqref{eq:55}). We use (\ref{eq:10}b)
to establish that $\omega_{\delta,j}$ is not empty.
\end{proof}
We conclude this section by showing that for $\mathcal O(\kappa)$ currents
(i.e. when $J$ is independent of $\kappa$) $\|\psi_\kappa\|_2$ must be small.
To this end we define $\Phi_n$ as the solution of (\ref{eq:11b}a,b), and
\begin{equation}
\label{eq:59}
\int_\Omega |\psi_\kappa|^2\Phi_n \,dx =0\, .
\end{equation}
The above condition is a natural choice as by \eqref{eq:33} we have
that
\begin{displaymath}
\int_\Omega |\psi_\kappa|^2\phi_\kappa \,dx =0\, .
\end{displaymath}
It can be easily verified from \eqref{eq:20} that
\begin{equation}\label{calconst}
\Phi_n=\phi_n+ C(\kappa,c)\,,
\end{equation}
where $\phi_n$ denotes the solution of \eqref{eq:11b}.
The constant can be extracted from \eqref{eq:59}:
$$
C (\kappa,c) = - \frac{\int_\Omega \phi_n |\psi_\kappa|^2\, dx }{\int_\Omega |\psi_\kappa|^2\, dx}\,,
$$
from which we get the following upper bound (independent of $\kappa$ and $c$)
\begin{equation}
\label{eq:60}
|C (\kappa,c) | \leq \|\phi_n\|_\infty < +\infty\,.
\end{equation}
\begin{proposition}
Under Assumptions \eqref{eq:2}-\eqref{eq:10} there exists \linebreak
$C(J,\Omega)>0$ and $\kappa_0>0$ such that for any $\kappa\geq \kappa_0$,
\begin{equation}
\label{eq:61}
\| \psi_\kappa \|_2 \leq C (J,\Omega) (1+c^{-1/2})^{1/3}\kappa^{-1/6} \,.
\end{equation}
\end{proposition}
\begin{proof}
Let $\Phi_{1,\kappa}=\phi_k-c\kappa\Phi_n$.
An immediate consequence of \eqref{eq:33} is that
\begin{equation}
\label{eq:62}
\begin{cases}
-\Delta\Phi_{1,\kappa} + c|\psi|^2_\kappa \Phi_{1,\kappa} = - c^2|\psi|^2_\kappa \kappa\Phi_n & \text{ in } \Omega\,, \\
\frac{\partial\Phi_{1,\kappa}}{\partial\nu}= 0 & \text{ on } \partial\Omega\,.
\end{cases}
\end{equation}
Taking the inner product with $\Phi_n$ yields, with the aid of (\ref{eq:59}),
\begin{equation}
\label{eq:63}
\aligned
& \kappa\| \psi_\kappa \Phi_n\|_2^2 = - \frac{1}{c^2}\langle\nabla\Phi_n,\nabla\Phi_{1,\kappa}\rangle +\frac{1}{c}\langle \langle|\psi_\kappa|
\Phi_n,|\psi_\kappa| (\Phi_{1,\kappa}-(\Phi_{1,\kappa})_\Omega \rangle\\
\leq& \frac{1}{c^2}\|\nabla\Phi_n\|_2\|\nabla\Phi_{1,\kappa}\|_2 + \frac{1}{c}\| \psi_\kappa
\Phi_n\|_2 \| \psi_\kappa(\Phi_{1,\kappa}-(\Phi_{1,\kappa})_\Omega )\|_2\,,
\endaligned
\end{equation}
where $(\Phi_{1,\kappa})_\Omega $ is the average of $\Phi_{1,\kappa}$ in $\Omega$.
With the aid of (\ref{eq:41}) (note that $\nabla \phi_{1,\kappa}=\nabla \Phi_{1,\kappa}$),
the fact that \break $|\psi_\kappa|\leq1$, and the Poincar\'e inequality we then
obtain
\begin{equation}
\label{eq:64}
\| \psi_\kappa \Phi_n\|_2 \leq C\kappa^{-1/2}(1+ c^{-1/2}) \,.
\end{equation}
We now set
\begin{displaymath}
{\mathcal U}_\kappa = \{ x\in\Omega \,:\, |\Phi_n(x) |< (1+c^{-1/2})^{2/3}\kappa^{-1/3}\,\} \,.
\end{displaymath}
By \eqref{eq:59} the level set $\Phi_n^{-1}(0)$ lies inside
$\Omega$.
Let $x_0\in\Phi_n^{-1}(\tau)$ for some $\tau\neq0$, and set
$$\Gamma_\perp=B_n^{-1}(B_n(x_0)).
$$
By (\ref{eq:11}a) $B_n$ is the conjugate harmonic function of $\Phi_n$,
and hence $\Gamma_\perp$ must be perpendicular to $\Phi_n^{-1}(\tau)$ at $x_0$. Note
that in \cite[(2.3)]{alhe14} we showed that $B_n^{-1}(\mu)$ is a simple
smooth curve connecting the two connected components of $\partial\Omega_c$ for any
$h_1<\mu<h_2$. We denote by $\tilde{\Gamma}_\perp$ the subcurve of $\Gamma_\perp$
originating from $x_0$ in the direction where $\Phi_n$ decreases if
$\tau>0$ or increases if $\tau<0$, and terminating either on $\Phi_n^{-1}(0)$ or
on the boundary. Clearly,
\begin{displaymath}
|\tilde{\Gamma}_\perp|\inf_{x\in\Omega}|\nabla\Phi_n|\leq \Big|\int_{\tilde{\Gamma}_\perp} \nabla\Phi_n ds\Big| \leq |\tau|\,,
\end{displaymath}
where $\int_\Gamma \vec{V}$ denotes the circulation of $\vec{V}$ along the path $\Gamma$.
In \cite[\S2.3]{alhe14} we have
established that $|\nabla\Phi_n|=|\nabla B_n|>0$ in $\bar{\Omega}$. It follows that
\begin{equation}
\label{eq:65}
d(x_0,\partial\Omega\cup\Phi_n^{-1}(0)) \leq |\tilde{\Gamma}_\perp| \leq C|\tau|\,.
\end{equation}
Let
\begin{displaymath}
\tilde{{\mathcal U}}_\kappa(r)= \{x\in\Omega \,| \, d(x,\partial\Omega\cup\Phi_n^{-1}(0)) \leq
r(1+c^{-1/2})^{2/3}\kappa^{-1/3}\}\,.
\end{displaymath}
By \eqref{eq:65} we obtain that for sufficiently large $r$ there
exists $\kappa_0(r)$ such that for all $\kappa>\kappa_0$ and
$c\in{\mathbb{R}}$ we have ${\mathcal U}_\kappa \subseteq \tilde{{\mathcal U}}_\kappa(r)$. Consequently,
\begin{equation}
\label{eq:66}
| {\mathcal U}_\kappa| \leq C(1+c^{-1/2})^{2/3}\kappa^{-1/3}(|\Phi_n^{-1}(0)|+|\partial\Omega|)\leq C(1+c^{-1/2})^{2/3}\kappa^{-1/3}\,.
\end{equation}
By \eqref{eq:64} we have that
\begin{displaymath}
\| \psi_\kappa\|_{L^2(\Omega\setminus{\mathcal U}_\kappa)} \leq C(1+c^{-1/2})^{1/3}\kappa^{-1/6} \,,
\end{displaymath}
whereas from \eqref{eq:66} and \eqref{eq:26} we learn that
\begin{displaymath}
\| \psi_\kappa\|_{L^2({\mathcal U}_\kappa)} \leq C(1+c^{-1/2})^{1/3}\kappa^{-1/6} \,.
\end{displaymath}
The proposition can now be readily verified.
\end{proof}
An immediate conclusion is that whenever $c\kappa\gg1$, $|\psi_\kappa|$ is small. If
$c=\mathcal O(\kappa^{-1})$, $|\psi_\kappa|$ may not tend to $0$ as $\kappa\to\infty$. Further research
is necessary to establish this point.
\begin{remark}
If, for some $0<\alpha<1$, we assume that $J=J(\cdot, \kappa)$ satisfies
$$
\|J\|\leq C\kappa^{-\alpha}\,,
$$
then (\ref{eq:63}) and
(\ref{eq:41}) remain valid. Assuming $c=1$, and using this time (\ref{eq:48}), we obtain
instead of (\ref{eq:64}), for any $0<\beta$, that
\begin{displaymath}
\| \psi_\kappa \Phi_n\|_2 \leq C_\beta \kappa^{-1/2}(\kappa^{\alpha/2}+\kappa^\beta\|\psi_\kappa\|_2^{1/2}) \,.
\end{displaymath}
Using the above, with sufficiently small $\beta$, we obtain, similarly to the derivation of (\ref{eq:61})
\begin{displaymath}
\| \psi_\kappa\|_2\leq C\kappa^{-(1-\alpha)/6}\,,
\end{displaymath}
which implies
$$\|\psi_\kappa\|_2\xrightarrow[\kappa\to + \infty]{}0\,.
$$
This result stands in sharp contrast with the behavior obtained
in the absence of electric potential \cite{sase03,attar2014ground}.
\end{remark}
\section{Time-Dependent Analysis}
\label{sec:4}
In this section we return to the time-dependent problem as introduced
in \eqref{eq:1}. For convenience we set here $$c=1\,.$$
\begin{subequations}
\label{eq:67}
\begin{alignat}{2}
& \frac{\partial\psi_\kappa}{\partial t}- \nabla_{\kappa A_\kappa}^2\psi_\kappa+i\kappa\phi_\kappa \psi_\kappa =\kappa^2(1-|\psi_\kappa|^2)\psi_\kappa & \text{ in } (0,+\infty)\times \Omega& \,,\\
&\frac{\partial A_\kappa}{\partial t}+\nabla\phi_\kappa +\text{\rm curl\,}^2A_\kappa
=\frac{1}{\kappa}\Im(\bar\psi_\kappa\nabla_{\kappa A_\kappa}\psi_\kappa) &
\text{ in } (0,+\infty)\times \Omega&\,, \\
&\psi =0 &\text{ on } (0,+\infty)\times \partial\Omega_c&\,, \\
& \nabla_{\kappa A_\kappa}\psi_\kappa\cdot\nu=0 & \text{ on } (0,+\infty)\times \partial\Omega_i&\,, \\
& \frac{\partial\phi_\kappa}{\partial\nu} = - \kappa J(x) & \text{ on }(0,+\infty)\times \partial\Omega_c&\,, \\
& \frac{\partial\phi_\kappa}{\partial\nu}=0 & \text{ on } (0,+\infty)\times \partial\Omega_i &\,, \\[1.2ex]
& \Xint-_{\partial\Omega}\text{\rm curl\,} A_\kappa \, ds = \kappa h_{ex} &\quad \text{ on } (0,+\infty)&\,, \\
& \psi(0,x)=\psi_0(x) & \text{ in } \Omega&\,, \\
&A(0,x)=A_0(x) & \text{ in } \Omega & \,.
\end{alignat}
\end{subequations}
We assume again \eqref{condinit}-(\ref{eq:6}), \eqref{eq:8}, and
\eqref{eq:10}. Since in the time dependent case $\phi_\kappa$ is determined
up to a constant in view of \eqref{eq:4}
and \eqref{eq:5}, we can
further impose
\begin{equation}\label{eq:68}
\int_\Omega\phi_\kappa (t,x)\,dx =0\,, \quad \forall t>0 \,.
\end{equation}
It follows from \eqref{condinit} by the maximum
principle (see \cite[Theorem 2.6] {alhe14}) that
\begin{equation}
\label{eq:69}
\|\psi_\kappa(t, \cdot)\|_\infty\leq 1 \,, \quad \forall t\geq0\,.
\end{equation}
We recall from \cite [Subsection 2.4] {alhe14} the following spectral entity
\begin{subequations}
\label{eq:70}
\begin{equation}
\lambda= \inf_{
\begin{subarray}{2}
V\in{\mathcal H}_d \\
\|V\|_2 =1
\end{subarray}} \| \text{\rm curl\,} V\|_2^2 \,,
\end{equation}
where
\begin{equation}
{\mathcal H}_d = \big\{ V\in H^1(\Omega,{\mathbb{R}}^2)\,:\, \text{\rm div\,} V=0 \,, V\big|_{\partial\Omega}\cdot\nu=0 \big\} \,.
\end{equation}
\end{subequations}
We further recall from \cite [Proposition 2.5] {alhe14} that, under condition $(R_1)$ on $\partial \Omega$,
\begin{displaymath}
\lambda=\lambda^D:=\inf_{
\begin{subarray}{2}
u\in H^1_0(\Omega) \\
\|u\|_2 =1
\end{subarray}} \| \nabla u\|_2^2>0 \,.
\end{displaymath}
We retain our definition of the normal fields $(A_n,\phi_n)$ via \eqref{eq:11}. For
the solution $(A_\kappa,\phi_\kappa)$ of \eqref{eq:67} we set
\begin{equation}\label{time-dep-AB} \aligned
&A_{1,\kappa}(t,x)=A_\kappa(t,x)-\kappa A_n(x),\\
&\phi_{1,\kappa}(t,x)=\phi_\kappa(t,x)-\kappa\phi_n(x),\\
&B_\kappa(t,x)=\text{\rm curl\,} A_\kappa(t,x),\\
&B_{1,\kappa}(t,x)=\text{\rm curl\,} A_{1,\kappa}(t,x).
\endaligned
\end{equation}
Clearly,
\begin{subequations}
\label{eq:71}
\begin{alignat}{2}
\frac{\partial A_{1,\kappa}}{\partial t}+\nabla\phi_{1,\kappa} + \text{\rm curl\,} B_{1,\kappa}
& = \frac{1}{\kappa}\Im(\bar\psi_\kappa\, \nabla_{\kappa A_\kappa}\psi_\kappa) & \text{ in }
(0,+\infty)\times \Omega\,,\\
\frac{\partial\phi_{1,\kappa}}{\partial\nu}&= 0 & \text{ on } (0,+\infty)\times \partial\Omega\,, \\
\Xint-_{\partial\Omega}B_{1,\kappa}(t,x)\,ds& = 0 & \text{ in } (0,+\infty) \,.
\end{alignat}
\end{subequations}
We begin by the following auxiliary estimate. We recall that \break $\|A(t,\cdot)\|_{1,2}=\|A(t,\cdot)\|_{H^1(\Omega,{\mathbb{R}}^2)}$.
\begin{lemma}
\label{lem:parabolic-A1}
Let $A_{1,\kappa}$ and $B_{1,\kappa}$ be defined by \eqref{time-dep-AB}.
Suppose that \linebreak $\|A_{1,\kappa}(\cdot,0)\|_2\leq M$ (where $M$ may depend on
$\kappa$). Then, under the above assumptions, there exists $ t^*(M)$ and a
constant $C=C(\Omega,t^*)>0$ such that for all $t>t^*$ and $\kappa\geq 1$ we
have
\begin{equation}
\label{eq:72}
\|A_{1,\kappa}(t,\cdot)\|_{1,2} + \|A_{1,\kappa}\|_{L^2(t,t+1,H^2(\Omega))} \leq C.
\end{equation}
\end{lemma}
\begin{proof}
By \cite[Lemma 5.3]{alhe14} there exists a constant $C=C(\Omega)>0$ such that for
sufficiently large
$\kappa$
\begin{displaymath}
\|A_{1,\kappa}(t,\cdot)\|_2^2 \leq \Big(\|A_{1,\kappa}(0,\cdot)\|_2^2+\frac{C}{\kappa^2}\Big)
e^{-\lambda t} + C \int_0^te^{-\lambda(t-\tau)}\|\psi_\kappa(\tau,\cdot)\|_2^2 \,d\tau \,.
\end{displaymath}
From \eqref{eq:69}, we then get
\begin{displaymath}
\|A_{1,\kappa}(t,\cdot)\|_2^2 \leq \Big(\|A_{1,\kappa}(0,\cdot)\|_2^2+\frac{C}{\kappa^2}\Big)\,
e^{-\lambda t} + \frac{C}{\lambda}
\end{displaymath}
We thus have
\begin{equation}\label{anytime}
\|A_{1,\kappa}(t,\cdot)\|_2^2 \leq (M+C) e^{-\lambda t} + \frac{C}{\lambda}
\end{equation}
Hence, there exists $t_0^*(M)$, such that for $t \geq t_0^*(M)$, we have
\begin{equation}
\label{eq:73}
\|A_{1,\kappa}(t,\cdot)\|_2 \leq \frac{2C}{\lambda} \,.
\end{equation}
Next, we apply \cite[Theorem C.1 (Formula C.4)]{alhe14} to the operator
$\mathcal L^{(1)}$ (as introduced there in Example (4) above this theorem) to
obtain that
\begin{equation}\label{consc1}
\aligned
& \|A_{1,\kappa}\|_{L^\infty(t_0,t_0+1,H^1(\Omega))}+
\|A_{1,\kappa}\|_{L^2(t_0,t_0+1,H^2(\Omega))}\\& \qquad
\leq \frac{C}{\kappa}\|\Im \{\bar{\psi}_\kappa\nabla_{\kappa
A_\kappa}\psi_\kappa\}\|_{L^2(t_0,t_0+1,L^2(\Omega))} +
C \|A_{1,\kappa}(t_0,\cdot)\|_{1,2} \,.
\endaligned
\end{equation}
with a constant $C$ independent of $t_0$.
Since from (\ref{eq:67}a) (cf. \cite{alhe14}) we can easily get
that
\begin{equation}
\label{eq:74}
\|\nabla_{\kappa A_\kappa}\psi_\kappa (t,\cdot)\|_2^2 \leq \kappa^2\|\psi_\kappa(t,\cdot)\|_2^2-\frac{1}{2}
\frac{d\|\psi_\kappa (t,\cdot)\|_2^2}{dt} \,,
\end{equation}
we obtain by integrating over $(t_0,t_0+1)$
\begin{equation}\label{eq:70a}
\|\nabla_{\kappa A_\kappa}\psi_\kappa \|^2_{L^2(t_0,t_0+1,L^2(\Omega))} \leq \kappa^2\|\psi_\kappa \|_{L^2(t_0,t_0+1,L^2(\Omega))}^2 + \frac 12 \|\psi_\kappa(t_0, \cdot)\|_2^2\,,
\end{equation}
and note for later reference that it implies
\begin{equation}\label{eq:70aa}
\|\nabla_{\kappa A_\kappa}\psi_\kappa \|_{L^2(t_0,t_0+1,L^2(\Omega))} \leq C(\Omega)\, \kappa\,.
\end{equation}
Implementing the upper bound \eqref{eq:70a} in \eqref{consc1}, yields
$$\aligned
& \|A_{1,\kappa}\|_{L^\infty(t_0,t_0+1,H^1(\Omega))}+ \|A_{1,\kappa}\|_{L^2(t_0,t_0+1,H^2(\Omega))} \\
\leq& C\Big[1+
\|\psi_\kappa\|_{L^2(t_0,t_0+1,L^2(\Omega))} +
\frac{1}{\kappa}\|\psi_\kappa(t_0, \cdot)\|_2+\|A_{1,\kappa}(t_0,\cdot)\|_{1,2}\Big] \,.
\endaligned
$$
We next apply \cite[Theorem C.1 (Formula C.2)]{alhe14} to obtain in
precisely the same manner
$$
\begin{array}{l}
\|A_{1,\kappa}\|_{L^\infty(t_0,t_0+1,H^1(\Omega))}\\
\qquad \leq C\Big[1+
\|\psi_\kappa\|_{L^2(t_0-1,t_0+1,L^2(\Omega))} +
\frac{1}{\kappa}\|\psi_\kappa(t_0, \cdot)\|_2 + \|A_{1,\kappa}(t_0-1,\cdot)\|_2 \Big] \,.
\end{array}
$$
The above together with \eqref{eq:69} and \eqref{eq:73} yields, for
$t_0\geq t_0^* +1$,
\begin{equation}
\label{est-of-B}
\|A_{1,\kappa}\|_{L^\infty(t_0,t_0+1,H^1(\Omega))}+ \|A_{1,\kappa}\|_{L^2(t_0,t_0+1,H^2(\Omega))}
\leq C\,,
\end{equation}
which implies \eqref{eq:72}, with $t^* = t_0^* (M) +1$.
\end{proof}
\begin{remark}\label{usefulrem}
Since our interest is in the limit as $t \to +\infty$, Lemma~\ref{lem:parabolic-A1} allows us to assume in the sequel, without
any loss of generality, that (\ref{eq:72}) is satisfied for all
$t\geq0$. We have just to make a translation $t \mapsto t -t^*$ and to
observe that $\psi_\kappa(t^*,\cdot)$ has the same properties as $\psi_0$.
\end{remark}
\begin{proposition}
\label{lem:asymp-exp}
Let $\omega_{\delta,j}$ ($j\in\{1,2\}$) be defined in
\eqref{eq:56}.
Suppose that
for some $j\in\{1,2\}$ we have that
\begin{equation}
\label{eq:75}
1<|h_j| \,.
\end{equation}
Then, there exist $C>0$ and $\delta_0>0$, and, for any $0<\delta<\delta_0$, $\kappa_0(\delta)\geq 1$ such that, for $\kappa\geq \kappa_0(\delta)$,
\begin{equation}
\label{eq:76}
\limsup_{t\to\infty}\int_{\omega_{\delta,j} }|\psi_\kappa|^2(t,x) \,dx \leq
\frac{C_\delta }{\kappa^2} \,.
\end{equation}
\end{proposition}
\begin{proof}
Without loss of generality we may assume $h_j>0\,$; otherwise we apply
to \eqref{eq:67} the transformation
$(\psi_\kappa,A_\kappa,\phi_\kappa)\to(\bar{\psi_\kappa},-A_\kappa,-\phi_\kappa)$.
\vspace{1ex}
{\it Step 1: Let, for $n\geq1$,
\begin{displaymath}
a_n = \|{\widehat \zeta}\psi_\kappa\|_{L^\infty(n-1, n,L^2(\Omega))} \,.
\end{displaymath}
Prove that for all $\delta \in (0,1)$ and $\kappa \geq \kappa_0(\delta)$,
\begin{equation}\label{inega}
a_n^2 \leq C \delta^{-3} \Big(\kappa^{-2} + \kappa^{-1} (a_{n} + a_{n-1})\Big)\,.
\end{equation}}
\vspace{2ex}
Let $\eta$ and $\eta_r$ be given by \eqref{eq:51} and \eqref{eq:52}
respectively. Then, set
\begin{equation}
\label{eq:77}
{\widehat \zeta}=\eta \, \eta_\delta\, .
\end{equation}
Multiplying (\ref{eq:67}a) by ${\widehat \zeta}^{\,2} \bar{\psi_\kappa}$ and integrating by
parts yields
\begin{displaymath}
\frac{1}{2}\frac{d}{dt} \left( \|{\widehat \zeta}\psi_\kappa (t,\cdot)\|_2^2\right) \, + \|\nabla_{\kappa A_\kappa}({\widehat \zeta}\psi_\kappa (t,\cdot))\|_2^2 \leq
\kappa^2\|{\widehat \zeta}\psi_\kappa (t,\cdot)\|_2^2 + \|\psi_\kappa (t,\cdot)\, \nabla{\widehat \zeta}\|_2^2 \,.
\end{displaymath}
By \cite[Theorem 2.9] {AHS},
we have
$$\aligned
\|\nabla_{\kappa A_\kappa}({\widehat \zeta}\psi_\kappa (t,\cdot))\|_2^2 \geq& \langle\kappa B_\kappa (t,\cdot) {\widehat \zeta}\psi_\kappa(t,\cdot),{\widehat \zeta}\psi_\kappa(t,\cdot)\rangle \\ =&
\kappa^2\langle B_n{\widehat \zeta}\psi_\kappa(t,\cdot),{\widehat \zeta}\psi_\kappa(t,\cdot) \rangle + \langle\kappa B_{1,\kappa}(t,\cdot)\,{\widehat \zeta}\psi_\kappa(t,\cdot),{\widehat \zeta}\psi_\kappa(t,\cdot)\rangle \\
\geq&
\kappa^2\Big(1+\frac{\delta}{2}\Big)\|{\widehat \zeta}\psi_\kappa(t,\cdot)\|_2^2+ \langle\kappa B_{1,\kappa}{\widehat \zeta}\psi_\kappa(t,\cdot)\,,\,{\widehat \zeta}\psi_\kappa(t,\cdot)\rangle \,.
\endaligned
$$
We can thus write
\begin{displaymath}
\begin{array}{l}
\frac{1}{2}\frac{d}{dt} \left( \|{\widehat \zeta}\psi_\kappa(t,\cdot) \|_2^2\right) + \frac{\kappa^2\delta}{2}\|{\widehat \zeta}\psi_\kappa (t,\cdot) \|_2^2\\ \quad \leq
\|\psi_\kappa (t,\cdot) \nabla\eta\|_2^2 + \|\psi_\kappa(t,\cdot) \nabla\eta_\delta\|_2^2- \langle\kappa B_{1,\kappa}(t,\cdot) {\widehat \zeta}\psi_\kappa(t,\cdot),{\widehat \zeta}\psi_\kappa(t,\cdot)\rangle \,.
\end{array}
\end{displaymath}
Since
\begin{displaymath}
\|\psi_\kappa (t,\cdot)\nabla\eta\|_2^2 + \|\psi_\kappa(t,\cdot) \nabla\eta_\delta\|_2^2 \leq \frac{C}{\delta^2} \,,
\end{displaymath}
we obtain that
\begin{equation}
\label{eq:78}
\begin{array}{l}
\frac{1}{2}\frac{d}{dt} \left( \|{\widehat \zeta}\psi_\kappa (t,\cdot)\|_2^2 \right)
+ \frac{\kappa^2\delta}{2}\|{\widehat \zeta}\psi_\kappa (t,\cdot)\|_2^2\\ \qquad
\leq \frac{C}{\delta^2} - \langle \kappa B_{1,\kappa} (t,\cdot) {\widehat \zeta}\psi_\kappa(t,\cdot)\,,\, {\widehat \zeta}\psi_\kappa (t,\cdot) \rangle \,.
\end{array} \end{equation}
From \eqref{eq:78} we can conclude that
\begin{equation}\label{eq:79}
\aligned
\|{\widehat \zeta}\psi_\kappa(t,\cdot)\|_2^2 \leq& \|{\widehat \zeta}\psi_0\|_2^2\, e^{-\delta\kappa^2t} +
\frac{C}{\delta^3\kappa^2}\\
&+2 \int_0^t
e^{-\delta\kappa^2(t-\tau)}\big|\langle\kappa B_{1,\kappa}(\tau,\cdot) {\widehat \zeta}\psi_\kappa(\tau,\cdot) ,{\widehat \zeta}\psi_\kappa(\tau,\cdot) \rangle \big|\,d\tau \,.
\endaligned
\end{equation}
To estimate the last term on the right-hand-side of \eqref{eq:79}, we start from
$$ \aligned
& \int_0^t e^{-\delta\kappa^2(t-\tau)}\big|\langle\kappa
B_{1,\kappa}(\tau,\cdot) {\widehat \zeta}\psi_\kappa (\tau,\cdot) ,{\widehat \zeta}\psi_\kappa(\tau, \cdot) \rangle\big|\,d\tau \\
\leq & \kappa\, \Big[\int_0^t e^{-\delta\kappa^2(t-\tau)}
{ \|B_{1,\kappa}(\tau, \cdot)\|_2^2 \,d\tau} \cdot \int_0^t e^{-\delta\kappa^2(t-\tau)}
\|{\widehat \zeta}{ \psi_\kappa(\tau, \cdot )\|_4^4}\,d\tau\Big]^{1/2}\,. \\
\endaligned
$$
With Remark \ref{usefulrem} in mind, we use \eqref{eq:69} to obtain
$$
\int_0^t e^{-\delta\kappa^2(t-\tau)}
\|B_{1,\kappa}(\tau, \cdot)\|_2^2 \,d\tau \leq \frac{C}{\delta \kappa^2} \,.
$$
Implementing the above estimate, we obtain
$$
\begin{array}{l}
\int_0^t e^{-\delta\kappa^2(t-\tau)}\big|\langle\kappa
B_{1,\kappa}(\tau,\cdot) {\widehat \zeta}\psi_\kappa (\tau,\cdot) ,{\widehat \zeta}\psi_\kappa(\tau, \cdot) \rangle\big|\,d\tau\\
\qquad
\leq C \delta^{-\frac 12} \Big[\int_0^t e^{-\delta\kappa^2(t-\tau)}
\|{\widehat \zeta}{\psi_\kappa(\tau, \cdot )\|_4^4}\,d\tau\Big]^{1/2}\,.
\end{array}
$$
To control of the right hand side we now write for $t\geq 1$
$$\aligned
& \Big[\int_0^t e^{-\delta\kappa^2(t-\tau)}
\|{\widehat \zeta}\psi_\kappa(\tau, \cdot )\|_4^4\,d\tau\Big]^{1/2} \\
& \leq \Big[\int_0^{t-1} e^{-\delta\kappa^2(t-\tau)}
\|{\widehat \zeta}\psi_\kappa(\tau, \cdot )\|_4^4\,d\tau\Big]^{1/2} + \Big[\int_{t-1} ^t e^{-\delta\kappa^2(t-\tau)}
\|{\widehat \zeta}\psi_\kappa(\tau, \cdot )\|_4^4\,d\tau\Big]^{1/2}\\
& \leq C \Big[\int_{t-1} ^t e^{-\delta\kappa^2(t-\tau)}
\|{\widehat \zeta}\psi_\kappa(\tau, \cdot )\|_2^2\,d\tau\Big]^{1/2} + C \delta^{-\frac 12} \kappa^{-1} e^{- \frac{\delta }{2} \kappa^2}\\
& \leq C \delta^{-\frac 12} \kappa^{-1} \|\widehat \zeta
\psi_\kappa\|_{L^\infty(t-1,t,L^2(\Omega)) } + C \delta^{-\frac 12} \kappa^{-1} e^{-
\frac{\delta }{2} \kappa^2} \,.
\endaligned
$$
Substituting the above into \eqref{eq:79} yields, with a new constant
$C$, for $\kappa$ large enough, and for $t\geq1\,$,
$$
\begin{array}{l}
\|{\widehat \zeta}\psi_\kappa(t,\cdot)\|_2^2 \\ \quad \leq C \delta^{-3} \kappa^{-2} + C \delta^{-1} \kappa^{-1} e^{ - \frac{\delta}{2} \kappa^2}
+ C \delta^{-1} \kappa^{-1} \|\widehat \zeta \psi_\kappa\|_{L^\infty(t-1,t,L^2(\Omega)) } \,.
\end{array}
$$
From which we easily obtain \eqref{inega}.
\vspace{1ex}
{\it Step 2: Prove \eqref{eq:76}.}
\vspace{1ex}
By (\ref{eq:69}) we have
$$
0 < a_n \leq C \,,
$$
which readily yields
$$
a_n \leq C \delta^{-\frac 32}\kappa^{-\frac 12}\,.
$$
We improve the above estimate by reimplementing \eqref{inega}.
To this end we set
$$\widehat C := C \delta^{- \frac 32 }\,,$$ and then let
\begin{displaymath}
a_n= \frac{\widehat C}{\kappa}\alpha_n \,.
\end{displaymath}
Substituting into (\ref{inega}) yields
\begin{equation}\label{inegaga}
\alpha_n^2 \leq 1 + \alpha_{n-1} +\alpha_n \,.
\end{equation}
Suppose that for some $N\geq0$, we have $\alpha_N\leq 1 + \sqrt{2} \,$, then\break $\alpha_{N+1}\leq 1 +\sqrt{2}$ and
hence $\alpha_n\leq 1 + \sqrt{2}$ for all $n\geq N$.
If $\alpha_{n-1} > 1+ \sqrt{2}$ for any $n$, we have, with $\hat \alpha_n = \alpha_n - \frac 12\,$,
\begin{displaymath}
\hat \alpha_n^2\leq \frac 74 + \hat \alpha_{n-1} < \hat \alpha_{n-1}^2 \,.
\end{displaymath}
Hence, $\hat \alpha_n<\hat \alpha_{n-1}$ which means that $\hat \alpha_n$
converges as a positive decreasing sequence, and necessarily to a
limit smaller than $1/2 + \sqrt{2}$. We thus conclude that
\begin{equation}
\label{eq:80}
\limsup \alpha_n \leq 1 + \sqrt{2} \,,
\end{equation}
and hence
\begin{equation}
\label{ineq-1} \limsup a_n \leq \frac{C (1 +\sqrt{2})}{\delta^{3/2}\kappa},
\end{equation}
from which \eqref{eq:76} can easily be deduced.
\end{proof}
We next obtain the following improvement over (\ref{eq:70aa}) for
$\nabla_{\kappa A_\kappa}(\widehat \zeta \psi_\kappa)$.
\begin{proposition}
Let $p \geq 2$. For any $\delta >0$, there exists $\kappa_0 (\delta)$ and $C(\delta)$
such that for $\kappa \geq \kappa (\delta)$ we have, with $u={\widehat \zeta}\psi_\kappa\,,
$ the following estimate:
\begin{equation}\label{eq:89a}
\|\nabla_{\kappa A_\kappa}u\|_{L^p(t_0,t_0+1; L^p(\Omega))}\leq C (\delta) \kappa^{6(1-2/p)}\,.
\end{equation}
\end{proposition}
\begin{proof} {\it Step 1: Prove that for some $C(\delta)>0$ we have, for
sufficiently large $\kappa$ that
\begin{equation}
\label{eq:87}
\|u\|_{L^2(t_0-1,t_0+1,H^2(\Omega))} \leq C\kappa^3\,.
\end{equation}}
\vspace{1ex}
We rewrite (\ref{eq:67}a, c, d) in the form
\begin{displaymath}
\begin{cases}
\frac{\partial\psi_\kappa}{\partial t}-\Delta\psi_\kappa = -2i\kappa A_\kappa\cdot\nabla\psi_\kappa - |\kappa A_\kappa|^2\psi_\kappa+ \kappa^2\psi_\kappa\big( 1 - |\psi_\kappa|^{2}
\big)\\
\qquad\qq\qquad \qquad - i\kappa\phi_\kappa\psi_\kappa&\text{ in } (0,+\infty)\times\Omega,\\
\psi_\kappa =0 &\text{ on } (0,+\infty)\times \partial\Omega_c \,, \\
\frac{\partial\psi_\kappa}{\partial\nu}=i\kappa A_\kappa\psi_\kappa\cdot\nu=0 &\text{ on }(0,+\infty)\times \partial\Omega_i \,.\\
\end{cases}
\end{displaymath}
Clearly, by our choice of $\widehat \zeta$,
\begin{displaymath}
\begin{cases}
\frac{\partial u}{\partial t}-\Delta u
= {\widehat \zeta}\big[-2i\kappa A_\kappa\cdot\nabla\psi_\kappa - |\kappa
A_\kappa|^2\psi_\kappa + \kappa^2\psi_\kappa\big( 1 - |\psi_\kappa|^{2}
\big)& \\
\qquad\qquad\quad - i\kappa\phi_\kappa\psi_\kappa\big] + 2\nabla{\widehat \zeta}\cdot\nabla\psi_\kappa
+ \psi_\kappa \Delta{\widehat \zeta} & \text{ in } (0,+\infty)\times \Omega\, ,\\
u =0 & \text{on } (0,+\infty)\times \partial\Omega \,.
\end{cases}
\end{displaymath}
By \cite[Theorem C.1]{alhe14} (this time applied to the Dirichlet
Laplacian in $\Omega$) in the interval $(t_0-1,t_0+1)$
\begin{equation}\label{eq:81}
\aligned
&\|u\|_{L^2(t_0,t_0+1,H^2(\Omega))} \\
\leq &\big\|{\widehat \zeta}\big[-2i\kappa A_\kappa\cdot\nabla\psi_\kappa - |\kappa
A_\kappa|^2\psi_\kappa + \kappa^2\psi_\kappa\big( 1 - |\psi_\kappa|^{2}
\big)- i\kappa\phi_\kappa\psi_\kappa\big]\big\| _{L^2(t_0-1,t_0+1,L^2(\Omega,{\mathbb{R}}^2))} \\
&+ \|2\nabla{\widehat \zeta}\cdot\nabla\psi_\kappa
+ \psi_\kappa \Delta{\widehat \zeta}\|_{L^2(t_0-1,t_0+1,L^2(\Omega,{\mathbb{R}}^2))} + C\|u(t_0-1,\cdot)\|_{2} \,.
\endaligned
\end{equation}
By \eqref{eq:69} we have that
\begin{equation}
\label{eq:82}
\|2\nabla{\widehat \zeta}\cdot\nabla\psi_\kappa + \psi_\kappa \Delta{\widehat \zeta}\|_{L^2(t_0-1,t_0+1,L^2(\Omega))}
\leq C(1+\|\nabla\psi_\kappa\|_{L^2(t_0-1,t_0+1,L^2(\Omega))}) \,.
\end{equation}
As
\begin{displaymath}\
\|\nabla\psi_\kappa\|_{L^2(t_0-1,t_0+1,L^2(\Omega))} \leq
\|\nabla_{\kappa A_\kappa}\psi_\kappa\|_{L^2(t_0-1,t_0+1,L^2(\Omega))}+\|\kappa A_\kappa\psi_\kappa\|_{L^2(t_0-1,t_0+1,L^2(\Omega))}\,,
\end{displaymath}
we obtain in view of (\ref{eq:70a}) and (\ref{eq:72}) that
\begin{equation}
\label{eq:83}
\|\nabla\psi_\kappa\|_{L^2(t_0-1,t_0+1,L^2(\Omega))} \leq C\kappa^2 \,.
\end{equation}
Substituting the above into (\ref{eq:82}) yields
\begin{equation}
\label{eq:84}
\|2\nabla{\widehat \zeta}\cdot\nabla\psi_\kappa + \psi_\kappa \Delta{\widehat \zeta}\|_{L^2(t_0-1,t_0+1,L^2(\Omega))} \leq C\kappa^2\,.
\end{equation}
We next observe that
\begin{displaymath}
\|{\widehat \zeta} |\kappa A_\kappa|^2\psi_\kappa \|_{L^2(t_0-1,t_0+1,L^2(\Omega))} \leq
\kappa^2\|A_\kappa\|_{L^4(t_0-1,t_0+1,L^\infty(\Omega,{\mathbb{R}}^2))}^2 \|{\widehat \zeta}\psi_\kappa \|_{L^2(t_0-1,t_0+1,L^2(\Omega))}\,.
\end{displaymath}
Since $\widehat \zeta$ is supported in the set $\omega_{\delta,j}$, we may use
\eqref{eq:76}, which together with Agmon's inequality \cite[Lemma
13.2]{ag65}, (\ref{time-dep-AB}), and \eqref{eq:72}, yield, for $t_0$ large enough,
\begin{equation}\label{eq:85} \aligned
& \|{\widehat \zeta} |\kappa A_\kappa|^2\psi_\kappa \|_{L^2(t_0-1,t_0+1,L^2(\Omega))}\\
\leq& C\kappa\|A_\kappa\|_{L^\infty(t_0-1,t_0+1,L^2(\Omega,{\mathbb{R}}^2))}
\|A_\kappa\|_{L^2(t_0-1,t_0+1,H^2(\Omega,{\mathbb{R}}^2))}\\
\leq& C\kappa^3\,.
\endaligned
\end{equation}
Similarly,
\begin{equation}\label{eq:86}
\aligned
& \|{\widehat \zeta}\kappa A_\kappa\cdot\nabla\psi_\kappa\|_{L^2(t_0-1,t_0+1,L^2(\Omega))}\\
\leq& \kappa\|A_\kappa\|_{L^4(t_0,t_0+1,L^\infty(\Omega,{\mathbb{R}}^2))}
\big[\|u\|_{L^4(t_0-1,t_0+1,H^1(\Omega))}
+\|\psi_\kappa\nabla{\widehat \zeta}\|_{L^4(t_0-1,t_0+1,L^2(\Omega))}\big]\\
\leq& C\kappa^2 +
C\kappa^2\|u\|^{1/2}_{L^2(t_0-1,t_0+1,H^2(\Omega))}\|u\|^\frac 12 _{L^\infty(t_0-1,t_0+1,L^2(\Omega))} \\
\leq & C\Big[\kappa^2 + \kappa^{3/2}\|u\|^\frac 12_{L^2(t_0-1,t_0+1,H^2(\Omega))}\Big]\,.
\endaligned
\end{equation}
Substituting \eqref{eq:86} together with \eqref{eq:84},
and
\eqref{eq:85} into \eqref{eq:81} yields with the aid of \eqref{eq:69}
\begin{equation}\label{eq:80a}
\|u\|_{L^2(t_0,t_0+1,H^2(\Omega))} \leq C\Big[\kappa^3 + \kappa^{3/2}\|u\|^{\frac 12}_{L^2(t_0-1,t_0+1,H^2(\Omega))}\Big]\,
\end{equation}
Proceeding as in the proof of Proposition \ref{lem:asymp-exp}, we can
assume $C\geq 1$ in \eqref{eq:80a} and set
$$
\alpha_n = C^{-1} \kappa^{-\frac 32} \|u\|_{L^2(n,n+1,H^2(\Omega))} ^\frac 12\,.
$$
We now can rewrite \eqref{eq:80a} in the form
$$
\alpha_n^2 \leq ( 1 +\alpha_{n-1} + \alpha_n)\,,
$$
which is precisely \eqref{inegaga}.
We can thus conclude (\ref{eq:80}), and hence, for a new value of $C$,
\eqref{eq:87} easily follows.
\vspace{1ex}
{\it Step 2: Prove that
\begin{equation}
\label{eq:89}
\|\nabla_{\kappa A_\kappa}u\|_{L^2(t_0,t_0+1,H^1(\Omega,{\mathbb{R}}^2))} \leq C \kappa^3 \,.
\end{equation}}
\vspace{2ex}
It can be easily verified that
\begin{equation}
\label{eq:88}
\|\nabla_{\kappa A_\kappa}u(t,\cdot)\|_{1,2}\leq \|u(t,\cdot)\|_{2,2} + \kappa\|\,|A_\kappa|\nabla u(t,\cdot)\|_2 +
\kappa\|u\nabla A_{\kappa}(t,\cdot)\|_2 \,.
\end{equation}
Furthermore, in the same manner we have obtained \eqref{eq:86} we
obtain, with the aid of \eqref{eq:87}
\begin{displaymath}
\kappa \|\,|A_\kappa|\nabla u\|_{L^2(t_0,t_0+1,L^2(\Omega,{\mathbb{R}}^2))} \leq C
\kappa^{3/2}\|u\|_{L^2(t_0,t_0+1,H^2(\Omega,{\mathbb{R}}^2))}^{1/2}\leq C\kappa^3 \,.
\end{displaymath}
By \eqref{eq:69} and \eqref{eq:72} we have that
\begin{displaymath}
\kappa\|u\nabla A_{\kappa}\|_{L^2(t_0,t_0+1,L^2(\Omega,{\mathbb{R}}^2))} \leq C\kappa^2 \,.
\end{displaymath}
We can now conclude \eqref{eq:89} from \eqref{eq:88}.
\vspace{1ex}
{\it Step 3: Prove \eqref{eq:89a}.}
\vspace{1ex}
In \cite[(5.35)]{alhe14} it was shown that
\begin{equation}
\label{eq:90}
\|\nabla_{\kappa A_\kappa}\psi_\kappa\|_{L^\infty(t_0,t_0+1; L^2(\Omega))}\leq C\kappa^3\,.
\end{equation}
(Note that while the setting in \cite{alhe14} is different then - in
particular, we assume there $J\sim\mathcal O(\kappa)$ - the estimate is
still valid in the present case because $c=1$.)
Hence, we get
\begin{equation}
\label{eq:91}
\|\nabla_{\kappa A_\kappa}u\|_{L^\infty(t_0,t_0+1; L^2(\Omega))}\leq C\kappa^3\,.
\end{equation}
We now use Gagliardo-Nirenberg interpolation inequality (see \cite{ni59}) to
obtain
$$
\aligned
& \|\nabla_{\kappa A_\kappa}u\|_{L^p(t_0,t_0+1; L^p(\Omega))}^p\qquad \\
& \leq C \int_{t_0}^{t_0+1}
\|\nabla_{\kappa A_\kappa}u(t,\cdot)\|_{1,2}^{p-2}\|\nabla_{\kappa A_\kappa}u(t,\cdot)\|_2^2 \,dt\\
& \leq C \|\nabla_{\kappa A_\kappa}u\|_{L^2(t_0,t_0+1; H^1(\Omega))}^{p-2} \|\nabla_{\kappa
A_\kappa}u\|_{L^{\frac{4}{4-p}}(t_0,t_0+1; L^2(\Omega))}^2 \,.
\endaligned
$$
Consequently,
\begin{equation}\label{eq:92}
\aligned
&\|\nabla_{\kappa A_\kappa}u\|_{L^p(t_0,t_0+1; L^p(\Omega))}^p \\
\leq &C \|\nabla_{\kappa A_\kappa}u\|_{L^2(t_0,t_0+1; H^1(\Omega))}^{p-2} \|\nabla_{\kappa
A_\kappa}u\|_{L^2(t_0,t_0+1; L^2(\Omega))}^{4-p} \|\nabla_{\kappa
A_\kappa}u\|_{L^\infty(t_0,t_0+1; L^2(\Omega))}^{p-2} \,.
\endaligned
\end{equation}
Multiplying (\ref{eq:67}a) by ${\widehat \zeta}^2\bar{\psi}_\kappa$ and integrating over
$\Omega$ we obtain for the real part that
\begin{displaymath}
\|\nabla_{\kappa A_\kappa(t,\cdot)}u(t,\cdot)\|_2^2 \leq \kappa^2\|u (t,\cdot)\|_2^2-\frac{1}{2}
\frac{d\|u (t,\cdot)\|_2^2}{dt}+ \|\psi_\kappa (t,\cdot) \nabla{\widehat \zeta}\|_2^2 \,,
\end{displaymath}
Integrating over $(t_0,t_0+1)$ and using \eqref{eq:76} we then obtain, for sufficiently large $t_0$,
\begin{displaymath}
\|\nabla_{\kappa A_\kappa}u\|_{L^2(t_0,t_0+1; L^2(\Omega))}^2 \leq C \,.
\end{displaymath}
Substituting the above, \eqref{eq:89}, and \eqref{eq:91}, into
\eqref{eq:92} we then obtain \eqref{eq:89a}.
\end{proof}
We can now obtain the following improved regularity for $B_{1,\kappa}$
\begin{proposition}
Let $2<p\leq12/5$. For $0<\delta<\delta_0$, there exists a constant $C=C(\Omega,\delta)>0$ such that for all $t_0>1$
and $\kappa>\kappa_0(\delta)$ we have
\begin{equation}
\label{eq:93}
\|B_{1,\kappa}\|_{L^p(t_0,t_0+1,W^{1,p}(\omega_{\delta,j}))} \leq C \,.
\end{equation}
\end{proposition}
\begin{proof}
Taking the curl of \eqref{eq:71} yields that $B_{1,\kappa}$ is a weak
solution of
\begin{equation}
\label{eq:94}
\begin{cases}
\frac{\partial B_{1,\kappa}}{\partial t} - \Delta B_{1,\kappa} = \frac{1}{\kappa}\text{\rm curl\,}
\Im(\bar{\psi}_\kappa\nabla_{\kappa A_\kappa}\psi_\kappa) & \text{ in } (0,+\infty)\times \Omega \\
B_{1,\kappa} =0 & \text{ on } (0,+\infty)\times \partial\Omega \,.
\end{cases}
\end{equation}
Let
\begin{displaymath}
\mathcal B_{\widehat \zeta} = {\widehat \zeta}B_{1,\kappa} \,,
\end{displaymath}
where the cutoff function $\widehat{\zeta}$ is defined by (\ref{eq:77}).
Let further $\hat{\Omega}(\delta)\subset\Omega$ be smooth and satisfy
$$
{\rm supp}\,\widehat{\zeta}\subset \hat{\Omega}(\delta)\,.$$
As for any $V\in H^1(\Omega,{\mathbb{R}}^2)$ we have that
\begin{displaymath}
\text{\rm curl\,} V = \text{\rm div\,} V_\perp\,,
\end{displaymath}
it can be easily verified
from (\ref{eq:94}) that
\begin{displaymath}
\frac{\partial \mathcal B_{\widehat \zeta}}{\partial t} - \, \Delta \mathcal B_{\widehat \zeta} =
\frac{1}{\kappa}{\widehat \zeta}\text{\rm div\,}\big( \Im(\bar{\psi}_\kappa\nabla_{\kappa
A_\kappa}\psi_\kappa)_\perp\big) -2\text{\rm div\,}(B_{1,\kappa}\nabla{\widehat \zeta)}) +
B_{1,\kappa}\Delta\widehat \zeta\,.
\end{displaymath}
Consequently,
\begin{equation}
\label{eq:95}
\left\{
\begin{array}{rll}
\frac{\partial \mathcal B_{\widehat \zeta}}{\partial t} - \, \Delta \mathcal B_{\widehat \zeta} &=
\frac{1}{\kappa}{\widehat \zeta}\, \text{\rm div\,}\,\big( \Im(\bar{\psi}_\kappa\nabla_{\kappa
A_\kappa}\psi_\kappa)_\perp\big)& \\
&\quad -\text{\rm div\,}(2 B_{1,\kappa}\nabla{\widehat \zeta} +
\nabla \Delta_D^{-1}(B_{1,\kappa}\Delta{\widehat \zeta}) ) & \text{in } (t_0-1,t_0+1)\times\hat{\Omega} \\
\mathcal B_{\widehat \zeta} &=0 & \text{on } (t_0-1,t_0+1)\times\partial\hat{\Omega} \\
\mathcal B_{\widehat \zeta} (t_0-1,\cdot) & = \widehat \zeta\, \mathcal B_{1,\kappa} (t_0-1,\cdot) &\text{in } \hat{\Omega} \,.
\end{array}
\right.
\end{equation}
In the above $\Delta_D^{-1}$ denotes the inverse Dirichlet Laplacian in $
\hat{\Omega}$.
In order to apply
\cite[Theorem 1.6]{by07} which is devoted to the case of parabolic
operators written in divergence form and with zero initial condition
we first decompose the solution of \eqref{eq:95} into two
Cauchy-Dirichlet problems.
The first of them is:
\begin{equation} \label{eq:117a}
\left\{
\begin{array}{rll}
\frac{\partial U_1}{\partial t} - \, \Delta U_1 &= \text{\rm div\,} f_1
& \text{ in } (t_0-1,t_0+1)\times\hat{\Omega}\,, \\
U_1 &=0 & \text{ on } (t_0-1,t_0+1)\times\partial\hat{\Omega} \,,\\
U_1 (t_0-1,\cdot)& = 0 &\text{ in } \hat{\Omega} \,,
\end{array}
\right.
\end{equation}
in which
\begin{equation}\label{eq:117aa}
f_1 = \frac{1}{\kappa}{\widehat \zeta}\, \,\big( \Im(\bar{\psi}_\kappa\nabla_{\kappa
A_\kappa}\psi_\kappa)_\perp\big) - 2 B_{1,\kappa}\nabla{\widehat \zeta} -
\nabla \Delta_D^{-1}(B_{1,\kappa}\Delta{\widehat \zeta}) \,.
\end{equation}
The second one is:
\begin{equation}
\label{eq:117b}
\left\{
\begin{array}{rll}
\frac{\partial U_2}{\partial t} - \, \Delta U_2 &= F_2
& \text{in } (t_0-1,t_0+1)\times\hat{\Omega} \,,\\
U_2 &=0 & \text{on } (t_0-1,t_0+1)\times\partial\hat{\Omega} \,,\\
U_2(t_0-1,\cdot )& = \widehat \zeta\, \mathcal B_{1,\kappa} (t_0-1,\cdot) &\text{in } \hat{\Omega} \,,
\end{array}
\right.
\end{equation}
where
\begin{equation} \label{eq:117ba}
F_2:= - \frac{1}{\kappa} \,\big( \Im(\bar{\psi}_\kappa \nabla {\widehat \zeta}\cdot (\nabla_{\kappa
A_\kappa}\psi_\kappa)_\perp) \big)\,.
\end{equation}
By uniqueness of the weak solution, we have
\begin{equation} \label{eq:117c}
\mathcal B_{\widehat \zeta} = U_1 + U_2 \text{ in } (t_0-1,t_0+1)\times\hat{\Omega}\,.
\end{equation}
We now separately estimate $U_1$ and $U_2$ in $L^p (t_0,t_0+1, W^{1,p}( \hat \Omega))\,$.
\vskip0.05in
{\it Estimate of $U_1$}
\vspace{1ex}
We apply \cite[Theorem 1.6]{by07} to obtain
\begin{equation} \label{eq:96}
\| U_1\|_{L^p(t_0,t_0+1,W^{1,p}(\Omega))}\\
\leq C \,\|f_1\|_{L^p(t_0,t_0+1,L^p(\Omega,{\mathbb{R}}^2)}\,.
\end{equation}
It can be easily verified, by the Gagliardo-Nirenberg interpolation
inequality \cite{ni59} that, for
all $2<p<4$, there exists a constant $C$ such that,
for any $\phi \in L^2(t_0,t_0+1,H^1(\Omega))\cap L^\infty (t_0,t_0+1,L^2(\Omega))$, we have \begin{multline}\label{eq:76a}
\|\phi \|_{L^p(t_0,t_0+1,L^p(\Omega,{\mathbb{R}}^2))}^p\leq C \int_{t_0}^{t_0+1}\|\phi (\tau,\cdot)
\|_2^2 \|\phi (\tau,\cdot) \|_{1,2}^{p-2}\,d\tau \\ \leq C \| \phi \|_{L^{\frac{4}{4-p}}(t_0,t_0+1,L^2(\Omega,{\mathbb{R}}^2))}^2\|\phi\|_{L^2(t_0,t_0+1,H^1(\Omega,{\mathbb{R}}^2))}^{p-2}\,.
\end{multline}
By \eqref{eq:76a}, \eqref{eq:72}, Remark~\ref{usefulrem}, and Sobolev embeddings we
have:
\begin{equation}\label{eq:97}
\| \mathcal B_{\widehat \zeta} (t_0,\cdot)\|_p +
\|\mathcal B_{1,\kappa} \nabla{\widehat \zeta}\|_{L^p(t_0,t_0+1,L^p(\Omega))} +
\|\mathcal B_{1,\kappa}
\Delta{\widehat \zeta}\|_{W^{-1,p}(t_0,t_0+1,L^p(\Omega,{\mathbb{R}}^2))}
\leq C\,.
\end{equation}
Furthermore, by \eqref{eq:69} we have that
\begin{displaymath}
\begin{array}{l}
\|{\widehat \zeta}\Im(\bar{\psi}_\kappa\nabla_{\kappa A_\kappa}\psi_\kappa)\|_{L^p(t_0,t_0+1,L^p(\Omega,{\mathbb{R}}^2))}\\
\qquad \leq
\|\psi_\kappa\nabla{\widehat \zeta}\|_{L^p(t_0,t_0+1,L^p(\Omega,{\mathbb{R}}^2))}+ \|\nabla_{\kappa
A_\kappa}({\widehat \zeta}\psi_\kappa)\|_{L^p(t_0,t_0+1,L^p(\Omega,{\mathbb{R}}^2))} \\
\qquad \leq \|\nabla_{\kappa A_\kappa}({\widehat \zeta}\psi_\kappa)\|_{L^p(t_0,t_0+1,L^p(\Omega,{\mathbb{R}}^2))} +
C\,.
\end{array}
\end{displaymath}
Substituting the above together with \eqref{eq:97}
into \eqref{eq:96} yields for $t_0$ large enough
\begin{equation}\label{eq:98}
\| U_1\|_{L^p(t_0,t_0+1,W^{1,p}(\Omega))} \leq C\Big(1 +\frac{1}{\kappa}
\|\nabla_{\kappa A_\kappa}({\widehat \zeta}\psi_\kappa)\|_{L^p(t_0,t_0+1,L^p(\Omega,{\mathbb{R}}^2))}\Big) \,.
\end{equation}
Substituting \eqref{eq:89a}
into \eqref{eq:98} yields
\begin{equation}\label{eq:79a}
\| U_1\|_{L^p(t_0,t_0+1,W^{1,p}(\Omega))}
\leq C(1+\kappa^{5-12/p})
\leq \widehat C\,,
\end{equation}
since $2< p\leq 12/5$.
\vspace*{2ex}
{\it Estimate of $U_2$}
\vspace{1ex}
Here we apply first $L^2$ estimates and then combine them with Sobolev's
estimates. It is in this part that we need the information on $F_2$ and $U_2$
in $[t_0-1,t_0+1)\times \hat \Omega$ in order to bound the various norms on
$(t_0,t_0+1)\times \hat \Omega$. We begin by applying once again \cite[Theorem
C.1]{alhe14} (combining $(C.1)$ and $(C.2)$ there) to obtain
$$
\begin{array}{l}
\| U_2\|_{ L^2(t_0,t_0+1, H^2)} + \| U_2 \|_{L^\infty (t_0,t_0+1,H^1)}\\ \qquad
\leq C \left( \| F_2\|_{L^2 ((t_0-1,t_0 +1)\times \hat \Omega)} + \| U_2(t_0-1,\cdot)\|_{L^2(\hat \Omega)} \right)\,,
\end{array}
$$
where $F_2$ and $U_2(t_0-1,\cdot)$ given in \eqref{eq:117ba} and
\eqref{eq:117b}. Applying Gagliardo-Nirenberg's inequality yields,
for $ 2 <p <4$,
\begin{equation}\label{eq:89b}
\| U_2\|_{ L^p(t_0,t_0+1,W^{1,p})} \leq C \left( \| F_2\|_{L^2
((t_0-1,t_0 +1)\times \hat \Omega)} + \| U_2(t_0-1,\cdot)\|_{L^2(\hat \Omega)}
\right)\,,
\end{equation}
By \eqref{eq:70aa} we have that
\begin{displaymath}
\| F_2\|_{L^2((t_0-1,t_0 +1)\times \hat \Omega)} \leq C \,.
\end{displaymath}
Furthermore, using \eqref{eq:72}, with Remark \ref{usefulrem} in mind
yields
\begin{displaymath}
\| U_2(t_0-1,\cdot)\|_2 \leq C \,.
\end{displaymath}
Consequently, by \eqref{eq:89b}, there exist, for any $2 <p <4$ and
any $\delta >0\,,$ constants $C(\delta)$ and $\kappa(\delta)$ such that for any $\kappa
\geq \kappa_0(\delta)$ and any $t_0 >1$ we have
\begin{equation}
\label{eq:89c}
\| U_2\|_{ L^p(t_0,t_0+1,W^{1,p})} \leq C (\delta)\,.
\end{equation}
The combination of \eqref{eq:79a} and \eqref{eq:89c} together with
\eqref{eq:117c} completes the proof of the proposition.
\end{proof}
We can now establish the exponential decay of $\psi_\kappa$.
\begin{proposition}
\label{asymp-exp-2}
Let $\omega_{\delta,j}$ ($j\in\{1,2\}$) be given by \eqref{eq:56}. Suppose
that for some $k\in\{1,2\}$ \eqref{eq:75} is satisfied. Then, there
exist $C>0$ and $\delta_0>0$, and, for any $0<\delta<\delta_0$,
$\kappa_0(\delta)$, such that for any $\kappa\geq \kappa_0(\delta)$ we have
\begin{equation}
\label{eq:99}
\limsup_{t\to\infty}\int_{\omega_{\delta,j}} \exp \Big(\delta^{1/2}
\kappa d(x,
\Gamma_{\delta,j})\Big) |\psi_\kappa|^2(t,x) \,dx \leq C \,.
\end{equation}
\end{proposition}
\begin{proof}
Without loss of generality we may, as before, assume $h_j>0$.
Let $\check{\chi}$ and $\check{\zeta}$ be defined by \eqref{check1} and
\eqref{check2}.
\vspace{1ex}
{\it Step 1: Prove that
\begin{equation}
\label{eq:101}
\|\check{\zeta}\psi_\kappa(t,\cdot)\|_2^2 \leq \|\check{\zeta}\psi_0\|_2^2e^{-2\gamma\kappa^2t} +
\frac{C(\delta)}{\kappa^4}+ \int_0^t
e^{-2\gamma\kappa^2(t-\tau)}\big|\langle\kappa B_{1,\kappa}\check{\zeta}\psi_\kappa,\check{\zeta}\psi_\kappa\rangle(\tau)\big|\,d\tau \,.
\end{equation}}
\vspace{1ex}
Multiplying
(\ref{eq:67}a) by $\check{\zeta}^2 \bar{\psi}$ and integrating by parts yields
\begin{displaymath}
\frac{1}{2}\frac{d}{dt}\left( \|\check{\zeta}\psi_\kappa(t,\cdot) \|_2^2 \right) + \|\nabla_{\kappa A_\kappa(t,\cdot)}(\check{\zeta}\psi_\kappa(t,\cdot))\|_2^2 \leq
\kappa^2\|\check{\zeta}\psi_\kappa(t,\cdot)\|_2^2 + \|\psi_\kappa (t,\cdot) \nabla\check{\zeta}\|_2^2 \,.
\end{displaymath}
By Theorem 2.9 in \cite{AHS} we have
$$\aligned
\|\nabla_{\kappa A_\kappa}(\check{\zeta}\psi_\kappa)\|_2^2
& \geq \langle\kappa B_\kappa(t,\cdot) \check{\zeta}\psi_\kappa(t,\cdot),\check{\zeta}\psi_\kappa (t,\cdot)\rangle\\ & =
\kappa^2\langle B_n\check{\zeta}\psi_\kappa,\check{\zeta}\psi_\kappa\rangle + \langle\kappa B_{1,\kappa}\check{\zeta}\psi_\kappa,\check{\zeta}\psi_\kappa\rangle \\
&\geq
\kappa^2\Big(1+\frac{\delta}{2}\Big)\|\check{\zeta}\psi_\kappa\|_2^2+ \langle\kappa B_{1,\kappa}\check{\zeta}\psi_\kappa,\check{\zeta}\psi_\kappa\rangle \,.
\endaligned
$$
We can thus write
\begin{equation}\label{eq3.32}
\aligned
&\frac{1}{2}\frac{d}{dt}\left( \|\check{\zeta}\psi_\kappa(t,\cdot)\|_2^2\right) +
\kappa^2\Big( \frac{\delta}{2}- \frac{\delta}{4}
\Big)\|\check{\zeta}\psi_\kappa(t,\cdot)\|_2^2\\
\leq&
\|\psi_\kappa (t,\cdot) \nabla\eta\|_2^2 + \|\tilde \chi\,\psi_\kappa (t,\cdot)\nabla\eta_\delta\|_2^2- \langle\kappa B_{1,\kappa}(t,\cdot)\check{\zeta}\psi_\kappa(t,\cdot),\check{\zeta}\psi_\kappa(t,\cdot)\rangle \,.
\endaligned
\end{equation}
By \eqref{eq:76}, for every $0<\delta\leq\delta_0$, we have
\begin{displaymath}
\|\psi_\kappa (t,\cdot) \nabla\eta\|_2^2 + \|\tilde \chi\, \psi_\kappa (t,\cdot) \,\nabla\eta_\delta\|_2^2 \leq \frac{C}{\kappa^2} \,,
\end{displaymath}
which when substituted into \eqref{eq3.32} yields
\begin{equation}
\label{eq:100}
\frac{1}{2}\frac{d}{dt}\left( \|\check{\zeta}\psi_\kappa (t,\cdot)\|_2^2\right) + \kappa^2\gamma\|\check{\zeta}\psi_\kappa (t,\cdot) \|_2^2
\leq \frac{C}{\kappa^2} - \langle\kappa B_{1,\kappa}(t,\cdot) \check{\zeta}\psi_\kappa(t,\cdot) ,\check{\zeta}\psi_\kappa (t,\cdot)\rangle \,,
\end{equation}
where $\gamma=\delta/4$. We now get \eqref{eq:101} from \eqref{eq:100}.
\vspace{1ex}
{\it Step 2: Prove that, for all $n\geq2$,
\begin{equation}
\label{eq3.33}
\aligned
&\|\check{\zeta}\psi_\kappa(\tau, \cdot )\|_{L^\infty(t^*+n-1,t^*+n,L^2(\Omega))}^2 \\
\leq& C \kappa^{-(p-2)/p}\|\check{\zeta}\psi_\kappa(\tau, \cdot )\|_{L^\infty(t^*+n-2,t^*+n,L^2(\Omega))}^2 +{C\over\kappa^4} \,,
\endaligned
\end{equation}
where $C$ is independent of $\kappa\,$.}
\vspace{1ex}
To prove \eqref{eq3.33} we need to estimate the last term on the
right-hand-side of \eqref{eq:101}. To this end we first write
\begin{multline*}
\int_0^t e^{-2\gamma\kappa^2(t-\tau)}\big|\langle\kappa
B_{1,\kappa}\check{\zeta}\psi_\kappa,\check{\zeta}\psi_\kappa\rangle (\tau) \big|\,d\tau \\
\leq \kappa \left(\int_{t-1}^t e^{-2\gamma\kappa^2(t-\tau)}
{ \|B_{1,\kappa}(\tau, \cdot
)\|_{L^\infty(\omega_{\delta/2,j})}}\,d\tau\right)\,\cdot\|\check{\zeta}\psi_\kappa\|_{L^\infty(t-1,t;L^2(\Omega))}\\
+ \kappa\int_0^{t-1}
e^{-2\gamma\kappa^2(t-\tau)}\|B_{1,\kappa}(\tau,\cdot)\|_1\,d\tau
\|\check{\zeta}\psi_\kappa\|_{L^\infty(0,t-1;L^2(\Omega))}\,.
\end{multline*}
For the last term on the right-hand-side we have in view of Remark~\ref{usefulrem}, \eqref{check1}, \eqref{check2}, and \eqref{eq:69} that
\begin{displaymath}
\kappa\int_0^{t-1}
e^{-2\gamma\kappa^2(t-\tau)}\|B_{1,\kappa}(\tau,\cdot)\|_1\,d\tau
\|\check{\zeta}\psi_\kappa\|_{L^\infty(0,t-1;L^2(\Omega))}\leq Ce^{-2\gamma\kappa^2}e^{C\kappa}\,.
\end{displaymath}
Hence for sufficiently large $\kappa$ we have
\begin{multline}
\label{eq:102}
\int_0^t e^{-2\gamma\kappa^2(t-\tau)}\big|\langle\kappa
B_{1,\kappa}\check{\zeta}\psi_\kappa,\check{\zeta}\psi_\kappa\rangle (\tau) \big|\,d\tau \leq \\
\kappa \left(\int_{t-1}^t e^{-2\gamma\kappa^2(t-\tau)} \|B_{1,\kappa}(\tau, \cdot
)\|_{L^\infty(\omega_{\delta/2,j})}\,d\tau\right)\,\cdot\|\check{\zeta}\psi_\kappa\|_{L^\infty(t-1,t;L^2(\Omega))}+ Ce^{-\gamma\kappa^2}\,.
\end{multline}
Since by Sobolev embeddings
\begin{displaymath}
\|B_{1,\kappa}\|_{L^p(t-1,t,L^\infty(\omega_{\delta/2,j}))} \leq C\|B_{1,\kappa}\|_{L^p(t-1,t,W^{1,p}(\omega_{\delta/2,j}))} \,,
\end{displaymath}
we can use \eqref{eq:93} to obtain, for
sufficiently large $t$ and $\kappa$, and for any $2<p\leq12/5\,$,
$$ \aligned
& \int_{t-1}^t e^{-2\gamma\kappa^2(t-\tau)}\|B_{1,\kappa}(\tau, \cdot)\|_{L^\infty(\omega_{\delta/2,j})}\,d\tau \\
\leq&
\|B_{1,\kappa}(\tau, \cdot)\|_{L^p(t-1,t;L^\infty(\omega_{\delta/2,j}))}\,\Big[\int_{t-1}^t
e^{-\frac{2p}{p-1}\gamma\kappa^2(t-\tau)}\,d\tau\Big]^{\frac{p-1}{p}} \\
\leq&
C\kappa^{-2(p-1)/p}\,.
\endaligned
$$
Substituting the above into \eqref{eq:102} yields
$$\aligned
&\int_0^t e^{-2\gamma\kappa^2(t-\tau)}\big|\langle\kappa B_{1,\kappa}\check{\zeta}\psi_\kappa,\check{\zeta}\psi_\kappa\rangle\big|\,d\tau \\
\leq& C\kappa^{-(p-2)/p}\|\check{\zeta}\psi_\kappa\|_{L^\infty(t-1,t;L^2(\Omega))} + Ce^{- \frac 14 \delta \kappa^2 } \,,
\endaligned
$$
which, when substituted into \eqref{eq:101}, yields \eqref{eq3.33}.
\\
{\it Step 3: Prove \eqref{eq:99}.}
\\
Let now
\begin{displaymath}
b_n = \|\check{\zeta}\psi_\kappa(\tau, \cdot )\|_{L^\infty(t^*+n-1,t^*+n,L^2(\Omega))}^2 \,.
\end{displaymath}
From \eqref{eq3.33} we get that if $p=12/5$ then for sufficiently large
$\kappa$ it holds that
\begin{displaymath}
b_n \leq C \left( \kappa^{-1/6}(b_{n-1} + b_n)+ 1\right)\,,
\end{displaymath}
where $C$ is independent of $\kappa$.
For another constant $\widehat C$, we get for sufficiently large $\kappa$,
\begin{displaymath}
b_n \leq \widehat C (\kappa^{-1/6} b_{n-1} + 1)\,.
\end{displaymath}
This immediately implies, for $\kappa$ large enough so that $\widehat C
\kappa^{-\frac 16} \leq \frac 12$, the upperbound
\begin{displaymath}
\limsup_{n\to\infty}b_n \leq C_0\,,
\end{displaymath}
where $C_0$ is independent of $\kappa$. Consequently,
$$
\limsup_{t\to\infty}\int_{\omega_{\delta,j}} \exp\Big(\delta^\frac 12 \kappa d(x,
\partial\omega_{\delta,j})\Big) |\psi_\kappa(t,x)|^2 \,dx \leq C_0\,,
$$
which readily yields \eqref{eq:99}.
\end{proof}
\newpage
\section{Large domains}
\label{sec:5}
The main goal of this section is to prove Proposition
\ref{largedomain}. To this end it is more convenient to consider \eqref{eq:18}
in a fixed domain. Assuming that $0\in\Omega$ we set $\epsilon=1/R$ (hence we have $\epsilon \ll 1$)
and apply the transformation,
\begin{equation}
\label{eq:103}
\psi_\epsilon(\epsilon x) = \psi(x) \, , \qquad A(x) = \epsilon^{-1}A_\epsilon(\epsilon x)\,, \qquad
\phi(x) = \phi_\epsilon(\epsilon x) \,,
\end{equation}
If we write $y=\epsilon x$, we have:
\begin{displaymath}
\text{\rm curl\,}_x^2A = \epsilon\,\text{\rm curl\,}_y^2A_\epsilon \quad ; \quad
\nabla_x\phi = \epsilon\,\nabla_y\phi_\epsilon \quad ;
\quad \nabla_A\psi = \epsilon\,\nabla_{\epsilon^{-2}A_\epsilon}\psi_\epsilon \,,
\end{displaymath}
leading to the following system for $(\psi_\epsilon, A_\epsilon, \phi_\epsilon)$
\begin{subequations}
\label{eq:104}
\begin{alignat}{2}
& \Delta_{\epsilon^{-2}A_\epsilon}\psi_\epsilon + \frac{\psi_\epsilon}{\epsilon^2} \big( 1 - |\psi_\epsilon|^{2}
\big)-\frac{i}{\epsilon^2}\phi_\epsilon\psi_\epsilon =0 & \quad \text{ in } \Omega\, ,\\
& \text{\rm curl\,}^2A_\epsilon + \nabla\phi_\epsilon = \Im\big(\bar\psi_\epsilon
\nabla_{\epsilon^{-2}A_\epsilon}\psi_\epsilon\big) & \quad \text{ in } \Omega\,,\\
&\psi_\epsilon=0 &\quad \text{ on } \partial\Omega_c \,, \\
& \nabla_{\epsilon^{-2}A_\epsilon}\psi_\epsilon\cdot\nu=0 & \quad \text{ on } \partial\Omega_i \,,\\
& \frac{\partial\phi_\epsilon}{\partial\nu}= f(\epsilon)J &\quad \text{ on } \partial\Omega \,,\\
& \frac{\partial\phi_\epsilon}{\partial\nu}= 0 &\quad \text{ on } \partial\Omega_i\,,\\[1.2ex]
&\Xint-_{\partial\Omega}\text{\rm curl\,} A_\epsilon(x)\,ds = f(\epsilon)h_{ex}\,.
\end{alignat}
\end{subequations}
In the above
$$
f(\epsilon)=F(1/\epsilon)= \epsilon^{-\alpha}\,.
$$
It follows from \eqref{eq:6} that
\begin{displaymath}
h_j = b_j\epsilon^{-\alpha}\,, \quad j=1,2\,,
\end{displaymath}
where $b_j$ is independent of $\epsilon$ for $j=1,2\,$.\\
We assume that $A_\epsilon$ is in the Coulomb gauge space \eqref{eq:5}, and
suppose that a weak solution $(\psi_\epsilon,A_\epsilon,\phi_\epsilon)\in H^1(\Omega,{\mathbb{C}})\times
H^1(\Omega,{\mathbb{R}}^2)\times L^2(\Omega)$ exists. Proposition \ref{largedomain} can now be
reformulated in the following way:
\begin{proposition}
\label{semiclassical}
Let $(\psi_\epsilon,A_\epsilon,\phi_\epsilon)$ denote a solution of \eqref{eq:104}, and let
$h$ be given by \eqref{eq:9}.
Suppose that for some $0<\gamma<1$ and $\epsilon_0>0$ we have
\begin{displaymath}
\epsilon^{-\gamma}<h\,, \quad \forall\, 0<\epsilon<\epsilon_0\,.
\end{displaymath}
Then, there exists a compact set $K\subset\Omega$, $C>0$, and $\alpha >0$,
such that for any $0\,<\,\epsilon\,<\epsilon_0\,$ we have
\begin{equation}
\label{eq:105}
\int_{K} |\psi_\epsilon(x)|^2 \,dx \leq C e^{-\alpha/\epsilon} \,.
\end{equation}
\end{proposition}
We split the proof of Proposition \ref{semiclassical} into several
steps, to each of them we dedicate a separate lemma. We begin by observing,
as in Section \ref{sec:3}, that
\begin{equation}
\label{eq:106}
\|\psi_\epsilon\|_\infty\leq 1\,.
\end{equation}
Let
\begin{equation}
\label{eq:107}
A_{1,\epsilon}=A_\epsilon-\epsilon^{-\alpha}A_n,\quad
\phi_{1,\epsilon}=\phi_\epsilon-\epsilon^{-\alpha}\phi_n,
\end{equation}
Set further
\begin{equation}
\label{eq:108}
B_{1,\epsilon}= \text{\rm curl\,} A_{1,\epsilon}\,; \quad B_\epsilon= \text{\rm curl\,} A_\epsilon \,.
\end{equation}
By (\ref{eq:104}b) and (\ref{eq:11}a), we then have
\begin{subequations}
\label{eq:109}
\begin{empheq}[left={\empheqlbrace}]{alignat=2}
&\nabla_\perp B_{1,\epsilon} + \nabla\phi_{1,\epsilon} = \Im(\bar\psi_\epsilon\, \nabla_{\epsilon^{-2}A_\epsilon}\psi_\epsilon) & \text{ in }
\Omega\,,\\
&\frac{\partial\phi_{1,\epsilon}}{\partial\nu}= 0 & \text{ on } \partial\Omega \,, \\
&\Xint-_{\partial\Omega}B_{1,\epsilon}(x)\,ds = 0 \,. &
\end{empheq}
\end{subequations}
Note that since $\partial B_{1,\epsilon}/\partial\tau=\partial\phi_{1,\epsilon}/\partial\nu=0$ on $\partial\Omega$
we must have by (\ref{eq:109}c) that
\begin{equation}
\label{eq:110}
B_{1,\epsilon}|_{\partial\Omega}\equiv 0\,.
\end{equation}
We begin with the following auxiliary estimate.
\begin{lemma}
\label{lem:magnetic-bound}
Let $w_\epsilon$ denote the solution of
\begin{equation}\label{eq:111}
\left\{\aligned
& \Delta w_\epsilon - \frac{1}{\epsilon^2}|\psi_\epsilon|^2w_\epsilon = 0\quad & \text{\rm in }\Omega\;, \\
& w_\epsilon = B_\epsilon - 1 \quad& \text{\rm on } \partial\Omega.
\endaligned
\right.
\end{equation}
Under the assumptions on $J$ and $\Omega$ in (\ref{eq:3})-(\ref{hyptopolo}) we have
\begin{equation}
\label{eq:112}
\|B_\epsilon-1-w_\epsilon\|_\infty \leq \frac{1}{2} \,.
\end{equation}
Furthermore, we have
\begin{equation}
\label{eq:113}
\|\nabla\phi_\epsilon\|_2+ \|\phi_\epsilon\|_\infty \leq C\epsilon^{-\alpha} \,.
\end{equation}
\end{lemma}
\begin{proof}
As can be easily verified from (\ref{eq:104}a) we have (see in
\cite{al02} (formula (2.4) in the case when $\phi_\epsilon=0$),
\begin{equation}
\label{eq:114}
\frac 12 \Delta|\psi_\epsilon|^2 = -\frac{|\psi_\epsilon|^2}{\epsilon^2}(1-|\psi_\epsilon|^2) +
|\nabla_{\epsilon^{-2}A_\epsilon}\psi_\epsilon|^2 \,.
\end{equation}
Furthermore, taking the curl of (\ref{eq:104}b) yields
\begin{displaymath}
\Delta B_\epsilon - \frac{1}{\epsilon^2}|\psi_\epsilon|^2B_\epsilon =
-\Im (\nabla_{\epsilon^{-2}A_\epsilon}\bar\psi_\epsilon\times\nabla_{\epsilon^{-2}A_\epsilon} \psi_\epsilon) \,.
\end{displaymath}
Note that
$$\aligned
&\text{\rm curl\,} \Im(\bar\psi_\epsilon\, \nabla_{\epsilon^{-2}A_\epsilon}\psi_\epsilon) \\
=&\Im(\nabla\bar\psi_\epsilon\times \nabla_{\epsilon^{-2}A_\epsilon}\psi_\epsilon
-i\epsilon^{-2}\bar\psi_\epsilon A_\epsilon\times\nabla\psi_\epsilon) - \frac{1}{\epsilon^2}|\psi_\epsilon|^2B_\epsilon\\
=&\Im (\nabla_{\epsilon^{-2}A_\epsilon}\bar \psi_\epsilon\times\nabla_{\epsilon^{-2}A_\epsilon} \psi_\epsilon) -
\frac{1}{\epsilon^2}|\psi_\epsilon|^2B_\epsilon \,.
\endaligned
$$
Let
$$u_\epsilon=B_\epsilon-1 + {|\psi_\epsilon|^2\over 2}-w_\epsilon\,.
$$
Combining the above and \eqref{eq:114}
yields that (cf. also \cite{al02})
\begin{displaymath}
\begin{cases}
\Delta u_\epsilon - \frac{1}{\epsilon^2}|\psi_\epsilon|^2u_\epsilon = |\nabla_{\epsilon^{-2}A_\epsilon}\psi_\epsilon|^2 -\Im (\nabla_{\epsilon^{-2}A_\epsilon}\bar \psi_\epsilon\times\nabla_{\epsilon^{-2}A_\epsilon} \psi_\epsilon)
+ \frac{1}{2\epsilon^2} |\psi_\epsilon|^4\geq 0 &
\text{in } \Omega, \\
u_\epsilon = \frac{|\psi_\epsilon|^2}{2} & \text{on } \partial\Omega.
\end{cases}
\end{displaymath}
By the weak maximum principle (cf. for instance \cite[Theorem
8.1]{GT}) and \eqref{eq:106} we obtain that for sufficiently small $\epsilon$
\begin{displaymath}
u_\epsilon (x)\leq \frac{1}{2}\,.
\end{displaymath}
The lower bound in \eqref{eq:112} follows easily by setting
$$\tilde{u}_\epsilon = -B_\epsilon-1 + {|\psi_\epsilon|^2\over 2}+w_\epsilon
$$
to obtain
\begin{displaymath}
\begin{cases}
\Delta \tilde{u}_\epsilon - \frac{1}{\epsilon^2}|\psi_\epsilon|^2\tilde{u}_\epsilon = |\nabla_{\epsilon^{-2}A_\epsilon}\psi_\epsilon|^2 +
\Im (\nabla_{\epsilon^{-2}A_\epsilon}\bar \psi_\epsilon\times\nabla_{\epsilon^{-2}A_\epsilon} \psi_\epsilon) + \frac{1}{2\epsilon^2} |\psi_\epsilon|^4 \geq 0 &
\text{in } \Omega, \\
\tilde{u}_\epsilon = \frac{|\psi_\epsilon|^2}{2} & \text{on } \partial\Omega,
\end{cases}
\end{displaymath}
upon which we use again the weak maximum principle.
To prove \eqref{eq:113} we first obtain for $\phi_\epsilon$, in the same manner
used to derive \eqref{eq:33}, the following problem:
\begin{equation}\label{eq:115}
\left\{
\aligned
-&\Delta\phi_\epsilon + \frac{\rho^2_\epsilon}{\epsilon^2}\phi_\epsilon =0 \quad& \text{\rm in } \Omega\,, \\
& \frac{\partial\phi_\epsilon}{\partial\nu}= \epsilon^{-\alpha}J \quad& \text{\rm on } \partial\Omega_c\,,\\
& \frac{\partial\phi_\epsilon}{\partial\nu}= 0 \quad& \text{\rm on } \partial\Omega_i\,.
\endaligned\right.
\end{equation}
Then, we follow the same steps as in the proof of \eqref{eq:34} to
obtain the bound $\|\phi_\epsilon\|_\infty\leq C \epsilon^{-\alpha}$. Multiplying \eqref{eq:115} by $\phi_\epsilon$ and
integrating by parts yields, using the preceding $L^\infty$ bound \eqref{eq:112}, we find that
\begin{displaymath}
\|\nabla\phi_\epsilon\|_2^2 \leq \int_{\partial\Omega} \phi_\epsilon \frac{\partial\phi_\epsilon}{\partial\nu}\,ds \leq C\epsilon^{-2\alpha}\,.
\end{displaymath}
\end{proof}
As a corollary we get:
\begin{corollary}
\begin{equation}
\label{eq:116}
\|B_\epsilon\|_\infty \leq \text{\rm max}(|b_1|,b_2)\epsilon^{-\alpha} + \frac{1}{2} \,.
\end{equation}
\end{corollary}
\begin{proof}
By \eqref{eq:112} and the maximum principle we have that
\begin{displaymath}
\|B_\epsilon-1\|_\infty\leq \|w_\epsilon\|_\infty +\frac{1}{2} \leq \text{\rm max}(|b_1|,b_2)\epsilon^{-\alpha} - \frac{1}{2}\,,
\end{displaymath}
which readily yields \eqref{eq:116}.
\end{proof}
We continue with the following auxiliary estimate:
\begin{lemma}\label{Lem3}
Let $\Omega$ and $J$ satisfy (\ref{eq:3})-(\ref{hyptopolo}), and $w_\epsilon$ be a solution of \eqref{eq:111}.
There exist positive constants $C$ and $\epsilon_0$, such that, for all $0<\epsilon<\epsilon_0$,
\begin{equation}
\label{eq:117}
\|\nabla w_\epsilon \|_\infty \leq
\frac{C}{\epsilon^{1+\alpha}} \,.
\end{equation}
\end{lemma}
\begin{proof}
For convenience of notation we drop the subscript $\epsilon$ in the proof
and bring only its main steps, as it rather standard. We first apply
the inverse transformation of \eqref{eq:103} to \eqref{eq:111} to
obtain
\begin{displaymath}
\begin{cases}
\Delta w - |\psi|^2w = 0 & \text{in }\Omega_R\,, \\
w = B - 1 & \text{on } \partial\Omega_R \,,
\end{cases}
\end{displaymath}
where $B=\text{\rm curl\,} A$.\\
We distinguish in the following between interior estimates and
boundary estimates. Let $x_0\in \partial \Omega_R$ and
$D_r=D_r(x_0)=B(x_0,r)\cap\Omega_R\,.
$\\
By the standard elliptic estimates we then have, in view of
\eqref{eq:106},
\begin{equation}
\label{eq:124}
\|w\|_{H^2(D_r)} \leq C(\|w\|_{L^2(D_{2r})}+ \| \epsilon^{-\alpha}B_n-1\|_{H^2(D_{2r})}) \,.
\end{equation}
To obtain the above we first observe that $B = \epsilon^{-\alpha} B_n$ on the
boundary, and then use the fact that the trace of $B_n$ in $H^\frac
32 (\partial\Omega)$ and is therefore bounded from above by a proper $H^2$ norm.
Similarly,
\begin{equation}
\label{eq:125}
\|w\|_{H^3(D_r)} \leq C\Big(\|w\|_{H^1(D_{2r})} + \|\epsilon^{-\alpha}B_n-1\|_{H^3(D_{2r})}
+\|w\|_{L^\infty(D_{2r})} \||\psi|^2\|_{H^1(D_{2r})}\Big) \,.
\end{equation}
Using Kato's inequality and (\ref{eq:18}a) yields
\begin{displaymath}
\|\nabla|\psi|\,\|_{L^2(D_r)} \leq \|\nabla_A\psi\|_{L^2(D_r)} \leq C\|\psi\|_{L^2(D_{2r})}\,,
\end{displaymath}
then we obtain by \eqref{eq:112} that
\begin{equation}\label{112aa}
\|w\|_{H^3(D_r)} \leq C\epsilon^{-\alpha}\,.
\end{equation}
An interior estimate is even easier. Consider $x_0\in \Omega_R$ such that
$D(x_0,2r) \subset \Omega_R$. There is no need in this case to include a
boundary term in \eqref{eq:124} and \eqref{eq:125}. We then obtain
\eqref{112aa} in this case in a similar manner.
The proof of \eqref{eq:117} now follows from Sobolev embeddings and
\eqref{eq:103}.
\end{proof}
We next define the following subdomain of $\Omega$:
\begin{displaymath}
{\mathcal D}_\delta(\epsilon) = \{ x\in\Omega \,:\, |B_\epsilon(x)|<\delta\epsilon^{-\alpha} \} \,.
\end{displaymath}
Let further
\begin{displaymath}
d_{\delta,j}(\epsilon) = d( {\mathcal D}_\delta(\epsilon) ,\partial\Omega_{i,j}),\quad j=1,2\,,
\end{displaymath}
where, as in the introduction, $\{\partial\Omega_{i,j}\}_{j=1}^{2}$ denotes the set of connected components of
$\partial\Omega_i$. We now obtain a lower bound of
\begin{equation}\label{d-delta}
d_\delta(\epsilon)=\text{\rm max}_{j\in\{1,2\}}d_{\delta,j}(\epsilon) .
\end{equation}
\begin{lemma}
Let
\begin{equation}
\label{eq:118}
\delta_0=\min(|b_1|,|b_2|)\,.
\end{equation}
Under the conditions of Lemma \ref{Lem3}, there exists, for any $0<\delta<\delta_0\,$, a
positive $C_\delta$ such that for sufficiently small $\epsilon$ we have
\begin{equation}
\label{eq:119}
d_\delta(\epsilon) \geq C_\delta\, \epsilon \,.
\end{equation}
\end{lemma}
\begin{proof}
Let $x\in\partial\Omega_i$ and $y\in{\mathcal D}_\delta(\epsilon)$. By \eqref{eq:117} we have
\begin{equation}
\label{eq:120}
|w_\epsilon(x)-w_\epsilon(y)| \leq \frac{C}{\epsilon^{1+\alpha}} |x-y|\,.
\end{equation}
By \eqref{eq:112} we have
\begin{displaymath}
|w_\epsilon(x)-w_\epsilon(y)| \geq |B_\epsilon(x)-B_\epsilon(y)| -\frac{1}{2}\geq (\delta_0-\delta)\epsilon^{-\alpha}-\frac{1}{2}\,.
\end{displaymath}
Combining the above with \eqref{eq:120} yields
\begin{displaymath}
|x-y| \geq (\delta_0-\delta)\epsilon-\frac{1}{2}\epsilon^{1+\alpha} \,,
\end{displaymath}
which readily yields \eqref{eq:119}.
\end{proof}
Next we show
\begin{lemma} Under the conditions of Lemma \ref{Lem3}, there exist $C>0$, $\epsilon_0>0$ and $\delta_0>0$ such that, for
$0<\epsilon\leq \epsilon_0 $ and $0< \delta \leq \delta_0\,$,
\begin{equation}
\label{eq:121}
\int_{\Omega\setminus{\mathcal D}_\delta(\epsilon)} \exp \left( [2\delta \Theta_0\epsilon^{-\alpha}]^{1/2}(4\epsilon)^{-1}d(x,{\mathcal D}_\delta(\epsilon))\right) |\psi_\epsilon|^2 \,dx \leq \frac{C}{\delta^{3/2}} \,.
\end{equation}
\end{lemma}
\begin{proof}
Let $\eta\in C^\infty(\Omega,[0,1])$ satisfy
\begin{displaymath}
\eta(x)=
\begin{cases}
1 & x\in \Omega\setminus {\mathcal D}_\delta(\epsilon)\,, \\
0 & x\in {\mathcal D}_{\delta/2}(\epsilon) \,.
\end{cases}
\end{displaymath}
With the aid of \eqref{eq:117} and \eqref{eq:112}, it can be easily
verified, for some $C=C(p,J,\Omega)>0$ which is independent of both $\delta$
and $\epsilon$, that for all $0<\delta<\delta_0$ we can construct $\eta$ with the
additional property
\begin{displaymath}
|\nabla\eta| \leq \frac{C}{\delta\epsilon} \,.
\end{displaymath}
Let further
$$\zeta=\chi\eta\,,
$$
where
\begin{displaymath}
\chi=
\begin{cases}
\exp\big(\alpha_\delta\epsilon^{-1-\alpha/2}d(x,{\mathcal D}_\delta(\epsilon))\big) &\text{if } x\in\Omega\setminus {\mathcal D}_\delta(\epsilon)\,, \\
1 &\text{if } x\in {\mathcal D}_\delta(\epsilon)\,.
\end{cases}
\end{displaymath}
We leave the determination of
$\alpha_\delta$ to a later stage.
Multiplying (\ref{eq:25}a) by $\zeta^2\bar{\psi_\epsilon}$ and integrating by parts yields
\begin{displaymath}
\|\nabla_{\epsilon^{-2}A_\epsilon}(\zeta\psi_\epsilon)\|_2^2 =\frac{1}{\epsilon^2}\big[ \|\zeta\psi_\epsilon\|_2^2 -
\|\zeta^{1/2}\psi_\epsilon\|_4^4\big] + \|\psi_\epsilon\nabla\zeta\|_2^2 \,.
\end{displaymath}
By \eqref{eq:22} we have that for sufficiently small $\epsilon$
\begin{displaymath}
\|\nabla_{\epsilon^{-2}A_\epsilon}(\zeta\psi_\epsilon)\|_2^2 \geq\frac{\Theta_0\delta}{2\epsilon^2}\epsilon^{-\alpha}
\|\zeta\psi_\epsilon\|_2^2 \,,
\end{displaymath}
where $\Theta_0$ is defined in \eqref{eq:20}. Consequently,
\begin{equation}
\label{eq:122}
\frac{\Theta_0\delta \epsilon^{-\alpha}-2}{2\epsilon^2}\|\zeta\psi_\epsilon\|_2^2 \leq \|\psi_\epsilon\nabla\zeta\|_2^2 \,.
\end{equation}
From \eqref{eq:122} we learn that
\begin{displaymath}
\|\zeta\psi_\epsilon\|_2 \leq \Big[\frac{2\epsilon^2}{\Theta_0\delta \epsilon^{-\alpha}-2}\Big]^{1/2} \Big(\alpha_\delta
\epsilon^{-(1+\alpha/2)}\|\zeta\psi_\epsilon\|_2 +
\|\psi_\epsilon\nabla\eta\|_2\Big) \,,
\end{displaymath}
where we have used the fact that $|\nabla d(x,{\mathcal D}_\delta(\epsilon))|\leq1$ a.e.
Choosing
\begin{displaymath}
\alpha_\delta=\frac{1}{2}\Big[\frac{\Theta_0\delta}{2}\Big]^{1/2}
\end{displaymath}
we obtain that for sufficiently small $\epsilon$
\begin{displaymath}
\|\zeta\psi_\epsilon\|_2 \leq 4\Big[\frac{2}{\delta}\Big]^{1/2} \epsilon^{1+\alpha/2}\|\psi_\epsilon\nabla\eta\|_2 \leq
\frac{C\epsilon^{\alpha/2}}{\delta^{3/2}}\|\psi_\epsilon\|_{L^2({\mathcal D}_\delta(\epsilon)\setminus {\mathcal D}_{\delta/2}(\epsilon))}\,.
\end{displaymath}
The proof of \eqref{eq:121} now follows from \eqref{eq:106}.
\end{proof}
We now obtain an improved lower bound.
\begin{proposition}
Let $\delta_0$ be given by \eqref{eq:118}. Under the conditions of Lemma
\ref{Lem3}, for any $0<\delta<\delta_0$, there exist positive constants $C_\delta$ and $\epsilon_\delta$ such that for
$0< \epsilon \leq \epsilon_\delta$ we have
\begin{equation}
\label{eq:123}
d_\delta(\epsilon) \geq C_\delta \,.
\end{equation}
\end{proposition}
\begin{proof}
We begin by noticing that by (\ref{eq:104}b), \eqref{eq:113}, and
\eqref{eq:121} we have
\begin{displaymath}
\|\nabla B_\epsilon\|_{L^2(\Omega\setminus{\mathcal D}_\delta(\epsilon))} \leq
\|\nabla\phi_\epsilon\|_{L^2(\Omega\setminus{\mathcal D}_\delta(\epsilon))}+Ce^{-1/\epsilon} \leq \widehat C \epsilon^{-\alpha} \,.
\end{displaymath}
Then, we write
\begin{displaymath}
(\delta_0-\delta)\epsilon^{-\alpha} \leq \|\nabla B_\epsilon\|_{L^1(\Omega\setminus{\mathcal D}_\delta(\epsilon))} \leq Cd_\delta^{1/2}
\|\nabla B_\epsilon\|_{L^2(\Omega\setminus{\mathcal D}_\delta(\epsilon))} \leq Cd_\delta^{1/2}\epsilon^{-\alpha}\,,
\end{displaymath}
from which \eqref{eq:123} readily follows.
\end{proof}
\begin{remark}
\label{rem:size}
Note that by \eqref{eq:121} and the above arguments, \eqref{eq:123}
holds even if we use $d_\delta$ to measure the distance of ${\mathcal D}_\delta(\epsilon)$ from
any point on $\partial\Omega$ where $B_\epsilon>\delta_1\epsilon^{-\alpha}$ with $\delta_1>\delta$.
\end{remark}
Proposition \ref{semiclassical} now follows from \eqref{eq:121} and \eqref{eq:123}.
\paragraph{\bf Acknowledgements}
The authors gratefully acknowledge Phuc Nguyen for introducing to them
Byun's result \cite{by07} and Monique Dauge for the idea behind the
proof of Lemma \ref{lem:Dirichlet-Neumann}. Y. Almog was partially
supported by NSF grant DMS-1109030 and by US-Israel BSF grant
no.~2010194. B. Helffer was partially supported by the ANR programme
NOSEVOL and Simons foundation visiting fellow at the Newton Institute
in Cambridge during the completion of this article. X.B. Pan was
partially supported by the National Natural Science Foundation of
China grant no. 11171111 and no. 11431005.
\vspace{2ex}
The authors declare that
they have no conflict of interest.
|
2,869,038,156,366 | arxiv | \section{Introduction}
\label{sec:intro}
Nanoparticle supercrystals are three-dimensional lattices of nanoparticles with long-range crystalline order.\cite{Murray1980,boles_self-assembly_2016,murray_synthesis_2000,Shevchenko2006,Coropceanu2019,Mueller2020,Schulz2020} They have long fascinated physicists, chemists and material scientists, because they promise material properties that cannot be found in nature.\cite{boles_self-assembly_2016} Supercrystals are tailored through particle-particle interaction and the choice of their building blocks. Initially, the discussion focused on collective vibrational and electronic modes of supercrystals made from semiconducting nanoparticles,\cite{Vovk2020} but interest recently turned to collective optical excitations,\cite{Baimuratov2013} especially, when using metallic nanoparticles as the supercrystal building blocks.\cite{Mueller2020,Schulz2020, Mueller2021,GarciaLojo2019,BlancoFormoso2020} The optical properties of metallic nanoparticle supercrystals are dominated by the response of their free electrons.
When light interacts with a metallic nanoparticle, it excites localized surface plasmons - the collective oscillation of the free electrons - that strongly absorb and scatter light.\cite{Kreibig,Kelly2003, MaierBook} Plasmons of different nanoparticles interact in supercrystals over various length scales creating coherent collective excitations that propagate through the lattice.\cite{Meinzer2014,Lamowski2018,Mueller2020} The collective plasmonic modes couple to photons forming hybrid quasi-particles called plasmon polaritons.\cite{Barnes2003} Supercrystal plasmon polaritons differ greatly in their properties from the excitations of the individual nanoparticles, they determine the optical response of metallic supercrystals, and open new pathways for scientific discoveries and technological development.\cite{Shalaev2008,Tame2013,Vovk2020,Mueller2020}
Recently, we showed that plasmonic supercrystals can be used to explore phenomena in the ultrastrong regime of light-matter coupling (USC).\cite{Mueller2020,Mueller2021,Baranov2020} In this regime the coupling strength is a considerable fraction of the bare frequency of the system, which leads to peculiar properties of the polariton states.\cite{Kockum2019,FornDiaz2019} For inter-particle gaps much smaller than the nanoparticle size, light-matter coupling even enters the regime of deep strong coupling (DSC),\cite{Mueller2020} where the interaction between light and matter exceeds the energies of the bare excitation. This means that the properties of plasmonic nanoparticle supercrystals can only be modeled when considering the existence of photonic states.\cite{Mueller2020} This is in sharp contrast to the standard treatment of light as an external perturbation.
The DSC regime, moreover, promotes a wide range of exquisite and interesting physical effects such as a decoupling of light and matter and the breakdown of the Purcell effect \cite{DeLiberato2014,Mueller2020}, the squeezing of the photonic components of the polaritons, super-Poissonian phonon and photon statistics \cite{Artoni1991}, and ground state electroluminescence.\cite{Cirio2016,Kockum2019,FornDiaz2019} Although most of these properties can be understood in terms of the Hopfield model for light-matter interaction \cite{Hopfield1958}, a microscopic model that is capable of making predictions for a specific plasmonic supercrystals is highly desirable.
Weick {\it et al.}\cite{Weick2015} and Lamowski {\it et al.}\cite{Lamowski2018} developed a quantum-based formalism that describes the polaritons of plasmonic supercrystals. The microscopic structure of the supercrystal and the dipole-dipole interactions between the nanoparticles turned out to be key for modelling the collective plasmon modes.\cite{Lamowski2018} The model uses two common approximations that appear very reasonable on first sight, but turn out to limit its applicability: It considers only the dipole excitation of plasmonic nanoparticles and neglects Umklapp processes when crossing the boundary of the Brillouin zone. The restriction to the dipole excitation is motivated by the small size ($<100\,$nm) of plasmonic nanoparticles that prohibits the excitation of higher-order electrical modes in individual particles.\cite{MaierBook,LeRuBook} This argument, while correct, misses that higher-order modes of individual nanoparticles combine into dipole-active eigenstates in plasmonic oligomers and supercrystals.\cite{Reich2020,LiJensen2006} These states may couple to the dipole-induced collective plasmons and the electromagnetic states affecting the final polariton dispersion. Umklapp processes are usually negligible for optical excitations, because the wavelength of visible light ($500\,$nm) is small compared to the translational periodicity in natural crystals ($0.1\,$nm). The unit cell of supercrystals, however, becomes a sizable fraction of the light wavelength, since nanoparticle diameters are several $10\,$ to $100\,$nm. The quasi-static approximation breaks down and Umklapp processes may turn out to be important.
In this work we develop a microscopic quantum model for plasmon-polaritons in metallic supercrystals that includes quadrupolar plasmonic excitations and Umklapp processes. We validate it by comparison to finite difference time domain (FDTD) simulations. Our model very well describes plasmon polaritons for supercrystals with low and high packing densities. We calculate the plasmon band structure for face centered cubic (FCC) supercrystals as a function of packing density showing that the quadrupole-derived collective eigenmodes cross and mix with the dipole-induced states for high packing densities. Including the coupling to the electromagnetic states results in plasmon polaritons in the USC and DSC regimes. The strongest contribution to light-matter coupling arise from the dipole plasmons; the quadrupole-photon coupling is an order of magnitude weaker than dipole-photon interaction. However, the quadrupole contribution is important for the polariton band structure, because the energies of the collective plasmons are overestimated by several 100\,meV at the Brillouin zone boundary when neglecting the quadrupole modes. We derive a closed expresssion for the reduced coupling strength and show that is mainly depends on the metal fill fraction. We extract the dipole, quadrupole, and photon contribution to all polariton states. The decoupling of light and matter clearly manifests as the three quasi-particles dominate distinct polariton branches in the DSC regime.
This paper is organized as follows: In Sec.~\ref{sec_QPPM}, we describe the theoretical framework and apply it to a three-dimensional Bravais lattice of spherical metallic nanoparticles. The theory for arbitrary crystal structures is given in the Supplementary Information. We compare the calculated polariton dispersion to FDTD simulations. In Sec.~\ref{sec:results} we calculate the plasmon and plasmon-polariton band structure of FCC supercrystals. We demonstrate how quadrupole modes and light-matter coupling affect the polariton dispersion. We discuss the properties of polaritons, the reduced coupling strength, and demonstrate the decoupling of light and plasmons in the DSC regime. In Sec.~\ref{sec:conclusions} we summarize the main findings of the paper.
\section{Quantum microscopic plasmon-polariton model\label{sec_QPPM}}
In this section we derive the microscopic model of plasmon polaritons in metallic supercrystals. We first present a general theoretical framework that describes individual and interacting nanoparticles with dipole and quadrupole excitations and their coupling to an electromagnetic field. This description is then applied to a Bravais lattic; we verify its validity and limitations by comparing to FDTD simulations of FCC nanoparticle crystals. Our microscopic quantum plasmon-polariton model contains the nanoparticle quadrupole in addition to their dipole excitations. Adding the quadrupole terms was challenging, because the the quantum description of light-matter interaction is based on the dipole approximation.
One difficulty we encountered was to find a proper description of the conjugate momenta for higher-order multipoles. This problem dates back to the description of nuclear excitations and was discussed first by Bohr and Mottelson.\cite{Bohr1953,Bohr1998} In 1978 Gulshani finally provided a formal development of canonically conjugate momenta for quadrupolar excitations,\cite{Gulshani1978} but no solution has been found for higher-order multipoles.
We start from the most general Hamiltonian for a set of charges distributed in space interacting with the electromagnetic field
\begin{equation}\label{eq_fullH_LM}
\mathcal H=\sum_n \frac{1}{2m}\left[\vec p_n - {\rm q}_n \vec A(\vec r_n, t)\right]^2 + V_{Coul} + \mathcal{H}_L,
\end{equation}
where $V_{Coul}$ is the Coulomb interaction between the different $n$ charges ${\rm q}_n$ in the system, $\mathcal{H}_L$ is the quantized free-electromagnetic field Hamiltonian and $\vec p_n$ is the conjugate momentum to $\vec r_n$. $\vec A$ is assumed to be in the Coulomb gauge.
We consider an isolated spherical metallic nanoparticle at the origin. The charges are bound by the nanoparticle volume, such that the summation $n$ in Eq.\ (\ref{eq_fullH_LM}) is restricted to the free electrons in the particle. We define a set of variables
\begin{equation}
h_{\sigma} =\frac{1}{N} \sum_{n} r_{n,\sigma},\:H_{\gamma} =\frac{1}{N\bar \rho} \sum_{n,\alpha,\beta}r'_{n,\alpha} r'_{n,\beta}\chi^\gamma_{\alpha\beta},
\end{equation}
where $h_{\sigma}$ represents the center of mass displacement along the $\sigma$ direction and is associated with the dipole moment of the charge distribution. $n=1,...,N_j$ runs through all the charges in a given nanoparticle. The second term $H_{\gamma}$ is associated to the quadrupole moment, with $\alpha$ and $\beta$ being different Cartesian directions. $\gamma$ specifies one of the five possible quadrupolar modes, and $r'_{n\alpha}=r_{n,\alpha}-h_{\alpha}$. $\bar \rho=\langle 1/N\sum_{n}r_{n}^2\rangle \sim \sqrt{\frac{2}{5}}\rho$,
with $\rho$ the radius of the spherical particle, is the expectation value of the diagonal term of the quadrupole moment. The respective conjugate momenta are
\begin{equation}
\pi_{\sigma}=\sum_{n} p_{n,\sigma} ,\: \Pi_{\gamma}= \frac{1}{\bar \rho}\sum_{n,\alpha,\beta} p'_{n,\alpha} r'_{n,\beta}\chi^\gamma_{\alpha\beta},
\end{equation}
with $p'_{n,\alpha}=p_{n,\alpha}-\pi_{\alpha}$. It should be mentioned that only the dipole part of this transformation is formally canonical. As discussed in the S.I., the quadrupole term can only be associated with a canonical transformation (and thus with the expected commutation relations) if the total angular momentum of the charge distribution is zero and if the charge displacements are small compared to the bulk charge.\cite{Gulshani1978} With these two considerations, the dynamical variable $\sum_n r_{n,\alpha}^2$ can be substituted by its expectation value $\bar \rho$ and the quadrupolar moments can be described in terms of the traceless and symmetric $3\times3$ matrices $\chi_\gamma$ which act as unit tensors within the dyadic double-dot product (see S.I.). With this approximation, we can write the following inverse transformation for the quadrupolar canonical variables
\begin{equation}
\left\{\begin{matrix}
\sum_{n,\alpha,\beta}r_{n,\alpha}r_{n,\beta}=N\bar\rho^2\left({\bf 1} + \frac{1}{\bar\rho}\sum_\gamma H_{\gamma}\chi_\gamma\right)\\
\sum_{n,\alpha,\beta}p_{n,\alpha}r_{n,\beta}=\bar\rho \sum_\gamma \Pi_{\gamma}\chi_\gamma
\end{matrix}\right. .
\end{equation}
Here, ${\bf 1}$ is the $3\times 3$ unit matrix.
We now consider a system of many nanoparticles. If there is no charge exchange between different particles, the full Hamiltonian of the system will be given by Eq.\ (\ref{eq_fullH_LM}) but with the summation in $n$ being extended to a summation for each particle $n\in j$ and a summation of the different particles in the system. With this, the matter Hamiltonian ($\mathcal{H}_M=\sum_n \frac{p^2_n}{2m} + V_{Coul}$) can be written as
\begin{equation}
\mathcal{H}_M=\mathcal{H}_M^D+\mathcal{H}_M^Q+\mathcal{H}_{\rm plpl}+\mathcal{H}^{HO}_M,
\end{equation}
where $\mathcal{H}_M^D + \mathcal{H}_M^Q$ represents the Hamiltonians for each nanoparticle in the system, including the kinetic energy associated with each of the canonical coordinates ($D$ for dipole and $Q$ for quadrupole) and the intra-particle Coulomb interactions. Both are assumed to be described in terms of harmonic oscillations with characteristic frequencies $\omega_{j,D}$ and $\omega_{j,Q}$, which can have different values for each nanoparticle $j$ in the system. The term $\mathcal{H}_M^{HO}$ represents the dynamics of the higher-order coordinates, which are disregarded in the present model.
The plasmon-plasmon interaction between the different particles in the system is
\begin{equation}
\mathcal{H}_{\rm plpl}=\mathcal{H}_{DD}+\mathcal{H}_{DQ}+\mathcal{H}_{QQ},
\end{equation}
corresponding to the dipole-dipole $\mathcal{H}_{DD}$, dipole-quadrupole $\mathcal{H}_{DQ}$ and quadrupole-quadrupole interaction $\mathcal{H}_{QQ}$. Explicit expressions for these terms are given in the Supplementary Information.
The characteristic size of plasmonic nanoparticles is $10-100\,$nm. The particles diameters are a considerable fraction of the light wavelength. The common approach for light-matter interaction of expanding the vector-potential in a Taylor series in the vicinity of the charge distribution and disregarding higher-order terms in $k$ will not be effective. We perform the expansion in a slightly different way; we start from the vector potential of a plane wave given by
\begin{equation}
A_\lambda(\vec r,t)=\sum_{\vec q}A_{\vec q,\lambda}(t)\exp(i\vec q\cdot\vec r).
\end{equation}
where $\lambda$ specifies the light polarization and $\vec q$ is a vector in reciprocal space - not to be confused with the charges $\rm q$. With this, the general light-matter interaction for a particle $j$ can be separated into two parts. The first-order part
\begin{equation}
\mathcal{H}_{LM}^{(1)}=-\frac{\rm q_e}{m_e}\sum_{j,(n\in j),\lambda,\vec q}p_{n\lambda} A_{\vec q\lambda}\exp(i\vec q\cdot\vec r_n),
\end{equation}
corresponds to the interaction of light with excitations of the electric charge distribution. The time dependence of $\vec A$ is implicit. The second-order part
\begin{equation}
\mathcal{H}_{LM}^{(2)}=\frac{\rm q^2_e}{2m_e}\sum_{j,n \in j,\lambda,\vec q,\vec q'}A^{\ast}_{\vec q'\lambda} A_{\vec q\lambda}\exp[i(\vec q-\vec q')\cdot\vec r_n],
\end{equation}
describes the back reaction of the electric field as it accelerates the electric charges.\cite{Selsto2007} This term
is known as the $A^2$ term in the light-matter Hamiltonian; it
becomes extremely important in the USC and DSC coupling regime.\cite{Kockum2019, Mueller2020}
Let us now take the spherical harmonic expansion of the plane wave around the position $\vec R_j$ of each nanoparticle in the system
\begin{equation}
\exp(i\vec q\cdot\vec r_j)\sim j_0(qr_j) + 3i \frac{j_1(qr_j)}{qr_j}\vec q\cdot \vec r_j,
\end{equation}
where $\vec r_j=\vec r-\vec R_j$ and we have retained only the lower order terms. We apply this expansion to the 1st order part of the light-matter interaction for each nanoparticle $j$ independently and sum to get the full Hamiltonian. It then has a dipole-like contribution
\begin{equation}
\mathcal{H}_{LM}^{(D,1)}=-\frac{\rm q_e}{m_e}\sum_{j,\lambda,\vec q}A_{\vec q\lambda}\exp(i\vec q\cdot \vec R_j)\sum_{n\in j}p_{n\lambda} j_0(qr_{n,j}),
\end{equation}
and a quadrupolar-like term
\begin{equation}
\begin{split}
\mathcal{H}_{LM}^{(Q,1)}=&
-\frac{\rm q_e}{m_e}\sum_{j,\lambda,\vec q,\alpha,\beta} iqA_{\vec q\lambda}\exp(i\vec q\cdot \vec R_j)\times \\&\times\left(\hat e_\lambda \hat e_q:\hat e_\alpha \hat e_\beta\right) \sum_{n\in j}\frac{3j_1(qr_n)}{qr_n} p_{n\alpha}r_{n\beta},
\end{split}
\end{equation}
where $(:)$ stands for a double-dot dyadic product.
We now substitute $j_0(qr_n)$ and $3j_1(qr_n)/qr_n$ by their mean values in each nanoparticle
\begin{equation}
{\rm f}_D=\langle j_0(qr_j) \rangle =\frac{3}{(q\rho_j)^3}\left[\sin(q\rho_j)-q\rho_j \cos(q\rho_j)\right]
\end{equation}
and
\begin{equation}
{\rm f}_Q=\langle \frac{3j_1(qr_n)}{qr_n}\rangle=\frac{9}{(q\rho_j)^3}\left[{\rm Si}(q\rho_j)-\sin(q\rho_j)\right],
\end{equation}
where Si$(x)$ is the sine integral function. ${\rm f}_D$ and ${\rm f}_Q$ are nanoparticle form factors somewhat similar to the atomic form factors in X-ray diffraction theory\cite{Kittel2005}.
The magnitude of the form factors decrease with increasing $q\rho$, effectively cutting off the contribution of photons with wavevectors much larger than $1/\rho$ to the light-matter interaction. This effect stems from the field retardation within the nanoparticle.
With this definition the 1st-order term in the light-matter interaction Hamiltonian becomes
\begin{equation}
\mathcal{H}_{LM}^{(D,1)}=-\sum_{j,\vec q}{\rm f}_D(q)\frac{{\rm Q}_{D}}{M}\vec \pi_j\cdot \vec A_{\vec q,\vec R_j},
\end{equation}
and
\begin{equation}
\mathcal{H}_{LM}^{(Q,1)}=\sum_{j,\vec q}{\rm f}_Q(q)\frac{ {\rm Q}_{Q}\bar \rho_j}{M} \Pi_{j,\nu}[\chi_\nu : \vec q\vec A_{\vec q,\vec R_j}],
\end{equation}
where $M$ is the total mass of the charges ($M=Nm_e$), ${\rm Q}_D$ and ${\rm Q}_Q$ are the screened dipole and quadrupole effective charges, which will depend on the relative permittivity of the surrounding medium, $\vec q \vec A_{\vec q,\vec R_j}$ is a dyadic, with $A_{\vec q,\vec R_j}=A_{\vec q}\exp(i\vec q\cdot\vec R_j)$.
Note that for small nanoparticle radii ($\rho$) the form factors approach unity and the plasmon-photon interaction obtained by applying Taylor's expansion is recovered. In that case, 1st order light-quadrupole interaction increases linearly with $q$. For the second-order term, associated to the $A^2$ light-matter interaction term, we have the two contributions
\begin{equation}
\mathcal{H}_{LM}^{(D,2)}=\sum_{j,\vec q,\vec q'}{\rm f}_D(q){\rm f}_D(q')\frac{{\rm Q}^2_{D}}{2M}\left[\vec A^\ast_{\vec q',\vec R_j}\cdot \vec A_{\vec q,\vec R_j}\right],
\end{equation}
\begin{equation}
\mathcal{H}_{LM}^{(Q,2)}=\sum_{j,\vec q,\vec q'} {\rm f}_Q(q){\rm f}_Q(q')\frac{{\rm Q}^2_{Q}\bar\rho_j^2}{2M}\left[\vec A^\ast_{\vec q',\vec R_j}\vec q'\cdot \vec q\vec A_{\vec q,\vec R_j}\right].
\end{equation}
The terms involving products between $\vec A_{\vec q}$ and $\bar\nabla \vec A_{\vec q}$ are disregarded, as the expectation value $\langle \sum_n\vec r_n\rangle=0$ vanishes. Also, all non-quadratic terms, involving more than two dynamical variables, are disregarded within this approximation.
\subsection{Plasmonic nanoparticle crystals}\label{sec:NPcrystals}
We now apply the proposed quantum mechanical description of light-matter coupling with dipole and quadrupole modes to a crystal of identical spherical nanoparticles placed in a Bravais lattice. The position of a nanoparticle in the crystal is determined by a lattice vector $\vec R$, and the index $j$ is dropped. The model can be trivially extended to crystals with an arbitrary basis - see Supplementary Information for equations.
Following the work of Weick~{\it et al.}\cite{Weick2015} and Lamowski~{\it et al.}\cite{Lamowski2018}, we expand the plasmonic and photonic dynamical variables into creation and annihilation operators defined in reciprocal space for a periodic arrangement of particles. We obtain the Hamiltonian for plasmon-plasmon interaction as
\begin{equation}
\begin{split}
\mathcal{H}_{\rm plpl}=&\sum_{\vec q,\nu,\nu'}\hbar\sqrt{\Lambda_{\bar\nu}\Lambda_{\bar\nu'}}S_{\nu,\nu'}^{\bar\nu,\bar\nu'}(\vec q)\left(b_{-\vec q,\nu}^\dag+b_{\vec q,\nu}\right)\times\\&\times\left(b_{\vec q',\nu'}^\dag+b_{-\vec q',\nu'}\right),
\end{split}
\label{eq_Hplpl}
\end{equation}
where $b_{\vec R,\nu}=1/\sqrt{N_\mathrm{cells}}\sum_{\vec q} b_{\vec q,\nu}\exp(i\vec q\cdot \vec R)$ is the annihilation operator for multipole oscillations of the nanoparticle in the unit cell defined by the lattice vector $\vec R$. $\bar\nu=D$ for $\nu=1-3$ which correspond to dipole modes while $\bar\nu=Q$ for the $\nu=4-8$ quadrupole modes.
$N_\mathrm{cells}$ is the number of unit cells.
The structure function $S_{\nu,\nu'}^{\bar\nu,\bar\nu'}$ in Eq.~\eqref{eq_Hplpl} depend only on the Bravais lattice; it is given by
\begin{widetext}
\begin{equation}
\label{eq_FDD}
S^{DD}_{\nu,\nu'}(\vec q)=\sum_{\vec{R}} \frac{1}{2}\frac{\delta_{\nu\nu'} -3 (\hat e_\nu\cdot\vec{n})(\hat e_{\nu'}\cdot\vec{n})}{(R/\bar R)^3}\exp(i\vec{q}\cdot\vec R),
\end{equation}
when both $\nu$ and $\nu'$ correspond to dipole modes,
\begin{equation}
\label{eq_FQQ}
S^{QQ}_{\nu,\nu'}(\vec q)=\sum_{\vec{R}}\frac{1}{6} \left\{35\frac{(\chi_\nu:\hat n \hat n)(\chi_{\nu'}:\hat n\hat n)}{(R/\bar R)^5} - 20\frac{(\chi_\nu\chi_{\nu'}:\hat n\hat n)}{(R/\bar R)^5}+2\frac{(\chi_\nu:\chi_{\nu'})}{(R/\bar R)^5}\right\} \exp(i\vec{q}\cdot\vec R),
\end{equation}
when both $\nu$ and $\nu'$ correspond to quadrupole modes, and
\begin{equation}
\label{eq_FDQ}
S^{DQ}_{\nu,\nu'}(\vec q)=\sum_{\vec{R}}\frac{1}{2}\left[
-5\left(\chi_{\nu'} : \frac{\hat n \hat n}{(R/\bar R)^4}\right)(\hat n \cdot \hat e_\nu) + 2 \left(\chi_{\nu'} : \frac{\hat n \hat e_\nu}{(R/\bar R)^4}\right)\right] \exp(i\vec{q}\cdot\vec R),
\end{equation}
\end{widetext}
when $\nu$ corresponds to a dipole mode and $\nu'$ corresponds to a quadrupole mode. Here $\hat a\hat b$ corresponds to dyadics formed by the two unit vectors $\hat a$ and $\hat b$. Also, $\hat n=\vec R/R$ and $\bar R = (V_{uc})^{1/3}$, with $V_{uc}$ being the volume of the unit cell. The dipole-dipole interaction does not converge for $q\rightarrow 0$ in a filled three-dimensional space.\cite{Lamowski2018,Cohen1955} For wavevectors below a cutoff value $q_c$, {\it i.e.}, $|q|<|q_c|$, the dipole-dipole structure function $S^{DD}$ is replaced by
$S^{DD}_{\nu,\nu'}=-2\pi\left[\delta_{\nu\nu'}-(\hat e_{\nu}\cdot \hat q)(\hat e_{\nu'}\cdot \hat q)\right]/3$. The value of $q_c$ that allows for a smooth dispersion relation depends on the Bravais lattice and on the number of unit cells considered. The plasmon-plasmon interaction Hamiltonian in Eq.~\eqref{eq_Hplpl} contains coupling factors $\Lambda_{\bar\nu}$; they are given by
\begin{equation}
\Lambda_{D}=\frac{ {\rm Q}_{D}^2}{8\pi\epsilon_0\epsilon_m M\omega_D V_{uc}},
\end{equation}
and
\begin{equation}
\Lambda_{Q}=\frac{({\rm Q}_{Q}\bar\rho)^2}{8\pi\epsilon_0\epsilon_m M\omega_QV_{uc}^{5/3}},
\end{equation}
where $\epsilon_m$ is the dielectric constant of the surrounding medium, which is assumed to be a positive constant.
We now turn to the interaction between plasmons and photons. The first-order part of the plasmon-photon coupling can be written as
\begin{equation}
\begin{split}
\mathcal{H}_{\rm plpt}^{(1)}=i\hbar\sum_{\vec q,\vec G,\lambda,\nu} &\omega_{\bar\nu}\xi^{\nu}_{\lambda,\vec G}\left(b_{-\vec q,\nu}^\dag -b_{\vec{q},\nu} \right)\times\\&\times\left(c_{-\vec{q}-\vec G,\lambda}+c_{\vec{q}+\vec G,\lambda}^\dag\right),
\end{split}
\end{equation}
where $\vec G$ runs through the reciprocal lattice vectors for the chosen lattice. We defined
\begin{equation}
\label{eq_xiD}
\xi^{\nu}_{\lambda,\vec G}(\vec q)=\rm{f}_D(|\vec q+\vec G|)\xi_0^D(\vec q)P^D_{\nu,\lambda}(\vec q+\vec G),
\end{equation}
for $\bar\nu=D$ and
\begin{equation}
\label{eq_xiQ}
\xi^{\nu}_{\lambda,\vec G}(\vec q)=i{\rm f}_Q(|\vec q+\vec G|)|\vec q+\vec G|\bar R\xi_0^Q(\vec q)P^Q_{\nu,\lambda}(\vec q+\vec G),
\end{equation}
for $\bar\nu=Q$, where
\begin{equation}
\xi_0^{\bar\nu}(\vec q)=\sqrt{\frac{2\pi\Lambda_{\bar\nu}}{\omega_{pt}(\vec q)}},
\end{equation}
and
\begin{equation}
\left\{
\begin{array}{ll}
P^D_{\nu,\lambda}(\vec q) = &\hat e_\nu \cdot \hat e_\lambda \\
P^Q_{\nu,\lambda}(\vec q) = &\frac{1}{2}[\chi_\nu :\hat e_q \hat e_\lambda+\hat e_\lambda \hat e_q]
\end{array}\right. ,
\end{equation}
with $\hat e_\lambda$ being functions of $\vec q$, since for both values of $\lambda$ the vector potential is perpendicular to the wavevector $\vec q$.
Finally, the second-order part of the plasmon-photon interaction is
\begin{equation}
\begin{split}
\mathcal{H}_{\rm plpt}^{(2)}=\hbar\sum_{\vec q,\lambda, \lambda', \vec G,\vec G'} &\Xi^{\lambda\lambda'}_{\vec G,\vec G'}(\vec q)\left(c_{-\vec{q}-\vec G',\lambda'}^\dag+c_{\vec{q}+\vec G',\lambda'}\right)\times \\\times & \left(c_{-\vec{q}-\vec G,\lambda}+c_{\vec{q}+\vec G,\lambda}^\dag \right),
\end{split}
\end{equation}
where
\begin{equation}
\label{eq_Xi}
\Xi^{\lambda\lambda'}_{\vec G, \vec G'}(\vec q)=\sum_{\nu}\omega_{\bar\nu}\xi^{\nu\ast}_{\lambda'\vec G'}(\vec q)\xi^{\nu}_{\lambda\vec G}(\vec q).
\end{equation}
It is interesting to note that for each photon mode the second-order photon coupling is obtained in terms of a sum involving the matrix elements of the first-order interactions. In the weak-coupling regime, this term can be obtained using the TRK sum rule and has the important effect of balancing out the 1st-order term as $q$ goes to zero, thus preventing super-radiant phase transitions.\cite{Hepp1973}
Recently, a generalized sum rule was obtained for the strong coupling regime.\cite{Savasta2020} It is written in terms of the eigenstates of the full Hamiltonian and cannot be directly applied to simplify our calculations. However, the existence of such a rule indicates that even in the strong coupling regime, the second-order terms perfectly balance out the first-order interactions. It is also noteworthy that the second-order term directly couples photon modes with different polarizations $\lambda$ and $\lambda'$, effectively mixing these two otherwise independent photon states and opening pathways for different types of chiral activity in strongly coupled systems.
To calculate the polaritonic modes, we can follow the work of Xiao\cite{Xiao2009}, and define a Bogoliubov vector operator
\begin{equation}
\Phi_{\vec q}=
\begin{pmatrix} \bar b_{\vec q} \\ \bar c_{\vec{q}}\\ \bar b_{-\vec q}^\dag \\ \bar c_{-\vec{q}}^\dag \end{pmatrix},
\end{equation}
where $\bar b_{\vec q}$, and $\bar b^\dag_{\vec q}$ are column vectors with each entry being an operator corresponding to a different plasmonic mode $\nu$.
$\bar c_{\vec q}$, and $\bar c^\dag_{\vec q}$ are column vectors with operators for each polarization $\lambda$
and each reciprocal lattice vector $\vec G$ considered. The values of $\vec q$ are limited to the positive half of the Brillouin zone. $\Phi_{\vec q}$ obeys the following dynamical equation \cite{Xiao2009}
\begin{equation}
i\hbar\frac{d}{dt}\Phi_{\vec q}=D_{\vec q}\Phi_{\vec q},
\end{equation}
with
\begin{equation}
D_{\vec q}=\hbar\begin{pmatrix}
\bar \alpha_{\vec q} & \bar \gamma_{\vec q}\\
-\bar \gamma^\dag_{\vec q} & -\bar \alpha_{\vec q}^\star
\end{pmatrix},
\end{equation}
where
\begin{equation}
\bar \alpha_{\vec q}=
\begin{pmatrix} \bar\omega_{pl}+\bar \Lambda (\bar S_{\vec q}+\bar S_{-\vec q}) & \bar\omega_{pl}\bar \xi_{\vec q} \\
\bar\omega_{pl}\bar \xi^\dag_{\vec q} &\bar\omega^{pt}_{\vec q}+2 \bar \Xi_{\vec q}
\end{pmatrix},
\end{equation}
and
\begin{equation}
\bar \gamma_{\vec q}=
\begin{pmatrix} \bar\Lambda(\bar{S}_{\vec q}+\bar{S}_{-\vec q}) & \bar\omega_{pl}\bar \xi_{\vec q}(\vec q) \\
-\bar\omega_{pl}\bar \xi^\dag_{\vec q} & 2 \bar \Xi_{\vec q}\end{pmatrix}.
\end{equation}
Here $\bar\omega_{pl}$ and $\bar\omega_{pt}$ are diagonal matrices with the energies of each of the plasmonic modes $\nu$ of the metallic nanoparticle and the photon modes (labelled by $\lambda$ and $\vec G$) that are taken into consideration. The matrix $\bar\Lambda$ is given by $\bar\Lambda=\sqrt{\Lambda_{\nu}\Lambda_{\nu'}}$, $\bar S_{\vec q}$ is given by Eqs.\ (\ref{eq_FDD})-(\ref{eq_FDQ}),
$\bar \xi_{\vec q}$ by Eqs.\ (\ref{eq_xiD})-(\ref{eq_xiQ}), and $\bar \Xi_{\vec q}$ by Eq.\ (\ref{eq_Xi}). A detailed description of these matrices is given in the Supporting Information.
The dynamical matrix $D_{\vec q}$ is diagonalized by a Bogoliubov-Valentin transformation $T_{\vec q}^{-1}D_{\vec q}T_{\vec q}$, which leads to a new set of creation and annihilation operators, $\Psi^\dag_{pp}(\vec q)$ and $\Psi_{pp}(\vec q)$, given by $\Psi_{pp}(\vec q)=T_{\vec q}\Phi_{\vec q}$ with eigenvalues $\hbar\omega_{pp}(\vec q)$.\cite{Xiao2009}
These operators correspond to the creation and annihilation of mixed excitations called plasmon-polaritons which have properties of both plasmons and photons.\cite{Lamowski2018} The eigenvalues can be associated with the plasmon-polariton dispersion. It should be mentioned that the transformation matrices $T_{\vec q}$ mix terms with both creation and annihilation operators of plasmons and photons, giving rise to many of the phenomena expected in the extreme regimes of light-matter coupling.\cite{Artoni1991,Cirio2016,Kockum2019,FornDiaz2019}
\subsection{Quasi-static approximation}\label{sec:quasistatic}
In this section we discuss the input parameters of our microscopic model. We want to calculate the plasmon-polariton dispersions of plasmonic supercrystals and compare it to experimental results as well as calculations performed within other techniques. To do so, we need the frequencies of the dipole $\omega_D$ and quadrupole $\omega_Q$ plasmon resonances in metallic nanoparticles as well as their coupling factors $\Lambda_D$ and $\Lambda_Q$.
The dipole and quadrupole frequencies are obtained within the quasi-static approximation. We consider a Drude metal with permittivity $\epsilon(\omega)=\epsilon_d-(\omega_p/\omega)^2$, neglecting losses. $\omega_p$ is the plasma frequency of the metal and $\epsilon_d$ a dielectric constant that accounts for the screening by bound charges.\cite{MaierBook} This yields the frequencies\cite{Kolwas2010,Shopa2010}
\begin{equation}
\omega_D=\frac{\omega_p}{\sqrt{\epsilon_d+2\epsilon_m}},
\end{equation}
and
\begin{equation}
\omega_Q=\frac{\omega_p}{\sqrt{\epsilon_d+(3/2)\epsilon_m}}.
\end{equation}
The coupling parameters are obtained by considering the screened effective charges
\begin{equation}
{\rm Q}_{l}=\frac{(2l+1)\epsilon_m}{l\epsilon(\omega)+(l+1)\epsilon_m} {\rm Q}_{l}^0,
\end{equation}
where $l=1$ for dipole and $l=2$ for quadrupole modes.\cite{Doerr2017} This leads to
\begin{equation}\label{eq_lambd}
\Lambda_D=\frac{ 9\epsilon_m \omega_D }{8\pi(\epsilon_{d}+2\epsilon_m)}F,
\end{equation}
and
\begin{equation}\label{eq_lambq}
\Lambda_Q=\left(\frac{3}{4\pi}\right)^{5/3}\frac{ 5\epsilon_m \omega_Q }{12(\epsilon_{d}+(3/2)\epsilon_m)}F^{5/3},
\end{equation}
where $F=4\pi\rho^3/3V_{uc}$ is the metal fill fraction, {\it i.e.}, the fraction of the unit volume cell that is filled by metal.
The expressions for $\Lambda_D$ and $\Lambda_Q$ allow a first estimate of the importance of the dipole and quadrupole contributions to the plasmon-polariton dispersion and light-matter coupling.
The ratio between the quadrupole and dipole coupling factors
\begin{equation}
\frac{\Lambda_Q}{\Lambda_D}=\frac{5}{18}\left(\frac{3}{4\pi}\right)^{2/3}\left(\frac{\epsilon_d+2\epsilon_m}{\epsilon_d+(3/2)\epsilon_m}\right)^{3/2}F^{2/3},
\end{equation}
scales with $F^{2/3}$. The prefactor ranges from $\sim 0.10$ for $\epsilon_d \gg \epsilon_m$ to $\sim 0.16$ for $\epsilon_m \gg \epsilon_d$. In simple crystals (Bravais lattices) of spherical nanoparticles $F \leq 0.74$, with the largest packing density for FCC and HCP lattices. The quadrupole-quadrupole (QQ) interaction is in this case limited to about 13$\%$ of the dipole-dipole (DD) interaction and the dipole-quadrupole (DQ) interaction to 36$\%$. Especially, for smaller packing fractions the dipole-derived terms are expected to dominate the polariton dispersion, but we expect important contributions of the quadrupole terms for high packing. Even larger metal fill fractions may be obtained with non-spherical nanoparticles and in supercrystals with more than one nanoparticle per unit cell.\cite{Murray1980,Coropceanu2019}
The Rabi frequency associated to the interaction of light with the dipole and quadrupole plasmons can be estimated as $\Omega_R^\nu=\omega_{\bar\nu}\xi_0^\nu(\omega_{\bar\nu}/c)$ where $\bar\nu=D$, $Q$ for dipoles and quadrupoles, respectively. Within the quasi-static approximation we obtain the explicit expressions
\begin{equation}\label{Eq_OmD}
\Omega_{R}^D=\omega_D\sqrt{\frac{3F_0}{4}\left(\frac{3\epsilon_m}{\epsilon_d+2\epsilon_m}\right)}f^{1/2},
\end{equation}
and
\begin{equation}\label{Eq_OmQ}
\Omega_{R}^Q=\omega_Q\sqrt{\frac{\pi}{3}\left(\frac{3F_0}{4\pi}\right)^{5/3}\left(\frac{5\epsilon_m}{2\epsilon_d+3\epsilon_m}\right)}f^{5/6},
\end{equation}
where $f=F/F_0$ and $F_0$ is the maximum fill factor for a given lattice. For an FCC supercrystal the Rabi frequencies are limited to $\Omega_{R}^D=0.91\omega_D$ and $\Omega_{R}^Q=0.31\omega_Q$, which is obtained by setting $\epsilon_m \gg \epsilon_d$, $f=1$, and $F_0=0.74$ in Eqs.\ (\ref{Eq_OmD}) and (\ref{Eq_OmQ}). This places the Rabi frequencies on the order of eV for high packing densities, in excellent agreement with our recent experimental results.\cite{Mueller2020}
\subsection{Validating the model}
Before discussing and analyzing the bandstructure and properties of plasmon polaritons, we demonstrate the validity of our model by comparing it to FDTD simulations. We first describe the parameters used in both simulations. We considered an FCC nanoparticle crystal and the high-symmetry $\Gamma L$ and $\Gamma K$ directions. The nanoparticle diameters were $d=50\,$nm with interparticle (center to center) distance of $a=65\,$nm, which yields a metal fill fraction $f=0.46$. The calculations were done with the Drude model using a plasma frequency $\hbar \omega_p=9\,$eV and $\epsilon_d=1$. The nanoparticles were placed in vacuum ($\epsilon_m=1$).
For the microscopic quantum calculation we considered Umklapp processes with $\vec G$ within up to six Brillouin zones. Plasmon-polariton energy differences of up to 10\% were obtained for some of the modes if Umklapp processes were neglected, see Fig.\,S1 for details.
The lattice vector summation in real space for calculating $S^{DD}$ was performed for $|\vec R|$ below a cutoff radius $R_{D}=60\bar R$. For $S^{DQ}$ and $S^{QQ}$ a cutoff radius of $R_Q=7\bar R$ sufficed to achieve full convergence. This reflects the fact that the dipole-quadrupole and quadrupole-quadrupole interactions fall off faster with distance than dipole-dipole coupling. A cutoff wavevector $q_c=0.3\pi/a$ was used for the lattice sums. These parameters were used throughout the paper, unless otherwise stated.
The FDTD simulations were done with the commercial software package Lumerical FDTD Solutions. We constructed the unit cell of an FCC crystal that is composed of spherical nanoparticles. The nanoparticles were assigned the dielectric function $\epsilon(\omega)=\epsilon_d-\omega_p^2/(\omega^2-i\gamma \omega)$ with a loss rate $\hbar \gamma = 65$\,meV (see above for the other parameters). We used a mesh size of 1\,nm to discretize space. To calculate the polariton dispersion we placed local emitters and point monitors inside the crystal.\cite{SunLin2018} We used point dipoles as light sources that radiated along the $[111]$ ($\Gamma L$) or $[110]$ ($\Gamma K$) direction. A 0.7\,fs light-pulse was injected and the electric field recorded in the time interval from 10 to 50\,fs by a point monitor. The frequency dependent electric field was obtained by a Fourier transformation. We used Bloch periodic boundary conditions to choose a specific wave vector. By running a sweep of simulations for different wave vectors we obtained the polaritonic band structure.
\begin{figure}[!htb]
\centering
\includegraphics[width=8.5cm]{Fig_FDTD_comp.pdf}
\caption{Polariton band structure of an FCC crystal of spherical nanoparticles along the (a) $\Gamma L$ and (b) $\Gamma K$ high-symmetry directions. Full lines were calculated with the microscopic quantum model. The colormap shows the magnitude (in log scale) of the electric field obtained in FDTD simulations as a function of energy and momentum. $\hbar \omega_p=9$ eV, $\epsilon_m=\epsilon_d=1$, $a=65$ nm and $f=0.46$.}
\label{fig:FDTD_MQPP}
\end{figure}
Figure \ref{fig:FDTD_MQPP} compares the band structure obtained with FDTD and the microscopic model. The background of the figure is a color map of the integrated intensity of the electric field as a function of $\omega$ and $q$, which corresponds to the polariton dispersion predicted by the FDTD simulations. The black lines show the plasmon-polariton dispersion calculated with the microscopic quantum model. Our model reproduces the FDTD dispersion very well. The far-field response of the supercrystal is dominated by the two dipole-derived bands.\cite{Mueller2020} These are the bands with lowest and highest energy in Fig.~\ref{fig:FDTD_MQPP}, which are excellently described by the quantum mechanical model. Along the $\Gamma L$ direction, Fig.~\ref{fig:FDTD_MQPP}(a), the quadrupole bands between the two dipole-derived states agree also between FDTD and the microscopic model. The FDTD simulation appears to contain more states, which originate from hexapole eigenmodes of the nanoparticles or artefacts of the simulation.
The $\Gamma K$ direction, Fig.~\ref{fig:FDTD_MQPP}(b), is the high-symmetry direction of the FCC lattice that is most strongly affected by the quadrupole modes. As discussed below, the dipole-only model strongly overestimates the energy of the lowest lying polariton band near the $K$ point, whereas the inclusion of the quadrupole modes results in pretty good agreement with the FDTD results.
The two FDTD simulations shown in Fig.~\ref{fig:FDTD_MQPP} took several hours each, whereas the microscopic quantum mechanical band structure was obtained in seconds. Our model allows a rapid screening of many supercrystal structures, fill factors, nanoparticle shapes, and so forth. Its true strength, however, goes beyond its computational capability: The microscopic model allows an in-depth study of the origin of the plasmon-polariton band structure and its properties as we will show in the following section.
\section{Results and discussion}
\label{sec:results}
We modeled the plasmon-polariton band structure of FCC nanoparticle supercrystals using our microscopic model.
With the simulations we can explain the contribution of the interaction between the nanoparticles and with the electromagnetic modes to the final polariton states. We are able to extract the coupling and mixing of dipole- and quadrupole-derived states in this particular Bravais lattice. Finally, we show how to extract the dipolar, quadrupolar, and photonic contribution to each polariton state. The results impressively reproduce the decoupling of light and matter in the USC and DSC regime.\cite{DeLiberato2014,Mueller2020}
\subsection{Collective plasmon modes}
\begin{figure*}[!htb]
\centering
\includegraphics[width=17.5cm]{Fig_Plasmonic_Bands.pdf}
\caption{Collective plasmon dispersion along the high symmetry directions of an FCC lattice for fill factors (a) $f=0.06$, (b) $0.31$, and (c) $0.90$. The blue (red) lines are induced by the dipole (quadrupole) plasmons of the nanoparticles neglecting dipole-quadrupole interactions. The black lines are a full calculation including dipole and quadrupole modes and their interaction. $\hbar \omega_p=9$ eV, $\epsilon_m=\epsilon_d=1$ and $a=62$~nm.}
\label{fig:PlasmonicBands}
\end{figure*}
We model the optical properties of plasmonic nanoparticle supercrystals in a step-by-step approach. We start with the interaction between dipole and quadrupole nanoparticle states that give rise to collective plasmon modes. This initial plasmonic band structure omits the coupling to electromagnetic states.\cite{Weick2015,Lamowski2018, Mueller2020} Including the photons will later create the supercrystal plasmon polaritons.
In Fig.~\ref{fig:PlasmonicBands} we show the plasmonic bandstructure of an FCC crystal considering both dipole and quadrupole nanoparticle excitations. In each panel we also show the bandstructures for the dipole (blue) and quadrupole (red) modes when turning off the interaction between the dipole and quadrupole states. For the lowest fill factor $f=0.06$ in Fig.~\ref{fig:PlasmonicBands}(a) the dipole and quadrupole states do not cross and are largely decoupled as can be seen by the agreement between the black and the blue/red lines. The lowest plasmonic state at the $\Gamma$ point [$\sim 5.1$ eV in Fig. \ref{fig:PlasmonicBands}(a)] is a two-fold degenerate dipole-induced state.\cite{Lamowski2018} It remains degenerate along the $\Gamma X$ and $\Gamma L$ directions but splits along $\Gamma K$. These two bands are associated with transverse oscillations of the plasmons, {\it i.e.}, the electrons oscillate perpendicular to the propagation direction. The uppermost dipole-induced band [$\sim 5.4$ eV at $\Gamma$ in Fig.~\ref{fig:PlasmonicBands}(a)] is a longitudinal oscillation that does not couple directly with light.
The quadrupole states are constant across the Brillouin zone for $f=0.06$, but become dispersive for the larger fill factors $f = 0.31$, Fig.~\ref{fig:PlasmonicBands}(b), and $0.90$, Fig.~\ref{fig:PlasmonicBands}(c). The quadrupole states consist of five bands that are two- and three-fold degenerate at the $\Gamma$ point [for example, at $5.62\,$ and $5.75\,$eV in Fig.~\ref{fig:PlasmonicBands}(c)]. Along the $\Gamma L$ direction the three-fold degenerate state splits into a two-fold and a non-degenerate band, while the lower branch remains two-fold degenerate. Along the high-symmetry lines the bands split and cross, but overall the quadrupole dispersion is much narrower (0.3 eV for $f=0.9$) than the dispersion of the dipole bands (5 eV for for $f=0.9$). The reason is that the dipole-dipole coupling is much stronger than the quadrupole-quadrupole coupling, with a ratio of $\Lambda_D/\Lambda_Q\sim 13$.
The dispersion of the dipole-derived plasmon band increases rapidly with metal fill fraction, Fig.~\ref{fig:PlasmonicBands}.
For $f>0.1$ the dipole band cross the energy of the quadrupole states. The two types of bands overlap and the dipole-quadrupole interaction strongly affects the plasmonic dispersion. The magnitude of this interaction can be qualitatively evaluated by observing the differences between the black lines (including $DQ$ interaction) and the blue and red lines in Fig.~\ref{fig:PlasmonicBands}. For $f=0.31$ and $0.90$ the differences is very pronounced, especially in the $\Gamma K$ and the $XWL$ directions. This is in contrast with the results for $f=0.06$, Fig.~\ref{fig:PlasmonicBands}(a), where the black and blue/red lines are identical throughout the Brillouin zone.
The dipole-quadrupole mixing depends on the symmetry of the plasmonic bands. For example, along the $\Gamma L$ direction the states remain unchanged by the coupling, indicating that dipole and quadrupole modes cannot couple in this high-symmetry direction. This explains why the dipole approximation worked very well for analysing the optical spectra of gold nanoparticle supercrystals where the light propagated normal to the (111) surface,\cite{Mueller2020} see discussion further below. In contrast, $DQ$ coupling is allowed along the $\Gamma K$ and the $XWL$ high-symmetry lines. The mixing prevents the crossing of the transverse and longitudinal dipole-derived bands at $W$ and close to $K$, see Fig.~\ref{fig:PlasmonicBands}.
Dipole-quadrupole coupling also reduces the splitting of the transverse states. For the $\Gamma K$ direction, two out of the five quadrupole bands are strongly mixed with the dipole modes, while the other three remain practically unchanged. We also note that the lowest transverse dipole bands are less affected by the dipole-quadrupole coupling, because of the larger energy difference between the states.
\subsection{Plasmon polaritons}
\begin{figure*}[!htb]
\centering
\includegraphics[width=17.5cm]{Fig_Plasmon_Polariton.pdf}
\caption{Plasmon-polariton dispersion along the high symmetry directions of an FCC crystal with metal fill factors (a) $f=0.06$, (b) $0.31$, and (c) $0.90$. Black lines are calculated including all terms and interactions, while green dashed lines show the dispersion of the bare plasmons. Yellow dashed lines show the bare photon dispersion. $\hbar \omega_p=9$ eV, $\epsilon_m=\epsilon_d=1$ and $a=62$ nm.}
\label{fig:Plasmon_Polariton}
\end{figure*}
After having examined the bare plasmon bands, we include the coupling to free-space photons and calculate the plasmon-polariton dispersion. The polaritons are coupled electronic and electromagnetic eigenstates of the nanoparticle supercrystals.\cite{LiJensen2006, HuangChengPing2010, Lamowski2018, Mueller2020} In experiments with gold nanoparticles supercrystals these excitations determined the optical response for energies below the interband transitions.\cite{Mueller2020} Figures \ref{fig:Plasmon_Polariton}(a)-(c) show the plasmon-polariton bands obtained by including the plasmon-photon interaction,
considering Umklapp processes up to the sixth Brillouin zone ($n_{BZ}=6$). To allow for a comparison, we also show as dashed lines the bare plasmon (black) and photon (yellow) energies. For the smallest fill fraction ($f=0.06$), light-matter interaction is determined mainly by the dipole excitations. The coupling of the transverse dipole bands and photons gives rise to a pronounced level anticrossing, resulting in plasmon-polariton bands with a dispersion $E_{pp}$ that is very different from the uncoupled states.\cite{LiJensen2006, HuangChengPing2010, Lamowski2018, Mueller2020} We observe two nearly degenerate parabolic bands centered at the $\Gamma$ point, which we will refer to as the upper plasmon-polariton (UPP) and two lower bands, the lower plasmon-polaritons (LPP). The LPPs start off as linear bands with vanishing energy at the $\Gamma$ point. They bend down and become almost flat at the zone edges. The quadrupole bands remain flat over the entire Brillouin zone in Fig.~\ref{fig:Plasmon_Polariton}(a), because the interaction with light is negligible at this metal fill fraction. The longitudinal dipole-derived bands do not couple with light and their polariton dispersion remains unchanged compared to the bare plasmons.
With increasing metal fill factor quadrupole modes mix with the dipole plasmons and the photons resulting in six intermediate plasmon-polaritons (IPPs), Fig.~\ref{fig:Plasmon_Polariton}(b) and (c). This occurs because the coupling between the plasmon and between plasmons and photons increases with metal fill fraction. The topmost IPP band corresponds mainly to the longitudinal dipole-derived band that does not couple directly with light and only weakly with the quadrupole modes. The five other bands are mainly composed of quadrupole-like oscillations which are downshifted by their interaction with the electromagnetic field. This downshift is different for each of the bands, effectively increasing the bandwidth of IPPs. For instance, for $f=0.90$, the energy difference between the lower and upper quadrupole-derived plasmon-polaritons at the $X$ point is $\sim 1.3$ eV. This is more than four times larger than for the plasmon bands and no coupling to photons, width $0.3$\,eV see dashed lines.
The UPP and LPP bands in the $\Gamma L$ direction are derived from the dipole modes without quadrupole mixing, due to the absence of dipole-quadrupole interaction along this high-symmetry direction. This explains why the polariton band structure observed along the $[111]$ direction of gold nanoparticle supercrystals was excellently described by a single band Hopfield model and a microscopic calculations within the dipole approximation\cite{Mueller2020}. The situation is different along the $\Gamma K$ and the $XWL$ directions, for which the dipole-quadrupole coupling is strong, as seen in Fig.~\ref{fig:PlasmonicBands}. Along these directions, the dipole-quadrupole and the light-quadrupole interactions lead to an anti-crossing between the quadrupole and dipole bands, thus effectively pushing down the topmost LPP. This result shows that a complete and accurate description of metallic supercrystals requires that quadrupole modes and Umklapp processes, see Supplementary Information, are included in the model. The strong dependence of the coupling on the direction in the Brillouin zone points towards symmetry-based selection rules for dipole-quadrupole and light-matter coupling, which would be interesting to study for various crystal symmetries.
We now examine the coupling and level anticrossing of the dipole and quadrupole modes and the photons in greater detail. In Fig. \ref{fig:WxF}(a) we show the energies of the bare plasmon and photon bands at $q=0.17\Gamma K$ as a function of the metal fill factor and in Fig. \ref{fig:WxF}(b) the corresponding energies of the plasmon-polariton bands. The colors indicate the magnitude of the contribution of dipole (blue) and quadrupole (red) modes and photons (yellow) to the states according to the color code triangle in (a). Without light-matter coupling, the longitudinal dipole-derived plasmon modes [blue lines in Fig.~\ref{fig:WxF}(a)] upshift almost linearly with filling, while the quadrupole energies remain nearly constant. At $f\sim 0.12$, the energies of the two sets of bands cross. For one of the quadrupole modes, the interaction with the longitudinal dipole band causes an avoided crossing with a gap on the order of $0.01\,$eV, see the enlarged panel in Fig.~\ref{fig:WxF}(c). The other quadrupole and dipole bands are only weakly affected by the $DQ$ interaction.
Without light-matter coupling the photon energy remains at $E=2.6$\,eV independent of filling. When light-matter coupling is "turned on", plasmon polaritons form. The UPP band is mainly composed of transverse dipole-derived plasmon states at $0.17\,\Gamma K$ and for vanishing metal content $f\sim 0$ . With increasing filling the UPPs become more photon-like (yellow color); at the same time, their energy increases in parallel with the longitudinal dipole mode (blue) that does not interact with light. The LPP shifts to smaller energies with increasing $f$ and obtains a strong dipole plasmon contribution.
The spectral range of the anticrossing of the UPPs and the quadrupole bands is shown at higher magnification in Fig.~\ref{fig:WxF}(d). Along the $\Gamma K$ direction, dipole plasmons, quadrupole plasmons, and photon all mix into polariton states. As the polarization dependence of the light-quadrupole and light-dipole interactions are different, we expect cross-polarized absorption and chiral activity, which will be studied in a future work. As the fill factor increases further, the UPP bands become increasingly photon-like and their interaction with the quadrupole modes causes the latter to downshift in energy, thus increasing the overall bandwidth of the quadrupole modes.
\begin{figure}[!htb]
\centering
\includegraphics[width=12cm]{Fig_weightxF.pdf}
\caption{Energies of (a) the bare plasmon and photon states and of (b) the plasmon-polariton states for $q=0.17\Gamma K$. The coloring indicates the dipole, quadrupole, and photon contribution to each state - see inset in (a). (c) and (d) show zoomed images of the rectangular areas in (a) and (b), respectively.}
\label{fig:WxF}
\end{figure}
\subsection{Normalized coupling strength}
The normalized coupling strength $\eta=\Omega_R/\omega_0$ compares the Rabi frequency $\Omega_R$ to the bare frequency $\omega_0$ of the system. $\Omega_R$ can be found from the minimum energy splitting between the LPP and UPP bands (divided by two) and $\omega_0$ from a calculation of the bare plasmon dispersion.\cite{Baranov2020,Kockum2019,FornDiaz2019} In this section we derive a close expression for $\eta$ as a function of our model parameters. It will facilitate chosing a nanoparticle supercrystal for a desired coupling. We will show that a wide range of USC and DSC light-matter interaction can be realized in plasmonic supercrystals.
The bare plasmon energies without coupling to the electromagnetic states depends on the plasmon coupling factors $\Lambda_{\bar \nu}$ ($\bar\nu=D,Q$) defined in Eqs. (\ref{eq_lambd}) and (\ref{eq_lambq}). The energies are well described by\cite{Mueller2020}
\begin{equation}\label{eq_omegapl}
E_{pl,\nu}(\vec q)=\hbar\omega_{\bar\nu}\sqrt{1+ s_{\nu,f}(q)\Lambda_{\bar \nu} },
\end{equation}
where $s_{\nu,f}(q)$ incorporates the effects of the lattice; it depends only weakly on $f$. $s_{\nu,f}(q)$ measures the enhancement of the effective plasmon-plasmon coupling due to the crystalline structure. This value can be calculated numerically for each of the plasmon bands at any given $q$.\cite{Coropceanu2019} However, as we are interested in an effective expression, we use a different approach and fit the energy of the lowest lying $D$ and $Q$ bands at the $\Gamma$ and $X$ points, respectively. These are the wavevectors of the largest bandwidths of the dipole- and quadrupole-derived bands, which gives us an overall measure for plasmon-plasmon coupling. Figure~\ref{fig:etaCalc}(a) shows the energies of the upper and lower dipole and quadrupole bands at the $\Gamma$ and $X$ point. These are fitted to Eq.(\ref{eq_omegapl}) and the fitting parameters are shown in Table \ref{tab:etalamb}.
\begin{table}[]
\centering
\begin{tabular}{cccc}\hline\hline
& label & $q$ & value \\\hline
upper dipole & $s_D^u$ & $\Gamma$ & 16.5 \\
lower dipole & $s_D^l$ & $\Gamma$ & -8.3 \\
upper quadrupole & $s_Q^u$ & $X$ & 14 \\
lower quadrupole & $s_Q^l$ & $X$ & -7\\\hline\hline
\end{tabular}
\caption{Fit parameters for the upper and lower dipole and quadrupolar states at $q=\Gamma$ and $X$, respectively. Calculated results were fitted to Eq.(\ref{eq_eta}) with one free parameter.}
\label{tab:etalamb}
\end{table}
\begin{figure}[!htb]
\centering
\includegraphics[width=8cm]{Fig_LambdaCalc.pdf}
\caption{(a) Energies of the upper and lower dipole (blue symbols) and quadrupole (red symbols) plasmon bands for different values of $f$ of an FCC crystal. The dipole (quadrupole) energies are evaluated at the $\Gamma$ ($X$) point. Red and blue lines are fits to the data points, using Eq.~\ref{eq_omegapl}, see Table\ \ref{tab:etalamb}. (b) Normalized coupling strength $\eta_t$ of the transverse dipole-derived plasmon band for the $\Gamma L$ (squares), $\Gamma X$ (triangles) and $\Gamma K$ (dots) high-symmetry directions. Blue and red lines are coupling strengths predicted with Eq. (\ref{eq_eta}) for the dipole and quadrupole modes, respectively.}
\label{fig:etaCalc}
\end{figure}
We now use the Rabi frequencies and dipole and quadrupole coupling strengths in Eqs.\ \eqref{Eq_OmD}-\eqref{Eq_OmQ} and combine it with the plasmon energy in Eq.~(\ref{eq_eta}). We find a compact expression for
the maximum reduced coupling
\begin{equation}\label{eq_eta}
\eta_\nu=\sqrt{\frac{2\pi\Lambda_{\bar \nu}}{(1+\bar s^l_\nu\Lambda_{\bar \nu})}},
\end{equation}
where $s^l_\nu$ are the fitting parameters for the lowest energy dipole and quadrupole bands at the chosen points, see Table \ref{tab:etalamb}. For a single plasmonic state $\eta$ can be found from Eq.~\eqref{eq_eta}. A more general expression is necessary for mixed dipole and quadrupole bands. As we are mainly interested in the maximum normalized coupling strength, we will focus on the coupling to the lowest energy dipole- and quadrupole-derived bands.
The solid lines in Fig.~\ref{fig:etaCalc}(b) show the dependence of the reduced dipole and quadrupole coupling in Eq.~\eqref{eq_eta} on the fill fraction. The symbols are the reduced coupling strengths of the transverse dipole-derived bands $\eta_t=\Delta E_\mathrm{UL}/(2E^t_{pl,D})$ from a microscopic quantum calculation evaluated at the crossing $|q_0|\sim E^t_{pl,D}/\hbar c$ of the bare dipole plasmon and the photon dispersion along $\Gamma L$, $\Gamma X$ and $\Gamma K$. $E^t_{pl,D}$ is the energy of the transverse dipole-induced plasmon and $\Delta E_{UL}\sim E_\mathrm{UPP}(q_0)-E_\mathrm{LPP}(q_0)$ is the energy difference between the upper and lower plasmon polariton branches. The normalized coupling strength obtained for the different directions correspond well with the value obtained by the expression for the dipole coupling strentgh $\eta\sim \eta_D$. The fact that this is true even for the $\Gamma K$ direction indicates that the quadrupole contribution the coupling strength of the transverse dipole is negligible. Furthermore, it shows that Eq.\ \eqref{eq_eta}, with the parameters in Table\ \ref{tab:etalamb}, can be used to estimate the maximum coupling strength in FCC supercrystals. With this, for a metal fill fraction of 3\% the maximum coupling strength is on the order of $\eta=0.13$, and therefore already in the USC regime ($\eta>0.1$), while the DSC regime is reached for fill fractions above 80\%, Fig.~\ref{fig:etaCalc}(b).
\subsection{Decoupling of light and matter the USC and DSC regimes}\label{sec:insights}
\begin{figure*}[!htb]
\centering
\includegraphics[width=17cm]{Fig_Decomp.pdf}
\caption{Decomposition of the plasmon-polariton bands into their dipole (blue), quadupole (red) and photon (yellow) contributions for fill factors (a)$f=$ 0.06,(b) 0.31, and (c) 0.90. The size of the data points shows the magnitude of the relative contributions of each of these bare excitations to the plasmon-polariton bands.}
\label{fig:decomp}
\end{figure*}
A fascinating signature of ultrastrong and deep strong coupling is the decoupling of light and matter in space and in frequency that leads to a breakdown of the Purcell effect.\cite{DeLiberato2014} The Purcell effect describes the increase in radiative damping with increasing light-matter coupling.\cite{Purcell1946} In the weak and strong coupling regimes ($\eta\ll 1$) the radiative damping scales with $\eta^2$. De Liberato \cite{DeLiberato2014} predicted that the Purcell effect saturates around $\eta\sim 0.5$ and radiative damping decreases for higher values of $\eta$. Mueller {\it et al.} \cite{Mueller2020} demonstrated the breakdown of the Purcell effect in plasmonic supercrystals for $\eta > 1$. Although this breakdown can be described classically,\cite{DeLiberato2014} the microscopic quantum description of the plasmon-polaritons allows us to directly observe the decoupling with increasing fill fraction, {\it i.e.}, increasing light-matter coupling.
We use the transformation matrix $T_{\vec q}$ introduced in Sect.~\ref{sec:NPcrystals} to decompose the plasmon-polariton states into the bare dipole and quadrupole plasmonic and the photonic components. In Fig.~\ref{fig:decomp} we show the plasmon-polariton states decomposed into dipole plasmons, quadrupole plasmons, and photons for three fill factors. For small filling $f=0.06$ the plasmonic characters are mainly localized in the energy regions of the dipole and quadrupole plasmon bands. The band with linear dispersion are photonic in character. There is no mixing between the dipole and quadrupole states as expected from our discussion of the bare plasmon dispersion. The dipole plasmon and the photon mix slightly at their crossing, so that the linear bands show a small dipole-plasmon character while the flat band at about $5.19\,$eV shows a weak photonic character.
With increasing $f$ the mixing between the three states becomes more apparent. For $f=0.31$ the dipole plasmonic character of the linearly dispersive bands is very pronounced. We also see a non-negligible mixing between the dipole and quadrupole states as well as the quadrupole modes and photons. Peaks related to quadrupole modes should start to appear in the absorption spectra and affect the overall dispersion of the polaritons. The top plasmon-polariton bands acquires a finite mass, because it is composed of photons, dipole plasmons and even a small contribution from quadrupole modes. Finally, the distribution of the plasmon and photon states becomes asymmetric for lower and upper polaritons: While the low-energy states have a stronger plasmonic component, the upper polaritons are photonic in character.
For the high metal fill fraction $f = 0.9$, the linear bands starting at zero energy are almost entirely composed of dipole states, having only weak photonic and quadrupole plasmonic character. The weakly dispersive bands in the gap between UPP and LPP remain strongly plasmonic in nature and are predominantly composed of quadrupole modes, but the states became mixed with photons and with dipole plasmons. This indicates that they should be accessible optically. The UPP branches developed into a pair of massive bands mainly composed of photons. Overall, it is striking that there is little overlap between the three different types of quasi-particles, which is a manifestation of the light-matter decoupling in the DSC regime.\cite{DeLiberato2014,Mueller2020} Indeed, the distribution of states for highest filling resembles the low-filling case: Each component - dipole, quadrupole, and photon - is concentrated in a portion of the polariton dispersion with little contribution to the other states.
The quantum model proposed here gives insight into the nature of the plasmon-polariton states in addition to its excellent description of the plasmon-polariton band structure. In future, it may be applied to calculate quantum related properties such as the squeezing of plasmons and photons, the population of the supercrystal ground state with photons and plasmons, and correlation functions of the bare excitations. \cite{Artoni1991,Ciuti2005, Cirio2016,Kockum2019,FornDiaz2019}
\section{Conclusions}
\label{sec:conclusions}
In conclusion, we proposed a microscopic model to calculate plasmon polaritons in nanoparticle supercrystals. Our model includes the dipole and quadrupole modes of the nanoparticle building blocks and their coupling to photons. We show that the mixing of the dipole and quadrupole-derived states is important for calculating the collective plasmon and plasmon-polariton sates. The microscopic quantum model leads to a closed expression for the reduced light-matter coupling strength of the dipole and quadrupole modes. The dipole derived states of FCC nanoparticle supercrystals are in the ultrastrong coupling regime for all realistic fill fractions and enter deep strong coupling for a fill fraction of $0.8$ (assuming vacuum between the nanoparticles in the crystal).
The quantum based calculations give insight into the unique properties of strongly coupled systems as we show for the example light-matter decoupling in the DSC regime. The model can be applied for different lattice structures including lattices with more than one particle in the basis. It will contribute to the study and optimization of the many supercrystal structures currently being developed.
\section*{Acknowledgements}
\label{sec:acknowledgements}
We thank J. Weick for useful discussions. E.B.B acknowledges financial support from CNPq, CAPES (finance code 001) and FUNCAP (PRONEX PR2-0101-00006.01.00/15). S.R. and N.S.M. acknowledge support by the European Research Council ERC under grant DarkSERS (grant number 772108). This work was supported by the Focus Area NanoScale of Freie Universit\"at Berlin.
\newpage
\bibliographystyle{unsrt}
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2,869,038,156,367 | arxiv |
\subsection{RSA Retrieve-then-predict Process\label{section: 4.1}}
\input{table/03_main_result.tex}
\subsection{Model Architectures\label{section: 4.1}}
The RSA model comprises of a neural sequence retriever $p(r|x)$, and a protein model that combines both original input and retrieved sequence to obtain prediction $p(y|x, r)$.
\subsubsection{RSA Retriever}
The retriever is defined as finding the sequences that are semantically close to the query. Denote retriever model as $G$ which encode protein sequence and output embeddings.
\begin{equation}
\begin{aligned}
p(r|x) &= \frac{\exp f(x,r)}{\sum_{r'\in \mathcal{R}} \exp f(x,r')}, \\ f(x,r) &= -||G(x)-G(r)||_2
\end{aligned}
\label{pzx}
\end{equation}
The similarity score $f(x,r)$ is defined as the negative L2 distance between the embedding of the two sequences. The distribution is the softmax distribution over similarity scores.
For protein retrieval, we aim to retrieve protein sequences that have similar structures or are homologous to the query sequence. Motivated by the k-nearest neighbor retrieval experiment with ESM-1b \citep{esm1b} pre-trained embeddings (as shown in Table \ref{table: retrieve experiment} and
Figure \ref{retrieval plot}), we implement the embedding functions using a 34-layer ESM-1b encoder. We obtain sequence embeddings by performing average pooling over token embeddings. Note that finding the most similar proteins from a large-scale sequence database is computationally heavy. To accelerate retrieval, we use Faiss indexing \citep{faiss}, which uses clustering of dense vectors and quantization to allow efficient similarity search at a massive scale.
\subsubsection{RSA Encoder}
\textbf{Retrieval Augmented Protein Encoder}
Given a sequence $x$ and a retrieved sequence $r$ with length $L$ and $M$ respectively, the protein encoder combines $x$ and $r$ for prediction $p(y|x, r)$. To make our model applicable to any protein learning task, we need to augment both sequence-level representation and token-level representation. To achieve this, we concatenate the two sequences before input into the transformer encoder, which uses self-attention to aggregate global information from the retrieved sequence $r$ into each token representation.
\begin{align*}
& {A} = \sigma(\frac{(H_{[x;r]}W^Q)(H_{[x;r]}W^K)^T} {\sqrt{d}}), A = [A_{x}; A_{r}] \\
& Attn(H_{[x;r]}) = (A_xH_{x}W^V + A_rH_{r}W^V)W^O
\end{align*}
where $H_{[x;r]} = [h_1^x, h_2^x, ..., h_L^x, h_1^r... h_M^r]^T$ denotes the input embedding of original and retrieved sequences. The output token representation $h_i$ automatically learns to select and combine the representation of retrieved tokens. This can also be considered a soft version of MSA alignment. After computing for each pair of $(x,r)$, we aggregate them by weight $p(r|x)$ defined in Eq. \ref{pzx}.
\subsection{RSA Training\label{section 4.2}}
\textbf{Training}
For downstream finetuning, we maximize $p(y|x)$ by performing training on the retrieval augmented protein encoder. We freeze the retriever parameters during training. For a query sequence with $N$ retrieved proteins, the computation cost is $N$ times the original model, $O(NL^2)$ for a transformer encoder layer, which is more efficient than the MSA Transformer with a $O(NL^2) + O(N^2L)$ computation cost. Also, the retrieval is performed on the fly.
\subsection{The Unified Framework \label{section unified}}
Taking one step further, we define a set of design dimensions to characterize the retrieving and aggregation processes. We detail the design dimensions below and illustrate how popular models (Appendix \ref{augmentation methods}) and our proposed methods (\S\ref{sec:rsa}) fall along them in Table \ref{table: pipeline}. These design choices includes:
\begin{itemize}
\item \textbf{Retriever Form} indicates the retriever type used. Multiple Sequence Alignment is a discrete retrieval method that uses E-value thresholds~\citep{ye2006blast} to find homologous sequences. Dense retrieval~\citep{johnson2019billion} has been introduced to accelerate discrete sequence retrieval. The method represents the database with dense vectors and retrieves the sequences that have top-$k$ vector similarity with the query.
\item \textbf{Alignment Form} indicates whether retrieved sequences are aligned, as illustrated in Appendix Figure \ref{fig: illustrated msa}.
\item \textbf{Weight Form} is the aggregation weight of homologous sequences, as the $p(r_n|x)$ in Eq. \ref{eq: probabilistic look}. Here we denote this weight as $\lambda_n$. Traditionally, aggregation methods consider the similarity of different homologous sequences to be the same and use average weighting. MSA Transformer also use a weighted pooling method though the weights of $\lambda_n$ use global attention and are dependent on all homologous sequences.
\item \textbf{Aggregation Function} is how the representations of homologous sequences are aggregated to the original sequence to form downstream prediction, as in $p(y|x,r)$. For example, considering the sequence classification problem, a fully connected layer maps representations to logits. MSA Transformer first aggregates the representations $R_n$ and then maps the aggregated representation to logits $y$, and the retrieval augmentation probabilistic form first maps each representation to logits $p(y|x,r_n)$ and then linearly weight the logits with $\lambda_n$ in Eq. \ref{eq: probabilistic look}.
\end{itemize}
Our discussion and formulation so far reach the conclusion that MSA augmentation methods intrinsically use the retrieval augmentation approach. This highlights the potential of RSA to replace MSA Augmentations as a computationally effective and more flexible method.
However, MSA-based methods claim a few advantages: the \textit{alignment} process can help the model capture column-wise residue evolution; and the \textit{MSA Retriever} uses a discrete, token-wise search criterion that ensures all retrieved sequences are homology. We propose two novel variants to help verify these claims.
\paragraph{Unaligned MSA Augmentation.} \label{accelerated msa} MSA modeling traditionally depends on the structured alignment between sequences to learn evolutionary information. However, deep models have the potential to learn patterns from unaligned sequences. \citet{riesselman2019accelerating} shows that the mutation effect can be learned from unaligned sequences using autoregressive models. Therefore, we first introduce this variant that uses the homologous sequences from MSA to augment representations without alignment.
\paragraph{Accelerated MSA Transformer.} This variant explores substituting the discrete retrieval process in MSA with a dense retriever. We use the K-nearest neighbor search to find the homologous sequences. We still align the sequences before input into MSA Transformer. We introduce this variant to find if MSA builder has an advantage over our pre-trained dense retriever in finding related sequences.
An empirical study of the performance of these models can be found in Subsection \ref{section: msa alignment experiments}.
\subsection{General Setup} \label{General Setup}
\textbf{Downstream tasks} In order to evaluate the performance of our trained model, six datasets are introduced, namely secondary structure prediction, contact prediction, remote homology prediction, subcellular localization prediction, stability prediction, and protein-protein interaction. Please refer to Appendix Table \ref{Table: table task} for more statistics of the datasets. The train-eval-test splits follow TAPE benchmark \citep{rao2019evaluating} for the first four tasks and PEER benchmark \citep{xu2022peer} for subcellular localization and protein-protein interaction. The introduction to datasets is in Appendix \ref{Introduction to the datasets}.
\textbf{Retriever and MSA Setup} Limited by available computation resources, we build a database on Pfam~\citep{pfam-gebali2018} sequences, which covers 77.2\% of the UniProtKB~\citep{apweiler2004uniprot} database and reaches the evolutionary scale. We generate ESM-1b pre-trained representations of 44 million sequences from Pfam-A and use Faiss~\citep{johnson2019billion}
to build the retrieval index. For a fair comparison, the MSA datasets are also built on the Pfam database. We use HHblits~\citep{remmert2012hhblits} to extract MSA. The details are shown in Appendix \ref{Introduction to the retrievers}.
\textbf{Baselines} We apply our retrieval method to both pre-trained and randomly initialized language models. Following \citet{rao2019evaluating} and \citet{rao2021msa}, we compare our model with vanilla protein representation models, including LSTM\citep{liu2017deep}, Transformers\citep{vaswani2017attention} and pre-trained models ESM-1b\citep{esm1b}, ProtBERT\citep{elnaggar2020prottrans}. We also compare with state-of-the-art knowledge-augmentation models: Potts Model\citep{balakrishnan2011learning}, MSA Transformer\citep{rao2021msa} that inject evolutionary knowledge through MSA, OntoProtein\citep{zhang2022ontoprotein} that uses gene ontology knowledge graph to augment protein representations and PMLM\citep{he2021pre} that uses pair-wise pretraining to improve co-evolution awareness. We use the reported results of LSTM from \citet{zhang2021co, xu2022peer}.
\textbf{Training and Evaluation}
Our RSA model is applicable to any global-aware encoders. To demonstrate RSA as a general method, we perform experiments both with a shallow transformer encoder, and a large pre-trained ProtBERT encoder. The Transformer model has 512 dimensions and 6 layers. All self-reported models use the same truncation strategy and perform parameter searches on the learning rate, warm-up rate, seed, and batch size. For evaluation, we choose the best-performing model on the validation set and perform prediction on the test set.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\linewidth]{img/denovo_compare_contact.png}
\caption{Contact Prediction of RSA and MSA Transformer on De Novo Proteins. We plot samples that RSA have better predictions under the diagonal line.}
\label{fig: de novo contact}
\vspace{-3mm}
\end{figure}
\subsection{Main Results}
\label{Main Results}
We show the result for downstream tasks in Table
\ref{main result}, including models with/without pretraining, and with/without knowledge augmentations. We form the following conclusion: \textbf{Retrieval Sequence Augmentations perform on par with or even better than other knowledge-augmented methods without additional pre-training.} The last two blocks compare our method with previous augmentation methods. Our method outperforms MSA Transformer on average by 5\% and performs on par with PMLM on structure and evolution prediction tasks. Notably, both MSA Transformer and PMLM perform additional pre-training with augmentations, while our method uses no additional pre-training. From the results, we can see that RSA combined transformer model also improves by 10\% than other shallow models, demonstrating the effectiveness of our augmentation to both shallow models and pre-trained models.
\input{table/08_domain_adpation.tex}
\subsection{Retrieval Augmentation for Domain Adaptation}
{We investigate the model's transfer performance in domains with distribution shifts. We train our model on the Remote Homology dataset, and test it on three testsets with increasing domain gaps: proteins that are within the same Family, Superfam, and Fold as the training set respectively. The results are in Table \ref{domain shift}. It is pertinent to note that MSA transformer's performance decreases dramatically when the gap between the domains increases. Our model surpasses MSA Transformer by a large margin on shifted domains, especially from 0.5032 to 0.6770 on Superfam. Our model proves to be more reliable for domain shifts, illustrating that retrieval facilitates the transfer across domains. }
Furthermore, we test our model on 108 out-of-domain De Novo proteins for the contact prediction task. De Novo proteins are synthesized by humans and have a different distribution from natural proteins. It can be seen in Figure \ref{fig: de novo contact} that, in addition to surpassing MSA transformer on average precision by 1\%, RSA also exceeds MSA transformer on 63.8\% of data, demonstrating that RSA is more capable of locating augmentations for out-of-distribution proteins.
We also test our model on the secondary structure task with new domain data, as shown in Appendix (Table~\ref{casp12} and Figure~\ref{fig:denovo ssp}). The results also show that our model surpasses MSA Transformer in transferring to unseen domains.
\input{table/04_alignment.tex}
\subsection{Retrieval Speed}
A severe speed bottleneck limits the use of previous MSA-based methods. In this part, we compare the computation time of RSA with MSA and an accelerated version of MSA as introduced in Section ~\ref{accelerated msa}. As shown in Figure \ref{fig: speed compare}, alignment time cost is much more intense than retrieval time. Even after reducing the alignment database size to 500, accelerated MSA still need 270 min to build MSA. At the same time RSA only uses dense retrieval, and is accelerated 373 times. Note that with extensive search, MSA can find \textit{all} available alignments in a database. However, this would be less beneficial to deep protein language models as the memory limit only suffices a few dozens of retrieved sequences.
\subsection{Retrieved Protein Interpretability}
\label{Retrieved Protein Interpretability}
The previous retrieval-augmented language models rely on a dense retriever to retrieve knowledge-relevant documents. However, it remains indistinct what constitutes knowledge for protein understanding and how retrieved sequences can be used for improving protein representations. In this section, we take a close look at the retrieved protein sequences to examine their homology and geometric properties.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\linewidth]{img/retrieval-plot.png}
\caption{Plot of the -log(E-values) of MSA and Dense Retriever obtained sequences on the test sets for six tasks. E-values of both methods are obtained with HHblits\citep{remmert2012hhblits}. Sequences with -log E-value \textgreater 10 are high-quality homologous sequences. We also show with bar plots the percentage of sequences in the test sets that have homologous sequences.}
\label{retrieval plot}
\vspace{-5mm}
\end{figure}
\textbf{Dense Retrievers Find Homologous Sequences.} One type of knowledge distinct to the protein domain is sequence homology, which infers knowledge on shared ancestry between proteins in evolution. Homologous sequences are more likely to share functions or similar structures. We analyze whether retrieved sequences are homologous.
{As illustrated in Figure \ref{retrieval plot} (right axis), across all six datasets, our dense retriever retrieved a high percentage of homologous proteins that can be aligned to the original protein sequence, comparable to traditional HMM-based MSA retrievers. We additionally plot each dataset's negative log E-values distribution in Figure \ref {retrieval plot}. Accordingly, pre-trained protein models can be used directly as dense retrieval of homologous sequences.}
\input{table/05_variants.tex}
\textbf{RSA Retriever Find Structurally Similar Protein}
Protein structures are also central to protein functions and properties. In this section, we analyze whether retrieved sequences are structurally similar. In Figure \ref{tm plot}, we plot the TM scores between the RSA retrieved protein and the origin protein on ProteinNet \citep{alquraishi2019proteinnet} test set. Using ESMFold\footnote{https://esmatlas.com/resources?action=fold}, we obtain the 3D structures of the top 5 retrieved proteins and then calculate the TM score between these proteins and the query protein. Most of the retrieved proteins exceed the 0.2 criteria, which indicates structural similarity, and about half are above the 0.5 criteria, which indicates high quality. Accordingly, this indicates that the dense retrieval algorithm is capable of finding proteins with structural knowledge.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{img/geometric.png}
\caption{Plot of the cumulative distribution of TM-scores for proteins from dense retrieval. The value at $a$ shows the probability that TM-score is larger than $a$. We also give a visual example of retrieved protein to illustrate similar structures.}
\label{tm plot}
\vspace{-3mm}
\end{figure}
\subsection{Ablation Study}
\label{Ablation Study}
\textbf{Ablation on Retriever: Unaligned MSA Augmentation. }
We ablate RSA retriever by using MSA retrieved proteins as augmentations to our model, denoted as Unaligned MSA Augmentation. The results are in Table \ref{Unaligned MSA Augmentation}. As the result shows, Unaligned MSA Augmentation performs worse than our RSA model, especially on the Stability dataset, where the performance drops from 0.778 to 0.7443. It thus confirms the ability of our dense retriever to provide more abundant knowledge for protein models.
{\textbf{Ablation on Retriever: Ablation on Retrieval Number }
Our study examines the effect of injected knowledge quantity for RSA and all retrieval baselines. The results are listed in Table \ref{augmentation number}. We select the Contact dataset because all baseline models are implemented on this dataset. RSA and all baselines perform consistently better as the retrieval number increases. Also, our model outperforms all baseline models for all augmentation numbers.}
\input{table/06_depth.tex}
\textbf{Ablation on aggregation: }We compare RSA with Accelerated MSA Transformer to evaluate whether our aggregation method is beneficial for learning protein representations. Note that only part of the retrieved sequences that satisfy homologous sequence criteria are selected and utilized during alignment. As shown in Table \ref{accelerated MSA}, the performance of the Accelerated MSA Transformer drops a lot compared to RSA. In contrast to MSA type aggregation, which is restricted by token alignment, our aggregation is more flexible and can accommodate proteins with variant knowledge.
\textbf{Is MSA retriever necessary? }Table \ref{accelerated MSA} illustrates that Accelerated MSA Transformer performs near to MSA Transformer (MSA N=16) for most datasets, except for Stability and PPI on which our retriever failed to find enough homologous sequences, as Figure \ref{retrieval plot} demonstrates. Our retriever is therefore capable of finding homologous sequences for most tasks and is able to replace the MSA retriever.
\textbf{Is MSA alignment necessary? }\label{section: msa alignment experiments}To support that MSA alignment is not necessary, we compare Unaligned MSA Augmentation to the original MSA transformer. As revealed by the results in Table \ref{Unaligned MSA Augmentation}. Unaligned MSA Augmentation performs close to the MSA transformer. This confirms our declaration that self-attention is capable of integrating protein sequences into representations.
\section{A Brief Recap on Proteins}
Proteins are the end products of the decoding process that starts with the information in cellular DNA. As workhorses of the cell, proteins compose structural and motor elements in the cell, and they serve as the catalysts for virtually every biochemical reaction that occurs in living things. This incredible array of functions derives from a startlingly simple code that specifies a hugely diverse set of structures.
In fact, each gene in cellular DNA contains the code for a unique protein structure. Not only are these proteins assembled with different amino acid sequences, but they also are held together by different bonds and folded into a variety of three-dimensional structures. The folded shape, or conformation, depends directly on the linear amino acid sequence of the protein.
\textbf{1. What are proteins made of?
}
20 kinds of amino acids. Within a protein, multiple amino acids are linked together by peptide bonds, thereby forming a long chain.
\textbf{2. Protein structures
}
There are four levels of structures:
\begin{itemize}
\item Primary structure: amino acids sequence
\item Secondary structure: stable folding patterns, including Alpha Helix, Beta Sheet.
\item Tertiary structure: ensemble of formations and folds in a single linear chain of amino acids
\item macromolecules with multiple polypeptide chains or subunits
\end{itemize}
\textbf{3. Protein Homology}
Protein homology is defined as shared ancestry in the evolutionary history of life. There exists different kinds of homology, including orthologous homology that may be similar function proteins across species (human and mice $\alpha$-goblin), and paralogous homology that is the result of mutations (human $\alpha$-goblin and $\beta$-goblin). Homologies result in conservative parts in protein sequences, or leads to similar structures and functions.
\textbf{4. Multiple Sequence Alignments}
A method used to determine conservative regions and find homologous sequences. An illustration is given here to show how sequences are aligned.
\section{Overview of Previous Protein Representation Augmentation Methods \label{augmentation methods}}
Below we introduce several state-of-the-art evolution augmentation methods for protein representation learning. These methods rely on MSA as input to extract representations. We use $x$ to denote a target protein and its MSA containing $N$ homologous proteins.
\textbf{Potts Model~\citep{balakrishnan2011learning}}. This line of research fits a Markov Random Field to the underlying MSA with likelihood maximization. This approach is different from other protein representation learning methods as it only learns a pairwise score for residues contact prediction. We will focus on other methods that augment protein representations that can be used for diverse downstream predictions.
\textbf{Co-evolution Aggregator~\citep{yang2020improved, ju2021copulanet}}. One way to build an evolution informed representation is to use a MSA encoder to obtain the co-evolution related statistics. By applying MSA encoder on the $n$-th homologous protein in the MSA, we can get a total of $L\times d$ embeddings $R_n$, each position is a $d$ channel {one-hot} embedding indicating the amino acid type. We use $w_n$ to denote the weight from $R_n$ when computing the token representation $h_i$:
\begin{align}
h_i = \frac{1}{M_{\textit{eff}}} \sum_{n=1}^N w_n R_n(i),
\end{align}
where $M_{\textit{eff}} = \sum_{n=1}^N w_n$ and $w_n = \frac{1}{N}$. For contact prediction, pair co-evolution representation are computed in a similar way from the hadamard product:
\begin{align}
h_{ij} = \frac{1}{M_{\textit{eff}}} \sum_{n=1}^N w_n R_n(i) \bigotimes R_n(j).
\end{align}
\textbf{Ensembling Over MSA~\citep{rao2020transformer}}. This approach aligns and ensembles representations of homologous sequences. Consider the encoder extract the same token representations for unaligned and aligned sequences.
The ensembled token representation is:
\begin{align}
h_i = \frac{1}{N}\sum_{n=1}^N R_n(i), h_{ij}= \frac{1}{N} \sum_{n=1}^N \sigma(\frac{R_n(i)W_Q(R_n(j)W_K)^T}{N\sqrt{d}}).
\end{align}
\textbf{MSA Transformer~\citep{rao2021msa}} In each transformer layer, a tied row attention encoder extracts the dense representation $R_n$, then a column attention encoder
\begin{align}
\label{eq: msa transformer}
R_s(i) = \sum_{n=1}^N \sigma(\frac{R_s(i)W_Q(R_n(i)W_K)^T}{N\sqrt{d}})R_n(i)W_V.
\end{align}
\section{Experiment Setups}
\subsection {Introduction to the datasets}
\label{Introduction to the datasets}
\textit{Secondary structure prediction (SSP, 8-class)} aims to predict the secondary structure of proteins, which indicates the local structures. \textit{Contact prediction} predicts the long-range (distance \textgreater 6) residue-residue contact, which measures the ability of models to capture global tertiary structures. Homology prediction aims to predict the fold label of any given protein, which indicates the evolutionary relationship of proteins. \textit{Stability} prediction is a protein engineering task, which measures the change in stability w.r.t.
residue mutations. \textit{Subcellular Localization (Loc)} prediction predicts the local environment of proteins in the cell, which is closely related to protein functions and roles in biological processes. \textit{Protein protein interaction (PPI)} predicts whether two proteins interact with each other, which is crucial for protein function understanding and drug discovery.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\linewidth]{img/illustrate_msa.png}
\caption{Illustrated difference of aligned and unaligned homologous sequences. }
\label{fig: illustrated msa}
\end{figure}
\subsection{Retriever and MSA Details \label{Introduction to the retrievers}}
We adopt Faiss~\citep{johnson2019billion} indexing to accelerate the retrieval process by clustering the pre-trained dense vectors. In our implementation, we use the Inverted file with Product Quantizer encoding Indexing and set the size of quantized vectors to 64, the number of centroids to 4096, and the number of probes to 8. During retrieval, L2 distances are used to measure sequence similarity. The index is first trained on $.5\%$ of all retrieval data and then add all vectors. For MSA datasets, We use HHblits~\citep{remmert2012hhblits} to perform alignment, and the iteration and E-value thresholds of HHblits are set as $3$ and $1$.
\section{Supplementary Experiment Analysis}
\subsection{Baselines}
\textbf{Protein representation learning benefits from knowledge augmentations.}
In this part, we examine the performance of three types of baseline models. As shown in Table \ref{main result}, structure and evolution-related tasks all benefit greatly from pre-training, with over 20\% improvement in contact prediction and over 40\% improvement in homology prediction. Also, we observe that all kinds of knowledge-augmentation methods improve performance on a few downstream tasks. Though based purely on MSA information, Potts model shows competitive performance to vanilla pre-trained models. MSA Transformer with depth=16 MSA input also sees 12\% improvement on its no-MSA input performance. OntoProtein also improves on homology prediction and stability prediction, since knowledge graph enhancement is more suitable to function prediction than structure understanding. PMLM is the SOTA model on both structure and evolution-related tasks through co-evolution pre-training on Pfam database. This trend shows that current scale ( \textless 1 Billion parameters) pre-trained models still need knowledge augmentations to reach SOTA, and evolutionary knowledge is especially important for downstream prediction.
\subsection{Domain Adaptation Analysis}
In this section, we perform additional analysis on secondary structure prediction tasks. We perform training on NetSurfP-2.0\citep{klausen2019netsurfp} training set and test on two datasets with domain gaps. On CASP12, RSA marginally outperforms other baselines, as shown in Table 8. We also test on 10 de novo proteins (6YWC, 2LUF, 7BPM, 7BPL, 7CBC, 1FSD, 1IC9, 5JI4, 5KWO, 6W6X). Since we didn't find secondary structure labels for these proteins, we provide visualization in Figure~\ref{fig:denovo ssp} which shows that our model has an obvious overhead over MSA Transformer on predicting geometric components.
\begin{table}[htbp]
\caption{The domain adaptation performance of models on CASP12 secondary structure prediction.}
\vspace{2mm}
\centering
\begin{tabular}{lccc}
\toprule
Method & CASP12 \\ \midrule
ProtBERT & 0.628 \\
MSA Transformer & 0.621 \\
Accelerated MSA Transformer & 0.620 \\
RSA (ProtBERT backbone) & \textbf{0.631} \\
\bottomrule
\end{tabular}
\label{casp12}
\end{table}
\begin{figure}
\centering
\resizebox{\linewidth}{!}{\includegraphics{img/denovo.png}}
\caption{Prediction of Secondary Structure on De Novo Dataset. Each color corresponds to a different secondary structure.}
\label{fig:denovo ssp}
\end{figure}
\section{Dataset details}
\subsection{Downstream tasks}
Table~\ref{Table: table task} gives the details for the datasets.
\input{table/02_tasks.tex}
\subsection{De Novo Protein Dataset}
We follow \citet{chowdhury2022single} to curate a de novo dataset of 108 proteins from Protein Data Bank~\citep{bankrcsb}. These proteins are originally designed de novo using computationally parametrized energy functions and are well-suited for out-of-domain tests. Note that different from orphan dataset, MSA can be built for this dataset, though showing a decline in quality.
\section{Additional Visualization of Retrieved Sequence 3D Structure}
\begin{figure}[htbp]
\centering
\resizebox{0.5\linewidth}{!}{\includegraphics{img/structural_results.png}}
\caption{Query and Retrieved Sequence Structures}
\label{fig:structures}
\end{figure}
As shown in Figure~\ref{fig:structures}, we random picked a few more examples to illustrate the structural similarity between query protein and retrieval proteins.
\section{Introduction}
\input{02_introduction}
\section{Related Work}
\input{03_related_work}
\section{Problem Statement and Notations}
\input{04_preliminaries}
\section{MSA Transformer as a Retrieval Augmentation Method}
\input{06_framework.tex}
\section{Retrieval Sequence Augmentations}
\label{sec:rsa}
\input{05_methodology}
\section{Experiments}
\input{07_experiments}
\section{Conclusions and Future Work}
\input{08_conclusions.tex}
|
2,869,038,156,368 | arxiv | \section{Introduction}
One of the interesting features of low--energy supergravity (SUGRA)
models is that the electroweak symmetry breaking can be a direct
consequence of supersymmetry (SUSY) breaking [1]. In the ordinary
SUGRA models, SUSY breaking takes place in
a hidden sector of the theory, so that the gravitino mass $m_{3/2}$
becomes of the electroweak scale order. Below the Planck mass,
$M_P$, one is left with a global SUSY Lagrangian plus some terms
(characterized by the $m_{3/2}$ scale) breaking explicitly, but softly,
global SUSY. As we will briefly review below, the breakdown of
$SU(2)\times U(1)_Y$ appears as an automatic consequence of the
radiative corrections to these terms. The so--called $\mu$ problem
[2] arises in this context.
Let us consider a SUGRA theory with superpotential $W(\phi_i)$
and canonical kinetic terms for the $\phi_i$ fields\footnote{We will
consider this case throughout the paper for simplicity. Our general
conclusions will not be modified by taking a more general case.}.
Then, the scalar potential takes the form [3]
\begin{eqnarray}
V = e^K \left[ \sum_i \left| \frac{\partial W}{\partial \phi_i}
+ \bar{\phi_i}W \right|^2 - 3|W|^2 \right]\;+\;\mathrm{D}\;\mathrm{terms}
\;\;,
\label{V}
\end{eqnarray}
where $K=\sum_i|\phi_i|^2$ is the K\"ahler potential. It is customary
to consider $W$ as a sum of two terms corresponding to the observable
sector $W^{obs}(\phi_i^{obs})$ and a hidden sector
$W^{hid}(\phi_i^{hid})$
\begin{eqnarray}
W(\phi_i^{obs},\phi_i^{hid})=
W^{obs}(\phi_i^{obs}) + W^{hid}(\phi_i^{hid})
\;\;.
\label{W}
\end{eqnarray}
$W^{hid}(\phi_i^{hid})$ is assumed to be
responsible for the SUSY breaking,
which implies that some of the $\phi_i^{hid}$ fields
acquire non--vanishing vacuum expectation values (VEVs) in the process.
Then, the form of the effective
observable scalar potential obtained from eq.(\ref{V}), assuming
vanishing cosmological constant, is [4]
\begin{eqnarray}
V^{obs}_{eff} = \sum_i \left| \frac{\partial \hat{W}^{obs}}
{\partial \phi_i^{obs}}\right|^2 &+& m_{3/2}^2 \sum_i | \phi_i^{obs}|^2
+ \left( Am_{3/2}\hat W^{obs}_t + Bm_{3/2}\hat W^{obs}_b\; +\;
\mathrm{h.c.} \right)
\nonumber \\
\;&+&\;\mathrm{D}\;\mathrm{terms}
\label{Vobs}
\end{eqnarray}
with
\begin{eqnarray}
m_{3/2}^2 = e^{K^{hid}}|W^{hid}|^2
\label{m32}
\end{eqnarray}
\begin{eqnarray}
B = A-1=\sum_i \left( | \phi_i^{hid}|^2 + \frac{\bar{\phi}^{hid}}
{\bar{W}^{hid}}\frac{\partial \bar{W}^{hid}}
{\partial \bar{\phi}_i^{hid}}\right)\;-\;1\;\;,
\label{AB}
\end{eqnarray}
where $K^{hid}=\sum_i |\phi_i^{hid}|^2$, $\hat W^{obs}$ is the
rescaled observable superpotential $\hat W^{obs}=e^{K^{hid}/2}
W^{obs}$, the subindex $t$($b$) denotes the trilinear (bilinear) part of
the superpotential, and $A$, $B$ are dimensionless numbers of $O(1)$,
which depend on the VEVs of the hidden fields. Since we are assuming
that SUSY breaking takes place at a right scale, the gravitino mass
given by eq.(\ref{m32}) is hierarchically smaller than the Planck
mass (i.e. of order the electroweak scale).
In the minimal supersymmetric standard model (MSSM) the matter content
consists of three generations of quark and lepton superfields plus
two Higgs doublets, $H_1$ and $H_2$, of opposite hypercharge.
Under these conditions the most general effective observable
superpotential has the form
\begin{eqnarray}
W^{obs}=\sum_{generations}(h_uQ_LH_2u_R +
h_dQ_LH_1d_R + h_eL_LH_1e_R )+ \mu H_1H_2\;\;.
\label{Wobs}
\end{eqnarray}
This includes the usual Yukawa couplings (in a self--explanatory
notation) plus a possible mass term for the Higgses, where $\mu$
is a free parameter. From eq.(\ref{Vobs}) the relevant Higgs
scalar potential along the neutral direction for the electroweak
breaking is readily obtained
\begin{eqnarray}
V(H_1,H_2)=\frac{1}{8}(g^2+g'^2)\left(|H_1|^2-|H_2|^2\right)^2
+ \mu_1^2|H_1|^2 + \mu_2^2|H_2|^2 -\mu_3^2(H_1H_2+\mathrm{h.c.})
\;\;,
\label{Vhiggs}
\end{eqnarray}
where
\begin{eqnarray}
\mu_{1,2}^2 &=& m_{3/2}^2 + \hat\mu^2
\nonumber \\
\mu_{3}^2 &=& -Bm_{3/2}\hat\mu
\nonumber \\
\hat\mu &\equiv & e^{K^{hid}/2}\mu\;\;.
\label{mus}
\end{eqnarray}
This is the SUSY version of the usual Higgs potential in the standard
model. In order for the potential to be bounded from below,
the condition
\begin{eqnarray}
\mu_{1}^2+\mu_2^2-2|\mu_3^2|>0
\label{condmus}
\end{eqnarray}
must be
imposed all over the energy range $[M_Z,M_P]$. This implies in particular
$\langle H_{1,2}\rangle = 0$ at the Planck scale. Below the Planck
scale, one
has to consider the radiative corrections to the scalar potential.
Then the boundary conditions of eq.(\ref{mus}) are substantially
modified in such a way that the determinant of the Higgs
mass--squared matrix
becomes negative, triggering $\langle H_{1,2}\rangle \neq 0$
and $SU(2)\times U(1)_Y$ symmetry breaking [1].
For this scheme to work, the presence of the last term in
eq.(\ref{Wobs}) is crucial. If $\mu=0$, then the form of the
renormalization group equations (RGEs) implies that such a term
is not generated at any $Q$ scale since $\mu(Q)\propto \mu$.
The same occurs for $\mu_3$, i.e. $\mu_3(Q)\propto \mu$.
Then, the minimum of the potential of eq.(\ref{Vhiggs}) occurs
for $H_1=0$ and, therefore, $d$--type quarks
and $e$--type leptons remain massless. Besides, the superpotential
of eq.(\ref{Wobs}) with $\mu=0$ possesses a spontaneously broken
Peccei--Quinn
symmetry [5] leading to the appearance of an unacceptable
Weinberg--Wilczek axion [6].
Once it is accepted that the presence of the $\mu$ term
in the superpotential is essential, there arises an inmediate
question: Is there any dynamical reason why $\mu$ should be
small, of the order of the electroweak scale? Note that,
to this respect, the $\mu$ term is different from the SUSY
soft--breaking terms,
which are characterized by the small scale $m_{3/2}$ once
we assume correct SUSY breaking. In principle the natural scale
of $\mu$ would be $M_P$, but this would reintroduce the
hierarchy problem since the Higgs scalars get a contribution
$\mu^2$ to their squared mass [see eq.(\ref{mus})]. Thus,
any complete explanation of the electroweak breaking scale
must justify the origin of $\mu$. This is the so--called
$\mu$ problem [2]. This problem has been considered by
several authors and different possible solutions
have been proposed [2,7,8].
In this letter we suggest a scenario in which $\mu$ is
generated by non--renormalizable terms
and its size is directly related to the gravitino mass.
A comparison with the scenarios of refs.[2,7,8] is also made.
\section{A natural solution to the $\mu$ problem}
Let us start with a simple scenario with superpotential
\begin{eqnarray}
W=W_o + \lambda W_oH_1H_2
\;\;.
\label{WWo}
\end{eqnarray}
where $W_o$ is the usual superpotential (including both observable
and hidden sectors) {\em without} a $\mu H_1H_2$ term. We have allowed
in (\ref{WWo}) a non--renormalizable term, characterized by the coupling
$\lambda=O(1)$ (in Planck units), which mixes the observable sector with
the hidden sector (other higher--order terms of this kind could also be
included, but they are not relevant for the present analysis).
The $\mu H_1H_2$ term must be absent from $W_o$ since, as was
mentioned above, the natural scale for $\mu$ would otherwise be $M_P$.
Certainly, this is technically possible in a supersymmetric theory,
since the non--renormalization theorems assure that this term cannot
be generated radiatively if initially $\mu=0$. One may wonder, however,
whether there is a theoretical reason for the absence of the
$\mu H_1H_2$ term from $W_o$ in eq.(\ref{WWo}), since it is not forbidden by
any
symmetry of the theory\footnote{The $\mu H_1H_2$ term can be
forbidden by invoking a Peccei--Quinn (PQ) symmetry [2,8]. This
is not possible here since (\ref{WWo}) does not possess any PQ
symmetry.}. It is quite remarkable here that this is provided
in the low--energy SUSY theory obtained
from superstrings. In this case mass terms (like $\mu H_1H_2$) are
forbidden in the superpotential. We will see in section 4 an explicit
example in this context. Finally, non-renormalizable terms
(like $\lambda W_oH_1H_2$) are in principle allowed in a generic
SUGRA theory.
Next, we show that
the $\lambda W_oH_1H_2$ term yields dynamically a $\mu$ parameter.
Using the general expression of eq.(\ref{V}), the scalar
potential $V$ generated by $W$ has the form
\begin{eqnarray}
V = &e^K& \left\{ \sum_i \left| \frac{\partial [W_o(1+\lambda
H_1H_2)]}{\partial \phi_i}
+ \bar{\phi_i}W_o (1+\lambda H_1H_2)\right|^2 - 3|W_o(1+\lambda
H_1H_2)|^2 \right\}
\nonumber\\
\;&+&\;\mathrm{D}\;\mathrm{terms}
\;\;,
\label{V2}
\end{eqnarray}
which can be written as
\begin{eqnarray}
V = V^{(1)}|1+\lambda H_1H_2|^2 &+&
e^K \left\{ \left| \frac{\partial [W_o(1+\lambda
H_1H_2)]}{\partial H_1}
+ \bar{H_1}W_o (1+\lambda H_1H_2)\right|^2 +(H_1\leftrightarrow H_2)
\right\}
\nonumber\\
\;&+&\;\mathrm{D}\;\mathrm{terms}
\;\;,
\label{V3}
\end{eqnarray}
where
\begin{eqnarray}
V^{(1)}\equiv e^K \left( \sum_i \left| \frac{\partial W_o}
{\partial \phi_i}
+ \bar{\phi_i}W_o \right|^2 - 3|W_o|^2 \right)\;
;\;\;\phi_i\neq H_{1,2}\;\;.
\label{V1}
\end{eqnarray}
Since $H_{1,2}$ enter in $W_o$ only through the ordinary Yukawa
couplings and we are assuming vanishing VEVs for the observable
scalar fields, it is clear (recall that $W_o$ does not contain
a $\mu H_1H_2$ coupling) that $\left. \frac{\partial W_o}
{\partial H_{1,2}}\right|_{min}=0$. Besides, the vanishing
of the cosmological constant implies $V^{(1)}=0$ at the minimum
of the potential.
So, we can extract from the second term in eq.(\ref{V3}) the soft
terms associated with $H_{1,2}$:
\begin{eqnarray}
V(H_1,H_2)=&\frac{1}{8}&(g^2+g'^2)\left(|H_1|^2-|H_2|^2\right)^2
+ m_{3/2}^2(1+\lambda^2)|H_1|^2 + m_{3/2}^2(1+\lambda^2)|H_2|^2
\nonumber\\
&+&2m_{3/2}^2\lambda(H_1H_2+\mathrm{h.c.})
\;\;.
\label{Vhiggs2}
\end{eqnarray}
Comparing eqs.(\ref{Wobs}--\ref{mus}) with
eqs.(\ref{WWo},\ref{Vhiggs2}) it is clear that $\lambda W_oH_1H_2$
behaves like a $\mu$ term when $W_o$ acquires a non--vanishing VEV
dynamically. Defining $\lambda\langle W_o\rangle\equiv\mu$ we
can write eq.(\ref{Vhiggs2}) as eqs.(\ref{Vhiggs},\ref{mus}) where
now the value of $B$ is
\begin{eqnarray}
B=2
\;\;.
\label{B}
\end{eqnarray}
The value of $A$ is still given by eq.(\ref{AB}), but the relation
$B=A-1$ is no longer true. The fact that the new
"$\mu$ parameter" is of the electroweak--scale order
is a consequence of our assumption of a correct SUSY--breaking
scale $m_{3/2}=e^{K/2}W=O(M_Z)$. Finally, note that the usual
condition for the potential to be bounded from below
(\ref{condmus}) is automatically satisfied by (\ref{Vhiggs2})
for any value of $\lambda$.
One may wonder how general is the simple scenario of eq.(\ref{WWo}).
First of all, let us note that the fact that $H_1H_2$ is not forbidden
by any symmetry of the theory is a key ingredient for this scenario to
work. An obvious generalization of (\ref{WWo}) arises when $W_o$
consists of several
terms $W_o=W_o^{(1)}+W_o^{(2)}+...$ and $H_1H_2$ couples with
a different strength to each term, i.e. $(\lambda_1W_o^{(1)}+
\lambda_2W_o^{(2)}+...)H_1H_2$. However, provided that the hierarchical
small value for $\langle W_o\rangle$ is not achieved by a fine--tuning
between the VEVs of the various terms $W_o^{(1)},W_o^{(2)},...$,
it is clear that the order of magnitude of $\mu$ continues being
$m_{3/2}$. Apart from this, it should be noticed that $\lambda_i=O(1)$
(in Planck units) is only natural if $W_o^{(i)}$ is not an operator
with a extremely small coupling constant. However, this would be a
naturalness problem by itself. This would happen, for instance,
for $W_o^{(i)}=m\Phi^2$ with $m<<M_P$. (These terms
are forbidden in string theories.)
To conclude this section, it is worth noticing that in the context of
supergravity theories there is another
possible solution to the $\mu$ problem. Since the K\"ahler potential
$K$ is an arbitrary real--analytic function of the scalar fields,
we can study for example a theory with the following
$K$
\begin{eqnarray}
K=\sum_i|\phi_i|^2 + f(g(\phi_j, \bar \phi_j)H_1H_2\ +\ \mathrm{h.c.})
\;\;,
\label{K}
\end{eqnarray}
where $\phi_j\neq H_{1,2}$ and $f$ and $g$ are generic functions
($\langle g(\phi_j, \bar\phi_j)\rangle= O(1)$). Then, although $W_o$
does not contain a $\mu$ term, this is generated in the
scalar potential. This is trivial to see for the simplest case
(i.e. $f(x)=x$, $g=$ const. $\equiv\lambda$). Then the theory is equivalent
to one with K\"ahler potential $\sum_i|\phi_i|^2$ and superpotential
$W_oe^{\lambda H_1H_2}$, since the function ${\cal G}=K+\log|W|^2$
that defines the SUGRA theory is the same for both. Expanding
the exponential, the first two terms coincide with eq.(\ref{WWo})
and hence we obtain the same $\mu$ term as in eq.(\ref{Vhiggs2}).
The possibility (\ref{K}) was examined in ref.[7] for $f(x)=x$ and
when $g$ is a non--trivial function of the hidden fields,
in particular for the simplest case $g(\phi_j, \bar\phi_j)=\bar\xi$,
where $\bar\xi$ is a hidden field. It remains to be explored whether
a K\"ahler potential similar to that of eq.(\ref{K}) can arise in
the context of superstring theories.
\section{Expectation values for the Higgses}
In the above analysed solution to the $\mu$ problem it is assumed
that the observable scalar fields have vanishing VEVs at the Planck
scale. Since the non--renormalizable term $\lambda W_oH_1H_2$
mixes observable and hidden fields, one may wonder whether that
assumption is still true for the Higgses. We will show now that
this is in fact the case.
We assume here that the initial superpotential $W_o$
gives a correct SUSY breaking,
i.e. small gravitino mass and vanishing cosmological
constant. This means that $V_o$, i.e. the scalar potential
derived from $W_o$, is vanishing at the minimum
$\left. V_{o}\right|_{min}=0$ and thus
positive--definite. Using
the general expression of eq.(\ref{V}), $V_o$
can be decomposed in three pieces
\begin{eqnarray}
V_o = V^{(1)}\;+\;e^K \left\{ \left| \frac{\partial W_o}{\partial H_1}
+ \bar{H_1}W_o \right|^2 + (H_1\rightarrow H_2)\right\}\;+\;
\mathrm{D}\;\mathrm{terms}
\;\;,
\label{Vo}
\end{eqnarray}
where $V^{(1)}$ is defined in eq.(\ref{V1}). Recalling that we
are assuming that $W_o$ does not contain a $\mu H_1H_2$ term
and that $\left. \frac{\partial W_o}{\partial H_{1,2}}\right|_{min}=0$
(since squarks and sleptons are supposed to have vanishing VEVs),
it is clear that $V^{(1)}$ is flat in $H_{1,2}$.
So, the minimum of the second piece of (\ref{Vo}) is zero and occurs
at $H_{1,2}=0$ (for any value of $W_o$). Therefore, necessarily
$\left. V^{(1)}\right|_{min}=0$, i.e. $V^{(1)}$ is also
positive--definite. All this is very ordinary: it simply means that
the hidden sector is entirely responsible for the breaking.
(Note that the $H_{1,2}$ F--terms are vanishing, while some
of the hidden fields F--terms must be different from zero.)
Notice also that from (\ref{Vo}) one obtains
$e^{K}|W_o|^2(|H_1|^2+|H_2|^2)=
m_{3/2}^2|H_1|^2+m_{3/2}^2|H_2|^2$
but, because of the absence of a $\mu H_1H_2$ term in $W_o$, there
is no $Bm_{3/2}\hat\mu H_1H_2$ term in the scalar potential.
Let us now study the impact
of doing, according to our approach, $W_o \rightarrow W = W_o + \lambda
W_oH_1H_2$. The corresponding scalar
potential, $V$, has already been written in eq.(\ref{V3}). Now, since
$V^{(1)}$ is positive--definite, so is $V$. In fact, the minimum of $V$
is for $V=0$ and occurs when the three pieces of (\ref{V3}) are
vanishing. Clearly, the minimum of the first and third pieces
of (\ref{V3}) coincides with that of eq.(\ref{Vo}) above, implying
$\left. V^{(1)}\right|_{min}=0$,\footnote{The only exception occurs if
$\lambda H_1H_2=-1$,
but then the second piece of (\ref{V3}), which is also
positive--definite, is different from zero, so this is not a
solution for the minimization of the whole potential.}
and thus the VEV of $W_o$ is the
same
as when we started with just $W_o$. Finally,
recalling that
$\left. \frac{\partial W_o}{\partial H_{1,2}}
\right|_{min}=0$, it is clear that the second piece of $V$
in eq.(\ref{V3}) has two possible minima
\begin{eqnarray}
H_1, H_2=0
\;\;,
\label{Hs1}
\end{eqnarray}
\begin{eqnarray}
\lambda H_2+(1+\lambda H_1H_2)\bar H_1&=&0
\nonumber\\
(H_1\leftrightarrow H_2)&=&0
\label{Hs2}
\end{eqnarray}
As was explained in section 1, the solution (\ref{Hs1}) is the
phenomenologically interesting one, whereas
the solution (\ref{Hs2}) leads to $H_{1,2}\sim M_P$,
so it is not phenomenologically viable. We can ignore this
solution since if $H_{1,2}$ are initially located at $H_{1,2}=0$
(e.g. by thermal effects) they will remain there as long as (\ref{Hs1})
continues to be a minimum solution. Of course, radiative corrections
will trigger non--zero VEVs of the correct size for $H_1$, $H_2$.
\section{A realistic example}
As we saw in section 2, the assumption of correct
SUSY breaking was crucial for obtaining the $\mu$ parameter
of the electroweak--scale order. As a matter of fact, gaugino
condensation effects in the hidden sector [9] are the most
satisfactory mechanism so far explored, able to break SUSY at
a scale hierarchically smaller than $M_P$ [10]. The reason
is that the scale of gaugino condensation corresponds to the
scale at which the gauge coupling becomes large, and this is
governed by the running of the coupling constant. Since
the running is only logarithmically dependent on the scale,
the gaugino condensation scale is suppressed relative to the
initial one by an exponentially small factor $\sim e^{-1/2\beta g^2}$
($\beta$ is the one--loop coefficient of the beta function
of the hidden sector gauge group $G$). This mechanism has been
intensively studied in the context of SUGRA theories coming from
superstrings [11,12], where
the gauge coupling is related to the VEV of the dilaton field
$S$ (more specifically Re$S=g^{-2}$). Recall that
we have argued in section 2 that superstring theories are precisely
a natural context where the solution of the $\mu$ problem presented
here can be implemented, since
mass terms, such as $\mu H_1 H_2$, appearing in the superpotential
are automatically
forbidden in superstrings. Besides, non--renormalizable terms
like $\lambda W_oH_1H_2$ in eq.(\ref{WWo})
are in principle allowed and, in fact, they are usually present [13].
In the absence of hidden matter, the condensation process is
correctly described by a non--perturbative effective
superpotential
\begin{eqnarray}
W_o\propto e^{-3S/2\beta_o}
\;\;,
\label{Wcond}
\end{eqnarray}
with $\beta_o=3C(G)/16\pi^2$, where $C(G)$ is the Casimir operator
in the adjoint representation of $G$. It is difficult to imagine,
however, how the mechanism expounded in section 2 could be implemented
here. More precisely, it is not clear
that we could have something like
$W=W_o+\lambda W_oH_1H_2$, due to the effective character
of (\ref{Wcond}).
Fortunately, things are different in the presence of hidden matter,
which is precisely the most frequent case in string constructions
[13]. There is not at present a generally accepted formalism
describing the condensation in the presence of massless matter,
but the case of massive matter is well understood [14]. For example,
in the case of $G=SU(N)$ with $M(N+\bar N)$ "quark" representations
$Q_\alpha$, $\bar Q_\alpha$,
$\alpha=1,...,M$, with a mass term given by
\begin{eqnarray}
W_o^{pert}=-\sum_{\alpha,\beta}{\cal M}_{\alpha,\beta}Q_{\alpha}
\bar Q_\beta
\;\;,
\label{Wpert}
\end{eqnarray}
the complete condensation superpotential can be written as [12]
\begin{eqnarray}
W_o\propto [\mathrm{det}{\cal M}]^{\frac{1}{N}} e^{-3S/2\beta_o}
\;\;.
\label{Wcond2}
\end{eqnarray}
It should be noticed here that, strictly speaking, there are no mass
terms like (\ref{Wpert}) in the context of string theories.
However the matter fields usually have trilinear couplings which
play the role of mass terms with a dynamical mass given by the VEV
of another matter field. The simplest case occurs when there is
an $SU(N)$ singlet field $A$ giving mass to all the quark representations.
Then (\ref{Wpert}) takes the form
\begin{eqnarray}
W_o^{pert}=-\sum_{\alpha=1}^M AQ_{\alpha}\bar Q_\alpha
\;\;,
\label{Wpert2}
\end{eqnarray}
and $\mathrm{det}{\cal M}=A^M$. Now, if $H_1H_2$ is an allowed
coupling from
all the symmetries of the theory, it is natural to promote
$W_o^{pert}$ to\footnote{We neglect here higher--order
non--renormalizable couplings since they do not contribute to the
$\mu$ term.}
\begin{eqnarray}
W^{pert}=-\sum_{\alpha}A(1+\lambda' H_1H_2)Q_{\alpha}\bar Q_\alpha
\;\;,
\label{Wpert3}
\end{eqnarray}
so that $\mathrm{det}{\cal M}=[A(1+\lambda' H_1H_2)]^M$, and
(\ref{Wcond2}) takes the form
\begin{eqnarray}
W_o\rightarrow W\propto
[A(1+\lambda' H_1H_2)]^{\frac{M}{N}} e^{-3S/2\beta_o}
\simeq A^{\frac{M}{N}}(1+\frac{M}{N}\lambda' H_1H_2) e^{-3S/2\beta_o}
\;\;.
\label{Wcond3}
\end{eqnarray}
Thus
\begin{eqnarray}
W=W_o+\lambda W_oH_1H_2
\;\;,
\label{WWocond2}
\end{eqnarray}
where we have defined $\lambda\equiv\frac{M}{N}\lambda'$. This is
precisely the kind of superpotential we wanted (see eq.(\ref{WWo}))
in order to generate the $\mu$ term dynamically.
In ref.[8] an interesting solution to the $\mu$ problem was proposed
in a similar context with a PQ symmetry, using the presence of a term
$H_1H_2Q\bar Q$
in the superpotential and assuming that the scalar components
of $Q$ and $\bar Q$ condense at a scale $\Lambda\simeq 10^{11}$ GeV.
As mentioned above, the only accepted formalism
describing the
condensation is in the presence of massive matter. Thus the previous
term
behaves as a dynamical mass term for the squarks
and the complete superpotential (\ref{Wcond2}) becomes
$W\propto (H_1H_2)^{\frac{1}{N}}
e^{-3S/2\beta_o}$. This is phenomenologically unviable since
the Higgses must have vanishing VEVs at $M_P$ for a correct
phenomenology, which would imply $\langle W\rangle=0$ and thus
no SUSY breaking. We can improve this model by including
a mass term for $Q\bar Q$. However, a genuine mass term for $Q\bar Q$
would break
the PQ symmetry, so one should consider something similar to
(\ref{Wpert2}). Then the perturbative superpotential is
\begin{eqnarray}
W^{pert}\sim AQ\bar Q + H_1H_2 Q\bar Q
\;\;,
\label{Wpert4}
\end{eqnarray}
and the scenario becomes much more similar to that given
by eq.(\ref{Wpert3}). However, there still is an important
difference. In eq.(\ref{Wpert3}) $H_1H_2$ couples to
$AQ\bar Q$ (which is the natural thing if $H_1H_2$ is invariant
under all the symmetries of the theory) instead of $Q\bar Q$;
thus there is no PQ symmetry. Moreover, (\ref{Wpert3}) leads
to (\ref{WWocond2}) in which the $\mu$ scale is directly given
by the $m_{3/2}$ scale ($\mu=O(m_{3/2})$). However from
(\ref{Wpert4}) the $\mu$ scale is given by the
squark condensation scale [12] $\langle Q\bar Q\rangle/M_P\simeq
m_{3/2}M_P/N\langle A\rangle$, so that the value of $\mu$ in this case
tends to be a bit too large.
\section{Summary and conclusions}
We have proposed a simple mechanism for solving the $\mu$ problem
in the context of minimal low--energy SUGRA models. This is based
on the appearance
of non--renormalizable couplings in the superpotential. In particular,
if $H_1H_2$ is an allowed operator by all the symmetries of the theory,
it is natural to promote the usual renormalizable superpotential $W_o$
to $W_o+\lambda W_o H_1H_2$, yielding an effective $\mu$ parameter
whose size is directly related to the gravitino mass once SUSY
is broken (this result
is essentially maintained if $H_1H_2$ couples with different strengths
to the various terms present in $W_o$).
On the other hand, the $\mu$ term must be absent in $W_o$, otherwise
the natural scale for $\mu$ would be $M_P$.
Certainly this is technically possible in a supersymmetric theory
since the non--renormalization theorems assure that
this term cannot be generated radiatively if initially $\mu=0$. Remarkably
enough, however,
a theoretical reason for the absence of the
$\mu H_1H_2$ term from $W_o$ is provided
in the low--energy SUSY theory obtained
from superstrings. In this case mass terms (such as $\mu H_1H_2$) are
forbidden in the superpotential (however, non--renormalizable terms
like $\lambda W_oH_1H_2$
are in principle allowed and, in fact, they are usually present).
We have also addressed other alternative solutions, comparing them
with the one proposed here. On the other hand, we have analysed
the $SU(2)\times U(1)$ breaking, finding that it takes place
satisfactorily.
Finally, we have given a realistic example in which SUSY
is broken by gaugino condensation in the
presence of hidden matter (which is the usual situation in strings),
and where the mechanism proposed for solving the $\mu$ problem
can be gracefully implemented.
\vspace{2cm}
\noindent{\bf ACKNOWLEDGEMENTS}
We gratefully acknowledge J. Louis for extremely useful discussions.
\vspace{1.7cm}
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2,869,038,156,369 | arxiv | \section{Introduction and background}
There have been promising advances in practical quantum computing in recent years and a range of prototype devices are currently in development \cite{qed_ibm,Brandl2016,Harty2014}. Quantum computing architectures can be broadly divided into two classes: gate model devices based on discrete operations, and continuous time devices for which the native dynamics of the quantum system is used to solve problems. For gate model quantum computing, the aim is to build a scalable and fault tolerant device capable of universal logic. In principle, this can be achieved through the use of quantum error correction protocols to achieve arbitrary suppression of noise at the logical level \cite{Preskill385,Kitaev1997,Aharonov:1997:FQC:258533.258579,Knill1998}. In contrast, continuous time quantum computers, in particular the highly successful subfield of quantum annealing, can operate without having to be error corrected \cite{brooke99a,johnson11a,denchev16a,lanting14a,boixo16a}. Quantum annealing devices have already been built with thousands of qubits \cite{D-Wave}. Whilst existing quantum annealing devices are not universal, they have been found to be useful for a wide variety of applications, for example in theoretical computer science \cite{chancellor16a}, finance \cite{marzec16a,Orus18a,Venturelli18a}, aerospace \cite{coxson14a}, machine learning \cite{amin16a,Benedetti16a,Benedetti16b,Adachi2015}, mathematics \cite{Li17a,Bian13a}, decoding of communications \cite{chancellor16b}, hydrology \cite{omalley18a}, and computational biology \cite{perdomo-ortiz12a}.
Quantum error correction protocols impose considerable overhead in the design of fault tolerant quantum computers. For example, the surface code will demand around four thousand physical qubits per logical qubit \cite{gidney18,fowler2012surface}. In addition to the qubit overhead, another consideration is how to efficiently interpret the output of a quantum error correction code in real-time. This task, referred to as decoding, is known in to be computationally demanding \cite{Breuckmann17a}, with data rates from some quantum codes expected to be of the order $100$Gbit/sec. In this present paper we explore the use of quantum annealers and related Ising model devices as specialised co-processors for decoding.
The task of decoding imposes a bottleneck on the successful operation of quantum error correction (especially in situations where non-Clifford operations are to be performed). Achieving even a small improvement in decoding could therefore lead to major in gains in performance of the quantum computer. As a result, the decoding problem is a high-value use case for quantum annealing devices. In addition, this work demonstrates how the two paradigms in quantum computing architecture can be combined in a hybrid setting.
\subsection{Quantum error correction}
In the gate model of quantum computation, universal computation is achieved through the application of discrete operations from a finite set of qubit gates. Because these gates are realised experimentally via the precise manipulation of fragile quantum systems, both control and memory faults are common. As such, quantum error correction protocols play an essential role in the design of any gate model quantum computer.
Adapting existing classical error correction protocols for use on quantum hardware is not straightforward. The No-Cloning theorem prohibits quantum data from being redundantly encoded via simple duplication \cite{Wootters1982}. Furthermore, quantum error correction protocols must be carefully designed to avoid compromising the encoded information via wavefunction collapse. A final complication that arises when designing quantum codes is that qubits are susceptible to multiple error types; whereas in classical error correction only bit-flips ($X$-errors) need to be considered, in quantum error correction an additional error type, phase-flips ($Z$-errors), must also be considered \cite{Devitt2013}.
The stabilizer framework for quantum error correction has been developed to allow quantum codes to be constructed within the above constraints \cite{Gottesman97,Gottesman:1998hu}. The essential idea is that the information encoded in a register of qubits is distributed across a larger entangled system of qubits. The extra degrees of freedom due to this expansion allows errors to do be detected using a series of `stabilizer measurements'. These stabilizer measurements reveal information about the parity of the register, whilst leaving the encoded quantum information unchanged. In general, a stabilizer code is labelled using the $[[n,k,d]]$ notation, where $n$ is the total number of qubits and $k$ is the number of logical qubits. The code distance $d$ is the minimum weight error that will go undetected by the code. We note also that the stabilizer framework and other error correction schemes can also be applied in a continuous time setting, where thermal dissipation can be used perform the corrections automatically (see e.g.~\cite{Pudenz14a,Jordan06a,sarovar05a,vinci15a,young13a,Sarovar13a}). However, this is not the subject of our current work.
The recently introduced coherent parity check (CPC) framework for quantum error correction \cite{chancellor16,roffe17,Roffe18a} provides a tool kit for the conversion of classical codes to quantum stabilizer codes. CPC codes are known to include large classes of stabilizer codes such as CSS codes, and are conjectured to be a superset of all known stabilizer codes. CPC codes have a specific structure in which the qubits are separated into two distinct types: data qubits and parity qubits. Error correction then proceeds via a three-part parity checking sequence. First, a set of bit parity checks are performed on the data qubits and the results copied to the parity qubits. This is followed by a second round of parity checks to detect phase-flips, with the results again being copied to the parity qubits. The final round of parity checks, referred to as cross-checks, takes place between the parity qubits themselves and fixes the code distance to the desired length. The specific advantage of the the CPC framework is that the parity checking sequences in each stage can be taken directly from existing classical codes.
The effect of a CPC encoder is to place the data and parity-check qubits in known stabilizer states, which can then be measured. Error propagation through such stabilizer measurements is identical to error propagation through a decoder (that is, the reverse of the encoder operations). One can therefore think either of stabilizer measurements on the encoded state, or, equivalently, the measurement of parity check qubits after a decoder. The results of all the stabilizer measurements can be combined to form a binary string called a syndrome. The role of a decoder in an error correction protocol is to infer the best recovery operation given the information provided by this syndrome. For classical codes, various methods exist that allow decoding to be performed efficiently \cite{MacKay1996,MacKay1999}. Unfortunately, owing to the fact that stabilizer codes must be able to detect two-error types simultaneously, it is not always possible to use these existing techniques directly. As a result, bespoke decoding strategies need to be developed for use with quantum codes.
\subsection{The decoding problem}
Whether in a classical or quantum setting, the problem of decoding amounts to answering the same fundamental question: ``given the data available, what actions would be most likely to preserve the encoded information?'' There are generally recognized to be two different techniques \cite{Jaynes57a,Jaynes68a,Frieden72a} by which such inference can be achieved:
\begin{enumerate}
\item Maximum likelihood estimation (MLE), for which it is assumed that the single most likely event, in this case the single most likely pattern of errors, has occurred.
\item Maximum entropy inference (MaxEnt), for which a distribution which maximizes entropy subject to constraints is used to draw probabilistic conclusions about the errors which make up the pattern.
\end{enumerate}
If all errors occur at the same rate and are less than $50\%$ likely to happen, MLE reduces to finding the the fewest number of errors which are consistent with the result of an error correction measurement (syndrome measurement). On the other hand, MaxEnt relies on finding the probability of every event, so in principle will always depend in detail on the error rate(s). We discuss later how both of these approaches can be mapped to specialized hardware.
Before continuing, it is worth considering a highly simplified example to demonstrate the difference between MLE and MaxEnt. Consider a simple (fictional) code where the observed parity measurements correspond to four distinct possibilities. In one of the four error patterns error $e_1$ is the only error, but in the other three patterns two errors other than $e_1$ have occurred. Based on MLE, we would always correct error $e_1$, since the single most likely error pattern (assuming all error rates are less than 50\%) is the one with the lowest weight. On the other hand the MaxEnt decoding depends on the error rate. Assuming all errors are equally likely and occur with a probability $p$, we can see that if $p<\frac{1}{3}$, then correcting $e_1$ will indeed reduce the error rate. However, if $\frac{1}{3}<e_1<\frac{1}{2}$, than it is more likely that the actual error pattern is one of the of the three cases in which error $e_1$ has \emph{not} occurred. In this case the MLE strategy actually does more harm than good, while the MaxEnt strategy does not perform the detrimental `correction'.
A practical decoding method is a key element in classical error correction code design. Leading classical error correction schemes, such as low density parity check (LDPC) codes and turbo codes, rely upon efficient approximate decoding strategies. For example, LDPC codes are decoded using belief propagation on a type of graphical model known as a factor graph \cite{MacKay1996,MacKay1999}.
A method for mapping quantum codes to a factor graphs has been demonstrated in \cite{Roffe18a}, but this technique required $Y$ errors to be represented as burst errors. In this work, as a by-product of our Ising model construction, we also prersent a factor graph construction which does does not require a multi-bit error model. While we focus on Ising model specialized hardware in this work, it is also worth noting that specialized hardware for belief propagation, for instance application specific circuits (ASICs), could also provide a promising route to improve decoding.
There is room for improvement in quantum decoding. As an example, a threshold of $p_c\approx10.3\%$ can be achieved on a 2D toric code using a minimum weight matching (a MLE technique) decoder \cite{Wang03a,Breuckmann17a}. However, statistical mechanics arguments based on an Ising model corresponding to these codes suggests a maximum threshold of $p_c\approx10.9\%$. Because this maximum threshold is calculated based on the statistical mechanical properties of an Ising model, a perfect thermal sampler on this model would be able to saturate the decoding bound \emph{by construction}. It is of course an open question whether such a high quality sampler could actually be practically implemented.
\subsection{Ising model computing}
Although it began as a model for magnetism \cite{Ising25a}, the Ising model has since been demonstrated to be a powerful tool in representing hard optimization and machine learning problems in specialized hardware. Finding the ground state (lowest energy) of an Ising spin glass is known to be NP-hard. Therefore all NP-hard problems can be mapped to it with only polynomial overhead. Moreover, practical mappings of many important problems are actually known, with examples include partitioning, covering, and satisfiability \cite{Lucas2014a,Choi10a,Chancellor16c}. In addition to the ground state problem being NP-hard, Ising models are known to be universal in the sense that \emph{any} classical spin model can be efficiently simulated by an Ising model \cite{De_las_Cuevas16a}. The algebraic similarity between Ising models and the decoding of codes was first noticed by Sourlas \cite{Sourlas1989a}, who demonstrated that under certain circumstances Ising spin glasses behave as optimal error correcting codes.
Probably the most well known specilized hardware which use Ising model based encodings are quantum annealers, such as those produced by D-Wave Systems Inc \cite{D-Wave}. Annealers, however, are not the only Ising model based computational machines. There have recently been efforts to produce other specialized Ising model based computing hardware \cite{McMahon16a,Inagaki16a,Yamaoka16a,Fujitsu_announce}, including some fully classical devices.
It is not only the ground states of Ising models which are computationally interesting. Thermal distributions over Ising models perform a constrained entropy maximization, which is useful for MaxEnt inference among other tasks. There has been much recent work for instance on how specialized annealing hardware may be used to realise Boltzmann machines by sampling a Boltzmann distribution \cite{Adachi2015,Amin2018a,Benedetti2016a,Benedetti2017a}. Additionally, Ising thermal distributions can be used to perform maximum entropy decoding of (classical) communications \cite{chancellor16b} via MaxEnt techniques.
Specialized Ising hardware can take an variety of forms, including fully classical CMOS devices which operate at room temperature \cite{Yamaoka16a,Fujitsu_announce}, and optimize over Ising models directly, rather than being arranged in a more traditional architecture. Using these devices directly for solid state quantum computing would mean a high rate of communication with a room temperature environment, and therefore a large heat and noise flux incident on the device. To avoid this problem, it would be preferable to perform the logical operations associated with decoding at deep crogenic, rather than room temperature. CMOS information processing devices can be operated at deep cryogenic temperatures, and the importance for quantum computing has been highlighted \cite{Homulle2018a,Patra2018a,Weinreb2007a,Homulle2018b,Conway-Lamb2016a}. For estimates of power and area requirements for controls in silicon qubits, see \cite{Geck19a}. Given that quite complex devices, including field programmable gate arrays \cite{Homulle2018b,Conway-Lamb2016a} can be operated at cryogenic temperatures of around $4$ K, a cryogenic CMOS implementation of the (fully classical) Ising model computers being explored by \cite{Yamaoka16a,Fujitsu_announce} could provide a promising path for practically implementable quantum error correction.
\section{Mapping Quantum Error Correction to Ising Machines}
It has recently been demonstrated in \cite{Roffe18a} that the decoding of many quantum error correction codes can be mapped to a classical factor graph via the so called coherent parity check (CPC) formalism, originally introduced in \cite{chancellor16}. It is in turn known that due to the nature of the parity checks in these codes, their factor graphs naturally map to Ising models, which is the preferred encoding style of an emerging family of specialized computing hardware \cite{McMahon16a,Inagaki16a,Yamaoka16a,Fujitsu_announce}, including quantum annealers \cite{D-Wave,johnson11a}. This mapping is valid for all codes which can be described within the CPC framework. Whilst it is an open question as to which codes can be described within this framework, it is known that the framework can at least describe all CSS codes \cite{chancellor16}. It is also suspected that the CPC framework may be able to describe all stabilizer codes up to local unitaries.
While we will not review the entire CPC formalism here, it is worth remarking on the key elements of the graphical construction, as it is important to understand how to map the decoding of quantum codes to Ising models. The construction of a classical factor graph using the graphical version of the CPC framework presented in \cite{Roffe18a} begins with a so called operational representation of the code, which represents how (unmeasured) data qubits interact with which (measured) parity check qubits. This operational representation must then be annotated with directed edges to represent indirect propagation of errors. As it is not directly relevant to the discussion here, we will not review how this annotation is done, but instead refer the reader to \cite{Roffe18a} for a simple set of graphical rules.
In the operational representation, unmeasured data qubits are represented by triangles \scalebox{.5}{\input{tikzit/data_qubit.tikz}} and measured parity check qubits are represented by stars \scalebox{.5}{\input{tikzit/parity_qubit.tikz}}. There are three types of edges between these qubits, representing the three ways errors can be transmitted: directly through bit error checks, directly through phase error checks, and through indirect propagation due to the quantum nature of the interactions. For the purposes of this paper, it is only necessary to review the translation of the qubits to classical factor graph representation and not how the edges translate. The data qubit translates to two classical data bits:
\begin{equation}
\input{tikzit/unmeasured_definition.tikz}.
\end{equation}
The parity check qubits on the other hand translates to a factor \footnote{We use the compressed factor graph notation used in \cite{Roffe18a}, where the single body elements of each factor are absorbed into the factor creating a `soft' constraint. This is consistent with the most compact and natural Ising model representation. While this differs from the traditional Tanner graph representation, it is unambiguous and a translation between the two consists of converting the factor to a hard constraint and adding an additional bit with the weight of the factor. }, representing the measured bit information, and a classical data bit representing the unmeasured phase information:
\begin{equation}
\input{tikzit/measurement_definition.tikz}.
\end{equation}
Finally, it is possible for a parity check bit to have a self loop, where its own phase information is propagated to its (measured) bit degree of freedom. In this case, the qubit again translates to a parity check node and a classical bit, but this time with an edge connecting them,
\begin{equation}
\input{tikzit/measurement_definition_self.tikz}.
\end{equation}
The classical factor graphs which represent these quantum error correcting codes will have two types of nodes: classical bits representing whether or not a degree of freedom was errored, and parity check nodes representing measurements of the bit degrees of freedom of the parity check qubits. These measurements effectively tell us the parity of errors (whether an odd or even number have occurred) over a given set of bit and phase degrees of freedom. If we assume a very simple error model where all qubits have an equal likelihood to have a bit (X) or phase (Z) error, and no other error types are allowed, then the energy with respect to the following Ising Hamiltonian is proportional to the number of errors up to an irrelevant offset
\begin{equation}
H^{\mathrm{err\,count}}=-\sum_{i=1}^{k+n} \sigma^z_i-\sum_{j=1}^{n-k}s_j\prod_{l \in Q_j}\sigma^z_l
\end{equation}
where $s_j\in \{+1,-1\}$ is the parity of the $j$th error measurement and $Q_j$ is the set of degrees of freedom (represented as classical bits) checked by that measurement. In this case, an Ising spin taking a $+1$ value indicates no error, while a $-1$ value indicates that an error has occurred. For the bit degrees of freedom of the parity checking qubits, a value of the parity checking measurement which is different from $s_j$ indicates an error. Since an $X$ or $Z$ error on any qubit increases the energy with respect to this Hamiltonian by $2$, it follows that this Hamiltonian effectively counts the number of errors necessary to give a syndrome measurement $\{s\}$, and further follows that the lowest energy state is the one with the fewest errors. Furthermore, if we assume that our errors have a probability less than $50\%$, than the error configuration(s) with the least errors are also the most likely. Finding the ground state of this Hamiltonian is therefore equivalent to performing maximum likelihood (MLE) inference.
This Hamiltonian, however, is not only a tool for MLE inference. Let us consider a Boltzmann distribution
\begin{equation}
p_a=\frac{\exp\left(-\frac{E_a}{T}\right)}{Z} \label{BD}
\end{equation}
where $E_a$ is the energy of configuration $a$, i.e. the value of the Hamiltonian, and $Z$ is the partition function which guarantees that the probability distribution is normalized. Recall that $E_a$ is proportional to the number of errors required to measure the syndrome given by $\{s\}$ with the error configuration $a$ on the unmeasured degrees of freedom up to an energy offset. The structure of the Boltzmann distribution means that for every additional error the probability of the configuration is decreased by a multiplicative factor of $\exp(-\frac{2}{T})$. Since the probability of an error configuration in the actual decoding process is also decreased by a constant multiplicative factor if the number of errors in the configuration is increased by one, it follows that sampling from the Boltzmann distribution over the Ising model is the same as sampling over error distributions at a finite error rate. Inferring the likelihood of specific errors at a known overall error rate is a form of maximum entropy (MaxEnt) inference, a powerful tool which allows knowledge of the error rate to help with decoding. There is a formally rigorous relationship between the error rate and the temperature of the distribution, namely, sampling at an error rate $p$ is equivalent to sampling the Boltzmann distribution at the Nishimori temperature \cite{Nishimori1980,Nishimori2001}
\begin{equation}
T_{Nish}=2\,\left(\ln\left(\frac{1-p}{p}\right)\right)^{-1}.
\end{equation}
If the error rate is known, at least approximately, as it almost always will be on real devices, then maximum entropy inference can use this additional information to perform better than maximum likelihood. One dramatic example of this is \cite{chancellor16b} where maximum entropy inference performed on a programmable quantum annealer was shown to be able to out-perform perfect maximum likelihood decoding.
While the example given here provides motivation for the possibility of using Ising machine based techniques, including maximum entropy inference to decode quantum error correcting codes, it has thus far been done in a rather unrealistic setting. Assuming equal error rates on every degree of freedom is natural in classical communications, where parity checks and data will all be sent through the same channel. The same assumption is somewhat artificial in the setting of quantum error correction, where there is no a priori reason to expect that bit and phase errors will occur at the same rate. Furthermore, we have thus far considered an error model which does not contain a separate mechanism for $Y$ errors, where the bit and phase degrees of freedom on a qubit flip preferentially at the same time. While these errors can be treated as burst errors in the factor graph, there is no obvious natural way to include burst errors directly in the Ising Hamiltonian. We show in the next section how arbitrary single qubit error models can be represented as Ising models, and in turn how they can be represented as weighted factor graphs where burst errors need not be explicitly included to represent arbitrary single qubit error models.
\subsection{Deducing the error pattern from Ising decoding}
We now briefly discuss how the logical error pattern can be deduced from the Ising model. This can be done in two ways. Firstly for MaxEnt the error pattern is deduced by examining the lowest energy configuration found (or one of the lowest energy configurations chosen at random in the event of a tie) and calculating what logical corrections are needed. For MaxEnt, given a collection of configurations sampled with Boltzmann weighted probabilities, logical errors can be deduced by taking a `vote' of all sampled configurations for each qubit, i.e. by tabulating the number for which no correction, an $X$ correction, a $Z$ correction, or a $Y$ correction is appropriate. The correction with the most `votes' is the one which is most likely to correct the information when averaging over all possible error patterns. If information about other kinds of errors is required, then a similar approach can be followed.
\subsection{Hamiltonian terms for $Y$ errors}
Recall that the maximum likelihood error configuration(s) needed for MLE decoding are recovered from MaxEnt (maximum entropy) inference by isolating the most probable (lowest energy) configuration(s). This can be shown directly by observing that at a sufficiently low temperature, the Boltzmann distribution on an Ising model will be dominated by the lowest energy configuration(s). It is therefore sufficient to demonstrate an Ising model construction where the Boltzmann distributions can be used to perform MaxEnt inference for arbitrary $X$, $Y$, and $Z$ error rates, as such a model will necessarily also be able to perform MLE inference by focusing only on the lowest energy states (equivalent to a $T=0$ Boltzmann distribution).
We need to consider both errors on the data qubits and the parity check qubits. Because it is conceptually simpler, we start with the case of a data qubit. In the factor graph representation, such a qubit is represented by two classical bit variables, one for the bit information of the qubit and one for the phase information. These bits can be represented as Ising spins. In an Ising model representation, these two spins can have a coupling between them. We therefore write the total Hamiltonian as follows
\begin{equation}
H^{\mathrm{data}}=-h_1\sigma^z_1-h_2\sigma^z_2-J\,\sigma^z_1\sigma^z_2. \label{H_data}
\end{equation}
To map a full $X$, $Z$, $Y$ error model we must map the probabilities of these spins being in the $+1,+1$, $-1,+1$, $+1,-1$ or $-1,-1$ state to the no error, bit flip error only ($p_x(1-p_z)$), phase flip error only ($p_z(1-p_x)$), and bit and phase flip error ($p_{xz}=p_xp_z+p_y$) probabilities respectively for a Boltzmann distribution \eqref{BD} obtained for this Hamiltonian at a finite temperature $T$.
To avoid having to calculate the partition function, we compare ratios of probabilities between the Boltzmann distribution and the distribution we are trying to emulate, for instance
\begin{eqnarray}
\frac{p_{(-1,+1)}}{p_{(+1,+1)}}&=&\exp\left(-2\frac{h_1+J}{T}\right) \nonumber \\
&=&\frac{p_x(1-p_z)}{1-p_x-p_{xz}-p_z+2p_xp_z}\equiv\bar{p}_x \label{data_x}
\end{eqnarray}
where the definition of $\bar{p}_x$ is for mathematical convenience in later calculations. We furthermore set
\begin{eqnarray}
\frac{p_{(+1,-1)}}{p_{(+1,+1)}}&=&\exp\left(-2\frac{h_2+J}{T}\right)\nonumber \\
&=&\frac{p_z(1-p_x)}{1-p_x-p_{xz}-p_z+2p_xp_z}\equiv\bar{p}_z, \label{data_z}
\end{eqnarray}
and
\begin{eqnarray}
\frac{p_{(-1,-1)}}{p_{(+1,+1)}}&=&\exp\left(-2\frac{h_1+h_2}{T}\right)\nonumber \\
&=&\frac{p_{xz}}{1-p_x-p_{xz}-p_z+2p_xp_z}\equiv\bar{p}_{xz}. \label{data_xz}
\end{eqnarray}
We now have three equations and three unknowns, solving for the terms in Eq. \ref{H_data}, we find that,
\begin{eqnarray}
J&=&\frac{T}{4}\left[-\ln(\bar{p_x})-\ln(\bar{p_z})+\ln(\bar{p}_{xz})\right] \nonumber \\
h_1&=&-\frac{T}{2}\,\ln(\bar{p}_x)-J \nonumber \\
h_2&=&-\frac{T}{2}\,\ln(\bar{p}_z)-J.
\end{eqnarray}
Now we turn to the parity checking qubits, each of which is represented by only a single bit corresponding to the phase degree of freedom. Whether or not the bit degree of freedom has been errored can be inferred by checking whether the measured parity value agrees with the parity of the errors it detected. Stated another way, if we deduce that an odd (even) number of bits in the parity check were errored and the parity check shows and even (odd) parity, then we conclude that the bit degree of freedom was corrupted. As discussed in the previous section, we encode the measured error value in the sign of the parity check over the bits $Q$. If no error has been detected, then the coupling strength in the Ising mapping will be negative (ferromagnetic). Otherwise it will be positive (anti-ferromagnetic). To take into account the effects of $Y$ errors on the parity check as well as the phase degree of freedom, we introduce an additional multi-body term which acts on the phase bit ($b_p$) as well as the bits within the parity check \footnote{Or, in the case where the phase bit is already contained in $Q$, we add a new term which acts on all of the bits in the parity check \emph{except} for the phase bit}. The resultant Hamiltonian parity check term takes the form
\begin{eqnarray}
H^{\mathrm{par}}_\pm(Q,b_p)=-h_{\pm}\sigma^z_{b_p}-a_{Q\pm} \prod_{i\in Q}\sigma^z_i - \nonumber \\
- a_{Q\,b_p\pm} \sigma^z_{b_p} \prod_{i\in Q}\sigma^z_i .
\end{eqnarray}
As was done for the data qubits, focusing first on the $+$ case, where an error is not detected, we solve for ratios of probabilities
\begin{align}
&\frac{p_{(-1,+1)}}{p_{(+1,+1)}}=\exp(-2\frac{a_{Q+}+ a_{Q\,b_p+} }{T})=\bar{p}_x \nonumber \\
&\frac{p_{(+1,-1)}}{p_{(+1,+1)}}=\exp(-2\frac{h_++ a_{Q\,b_p+}}{T})=\bar{p}_z \nonumber \\
&\frac{p_{(-1,-1)}}{p_{(+1,+1)}}=\exp(-2\frac{h_++ a_{Q+}}{T})=\bar{p}_{xz}.
\end{align}
These equations have the same mathematical structure as Eqs. \eqref{data_x}, \eqref{data_z}, and \eqref{data_xz}, it follows immediately that
\begin{eqnarray}
a_{Q\,b_p+}&=&\frac{T}{4}\left[-\ln(\bar{p_x})-\ln(\bar{p_z})+\ln(\bar{p}_{xz})\right] \nonumber \\
a_{Q+}&=&-\frac{T}{2}\,\ln(\bar{p}_x)-a_{Q\,b_p+} \nonumber \\
h_+&=&-\frac{T}{2}\,\ln(\bar{p}_z)-a_{Q\,b_p+}.
\end{eqnarray}
The task now remains to find the values for the $-$ case, in other words when an error has been detected. In this case we have
\begin{eqnarray}
\frac{p_{(-1,+1)}}{p_{(+1,+1)}}&=&\exp(-2\frac{a_{Q-}+a_{Q\,b_p-} }{T})=\bar{p}_x^{-1} \nonumber \\
\frac{p_{(+1,-1)}}{p_{(+1,+1)}}&=&\exp(-2\frac{h_-+ a_{Q\,b_p-}}{T})=\bar{p}_{xz}\bar{p}_x^{-1} \nonumber \\
\frac{p_{(-1,-1)}}{p_{(+1,+1)}}&=&\exp(-2\frac{h_-+a_{Q-}}{T})=\bar{p}_{z}\bar{p}_x^{-1}.
\end{eqnarray}
Except for the difference in the RHS, these equations again have the same mathematical structure. We can solve for the relevant terms as follows
\begin{eqnarray}
a_{Q\,b_p-}&=&\frac{T}{4}\left[\ln(\bar{p}_{x})-\ln(\bar{p}_{xz})+\ln(\bar{p}_{z}))\right] \nonumber \\
a_{Q-}&=&\frac{T}{2}\,\ln(\bar{p}_x)-a_{Q\,b_p-} \nonumber \\
h_-&=&-\frac{T}{2}\,\left[\ln(\bar{p}_{xz})-\ln(\bar{p}_{x})\right]-a_{Q\,b_p-}.
\end{eqnarray}
With the above, we have now completed all of the necessary terms to construct a complete Ising Hamiltonian for which the Boltzmann distribution performs maximum entropy inference on a general quantum error correcting code. In the next subsection we discuss the full Hamiltonian, as well as its expression as a factor graph.
\subsection{Full Hamiltonian and factor graph representation}
Now that we have constructed all of the necessary pieces, lets construct the whole Hamiltonian for maximum entropy inference on a CPC code. This Hamiltonian will involve $k$ Ising spins which represent the bit degrees of freedom on the data qubits $b$, $k$ spins which represent the phase information $p$, and $n-k$ spins which represent the phase degrees of freedom on parity check qubits. The Hamiltonian takes the form
\begin{eqnarray}
H_{\mathrm{decode}}&=&\sum^k_{i=1}(-h_1\sigma^z_{b_i}-h_2\sigma^z_{p_i}-J\sigma^z_{b_i}\sigma^z_{p_i}) + \nonumber \\
&&+\sum^{n-k}_{j=1}H^{\mathrm{par}}_{s_j}(Q_j,p'_j),
\end{eqnarray}
where $Q_j$ is the set of degrees of freedom (represented by Ising spins) checked by the $j$th parity checking qubit, and $s_j\in\{+1,-1\}$ is the syndrome measurement for that qubit.
Because this Hamiltonian has been constructed so that Boltzmann distributions with respect to it reproduce error probabilities, the probability of an individual error configuration will be proportional to $\exp(-E/T)$, the exponent of that configuration's energy. We therefore deduce by the fact that this function decreases monotonically, that the lowest energy configuration with respect to this Hamiltonian is also the single most likely one. Therefore for a Hamiltonian constructed at \emph{any} finite positive temperature $T$, the maximum likelihood configuration is the ground state.
In addition to being represented as a Hamiltonian, the decoding problem including $Y$ errors can also be represented as a (weighted) factor graph. This can be done by examining the interactions in the Hamiltonian and constructing a graphical model from them. This model is similar to the original factor graph considered in \cite{Roffe18a}, but has one additional factor per qubit to adjust the error probabilities to include $Y$. The rules given in \cite{Roffe18a} can be modified as follow to explicitly include a $Y$ error in the factor graph, rather than treating it as a burst error on the code. The first modification is that when mapping a data qubit (represented by a triangle, \scalebox{.5}{\input{tikzit/data_qubit.tikz}} in the graphical language of that paper), an additional weighted factor needs to be added to include correlations between the bit and phase degrees of freedom of that qubit. The translation of a data qubit in the operational representation is therefore
\begin{equation}
\input{tikzit/unmeasured_definition_Y.tikz},
\end{equation}
where the color of the factor is added to emphasize that it is weighted differently \footnote{or if considered as a hard constriant that the isolated data bit which the node connects to is weighted differently.} from the parity checks. For the parity check qubits, (denoted by a star \scalebox{.5}{\input{tikzit/parity_qubit.tikz}}) two separate cases need to be considered, both the cases with and without a self loop. Without the self loop the definition is
\begin{equation}
\input{tikzit/measurement_definition_Y.tikz},
\end{equation}
where edges extending off the edge of the figure correspond to the parity checks which the qubit performs on other qubits. The magenta color indicates the additional weighted factor to take the $Y$ errors into account. Edges representing other qubits checking the phase qubit have been omitted for visual clarity. The final case we need to consider is the case where the parity check qubit has a self-loop in the annotated operational representation. In this case the additional factor interacts with all of the qubits the parity qubit checks, but not the phase bit
\begin{equation}
\input{tikzit/measurement_definition_Yself.tikz}.
\end{equation}
Edges representing other qubits checking the parity check qubit have again been omitted for visual clarity. Aside from these variations of the node definitions, the explicit factor graph construction for a model including $Y$ errors is exactly the same as the one given in \cite{Roffe18a}.
\subsection{Decoding over time}
The foregoing discussion has implicitly assumed that the gates comprising the decoder/stabilizer measurements do not themselves introduce error. To be fully general, all quantum decoders must be able to tolerate gate errors. The standard solution in QEC to process syndrome data fully fault-tolerantly is for decoding to happen over time as well as space (see for example \cite{Dennis2001} in the context of surface codes). The idea here is that not only is the syndrome measurement at one time used to determine the likely event, but rather an entire sequence of syndrome measurements is used to reconstruct the most likely series of events. The Ising and factor graph approaches can be extended to this kind of decoding as well. If we first consider that each qubit starts out in the unerrored state, we begin with the standard factor graph or Ising model for the single-shot decoding from the first round of syndrome gathering. For the second round however, rather than single body terms corresponding to whether or not a qubit has been errored, another copy of the Ising model description of the code should be added. However, rather than having single body terms, it should have couplings to the previous copy. If an error has occurred in the previous round and persists, it should trigger the same syndromes. The single body terms on data qubits thus become two body terms coupling to the previous round, and likewise the two body terms become four body terms. The terms relating to parity check qubits remain unchanged, as syndrome information is measured and not carried forward from round to round.
Mathematically, the transformation from single round decoding to decoding over time transforms Eq.~\ref{H_data} to a comparison with the values observed at the previous time,
\begin{eqnarray}
H^{\mathrm{data}}=h_1\sigma^z_{1,t-1}\sigma^z_{1,t}+h_2\sigma^z_{2,t-1}\sigma^z_{2,t}- \nonumber \\
-J\,\sigma^z_{1,t-1}\sigma^z_{2,t-1}\sigma^z_{1,t}\sigma^z_{2,t}, \label{H_data_time}
\end{eqnarray}
where $t$ and $t-1$ are indices to indicate slices representing syndrome measurements at different times. The quantities $h_1$, $h_2$, and $J$ are the same as in Eq.~\ref{H_data} because the statistics of the errors is agnostic to how the correction is performed. As with decoding at a single time, the interactions on a single data qubit can be expressed as a factor graph, with a pair of bits per measurement cycle,
\begin{equation}
\input{tikzit/over_time.tikz}.
\end{equation}
Each factor is labelled with its weight, and edges which do not connect to vertexes indicate interactions with the parity checks within a time slice. Note that except for the leftmost set of classical bits, there are no single body terms.
\section{Numerical results}
\begin{figure}
\includegraphics[width=6cm]{5_1_3_thresh}
\begin{centering}
\caption{\label{fig:5_1_3_thresh} Probability of a logical $X$, $Z$ or $Y$ in a $[[5,1,3]]$ code for $p_x=p_z=p$ and $p_y=0$. Blue is maximum likelihood decoding, green is maximum entropy, and red is for data which is encoded but no corrective action is taken. The black line is the raw averaged error probability of an unprotected qubit subject to the same errors. Dotted lines are guides to the eye to show the threshold under the two decoding strategies. The upper inset is a zoom on the threshold while the lower inset shows a wider range of $p$.}
\par
\end{centering}
\end{figure}
We now compare numerical results of maximum entropy and maximum likelihood inference for two codes, a $[[5,1,3]]$ code and a $[[9,3,3]]$ code, the CPC matrices for which can be found in the appendix. We choose small codes where the probabilities can be calculated exhaustively and so that there is no statistical error in our calculations and thresholds can readily be calculated by bisection. We consider a simple error model with a single round of error correction and perfect syndrome collection. This is appropriate as the goal of these calculations is to provide proof-of-principle for our decoding methods, not to demonstrate the practicality of the codes.
For the $[[5,1,3]]$ code, we consider a situation where $p_x=p_z=p$ and $p_y=0$, in other words equal probability of dephasing or bit flip errors, but no error which independently implements a combined flip and dephasing error. The results of this decoding can be found in Fig.~\ref{fig:5_1_3_thresh}. This figure demonstrates that not only does the maximum entropy strategy reduce the probability of a logical error compared to maximum likelihood, it also moves the threshold, defined as the point for which the probability of an error exceeds the probability for an unprotected qubit, by approximately $0.5\%$. For a summary of threshold values, see table \ref{tab:thresholds}.
\begin{figure}
\includegraphics[width=6cm]{9_3_3_thresh}
\begin{centering}
\caption{\label{fig:9_3_3_thresh} Probability of a logical $X$, $Z$ or $Y$ averaged over the logical degrees of freedom for a $[[9,3,3]]$ code for $p_x=p_z=p$ and $p_y=0.1 p$ (rates with no independent $Y$ error are depicted as dashed lines for comparison). Blue is maximum likelihood decoding, green is maximum entropy, and red is for data which is encoded but no corrective action is taken. The black line is the raw averaged error probability of an unprotected qubit subject to the same errors. Dotted lines are guides to the eye to show the threshold under the two decoding strategies. The inset is a zoom on the threshold.}
\par
\end{centering}
\end{figure}
As a more sophisticated example, we consider a $[[9,3,3]]$ code with $p_x=p_z=p$ and $p_y=0.1 p$, the results for which are shown in Fig.~ \ref{fig:9_3_3_thresh}. This more sophisticated multi-qubit code again shows better decoding from a maximum entropy strategy and a shift in the threshold. All threshold values are depicted in table \ref{tab:thresholds}.
One final case which is often considered theoretically is the case of completely isotropic error, $p_x=p_z=p_{xz}=p$. Solving for $p_y$ we find that this case corresponds to $p_y=p-2\,p^2$. The results for the $[[9,3,3]]$ code with isotropic error appear in Fig.~\ref{fig:9_3_3_iso}.
\begin{figure}
\includegraphics[width=6cm]{9_3_3_iso}
\begin{centering}
\caption{\label{fig:9_3_3_iso} Probability of a logical $X$, $Z$ or $Y$ averaged over the logical degrees of freedom for a $[[9,3,3]]$ code for $p_x=p_z=p$ and $p_y=p-2\,p^2$, corresponding to isotropic errors (rates with no independent $Y$ error are depicted as dashed lines for comparison). Blue is maximum likelihood decoding, green is maximum entropy, and red is for data which is encoded but no corrective action is taken. The black line is the raw averaged error probability of an unprotected qubit subject to the same errors. Dotted lines are guides to the eye to show the threshold under the two decoding strategies.}
\par
\end{centering}
\end{figure}
\begin{table*}
\begin{tabular}{|c|c|c|c|}
\hline
code type & MLE threshold & MaxEnt threshold & difference\tabularnewline
\hline
\hline
$[[5,1,3]]$, $p_{y}=0$ & $p=0.07989$ & $p=0.08460$ & $0.004708$\tabularnewline
\hline
$[[9,3,3]]$, $p_{y}=0$ & $p=0.03005$ & $p=0.03358$ & $0.003534$\tabularnewline
\hline
$[[9,3,3]]$, $p_{y}=0.1p$ & $p=0.02760$ & $p=0.03000$ & $0.002401$\tabularnewline
\hline
$[[9,3,3]]$, $p_{y}=p-2\,p^2$ & $p=0.01701$ & $p=0.01880$ & $0.001789$\tabularnewline
\hline
\end{tabular}
\caption{\label{tab:thresholds} Thresholds of maximum entropy (MaxEnt) and maximum likelihood (MLE) strategies for different codes considered in this paper.}
\end{table*}
\section{Implementation}
The Ising model representation of quantum error correcting codes has a fixed structure, in other words, the measured syndrome values $\{s\}$ affect the magnitude and sign of different coupling elements, but not which variables interact with which other variables. This structure means that any decoder could be constructed in an \emph{application specific} way, much in the way that an application specific integrated circuit (ASIC) approach is used in some cases in electrical engineering. In fact for superconducting circuit annealers and many quantum inspired implementations, the decoder itself would actually be an ASIC, but in the interest of generality, we refer to hardware which is designed to only perform quantum error correction as an application specific processor.
The application specific approach affords several advantages over using general purpose Ising hardware. Probably most importantly, the hardware can be designed to minimize or eliminate the costs associated with embedding the problem. It is generally recognized that the embedding overheads are a major obstacle for current quantum annealer technology \cite{Biswas17a,Rieffel15a,Hammerly18a}. For a quantum annealer, this would require multi-body interactions, however there are several ideas currently in the literature on how this can be done in hardware \cite{Chancellor17b,Leib16a,Rocchetto201a,Lechner2015a,Zhao17a,MelansonAQC2018}. It could also be done using a variety of mapping techniques from multi-body to two-body couplers (a process known as `quadratization'). For a review of these techniques see \cite{Dattani18a}. Additionally, if the decoding is made with the same type of hardware as the quantum computer it is correcting (superconducting circuits or trapped atoms/ions for instance), it is likely that the connectivity constraints on the two devices will be similar and that an efficient embedding of the decoding problem can be found.
An additional benefit of an application specific approach is that the control requirements would be much less demanding, as the device would only have to be able to have a number of user controlled binary variables equal to the number of syndromes, rather than to approximate a continuum of strengths for each qubit and coupler. On currently implemented quantum annealing hardware, the area taken up by the classical control digital to analog converters (DACs) can be a limiting factor in chip design \cite{Bunyk2014a}, reducing the the complexity of the controls would free up area for qubits and couplers and allow for more powerful devices.
Another benefit of using specialized hardware to decode is that the decoding hardware, whether quantum or classical, can be directly integrated with the quantum computer it is correcting. For a superconducting circuit architecture this would mean having quantum and/or classical error correction hardware in the same cryostat, thus cutting down on requirements to communicate with devices at room temperature and reducing the heat flux incident on the device. This approach could also reduce communication latency, and potential delays waiting for results to be returned from more traditional classical computing software.
Decoding could also potentially be performed using hybrid quantum/classical techniques, for instance a superconducting quantum annealer could be paired with a cryo-CMOS or classical superconducting ASIC which uses belief propagation, or other classical techniques to find high quality solutions, but which are not distributed in a Boltzmann distribution. Reverse annealing \cite{Chancellor17a,RA_whitepaper,Perdomo-Ortiz11} could then use these as a starting point to find an approximate Boltzmann distribution for use in MaxEnt inference. In particular, it has recently been shown that two orders of magninitude improvement in solution time is possible through reverse annealing even when starting from a solution found by a simple greedy search \cite{Venturelli18a}. A graphical representation of information flow in an application specific annealer accelerated error correction unit appears in Fig.~\ref{fig:ASIC_annealer_block}.
\begin{figure}[t]
\begin{centering}
\includegraphics[width=8cm]{ASIC_annealer_block}
\end{centering}
\caption{ Information flow in a hybrid quantum/classical annealer accelerated error correction setup. Classical hardware finds high probability error configuration which are then converted an approximate sampling of a Boltzmann distribution with the aid of a quantum annealer. \label{fig:ASIC_annealer_block}}
\end{figure}
\section{Discussion and outlook}
In this work we have developed a general way to map the problem of decoding quantum error correction to specialized Ising model hardware, which works for any code describable in the CPC formalism (at least all CSS codes, if not all stabilizer codes). Our methods are general in the sense that depending on the capabilities of the Ising hardware they can perform either maximum likelihood or maximum entropy decoding. We have demonstrated numerically that maximum entropy decoding using our formalism can improve the decoding of a small code.
Since large scale quantum annealers actually exist, it would be feasible to test the performance of these devices with simulated large scale quantum error correction. Similar studies have already been performed for classical error correction in \cite{chancellor16b}, and an extension to more complex codes is forthcoming \cite{DWclassicalECinPrep}. The choice to map small codes in this paper was only made for presentation and to make the exact thermal distribution calculable. The complexity of our mapping from quantum codes to Ising models scales mildly with code size, and therefore would be feasible even for very large quantum error correction codes.
\acknowledgements
JR acknowledges the support of the QCDA project which
has received funding from the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union's Horizon 2020 Programme. SZ acknowledges support from the Oxford-Man Institute for Quantitative Finance. DH acknowledges support from the ``Investissements d'avenir'' (ANR-15-IDEX-02) program of the French National Research Agency. NC acknowledges support from from EPSRC grant refs EP/L022303/1 and EP/S00114X/1.
|
2,869,038,156,370 | arxiv | \section{Introduction}
One of the interesting aspects of developmental processes is that one could get multiple heritable cell fates without irreversible changes to the genetic information. Heritable differences in phenotype, despite having the same genetic information, goes by the name of epigenetic phenomenon. Apart from its fundamental role in development, epigenetic effects are of great importance in certain diseases like cancer \cite{Epigenetics04, EpigeneticsAllis}. There are many mechanisms that could lead to epigenetic effects. One of these mechanisms is transcriptional silencing.
Regions of eukaryotic chromosomes could be divided into euchromatin and heterochromatin, based on the degree of condensation during interphase, the period between nuclear divisions.
Heterochromatin refers to condensed domains where the nucleosomes, with DNA spooled around, are packed into higher order structures. Genes in this area, as opposed to genes in less condensed euchromatin areas, are not normally transcriptionally active. Therefore, the formation of heterochromatin is a way of silencing the expression of a number of adjacent genes. Furthermore, in many circumstances, the organization of chromosomes into heterochromatin and euchromatin regions will be inherited by the new cells generated from cell divisions \cite{MBoC}. As a result, heterochromatin formation plays a crucial role in multi-cellular development by stabilizing gene expression patterns in specialized cells. One example of this is the cell type dependent silencing of Hox genes, important in development of body plans, by the Polycomb group of proteins \cite{Gilbert}.
One might distill some overarching similarities from various mechanisms proposed for silencing in diverse organisms \cite{GrewalMoazed}.
In the general model, there is usually a region that nucleates silencing by recruiting a silencing complex incorporating a histone modifying enzyme (figure 1). Modification of some of the lysines in the histone tails leads to binding by components of silencing complex, which, in turn, recruits further histone modifying enzymes. That is how the process propagates till it meets some boundary element (or the system reaches a stationary state due to exhaustion of one of the components of the silencing complex).
Budding yeast, \emph{S.\ cerevisiae}, played an important role in understanding how chromatin silencing works.
In the budding yeast, there are three kinds of regions that are silenced: the telomeres, the ribosomal DNA and the silent mating type loci. There are two silent mating type loci on chromosome III, \emph{HML} and \emph{HMR}, flanking the active mating-type locus. \emph{HML} and \emph{HMR} contain copies of genes that decide $\alpha$-type and a-type identity, respectively. Information from the silent mating-type loci gets copied into the active mating locus through a recombination mediated process called mating type switching. The recombination event is initiated by a double strand break in the mating locus by the HO endonulease in haploid yeasts. The study of mating type in yeast shed light on many fundamental biological questions \cite{MBoC, MCB}.
The mechanism by which silencing nucleates and spreads in budding yeast is relatively well investigated \cite{Rusche, Moazed} and provides a concrete example of the more general model mentioned before. It is known that the Silenced Information Regulator (SIR) proteins are the main players in gene silencing at telomeres and mating type loci in yeast. As discussed above, the model for step-wise gene silencing in S. cerevisiae (figure 1), also posits that silencing happens in two distinct steps: \emph{nucleation} and \emph{spreading}. To be concrete, let us discuss silencing at the silent mating loci. In the nucleation step, with the help of site-specific DNA binding proteins (like Rap1) and with Sir1 as a tether, Sir2, Sir3 and Sir4 will form a Sir Complex on the nucleation site. Deacetylation of certain lysines on the neighboring histones H3 and H4, by Sir2 (a $NAD^+$ dependent histone deacetylase (HDAC)) will make binding of Sir3/Sir4 complex easier in the neighborhood of the original nucleation site. Sir3/Sir4, in turn, would recruit more Sir2. Hence, the spreading starts. More deacetylation of histone tails improves the recruitment of other Sir proteins and formation of more stable complexes on neighboring sites. If histone deacetylation is transferred further on, it will result in spreading of silencing to even distal sites. Although the nucleation step is different in telomeric silencing, the process of spreading seems to be very similar \cite{Aparicio}.
However, in the wild type budding yeast, the regions that are silenced are, typically, always silenced. To see epigenetic effects, one needs to ``weaken" the system.
As mentioned above, Sir1 is required for efficient nucleation of silencing at \emph{HMR}/\emph{HML} loci. Experiments on {\it sir1} mutants (where the nucleation effect is defective if not absent) show that the genes at \emph{HML} loci can be either repressed or derepressed in individual cells, representing two phenotypically distinguishable cells \cite{PillusRine}. Both states are stable to small fluctuations and are typically preserved in cell divisions - there is only a small probability of transitions back and forth between states - suggesting that the system is actually in a bistable regime.
In the systems biology community, epigenetic switches in prokaryotes have received quite a bit of attention. Multiple phenotypes are usually represented as multiple stable attractors in deterministic descriptions of the biochemical dynamics. Computational modeling of lambda phage \cite{Ptashne} has played a crucial role in the development of systems biology \cite{sheaackers, arkin}. From the response of lac operon in the presence of TMG \cite{NovickWeiner, ouden, Guet} to synthetic genetic networks like the toggle switch \cite{toggle}, mathematical analysis has been an integral part of understanding such phenomena. In particular, the biological model, in each of these examples, provides a mechanism of positive feedback. However, positive feedback is not sufficient to guarantee multistability, essential for giving rise to non-trivial epigenetic states.
The crucial aspect of the analysis of the mathematical model is computing the bifurcation diagram telling us which region in the space of control parameters is actually associated with bistability. The bifurcation diagram also indicates the qualitative behavior of the system when perturbed (or mutated) in a particular manner. In contrast to prokaryotic epigenetic switches just mentioned, modeling eukaryotic epigenetic silencing involves studying a spatially extended bistable system. Hence, the system shows interesting phenomena, like front propagation, allowing for a richer bifurcation diagram.
In this paper, we introduce a mathematical model of step-wise heterochromatin silencing. A mean field description of the dynamics explains many features of the real system. Epigenetic states, in the absence of efficient nucleation, can be explained as a consequence of the existence of two stable uniform static solutions: the hyper-acetylated state and silenced states on DNA. Studying the conditions under which the positive reinforcement inherent in the proposed silencing mechanism is strong enough to give rise to bistability is one of the main goals of our paper. In addition, the conditions required for static fronts will set additional constraints on the model. At the end, we propose experiments designed to test the ideas discussed here.
\section{Methods}
\subsection{Mathematical formulation of a model of silencing}
We formulate a quantitative version of the conventional biological model of silencing \cite{GrewalMoazed, Moazed}. The main variables involved in final equations are \emph{the local degree of acetylation} and \emph{the local probability of occupation by Sir complex}, both of which could depend on time, as well as on the position of nucleosomes on DNA, represented as a one-dimensional lattice. We define function, $S_{i}(t)$ on this lattice, as a number between $0$ and $1$, to represent fractional number of Sir complexes at site \emph{i}. Fractional degree of acetylation, $A_{i}(t)$, is defined in the same way too. Writing chemical equations,
in the mean-field treatment of the system, we get,
\begin{eqnarray}
\frac{dS_{i}(t)}{dt}
&=&\rho_{i}(t)(1-S_{i}(t))f(1-A_{i}(t))-\eta S_{i}(t)\label{SirEq}\\
\frac{dA_{i}(t)}{dt}
&=&\alpha(1-A_{i}(t))(1-S_{i}(t))-(\lambda+\sum_{j}\gamma_{ij}S_{j}(t)) A_{i}(t)\label{AceEq}
\end{eqnarray}
In equation (\ref{SirEq}), the first term on the right hand side is the Sir complex binding rate and the next term is the ``fall off" rate. The 3D concentration of ambient Sir complex at site $i$, which is generally a function of time, is denoted by $\rho_{i}(t)$. Since free Sir proteins in the environment do not form Sir complexes by themselves, this quantity actually represents a function of concentrations of all components (Sir3, Sir2 and Sir4) that are ready to make a Sir complex on the site. For example, in the simplest case, when each protein is in low abundance, this function would be proportional to the product of the three concentrations. However, throughout this paper we will not need to go into the details of this function. The function $f(x)$ dictates the cooperativity in Sir complex binding and should be a monotonically increasing function of $x$, $0\leq x\leq 1$. We use $f(x)=x^n$, where $n$ is the degree of cooperativity between deacetylated histone tails in recruiting Sir proteins. At last, $\eta$ is degradation rate of bound Sir complexes. In equation (\ref{AceEq}), as in equation (\ref{SirEq}), on \emph{RHS}, the first term advocates creation and next one degradation. The parameter $\alpha$ represents the constant acetylation rate\footnote{More generally, the acetylation rate could be $\alpha(1-A_i)(1+\sigma-S_i)$ allowing acetylation of histones in silencing complex bound nucleosomes, but adding this process does not make a qualitative difference.}. In the second term, the summation accounts for the contribution of adjacent Sir complexes in deacetylation of site $i$. Since Sir complex is only capable of deacetylation of sites in its neighborhood, $\gamma_{ij}$ is assumed to be symmetric with respect to its indices and drop significantly as $|i-j|$ gets large. Finally, $\lambda$ is the rate of deacetylation from the rest of deacetylase proteins. This rate is assumed to be a constant both in time and position.
\subsection{Generalized model including feedback from modulated transcription rate}
In the model formulated in the previous section, we allowed a certain degree of cooperativity in how deacetylated histone tails recruit the silencing complex. As we will see, this cooperativity would essential for having multistability within this model. However, the cooperativity in that particular interaction is not absolutely essential when we have other nonlinear effects in play.
One rather plausible effect is as follows. Transcription of a gene is often associated with a higher rate of acetylation of histone tails. It is believed to be one of the reasons why highly transcribed genes are hard to silence. For example, a tRNA gene, usually producing a large amount of RNA, has been found to have an important role in a silencing boundary \cite{Kamakaka}. One might therefore imagine that silencing, which affects local transcription rates, indirectly affects the local acetylation rate. One way to model this is to introduce an additional function $g(1-S_i)$ in the local acetylation rate making it
$\alpha(1-A_{i}(t))(1-S_{i}(t))g(1-S_i(t))$. If there is no such feedback from silencing, we could have $g(y)=1$. We will consider $g(y)\sim y^{m-1}$, $m=1$ being the case of no feedback, where as the simplest models of feedback would lead to $m=2$. For a general value of $m$ (and $n$) our model now would be given by the following equations.
\begin{eqnarray}
\frac{dS_{i}(t)}{dt}
&=&\rho_{i}(t)(1-S_{i}(t))(1-A_{i}(t))^n-\eta S_{i}(t)\label{SirEq-m}\\
\frac{dA_{i}(t)}{dt}
&=&\alpha(1-A_{i}(t))(1-S_{i}(t))^m-(\lambda+\sum_{j}\gamma_{ij}S_{j}(t)) A_{i}(t)\label{AceEq-m}
\end{eqnarray}
Thus, the nature of nonlinearity in these models is characterized by a number doublet $(m,n)$.
\subsection{Determining the nullclines and the bifurcation diagram}
One could analyze the uniform static solutions of equations and study the stability. The stationary states are obtained by solving the algebraic equations produced by setting time derivatives to zero. We analyze first the case where \emph{available} SIR concentrations are kept at a constant level. This means $\rho_i(t)=\rho,$ a time (and position) independent number.
Dropping all $i$ indices and replacing the non-local term $\sum_{j}\gamma_{ij}S_{j}$ with $\Gamma_0S$, we can rewrite equations as:
\begin{eqnarray}
\frac{dS(t)}{dt}
&=&\rho (1-S(t))f(1-A(t))-\eta S(t)\label{Uniform0}\\
\frac{dA(t)}{dt}
&=&\alpha(1-A(t))(1-S(t))g(1-S(t))-(\lambda+\Gamma_0 S(t)) A(t)
\label{Uniform}
\end{eqnarray}
Setting time derivatives to zero gives the nullclines. Taking derivatives of nullclines with respect to $A$ (or $S$) and setting these derivatives
equal to each other provides us with the condition for the saddle-node bifurcation determining the boundary of bistability region. Finally we write $\alpha$ and $\rho$, at this boundary, parametrically in terms of $S$. For example, for the $(1,n)$ models, where $n$ is the degree of cooperativity ($f(x)=x^n$), we have:
\begin{eqnarray}
\alpha
&=&\frac{(\lambda+\Gamma_0 S)^2}{(1-S)[(\Gamma_0+\lambda)(n-1)S-\lambda(1-S)]}\\
\rho
&=&\frac{\eta S}{(1-S)}\left(\frac{n(\Gamma_0+\lambda)S}{(\Gamma_0+\lambda)(n-1)S-\lambda(1-S)}\right)^n
\label{Nullclines}
\end{eqnarray}
\subsection{Study of non-uniform solutions}
In the following subsections, we go beyond analyzing the stable uniform solutions. In the region of parameter space where the system is bistable, it is possible to study how fronts between a silenced region and an unsilenced region move. In a system with a well defined free energy function, the average motion of a front or interface is determined by the difference of free energies of the two states across the front. Deterministically speaking the lower free energy state (usually called the stable state) would invade the metastable state with higher free energy. At the points where the two free energies are the same, the average front velocity is zero. Although, in non-equilibrium systems, like the one at hand, there is no useful free energy to be defined, one might still explore the region of parameter space where silenced state invades the unsilenced ones and vice versa (and the line in between where the front becomes stationary).
We study the motion of boundary between the two stable phases in the bistable parameter region both in the current discrete model and in a local continuum version of the model where the lattice is replaced by a continuous 1d system.
\subsection{Formulation of the continuum version of the model}
In the continuum formulation, we replace the index $i$ by a continuous variable $x$, $x=ia$, where $a$ is the lattice spacing. The variables $A_i,S_i$ become functions of $x$, namely $A(x), S(x)$. If we define $\gamma(z)$ to be $\gamma_{ij}/a$, where $z=(j-i)a$.
When $a$ tends to zero the expression $\sum_j\gamma_{ij}S_j$ becomes $\int dz \gamma(z)S(x+z)$. By Taylor expanding $S(x+z)$ in $z$ and keeping up to the second order term (note that, this approximation is valid since $\gamma(z)$ drops to zero as $z/a$ increases), we get
the local continuum versions of equations (\ref{SirEq}) and (\ref{AceEq}) which become
\begin{eqnarray}
\frac{\partial S(x,t)}{\partial t}&=&\rho(1-S(x,t))f(1-A(x,t))-\eta S(x,t)\\
\frac{\partial A(x,t)}{\partial t}&=&\alpha(1-A(x,t))(1-S(x,t))g(1-S(x,t))-\nonumber\\
& &\left(\lambda+\Gamma_{0}S(x,t)+\Gamma_{2}\frac{\partial^2 S(x,t)}{\partial x^2}\right)A(x,t)
\end{eqnarray}
where $\Gamma_{0}=\int\gamma(z)dz$ and $\Gamma_{2}=1/2\int\gamma(z)z^2dz$.
For each set of parameters, there is a front velocity \cite{AronsonWeinberger, CrossHohenberg, CompCellBio}, $c$, for which there is only ``one" (or none) continuous solution that represents a transition between the stationary states representing euchromatin and silenced heterochromatin.
\subsection{Computing the zero velocity line in the continuum model}
The analysis of the continuum system follows the standard route \cite{AronsonWeinberger, CrossHohenberg, CompCellBio}. Assuming $A(x,t)=A(x-ct)$ and $S(x,t)=S(x-ct)$, and using $u=x-ct$:
\begin{eqnarray}
c\frac{\ dS(u)}{\ du}+\rho(1-S(u))f(1-A(u))-\eta S(u)&=&0\label{SirEqContSimple}\\
c\frac{\ dA(u)}{\ du}+\alpha(1-A(u))(1-S(u))g(1-S(u))-& &\nonumber\\
\left(\lambda+\Gamma_{0}S(u)+\Gamma_{2}\frac{\ d^2 S(u)}{\ du^2}\right)A(u)&=&0\label{AceEqContSimple}
\end{eqnarray}
The analysis of problem for $c=0$ is considerably simpler, since equation (\ref{SirEqContSimple}) turns out to be an algebraic equation allowing $A$ to be expressed in terms of $S$, namely, $$A(S)=1-f^{-1}(\frac{\eta S}{\rho (1-S)}).$$ We define a potential $V(S)=-\lambda S-\Gamma_0 S^2/2+\alpha \int dS (1-A(S))(1-S)g(1-S)/A(S)$, so that other equation, namely equation (\ref{AceEqContSimple}), could be written as
\begin{equation}
\Gamma_{2}\frac{\ d^2 S}{\ dz^2}=\frac{\ dV(S)}{\ dS}
\end{equation}
The values of parameters, for which the potential $V(S)$ has two local minima with equal potential values, correspond to existence of a zero velocity front. Note that we could use this potential only to describe zero velocity fronts, and not for the general traveling solution.
\subsection{Computing the zero velocity line in the discrete model}
Rewriting nullclines for discrete model:
\begin{eqnarray}
S_{i}
&=&\frac{\rho(1-A_{i})^n}{\eta+\rho(1-A_{i})^n}\label{Uniform1}\\
A_{i}
&=&\frac{\alpha(1-S_{i})^{m}}{\lambda+\sum_{j}\gamma_{ij}S_{j}+\alpha(1-S_{i})^{m}}
\label{Uniform2}
\end{eqnarray}
we checked whether above equations allow a zero velocity barrier as a fixed point. For each pair of $\alpha$ and $\rho$ inside the bistable region, starting with an array of $i_{max}=100$ sites,
with fixed boundary conditions and initial condition of half recursive sites silenced/deacetylated and the remaining unsilenced/acetylated (with values of $S$
and $A$ obtained from fixed points of corresponding \emph{uniform} nullclines, obtained by setting the right hand sides of the equations (\ref{Uniform0}) and (\ref{Uniform}) to zero), we applied equations (\ref{Uniform1}) and (\ref{Uniform2}) respectively. By iteration we let
the system evolve until it reached its fixed point (within an error of 0.01\%).
\subsection{Dynamics with Sir protein depletion}
Finally, we consider the effect of $\rho(t)$ not being constant. We model the limited supply of Sir proteins by putting a constraint on total number of Sir complexes, in solution and on the DNA.
Going back to original equations ( \ref{SirEq}) and (\ref{AceEq}), we need to replace the constant $\rho_{i}(t)$ by a $\rho(t)$ which is given as follows:
\begin{equation}
\rho(t)=(S_{total}-\sum_{k}S_{k}(t))/V
\end{equation}
where $S_{total}$ is total number of functional Sir complexes in the system, which is constant and $V$ represents the average volume.
\section{Results}
\subsection{Bifurcation analysis of the model of silencing}
One could analyze the uniform static solutions of equations and study the stability. The stationary states are obtained by solving the algebraic equations produced by setting time derivatives to zero. We analyze first the case where \emph{available} SIR concentrations are kept at a constant level. This means $\rho_i(t)=\rho,$ a time (and position) independent number. One can see that, for $f(x)=x^n, g(y)=1$, depending on chemical parameters, one can get either one or three fixed points (figure 2), provided $n>1$. The three fixed point case always includes two stable points enclosing the other unstable saddle point, so the system is actually in a bistable state as could be seen by local analysis (figure 3). One of the two states has low acetylation and higher chance of repression, while the other state has a high degree of acetylation and higher chance of derepression (figure 3).
The bifurcation diagram, indicating regions in the parameter space of $\rho$ and $\alpha$ leading to monostability and to bistability, is shown in figure 4 (solid lines). Note that the critical point of this bifurcation is at low availability silencing factors coupled with low rate of acetylation. This feature will have an important implication when we consider mutants lacking particular acetyltransferases.
The constraint that $n$ needs to be greater than one suggests that in the simplest models (namely the $(1,n)$ models), we need a certain degree of cooperativity in recruitment of Sir complexes. Currently there in not much evidence for or against such a cooperative effect from deacetylated histone side chains. In the next subsection, we point out that in a more generalized model, such an effect is not essential.
\subsection{Cooperativity versus feedback from modulated transcription rate}
We discussed $(1,n)$ models in the previous section and found that we need $n$ to be greater than one for these subclass of models. Therefore, we analyzed the more general model with transcriptional feedback, described by equations (\ref{SirEq-m}) and (\ref{AceEq-m}). We also found that if we let $m=2$, we could get bistability with $n=1$ (results not shown). Moreover, the structure of the bifurcation diagram is essentially the same. Since both kinds of models, those with Sir binding depending strongly nonlinearly on the degree of deacetylation, as well as those where the effect of silencing on local transcription feeds back on the acetylation rate, provide qualitatively similar results for many of the predictions to be made in later sections, we will continue to show the results of the $(1,n)$ models, fully keeping in mind that there is a broader class of models leading to the same qualitative predictions.
\subsection{Nonuniform solutions and front propagation}
For the purpose of this paper, we would focus on the part of the parameter space where the front velocity is zero. The zero velocity line is represented by the `dashed' line in figure 3. This line divides the bistable region into two parts. In the upper half, a front would move abolishing silencing, whereas, in the lower half, the movement would spread silencing.
We have also studied the discrete model directly. As expected, discrete model gives a band in the parameter space for front propagation failure \cite{Keener}. The boundaries of the band are represented by the dotted lines in figure 3.
This band shrinks to the zero velocity line as one takes the continuum limit. One might ask which of these descriptions are closer to reality. If we count each nucleosome as a unit and expect one silencing complex per nucleosome, then that provides us with a natural lattice spacing. However, the nucleosomes are not quite static. They could move around or disappear (if the histone octamer falls off DNA). If the time scale of nucleosome dynamics is much slower than that of the silencing process, then we are justified in taking the nucleosome array as a lattice to operate upon. If the time scales are the other way around, we might average out the nucleosome fluctuations and get an effective continuum description. The truth probably is somewhere in between, leading to a fuzzy region of low front mobility crossing over to high front mobility regions above and below in the bifurcation diagram.
\subsection{The role of finite supply of Sir proteins}
The previous discussion assumed that the available ambient concentrations of Sir proteins were constant, reflected in $\rho$ being held constant. We could use our insights, into the bifurcation diagram, to infer what would happen if the total number of Sir proteins (the sum of those in solution and those bound to DNA) were fixed. This is particularly interesting in the bistable region.
Our treatment is very similar to studying phase equilibrium with a fixed number of particles. For example, consider a liquid gas mixture at a constant temperature in a fixed volume with fixed number of particles, and imagine that there is an interface between the two states. The interface moves, and the fractions of particles in the different states change till chemical potential of the two states become equal. Under this final condition, the interface does not move anymore, apart from thermal fluctuations around the average position. As we noted before, in our problem, we may not define a free energy, but we could indeed talk about average movement of interface between two states, namely the front, and the condition under which the interface stops moving.
Depending upon the size of the silenced region, one would get interesting titration effects in this model. Suppose we are in the bistable region of the parameter space and start from the locally stable uniform unsilenced solution. Let us ask, what happens if we nucleate a small region of silencing, say, by tethering a protein that recruits the silencing factors locally. If we are in the upper half of the bistable region (Region I in figure 4), high acetylation rate makes difficult for silenced region to spread into the unsilenced region. In that case, in the deterministic model, silencing is going to remain localized around the region of recruitment.
Now, if the acetylation rate, $\alpha$, is tuned down, say, by knocking off an acetyltransferase; we go into the region where the silencing can spread into unsilenced DNA (Region II in figure 4). So, naively, we expect much more silencing.
However, since $\rho$ is no longer fixed, as silencing spreads, $\rho(t)$ reduces. At this point one of the two following things could happen. The silencing could stop at special boundary elements on DNA where some other process stops the spread of silencing \cite{BiBroach, Kamakaka}. Alternatively, the front could stop because $\rho(t)$ reduces enough to reach a point on bifurcation diagram where the propagation velocity is zero. Thus, in this case, the effect of reducing $\alpha$ is to effectively reduce $\rho$ as well, so that the system always stays on the zero velocity region. Note that, if there are more than one region in DNA where silencing spreads by the same mechanism and if at least one of these regions does not possess a boundary element, then we are led to the same situation, namely $\rho$ reducing enough to stop front movement. We will explore the biological consequence of this observation next.
\subsection{Predictions from the Model}
The bifurcation diagram presents a classification of qualitatively different kinds of dynamics possible within the model. It provides us with a more precise vocabulary for discussing qualitative consequences of alternative models. Combining this with experimental facts, we should be able to place the wild type yeast and various mutants in this diagram. Outside the bistable region, the dynamics decides a self-consistent level of silencing. For instance, for parameters chosen from above the bistable region in figure 4, recruitment of silencing complex at one place only affects a small region around it, with the effects dying off exponentially with distance from the nucleation center. The upper part of the bistable region, with higher values of $\alpha$ (Region I), is not qualitatively very different in that regard.
The only difference comes in, when one considers stochastic dynamics, which allows for occasional formation of silencing in the whole region.
In the lower half of the region, Region II, nucleation leads to spreading. This is the region where the naive expectation from the popular biological model matches the results of mathematical analysis. We have argued, that under some conditions, the dynamics of Sir depletion would lead the systems starting in this region into the border of the two regions (zero front velocity curve, figure 4). A locus of DNA, described by parameters of Region II, could possibly see non-specific silencing induced by stochastic nucleation of silencing.
In this bifurcation diagram, where is the point corresponding to silencing dynamics in silent mating loci in wild type yeast? The fact that the silent loci in the {\it sir1} mutants could be in either state, suggests that one is in the region allowing bistability. A tougher question to answer is which part of the bistable region it is in. In Region II, there is quite some chance of getting undesirable non-specific silencing. As one moves away from the cusp point (or critical point, figure 4), the rate of stochastic switching to silenced state can in principle become very small for regions close to the zero velocity line. However, it is really a matter of numbers. Given that the genome size is large one needs the probability of non-specific nucleation to be small enough to prevent occurrences of inadvertent local silencing. On the other hand, in Region I, typically, the silencing spreads very little from the nucleation center, unless the system is close to the cusp point. In fact, if one defines a length scale by how far the effect of silencing of local nucleation spreads, that length scale diverges exactly at the cusp point. So the system could also operate at a point where this length scale is large but not too close to the critical point where the switching rate is high.
The dynamics represented in the popular cartoon model of silencing, reviewed in \cite{GrewalMoazed},
corresponds to the behavior in Region II. Such models come with explicit requirement of boundary elements to stop the spreading. On top of that, as mentioned before, there should be an argument why chance nucleation in somewhere else in the genome does not cause spontaneous non-specific silencing. However, if the system is in Region I, then one could observe a reduction in silencing with increasing the distance from the nucleation center, namely the silencer. Such a claim has been made by some researchers \cite{BroachPNAS}.
Perhaps the most crucial result of our analysis is the elaboration of these two distinct possibilities within the same molecular model. It may not be crucial to decide in favor of one or the other scenario, given the likelihood of spatial inhomogeneity of the parameters. If the value of effective parameters like $\alpha$ varies in space, different sections of chromosome can demonstrate different silencing behaviors depending on what regions of bifurcation diagram they corresond to. Therefore more investigations are required to decide where on bifurcation diagram the whole or different sections of wild type chromosome are actually located. As a result, the qualitative picture that our model suggests sheds more light on the direction of future investigations in this matter.
Independent of where the wild type yeast is located in parameter space, we could discuss the consequences of lowering the acetylation rate as it happens in, say, the {\it sas2} mutant \cite{Horikoshi, Grunstein}. We argued that if there are fronts of silencing that are not pinned down by boundary elements somewhere in the genome, then our argument about $\rho$ (Sir binding rate) reducing and moving the system back to the zero velocity line/region, applies. This is indeed a possibility in yeast. Although the silent mating loci have well defined boundary elements, the same may not be true of all the telomeric regions. This result might explain certain counterintuitive features of mutants of certain genes like {\it sas2 } which code for acetyltransferases.
If the reduced acetylation rate in {\it sas2} mutant is within a certain range, the system will {\it always} flow back closer to the cusp point at the tip of the bistable region thanks to Sir titration effects. Near the cusp point, the degree of silencing changes very sharply with the changes of Sir availability (see figure 5). We believe that the resulting system becomes extremely susceptible to cellular noise and would display a wide distribution of expression. Thus, as opposed to the naive expectation that {\it SAS2} deletion will just make every thing more transcriptionally silent, one should find individual cells that show good expression from the ``silent" loci.
We speculate, whether this is the reason why the {\it SAS genes} may have been picked up in an assay looking for defects in silencing. Recent single cell measurements observation for GFP expression from {\it sir1sas2} cells show a wide but unimodal distribution of expression in a cell population, where as {\it sir1} cell population show bimodal distribution, characteristic of epigenetic states \cite{SingleCell}.
Another simple consequence of the bifurcation diagram is that one could say qualitative things about the epigenetic switching rate in different parts of the the bifurcation diagram. For example, we expect the switching rate to get faster near the cusp point. We expect as the level of Sas2 is lowered continuously,
we will see a rise in switching rate, as the system would move toward the cusp point.
\section{Discussion}
We have formulated a mathematical version of the model of silencing and computed the bifurcation diagram of the system. This diagram is consistent with several observations about mutants. It is, in principle, possible to explore the whole two dimensional control parameter space experimentally. For example, one could study single cell fluorescent protein expressions from reporters in \emph{HML} and in \emph{HMR} while modifying $\rho$ by regulating Sir proteins, and modulating $\alpha$ via changing the level of Sas2.
In addition to the {\it sas2} mutant, which we discussed extensively, one of the mutants that we want to understand is {\it dot1}. Part of the reason to study this mathematical model is the apparent paradox: if the Sir2,3,4 system itself can propagate further from region with stochastic nucleation of silencing, why many other regions, not contiguous to silencing at nucleation sites, do not show occasional heritable silencing? In fact, a screen for high copy disruptors of telomeric silence \cite{Gottschling}, produced, among others, a gene called {\it DOT1} whose deletion cause nonspecific silencing. Understanding how Dot1 affects silencing requires us to consider additional states like methylation of histones \cite{Dot1}. Based on our preliminary study of a full model of the system with additional states it seems that the simpler model studied in this paper, with some change of parameters, could effectively capture the effect of Dot1. This is one future direction that we are pursuing.
We have touched upon the effects of noise but have not explicitly made a stochastic model. Fluctuation in bio-molecular networks has been the subject of many
research activities recently \cite{rao}. To analyze single cell data, one needs to know not only how the deterministic model behaves but also how noise in various quantities affects expression. A stochastic version of the model, a lattice model with local states of acetylation, and Sir occupancy, could be studied by direct simulation. However, as seen in studies of yeast gene expression \cite{euknoise, OShea}, extrinsic noise, equivalent to fluctuations in the parameters themselves, often dominates over intrinsic fluctuations of the processes described here with fixed parameters. Hence, to study this properly, we will need to add a free parameter each characterizing the slow noise in the control parameters $\rho$ and $\alpha$ for modeling the effect of cell to cell variation of Sir proteins and acetyltransferases. However, we need to be careful to avoid overfitting the data. Another interesting direction involves modeling of noise induced transition between epigenetic states.
We finally mention two issues not dealt at all within this paper that needs further attention. One is that our model of DNA, as a one dimensional system, may be called into question if the heterochromatin formation happens very fast (compared to the speed with which silencing spreads), making the DNA fold up into higher order organization quickly. The other interesting issue is inheritance of silencing. Could we have our model capture inheritance in a coarse grained manner, or do we stand to gain something by modeling the probable silencing of duplicated DNA explicitly? Of course, for any biological model, there are many ways of making it more realistic. However, not many of these `improvements' change the qualitative properties of the bifurcation diagram. We believe our model includes enough features of the biological phenomena to be a good starting point for more refined discussion of the qualitative behavior of this system.
\section{Conclusions and Outlook}
After submitting this manuscript for publication, we noted a recently published paper \cite{Dodd} on a model for chromatin silencing in {\it S. pombe}. An interesting point noted in that paper is that if there are more than two states of the histone modification, it is possible to arrange the parameters system to have stable epigenetic states without cooperativity in any other process. Given that, in {\it S. cerevisiae}, particular histone methylations play a role in activation \cite{Dot1, Madhani}, it is worth looking at their role in bistability.
In biological context, the discussion of nucleation of silencing is mostly focussed on the process of assisted nucleation. The propensity of the system (or the lack thereof) to have spontaneous nucleation has not received as much attention. For example, finding mutations which enhance the probability of spontaneous nucleation would be of great interest. The study on disruptors of telomeric silencing has possibly already unearthed some of these mutants \cite{Gottschling} in {\it S. cerevisiae}. However, there could be much more to the mechanism of control of silencing. This is one place where theoretical studies could possibly suggest what specific signature to look for, spurring on further experimental study.
The issue of spontaneous nucleation is intimately tied to the question of noise induced switching of epigenetic states. In the context of epigenetic switches involving feedback through regulatory proteins, there has been much theoretical work done \cite{AurellSneppen, AurellBrown, Roma, Hornos, KimWang}. However, for chromosomal epigenetic mechanisms one faces a new class of problems. A crucial issue is whether the action of the histone modifying enzyme is very local and the silencing spreads nucleosome to nucleosome along the length of DNA or is so non-local enough so that every nucleosome in the locus affects each other and that we are essentially in a mean-field regime. Answers to these questions are currently unknown.
Our understanding of the role of epigenetic effects in the control of human embryonic cell fate is going through a revolution at this moment \cite{YoungCell, LanderCell}. As the key molecular players and their interactions with the chromatin become well specified, we need a sophisticated model to understand cellular memory and its location specificity in the genome. Our experience with modeling the Sir-dependent silencing in yeast, and our ability to make refined predictions would be an invaluable guide in dealing with the complexities of metazoan development.
\ack
We acknowledge many informative discussions with James Broach, Bradley Cairns, Marc Gartenberg, Vincenzo Pirrotta and John Widom. One of the authors (A.\ S.) thanks Daniel Gottschling for explaining the role of {\it DOT} genes. We are grateful to Vijayalakshmi Nagaraj and Andrei Ruckenstein for their comments on the manuscript. Anirvan Sengupta was supported by NGHRI grant R01HG03470. Mohammad Sedighi was partially suppoerted by the the NIH workforce training grant R90DK071502. This research benefitted from several visits to KITP programs which were supported in part by the National Science Foundation under Grant No. PHY99-07949.
\section*{Glossary}
\renewcommand{\descriptionlabel}[1]%
{\hspace{\labelsep}\textit{#1}}
\begin{description}
\item[Epigenetics.] The study of heritable changes in a gene's functioning that occur without irreversible changes in the DNA sequence.
\item[Chromatin.] The complex of DNA and protein making up chromosomes
\item[Euchromatin.] A lightly packed form of chromatin, often actively transcribed.
\item[Heterochromatin.] A tightly packed form of chromatin, usually with limited transcription.
\item[Gene silencing.] Switching off a gene by an epigenetic mechanism.
\item[Sir proteins.] A set of budding yeast proteins involved in {\bf S}ilent mating type {\bf I}nformation {\bf R}egulation.
\end{description}
\section*{References}
|
2,869,038,156,371 | arxiv |
\section{Introduction}
For integers $k\geq2$ and $m$ we consider the equation
\begin{equation}\label{zkeq}
x^2+y^2+z^k=m.
\end{equation}
For $k=2$ the famous theorem of Gauß about sums of three squares says that (\ref{zkeq}) has an integral solution if and only if $m\geq0$ and $m$ is not of the form $4^u(8l+7)$ for non-negative integers $u$ and $l$. The non-existence of integral solutions can in this case always be explained by the non-existence of real or 2-adic solutions.
Vaughan conjectured in \cite[Chapter 8]{vaughan} that for sufficiently large $m$ there is an integral solution to (\ref{zkeq}) satisfying $x,y,z\geq0$ whenever for each prime $q$ there is some solution $(x,y,z)\in\mathbb{Z}_q$ to the above equation such that $q\nmid\gcd(x,y,z)$. For odd $k\geq3$ such local solutions always exist. His conjecture would then imply the integral Hasse principle for sufficiently large $m$.
This was however disproved by Jagy and Kaplanski in \cite{jagykaplansky}. They gave an elementary proof using quadratic reciprocity that there is no integral solution if $k=9$ and $m=(6p)^3$ for some prime $p\equiv1\mod4$. The remark following their theorem mentions that if $k$ is an odd composite integer, then for infinitely many $m\in\mathbb{Z}$ \cref{zkeq} has no solution.
Dietmann and Elsholtz gave examples of failures of strong approximation in \cite{dietmannpub} for $k=4$ and more general ones in \cite{dietmann} for arbitrary $k\geq2$.
Brauer--Manin obstructions were originally introduced by Manin to explain failures of the Hasse principle for rational points on certain cubic surfaces (see for example \cite{manin}). For an overview of further developments of Brauer--Manin obstructions for the Hasse principle and weak approximation for rational points see \cite{peyre}.
This method was adapted to integral points and applied to quadratic forms such as $x^2+y^2+z^2=m$ by Colliot-Thél\`ene and Xu in \cite{ctxu1}. Further examples of failures of the integral Hasse principle and strong approximation explained by Brauer--Manin obstructions are given in \cite{kreschtschinkel}, \cite{ct7}, \cite{ct8} and \cite{ct9}.
We show that the counterexample to the integral Hasse principle given in \cite{jagykaplansky} can be explained by a Brauer--Manin obstruction (see \Cref{jagythm}).
Furthermore, we systematically find new counterexamples to the integral Hasse principle and strong approximation:
\begin{mtheorem}\label{nweakgeneralm}
The following equations do not fulfill strong approximation away from $\infty$ due to Brauer--Manin obstructions:
\begin{align*}
x^2+y^2+z^k=n^k,&\qquad\text{$k\geq3$ odd, $n\equiv1\mod4$}\\
x^2+y^2+z^k=n^k,&\qquad\text{$k\geq2$ even, $n>0$}
\end{align*}
\end{mtheorem}
\begin{proof}
See \Cref{nweakgeneral}.
\end{proof}
Our second goal is to show the fulfillment of the integral Hasse principle and strong approximation away from $\infty$ at the variable $z$ in case there is no Brauer--Manin obstruction via certain elements of the Brauer group. Unfortunately, we can only do this under assumption of Schinzel's hypothesis (H), a generalization of Dirichlet's theorem on primes within arithmetic progressions to prime values of polynomials.
Schinzel's hypothesis (H) has been employed by Colliot-Th\'el\`ene and Sansuc in \cite{ct2} to prove the Hasse principle and weak approximation for rational solutions of equations similar to (\ref{zkeq}). This technique has subsequently been used for example in \cite{ct3}, \cite{ct6}, \cite{ct4}, \cite{witt} and \cite{wei}.
However, as far as we know, for integral points the potential use of Schinzel's hypothesis (H) was so far only briefly mentioned in Remark (v) on pages 618--619 of \cite{ct6}.
\begin{mtheorem}\label{schinzelthmm}
Let $k\geq3$ be an odd integer. Under Schinzel's hypothesis (H) each solution $(x_v,y_v,z_v)_v\in\prod_v\mathbb{Z}_v^3$ to \cref{zkeq} without any Brauer--Manin obstruction generated by Azumaya algebras of the form described in \Cref{azuconst} can be approximated with respect to the variable $z$ by integral solutions to \cref{zkeq}.
\end{mtheorem}
\begin{proof}
See \Cref{schinzelthm}.
\end{proof}
Jagy and Kaplanski conjectured in \cite{jagykaplansky} that (\ref{zkeq}) has an integral solution whenever $k$ is an odd prime.
\begin{mtheorem}\label{primekm}
Let $k$ be an odd prime. Under Schinzel's hypothesis (H) every integer is of the form $x^2+y^2+z^k$ for integral $x,y,z\in\mathbb{Z}$.
\end{mtheorem}
\begin{proof}
See \Cref{primek}.
\end{proof}
For each prime $p$ and $x\in\mathbb{Q}_p^\times$ let $r_p(x):=\frac x{p^{v_p(x)}}$.
\begin{mtheorem}\label{abthmm}
Let $k$ be the product of two primes $a,b\equiv1\mod4$ and let $m\in\mathbb{Z}\setminus\{0\}$.
For the existence of integral solutions to \cref{zkeq}, it is necessary and under Schinzel's hypothesis (H) also sufficient that the following two statements are both true.
\begin{itemize}
\item There is no $n\equiv6\mod8$ such that $m=n^a$ and for each prime $p\equiv3\mod4$ dividing $n$:
$b\nmid v_p(n)$ or $2\mid v_p(n)$ or there is no $z'\in\{0,\dots,p-1\}$ such that
\[p\mid r_p(n)^{a-1}+\dots+z'^{(a-1)b}.
\]
\item There is no $n\equiv6\mod8$ such that $m=n^b$ and for each prime $p\equiv3\mod4$ dividing $n$:
$a\nmid v_p(n)$ or $2\mid v_p(n)$ or there is no $z'\in\{0,\dots,p-1\}$ such that
\[p\mid r_p(n)^{b-1}+\dots+z'^{(b-1)a}.
\]
\end{itemize}
\end{mtheorem}
\begin{proof}
See \Cref{abthm}.
\end{proof}
For $m\in\mathbb{Z}$ and odd $k\geq1$ an algorithm is given in \Cref{algosection}, which, using Schinzel's hypothesis (H), determines whether $m$ is of the form $x^2+y^2+z^k$.
Finally, lists of small positive integers not of the form $x^2+y^2+z^k$ are given for small odd $k$.
\textbf{Acknowledgements.} We thank Jean-Louis Colliot-Thélène, Christian Elsholtz, Dasheng Wei and the referee for their comments.
\section{Preliminaries}
From now on, let $K$ be a number field, $\Omega$ the set of places of $K$ and $\Omega_\infty\subseteq\Omega$ the set of archimedian places of $K$. Let $K_v$ be the completion of $K$ with respect to $v$ for each $v\in\Omega$. Let $\O_v$ be the corresponding valuation ring for each $v\in\Omega\setminus\Omega_\infty$ and let $\O_v:=K_v$ for each $v\in\Omega_\infty$. The valuation associated to $v\in\Omega\setminus\Omega_\infty$ is called $v_v$. The ring $\O:=\{x\in K\mid v_v(x)\geq0\ \forall v\in\Omega\setminus\Omega_\infty\}$ is called the ring of integers of $K$.
In this section, let $X$ be a variety over $K$.
For topological rings $R$ over $K$, the set of $R$-rational points $X(R)$ obtains the induced topology.
Given a class of varieties, one often wants to know whether the existence of local solutions implies the existence of global solutions, or, even better, whether the existence of local integral solutions implies the existence of integral solutions. This leads to
\begin{definition}\label{adeledef}
Let $S$ be a subset of $\Omega$.
The set of $S$-adeles
\[
\mathbb{A}_S:=\Big\{(x_v)_{v\in\Omega\setminus S}\in\prod_{v\in\Omega\setminus S}K_v \;\Big\vert\; x_v\in\O_v\text{ for almost every }v\in\Omega\setminus S\Big\}
\]
is a ring by coordinatewise addition and multiplication.
The ring $\mathbb{A}:=\mathbb{A}_\emptyset$ is called the \emph{adele ring of $K$}.
The sets
\[
\Big\{\prod_{v\in\Omega\setminus S}A_v\;\Big\vert\; A_v=\O_v\text{ for almost all }v\in\Omega\setminus S\text{ and $A_v$ open in $K_v$ for all }v\in\Omega\setminus S\Big\}
\]
define a basis for the topology on $\mathbb{A}_S$.
For $S\subsetneq\Omega$ the field $K$ may be diagonally embedded into $\mathbb{A}_S$ as for every $x\in K$ there are only finitely many $v$ such that $x\not\in\O_v$. Below, the images of these embeddings are identified with $K$.
\end{definition}
Given a variety $X$ and some $S\subsetneq\Omega$, obviously $X(K)\subseteq X(\mathbb{A}_S)$. It is of interest how $X(K)$ relates to $X(\mathbb{A}_S)$.
\begin{definition}
We say that the variety $X$ satisfies \emph{strong approximation away from $S\subset\Omega$} if $\overline{X(K)}=X(\mathbb{A}_S)$ (where the closure is taken inside $X(\mathbb{A}_S)$), i.e., if $X(K)$ is dense in $X(\mathbb{A}_S)$.
\end{definition}
An introduction to Brauer--Manin obstructions can be found in \cite{skorobogatov}.
\begin{definition}[{\cite[Chapter IV]{milneetale}}]
An $\O_X$-algebra $A$ is called an \emph{Azumaya algebra over $X$} if it is coherent (i.e., there is some open covering by affine schemes $U_i\cong\Spec A_i$, such that $A|_{U_i}\cong \widetilde M_i$ for some finitely generated $A_i$-module $M_i$ for each $i$) and if $A_x\otimes_{\O_{X,x}}\kappa(x)$ is a central simple algebra over the residue field $\kappa(x)$ for every $x\in X$.
If furthermore $k$ is a field extension of $K$, then for each $x\in X(k)$ (i.e., each morphism $x: \Spec k\rightarrow X$ of $K$-schemes) let $A(x):=A_{x(\eta)}\otimes_{\O_{X,x(\eta)}}k$.
\end{definition}
\begin{remark}
If $A$ is an Azumaya algebra over $X$, then $A(x)$ is a central simple $k$\nobreakdash-algebra for each $x\in X(k)$, as $A_{x(\eta)}\otimes_{\O_{X,x(\eta)}}\kappa(x(\eta))$ is a central simple $\kappa(x(\eta))$-algebra and $k$ is a $\kappa(x(\eta))$-algebra by the morphism $x$, such that
\[A(x)\cong \left(A_{x(\eta)}\otimes_{\O_{X,x(\eta)}}\kappa(x(\eta))\right)\otimes_{\kappa(x(\eta))}k.
\]
\end{remark}
\begin{definition}
For $v\in\Omega$ let $\inv_v:\Br(K_v)\rightarrow\mathbb{Q}/\mathbb{Z}$ be the invariant map from local class field theory. For simplicity, we will refer to the class of $a\in\mathbb{Q}$ in $\mathbb{Q}/\mathbb{Z}$ by $a$, too.
\end{definition}
\begin{theorem}[Brauer--Manin obstruction, {\cite[Chapter 5.2]{skorobogatov}} and {\cite[Section 2]{ct8}}]\label{skorothm}
For every Azumaya algebra $A$ over $X$ the set
\[
X(\mathbb{A})^A:=\bigg\{(x_v)_v\in X(\mathbb{A})\;\bigg|\;\sum_v\inv_v(A_v(x_v))=0\bigg\}
\]
contains $\overline{X(K)}$.
Hence, for each $S\subsetneq\Omega$ the set $X(\mathbb{A}_S)^A\subseteq X(\mathbb{A}_S)$ defined as
\[
\bigg\{(x_v)_{v\in\Omega\setminus S}\in X(\mathbb{A}_S)\;\bigg|\;\exists(x_v)_{v\in S}\in X(\mathbb{A}_{\Omega\setminus S})\textnormal{ such that }\sum_v\inv_v(A_v(x_v))=0\bigg\}
\]
contains $\overline{X(K)}$.
\end{theorem}
\section{Azumaya algebra}
In this section, we will define an Azumaya algebra over the scheme defined by \cref{zkeq}. We will then compute its local invariants.
\subsection{Construction}\label{azuconst}
Let $K=\mathbb{Q}$. Recall, that for each prime $p$ and $x\in\mathbb{Q}_p^\times$ we defined $r_p(x):=\frac x{p^{v_p(x)}}$.
For each $v\in\Omega$ let $(\cdot,\cdot):\mathbb{Q}_v^\times\times\mathbb{Q}_v^\times\rightarrow\{\pm1\}$ denote the Hilbert symbol of degree 2 (i.e., $(a,b)=1$ if and only if there exist $x,y\in\mathbb{Q}_v$ such that $a=y^2-bx^2$).
For each ring $R$ of characteristic different from 2 and $a,b\in R^\times$ let $\leg{a,b}{R}$ denote the quaternion algebra over $R$ with parameters $a,b$ (i.e., it is a free $R$-module with basis $1,i,j,ij$ such that $i^2=a$, $j^2=b$ and $ji=-ij$).
\begin{lemma}\label{quathilblemma}
For all $a\in\mathbb{Q}_v^\times$, we have:
\begin{align*}
(a,-1)=1&\Leftrightarrow\exists x,y\in\mathbb{Q}_v:a=x^2+y^2\\
&\Leftrightarrow\inv_v\leg{a,-1}{\mathbb{Q}_v}=0\\
&\Leftrightarrow\begin{cases}
a>0,&v=\infty,\\
r_2(a)\equiv1\mod4,&v=2,\\
0=0,&v\equiv1\mod4,\\
2\mid v_v(a),&v\equiv3\mod4.
\end{cases}
\end{align*}
For $v$-adic integers $a\in\mathbb{Z}_v^\times$, we even have:
\[
(a,-1)=1\Leftrightarrow\exists x,y\in\mathbb{Z}_v:a=x^2+y^2.
\]
\end{lemma}
\begin{proof}
See \cite[III.1, Theorem 1]{serre} and \cite[Proposition 1.1.7]{gille-tamas}.
The last equivalence is trivial if $v=\infty$, so let $p:=v$ be prime. The implication from right to left is obvious. Conversely, remark that there are at least $x',y'\in\mathbb{Q}_v$ such that $a=x'^2+y'^2$. Let $t:=\max(-v_p(x'),-v_p(y'))$. If $t\leq0$, then $x',y'\in\mathbb{Z}_v$, so assume $t>0$. Then we have $(x'p^t)^2+(y'p^t)^2=ap^{2t}$ with $x'p^t,y'p^t\in\mathbb{Z}_v$. Therefore, $(x'p^t)^2\equiv-(y'p^t)^2\mod p^2$. As $x'p^t$ and $y'p^t$ are not both divisible by $p$, this implies that $-1$ is a quadratic residue modulo $p^2$. Hence, $p\equiv1\mod4$, so there are $x'',y''\in\mathbb{Z}$ such that $p=x''^2+y''^2$. According to the pigeonhole principle, $r_p(a)\mod p$ is the sum of two quadratic residues, which we can lift to $x''',y'''\in\mathbb{Z}_p$ satisfying $r_p(a)\equiv x'''^2+y'''^2$ using Hensel's Lemma (as $p\neq2$ and $p\nmid r_p(a)$). Finally, repeated application of Brahmagupta's identity
\[
(xx''-yy'')^2+(xy''+yx'')^2=(x^2+y^2)(x''^2+y''^2)=p(x^2+y^2)
\]
yields $x,y\in\mathbb{Z}_p$ such that $a=p^{v_p(a)}r_p(a)=x^2+y^2$.
\end{proof}
Let $n\in\mathbb{Z}\setminus\{0\}$ and $a,b>0$ be integers such that $n>0$ or $2\nmid ab$. Consider the equation
\begin{equation}\label{zabnagl}
x^2+y^2+z^{ab}=n^a.
\end{equation}
Let
\[\mathfrak{X}:=\Spec\mathbb{Z}[X,Y,Z]/(X^2+Y^2+Z^{ab}-n^a)
\]
and
\[\mathfrak X_\mathbb{Q}:=\mathfrak{X}\otimes_\mathbb{Z}\mathbb{Q}=\Spec\mathbb{Q}[X,Y,Z]/(X^2+Y^2+Z^{ab}-n^a).
\]
The variety $\mathfrak{X}_\mathbb{Q}$ is covered by the principal open subsets $U_1:=D(n-Z^b)\subseteq \mathfrak X_\mathbb{Q}$ and $U_2:=D(n^{a-1}+n^{a-2}Z^b+\dots+nZ^{(a-2)b}+Z^{(a-1)b})\subseteq \mathfrak X_\mathbb{Q}$. Indeed, as $a,n\in\mathbb{Q}^\times$, we have
\begin{align*}
V(n-Z^b)\cap V(n^{a-1}+\dots+Z^{(a-1)b})
&=V(n-Z^b,n^{a-1}+\dots+Z^{(a-1)b})\\
&=V(n-Z^b,an^{a-1})
=\emptyset.
\end{align*}
Consider the $\O_{\mathfrak X_\mathbb{Q}}|_{U_1}$-algebra
\[
A_1:=\leg{n-Z^b,-1}{\O_{\mathfrak X_\mathbb{Q}}(U_1)}^\sim
\]
and the $\O_{\mathfrak X_\mathbb{Q}}|_{U_2}$-algebra
\[
A_2:=\leg{n^{a-1}+\dots+Z^{(a-1)b},-1}{\O_{\mathfrak X_\mathbb{Q}}(U_2)}^\sim.
\]
There is an $\O_{\mathfrak X_\mathbb{Q}}|_{U_1\cap U_2}$-algebra isomorphism $A_1|_{U_1\cap U_2}\xrightarrow\sim A_2|_{U_1\cap U_2}$ induced by
\begin{align*}
i&\mapsto\frac{Xi'+Yi'j'}{n^{a-1}+\dots+Z^{(a-1)b}}\\
j&\mapsto j'
\end{align*}
where $i,j$ and $i',j'$ are the canonical generators of $A_1|_{U_1\cap U_2}$ and $A_2|_{U_1\cap U_2}$, respectively (this is an $\O_{\mathfrak X_\mathbb{Q}}|_{U_1\cap U_2}$-algebra isomorphism as $X^2-(-1)Y^2=(n-Z^a)(n^{a-1}+\dots+Z^{(a-1)b})$).
Hence $A_1$ and $A_2$ can be glued along $U_1\cap U_2$ to obtain an $\O_{\mathfrak X_\mathbb{Q}}$-algebra $A$ such that $A|_{U_1}\cong A_1$ and $A|_{U_2}\cong A_2$.
Quaternion algebras over fields (with nonzero arguments) are central simple algebras, so $A$ is an Azumaya algebra.
In the following, we are interested in strong approximation ``at $Z$'' away from $\infty$. To this end, we choose a suitable topology: In $\mathbb{Q}_v^3$ we equip the first two components (i.e., those belonging to the variables $X$ and~$Y$) with the trivial topology (sometimes called indiscrete topology) and the last one (i.e., that belonging to the variable~$Z$) with the usual topology on $\mathbb{Q}_v$. Accordingly, the sets $\mathfrak X(\mathbb{A})\subseteq\prod_v\mathbb{Q}_v^3$, $\mathfrak{X}(\mathbb{Z}_v)\subseteq\mathbb{Z}_v^3\subseteq\mathbb{Q}_v^3$, etc. obtain the induced topologies.
Strong approximation with respect to the usual topology (i.e., at $X$, $Y$ and $Z$) seems more difficult, as for fixed $p,r,s\in\mathbb{Z}^+$ the equation $(px+r)^2+y^2=s$ does not fulfill the integral Hasse principle (unlike the equation $x^2+y^2=s$).
If strong approximation ``at $Z$'' away from $\infty$ is not fulfilled, then strong approximation away from $\infty$ with respect to the usual topology is not fulfilled, either.
\begin{lemma}
Let
\[
U:=U_1\cap U_2=D(n^a-Z^{ab})
\]
and
\[
I_v:=\inv_v(A(\mathfrak{X}(\mathbb{Z}_v))).
\]
for any $v\in\Omega$. Then $I_v\subseteq\{0,1/2\}$ and
\[
I_v=\inv_v(A(U(\mathbb{Q}_v)\cap\mathfrak{X}(\mathbb{Z}_v))).
\]
For $(x,y,z)\in \mathfrak X(\mathbb{Q}_v)$ we have
\begin{align*}
\inv_v(A(x,y,z))=0 &\Leftrightarrow (n-z^b,-1)=1&&\textnormal{ if }n-z^b\neq0,\\
\inv_v(A(x,y,z))=0 &\Leftrightarrow (n^{a-1}+\dots+z^{(a-1)b},-1)=1&&\textnormal{ if }n^{a-1}+\dots+z^{(a-1)b}\neq0,
\end{align*}
\begin{equation*}
w\in I_v\Leftrightarrow\exists z\in\mathbb{Z}_v\textnormal{ such that }(n^a-z^{ab},-1)=1\textnormal{ and }(n-z^b,-1)=\begin{cases}1,&w=0,\\-1,&w=1/2.\end{cases}
\end{equation*}
\end{lemma}
\begin{proof}
The inclusion $I_v\subseteq\{0,1/2\}$ follows from the fact that quaternion algebras over fields $k$ have order 2 in the Brauer group $\Br(k)$.
The first two equivalences follow straight from the definition of $A$ and \Cref{quathilblemma}.
It is easy to see that $U(\mathbb{Q}_v)$ is dense in $\mathfrak X(\mathbb{Q}_v)$. Hence, $U(\mathbb{Q}_v)\cap \mathfrak{X}(\mathbb{Z}_v)$ is dense in $\mathfrak{X}(\mathbb{Z}_v)$ because $\mathfrak{X}(\mathbb{Z}_v)$ is an open subset of $\mathfrak{X}(\mathbb{Q}_v)$. As $\inv_v\circ A:\mathfrak X(\mathbb{Q}_v)\rightarrow\mathbb{Q}/\mathbb{Z}$ is locally constant (even in the topology chosen above!), this implies that $I_v=\inv_v(A(U(\mathbb{Q}_v)\cap\mathfrak{X}(\mathbb{Z}_v)))$.
For each $z\in\mathbb{Z}_v$ satisfying $n^a-z^{ab}\neq0$, there exist $x,y\in\mathbb{Z}_v$ such that $x^2+y^2=n^a-z^{ab}$ if and only if ${(n^a-z^{ab},-1)=1}$. Together with the first equivalence and $I_v=\inv_v(A(U(\mathbb{Q}_v)\cap\mathfrak{X}(\mathbb{Z}_v)))$ this proves the final one.
\end{proof}
\subsection{Place \texorpdfstring{$\infty$}{∞}}
\begin{lemma}\label{real}
We have $I_\infty=\{0\}$.
\end{lemma}
\begin{proof}
A real solution is $(0,0,\sqrt[b]n)$ (as $n>0$ or $2\nmid b$), so $I_\infty\neq\emptyset$.
If $(x,y,z)\in U(\mathbb{R})$, then $n^a-z^{ab}=x^2+y^2\geq0$ and $n>0$ or $2\nmid a$, so $n\geq z^b$. As $(x,y,z)\in U_1(\mathbb{R})$ we have $n\neq z^b$, i.e., $n-z^b>0$, so $(n-z^b,-1)=1$. Hence $\inv_\infty(A(x,y,z))=0$.
\end{proof}
\subsection{Place 2}
\begin{lemma}\label{a1I2}
If $a=1$ and $b$ is odd, then $I_2=\{0\}$.
\end{lemma}
\begin{proof}
Due to $(n^a-z^{ab},-1)=(n-z^b,-1)$ for $z\in\mathbb{Z}_2$ it is obvious that $I_2\subseteq\{0\}$. The set $I_2$ is nonempty as there is some odd $z\in\mathbb{Z}$ such that $n-z\equiv1\text{ or }2\mod8$ and this fulfills $n^a-z^{ab}\equiv n-z^b\equiv n-z\equiv1\text{ or }2\mod8$ (as $b$ and $z$ are odd), so $(n^a-z^{ab},-1)=1$.
\end{proof}
\begin{lemma}\label{2notempty}
If $r_2(n)^a\equiv1\mod4$, then $I_2\neq\emptyset$.
\end{lemma}
\begin{proof}
Let $z:=0$. Then $r_2(n^a-z^{ab})\equiv r_2(n)^a\equiv1\mod4$, so $(n^a-z^{ab},-1)=1$.
\end{proof}
\begin{lemma}\label{2bnmid}
If $a\geq2$ and $r_2(n)^a\equiv1\mod4$ and $b\mid v_2(n)+1$, then $I_2=\{0,1/2\}$.
\end{lemma}
\begin{proof}
Let $z_1:=0$ and $z_2:=2^{(v_2(n)+1)/b}$. Now $r_2(n^a-z_1^{ab})\equiv r_2(n)^a\equiv1\mod4$ and $r_2(n^a-z_2^{ab})\equiv r_2(r_2(n)^a-2^a)\equiv r_2(n)^a\equiv1\mod4$, so $(n^a-z_1^{ab},-1)=(n^a-z_2^{ab},-1)=1$.
Furthermore $r_2(n-z_1^b)\equiv r_2(n)\not\equiv r_2(n)-2\equiv r_2(r_2(n)-2)\equiv r_2(n-z_2^b)\mod4$, so $(n-z_1^b,-1)\neq(n-z_2^b,-1)$.
\end{proof}
\begin{lemma}\label{6mod8ne}
If $a\geq3$ and $b$ are odd and $n\equiv6\mod8$, then $1/2\in I_2$.
\end{lemma}
\begin{proof}
Let $z:=-1$. Then $n^a-z^{ab}\equiv n^a+1\equiv1\mod4$ and $n-z^b\equiv n+1\equiv3\mod4$, so $1/2\in I_2$.
\end{proof}
\begin{lemma}\label{6mod8}
If $a,b\geq3$ are odd and $n\equiv6\mod8$, then $I_2=\{1/2\}$.
\end{lemma}
\begin{proof}
We know $1/2\in I_2$ from the previous lemma.
Let $(x,y,z)\in U(\mathbb{Q}_2)\cap\mathfrak{X}(\mathbb{Z}_2)$.
If $z$ is odd, then $n^{a-1}+\dots+z^{(a-1)b}\equiv nz^{(a-2)b}+z^{(a-1)b}\equiv2+1\equiv3\mod4$, so $(n^{a-1}+\dots+z^{(a-1)b},-1)=-1$.
If $z$ is even, then
\[
1\equiv r_2(n^a-z^{ab})\equiv r_2((n/2)^a-2^{a(b-1)}(z/2)^{ab})\equiv r_2(n/2)^a\equiv3\mod4
\]
yields a contradiction.
\end{proof}
\begin{lemma}\label{not6mod8}
If $a,b$ are odd and $n\not\equiv6\mod8$, then $0\in I_2$.
\end{lemma}
\begin{proof}
The values of $z$ in the following table fulfill $r_2(n^a-z^{ab})\equiv r_2(n-z^b)\equiv1\mod4$:
\begin{tabular}{c|c|c|c|c|c|c|c}
$n\mod8$&0&1&2&3&4&5&7\\\hline
$z$&-1&0&0&1&$-1$&3&5
\end{tabular}
\end{proof}
\subsection{Odd places}\label{oddplacessection}
\begin{lemma}\label{odd}
We have $0\in I_p$ for all odd primes $p$.
\end{lemma}
\begin{proof}
One of the numbers $n^a$ or $n^a-1$ is not divisible by $p$. Set $z:=0$ or $z:=1$, respectively. Then $n^a-z^{ab}$ and hence $n-z^b$ are not divisible by $p$, so $(n^a-z^{ab},-1)=(n-z^b,-1)=1$.
\end{proof}
Hence it is only interesting whether $1/2\in I_p$.
\begin{lemma}\label{1mod4}
We have $I_p=\{0\}$ for all primes $p\equiv1\mod4$.
\end{lemma}
\begin{proof}
The previous lemma implies $I_p\neq\emptyset$. Furthermore, we have $(t,-1)=1$ for all $t\in\mathbb{Q}_p^\times$.
\end{proof}
Hence only the case $p\equiv3\mod4$ is interesting, so let $p\equiv3\mod4$ be prime for the rest of \Cref{oddplacessection}.
\begin{lemma}\label{pfull}
If $2\mid a$ and $2\nmid v_p(n)$, then $I_p=\{0,1/2\}$.
\end{lemma}
\begin{proof}
Take $z_1:=1$ and $z_2:=0$. Then
\begin{align*}
(n^a-z_1^{ab},-1)&=\phantom{-}1&&\text{ (as $2\mid0=v_p(n^a-z_1^{ab})$)}\\
(n^a-z_2^{ab},-1)&=\phantom{-}1&&\text{ (as $2\mid av_p(n)=v_p(n^a-z_2^{ab})$)}\\
(n-z_1^b,-1)&=\phantom{-}1&&\text{ (as $2\mid0=v_p(n-z_1^b)$)}\\
(n-z_2^b,-1)&=-1&&\text{ (as $2\nmid v_p(n)=v_p(n-z_2^b)$)}.
\end{align*}
\end{proof}
Let $a,b$ be odd for the rest of \Cref{oddplacessection}.
Then the following lemma simplifies the analysis of $I_p$.
\begin{lemma}\label{vnvzb}
Let $z\in\mathbb{Z}_p$ such that $n^a-z^{ab}\neq0$. Then the following statements are equivalent:
\begin{enumerate}[a)]
\item There are $x,y\in\mathbb{Z}_p$ such that $(x,y,z)\in \mathfrak{X}(\mathbb{Z}_p)$ and $\inv_v(A(x,y,z))=1/2$ (hence $1/2\in I_p$).
\item $v_p(n)=v_p(z^b)$ and
\begin{align*}
1&\equiv v_p(r_p(n)^{a-1}+\dots+r_p(z)^{(a-1)b})&\mod2\\
1+v_p(n)&\equiv v_p(r_p(n)-r_p(z)^b)&\mod2\\
v_p(n)&\equiv v_p(r_p(n)^a-r_p(z)^{ab})&\mod2&.
\end{align*}
\end{enumerate}
\end{lemma}
\begin{remark}
The sum of the first two congruences in statement b) is the third one, so only two of them have to be proved.
\end{remark}
\begin{proof}[Proof of the lemma]
Assume a). Then (as $(n^{a-1}+\dots+z^{(a-1)b},-1)=-1$):
\[
2\nmid v_p(n^{a-1}+\dots+z^{(a-1)b})
\]
If $v_p(n)<v_p(z^b)$, then $2\mid(a-1)v_p(n)=v_p(n^{a-1}+\dots+z^{(a-1)b})$ yields a contradiction.
If $v_p(n)>v_p(z^b)$, then $2\mid(a-1)v_p(z^b)=v_p(n^{a-1}+\dots+z^{(a-1)b})$ yields a contradiction.
Hence $v_p(n)=v_p(z^b)$.
Then
\begin{align*}
1&\equiv v_p(n^{a-1}+\dots+z^{(a-1)b})\\
&\equiv(a-1)v_p(n)+v_p(r_p(n)^{a-1}+\dots+r_p(z)^{(a-1)b})\\
&\equiv v_p(r_p(n)^{a-1}+\dots+r_p(z)^{(a-1)b})\mod2
\end{align*}
and (as $(n-z^b,-1)=-1$)
\begin{align*}
1+v_p(n)
&\equiv v_p(n)+v_p(n-z^b)\\
&\equiv 2v_p(n)+v_p(r_p(n)-r_p(z)^b)\\
&\equiv v_p(r_p(n)-r_p(z)^b)\mod2.
\end{align*}
Conversely, b) implies
\begin{align*}
v_p(n^a-z^{ab})
&\equiv av_p(n)+v_p(r_p(n)^a-r_p(z)^{ab})\\
&\equiv(a+1)v_p(n)\equiv0\mod2,
\end{align*}
so there are $x,y\in\mathbb{Z}_p$ such that $(x,y,z)\in \mathfrak{X}(\mathbb{Z}_p)$.
Furthermore
\[
v_p(n-z^b)\equiv v_p(n)+v_p(r_p(n)-r_p(z)^b)\equiv1\mod2,
\]
so $\inv_v(A(x,y,z))=1/2$.
\end{proof}
\begin{lemma}\label{jagypre1}
Assume $p\nmid an$. Then $1/2\not\in I_p$.
\end{lemma}
\begin{proof}
Suppose $(x,y,z)\in U(\mathbb{Q}_p)\cap\mathfrak{X}(\mathbb{Z}_p)$ and $\inv_p(A(x,y,z))=1/2$. Then $v_p(n-z^b)$ and $v_p(n^{a-1}+\dots+z^{(a-1)b})$ are odd. Hence $n-z^b$ and $n^{a-1}+\dots+z^{(a-1)b}$ have to be divisible by $p$, so together $p\mid an^{a-1}$. Therefore $p\mid an$.
\end{proof}
\begin{lemma}\label{jagypre2}
Assume $b\nmid v_p(n)$. Then $1/2\not\in I_p$.
\end{lemma}
\begin{proof}
Suppose $(x,y,z)\in U(\mathbb{Q}_p)\cap\mathfrak{X}(\mathbb{Z}_p)$ and $\inv_p(A(x,y,z))=1/2$. According to \Cref{vnvzb} we have $v_p(n)=v_p(z^b)$, so $b\mid v_p(n)$.
\end{proof}
\begin{lemma}\label{12insidenmid}
Let $p\nmid ab$. Then the following two statements are equivalent:
\begin{enumerate}[a)]
\item $1/2\in I_p$.
\item $b\mid v_p(n)$ and $2\nmid v_p(n)$ and there is some $z'\in\mathbb{Z}$ such that $p\mid r_p(n)^{a-1}+\dots+z'^{(a-1)b}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Assume a). Let $(x,y,z)\in U(\mathbb{Q}_p)\cap\mathfrak{X}(\mathbb{Z}_p)$ such that $\inv_p(A(x,y,z))=1/2$.
\Cref{vnvzb} shows that $v_p(n)=v_p(z^b)$ (so $b\mid v_p(n)$).
It also shows that $2\nmid v_p(r_p(n)^{a-1}+\dots+r_p(z)^{(a-1)b})$, so
$p\mid r_p(n)^{a-1}+\dots+r_p(z)^{(a-1)b}$.
If $2\mid v_p(n)$, then $2\nmid v_p(r_p(n)-r_p(z)^b)$ according to \Cref{vnvzb}. Therefore ${p\mid r_p(n)-r_p(z)^b}$ and $p\mid r_p(n)^{a-1}+\dots+r_p(z)^{(a-1)b}$. Together $p\mid a r_p(n)^{a-1}$, which is obviously impossible.
Therefore $2\nmid v_p(n)$.
Conversely, assume b). As $p\nmid ab$ and obviously $p\nmid z'$, we have
\[r_p(n)^a-z'^{ab}\not\equiv r_p(n)^a-z'^{ab}-abz'^{ab-1}p\equiv r_p(n)^a-(z'+p)^{ab}\mod p^2.
\]
Hence $v_p(r_p(n)^a-z'^{ab})\leq 1$ or $v_p(r_p(n)^a-(z'+p)^{ab})\leq1$. Let $z:=p^{v_p(n)/b}z'$ or $z:=p^{v_p(n)/b}(z'+p)$, respectively.
Therefore (as $p\mid r_p(n)^{a-1}+\dots+r_p(z)^{(a-1)b}$)
\[
1\leq v_p(r_p(n)^{a-1}+\dots+r_p(z)^{(a-1)b})\leq v_p(r_p(n)^a-r_p(z)^{ab})\leq1,
\]
so $v_p(r_p(n)^{a-1}+\dots+r_p(z)^{(a-1)b})=1$ and $v_p(r_p(n)^a-r_p(z)^{ab})\equiv1\equiv v_p(n)\mod2$.
Then \Cref{vnvzb} (together with its remark) shows that $1/2\in I_p$.
\end{proof}
\section{Failure of strong approximation and the integral Hasse principle}
In this section, we use the computations of local invariants of the Azumaya algebra $A$ over $\mathfrak{X}_\mathbb{Q}$ defined in the previous section to obtain counterexamples to strong approximation and the integral Hasse principle.
\begin{lemma}
Let $a,b$ be odd. Then \cref{zabnagl} has $v$-adic integral solutions for each place~$v$.
\end{lemma}
\begin{proof}
See \Cref{real,odd,6mod8ne,not6mod8,a1I2}.
\end{proof}
The following theorem explains failures of strong approximation away from $\infty$. Not all $I_v$ have to be explicitly known to be able to apply it.
\begin{theorem}\label{notconstthm}
If $\mathfrak{X}(\mathbb{Z}_v)\neq\emptyset$ for each $v\in\Omega$ and if $|I_w|=2$ for some $w\in\Omega$, then strong approximation ``at $Z$'' away from $\infty$ fails for the equation (\ref{zabnagl}) due to a Brauer--Manin obstruction.
\end{theorem}
\begin{proof}
Let $L_v=L_v'\in \mathfrak{X}(\mathbb{Z}_v)$ for all $v\in\Omega\setminus\{w\}$ and $L_w,L_w'\in \mathfrak{X}(\mathbb{Z}_w)$ such that $\inv_w(A(L_w))\neq\inv_w(A(L_w'))$. Then $\sum_{v\in\Omega}\inv_v(A(L_v))\neq\sum_{v\in\Omega}\inv_v(A(L_v'))$. Hence $(L_v)_v$ or $(L_v')_v\not\in \mathfrak{X}(\mathbb{A})^A$, i.e., $(L_v)_v$ or $(L_v')_v\not\in\overline{\mathfrak{X}(\mathbb{Q})}$ (where the closure is taken with respect to the topology defined in \Cref{azuconst}) although $(L_v)_v,(L'_v)_v\in \mathfrak{X}(\mathbb{A}_{\{\infty\}})$.
\end{proof}
\begin{corollary}
If $a\geq2$ and $r_2(n)^a\equiv1\mod4$ and $b\mid v_2(n)+1$, then strong approximation ``at $Z$'' away from $\infty$ fails for (\ref{zabnagl}) due to a Brauer--Manin obstruction.
\end{corollary}
\begin{proof}
$\mathfrak{X}(\mathbb{Z}_v)\neq\emptyset$ for all $v\neq2$ according to \Cref{real,odd}. $|I_2|=2$ according to \Cref{2bnmid}.
\end{proof}
\begin{corollary}[cf.\ \cref{nweakgeneralm}]\label{nweakgeneral}
According to the previous corollary the following equations do not fulfill strong approximation ``at $Z$'' away from $\infty$:
\begin{align*}
x^2+y^2+z^a=n^a,&\qquad\text{$a\geq3$ odd, $n\equiv1\mod4$}\\
x^2+y^2+z^a=n^a,&\qquad\text{$a\geq2$ even, $n>0$}
\end{align*}
\end{corollary}
\begin{remark}
Dietmann and Elsholtz showed in \cite{dietmann} and for the case $a=4$ in \cite{dietmannpub} that for $a\geq2$ and sufficiently large $N$ the number of integers $0<m\leq N$ such that $x^2+y^2+z^a=m$ does not fulfill strong approximation ``at $Z$'' away from $\infty$ is at least
\[
\begin{cases}
\frac{aN^{1/(2a)}}{\varphi(a)\log(N)},&a\text{ odd}\\
\frac{N^{1/2}}{2\log(N)},&a\text{ even}
\end{cases}
\]
The above example shows that this number is at least
\[
\begin{cases}
\frac{[N^{1/a}]+3}4,&a\text{ odd}\\
[N^{1/a}],&a\text{ even}
\end{cases}
\]
The following corollary gives a better estimate for even $a$.
\end{remark}
\begin{corollary}
If $n>0$ and $2\mid a$ and $n$ is not a sum of two squares, then strong approximation ``at $Z$'' away from $\infty$ fails for (\ref{zabnagl}) due to a Brauer--Manin obstruction.
\end{corollary}
\begin{proof}
There has to be some prime $p\equiv3\mod4$ such that $2\nmid v_p(n)$. Now $\mathfrak{X}(\mathbb{Z}_v)\neq\emptyset$ for all $v\in\Omega$ according to \Cref{real,odd,2notempty} and $|I_p|=2$ according to \Cref{pfull}.
\end{proof}
Unfortunately, \Cref{notconstthm} cannot explain the overall absence of integral solutions ($\mathfrak{X}(\mathbb{A})^A\neq\emptyset$ whenever its conditions are satisfied) and it does not return any explicit points which are not contained in $\mathfrak{X}(\mathbb{A})^A$. To accomplish this, $I_v$ has to be explicitly computed for every $v\in\Omega$.
The following theorem is a generalization of the theorem in \cite{jagykaplansky} where an elementary proof for the case $a=b=3$ and $n=6q$ for primes $q\equiv1\mod4$ is given.
\begin{theorem}\label{jagythm}
Let $a,b\geq3$ be odd integers and $n\equiv6\mod8$ such that $b\nmid v_p(n)$ for all prime divisors $p\equiv3\mod4$ of $an$.
Then (\ref{zabnagl}) has no solutions in $\mathbb{Z}$ although it has $v$-adic integral solutions for each place $v$ and this is explained by a Brauer--Manin obstruction.
In particular, the integral Hasse principle fails.
\end{theorem}
\begin{proof}
We have $I_\infty=\{0\}$ according to \Cref{real} and $I_p=\{0\}$ for all primes $p\equiv1\mod4$ according to \Cref{1mod4}. Moreover, $I_2=\{1/2\}$ according to \Cref{6mod8}. Finally, $I_p=\{0\}$ for all primes $p\equiv3\mod4$ according to \Cref{odd,jagypre1,jagypre2}.
Hence there are $v$-adic integral solutions for each $v$ and $\sum_v\inv_v(A_v(x_v,y_v,z_v))=1/2$ for each $(x_v,y_v,z_v)_v\in\prod_v\mathfrak{X}(\mathbb{Z}_v)$. This implies $\mathfrak{X}(\mathbb{Z})\subseteq \mathfrak{X}(\mathbb{A})^A\cap \mathfrak{X}(\mathbb{Z})=\emptyset$.
\end{proof}
\begin{remark}
For odd $a,b\geq3$ there are always infinitely many integers $n\equiv6\mod8$ such that $b\nmid v_p(n)$ for all prime divisors $p\equiv3\mod4$ of $an$, so there are infinitely many integers $n$ such that (\ref{zabnagl}) has no integral solutions. This confirms part b) of the remark following the Theorem in \cite{jagykaplansky}.
\end{remark}
\begin{proof}
Take $n:=2l\prod_{p\mid a}p$ where $l$ is the product of distinct primes such that $l\equiv1\mod4$ if $\prod_{p\mid a}p\equiv3\mod4$ and $l\equiv3\mod4$ otherwise.
\end{proof}
Dietmann and Elsholtz proved in \cite{dietmannpub} and \cite{dietmann} that (\ref{zabnagl}) does not fulfill strong approximation away from $\infty$ if
\begin{itemize}
\item $a=2$ and $n\equiv7\mod8$ is prime or
\item $a\geq3$ is odd, $b=1$ and $p\equiv1\mod4a$ is a prime such that $n=p^2$.
\end{itemize}
This can also be proved using the same strategy as above.
\section{Fulfillment of strong approximation}
Let $k\geq3$ be an odd integer and $m\in\mathbb{Z}\setminus\{0\}$.
Davenport and Heilbronn showed in \cite{davenportheilbronn}, that for all except $o(N)$ integers $1\leq m\leq N$ the equation
\begin{equation}\tag{1}
x^2+y^2+z^k=m\label{zk2}
\end{equation}
has a solution with $x,y,z\in\mathbb{Z}$.
As above, let
\[
\mathfrak{X}:=\Spec\mathbb{Z}[X,Y,Z]/(X^2+Y^2+Z^k-m)
\]
and
\[
\mathfrak{X}_\mathbb{Q}:=\mathfrak{X}\otimes_\mathbb{Z}\mathbb{Q}=\Spec\mathbb{Q}[X,Y,Z]/(X^2+Y^2+Z^k-m)
\]
and
\[
U:=D(m-Z^k)\subseteq \mathfrak{X}_\mathbb{Q}.
\]
Given $k$ and $m$ there may be multiple triples $(a,b,n)$ of integers with $a,b>0$ such that $k=ab$ and $m=n^a$, i.e., there may be multiple Azumaya algebras to consider for Brauer--Manin obstruction.
To this end, let $S(a,b,n):=\left(\prod_v\mathfrak{X}(\mathbb{Z}_v)\right)^A$ with $A$ defined as in \Cref{azuconst} and let $I_v(a,b,n):=I_v$ as defined in \Cref{azuconst}.
Then we can define the subset $L$ of the solutions in $\mathbb{A}$ to \cref{zk2} for which there is no Brauer--Manin obstruction corresponding to any Azumaya algebra from \Cref{azuconst}:
\[
L:=\bigcap\limits_{{{\scriptstyle a,b,n\in\mathbb{Z}:\atop\scriptstyle a,b>0,}\atop\scriptstyle k=ab,}\atop\scriptstyle m=n^a}S(a,b,n)
\]
Of course, $\mathfrak{X}(\mathbb{Z})\subseteq L$.
The next theorem will show that Brauer--Manin obstructions with such Azumaya algebras explain all failures of strong approximation ``at $Z$'' away from $\infty$ if Schinzel's hypothesis (H) is true.
\begin{lemma}\label{rootdeg}
Let $K$ be a field, $k\geq1$ an odd integer and $u\in K$ such that $u$ is not a $p$-th power in $K$ for any prime divisor $p$ of $k$.
Then $[K(\sqrt[k]u):K]=k$.
\end{lemma}
\begin{proof}
See \cite[Thm.\ VI.9.1]{lang}.
\end{proof}
\begin{lemma}\label{phiirred}
Let $d,b\geq1$ be odd integers and $n\in\mathbb{Q}$ such that $n$ is not a $p$-th power for any prime divisor $p$ of $b$.
Then $\phi_d(X^b/n)\in\mathbb{Q}[X]$ is irreducible (where $\phi_d$ is the $d$-th cyclotomic polynomial).
\end{lemma}
\begin{proof}
For any positive integer $s$, let $\zeta_s$ denote a primitive $s$-th root of unity. The polynomial $\phi_d(X^b/n)$ has a root $\sqrt[b]n\cdot\zeta_{bd}$. Assume $\phi_d(X^b/n)$ is reducible. Hence (as $\deg(\phi_d(X^b/n))=b\varphi(d)$)
\begin{align*}
[\mathbb{Q}(\sqrt[b]n\cdot\zeta_{bd}):\mathbb{Q}(\zeta_d)]\varphi(d)
&=[\mathbb{Q}(\sqrt[b]n\cdot\zeta_{bd}):\mathbb{Q}(\zeta_d)]\cdot[\mathbb{Q}(\zeta_d):\mathbb{Q}]\\
&=[\mathbb{Q}(\sqrt[b]n\cdot\zeta_{bd}):\mathbb{Q}]\\
&<b\varphi(d),
\end{align*}
so $[\mathbb{Q}(\sqrt[b]n\cdot\zeta_{bd}):\mathbb{Q}(\zeta_d)]<b$. \Cref{rootdeg} therefore implies that $n\cdot\zeta_d$ is a $p$-th power in $\mathbb{Q}(\zeta_d)$ for some prime divisor $p$ of $b$, say $x\in\mathbb{Q}(\zeta_d)$ and $x^p=n\cdot\zeta_d$.
If $p\mid d$, then $(x/\overline x)^p=\zeta_d^2$, but this is impossible as the roots of unity in $\mathbb{Q}(\zeta_d)$ are $\mu_{2d}$ but $\mu_{2d}^p=\mu_{2d/p}\not\ni\zeta_d^2$.
Hence $p\nmid d$. Let then $r\in\mathbb{Z}$ such that $rp\equiv1\mod d$. Hence for $y:=x/\zeta_d^r$ we have
\[
y^p=\frac{x^p}{\zeta_d^{rp}}=\frac{n\cdot\zeta_d}{\zeta_d}=n,
\]
so in particular every $\tau\in\Gal(\mathbb{Q}(\zeta_d)|\mathbb{Q})$ fulfills $(\tau y)^p=\tau y^p=y^p$. Therefore $(\tau y/y)^p=1$ but this is only possible if $\tau y=y$ as $\mathbb{Q}(\zeta_d)$ contains no primitive $p$-th root of unity. Hence we conclude that $y\in\mathbb{Q}$. This yields a contradiction as $y^p=n$ but $n$ is not a $p$-th power in $\mathbb{Q}$ by assumption.
\end{proof}
Recall the statement of Schinzel's hypothesis (H):
\begin{hypothesis}[H]
Let $f_1,\dots,f_s\in\mathbb{Z}[X]$ be polynomials irreducible in $\mathbb{Q}[X]$ such that
\[\gcd\{f_1(x)\cdots f_s(x)\mid x\in\mathbb{Z}\}=1
\]
and $f_i(x)\rightarrow\infty$ for $x\rightarrow\infty$ for each $1\leq i\leq s$. Then there is some $x\in\mathbb{Z}$ such that $f_i(x)$ is prime for each $i$.
\end{hypothesis}
Below, we will use the following consequence of Schinzel's hypothesis (H).
\begin{lemma}\label{schinzelhelp}
Let $f_1,\dots,f_s\in\mathbb{Z}[X]$ be polynomials irreducible in $\mathbb{Q}[X]$ such that
\[\gcd\{f_1(x)\cdots f_s(x)\mid x\in\mathbb{Z}\}=1
\]
and $f_i(x)\rightarrow\infty$ for $x\rightarrow\infty$ for each $1\leq i\leq s$. Let furthermore $c,e\in\mathbb{Z}$ such that $v_p(f_i(c))\leq v_p(e)$ for all prime divisors $p$ of $e$ and each $i$. Assume Schinzel's hypothesis~(H) is true. Then there is some $x\equiv c\mod e$ such that $\frac{f_i(x)}{\gcd(f_i(c),e)}$ is prime for each $i$.
\end{lemma}
\begin{proof}
Let $g_i(X):=\frac{f_i(eX+c)}{\gcd(f_i(c),e)}$. Obviously $g_i\in\mathbb{Z}[X]$ and $g_i$ is irreducible in $\mathbb{Q}[X]$.
Assume that $p$ is a prime divisor of $\gcd\{g_1(x)\cdots g_s(x)\mid x\in\mathbb{Z}\}$. There must be some $y\in\mathbb{Z}$ such that $p\nmid f_1(y)\cdots f_s(y)$.
For $p\nmid e$ there is some $r\in\mathbb{Z}$ such that $er+c\equiv y\mod p$. Then
\[p\mid g_1(r)\cdots g_s(r)\mid f_1(er+c)\cdots f_s(er+c),
\]
so $0\equiv f_1(er+c)\cdots f_s(er+c)\equiv f_1(y)\cdots f_s(y)\mod p$ yields a contradiction.
For $p\mid e$ we have $v_p(f_i(c))\leq v_p(e)$ and hence $p\nmid \frac{f_i(c)}{\gcd(f_i(c),e)}=g_i(0)$ for each $i$. Therefore $p\nmid g_1(0)\cdots g_s(0)$, which is a contradiction too.
Hence $\gcd\{g_1(x)\cdots g_s(x)\mid x\in\mathbb{Z}\}=1$ and $g_i\in\mathbb{Z}[X]$ is irreducible in $\mathbb{Q}[X]$ and $g_i(x)\rightarrow\infty$ for $x\rightarrow\infty$ for each $i$, so Schinzel's hypothesis (H) implies that there is some $z\in\mathbb{Z}$ such that $g_i(z)=\frac{f_i(ez+c)}{\gcd(f_i(c),e)}$ is prime for each $i$. The claim follows with $x:=ez+c$.
\end{proof}
\begin{theorem}[cf.\ \cref{schinzelthmm}]\label{schinzelthm}
If Schinzel's hypothesis (H) is true, then $\overline{\mathfrak{X}(\mathbb{Z})}=L$ (where the closure is taken with respect to the topology defined in \Cref{azuconst}).
\end{theorem}
\begin{proof}
Let $a$ be the largest divisor of $k$ such that $m$ is an $a$-th power. Let $n:=\sqrt[a]m$ and $b:=\frac ka$.
Consider the factorization\footnote{From now on, ``divisor'' will mean ``\emph{positive} divisor''.}
\begin{align*}
X^{ab}+n^a
&=-n^a\left(\left(-\frac{X^b}n\right)^a-1\right)
=-n^a\prod_{d\mid a}\phi_d\left(-\frac{X^b}n\right)\\
&=\prod_{d\mid a}(-n)^{\varphi(d)}\phi_d\left(-\frac{X^b}n\right).
\end{align*}
The last equality follows from the fact that $\sum_{d\mid a}\varphi(d)=a$.
Let $f_d(X):=(-n)^{\varphi(d)}\phi_d\left(-\frac{X^b}n\right)$ for each divisor $d$ of $a$.
The polynomials $f_d(X)\in\mathbb{Q}[X]$ are irreducible according to \Cref{phiirred} and the choice of $a$.
Moreover $f_d(X)\in\mathbb{Z}[X]$ as $\phi_d(Y)$ has integral coefficients and degree $\varphi(d)$.
Take $(x_v,y_v,z_v)_v\in L$, a finite set $T\subseteq\Omega\setminus\{\infty\}$ and $t\geq0$. We have to show that there is some $(x,y,z)\in \mathfrak{X}(\mathbb{Z})$ such that $v_p(z_p-z)\geq t$ for all $p\in T$.
The set $L$ is open, as it is the intersection of finitely many sets $S(a,b,n)$, which are themselves open as the map $\prod_v\mathfrak{X}(\mathbb{Z}_v)\rightarrow\{0,1/2\}$ given by $(x_v,y_v,z_v)_v\mapsto\sum_v\inv_v(A(x_v,y_v,z_v))$ is locally constant. As $U(\mathbb{Q}_v)\cap \mathfrak{X}(\mathbb{Z}_v)$ is dense in $\mathfrak{X}(\mathbb{Z}_v)$ for each $v\in\Omega$, we can hence assume that $n^a-z_p^{ab}\neq0$ for all primes $p$.
{\parindent=1cm
\hangindent=1cm
\textbf{Claim.} For each $d\mid a$ we have $\prod_p(f_d(-z_p),-1)=1$ where the product runs over all primes $p$ (in particular $(f_d(-z_p),-1)=1$ for almost all primes $p$).
\emph{Proof.} The factors $(f_d(-z_p),-1)$ are well-defined as $f_d(-z_p)\neq0$ due to $n^a-z_p^{ab}\neq0$.
As $(x_v,y_v,z_v)_v\in S(\frac ad,db,n^d)$, we conclude that $\prod_p(n^d-z_p^{db},-1)=1$. But
\[
X^{db}+n^d=\prod_{d'\mid d}(-n)^{\varphi(d')}\phi_{d'}\left(-\frac{X^b}n\right)=\prod_{d'\mid d}f_{d'}(X),
\]
so
\[
1=\prod_p(n^d-z_p^{db},-1)=\prod_p\prod_{d'\mid d}(f_{d'}(-z_p),-1).
\]
The result follows by induction by $d$.
\qed
}
Assume without loss of generality that $2\in T$ and that for each prime $p$: if $p\in T$, then $t\geq v_p(f_d(-z_p))+2$ for all $d\mid a$ and if $(f_d(-z_p),-1)\neq1$ for some $d\mid a$, then $p\in T$.
The leading coefficient of $f_d(X)$ is 1 and its degree is $b\varphi(d)>0$, so $f_d(u)\rightarrow\infty$ for $u\rightarrow\infty$.
Furthermore $\gcd\{\prod_{d\mid a}f_d(u)\mid u\in\mathbb{Z}\}=\gcd\{u^{ab}+n^a\mid u\in\mathbb{Z}\}=1$, as it divides $\gcd(n^a,1+n^a)=1$.
The Chinese remainder theorem shows that there is some $c\in\mathbb{Z}$ such that $c\equiv -z_p\mod p^t$ for each $p\in T$.
Let $e:=\prod_{p\in T}p^t$.
Now $v_p(f_d(c))=v_p(f_d(-z_p))\leq t-2<t=v_p(e)$ for each $p\in T$ and each $d\mid a$.
Hence applying \Cref{schinzelhelp} proves that there is some $z\in\mathbb{Z}$ such that $-z\equiv c\mod e$ (in particular $v_p(z_p-z)\geq\min(v_p(z_p+c),v_p(-z-c))\geq t$ for each prime $p\in T$) and $\frac{f_d(-z)}{\gcd(f_d(c),e)}$ is prime for each $d\mid a$.
Therefore $f_d(-z)\equiv f_d(c)\equiv f_d(-z_p)\mod p^t$, so in particular
\[
v_p(f_d(-z))=v_p(f_d(-z_p))
\]
and
\[
r_p(f_d(-z))\equiv r_p(f_d(-z_p))\mod p^2
\]
as $t\geq v_p(f_d(-z_p))+2$ for each $p\in T$.
Hence $r_2(f_d(-z))\equiv r_2(f_d(-z_2))\mod4$. As $\prod_p(f_d(-z_p),-1)=1$ and moreover ${p\in T}$ whenever $(f_d(-z_p),-1)\neq1$ (i.e., whenever $p^{v_p(f_d(-z_p))}\not\equiv1\mod4$), the following congruence holds:
\[
r_2(f_d(-z_2))\equiv\prod_{p\neq2}p^{v_p(f_d(-z_p))}\equiv\prod_{p\in T\setminus\{2\}}p^{v_p(f_d(-z_p))}\equiv r_2(\gcd(f_d(c),e))\mod4.
\]
Together we get $r_2(f_d(-z))\equiv r_2(\gcd(f_d(c),e))\mod4$, so $r_2\left(\frac{f_d(-z)}{\gcd(f_d(c),e)}\right)\equiv1\mod4$. As $\frac{f_d(-z)}{\gcd(f_d(c),e)}$ is prime, it is therefore a sum of two squares.
The product
\[
\prod_{d\mid a}\gcd(f_d(c),e)=\prod_{p\in T}\prod_{d\mid a}p^{v_p(f_d(-z_p))}=\prod_{p\in T}p^{v_p(n^a-z_p^{ab})}
\]
is also a sum of two squares as all primes $p\equiv3\mod4$ occur an even number of times in it because they do so in $n^a-z_p^{ab}=x^2+y^2$.
Now in the factorization
\[
n^a-z^{ab}=\prod_{d\mid a}f_d(-z)=\Bigg(\prod_{d\mid a}\gcd(f_d(c),e)\Bigg)\cdot\Bigg(\prod_{d\mid a}\frac{f_d(-z)}{\gcd(f_d(c),e)}\Bigg)
\]
the first product and each factor of the second product are sums of two squares, so $n^a-z^{ab}$ is a sum of two squares, too.
\end{proof}
\begin{lemma}\label{S1kn}
We have $S(1,k,m)=\prod_v\mathfrak{X}(\mathbb{Z}_v)$.
\end{lemma}
\begin{proof}
For each place $v$ and $(x_v,y_v,z_v)\in U(\mathbb{Q}_v)\cap\mathfrak{X}(\mathbb{Z}_v)$ we have
\[(m-z_v^k,-1)=(m^1-z_v^{1\cdot k},-1)=1,
\]
so $\inv_v(A(x_v,y_v,z_v))=0$.
\end{proof}
\begin{remark}
\Cref{schinzelthm} does not hold for arbitrary even $k$. For example \cref{zk2} does not have an integral solution for $k=4$ and $m=22$ but it has local solutions $(x_v,y_v,z_v)_v\in \prod_v\mathfrak{X}(\mathbb{Z}_v)=S(1,k,m)=L$ given by
\[
z_v=
\begin{cases}
1,&v=2\text{ or }11\\
0,&\text{else}.
\end{cases}
\]
\end{remark}
\begin{corollary}\label{bunyathm}
Assume Schinzel's hypothesis (H) is true.
Let $k$ be an odd positive integer and assume that $m$ is not a $p$-th power for any prime $p\mid k$. Then there exists an integral solution to \cref{zk2}.
\end{corollary}
\begin{proof}
Then $\overline{\mathfrak{X}(\mathbb{Z})}=L=S(1,k,n)=\prod_v\mathfrak{X}(\mathbb{Z}_v)\neq\emptyset$ according to \Cref{real,odd,a1I2}.
\end{proof}
\begin{remark}
The proof of \Cref{bunyathm} needs Schinzel's hypothesis (H) only in the case of one polynomial, also known as Bunyakovsky's conjecture (cf. \cite[Conjecture 1]{moroz}).
Under this assumption, this corollary includes the result of Davenport and Heilbronn mentioned above.
\end{remark}
\begin{corollary}[cf.\ \cref{primekm}]\label{primek}
Assume Bunyakovsky's conjecture is true.
Let $k$ be an odd prime. Then there exists an integral solution to \cref{zk2}.
\end{corollary}
\begin{proof}
If $m$ is a $k$-th power, then $(0,0,\sqrt[k]m)$ is a solution. Otherwise the previous corollary applies.
\end{proof}
\begin{lemma}\label{abgeneral}
Let $k$ be the product of two odd primes $a$ and $b$ and let $m\in\mathbb{Z}\setminus\{0\}$.
For the existence of integral solutions to \cref{zk2}, it is necessary and under Schinzel's hypothesis (H) also sufficient that the following two statements are both true.
\begin{itemize}
\item There is no $n\in\mathbb{Z}$ such that $m=n^a$ and $0\not\in I_2(a,b,n)$ and $1/2\not\in I_p(a,b,n)$ for each prime $p\equiv3\mod4$.
\item There is no $n\in\mathbb{Z}$ such that $m=n^b$ and $0\not\in I_2(b,a,n)$ and $1/2\not\in I_p(b,a,n)$ for each prime $p\equiv3\mod4$.
\end{itemize}
\end{lemma}
\begin{proof}
For necessity, let $n\in\mathbb{Z}$ and $m=n^a$ and $0\not\in I_2(a,b,n)$ and $1/2\not\in I_p(a,b,n)$ for each prime $p\equiv3\mod4$. Then $S(a,b,n)=\emptyset$ according to \Cref{real,1mod4}. Hence $\mathfrak{X}(\mathbb{Z})\subseteq S(a,b,n)=\emptyset$.
Conversely, if $m$ is an $ab$-th power, then $(0,0,\sqrt[ab]m)$ is a solution.
If $m$ is neither an $a$-th power nor a $b$-th power, then \Cref{bunyathm} proves the claim.
Let therefore without loss of generality $m$ be an $a$-th power but not an $ab$-th power, so there is some $n\in\mathbb{Z}$ such that $m=n^a$. Now the first statement given above implies that $0\in I_2(a,b,n)$ or $1/2\in I_p(a,b,n)$ for some prime $p\equiv3\mod4$. Together with \Cref{real,odd,6mod8,not6mod8} this shows that $S(a,b,n)\neq\emptyset$.
Moreover $S(1,ab,m)=\prod_v\mathfrak{X}(\mathbb{Z}_v)$ according to \Cref{S1kn}.
Hence $\overline{\mathfrak{X}(\mathbb{Z})}=S(1,ab,m)\cap S(a,b,n)=S(a,b,n)\neq\emptyset$.
\end{proof}
\begin{theorem}[cf.\ \cref{abthmm}]\label{abthm}
Let $k$ be the product of two primes $a,b\equiv1\mod4$ and let $m\in\mathbb{Z}\setminus\{0\}$.
For the existence of integral solutions to \cref{zk2}, it is necessary and under Schinzel's hypothesis (H) also sufficient that the following two statements are both true.
\begin{itemize}
\item There is no $n\equiv6\mod8$ such that $m=n^a$ and for each prime $p\equiv3\mod4$ dividing $n$:
$b\nmid v_p(n)$ or $2\mid v_p(n)$ or there is no $z'\in\{0,\dots,p-1\}$ such that
\[p\mid r_p(n)^{a-1}+\dots+z'^{(a-1)b}.
\]
\item There is no $n\equiv6\mod8$ such that $m=n^b$ and for each prime $p\equiv3\mod4$ dividing $n$:
$a\nmid v_p(n)$ or $2\mid v_p(n)$ or there is no $z'\in\{0,\dots,p-1\}$ such that
\[p\mid r_p(n)^{b-1}+\dots+z'^{(b-1)a}.
\]
\end{itemize}
\end{theorem}
\begin{proof}
The condition of \Cref{abgeneral} is equivalent to the condition of this theorem according to \Cref{6mod8,not6mod8,jagypre1,12insidenmid}.
\end{proof}
\section{Algorithm}\label{algosection}
We give an algorithm to decide for given $m\in\mathbb{Z}$ and odd $k>0$ if the number $m$ is of the form $x^2+y^2+z^k$.
\begin{algorithmic}[1]
\Function{combi}{$a,b,n,p$}
\State Consider all Hilbert symbols over $\mathbb{Q}_p$.\label{nots}
\State Let $f_d(Z):=(-n)^{\varphi(d)}\phi_d(-Z^b/n)$.
\State For each $z\in\mathbb{Z}$ let $w_z:=\{d\text{ divisor of }a\mid (f_d(-z),-1)\neq1\}$.
\State For each $z\in\mathbb{Z}$ and $t\geq0$ let $G_{t,z}:=\{d\text{ divisor of }a\mid v_p(f_d(-z))+1\geq t\}$.\label{note}
\If{$p\equiv1\mod4$}
\State\Return$\{\emptyset\}$
\Else
\State $W\gets\emptyset$
\State $S_0\gets\{0\}$
\State $t\gets0$
\While{$S_t\neq\emptyset$}
\State $S_{t+1}\gets\emptyset$
\ForAll{$z\in S_t$}
\If{$(n^a-z^{ab},-1)=1$}
\State $W\gets W\cup\{w_z\}$
\EndIf\label{Rpos1}
\If{$|G_{t,z}|>1$ or\Statex[6] ($|G_{t,z}|=1$ and $w_z\setminus G_{t,z}\not\in W$ and $w_z\cup G_{t,z}\not\in W$)}
\State $S_{t+1}\gets S_{t+1}\cup\{z'\in[0,p^{t+1}-1]\mid z'\equiv z\mod p^t\}$
\EndIf
\EndFor
\State $t\gets t+1$
\EndWhile
\State\Return $W$
\EndIf
\EndFunction
\algstore{alg1}
\end{algorithmic}
\begin{lemma}
Let $a\geq1$ and $b\geq3$ be odd integers, $n$ an integer such that $n$ is not a $q$-th power for any prime divisor $q$ of $b$ (then $n^a-z^{ab}\neq0$ for all $z\in\mathbb{Z}$) and let $p$ be prime.
Let $f_d(Z)$, $w_z$ and $G_{t,z}$ be defined as in lines \ref{nots} to \ref{note}.
Then \Call{combi}{$a,b,n,p$} terminates and returns
\[
C:=\{w_z\mid z\in\{0,1,2,\dots\}\textnormal{ such that }(n^a-z^{ab},-1)=1\}.
\]
\end{lemma}
\begin{proof}
For $p\equiv1\mod4$ the result is immediate, so let $p\not\equiv1\mod4$.
The algorithm describes a pruned breadth-first search\footnote{i.e., a breadth-first search in which insignificant branches are ignored} on the infinite directed graph with the following node set $V$ and edge set $E$:
\[
V:=\{(t,z)\mid t\geq0\text{ and }0\leq z\leq p^t-1\}\]
\[E:=\{((t,z),(t+1,z'))\in V^2\mid z\equiv z'\mod p^t\}.
\]
Each time a node $(t,z)$ with $(n^a-z^{ab},-1)=1$ is visited, $w_z$ is appended to $W\subseteq C$.
In this graph there is a path from $(t,z)\in V$ to $(t',z')\in V$ if and only if $t\leq t'$ and ${z\equiv z'\mod p^t}$.
Obviously every node can be reached from $(0,0)$.
Therefore a complete breadth-first search would eventually find every $c\in C$.
Let $(t,z)$ and $(t',z')$ be nodes such that $(t',z')$ is reachable from $(t,z)$, i.e., such that ${z'\equiv z\mod p^t}$. If $d\not\in G_{t,z}$ is a divisor of $a$, then $v_p(f_d(-z))<t-1$. Then $f_d(-z')\equiv f_d(-z)\mod p^t$ implies $v_p(f_d(-z'))=v_p(f_d(-z))$ and $r_p(f_d(-z'))\equiv r_p(f_d(-z))\mod p^2$, so $(f_d(-z'),-1)=(f_d(-z),-1)$. Hence $w_{z'}\setminus G_{t,z}=w_z\setminus G_{t,z}$.
In particular $w_{z'}=w_z$ if $G_{t,z}=\emptyset$. Therefore the breadth-first search does not have to be continued from $(t,z)$ on if $G_{t,z}=\emptyset$.
Every set $c\in C$ has an even number of elements as $\prod_{d\mid a}(f_d(-z),-1)=(n^a-z^{ab},-1)$ for each $z\in\mathbb{Z}$. Hence for each subset $w'$ of the set of divisors of $a$ and each divisor $g$ of $a$ at least one of the sets $w'\setminus\{g\}$ and $w'\cup\{g\}$ is not contained in $C$.
Therefore, if $G_{t,z}=\{g\}$ for some divisor $g$ of $a$ and $w_z\setminus\{g\}$ or $w_z\cup\{g\}$ has already been found (and is therefore contained in $C$), then the breadth-first search does not have to be continued from $(t,z)$, either.
Hence altogether the above algorithm finds every element of $C$, so the only remaining question is whether it terminates in a finite amount of time.
Assume it does not.
Then there has to be an infinite path $(0,z_0)\rightarrow(1,z_1)\rightarrow(2,z_2)\rightarrow\dots$ of which every edge is visited during the breadth-first search.
The definition of the edge set $E$ proves that $z_t$ converges to some $\overline z\in\mathbb{Z}_p$ such that ${\overline z\equiv z_t\mod p^t}$ for each $t\geq0$.
If $d\in G_{t+1,z_{t+1}}$, then $f_d(-z_t)\equiv f_d(-z_{t+1})\equiv0\mod p^t$, so $d\in G_{t,z_t}$.
Hence $G_{0,z_0}\supseteq G_{1,z_1}\supseteq G_{2,z_2}\supseteq\dots$.
If $G_{t,z_t}$ was empty for some $t$, then the breadth-first search would not continue from the node $(t,z_t)$ on.
Hence $\bigcap_{t\geq0}G_{t,z_t}\neq\emptyset$.
If $g\in\bigcap_{t\geq0}G_{t,z_t}$, then $f_g(-\overline z)\equiv f_g(-z_t)\equiv0\mod p^{t-1}$ for all $t\geq1$, so $f_g(-\overline z)=0$. However, the polynomials $f_d(Z)$ with $d\mid a$ are irreducible (according to \Cref{phiirred}) and pairwise distinct, so no two of them have any common roots. Hence $\bigcap_{t\geq0}G_{t,z_t}$ contains exactly one element $g$. Let $T\geq0$ such that $G_{T,z_T}=\{g\}$.
Then $w_{z_t}\setminus\{g\}=w_{z_T}\setminus\{g\}$ for each $t\geq T$.
As the breadth-first search continues at every node $(t,z_t)$, it follows that neither $w_{z_T}\setminus\{g\}$ nor $w_{z_T}\cup\{g\}$ are found. As every element of $C$ is eventually found, this shows that $w_{z_T}\setminus\{g\},w_{z_T}\cup\{g\}\not\in C$.
However,
\[n^a-(\overline z\pm p^s)^{ab}\equiv n^a-\overline z^{ab}\mp ab\overline z^{ab-1}p^s\equiv\mp ab\overline z^{ab-1}p^s\mod p^{2s}
\]
(as $f_g(-\overline z)=0$ and $f_g(Z)$ divides $n^a-Z^{ab}$). Therefore (as $\overline z\neq0$ due to $n^a-\overline z^{ab}=0$), by choosing $s$ sufficiently large and of the correct parity and the appropriate sign, we get some $z'\equiv\overline z\mod p^T$ such that $(n^a-z'^{ab},-1)=1$, so $w_{z'}\in C$. Moreover, $w_{z'}\setminus\{g\}=w_{z_T}\setminus\{g\}$ because of $G_{t,z_t}=\{g\}$. This proves that $w_{z_T}\setminus\{g\}\in C$ or $w_{z_T}\cup\{g\}\in C$, which is a contradiction.
Therefore the algorithm terminates in a finite amount of time.
\end{proof}
Let $\bigtriangleup$ denote the symmetric difference (i.e., $A\bigtriangleup B=(A\setminus B)\cup(B\setminus A)$).
Consider now the following algorithm:
\begin{algorithmic}[1]
\algrestore{alg1}
\Function{ispossible}{$k,m$}
\If{$\sqrt[k]m\in\mathbb{Z}$}
\State \Return true
\Else
\State $a\gets\max\{d\text{ divisor of }k\mid m\text{ is $d$-th power}\}$
\State $b\gets \frac ka$
\State $n\gets\sqrt[a]m$
\State $T\gets\{\emptyset\}$
\ForAll{$p\mid 2an$ prime}
\State $W\gets$ \Call{combi}{$a,b,n,p$}
\State $T\gets\{t\bigtriangleup w\mid t\in T, w\in W\}$
\EndFor\label{Tfinished}
\State \Return $\emptyset\in T$
\EndIf
\EndFunction
\end{algorithmic}
\begin{theorem}
Let $k\geq1$ be odd and $m\in\mathbb{Z}$.
Then \Call{ispossible}{$k,m$} always terminates. If it returns ``false'', then $m$ is not of the form $x^2+y^2+z^k$ for integers $x,y,z$. If Schinzel's hypothesis (H) is true, then the converse also holds.
\end{theorem}
\begin{proof}
The case $\sqrt[k]m\in\mathbb{Z}$ is obvious, so assume $\sqrt[k]m\not\in\mathbb{Z}$.
According to the previous lemma (and as $\{0,1,2,\dots\}$ is dense in $\mathbb{Z}_p$ and the map $\mathbb{Q}_p^\times\rightarrow\{\pm1\}$ defined by $u\mapsto (u,-1)$ is locally constant for each prime $p$), the set $T$ can after line \ref{Tfinished} be described as follows:
\[
T=\bigg\{\mathop{\bigtriangleup}\limits_{p\mid 2an}w_{z_p}\ \bigg|\ (x_p,y_p,z_p)_p\in\prod_{p\mid 2an}U(\mathbb{Q}_p)\cap\mathfrak{X}(\mathbb{Z}_p)\bigg\}.
\]
Hence $\emptyset\in T$ if and only if there is some $(x_p,y_p,z_p)_p\in\prod_{p\mid 2an}U(\mathbb{Q}_p)\cap\mathfrak{X}(\mathbb{Z}_p)$ such that $\prod_{p\mid 2an}(f_d(-z_p),-1)=1$ for each divisor $d$ of $a$.
If $v$ is a prime not dividing $2an$ or $v=\infty$ and $(x_v,y_v,z_v)\in U(\mathbb{Q}_v)\cap\mathfrak{X}(\mathbb{Z}_v)$, then, according to \Cref{real,1mod4,jagypre1}, $(n^d-z_v^{db},-1)=1$ for each $d\mid a$.
Furthermore, for each such place $v$ the set $U(\mathbb{Q}_v)\cap\mathfrak{X}(\mathbb{Z}_v)$ is nonempty according to \Cref{real,odd}.
The equation
\[
(n^d-z_v^{db},-1)=\prod_{d'\mid d}(f_{d'}(-z_v),-1)
\]
therefore shows (as in the proof of the claim in the proof of \Cref{schinzelthm}) that there is some $(x_v,y_v,z_v)_v\in\prod_vU(\mathbb{Q}_v)\cap\mathfrak{X}(\mathbb{Z}_v)$ such that $\prod_v(n^d-z_v^{bd},-1)=1$ for each $d\mid a$ if and only if $\emptyset\in T$.
Therefore $L\neq\emptyset$ if and only if $\emptyset\in T$, so the claim follows with \Cref{schinzelthm}.
\end{proof}
For each odd composite integer $1<k<50$, \Cref{listexceptions} lists values of positive integers $m\leq10^9$ such that \cref{zk2} has no integral solution, determined using our algorithm. The lists might be incomplete if Schinzel's hypothesis (H) is false.
\renewcommand{\arraystretch}{1.17}
\begin{table}[ht]
\caption{List of integers without integral solution}\label{listexceptions}
\begin{tabular}{c|p{11.5cm}}
\textbf{$k$}&\textbf{List of integers $1\leq m\leq10^9$ without integral solution}\\\hline
\input{bigtable.tex}
\end{tabular}
\end{table}
\renewcommand{\arraystretch}{1}
\bibliographystyle{alpha}
|
2,869,038,156,372 | arxiv | \section*{Introduction}
\qquad \emph{Exact sequences} and \emph{exact functors} are important tools
in Homological Algebra which was developed first in the categories of
modules over rings \cite{CE1956} and generalized later to arbitrary Abelian
categories (e.g. \cite{Hel1958}). Different sets of axioms characterizing
\emph{additive} abstract categories which can be considered -- in some sense
-- \emph{natural home} for exact sequences were developed over time; such
categories were called \emph{exact} (e.g. \emph{Buchsbaum-exact categories}
\cite{Buc1955}, \emph{Quillen-exact categories} \cite{Qui1973}). For these
categories, the defining axioms are usually based on a distinguished class
of sequences, called an \emph{exact structure}, which is used to define the
(short and long) exact sequences in the resulting exact category as well as
exact functors between such exact categories. On the other hand, the
so-called \emph{Barr-exact categories }\cite{Bar1971}, which are \emph
regular categories }with canonical $(\mathbf{RegEpi},\mathbf{Mono})
-factorization structures (e.g. \cite{Gri1971}, \cite[14.E]{AHS2004}, \cit
{Bor1994b}), provide an alternative notion of exactness in possibly \emph
non-additive} categories. In such categories, the role of exact sequences is
played by the so-called \emph{exact forks} which are also used to define
exact functors between Barr-exact categories. For a systematic study and
comprehensive exposition of these and other notions of exact categories, the
interested reader is advised to consult \cite{Bue2010}.
An elegant notion of exact categories to which we refer often in this
manuscript is duo to Puppe \cite{Pup1962} (see also Mitchell \cite{Mit1965
). We call a category $\mathfrak{C}$ a \emph{Puppe-exact category}\textit{\
iff it is pointed (i.e. $\mathrm{Hom}_{\mathfrak{C}}(A,B)$ has a zero
morphism for each $A,B\in \mathrm{Obj}(\mathfrak{C})$) and has a $(\mathbf
NormalEpi},\mathbf{NormalMono})$\emph{-factorization structure} (e.g. \cite
14.F]{AHS2004}); such a category is \emph{additive} if and only if it is
Abelian (cf. \cite[3.2]{BP1969}). By \cite[13.1.3]{Sch1972}, any Puppe-exact
category has kernels and cokernels; moreover it is \emph{normal} (i.e. every
monomorphism is a kernel) and \emph{conormal} (i.e. every epimorphism is a
cokernel). The \emph{image} (\emph{coimage}) of a morphism $\gamma $ in a
Puppe-exact category $\mathfrak{C}$ is defined as $\func{Im}(\gamma ):
\mathrm{Ker}(\mathrm{coker}(\gamma )$ ($\mathrm{Coim}(\gamma ):=\mathrm{Coke
}(\mathrm{ker}(\gamma ))$) and a sequence $A\overset{f}{\longrightarrow }
\overset{g}{\longrightarrow }C$ in $\mathfrak{C}$ is said to be \emph{exact}
iff $\func{Im}(f)\simeq \mathrm{Ker}(g)$ or equivalently $\mathrm{Coim
(g)\simeq \mathrm{Coker}(f)$ \cite[12.4.9, 13.1.3]{Sch1972}.
Many interesting pointed categories are not Puppe-exact, (e.g. some
varieties of Universal Algebra like the variety $\mathbf{Grp}$ of groups,
the variety $\mathbf{Mon}$ of monoids and the variety $\mathbf{pSet}$ of
pointed sets). Thus, the following question arises naturally
\begin{equation*}
\text{\textbf{Question:}}\mathbf{\ }\emph{When\ is\ an\ exact\ sequence\ }
\overset{f}{\longrightarrow }B\overset{g}{\longrightarrow }C\emph{\ in\ a\
pointed\ category\ exact?}
\end{equation*
The main goal of this article is providing an answer to the above mentioned
question. Our approach is based on analyzing the notion of exact sequences
in Puppe-exact categories and then generalizing it to any pointed category
\mathfrak{C}$ relative to a given $(\mathbf{E},\mathbf{M})$\emph
-factorization structure}, which always exists \cite[Section 14]{AHS2004}
(see also \cite{Bar2002}): we say that a sequence $A\overset{f}
\longrightarrow }B\overset{g}{\longrightarrow }C$ is $(\mathbf{E},\mathbf{M
) $\emph{-exact} iff there exist $f^{\prime }\in \mathbf{E}$ and $g^{\prime
\prime }\in \mathbf{M}$ such that $f=\mathrm{ker}(g)\circ f^{\prime }$ and
g=g^{\prime \prime }\circ \mathrm{coker}(f)$ are the \emph{essentially uniqu
} $(\mathbf{E},\mathbf{M})$-factorizations of $f$ and $g$ in $\mathfrak{C}.$
To illustrate this notion of exactness, we introduce a restricted version of
the Short Exact Lemma and introduce a class of \emph{relative homological
categories} which generalizes the notion of \emph{homological categories} in
the sense of Borceux and Bourn \cite[Chapter 4]{BB2004}.
Before we proceed, we find it suitable to include the following
clarification. A successful notion of exact sequences already exists in
several pointed categories which are not Puppe-exact (e.g. in $\mathbf{Grp}
):\ A sequence $A\overset{f}{\longrightarrow }B\overset{g}{\longrightarrow
C $ of groups is exact iff $\func{Im}(f)\simeq \mathrm{Ker}(g).$ While used
in many papers, and even considered \emph{standard}, this notion of
exactness is not necessarily appropriate in other pointed categories (e.g.
in $\mathbf{Mon}$). We briefly demonstrate why we believe this is the case.
Firstly, one should be careful about the definition of the image (coimage)
of a morphisms $\gamma $ in a category which is not Puppe-exact: although
several authors define $\func{Im}(\gamma ):=\mathrm{Ker}(\mathrm{coker
(\gamma ))$ ($\mathrm{Coim}(\gamma ):=\mathrm{Coker}(\mathrm{ker}(\gamma ))
), this might not be the appropriate notion in a category which is not
Puppe-exact as it does not necessarily satisfy the universal property that
an image (coimage) is supposed to satisfy (cf. \cite[5.8.7]{Fai1973} and
\cite{EW1987}). Secondly, even if the appropriate image (coimage) is used,
one has to take into consideration the natural dual condition of exactness,
namely $\mathrm{Coim}(g)\simeq \mathrm{Coker}(f).$ This \emph{hidden}
condition is equivalent to $\func{Im}(f)\simeq \mathrm{Ker}(g)$ in
Puppe-exact categories \cite[Lemma 13.1.4]{Sch1972}; however, this is not
necessarily the case in categories which are not Puppe-exact. So, one might
end up with two different notions: \emph{left-exact sequences} for which
\func{Im}(f)\simeq \mathrm{Ker}(g)$ and \emph{right-exact sequences} for
which $\mathrm{Coim}(g)\simeq \mathrm{Coker}(f),$ while exact sequences have
to be defined as those which are left-exact and right-exact.
An example that demonstrates how adopting the definition of exact sequences
in Puppe-exact categories to arbitrary pointed categories might create
serious problems is the notion of exact sequences of semimodules over
semirings due to Takahashi \cite{Tak1981}. An unfortunate choice of a notion
of exactness and an inappropriate choice of a \emph{tensor functor} which is
not left adjoint of the Hom functor, in addition to the \textit{bad} nature
of monoids (in contrast with the \textit{good} nature of groups), are among
the main reasons for failing to develop a satisfactory homological theory
for semimodules or commutative monoids so far (there are indeed many
successful investigations related to the homology of monoids, e.g. \cit
{KKM2000}).
This manuscript is divided as follows. After this introduction, and for the
convention of the reader, we recall in Section 1 some terminology and
notions from Category Theory. In particular, we analyze the notion of exact
sequences in \emph{Puppe-exact categories} and use that analysis to
introduce a new notion of exact sequences in arbitrary pointed categories.
Moreover, we present some special classes of morphisms which play an
important role in the sequel. In Section 2, we collect some definitions and
results on semirings and semimodules and clarify the differences between the
terminology used in this paper and the classical terminology; we also
clarify the reason for changing some terminology. In Section 3, we apply our
general definition of exactness to obtain a new notion of exact sequences of
semimodules over semirings. We demonstrate how this notion enables us to
characterize in a very simple way, similar to that in homological
categories, different classes of morphisms (e.g. monomorphisms, regular
epimorphisms, isomorphisms). In Section 4, we illustrate the advantage of
our notion of exactness over the existing ones by showing how it enables us
to prove some of the elementary diagram lemmas for semimodules over
semirings. Moreover, we introduce a restricted version of the \emph{Short
Five Lemma \ref{short-5}}, which characterizes the homological categories
among the pointed regular ones, and use it to introduce a new class of \emph
relative homological categories} w.r.t. a given factorization structure and
a special class of morphisms. The category of cancellative semimodules over
semirings, in particular the category of cancellative commutative monoids,
is introduced as a prototype of such categories. Moreover, we prove a
restricted version of the \emph{Snake Lemma \ref{snake}} for cancellative
semimodules (cancellative commutative monoids) which opens the door for
introducing and investigating \emph{homology objects} in such categories.
\section{Exact Sequences in Pointed Categories}
\qquad Throughout, and unless otherwise explicitly mentioned, $\mathfrak{C}$
is an arbitrary \emph{pointed} category (i.e. $\mathrm{Hom}_{\mathfrak{C
}(A,B)$ has a zero morphism); all objects and morphisms are assumed to be in
$\mathfrak{C}.$ When clear from the context, we may drop $\mathfrak{C}.$ Our
main references in Category Theory are \cite{AHS2004} and \cite{Mac1998}.
\begin{punto}
A \emph{monomorphism} in $\mathfrak{C}$ is a morphism $m$ such that for any
morphisms $f_{1},f_{2}:
\begin{equation*}
m\circ f_{1}=m\circ f_{2}\Rightarrow f_{1}=f_{2}.
\end{equation*
An \emph{equalizer} of a family of morphisms $(f_{\lambda }:A\rightarrow
B)_{\Lambda }$ in $\mathfrak{C}$ is a morphism $g:A^{\prime }\rightarrow A$
in $\mathfrak{C}$ such that $f_{\lambda }\circ g=f_{\lambda ^{\prime }}\circ
g$ for all $\lambda ,\lambda ^{\prime }\in \Lambda $ and whenever there
exists $g^{\prime }:A^{\prime \prime }\rightarrow A$ with $f_{\lambda }\circ
g^{\prime }=f_{\lambda ^{\prime }}\circ g^{\prime }$ for all $\lambda
,\lambda ^{\prime }\in \Lambda $ then there exists a \emph{unique} morphism
\widetilde{g}:A^{\prime \prime }\rightarrow A^{\prime }$ such that $g\circ
\widetilde{g}=g^{\prime }:$
\begin{equation*}
\xymatrix{& A'' \ar@{.>}_{\tilde{g}}[dl] \ar^{g'}[d] & & \\ A' \ar_{g}[r] &
A \ar@<1ex>^{f_{\lambda}}[r] \ar[r] & B}
\end{equation*
With $\mathrm{Equ}((f_{\lambda })_{\lambda \in \Lambda })$ we denote the
domain of the \emph{essentially unique} equalizer of $(f_{\lambda
})_{\lambda \in \Lambda },$ if it exists. A morphism $g$ in in $\mathfrak{C}$
is said to be a \emph{regular monomorphism} iff $g=\mathrm{equ}(f_{1},f_{2})$
for two morphisms $f_{1},f_{2}$ in $\mathfrak{C}.$
\end{punto}
\begin{punto}
Let $g$ be a morphism in $\mathfrak{C}.$ We call $\mathrm{ker}(f):=\mathrm
Equ}(f,0)$ the \emph{kernel} of $f.$ We say that $g$ is a \emph{normal
monomorphism} iff $g=\mathrm{ker}(f)$ for some morphism $f$ in $\mathfrak{C
. $ The category $\mathfrak{C}$ is said to be \emph{normal} iff every
monomorphism in $\mathfrak{C}$ is normal.
\end{punto}
\begin{punto}
An \emph{epimorphism} in $\mathfrak{C}$ is a morphism $e$ such that for any
morphisms $f_{1},f_{2}:
\begin{equation*}
f_{1}\circ e=f_{2}\circ e\Rightarrow f_{1}=f_{2}.
\end{equation*
A coequalizer of a family of morphisms $(f_{\lambda }:A\rightarrow
B)_{\Lambda }$ in $\mathfrak{C}$ is a morphism $g:B\rightarrow B^{\prime }$
in $\mathfrak{C}$ such that $g\circ f_{\lambda }=g\circ f_{\lambda ^{\prime
}}$ for all $\lambda ,\lambda ^{\prime }\in \Lambda $ and whenever there
exists $g^{\prime }:B\rightarrow B^{\prime \prime }$ with $g^{\prime }\circ
f_{\lambda }=g^{\prime }\circ f_{\lambda ^{\prime }}$ for all $\lambda
,\lambda ^{\prime }\in \Lambda $ then there exists a \emph{unique} morphism
\widetilde{g}:B^{\prime }\rightarrow B^{\prime \prime }$ such that
\widetilde{g}\circ g=g^{\prime }:
\begin{equation*}
\xymatrix{ A \ar@<1ex>^{f_{\lambda}}[r] \ar[r] & B \ar_{g'}[d] \ar^{g}[r] &
B' \ar@{.>}^{\tilde{g}}[dl] \\& B'' & &}
\end{equation*
With $\mathrm{Coequ}((f_{\lambda })_{\lambda \in \Lambda })$ we denote the
codomain of the essentially unique coequalizer of $(f_{\lambda })_{\lambda
\in \Lambda },$ if it exists. A morphism $g$ is said to be a \emph{regular
epimorphism} iff $g=\mathrm{Coequ}(f_{1},f_{2})$ for two morphisms
f_{1},f_{2}$ in $\mathfrak{C}.$
\end{punto}
\begin{punto}
We call $\mathrm{Coker}(f):=\mathrm{Coequ}(f,0)$ the \emph{cokernel} of $f.$
A morphism $g$ is said to be a \emph{conormal epimorphism} iff $g=\mathrm
coker}(f)$ for some morphism $f$ in $\mathfrak{C}.$ The category $\mathfrak{
}$ is said to be \emph{conormal} iff every epimorphism in $\mathfrak{C}$ is
conormal.
\end{punto}
\begin{notation}
We fix some notation:
\begin{itemize}
\item With $\mathbf{Mono}$\textbf{$($}$\mathfrak{C)}$ ($\mathbf{RegMono}
\textbf{$($}$\mathfrak{C)}$) we denote the class of (regular) monomorphisms
in $\mathfrak{C}$ and by $\mathbf{Epi}$\textbf{$($}$\mathfrak{C)}$ ($\mathbf
RegEpi}$\textbf{$($}$\mathfrak{C)}$) the class of (regular) epimorphisms in
\mathfrak{C}.$ We denote by $\mathbf{NormMono}$\textbf{$($}$\mathfrak{C)
\subseteq \mathbf{RegMono(\mathfrak{C)}\ }$($\mathbf{NormEpi(\mathfrak{C)}
\subseteq \mathbf{RegEpi(\mathfrak{C)}}$) the class of normal monomorphisms
(normal epimorphisms) in $\mathfrak{C}.$
\item With $\mathbf{Iso}$\textbf{$($}$\mathfrak{C)}$ we denote the class of
isomorphisms and with $\mathbf{Bimor}$\textbf{$($}$\mathfrak{C)}$ the class
of bimorphisms (i.e. monomorphisms and epimorphisms) in $\mathfrak{C}.$
\item Let $\mathfrak{C}$ be concrete (over the category $\mathbf{Set}$ of
sets) with underlying functor $U:\mathfrak{C}\longrightarrow \mathbf{Set}.$
We denote by $\mathbf{Inj}$\textbf{$($}$\mathfrak{C)}$ ($\mathbf{Surj(
\mathfrak{C)}$) the class of morphisms $\gamma $ in $\mathfrak{C}$ such that
$U(\gamma )$ is an injective (surjective)\ map.
\end{itemize}
\end{notation}
\begin{remark}
(e.g. \cite[7.76]{AHS2004}) We hav
\begin{equation*}
\mathbf{Iso(\mathfrak{C)}}\subseteq \mathbf{NormMono(}\mathfrak{C)}\subseteq
\mathbf{RegMono(\mathfrak{C)}}\subseteq \mathbf{Mono(\mathfrak{C)}}
\end{equation*
an
\begin{equation*}
\mathbf{Iso(\mathfrak{C}})\subseteq \mathbf{NormEpi(}\mathfrak{C)}\subseteq
\mathbf{RegEpi(\mathfrak{C)}}\subseteq \mathbf{Epi(\mathfrak{C)}}.
\end{equation*}
\end{remark}
\begin{definition}
(Compare with \cite[5.8.7]{Fai1973}, \cite{EW1987}, \cite{Bar2002}) Let
\mathbf{E}$ and $\mathbf{M}$ be classes of morphisms in $\mathbf{\mathfrak{C
}$ and $\gamma :X\longrightarrow Y$ a morphism in $\mathfrak{C}.$
\begin{enumerate}
\item The $\mathbf{M}$\emph{-image} of $\gamma $ is $\mathrm{im}(\gamma )
\func{Im}(\gamma )\longrightarrow Y$ in $\mathbf{M}$ such that $\gamma
\mathrm{im}(\gamma )\circ \iota _{\gamma }$ for some morphism $\iota
_{\gamma }$ and if $\gamma =m\circ \iota $ for some $m\in \mathbf{M},$ then
there exists a unique morphism $\alpha _{m}:\func{Im}(\gamma
)\longrightarrow Z$ such that $m\circ \alpha _{m}=\mathrm{im}(\gamma )$ and
\alpha _{m}\circ \iota _{\gamma }=\iota .$
\item The $\mathbf{E}$\emph{-coimage} of $\gamma $ is $\mathrm{coim}(\gamma
):X\longrightarrow \mathrm{Coim}(\gamma )$ in $\mathbf{E}$ such that $\gamma
=c_{\gamma }\circ \mathrm{coim}(\gamma )$ for some morphism $c_{\gamma }$
and if $\gamma =c\circ e$ for some $e\in \mathbf{E},$ then there exists a
unique morphism $\beta _{e}:Z\longrightarrow \mathrm{Coim}(\gamma )$ such
that $\beta _{e}\circ e=\mathrm{coim}(\gamma )$ and $c_{\gamma }\circ \beta
_{e}=c.
\begin{equation*}
\begin{tabular}{lll}
$\xymatrix{ & & {\rm Im}(\gamma) \ar@{.>}^{{\rm im}(\gamma)}[ddddrr]
\ar@{-->}_(.65){\alpha_m}[dd] & & \\ & & & & \\ & & Z \ar_{m}[ddrr] & & \\
\\ X \ar_{\iota}[uurr] \ar_(.6){\gamma}[rrrr] \ar^{\iota_{\gamma}}[uuuurr] &
& & & Y}$ & & $\xymatrix{ & & Z \ar@{-->}_(.65){\beta_e}[dd]
\ar^{c}[ddddrr] & & \\ & & & & \\ & & {\rm Coim}(\gamma)
\ar_{c_{\gamma}}[ddrr] & & \\ \\ X \ar^{e}[uuuurr] \ar_(.6){\gamma}[rrrr]
\ar@{.>}_{{\rm coim}(\gamma)}[uurr] & & & & Y}
\end{tabular
\end{equation*}
\end{enumerate}
\end{definition}
\begin{definition}
(\cite[14.1]{AHS2004})\ Let $\mathbf{E}$ and $\mathbf{M}$ be classes of
morphisms in $\mathbf{\mathfrak{C}}.$ The pair $(\mathbf{E},\mathbf{M})$ is
called a \emph{factorization structure }(\emph{for morphisms} in) $\mathbf
\mathfrak{C}},$ and $\mathbf{\mathfrak{C}}$ is said to be $(\mathbf{E}
\mathbf{M})$\emph{-structured,} provided that
\begin{enumerate}
\item $\mathbf{E}$ and $\mathbf{M}$ are closed under composition with
isomorphisms.
\item $\mathbf{\mathfrak{C}}$ has $(\mathbf{E},\mathbf{M})$-factorizations,
\emph{i.e.} each morphism $f$ in $\mathbf{\mathfrak{C}}$ has a factorization
$f=m\circ e$ with $m\in \mathbf{M}$ and $e\in \mathbf{E}.$
\item $\mathbf{\mathfrak{C}}$ has the \emph{unique} $(\mathbf{E},\mathbf{M})
\emph{-diagonalization property} (or the \emph{diagonal-fill-in property})
\emph{i.e. }for each commutative squar
\begin{equation*}
\xymatrix{A \ar^{e}[r] \ar_{f}[d] & B \ar^{g}[d] \ar@{.>}^ {d}[ld] \\ C
\ar_{m}[r] & D }
\end{equation*
with $e\in \mathbf{E}$ and $m\in \mathbf{M},$ there exists a \emph{unique}
morphism $d:B\longrightarrow C$ such that $d\circ e=f$ and $m\circ d=g.$
\end{enumerate}
\end{definition}
\begin{punto}
Let $\mathfrak{C}$ be an $(\mathbf{E},\mathbf{M})$-structured category and
\gamma :X\longrightarrow Y$ be a morphism in $\mathfrak{C}$ with $(\mathbf{E
,\mathbf{M})$-factorization $\gamma :X\overset{e}{\longrightarrow }U\overset
m}{\longrightarrow }Y.$ Let $\mathrm{Coim}(\gamma )$ and $\func{Im}(\gamma )$
be the the $\mathbf{E}$-coimage of $\gamma $ and the $\mathbf{M}$-image of
\gamma ,$ respectively. Then there exist isomorphisms $\mathrm{Coim}(\gamma
\overset{d_{1}}{\simeq }U\overset{d_{2}}{\simeq }\func{Im}(\gamma )$ such
that $d_{2}\circ d_{1}$ is the canonical morphism $d_{\gamma }:\mathrm{Coim
(\gamma )\longrightarrow \func{Im}(\gamma ),$ which is in this case an
isomorphism
\begin{equation*}
\xymatrix{ & & {\rm Coim}(\gamma) \ar@{-->}_(.65){d_{\gamma}}[dd]
\ar^{c_{\gamma}}[ddddrr] & & \\ & & & & \\ & & {\rm Im}(\gamma)
\ar@{.>}_{{\rm im}(\gamma)}[ddrr] & & \\ \\ X \ar@{.>}^(.6){{\rm
coim}(\gamma)}[uuuurr] \ar_{\iota_{\gamma}}[uurr] \ar_{\gamma}[rrrr] & & & &
Y}
\end{equation*}
\end{punto}
\begin{remarks}
\begin{enumerate}
\item \label{unique}For any category, $(\mathbf{Iso},\mathbf{Mor})$ and $
\mathbf{Mor},\mathbf{Iso})$ are \emph{trivial} factorization structures.
\item Some authors assume that $\mathbf{E}\subseteq \mathbf{Epi}(\mathfrak{C
)$ and $\mathbf{M}\subseteq \mathbf{Mon}(\mathfrak{C})$ (e.g. \cite{Bar2002
).
\item If $(\mathbf{E},\mathbf{M})$ is a factorization structure for
\mathfrak{C},$ then $\mathbf{E}\cap \mathbf{M}=\mathbf{Iso}(\mathfrak{C}).$
\item As a result of the unique diagonalization property, any $(\mathbf{E}
\mathbf{M})$-factorization in an $(\mathbf{E},\mathbf{M})$-structured
category is \emph{essentially} unique (compare with \cite[Proposition 14.4
{AHS2004}). Suppose that $m_{1}\circ e_{1}=\gamma =m_{2}\circ e_{2}$ are two
$(\mathbf{E},\mathbf{M})$-factorizations of a morphism $\gamma
:A\longrightarrow B$ in $\mathfrak{C}
\begin{equation*}
\xymatrix{A \ar^{e_1}[r] \ar_{e_2}[d] & C_1 \ar^{m_1}[d] \ar@{.>}_{h}[dl]\\
C_2 \ar_{m_2}[r] & B}
\end{equation*
Then there exists a (unique) isomorphism $h:C_{1}\longrightarrow C_{2}$ s.t.
the above diagram commutes.
\end{enumerate}
\end{remarks}
\subsection*{Exact Categories}
\qquad There are several notions of \emph{exact sequences} and \emph{exact
categories} in the literature (e.g. \cite{Buc1955}, \cite{Qui1973}, \cit
{Pup1962}, \cite{Bar1971}).
\begin{punto}
Call $\mathfrak{C}$ a\emph{\ Puppe-exact category} iff it is pointed and has
a $(\mathbf{NormalEpi},\mathbf{NormalMono})$-factorization structure. By
\cite[14.F (a)]{AHS2004}, a pointed category is Puppe-exact if and only if
it has $(\mathbf{NormalEpi},\mathbf{NormalMono})$\emph{-factorizations},
i.e. every morphism $\gamma $ admits a -- necessarily \emph{unique} --
factorization $\gamma =\gamma ^{\prime \prime }\circ \gamma ^{\prime }$ such
that $\gamma ^{\prime }$ is a cokernel and $\gamma ^{\prime \prime }$ is a
kernel. The image and the coimage of a morphism $\gamma :X\longrightarrow Y$
in such a categories are given by $\func{Im}(\gamma ):=\mathrm{Ker}(\mathrm
coker}(\gamma ))$ and $\mathrm{Coim}(\gamma ):=\mathrm{Coker}(\mathrm{ker
(\gamma )),$ respectively. Moreover, a sequence $A\overset{f}
\longrightarrow }B\overset{g}{\longrightarrow }C$ is said to be \emph{exact}
iff $\func{Im}(f)\simeq \mathrm{Ker}(g).$
\end{punto}
\begin{remarks}
\begin{enumerate}
\item Every non-empty Puppe-exact category has a zero-object, kernels and
cokernels, is normal, conormal and has equalizers.
\item Let $\mathfrak{C}$ be a category with a zero-object, kernels,
cokernels and equalizers. If $\mathfrak{C}$ is normal, then $\mathrm{Coker}
\mathrm{ker}(\gamma ))\simeq \mathrm{Ker}(\mathrm{coker}(\gamma ))$ for any
morphism $\gamma $ in $\mathfrak{C}$ \cite[Proposition 5.20]{Fai1973}.
\end{enumerate}
\end{remarks}
\qquad In light of the previous remarks, \cite[Lemma 31.14]{Sch1972} can be
restated as follows:
\begin{lemma}
\label{P-exact}Let $\mathfrak{C}$ be a Puppe-exact category, $A\overset{f}
\longrightarrow }B\overset{g}{\longrightarrow }C$ a sequence in $\mathfrak{C}
$ with $g\circ f=0$ and consider the following commutative diagram with
canonical and induced factorizations
\begin{equation*}
\xymatrix{ & & & {\rm Coim}(f) \ar|-{d_f}[dd] \ar|-{c_f}[ddddddrrr] & & & &
& & {\rm Coker}(f) \ar@{.>}|-{\beta_{{\rm coker}(f)}}[dd]
\ar@{-->}|-{g''}[ddddddrrr] & & & \\ & & & & & & & & & & & & \\ & & & {\rm
Im}(f) \ar@{.>}|-{\alpha_{{\rm im}(f)}}[dd] \ar|-{{\rm im}(f)} [ddddrrr] & &
& & & & {\rm Coim}(g) \ar|-{d_g}[dd] \ar|-{c_g}[ddddrrr] & & & \\ & & & & &
& & & & & & & \\ & & & {\rm Ker}(g) \ar[ddrrr]|-{{\rm ker}(g)} & & & & & &
{\rm Im}(g) \ar[ddrrr]|-{{\rm im}(g)} & & & \\ & & & & & & & & & & & & \\ A
\ar@{-->}|-{f'}[uurrr] \ar|-{f}[rrrrrr] \ar|-{\iota_f}[rrruuuu]
\ar[uuuuuurrr]|-{{\rm coim}(f)} & & & & & & B \ar|-{g}[rrrrrr]
\ar[uuuuuurrr]|-{{\rm coker}(f)} \ar[rrruuuu]|-{{\rm coim}(g)}
\ar|-{\iota_g}[uurrr] & & & & & & C }
\end{equation*
The following are equivalent:
\begin{enumerate}
\item $A\overset{f}{\longrightarrow }B\overset{g}{\longrightarrow }C$ is
exact \emph{(}i.e. $\func{Im}(f)\overset{\alpha _{\mathrm{im}(f)}}{\simeq
\mathrm{Ker}(g)$\emph{)};
\item $\mathrm{Coker}(f)\overset{\beta _{\mathrm{coker}(f)}}{\simeq }\mathrm
Coim}(g);$
\item $\mathrm{Coim}(f)\simeq \mathrm{Ker}(g);$
\item $\mathrm{Coker}(f)\simeq \func{Im}(g);$
\item $\mathrm{Im}(f)\simeq \mathrm{Ker}(\mathrm{coim}(g));$
\item $\mathrm{Coim}(g)\simeq \mathrm{Coker}(\mathrm{im}(f));$
\item $f^{\prime }$ is a (normal) epimorphism;
\item $g^{\prime \prime }$ is a (normal) monomorphism.
\end{enumerate}
\end{lemma}
Inspired by the previous lemma, we introduce a notion of exact sequences in
any pointed category:
\begin{definition}
Let $\mathfrak{C}$ be any pointed category and fix a factorization structure
$(\mathbf{E},\mathbf{M})\ $for $\mathfrak{C}.$ We call a sequenc
\begin{equation}
A\overset{f}{\longrightarrow }B\overset{g}{\longrightarrow }C \label{ABC}
\end{equation
\emph{exact} w.r.t. $(\mathbf{E},\mathbf{M})$ iff $f$ and $g$ have
factorizations
\begin{equation*}
f=\mathrm{ker}(g)\circ f^{\prime }\text{ and }g=g^{\prime \prime }\circ
\mathrm{coker}(f)\text{ with }(f^{\prime },\mathrm{ker}(g)),(\mathrm{coker
(f),g^{\prime \prime })\in \mathbf{E}\times \mathbf{M}.
\end{equation*
\begin{equation*}
\xymatrix{& A \ar@{.>}_{f'}[dl] \ar^{f}[d] & \\ {\rm Ker}(g) \ar_{{\rm
ker}(g)}[r] & B \ar_{g}[d] \ar^(0.4){{\rm coker}(f)}[r] & {\rm Coker}(f)
\ar@{.>}^{g''}[dl] \\ & C & }
\end{equation*
When the factorization structure is clear from the context we drop it. We
call a sequenc
\begin{equation*}
\cdots \longrightarrow A_{i-1}\overset{f_{i-1}}{\longrightarrow }A_{i
\overset{f_{i}}{\longrightarrow }A_{i+1}\longrightarrow \cdots
\end{equation*
\emph{exact} at $A_{i}$ iff $A_{i-1}\overset{f_{i-1}}{\longrightarrow }A_{i
\overset{f_{i}}{\longrightarrow }A_{i+1}$ is exact; moreover, we call this
sequence \emph{exact} iff it is exact at $A_{i}$ for every $i.$ An exact
sequenc
\begin{equation}
0\longrightarrow A\overset{f}{\longrightarrow }B\overset{g}{\longrightarrow
C\longrightarrow 0 \label{ses}
\end{equation
is called a \emph{short exact sequence}.
\end{definition}
\begin{remark}
Let $\mathfrak{C}$ be a pointed category and fix a factorization structure $
\mathbf{E},\mathbf{M})\ $for $\mathfrak{C}.$ It follows immediately from the
definition that (\ref{ses}) is a short exact sequence w.r.t. $(\mathbf{E}
\mathbf{M})$ if and only if $\mathrm{Coker}(f)\in \mathbf{E},$ $\mathrm{Ker
(g)\in \mathbf{M},$ $f=\mathrm{Ker}(g)$ and $g=\mathrm{Coker}(f).$
\end{remark}
\begin{ex}
Let $\mathfrak{C}$ be a Puppe exact category. A sequence $A\overset{f}
\longrightarrow }B\overset{g}{\longrightarrow }C$ is \emph{exact} if and
only if $f=\mathrm{ker}(g)\circ f^{\prime },$ $g=g^{\prime \prime }\circ
\mathrm{coker}(f)$ with $(f^{\prime },\mathrm{ker}(g)),$ $(\mathrm{coker
(f),g^{\prime \prime })\in \mathbf{NormalEpi}\times \mathbf{NormalMono}.$
Notice that, by Lemma \ref{P-exact}, this is equivalent to the classical
notion of exacts sequences in Puppe-exact categories, namely $\func{Im
(f)\simeq \mathrm{Ker}(g).$ This applies in particular to the categories of
modules over rings (e.g. the category $\mathbf{Ab}$ of Abelian groups).
\end{ex}
\begin{ex}
Let $\mathfrak{C}$ be a pointed $(\mathbf{E},\mathbf{M})$-structure category
with $\mathbf{NormalEpi}\subseteq \mathbf{E}\subseteq \mathbf{Epi}$ and
\mathbf{NormalMono}\subseteq \mathbf{M}\subseteq \mathbf{Mono}.$ Then a
sequence $A\overset{f}{\longrightarrow }B\overset{g}{\longrightarrow }C$ in
\mathfrak{C}$ is exact if and only if the essentially unique $(\mathbf{E}
\mathbf{M})$ factorizations $f=m_{1}\circ e_{1},$ $g=m_{2}\circ e_{2}$ can
be chosen so that $m_{1}=\mathrm{ker}(e_{2})$ and $e_{2}=\mathrm{coker
(m_{1}).$ Moreover, a sequence $0\longrightarrow A\overset{f}
\longrightarrow }B\overset{g}{\longrightarrow }C\longrightarrow 0$ is exact
if and only if $f=\mathrm{Ker}(g)$ and $g=\mathrm{Coker}(f).$ This applies
to general pointed categories which are $(\mathbf{RegEpi},\mathbf{Mono})
-structured or $(\mathbf{Epi},\mathbf{RegMono})$-structured. In particular,
this applies to pointed regular categories (compare with \cite[Definition
4.1.7]{BB2004}).
\end{ex}
\begin{ex}
Let $\mathfrak{C}$ be a pointed protomodular category (in the sense of D.
Bourn \cite{Bou1991}) with finite limits. By \cite[Proposition 3.1.23
{BB2004}, $g\in \mathbf{RegEpi}(\mathfrak{C})$ if and only if $g=\mathrm
coker}(\mathrm{ker}(g)).$ If $\mathfrak{C}$ is $(\mathbf{RegEpi},\mathbf{Mon
})$-structured or $(\mathbf{Epi},\mathbf{RegMono})$-structured, then it
follows that a sequence $0\longrightarrow A\overset{f}{\longrightarrow }
\overset{g}{\longrightarrow }C\longrightarrow 0$ in $\mathfrak{C}$ is exact
if and only if $f=\mathrm{ker}(g)$ and $g$ is a regular epimorphism. This
applies in particular to homological categories, which are precisely pointed
and protomodular regular categories \cite{BB2004}.
\end{ex}
\begin{ex}
Let $(\mathfrak{C};\mathbf{E})$ be a \emph{relative homological category} in
the sense of \cite{Jan2006}, where $\mathbf{E}$ is a distinguished class of
normal epimorphisms and assume that $\mathfrak{C}$ is $(\mathbf{E},\mathbf
Mono})$-structured (which is not actually assumed in the defining axioms of
such categories). Analyzing Condition (a) on $g^{\prime }$ (page 192), which
was assumed to prove the so called \emph{Relative Snake Lemma, }shows that
this assumption and along with the assumptions on $f^{\prime }$ are
essentially equivalent to assuming that the row $0\longrightarrow A^{\prime
\overset{f^{\prime }}{\longrightarrow }B^{\prime }\overset{g^{\prime }}
\longrightarrow }C^{\prime }$ is $(\mathbf{E},\mathbf{Mono})$-exact.
\end{ex}
\subsection*{Steady Morphisms}
\qquad In what follows, we consider a special class of categories to which
there is a natural transfer of the notion of exact sequences in Puppe-exact
categories.
\begin{definition}
Let $\mathfrak{C}$ be a pointed $(\mathbf{E},\mathbf{M})$-structured
category. We say that a morphism $\gamma :X\longrightarrow Y$ in $\mathfrak{
}$ is:
\emph{steady} w.r.t. $(\mathbf{E},\mathbf{M})$ iff $\mathrm{Ker}(\gamma ),$
\mathrm{Coker}(\mathrm{ker}(\gamma ))$ exist in $\mathfrak{C}$ and $\gamma $
admits an $(\mathbf{E},\mathbf{M})$- factorization $\gamma =\gamma ^{\prime
\prime }\circ \mathrm{coker}(\mathrm{ker}(\gamma )),$ equivalently $\mathrm
Coker}(\mathrm{ker}(\gamma ))\simeq \mathrm{Coim}(\gamma );$
\emph{costeady} w.r.t. $(\mathbf{E},\mathbf{M})$ iff $\mathrm{Coker}(\gamma
),$ $\mathrm{Ker}(\mathrm{coker}(\gamma ))$ exist in $\mathfrak{C}$ and
\gamma $ admits an $(\mathbf{E},\mathbf{M})$- factorization $\gamma =\mathrm
ker}(\mathrm{coker}(\gamma ))\circ \gamma ^{\prime },$ equivalently $\mathrm
Ker}(\mathrm{coker}(\gamma ))\simeq \func{Im}(\gamma );$
\emph{bisteady} w.r.t. $(\mathbf{E},\mathbf{M})$ iff $\gamma $ is steady and
costeady w.r.t. $(\mathbf{E},\mathbf{M}),$ equivalently $\mathrm{Coker}
\mathrm{ker}(\gamma ))\simeq \mathrm{Coim}(\gamma )\overset{d_{\gamma }}
\simeq }\func{Im}(\gamma )\simeq \mathrm{Ker}(\mathrm{ker}(\gamma )).$
\begin{equation*}
\xymatrix{ & & & & {\rm Coker}({\rm ker}(\gamma)) \ar@{.>}^{\gamma
''}[ddddrr] \ar@{-->}_(.65){\overline{\gamma}}[dd] & & & & \\ & & & & & & &
& \\ & & & & {\rm Ker} ({\rm coker}(\gamma)) \ar|-{{\rm ker} ({\rm
coker}(\gamma))}[ddrr] & & & & \\ & & & & & & & & \\ {\rm Ker}(\gamma)
\ar_{{\rm ker}(\gamma)}[rr] & & X \ar@{.>}|-{\gamma '}[uurr]
\ar_(.6){\gamma}[rrrr] \ar^{{\rm coker} ({\rm ker}(\gamma))}[uuuurr] & & & &
Y \ar_{{\rm coker}(\gamma)}[rr] & & {\rm Coker}(\gamma)}
\end{equation*
We call $\mathfrak{C}$ \emph{steady }(resp. \emph{costeady, bisteady})
w.r.t. $(\mathbf{E},\mathbf{M})$ iff all morphisms in $\mathfrak{C}$ are
steady (resp. costeady, bisteady) w.r.t. $(\mathbf{E},\mathbf{M}).$
\end{definition}
\begin{remark}
Let $\mathfrak{C}$ be a pointed $(\mathbf{E},\mathbf{M})$-structured
category. If $\mathfrak{C}$ is bisteady w.r.t. $(\mathbf{E},\mathbf{M}),$
then $\mathfrak{C}$ is Puppe-exact: in this case, every morphism in
\mathfrak{C}$ has a $(\mathbf{NormalEpi},\mathbf{NormalMono})
-factorization, whence $\mathfrak{C}$ is Puppe-exact \cite[14.F]{AHS2004}.
Moreover, if $\mathbf{NormalEpi}\subseteq \mathbf{E}$ and $\mathbf{NormalMon
}\subseteq \mathbf{M}$ then $\mathfrak{C}$ is bisteady w.r.t. $(\mathbf{E}
\mathbf{M})$ if and only if $\mathfrak{C}$ is Puppe-exact.
\end{remark}
\begin{punto}
\label{UA}All varieties -- in the sense of Universal Algebra -- are $
\mathbf{RegEpi},\mathbf{Mono})$-structured. Moreover, the class of regular
epimorphisms coincides with that of surjective morphisms, and the class of
monomorphisms coincides with that of injective morphisms. Let $\mathcal{V}$
be a pointed variety. We say that a morphism $\gamma :X\longrightarrow Y$ in
$\mathcal{V}$ is \emph{steady} (resp. \emph{costeady}, \emph{bisteady}) iff
\gamma $ is steady (resp. costeady, bisteady) w.r.t. $(\mathbf{Surj},\mathbf
Inj}).$ With $\func{Im}(\gamma )$ ($\mathrm{Coim}(\gamma )$) we mean the
\mathbf{Inj}$-image (the $\mathbf{Surj}$-coimage) of $\gamma .$ Moreover, we
say that a sequence $X\overset{f}{\longrightarrow }Y\overset{g}
\longrightarrow }Z$ in $\mathcal{V}$ is \emph{exact} iff it is $(\mathbf{Sur
},\mathbf{Inj})$-exact.
\end{punto}
\begin{ex}
The variety $\mathbf{Grp}$ of all (Abelian and non-Abelian) groups is
steady. Let $\gamma :X\longrightarrow Y$ be any morphism of groups. Notice
that $\mathrm{Ker}(\gamma )=\{x\in X\mid \gamma (x)=1_{Y}\}$ while $\mathrm
Coker}(\gamma )=Y/N_{\gamma },$ where $N_{\gamma }$ is the \emph{normal
closure} of $\gamma (X).$ Consider the canonical $(\mathbf{Surj},\mathbf{Inj
)$-factorization $\gamma :X\overset{\mathrm{im}(\gamma )}{\longrightarrow
\gamma (X)\overset{\iota }{\longrightarrow }Y$ where $\iota $ is the
canonical embedding. Consider also the factorization $\gamma :X\overset
\mathrm{coker}(\mathrm{ker}(\gamma ))}{\longrightarrow }X/\mathrm{Ker
(\gamma )\overset{\gamma ^{\prime \prime }}{\longrightarrow }Y.$ Clearly,
\gamma $ is steady if and only if $\gamma ^{\prime \prime }$ is injective.
Indeed, if $\gamma ^{\prime \prime }([x_{1}])=\gamma ^{\prime \prime
}([x_{2}])$ for some $x_{1},x_{2}\in X,$ then $\gamma (x_{1})=\gamma (x_{2})$
whence $\gamma (x_{1}^{-1}x_{2})=1_{Y}$ and it follows that
x_{1}^{-1}x_{2}=k$ for some $k\in \mathrm{Ker}(\gamma ),$ i.e.
[x_{1}]=[x_{2}].$ Consequently, $\gamma ^{\prime \prime }\in \mathbf{Inj}.$
On the other hand, consider the factorization $\gamma :X\overset{\gamma
^{\prime }}{\longrightarrow }N_{\gamma }\overset{\mathrm{ker}(\mathrm{coker
(\gamma ))}{\longrightarrow }Y.$ Then $\gamma $ is costeady if and only if
\gamma (X)=N_{\gamma }$ if and only if $\gamma (X)\leq G$ is a normal
subgroup. Clearly, $\mathbf{Grp}$ is not costeady: Let $G$ be a group, $H$ a
subgroup that is not normal in $G$ and let $\gamma :H\hookrightarrow G$ be
the embedding. Indeed, $H=\gamma (H)\neq N_{\gamma },$ \emph{i.e.} $\gamma $
is not costeady. Consequently, $\mathbf{Grp}$ is not a bisteady category.
\end{ex}
\qquad In the following example, we demonstrate how the \emph{classical
notion of exact sequences of groups is consistent with our new definition of
exact sequences in arbitrary pointed categories.
\begin{ex}
\label{G-exact}Let $A\overset{f}{\longrightarrow }B\overset{g}
\longrightarrow }C$ be a sequence of groups and consider the canonical
factorizations of $f:A\overset{\mathrm{im}(f)}{\longrightarrow }f(A)\overset
\iota _{1}}{\hookrightarrow }B$ and $g:B\overset{\mathrm{im}(g)}
\longrightarrow }g(B)\overset{\iota _{2}}{\hookrightarrow }C.$ If the given
sequence is exact, then $f=\mathrm{ker}(g)\circ f^{\prime }$ with $f^{\prime
}$ surjective. This implies that $f(A)=\mathrm{Ker}(g).$ On the other hand,
assume that $f(A)=\mathrm{Ker}(g).$ Then $f$ has a an $(\mathbf{Inj},\mathbf
Surj})$-factorization as $f=\mathrm{ker}(g)\circ \mathrm{im}(f).$ Moreover,
it is evident that there is an isomorphism of groups $B/\mathrm{Ker}(g
\overset{\gamma }{\simeq }g(B).$ So, $g$ has an $(\mathbf{Inj},\mathbf{Surj
) $-factorization $g=(\iota _{2}\circ \gamma )\circ \mathrm{coker}(g).$ It
follows that $A\overset{f}{\longrightarrow }B\overset{g}{\longrightarrow }C$
is exact if and only if $f(A)=\mathrm{Ker}(g).$
\end{ex}
\section{Semirings and Semimodules}
\qquad \emph{Semirings} (\emph{semimodules}) are roughly speaking, rings
(modules) without subtraction. Semirings were studied independently by
several algebraists, especially by H. S. Vandiver \cite{Van1934} who
considered them as the \emph{best} algebraic structures which unify rings
and bounded distributive lattices. Since the sixties of the last century,
semirings were shown to have significant applications in several areas as
Automata Theory (e.g. \cite{Eil1974}, \cite{Eil1976}, \cite{KS1986}),
Theoretical Computer Science (e.g. \cite{HW1998}) and many classical areas
of mathematics (e.g. \cite{Go19l99a}, \cite{Gol1999b}).
Recently, semirings played an important role in several emerging areas of
research like Idempotent Analysis (e.g. \cite{KM1997}, \cite{LMS2001}, \cit
{Lit2007}), Tropical Geometry (e.g. \cite{R-GST2005}, \cite{Mik2006}) and
many aspects of modern Mathematics and Mathematical Physics (e.g. \cit
{Gol2003}, \cite{LM2005}). In his dissertation \cite{Dur2007}, N. Durov
demonstrated that semirings are in one-to-one correspondence with what he
called \emph{algebraic additive monads} on the category $\mathbf{Set}$ of
sets. Moreover, a connection between semirings and the so-called $\mathbb{F}
-rings, where $\mathbb{F}$ is the field with one element, was pointed out in
\cite[1.3 -- 1.4]{PL}.
The theory of semimodules was developed mainly by M. Takahashi, who
published several fundamental papers on this topic (cf. \cite{Tak1979} --
\cite{Tak1985}) and to whom research in the theory of semimodules over
semirings is indeed indebted. However, it seems to us that there are some
gaps in his theory of semimodules which led to confusion and sometimes
conceptual misunderstandings. Instead of introducing appropriate definitions
and notions that fit well with the special properties of the category of
semimodules over semirings, some definitions and notions which are fine in
\emph{Puppe-exact categories }in general, and in categories of modules over
rings in particular, were enforced in a category which is, in general, far
away from being Puppe-exact.
A systematic development of the homological theory of semirings and
semimodules has been initiated recently in a series of papers by Y. Katsov
\cite{Kat1997} and carried out in a continuing series of papers (e.g. \cit
{Kat2004a}, \cite{Kat2004b}, \cite{KTN2009}, \cite{KN2011}, \cite{IK2011},
\cite{IK2011}). Another approach that is worth mentioning was initiated by
A. Patchkoria in \cite{Pat1998} and continued in a series of papers (e.g.
\cite{Pat2000a}, \cite{Pat2000b}, \cite{Pat2003}, \cite{Pat2006}, \cit
{Pat2009}).
In what follows, we revisit the category of semimodules over a semiring. In
particular, we adopt a new definition of exact sequences of semimodules and
investigate it. We also introduce some terminology that will be needed in
the sequel.
\begin{punto}
\label{semig}Let $(S,\ast )$ be a semigroup. We call $s\in S$ \emph
cancellable} iff for any $s_{1},s_{2}\in S:
\begin{equation*}
s_{1}\ast s=s_{2}\ast s\Longrightarrow s_{1}=s_{2}\text{ and }s\ast
s_{1}=s\ast s_{2}\Longrightarrow s_{1}=s_{2}.
\end{equation*
We call $S$ \emph{cancellative} iff all elements of $S$ are cancellable. We
say that a morphism of semigroups $f:S\longrightarrow S^{\prime }$ is \emph
cancellative} iff $f(s)\in S^{\prime }$ is cancellable for every $s\in S.$
We call $S$ an \emph{idempotent semigroup} iff $s\ast s=s$ for every $s\in
S. $
\end{punto}
\begin{punto}
Let $(S,+)$ be an Abelian additive semigroup. A subset $X\subseteq S$ is
said to be \emph{subtractive }iff for any $s\in S$ and $x\in X$ we have:
s+x\in X\Longrightarrow s\in X.$ The\emph{\ subtractive closure} of a
non-empty subset $X\subseteq S$ is given b
\begin{equation*}
\overline{X}:=\{s\in S\mid s+x_{1}=x_{2}\text{ for some }x_{1},x_{2}\in X\}.
\end{equation*
If $X$ is a subsemigroup of $S,$ then indeed $X$ is subtractive if and only
if $X=\overline{X}.$ We call a morphism of Abelian semigroups
f:S\longrightarrow S^{\prime }$ \emph{subtractive} iff $f(S)\subseteq
S^{\prime }$ is subtractive, equivalently if
\begin{equation*}
f(S)=\{s^{\prime }\in S^{\prime }\mid \text{ }s^{\prime }+f(s_{1})=f(s_{2}
\text{ for some }s_{1},s_{2}\in S\}.
\end{equation*}
\end{punto}
\begin{punto}
A \emph{semiring} is an algebraic structure $(S,+,\cdot ,0,1)$ consisting of
a non-empty set $S$ with two binary operations \textquotedblleft $+
\textquotedblright\ (addition) and \textquotedblleft $\cdot
\textquotedblright\ (multiplication) satisfying the following conditions:
\begin{enumerate}
\item $(S,+,0)$ is an Abelian monoid with neutral element $0_{S};$
\item $(S,\cdot ,1)$ is a monoid with neutral element $1;$
\item $x\cdot (y+z)=x\cdot y+x\cdot z$ and $(y+z)\cdot x=y\cdot x+z\cdot x$
for all $x,y,z\in S;$
\item $0\cdot s=0=s\cdot 0$ for every $s\in S$ (i.e. $0$ is \emph{absorbing
).
\end{enumerate}
Let $S,S^{\prime }$ be semirings. A map $f:S\rightarrow S^{\prime }$ is said
to be a \emph{morphism of semirings} iff for all $s_{1},s_{2}\in S:
\begin{equation*}
f(s_{1}+s_{2})=f(s_{1})+f(s_{2}),\text{ }f(s_{1}s_{2})=f(s_{1})f(s_{2})
\text{ }f(0_{S})=0_{S^{\prime }}\text{ and }f(1_{S})=1_{S^{\prime }}.
\end{equation*
The category of semirings is denoted by $\mathbf{SRng}.$
\end{punto}
\begin{punto}
Let $(S,+,\cdot )$ be a semiring. We say that $S$ is
\emph{cancellative} iff the additive semigroup $(S,+)$ is cancellative;
\emph{commutative} iff the multiplicative semigroup $(S,\cdot )$ is
commutative;
\emph{semifield} iff $(S\backslash \{0\},\cdot ,1)$ is a commutative group.
\end{punto}
\begin{exs}
Rings are indeed semirings. A trivial, but important, example of a \emph
commutative }semiring is $(\mathbb{N}_{0},+,\cdot )$ (the set of
non-negative integers). Indeed, $(\mathbb{R}_{0}^{+},+,\cdot )$ and $
\mathbb{Q}_{0}^{+},+,\cdot )$ are semifields. A more interesting example is
the semi-ring $(\mathrm{ideal}(R),+,\cdot )$ consisting of all ideals of a
(not necessarily commutative) ring; this appeared first in the work of \emph
Dedekind} \cite{Ded1894}. On the other hand, for an integral domain $R,$ $
\mathrm{ideal}(R),+,\cap )$ is a semiring if and only if $R$ is a Pr\"{u}fer
domain. Every bounded distributive lattice $(R,\vee ,\wedge )$ is a
commutative (additively) idempotent semiring. The \emph{additively idempoten
} semirings $\mathbb{R}_{\max }:=(\mathbb{R}\cup \{-\infty \},\max ,+)$ and
\mathbb{R}_{\min }:=(\mathbb{R}\cup \{\infty \},\min ,+)$ play an important
role in idempotent and tropical mathematics (e.g. \cite{Lit2007}); the
subsemirings $\mathbb{N}_{\max }:=(\mathbb{N}\cup \{-\infty \},\max ,+)$ and
$\mathbb{N}_{\min }:=(\mathbb{N}\cup \{\infty \},\min ,+)$ played an
important role in Automata Theory (e.g. \cite{Eil1974}, \cite{Eil1976}). The
singleton set $S=\{0\}$ is a semiring with the obvious addition and
multiplication. In the sequel, we always assume that $0_{S}\neq 1_{S}$ so
that $S\neq \{0\},$ the \emph{zero semiring}.
\end{exs}
\begin{punto}
Let $S$ be a semiring. A \emph{right }$S$\emph{-semimodule} is an algebraic
structure $(M,+,0_{M};\leftharpoondown )$ consisting of a non-empty set $M,$
a binary operation \textquotedblleft $+$\textquotedblright\ along with a
right $S$-actio
\begin{equation*}
M\times S\longrightarrow M,\text{ }(m,s)\mapsto ms,
\end{equation*
such that:
\begin{enumerate}
\item $(M,+,0_{M})$ is an Abelian monoid with neutral element $0_{M};$
\item $(ms)s^{\prime }=m(ss^{\prime }),$ $(m+m^{\prime })s=ms+m^{\prime }s$
and $m(s+s^{\prime })=ms+ms^{\prime }$ for all $s,s^{\prime }\in S$ and
m,m^{\prime }\in M;$
\item $m1_{S}=m$ and $m0_{S}=0_{M}=0_{M}s$ for all $m\in M$ and $s\in S.$
Let $M,M^{\prime }$ be right $S$-semimodules. A map $f:M\rightarrow
M^{\prime }$ is said to be a \emph{morphism of right }$S$\emph{-semimodules}
(or $S$\emph{-linear}) iff for all $m_{1},m_{2}\in M$ and $s\in S:
\begin{equation*}
f(m_{1}+m_{2})=f(m_{1})+f(m_{2})\text{ and }f(ms)=f(m)s.
\end{equation*
The set $\mathrm{Hom}_{S}(M,M^{\prime })$ of $S$-linear maps from $M$ to
M^{\prime }$ is clearly a monoid under addition. The category of right $S
-semimodules is denoted by $\mathbb{S}_{S}.$ Similarly, one can define the
category of left $S$-semimodules $_{S}\mathbb{S}.$ A right $S$-semimodule
M_{S}$ is said to be \emph{cancellative} iff the semigroup $(M,+)$ is
cancellative. With $\mathbb{CS}_{S}\subseteq \mathbb{S}_{S}$ (resp. $_{S
\mathbb{CS}\subseteq $ $_{S}\mathbb{S})$ we denote the full subcategory of
cancellative right (left) $S$-semimodules.
\end{enumerate}
\end{punto}
\begin{punto}
Let $M$ be a right $S$-semimodule. A non-empty subset $L\subseteq M$ is said
to be an $S$\emph{-subsemimodule, }and we write $L\leq _{S}M,$ iff $L$ is
closed under \textquotedblleft $+_{M}$\textquotedblright\ and $ls\in L$ for
all $l\in L$ and $s\in S.$
\end{punto}
\begin{ex}
Every Abelian monoid $(M,+,0_{M})$ is an $\mathbb{N}_{0}$-semimodule in the
obvious way. Moreover, the categories $\mathbf{CMon}$ of commutative monoids
and the category $\mathbb{S}_{\mathbb{N}_{0}}$ of $\mathbb{N}_{0}
-semimodules are isomorphic.
\end{ex}
\subsection*{Congruences}
\begin{punto}
\label{zuka}Let $M$ be an $S$-semimodule. An equivalence relation
\textquotedblleft $\equiv $\textquotedblright\ on $M$ is a said to be an $S
\emph{-congruence on }$M$ iff for any $m,m^{\prime },m_{1},m_{1}^{\prime
},m_{2},m_{2}^{\prime }\in M$ and $s\in S$ we have
\begin{equation*}
\lbrack m_{1}\equiv m_{1}^{\prime }\text{ and }m_{2}\equiv m_{2}^{\prime
}\Rightarrow \lbrack m_{1}+m_{2}\equiv m_{1}^{\prime }+m_{2}^{\prime }]\text{
and }[m\equiv m^{\prime }\Rightarrow ms\equiv m^{\prime }s].
\end{equation*
The set $M/\equiv $ of equivalence classes inherit a structure of an $S
-semimodule in the obvious way and there is a canonical surjection of $S
-semimodules $\pi _{\equiv }:M\rightarrow M/\equiv .$
\end{punto}
\begin{punto}
Let $M$ be an $S$-semimodule. Every $S$-subsemimodule $L\leq _{S}M$ induces
two $S$-congruences on $M:$ the \emph{Bourne relation
\begin{equation*}
m_{1}\equiv _{L}m_{2}\Leftrightarrow m_{1}+l_{1}=m_{2}+l_{2}\text{ for some
l_{1},l_{2}\in L;
\end{equation*
and the \emph{Iizuka relation
\begin{equation*}
m_{1}[\equiv ]_{L}m_{2}\Leftrightarrow m_{1}+l_{1}+m^{\prime
}=m_{2}+l_{2}+m^{\prime }\text{ for some }l_{1},l_{2}\in L\text{ and
m^{\prime }\in M.
\end{equation*
We call the $S$-semimodule $M/L:=M/_{\equiv _{L}}$ the \emph{quotient of }$M
\emph{\ by }$L$ or the \emph{factor semimodule} of $M$ by $L.$ One can
easily check that $M/L=M/\overline{L}.$ If $M$ is cancellative, then $L$ and
$M/L$ are cancellative. On the other hand, the $S$-semimodule $M/[\equiv
]_{L}$ is cancellative.
\end{punto}
\begin{proposition}
The category $\mathbb{S}_{S}$ and its full subcategory $\mathbb{CS}_{S}$
have kernels and cokernels, where for any morphism of $S$-semimodules
f:M\rightarrow N$ we hav
\begin{equation*}
\mathrm{Ker}(f)=\{m\in M\mid f(m)=0\}\text{ and }\mathrm{\mathrm{Coker}
(f)=N/f(M).
\end{equation*}
\end{proposition}
Taking into account the fact that $\mathbb{S}_{S}$ is a variety (in the
sense of Universal Algebra) we have:
\begin{proposition}
\label{cc}\emph{(\cite{Tak1982b}, \cite{Tak1982c}, \cite{TW1989})}\ The
category of semimodules is
\begin{enumerate}
\item complete (i.e. has equalizers $\&$ products);
\item cocomplete (i.e. has coequalizers $\&\ $coproducts);
\item Barr-exact categories \emph{\cite{Bar1971}}.
\end{enumerate}
\end{proposition}
\begin{remark}
In \cite{Tak1982c}, Takahashi proved that the category of semimodules over a
semiring is $c$\emph{-cocomplete}, which is a \emph{relaxed} notion of
cocompleteness which he introduced. However, it was pointed to the author by
F. Linton (and other colleagues from the Category List) that such a category
is indeed cocomplete in the classical sense since it is a variety.
\end{remark}
\begin{punto}
As a variety, the category of $S$-semimodules is regular; in particular,
\mathbb{S}_{S}$ has a $(\mathbf{RegEpi},\mathbf{Mono})$-factorization
structure. Let $\gamma :X\longrightarrow Y$ be a morphism of $S
-semimodules. Then $\func{Im}(\gamma )=\gamma (X)$ and $\mathrm{Coim}(\gamma
)=X/f,$ where $X/f$ is the quotient semimodule $X/\equiv _{f}$ given by
x\equiv _{f}x^{\prime }$ iff $f(x)=f(x^{\prime }).$ Indeed, we have a
canonical isomorphis
\begin{equation*}
d_{\gamma }:\mathrm{Coim}(\gamma )\simeq \func{Im}(\gamma ),\text{
[x]\mapsto \gamma (x).
\end{equation*}
\end{punto}
\begin{remark}
Takahashi defined the \emph{image} of a morphism $\gamma :X\rightarrow Y$ of
$S$-semimodules as $\mathrm{Ker}(\mathrm{coker}(\gamma ))$ and the \emph
proper image} as $\gamma (X).$ In fact, $\gamma (X)$ is the \emph{image} of
\gamma $ in the categorical sense (e.g. \cite[5.8.7]{Fai1973}, \cite{EW1987
).
\end{remark}
\begin{punto}
We call a morphism of $S$-semimodules $\gamma :M\longrightarrow N:$
\emph{subtractive} iff $\gamma (M)\subseteq N$ is subtractive;
\emph{strong} iff $\gamma (M)\subseteq N$ is strong;
$k$\emph{-uniform} iff for any $x_{1},x_{2}\in X:
\begin{equation}
\gamma (x_{1})=\gamma (x_{2})\Longrightarrow \text{ }\exists \text{
k_{1},k_{2}\in \mathrm{Ker}(\gamma )\text{ s.t. }x_{1}+k_{1}=x_{2}+k_{2};
\label{k-steady}
\end{equation}
$i$\emph{-uniform} iff $\gamma (X)=\overline{\gamma (X)}:=\{y\in Y\mid
y+\gamma (x_{1})=\gamma (x_{2})$ for some $x_{1},x_{2}\in X\};$
\emph{uniform }iff $\gamma $ is $k$-uniform and $i$-uniform;
\emph{semi-monomorphism} iff $\mathrm{Ker}(\gamma )=0;$
\emph{semi-epimorphism} iff $\overline{\gamma (X)}=Y;$
\emph{semi-isomorphism} iff $\mathrm{Ker}(\gamma )=0$ and$\ \overline{\gamma
(X)}=Y.$
\end{punto}
\begin{remark}
The uniform ($k$-uniform, $i$-uniform) morphisms of semimodules were called
\emph{regular} ($k$\emph{-regular, }$i$-\emph{regular}) by Takahashi \cit
{Tak1982c}. We think that our terminology avoids confusion sine a regular
monomorphism (regular epimorphism) has a different well-established meaning
in the language of Category Theory.
\end{remark}
\begin{lemma}
\label{co-stead}Let $\gamma :X\longrightarrow Y$ be a morphism of $S
-semimodules.
\begin{enumerate}
\item The following are equivalent:
\begin{enumerate}
\item $\gamma $ is steady;
\item $\mathrm{Coker}(\mathrm{ker}(\gamma ))\simeq \mathrm{Coim}(\gamma );$
\item $X/\mathrm{Ker}(\gamma )\simeq \gamma (X);$
\item $\gamma $ is $k$-uniform.
\end{enumerate}
\item The following are equivalent:
\begin{enumerate}
\item $\gamma $ is costeady;
\item $\mathrm{Ker}(\mathrm{coker}(\gamma ))\simeq \func{Im}(\gamma );$
\item $\overline{\gamma (X)}=\gamma (X);$
\item $\gamma $ is $i$-uniform (subtractive).
\end{enumerate}
\item The following are equivalent:
\begin{enumerate}
\item $\gamma $ is bisteady;
\item $\mathrm{Coker}(\mathrm{ker}(\gamma ))\simeq \mathrm{Ker}(\mathrm{Coke
}(\gamma ));$
\item $X/\mathrm{Ker}(\gamma )\simeq \overline{\gamma (X)};$
\item $\gamma $ is uniform;
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{Beweis}
Notice that the canonical $(\mathbf{Surj},\mathbf{Mono})$- factorization of
\gamma $ is given by $\gamma :X\overset{\mathrm{coim}(\gamma )}
\longrightarrow }\gamma (X)\overset{\mathrm{im}(\gamma )}{\hookrightarrow
Y. $
\begin{enumerate}
\item By definition, $\gamma $ is steady iff $\gamma $ admits a $(\mathbf
Surj},\mathbf{Mono})$- factorization $\gamma =m_{1}\circ \mathrm{coker}
\mathrm{ker}(\gamma )).$ It follows that $\gamma $ is steady if and only if
\mathrm{Coker}(\mathrm{ker}(\gamma ))\simeq \mathrm{Coim}(\gamma )$ if and
only if $X/\mathrm{Ker}(\gamma )\simeq \gamma (X)$ which is equivalent to
\gamma $ being $k$-uniform.
\item By definition $\gamma $ is costeady if and only if $\gamma $ admits a
(\mathbf{Surj},\mathbf{Mono})$- factorization $\gamma =\mathrm{ker}(\mathrm
coker}(\gamma ))\circ e_{2}.$ It follows that $\gamma $ is costeady if and
only if $\gamma (X)=\mathrm{Ker}(\mathrm{coker}(\gamma )).$ Notice tha
\begin{equation*}
\begin{tabular}{lll}
$\mathrm{Ker}(\mathrm{\mathrm{coker}}(\gamma ))$ & $=$ & $\{y\in Y\mid
y\equiv _{\gamma (X)}0\}$ \\
& $=$ & $\{y\in Y\mid y+\gamma (x_{1})=\gamma (x_{2})$ for some
x_{1},x_{2}\in X\}$ \\
& $=$ & $\overline{\gamma (X)}.
\end{tabular
\end{equation*
It follows that $\gamma $ is costeady if and only if $\gamma (X)=\overline
\gamma (X)}$ which is equivalent to $\gamma $ being subtractive.
\item This is a combination of \textquotedblleft 1\textquotedblright\ and
\textquotedblleft 2\textquotedblright .$\blacksquare $
\end{enumerate}
\end{Beweis}
\begin{punto}
Let $M$ be an $S$-semimodule, $L\leq _{S}M$ an $S$-subsemimodule and
consider the factor semimodule $M/L.$ Then we have a surjective morphism of
S$-semimodule
\begin{equation*}
\pi _{L}:=M\rightarrow M/L,\text{ }m\mapsto \lbrack m]
\end{equation*
wit
\begin{equation*}
\mathrm{Ker}(\pi _{L})=\{m\in M\mid m+l_{1}=l_{2}\text{ for some
l_{1},l_{2}\in L\}=\overline{L};
\end{equation*
in particular, $L=\mathrm{Ker}(\pi _{L})$ if and only if $L\subseteq M$ is
subtractive.
\end{punto}
\section{Exact Sequences of Semimodules}
\qquad Throughout this section, $S$ is a ring, an $S$-semimodule is a right
S$-semimodule unless otherwise explicitly specified. Moreover, $\mathbb{S
_{S}$ ($\mathbb{CS}_{S}$) denotes the category of (cancellative) right $S
-semimodules.
The notion of \emph{exact sequences} of semimodules adopted by Takahashi
\cite{Tak1981} ($L\overset{f}{\longrightarrow }M\overset{g}{\longrightarrow
N$ is exact iff $\overline{f(M)}=\mathrm{Ker}(g)$) seems to be inspired by
the definition of exact sequences in Puppe-exact categories. We believe it
is inappropriate. The reason for this is that neither $\mathrm{Ker}(\mathrm
coker}(f))=\overline{f(L)}$ is the appropriate \emph{image} of $f$ nor is
\mathrm{Coker}(\mathrm{ker}(g))=B/\mathrm{Ker}(g)$ the appropriate \emph
coimage} of $g.$
Being a Barr-exact category, a natural tool to study exactness in the
category of semimodules is that of an \emph{exact fork}, introduced in \cit
{Bar1971} and applied to study exact functors between categories of
semimodules by Katsov et al. in \cite{KN2011}. However, since the category
of semimodules has additional features, one still expects to deal with exact
sequences rather than the more complicated exact forks.
In addition to Takahashi's classical definition of exact sequences of
semimodules, two different notions of exactness for sequences of semimodules
over semirings were introduced recently. The first is due to Patchkoria \cit
{Pat2003} ($L\overset{f}{\longrightarrow }M\overset{g}{\longrightarrow }N$
is exact iff $f(L)=\mathrm{Ker}(g)$) and the second is due to Patil and
Deore \cite{PD2006} ($L\overset{f}{\longrightarrow }M\overset{g}
\longrightarrow }N$ is exact iff ${\overline{f(L)}}=\mathrm{Ker}(g)$ and $g$
is \emph{steady}). Each of these definitions is stronger than Takahashi's
notion of exactness and each proved to be more efficient in establishing
some nice homological results for semimodules over semirings. However, no
clear \emph{categorical} justification for choosing either of these two
definitions was provided. A closer look at these definitions shows that they
are in fact dual to each other in some sense, and so no it not suitable --
in our opinion -- to choose one of them and drop the other. This motivated
us to introduce in Section one a new notion of exact sequences in general
pointed varieties. Applied to categories of semimodules, it turned out that
our notion of exact sequences of semimodules is in fact a combination of the
two notions of exact sequences of semimodules in the sense of \cite{Pat2003}
and \cite{PD2006}. For the sake of completeness, we mention here that there
is another notion of exact sequences of semimodules that was introduced in
\cite{AM2002}. However, the definition is rather technical and introduced
new definitions of \emph{epic} and \emph{monic} morphisms that are different
from the classical ones.
As indicated for general varieties in \ref{UA}, the category of semimodules
is $(\mathbf{RegEpi},\mathbf{Mono})$-structured, $\mathbf{RegEpi}=\mathbf
Surj}$ and $\mathbf{Mono}=\mathbf{Inj}.$ We say that a morphism of
semimodules $\gamma :X\longrightarrow Y$ is \emph{steady} (resp. \emph
costeady}, \emph{bisteady}) iff $\gamma $ is steady (resp. costeady,
bisteady) w.r.t. $(\mathbf{Surj},\mathbf{Inj}).$ Moreover, we say that a
sequence $X\overset{f}{\longrightarrow }Y\overset{g}{\longrightarrow }Z$ of
semimodules is \emph{exact} iff it is $(\mathbf{Surj},\mathbf{Inj})$-exact.
\begin{lemma}
\label{S-mon-epi}Le
\begin{equation}
L\overset{f}{\rightarrow }M\overset{g}{\rightarrow }N \label{lmn}
\end{equation
be a sequence of $S$-semimodules with $g\circ f=0$ and consider the induced
morphisms $f^{\prime }:L\rightarrow \mathrm{Ker}(g)$ and $g^{\prime \prime }
\mathrm{Coker}(f)\rightarrow N.$
\begin{enumerate}
\item If $f^{\prime }$ is an epimorphism, then $\overline{f(L)}=\mathrm{Ker
(g).$
\item $f^{\prime }$ is a regular epimorphism (surjective) if and only if
f(L)=\mathrm{Ker}(g)$ if and only if $\overline{f(L)}=\mathrm{Ker}(g)$ and
f $ is $i$-uniform.
\item $g^{\prime \prime }:\mathrm{Coker}(f)\rightarrow N$ is a monomorphism
if and only if $\overline{f(L)}=\mathrm{Ker}(g)$ and $g$ is $k$-uniform.
\end{enumerate}
\end{lemma}
\begin{Beweis}
Since $g\circ f=0,$ we have $f(L)\subseteq \overline{f(L)}\subseteq \mathrm
Ker}(g).$
\begin{enumerate}
\item Assume that $f^{\prime }:L\rightarrow \mathrm{Ker}(g)$ is an
epimorphism. Suppose that $\overline{f(L)}\subsetneqq \mathrm{Ker}(g),$ so
that there exist $m^{\prime }\in \mathrm{Ker}(g)\backslash f(L).$ Consider
the $S$-linear maps
\begin{equation*}
L\overset{\widetilde{f}}{\rightarrow }\mathrm{Ker}(g)\overset{f_{1}}
\underset{f_{2}}{\rightrightarrows }}\mathrm{Ker}(g)/f(L),
\end{equation*
where $f_{1}(m)=[m]$ and $f_{2}(m)=[0]$ for all $m\in \mathrm{Ker}(g).$ For
each $l\in L$ we have
\begin{equation*}
(f_{1}\circ f^{\prime })(l)=[f(l)]=[0]=(f_{2}\circ f^{\prime })(l).
\end{equation*
Whence, $f_{1}\circ f^{\prime }=f_{2}\circ f^{\prime }$ while $f_{1}\neq
f_{2}$ (since $f_{1}(m^{\prime })=[m^{\prime }]\neq \lbrack
0]=f_{2}(m^{\prime });$ otherwise $m^{\prime }+f(l_{1})=f(l_{2})$ for some
l_{1},l_{1}^{\prime }\in L$ and $m^{\prime }\in \overline{f(L)}$ which
contradicts our assumption). So, $f^{\prime }$ is not an epimorphism, a
contradiction. Consequently, $\overline{f(L)}=\mathrm{Ker}(g).$
\item Clear.
\item $(\Rightarrow )$ Assume that $g^{\prime \prime }:\mathrm{Coker
(f)\rightarrow N$ is a monomorphism. Let $m\in \mathrm{Ker}(g),$ so that
g(m)=0.$ Then $g^{\prime \prime }([m])=0.$ Since $g^{\prime \prime }$ is a
monomorphism, we have $[m]=[0]$ and so $m+f(l)=f(l^{\prime })$ for some
l,l^{\prime }\in L,$ whence $m\in \overline{f(L)}.$ Suppose now that
g(m)=g(m^{\prime })$ for some $m,m^{\prime }\in M.$ Then $g^{\prime \prime
}([m])=g^{\prime \prime }([m^{\prime }])$ and it follows, by the injectivity
of $g^{\prime \prime },$ that $[m]=[m^{\prime }]$ which implies that
m_{1}+m_{1}=m^{\prime }+m_{1}^{\prime }$ for some $m_{1},m_{1}^{\prime }\in
\overline{f(L)}=\mathrm{Ker}(g).$ So, $g$ is $k$-uniform.
$(\Leftarrow )$ Assume that $\overline{f(L)}=\mathrm{Ker}(g)$ and that $g$
is $k$-uniform. Suppose that $g^{\prime \prime }([m])=g^{\prime \prime
}([m^{\prime \prime }])$ for some $m_{1},m_{2}\in M.$ Then $g(m)=g(m^{\prime
}).$ Since $g$ is $k$-uniform, we $m+k=m^{\prime }+k^{\prime }$ for some
k,k^{\prime }\in \mathrm{Ker}(g)=\overline{f(L)}$ and it follows that
[m]=[m^{\prime }]$ (notice that $M/f(L)=M/\overline{f(L)}$).$\blacksquare $
\end{enumerate}
\end{Beweis}
\begin{corollary}
A sequence of semimodules $L\overset{f}{\rightarrow }M\overset{g}
\rightarrow }N$ is exact if and only if $f(L)=\mathrm{Ker}(g)$ and $g$ is $k
-uniform.
\end{corollary}
\begin{remarks}
\label{CS-mon-epi}
\begin{enumerate}
\item A morphism of cancellative semimodules $h:X\rightarrow Y$ is an
epimorphism in $\mathbb{CS}_{S}$ if and only if $\overline{h(X)}=Y.$ Indeed,
if $h$ is an epimorphism, then it follows by Lemma \ref{S-mon-epi} that
\overline{h(X)}=Y$ (take $g:Y\rightarrow 0$ as the zero-morphism). On the
other hand, assume that $\overline{h(L)}=Y.$ Let $Z$ be any cancellative
semimodule and consider any $S$-linear maps
\begin{equation*}
X\overset{h}{\rightarrow }Y\underset{h_{2}}{\overset{h_{1}}
\rightrightarrows }}Z
\end{equation*
with $h_{1}\circ h=h_{2}\circ h.$ Let $y\in Y$ be arbitrary. By assumption,
y+h(x_{1})=h(x_{2})$ for some $x_{1},x_{2}\in X,$ whenc
\begin{equation*}
h_{1}(y)+(h_{1}\circ h)(x_{1})=(h_{1}\circ h)(x_{2})=(h_{2}\circ
h)(x_{2})=h_{2}(y)+(h_{2}\circ h)(x_{1}).
\end{equation*
Since $Z$ is cancellative, we conclude that $h_{1}(y)=h_{2}(y).$
\item Consider the embedding $\iota :\mathbb{N}_{0}\hookrightarrow \mathbb{Z}
$ in $\mathbb{CS}_{\mathbb{N}_{0}}.$ Indeed, $\overline{\mathbb{N}_{0}}
\mathbb{Z},$ whence $\iota $ is an epimorphism which is not regular.
\item Let $L\overset{f}{\rightarrow }M\overset{g}{\rightarrow }N$ be a
sequence in $\mathbb{CS}_{S}$ with $g\circ f=0.$ By \textquotedblleft
1\textquotedblright , the induced morphism $f^{\prime }:L\rightarrow \mathrm
Ker}(g)$ is an epimorphism if and only if $\overline{f(L)}=\mathrm{Ker}(g).$
\end{enumerate}
\end{remarks}
\begin{punto}
\label{def-exact}We call a sequence of $S$-semimodules $L\overset{f}
\rightarrow }M\overset{g}{\rightarrow }N:$
\emph{proper-exact} iff $f(L)=\mathrm{Ker}(g);$
\emph{semi-exact} iff $\overline{f(L)}=\mathrm{Ker}(g);$
\emph{quasi-exact} iff $\overline{f(L)}=\mathrm{Ker}(g)$ and $g$ is $k
-uniform;
\emph{uniform} (resp. $k$\emph{-uniform}, $i$\emph{-uniform}) iff $f$ and $g$
are uniform (resp. $k$-uniform, $i$-uniform).
\end{punto}
\begin{punto}
We call a (possibly infinite) sequence of $S$-semimodules
\begin{equation}
\cdots \rightarrow M_{i-1}\overset{f_{i-1}}{\rightarrow }M_{i}\overset{f_{i}
{\rightarrow }M_{i+1}\overset{f_{i+1}}{\rightarrow }M_{i+2}\rightarrow \cdots
\label{chain}
\end{equation}
\emph{chain complex} iff $f_{j+1}\circ f_{j}=0$ for every $j;$
\emph{exact} (resp. \emph{proper-exact}, \emph{semi-exact}) iff each partial
sequence with three terms $M_{j}\overset{f_{j}}{\rightarrow }M_{j+1}\overset
f_{j+1}}{\rightarrow }M_{j+2}$ is exact (resp. proper-exact, semi-exact);
\emph{uniform }(resp. $k$-\emph{uniform}, $i$-\emph{uniform}) iff $f_{j}$ is
uniform (resp. $k$-uniform, $i$-uniform) for every $j.$
\end{punto}
\begin{definition}
Let $M$ be an $S$-semimodule.
\begin{enumerate}
\item A subsemimodule $L\leq _{S}M$ is said to be a \emph{uniform }(\emph
normal})\emph{\ }$S$-\emph{subsemimodule} iff the embedding
0\longrightarrow L\overset{\iota }{\rightarrow }M$ is uniform (normal).
\item A quotient $M/\rho ,$ where $\rho $ is an $S$-congruence relation on
M,$ is is said to be a \emph{uniform }(\emph{conormal})\emph{\ quotient} iff
the surijection $\pi _{L}:M\rightarrow M/\rho $ is uniform (conormal).
\end{enumerate}
\end{definition}
\begin{remark}
Every normal subsemimodule (normal quotient) is uniform.
\end{remark}
\qquad The following result can be easily verified.
\begin{lemma}
\label{i-uniform}Let $L\overset{f}{\rightarrow }M\overset{g}{\rightarrow }N$
be a sequence of semimodules.
\begin{enumerate}
\item Let $g$ be injective.
\begin{enumerate}
\item $f$ is $k$-uniform if and only if $g\circ f$ is $k$-uniform.
\item If $g\circ f$ is $i$-uniform (uniform), then $f$ is $i$-uniform
(uniform).
\item Assume that $g$ is $i$-uniform. Then $f$ is $i$-uniform (uniform) if
and only if $g\circ f$ is $i$-uniform (uniform).
\end{enumerate}
\item Let $f$ be surjective.
\begin{enumerate}
\item $g$ is $i$-uniform if and only if $g\circ f$ is $i$-uniform.
\item If $g\circ f$ is $k$-uniform (uniform), then $g$ is $k$-uniform
(uniform).
\item Assume that $f$ is $k$-uniform. Then $g$ is $k$-uniform (uniform) if
and only if $g\circ f$ is $k$-uniform (uniform).
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{Beweis}
\begin{enumerate}
\item Let $g$ be injective; in particular, $g$ is $k$-uniform.
\begin{enumerate}
\item Assume that $f$ is $k$-uniform. Suppose that $(g\circ
f)(l_{1})=(g\circ f)(l_{2})$ for some $l_{1},l_{2}\in L.$ Since $g$ is
injective, $f(l_{1})=f(l_{2}).$ By assumption, there exist $k_{1},k_{2}\in
\mathrm{Ker}(f)$ such that $l_{1}+k_{1}=l_{2}+k_{2}.$ Since $\mathrm{Ker
(f)\subseteq \mathrm{Ker}(g\circ f),$ we conclude that $g\circ f$ is $k
-uniform. On the other hand, assume that $g\circ f$ is $k$-uniform. Suppose
that $f(l_{1})=f(l_{2})$ for some $l_{1},l_{2}\in L.$ Then $(g\circ
f)(l_{1})=(g\circ f)(l_{2})$ and so there exist $k_{1},k_{2}\in \mathrm{Ker
(g\circ f)$ such that $l_{1}+k_{1}=l_{2}+k_{2}.$ Since $g$ is injective,
\mathrm{Ker}(g\circ f)=\mathrm{Ker}(f)$ whence $f$ is $k$-uniform.
\item Assume that $g\circ f$ is $i$-uniform. Let $m\in \overline{f(L)},$ so
that $m+f(l_{1})=f(l_{2})$ for some $l_{1},l_{2}\in L.$ Then $g(m)\in
\overline{(g\circ f)(L)}=(g\circ f)(L).$ Since $g$ is injective, $m\in f(L).$
So, $f$ is $i$-uniform.
\item Assume that $g$ and $f$ are $i$-uniform. Let $n\in \overline{(g\circ
f)(L)},$ so that $n+g(f(l_{1}))=g(f(l_{2}))$ for some $l_{1},l_{2}\in L.$
Since $g$ is $i$-uniform, $n\in g(M)$ say $n=g(m)$ for some $m\in M.$ But $g$
is injective, whence $m+f(l_{1})=f(l_{2}),$ i.e. $m\in \overline{f(L)}=f(L)$
since $f$ is $i$-uniform. So, $n=g(m)\in (g\circ f)(L).$ We conclude that
g\circ f$ is $i$-uniform.
\end{enumerate}
\item Let $f$ be surjective; in particular, $f$ is $i$-uniform.
\begin{enumerate}
\item Assume that $g$ is $i$-uniform. Let $n\in \overline{(g\circ f)(L)}$ so
that $n+g(f(l_{1}))=g(f(l_{2}))$ for some $l_{1},l_{2}\in L.$ Since $g$ is
i $-uniform, $n=g(m)$ for some $m\in M.$ Since $f$ is surjective, $n=g(m)\in
(g\circ f)(L).$ So, $g\circ f$ is $i$-uniform.
On the other hand, assume that $g\circ f$ is $i$-uniform. Let $n\in
\overline{g(M)},$ so that $n+g(m_{1})=g(m_{2})$ for some $m_{1},m_{2}\in M.$
Sine $f$ is surjective, there exist $l_{1},l_{2}\in L$ such that
f(l_{1})=m_{1}$ and $f(l_{2})=m_{2}.$ Then, $n+(g\circ f)(l_{1})=(g\circ
f)(l_{2}),$ i.e. $n\in \overline{(g\circ f)(L)}=(g\circ f)(L)\subseteq g(M).$
So, $g$ is $i$-uniform.
\item Assume that $g\circ f$ is $k$-uniform. Suppose that $g(m_{1})=g(m_{2})$
for some $m_{1},m_{2}\in M.$ Since $f$ is surjective, we have $(g\circ
f)(l_{1})=(g\circ f)(l_{2})$ for some $l_{1},l_{2}\in L.$ By assumption,
g\circ f$ is $k$-uniform and so there exist $k_{1},k_{2}\in \mathrm{Ker
(g\circ f)$ such that $l_{1}+k_{1}=l_{2}+k_{2}$ whence
m_{1}+f(k_{1})=m_{2}+f(k_{2}).$ Indeed, $f(k_{1}),f(k_{2})\in \mathrm{Ker
(g).$ i.e. $g$ is $k$-uniform.
\item Assume that $f$ and $g$ are $k$-uniform. Suppose that $(g\circ
f)(l_{1})=(g\circ f)(l_{2})$ for some $l_{1},l_{2}\in L.$ Since $g$ is $k
-uniform, we have $f(l_{1})+k_{1}=f(l_{2})+k_{2}$ for some $k_{1},k_{2}\in
\mathrm{Ker}(g).$ But $f$ is surjective; whence $k_{1}=f(l_{1}^{\prime })$
and $k_{2}=f(l_{2}^{\prime })$ for some $l_{1},l_{2}\in L,$ i.e.
f(l_{1}+l_{1}^{\prime })=f(l_{2}+l_{2}^{\prime }).$ Since $f$ is $k
-uniform, $l_{1}+l_{1}^{\prime }+k_{1}^{\prime }=l_{2}+l_{2}^{\prime
}+k_{2}^{\prime }$ for some $k_{1}^{\prime },k_{2}^{\prime }\in \mathrm{Ker
(f).$ Indeed, $l_{1}^{\prime }+k_{1}^{\prime },l_{2}^{\prime }+k_{2}^{\prime
}\in \mathrm{Ker}(g\circ f).$ We conclude that $g\circ f$ is $k$-uniform.
\blacksquare $
\end{enumerate}
\end{enumerate}
\end{Beweis}
\begin{remark}
Let $L\leq _{S}M\leq _{S}N$ be $S$-semimodules. It follows directly from the
previous lemma that if $L$ is uniform in $N,$ then $L$ is a uniform in $M$
as well. Moreover, if $M$ is uniform in $N,$ then $L$ is uniform in $N$ if
and only if $L$ is uniform in $M.$
\end{remark}
Our notion of exactness allows characterization of special classes of
morphisms in a way similar to that in homological categories (compare with
\cite[Proposition 4.1.9]{BB2004}, \cite[Propositions 4.4, 4.6]{Tak1981},
\cite[Proposition 15.15]{Go19l99a}):
\begin{proposition}
\label{inj-surj}Consider a sequence of semimodule
\begin{equation*}
0\longrightarrow L\overset{f}{\longrightarrow }M\overset{g}{\longrightarrow
N\longrightarrow 0.
\end{equation*}
\begin{enumerate}
\item The following are equivalent:
\begin{enumerate}
\item $0\longrightarrow L\overset{f}{\rightarrow }M$ is exact;
\item $\mathrm{Ker}(f)=0$ and $f$ is steady;
\item $f$ is semi-monomorphism and $k$-uniform;
\item $f$ is injective;
\item $f$ is a monomorphism.
\end{enumerate}
\item $0\longrightarrow L\overset{f}{\longrightarrow }M\overset{g}
\longrightarrow }N$ is semi-exact and $f$ is uniform if and only if $L\simeq
\mathrm{Ker}(g).$
\item $0\longrightarrow L\overset{f}{\longrightarrow }M\overset{g}
\longrightarrow }N$ is exact if and only if $L\simeq \mathrm{Ker}(g)$ and $g$
is $k$-uniform.
\item The following are equivalent:
\begin{enumerate}
\item $M\overset{\gamma }{\rightarrow }N\rightarrow 0$ is exact;
\item $\mathrm{Coker}(\gamma )=0$ and $\gamma $ is costeady;
\item $\gamma $ is semi-epimorphism and $i$-uniform;
\item $\gamma $ is surjective;
\item $\gamma $ is a regular epimorphism;
\item $\gamma $ is a subtractive epimorphism
\end{enumerate}
\item $L\overset{f}{\rightarrow }M\overset{g}{\rightarrow }N\rightarrow 0$
is semi-exact and $g$ is uniform if and only if $N\simeq \mathrm{Coker}(f).$
\item $L\overset{f}{\longrightarrow }M\overset{g}{\longrightarrow
N\longrightarrow 0$ is exact if and only if $N\simeq \mathrm{Coker}(f)$ and
f$ is $i$-uniform.
\end{enumerate}
\end{proposition}
\begin{corollary}
The following are equivalent:
\begin{enumerate}
\item $0\rightarrow L\overset{f}{\rightarrow }M\overset{g}{\rightarrow
N\rightarrow 0$ is a exact sequence of $S$-semimodules;
\item $L\simeq \mathrm{Ker}(g)$ and $\mathrm{\mathrm{\mathrm{\mathrm{Coker}}
}(f)\simeq N;$
\item $f$ is injective, $f(L)=\mathrm{Ker}(g),$ $g$ is surjective and ($k
-)uniform.
In this case, $f$ and $g$ are uniform morphisms.
\end{enumerate}
\end{corollary}
\begin{remark}
A morphism of semimodules $\gamma :X\longrightarrow Y$ is an isomorphism if
and only if $0\longrightarrow X\longrightarrow Y\longrightarrow 0$ is exact
if and only if $\gamma $ is a uniform bimorphism. The assumption on $\gamma $
to be uniform cannot be removed here. For example, the embedding $\iota
\mathbb{N}_{0}\longrightarrow \mathbb{Z}$ is a bimorphism of commutative
monoids ($\mathbb{N}_{0}$-semimodules) which is not an isomorphism. Notice
that $\iota $ is not $i$-uniform; in fact $\overline{\iota (\mathbb{N}_{0}})
\mathbb{Z}.$
\end{remark}
\begin{lemma}
\label{1st-IT}\emph{(Compare with \cite[Proposition 4.3.]{Tak1981})} Let
\gamma :X\rightarrow Y$ be a morphism of $S$-semimodules.
\begin{enumerate}
\item The sequenc
\begin{equation}
0\rightarrow \mathrm{Ker}(\gamma )\overset{\mathrm{\ker }(\gamma )}
\longrightarrow }X\overset{\gamma }{\rightarrow }Y\overset{\mathrm{\mathrm
coker}}(\gamma )}{\longrightarrow }\mathrm{\mathrm{\mathrm{\mathrm{\mathrm
Coker}}}}}(\gamma )\rightarrow 0 \label{ker-coker}
\end{equation}
is semi-exact. Moreover, (\ref{ker-coker}) is exact if and only if $\gamma $
is uniform.
\item We have two exact sequence
\begin{equation*}
0\rightarrow \overline{\gamma (X)}\overset{\mathrm{ker}(\mathrm{\mathrm{coke
}}(\gamma ))}{\longrightarrow }Y\overset{\mathrm{\mathrm{coker}}(\gamma )}
\longrightarrow }Y/\gamma (X)\rightarrow 0.
\end{equation*
an
\begin{equation*}
0\rightarrow \mathrm{Ker}(\gamma )\overset{\mathrm{ker}(\gamma )}
\longrightarrow }X\overset{\mathrm{\mathrm{coker}}(\mathrm{ker}(\gamma ))}
\longrightarrow }X/\mathrm{Ker}(\gamma )\rightarrow 0.
\end{equation*}
\end{enumerate}
\end{lemma}
\begin{corollary}
\label{reg-sub}\emph{(Compare with \cite[Proposition 4.8.]{Tak1981}) }Let $M$
be an $S$-semimodule.
\begin{enumerate}
\item Let $\rho $ an $S$-congruence relation on $M$ and consider the
sequence of $S$-semimodule
\begin{equation*}
0\longrightarrow \mathrm{Ker}(\pi _{\rho })\overset{\iota _{\rho }}
\longrightarrow }M\overset{\rho }{\longrightarrow }M/\rho \longrightarrow 0.
\end{equation*}
\begin{enumerate}
\item $0\rightarrow \mathrm{Ker}(\pi _{\rho })\overset{\iota _{\rho }}
\longrightarrow }M\overset{\pi _{\rho }}{\longrightarrow }M/\rho \rightarrow
0$ is exact.
\item $M/\rho =\mathrm{Coker}(\iota _{\rho }),$ whence $M/\rho $ is a normal
quotient.
\end{enumerate}
\item Let $L\leq _{S}M$ an $S$-subsemimodule.
\begin{enumerate}
\item The sequence $0\rightarrow L\overset{\iota }{\longrightarrow }M\overse
{\pi _{L}}{\longrightarrow }M/L\rightarrow 0$ is semi-exact.
\item $0\rightarrow \overline{L}\overset{\iota }{\longrightarrow }M\overset
\pi _{L}}{\longrightarrow }M/L\rightarrow 0$ is exact.
\item The following are equivalent:
\begin{enumerate}
\item $0\rightarrow L\overset{\iota }{\longrightarrow }M\overset{\pi _{L}}
\longrightarrow }M/L\rightarrow 0$ is exact;
\item $L\simeq \mathrm{Ker}(\pi _{L});$
\item $0\longrightarrow L\overset{\iota }{\longrightarrow }\overline{L
\longrightarrow 0$ is exact;
\item $L$ is a uniform subsemimodule;
\item $L$ is a normal subsemimodule.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{corollary}
\section{Homological lemmas}
\qquad In this section we prove some elementary diagram lemma for
semimodules over semirings. These apply in particular to commutative
monoids, considered as semimodules over the semiring of non-negative
integers. Recall that a sequence $A\overset{f}{\longrightarrow }B\overset{g}
\longrightarrow }C$ of semimodules is exact iff $f(A)=\mathrm{Ker}(g)$ and
g $ is is $k$-uniform (equivalently, $f(A)=\mathrm{Ker}(g)$ and
g(b)=g(b^{\prime })\Longrightarrow b+f(a)=b^{\prime }+f(a^{\prime })$ for
some $a,a^{\prime }\in A$).
\qquad The following result can be easily proved using \emph{diagram chasing}
(compare \textquotedblleft 2\textquotedblright\ with \cite[Lemma 1.9
{Pat2006}).
\begin{lemma}
\label{short}Consider the following commutative diagram of semimodule
\begin{equation*}
\xymatrix{ & & 0 \ar[d] \\ L_1 \ar[r]^{f_1} \ar[d]_{\alpha_1} & M_1
\ar[r]^{g_1} \ar[d]_{\alpha_2} & N_1 \ar[d]_{\alpha_3} \\ L_2 \ar[r]^{f_2}
\ar[d] & M_2 \ar[r]^{g_2} & N_2 \\ 0 & & }
\end{equation*
and assume that the first and the third columns are exact (i.e. $\alpha _{1}$
is surjective and $\alpha _{3}$ is injective).
\begin{enumerate}
\item Let $\alpha _{2}$ be surjective. If the first row is exact, then the
second row is exact.
\item Let $\alpha _{2}$ be injective. If the second row is exact, then the
first row is exact.
\item Let $a_{2}$ be an isomorphism. The first row is exact if and only if
the second row is exact.
\end{enumerate}
\end{lemma}
\begin{Beweis}
\begin{enumerate}
\item Let $\alpha _{2}$ be surjective and assume that the first row is exact.
\begin{itemize}
\item $f_{2}(L_{2})=\mathrm{Ker}(g_{2}).$
Notice that $g_{2}\circ f_{2}\circ \alpha _{1}=g_{2}\circ \alpha _{2}\circ
f_{1}=\alpha _{3}\circ g_{1}\circ f_{1}=0.$ Since $\alpha _{1}$ is an
epimorphism, we conclude that $g_{2}\circ f_{2}=0;$ in particular,
f_{2}(L_{2})\subseteq \mathrm{Ker}(g_{2}).$ On the other hand, let $m_{2}\in
\mathrm{Ker}(g_{2}).$ Since $\alpha _{2}$ is surjective, there exists
m_{1}\in M_{1}$ such that $\alpha _{2}(m_{1})=m_{2}.$ Since $\alpha _{3}$ is
a semi-monomorphism and $(\alpha _{3}\circ g_{1})(m_{1})=(g_{2}\circ \alpha
_{2})(m_{1})=0,$ we conclude that $g_{1}(m_{1})=0.$ Since the first row is
exact, there exists $l_{1}\in L_{1}$ such that $m_{1}=f_{1}(l_{1}).$ It
follows that $m_{2}=(\alpha _{2}\circ f_{1})(l_{1})=f_{2}(\alpha
_{1}(l_{1}))\in f_{2}(L_{2}).$
\item $g_{2}$ is $k$-uniform.
Suppose that $g_{2}(m_{2})=g_{2}(m_{2}^{\prime }).$ Since $\alpha _{2}$ is
surjective, there exist $m_{1},m_{1}^{\prime }\in M_{1}$ such that $\alpha
_{2}(m_{1})=m_{2}$ and $\alpha _{2}(m_{1}^{\prime })=m_{2}^{\prime }.$ Since
$\alpha _{3}$ is injective and $(\alpha _{3}\circ g_{1})(m_{1})=(g_{2}\circ
\alpha _{2})(m_{1})=(g_{2}\circ \alpha _{2})(m_{1}^{\prime })=(\alpha
_{3}\circ g_{1})(m_{1}^{\prime })$ we have $g_{1}(m_{1})=g_{1}(m_{1}^{\prime
}).$ Since $g_{1}$ is $k$-uniform and $f_{1}(L_{1})=\mathrm{Ker}(g_{1})$
there exist $l_{1},l_{1}^{\prime }\in L_{1}$ such that
m_{1}+f_{1}(l_{1})=m_{1}^{\prime }+f_{1}(l_{1}^{\prime }).$ It follows that
m_{2}+(\alpha _{2}\circ f_{1})(l_{1})=m_{2}^{\prime }+(\alpha _{2}\circ
f_{1})(l_{1}^{\prime })$ whence $m_{2}+f_{2}(\alpha
_{1}(l_{1}))=m_{2}^{\prime }+f_{2}(\alpha _{1}(l_{1}^{\prime })).$ Since
f_{2}(L_{2})\subseteq \mathrm{Ker}(g_{2}),$ we conclude that $g_{2}$ is $k
-uniform.
\end{itemize}
\item Let $\alpha _{2}$ be injective and assume that the second row is exact.
\begin{itemize}
\item $f_{1}(L_{1})=\mathrm{Ker}(g_{1}).$
Notice that $\alpha _{3}\circ g_{1}\circ f_{1}=g_{2}\circ \alpha _{2}\circ
f_{1}=g_{2}\circ f_{2}\circ \alpha _{1}=0.$ Since $\alpha _{3}$ is a
monomorphism, we conclude that $g_{1}\circ f_{1}=0,$ i.e.
f_{1}(L_{1})\subseteq \mathrm{Ker}(g_{1}).$ Let $m_{1}\in \mathrm{Ker
(g_{1}).$ Then $(g_{2}\circ \alpha _{2})(m_{1})=(\alpha _{3}\circ
g_{1})(m_{1})=0.$ Since the second row is exact, there exist $l_{2}\in L_{2}$
such that $f_{2}(l_{2})=\alpha _{2}(m_{1}).$ Since $\alpha _{1}$ is
surjective, there exists $l_{1}\in L_{1}$ such that $\alpha
_{2}(m_{1})=f_{2}(l_{2})=f_{2}(\alpha _{1}(l_{1}))=(\alpha _{2}\circ
f_{1})(l_{1}).$ Since $\alpha _{2}$ is injective, $m_{1}=f_{1}(l_{1}).$
\item $g_{1}$ is $k$-uniform.
Suppose that $g_{1}(m_{1})=g_{1}(m_{1}^{\prime })$ for some
m_{1},m_{1}^{\prime }\in M_{1}.$ Then we have $(g_{2}\circ \alpha
_{2})(m_{1})=(\alpha _{3}\circ g_{1})(m_{1})=(\alpha _{3}\circ
g_{1})(m_{1}^{\prime })=(g_{2}\circ \alpha _{2})(m_{1}^{\prime }).$ Since
g_{2}$ is $k$-uniform and $f_{2}(L_{2})=\mathrm{Ker}(g_{2}),$ there exist
l_{2},l_{2}^{\prime }\in L_{2}$ such that $\alpha
_{2}(m_{1})+f_{2}(l_{2})=\alpha _{2}(m_{1}^{\prime })+f_{2}(l_{2}^{\prime
}). $ Since $\alpha _{1}$ is surjective, there exist $l_{1},l_{1}^{\prime
}\in L_{1}$ such that $\alpha _{2}(m_{1}+f_{1}(l_{1}))=\alpha
_{2}(m_{1})+(f_{2}\circ \alpha _{1})(l_{1})=\alpha _{2}(m_{1}^{\prime
})+(f_{2}\circ \alpha _{1})(l_{1}^{\prime })=\alpha _{2}(m_{1}^{\prime
}+f_{1}(l_{1}^{\prime })).$ Since $\alpha _{2}$ is injective, we have
m_{1}+f_{1}(l_{1})=m_{2}+f_{1}(l_{1}^{\prime })$ and we are done since
f_{1}(L_{1})\subseteq \mathrm{Ker}(g_{1}).$
\end{itemize}
\item This is a combination of \textquotedblleft 1\textquotedblright\ and
\textquotedblleft 2\textquotedblright .$\blacksquare $
\end{enumerate}
\end{Beweis}
\subsection*{$\mathcal{R}$-Homological Categories}
\qquad It is well-known that the category of groups, despite being
non-Abelian (in fact not even Puppe-exact, but semiabelian in the sense of
Janelidze et al. \cite{JMT2002}), satisfies the so-called \emph{Short Five
Lemma}. It was shown in \cite[Theorem 4.1.10]{BB2004} that satisfying this
lemma characterizes the so-called \emph{protomodular categories}, whence the
\emph{homological categories}, among the pointed regular ones. Inspired by
this, we introduce in what follows a notion of (\emph{weak}) relative
homological categories with prototype the category of cancellative
commutative monoids, or more generally, the categories of cancellative
semimodules over semirings.
\begin{definition}
Let $\mathfrak{C}$ be a pointed category and $\mathcal{R}=((\mathbf{E}
\mathbf{M});\mathcal{A})$ where $(\mathbf{E},\mathbf{M})$ is a factorization
structure for $\mathfrak{C}$ and $\mathcal{A}\subseteq \mathrm{Mor}
\mathfrak{C}).$ We say that $\mathfrak{C}$ satisfies the \emph{Short }
\mathcal{R}$\emph{-Five Lemma} iff for every commutative diagram with $
\mathbf{E},\mathbf{M})$-exact rows and $\alpha _{2}\in \mathcal{A}:
\begin{equation*}
\xymatrix{0 \ar[r] & L_1 \ar[r]^{f_1} \ar[d]_{\alpha_1} & M_1 \ar[r]^{g_1}
\ar[d]_{\alpha_2} & N_1 \ar[d]_{\alpha_3} \ar[r] & 0\\ 0 \ar[r] & L_2
\ar[r]^{f_2} & M_2 \ar[r]^{g_2} & N_2 \ar[r] & 0}
\end{equation*
if $\alpha _{1}$ and $\alpha _{3}$ are isomorphisms, then $\alpha _{2}$ is
an isomorphism.
\end{definition}
\begin{definition}
Let $\mathfrak{C}$ be a category and $\mathcal{R}=((\mathbf{E},\mathbf{M})
\mathcal{A})$ where $\mathbf{E},\mathbf{M},\mathcal{A}\subseteq \mathrm{Mor}
\mathfrak{C}).$ We say that $\mathfrak{C}$ is
\begin{enumerate}
\item $(\mathbf{E},\mathbf{M})$\emph{-regular} iff $\mathfrak{C}$ has finite
limits, is $(\mathbf{E},\mathbf{M})$-structured and the morphisms in
\mathbf{E}$ are \emph{pullback stable}.
\item $\mathcal{R}$\emph{-homological category} iff $\mathfrak{C}$ is $
\mathbf{E},\mathbf{M})$-regular and satisfies the Short $\mathcal{R}$-Five
Lemma.
\end{enumerate}
\end{definition}
\begin{ex}
One recovers the homological categories in the sense of \cite{BB2004} (i.e.
those which are pointed, regular and protomodular) as follows: a pointed
category $\mathfrak{C}$ is homological iff $\mathfrak{C}$ is $\mathcal{R}
-homological where $\mathcal{R}=((\mathbf{RegEpi},\mathbf{Mono});\mathrm{Mor
(\mathfrak{C})).$
\end{ex}
\begin{lemma}
\label{diagram}Consider the following commutative diagram of semimodules
with exact row
\begin{equation*}
\xymatrix{L_1 \ar[r]^{f_1} \ar[d]_{\alpha_1} & M_1 \ar[r]^{g_1}
\ar[d]_{\alpha_2} & N_1 \ar[d]_{\alpha_3} \\ L_2 \ar[r]^{f_2} & M_2
\ar[r]^{g_2} & N_2}
\end{equation*}
\begin{enumerate}
\item We have:
\begin{enumerate}
\item Let $g_{1}$ and $\alpha _{1}$ be surjective. If $\alpha _{2}$ is
injective, then $\alpha _{3}$ is injective.
\item Let $f_{2}$ be injective and $\alpha _{3}$ a semi-monomorphism. If
\alpha _{2}$ is surjective, then $\alpha _{1}$ is surjective.
\end{enumerate}
\item Let $f_{2}$ be a semi-monomorphism.
\begin{enumerate}
\item If $\alpha _{1}$ and $\alpha _{3}$ are semi-monomorphisms, then
\alpha _{2}$ is a semi-monomorphism.
\item Let $f_{1},$ $\alpha _{2}$ be cancellative and $f_{2}$ be $k$-uniform.
If $\alpha _{1}$ and $\alpha _{3}$ are injective, then $\alpha _{2}$ is
injective.
\item If $g_{1},$ $\alpha _{1},$ $\alpha _{3}$ are surjective (and $\alpha
_{2}$ is $i$-uniform), then $\alpha _{2}$ is a semi-epimorphism (surjective).
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{Beweis}
\begin{enumerate}
\item Consider the given commutative diagram.
\begin{enumerate}
\item $\alpha _{3}$ is injective.
Suppose that $\alpha _{3}(n_{1})=\alpha _{3}(n_{1}^{\prime })$ for some
n_{1},n_{1}^{\prime }\in N_{1}.$ Since $g_{1}$ is surjective,
n_{1}=g_{1}(m_{1})$ and $n_{1}^{\prime }=g_{1}(m_{1}^{\prime })$ for some
m_{1},m_{1}^{\prime }\in M_{1}.$ It follows that $(g_{2}\circ \alpha
_{2})(m_{1})=(g_{2}\circ \alpha _{2})(m_{1}^{\prime }).$ Since $g_{2}$ is $k
-uniform and $f_{2}(L_{2})=\mathrm{Ker}(g_{2}),$ there exist
l_{2},l_{2}^{\prime }\in L_{2}$ such that $\alpha
_{2}(m_{1})+f_{2}(l_{2})=\alpha _{2}(m_{1}^{\prime })+f_{2}(l_{2}^{\prime
}). $ By assumption, $\alpha _{1}$ is surjective and so there exist
l_{1},l_{1}^{\prime }\in L_{1}$ such that $\alpha _{1}(l_{1})=l_{2}$ and
\alpha _{1}(l_{1}^{\prime })=l_{2}^{\prime }.$ It follows tha
\begin{equation*}
\begin{array}{rclc}
\alpha _{2}(m_{1})+(f_{2}\circ \alpha _{1})(l_{1}) & = & \alpha
_{2}(m_{1}^{\prime })+(f_{2}\circ \alpha _{1})(l_{1}^{\prime }) & \\
\alpha _{2}(m_{1})+(\alpha _{2}\circ f_{1})(l_{1}) & = & \alpha
_{2}(m_{1}^{\prime })+(\alpha _{2}\circ f_{1})(l_{1}^{\prime }) & \\
m_{1}+f_{1}(l_{1}) & = & m_{1}^{\prime }+f_{1}(l_{1}^{\prime }) & \text{(
\alpha _{2}\text{ is injective)} \\
g_{1}(m_{1}) & = & g_{1}(m_{1}) & \text{(}g_{1}\circ f_{1}=0\text{)} \\
n_{1} & = & n_{1}^{\prime } &
\end{array
\end{equation*}
\item $\alpha _{1}$ is surjective.
Let $l_{2}\in L_{2}.$ Since $\alpha _{2}$ is surjective, there exists
m_{1}\in M_{1}$ such that $f_{2}(l_{2})=\alpha _{2}(m_{1}).$ It follows that
$0=(g_{2}\circ f_{2})(l_{2})=(g_{2}\circ \alpha _{2})(m_{1})=(\alpha
_{3}\circ g_{1})(m_{1}),$ whence $g_{1}(m_{1})=0$ (since $\alpha _{3}$ is a
semi-monomorphism). Since the first row is exact, $m_{1}=f_{1}(l_{1})$ for
some $l_{1}\in L_{1}$ and so $f_{2}(l_{2})=\alpha _{2}(m_{1})=(\alpha
_{2}\circ f_{1})(l_{1})=(f_{2}\circ \alpha _{1})(l_{1}).$ Since $f_{2}$ is
injective, we have $l_{2}=\alpha _{1}(l_{1}).$
\end{enumerate}
\item Let $f_{2}$ be a semi-monomorphism, \emph{i.e.} $\mathrm{Ker
(f_{2})=0. $
\begin{enumerate}
\item We claim that $\alpha _{2}$ is a semi-monomorphism.
Suppose that $\alpha _{2}(m_{1})=0$ for some $m_{1}\in M_{1}.$ Then $(\alpha
_{3}\circ g_{1})(m_{1})=(g_{2}\circ \alpha _{2})(m_{1})=0,$ whence
g_{1}(m_{1})=0$ since $\mathrm{Ker}(\alpha _{3})=0.$ It follows that
m_{1}=f_{1}(l_{1})$ for some $l_{1}\in L_{1}.$ So, $0=\alpha
_{2}(m_{1})=(\alpha _{2}\circ f_{1})(l_{1})=(f_{2}\circ \alpha _{1})(l_{1}),$
whence $l_{1}=0$ since both $f_{2}$ and $\alpha _{1}$ are
semi-monormorphisms; consequently, $m_{1}=f_{1}(l_{1})=0.$
\item We claim that $\alpha _{2}$ is injective.
Suppose that $\alpha _{2}(m_{1})=\alpha _{2}(m_{1}^{\prime })$ for some
m_{1},m_{1}^{\prime }\in M_{1}.$ Then $(\alpha _{3}\circ
g_{1})(m_{1})=(g_{2}\circ \alpha _{2})(m_{1})=(g_{2}\circ \alpha
_{2})(m_{1}^{\prime })=(\alpha _{3}\circ g_{1})(m_{1}^{\prime }),$ whence
g_{1}(m_{1})=g_{1}(m_{1}^{\prime })$ since $\alpha _{3}$ is injective. Since
$g_{1}$ is $k$-uniform and $\mathrm{Ker}(g_{1})=f_{1}(L_{1}),$ there exist
l_{1},l_{1}^{\prime }\in L_{1}$ such that $m_{1}+f_{1}(l_{1})=m_{1}^{\prime
}+f_{1}(l_{1}^{\prime }).$ Then we hav
\begin{equation*}
\begin{array}{rclc}
\alpha _{2}(m_{1})+(\alpha _{2}\circ f_{1})(l_{1}) & = & \alpha
_{2}(m_{1}^{\prime })+(\alpha _{2}\circ f_{1})(l_{1}^{\prime }) & \\
\alpha _{2}(m_{1}^{\prime })+(f_{2}\circ \alpha _{1})(l_{1}) & = & \alpha
_{2}(m_{1}^{\prime })+(f_{2}\circ \alpha _{1})(l_{1}^{\prime }) & \\
(f_{2}\circ \alpha _{1})(l_{1}) & = & (f_{2}\circ \alpha _{1})(l_{1}^{\prime
}) & \text{(}\alpha _{2}\text{ is cancellative)} \\
l_{1} & = & l_{1}^{\prime } & \text{(}f_{2}\text{ and }\alpha _{1}\text{ are
injective)} \\
m_{1}+f_{1}(l_{1}^{\prime }) & = & m_{1}^{\prime }+f_{1}(l_{1}^{\prime }) &
\text{(}f_{1}\text{ is cancellative)} \\
m_{1} & = & m_{1}^{\prime } &
\end{array
\end{equation*}
\item We claim that $\alpha _{2}$ is a semi-epimorphism.
Let $m_{2}\in M_{2}.$ Since $\alpha _{3}$ and $g_{1}$ are surjective, there
exists $m_{1}\in M_{1}$ such that $g_{2}(m_{2})=(\alpha _{3}\circ
g_{1})(m_{1})=(g_{2}\circ \alpha _{2})(m_{1}).$ Since $g_{2}$ is $k
-uniform, $f_{2}(L_{2})=\mathrm{Ker}(g_{2})$ and $\alpha _{1}$ is
surjective, there exist $l_{1},l_{1}^{\prime }\in L_{1}$ such tha
\begin{eqnarray*}
m_{2}+(f_{2}\circ \alpha _{1})(l_{1}) &=&\alpha _{2}(m_{1})+(f_{2}\circ
\alpha _{1})(l_{1}^{\prime }) \\
m_{2}+\alpha _{2}(f_{1}(l_{1})) &=&\alpha _{2}(m_{1}+f_{1}(l_{1}^{\prime })).
\end{eqnarray*
Consequently, $M_{2}=\overline{\alpha _{2}(M_{1})},$ \emph{i.e.} $\alpha
_{2} $ is a semi-epimorphism. If $\alpha _{2}$ is $i$-uniform, then $M_{2}
\overline{\alpha _{2}(M_{1})}=\alpha _{2}(M_{1}),$ whence $\alpha _{2}$ is
surjective.$\blacksquare $
\end{enumerate}
\end{enumerate}
\end{Beweis}
\begin{corollary}
\label{cor-short5}Consider the following commutative diagram of semimodules
with exact rows and assume that $M_{1}$ and $M_{2}$ are cancellativ
\begin{equation*}
\xymatrix{& L_1 \ar[r]^{f_1} \ar[d]_{\alpha_1} & M_1 \ar[r]^{g_1}
\ar[d]_{\alpha_2} & N_1 \ar[d]_{\alpha_3} \ar[r] & 0\\ 0 \ar[r] & L_2
\ar[r]^{f_2} & M_2 \ar[r]^{g_2} & N_2 & }
\end{equation*}
\begin{enumerate}
\item Let $\alpha _{2}$ be an isomorphism. Then $\alpha _{1}$ is surjective
if and only if $\alpha _{3}$ is injective.
\item Let $\alpha _{2}$ be $i$-uniform. If $\alpha _{1}$ and $\alpha _{3}$
are isomorphisms, then $\alpha _{2}$ is an isomorphism.
\end{enumerate}
\end{corollary}
\begin{proposition}
\label{short-5}\emph{(The Short Five Lemma)} Consider the following
commutative diagram of semimodules with $M_{1},M_{2}$ cancellativ
\begin{equation*}
\xymatrix{0 \ar[r] & L_1 \ar[r]^{f_1} \ar[d]_{\alpha_1} & M_1 \ar[r]^{g_1}
\ar[d]_{\alpha_2} & N_1 \ar[d]_{\alpha_3} \ar[r] & 0\\ 0 \ar[r] & L_2
\ar[r]^{f_2} & M_2 \ar[r]^{g_2} & N_2 \ar[r] & 0}
\end{equation*
Then $\alpha _{1},$ $\alpha _{3}$ are isomorphisms and $\alpha _{2}$ is $i
-uniform if and only if $\alpha _{2}$ is an isomorphism. In particular, the
category $\mathbb{CS}_{S}$ of cancellative right $S$-semimodules is
\mathcal{R}$-homological, where $\mathcal{R}=((\mathbf{Surj},\mathbf{Inj})
\mathcal{I})$ and $\mathcal{I}$ is the class of $i$-uniform morphisms.
\end{proposition}
\begin{lemma}
\label{5-details}Consider the following commutative diagram of semimodules
with exact row
\begin{equation*}
\xymatrix{U_1 \ar[r]^{e_1} \ar[d]_{\gamma} & L_1 \ar[r]^{f_1}
\ar[d]_{\alpha_1} & M_1 \ar[r]^{g_1} \ar[d]_{\alpha_2} & N_1 \ar[r]
\ar[d]_{\alpha_3} \ar[r]^{h_1} & V_1 \ar[d]_{\delta} \\ U_2 \ar[r]^{e_2} &
L_2 \ar[r]^{f_2} & M_2 \ar[r]^{g_2} & N_2 \ar[r]^{h_2} & V_2}
\end{equation*}
\begin{enumerate}
\item Let $\gamma $ be surjective.
\begin{enumerate}
\item If $\alpha _{1}$ is injective and $\alpha _{3}$ is a
semi-monomorphisms, then $\alpha _{2}$ is a semi-monomorphism.
\item Assume that $f_{1}$ and $\alpha _{2}$ are cancellative. If $\alpha
_{1} $ and $\alpha _{3}$ are injective, then $\alpha _{2}$ is injective.
\end{enumerate}
\item Let $\delta $ be a semi-monomorphism. If $\alpha _{1},$ $\alpha _{3}$
are surjective (and $\alpha _{2}$ is $i$-uniform), then $\alpha _{2}$ is a
semi-epimorphism (surjective).
\item Let $f_{1},\alpha _{2}$ be cancellative, $\gamma $ be surjective and
\delta $ be injective. If $\alpha _{1}$ and $\alpha _{3}$ are isomorphisms,
then $\alpha _{2}$ is injective and a semi-epimorphism.
\end{enumerate}
\end{lemma}
\begin{Beweis}
Assume that the diagram is commutative and that the two rows are exact.
\begin{enumerate}
\item Let $\gamma $ be surjective.
\begin{enumerate}
\item Assume that $\alpha _{1}$ is injective and that $\alpha _{3}$ is a
semi-isomorphism. We claim that $\alpha _{2}$ is a semi-monomorphism.
Suppose that $\alpha _{2}(m_{1})=0$ for some $m_{1}\in M_{1}$ so that
(\alpha _{3}\circ g_{1})(m_{1})=(g_{2}\circ \alpha _{2})(m_{1})=0.$ Since
\alpha _{3}$ is a semi-monomorphism $g_{1}(m_{1})=0,$ whence
m_{1}=f_{1}(l_{1})$ for some $l_{1}\in L_{1}.$ So, $0=\alpha
_{2}(m_{1})=(\alpha _{2}\circ f_{1})(l_{1})=(f_{2}\circ \alpha _{1})(l_{1}),$
whence $\alpha _{1}(l_{1})=(e_{2}\circ \gamma )(u_{1})=(\alpha _{1}\circ
e_{1})(u_{1})$ for some $u_{1}\in U_{1}$ (since $\gamma $ is surjective and
\mathrm{Ker}(f_{2})=e_{2}(U_{2})$). Since $\alpha _{1}$ is injective, it
follows that $l_{1}=e_{1}(u_{1})$ whence $m_{1}=f_{1}(l_{1})=(f_{1}\circ
e_{1})(u_{1})=0.$
\item Assume that $f_{1},\alpha _{2}$ are cancellative and $\alpha _{1},$
\alpha _{3}$ are injective. We claim that $\alpha _{2}$ is injective.
Suppose that $\alpha _{2}(m_{1})=\alpha _{2}(m_{1}^{\prime })$ for some
m_{1},m_{1}^{\prime }\in M_{1}.$ Then $(\alpha _{3}\circ
g_{1})(m_{1})=(g_{2}\circ \alpha _{2})(m_{1})=(g_{2}\circ \alpha
_{2})(m_{1}^{\prime })=(\alpha _{3}\circ g_{1})(m_{1}^{\prime }),$ whence
g_{1}(m_{1})=g_{1}(m_{1}^{\prime })$ (notice that $\alpha _{3}$ is
injective). Since $g_{1}$ is $k$-uniform and $\mathrm{Ker
(g_{1})=f_{1}(L_{1}),$ there exist $l_{1},l_{1}^{\prime }\in L_{1}$ such
that $m_{1}+f_{1}(l_{1})=m_{1}^{\prime }+f_{1}(l_{1}^{\prime }).$ Then we
hav
\begin{equation*}
\begin{array}{rclc}
\alpha _{2}(m_{1})+(\alpha _{2}\circ f_{1})(l_{1}) & = & \alpha
_{2}(m_{1}^{\prime })+(\alpha _{2}\circ f_{1})(l_{1}^{\prime }) & \\
\alpha _{2}(m_{1}^{\prime })+(f_{2}\circ \alpha _{1})(l_{1}) & = & \alpha
_{2}(m_{1}^{\prime })+(f_{2}\circ \alpha _{1})(l_{1}^{\prime }) & \\
f_{2}(\alpha _{1}(l_{1})) & = & f_{2}(\alpha _{1}(l_{1}^{\prime })) & \text{
}\alpha _{2}\text{ is cancellative)} \\
\alpha _{1}(l_{1})+k_{2} & = & \alpha _{1}(l_{1}^{\prime })+k_{2}^{\prime }
& \text{(}f_{2}\text{ is $k$-uniform)} \\
\alpha _{1}(l_{1})+(e_{2}\circ \gamma )(u_{1}) & = & \alpha
_{1}(l_{1}^{\prime })+(e_{2}\circ \gamma )(u_{1}^{\prime }) & \text{(}\gamma
\text{ is surjective)} \\
\alpha _{1}(l_{1})+(\alpha _{1}\circ e_{1})(u_{1}) & = & \alpha
_{1}(l_{1}^{\prime })+(\alpha _{1}\circ e_{1})(u_{1}^{\prime }) & \\
l_{1}+e_{1}(u_{1}) & = & l_{1}^{\prime }+e_{1}(u_{1}^{\prime }) & \text{(
\alpha _{1}\text{ is injective)} \\
f_{1}(l_{1}) & = & f_{1}(l_{1}^{\prime }) & \text{(}f_{1}\circ e_{1}=0\text{
} \\
m_{1}+f_{1}(l_{1}) & = & m_{1}+f_{1}(l_{1}^{\prime }) & \\
m_{1}^{\prime }+f_{1}(l_{1}^{\prime }) & = & m_{1}+f_{1}(l_{1}^{\prime }) &
\\
m_{1}^{\prime } & = & m_{1} & \text{(}f_{1}\text{ is cancellative)
\end{array
\end{equation*}
\end{enumerate}
\item Let $\delta $ be a semi-monomorphism. Assume that $\alpha _{1}$ and
\alpha _{3}$ are surjective. Let $m_{2}\in M_{2}.$ Since $\alpha _{3}$ is
surjective, there exists $n_{1}\in N_{1}$ such that $g_{2}(m_{2})=\alpha
_{3}(n_{1}).$ It follows that $0=(h_{2}\circ g_{2})(m_{2})=(h_{2}\circ
\alpha _{3})(n_{1})=(\delta \circ h_{1})(n_{1}),$ whence $h_{1}(n_{1})=0$
since $\delta $ is a semi-monomorphism. Since $g_{1}(M_{1})=\mathrm{Ker
(h_{1}),$ we have $n_{1}=g_{1}(m_{1})$ for some $m_{1}\in M_{1}.$ Notice
that $(g_{2}\circ \alpha _{2})(m_{1})=(\alpha _{3}\circ g_{1})(m_{1})=\alpha
_{3}(n_{1})=g_{2}(m_{2}).$ Since $g_{2}$ is $k$-uniform, $f_{2}(L_{2})
\mathrm{Ker}(g_{2})$ and $\alpha _{1}$ is surjective, there exists
l_{1},l_{1}^{\prime }\in L_{1}$ such tha
\begin{eqnarray*}
m_{2}+(f_{2}\circ \alpha _{1})(l_{1}) &=&\alpha _{2}(m_{1})+(f_{2}\circ
\alpha _{1})(l_{1}^{\prime }) \\
m_{2}+\alpha _{2}(f_{1}(l_{1})) &=&\alpha _{2}(m_{1}+f_{1}(l_{1}^{\prime })),
\end{eqnarray*
i.e. $m_{2}\in \overline{\alpha _{2}(M_{1})}.$ Consequently, $M_{2}
\overline{\alpha _{2}(M_{1})}.$ If $\alpha _{2}$ is $i$-uniform, then
\alpha _{2}(M)=\overline{\alpha _{2}(M_{1})}=M_{2},$ \emph{i.e. }$\alpha
_{2} $ is surjective.
\item This is a combination of \textquotedblleft 1\textquotedblright\ and
\textquotedblleft 2\textquotedblright .$\blacksquare $
\end{enumerate}
\end{Beweis}
\begin{corollary}
\label{5-lemma}\emph{(The Five Lemma) }Consider the following commutative
diagram of semimodules with exact rows and columns and assume that $f_{1}$
and $\alpha _{2}$ are cancellativ
\begin{equation*}
\xymatrix{ & & & & 0 \ar[d] \\ U_1 \ar[r]^{e_1} \ar[d]_{\gamma} & L_1
\ar[r]^{f_1} \ar[d]_{\alpha_1} & M_1 \ar[r]^{g_1} \ar[d]_{\alpha_2} & N_1
\ar[r] \ar[d]_{\alpha_3} \ar[r]^{h_1} & V_1 \ar[d]_{\delta} \\ U_2
\ar[r]^{e_2} \ar[d] & L_2 \ar[r]^{f_2} & M_2 \ar[r]^{g_2} & N_2 \ar[r]^{h_2}
& V_2 \\ 0 & & & & }
\end{equation*}
\begin{enumerate}
\item If $\alpha _{1}$ and $\alpha _{3}$ are injective, then $\alpha _{2}$
is injective.
\item Let $\alpha _{2}$ be $i$-uniform. If $\alpha _{1}$ and $\alpha _{3}$
are surjective, then $\alpha _{2}$ is surjective.
\item Let $\alpha _{2}$ be $i$-uniform. If $\alpha _{1}$ and $\alpha _{3}$
are isomorphisms, then $\alpha _{2}$ is an isomorphism.
\end{enumerate}
\end{corollary}
\subsection*{The Snake Lemma}
\qquad One of the basic homological lemmas that are proved usually in
categories of modules (e.g. \cite{Wis1991}), or more generally in Abelian
categories, is the so called \emph{Kernel-Cokernel Lemma} (\emph{Snake Lemma
). Several versions of this lemma were proved also in non-abelian categories
(e.g. \emph{homological categories} \cite{BB2004}, \emph{relative
homological categories} \cite{Jan2006} and incomplete relative homological
categories \cite{Jan2010b}).
\begin{lemma}
\label{9-1}Consider the following commutative diagram with exact columns and
assume that the second row is exact
\begin{equation*}
\xymatrix{ & & 0 \ar[d] & 0 \ar[d] & \\ & L_1 \ar[r]^{f_1} \ar[d]_{\alpha_1}
& M_1 \ar[r]^{g_1} \ar[d]_{\alpha_2} & N_1 \ar[d]_{\alpha_3} & \\ & L_2
\ar[r]^{f_2} \ar[d]_{\beta_1} & M_2 \ar[r]^{g_2} \ar[d]_{\beta_2} & N_2
\ar[d]_{\beta_3} &\\ & L_3 \ar[r]^{f_3} & M_3 \ar[r]^{g_3} & N_3 & }
\end{equation*}
\begin{enumerate}
\item If $f_{3}$ is injective and $f_{2}$ is cancellative, then the first
row is exact.
\item If $g_{2},$ $\beta _{1}$ are surjective, the third row is exact (and
g_{1}$ is $i$-uniform), then $g_{1}$ is a semi-epimorphism (surjective).
\end{enumerate}
\end{lemma}
\begin{Beweis}
Assume that the second row is exact.
\begin{enumerate}
\item Notice that $\alpha _{3}\circ g_{1}\circ f_{1}=g_{2}\circ \alpha
_{2}\circ f_{1}=g_{2}\circ f_{2}\circ \alpha _{1}=0,$ whence $g_{1}\circ
f_{1}=0$ since $\alpha _{3}$ is a monomorphism. In particular,
f_{1}(L_{1})\subseteq \mathrm{Ker}(g_{1}).$
\begin{itemize}
\item We claim that $f_{1}(L_{1})=\mathrm{Ker}(g_{1}).$
Let $m_{1}\in \mathrm{Ker}(g_{1}),$ so that $g_{1}(m_{1})=0.$ It follows tha
\begin{equation*}
\begin{array}{rclc}
(\alpha _{3}\circ g_{1})(m_{1}) & = & 0 & \\
(g_{2}\circ \alpha _{2})(m_{1}) & = & 0 & \\
\alpha _{2}(m_{1}) & = & f_{2}(l_{2}) & \text{(2nd row is proper exact)} \\
0 & = & (\beta _{2}\circ f_{2})(l_{2}) & \text{(}\beta _{2}\circ \alpha
_{2}=0\text{)} \\
0 & = & (f_{3}\circ \beta _{1})(l_{2}) & \\
\beta _{1}(l_{2}) & = & 0 & \text{(}f_{3}\text{ is a semi-monomorphism)} \\
l_{2} & = & \alpha _{1}(l_{1}) & \text{(1st column is proper exact)} \\
f_{2}(l_{2}) & = & (f_{2}\circ \alpha _{1})(l_{1}) & \\
\alpha _{2}(m_{1}) & = & \alpha _{2}(f_{1}(l_{1})) & \\
m_{1} & = & f_{1}(l_{1}) & \text{(}\alpha _{2}\text{ is injective)
\end{array
\end{equation*}
\item We claim that $g_{1}$ is $k$-uniform.
Suppose that $g_{1}(m_{1})=g_{1}(m_{1}^{\prime })$ for some
m_{1},m_{1}^{\prime }\in M_{1}.$ It follows tha
\begin{equation*}
\begin{array}{rclc}
(\alpha _{3}\circ g_{1})(m_{1}) & = & (\alpha _{3}\circ g_{1})(m_{1}^{\prime
}) & \\
(g_{2}\circ \alpha _{2})(m_{1}) & = & (g_{2}\circ \alpha _{2})(m_{1}^{\prime
}) & \\
\alpha _{2}(m_{1})+f_{2}(l_{2}) & = & \alpha _{2}(m_{1}^{\prime
})+f_{2}(l_{2}^{\prime })\text{ (2nd row is exact)} & \\
(\beta _{2}\circ f_{2})(l_{2}) & = & (\beta _{2}\circ f_{2})(l_{2}^{\prime }
\text{ (}\beta _{2}\circ \alpha _{2}=0\text{)} & \\
(f_{3}\circ \beta _{1})(l_{2}) & = & (f_{3}\circ \beta _{1})(l_{2}^{\prime })
& \\
\beta _{1}(l_{2}) & = & \beta _{1}(l_{2}^{\prime })\text{ (}f_{3}\text{ is
injective)} & \\
l_{2}+\alpha _{1}(l_{1}) & = & l_{2}^{\prime }+\alpha _{1}(l_{1}^{\prime }
\text{ (first column is exact)} & \\
f_{2}(l_{2})+(f_{2}\circ \alpha _{1})(l_{1}) & = & f_{2}(l_{2}^{\prime
})+(f_{2}\circ \alpha _{1})(l_{1}^{\prime }) & \\
f_{2}(l_{2})+(\alpha _{2}\circ f_{1})(l_{1}) & = & f_{2}(l_{2}^{\prime
})+(\alpha _{2}\circ f_{1})(l_{1}^{\prime }) & \\
\alpha _{2}(m_{1})+f_{2}(l_{2})+(\alpha _{2}\circ f_{1})(l_{1}) & = & \alpha
_{2}(m_{1})+f_{2}(l_{2}^{\prime })+(\alpha _{2}\circ f_{1})(l_{1}^{\prime })
& \\
f_{2}(l_{2}^{\prime })+\alpha _{2}(m_{1}^{\prime }+f_{1}(l_{1})) & = &
f_{2}(l_{2}^{\prime })+\alpha _{2}(m_{1}+f_{1}(l_{1}^{\prime }))\text{ (
f_{2}\text{ is cancellative)} & \\
m_{1}^{\prime }+f_{1}(l_{1}) & = & m_{1}+f_{1}(l_{1}^{\prime })\text{ (
\alpha _{2}\text{ is injective)} &
\end{array
\end{equation*
Since $f_{1}(L_{1})\subseteq \mathrm{Ker}(g_{1}),$ it follows that $g_{1}$
is $k$-uniform.
\end{itemize}
\item We claim that $g_{1}$ is a semi-epimorphism.
Let $n_{1}\in N_{1}.$ Let $m_{2}\in M_{2}$ be such that $g_{2}(m_{2})=\alpha
_{3}(n_{1}).$ The
\begin{equation*}
\begin{array}{rclc}
g_{3}(\beta _{2}(m_{2})) & = & \beta _{3}(g_{2}(m_{2})) & \\
& = & (\beta _{3}\circ \alpha _{3})(m_{2}) & \\
& = & 0 & \text{(}\beta _{3}\circ \alpha _{3}=0\text{)} \\
\beta _{2}(m_{2}) & = & f_{3}(l_{3}) & \text{(3rd row is exact)} \\
& = & f_{3}(\beta _{1}(l_{2})) & \text{(}\beta _{1}\text{ is surjective)} \\
& = & \beta _{2}(f_{2}(l_{2})) & \\
m_{2}+\alpha _{2}(m_{1}) & = & f_{2}(l_{2})+\alpha _{2}(m_{1}^{\prime }) &
\text{(2nd column is exact)} \\
g_{2}(m_{2})+(g_{2}\circ \alpha _{2})(m_{1}) & = & (g_{2}\circ \alpha
_{2})(m_{1}) & \text{(}g_{2}\circ f_{2}=0\text{)} \\
\alpha _{3}(n_{1}+g_{1}(m_{1})) & = & \alpha _{3}(g_{1}(m_{1}^{\prime })) &
\\
n_{1}+g_{1}(m_{1}) & = & g_{1}(m_{1}^{\prime }) & \text{(}\alpha _{3}\text{
is injective)
\end{array
\end{equation*
Consequently, $N_{1}=\overline{g_{1}(M_{1})}$ ($=$ $g_{1}(M_{1})$ if $g_{1}$
is assumed to be $i$-uniform).$\blacksquare $
\end{enumerate}
\end{Beweis}
\qquad Similarly, one can prove the following result.
\begin{lemma}
\label{9-3}Consider the following commutative diagram with exact columns and
assume that the second row is exac
\begin{equation*}
\xymatrix{ & L_1 \ar[r]^{f_1} \ar[d]_{\alpha_1} & M_1 \ar[r]^{g_1}
\ar[d]_{\alpha_2} & N_1 \ar[d]_{\alpha_3} & \\& L_2 \ar[r]^{f_2}
\ar[d]_{\beta_1} & M_2 \ar[r]^{g_2} \ar[d]_{\beta_2} & N_2 \ar[d]_{\beta_3}
&\\ & L_3 \ar[r]^{f_3} \ar[d] & M_3 \ar[r]^{g_3} \ar[d] & N_3 & \\ & 0 & 0 &
}
\end{equation*}
\begin{enumerate}
\item If $g_{1}$ is surjective and $f_{3}$ is $i$-uniform, then the third
row is exact.
\item If $f_{2},$ $\alpha _{3}$ are injective, $\alpha _{2}$ is cancellative
and the first row is exact, then $f_{3}$ is injective.
\end{enumerate}
\end{lemma}
\begin{Beweis}
Assume that the second row is exact.
\begin{enumerate}
\item Notice that $g_{3}\circ f_{3}\circ \beta _{1}=g_{3}\circ \beta
_{2}\circ f_{2}=\beta _{3}\circ g_{2}\circ f_{2}=0.$ Since $\beta _{1}$ is
an epimorphism, we have $g_{3}\circ f_{3}=0$ (i.e. $f_{3}(L_{3})\subseteq
\mathrm{Ker}(g_{3})$).
\begin{itemize}
\item We claim that $f_{3}(L_{3})=\mathrm{Ker}(g_{3}).$ Let $m_{3}\in
\mathrm{Ker}(g_{3}).$
Since $\beta _{2}$ is surjective, $m_{3}=\beta _{2}(m_{2})$ for some
m_{2}\in M_{2}.$ It follows that $0=(g_{3}\circ \beta _{2})(m_{2})=(\beta
_{3}\circ g_{2})(m_{2}),$ i.e. $g_{2}(m_{2})\in \mathrm{Ker}(\beta
_{3})=\alpha _{3}(N_{1}).$ We hav
\begin{equation*}
\begin{array}{rclc}
g_{2}(m_{2}) & = & \alpha _{3}(n_{1}) & \\
& = & (\alpha _{3}\circ g_{1})(m_{1}) & \text{(}g_{1}\text{ is surjective)}
\\
& = & (g_{2}\circ \alpha _{2})(m_{1}) & \\
m_{2}+f_{2}(l_{2}) & = & \alpha _{2}(m_{1})+f_{2}(l_{2}^{\prime }) & \text
(2nd row is exact)} \\
\beta _{2}(m_{2})+(\beta _{2}\circ f_{2})(l_{2}) & = & (\beta _{2}\circ
f_{2})(l_{2}^{\prime }) & \text{(}\beta _{2}\circ \alpha _{2}=0\text{)} \\
m_{3}+(f_{3}\circ \beta _{1})(l_{2}) & = & (f_{3}\circ \beta
_{1})(l_{2}^{\prime }) &
\end{array
\end{equation*
We conclude that $\mathrm{Ker}(g_{3})=\overline{f_{3}(L_{3})}=f_{3}(L_{3}).$
\item We claim that $g_{3}$ is $k$-uniform.
Suppose that $g_{3}(m_{3})=g_{3}(m_{3}^{\prime })$ for some
m_{3},m_{3}^{\prime }\in M_{3}.$ Since $\beta _{2}$ is surjective, there
exist $m_{2},m_{2}^{\prime }\in M$ such that $\beta _{2}(m_{2})=m_{3}$ and
\beta _{2}(m_{2}^{\prime })=m_{3}^{\prime }.$ The
\begin{equation*}
\begin{array}{rclc}
(g_{3}\circ \beta _{2})(m_{2}) & = & (g_{3}\circ \beta _{2})(m_{2}^{\prime })
& \\
(\beta _{3}\circ g_{2})(m_{2}) & = & (\beta _{3}\circ g_{2})(m_{2}^{\prime })
& \\
g_{2}(m_{2})+\alpha _{3}(n_{1}) & = & g_{2}(m_{2}^{\prime })+\alpha
_{3}(n_{1}^{\prime }) & \text{(3rd column is exact)} \\
g_{2}(m_{2})+(\alpha _{3}\circ g_{1})(m_{1}) & = & g_{2}(m_{2}^{\prime
})+(\alpha _{3}\circ g_{1})(m_{1}^{\prime }) & \text{(}g_{1}\text{ is
surjective)} \\
g_{2}(m_{2})+(g_{2}\circ \alpha _{2})(m_{1}) & = & g_{2}(m_{2}^{\prime
})+(g_{2}\circ \alpha _{2})(m_{1}^{\prime }) & \\
m_{2}+\alpha _{2}(m_{1})+f_{2}(l_{2}) & = & m_{2}^{\prime }+\alpha
_{2}(m_{1}^{\prime })+f_{2}(l_{2}^{\prime }) & \text{(2nd row is exact)} \\
\beta _{2}(m_{2})+(\beta _{2}\circ f_{2})(l_{2}) & = & \beta
_{2}(m_{2}^{\prime })+(\beta _{2}\circ f_{2})(l_{2}^{\prime }) & \text{(
\beta _{2}\circ \alpha _{2}=0\text{)} \\
m_{3}+(f_{3}\circ \beta _{1})(l_{2}) & = & m_{3}^{\prime }+(f_{3}\circ \beta
_{1})(l_{2}^{\prime }) &
\end{array
\end{equation*
Since $f_{3}(L_{3})\subseteq \mathrm{Ker}(g_{3}),$ we conclude that $g_{3}$
is $k$-uniform.
\end{itemize}
\item We claim that $f_{3}$ is injective.
Suppose that $f_{3}(l_{3})=f_{3}(l_{3}^{\prime })$ for some
l_{3},l_{3}^{\prime }\in L_{3}.$ Since $\beta _{1}$ is surjective, there
exist $l_{2},l_{2}^{\prime }\in L_{2}$ such that $\beta _{1}(l_{2})=l_{3}$
and $\beta _{1}(l_{2}^{\prime })=l_{3}^{\prime }.$ Then we hav
\begin{equation*}
\begin{array}{rcll}
(f_{3}\circ \beta _{1})(l_{2}) & = & (f_{3}\circ \beta _{1})(l_{2}^{\prime })
& \\
(\beta _{2}\circ f_{2})(l_{2}) & = & (\beta _{2}\circ f_{2})(l_{2}^{\prime })
& \\
f_{2}(l_{2})+\alpha _{2}(m_{1}) & = & f_{2}(l_{2}^{\prime })+\alpha
_{2}(m_{1}^{\prime })\text{ (2nd column is exact)} & \\
(g_{2}\circ \alpha _{2})(m_{1}) & = & (g_{2}\circ \alpha _{2})(m_{1}^{\prime
})\text{ (}g_{2}\circ f_{2}=0\text{)} & \\
(\alpha _{3}\circ g_{1})(m_{1}) & = & (\alpha _{3}\circ g_{1})(m_{1}^{\prime
}) & \\
g_{1}(m_{1}) & = & g_{1}(m_{1}^{\prime })\text{ (}\alpha _{3}\text{ is
injective)} & \\
m_{1}+f_{1}(l_{1}) & = & m_{1}^{\prime }+f_{1}(l_{1}^{\prime })\text{ (1st
row is exact)} & \\
\alpha _{2}(m_{1})+(\alpha _{2}\circ f_{1})(l_{1}) & = & \alpha
_{2}(m_{1}^{\prime })+(\alpha _{2}\circ f_{1})(l_{1}^{\prime }) & \\
\alpha _{2}(m_{1})+(f_{2}\circ \alpha _{1})(l_{1}) & = & \alpha
_{2}(m_{1}^{\prime })+(f_{2}\circ \alpha _{1})(l_{1}^{\prime }) & \\
f_{2}(l_{2})+\alpha _{2}(m_{1})+(f_{2}\circ \alpha _{1})(l_{1}) & = &
f_{2}(l_{2})+\alpha _{2}(m_{1}^{\prime })+(f_{2}\circ \alpha
_{1})(l_{1}^{\prime }) & \\
f_{2}(l_{2}^{\prime })+\alpha _{2}(m_{1}^{\prime })+(f_{2}\circ \alpha
_{1})(l_{1}) & = & f_{2}(l_{2})+\alpha _{2}(m_{1}^{\prime })+(f_{2}\circ
\alpha _{1})(l_{1}^{\prime }) & \\
f_{2}(l_{2}^{\prime }+\alpha _{1}(l_{1})) & = & f_{2}(l_{2}+\alpha
_{1}(l_{1}^{\prime }))\text{ (}\alpha _{2}\text{ is cancellative)} & \\
l_{2}^{\prime }+\alpha _{1}(l_{1}) & = & l_{2}+\alpha _{1}(l_{1}^{\prime }
\text{ (}f_{2}\text{ is injective)} & \\
\beta _{1}(l_{2}^{\prime }) & = & \beta _{1}(l_{2})\text{ (}\beta _{1}\circ
\alpha _{1}=0\text{)} & \\
l_{3}^{\prime } & = & l_{3}.\blacksquare &
\end{array
\newline
\end{equation*}
\end{enumerate}
\end{Beweis}
\begin{proposition}
\label{9}\emph{(The Nine Lemma) }Consider the following commutative diagram
with exact columns and assume that the second row is exact, $\alpha
_{2},f_{2}$ are cancellative and $f_{3},g_{1}$ are $i$-unifor
\begin{equation*}
\xymatrix{ & 0 \ar@{.>}[d] & 0 \ar[d] & 0 \ar[d] & \\ 0 \ar@{.>}[r] & L_1
\ar[r]^{f_1} \ar[d]_{\alpha_1} & M_1 \ar[r]^{g_1} \ar[d]_{\alpha_2} & N_1
\ar[r] \ar[d]_{\alpha_3} & 0 \\ 0 \ar[r] & L_2 \ar[r]^{f_2} \ar[d]_{\beta_1}
& M_2 \ar[r]^{g_2} \ar[d]_{\beta_2} & N_2 \ar[r] \ar[d]_{\beta_3} & 0 \\ 0
\ar[r] & L_3 \ar[r]^{f_3} \ar[d] & M_3 \ar[r]^{g_3} \ar[d] & N_3
\ar@{-->}[r] \ar@{-->}[d] & 0 \\ & 0 & 0 & 0}
\end{equation*
Then the first row is exact if and only if the third row is exact.
\end{proposition}
\begin{proposition}
\label{snake}\emph{(The Snake Lemma) }Consider the following diagram of
semimodules in which the two middle squares are commutative and the two
middle rows are exact. Assume also that the columns are exact (or more
generally that $\alpha _{1},\alpha _{3}$ are $k$-uniform and $\alpha _{2}$
is uniform
\begin{equation*}
\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] & \\ & {\rm Ker}(\alpha_1)
\ar[d]_{{\rm ker}(\alpha_1)} \ar@{.>}[r]^{f_K} & {\rm Ker}(\alpha_2)
\ar[d]_{{\rm ker}(\alpha_2)} \ar@{.>}[r]^{g_K} & {\rm Ker}(\alpha_3)
\ar[d]_{{\rm ker}(\alpha_3)} \ar@{-->}[dddll]^{\delta} & \\ & L_1
\ar[r]^{f_1} \ar[d]_{\alpha_1} & M_1 \ar[r]^{g_1} \ar[d]_{\alpha_2} & N_1
\ar[r] \ar[d]_{\alpha_3} & 0 \\ 0 \ar[r] & L_2 \ar[r]^{f_2} \ar[d]_{{\rm
coker}(\alpha_1)} & M_2 \ar[r]^{g_2} \ar[d]_{{\rm coker}(\alpha_2)} & N_2
\ar[d]_{{\rm coker}(\alpha_3)} & \\ & {\rm Coker}(\alpha_1)
\ar@{.>}[r]_{f_C} \ar[d] & {\rm Coker}(\alpha_2) \ar@{.>}[r]_{g_C} \ar[d] &
{\rm Coker}(\alpha_3) \ar[d] & \\ & 0 & 0 & 0}
\end{equation*}
\begin{enumerate}
\item There exist unique morphisms $f_{K},g_{K},f_{C}$ and $g_{C}$ which
extend the diagram commutatively.
\item If $f_{1}$ is cancellative, then the first row is exact.
\item If $f_{C}$ is $i$-uniform, then the last row is exact.
\item There exists a $k$-uniform \emph{connecting morphism} $\delta :\mathrm
Ker}(\alpha _{3})\longrightarrow \mathrm{Coker}(\alpha _{1})$ such that
\mathrm{Ker}(\delta )=\overline{g_{K}(\mathrm{Ker}(\alpha _{2}))}$ and
\delta (\mathrm{Ker}(\alpha _{3}))=\mathrm{Ker}(f_{C}).$
\item If $\alpha _{2}$ is cancellative and $g_{K}$ is $i$-uniform, then the
following sequence is exac
\begin{equation*}
\xymatrix{ & {\rm Ker}(\alpha_2) \ar@{.>}[r]^{g_K} & {\rm Ker}(\alpha_3)
\ar@{-->}[r]^{\delta} & {\rm Coker}(\alpha_1) \ar@{.>}[r]^{f_C} & {\rm
Coker}(\alpha_2) &}
\end{equation*}
\end{enumerate}
\end{proposition}
\begin{Beweis}
\begin{enumerate}
\item The existence and uniqueness of the morphisms $f_{K},g_{K},f_{C}$ and
g_{C}$ is guaranteed by the definition of the (co)kernels and the
commutativity of the middle two squares.
\item This follows from Lemma \ref{9-1} applied to the first three rows.
\item This follows from Lemma \ref{9-3} applied to the last three rows.
\item We show first that $\delta $ exists and is well-defined.
\begin{itemize}
\item We define $\delta $ as follows. Let $k_{3}\in \mathrm{Ker}(\alpha
_{3}).$ Choose $m_{1}\in M_{1}$ and $l_{2}\in L_{2}$ such that
g_{1}(m_{1})=k_{3}$ and $f_{2}(l_{2})=\alpha _{2}(m_{1});$ notice that this
is possible since $g_{1}$ is surjective and $(g_{2}\circ \alpha
_{2})(m_{1})=(\alpha _{3}\circ g_{1})(m_{1})=\alpha _{3}(k_{3})=0$ whence
\alpha _{2}(m_{1})\in \mathrm{Ker}(g_{2})=f_{2}(L_{2}).$ Define $\delta
(k_{3}):=\mathrm{coker}(\alpha _{1})(l_{2})=[l_{2}],$ the coset of
L_{2}/\alpha _{1}(L_{1})$ which contains $l_{2}.$
\item $\delta $ is well-defined, i.e. $\delta (k_{3})$ is independent of our
choice of $m_{1}\in M_{1}$ and $l_{2}\in L_{2}$ satisfying the stated
conditions.
Suppose that $g_{1}(m_{1})=k_{3}=g_{1}(m_{1}^{\prime }).$ Since the second
row is exact, there exist $l_{1},l_{1}^{\prime }\in L_{1}$ such that
m_{1}+f_{1}(l_{1})=m_{2}+f_{1}(l_{1}^{\prime }).$ It follows tha
\begin{equation*}
\begin{array}{rclc}
\alpha _{2}(m_{1})+(\alpha _{2}\circ f_{1})(l_{1}) & = & \alpha
_{2}(m_{1}^{\prime })+(\alpha _{2}\circ f_{1})(l_{1}^{\prime }) & \\
f_{2}(l_{2})+(f_{2}\circ \alpha _{1})(l_{1}) & = & f_{2}(l_{2}^{\prime
})+(f_{2}\circ \alpha _{1})(l_{1}^{\prime }) & \\
f_{2}(l_{2}+\alpha _{1}(l_{1})) & = & f_{2}(l_{2}^{\prime }+\alpha
_{1}(l_{1}^{\prime })) & \\
l_{2}+\alpha _{1}(l_{1}) & = & l_{2}^{\prime }+\alpha _{1}(l_{1}^{\prime })
& \text{(}f_{2}\text{ is injective)} \\
\lbrack l_{2}] & = & [l_{2}^{\prime }] &
\end{array
\end{equation*
Thus $l_{2}$ and $l_{2}^{\prime }$ lie in the same coset of $L_{2}/\alpha
_{1}(L_{1}),$ \emph{i.e.} $\delta $ is well-defined.
\item Clearly $\overline{g_{K}(\mathrm{Ker}(\alpha _{2}))}\subseteq \mathrm
Ker}(\delta )$ (notice that $f_{2}$ is a semi-monomorphism). We claim that
\overline{g_{K}(\mathrm{Ker}(\alpha _{2}))}=\mathrm{Ker}(\delta ).$
Suppose that $k_{3}\in \mathrm{Ker}(\delta )$ for some $k_{3}\in \mathrm{Ker
(\alpha _{3}).$ Let $m_{1}\in M_{1}$ be such that $g_{1}(m_{1})=k_{3}$ and
consider $l_{2}\in L_{2}$ such that $f_{2}(l_{2})=\alpha _{2}(m_{1}).$ By
assumption, $[l_{2}]=\delta (k_{3})=0,$ \emph{i.e.} $l_{2}+\alpha
_{1}(l_{1})=\alpha _{1}(l_{1}^{\prime })$ for some $l_{1},l_{1}^{\prime }\in
L_{1}.$Then we hav
\begin{equation*}
\begin{array}{rclc}
f_{2}(l_{2})+(f_{2}\circ \alpha _{1})(l_{1}) & = & (f_{2}\circ \alpha
_{1})(l_{1}^{\prime }) & \\
\alpha _{2}(m_{1})+\alpha _{2}(f_{1}(l_{1})) & = & \alpha
_{2}(f_{1}(l_{1}^{\prime })) & \\
m_{1}+f_{1}(l_{1})+k_{2} & = & f_{1}(l_{1}^{\prime })+k_{2}^{\prime } &
\text{(}\alpha _{2}\text{ is }k\text{-uniform)} \\
k_{3}+g_{K}(k_{2}) & = & g_{K}(k_{2}^{\prime }) & \text{(}g_{1}\circ f_{1}=
\text{)
\end{array
\end{equation*
Consequently, $\overline{g_{K}(\mathrm{Ker}(\alpha _{2}))}=\mathrm{Ker
(\delta ).$
\item Notice that for any $k_{3}\in \mathrm{Ker}(\alpha _{3}),$ we have
(f_{C}\circ \delta )(k_{3})=f_{C}([l_{2}])$ where $g_{1}(m_{1})=k_{3}$ and
f_{2}(l_{2})=\alpha _{2}(m_{1}).$ It follows that
\begin{equation*}
(f_{C}\circ \delta )(k_{3})=f_{C}(l_{2})=[f_{2}(l_{2})]=[\alpha
_{2}(m_{1})]=[0].
\end{equation*
Consequently, $\delta (\mathrm{Ker}(\alpha _{3}))\subseteq \mathrm{Ker
(f_{C}).$ We claim that $\delta (\mathrm{Ker}(\alpha _{3}))=\mathrm{Ker
(f_{C}).$
Let $[l_{2}]\in \mathrm{Ker}(f_{2}),$ \emph{i.e.}
[f_{2}(l_{2})]=f_{C}([l_{2}])=[0],$ for some $l_{2}\in L_{2}.$ Then there
exist $m_{1},m_{1}^{\prime }\in M_{1}$ such that $f_{2}(l_{2})+\alpha
_{2}(m_{1})=\alpha _{2}(m_{1}^{\prime }).$ By assumption, $\alpha _{2}$ is
i $-uniform, whence there exists $\mathbf{m}_{1}\in M_{1}$ such that $\alpha
_{2}(\mathbf{m}_{1})=f_{2}(l_{2}).$ It follows that $(\alpha _{3}\circ
g_{1})(\mathbf{m}_{1})=(g_{2}\circ \alpha _{2})(\mathbf{m}_{1})=(g_{2}\circ
f_{2})(l_{2})=0.$ So, $g_{1}(\mathbf{m}_{1})\in \mathrm{Ker}(\alpha _{3})$
and $\delta (g_{1}(\mathbf{m}_{1}))=[l_{2}].$ Consequently, $\mathrm{Ker
(f_{C})=\delta (\mathrm{Ker}(\alpha _{3})).$
\item We claim that $\delta $ is $k$-uniform.
Suppose that $\delta (k_{3})=\delta (k_{3}^{\prime })$ for some
k_{3},k_{3}^{\prime }\in \mathrm{Ker}(\alpha _{3}).$ Let
m_{1},m_{1}^{\prime }\in M_{1}$ and $l_{2},l_{2}^{\prime }\in L_{2}$ be such
that $g_{1}(m_{1})=k_{3},$ $g_{1}(m_{1}^{\prime })=k_{3}^{\prime },$ $\alpha
_{2}(m_{1})=f_{2}(l_{2})$ and $\alpha _{2}(m_{1}^{\prime
})=f_{2}(l_{2}^{\prime }).$ By assumption, $[l_{2}]=[l_{2}^{\prime }],$
\emph{i.e. }$l_{2}+\alpha _{1}(l_{1})=l_{2}+\alpha _{1}(l_{1}^{\prime })$
for some $l_{1},l_{1}^{\prime }\in L_{1}.$ Notice tha
\begin{equation*}
\begin{array}{rclc}
f_{2}(l_{2})+(f_{2}\circ \alpha _{1})(l_{1}) & = & f_{2}(l_{2}^{\prime
})+(f_{2}\circ \alpha _{1})(l_{1}^{\prime }) & \\
\alpha _{2}(m_{1})+(\alpha _{2}\circ f_{1})(l_{1}) & = & \alpha
_{2}(m_{2}^{\prime })+(\alpha _{2}\circ f_{1})(l_{1}^{\prime }) & \\
m_{1}+f_{1}(l_{1})+k_{2} & = & m_{1}^{\prime }+f_{1}(l_{1}^{\prime
})+k_{2}^{\prime } & \text{(}\alpha _{2}\text{ is }k\text{-uniform)} \\
g_{1}(m_{1})+g_{K}(k_{2}) & = & g_{1}(m_{1}^{\prime })+g_{K}(k_{2}^{\prime })
& \text{(}g_{1}\circ f_{1}=0\text{)} \\
k_{3}+g_{K}(m_{1}) & = & k_{3}^{\prime }+g_{K}(m_{1}) &
\end{array
\end{equation*
\qquad Since $g_{K}(\mathrm{Ker}(\alpha _{2}))\subseteq \mathrm{Ker}(\delta
) $ we conclude that $\delta $ is $k$-uniform.
\end{itemize}
\item If $g_{K}$ is $i$-uniform, then we have $\mathrm{Ker}(\delta )
\overline{g_{K}(\mathrm{Ker}(\alpha _{2}))}=g_{K}(\mathrm{Ker}(\alpha _{2}))$
and it remains only to prove that $f_{C}$ is $k$-uniform.
Suppose that $f_{C}[l_{2}]=f_{C}[l_{2}^{\prime }]$ for some
l_{2},l_{2}^{\prime }\in L_{2}.$ Then there exist $m_{1},m_{1}^{\prime }\in
M_{1}$ such that $f_{2}(l_{2})+\alpha _{2}(m_{1})=f_{2}(l_{2}^{\prime
})+\alpha _{2}(m_{1}^{\prime }).$ It follows tha
\begin{equation*}
\begin{array}{rclc}
(g_{2}\circ \alpha _{2})(m_{1}) & = & (g_{2}\circ \alpha _{2})(m_{1}^{\prime
})\text{ (}g_{2}\circ f_{2}=0\text{)} & \\
(\alpha _{3}\circ g_{1})(m_{1}) & = & (\alpha _{3}\circ g_{1})(m_{1}^{\prime
}) & \\
g_{1}(m_{1})+k_{3} & = & g_{1}(m_{1}^{\prime })+k_{3}^{\prime }\text{ (
\alpha _{3}\text{ is }k\text{-uniform)} & \\
g_{1}(m_{1}+\mathbf{m}_{1}) & = & g_{1}(m_{1}^{\prime }+\mathbf{m
_{1}^{\prime })\text{ (}g_{1}\text{ is surjective)} & \\
m_{1}+\mathbf{m}_{1}+f_{1}(\mathbf{l}_{1}) & = & m_{1}^{\prime }+\mathbf{m
_{1}+f_{1}(\mathbf{l}_{1}^{\prime })\text{ (2nd row is exact)} & \\
\alpha _{2}(m_{1})+\alpha _{2}(\mathbf{m}_{1})+(\alpha _{2}\circ f_{1})
\mathbf{l}_{1}) & = & \alpha _{2}(m_{1}^{\prime })+\alpha _{2}(\mathbf{m
_{1}^{\prime })+(\alpha _{2}\circ f_{1})(\mathbf{l}_{1}^{\prime }) & \\
f_{2}(l_{2}^{\prime })+\alpha _{2}(m_{1})+\alpha _{2}(\mathbf{m
_{1})+(f_{2}\circ \alpha _{1})(\mathbf{l}_{1}) & = & [f_{2}(l_{2}^{\prime
})+\alpha _{2}(m_{1}^{\prime })]+\alpha _{2}(\mathbf{m}_{1}^{\prime
})+(f_{2}\circ \alpha _{1})(\mathbf{l}_{1}^{\prime }) & \\
f_{2}(l_{2}^{\prime })+\alpha _{2}(m_{1})+\alpha _{2}(\mathbf{m
_{1})+(f_{2}\circ \alpha _{1})(\mathbf{l}_{1}) & = & f_{2}(l_{2})+\alpha
_{2}(m_{1})+\alpha _{2}(\mathbf{m}_{1}^{\prime })+(f_{2}\circ \alpha _{1})
\mathbf{l}_{1}^{\prime }) & \\
f_{2}(l_{2}^{\prime })+\alpha _{2}(\mathbf{m}_{1})+(f_{2}\circ \alpha _{1})
\mathbf{l}_{1}) & = & f_{2}(l_{2})+\alpha _{2}(\mathbf{m}_{1}^{\prime
})+(f_{2}\circ \alpha _{1})(\mathbf{l}_{1}^{\prime })\text{ (}\alpha _{2
\text{ is cancellative)} & \\
f_{2}(l_{2}^{\prime }+\mathbf{l}_{2}+\alpha _{1}(\mathbf{l}_{1})) & = &
f_{2}(l_{2}+\mathbf{l}_{2}^{\prime }+\alpha _{1}(\mathbf{l}_{1}^{\prime })
\text{ (third row is exact)} & \\
l_{2}^{\prime }+\mathbf{l}_{2}+\alpha _{1}(\mathbf{l}_{1}) & = & l_{2}
\mathbf{l}_{2}^{\prime }+\alpha _{1}(\mathbf{l}_{1}^{\prime })\text{ (}f_{2
\text{ is injective)} & \\
\lbrack l_{2}^{\prime }]+[\mathbf{l}_{2}] & = & [l_{2}]+[\mathbf{l
_{2}^{\prime }] & \\
\lbrack l_{2}^{\prime }]+\delta (k_{3}) & = & [l_{2}]+\delta (k_{3}^{\prime
}) &
\end{array
\end{equation*
Since $\delta (\mathrm{Ker}(\alpha _{3}))\subseteq \mathrm{Ker}(f_{C}),$ we
conclude that $f_{C}$ is $k$-uniform.$\blacksquare $
\end{enumerate}
\end{Beweis}
\textbf{Acknowledgments.} The author thanks all mathematicians who clarified
to him some issues related to the nature of the categories of semimodules
and exact sequences or sent to him related manuscripts especially F. Linton,
G. Janelidze, Y. Katsov, A. Patchkoria, H. Porst and R. Wisbauer.
|
2,869,038,156,373 | arxiv | \section{introduction}
Recently, a scheme for inducing gravitation-like interatomic
potentials has been proposed \cite{DODell2000}, opening up the
possibility to create self-bound Bose-Einstein condensates (BECs)
\cite{DODell2001,DODell2002,Choi,Hu,Giovanazzi}. In such a scheme, the
gravitation-like interatomic potential can be
achieved by irradiating the atoms with intense, extremely off-resonant
electromagnetic fields. The usual strong anisotropy due to
dipole-dipole interactions can, in fact, be averaged out \cite{Thiru}
by the proper combination of laser beams, leaving a $-u/r$ potential,
where $u$ is the strength of the potential (the analogue of $GMm$,
where $G$ is the Newton's constant and $M$ and $m$ are the masses) and
$r$ is the interparticle distance. The strength $u$ of the potential
can be adjusted by changing the laser intensity \cite{DODell2000}.
In ultracold atomic gases, two interesting regimes for self-bound BECs
have been predicted, assuming that the short-range interatomic
interactions can be independently tuned by, say, applying a magnetic
field near a Feshbach resonance \cite{Fesh}.
In one regime, the
attractive $1/r$ interactions are balanced by the repulsive mean field
interation assuming a positive two-body scattering length and
negligible kinetic energy. In the other regime, the balancing factor
is kinetic energy, assuming negligible mean field interactions. In
both regimes, the resulting BEC is self-bound.
From a broader point of view,
the induced gravitation-like interaction might make possible
experimental emulation of boson stars (a system of self-gravitating
bosons) in the regime where the kinetic energy balances the $-u/r$
potential \cite{Wang,IMMoroz}. Moreover, purely attractive $1/r$
potentials constitute an interesting contrast to the attractive {\em and}
repulsive Coulomb potentials atomic physics are used to.
The existence of a lower bound for the ground state energy in
many-body systems interacting through attractive $1/r$ potentials is
of fundamental importance in order to prove the existence of the
thermodynamical limit and the stability of normal matter
\cite{Fisher}. It was shown in Ref.~\cite{JLBasdevant} that for a
system of $N$ identical, spinless (or spin stretched)
bosons of mass $m$ interacting gravitationally, the lower and upper
bounds for the ground state energy are, respectively,
$-\frac{1}{16}N^2(N-1)\:G^2m^5/\hbar^2$ and
$-0.0542N(N-1)^2\:G^2m^5/\hbar^2$, where the upper bound was obtained
variationally.
For small $N$, however, the discrepancy between the
lower and upper bounds becomes large. For $N=3$, using a more
refined trial function, they obtained $-0.95492\:G^2m^5/\hbar^2$
for the upper bound, representing a difference of about
$15\%$ between the upper and lower ($-\frac{9}{8}\:G^2m^5/\hbar^2$),
bounds and a ground state energy equal to
$E_{0}\cong -1.067\:G^2m^5/\hbar^2$.
In this paper, we have used the adiabatic hyperspherical
representation for this systm to obtain effective three-body
potentials, the corresponding channel functions, and the nonadiabatic
couplings. Using these, we calculate
the ground state and low-lying $0^{+}$ excited state energies
converged to seven digits. We have also used the adiabatic
hyperspherical representation to obtain lower and upper bounds that
differ by about $0.1\%$, indicating that a simple single-channel
description offers a quite accurate description of such systems.
\section{The adiabatic hyperspherical representation}
We have solved the Schr\"odinger equation in hyperspherical
coordinates. After separation of the center-of-mass motion, the system
is described by the hyperradius $R$ which gives the
overall size; three Euler angles $\alpha$, $\beta$ and
$\gamma$, specifying the orientation of the plane containing the three
particles relative to the space-fixed frame; and other two hyperangles
$\varphi$ and $\theta$, describing the internal relative motion
between the particles. We have defined $\varphi$ and $\theta$ as a
modification of Smith-Whitten coordinates
\cite{Esry-02,Esry-03,Smith}.
The key to the adiabatic hyperspherical representation is that the
dynamics of the three-body system is reduced to collective motion
under the influence of one-dimensional effective potentials in $R$,
which is governed by a system of ordinary differential equations.
The hyperspherical coordinates are introduced through the mass-scaled
Jacobi coordinates $\vec{\rho}_{1}$ and $\vec{\rho}_{2}$ (see
Fig. \ref{JacobiCoords}) defined as
\begin{eqnarray}
&&\vec{\rho}_{1}=(\vec{r}_{2}-\vec{r}_{1})/d, \nonumber \\
&&\vec{\rho}_{2}=d\left(\vec{r}_{3}
-\frac{m_{1}\vec{r}_{1}+m_{2}\vec{r}_{2}}{m_{1}+m_{2}}\right).
\end{eqnarray}
\noindent
In the above equations $\vec{r}_{i}$ is the position of the particle $i$
(of mass $m_{i}$) relative to a space-fixed frame. For three identical
particles of mass $m$, we define a three-body reduced mass as
$\mu=m/\sqrt{3}$ which gives $d=2^{1/2}/3^{1/4}$ \cite{Esry-02,Esry-03}. It is
important to note that the hyperradius,
\begin{equation}
R^2=\rho_{1}^2+\rho_{2}^2, \hspace{0.25in}R\in[0,\infty),
\end{equation}
\noindent
is an invariant quantity, i.e., it does not depend on the particular
choice of the hyperangles or labels of the particles.
\begin{figure}
\includegraphics[width=2.in,angle=0,clip=true]{Fig.01.arXiv.ps}
\caption{The mass-scaled Jacobi coordinates for systems with three
particles.}
\label{JacobiCoords}
\end{figure}
The Schr\"odinger equation can be more conveniently written in terms
of the rescaled wave function $\psi=R^{5/2}\Psi$, as
\begin{eqnarray}
\left[-\frac{\hbar^2}{2\mu}\frac{\partial^2}{\partial R^2}
+H_{\rm ad}(R,\Omega)\right]\psi(R,\Omega)=E\psi(R,\Omega),\label{schr}
\end{eqnarray}
\noindent
where $E$ is the total energy and $H_{\rm ad}$ is the adiabatic
Hamiltonian given by
\begin{eqnarray}
H_{\rm ad}(R,\Omega)=\frac{\hbar^2}{2\mu R^2}
\left[{\Lambda^{2}(\Omega)+\frac{15}{4}}\right]+V(R,\varphi,\theta).
\label{had}
\end{eqnarray}
\noindent
The adiabatic Hamiltonian $H_{\rm ad}$
contains all hyperangular dependence, represented collectively by
$\Omega\equiv\{\varphi,\theta,\alpha,\beta,\gamma\}$, and includes the
hyperangular kinetic energy in the grand angular momentum operator
$\Lambda^{2}$ as well as all interparticle interactions $V$.
In the adiabatic hyperspherical representation, the total wave
function is expanded in terms of the channel functions
$\Phi_{\nu}(R;\Omega)$,
\begin{equation}
\psi_{n}(R,\Omega)=\sum_{\nu}F_{n\nu}(R)\Phi_{\nu}(R;\Omega),
\label{chfun}
\end{equation}
\noindent
where $F_{n\nu}(R)$ are the hyperradial wavefunctions, $n$ labels the
different energy eigenstates for a given $\nu$, and $\nu$ represents
all remaining quantum numbers necessary to specify each
channel. The channel functions $\Phi_{\nu}(R;\Omega)$ form a complete
set of orthonormal functions at each value of $R$ and are
eigenfunctions of the adiabatic Hamiltonian:
\begin{equation}
H_{\rm ad}(R,\Omega)\Phi_{\nu}(R;\Omega)
=U_{\nu}(R)\Phi_{\nu}(R;\Omega).\label{poteq}
\end{equation}
\noindent
The eigenvalues $U_{\nu}(R)$ help define
effective three-body potentials
for the hyperradial motion.
Substituting Eq.~(\ref{chfun}) into the Schr\"odinger equation
(\ref{schr}) and projecting out $\Phi_{\nu'}$,
we obtain the hyperradial Schr\"odinger equation
\begin{widetext}
\begin{eqnarray}
\left[-\frac{\hbar^2}{2\mu}\frac{d^2}{dR^2}+U_{\nu}(R)\right]F_{\nu}(R)
-\frac{\hbar^2}{2\mu}\sum_{\nu'}
\left[2P_{\nu\nu'}(R)\frac{d}{dR}+Q_{\nu\nu'}(R)\right]F_{\nu'}(R)
=EF_{\nu}(R),\label{radeq}
\end{eqnarray}
\end{widetext}
\noindent
which describes the motion of the three-body system under
the influence of the effective potentials
$U_{\nu}(R)-Q_{\nu\nu}(R)/2\mu$. The nonadiabatic coupling terms
$P_{\nu\nu'}(R)$ and $Q_{\nu\nu'}(R)$ drive inelastic collisions
three-body scattering processes and are defined as
\begin{eqnarray}
P_{\nu\nu'}(R) &=&
\Big\langle\hspace{-0.15cm}\Big\langle\Phi_{\nu}\Big|
\frac{d}{dR}\Big|\Phi_{\nu'}\Big\rangle\hspace{-0.15cm}\Big\rangle
\label{puv}
\end{eqnarray}
\noindent
and
\begin{eqnarray}
Q_{\nu\nu'}(R) &=&
\Big\langle\hspace{-0.15cm}\Big\langle\Phi_{\nu}\Big|
\frac{d^2}{dR^2}\Big|\Phi_{\nu'}\Big\rangle\hspace{-0.15cm}\Big\rangle,
\label{quv}
\end{eqnarray}
\noindent
where the double brackets denote integration over the angular
coordinates $\Omega$ only. As it stands, Eq.~(\ref{radeq}) is
exact. In practice, of course, the sum over channels must be
truncated. In fact, the accuracy of the solutions can be monitored
with successively larger truncations since the bound state
energies obtained at each stage are an upper bound by the variational
principle.
In this paper, we explore the solutions of the system of differential
equations (\ref{radeq}) for three particles with attractive $1/r$
interactions. We determine the effective potentials and couplings by
solving Eq.~(\ref{poteq}) for the $J^{\pi}=0^{+}$ symmetry, where $J$
is the total orbital angular momentum and $\pi$ is the total
parity. The low-lying bound state energies are then determined by
solving Eq.~(\ref{radeq}).
In order to solve the adiabatic equation (\ref{poteq}), we have
expanded the channel functions $\Phi_{\nu}(R;\Omega)$ in terms of the
Wigner $D$ functions \cite{Esry-02,Parker1987,RoseBook},
\begin{equation}
\Phi_{\nu}^{JM\pi}(R;\Omega)=\sum_{K}\phi_{K\nu}(R;\theta,\varphi)
D^J_{KM}(\alpha,\beta,\gamma),
\end{equation}
where $K$ and $M$ are the projection of $\vec{J}$ onto the body-fixed
and space-fixed $z-$axes, respectively. After projecting out the $D$
functions, the resulting coupled system of partial differential
equations for $\phi_{K\nu}(R;\theta,\varphi)$ is solved (for each value
of $R$) by expanding $\phi_{K\nu}(R;\theta,\varphi)$ on a direct product
of fifth order basis splines \cite{Boor1978} in the hyperangles
$\theta$ and $\varphi$ \cite{Esry-02,Esry-03}. For $J^{\pi}=0^+$, of course,
the sum involves only one term, requiring the solution of a single
two-dimensional partial differential equation.
The potential $V$ in Eq.~(\ref{had}) is given by a pairwise sum of
attractive $1/r$ potentials,
\begin{equation}
V(R,\theta,\varphi)=-\frac{u}{r_{12}}-\frac{u}{r_{23}}-\frac{u}{r_{31}},
\end{equation}
where $u$ is the gravitation-like coupling. The interparticle
distances $r_{ij}$ are given in terms of the hyperspherical
coordinates by
\begin{eqnarray}
r_{12}&=&
{3^{-1/4}}{R}\left[1+\sin{\theta}\sin(\varphi-{\pi}/{6})\right]^{1/2},
\nonumber \\
r_{23}&=&
{3^{-1/4}}{R}\left[1+\sin{\theta}\sin(\varphi-5{\pi}/{6})\right]^{1/2},
\nonumber \\
r_{31}&=&
{3^{-1/4}}{R}\left[1+\sin{\theta}\sin(\varphi+{\pi}/{2})\right]^{1/2}.
\end{eqnarray}
Figure~\ref{Potential} shows the potential $V(R,\theta,\varphi)$ as a
function of $\theta$ and $\varphi$ at
$R=100$. The singular points at $\theta=\pi/2$ and
$\varphi=\pi/3$, $\pi$ and $5\pi/3$ are the points where $r_{23}=0$,
$r_{31}=0$ and $r_{12}=0$, respectively. Notice that our choice for
the hyperangles $\theta$ and $\varphi$ \cite{Esry-02,Esry-03} --- like any of
the so-caller democratic or Smith-Whitten coordinates --- implies a
periodicity of the potential in $\varphi$ for identical particles,
which substantially simplifies the numerical solution of
Eq.~(\ref{poteq}) when symmetrizing the wave function. The
two-dimensional equation for $J^{\pi}=0^{+}$ thus needs to be solved
only from $\varphi=0$ to $2\pi/3$, with the requirement that the
derivative of $\phi(R;\theta,\varphi)$ with respect to $\varphi$ is
zero at each boundary, followed by postsymmetrization to extract the
completely symmetric solutions. We have treated the cusp due the
$1/r$ divergence as $r\rightarrow0$ by including more spline functions
\cite{Boor1978} at $\varphi=\pi/3$. For other $J^{\pi}$ the solutions
can be determined in a similar way \cite{Esry-02,Esry-03}.
\begin{figure}
\includegraphics[width=2.3in,angle=270,clip=true]{Fig.02.arXiv.ps}
\caption{The potential $V(R,\theta,\varphi)$ at
$R=100$. Due to the symmetry properties of the three identical
particles, the adiabatic equation [Eq.~(\ref{poteq})] is solved only for
$\theta=0$ to $\pi/2$ and $\varphi=0$ to $2\pi/3$}
\label{Potential}
\end{figure}
In the atomic case, suggested in Ref.~\cite{DODell2000}, $u$ is given
by (in S.I. units)
\begin{equation}
u=\frac{11}{4\pi}\frac{Iq^2\alpha_{p}^2}{c\varepsilon_{0}^2}
\end{equation}
\noindent
where $I$ is the laser intensity, $q$ the photon wave number, and
$\alpha_{p}$ the atomic dynamic polarizability. So, under the
conditions discussed in Ref.~\cite{DODell2000}, the strength of the
interaction is controllable via the laser's parameters. For the
present calculations, however, we base our units on $u$, thus
producing a unitless equation. Our length units are $2\hbar^2/mu$, and
our energy units are $mu^2/2\hbar^2$. These yield, in analogy to
atomic units, a two-body energy spectrum $E_{n}=-1/2n^2$.
\section{Results and Discussion}
Figure \ref{AdiaPot} shows the effective three-body potentials in the
form $[-2U_{\nu}(R)]^{-1/2}$ such that for large values of $R$ they
converge to the principal quantum number $n_{\rm 2b}$ associated with the
two-body hydrogen-like subsystems (for $R\rightarrow\infty$ one
particle is far from the others). The lowest potential in
Fig.~\ref{AdiaPot}, converging to $n_{\rm 2b}=1$, supports the $0^+$ bound
states (or $S^e$, in analogy with atomic spectroscopic notation).
In particular, it contains both the ground state and the $S^e$ series
of singly excited states, using the language of atomic structure. The
higher potentials (converging to $n_{\rm 2b}>1$) support series of
doubly excited states that are coupled to and can decay to the
continuum of the lowest potential and are thus metastable.
These comments draw on the similarity of these potentials to those for
three-body systems like He and H$^-$ \cite{CDLin}. There are minor
differences, of course, due to the different permutational symmetries
of the two systems and to the absence of the Coulomb repulsion for the
present case.
\begin{figure}
\includegraphics[width=2.8in,angle=270,clip=true]{Fig.03.arXiv.ps}
\caption{The hyperspherical potentials for three identical bosons with
attractive $1/r$ interactions. For large values of
$R$, $[-2U_{\nu}(R)]^{-1/2}$ converges to the principal
quantum number $n_{\rm 2b}$ for the hydrogen-like subsystems.}
\label{AdiaPot}
\end{figure}
Since the $J^{\pi}=0^{+}$ channel functions depend only on the
hyperangles $\theta$ and $\varphi$ they can be plotted in their
entirety for each value of $R$. Figures \ref{RadWFRminV1} and
\ref{RadWFRminV2} show the channel functions for the two lowest
channels at $R=0.69$ and $100$ for $\nu=1$, and $R=5.75$ and
$100$ for $\nu=2$. The first $R$ value lies near the respective
potential minima; and the second, in the asymptotic region.
For small $R$, we plot $\Phi_{\nu}$ in the range $0\le\varphi\le
2\pi/3$ from which the function in the whole range $0\le\varphi\le
2\pi$ can be obtained by symmetry (translation of the plotted portion
by $2\pi/3$ and $4\pi/3$). For large $R$, however, we plotted the
solution only in the range $\pi/6\le\varphi\le\pi/2$ to emphasize the
two-body character of the solution.
The hyperangular distributions are useful because they reveal the
geometry of the
system. At $\theta=0$, for instance, the atoms form an equilateral
triangle, while for $\theta=\pi/2$ they lie along a line.
Figure \ref{RadWFRminV1}(a) shows that near the $\nu=1$ potential
minimum (see Fig.~\ref{AdiaPot}) the lowest channel function is spread
out over the entire hyperangular plane with an increased
amplitude near the two-body coalescence point ($r_{31}=0$) at
$\theta=\pi/2$ and $\varphi=\pi/3$. This point corresponds to a linear
configuration with two of the particles closer to each other than to
the third particle, representing strong two-body correlations. Since
this $R$ is the equilibrium distance, the three particles in the
ground state thus assume all triangular shapes between linear and
equilateral with a preference for linear shapes having two particles
close to each other (taking into account the $sin2\theta$ volume
element). As $R$ increases [Fig.~\ref{RadWFRminV1}(b)] the
channel function ``collapses'' to the region around the coalescence
points, displaying the two-body character it must have for
$R\rightarrow\infty$ --- in this case the $1s$ state. Figures
\ref{RadWFRminV2}(a) and \ref{RadWFRminV2}(b) show the $\nu=2$ channel
function. They show much the same behavior as $\nu=1$ except with the
necessary addition of a node. Figure \ref{RadWFRminV2}(b) shows that
$\nu=2$ converges to the $2s$ state of the two-body subsystem. Note
that for identical spinless (or spin-stretched) bosons, $2p$ two-body
states are not allowed by symmetry, so there is only a single
potential correlating to $n_{\rm 2b}=2$ at $R\rightarrow\infty$.
\begin{figure}
\includegraphics[width=2.3in,angle=270,clip=true]{Fig.04.a.arXiv.ps}
\includegraphics[width=2.3in,angle=270,clip=true]{Fig.04.b.arXiv.ps}
\caption{The lowest $J^{\pi}=0^+$ channel function ($\nu =1$) as a
function of $\theta$ and $\varphi$ at (a) $R=0.69$ and (b)
$R=100$.}
\label{RadWFRminV1}
\end{figure}
\begin{figure}
\includegraphics[width=2.3in,angle=270,clip=true]{Fig.05.a.arXiv.ps}
\includegraphics[width=2.3in,angle=270,clip=true]{Fig.05.b.arXiv.ps}
\caption{The first excited $J^{\pi}=0^+$ channel function ($\nu =2$)
as a function of $\theta$ and $\varphi$ at (a) $R=5.75$ and (b)
$R=100$.}
\label{RadWFRminV2}
\end{figure}
In Figure~\ref{Coupling} we show the nonadiabatic couplings
$P_{\nu\nu'}(R)$ between the lowest channel and the next three
channels ($P_{\nu\nu}=0$ and $P_{\nu\nu'}=-P_{\nu'\nu}$) calculated
from Eq.~(\ref{quv}). The fact that the coupling $P_{12}$ in
Fig.~\ref{Coupling} is substantially larger than the couplings with
higher channels implies rapid convergence for the bound state energies
as a function of the number of channels. Further, $P_{12}$ peaks
around $R=6$ which correlates roughly with the location of an avoided
crossing between the corresponding potential curves as expected (see
Fig.\ref{AdiaPot}).
\begin{figure}
\includegraphics[width=2.25in,angle=270,clip=true]{Fig.06.arXiv.ps}
\caption{Nonadiabatic coupling $P_{\nu \nu \prime}$ between the
lowest channel and the next three channels.}
\label{Coupling}
\end{figure}
We have solved Eq.~(\ref{radeq}) and determined the bound state
energies and hyperradial wavefunctions $F_{n\nu}(R)$ including up to
15 channels. For example, Figure~\ref{WF} shows the ground state wave
function for a three channel calculation.
\begin{figure}
\includegraphics[width=2.25in,angle=270,clip=true]{Fig.07.arXiv.ps}
\caption{First three components of the ground state hyperradial
wavefunction $F_{n\nu}$ ($n=1$) associated with the first three
channels $\nu=1$, $2$, and $3$.}
\label{WF}
\end{figure}
The $\nu=1$ component is associated with the lowest adiabatic channel,
while the $\nu=2$ and $\nu=3$ are related to the next two adiabatic
channels and are present due to the coupling between the
channels. Note that the probability given by
${\cal P}_{n\nu}=\int_{0}^{\infty}|F_{n\nu}(R)|^2dR$ due to the first
term dominates the other contributions. In Table~\ref{ProRadialWF} we
show ${\cal P}_{n\nu=1}$ for the ground state ($n=1$) and
the four lowest excited states for a 15 channel calculation. The
$\nu=1$ adiabatic channel represents roughly $99\%$ or more of the
probability for each state, showing that the adiabatic expansion is,
in fact, quite good.
\begin{table}[htbp]
\begin{ruledtabular}
\caption{The probability ${\cal P}_{n\nu}$ associated with the $\nu=1$
term of the expansion (\ref{chfun}) for the ground state ($n=1$) and
the next four excited states.\label{ProRadialWF}}
\begin{tabular}{cccc}
$n$ & ${\cal P}_{n1}=\int_{0}^{\infty}|F_{n1}|^2dR$ \\ \hline
1 & 99.98$\%$ \\
2 & 99.80$\%$ \\
3 & 99.15$\%$ \\
4 & 98.90$\%$ \\
5 & 99.21$\%$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
In Table~\ref{EnergyChannels} we show the ground state energy as a
function of the number of channels, further demonstrating the expected
rapid convergence of the adiabatic expansion. The ground state energy
for this system has certanly been calculated before. In
Ref.~\cite{JLBasdevant}, for instance, a ground
state energy of $E_0\cong -1.067\:G^2m^5/\hbar^2=-2.134\:mu^2/\hbar^2$
(in our units) was obtained. Comparison with
Table~\ref{EnergyChannels} shows that since both calculations are
variational, our single channel calculation already gives a more
precise result. We speculate that the large differences in the
potential energies shown in Fig.~\ref{AdiaPot} are the main reason
that a single channel already gives such a good result. In fact, this
channel separation is closely related to
the small magnitude of the coupling terms shown in Fig.~\ref{Coupling}.
Table~\ref{EnergyChannels} also shows that our six channel
approximation for the ground state gives a result converged to seven
digits.
\begin{table}[htbp]
\caption{Convergence of the ground state energy as a function of the
number of channels included in Eq.~(\ref{radeq}).\label{EnergyChannels}}
\begin{ruledtabular}
\begin{tabular}{cccc}
Number of channels & Ground state energy ($\:mu^2/\hbar^2$) & \\ \hline
1 & -2.136\:033 & \\
2 & -2.136\:481 & \\
3 & -2.136\:523 & \\
4 & -2.136\:525 & \\
5 & -2.136\:526 & \\
6 & -2.136\:527 & \\
15 & -2.136\:527 & \\
\end{tabular}
\end{ruledtabular}
\end{table}
In Table~\ref{EnergyBoundn} we give our converged results for the
ground and first four excited states (calculated with 15 channels).
Is well known \cite{Starace} that the hyperspherical energy obtained
disregarding all couplings and the energy obtained considering only
the diagonal coupling in Eq.~(\ref{radeq}) are, in fact, lower and upper
bounds for the ground state energy, respectively. In these
approximations, we
obtained a lower bound corresponding to $-2.138\:650\:mu^2/\hbar^2$
and an upper bound of $-2.136\:033\:mu^2/\hbar^2$.
The difference between them is about $0.1\%$
while the results obtained in Ref.~\cite{JLBasdevant} give a
difference of about $10\%$. By comparison, for a system like the He
atom \cite{Groote}, where the electronic repulsion plays an important
rule, the relative difference between the lower and upper bounds
estimated from hyperspherical potential curves is about $1\%$.
\begin{table}[htbp]
\caption{Ground state and excited states energies $E_{n,\nu}$
($\nu=1$) calculated using 15 coupled channels.\label{EnergyBoundn}}
\begin{ruledtabular}
\begin{tabular}{cccc}
$n$ & $E_{n,\nu}$ ($\:mu^2/\hbar^2$) \\ \hline
1 & -2.136\:527 \\
2 & -1.145\:881 \\
3 & -0.786\:454 \\
4 & -0.661\:162 \\
5 & -0.603\:740 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\section{summary}
We have used the adiabatic hyperspherical representation to describe
system of three identical bosons with attractive $1/r$
potentials. Such a system might eventually be created experimentally
by irradiating ultracold atoms with intense, extremely off-resonant
lasers.
We calculated the ground state and excited state energies converged
to seven digits which represents a substantial improvement over
previous results. Our method is essentially exact, with the
only approximation being the truncation of the number of channels
used in the expansion of the total wave function.
Although other methods, such as Hylleraas
variational techniques, might provide much better
bound states energies, as we have shown here, the adiabatic
hyperspherical representation naturally offers qualitative information
along with quantitative results.
\acknowledgments
This work was supported by the National Science Foundation.
|
2,869,038,156,374 | arxiv | \section{Introduction}
Conformal field theories (CFT)~\cite{di1996conformal} often serve as laboratories for interesting phenomena in physics. High degree of symmetry imposes strong constraints not only on the observables of a given theory, but also on its self-consistency. Such constraints allow better analytic control and qualitative understanding of complex phenomena, notably in the strong coupling regime. It is sometimes also possible to generalize the CFT results and techniques beyond the conformal case.
One particular class of interesting problems successfully tackled by the CFT methods in the last couple of decades is the problems related to out-of-equilibrium dynamics of quantum systems~\cite{polkovnikov2011colloquium,eisert2015quantum,d2016quantum,gogolin2016equilibration,altman2015non,Calabrese:2016xau}. Quite often one is interested in the quantum evolution of otherwise stationary system after a quench -- a non-adiabatic change of the Hamiltonian. Besides the general importance of such problems the recent interest was also stimulated by the progress in experimental techniques, such as the control of dynamics of cold atoms~\cite{greiner2002collapse,trotzky2012probing,cheneau2012light,bloch2012quantum,blatt2012quantum,meinert2013quantum,langen2013local,islam2015measuring}, which allowed to test various theoretical model predictions.
CFT methods are particularly powerful in two dimensions. Therefore 1+1 dimensional systems turned a natural subject of the early studies. For example, it is relatively straightforward to analyze time dependence of correlation functions in the quenched systems. Some interesting features of such systems, initially observed in the CFT models, include light-cone spreading of the correlations~\cite{calabrese2006time,Calabrese:2007rg}, linear growth of the entanglement entropy~\cite{Calabrese:2005in} and quantum revivals in finite systems~\cite{Cardy:2014rqa}.
AdS/CFT correspondence~\cite{AdS/CFT,Gubser:1998bc,Witten:1998qj} provides another powerful tool to analyze complex systems, characterized by strong coupling. In low dimension this correspondence can reproduce some characteristic features of conformal theories, especially in two dimensions. A famous example is the holographic formula of Ryu and Takayanagi~\cite{Ryu:2006bv} that expresses the entanglement entropy of a subsystem in terms of the dual geometry. In 1+1 dimensional theory, the entanglement entropy of an interval of length $l$ has a universal piece~\cite{Cardy:1988tk,Holzhey:1994we,Korepin:2004zz,CC}
\begin{equation}\bea\ds
\label{RT2D}
S_{\rm E} \ = \ \frac{c}{3}\log\frac{l}{\epsilon} + \ldots\,,
\ea\end{equation}
where $c$ is the CFT central charge, $\epsilon$ is the UV cutoff and dots stand
for the non-universal constant piece. In AdS/CFT the logarithm is a length of
the geodesic line in the three-dimensional anti de Sitter space, connecting the
endpoints of the interval. AdS/CFT was also successfully applied to the
discussion of the quenched dynamics.
See~\cite{Danielsson:1999fa,Janik:2006gp,AbajoArrastia:2010yt,Albash:2010mv,
Balasubramanian:2010ce,Aparicio:2011zy,Keranen:2011xs,Allais:2011ys,
Buchel:2012gw,Buchel:2013gba,Hartman:2013qma,Liu:2013qca,Bhaseen:2013ypa,
Abajo-Arrastia:2014fma,daSilva:2014zva,Caputa:2013eka} for an
incomplete set of references. Behavior of correlators and characteristic
features of the entanglement evolution were reproduced in those studies.
In this work we will be focusing on the discussion of time-dependent dynamics from the point of view of a specific setup in the AdS/CFT correspondence introduced by Takayanagi~\cite{Takayanagi:2011zk}. This setup was dubbed AdS/BCFT as it refers to a gravity dual description of systems with boundaries, which are also amenable to treatment by means of boundary conformal field theories, or BCFTs~\cite{Cardy:1989ir,Cardy:2004hm}. AdS/BCFT exhibits some characteristic features of BCFTs~\cite{Takayanagi:2011zk,AdS/BCFT2,Cavalcanti:2018pta}, although the precise correspondence between the dynamical elements of two approaches has not been established in general.
In AdS/BCFT the dynamics of the boundary of the CFT is encoded in the dynamics
of codimension one hypersurface ending on that boundary. In what follows we will
describe new solutions of the AdS/BCFT, which are both time and temperature
dependent and propose some applications. In particular, we will demonstrate how
these solutions can be discussed in the context of the evolution of the
entanglement entropy after a local quench. The quench protocol we will consider
is similar to the so-called cut and glue quenches. Holographic models of such
quenches were perhaps originally considered
in~\cite{Ugajin:2013xxa,Nozaki:2013wia} and more recently
in~\cite{Mandal:2016cdw,
Shimaji:2018czt,Caputa:2013eka,Ageev:2019fjf,Kudler-Flam:2020url}.
Holographic models reproduce well the behavior of the entanglement entropy
observed in the CFT
calculations~\cite{Calabrese:2007mtj,2008JSMTE..01..023E,Stephan:2011,
Stephan:2013,Asplund:2013zba,Asplund:2014coa}. In particular, the entanglement
entropy grows at initial times, but decays at later ones.
The quench that we will consider exhibits a different entropy behavior. This is related to the fact the initial state is prepared differently. It corresponds to a nucleation of a Euclidean bubble at $t=0$. For $t>0$ the bubble expands. For this reason we will refer to such a protocol as to a bubble quench.
For early times ${\epsilon}\ll t \ll l$ the behaviour of the entropy is consistent with the cut and glue quench analysis, as in~\cite{Calabrese:2007mtj}:
\begin{equation}\bea\ds
S_{\rm E}(t) \ \sim \ \frac{c}{3}\log\frac{t^2}{\epsilon} + k\,,
\ea\end{equation}
where $k$ is a non-universal part equal to $k=-(c/3)\log(l/2)$ in our model. At finite temperature the early time behavior is replaced by
\begin{equation}\bea\ds
S_{\rm E}(t) \ \sim \ \frac{c}{3}\log \left[ \frac{%
2\pi T}{\epsilon }\frac{t^{2}}{\sinh \left(\pi Tl\right)} \right] \,,
\ea\end{equation}
while at intermediate times, $T^{-1}\ll t < l$, it exhibits linear behavior,
\begin{equation}\bea\ds
S_E\sim \frac{c}{3}2\pi T\left(t-\frac{l}{2}\right) + \frac{c}{3}\log\frac{1}{\pi\epsilon T}\,,
\ea\end{equation}
usually observed in the case of global quenches.
From the standard holographic considerations we argue that at late times $t>\ell$ the entropy should saturate. This occurs in a non-analytic manner (phase transition) in the present model. Such effect is also well known in the analysis of global quenches~\cite{Calabrese:2005in}. It is related to the finite speed of the quasiparticle propagation (light-cone spreading).
This paper is organized as follows. In section~\ref{sec:model} we give a very brief introduction to the AdS/BCFT construction. In section~\ref{sec:solutions} we apply a diffeormphism to construct time-dependent solutions of AdS/BCFT. In section~\ref{sec:examples}
we discuss applications of the solutions to the description of quantum quenches and derive formulae for the time-dependent entanglement entropy and discuss some effects of finite temperature. We summarize our results and observations in section~\ref{sec:conclusions}.
\paragraph*{Note added in the second version.} After the first version of the paper came out we learned about important earlier work on cut and glue quenches, both in the CFT and holographic context. Since our protocol is different from those conventionally used in the cut and glue protocols, we changed the title of the paper and referred to the type of quench studied in this paper as to a bubble quench.
\section{The model}
\label{sec:model}
In this section we are going to briefly define the AdS/BCFT construction of Takayanagi~\cite{Takayanagi:2011zk}. See~\cite{AdS/BCFT2} for a more complete review.
The idea of the construction is to provide a holographic dual description of a system that has a boundary. The boundary can be introduced through boundary conditions imposed on the degrees of freedom. The dual holographic theory must encode the degrees of freedom of the CFT with a boundary, so one does not need the whole of anti de Sitter space to encode the smaller system. Instead one extends the boundary of the CFT into the bulk and introduces additional boundary conditions in the AdS bulk region, which should be compatible with the boundary conditions imposed on the original CFT.
The setup is illustrated by figure~\ref{AdSBCFT}. Let the CFT be defined in space $M$ of some dimension $d$, whose boundary is $\partial M=P$. The CFT is complemented with appropriate boundary conditions on $P$. A dual gravity theory will reside in space $N$ with dimension $d + 1$, whose total boundary $\partial N = Q \cup M$ includes a hypersurface $Q$ in the bulk of gravity, such that $\partial Q = \partial M = P$. Such constructions can be realized in full (top-down) string theory, where the role of hypersurface $Q$ of boundary conditions is played by appropriate $D$-branes~(see \cite{Karch:2000gx,DeWolfe:2001pq,Bak:2003jk,DHoker:2007zhm,DHoker:2007hhe,Aharony:2011yc} for some examples), so in full string theory the hypersurface $Q$ is dynamically fixed.
\begin{figure}[tbh]
\centering
\includegraphics[height=6cm]{AdSBCFT.pdf} \quad \quad
\caption{In AdS/BCFT construction of~\cite{Takayanagi:2011zk} gravity theory in $d + 1$-dimensional space $N$ with boundary $Q$ is expected to be dual to a CFT defined in $d$-dimensional space $M$, with boundary $P=\partial M=\partial Q$.}
\label{AdSBCFT}
\end{figure}
In the proposal of~\cite{Takayanagi:2011zk} the dynamics of branes is replaced by Neumann boundary condition, which is expected to correctly account for the backreaction of the part of the gravity theory beyond the end-of-the-world $Q$:
\begin{equation}
K_{ab}-Kh_{ab} \ = \ \kappa T_{ab}-\Sigma h_{ab} \,.
\label{Junc-Cond}
\end{equation}
This is in general a second-order equation for the induced metric $h_{ab}$ on $Q$, which defines the embedding of $Q$ in the bulk space $N$ (solving it for $h_{ab}$ is equivalent to solving it for $z(x)$ which determines the embedding of $Q$ in terms of figure~\ref{AdSBCFT}). $K_{ab}$ is the pullback of the extrinsic curvature on $Q$ ($K$ being its scalar). The right hand side of the equation is the stress-energy tensor of the matter placed on $Q$ with units set by $\kappa=8\pi G$, where $G$ being the $d+1$-dimensional Newton's constant. Part of the stress-energy tensor corresponding to a constant energy density $\Sigma$ is separated. $\Sigma$ may also be referred as to the surface tension, or equivalently, cosmological constant on $Q$.
In the next section we will describe some solutions to equation~(\ref{Junc-Cond}) with $T_{ab}=0$. Since the Neumann boundary condition should reflect the choice of the dual boundary condition on $P$, $\Sigma$ should have a meaning in terms of the CFT data. The exact meaning has not been established so far, but some insight can be obtained by studying the entropy of the defect created by $P$, as in~\cite{Takayanagi:2011zk,Magan:2014dwa,Cavalcanti:2018pta}.
We will restrict our interest to 1+1 CFT examples and to asimptotically $AdS_3$ bulk geometries in the Poincaré patch. We will use $x^\mu=(t,x)$ as the CFT coordinates and $z$ as the coordinate in the gravity bulk.
The main player in our game will be the asimptotically anti de Sitter geometry given by the metric
\begin{equation}
ds^{2}=\frac{L^{2}}{z^{2}}\left(
-f(z)dt^{2} + dx^{2} + \frac{dz^{2}}{f(z)}\right)\ . \label{AdSMetric}
\end{equation}
Pure AdS space is represented by $f(z) = 1$, while the BTZ black hole geometry is obtained when $f(z)=1-z^{2}/z_{h}^{2}$. The latter geometry is dual to a finite temperature CFT state, with temperature given by $T=1/(2\pi z_{h})$, where $z_h$ is the coordinate of the horizon of the black hole. The two solutions can be related to each other by a (large) diffeomorphism. We will make use of this fact in the next section.
The simplest static solution of boundary condition~(\ref{Junc-Cond}) is obtained for the half-plane configuration. We choose boundary $P$ to be parameterized by equation $x=0$ (as in figure~\ref{AdSBCFT}). The embedding of $Q$ can then be parameterized by $x=x(z)$. For $T_{ab}=0$ and empty $AdS_3$ equation~(\ref{Junc-Cond}) is solved by a straight line embedding~\cite{Takayanagi:2011zk},
\begin{equation}
x\left(z\right) \ = \ z\cot \theta \,, \qquad \text{where} \qquad \cos \theta =L \Sigma .
\label{7}
\end{equation}
Tension $\Sigma$ defines the angle, at which plane $Q$ intersects the asymptotic AdS boundary. We define $\theta$ as the angle external to region $N$ encoding physics in $M$. It can be seen that the case $0\leq \theta <\pi /2$ corresponds to $\Sigma>0$. The tension is negative for $\pi /2<\theta \leq \pi $. In both cases the tension is bounded: $\left\vert \Sigma \right\vert \leq 1/L$.
In the finite temperature geometry the solution is slightly more involved,
\begin{equation}
x\left( z\right) =z_{h}\text{arsinh}\left( \frac{z}{z_{h}}\cot \theta
\right) \text{ }. \label{8}
\end{equation}
Angle $\theta$, again, is the angle at which $Q$ crosses the boundary at $z = 0$, external to subspace $N$. In the $z\rightarrow 0$ ($f(z)\to 1$) limit one recovers the pure AdS result (\ref{7}).
Some other solutions to boundary conditions~(\ref{Junc-Cond}) were considered in~\cite{Nozaki_2012,Magan:2014dwa,Erdmenger_2015,Seminara:2017hhh,Seminara:2018pmr,Cavalcanti:2018pta,Shashi:2020mkd,Sato:2020upl}. In the next section we will generate time dependent finite temperature solutions applying a conformal transformation.
\section{Time-dependent AdS/BCFT solutions}
\label{sec:solutions}
Other solutions of equation~(\ref{Junc-Cond}) can be generated by applying isometries to the basic solution (\ref{7}). The AdS$_{d+1}$ metric is invariant under the $d$-dimensional conformal group. For example, boosts create lines $x=z\cot\theta(\eta)$ moving with constant velocity $\eta$ in the $AdS_3$ bulk, intercepting the plane $z=0$ at an $\eta$-dependent angle~\cite{Cavalcanti:2018pta}.
More interesting configurations can be obtained from the special conformal transformations~\cite{Takayanagi:2011zk}. In the Euclidean space ($t \rightarrow it_E$) the half-plane $x>0$ on the boundary can be mapped to the interior of a disc by a global transformation~\cite{Berenstein:1998ij}
\begin{equation}
x_{\mu }^{\prime }=\frac{x_{\mu }+c_{\mu }x^{2}}{1+2\left( c\cdot x\right)
+c^{2}x^{2}} \ , \label{SCTx}
\end{equation}
where $c_{\mu}$ is a constant vector and $x^\mu = (x,t_E)$. The map of the half-plane $x=0$ to the disc of radius $R$ corresponds to the choice $c_{\mu}=(1/2R,0)$. The AdS metric is invariant under this transformation provided the coordinate $z$ is transformed as
\begin{equation}
z^{\prime }=\frac{z}{1+2\left( c\cdot x\right) +c^{2}x^{2}} \ . \label{SCTz}
\end{equation}
In the bulk, the transformation maps the two-dimensional Euclidean AdS$_2$ slices, including $Q$~(\ref{7}) into spherical domes sitting on $M$. The new $Q$ is defined by equation
\begin{equation}
t_{E}^{2}+\left( x-R\right) ^{2}+\left( z-R\cot \theta \right)^2 =R^{2}\csc
^{2}\theta \ . \label{esferaAdS}
\end{equation}
As before, $\theta$ is the external intersection angle of the spherical surface with the $z=0$ boundary. When tension $\Sigma = 0$, or $\theta = \pi /2$, $Q$ is exactly a hemisphere. One can also consider the analytic continuation of the spherical solution to the Minkowski space.
\begin{equation}
-t^{2}+\left( x-R\right) ^{2}+\left( z-R\cot \theta \right)^2 =R^{2}\csc
^{2}\theta \ . \label{hiperboloideAdS}
\end{equation}
The real-time solution describes a compact space with expanding walls.
One possible application of such solutions is in the context of the dynamics of phase transitions, or the problem of the decay of a false vacuum. Euclidean solution~(\ref{esferaAdS}) describes an imaginary time nucleation of a bubble of a new phase, while Minkowskian solution~(\ref{hiperboloideAdS}) -- the expansion of the bubble after the nucleation (figure~\ref{figure1}).
In this setup, the anti-de Sitter space represents the true vacuum, while the effect of the false vacuum is effectively described through the non-zero surface tension. As we shall see, in this model, temperature effects accelerate the expansion, and there is no finite temperature phase transition.
\begin{figure}[tbh]
\centering
\includegraphics[height=6cm]{fig1.pdf}\quad \quad \quad \quad
\caption{Nucleation of a Euclidean bubble of anti-de Sitter space $t<0$, creation ($t=0$) and evolution of the real bubble ($t>0$).} \label{figure1}
\end{figure}
To generalize the above solutions to the case of finite temperature we are going to apply a bulk diffeomorphism, which relates the empty $AdS_3$ geometry with that of the BTZ black hole. Since equation~(\ref{Junc-Cond}) is a tensor equation, we expect it to transform covariantly under the diffeomorphism. That is, it maps solutions of the equation to other solutions of the equation.
Let us consider a general transformation of the metric
\begin{equation}
g_{\mu \nu }^{\prime }=\frac{\partial x^{\alpha }}{\partial x^{\prime \mu }}
\frac{\partial x^{\beta }}{\partial x^{\prime \nu }}g_{\alpha \beta } \ .
\label{MetricTrans}
\end{equation}
To distinguish between AdS and BTZ coordinates in the above equation, we reserve the original coordinates $\left( t,x,z\right)$ for the empty AdS, and use primed coordinates, $\left( t^{\prime },x^{\prime },z^{\prime }\right)$ for the BTZ. With these definitions, we obtain the following coordinate transformation
\begin{eqnarray}
t &=&-z_{h}+\frac{z_{h}^{2}\cosh \left( x^{\prime }/z_{h}\right) \text{e}%
^{t^{\prime }/z_{h}}}{\sqrt{z_{h}^{2}-z^{\prime 2}}} \ , \nonumber \\
x &=&\frac{z_{h}^{2}\sinh \left( x^{\prime }/z_{h}\right) \text{e}%
^{t^{\prime }/z_{h}}}{\sqrt{z_{h}^{2}-z^{\prime 2}}} \ , \label{AdS-BTZ-trans} \\
z &=&\frac{z_{h}z^{\prime }\text{e}^{t^{\prime }/z_{h}}}{\sqrt{%
z_{h}^{2}-z^{\prime 2}}} \ . \nonumber
\end{eqnarray}%
In deriving these formulae, the integration constants were fixed by imposing the condition that in the limit $z_{h}\rightarrow \infty$ we must recover $t=t^{\prime },$ $x=x^{\prime }$ e $z=z^{\prime }$.
Transformations~(\ref{AdS-BTZ-trans}) can be readily used on equation~(\ref{esferaAdS}) to obtain a new, temperature and time dependent solution of the AdS/BCFT boundary condition~(\ref{Junc-Cond}):
\begin{eqnarray}
t_{E}^{\prime }=z_{h}\text{arccos}\left\{ \frac{1}{z_{h}\sqrt{f\left(
z^{\prime }\right) }}\left[ z_{h}\cosh \left( x^{\prime }/z_{h}\right)-R\frac{z^{\prime }}{z_{h}}\cot \theta -R\sinh \left( x^{\prime }/z_{h}\right)
\right] \right\} . \label{Esfera-BTZ}
\end{eqnarray}
The analytical continuation to the Minkowski space is performed by doing a Wick rotation $t\rightarrow it_{E}$ and $t'\rightarrow it'_{E}$, so we also find a hyperboloid-like solution
\begin{eqnarray}
t^{\prime }=z_{h}\text{arccosh}\left\{ \frac{1}{z_{h}\sqrt{f\left(
z^{\prime }\right) }}\left[ z_{h}\cosh \left( x^{\prime }/z_{h}\right)-R\frac{z^{\prime }}{z_{h}}\cot \theta -R\sinh \left( x^{\prime }/z_{h}\right)
\right] \right\} . \label{Hiperb-BTZ}
\end{eqnarray}
The parameter $R$ in the transformed solutions becomes the height of the profile in the $z$ direction, $0\leq R\leq z_h$. If one, however, analytically continues to $R>z_h$ one would get a class single-boundary solutions, whose $R\to\infty$ limit is static solution~(\ref{8}). We will not consider this branch here.
The characteristic shape of the hypersurface $Q$ is demonstrated by figure~\ref{figure2}, where again, the Euclidean solution is glued with the Minkowskian one at $t=0$. In comparison with configuration shown on figure~\ref{figure1}, the new $Q$ is bounded in $z$ dimension by the horizon of the black hole. One can also check that (\ref{Esfera-BTZ}) and (\ref{Hiperb-BTZ}) recover shapes~(\ref{esferaAdS}) and (\ref{hiperboloideAdS}) respectively, in the $z_h\rightarrow\infty$ limit.
\begin{figure}[tbh]
\centering
\includegraphics[height=7cm]{fig2.pdf}
\caption{Nucleation of a Euclidean bubble of the BTZ spacetime $t<0$. Creation ($t=0$) and evolution of the real bubble ($t>0$), for $\theta = \pi /2$.} \label{figure2}
\end{figure}
In the finite temperature case the parameter $R$ is related to the spatial radius $\rho$ at time $t=0$ through
\begin{eqnarray}
\rho
\ = \ z_h{\rm arctanh}\frac{R}{z_h} \,,
\label{bubblesize}
\end{eqnarray}
which makes sense only when $R\leq z_h$, and for $z_h\to \infty$ reduces to $\rho=R$. We also see that the profile intercepts the horizon at infinite radius $\rho\rightarrow\infty$, or $R \rightarrow z_{h}$. The profiles of $t=0$ configurations is shown on figure~\ref{figure3}.
\begin{figure}[tbh]
\centering
\includegraphics[height=6cm]{fig3.pdf}
\caption{Profiles of $Q$ in the BTZ geometry for different values of $R$ (height of the bubble), at $t = 0$ and $\theta =\pi/2$. As $R$ approaches the horizon, the width of the bubble becomes infinite, $l=2\rho\to\infty$.} \label{figure3}
\end{figure}
It is also interesting to compare the rate of expansion of the bubbles at zero and finite temperature. On figure~\ref{figura5}, we show the trajectories of walls of two bubbles in zero (blue) and non-zero (red) temperature geometries, as well as the velocities of the walls. Both bubbles have the same size at the moment of creation and their walls are at rest. At late times the walls of either bubble approach the speed of light. However, at finite temperature the walls have larger acceleration. Although the plots are shown for $\Sigma=0$, this effect does not depend on the tension. Consequently, in this model one does not observe a critical (stationary) bubble at any temperature.
\begin{figure}[tbh]
\includegraphics[height=0.3\linewidth]{fig6.pdf}
\hfill{
\includegraphics[height=0.3\linewidth]{fig7.pdf}
}
\caption{(Left) trajectories of the $T=0$ (\ref{hiperboloideAdS}) (blue curve) and $T\neq 0$ (\ref{Hiperb-BTZ}) (red curve) bubble walls. (Right) velocities of the two types of walls as a function of $x$.}
\label{figura5}
\end{figure}
\section{Local quantum quench}
\label{sec:examples}
The time-dependent solutions of the AdS/BCFT problem can also be discussed in a more general context of the dynamics of quantum systems out of equilibrium. One can apply an abrupt local, or global perturbation (quench) to the system and study the time behavior of the correlators. Entanglement entropy is a measure of quantum correlations. For this reason it is an interesting object to study in the non-equilibrium dynamics.
One distinguishes global and local quenches. In global quenches one perturbs the system as a whole, for example, by tuning its Hamiltonian. In local quenches, one only perturbs a part of the system. A well-established result in the case of global quenches is the linear growth of the entanglement entropy~\cite{Calabrese:2005in}. The linear growth saturates for a finite system and for late times, the entropy is constant. These observations are true also beyond the conformal case~\cite{Calabrese:2016xau}.
We would like to compare the AdS/BCFT bubbles constructed in the previous section with local quenches. In \emph{cut and glue} quench protocol two complimentary subsystems are prepared unentangled in their respective ground states. Then the two subsystems are brought together and their joint evolution is investigated. Alternatively, in such a quench, the global system is disconnected at certain moment of time and the dynamics of the isolated subsystems is watched. In $1+1$ CFT this quench protocol can be treated by appropriate conformal transformations, mapping to a simple BCFT configuration.
From the holographic point of view, our prescription is close to the double
quenches considered in~\cite{Caputa:2019avh}, where two slits in the initial
state correspond to the walls of our bubble. Here the walls of the bubble are
entangled through the Euclidean nucleation protocol (see a more recent
model~\cite{Akal:2020wfl}, where the bubble does not reconnect with the AdS
boundary). Also, in our analysis, the
exterior of the bubble will always remain unentangled and disconnected. In such
a case the problem is solved without invoking a conformal map.
In terms of the holographic Ryu-Takayanagi prescription~\cite{Ryu:2006bv}, which in our case coincides with the more general covariant one of Hubeny, Rangamani and Takayanagi~\cite{Hubeny:2007xt}, either of the two bubbles~(\ref{esferaAdS}), or~(\ref{Esfera-BTZ}) at $t=0$, have zero entanglement with the exterior, as in the cut and glue protocol. This is because by the prescription, the entanglement entropy is proportional to the area (length in 3D) of the minimal area spacelike hypersurface (geodesic line in 3D) $\gamma_{\min}$ that connects the endpoints (walls) of the bubble in the gravity bulk,
\begin{equation}\bea\ds
\label{RTeq}
S_{\rm E }\ = \ \frac{{\rm Area}[\gamma_{\min}]}{4G}\,.
\ea\end{equation}
However, in the presence of the end-of-the-world brane $Q$ the geodesic is allowed to end on $Q$. Since $Q$ also ends on the walls of the bubble, the geodesic has zero length and the entanglement entropy is zero. We will illustrate this argument in an example that follows. See also~\cite{Ugajin:2013xxa,Erdmenger_2015,Erdmenger_2016entang,Erdmenger_2016holog,Miao_2017,Seminara:2017hhh,Cavalcanti:2018pta} for a similar discussion.
During the expansion, for $t>0$, the bubble will continue having zero entanglement entropy. By analogy, with quantum quenches, the walls of the bubble follow propagation of the front of the quasiparticles, and there is no entanglement with anything outside of the bubble. (The walls will eventually form a light cone, analogous to the lightcone of the quenched systems~\cite{calabrese2006time,Calabrese:2007rg}.) Below we will focus on the entanglement of other subsystems and compute their entropy.
The first configuration we are going to analyze is shown on figure~\ref{setup}: we would like to study the entanglement of the two halves of the bubble. In the figure, the black curves correspond to boundary $Q$ in AdS (left) and BTZ (right) spaces, given by equations (\ref{hiperboloideAdS}) and (\ref{Hiperb-BTZ}), respectively. The profiles with $\theta = \pi /2$ at $t = 0$ are shown.
Using equation~(\ref{RTeq}) we compute the entanglement entropy of any of the halves of the bubble as the length of a geodesic line connecting the walls of the bubble in the gravity bulk. There are two options in this case (shown as blue and green lines on figure~\ref{setup}): one line connects the endpoints of the interval representing a half of the bubble, while the other option connects the center of the bubble with the curve $Q$. It turns out that the second geodesic (green vertical line on figure~\ref{setup}) is always shorter in either geometry.
\begin{figure}[tbh]
\includegraphics[height=0.23\linewidth]{fig4.pdf}
\hfill{
\includegraphics[height=0.23\linewidth]{fig5.pdf}
}
\caption{(Left) geodesics (blue and green lines) and end-of-the-world surface $Q$ (black) in the empty AdS geometry, $\theta = \pi /2$. (Right) same in the BTZ geometry.}
\label{setup}
\end{figure}
In the case of empty AdS (left panel on figure~\ref{setup}) the boundary $Q$ is a circular arc of radius $R\csc\theta$ centered at $z=R\cot\theta$, equation~(\ref{hiperboloideAdS}). For the bubble of radius $r(t)$ at time $t$, $r(t)^{2}=R^2 + t^2$. Calculation then yields the length of the geodesic in geometry~(\ref{AdSMetric}) connecting the middle of the bubble with $Q$ as
\begin{equation}
\ell_0 \ = \ L\int _{\epsilon}^{z_\ast} \frac{dz}{z} \ = \ L\log \left(\frac{z_\ast}{\epsilon}\right) \ = \ L\log \left(\frac{\sqrt{R^2\csc ^2\theta + t^2}+R\cot\theta}{\epsilon}\right)\,,
\label{ComVertADS}
\end{equation}
where $z_\ast$ is the height of the arc $Q$. As usual, $\epsilon$ is a UV cut-off introduced to make the length finite.
Meanwhile the length of the geodesic connecting the two endpoints of the interval of length $r(t)$ (blue circle on figure~\ref{setup}) is given by
\begin{equation}
\ell \ = \ 2\times L\int _{\epsilon}^{r/2} \frac{rdz}{z\sqrt{r^2-4z^2}} \ = \ 2L \log \left(\frac{\sqrt{R^2 + t^2}}{\epsilon}\right)+O(\epsilon^2).
\end{equation}
By choosing the cutoff $\epsilon$ sufficiently small one can always make $\ell>\ell_0$. The reason for that is that the geodesic $\ell_0$ has only one endpoint on the boundary $z=0$, where the distance must be regulated. Therefore we use $\ell_0$ in equation~(\ref{RTeq}). The standard conversion between the 3D gravity and CFT parameters is~\cite{Brown:1986nw}
\begin{equation}\bea\ds
c \ = \ \frac{3L}{2G}\,.
\ea\end{equation}
At late times the entanglement entropy of a half of the bubble behaves as
\begin{equation}\bea\ds
S_{\rm E} \ \sim \ \frac{c}{6}\log\frac{t}{\epsilon}\,.
\ea\end{equation}
On figure~\ref{ComprimentoDaLinhaReta} (left) we plot the behavior of the entropy for different values of the initial bubble size. The entropy interpolates between the initial value, cf.~\cite{Cavalcanti:2018pta}
\begin{equation}\bea\ds
\frac{c}{6}\log \left(\frac{2R}{\epsilon}\cot\frac{\theta}{2}\right)
\ea\end{equation}
controlled by $R$ and $\theta$ and the late time logarithmic growth, independent from those parameters.
A similar calculation can be done in the finite temperature case, as on the right of figure~\ref{setup}. For the qualitative understanding it is sufficient to consider the case $\theta = \pi /2$. As for zero temperature, the geodesic that connects the center of the bubble with $Q$ (green line on figure~\ref{setup}) can always be made shorter if $\epsilon$ is sufficiently small. The turning-point $z_{*}$ (the height) of the $Q$ profile is given by the expression, cf.~(\ref{Hiperb-BTZ})
\begin{equation}
z_{*} \ = \ \text{sech}(t/z_h) \ \sqrt{z_h^2\text{sech}^2 (t/z_h) +R^2} ~.
\end{equation}
Therefore, the temporal behavior of geodetic $\ell_0$ is
\begin{equation}
\ell_{0}\ =\ L\int_{\epsilon }^{z_{\ast }}\frac{dz}{z\sqrt{1-\frac{%
z^{2}}{z_{h}^{2}}}}\ =\ L\log \left[ \frac{2z_{h}}{\epsilon }\frac{\text{sech}%
\left( t/z_{h}\right) \sqrt{R^{2}+z_{h}^{2}\sinh ^{2}\left( t/z_{h}\right) }%
}{z_{h}+\sqrt{z_{h}^{2}-\text{sech}^{2}\left( t/z_{h}\right) \left[
R^{2}+z_{h}^{2}\sinh ^{2}\left( t/z_{h}\right) \right] }}\right] ~.
\label{ComVertBTZ}
\end{equation}
For $t\rightarrow\infty$, the length $\ell_0$ tends to an asymptotic value equal to $L\log(2z_h/\epsilon)$, as can be also seen on figure~\ref{ComprimentoDaLinhaReta} (right). The asymptotic value of the entropy is
\begin{equation}\bea\ds
\label{finiteTentropy}
S_{\rm E} \to \frac{c}{6}\log\left(\frac{1}{\pi\epsilon T}\right)\,.
\ea\end{equation}
As in the $T=0$ case, the values of $R$ and $\theta$ only affect the early time behavior of the entropy, for example, the $R$-dependence at $t=0$ is given by
\begin{equation}\bea\ds
S_{\rm E} \ \sim \ \frac{c}{6}\log \left[\frac{1}{\pi\epsilon T}\frac{\sinh2\pi Tl}{1+\cosh2\pi Tl}\right]\,,
\ea\end{equation}
when $t\ll 1/T$. Here we used relation~(\ref{bubblesize}) between parameter $R$ and the initial radius $\rho\equiv l$ of the bubble.
\begin{figure}[tbh]
\includegraphics[height=0.3\linewidth]{ComprimentoAdSs.pdf}
\hfill{
\includegraphics[height=0.3\linewidth]{ComprimentoBTZz.pdf}
}
\caption{Plot of the time dependence of the entanglement entropy of the two halves of the expanding bubble for different initial sizes of the bubble. The left plot corresponds to empty AdS geometry shows asymptotic logarithmic growth. The right plot, for the finite temperature geometry, shows saturation of the entropy at a value independent from the initial parameters of the bubble.}
\label{ComprimentoDaLinhaReta}
\end{figure}
A more common configuration to study in the context of local quenches is the evolution of the entanglement of the interval, corresponding to the $t=0$ bubble with the remainder of the system. Recall that at time $t=0$, the bubble is unentangled with the exterior. We will consider $\theta=\pi/2$ and first treat the case of zero temperature.
The setup we are going to study is shown on figure~\ref{SetupDiferente}. The dashed black arc of radius $R$ is the $Q$-profile of the initial bubble created at $t = 0$. The bubble begins to expand, and the profile of $Q$ at some later time $t$ is shown as a continuous black arc on figure~\ref{SetupDiferente}. At this time the radius of $Q$ (and the radius of the bubble) is given by $l=\sqrt{t^2+R^2}$.
We would like to know the entanglement of the initial interval of size $l=2R$ with the rest of the system. For this we are going to use equation~(\ref{RTeq}) with an appropriate minimal geodesic line. The blue curves of radius $r$ on figure~\ref{SetupDiferente} correspond to the natural choice of the minimal surface at early times, after the beginning of the expansion. An alternative choice would be a geodesic connecting the endpoints of the interval, which in this case coincides with the dashed line. Clearly the second geodesic is longer. The two blue pieces have zero length at $t=0$ and the entanglement is zero as claimed in the beginning of this section.
In order to apply equation~(\ref{RTeq}) we have to calculate the length of the blue curves, which is inside boundary $Q$, from the endpoints of the interval until the intersection point $z_{0}$ with the black curve. The geodesics must satisfy Dirichlet boundary condition at the endpoints and they must be perpendicular to the black curve at the intersection point (Neumann boundary conditions). The blue geodesics are also circular arcs with
\begin{equation}
r=\frac{l^2-R^2}{2R} \qquad \text{and} \qquad z_0 = \frac{r\sqrt{R(R+2r)}}{R+r}~.
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[height=5.5cm]{setupdiferente.pdf}
\caption{Calculation of the entanglement entropy of the finite interval bounded by the dashed arc with the rest of the system after a cut and glue quench.}
\label{SetupDiferente}
\end{figure}
Therefore, the length of the relevant pieces of the geodesics is
\begin{equation}
\ell \ = \ 2\times L\int _{\epsilon}^{z_0}\frac{dz}{z}\frac{r}{\sqrt{r^2-z^2}}
\ = \ 2L\log \left(\frac{t^2}{\epsilon\sqrt{t^2+R^2}}\right) + O(\epsilon^2)\,.
\label{9}
\end{equation}
However, as the continuous black arc expands over time, the blue geodesic also expands in size so that there will be a certain instant of time that its length $\ell$ is greater than the length of the dashed black curve $\ell_0$. At this moment we have to switch to $\ell_0$ in equation~(\ref{RTeq}). In this model the change is non-analytic. The length of the dashed black circle is given by
\begin{equation}
\ell_0 \ = \ 2\times L\int _{\epsilon}^{R}\frac{dz}{z}\frac{R}{\sqrt{R^2-z^2}} \ = \ 2L\log \left(\frac{2R}{\epsilon}\right) + O(\epsilon^2)\,.
\label{10}
\end{equation}
The phase transition occurs at $t_c = R\sqrt{2(1+\sqrt{2})}$.
At initial times the entropy of the system grows logarithmically,
\begin{equation}\bea\ds
\label{CAGentropy}
S_{\rm E} \ \sim \ \frac{c}{3}\log\frac{t^2}{\epsilon R}\,, \qquad t\ll R\,.
\ea\end{equation}
At later times it saturates at the standard universal value of the entropy of a finite interval. The plot of the entropy and of the phase transition is shown on figure~\ref{PhaseTransition}~(left).
\begin{figure}[thb]
\includegraphics[height=0.36\linewidth]{TransicaodefaseAdS.pdf}
\hfill{
\includegraphics[height=0.36\linewidth]{TransicaodefaseBTZ.pdf}
}
\caption{(Left) Evolution of the entanglement entropy of a finite interval in a bubble quench from the AdS/BCFT calculation (configuration of figure~\ref{SetupDiferente}). The blue curve shows the logarithmic growth from equation (\ref{9}). The growth saturates at $S_0$ (red line), equation (\ref{10}). The gray line is the behavior predicted in a local cut and glue quench, equation~(\ref{logsine}) (Right) Similar plot for $T\neq 0$. The blue line has a linear segment described by equation~(\ref{lineargrowth}). The gray line illustrates the $T=0$ curve.}
\label{PhaseTransition}
\end{figure}
{To find the entropy at finite temperature one can apply transformations~(\ref{AdS-BTZ-trans}). Consequently, one derives the following analytical formula}
\begin{equation}
S_E=\frac{c}{3}\log \left[ \frac{2z_{h}\left( \cosh \left( t/z_{h}\right) -1\right) \text{%
e}^{t/z_{h}}}{\epsilon \sqrt{2\left( \cosh \left( t/z_{h}\right) -1\right)
\text{e}^{t/z_{h}}+\sinh ^{2}\left( l/2z_{h}\right) \text{e}^{2t/z_{h}}}}%
\right], \label{A}
\end{equation}
where $l$ is the length (rather than radius) of the initial bubble. For short times the generalization of equation~(\ref{CAGentropy}) is
\begin{equation}
S_E\sim \frac{c}{3}\log \left[ \frac{%
2\pi T}{\epsilon }\frac{t^{2}}{\sinh \left( \pi Tl\right)} \right] \,.
\label{tpequenoBTZ}
\end{equation}
It is interesting that at intermediate times, $T^{-1}\ll t <l$, the entropy grows linearly,
\begin{equation}\bea\ds
\label{lineargrowth}
S_E\sim \frac{c}{3}2\pi T\left(t-\frac{l}{2}\right) + \frac{c}{3}\log\frac{1}{\pi\epsilon T}\,.
\ea\end{equation}
We note that the entropy is twice the value of equation~(\ref{finiteTentropy}) at $t=l/2$.
For $t>l$ one expects the saturation phase transition to the value of the entropy of an interval at finite temperature:
\begin{equation}\bea\ds
S_0 \ = \ \frac{c}{3}\log\left[\frac{1}{\pi T\epsilon}\sinh\left(2\pi Tl\right)\right]\,.
\ea\end{equation}
This finite temperature behavior is illustrated by the right plot of figure~\ref{PhaseTransition}.
\section{Conclusions}
\label{sec:conclusions}
In this paper we obtained new time-dependent solutions to the AdS/BCFT problem of~\cite{Takayanagi:2011zk}. These new solutions correspond to expanding walls in anti-de Sitter space and are natural to discuss in the context of the process of nucleation and expansion of new phases.
We demonstrated that the time-dependent solutions can be suitable for the
discussion of quenched dynamics, in a setup similar to the local cut and glue
protocol. This observation is supported by the behavior of the entanglement
entropy, which grows logarithmically at early times. However, our protocol is
slightly different from the conventional cut and glue quench, and the late
behavior of the entropy is different.
In the bubble protocol one can observe other characteristic features of quenched dynamics. The solutions are compatible with the light-cone expansion of the correlations. The entanglement of a finite interval saturates at finite time. This effect is typically discussed in the case of the global quench~\cite{Calabrese:2005in}, and since we disregard the exterior of the bubble, it is perhaps not so surprising that the linear behavior also occurs in the bubble quench.
Some analytical results for local quenches of finite intervals are harder to obtain using the CFT techniques~\cite{Calabrese:2007rg}, because the appropriate conformal maps become non-invertible. Similar problems can occur in the geometric analysis, although we have been able to get some exact analytical results. In particular, at early times we are able to derive equations~(\ref{9}) and~(\ref{CAGentropy}) for the entropy. The early time behavior is compatible with
\begin{equation}\bea\ds
\label{logsine}
S_{\rm E} \ = \ \frac{c}{3} \log\left(\frac{2t}{\pi\epsilon}l\sin\frac{\pi t}{l}\right)+k'\,.
\ea\end{equation}
which is the cut and glue quench result in
CFT~\cite{Calabrese:2007mtj,2008JSMTE..01..023E} and in the holographic
models~\cite{Ugajin:2013xxa,Asplund:2013zba,Asplund:2014coa,Mandal:2016cdw,
Shimaji:2018czt, Caputa:2013eka,Ageev:2019fjf,Kudler-Flam:2020url}. First, the
match is up to the factor of the length $l$. Naively, the argument of the
logarithm in equation~(\ref{logsine}) is not dimensionless, which means that
there is a dimensionful scale hidden in the non-universal part $k'$. The
holographic derivation automatically gives the correct dimension removing one
$l$ factor. The saturation point in the bubble quench occurs at $t\sim 1.05 l$,
which is very close to the point, where the argument of the logarithm
in~(\ref{logsine}) vanishes, as can also be seen on figure~\ref{PhaseTransition}
(left).
Finally we derive equations for the evolution of the entropy of a finite interval at finite temperature. For early times we derive equation~(\ref{tpequenoBTZ}), which could be tested by CFT techniques, cf.~\cite{Wen:2018svb}. As already mentioned, at finite $T$, the entropy tends to show linear growth~(\ref{lineargrowth}) for intermediate times $T^{-1}\ll t < l$, before the saturation phase transition.
We believe that some of the outstanding issues of the present analysis, as well as other interesting questions of the quenched dynamics can be further addressed in the AdS/BCFT formalism. We leave this for a future work. One important question, is whether one can make the AdS/BCFT correspondence more precise by relating $\theta$ (or $\Sigma$) to CFT quantities, or finding appropriate forms of $T_{ab}$ in equation~(\ref{Junc-Cond}).
\paragraph*{Acknowledgements.} We would like to thank Madson R.~O.~Silva for
collaboration on parts of this project. We also grateful to J\'er\^ome Dubail
and, especially, to Zixia Wei for useful correspondence regarding the first
version of the paper. The work of AC was supported by the Brazilian Ministry of
Education (MEC), the work of DM was supported by the Russian Science Foundation
grant No.~16-12-10344.
\bibliographystyle{hieeetr}
|
2,869,038,156,375 | arxiv | \section{Introduction}
The study of globular clusters (GCs) offers a rich platform for the validation of stellar evolution theories and the formulation of dynamical evolution scenarios of stellar systems.~In particular, recent discoveries \citep[e.g.][]{fer12,knigge09} have prompted Blue Straggler Stars (BSSs) as key elements of the puzzle, given their tight relation with these two fields.~BSSs are empirically defined as stars that appear bluer and brighter than turn-off stars in the color-magnitude diagram (CMD) of a GC. Although their definite physical nature is still a matter of debate, the most popular formation channels for BSSs are the ones explaining them as products of mass-transfer/mergers in binary stellar systems \citep{mccrea64} and as the results of direct stellar collisions \citep{hillsday76}.~Both are strongly dependent on the local environment's dynamical state, and, therefore, BSSs can be used as tracers of past dynamical events as well as of the current dynamical state, given their strong response to two-body relaxation effects due to their relatively high stellar masses.~\cite{fer12} proposed that the current BSS radial density profile can be used as a ``dynamical clock" to estimate the dynamical age of a GC, and continued to classify a large set of GCs in three different classes according to their dynamical state.~The dynamically oldest category contains, among others, GCs M\,30 and NGC\,362, which have been studied in this context more in detail by \citet[hereafter F09]{fer09} and \citet[hereafter D13]{dal13}.~These studies have found that the BSS population is indeed consistent with dynamically old GCs.~Furthermore, their CMDs reveal well defined double BSS sequences, which the authors claim are the result of single short-lived dynamical events, such as core-collapse.~It is believed that during such core contraction the stellar collision rate would become enhanced, producing the bluer BSS sequence in the CMD, while the boosted binary interaction rate would lead to enhanced Roche-lobe overflows, thereby producing the redder BSS sequence.~It is, therefore, clear that observational properties of BSSs can provide valuable information for the understating of the dynamical evolution of GCs.~Here, we present a study based principally on high-quality HST photometry of the inner BSS population in the poorly studied Galactic GC NGC\,1261, which contains evidence for a very particular dynamical history, making it a similar, yet unique case among other GCs with well defined BSS features, such as M\,30 and NGC\,362.
\section{Data Description}
The inner region photometric catalog comes from the HST/ACS Galactic Globular Cluster Survey \citep{sar07}.~It consists of $\sim$30 min.~exposures in the F606W ($\sim\!V$) and F814W ($\sim\!I$) bands for the central $3.4\arcmin\times3.4\arcmin$ field of NGC\,1261.~The photometry was corrected to account for updated HST/ACS WFC zero points and calibrated in the Vega photometric system.~The catalog provides high quality photometry down to $\sim\!6$ mag below the main-sequence turn-off. Additionally, we performed PSF photometry using the DoPHOT software package \citep{sch93, alo12} on HST/WFC3 data taken with the F336W ($\sim\!U$) band, available from the Hubble Legacy Archive (PI: Piotto, Proposal ID: 13297).~The photometry was calibrated using Stetson standards \citep{ste00} from the NGC\,1261 field. The astrometry was refined with the HST/ACS optical catalog using bright isolated stars, after which we reach a median accuracy of $\sim\!0.002\arcsec$ between the optical and F336W-band catalogs.~We also use wide-field photometry from the catalog published by \cite{kra10}, built from observations at the 1.3-m Warsaw telescope at Las Campanas Observatory, using a set of $UBVI$ filters and a $14\arcmin\times14\arcmin$ field of view. The photometry is calibrated to \cite{ste00} and the median error is $\leqslant$0.04 mag for all filters and colors down to $V=20$ mag.~The complete description of their data reduction and photometric calibration can be found in \cite{kra10}.
\section{The central BSS population in NGC\,1261}
BSSs are selected through their position in a CMD. Having three filters, we can use the additional color information to remove contaminants. First, we cross-match the optical ACS catalog with the F336W-band catalog and keep matched sources with separations $<$ 0.02\arcsec\ ($\sim\!0.5$ pix) and reported errors\footnote{Taken from the HST/ACS catalog. More information on the errors is found on the catalog's README file.} $<$ 0.03 mag in the F606W and F814W filter.~This results in $\sim\!25000$ sources in a $\sim\!2.7\arcmin\!\times\!2.7\arcmin$ field centered on NGC\,1261 (see Figure~\ref{cmd1}).~We use a F814W\,$<$\,19.5 mag limit for the BSS selection criteria from \cite{lei11}, who define the BSS region based on magnitude and color cuts in the (F606W-F814W)\,vs.\,F814W CMD, shown by the polygon in Figure~\ref{cmd1}.~All stars inside the region are considered to be BSS candidates, and used in the following analysis (unless removed from the sample; see further down in the text).
\begin{figure}[t!]
\centering
\includegraphics[width=9cm]{fig1.pdf}
\caption{\small{(F606W-F814W)\,vs.\,F814W CMD of the NGC\,1261 inner region. All detections come from the final matched HST catalog.~The R-BSS (red triangles), B-BSS (blue squares) and eB-BSS (black circles; filled circle is explained in Fig~\ref{Ucmd}) sub-samples are marked; crosses mark the rest of the BSS sample (except for diagonal crosses, see Fig~\ref{Ucmd}).~Pentagon symbols mark special cases, see Fig~\ref{Ucmd} .~The inset panel shows the distribution of BSS perpendicular distances from the best-fit line to the B-BSS sequence (shown as a dashed line) in mag units.~The solid red line shows a non-parametric Epanech\-nikov-kernel probability density estimate with 90\% confidence limits represented by the dotted pink lines.} }
\label{cmd1}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=9cm]{fig2.pdf}
\caption{\small{(F336W-F814W)\,vs.\,F336W CMD of the NGC\,1261 inner region.~Filled and $\times$-shape symbols indicate BSS candidates that were rejected from our BSS sample.~Pentagon symbols mark BSS candidates that appear slightly faint in the F336W band, relative to their parent subsamples.} }
\label{Ucmd}
\end{figure}
We identify two prominent BSS sequences in the CMD, each containing $\sim\!20$ BSSs (see inset panel).~We label them the blue-sequence-BSSs (B-BSSs; shown as blue squares) and the red-sequence-BSSs (R-BSSs, red triangles).~Similar to M\,30 and NGC\,362\footnote{Note that in those studies, the HST F555W filter is used as a V band proxy, while we use the F606W filter.}, the B-BSS sequence is narrower and better defined, while the R-BSS sequence is dispersed towards redder colours.~We note the appearance of a group of BSSs lying bluer from the B-BSS sequence in the CMD. Since they cannot be directly associated with the B-BSS sequence, as they are clearly separated in the diagram, we choose to label them as extremely-blue-BSSs (eB-BSSs, black circles) and we later discuss their likelihood of being associated to a particular BSS population.~This bluer component has neither been observed in M\,30 nor in NGC\,362, and, if confirmed real, requires a given BSS formation scenario, particular to the history of NGC\,1261, that is not present in the other two GCs.
We now introduce the F336W-band photometry for contaminant detection.~We show in Figure~\ref{Ucmd} the (F336W-F814W)\,vs.\,F336W CMD and use the same symbols for the BSSs as in Figure~\ref{cmd1}.~Three BSS candidates plus another star from the eB-BSS group cannot be distinguished from $normal$ MS/SGB stars based on their location in the (F336W-F814W)\,vs.\,F336W CMD (shown as diagonal crosses and a filled circle, respectively), and hence we exclude them from the full BSS sample and subsequent analysis.~It is worth noting that two R-BSSs and another two eB-BSSs in Figure~\ref{Ucmd} (additionally marked with a pentagon) also show significantly redder (F336W-F814W) colors than the rest of their groups.~This result may point into photometric variability (since the F336W and F814W-band images on HST were taken $\sim\!7$ years apart, while the F606W and F814W-band exposures are less than an hour apart).~Indeed W\,UMa eclipsing binary systems are frequent among BSSs and have been detected in the double BSS sequences of NGC 362 (D13) and M\,30 (F09).~In particular, the W\,UMa BSSs detected in NGC\,362 were found to show a typical variability of 0.3 mag in all F390W, F555W and F814W bands (see Figure~10 in D13) which could account for the deviations found in (F336W-F814W) colors for these BSSs in NGC\,1261.~Another explanation could be the blend of cooler stars, which would explain the apparent missing flux in the F336W band.~However, the fact that the source centroids were requested to match within 0.02\arcsec\ ($\sim$0.2 FWHM in the optical) in the optical and near-ultraviolet, along with the conservative 0.03 mag error limit in the optical bands, points rather towards a well-fitted single PSF detection.~We compare now the identified sub-samples with isochrones from stellar collisional models of \cite{sil09}.~The models have a metallicity of $Z\!=\!0.001$ ([Fe/H]~$\!=\!-1.27$) and solar-scaled chemical composition ([$\alpha$/Fe]~$\!=\!0$).~Pairs of stars with masses between 0.4 and 0.8\,$M_\odot$ are collided using the MMAS software package \citep{lom02}.~The parent stars are assumed to be non-rotating, and 10 Gyr old at the time of the collision.~The collision products are evolved using the Monash stellar evolution code, as described in \cite{sil09}.~The time of the collision was taken to be $t\!=\!0$, and isochrones of various ages were calculated by interpolating along the tracks to determine the stellar properties (effective temperature, luminosity, etc.) at ages between 0.2 and 5 Gyr.~For the isochrones, we used collision products of the following stellar mass combinations: $0.4\!+\!0.4, 0.4\!+\!0.5, 0.4\!+\!0.6, 0.5\!+\!0.6, 0.6\!+\!0.6$, and $0.8\!+\!0.8\,M_\odot$.~We have adopted the distance modulus and reddening values of NGC\,1261 from \cite{dot10} and augmented the distance modulus by 0.25 mag in order to match the low-mass end of the collisional isochrone to the location of the main-sequence fiducial line in the CMD\footnote{This is mostly caused by the different metallicity/alpha enhancement between our models and the cluster, which has a metallicity closer to [Fe/H]$\sim-$1.35...$-$1.38 according to the latest studies \citep{kra10,pau10,dot10}, and a large $\alpha$-enhancement as it is expected for GCs in the halo \citep{pritzl05,woodley10} .}.~We find that the location of the B-BSSs and eB-BSSs can be reproduced by 2 Gyr and 200 Myr old isochrones of the stellar collisional models, respectively, as seen in Figure~\ref{cmd_iso}.~We note that the agreement of the model with the B-BSSs is remarkably good.~The eB-BSS sub-sample, although much less populated, is also somewhat consistent with the collisional 0.2\,Gyr isochrone model.~This enlarges the likelihood of this BSS feature being a real distinct population.~The R-BSSs need further explanation.~F09 and D13 found for M\,30 and NGC\,362 that the lower bound of the R-BSSs could be fairly well bracketed by the zero-age main-sequence (ZAMS) shifted by 0.75 mag towards brighter luminosities in the $V$ (F555W in their case) band, which approximately indicates the region populated by mass-transfer binaries, as predicted by \cite{tia06}.~In our case we use the 0.25 Gyr old Dartmouth isochrones \citep{dot08}, and find that the R-BSS sequence can only be reproduced by a region bracketed by the 0.25 Gyr old isochrone shifted by 0.45 mag and 0.75 mag to brighter F606W-band luminosities (grey region in the Figure). This shift is less than that required for M\,30 and NGC\,362 , and we note that a small difference is expected due to slightly different HST filters (F606W vs. F555W) as a $V$ band proxy. However, the prediction by \citeauthor{tia06} is based only on Case A (main-sequence donor) mass-transfer models.~\cite{lu10} showed that case B (red-giant donor) mass-transfer products lie indeed in a bluer region than the ZAMS+0.75 mag boundary.~Our current understanding of binary mass transfer is limited and, hence, our observations could help putting constraints on future binary stellar evolution models.
\begin{figure}[t!]
\centering
\includegraphics[width=9cm]{fig3.pdf}
\caption{\small{(F606W-F814W)\,vs.\,F606W CMD of the NGC\,1261 inner region with overplotted collisional isochrones of 2 Gyr and 200 Myr old, up to 1.3 $M_\odot$ and 1.6 $M_\odot$, respectively.~The grey band shows the zero-age main sequence isochrone, shifted by 0.45 and 0.75\,mag to brighter luminosities, to match the R-BSS sequence.~The red line is a Dartmouth isochrone with cluster parameters adopted from \cite{dot10}.~The representative photometric errors as obtained in the HST/ACS catalog are plotted in red bars.} }
\label{cmd_iso}
\end{figure}
\section{Dynamical State of NGC\,1261}
F09 and D13 have demonstrated that M\,30 and NGC\,362 show signs of being in an advanced state of dynamical evolution, as revealed by their centrally segregated BSS radial profiles \citep[which puts them in the {\sc Family~III} group in][]{fer12} and by their centrally peaked radial stellar density profiles.~Likewise, we plot in Figure~\ref{BSS_norm_prof} the normalized, cumulative BSS radial density profile for all BSS candidates and for the R-BSS and B-BSS sequences in NGC\,1261 out to 3.8 core radii, i.e.~the extent of the HST observations.~As expected, the whole BSS sample, as well as each BSS sequence, are more centrally concentrated than the reference SGB population\footnote{We choose SGB stars from the CMD inside the 19.2$<$F606W$<$19.5 mag range.}.~However, contrary to what was found in M\,30 and NGC\,362, we find the B-BSS component more centrally concentrated than the R-BSS component.~A K-S test shows that the null-hypothesis of these stars being drawn from the same radial distribution has a p-value of 0.33, which implies a non-negligible likelihood for common parent distributions.~Nevertheless, the difference between these profiles and the ones for M\,30 and NGC\,362 is still significant and motivates a discussion on possible qualitative differences of BSS formation history in NGC\,1261 versus the other GCs.~We do not find any B-BSSs inside $\sim\!0.5\,r_c$ (or about 10\arcsec), in agreement with F09 and D13, who as well report no B-BSSs within the inner 5-6\arcsec ($\sim\!1.5\,r_c$ and $\sim\!0.5\,r_c$ respectively).~These authors suggest that dynamical kicks are responsible for clearing the innermost region of any B-BSSs.~The detailed process is, however, unknown as accurate dynamical models are numerically expensive and, thus, still lacking.
\begin{figure}[t!]
\centering
\includegraphics[width=9cm]{fig4.pdf}
\caption{\small{Normalized cumulative radial distribution of BSSs in the central 3.8$\,r_c$ of NGC\,1261.~The full BSS sample (solid black line), B-BSSs (long-dashed blue line) and R-BSSs (short-dashed red line) are plotted.~The corresponding cumulative distribution of the reference SGB population is shown as the shaded region.~We use the centre of gravity coordinates RA~$\!=\!$~03h\,12m\,16.21s, DEC~$\!=\!-55^{\rm o}\,12\arcmin\,58.4\arcsec$ given by \cite{gol10}, and $r_c=0.35'$ \citep{har96}.}}
\label{BSS_norm_prof}
\end{figure}
The results from Figure~\ref{BSS_norm_prof} alone cannot be used to suggest an advanced dynamical state in NGC\,1261, as the central concentration of BSSs is now known to be ubiquitous among all studied GCs.~A more complete understanding can be obtained by looking at the BSS radial profile at larger cluster-centric distances, as the radial distance of the normalized BSS density minimum will depend on the cluster's dynamical age.~\cite{fer12} showed that as a GC evolves dynamically the BSS radial density profile takes the form of a central high maximum and a secondary peak at larger radii with a minimum in between. In order to follow this approach we include the wide-field photometry catalog in our analysis and carefully merge it \footnote{The resulting merged SGB radial profile distribution is checked to be smooth and continuous, therefore, ruling out severe completeness issues for the BSSs, which are in a similar magnitude range.~The merging point is chosen at 3.8 $r_c$.} with the inner-region ACS catalog, providing us with a sampling of NGC\,1261 out to $r\!>\!30\,r_c$.
We plot in Figure~\ref{BSS_rad_frac} the relative fraction of BSSs to SGBs\footnote{They were selected using a $19.2\!<\!V\!<\!19.5$ mag range cut. We use the ``$V$ $ground$" magnitude on the ACS catalog in order to make them compatible with the wide-field gound-based photometry.} as a function of cluster-centric radius, assuming Poissonian noise for the error bars.~The BSSs in the wide-field catalog are the ones selected by \cite{kra10}, i.e. stars on the BSS region of all ($B$-$V$), ($V$-$I$) and ($B$-$I$) CMDs.~We then apply an additional magnitude cut ($I<19.45$ mag) in order to hold the same faint magnitude limit used with the inner sample (see Fig~\ref{cmd1})\footnote{The F814W$<$19.5 mag limit from Fig.~1 is checked using the ``$I$ $ground$" magnitude on the ACS catalog and is found to translate to $I < 19.45$ mag.}.~Considering these selection steps, combining both samples is hence acceptable for our purposes.~We find that the BSS fraction is maximal in the center and drops rapidly with radius reaching near to zero at $r\!\sim\!10\,r_c$.~A subsequent rising in the fraction profile is discernible, although this is only caused by the detection of two BSSs alone (see the inset panel) and therefore not strongly supported statistically.~The absence of a clear outer layer suggests that the majority of the BSS population has already been affected by dynamical friction. Hence, in the framework constructed by \cite{fer12}, NGC\,1261 would classify as a dynamically old cluster and would be grouped in late-{\sc Family\,II}/{\sc Family\,III} , not surprisingly similar to M\,30 and NGC\,362.~Moreover, the half-mass relaxation time is about $10^8$ years, which is actually shorter than that of NGC\,362, $\log t=8.7$ and M\,30, $\log t=9.2$, as found in \cite{pau10}.
\begin{figure}[t!]
\centering
\includegraphics[width=9cm]{fig5.pdf}
\caption{\small{Ratio of BSS to SGB stars as a function of radial distance.~The inset panel shows the number of BSSs and SGB stars. Both panels have a dashed vertical line indicating the approximate location of the cluster's tidal radius, according to \cite{pau10}}.}
\label{BSS_rad_frac}
\end{figure}
However, the core structure of NGC 1261 does not show signatures of core collapse.~It is well approximated by a King model with a concentration parameter $c\!\approx\!1.2$ \citep{har96, pau10}.~It has a central luminosity density of 2.22 $L_{\odot}$pc$^{-3}$ \citep{pau10}, which is relatively low for GCs.~Moreover, the binary fraction radial profile found by \cite{mil12} shows a flat distribution, i.e.~without signs of mass segregation, contrary to the ones in M\,30 and NGC\,362, which are centrally peaked.~Therefore it is not surprising that the binary merger products, i.e.~the R-BSS population, in these GCs are also more centrally segregated than the R-BSS population in NGC\,1261.~This apparent evolutionary contradiction may not be as problematic as it first seems.~According to both Monte-Carlo dynamical models \citep[e.g.][]{heg08} and direct-integration N-body models \citep[e.g.][]{hur12}, clusters can go through core-collapse and then, if there is some energy source in the core, stay or pass through a long-lived post-core-collapse bounce state in which they do not show classic post core-collapse signatures.~The \citeauthor{heg08} models of M\,4 all went through core collapse at $t\!=\!8$\,Gyr, and then remained in a non-collapsed state for another $\sim\!2\!-\!3$ Gyr due to ``binary burning" in the core. \citeauthor{hur12} propose a binary black hole as an alternative central potential energy source.~We know today that black holes may be common in GCs \citep[e.g.][]{str12, cho13}, so that a binary black hole in the core of NGC\,1261 may be considered a valid possibility.\\
\section{Summary and Conclusions}
We find that the inner BSS population in NGC\,1261 includes at least two distinct well defined sequences similar to what was found in M\,30 and NGC\,362, and as well includes a smaller group of BSSs that have unusually blue colours in the CMD, and which could be associated with a distinct coeval BSS population, if confirmed real.~The comparison with collisional stellar evolution models reveals that the B-BSS and eB-BSS sub-samples are consistent with a 2 Gyr and 0.2 Gyr old stellar collision-product population, respectively. This provides the grounds for considering NGC\,1261 an extremely valuable test laboratory for stellar collision and BSS formation models.~This observation along with evidence collected from the literature suggest as a preliminary interpretation for the dynamical history of NGC\,1261 that the cluster experienced a core-collapse phase about 2 Gyr ago, and that since then it has bounced through core oscillations.~The subsequent core oscillations occasionally created more BSSs during these short timescale processes when the central stellar density was particularly enhanced -- one such likely 0.2 Gyr ago. During these periods of core density enhancements the cluster likely burned some of its core binaries, thereby flattening their radial density distribution profile, and currently the cluster is likely in a post core-collapsed state which, according to different simulations, may appear as an unevolved GC. Follow-up spectroscopic characterization of the BSS sequences in NGC\,1261 is of utmost importance in order to confirm and better understand their origins and formation mechanisms, and in particular test the chemical abundance predictions related to different BSS formation models.
\acknowledgments
We thank the anonymous referee for comments that improved the presentation and quality of our results.~MS and THP gratefully acknowledge support from CONICYT through the ALMA-CONICYT Project No.~37070887, FONDECYT Regular Project No.~1121005, FONDAP Center for Astrophysics (15010003), and BASAL Center for Astrophysics and Associated Technologies (PFB-06), as well as support from {\it Deutscher Akademischer Austauschdienst} (DAAD). AS is supported by NSERC.~This research has made use of the Aladin plot tool and the TOPCAT table manipulation software, found at http://www.starlink.ac.uk/topcat/.
{\it Facilities:} \facility{HST (ACS)}.
|
2,869,038,156,376 | arxiv | \section{Introduction}
\label{section_introduction}
The adhesion of cells is mediated by the specific binding of receptor and ligand molecules anchored in the cell membranes. Cell adhesion processes are essential for the distinction of self and foreign in immune responses, the formation of tissues, or the signal transduction across the synaptic cleft of neurons \cite{Alberts02}. These processes have therefore been studied intensively with a variety of experimental methods \cite{Alon95,Grakoui99,Delanoe04,Arnold04,Mossman05}. In addition, experiments on lipid vesicles with membrane-anchored receptor and ligand molecules aim to mimic the specific membrane binding processes leading to cell adhesion \cite{Albersdoerfer97,Maier01,Smith08}.
In many adhesion processes, the anchored receptor and ligand molecules can still diffuse, at least to some extent, within the contact area of the adhering membranes \cite{Grakoui99,Delanoe04,Mossman05}. As a consequence, the receptor-ligand complexes may form different spatial patterns such as clusters or extended domains in the contact area \cite{Grakoui99,Mossman05,Monks98,Davis04}. These pattern formation processes can be understood in the framework of discrete models in which the membranes are divided into small patches, and the receptors and ligands are described as single molecules that are either present or absent in the patches \cite{Lipowsky96,Weikl00,Weikl04,Asfaw06,Weikl06}. These discrete models are lattice models on elastic surfaces, and have two advantages: (i) They automatically incorporate the mutual exclusion of receptor or ligand molecules anchored within the same membrane; and (ii) they lead to effective membrane adhesion potentials that provide an intuitive understanding of the observed behavior in terms of nucleation and growth processes.
Cell adhesion involves many different length scales, from nanometers to tens of micrometers. The largest length scales of micrometers correspond to the diameter of the cell and the diameter of the contact zone in which the cell is bound to another cell or to a supported membrane. The separation of the two membranes in the cell contact zone is orders of magnitude smaller. The membrane separation is comparable to the length of the receptor-ligand complexes, which have a typical extension between 15 and 40 nanometers \cite{Dustin00}. Finally, the smallest relevant length scale is the binding range of a receptor and a ligand molecule, i.e.~the difference between the smallest and the largest local membrane separation at which the molecules can bind. The binding range reflects (i) the range of the lock-and-key interaction, (ii) the flexibility of the two binding partners, and (iii) the flexibility of the membrane anchoring. For the rather rigid protein receptors and ligands that typically mediate cell adhesion, the interaction range is around 1 nanometer. In contrast, the interaction range of surface-anchored flexible tether molecules with specific binding sites is significantly larger \cite{Jeppesen01,Morreira03,Moore06,Martin06}.
The wide range of length scales has important consequences for modeling cell adhesion. In general, the elasticity of the cell membrane is affected by the bending rigidity $\kappa$ of the membrane \cite{Helfrich73}, the membrane tension $\sigma$, and the cytoskeleton that is coupled to the membrane \cite{Gov03,Fournier04,Lin04,Auth07}. The tension dominates over the bending elasticity on length scales larger than the crossover length $\sqrt{\kappa/\sigma}$ \cite{Lipowsky95}, which is of the order of several hundred nanometers for cell membranes \cite{Krobath07}, while the bending elasticity dominates on length scales smaller than the crossover length. The elastic contribution of the cytoskeleton is relevant on length scales larger than the average distance of the cytoskeletal membrane anchors, which is around 100 nanometers \cite{Alberts02}. The overall shape of the cell membrane on micrometer scales therefore is governed by the cytoskeletal elasticity and membrane tension. In the cell adhesion zone, however, the relevant shape deformations and fluctuations of the membranes occur on length scales up to the average distance of the receptor-ligand bonds, since the bonds locally constrain the membrane separation. The average distance of the bonds roughly varies between 50 and 100 nanometers for typical bond concentrations in cell adhesion zones \cite{Grakoui99}. The relevant membrane shape deformations and fluctuations in the cell contact zone are therefore dominated by the bending rigidity of the membranes.
The adhesion of cells is mediated by a multitude of different receptor and ligand molecules. Some of these molecules can be strongly coupled to the cytoskeleton. In focal adhesions of cells, for example, clusters of integrin molecules are tightly coupled to the cytoskeleton {\em via} supramolecular assemblies that impose constraints on the lateral separation of the integrins \cite{Arnold04,Selhuber08}. Through focal adhesions, cells exert and sense forces \cite{Geiger01,Discher05,Bershadsky06,Girard07,Schwarz07,De08}. Other receptor and ligand molecules are not \cite{Delanoe04} or only weakly \cite{DeMond08} coupled to the cytoskeleton. These molecules are mobile and diffuse within the membranes. The diffusion process can be observed with single-molecule experiments \cite{Schuetz00,Sako00}.
The adhesion of membranes {\em via} mobile receptor and ligand molecules has been studied theoretically with a variety of models. These models can be grouped into two classes. In both classes, the membranes are described as thin elastic sheets. In the first class of models, the description is continuous in space, and the distribution of the membrane-anchored receptor and ligand molecules on the membranes are given by continuous concentration profiles \cite{Bell78,Bell84,Komura00,Bruinsma00,Chen03,Coombs04,Shenoy05,Wu06}. Dynamic, time-dependent properties of such models have been studied by numerical solution of reaction-diffusion equations \cite{Qi01,Raychaudhuri03,Burroughs02,Shenoy05}. In the second, more recent class of models, the membranes are discretized, and the receptors and ligands are described as single molecules \cite{Lipowsky96,Weikl00,Weikl01,Weikl02b,Asfaw06,Weikl06,Smith05,Krobath07,Tsourkas07,Tsourkas08,Reister08}. The dynamic properties can be numerically studied with Monte Carlo simulations \cite{Weikl02a,Weikl04,Krobath07,Tsourkas07,Tsourkas08}, and central aspects of the equilibrium behavior can be directly inferred from the partition function \cite{Weikl00,Weikl01,Weikl06,Asfaw06}.
\section{Effective adhesion potential}
\label{section_effective_adhesion_potential}
\begin{figure}[t]
\begin{center}
\resizebox{\columnwidth}{!}{\includegraphics{figure1small}}
\caption{(a) A membrane segment with receptor molecules (top) interacting with ligands embedded in an apposing membrane (bottom). A receptor can bind a ligand molecule if the local separation of the membranes is close to the length of the receptor-ligand complex. -- (b) The attractive interactions between the receptor and ligand molecules lead to an effective single-well adhesion potential $V_\text{ef}$ of the membranes. The depth $U_\text{ef}$ of the potential well depends on the concentrations and binding affinity of receptors and ligands, see eq.~(\ref{Uef}). The width $l_\text{we}$ of the binding well is equal to the binding range of the receptor-ligand interaction.
}
\label{figure_model_one}
\end{center}
\end{figure}
In discrete models, the two apposing membranes in the contact zone of cells or vesicles are divided into small patches \cite{Lipowsky96,Weikl00,Weikl01,Weikl02b,Asfaw06,Weikl06,Smith05,Krobath07,Tsourkas07,Tsourkas08,Reister08}. These patches can contain a single receptor or ligand molecule. Mobile receptor and ligand molecules diffuse by `hopping' from patch to patch, and the thermal fluctuations of the membranes are reflected in variations of the local separation of apposing membrane patches. A receptor can bind to a ligand molecule if the ligand is located in the membrane patch apposing the receptor, and if the local separation of the membranes is close to the length of the receptor-ligand complex. In these models, the linear size $a$ of the membrane patches is typically chosen around 5 nm to capture the whole spectrum of bending deformations of the lipid membranes \cite{Goetz99}.
Cells can interact {\em via} a multitude of different receptors and ligands. However, it is instructive to start with the relatively simple situation in which the adhesion is mediated by a single type of receptor-ligand bonds as in fig.~\ref{figure_model_one}(a). Such a situation occurs if a cell adheres to a supported membane with a single type of ligands, or if a vesicle with membrane-anchored receptors adheres to a membrane with complementary ligands. The effective membrane adhesion potential mediated by the receptor and ligands can be calculated by integrating over all possible positions of the receptors and ligands in the partition function of the models \cite{Weikl06,Weikl01}. In the case of a single type of receptors and ligands, the effective adhesion potential of the membranes is a single-well potential with the same range $l_\text{we}$ as the receptor-ligand interaction, but an effective binding energy $U_\text{ef}$ that depends on the concentrations and binding energy $U$ of receptors and ligands, see fig.~\ref{figure_model_one}(b). For typical concentrations of receptors and ligands in cell membranes, which are more than two orders of magnitude smaller than the maximum concentration $1/a^2\simeq 4\cdot 10^4/ \mu\text{m}^2$ in our discretized membranes with patch size $a\simeq 5$ nm, the effective potential depth is \cite{KrobathPreprint}
\begin{equation}
U_{\rm ef} \approx k_B T \, [R] [L ]\, a^2 e^{U/k_BT}
\label{Uef}
\end{equation}
where $[R]$ and $[L]$ are the area concentrations of unbound receptors and ligands. The quantity
\begin{equation}
K_\text{pl} \equiv a^2 e^{U/k_BT}
\label{Kpl}
\end{equation}
in eq.~(\ref{Uef}) can be interpreted as the binding equilibrium constant of the receptors and ligands in the case of two planar and parallel membranes with a separation equal to the length of the receptor-ligand bonds. The equilibrium constant characterizes the binding affinity of the molecules and can, in principle, be measured with the surface force apparatus in which the apposing membranes are supported on rigid substrates \cite{Israelachvili92,Bayas07}. In the case of flexible membranes, the binding affinity of the receptors and ligands is more difficult to capture, see next section.
\begin{figure}[t]
\begin{center}
\resizebox{\columnwidth}{!}{\includegraphics{figure2small}}
\caption{(a) Two membranes interacting {\it via} long (red) and short (green) receptor-ligand complexes. -- (b) The attractive interactions between the two types of receptors and ligands lead to an effective double-well adhesion potential $V_\text{ef}$ of the membranes. The potential well 1 at small membrane separations $l$ reflects the interactions of the short receptor-ligand complexes, and the potential well 2 at larger membrane separations the interactions of the long receptor-ligand complexes. The depths $U_1^\text{ef}$ and $U_2^\text{ef}$ of the two potential wells depend on the concentrations and binding energies of the two types of receptors and ligands, see eqs.~(\ref{U1ef}) and (\ref{U2ef}). }
\label{figure_model_two}
\end{center}
\end{figure}
The interaction of cells is often mediated by several types of receptor-ligand complexes that differ in their length. For two types of receptors and ligands as in fig.~\ref{figure_model_two}, the effective adhesion potential of the membranes is a double-well potential \cite{Asfaw06}. The depths of the two wells
\begin{eqnarray}
U_1^\text{ef} \approx k_BT \, [R_1][L_1] \, a^2 e^{U_1/k_BT}
\label{U1ef}\\
U_2^\text{ef} \approx k_BT \, [R_2][L_2] \, a^2 e^{U_2/k_BT}
\label{U2ef}
\end{eqnarray}
depend on the concentrations and binding energies $U_1$ and $U_2$ of the different types of receptors and ligands \cite{KrobathPreprint}. In analogy to eq.~(\ref{Kpl}), the quantities $K_\text{pl,1}\equiv a^2 e^{U_1/k_BT}$ and $K_\text{pl,2}\equiv a^2 e^{U_2/k_BT}$ can be interpreted as binding equilibrium constants in the case of planar membranes with a separation equal to the lengths $l_1$ or $l_2$ of the receptor-ligand complexes.
Repulsive membrane-anchored molecules such as anchored polymers or glycoproteins can lead to additional barriers in the effective adhesion potential \cite{Weikl01,Weikl02b,Weikl06}. The effective adhesion potentials simplify the characterization of the equilibrium properties of the membranes, and lead to an intuitive understanding of these properties, see next sections.
\section{Binding cooperativity}
\label{section_binding_cooperativity}
A receptor molecule can only bind an apposing ligand if the local membrane separation is comparable to the length of the receptor-ligand complex. A central quantity therefore is the fraction $P_b$ of the apposing membranes with a separation within the binding range of the receptor-ligand interaction. The concentration of bound receptor-ligand complexes
\begin{equation}
[RL] \approx P_b\, K_\text{pl}\, [R] [L]
\label{RL}
\end{equation}
is proportional to $P_b$ as well as to the concentrations $[R]$ and $[L]$ of unbound receptors and ligands \cite{KrobathPreprint}.
Thermal shape fluctuations of the membranes on nanometer scales in general lead to values of $P_b$ smaller than 1. For cell membranes, these nanometer scale fluctuations are not, or only weakly, suppressed by the cell cytoskeleton, in contrast to large-scale shape fluctuations. For simplicity, we assume now that the adhesion of the membranes is mediated by a single type of receptor and ligand molecules as in
fig.~\ref{figure_model_one}(a). The precise value of $P_b$ then depends on the well depth $U_\text{ef}$ of the effective adhesion potential shown in fig.~\ref{figure_model_one}(b), and on the bending rigidities of the membranes. For typical lengths and concentrations of receptors and ligands in cell adhesion zones, the fraction $P_b$ of the membranes within binding range of the receptors and ligands turns out to be much smaller than 1, and scaling analysis and Monte Carlo simulations lead to the relation \cite{KrobathPreprint}
\begin{equation}
P_b \approx c \, \kappa\, l_\text{we}^2 \, U_\text{ef} / (k_B T)^2
\label{Pb_approx}
\end{equation}
with prefactor $c=13 \pm 1$. With eqs.~(\ref{Uef}) and (\ref{Kpl}), we obtain
\begin{equation}
P_b \approx c \, (\kappa/k_B T) l_\text{we}^2 K_\text{pl }[R][L]
\label{Pb_approx_II}
\end{equation}
which shows that the membrane fraction $P_b$ within the binding range of the receptors and ligands is proportional to $[R]$ and $[L]$. Inserting eq.~(\ref{Pb_approx_II}) into eq.~(\ref{RL}) leads to
\begin{equation}
[RL] \approx c (\kappa/k_B T) l_\text{we}^2 K_\text{pl}^2 [R]^2 [L]^2
\label{central_equation}
\end{equation}
The concentration $[RL]$ of receptor-ligand complexes in the adhesion zone thus depends quadratically on the concentrations $[R]$ and $[L]$ of unbound receptors and ligands, which indicates cooperative binding. The binding cooperativity results from a `smoothening' of the thermally rough membranes and, thus, an increase of $P_b$ with increasing concentrations $[R]$ and $[L]$ of receptors and ligands, which facilitates the formation of additional receptor-ligand complexes. The relations (\ref{Pb_approx_II}) and (\ref{central_equation}) are good approximations up to $P_b\lesssim 0.2$, and can be extended to larger values of $P_b$ \cite{KrobathPreprint}.
For {\em soluble} receptor and ligand molecules, in contrast, the volume concentration
of the bound receptor-ligand complexes
\begin{equation}
[RL]_\text{3D} = K_\text{3D} [R]_\text{3D} [L]_\text{3D}
\label{equilibrium_constant}
\end{equation}
is proportional to the volume concentrations $[R]_\text{3D}$ and $[L]_\text{3D}$ of unbound receptors and unbound ligands in the solution. The binding affinity of the molecules then can be characterized by the equilibrium constant $K_\text{3D}$, which depends on the binding free energy of the complex \cite{Schuck97,Rich00,McDonnell01}. In analogy to eq.~(\ref{equilibrium_constant}), the binding affinity of membrane-anchored receptors and ligands is often quantified by
\begin{equation}
K_\text{2D} \equiv \frac{[RL]}{[R][L]}
\label{K2D}
\end{equation}
where $[RL]$, $[R]$, and $[L]$ are the area concentrations of bound receptor-ligand complexes, unbound receptors, and unbound ligands \cite{Orsello01,Dustin01,Williams01}.
However, it follows from our relation (\ref{central_equation}) that $K_\text{2D}$ is not constant, but depends on the concentrations of the receptors and ligands. From the eqs.~(\ref{Kpl}) and (\ref{RL}), we obtain the general relation
\begin{equation}
K_\text{2D} = P_b K_\text{pl}
\label{K2DKpl}
\end{equation}
As mentioned in the previous section, $K_\text{pl}$ is the well-defined two-dimensional equilibrium constant of the receptors and ligands in the case of planar membranes with $P_b=1$, e.g.~two supported membranes in the surface force apparatus with a separation equal to the length of the receptor-ligand complex \cite{Israelachvili92,Bayas07}.
The relation (\ref{K2DKpl}) also helps to understand why different experimental methods for measuring $K_\text{2D}$ have led to values that differ by several orders of magnitude \cite{Dustin01}. In fluorescence recovery experiments, $K_\text{2D}$ is measured in the equilibrated contact zone of a cell adhering to a supported membrane with fluorescently labeled ligands \cite{Dustin96,Dustin97,Zhu07,Tolentino08}. In micropipette experiments, in contrast, $K_\text{2D}$ is measured for initial contacts between two cells \cite{Chesla98,Williams01,Huang04}, for which $P_b$ can be several orders of magnitude smaller than in equilibrium \cite{KrobathPreprint}.
\section{Adhesion of vesicles}
\label{section_adhesion_of_vesicles}
%
\begin{figure}[t]
\begin{center}
\resizebox{0.95\columnwidth}{!}{\includegraphics{figure3small}}
\caption{Concentrations $[RL]$ and $[R]$ of bound and unbound receptors as a function of the receptor number $N_R$ of a vesicle adhering to a supported membrane, see eqs.~(\ref{Rvesicle}) and (\ref{RLvesicle}). In this numerical example, we have chosen the bending rigidity $\kappa = 25 k_B T$, the binding range $l_\text{we}=1$ nm, the binding equilibrium constant $K_\text{pl} = 1\, \mu \text{m}^2$ for planar membranes, and a ligand concentration $[L] = 20/ \mu \text{m}^2$ for the supported membrane, which results in the value $0.14/\mu \text{m}^2$ for the parameter $b$ in eqs.~(\ref{Rvesicle}) and (\ref{RLvesicle}). The total area of the vesicle is $100\, \mu \text{m}^2$, and the area of the contact zone is $30\,\mu\text{m}^2$. The fraction $P_b$ of the vesicle membrane in the contact zone with a separation within binding range of the receptors and ligands varies with $[R]$, see eq.~\ref{Pb_approx_II}, and attains the maximum value $P_b = 0.17$ for $N_R=5000$ in this example.
}
\label{figure_vesicle}
\end{center}
\end{figure}
Important aspects of cell adhesion can be mimicked by lipid vesicles with anchored receptor molecules \cite{Albersdoerfer97,Kloboucek99,Maier01,Smith06,Lorz07,Purrucker07,Smith08}. We focus here on a vesicle adhering to a supported membrane with complementary ligands. In the strong adhesion limit, the shape of the vesicle can be approximated by a spherical cap \cite{Lipowsky05,Seifert90}. The volume of the cap depends on the osmotic pressure balance between the outside and the interior of the vesicle.
If this volume is nearly constant, the contact area $A_c$ is nearly independent of the adhesion free energy \cite{Seifert90,Tordeux02,Gruhn05,Smith05}.
Since the total number $N_R$ of receptors in the vesicle membrane is fixed, we have
\begin{equation}
N_R = [R] A + [RL] A_c
\end{equation}
where $A$ is the total area of the vesicle, and $[RL]$ is the concentration of receptor-ligand complexes in the contact area. For typical small concentrations of receptors and ligands, the concentration $[R]$ of unbound receptors within the contact area and within the non-adhering membrane section of the vesicle are approximately equal in equilibrium since the excluded volume of the receptor-ligand complexes in the contact area is negligible. With eq.~(\ref{central_equation}), we then obtain
\begin{equation}
[R] = \frac{\sqrt{A^2 + 4 b A_c N_R}-A}{2 b A_c}
\label{Rvesicle}
\end{equation}
and
\begin{equation}
[RL] = \frac{\left(\sqrt{A^2 + 4 b A_c N_R}-A\right)^2}{4 b A_c^2}
\label{RLvesicle}
\end{equation}
with $b = c (\kappa/k_B T) l_\text{we}^2 K_\text{pl}^2 [L]^2$. Here, $[L]$ is the concentration of the unbound ligands in the supported membrane, which is nearly independent of the binding state of the vesicle if the membrane is large. Because of the binding cooperativity of the receptors and ligands, the concentrations $[R]$ and $[RL]$ are not linear in $N_R$, see fig.~\ref{figure_vesicle} for a numerical example.
\section{Domains of long and short receptor-ligand complexes}
\subsection{Critical concentrations for domain formation}
\label{subsection_critical_concentrations}
\begin{figure*}[t]
\begin{center}
\resizebox{2\columnwidth}{!}{\includegraphics{figure4small}}
\caption{Phase diagram of membranes adhering via long and short receptor/ligand complexes.
The membranes are unbound for small well depths $U_1^\text{ef}$ and $U_2^\text{ef}$ of the effective interaction potential shown in Fig.~\ref{figure_model_two}(b), i.e.~for small concentrations or binding energies of receptors and ligands, see
eqs.~(\ref{U1ef}) and (\ref{U2ef}). At large values of $U_1^\text{ef}$ and $U_2^\text{ef}$, the membranes are either bound in well 1 or well 2, i.e.~they are either bound by the short or by the long receptor/ligand complexes. At intermediate well depths $U_1^\text{ef}$ and $U_2^\text{ef}$, the membranes are bound in both potential wells. The critical point for the lateral phase separation (star) follows from eq.~(\ref{Uc}). For typical dimensions of cell receptors and ligands, the critical well depth $U_c^\text{ef}$ for lateral phase separation is significantly larger than the critical depths of unbinding \cite{Asfaw06}. In the absence of other repulsive interactions as assumed here, the membranes unbind due to steric repulsion.
}
\label{figure_phase_diagram}
\end{center}
\end{figure*}
Cells often interact {\em via} receptor-ligand complexes that differ significantly in size. For example, two important complexes in T-cell adhesion are the complexes of the T-cell receptor (TCR) with a length of 15 nm and integrin complexes with a length of 40 nm \cite{Dustin00}. The length mismatch induces a membrane-mediated repulsion between the different complexes because the membranes have to be curved to compensate the mismatch, which costs bending energy.
The equilibrium behavior of two membranes adhering {\em via} long and short receptor-ligand complexes is determined by the effective double-well adhesion potential shown in fig.~\ref{figure_model_two}(b), and by the bending rigidities of the membranes. The depths of the two wells reflect the concentrations and binding affinity of the two different types of receptors and ligands, see eqs.~(\ref{U1ef}) and (\ref{U2ef}) in section \ref{section_effective_adhesion_potential}. If the two wells are relatively shallow, membrane segments can easily cross the barrier between the wells, driven by thermal fluctuations. If the two wells are deep, the crossing from one well to the other well is suppressed by the potential barrier of width $l_\text{ba}$ between the wells. The potential barrier induces a line tension between a membrane segment that is bound in one of the wells and an adjacent membrane segment bound in the other well. Beyond a critical depth of the potential wells, the line tension leads to the formation of large membrane domains that are bound in well one or well two, see fig.~\ref{figure_phase_diagram}. Within each domain, the adhesion of the membranes is predominantly mediated by one of the two types of receptor-ligand complexes.
Scaling arguments indicate that domains bound in either well 1 or well 2 are formed if the depths $U_\text{1}^\text{ef}$ and $U_\text{2}^\text{ef}$ of the two potential wells exceed the critical potential depth \cite{Asfaw06}
\begin{equation}
U_c^\text{ef} = \frac{c (k_BT)^2}{\kappa l_\text{we} l_\text{ba}} \label{Uc}
\end{equation}
Numerical results from Monte Carlo simulations confirm eq.~(\ref{Uc}) and lead to the value $c = 0.225 \pm 0.02$ for the dimensionless prefactor \cite{Asfaw06}. The critical potential depth thus depends on the effective rigidity $\kappa=\kappa_1\kappa_2/(\kappa_1+\kappa_2)$ of two membranes with bending rigidities $\kappa_1$ and $\kappa_2$ and the width $l_\text{we}$ and separation $l_\text{ba}$ of the two potential wells, see fig.~\ref{figure_model_two}(b). The separation $l_\text{ba}$ of the wells is close to the length mismatch of the different types of receptor-ligand complexes, which is 25 nm for T cells. A reasonable estimate for the interaction range $l_\text{we}$ of the protein receptors and ligands that mediate cell adhesion is 1 nm, see section \ref{section_introduction}. With an effective bending rigidity $\kappa=\kappa_1\kappa_2/(\kappa_1+\kappa_2)$ of, e.g., 25 $k_B T$, we obtain the estimate $U_c^\text{ef} \simeq 360\,k_B T/\mu\text{m}^2$ for the critical potential depth of domain formation during T-cell adhesion. For planar-membrane equilibrium constants $K_\text{pl,1}$ and $K_\text{pl,2}$ around $1\,\mu\text{m}^2$, for example, the effective potential depths (\ref{U1ef}) and (\ref{U2ef}) exceed this critical potential depth if the concentrations of unbound receptors and ligands are larger than $20/\mu\text{m}^2$.
\subsection{Domain patterns during immune cell adhesion}
\label{section_domain_patterns}
\begin{figure}[t]
\begin{center}
\resizebox{0.6\columnwidth}{!}{\includegraphics{figure5small}}
\caption{Domain patterns in the T-cell contact zone: (a) Final pattern of helper T cells with a central TCR domain (green) surrounded by an integrin domain (red) \cite{Monks98,Grakoui99}. The pattern results from cytoskeletal transport of TCRs towards the contact zone center \cite{Mossman05,Weikl04}. -- (b) Simulated final pattern in the absence of TCR transport \cite{Weikl04}. The length of the boundary line between the TCR and the integrin domain is minimal in this pattern. -- (c) and (d) The two types of intermediate patterns observed in the first minutes of adhesion \cite{Davis04}. In simulations, both patterns result from the nucleation of TCR clusters in the first seconds of adhesion and the subsequent diffusion of unbound TCR and MHC-peptide ligands in the contact zone \cite{Weikl04}. The closed TCR ring in pattern (c) forms from fast-growing TCR clusters in the periphery of the contact zone at sufficiently large TCR-MHC-peptide concentrations. The pattern (d) forms at smaller TCR-MHC-peptide concentrations.}
\label{figure_patterns}
\end{center}
\end{figure}
The domains of long and short receptor-ligand complexes formed during the adhesion of T cells and other immune cells such as natural killer cells evolve in characteristic patterns. For T cells, the domains either contain complexes of TCR and its ligand MHC-peptide, or integrin complexes. The final domain pattern in the T-cell contact zone is formed within 15 to 30 minutes and consists of a central TCR domain surrounded by a ring-shaped integrin domain \cite{Monks98,Grakoui99}, see fig.~\ref{figure_patterns}(a). Interestingly, the intermediate patterns formed within the first minutes of T-cell adhesion are quite different \cite{Grakoui99,Davis04}. They are either inverse to the final pattern, with a central integrin domain surrounded by a ring-shaped TCR domain, see fig.~\ref{figure_patterns}(c), or exhibit several nearly circular TCR domains in the contact zone, see fig.~\ref{figure_patterns}(d).
To understand these patterns, several groups have modeled and simulated the adhesion of T cells and other immune cells \cite{Qi01,Weikl02a,Burroughs02,Lee03,Raychaudhuri03,Weikl04,Coombs04,Figge06,Tsourkas07,Tsourkas08}. One open question concerned the role of the T-cell cytoskeleton, which polarizes during adhesion, with a focal point in the center of the contact zone \cite{Alberts02,Dustin98}. Some groups have found that the final T-cell pattern with a central TCR domain can emerge independently of cytoskeletal processes \cite{Qi01,Lee03}. In contrast, Monte Carlo simulations of discrete models indicate that the central TCR cluster is only formed if TCR molecules are actively transported by the cytoskeleton towards the center of the contact zone \cite{Weikl04}. The active transport has been simulated by a biased diffusion of TCRs towards the contact zone center, which implies a weak coupling of TCRs to the cytoskeleton. In the absence of active TCR transport, the Monte Carlo simulations lead to the final, equilibrium pattern shown in fig.~\ref{figure_patterns}(b), which minimizes the energy of the boundary line between the TCR and the integrin domain \cite{Weikl04}. In agreement with these simulations, recent T-cell adhesion experiments on patterned substrates reveal cytoskeletal forces that drive the TCRs towards the center of the contact zone \cite{Mossman05,DeMond08}. The experiments indicate a weak frictional coupling of the TCRs to the cytoskeletal flow \cite{DeMond08}.
The intermediate patterns formed in the Monte Carlo simulations closely resemble the intermediate immune-cell patterns shown in figs.~\ref{figure_patterns}(c) and (d). In the first seconds of adhesion, the Monte Carlo patterns exhibit small TCR clusters \cite{Weikl04}. In the following seconds, the diffusion of free TCR and MHC-peptide molecules into the contact zone lead to faster growth of TCR clusters close to the periphery of the contact zone \cite{footnote}. For sufficiently large TCR-MHC-peptide concentrations, the peripheral TCR clusters grow into the ring-shaped domain of fig.~\ref{figure_patterns}(c). At smaller TCR-MHC-peptide concentration, the initial clusters evolve into the pattern of fig.~\ref{figure_patterns}(d). In agreement with experimental observations \cite{Davis04}, only these two types of intermediate patterns are formed in the simulations. The simulated patterns emerge spontaneously from the nucleation of TCR clusters and the diffusion of unbound TCR and MHC-peptide into the contact zone.
\subsection{Implications for T-cell activation}
\begin{figure}[b]
\begin{center}
\resizebox{0.75\columnwidth}{!}{\includegraphics{figure6small}}
\caption{Schematic diagram for the joint role of foreign and self MHC-peptides in TCR cluster formation and T-cell activation. Here, $[M_f]$ is the concentration of foreign MHC-peptides, and $[M_s]$ is the concentration of self MHC-peptides. The solid line represents the threshold for TCR cluster formation given by eq.~(\ref{threshold}). The slope of this line is the negative ratio $K_\text{TMs}/K_\text{TMf}$ of the binding equilibrium constants for the interaction of TCR with self MHC-peptide and with foreign MHC-peptide. For simplicity, we have assumed here a single, dominant type of self MHC-peptides.
}
\label{figure_TCRclusters}
\end{center}
\end{figure}
T cells mediate immune responses by adhering to cells that display foreign peptides on their surfaces \cite{Alberts02,Janeway07}. The peptides are presented by MHC proteins on the cell surfaces, and are recognized by the T-cell receptors (TCR). T-cell activation requires the binding of TCRs to the MHC-peptide complexes. But how precisely these binding events trigger T-cell activation still is a current focus of immunology (for reviews, see refs.~\cite{Choudhuri07,Davis06,Krogsgaard07}). Recent experiments indicate that the first T-cell activation signals coincide with the formation of TCR microclusters within the first seconds of T-cell adhesion \cite{Campi05,Yokosuka05,Bunnell02,Varma06,Yokosuka08,Yokosuka09}.
In the discrete model introduced in section \ref{section_effective_adhesion_potential} and fig.~\ref{figure_model_two}, TCR clusters in the T-cell contact zone can only form if two conditions are met. First, the effective potential depth $U_1^\text{ef}$ for the short TCR-MHC-peptide complexes and the depth $U_2^\text{ef}$ for the long integrin complexes have to exceed the critical depth (\ref{Uc}). Second, the effective potential $U_1^\text{ef}$ for the TCR complexes has to be larger than the effective depth $U_2^\text{ef}$ in the situation where no TCRs are bound. To understand the second condition, one has to realize that the concentrations of unbound TCRs and unbound integrins depend on the area fractions of the TCR and integrin clusters and domains in the contact zone. If no TCRs are bound, i.e.~if the whole contact zone is occupied by an integrin domain, the concentration of unbound TCRs is maximal. Hence, also the effective depth $U_1^\text{ef}$ for the TCRs is maximal in this situation, see eq.~(\ref{U1ef}). TCR clusters now form if $U_1^\text{ef}$ is larger than $U_2^\text{ef}$, which leads to a decrease in the concentration of unbound TCRs and, thus, to a decrease in $U_1^\text{ef}$. The area fraction of the TCR clusters grows until the equilibrium situation with $U_1^\text{ef}= U_2^
\text{ef}$ is reached \cite{Asfaw06}.
T-cell activation requires a threshold concentration of foreign MHC-peptide complexes. Interestingly, the threshold concentration of foreign MHC-peptide depends on the concentration of self MHC-peptide complexes, i.e.~of complexes between MHC and self peptides derived from proteins of the host cell \cite{Irvine02,Purbhoo04}. The foreign MHC-peptide complexes, in contrast, are complexes of MHC with peptides derived from viral or bacterial proteins. Self MHC-peptides typically bind weakly to TCR, since strong binding can result in autoimmune reactions. However, the number of self MHC-peptide complexes typically greatly exceed the number of foreign MHC-peptide on cell surfaces. Both self and foreign MHC-peptide complexes contribute to the effective potential depth $U_1^\text{ef}$ of the TCR-MHC-peptide interaction. For simplicity, we assume here a single, dominant type of self MHC-peptides with concentrations $[M_s]$. The effective potential depth then is
\begin{equation}
U_1^\text{ef} = k_B T \left([T][M_f] K_\text{TMf} + [T][M_s] K_\text{TMs} + \ldots \right) \label{U1efT}
\end{equation}
where $[T]$ and $[M_f]$ are the concentrations of unbound TCR and foreign MHC-peptide, and $K_\text{TMf}$ and $K_\text{TMs}$ are the binding equilibrium constants of foreign and self TCR-MHC-peptide complexes in the case of planar membranes, see eq.~(\ref{Kpl}). The dots in eq.~(\ref{U1efT}) indicate possible contributions from other receptor-ligand complexes with the same length as the TCR-MHC-peptide complex, e.g.~from the CD2-CD58 complex \cite{Dustin00}. In addition, repulsive glycoproteins with a length larger than TCR-MHC-peptide complex can affect $U_1^\text{ef}$ \cite{Weikl04,Weikl02a}. Similarly, the depth $U_2^\text{ef}$ of the second well depends on the concentrations and binding equilibrium constants of integrins and its ligands.
Let us now suppose that the numbers of TCRs, co-receptors such as CD2, integrins, and glycoproteins are approximately equal for different T cells and apposing cells, while the numbers of foreign and self MHC-peptides vary. The second condition $U_1^\text{ef}> U_2^\text{ef}$ for TCR cluster formation then leads to
\begin{equation}
[M_f] K_\text{TMf} + [M_s] K_\text{TMs} > c_t \label{threshold}
\end{equation}
where $c_t$ is a dimensionless threshold that depends on the TCR, co-receptor, integrin, and glycoprotein concentrations, etc. The threshold concentration of foreign MHC-peptide complexes for TCR cluster formation thus depends on the concentration of self MHC-peptide complexes, see fig.~\ref{figure_TCRclusters}. If the formation of TCR microclusters coincides with early activation signals as suggested in Refs.~\cite{Campi05,Yokosuka05}, the inequality (\ref{threshold}) also helps to understand the joint role of foreign and self MHC-peptides in T-cell activation.
\section{Adhesion {\em via} crosslinker molecules or adsorbed particles}
\label{section_adsorbed_particles}
\begin{figure}[b]
\begin{center}
\resizebox{0.7\columnwidth}{!}{\includegraphics{figure7small}}
\caption{
(a) Two membranes with receptors binding to solute molecules. At small membrane separations, the molecules can bind to two apposing receptors and, thus, crosslink the membranes. -- (b) Two membranes in contact with a solution of adhesive molecules or particles. A particle can bind the two membranes together for membrane separations slightly larger than the particle diameter. At larger separations, the particles can only bind to one of the membranes. -- (c) Effective adhesion potential $V$ of the membranes in subfigure (b) as a function of the membrane separation $l$ for small concentrations of the particles \cite{Rozycki08b}. The effective potential has a minimum at the separation $l=d+r$ where $d$ is the particle diameter, and $r$ is the range of the adhesive interaction between the particles and the membranes. At this separation, the particles are bound to both membranes. The effective potential is constant for large separations at which the particles can only bind to one of the membranes. The potential barrier at intermediate separations $d+ 2r < l < 2d$ results from the fact that a particle bound to one of the membranes locally `blocks' the binding of other particles to the apposing membrane.
}
\label{figure_linkers}
\end{center}
\end{figure}
\begin{figure*}[t]
\begin{center}
\resizebox{2.1\columnwidth}{!}{\includegraphics{figure8small}}
\caption{Effective adhesion energy $U_{\rm ef}$, given in eq.~(\ref{Uef_linkers}), as a function of the volume fraction $\phi$ of the adhesive particles for the binding energy $U= 8k_BT$ and $q = 0.25$. The effective adhesion energy is maximal at the optimal volume fraction $\phi^{\star} \approx e^{-U/k_BT}/ q \simeq 1.34 \cdot 10^{-3}$. At the optimal volume fraction, the particle coverage of two planar parallel membranes is close to 50\% for large separations, and almost 100\% for small separations at which the particles can bind to both surfaces \cite{Rozycki08b}. }
\label{figure_linkers_II}
\end{center}
\end{figure*}
The binding of receptor molecules on apposing membranes or surfaces is sometimes mediated by linker or connector molecules, see fig.~\ref{figure_linkers}(a). Biotinylated lipids in apposing membranes, for example, can be crosslinked by the connector molecule streptavidin \cite{Albersdoerfer97,Leckband94}. The effective binding affinity of the membranes then depends both on the area concentrations of the membrane receptors and the volume concentration of the linker molecules. A similar situation arises if adhesive molecules or particles directly bind to lipid bilayers \cite{Hu04,Baksh04,Winter06}. The adhesive particles can crosslink two apposing membranes if the membrane separation is close to the particle diameter, see fig.~\ref{figure_linkers}(b). At large membrane separations, the particles can only bind to one of the membranes.
The effective, particle-mediated adhesion potential of the membranes can be determined by integrating over all possible positions of the adhesive particles or linker molecules in the partition function of the considered model. Conceptually, this is similar to the calculation of the effective adhesion potential for membranes interacting {\em via} anchored receptors and ligands, which requires an integration over all positions of the receptor and ligand molecules in the membranes, see section \ref{section_effective_adhesion_potential}. For simplicity, we consider here the adhesive particles of fig.~\ref{figure_linkers}(b), which interact directly with the lipid bilayers. The explicit integration over the particle positions requires spatial discretizations. In a lattice model, the space between the apposing membranes is discretized into a cubic lattice with a lattice spacing equal to the particle diameter $d$ \cite{Rozycki08b}. In an alternative semi-continuous model, only the two spatial directions parallel to the membranes are discretized, while the third spatial direction perpendicular to the membranes is continuous \cite{Rozycki08b}. In both models, the effective, particle-mediated adhesion potential at large membrane separations has the form
\begin{equation}
V_\infty \approx - 2\frac{k_B T}{d^2} \ln\left[1+q \phi e^{U/k_B T}\right]
\label{Vinfty}
\end{equation}
for small volume fractions $\phi$ and large binding energies $U$ of the particles. At small separations close to particle diameter, the adhesion potential exhibits a minimum
\begin{equation}
V_\text{min} \approx - \frac{k_B T}{d^2} \ln\left[1+q \phi e^{2 U/k_B T}\right]
\label{Vmin}
\end{equation}
The model-dependent factor $q$ in eqs.~(\ref{Vinfty}) and (\ref{Vmin}) has the value 1 in the lattice gas model and the value $r/d$ in the semi-continous model with interaction range $r$ of the adhesive particles. In the semi-continuous model, the potential minimum is located at the membrane separation $l=d+r$, see fig.~\ref{figure_linkers}(c).
Independent of these two models, the eqs.~(\ref{Vinfty}) and (\ref{Vmin}) can also be understood as Langmuir adsorption free energies per binding site. Eq.~(\ref{Vmin}) can be interpreted as the Langmuir adsorption free energy for small membrane separations at which a particle binds both membranes with total binding energy $2U$, and eq.~(\ref{Vinfty}) as the Langmuir adsorption free energy for large separations. The Langmuir adsorption free energies result from a simple two-state model in which a particle is either absent (Boltzmann weight $1$) or present (Boltzmann weights $q\, \phi \, e^{2U/k_BT}$ and $q\, \phi \, e^{U/k_BT}$, respectively) at a given binding site.
The effective, particle-mediated adhesion energy of the membranes can be defined as
\begin{equation}
U_\text{ef} \equiv V_\infty - V_\text{min} \approx \frac{k_B T}{d^2} \ln \frac{1+q \phi e^{2 U/k_B T}}{\left(1+q \phi e^{U/k_B T}\right)^2}
\label{Uef_linkers}
\end{equation}
Interestingly, the effective adhesion energy is maximal at the volume fraction $\phi^{\star} \simeq e^{-U/k_BT} / q$, and considerably smaller at smaller or larger volume fractions, see fig.~\ref{figure_linkers_II}. At this optimal volume fraction, the particle coverage $c_{\infty} = -(d^2 /2) (\partial V_{\infty} / \partial U) \approx \phi /( \phi + \phi^{\star})$ of the unbound membranes is 50\%. In contrast, the particle coverage $c_{\rm min} = - (d^2 /2) (\partial V_{\rm min} / \partial U) \approx \phi /( \phi + \phi^{\star} e^{-U/k_BT})$ of the bound membranes is close to 100\% at $\phi = \phi^{\star}$. Bringing the surfaces from large separations within binding separations thus does not `require' desorption or adsorption of particles at the optimal volume fraction. The existence of an optimal particle volume fraction has important implications that are accessible to experiments, such as `re-entrant transitions' in which surfaces or colloidal objects first bind with increasing concentration of adhesive particles, and unbind again when the concentration is further increased beyond the optimal concentration.
\section{Active switching of adhesion receptors}
\label{section_active_switching}
Some adhesion receptors can be switched between different conformations. A biological example of switchable, membrane-anchored adhesion receptors are integrins. In one of their conformations, the integrin molecules are extended and can bind to apposing ligands \cite{Takagi02,Kim03,Dustin04}. In another conformation, the molecules are bent and, thus, deactivated. The transitions between these conformations are triggered by signaling cascades in the cells, which typically require energy input, e.g.~{\em via} ATP. Because of this energy input, the switching process is an active, non-equilibrium process. In biomimetic applications with designed molecules, active conformational transitions may also be triggered by light \cite{Moeller98,Ichimura00}. Other active processes of biomembranes include the forces exerted by embedded ion pumps \cite{Prost96,Manneville99,Ramaswamy00,Gov04,Lin06,ElAlaouiFaris09} or by the cell cytoskeleton \cite{Gov05,Zhang08b}, see also section \ref{section_domain_patterns}. Active conformational transitions of membrane proteins have also been suggested to couple to the local thickness \cite{Sabra98} or curvature \cite{Chen04} of the membranes.
In the absence of active processes, the adhesiveness of two membranes with complementary receptor and ligand molecules depends on the concentration and binding energies of the molecules, and can be captured by effective adhesion potentials, see section \ref{section_effective_adhesion_potential}. The adhesiveness of membranes with actively switched receptors, in contrast, depends also on the switching rates of the receptors. In the example illustrated in fig.~\ref{figure_active_switching}, the anchored receptors can be switched between two states: an extended `on'-state in which the receptors can bind to ligands anchored in the apposing membrane, and an `off'-state in which the receptors can't bind. In this example, the switching process from the on- to the off-state requires energy input, e.g.~in the form of ATP. As a consequence, the rate $\omega_{-}$ for this process does not depend on wether the receptor is bound or not, in contrast to equilibrium situations without energy input. In an equilibrium situation, the rate for the transition from the on- to the off-state depends on the binding state and binding energy of a receptor.
\begin{figure}[t]
\begin{center}
\resizebox{\columnwidth}{!}{\includegraphics{figure9small}}
\caption{(Top) A membrane with switchable receptors adhering to a second membrane with complementary ligands. The receptors are switched between a stretched, active conformation and a bent, inactive conformation. In the stretched conformation, the adhesion molecules can bind to their ligands in the apposing membrane. --
(Bottom) Monte Carlo data for the average rescaled membrane separation $\bar{z} = \bar{l}/a \sqrt{\kappa/k_B T}$ as a function of the switching rate $\omega=\omega_{+}=\omega_{-}$ of the receptors. Here, $\omega_{+}$ and $\omega_{-}$ are the on- and off-switching rates of the receptors, and $\tau$ is the characteristic relaxation time of a membrane segment with a linear size equal to the mean distance of the receptors.
The active switching leads to a stochastic resonance with increased membrane separations at intermediate switching rates. The details of the Monte Carlo simulations are described in Ref.~\cite{Rozycki06a}. In this example, the binding energy of the receptors and ligands is $U=2.8 k_B T$.
}
\label{figure_active_switching}
\end{center}
\end{figure}
The active switching of the receptors enhances the shape fluctuations of the membranes \cite{Rozycki06a,Rozycki06b,Rozycki07}. Since the steric repulsion of the membranes increases with the shape fluctuations, this enhancement of shape fluctuations leads to larger membrane separations. In Fig.~\ref{figure_active_switching}(b), the average membrane separation is shown as a function of the switching rates for equal on- and off-rates $\omega_{+}=\omega_{-}$. In this example, the fractions of receptors in the on- and off-state are constant and equal to $0.5$. The active switching leads to a stochastic resonance of the membrane shape fluctuations, with a maximum of the membrane separation at intermediate switching rates. At the resonance point, the switching rates are close to the fluctuation relaxation rate $1/\tau$ of a membrane segment with a linear size equal to the average separation of the receptors \cite{Rozycki06a}.
\section{Discussion and conclusions}
We have reviewed theoretical models for the adhesion of biomimetic membranes and cells {\em via} anchored but mobile receptor and ligand molecules. In these models, the membranes are described as elastic surfaces, and the receptors and ligands as single molecules.
We have argued in the introduction that the elasticity of the membranes is dominated by their bending energy on the relevant lateral length scales up to average separation of the receptor-ligand complexes, which is between 50 and 100 nm for typical concentrations of the complexes in cell adhesion zones \cite{Grakoui99}. The crossover length $\sqrt{\kappa/\sigma}$, above which the tension $\sigma$ dominates over the bending rigidity $\kappa$, is clearly larger for typical membrane tensions $\sigma$, see introduction. However, the average separation of cytoskeletal anchors in cell membranes may be close to the average separation of the receptor-ligand complexes. In the absence of active processes, the coupling of the membrane to the cytoskeleton may lead to a suppression of membrane shape fluctuations on length scales larger than the average separation of the anchors. In the presence of active cytoskeletal processes, the membrane shape fluctuations may even be increased \cite{Gov03,Auth07}. In the models reviewed here, the cytoskeletal elasticity is neglected since the relevant lateral length scales up to 50 or 100 nm are taken to be smaller than the average separation of the cytoskeletal anchors. However, the active transport of T cell receptors {\em via} a weak coupling to the cytoskeleton has been taken into account in section \ref{section_domain_patterns}, and the active switching of receptors has been considered in section \ref{section_active_switching}.
The characterization of the membrane elasticity by a uniform bending rigidity $\kappa$ is justified on length scales larger than the molecular components of the membranes, i.e.~on length scales larger than 5 or 10 nm. Molecular inhomogeneities within the membranes average out on these length scales, and the presence of anchored or embedded proteins leads to an increased bending rigidity, compared to pure lipid bilayers.
Important length scales in the direction perpendicular to the membranes are the length of the receptor-ligand complexes, and the binding range of the receptors and ligands. The binding range is the difference between the smallest and largest local membrane separation at which the molecules can bind, and depends on the interaction range of the molecular groups that stick together, the flexibility of the receptor and ligand molecules, and the flexibility of the membrane anchoring. In principle, the binding range may be measured experimentally, or inferred from simulations with atomistic membrane models in a multi-scale modeling approach. An important quantity is the fraction $P_b$ of the membranes with a local separation within the binding range of the receptors and ligands, see section \ref{section_binding_cooperativity}. The membrane fraction $P_b$ depends on the membrane shape fluctuations on the relevant nanoscales, and thus on the concentrations of the receptors and ligands, which constrain the shape fluctuations as bound complexes. The dependence of $P_b$ on the molecular concentrations leads to cooperative binding.
As reviewed in section \ref{section_effective_adhesion_potential}, the integration over all possible positions of the receptor and ligand molecules in the partition function of the models leads to effective adhesion potentials for the membranes. These effective adhesion potentials greatly simplify the characterization of the adhesion behavior. In the case of long and short receptor-ligand complexes, for example, the effective adhesion potential allows a general characterization of the critical point for phase separation, see section \ref{subsection_critical_concentrations}. If the adhesion is mediated by adsorbed particles, a similar integration over the degrees of freedom of these particles leads to an effective adhesion energy that is maximal at an optimal particle concentration, see section \ref{section_adsorbed_particles}.
|
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If your version of \LaTeX{} produces Postscript files, \texttt{\small ps2pdf} or \texttt{\small dvipdf} can convert these to PDF.
To ensure A4 format in \LaTeX, use the command {\small\verb|\special{papersize=210mm,297mm}|}
in the \LaTeX{} preamble (below the {\small\verb|\usepackage|} commands) and use \texttt{\small dvipdf} and/or \texttt{\small pdflatex}; or specify \texttt{\small -t a4} when working with \texttt{\small dvips}.
\subsection{Layout}
\label{ssec:layout}
Format manuscripts two columns to a page, in the manner these
instructions are formatted.
The exact dimensions for a page on A4 paper are:
\begin{itemize}
\item Left and right margins: 2.5 cm
\item Top margin: 2.5 cm
\item Bottom margin: 2.5 cm
\item Column width: 7.7 cm
\item Column height: 24.7 cm
\item Gap between columns: 0.6 cm
\end{itemize}
\noindent Papers should not be submitted on any other paper size.
If you cannot meet the above requirements about the production of your electronic submission, please contact the publication chairs above as soon as possible.
\subsection{Fonts}
For reasons of uniformity, Adobe's \textbf{Times Roman} font should be used.
If Times Roman is unavailable, you may use Times New Roman or \textbf{Computer Modern Roman}.
Table~\ref{font-table} specifies what font sizes and styles must be used for each type of text in the manuscript.
\begin{table}
\centering
\begin{tabular}{lrl}
\hline \textbf{Type of Text} & \textbf{Font Size} & \textbf{Style} \\ \hline
paper title & 15 pt & bold \\
author names & 12 pt & bold \\
author affiliation & 12 pt & \\
the word ``Abstract'' & 12 pt & bold \\
section titles & 12 pt & bold \\
subsection titles & 11 pt & bold \\
document text & 11 pt &\\
captions & 10 pt & \\
abstract text & 10 pt & \\
bibliography & 10 pt & \\
footnotes & 9 pt & \\
\hline
\end{tabular}
\caption{\label{font-table} Font guide. }
\end{table}
\paragraph{\LaTeX-specific details:}
To use Times Roman in \LaTeX2e{}, put the following in the preamble:
\begin{quote}
\small
\begin{verbatim}
\usepackage{times}
\usepackage{latexsym}
\end{verbatim}
\end{quote}
\subsection{Ruler}
A printed ruler (line numbers in the left and right margins of the article) should be presented in the version submitted for review, so that reviewers may comment on particular lines in the paper without circumlocution.
The presence or absence of the ruler should not change the appearance of any other content on the page.
The camera ready copy should not contain a ruler.
\paragraph{Reviewers:}
note that the ruler measurements may not align well with lines in the paper -- this turns out to be very difficult to do well when the paper contains many figures and equations, and, when done, looks ugly.
In most cases one would expect that the approximate location will be adequate, although you can also use fractional references (\emph{e.g.}, this line ends at mark $295.5$).
\paragraph{\LaTeX-specific details:}
The style files will generate the ruler when {\small\verb|\aclfinalcopy|} is commented out, and remove it otherwise.
\subsection{Title and Authors}
\label{ssec:title-authors}
Center the title, author's name(s) and affiliation(s) across both columns.
Do not use footnotes for affiliations.
Place the title centered at the top of the first page, in a 15-point bold font.
Long titles should be typed on two lines without a blank line intervening.
Put the title 2.5 cm from the top of the page, followed by a blank line, then the author's names(s), and the affiliation on the following line.
Do not use only initials for given names (middle initials are allowed).
Do not format surnames in all capitals (\emph{e.g.}, use ``Mitchell'' not ``MITCHELL'').
Do not format title and section headings in all capitals except for proper names (such as ``BLEU'') that are
conventionally in all capitals.
The affiliation should contain the author's complete address, and if possible, an electronic mail address.
The title, author names and addresses should be completely identical to those entered to the electronical paper submission website in order to maintain the consistency of author information among all publications of the conference.
If they are different, the publication chairs may resolve the difference without consulting with you; so it is in your own interest to double-check that the information is consistent.
Start the body of the first page 7.5 cm from the top of the page.
\textbf{Even in the anonymous version of the paper, you should maintain space for names and addresses so that they will fit in the final (accepted) version.}
\subsection{Abstract}
Use two-column format when you begin the abstract.
Type the abstract at the beginning of the first column.
The width of the abstract text should be smaller than the
width of the columns for the text in the body of the paper by 0.6 cm on each side.
Center the word \textbf{Abstract} in a 12 point bold font above the body of the abstract.
The abstract should be a concise summary of the general thesis and conclusions of the paper.
It should be no longer than 200 words.
The abstract text should be in 10 point font.
\subsection{Text}
Begin typing the main body of the text immediately after the abstract, observing the two-column format as shown in the present document.
Indent 0.4 cm when starting a new paragraph.
\subsection{Sections}
Format section and subsection headings in the style shown on the present document.
Use numbered sections (Arabic numerals) to facilitate cross references.
Number subsections with the section number and the subsection number separated by a dot, in Arabic numerals.
\subsection{Footnotes}
Put footnotes at the bottom of the page and use 9 point font.
They may be numbered or referred to by asterisks or other symbols.\footnote{This is how a footnote should appear.}
Footnotes should be separated from the text by a line.\footnote{Note the line separating the footnotes from the text.}
\subsection{Graphics}
Place figures, tables, and photographs in the paper near where they are first discussed, rather than at the end, if possible.
Wide illustrations may run across both columns.
Color is allowed, but adhere to Section~\ref{ssec:accessibility}'s guidelines on accessibility.
\paragraph{Captions:}
Provide a caption for every illustration; number each one sequentially in the form:
``Figure 1. Caption of the Figure.''
``Table 1. Caption of the Table.''
Type the captions of the figures and tables below the body, using 10 point text.
Captions should be placed below illustrations.
Captions that are one line are centered (see Table~\ref{font-table}).
Captions longer than one line are left-aligned (see Table~\ref{tab:accents}).
\begin{table}
\centering
\begin{tabular}{lc}
\hline
\textbf{Command} & \textbf{Output}\\
\hline
\verb|{\"a}| & {\"a} \\
\verb|{\^e}| & {\^e} \\
\verb|{\`i}| & {\`i} \\
\verb|{\.I}| & {\.I} \\
\verb|{\o}| & {\o} \\
\verb|{\'u}| & {\'u} \\
\verb|{\aa}| & {\aa} \\\hline
\end{tabular}
\begin{tabular}{lc}
\hline
\textbf{Command} & \textbf{Output}\\
\hline
\verb|{\c c}| & {\c c} \\
\verb|{\u g}| & {\u g} \\
\verb|{\l}| & {\l} \\
\verb|{\~n}| & {\~n} \\
\verb|{\H o}| & {\H o} \\
\verb|{\v r}| & {\v r} \\
\verb|{\ss}| & {\ss} \\
\hline
\end{tabular}
\caption{Example commands for accented characters, to be used in, \emph{e.g.}, \BibTeX\ names.}\label{tab:accents}
\end{table}
\paragraph{\LaTeX-specific details:}
The style files are compatible with the caption and subcaption packages; do not add optional arguments.
\textbf{Do not override the default caption sizes.}
\subsection{Hyperlinks}
Within-document and external hyperlinks are indicated with Dark Blue text, Color Hex \#000099.
\subsection{Citations}
Citations within the text appear in parentheses as~\citep{Gusfield:97} or, if the author's name appears in the text itself, as \citet{Gusfield:97}.
Append lowercase letters to the year in cases of ambiguities.
Treat double authors as in~\citep{Aho:72}, but write as in~\citep{Chandra:81} when more than two authors are involved. Collapse multiple citations as in~\citep{Gusfield:97,Aho:72}.
Refrain from using full citations as sentence constituents.
Instead of
\begin{quote}
``\citep{Gusfield:97} showed that ...''
\end{quote}
write
\begin{quote}
``\citet{Gusfield:97} showed that ...''
\end{quote}
\begin{table*}
\centering
\begin{tabular}{lll}
\hline
\textbf{Output} & \textbf{natbib command} & \textbf{Old ACL-style command}\\
\hline
\citep{Gusfield:97} & \small\verb|\citep| & \small\verb|\cite| \\
\citealp{Gusfield:97} & \small\verb|\citealp| & no equivalent \\
\citet{Gusfield:97} & \small\verb|\citet| & \small\verb|\newcite| \\
\citeyearpar{Gusfield:97} & \small\verb|\citeyearpar| & \small\verb|\shortcite| \\
\hline
\end{tabular}
\caption{\label{citation-guide}
Citation commands supported by the style file.
The style is based on the natbib package and supports all natbib citation commands.
It also supports commands defined in previous ACL style files for compatibility.
}
\end{table*}
\paragraph{\LaTeX-specific details:}
Table~\ref{citation-guide} shows the syntax supported by the style files.
We encourage you to use the natbib styles.
You can use the command {\small\verb|\citet|} (cite in text) to get ``author (year)'' citations as in \citet{Gusfield:97}.
You can use the command {\small\verb|\citep|} (cite in parentheses) to get ``(author, year)'' citations as in \citep{Gusfield:97}.
You can use the command {\small\verb|\citealp|} (alternative cite without parentheses) to get ``author year'' citations (which is useful for using citations within parentheses, as in \citealp{Gusfield:97}).
\subsection{References}
Gather the full set of references together under the heading \textbf{References}; place the section before any Appendices.
Arrange the references alphabetically by first author, rather than by order of occurrence in the text.
Provide as complete a citation as possible, using a consistent format, such as the one for \emph{Computational Linguistics\/} or the one in the \emph{Publication Manual of the American
Psychological Association\/}~\citep{APA:83}.
Use full names for authors, not just initials.
Submissions should accurately reference prior and related work, including code and data.
If a piece of prior work appeared in multiple venues, the version that appeared in a refereed, archival venue should be referenced.
If multiple versions of a piece of prior work exist, the one used by the authors should be referenced.
Authors should not rely on automated citation indices to provide accurate references for prior and related work.
The following text cites various types of articles so that the references section of the present document will include them.
\begin{itemize}
\item Example article in journal: \citep{Ando2005}.
\item Example article in proceedings, with location: \citep{borschinger-johnson-2011-particle}.
\item Example article in proceedings, without location: \citep{andrew2007scalable}.
\item Example arxiv paper: \citep{rasooli-tetrault-2015}.
\end{itemize}
\paragraph{\LaTeX-specific details:}
The \LaTeX{} and Bib\TeX{} style files provided roughly follow the American Psychological Association format.
If your own bib file is named \texttt{\small acl2020.bib}, then placing the following before any appendices in your \LaTeX{} file will generate the references section for you:
\begin{quote}\small
\verb|\bibliographystyle{acl_natbib}|\\
\verb|
\section{Introduction}
\label{sec:intro}
NLP has always been an applied discipline, with inspiration drawn both from basic reseach in computational linguistics, computer science, and cognitive science, as well as from business problems and applications in industry and the public sector. Even if some NLP researchers prefer to work at lower Technology Readiness Levels (TRLs), while others operate at a fairly high TRL range, most of the work in NLP has the potential to climb the TRL scale up to the more practical levels (i.e. from TRL level 7 upwards) \cite{NasaTrl2017}.
For those of us who work with external clients and habitually deliver results in the form of demonstrators and prototypes at TRL 6 or 7, a consistent and significant challenge is the state of the data available. In our experience, challenges regarding data are much more common in client-facing projects than challenges relating to the technical nature of models or algorithms. We argue that the lack of readiness with respect to data has become a serious obstacle when transferring findings from research to an applied setting. Even if the research problem is sufficiently well defined, and the business value of the proposed solution is well described, it is often not clear what type of data is required, if it is available, or if it at all exists.
The border between academic research in NLP and the application of the research in practical non-academic settings is becoming increasingly blurred with the convergence of NLP research to more or less production-ready frameworks and implementations.
On the one hand, research results have never been more accessible, and it has never been easier to obtain and adjust the architecture of, e.g., a state of the art language model to accommodate a new use case, and to construct a prototype showing the value the model would contribute to an external stakeholder.
On the other hand, while the technical maturity of the research community has improved, the understanding of business value, and by extension, the understanding of the impact and importance of data are still largely lacking. It is our firm belief that in order for NLP, and in particular the research community that targets the lower TRLs, to become even more relevant and thus also benefit from feedback from parties outside the field, we have to assume a more holistic approach to the entire life cycle of applied research, with a particular eye on {\bf data readiness}.
The intention for this paper is therefore to raise awareness of data readiness for NLP among researchers and practitioners alike, and to initiate and nurture much-needed discussions with respect to the questions that arise when addressing real-world challenges with state of the art academic research.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{data_per_year.png}
\caption{Relative frequency of publications in the ACL Anthology that mention the term ``data'' in the title for the last 20 years.}
\label{fig:data}
\end{figure}
\section{Related work}
\label{sec:related-work}
As is evident from Figure \ref{fig:data}, which shows the relative frequency of publications in the ACL Anthology\footnote{\url{www.aclweb.org/anthology/}} that mention the term ``data'' in the title over the last 20 years, there is an increasing interest in questions relating to data within our field. While there are a lot of activity related to data in the research community, few attempts at fostering a discussion of the whole process, from business problem to data access, has been made. Relevant areas of academic research include the following.
{\bf Access to unlabelled training data.} Efforts to collect and distribute text data at scale include corpora originating from CommonCrawl, e.g., \cite{cc:MackenzieBenhamPetriTrippasEtAl:2020:cc-news-en,cc:ElKishkyChaudharyGuzmanKoehn:2020:ccaligned}, along with tools to facilitate corpus creation \cite{wenzek_ccnet_2020}, as well as academic initiatives such as ELRA\footnote{\url{http://www.elra.info/}} and LDC\footnote{\url{https://www.ldc.upenn.edu/}}.
{\bf Creation of labelled training data.} Research in Active Learning, e.g., \cite{settles_active_2012,siddhant_deep_2018,liang_alice_2020,ein-dor_active_2020}, as well as Zero and Few-shot Learning \cite{srivastava_zero-shot_2018,ye_zero-shot_2020,pelicon_zero-shot_2020} allows for the utilization of pre-compiled knowledge, and human-in-the-loop approaches to efficient data labelling. However, these approaches assume that there is a clear objective to address, and that unlabelled data and expertise are available. In our experience, this is rarely the case.
{\bf Bias, transparency, fairness} are all areas that have bearing towards the state of data. Much of the research, however, is concerned with situations in which the data has already been collected. Most notable recent efforts include
The {\em Dataset Nutrition Label}, which is a diagnostic framework for enabling standardized data analysis \cite{holland_dataset_2018}; {\em Data Statements for NLP} that allows for addressing exclusion and bias in the field \cite{bender_data_2018}; {\em FactSheets} intended for increasing consumers' trust in AI services \cite{arnold_factsheets_2019}; and {\em Datasheets for Datasets} for facilitating better communication between dataset creators and consumers \cite{gebru_datasheets_2020}.
{\bf Model deployment.}
Tangential to our efforts to provide stakeholders with prototypes at TRL 6-7 is the work of deploying machine learning models to a production environment. Research in this area that also touches on data readiness in some form include that of \citet{polyzotis_data_2018} and \citet{paleyes_challenges_2020}.
Work on {\bf data readiness} related to other modalities than text include \citet{van2019quality} and \citet{harvey2019standardised} that both deal with data quality in medical imaging.
\citet{austin_path_2018} outlines practical solutions to common problems with data readiness when integrating diverse datasets from heterogeneous sources. \citet{afzal_data_2020} introduces the concept {\em Data Readiness Report} as a means to document data quality across a range of standardized dimensions.
We have not found any work that focuses specifically on data readiness in the context of NLP. Our contribution is therefore a set of questions that we have found valuable to bring up in discussions with new stakeholders in order to allow us, and them, to form an understanding of the state of the data involved in the particular challenge.
\section{Data Readiness Levels}
The notion of Data Readiness Levels (DRLs) provides a way of talking about data much in the same way TRLs facilitate communication regarding the maturity of technology \cite{lawrence2017data}. DRLs is a framework suitable for exchanging information with stakeholders regarding data {\em accessibility}, {\em validity}, and {\em utility}.
There are three different major Bands of the DRLs, and each band can be thought of as consisting of multiple levels. The state of data is usually a progress from Band C towards Band A, with a particular business goal in mind. Figure \ref{fig:drl-table} illustrates the three bands of the Data Readiness Levels.
\begin{figure*}[!ht]
\begin{center}
\includegraphics[width=0.85\linewidth]{data-readiness-levels.png}
\caption{An overview of the different bands of Data Readiness Levels.}
\label{fig:drl-table}
\end{center}
\end{figure*}
{\bf Band C} concerns the {\em accessibility} of data. All work at this level serves to grant the team intended to work with the data access to it; once access is provided, the data is considered to be at Band C - Level C-1, and ready to be brought into Band B. Issues that fall under Band C include: the existence of data; format conversion and encoding; legal aspects of accessibility; and programmatic aspects of accessibility.
{\bf Band B} concerns the {\em validity} of data. In order to pass Band B, the data has to be valid in the sense that it is representative of the task at hand. Furthermore, the data should be deduplicated, noise should be identified, missing values should be characterized, etc. At the top level of Band B, the data should be suitable for exploratory analysis, and the forming of working hypotheses.
{\bf Band A} concerns the {\em utility} of data. The utility of the data concerns the way in which the data is intended to be used: Is the data enough to solve the task at hand? A project should strive for data readiness at Band A - Level A-1. Note that the Data Readiness Levels should be interpreted with respect to a given task.
\section{Examples of challenges}
\label{sec:examples}
The following are examples of typical challenges we have encountered, framed as belonging to the different DRLs.\footnote{The examples are deliberately kept vague since we do not want to disclose the corresponding external stakeholder.}
\subsection{DRL Band C -- Accessibility}
{\bf Example 1: Data licensing.} It was assumed that the data to work with in the project was in the public domain and readily available. It turned out the data was a proprietary news feed under license restrictions. The consequences of this were two-fold: not having access to the data generation process meant we could not address one of the stakeholder's major problems (de-duplication, relevance assessment); and, the license restrictions prevented the project from publishing the dataset along with the research findings.
{\bf Example 2: Company culture.} The ownership of data was clearly specified, but the staff did not adhere to management's request to release the data due to the uncertainty of the result of the project. This resulted in delays. The fear of the loss of jobs may impact availability of data -- data readiness depends on the overall introduction of data-driven techniques in a new organization.
{\bf Example 3: Data format.} Raw data was stored as PDF files, generated by different sources. PDF is an output format, not an input format. Projects working with PDF files will always face challenges having to do with data conversion since there is currently no way of reliably converting an arbitrary PDF file into a format useful for NLP.
\subsection{DRL Band B -- Validity}
{\bf Example 4: Annotation guidelines.} The existing annotation guidelines were elaborate, but fell short in practical applicability. Partly due to the misalignment between the annotation task and the guidelines, it became very time consuming to annotate data which resulted in a small dataset to work with. In turn, this affected the range of possible NLP techniques applicable to the problem at hand. Annotation guidelines have to be unambiguous, precise, and possible for an annotator to remember.
{\bf Example 5: Annotation quality.} The data was assumed to be of good quality, but additional investigations revealed a low inter-annotator agreement. The consequence was that the existing data could not be trusted, and the annotation work had to be re-done. If the definition of a task is too hard for human annotators to agree on, a machine trained on the data will perform poorly.
{\bf Example 6: Annotation quality.} Existing information produced by the stakeholder was assumed to be useful in creating an annotated dataset for the specific task at hand, but it turned out that the information was incomplete and insufficient. The consequence of not being able to leverage existing data for distant supervision was that the range of applicable techniques for addressing the stakeholder's problem became severely limited.
\subsection{DRL Band A -- Utility}
{\bf Example 7: Annotation expectations.} It was known that the data to work with was annotated, but the way the annotations had been made had not been communicated. Instead of sequence level annotations, the data was annotated at the document level. As a consequence, we could not explore the type of information extraction techniques we had expected, but had to resort to document classification instead.
{\bf Example 8: Data sparseness.} The overall amount and velocity of data were assumed to be of sufficient quantity, but when aligning data availability with use case requirements, it turned out the data was too sparse. The task could not be pursued.
{\bf Example 9: Project scope.} The stakeholder's team and the unannotated data they provided to the project were at an exceptionally high DRL, but annotations for training, validation, and testing were very hard to obtain since the project had not planned for annotation work. As a consequence, we implemented a solution based on unsupervised learning instead of a supervised one.
\section{A method for DRL assessment}
\label{sec:questions}
We introduce a method for gaining rapid and rough assessment of the data readiness levels of a given project. The method consists of a range of questions, intended to fuel the discussions between the stakeholders involved in a project with respect to its means and goals, as well as a simple way of visualizing the responses to the questions in order to bring attention to the areas that need more work. We expect to evolve the method in coming projects. So far, it has helped us to preemptively address some of the issues exemplified in Section \ref{sec:examples}; they are a good starting point in reaching the appropriate data readiness for solving real-world problems related to NLP.
\subsection{Pre-requisites}
The pre-requisites for applying the method are the following: there should be a clear business or research-related objective for the project to achieve; the objective lends itself to a data-driven solution; and, there is data available that presumably is relevant for the task.
The method should be scheduled for application at suitable points in time for the project, i.e., anytime the project enters a phase that relies on data and experimentation to make progress in the project plan. We suggest to apply the method at the very beginning of the project, as well as (at least) before entering the first round of empirical experiments with respect to the data at hand and the project's objective.
\subsection{Post-conditions}
The outcome of the method is two-fold: a visual representation of the Data Readiness Levels of the project at a specific point-in-time (as exemplified in Section \ref{sec:method-example}); and, the insight into the state of data achieved by venting the questions among the project's stakeholders.
\subsection{The questions}
The purpose of each question is to draw the stakeholders' attention to one aspect of the data readiness of the project. However, since not all questions are relevant to all types of projects some may be omitted depending on the characteristics of the project at hand.
Each of the fifteen questions below can be answered by one of four options: {\bf Don't know}, {\bf No}, {\bf Partially}, and {\bf Yes}, where {\bf Don't know} is always considered the worst possible answer, and {\bf Yes} as the answer to strive for. The admittedly very coarse grained answer scale is intended to serve as a guide in assessing the state of the project's data readiness, rather than as a definitive and elaborate tool for detailed assessment.
\subsubsection{Questions related to Band C}
Band C, that concerns the accessibility of data, is the band in that is the least dependent on the actual objective of the project, but clearing it is still required in order to make the project successful.
\begin{itemize}
\item[Q1] {\bf Do you have programmatic access to the data?} The data should be made accessible to the people who are going to work with it, in a way that makes their work as easy as possible. This usually means programmatic access via an API, database, or spreadsheet.
\item[Q2] {\bf Are your licenses in order?} In the case you plan on using data from a third-party provider, either commercial or via open access, ensure that the licences for the data permit the kind of usage that is needed for the current project. Furthermore, make sure you follow the Terms of Service set out by the provider.
\item[Q3] {\bf Do you have lawful access to the data?} Make sure you involve the appropriate legal competence early on in your project. Matters regarding, e.g., personal identifiable information, and GDPR have to be handled correctly. Failing to do so may result in a project failure, even though all technical aspects of the project are perfectly sound.
\item[Q4] {\bf Has there been an ethics assessment of the data?} In some use cases, such as when dealing with individuals' medical information, the objectives of the project require an ethics assessment. The rules for such a probe into the data are governed by strict rules, and you should consult appropriate legal advisors to make sure your project adheres to them.
\item[Q5] {\bf Is the data converted to an appropriate format?} Apart from being accessible programmatically, and assessed with respect to licenses, laws, and ethics, the data should also be converted to a format appropriate for the potential technical solutions to the problem at hand. One particular challenge we have encountered numerous times, is that the data is on the format of PDF files. PDF is an excellent output format for rendering contents on screen or in print, but it is a terrible input format for data-driven automated processes (see, e.g., \cite{Panait2020} for examples).
\end{itemize}
\subsubsection{Questions related to Band B}
Band B concerns the validity of data. In pursuing projects with external parties, we have so far seen fairly few issues having to do with the validity of data. In essence, Band B is about trusting that the data format is what you expect it to be.
\begin{itemize}
\item[Q6] {\bf Are the characteristics of the data known?} Are the typical traits and features of the data known? Perform an exploratory data analysis, and run it by all stakeholders in the project. Make sure to exemplify typical and extreme values in the data, and encourage the project participants to manually look into the data.
\item[Q7] {\bf Is the data validated?} Ensure that the traits and features of the data make sense, and, e.g., records are deduplicated, noise is catered for, and that null values are taken care of.
\end{itemize}
\subsubsection{Questions related to Band A}
Band A concerns the utility of data. As such, it is tightly coupled to the objective of the project. In our experience, this is the most elusive data readiness level in that it requires attention every time the goal of a project changes.
\begin{itemize}
\item[Q8] {\bf Do stakeholders agree on the objective of the current use case?} What problem are you trying to solve? The problem formulation should be intimately tied to a tangible business value or research hypothesis. When specifying the problem, make sure to focus on the actual need instead of a potentially interesting technology. The characteristics of the problem dictates the requirements on the data. Thus, the specification is crucial for understanding the requirements on the data in terms of, e.g., training data, and the need for manual labelling of evaluation or validation data. Only when you know the characteristics of the data, it will be possible to come up with a candidate technological approach to solve the problem.
\item[Q9] {\bf Is the purpose of using the data clear to all stakeholders?} Ensure that all people involved in the project understands the role and importance of the data to be used. This is to solidify the efforts made by the people responsible for relevant data sources to produce data that is appropriate for the project's objective {\em and} the potential technical solution to address the objective.
\item[Q10] {\bf Is the data sufficient for the current use case?} Given the insight into what data is available, consider the questions: What data is needed to solve the problem? Is that a subset of the data that is already available? If not: is there a way of getting all the data needed? If there is a discrepancy between the data available, and the data required to solve the problem, that discrepancy has to be mitigated. If it is not possible to align the data available with what is needed, then this is a cue to go back to the drawing board and either iterate on the problem specification, or collect suitable data.
\item[Q11] {\bf Are the steps required to evaluate a potential solution clear?} How do you know if you have succeeded? The type of data required to evaluate a solution is often tightly connected to the way the solution is implemented: if the solution is based on supervised machine learning, i.e., requiring labelled examples, then the evaluation of the solution will also require labelled data. If the solution depends on labelled training data, the process of annotation usually also results in the appropriate evaluation data. Any annotation effort should take into account the quality of the annotations, e.g., the inter-annotator agreement; temporal aspects of the data characteristics, e.g., information on when we need to obtain newly annotated data to mitigate model drift; and, the representativity of the data. \citet{tseng_best_2020} provide a comprehensive set of best-practices for managing annotation projects.
\item[Q12] {\bf Is your organization prepared to handle more data like this beyond the scope of the project?} Even if the data processing in your organization is not perfect with respect to the requirements of machine learning, each project you pursue has the opportunity to articulate improvements to your organization's data storage processes. Ask yourself the questions: How does my organization store incoming data? Is that process a good fit for automatic processing of the data in the context of an NLP project, that is, is the data stored on a format that brings it beyond Band C ({\bf accessibility}) of the Data Readiness Levels? If not; what changes would need to be made to make the storage better?
\item[Q13] {\bf Is the data secured?} Ensure that the data used in the project is secured in such a way that it is only accessible to the right people, and thus not accessible by unauthorized users. Depending on the sensitivity of the project, and thus the data, there might be a need to classify the data according to the security standards of your organization (e.g., ISO 27001), and implement the appropriate mechanisms to protect the data and project outcome.
\item[Q14] {\bf Is it safe to share the data with others?} In case the project aims to share its data with others, the risks of leaking sensitive data about, e.g., your organization's business plans or abilities have to be addressed prior to sharing it.
\item[Q15] {\bf Are you allowed to share the data with others?} In case the project wishes to share its data, make sure you are allowed to do so according to the licenses, laws, and ethics previously addressed in the project.
\end{itemize}
\section{Example application of the method}
\label{sec:method-example}
For the purpose of exemplifying the use of the method described above, consider the fictitious case of project {\em Project}, an undertaking of a large organization with good experience of running conventional ICT projects, but little knowledge about data-driven NLP-based analysis. The actual subject matter and scope of {\em Project} is not important.
\begin{figure*}[!ht]
\begin{center}
\includegraphics[width=\linewidth]{drl-assessment-stage-1.png}
\caption{The figure illustrates the state of data readiness at the beginning of the fictitious project.}
\label{fig:drl-assessment-ex-1}
\end{center}
\end{figure*}
\subsection{First application of the method}
When the project is initiated, the project manager involves its members in a session in which they recognize all fifteen questions as relevant for the project's objectives, then discusses each question, and agrees on the appropriate answer. When the session is over, the project manager plots the responses in a radar chart, as displayed in Figure \ref{fig:drl-assessment-ex-1}, in such a way that each of the questions answered is present in the chart, starting with Q1 ({\em Programmatic access to data}) up north, then progressing clock-wise with each question.\footnote{The code for generating radar charts as part of assessing the data readiness of your own project is available here: \url{https://github.com/fredriko/draviz}} The responses are plotted such that {\bf Don't know} (the worst answer) is located at the center of the chart, while {\bf Yes} (the best answer) is closest to the chart's edge. The aim of the assessment method is for the surface constituted by the enclosed responses to cover as large an area as possible. The reasons are simple; first, all stakeholders, in particular those in executive positions, will gain immediate visual insight into the state of the data for the project and, hopefully, feel the urge to act to increase the area; second, it is easy to visualize the project's progress at discrete points in time by overlaying two (or more) radar charts.\footnote{Overlaying radar charts with the purpose of comparing them only works when a small number of charts are involved; we are experimenting with parallel plots when the number of distinct charts exceeds 3.}
From Figure \ref{fig:drl-assessment-ex-1}, it can be seen that {\em Project}, at its incarnation, is not very mature with respect to data readiness. The area covered by the enclosed responses is small, and the number of unknowns are large. The only certainties resulting by the initial assessment of {\em Project}, are that: there has been no ethical assessment made of the data; the data has not been converted to a suitable format; no characteristics of the data are known; the data has not been validated; there is not sufficient data for the use case; and the way to evaluate the success of the project has yet to be defined. On the bright side, the stakeholder partially agrees on the objective of the project, and the purpose of the data.
\subsection{Second application of the method}
Fast forward to the second data readiness assessment of {\em Project}. In this case, it is scheduled to take place prior to the project embarking on the first rounds of empirical investigations of the state of data in relation to the project's business objective. The purpose of looking into the data readiness of the project at this stage, is to support the project manager in their work regarding prioritization, management of resources, and handling of expectations in relation to the project's progression and ability to reach its goals.
\begin{figure*}[!ht]
\begin{center}
\includegraphics[width=\linewidth]{drl-assessment-stage-2.png}
\caption{The figure shows the corresponding state at the time where the project is ready to start making experiments based on the data.}
\label{fig:drl-assessment-ex-2}
\end{center}
\end{figure*}
Again, after all stakeholders have agreed on responses to the questions of the method, they are plotted in a radar chart. Figure \ref{fig:drl-assessment-ex-2} shows the state of the project after the second data readiness assessment. Progress has been made; the area covered by the responses is larger than it was at the initial assessment (Figure \ref{fig:drl-assessment-ex-1}). There are no unknowns left among the responses. Data is available and converted to a suitable format, its characteristics are known and the data format is generally trusted within the project. The fact that licenses, legal aspects of access, and ethics are not quite there yet does not, technically speaking, prohibit the project from moving on with the empirical investigation. However, these issues should be properly addressed before the project results are deployed to a general audience.
The stakeholders are still not in full agreement on the project's business objective, but they are aware of the purpose of the data, which has been deemed sufficient for the use case. Given the uncertainty with respect to the business objective, the steps required to evaluate proposed solutions are also unclear.
Beyond the scope of the project, the organization is not yet set up to in a way that is required to repeat and reproduce the findings of {\em Project} to future data, and data security is still work in progress. The project is allowed to share the data if it wishes to to so, but since management has decided to play it safe with respect to giving away too much information regarding the organization's future plans in doing so, it has been decided that data should not be share with external parties.
\section{Conclusions}
\label{sec:conclusions}
Research in NLP has never been more accessible; the impact of new results has the potential to reach far beyond the academic sphere. But with great power comes great responsibility. How can we
foster a better uptake of research among public agencies, and in industry, and thereby gain valuable insight into the research directions that really matter?
We introduce a method for assessing the Data Readiness Levels of a project, consisting of fifteen questions, and the accompanying means for visualizing the responses. We have utilized the proposed method and visualization technique in several projects with stakeholders in both the private and public sectors, and it has proven to be a very useful tool to improve the potential for successful application of NLP solutions to solve concrete business problems.
The method is a work-in-progress, and we thus invite researchers and practitioners in the NLP community to share their experience with respect to applied NLP research and data readiness at the following
GitHub repository \url{https://github.com/fredriko/nlp-data-readiness}.
\bibliographystyle{acl_natbib}
|
2,869,038,156,378 | arxiv | \section*{Introduction}
It is well-known that many methods from the classical model theory of first-order logic, most importantly compactness, fail when we restrict attention to finite structures. In this paper we explore the applicability of ultraproducts -- a familiar construction in (infinite) model theory -- to known inexpressibility results in finite and infinite model theory. This complements the more commonly used approach through game techniques. In particular, we show new proofs for variants of the well-known locality theorems of Hanf and Gaifman with little or no use of games.
The idea of using infinitary methods in general and ultraproducts in particular for questions in finite model theory was explored by Väänänen in \cite{vaananen}, unknown to me at the time of writing. Some of our results are close to those in the preprint \cite{lindellWeinstein} by Lindell, Towsner and Weinstein.
\subsection*{Acknowledgements}
This paper is a condensed and revised version of my Bachelor's thesis at Technische Universität Darmstadt from June 2013. I am deeply indebted to my supervisor Martin Otto for helpful ideas, advice and corrections. I would also like to thank Steven Lindell, Henry Towsner and Scott Weinstein for a preprint of their paper \cite{lindellWeinstein}.
\section{General results for ultraproducts}
We assume that the reader is familiar with the ultraproduct construction and \loz's theorem.
In this section we review some results about the structure of ultraproducts. Some of their distinctive properties make them useful for inexpressibility results in the upcoming sections.
\begin{proposition}\label{propositionCardinalityOfUltraproducts}
Let $(A_i)_{i \in \N}$ be a sequence of non-empty sets and consider them as structures over the empty signature. Let furthermore $\calU$ be an arbitrary non-principal ultrafilter on $\N$. Then the cardinality of the ultraproduct $\prod_i A_i / \calU$ is finite iff there is an $n \in \N$ such that $\abs{A_i} \leq n$ for $\calU$-many $i$. If the cardinality is not finite, then it is at least $2^\omega$.
\end{proposition}
\begin{proof}
The ultraproduct has at most $n$ elements iff $\calU$-many factors have at most $n$ elements, since this property can be expressed by a first-order formula.
Now assume conversely that for each $n \in \N$ we have $\calU$-many sets $A_i$ with more than $n$ elements. We show that there is an injective embedding of $\{ 0, 1 \}^\N$ into $\prod_i A_i / \calU$. To do this, we partition the index set $\N$ into disjoint sets $I_k$, $k \in \N$, such that $I_k \notin \calU$ and $\abs{A_i} \geq 2^k$ for all $i \in I_k$. This is always possible:
If $\calU$-many $A_i$ are finite, we simply put \[ I_k := \{ i \in \N \colon 2^k \leq \abs{A_i} < 2^{k + 1} \} \] for $k > 0$ and \[ I_0 := \{ i \in \N \colon \text{$A_i$ is infinite or has exactly one element } \} ;\] if $\calU$-many $A_i$ are infinite, we gather all $i$ such that $A_i$ is finite in $I_0$ and make all other $I_k$ one-element sets such that we eventually enumerate all $i$.
Having found such a partition into sets $I_k$, we define the map
\begin{align*}
f \colon \{ 0, 1 \}^\N &\to \prod_i A_i \\
f\big( (c_n)_n \big)_i &= \sum_{l = 0}^{k - 1} c_l 2^l \in \{ 0, \dotsc, 2^k - 1 \} \subseteq A_i \text{ if $i \in I_k$} ,
\end{align*}
where we identify $\{ 0, \dotsc, 2^k - 1 \}$ with a subset of $A_i$. We want to show that $f$ composed with the canonical projection $\pi \colon \prod_i A_i \to \prod_i A_i / \calU$ is still injective. Let $(c_n)$ and $(d_n)$ be two $0$-$1$-sequences differing at some index $m \in \N$. Then $f\big((c_n)_n\big)_i \neq f\big((d_n)_n\big)_i$ for all $i \in I_k$ with $k > m$, \ie $f\big((c_n)_n\big)$ differs from $f\big((d_n)_n\big)$ at $\calU$-many indices and hence $\pi\big(f\big((c_n)_n\big)\big) \neq \pi\big(f\big((d_n)_n\big)\big)$. This proves that \[ \pi \circ f \colon \{ 0, 1 \}^\N \to \prod_i A_i / \calU \] is injective.
\end{proof}
\begin{remark}
One could also use a simpler diagonalisation argument in the spirit of Cantor to show that the ultraproduct cannot be countably infinite. However, we often do not want to assume the continuum hypothesis and will need the full strength of the proposition.
\end{remark}
\begin{corollary}
Let $(A_i)_{i \in \N}$ be a sequence of non-empty sets of cardinality at most $2^\omega$ and $\calU$ a non-principal ultrafilter on $\N$. Then the ultraproduct $\prod_i A_i / \calU$ is either finite or its cardinality is exactly $2^\omega$.
\end{corollary}
\begin{proof}
The set $\prod_i A_i$ has at most cardinality $(2^\omega)^\omega = 2^\omega$. Hence its image $\prod_i A_i / \calU$ under the canonical projection has cardinality at most $2^\omega$.
\end{proof}
For the investigation of other ultraproduct properties we use the familiar model-theoretic notion of \emph{types}. In the following we will always use partial $1$-types with parameters: a type of a structure $\frakA$ (with parameters from $X \subseteq A$) is therefore a set $\Phi \subseteq \FO_1(\sigma \mathop{\dot\cup} \{ c_x \colon x \in X \})$, where the signature is expanded by new constant symbols standing for elements of $X$, such that there is an elementary extension $\frakB \succeq \frakA$ and an element $b \in B$ with $\frakB, b \models \Phi$ where $\frakB$ is understood to interpret $c_x$ as $x \in A \subseteq B$ for all $x \in X$.
We recall that a set of formulae $\Phi$ is a type of $\frakA$ iff it is finitely realised, \ie if for all finite $\Phi_0 \subseteq \Phi$ we have \[ \frakA \models \exists x \bigwedge \Phi_0(x) .\]
We also recall the definition of saturation and an important property of saturated structures.
\begin{definition}
Let $\frakA$ be a $\sigma$-structure and $\lambda$ an infinite cardinal. If every type of $\frakA$ with fewer than $\lambda$ parameters is realised in $\frakA$, we call $\frakA$ \emph{$\lambda$-saturated}. If $\frakA$ is $\abs A$-saturated, we simply call $\frakA$ \emph{saturated}.
\end{definition}
\begin{proposition}[{\cite[Theorem 8.1.8]{hodges}}] \label{propElemEquivSatStrAreIsom}
Let $\frakA$ and $\frakB$ be $\sigma$-structures of the same cardinality $\lambda$ such that both $\frakA$ and $\frakB$ are ($\lambda$-)saturated. If $\frakA$ and $\frakB$ are elementarily equivalent, then they are isomorphic.
\end{proposition}
We now show that ultraproducts are $\omega_1$-saturated in important cases.
\begin{proposition}\label{propCountableTypesAreRealised}
Let $(\frakA_i)_{i \in \N}$ be a sequence of $\sigma$-structures, $\calU$ a non-principal ultrafilter on $\N$ and $\Phi$ a type of $\frakA :=\prod_i \frakA_i / \calU$ with $\abs\Phi \leq \omega$. Then $\Phi$ is realised in $\frakA$. Consider the following additional condition:
\[ \text{For all $m \in \N$ and all finite subsets $\Phi_0 \subseteq \Phi$ it holds that $\frakA \models \exists^{\geq m} x \bigwedge \Phi_0(x)$.} \tag{$\ast$} \]
If condition $(\ast)$ holds, then there are at least $2^\omega$ elements of $\frakA$ realising $\Phi$; otherwise, the number of elements of $\frakA$ realising $\Phi$ is equal to $m_\text{max}$, the maximal $m \in \N$ for which $(\ast)$ holds for all finite subsets $\Phi_0 \subseteq \Phi$, and therefore finite.
\end{proposition}
\begin{proof}
Since we can treat any parameters as additional constants in the signature, it suffices to consider $\Phi$ without parameters. Enumerate $\Phi$ as $(\phi_n)_{n \in \N}$. We now define new formulae $\psi_n$. If condition $(\ast)$ holds, let \[ \psi_n := \exists^{\geq n}x \bigwedge_{k < n} \phi_k(x). \] Otherwise, let \[ \psi_n := \exists^{\geq m_\text{max}} x \bigwedge_{k < n} \phi_k(x). \] In both cases we have $\psi_0 \equiv \top$ by definition of the empty conjunction. For each $i \in \N$ we choose the maximal $m_i \leq i$ such that $\frakA_i \models \psi_{m_i}$. For all $n \in \N$ we now have $\frakA \models \psi_n$ and therefore $\calU$-many $i \geq n$ with $\frakA_i \models \psi_n$ and hence $m_i \geq n$. Let now $a := [(a_i)_i] \in A$ such that $\frakA_i, a_i \models \bigwedge_{k < m_i} \phi_k$. Then \loz's theorem implies that $\frakA, a \models \bigwedge_{k < n} \phi_k$ for all $n$; hence $a$ realises $\Phi$ in $\frakA$.
The number of witnesses that we get in this manner depends on our choice of $\psi_n$. If condition $(\ast)$ holds and therefore $\psi_n = \exists^{\geq n}x \bigwedge_{k < n} \phi_k(x)$, then $\frakA_i$ has at least $m_i$ witnesses for $\bigwedge_{k < m_i} \phi_k(x)$. Now Proposition \ref{propositionCardinalityOfUltraproducts} implies that we have constructed at least $2^\omega$ realisations of $\Phi$. But if condition $(\ast)$ is violated by $m_\text{max} + 1$ and some $\Phi_0 \subseteq \Phi$, clearly $\Phi$ itself cannot be realised more than $m_\text{max}$ times. On the other hand, all $\frakA_i$ have at least $m_\text{max}$ many witnesses for $\bigwedge_{k < m_i} \phi_k(x)$. This yields $m_\text{max}$ realisations of $\Phi$ in $\frakA$.
\end{proof}
\begin{corollary}
Let $(\frakA_i)_{i \in \N}$ be a sequence of structures over a countable signature $\sigma$ and $\calU$ a non-principal ultrafilter on $\N$. Then the ultraproduct $\prod_i \frakA_i / \calU$ is $\omega_1$-saturated.
\end{corollary}
\begin{proof}
Let $\Phi$ be a type with countably many parameters. The signature remains countable when we add constant symbols for these parameters. Since the set of first-order formulae over a countable signature is itself countable, $\Phi$ must be countable as well. By the preceding proposition, $\Phi$ is realised in the ultraproduct.
\end{proof}
\begin{corollary}\label{corollaryElemEquivUltraprodsAreIsomAssumingCH}
Assume the continuum hypothesis and let $(\frakA_i)_{i \in \N}$ and $(\frakB_i)_{i \in \N}$ be two sequences of structures over a countable signature $\sigma$ such that each individual structure has cardinality at most $2^\omega$. Let furthermore $\calU_1$ and $\calU_2$ be two non-principal ultrafilters on $\N$ and consider the ultraproducts $\frakA := \prod_i \frakA_i / \calU_1$ and $\frakB := \prod_i \frakB_i / \calU_2$. Then these ultraproducts are isomorphic iff they are elementarily equivalent.
\end{corollary}
\begin{proof}
Both ultraproducts have cardinality at most $2^\omega = \omega_1$. Hence Proposition \ref{propElemEquivSatStrAreIsom} is applicable, so the ultraproducts are already isomorphic if they are elementarily equivalent.
\end{proof}
\begin{remark}
This corollary shows that for $\N$-fold ultraproducts some usually distinct notions of equivalence between structures collapse; we will see one more example of this in the next section. In the other sections we will prove isomorphy of some ultraproducts without requiring the continuum hypothesis.
\end{remark}
In the remainder of the section we will show that Corollary \ref{corollaryElemEquivUltraprodsAreIsomAssumingCH} may fail without the continuum hypothesis. This is related to work by Shelah (\cite{viveLaDifference}), although our counterexamples have been constructed without reference to it.
We first construct a non-saturated ultraproduct. Consider the signature $\sigma = \{ E \}$ with a single binary relation and a two-sorted structure $\frakA$ given by $\omega_1$ and the set of its finite subsets, \ie \[ A := (\{ 0 \} \times \omega_1) \cup (\{ 1 \} \times \powfin(\omega_1)) \] with the relation interpreted by \begin{multline*} (a, b) \in E^\frakA :\iff \\ \text{ there exist $a_0 \in \omega_1, b_0 \in \powfin(\omega_1)$ such that $a = (0, a_0)$, $b = (1, b_0)$ and $a_0 \in b_0$}, \end{multline*} that is, we take $E$ to simply stand for elementhood. Now we consider a non-principal ultrafilter $\calU$ on $\N$ and form the ultrapower $\frakA_\infty := \frakA^\N / \calU$. Let \[ \iota \colon A \to A_\infty, a \mapsto [(a)_{n \in \N}] \] be the diagonal embedding. Consider the type \[ \Phi := \{ c_a E x \colon a \in \iota(\{ 0 \} \times \omega_1) \} \] where $c_a$ is the constant representing the parameter $a$. We easily see that $\Phi$ is finitely realised in $\frakA_\infty$ since for any finite subset $\Phi_0 \subseteq \Phi$ we can construct a witness componentwise. But for every $b = [ (b_i)_{i \in \N} ] \in A_\infty$ the set \[ I := \{ a \in \{ 0 \} \times \omega_1 \colon \text{ there exists an $i \in \N$ such that $a E^\frakA b_i$} \} \] is countable (since the set of all $a$ satisfying $a E^\frakA b_i$ for some fixed $i$ is finite); therefore its subset \[ \{ a \in \{ 0 \} \times \omega_1 \colon \iota(a) E^{\frakA_\infty} b \} \] is also countable and therefore not all of $\{ 0 \} \times \omega_1$, implying that $\Phi$ is not realised in $\frakA_\infty$. The type $\Phi$ has exactly $\omega_1$ parameters, so $\frakA_\infty$ is not $\omega_2$-saturated. However, $\frakA_\infty$ has cardinality $2^\omega$, which is at least $\omega_2$ if we assume the negation of the continuum hypothesis. Therefore $\frakA_\infty$ is not saturated under this assumption.
\begin{proposition}\label{propositionNonIsomElemEquivUltraprods}
It is consistent with ZFC to have two $\N$-fold ultraproducts, both of cardinality $2^\omega$, which are elementarily equivalent but not isomorphic.
\end{proposition}
\begin{proof}
By Easton's theorem \cite{easton} it is compatible with ZFC to require $2^{\omega_2} = 2^\omega$. Consider the structure $\frakA$ from the observation above. By Theorem 8.2.1 in \cite{hodges} we can find an $\omega_2$-saturated elementary extension $\frakB$ of $\frakA$ of cardinality no greater than \[ \omega_1^{< \omega_2} \leq \omega_1^{\omega_2} \leq (2^\omega)^{\omega_2} = 2^{\omega \omega_2} = 2^{\omega_2} = 2^\omega. \]
Let now $\calU_1$ and $\calU_2$ be arbitrary (not necessarily distinct) non-principal ultrafilters on $\N$ and consider the ultrapowers $\frakA_\infty = \frakA^\N / \calU_1$ and $\frakB_\infty = \frakB^\N / \calU_2$. Assume that there exists an isomorphism $f \colon \frakA_\infty \to \frakB_\infty$. We know that the type
\[ \Phi := \{ c_a E x \colon a \in \iota(\{ 0 \} \times \omega_1) \} \]
of $\frakA_\infty$ is not realised in $\frakA_\infty$. If we can show that there is a $b$ in $\frakB_\infty$ that realises
\[ \Phi' := \{ c_{f(a)} E x \colon a \in \iota(\{ 0 \} \times \omega_1) \}, \]
we have a contradiction, since then $f^{-1}(b)$ would necessarily realise $\Phi$ in $\frakA_\infty$. But finding $b$ is easy: For every $a \in \iota(\{ 0 \} \times \omega_1)$ pick a representative $a' = (a'_i)_i \in B^\N$ of $f(a)$. Now for every $i \in \N$ we can choose $b_i$ to satisfy
\[ \Phi'_i := \{ c_{a'_i} E x \colon a \in \iota(\{ 0 \} \times \omega_1) \}: \] $\Phi'_i$ is finitely realised in $\frakB$ because $\frakB \equiv \frakA$ and hence there must exist such a $b_i \in B$ by $\omega_2$-saturation of $\frakB$. It is now clear that $b := [(b_i)_{i \in \N}]$ realises the type $\Phi'$.
We have therefore shown that $\frakA_\infty$ and $\frakB_\infty$ are not isomorphic. However, they are elementarily equivalent since $\frakB \succeq \frakA$.
\end{proof}
\section{Notions from games: ultraproducts as a limit of their factors}
The general usefulness of ultraproducts stems from \loz's theorem; it allows us to interpret an ultraproduct as a form of ``limit'' of its factors, at least \wrt first-order properties. In this section we want to look at the purely logical properties of ultraproducts; this will also lead us to Ehrenfeucht games.
We recall that two structures are called \emph{$m$-equivalent}, denoted by $\equiv_m$, if they satisfy the same first-order sentences of quantifier-rank up to $m$.
We also recall the following facts:
\begin{proposition}
Let $\sigma$ be a finite relational signature and $m \in \N$.
\begin{enumerate}
\item There are only finitely many first-order formulae in signature $\sigma$ up to quantifier rank $m$.
\item For every $\sigma$-structure $\frakA$ there is a sentence $\chi$ of quantifier-rank $m$ that axiomatises the $\equiv_m$-equivalence class of $\frakA$, \ie for any $\sigma$-structure $\frakB$ it holds that $\frakA \equiv_m \frakB \iff \frakB \models \chi$.
\end{enumerate}
\end{proposition}
The following lemma is a useful starting point for investigating the behaviour of ultraproducts as limits of their factors.
\begin{lemma}\label{lemmaUltraproductsElEquivIffMostFactorsMEquiv}
Let $(\frakA_i)_{i \in I}$ and $(\frakB_i)_{i \in I}$ be two families of $\sigma$-structures and $\calU$ an ultrafilter on $I$. Consider the following two statements:
\begin{enumerate}
\item For all $m \in \N$, there are $\calU$-many indices $i \in I$ with $\frakA_i \equiv_m \frakB_i$.
\item $\prod_i \frakA_i / \calU \equiv \prod_i \frakB_i / \calU$
\end{enumerate}
Then the first statement implies the second one. If the signature $\sigma$ is finite and relational, then the converse is also true.
\end{lemma}
\begin{proof}
We first assume the first statement and prove the second one.
Let $\phi \in \FO(\sigma)$ and let $m$ be the quantifier-rank of $\phi$. Then
\begin{align*}
&{\prod}_i \frakA_i / \calU \models \phi \\
\iff &\{ i \in I \colon \frakA_i \models \phi \} \in \calU \\
\iff &\{ i \in I \colon \frakA_i \models \phi \text{ and } \frakA_i \equiv_m \frakB_i \} \in \calU \\
\iff &\{ i \in I \colon \frakB_i \models \phi \text{ and } \frakA_i \equiv_m \frakB_i \} \in \calU \\
\iff &\{ i \in I \colon \frakB_i \models \phi\} \in \calU \\
\iff &{\prod}_i \frakB_i / \calU \models \phi .
\end{align*}
Now assume conversely that the two ultraproducts $\prod_i \frakA_i / \calU$ and $\prod_i \frakB_i / \calU$ are elementarily equivalent. Let $m \in \N$. If the signature $\sigma$ is finite and relational, there is a formula $\chi$ of quantifier rank at most $m$ which axiomatises the class of all structures which are $m$-equivalent to the ultraproducts. Since the ultraproducts both satisfy $\chi$, there are $\calU$-many indices $i$ such that $\frakA_i \models \chi$ and $\frakB_i \models \chi$, which implies $\frakA_i \equiv_m \frakB_i$.
\end{proof}
\begin{remark}
If the signature $\sigma$ is infinite, the second implication of the lemma becomes false in general. Take $\sigma = \{ c \} \cup \{ P_i \colon i \in \N \}$ with a constant symbol and a countably infinite number of unary predicate symbols and consider structures $\frakA_n$ and $\frakB_n$, all with universe $\{ 0 \}$, such that the constant symbol is interpreted by $0$ and \[ \frakA_n \models P_i c \iff i \leq 2n \quad \text{and} \quad \frakB_n \models P_i c \iff i \leq 2n + 1. \] Then none of the structures are equivalent even when we only consider quantifier-free formulae, but the ultraproducts \wrt non-principal ultrafilters are isomorphic.
\end{remark}
In the rest of the section, we assume the signature $\sigma$ to be finite and relational.
It turns out that we can actually get even more than the preceding lemma when we consider the game-theoretic interpretation of elementary equivalence due to Ehrenfeucht and Fra\"issé. We assume that the reader is familiar with the concept of Ehrenfeucht games.
The basic result on these games is the following (\cite[Theorem 2.2.8]{finiteModelTheory}):
\begin{theorem}[Ehrenfeucht]
The following are equivalent for any $m \in \N$ and two $\sigma$-structures $\frakA$ and $\frakB$:
\begin{itemize}
\item The duplicator has a winning strategy for the $m$-round Ehrenfeucht game on $\frakA$ and $\frakB$.
\item $\frakA \equiv_m \frakB$
\end{itemize}
\end{theorem}
As an immediate consequence, the structures $\frakA$ and $\frakB$ are elementarily equivalent iff the duplicator has a winning strategy for an Ehrenfeucht game of any pre-announced length. This is not the same as saying that the duplicator has a strategy for playing ``forever'', \ie playing a countably infinite number of rounds. If we even have his stronger condition of equivalence of $\frakA$ and $\frakB$, we also call the two structures \emph{partially isomorphic}, written $\frakA \simeq_\text{part} \frakB$.
We will also sometimes use the extension \[ \frakA, a_1, \dotsc, a_n \simeq_\text{part} \frakB, b_1, \dotsc, b_n \] to mean that we can start the infinite Ehrenfeucht game with the elements $a_1, \dotsc, a_n$ and $b_1, \dotsc, b_n$ already selected and the duplicator still wins.
Partial isomorphy also has a characterisation in terms of logical formulae. The right logic to use this time is $L_{\infty\omega}(\sigma)$, the infinitary variant of $ \FO(\sigma)$ in which infinite disjunction and conjunction are allowed. We now get the following result (\cite[Theorem 3.2.7]{finiteModelTheory}):
\begin{theorem}[Karp]
The following are equivalent for any $m \in \N$ and two $\sigma$-structures $\frakA$ and $\frakB$:
\begin{itemize}
\item $\frakA \simeq_\text{part} \frakB$
\item $\frakA \equiv^{L_{\infty\omega}} \frakB$, \ie for all sentences $\phi \in L_{\infty\omega}(\sigma)$ we have $\frakA \models \phi \iff \frakB \models \phi$ .
\end{itemize}
\end{theorem}
\begin{lemma}\label{lemmaUltraproductsPartiallyIsomorphic}
Let $(\frakA_i)_{i \in \N}$ and $(\frakB_i)_{i \in \N}$ be two sequences of $\sigma$-structures and $\calU$ a non-principal ultrafilter on $\N$ such that for all $m \in \N$ there are $\calU$-many indices $i \in \N$ with $\frakA_i \equiv_m \frakB_i$. Then the ultraproducts $\frakA := \prod_i \frakA_i / \calU$ and $\frakB := \prod_i \frakB_i / \calU$ are partially isomorphic.
\end{lemma}
\begin{proof}
We give a winning strategy for the duplicator in the unbounded Ehrenfeucht game. The game is played with equivalence classes of sequences. Associating each equivalence class with an arbitrary representative, we can think of the game on $\frakA$ and $\frakB$ to consist of a sequence of games on the $\frakA_i$ and $\frakB_i$. For the duplicator to win the game on the ultraproducts, it suffices not to lose on $\calU$-many components at any given point in time. (For the duplicator to lose the game, there must exist an atomic formula with parameters which exhibits different behaviour on the ultraproduct; by \loz's theorem, this implies losing the game on $\calU$-many components.)
For $i \in \N$ define \[ n_i := \max \{ k \leq i \colon \frakA_i \equiv_k \frakB_i \} .\] By the assumptions on $(\frakA_i)$ and $(\frakB_i)$ and non-principality of the ultrafilter $\calU$, we have $n_i \geq m$ for $\calU$-many indices $i$ when $m$ is fixed. Now the duplicator has the following winning strategy for the game on $\frakA$ and $\frakB$: When the spoiler marks the $(m + 1)$-st element in the game, she interprets this as the $(m + 1)$-st elements in the games on all $\frakA_i$ and $\frakB_i$. For indices $i$ with $n_i > m$, she answers by making a move that wins the $n_i$-round game on $\frakA_i$ and $\frakB_i$. For all other indices $i$, she just makes an arbitrary move. Then she will not have lost for one more round on $\calU$-many factors of the ultraproduct.
\end{proof}
\begin{corollary}
Let $(\frakA_i)_{i \in \N}$ and $(\frakB_i)_{i \in \N}$ be two sequences of $\sigma$-structures and $\calU$ a non-principal ultrafilter on $\N$. Then the two ultraproducts $\frakA := \prod_i \frakA_i / \calU$ and $\frakB := \prod_i \frakB_i / \calU$ exhibit the same behaviour \wrt the logic $\mathrm{L}_{\infty\omega}$ if they are elementarily equivalent.
\end{corollary}
\begin{remark}
Lemma \ref{lemmaUltraproductsPartiallyIsomorphic} and therefore also the preceding corollary are true for all $\omega$-saturated structures, as can readily be seen. Nevertheless, we consider our proof via a game on pairs of ultraproduct factors an interesting new approach to this standard result.
\end{remark}
\section{Hanf sequences}
In this section, we assume the signature $\sigma$ to be finite and purely relational.
We first recall some notions from model theory.
\begin{definition}
Let $\frakA$ be a $\sigma$-structure with universe $A$.
\begin{itemize}
\item The \emph{Gaifman graph} of $\frakA$ is the graph with node set $A$ where there is an edge between two distinct nodes $a, b \in A$ iff there is a relation symbol $R \in \sigma$ and a tuple $\mathbf{a} \in R^\frakA$ such that $a$ and $b$ are elements of $\mathbf{a}$.
\item Let $m \in \N$ and $a \in A$. The \emph{$m$-ball of $a$}, denoted $S_m(a)$, is the set of all those $a' \in A$ such that the distance between $a$ and $a'$ in the Gaifman graph of $\frakA$ is at most $m$. We also use $S_m(a)$ synonymously with the induced substructure $\frakA \restriction S_m(a)$.
\item Let $m \in \N$ and $a \in A$. The \emph{$m$-ball type of $a$} is the isomorphism type of the structure $(\frakA \restriction S_m(a), a)$ understood as a substructure of $\frakA$ expanded by an additional constant symbol interpreted as $a$.
\end{itemize}
\end{definition}
We note that the edge relation in the Gaifman graph is $\FO(\sigma)$-definable and therefore $m$-balls are $\FO(\sigma)$-definable. In connection with the relativisation property of first-order logic, we observe the following:
\begin{observation}
For every formula $\phi(x) \in \FO_1(\sigma)$ and every $m \in \N$ there is a formula $\psi(x) \in \FO_1(\sigma)$ such that $\frakA, a \models \psi$ iff $S_m(a), a \models \phi$. In particular, we note that characteristic formulae axiomatising structures up to $n$-equivalence or up to isomorphism (for finite structures) can be relativised to $m$-balls in this manner.
\end{observation}
Furthermore we observe that if the Gaifman graph of a structure is disconnected, there is no relation that connects elements of different connected components. When we investigate homomorphisms and isomorphisms and also game strategies, we can therefore usually focus on one connected component:
\begin{observation}\label{observationIsomorphismFromConnComps}
Let $\frakA$ and $\frakB$ be two $\sigma$-structures. If we can find a bijection between the connected components of the Gaifman graphs of $\frakA$ and $\frakB$ such that associated components are isomorphic, then we can construct an isomorphism of $\frakA$ and $\frakB$ by joining isomorphisms of the connected components.
\end{observation}
Gaifman graphs can be used to phrase the following standard result (originally in \cite{hanf}, quoted as in \cite[Theorem 2.4.1]{finiteModelTheory}):
\begin{theorem}[Hanf]
Let $\frakA$ and $\frakB$ be $\sigma$-structures and let $m \in \N$. Suppose that for some $e \in \N$ all $3^m$-balls in $\frakA$ and $\frakB$ have less than $e$ elements and that for each $3^m$-ball type $\iota$, one of the following conditions holds:
\begin{enumerate}
\item Both $\frakA$ and $\frakB$ have the same number of elements of $3^m$-ball type $\iota$.
\item Both $\frakA$ and $\frakB$ have more than $m \cdot e$ elements of $3^m$-ball type $\iota$.
\end{enumerate}
Then the duplicator has a winning strategy in the $m$-round Ehrenfeucht game on $\frakA$ and $\frakB$, \ie $\frakA \equiv_m \frakB$.
\end{theorem}
The theorem is usually proved by explicitly giving the winning strategy. It is a useful tool for proving first-order inexpressibility of some property: it suffices to find two sequences $(\frakA_n)_n$ and $(\frakB_n)_n$ such that $\frakA_n$ is $n$-Hanf equivalent to $\frakB_n$ but the property in question is present in exactly one of the two structures.
We now want to present a new approach to this technique for proving inexpressibility. Our requirements will be similar to those of Hanf's theorem, but we will use ultraproducts instead of games.
\begin{definition}
A sequence of $\sigma$-structures $(\frakA_i)_{i \in \N}$ such that each of the structures has cardinality $\leq 2^\omega$ is called a \emph{Hanf sequence} if the following conditions are satisfied:
\begin{enumerate}
\item For every $m \in \N$ there is an $e \in \N$ such that every $m$-ball in each of the structures contains at most $e$ elements.
\item For every $m \in \N$ and every $m$-ball type $\iota$, one of the following two conditions holds:
\begin{enumerate}
\item For some $c \in \N$ there is an $i_0 \in \N$ such that for all $i \geq i_0$, $\frakA_i$ has exactly $c$ elements of $m$-ball type $\iota$.
\item For every $c \in \N$ there is an $i_0 \in \N$ such that for all $i \geq i_0$, $\frakA_i$ contains at least $c$ elements of $m$-ball type $\iota$.
\end{enumerate}
\end{enumerate}
\end{definition}
It is useful to compare the conditions for a Hanf sequence to the requirements on two structures for Hanf's Theorem: The first condition is also required there for one $m = 3^n$. The second condition in both cases can loosely be read as ``The number of elements of a specific $m$-ball type is either large or equal in all structures considered''.
\begin{remark}
Both conditions can be rephrased. It is easy to see that for the first condition it is both necessary and sufficient to have a uniform bound for the degrees in the Gaifman graphs of all $\frakA_i$.
The second condition can be rewritten in more abstract terms: the sequence $(n^\iota_i)_i$ given by
\[
n^\iota_i := n^\iota(\frakA_i) := \abs{\{ a \in \frakA_i \colon \text{ $a$ has $m$-ball type $\iota$ } \}} \in \N \cup \{ \infty \}
\]
must converge in the topological space $\N \cup \{ \infty \}$ (given as the Alexandroff compactification of the discrete space $\N$). We will later see how we can use other notions of convergence.
\end{remark}
Our main aim for this section is the following theorem:
\begin{theorem}\label{theoremHanfSequences}
Let $(\frakA_i)_{i \in \N}$ be a Hanf sequence. Let furthermore $\calU_1$ and $\calU_2$ be two non-principal ultrafilters on $\N$. Then the ultraproducts $\frakA^{\calU_1} := \prod_i \frakA_i / \calU_1$ and $\frakA^{\calU_2} := \prod_i \frakA_i / \calU_2$ are isomorphic.
\end{theorem}
To prove this theorem we will first look at the connected components of the Gaifman graphs of the two ultraproducts.
The connected component of an element $a$ in the Gaifman graph of a $\sigma$-structure can be written as $\bigcup_{i \in \N} S_i(a)$. We will also suggestively denote it by $S_\omega(a)$ (the \emph{$\omega$-ball of $a$}). The isomorphism type of $(S_\omega(a), a)$ will be the $\omega$-ball type of $a$.
We note that the $\omega$-ball type of $a$ encodes the full isomorphism type of the connected component of $a$ and additionally the position of $a$ in this connected component.
\begin{lemma}\label{lemmaIsomorphyOfOmegaBalls}
Let $\frakA$ and $\frakB$ be $\sigma$-structures. Let $a \in A$, $b \in B$ such that for each $m \in \N$ we have \[ (\frakA \restriction S_m(a), a) \simeq (\frakB \restriction S_m(b), b) \] and each of these $m$-balls is finite. Then \[ (\frakA \restriction S_\omega(a), a) \simeq (\frakB \restriction S_\omega(b), b), \] \ie $a$ and $b$ have the same $\omega$-ball type.
\end{lemma}
\begin{proof}
The statement can be seen as an application of König's lemma: consider the set \[ M := \{ p \colon S_m(a) \to S_m(b) \colon m \in \N, \text{$p$ is an isomorphism of $(S_m(a), a)$ and $(S_m(b), b)$} \} .\] We establish a graph structure on $M$ by putting an edge between maps $p \colon S_m(a) \to S_m(b)$ and $q \colon S_{m + 1}(a) \to S_{m + 1}(b)$ iff $q$ is an extension of $p$. Arranging the vertices in $M$ in levels by the number $m$, we naturally get a tree with root $S_0(a) \to S_0(b), a \mapsto b$. (The assumption on isomorphy of the $m$-balls of $a$ and $b$ guarantees that each level is non-empty.) Since every $m$-ball is finite, there are only finitely many maps $S_m(a) \to S_m(b)$ in each level and hence every vertex has finite degree. By König's lemma, there must exist an infinite path of vertices, \ie a chain $(p_i)_{i \in \N}$ where $p_i \colon S_i(a) \to S_i(b)$ is an isomorphism and $p_{i + 1}$ is an extension of $p_i$. We can therefore form the limit $p := \bigcup_i p_i$ and easily convince ourselves that this is an isomorphism from $(S_\omega(a), a)$ to $(S_\omega(b), b)$.
\end{proof}
\begin{remark}
Without the assumption about finiteness of the $m$-balls, the statement of the lemma can become false. In the signature of graphs consider a tree consisting of infinitely many disjoint finite paths of unbounded length starting from the root. Adding one path of infinite length then changes the isomorphism type, but does not change the isomorphism type of the $m$-balls of the root.
\end{remark}
\begin{lemma}\label{lemmaForHanfSequences}
Let $(\frakA_i)_{i \in \N}$ and $(\frakB_i)_{i \in \N}$ be two sequences of $\sigma$-structures, each of cardinality at most $2^\omega$, such that for every $m \in \N$ there is an $e \in \N$ such that every $m$-ball in any of the structures contains at most $e$ elements. Let furthermore $\calU_1, \calU_2$ be two non-principal ultrafilters on $\N$ such that for each $m \in \N$ and every $m$-ball type $\iota$, one of the following holds:
\begin{enumerate}
\item[(a)] There is a $c \in \N$ such that
\[ \{ i \in \N \colon n^\iota(\frakA_i) = c \} \in \calU_1 \quad \text{and} \quad \{ i \in \N \colon n^\iota(\frakB_i) = c \} \in \calU_2 .\]
\item[(b)] For every $c \in \N$ it holds that
\[ \{ i \in \N \colon n^\iota(\frakA_i) > c \} \in \calU_1 \quad \text{and} \quad \{ i \in \N \colon n^\iota(\frakB_i) > c \} \in \calU_2 .\]
\end{enumerate}
Then the ultraproducts $\frakA_\infty := \prod_i \frakA_i / \calU_1$ and $\frakB_\infty := \prod_i \frakB_i / \calU_2$ are isomorphic.
\end{lemma}
\begin{proof}
For every $m \in \N$, there is a bound on the size of $m$-balls in $\frakA_\infty$ and $\frakB_\infty$ (The property of having $m$-balls with at most $e$ elements is first-order definable.) Therefore each $\omega$-ball in $\frakA_\infty$ is countable. Furthermore, the set of elements of $\frakA_\infty$ of specific $\omega$-ball type is given by a countable type -- by the last lemma we simply have to prescribe all $m$-ball types and each $m$-ball type is an isomorphism type of a finite structure and therefore axiomatisable by a single formula. By Proposition \ref{propCountableTypesAreRealised} this means that the number of occurrences of every $\omega$-ball type is either finite or exactly $2^\omega$. (Since $\abs{\frakA_\infty} \leq 2^\omega$, more occurrences are impossible.) If some $m$-ball subtype $\iota'$ of an $\omega$-ball type $\iota$ is realised exactly $c \in \N$ times in $\calU_1$-many $\frakA_i$, then $\iota'$ will be realised $c$ times in $\frakA_\infty$. But if this is not the case for any $c$ and any subtype of $\iota$, then there will be $2^\omega$ occurrences of $\iota$. We especially note that the number of occurrences of $\iota$ is the same in $\frakA_\infty$ and $\frakB_\infty$.
We now want to show that each isomorphism type of connected components occurs the same number of times in both $\frakA_\infty$ and $\frakB_\infty$. Fix a connected component $C$ of the Gaifman graph of $\frakA_\infty$ or $\frakB_\infty$ and look for other occurrences of the isomorphism type of $C$.
Select an $\omega$-ball type $\iota$ occurring in $C$. As $C$ is countable, $\iota$ can occur at most count\-ably many times in $C$; in all of $\frakA_\infty$ and $\frakB_\infty$, $\iota$ can only occur in connected components isomorphic to $C$. Since $\iota$ occurs either finitely often or $2^\omega$ times in both $\frakA_\infty$ and $\frakB_\infty$, the number of occurrences of $\iota$ in one of the structures determines the number of occurrences of the isomorphism type of $C$: If $\iota$ occurs $2^\omega$ times, then the isomorphism type must occur $2^\omega$ times, but if $\iota$ occurs finitely often, then we can simply compute the number of occurrences of the isomorphism type of $C$ as we know that each occurrence of the latter corresponds to a fixed number of occurrences of $\iota$. As $\iota$ occurs the same number of times in both $\frakA_\infty$ and $\frakB_\infty$, the isomorphism type of $C$ must also occur the same number of times in $\frakA_\infty$ and $\frakB_\infty$.
It is therefore possible to match the connected components of the Gaifman graph of $\frakA_\infty$ one-to-one with the connected components of the Gaifman graph of $\frakB_\infty$ such that associated components are isomorphic.
In the light of Observation \ref{observationIsomorphismFromConnComps} this suffices to obtain an isomorphism of $\frakA_\infty$ and $\frakB_\infty$.
\end{proof}
\begin{proof}[Proof of the theorem]
This is now immediate. Since every cofinite set is contained in every non-principal ultrafilter, the conditions on Hanf sequences guarantee the applicability of the last lemma for any pair of non-principal ultrafilters on $\N$.
\end{proof}
\begin{remark}
We note that the slightly unwieldy conditions in the main lemma and the definition of Hanf sequences can be beautifully rephrased in topological terms. For this consider a sequence $(x_i)_{i \in \N}$ in a topological space, in this case $\N \cup \{ \infty \}$, and an arbitrary filter $\calF$ on $\N$. We define the sequence to converge to $x_\infty$ \wrt $\calF$, written $\calF{\!-\!}\lim_i x_i = x_\infty$, if for all neighbourhoods $U$ of $x_\infty$ we have \[ \{ i \in \N \colon x_i \in U \} \in \calF .\] Then the second condition on Hanf sequences simply states that $\calF_{\text{Fréchet}}{\!-\!}\lim_i n^\iota(\frakA_i)$ must exist (where $\calF_{\text{Fréchet}}$ is the cofinite filter). The condition in the main lemma reads as \[ \calU_1{\!-\!}\lim_i n^\iota(\frakA_i) = \calU_2{\!-\!}\lim_i n^\iota(\frakB_i) .\] It can be seen that both limits always exist; the remaining condition is equality of the limits. (Why do both limits exist? Consider a sequence $(x_i)$ in a compact space $X$ as a map $\mathbf x \colon \N \to X$. Then for any ultrafilter $\calU$ on $\N$, the image filter $\mathbf x(\calU)$ is an ultrafilter on $X$ and therefore converges to some $x_\infty \in X$. This is the limit of $(x_i)$ \wrt $\calU$.)
\end{remark}
\begin{corollary}
Let $(\frakA_i)_i$ be a Hanf sequence and $m \in \N$. Then almost all $\frakA_i$ are $m$-equivalent, \ie there exists $N \in \N$ such that \[ \frakA_n \equiv_m \frakA_N \] for all $n \geq N$.
\end{corollary}
\begin{proof}
Consider an arbitrary non-principal ultrafilter $\calU_1$ on $\N$. Since the signature $\sigma$ is finite, there is a formula $\chi$ which axiomatises $\prod_i \frakA_i / \calU_1$ up to $m$-equivalence. It now suffices to prove that only finitely many $\frakA_i$ do not satisfy $\chi$ -- all other $\frakA_i$ will then be $m$-equivalent to $\prod_i \frakA_i / \calU_1$. Assume to the contrary that there is an infinite set $I \subseteq \N$ of indices such that $\frakA_i \models \neg \chi$ for all $i \in I$. Then $I$ together with the Fréchet filter, \ie the set \[ \{ I \} \cup \{ M \subseteq \N \colon \text{ $\N \setminus M$ is finite} \} \] has the finite intersection property, hence there is a non-principal ultrafilter $\calU_2$ on $\N$ with $I \in \calU_2$. Now we have $\prod_i \frakA_i / \calU_2 \models \neg \chi$ but $\prod_i \frakA_i / \calU_1 \models \chi$ in contradiction to the last theorem.
\end{proof}
\begin{remark}
This corollary is the primary use of the main theorem. It can also be immediately derived from Hanf's theorem. The statement itself is therefore unremarkable. However, we are unaware of any previous proof not using games.
\end{remark}
We finish this section with some applications of the method of Hanf sequences. We will mostly consider undirected and directed graphs in the signature $\sigma_G = \{ E \}$ with binary edge relation $E$, as the transition from a structure to its Gaifman graph is very simple in these cases.
\begin{proposition}
For $n \in \N$, let $\frakA_n$ be the directed path graph on $n$ nodes, \ie the $\sigma_G$-structure with universe $\{ 1, \dotsc, n \}$ and edge relation \[ E^{\frakA_n} = \{ (i, i + 1) \colon 1 \leq i < n \} .\] Then $(\frakA_n)_n$ is a Hanf sequence. In particular, for any given finite $m$, almost all $\frakA_n$ are $m$-equivalent. This implies that the property of having an even number of elements is not definable by a single first-order formula on this class of structures.
\end{proposition}
\begin{proof}
Since every node has degree at most two, the first condition for Hanf sequences is easily satisfied. Let $m \in \N$ be given and consider all $m$-ball types occuring in any of the structures. For $n > 2m$, the elements $m + 1, m + 2, \dotsc, n - m$ in $\frakA_n$ all have the same $m$-ball type. All other $m$-ball types occur exactly once since they are ``cut off'' either at the upper or the lower end. This proves that $(\frakA_i)_i$ is a Hanf sequence.
\end{proof}
\begin{proposition}
The class of planar graphs is not definable by a first-order formula among all finite graphs.
\end{proposition}
\begin{proof}
Let $\frakA_0 = K_5$ be the complete graph on $5$ nodes with $10$ edges. Let $\frakA_n$ be constructed from $\frakA_0$ by inserting $n$ additional nodes in each edge. This yields a graph on $5 + 10n$ nodes with $10n$ nodes of degree two and $5$ distinguished nodes of degree four such that each pair among the latter nodes is connected by a path of length $n + 1$. It is clear that $\frakA_0$ is a minor of every $\frakA_n$, hence all $\frakA_n$ are not planar.
On the other hand, consider a graph $G_n$ with the following structure:
\[
\xymatrix{
{\bullet} \ar@{-}[r] & {\bullet} \ar@{.}[r] & {\hdots} \ar@{.}[r] &
{\bullet} \ar@{-}[r] & {\bullet} & \text{$n$ nodes} \\
& & {\bullet} \ar@{-}[ull] \ar@{-}[dll] \ar@{-}[urr] \ar@{-}[drr] \\
{\bullet} \ar@{-}[r] & {\bullet} \ar@{.}[r] & {\hdots} \ar@{.}[r] &
{\bullet} \ar@{-}[r] & {\bullet} & \text{$n$ nodes}
}
\]
This is a graph on $2n + 1$ nodes with one node of degree four and $2n$ nodes of degree two. Let now $\frakB_n := G_n \mathop{\dot\cup} G_n \mathop{\dot\cup} G_n \mathop{\dot\cup} G_n \mathop{\dot\cup} G_n$. Then $\frakB_n$ is obviously a planar graph since each $G_n$ is planar. Furthermore, we easily convince ourselves that the sequence $\frakA_2, \frakB_2, \frakA_3, \frakB_3, \dotsc$ is a Hanf sequence. As above, it follows that planarity is not definable by a single first-order formula.
\end{proof}
In the same manner we could now easily prove that connectivity of finite graphs is not first-order definable, as a sequence of structures consisting of cycle graphs and disjoint unions of two cycle graphs with increasing number of elements is a Hanf sequences. Instead, we prove the following stronger result. It was originally proved by Fagin in \cite{fagin} with a game argument. Nowadays it is often proved using Hanf's theorem. We slightly adapt the latter proof.
\begin{proposition}
The property of being connected is not definable in existential monadic second order logic on the class of finite graphs.
\end{proposition}
\begin{proof}
Assume to the contrary that the $\exists$-MSO sentence \[ \psi = \exists P_1 \dotsc \exists P_K \phi \] with $\phi \in \FO_0(\{ E, P_1, \dotsc P_K \})$ does define connectivity on finite graphs.
For every $n \in \N$, let $\frakC_n$ be a cycle graph coloured in such a way as to satisfy $\phi$, \ie a $\{ E, P_1, \dotsc P_K \}$-structure satisfying $\phi$ in which every element has degree $2$, and let $\frakC_n'$ be a a disjoint union of two coloured cycle graphs such that the following condition is satisfied:
\[ \text{For every $n$-ball type $\iota$, both $\frakC_n$ and $\frakC_n'$ have the same number of elements of $n$-ball type $\iota$.} \]
We will need to argue later that this condition can be satisfied. Note that $\frakC_n'$ cannot satisfy $\phi$ since a disjoint union of cycle graphs is not connected. We could now consider the sequence of structures $\frakC_1, \frakC_1', \frakC_2, \frakC_2', \dotsc$, but it is not clear whether this is a Hanf sequence. The original Lemma \ref{lemmaForHanfSequences} comes to the rescue. Let $\calU$ be a non-principal ultrafilter on $\N$ and consider $\frakC_\infty := \prod_i \frakC_i / \calU$ and $\frakC_\infty' := \prod_i \frakC_i' / \calU$. By construction $\frakC_\infty \models \phi$ and $\frakC_\infty' \models \neg\phi$.
However, for all $m \in \N$ and all $m$-ball types $\iota$ we have $n^\iota(\frakC_k) = n^\iota(\frakC_k')$ whenever $k \geq m$. From this we quickly derive \[ \calU{\!-\!}\lim_i n^\iota(\frakC_i) = \calU{\!-\!}\lim_i n^\iota(\frakC_i') .\] By Lemma \ref{lemmaForHanfSequences}, this implies $\frakC_\infty \simeq \frakC_\infty'$, which contradicts the behaviour of $\phi$.
It remains to see that we can find $\frakC_n$ and $\frakC_n'$ with the desired condition. This is exactly as in the standard proof of this proposition using Hanf's theorem. Consider a cycle graph on $N \geq 2n + 2$ nodes. As this satisfies $\psi$, there exists a colouring such that the coloured graph satisfies $\phi$. Since the $n$-ball of an element has exactly $2n + 1$ elements, at most $(2^K)^{2n + 1}$ $n$-ball types occur. For large $N$ we can force the existence of at two elements $a, b$ having the same $m$-ball type, even when taking cycle orientation into account, and having distance at least $2n + 1$. We can now modify two edges starting at $a$ and $b$ such that the cycle splits into two cycles, but without modifying the $m$-ball type of any element: Let $a'$ and $b'$ be neighbours of $a$ and $b$, respectively, in the same direction, \ie a traversal of the cycle starting at $a$ and $a'$ yields the nodes in order $a, a', b, b'$. Removing the edges $\{ a, a' \}$ and $\{ b, b' \}$ and adding the edges $\{ a, b' \}$ and $\{ b', a' \}$ results in two disjoint cycles, but does not change $n$-ball types due to choice of $a$ and $b$. Putting $\frakC_n$ to be the original coloured cycle of length $N$ and $\frakC_n'$ to be the modified graph fulfills the desired conditions.
\[
\xymatrix{
a' \ar@(l,l)@{.}[dd] & a \ar@{-}[l] & & & & a' \ar@(l,l)@{.}[dd] & a \ar@{-}[dd]\\
& & \ar@{-->}[rr] & & \\
b \ar@{-}[r] & b' \ar@(r, r)@{.}[uu] & & & & b \ar@{-}[uu] & b' \ar@(r, r)@{.}[uu]
}
\]
\end{proof}
\section{Gaifman's theorem via ultraproducts}
We again restrict our attention to finite relational signatures in this section.
Besides Hanf's theorem there is one more well-known locality theorem in first-order logic, which is due to Gaifman.
\begin{definition}
A formula $\phi(x) \in \FO_1(\sigma)$ is \emph{$l$-local} for $l \in \N$ if for all $\sigma$-structures $\frakA$ and all $a \in A$ we have \[ \frakA, a \models \phi \iff \frakA \restriction S_l(a), a \models \phi .\]
A sentence of the form
\[ \exists x_1, \dotsc, x_n \Big( \bigwedge_{1 \leq i < j \leq n} d_{> 2l}(x_i, x_j) \land \bigwedge_{1 \leq i \leq n} \psi(x_i) \Big) \]
where $\psi(x) \in \FO_1(\sigma)$ is an $l$-local formula is called a \emph{basic local sentence}. Here $d_{>2l}(x_i, x_j)$ is to be understood as a first-order formula asserting that $x_i$ and $x_j$ have distance greater than $2 l$ in the Gaifman graph.
\end{definition}
These definitions are useful because of the following locality result (\cite{gaifman}):
\begin{theorem}[Gaifman]
Every first-order sentence is logically equivalent to a boolean combination of basic local sentences.
\end{theorem}
This result is frequently proved by an Ehrenfeucht-Fra\"issé argument, although this is not how Gaifman originally proved it. We want to prove a variant of the theorem using ultraproducts. In our version we need an additional constraint on the size of $m$-balls as we know it from Hanf's theorem. We will also prove the full statement of Gaifman's theorem using a combination of game techniques, ultraproducts and compactness.
For both versions of the theorem we need the following consequence of the saturation properties of ultraproducts.
\begin{lemma}\label{lemmaOmegaSaturationImpliesNTypesAreRealised}
Let $\frakA$ be $\omega$-saturated and $\Psi \subseteq \FO_n(\sigma)$ such that \[ \frakA \models \exists x_1 \dotsm \exists x_n \bigwedge \Psi_0(x_1, \dotsc, x_n) \] for all finite subsets $\Psi_0 \subseteq \Psi$. Then there are elements $a_1, \dotsc, a_n \in A$ such that \[ \frakA, a_1, \dotsc, a_n \models \Psi .\]
\end{lemma}
We can now prove our first version of Gaifman's theorem.
\begin{theorem}\label{theoremGaifmanVariantIsomorphy}
Let $\frakA, \frakB$ be two $\N$-fold ultraproducts of cardinality $\leq 2^\omega$ \wrt non-principal ultrafilters such that the following condition is satisfied: For each $m \in \N$ there is an $e \in \N$ such that every $m$-ball in $\frakA$ and $\frakB$ has at most $e$ elements. If $\frakA$ and $\frakB$ satisfy the same basic local sentences, then they are isomorphic.
\end{theorem}
\begin{proof}
As in the proof of Theorem \ref{theoremHanfSequences} on Hanf sequences we want to show that each isomorphism type of connected components occurs as many times in $\frakB$ as it occurs in $\frakA$. By Observation \ref{observationIsomorphismFromConnComps} this suffices to prove that the structures are globally isomorphic. We recall that each connected component is described by any of the $\omega$-ball types of elements in it. Since each connected component is countable by the size restriction on $m$-balls and each $\omega$-ball type is either realised finitely often or exactly $2^\omega$ times, we find that each isomorphism type of connected components occurs finitely often or exactly $2^\omega$ times.
Let now $\iota$ be an arbitrary $\omega$-ball type. Assume that there are $n$ disjoint connected components in the Gaifman graph of $\frakA$ in which $\iota$ occurs; we show that there are at least $n$ such components in $\frakB$. Since the situation is symmetric in $\frakA$ and $\frakB$, this suffices to prove the claim.
Let $a_1, \dotsc, a_n \in A$ with $\omega$-ball type $\iota$ in disjoint connected components of the Gaifman graph of $\frakA$. For $m \in \N$ let $\chi_m(x) \in \FO_1(\sigma)$ be a formula asserting that \[ (S_m(x), x) \simeq (S_m(a_1), a_1) .\] Such a formula exists because $S_m(a_1)$ is finite by assumption. By definition $\chi_m$ is $m$-local. Now consider the sequence $(\psi_m)_m$ of formulae in $n$ free variables $x_1, \dotsc, x_n$ given by \[ \psi_m := \bigwedge_{1 \leq i < j \leq n} d_{>2m}(x_i, x_j) \land \bigwedge_{1 \leq i \leq n} \chi_m(x_i) .\] Clearly we have \[ \frakA, a_1, \dotsc, a_n \models \psi_m \] for all $m$. Hence each of the basic local sentences \[ \exists x_1 \dotsm \exists x_n \psi_m \] is satisfied in $\frakA$ and therefore also in $\frakB$. Since $\psi_{m + 1} \models \psi_m$, this means that every formula of the form \[ \exists x_1 \dotsm \exists x_n \bigwedge_{m \leq M} \psi_m \] with $M \in \N$ is true in $\frakB$. By the last lemma we therefore find elements $b_1, b_2, \dotsc, b_n$ such that $\frakB, b_1, \dotsc, b_n \models \psi_m$ for all $m$. The $b_i$ must pairwise have infinite distance since $\psi_m$ asserts that they have distance greater than $2 m$. Furthermore each $b_i$ has $\omega$-ball type $\iota$ by Lemma \ref{lemmaIsomorphyOfOmegaBalls}.
We have therefore found $n$ disjoint connected components of $\frakB$ in which the $\omega$-ball type $\iota$ occurs.
\end{proof}
We note that in the proof we did not actually use the factor structures of the ultraproducts. We only used the saturation properties of ultraproducts proved in the first section.
As a corollary we get a weak version of the Keisler-Shelah isomorphism theorem (\cite{keisler, shelahForKeislerShelah}), which in its full version is notoriously difficult to prove.
\begin{corollary}
Let $\frakA$ and $\frakB$ be elementarily equivalent $\sigma$-structures of cardinality $\leq 2^\omega$ such that for every $m \in \N$ there is an $e \in \N$ such that every $m$-ball in $\frakA$ and $\frakB$ has at most $e$ elements.
Let $\calU_1$ and $\calU_2$ be two arbitrary non-principal ultrafilters on $\N$. Then we have \[ \frakA^\N / \calU_1 \simeq \frakB^\N / \calU_2 .\]
\end{corollary}
\begin{proof}
Both the cardinality requirements and the other conditions expressible as first-order statements on $\frakA$ and $\frakB$ translate to their ultrapowers. Then the previous theorem is applicable.
\end{proof}
We are now almost in a position to prove the usual version of Gaifman's theorem. If we can get rid of the size restrictions on $m$-balls in Theorem \ref{theoremGaifmanVariantIsomorphy}, a simple compactness argument will yield Gaifman's theorem. We now join forces with the game technique to prove the following two lemmas. These are analogous to Lemma \ref{lemmaIsomorphyOfOmegaBalls} and Observation \ref{observationIsomorphismFromConnComps}.
\begin{lemma}\label{lemmaElementaryEquivalenceOfOmegaBalls}
Let $\frakA, \frakB$ be two $\omega$-saturated structures in signature $\sigma$ and $a \in A$, $b \in B$ such that
\[ (S_k(a), a) \equiv (S_k(b), b) \] for all $k \in \N$. Then \[ S_\omega(a), a \simeq_\text{part} S_\omega(b), b .\]
\end{lemma}
\begin{proof}
We give a winning strategy for the duplicator in the infinite Ehrenfeucht game. During the game we always maintain the following invariant: When $m$-tuples of elements $a_1, \dotsc, a_m \in S_\omega(a)$ and $b_1, \dotsc, b_m \in S_\omega(b)$ have been played after $m$ rounds, it holds that \[ (S_k(a), a, a_1, \dotsc, a_m) \equiv (S_k(b), b, b_1, \dotsc, b_m) \] for all $k$ such that $a_1, \dotsc, a_m \in S_k(a)$. If we manage to maintain this condition, it is clear that the duplicator will never lose the game.
Assume that $m$ rounds have been played. Let the spoiler choose a new element $a_{m + 1} \in S_\omega(a)$. (By symmetry of the situation it does not matter whether the spoiler chooses an element in $S_\omega(a)$ or in $S_\omega(b)$.) For $n \in \N$ and sufficiently large $k$, let $\phi_{k, n} \in \FO_{m + 2}(\sigma)$ be a formula such that
\[ \mathfrak{C}, c, c_1, \dotsc, c_{m + 1} \models \phi_{k, n} \iff (S_k(c), c, c_1, \dotsc, c_{m + 1}) \equiv_n (S_k(a), a, a_1, \dotsc, a_{m + 1}) \]
for any $\sigma$-structure $\mathfrak{C}$ and elements $c, c_1, \dotsc, c_{m + 1}$; this is possible as we can axiomatise the ball $(S_k(a), a, a_1, \dotsc, a_n)$ up to $n$-equivalence with a single formula since the signature $\sigma$ is finite. To maintain our invariant it suffices to find $b_{m + 1} \in S_\omega(b)$ such that \[ \frakB, b, b_1, \dotsc, b_m, b_{m + 1} \models \phi_{k, n} \] for all $k$ and all $n$. But we can interpret the set $\Phi$ consisting of all $\phi_{k, n}$ as a countable type with parameters $b, b_1, \dotsc, b_m$ since $\Phi$ is finitely realised in $\frakB$: Let $\Phi_0 \subseteq \Phi$ be a finite subset with the value $k$ of all $\phi_{k, n}$ in it bounded by some $K$. Then $a_{m + 1}$ realises $\Phi_0$ in $S_K(a)$ where we use $a, a_1, \dotsc, a_m$ as parameters; hence \[ \exists x \bigwedge\Phi_0(b, b_1, \dotsc, b_m, x) \] is also true in $S_K(b)$ (and therefore in $\frakB$) because of elementary equivalence.
Because of $\omega$-saturation we therefore find $b_{m + 1} \in B$ realising $\Phi$ with parameters $b, b_1, \dotsc, b_m$. Since the set of all $\phi_{kn}$ also fixes the distance of $b$ and $b_{m + 1}$, we even get $b_{m + 1} \in S_\omega(b)$. This establishes the invariant for one more round.
\end{proof}
\begin{remark}
Readers familiar with Fra\"issé's algebraic characterisation of partial isomorphy will recognise that we have essentially proved the existence of a back \& forth system.
\end{remark}
\begin{lemma}
Let $\frakA$ and $\frakB$ be $\sigma$-structures. Assume that for any finite number of disjoint connected components $A_1, \dotsc, A_n$ of the Gaifman graph of $\frakA$ with \[ \frakA \restriction A_1 \equiv \dotsb \equiv \frakA \restriction A_n \] there is an equal number $B_1, \dotsc, B_n$ of disjoint connected components of the Gaifman graph of $\frakB$ with \[ \frakA \restriction A_1 \equiv \frakB \restriction B_1 \equiv \dotsb \equiv \frakB \restriction B_n \] and vice versa with $\frakA$ and $\frakB$ exchanged; in other words, each $\equiv$-class of connected components either occurs the same finite number of times in both $\frakA$ and $\frakB$ or occurs infinitely often in both structures.
In this case, $\frakA$ and $\frakB$ are elementarily equivalent.
\end{lemma}
\begin{proof}
There is a winning strategy for the duplicator in an $m$-round Ehrenfeucht game on $\frakA$ and $\frakB$ for any $m \in \N$: For any move of the spoiler in a connected component of $\frakA$ or $\frakB$ in which no element has been picked yet, the duplicator chooses a new elementarily equivalent connected component of $\frakB$ or $\frakA$, respectively, and plays as if an $m$-round game were played on the two connected components. If the spoiler picks an element of a connected component of $\frakA$ or $\frakB$ in which an element has been picked before, the duplicator continues with the sub-game on the associated elementarily equivalent component of $\frakB$ or $\frakA$, respectively. Since we can ensure that no sub-game is lost by the duplicator -- we only select pairs of elementarily equivalent connected components -- the duplicator will not lose the global game.
\end{proof}
As the main ingredient for the proof of Gaifman's theorem we use the following lemma. Here we don't even need $\frakA$ and $\frakB$ to satisfy the specific ultraproduct saturation properties (e.g.\ that a countable type is realized by either finitely many or at least $2^\omega$ many elements), simple $\omega$-saturation suffices.
\begin{lemma}
Let $\frakA, \frakB$ be two $\omega$-saturated $\sigma$-structures. If $\frakA$ and $\frakB$ satisfy the same basic local sentences, then they are elementarily equivalent.
\end{lemma}
\begin{proof}
The proof is similar to the proof of Theorem \ref{theoremGaifmanVariantIsomorphy} except that elementary equivalence and partial isomorphy replace full isomorphy.
We want to apply the last lemma. Since partial isomorphy is a finer equivalence relation than elementary equivalence, it suffices to show that each $\simeq_\text{part}$-class of connected components either occurs the same finite number of times in both $\frakA$ and $\frakB$ or occurs infinitely often in both. Therefore choose $n$ partially isomorphic disjoint connected components $A_1, \dotsc, A_n$ of $\frakA$ and show that there are at least $n$ connected components of $\frakB$ of the same partial isomorphism type. (The situation is symmetric in $\frakA$ and $\frakB$.)
We pick an arbitrary element $a_1 \in A_1$. By playing the first round of an infinite Ehrenfeucht game, we find elements $a_i \in A_i$ such that \[ A_1, a_1 \simeq_\text{part} A_2, a_2 \simeq_\text{part} \dotsb \simeq_\text{part} A_n, a_n .\] For $k, m \in \N$ let $\chi_{k, m}(x) \in \FO_1(\sigma)$ be a formula asserting that \[ (S_k(x), x) \equiv_m (S_k(a_1), a) .\] This is obviously a $k$-local formula. We now consider the family $(\psi_{k, m})_{k, m \in \N}$ of formulae in $n$ free variables $x_1, \dotsc, x_n$ given by \[ \psi_{k, m} := \bigwedge_{1 \leq i < j \leq n} d_{> 2m}(x_i, x_j) \land \bigwedge_{1 \leq i \leq n} \chi_{k, m}(x_i) .\] Since each of the sentences \[ \exists x_1 \dotsm \exists x_n \bigwedge\Psi_0(x_1, \dotsc, x_n), \] where $\Psi_0$ is a finite collection of formulae of the form $\psi_{k, m}$, is equivalent to a basic local sentence and satisfied in $\frakA$ (by $a_1, \dotsc, a_n$) and therefore also in $\frakB$, Lemma \ref{lemmaOmegaSaturationImpliesNTypesAreRealised} implies the existence of elements $b_1, \dotsc, b_n$ such that \[ \frakB, b_1, \dotsc, b_n \models \psi_{k, m} \] for all $k, m$. This means that the $b_i$ have pairwise infinite distance, \ie are in different connected components of $\frakB$, and each satisfy all $\chi_{k, m}$, implying \[ (S_k(b_i), b_i) \equiv (S_k(a_1), a_1) \] and therefore, by Lemma \ref{lemmaElementaryEquivalenceOfOmegaBalls}, \[ (S_\omega(b_i), b_i) \simeq_\text{part} (S_\omega(a_1), a_1) .\]
We have therefore found $n$ disjoint connected components of $\frakB$ of the same partial isomorphism type as the $A_i$.
\end{proof}
We can now use compactness to prove Gaifman's theorem.
\begin{proof}[Proof of Gaifman's theorem]
Let $\phi \in \FO_0(\sigma)$ be a sentence. Let
\begin{align*}
\Psi &:= \{ \psi \in \FO_0(\sigma) \colon \text{ $\psi$ is a boolean combination of basic local sentences} \}, \\
\Psi_\phi &:= \{ \psi \in \Psi \colon \phi \models \psi \}
.
\end{align*} By compactness it suffices to show that $\Psi_\phi \models \phi$ since we then have $\bigwedge \Psi^0 \equiv \phi$ for some finite subset $\Psi^0 \subseteq \Psi_\phi$. Assume to the contrary that $\Psi_\phi \cup \{ \neg \phi \}$ has a model $\frakA$. Let \[ \Psi_\frakA := \{ \psi \in \Psi \colon \frakA \models \psi \} ;\] we claim that $\Psi_\frakA \cup \{ \phi \}$ is satisfiable. If it were not satisfiable, there would be a finite subset $\Psi_\frakA^0 \subseteq \Psi_\frakA$ such that $\phi \models \neg \bigwedge \Psi_\frakA^0$, but then $\neg \bigwedge \Psi_\frakA^0 \in \Psi_\phi$ and therefore $\frakA \models \neg \bigwedge \Psi_\frakA^0$, which is a contradiction to $\frakA \models \Psi_\frakA$. Hence we find a model $\frakB$ of $\Psi_\frakA \cup \{ \phi \}$.
Now the two structures $\frakA$ and $\frakB$ satisfy the same basic local sentences, but disagree on $\phi$. Forming $\N$-fold ultrapowers (or getting $\omega$-saturated elementary extension in some other way) we get a contradiction to the last lemma.
\end{proof}
\begin{remark}
This new proof for Gaifman's theorem has been found independently by Lindell, Towsner and Weinstein (\cite[Theorem (11)]{lindellWeinstein}).
\end{remark}
\section{Ultraproducts of finite linear orderings}
In the preceeding sections we have been reasonably successful in proving inexpressibility results for some classes of finite structures, especially graphs, using variants of the locality theorems of Hanf and Gaifman. In this section we look at one more class of structures, namely finite linear orderings. The results obtained in the previous two sections are not applicable as the Gaifman graph of a linearly ordered structure is a complete graph -- locality is therefore not a useful concept.
As a starting point, we look at the following theorem. It will turn out that we can use it to prove more general inexpressibility results.
\begin{theorem}\label{theoremEvenNumberOfEltsIsNotDefinableForLinOrds}
The property of having an even number of elements is not $\FO(\{ < \})$-definable in the class of all finite linear orderings.
\end{theorem}
To show this, let $\frakA_n$ be the canonical linear ordering on the elements $\{ 1, \dotsc, n \}$. Let $\calU$ be an arbitrary non-principal ultrafilter on $\N$. To prove the theorem, it suffices to show that \[ \frakA_\text{even} := \prod_{n \in \N} \frakA_{2n + 2} / \calU \equiv \prod_{n \in \N} \frakA_{2n + 1} / \calU =: \frakA_\text{odd}, \] since a single formula axiomatising even cardinality would separate $\frakA_\text{even}$ from $\frakA_\text{odd}$. It turns out that we can prove an even stronger statement with a simple structural argument.
\begin{lemma}
The two structures $\frakA_\text{even}$ and $\frakA_\text{odd}$ are isomorphic.
\end{lemma}
\begin{proof}
Both structures are discrete linear orderings with minimal and maximal elements since these are first-order definable properties. Consider the element \[ a_\text{max} := [ (2n + 2)_{n \in \N} ] \in \frakA_\text{even} ;\] it is clearly maximal in $\frakA_\text{even}$. As any element $[(a_n)_n]$ of $\frakA_\text{even}$ not equal to $a_\text{max}$ has a representative such that $a_n < 2n + 2$ for every $n$, we get a canonical isomorphism $\frakA_\text{odd} \simeq \frakA_\text{even} \setminus \{ a_\text{max} \}$. But since $\frakA_\text{even}$ is infinite (due to non-principality of $\calU$), its order type can be written as an ordered sum of a copy of $\omega$, some copies of $\omega^\ast + \omega$ and a copy of $\omega^\ast$. ($\omega^\ast$ denotes the reverse order type of $\omega$.) Removing any element clearly does not change the order type, \ie \[ \frakA_\text{even} \simeq \frakA_\text{even} \setminus \{ a_\text{max} \} \simeq \frakA_\text{odd}. \qedhere \]
\end{proof}
\begin{corollary}
Let $m \in \N$. Then there is an $N \in \N$ such that $\frakA_{2n + 1} \equiv_m \frakA_{2n + 2}$ for all $n \geq N$.
\end{corollary}
\begin{proof}
Assume there were infinitely many $n$ such that $\frakA_{2n + 1} \not\equiv_m \frakA_{2n + 2}$. Then we could construct a non-principal ultrafilter containing the set of all such $n$. In the light of the preceding lemma, this is a contradiction to Lemma \ref{lemmaUltraproductsElEquivIffMostFactorsMEquiv}.
\end{proof}
\begin{corollary}
Let $m \in \N$. Then almost all $\frakA_n$ are $m$-equivalent, \ie there exists $N \in \N$ such that \[ \frakA_n \equiv_m \frakA_N \] for all $n \geq N$.
\end{corollary}
\begin{proof}
We can repeat the proof of the lemma to get
\[ \prod_n \frakA_{2n + 2} / \calU \simeq \prod_n \frakA_{2n + 3} / \calU .\]
As in the preceding corollary, this yields $\frakA_{2n + 2} \equiv_m \frakA_{2n + 3}$ for almost all $n$. Combining this with the preceding corollary we get $\frakA_n \equiv_m \frakA_{n + 1}$ for all $n$ beyond some finite $N$.
\end{proof}
\begin{remark}
This last corollary is a much more general inexpressibility result than the theorem at the start of the section. It is also a standard result in first-order logic, which is usually obtained by giving a winning strategy for the duplicator in an $m$-round Ehrenfeucht game. Since locality theorems are not applicable -- the Gaifman graph of a linear ordering is always a complete graph -- this is normally done explicitly. We did not need games at all and used ultraproducts instead.
\end{remark}
We note that our results for finite directed path graphs are stronger than those for finite linear orderings. We showed that the sequence of finite directed path graphs is a Hanf sequence, therefore any two ultraproducts \wrt non-principal ultrafilters are isomorphic. For finite linear orderings, we have only shown that the ultraproduct does not change when we shift the ultrafilter by one position (corresponding to an ultrafilter on the odd numbers versus one on the even numbers). Only under the continuum hypothesis do we get isomorphy for two arbitrary ultrafilters due to \ref{corollaryElemEquivUltraprodsAreIsomAssumingCH}.
This difference between finite linear orderings and directed path graphs should not surprise us as it is a familiar observation that a successor relation is first-order definable from an ordering but not vice versa. When we leave the finite by means of an ultraproduct (preserving not much more than first-order properties), the natural correspondence of successor structure and ordering breaks down.
\section{Conclusion}
We have explored the ultraproduct method as an independent approach to first-order inexpressibility results. We have proved well-known inexpressibility results for finite linear orderings and finite graphs without resorting to either compactness or Ehren\-feucht-Fra\"issé arguments. Only for our new proof of Gaifman's theorem did we need games. Hence we have found ultraproducts to be an unexpectedly useful tool for the investigation of first-order-properties in finite model theory.
Some obvious structural questions about ultraproducts could not be answered by this paper. We have already seen in the first section that properties of ultraproducts may depend on the continuum hypothesis. It would be desirable to further investigate in which ways elementarily equivalent $\N$-fold ultraproducts of cardinality $2^\omega$ may fail to be isomorphic and whether such failures must always occur when we assume the negation of the continuum hypothesis. Furthermore, we would like to see an example for a single structure such that the isomorphism types of its $\N$-fold ultrapowers depend on the choice of non-principal ultrafilters. Lastly, we would like to see whether there are any ``natural'' examples for non-isomorphic elementarily equivalent ultraproducts -- for example, a closer investigation of the ultraproducts of finite linear orderings would be helpful, as in the last section we could only show elementary equivalence of different ultraproducts.
\bibliographystyle{alpha}
|
2,869,038,156,379 | arxiv | \section{Introduction}
\label{intro}
Environmental effects are believed to be the primary process of ceasing star
formation in low-mass galaxies with stellar masses (${\rm M_{*}}$) lower than
$10^{9.5}$${\rm\,M_\odot}$\ (or dwarf galaxies). Field low-mass galaxies may temporarily
quench their star formation through supernova feedback, but new gas accretion
and recycling would induce new starbursts with periods of tens of megayears
\citep[e.g.,][]{ycguo16bursty,sparre17}. \citet[][hereafter G12]{geha12} found
that the quenched fraction of galaxies with ${\rm M_{*}}$$<10^{9}$${\rm\,M_\odot}$\ drops rapidly
as a function of distance to massive host galaxies and that essentially all
local field galaxies in this mass regime are forming stars.
The environmental quenching of low-mass galaxies beyond the local universe,
however, is rarely investigated because of these galaxies' faint luminosity.
Most studies \citep[e.g., G12;][etc.]{quadri12, tal13, tal14, balogh16,
nancy16, fossati17} start from central galaxies and measure the quenched
fraction of their satellites. This method requires a complete sample of
satellites, which limits these studies to the local universe and/or to
intermediate-mass (${\rm M_{*}}$$\gtrsim 10^{9.5}$${\rm\,M_\odot}$) satellites.
In this letter, we use CANDELS data \citep{candelsoverview,candelshst} to
detect the effects of environmental quenching beyond $z\sim1$. Our
approach is different from but complementary to other studies. We start from
the ``victims'' --- quenched dwarf galaxies --- and search for their massive
neighbors, which are tracers of massive dark matter halos.
The concept of our approach is simple --- if environmental effects are solely
responsible for quenching all dwarf galaxies, all low-mass quenched galaxies
(QGs) should live close to a massive central galaxy in a massive halo. In
contrast, star-forming galaxies (SFGs) can live far away from massive dark
matter halos. Therefore, on average, QGs should have systematically shorter
distances to their massive neighbors than SFGs should. This systematic
difference between the two populations is evidence of the dwarf QG--massive
central galaxy connection and therefore a demonstration of environmental
quenching. Because our goal is to investigate {\it whether or not} such a dwarf
QG--massive central connection has been established, we only need to detect a
statistically meaningful signal, rather than to find all signals, to rule out
the null hypothesis of no environmental effects. This advantage allows us to
use an incomplete dwarf sample to study this topic beyond the local universe.
We adopt a flat ${\rm \Lambda CDM}$ cosmology with $\Omega_m=0.3$,
$\Omega_{\Lambda}=0.7$, and the Hubble constant $h\equiv H_0/100\ {\rm
km~s^{-1}~Mpc^{-1}}=0.70$. We use the AB magnitude scale \citep{oke74} and a
\citet{chabrier03} IMF.
\begin{figure*}[htbp]
\hspace*{-0.55cm}\includegraphics[scale=0.22, angle=0]{./fig1a.ps}
\hspace*{-0.55cm}\includegraphics[scale=0.22, angle=0]{./fig1b.ps}
\hspace*{-0.55cm}\includegraphics[scale=0.22, angle=0]{./fig1c.ps}
\caption[]{Examples of sample selection and environment measurement. Each
column shows a given ($z$, ${\rm M_{*}}$) bin as the title shows. In each column,
Panel (a) shows the selected QGs (red) and SFGs (blue) in the UVJ diagram.
Black dots show all galaxies (with $H_{F160W}<26$ and CLASS\_STAR$<$0.8) in
this bin. Panel (b) shows the PDFs of $d_{proj}$ of the QGs (red) and SFGs
(blue) galaxies. The dark, medium, and light gray regions show the 1$\sigma$,
2$\sigma$, and 3$\sigma$ levels of 3000 times of bootstrapping of SFGs to match
the number of the QGs. The red and blue circles show the medians of $d_{proj}$
of the QGs and SFGs. The gray bar shows the 1$\sigma$ (dark), 2$\sigma$
(medium), and 3$\sigma$ (light) levels of the medians of the bootstrapping. To
show the difference clearly, all median values are normalized so that the
median of the SFGs (blue circle) is equal to 1.5 Mpc. The number below the gray
bar shows the confidence level to which the null hypothesis that the QGs (red)
and SFGs (blue) have the same $d_{proj}$ medians is ruled out. Panel (c) shows
the CDFs of $d_{proj}$ of the QGs (red) and SFGs (blue) normalized by $R_{Vir}$
of the halos of their massive neighbors. The dark, medium, and light gray
regions show the 1$\sigma$, 2$\sigma$, and 3$\sigma$ levels of the
bootstrapping. All columns in this figure are at $0.5<z<0.75$.
\label{fig:example1}}
\end{figure*}
\begin{figure*}[htbp]
\hspace*{-0.55cm}\includegraphics[scale=0.22, angle=0]{./fig2a.ps}
\hspace*{-0.55cm}\includegraphics[scale=0.22, angle=0]{./fig2b.ps}
\hspace*{-0.55cm}\includegraphics[scale=0.22, angle=0]{./fig2c.ps}
\caption[]{Same as Figure \ref{fig:example1}, but showing three ${\rm M_{*}}$\ bins at
$0.75<z<1.00$.
\label{fig:example2}}
\end{figure*}
\begin{figure*}[t!]
\hspace*{-0.55cm}\includegraphics[scale=0.22, angle=0]{./fig3a.ps}
\hspace*{-0.55cm}\includegraphics[scale=0.22, angle=0]{./fig3b.ps}
\hspace*{-0.55cm}\includegraphics[scale=0.22, angle=0]{./fig3c.ps}
\caption[]{Same as Figure \ref{fig:example1}, but showing three ${\rm M_{*}}$\ bins at
$1.00<z<1.25$.
\label{fig:example3}}
\end{figure*}
\section{Data}
\label{data}
We use the photometric redshift (photo-z), ${\rm M_{*}}$, and rest-frame color
catalogs of four CANDELS fields: GOODS-S \citep{ycguo13goodss}, UDS
\citep{galametz13uds}, GOODS-N (G. Barro et al., in preparation) and COSMOS
\citep{nayyeri17cosmos}.
The photo-z measurement is described in \citet{dahlen13}.
For GOODS-S galaxies at $0.5<z<2.0$ and
$H<26$ AB, the 1$\sigma$ scatter of $|\Delta z|/(1+z)$ is 0.026 and the outlier
fraction (defined as $|\Delta z|/(1+z)>0.1$) is 8.3\%. We also divide the test
sample into low-mass (${\rm M_{*}}$$<10^{9}$${\rm\,M_\odot}$) and massive (${\rm M_{*}}$$>10^{9}$${\rm\,M_\odot}$)
sub-samples. The 1$\sigma$ scatter and outlier fraction of the low-mass (and
massive) sub-sample are 0.033 (0.024) and 13.7\% (7.2\%).
The ${\rm M_{*}}$\ measurement is described in \citet{santini15}, where each galaxy
is fit by 12 SED-fitting codes with different combinations of synthetic
stellar population models, star formation histories, fitting methods, etc. For
each galaxy, we use the median of the 12 best-fit ${\rm M_{*}}$\ as its ${\rm M_{*}}$. The
typical uncertainty of ${\rm M_{*}}$\ measurement is $\sim$0.15 dex. Rest-frame colors
are measured by using EAZY \citep{brammer08}.
\begin{figure}[htbp]
\center{\vspace*{-0.20cm}\hspace*{-0.0cm}\includegraphics[scale=0.17, angle=0]{./fig4.ps}}
\caption[]{{\it Panel (a)}: Statistics of the quenching--environment
connection. In each ($z$, ${\rm M_{*}}$) bin, the deviation between the medians of
$d_{proj}^{Q}$ and $d_{proj}^{SF,sub}$ is the upper number, while the numbers
of the QGs and SFGs are the two lower numbers. We choose the deviation
$\geq$3$\sigma$ as the threshold of the quenching--environment connection being
observed. All such bins are red, while others with the deviation $<$3$\sigma$
are cyan. Gray bins cannot be accessed by our current dataset. {\it Panel (b)}:
Fraction of a population of galaxies within $2 R_{Vir}$ of massive halos.
Different symbols show different populations. {\it Panel (c)}: Median $d_{proj}
/ R_{Vir}$ of different samples. {\it Panel (d)}: Inferred quenching timescales
in two redshift bins. The colors and symbols in Panel (b), (c), and (d) are the
same.
\label{fig:spatial}}
\end{figure}
\section{Method}
\label{method}
\subsection{Sample Selection}
\label{sub:sample}
Our sample consists of sources with F160W $H<26$ AB, PHOTFLAG=0 (no suspicious
photometry), and SExtractor CLASS\_STAR$<$0.8. The magnitude limit of $H=26$ AB
is approximately the 50\% completeness limit of CANDELS wide regions
\citep{ycguo13goodss} and it is corresponding to a galaxy of ${\rm M_{*}}$$\sim
10^{8}$${\rm\,M_\odot}$\ at $z \sim 0.5$ with a single stellar population that is 5 Gyr
old. We divided the whole sample into different $z$ and ${\rm M_{*}}$\ bins: $z =
0.5-2$ with $\Delta z=0.25$ and ${\rm log(M_{*})} = 8.0-10.5$ with ${\rm \Delta
log(M_{*})} = 0.5$.
In each ($z$, ${\rm M_{*}}$) bin, we use the UVJ diagram \citep{williams09,muzzin13}
to select QGs and SFGs. To avoid the contamination of
misidentified stars and sources with suspicious colors, we add one criterion
to refine the quenching region (the diagonal light brown line within the
original UVJ quenched region in Panel (a)s of Figures
\ref{fig:example1}--\ref{fig:example3}). This extra criterion may exclude some
very compact QGs \citep{barro13a}, but since our goal is to
obtain a clean and statistically meaningful sample rather than a complete one,
such exclusion is necessary and does not affect our results.
For SFGs, instead of using all galaxies in the UVJ star-forming region, we
measure the median and $\pm 1.5 \sigma$ level of the star-forming locus
(calculated in the directions parallel and perpendicular to the reddening
vector) and use them as the selection boundary. The selected SFGs are plotted
as blue points in Panel (a)s of Figures \ref{fig:example1}--\ref{fig:example3}.
Again, although many galaxies in the original UVJ star-forming locus are
excluded, we aim at constructing a clean rather than complete sample.
\subsection{Detecting Environmental Quenching Effects}
\label{sub:neighbor}
For each low-mass galaxy, we search for its nearest massive neighbor in sky
(projected distance). The massive sample is selected to have CLASS\_STAR$<$0.8,
PHOTFLAG=0, ${\rm M_{*}}$$^{massive}>10^{10.5}$${\rm\,M_\odot}$, and
${\rm M_{*}}$$^{massive}>$${\rm M_{*}}$$^{low-mass}$+0.5 dex\footnote{The last requirement
only affects galaxies with
$10^{10}$${\rm\,M_\odot}$$<$${\rm M_{*}}$$^{low-mass}$$<$$10^{10.5}$${\rm\,M_\odot}$. The ${\rm M_{*}}$$^{massive}$
threshold of our massive sample corresponds to dark matter halos of
${\rm M_{halo}}$$\gtrsim10^{12}$${\rm\,M_\odot}$. Since the ${\rm M_{*}}$--${\rm M_{halo}}$\ relation evolves little
with redshift in this mass regime \citep{behroozi13}, our choice of a fixed
${\rm M_{*}}$$^{massive}$ threshold at different redshifts allows us to investigate
the environmental effects of similar ${\rm M_{halo}}$\ at different cosmic times.}. The
redshift range of the massive sample is limited to $|z_{massive} -
z_{low-mass}|/(1+z_{low-mass})<0.10$, which is about 3$\sigma$ of our photo-z
accuracy. We calculate the projected distances between the low-mass galaxy and
the selected massive galaxies. The massive galaxy with the smallest
projected distance is chosen as the central galaxy of the low-mass galaxy. We
use $d_{proj}$ to denote this smallest projected distance.
Because of projection effects, a massive neighbor found through this method may
not be the real massive galaxy whose dark matter halo was responsible for
quenching the low-mass galaxy. But if environmental effects are the primary way
of quenching a population of low-mass galaxies,
statistically, QGs should be located closer to massive companions
than SFGs should. As a result, the $d_{proj}$ distribution of
a quenched population ($d_{proj}^Q$) should be skewed toward lower values than
that of SFGs ($d_{proj}^{SF}$).
Many studies \citep[e.g.,][]{scoville13,davies16} used the local overdensity
field constructed by Voronoi tessellation or the nearest neighbor method to
measure environments. Since the local overdensity of a satellite galaxy is
correlated with $d_{proj}$, our method is similar to those using a density
field.
While our simple method provides necessary information to test the {\it whether
or not} question of our particular interest, future work with the density field
approach could provide more accurate and detailed measurements of environmental
quenching.
We test whether $d_{proj}^Q$ is systematically and significantly smaller than
$d_{proj}^{SF}$ in each ($z$, ${\rm M_{*}}$) bin. Because in most ($z$, ${\rm M_{*}}$) bins,
the number of QGs is much smaller than that of SFGs, the small number
statistics needs to be taken into account. In each of these bins, we randomly
draw a sub-sample of the SFGs to match the number of the QGs and calculate the
median, probability distribution function (PDF), and cumulative distribution
functions (CDF) of $d_{proj}$ of the sub-sample ($d_{proj}^{SF,sub}$). We
repeat this bootstrapping sampling 3000 times, obtaining 3000 distributions of
$d_{proj}^{SF,sub}$. To exclude the null hypothesis, we ask the median
$d_{proj}^{Q}$ to be 3$\sigma$ smaller than the median of $d_{proj}^{SF,sub}$.
Panels (b) in Figures \ref{fig:example1}--\ref{fig:example3} show some examples
of our results. The ($z$, ${\rm M_{*}}$) bins labeled with $\geq$3$\sigma$ values
(i.e., median $d_{proj}^{Q}$ is 3$\sigma$ smaller than median
$d_{proj}^{SF,sub}$) are considered to have an established
quenching--environment connection.
In contrast, bins with $<$3$\sigma$ values cannot rule out the null hypothesis
of the two populations having the same $d_{proj}$ distributions with 3$\sigma$
confidence.
\begin{figure}[htbp]
\center{\vspace*{-0.20cm}\hspace*{-0.0cm}\includegraphics[scale=0.22, angle=0]{./fig5.ps}}
\caption[]{Quenching timescale at different redshifts. The red symbols are
calculated by this work, while the black and brown lines and symbols are taken
from the literature. The red dashed lines in the first two panels are the solid
red line in the third panel (i.e., $T_{Q}$\ at $0.8\leq z<1.2$) scaled up by
$(1+z)^{1.5}$.
\label{fig:timescale}}
\end{figure}
\section{Results}
\label{results}
Panel (a) of Figure \ref{fig:spatial} shows the deviation from $d_{proj}^{Q}$
to $d_{proj}^{SF,sub}$ in each ($z$, ${\rm M_{*}}$) bin. Our criterion of
environmental quenching being observed is that the median of $d_{proj}^{Q}$ is
3$\sigma$ smaller than that of $d_{proj}^{SF,sub}$. For galaxies with
$10^8$${\rm\,M_\odot}$$<$${\rm M_{*}}$$<10^{10}$${\rm\,M_\odot}$, such a quenching--environment connection
is observed up to $z \sim 1$, as shown by the larger-than-3$\sigma$ deviations.
This result is consistent with the quick emergence of low-mass QGs from the
measurement of stellar mass functions at $z\sim1$
\citep[e.g.,][]{ilbert13,huangjs13}. For galaxies with
$10^{10.0}$${\rm\,M_\odot}$$<$${\rm M_{*}}$$<10^{10.5}$${\rm\,M_\odot}$, the connection was established at a
lower redshift.
For those ($z$, ${\rm M_{*}}$) bins with $<$3$\sigma$ deviation, we cannot rule out
the null hypothesis of QGs and SFGs having the same $d_{proj}$ distributions
with more than 3$\sigma$ confidence. This may imply that the
quenching--environment connection has not been established in these bins. This
interpretation is at least consistent with some other studies for massive
galaxies (${\rm M_{*}}$$>$$10^{10}$${\rm\,M_\odot}$), which claimed that massive quiescent
galaxies at $z>1$ are not necessarily located in high-density environments
\citep[e.g.,][]{darvish15,darvish16,lin16}. For lower-mass galaxies at
$z\gtrsim1.5$, however, due to projection effects or small number statistics,
our method may have failed to detect existing quenching--environment
connections.
\section{Discussion}
\label{discussion}
\subsection{Spatial Distribution of Quenched Galaxies}
\label{discussion:spatial}
G12 found that 87\% (and 97\%) of dwarf QGs in their SDSS sample are within 2
$R_{Vir}$ (and 4 $R_{Vir}$) of a massive host galaxy.
We find similar results in our sample at $0.5<z<1.0$ (Panel (b) of Figure
\ref{fig:spatial}). About 90\% of the QGs below $10^{10}$${\rm\,M_\odot}$\ in our sample
are within 2 $R_{Vir}$. The fraction drops quickly to about 70\% for galaxies
above $10^{10}$${\rm\,M_\odot}$.
To calculate $R_{Vir}$, we first use the ${\rm M_{*}}$--${\rm M_{halo}}$\ relation of
\citet{behroozi13} to obtain ${\rm M_{halo}}$\ ($M_{\rm vir}$) of the massive neighbors.
Then, we derive $R_{Vir}$ through $M_{\rm vir}(z) = {4\pi \over 3} \Delta_{\rm
c}(z) \rho_{\rm crit}(z) R_{\rm vir}(z)^3$, where $\rho_{\rm crit}(z)$ is the
critical density of the universe at $z$, and $\Delta_{\rm c}(z)$ is calculated
by following \citet{bryan98}.
SFGs in our sample are also almost within 4 $R_{Vir}$. This could be a
projection effect. Since we search for massive neighbors within a long
line-of-sight distance ($|\Delta z|/(1+z)<0.10$), an SFG has a high chance of
being located within the projected 4 $R_{Vir}$ of a massive galaxy, even though
the massive galaxy is not its real central galaxy. In contrast, with a more
accurate redshift measurement, G12 found that only $\sim$50\% of the $z\sim0$
SFGs are within 4 $R_{Vir}$, suggesting that a large fraction of SFGs are
intrinsically outside 4 $R_{Vir}$. QGs in our sample may suffer from the same
projection effect. However, as discussed in Section \ref{sub:neighbor}, this
effect would not affect our {\it statistical} results.
A non-negligible fraction (10\%) of low-mass QGs are located between 2 and 4
$R_{Vir}$. They are likely central galaxies quenched by mechanisms not related
to environment, e.g., AGN and stellar feedback. They, however, may also be
evidence for quenching processes acting at large distances of massive halos
\citep[e.g.,][]{cen14}. Y. Lu et al. (2017, in preparation) found that, to
match the observed ${\rm M_{*}}$\ and stellar-phase metallicity simultaneously, gas
accretion of Milky Way (MW) satellite galaxies need to be largely reduced way
before they fall into $R_{Vir}$ of the MW halo, possibly by heating up the
intergalactic medium in the MW halo vicinity to $10^5$K. Alternatively,
\citet{slater13} used simulations to show that environmental effects are
prominent out to 2--3 $R_{Vir}$: satellites with very distant apocenters can be
quenched by tidal stripping and ram pressure stripping following a close
passage to the host galaxy.
We also extend the local results of G12 to higher ${\rm M_{*}}$\ by repeating our
measurements on the SDSS sample of \citet[][]{aldo15}. The results (red squares
in Panel (b) of Figure \ref{fig:spatial}), together with G12, suggest that the
fraction of QGs within 2 $R_{Vir}$ has almost no redshift dependence. This
constant fraction suggests that, at all redshifts, environment (especially
within 2 $R_{Vir}$) dominates the quenching of low-mass galaxies.
We also study the median distance of galaxies to their massive neighbors scaled
by $R_{Vir}$ (Panel (c) of Figure \ref{fig:spatial}). SFGs have a constant
median distance of $\sim$1.3 $R_{Vir}$ over a wide ${\rm M_{*}}$\ range. QGs are
closer to massive neighbors, but their median distance depends on ${\rm M_{*}}$: it
decreases from 1 $R_{Vir}$ at $10^8$${\rm\,M_\odot}$\ to 0.5 $R_{Vir}$ at $10^{9.5}$${\rm\,M_\odot}$,
then increases to $>1 R_{Vir}$ at $10^{10.5}$${\rm\,M_\odot}$. Also, we find no
significant difference between different redshifts.
\subsection{Quenching Timescale}
\label{discussion:timescale}
Quenching timescale ($T_{Q}$) is important to constrain quenching mechanisms. In
the local universe, at ${\rm M_{*}}$$>10^{10}$${\rm\,M_\odot}$, quenching likely occurs through
starvation, whose timescale (4--6 Gyr) is comparable to gas depletion
timescales \citep{fillingham15,peng15}. At ${\rm M_{*}}$$<10^{8}$${\rm\,M_\odot}$, ram pressure
stripping is likely the dominant mechanism
\citep{slater14,fillingham15,weisz15}. Its timescale (2 Gyr) is much shorter
and comparable to the dynamical timescale of the host dark matter halo.
The dominant quenching mechanism may change around a characteristic ${\rm M_{*}}$.
To infer $T_{Q}$, we assume all galaxies start quenching at 4 $R_{Vir}$. We choose
4 $R_{Vir}$ because G12 shows that beyond 4 $R_{Vir}$ the fraction of QGs is
almost zero, while the fraction of SFGs is still high. Theoretically,
\citet{cen14} also predicted the onset of quenching at a similar large halo
distance. Galaxies fall into massive halos while their star formation rates are
being reduced. They become fully quenched when they arrive at the observed
location. Therefore, $T_{Q}$\ is the time they spent on traveling from 4 $R_{Vir}$
to the observed location with an infall velocity (using circular velocity $V(R)
= \sqrt{ G M(<R) \over R}$ at 2 $R_{Vir}$ as an approximation).
Our method of measuring $T_{Q}$\ is different from most studies in the literature,
e.g., \citet{wetzel13, wheeler14,fillingham15,balogh16,fossati17}. They used
numerical simulations or semi-analytic models to match the basic demographics
(e.g., quenched fraction) of QGs. Our method is purely empirical, but relies
on the assumptions of the starting and end points (i.e., 4 $R_{Vir}$ and the
observed location, respectively) of quenching. Our $T_{Q}$\ definition, however,
characterizes the same physical quantity as other methods, i.e., the timescale
upon which satellites must quench following infalling into the vicinity of
their massive hosts.
The inferred $T_{Q}$\ is shown in Panel (d) of Figure \ref{fig:spatial}. Overall,
the $T_{Q}$\ dependence on ${\rm M_{*}}$\ is, if any, very weak between
$10^8$ and $10^{10}$${\rm\,M_\odot}$. Lower-redshift galaxies have longer $T_{Q}$, because
dynamical timescale decreases with redshift: lower-redshift galaxies need more
time to travel the same $d_{proj}^Q / R_{Vir}$.
Our measurements show excellent agreement with those of \citet{fossati17} and
\citet{balogh16} at ${\rm M_{*}}$$> 10^{9.5}$${\rm\,M_\odot}$\ (Figure \ref{fig:timescale}).
\citet{fossati17} used 3D-HST data
\citep{skelton143dhstcat,momcheva163dhstline} to study the environments of
galaxies with ${\rm M_{*}}$$\gtrsim 10^{9.5}$${\rm\,M_\odot}$\ in CANDELS fields. Agreement with
these detailed studies provides an assurance to our method: although built upon
simplified assumptions, it is able to catch the basic physical principles of
environmental quenching. Moreover, the good agreement also implies that
the projection effect discussed in Section \ref{discussion:spatial} does not
significantly bias our measurement.
Our results, together with the measurements of \citet{fossati17} and
\citet{balogh16}, imply a smooth $T_{Q}$\ transition -- and hence a quenching
mechanism transition -- around ${\rm M_{*}}$$\sim 10^{9.5}$${\rm\,M_\odot}$, which is broadly
consistent with other studies \citep[e.g.,][]{cybulski14,joshualee15}.
At ${\rm M_{*}}$$\gtrsim 10^{10}$${\rm\,M_\odot}$, starvation is likely to be responsible for
environmental quenching \citep{fillingham15}. Alternatively, however, these
galaxies could actually be centrals or recently quenched before becoming
satellites. For them, internal mechanisms (e.g., AGN and star formation
feedback) are likely dominating the quenching, as demonstrated by the
correlation between star formation and internal structures (e.g., central mass
density within 1 kpc discussed in \citet{fang13,barro17,woo17}).
The quenching mechanisms at ${\rm M_{*}}$$<10^{9.5}$${\rm\,M_\odot}$\ are still uncertain. Our
results suggest that $T_{Q}$\ mildly increases with ${\rm M_{*}}$\ at $0.5\leq z<0.8$.
Other studies of the local universe
\citep[e.g.,][]{slater14,fillingham15,wetzel15a} suggest a much stronger
${\rm M_{*}}$\ dependence of $T_{Q}$. For example, \citet{fillingham16} argued that $T_{Q}$\
drops quickly to $\sim$2 Gyr for galaxies with ${\rm M_{*}}$$\lesssim 10^{8}$${\rm\,M_\odot}$\ at
$z\sim0$ because of ram pressure stripping.
At ${\rm M_{*}}$$> 10^{9.5}$${\rm\,M_\odot}$, the redshift dependence of $T_{Q}$\ can be explained by
the change of the dynamical timescale. We scale up $T_{Q}$\ at $0.8\leq z<1.2$ by a
factor of $(1+z)^{1.5}$ to account for the redshift dependence of dynamical
timescale (see \citet{tinker10}). This scaled $T_{Q}$\ (red dashed lines in the
first two panels of Figure \ref{fig:timescale}) matches the actual $T_{Q}$\
measurements very well at ${\rm M_{*}}$$\gtrsim 10^{9.5}$${\rm\,M_\odot}$. However, it deviates
from the $T_{Q}$\ measurements at ${\rm M_{*}}$$<10^{9.0}$${\rm\,M_\odot}$.
At $z\sim0$ (7 Gyr after $z\sim1.0$), the scaled $T_{Q}$\ is significantly larger
than the $T_{Q}$\ measured by \citet{fillingham15}. \cite{balogh16} also found
similar results: at $z\sim1$, their $T_{Q}$\ of galaxies with
${\rm M_{*}}$$<10^{10}$${\rm\,M_\odot}$\ is longer than the $z\sim0$ $T_{Q}$\ scaled down by
$(1+z)^{1.5}$. Future work is needed to more quantitatively determine the
redshift dependence of the $T_{Q}$\ of low-mass galaxies.
\section{Conclusions}
\label{conclusion}
CANDELS allows us to investigate evidence of environmental quenching of dwarf
galaxies beyond the local universe. At $0.5<z\lesssim1.0$, we find that for
$10^{8}$${\rm\,M_\odot}$$<$${\rm M_{*}}$$<10^{10}$${\rm\,M_\odot}$, QGs are significantly closer to their
nearest massive companions than SFGs are, demonstrating that environment plays
a dominant role in quenching low-mass galaxies.
We also find that about 10\% of the QGs in our sample are located between two
and four $R_{Vir}$ of the massive halos.
The median projected distance from the QGs to their massive neighbors
($d_{proj}^Q/R_{Vir}$) decreases with satellite ${\rm M_{*}}$\ at ${\rm M_{*}}$$\lesssim
10^{9.5}$${\rm\,M_\odot}$, but increases with satellite ${\rm M_{*}}$\ at ${\rm M_{*}}$$\gtrsim
10^{9.5}$${\rm\,M_\odot}$. This trend suggests a smooth, if any, transition of $T_{Q}$\ around
${\rm M_{*}}$$\sim 10^{9.5}$${\rm\,M_\odot}$\ at $0.5<z<1.0$.
\
Y.G., D.C.K., and S.M.F. acknowledge support from NSF grant AST-0808133.
Support for Program HST-AR-13891 and HST-GO-12060 were provided by NASA through
a grant from the Space Telescope Science Institute, operated by the Association
of Universities for Research in Astronomy, Incorporated, under NASA contract
NAS5-26555. Z.C. acknowledges support from NSFC grants 11403016 \& 11433003.
|
2,869,038,156,380 | arxiv |
\section{Conclusion}
\label{sec:conclusion}
A problem with the implementation of the Nemenyi post-hoc test led to incorrect
results being published in our benchmark paper on cross-project defect
prediction. The mistake only affected research question RQ1, the other three
research questions were not affected. Within this correction paper, we explained
the problem in the statistical test, how this problem affected our results,
presented the corrected, and explained the changes that occoured. The
major findings regarding RQ1 are not changed, including the best performing approach, the result that the na\"{i}ve
baseline of using all data outperforms most proposed transfer learning
approaches, as well as that cross-validation can be outperformed by \ac{CPDP}.
Thus, the contributions of the article are still valid. Still, the correction
leads to differences in the rankings which are properly corrected and
discussed here. We apologize for this mistake and hope that this timely
correction mitigates the potential negative impact the wrong results may have.
\section*{Acknowledgements}
We want to thank Yuming Zhou from Nanjing University for pointing out the
inconsistencies in the results to us so fast, as well as the editors of this
journals who helped to determine how we should communicate this problem to the
community within days.
\section{Introduction}
\label{sec:introduction}
Unfortunately, the article ``A Comparative Study to Benchmark Cross-project
Defect Prediction Approaches''~\cite{Herbold2017b} has a problem in the
statistical analysis performed to rank \ac{CPDP} approaches.
Prof. Yuming Zhou from Nanjing University
pointed out an inconsistency in Table 8 of the the article. He noted that in
some cases the $rankscores$ are worse even if the mean values for
performance metrics are better. While this is possible in theory, with the Friedman
test~\cite{Friedman1940} with post-hoc Nemenyi test~\cite{Nemenyi1963}, such inconsistencies are unlikely.
Therefore, we immediately proceeded to check our results. These checks revealed that the inconsistencies are due to a problem
with our statistical analysis for the Research Question 1 (RQ1) ``Which
CPDP approaches perform best in terms of F-measure, G-measure, AUC, and MCC?''. None
of the raw results of the benchmark, nor any of the other research questions are
affected by the problem.
We will describe the problem and how we solved in in Section~\ref{sec:problem}.
Then, we will show the updated results regarding RQ1 and discuss the changes in
Section~\ref{sec:results}. Afterwards, we analyze the reasons for the changes in
Section~\ref{sec:reasons} to determine if all changes due to the correction are
plausible and the correction resolves the inconsistencies reported by Y. Zhou.
In Section~\ref{sec:replication}, we describe how we updated our replication kit
as part of this correction. Finally, we will conclude in
Section~\ref{sec:conclusion}.
Please note, that we assume that readers have read to the original article and are
familiar with the terminology used. We do not re-introduce any of the
terminology in this correction.
\section*{Acknowledgments}
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
\section{Problem with the Nemenyi test implementation}
\label{sec:problem}
On July 15th 2017, Y. Zhou imformed us that he found an inconsistency between
the results of CV-NET and CamargoCruz09-DT for the RELINK data for the performance
metric AUC. He noted that the mean value for CV-NET was higher than for
CamargoCruz09-DT, but the $rankscore$ was lower. He went to the raw data
provided as part of the replication kit~\cite{Herbold2017a} and confirmed that
the mean values were correct, and the AUC for CV-NET was higher for all three
products of the RELINK data. Based on this observervation, we re-checked our
statistical analysis of the results. We found the problem in our implementation
of the Nemenyi post-hoc test.
\subsection{Summary of the Friedman and Nemenyi tests}
To understand the problem, we briefly recap how the Friedman test with
post-hoc Nemenyi test works. The Friedman test determines if there are
stastical significant differences between populations. This is done using
pair-wise comparisons between the rankings of populations. If the Friedman test
determines significant differences, the Nemenyi post-hoc test compares the
populations to each other to determine the statistically significantly
different ones. The analysis with the Nemenyi test is based on two parameters:
the \ac{CD} and the average ranks of the populations in the pair-wise comparisons
between all populations on each data set.
Following the description by Dem\v{s}ar~\cite{Demsar2006}, \ac{CD} is defined as
\begin{equation}
CD = q_\alpha\sqrt{\frac{k(k+1)}{6N}}
\end{equation}
where $q_\alpha = \frac{qtukey(\alpha, N, \inf)}{\sqrt{2}}$ is the studentized
range distribution with infinite degrees of freedom divided\footnote{For simplicity, we refer to the studentized range
distribution as $qtukey(\alpha, N)$, following the name of the related method in
R} by $\sqrt{2}$, $\alpha$ the significance level, $k$ the number of populations
compared and $N$ the number of data sets. We can thus rewrite \ac{CD} as
\begin{equation}
\begin{split}
CD &= \frac{qtukey(\alpha, N, \inf)}{\sqrt{2}}\sqrt{\frac{k(k+1)}{6N}} \\
&= qtukey(\alpha, N, \inf)\sqrt{\frac{k(k+1)}{12N}}
\end{split}
\end{equation}
If we now assume that $R_i, R_j$ are the average ranks of population
$i,j \in \{1,\ldots,N\}$, the two populations are stastically significantly
different if
\begin{equation}
|R_i-R_j| > CD.
\end{equation}
In case a control population is available, it is possible to use a procedure
like Bonferroni correction~\cite{Dunn1961}. In this case, all populations are compared
to the control classifier instead of to each other. This greatly reduces the
number of pair-wise comparisons and can make the test more powerful. In this
case, for each pair a $z$-value is computed as
\begin{equation}
z = (R_i-R_j)/\sqrt{\frac{k(k+1)}{6N}}.
\end{equation}
The $z$-values are then used to rank classifiers. Since we do not have a control classifier and have to do pair-wise
comparisons with the \ac{CD} and cannot make use of the $z$-values. However, the
$z$-values play an important role when it comes to the problem with our analysis.
\subsection{z-values instead of Ranks}
Now that the concepts of the statistical tests are introduced, we can discuss
the actual problem in our implementation. We used the
\texttt{posthoc.friedman.nemenyi.test} function of the PMCMR
package~\cite{Pohlert2014} to implement the test. As part of the return values,
the function returns a matrix called \texttt{PSTAT}. Without checking directly
in the source code, we assumed these were the average ranks for each population, based on the documention of the package.
However, these are actually the absolute $z$-values multiplied with $\sqrt{2}$,
i.e.,
\begin{equation}
z' = |R_i-R_j|/\sqrt{\frac{k(k+1)}{6N}\cdot\sqrt{2}}.
\end{equation}
Thus, when we compared ranks, we did not actually compare the average ranks, but the
mean $z'$-values. This led to a wrong determination of ranks, which explain the
inconsistencies found by Y. Zhou.
\subsection{The solution}
To resolve the problem, we adopted the code from the PMCMR package that
determines the average ranks. We cross-checked our code with another
implementation of the Nemenyi test~\cite{Svetunkov2017}, to ensure that
the new code solves the problem.\footnote{Both implementations of the test do not return the raw
pair-wise comparisons and can, therefore, not be used directly.} We then used
the mean averages from that code, instead of the $z$-values that were returned
from the PMCMR package. As a result, the Nemenyi-test became much more
sensitive, because the scale of the average ranks is different than the scale of the $z$-values. Let us consider the scales for our experiments
with the JURECZKO data. Here, we have $N=62$ data sets and $k=135$ populations,
i.e., \ac{CPDP} approach and classifier combinations. The best possible average
rank is 135 (always wins), the worst possible is 1 (always loses). Thus, the
average ranks are on a scale from 1 to 135. In comparison, the highest possible
$z'$-value is
\begin{equation}
z = (135-1)/\sqrt{\frac{135\cdot136}{6\cdot62}}\cdot\sqrt{2} \approx 26.97,
\end{equation}
i.e., the scale is just from 0 (no difference in average ranks) and 26.97. Thus,
the scale of the $z'$-values has only about a fifth of the range the scale of
the average ranks has. Basically, with $z'$-values, 135
populations are fit into the scale 0 to 26.97, with rankings in the scale 1 to 135. This means that the
average distance between approaches is 0.2 with $z'$-values and 1 in case of
average ranks. Considering that we have a $CD \approx 1.26$ for $\alpha=0.95$
in this example, this makes a huge difference. With $z'$-values, it is unlikely
that two subsequently ranked approaches to have
a distance greater than CD, because $CD$ was more than 6.3 times higher than
average distance expected on the scale. This changes if the real scale with ranks is used. If you
have 135 cases with an average distance of 1, it is quite likely that a few of
these distances will be greater than 1.26, i.e., the CD.
We discuss this change in scales in this detail, because it requires a small
change in the ranking of approaches based on the Nemenyi test. Before, we considered three distinct ranks for
the calculation of the rankscore to achieve non-overlapping ranks:
\begin{itemize}
\item The populations that are within the CD of the best average ranking
population (top rank 1).
\item The populations that are within the CD of the worst average ranking
population (bottom rank~3).
\item The populations that are neither (middle rank 2).
\end{itemize}
This was the only way to deal with the small differences that resulted from
using the $z'$-values. However, this approach breaks on the larger scale,
because the distances now become larger, meaning fewer results are within the CD from
the best/worst ranking. For example, for the JURECZKO data and the performance
metric AUC, only two approaches would be on the first rank, i.e., only one
approach is within the CD of the best approach. Similarly, only six approaches
would be on the third rank, i.e., only five approaches are within the CD of the
worst approach. This would leave us with 127 approaches on the middle rank. This
ranking would be to coarse, and not show actual differences between approaches
anymore. To deal with this larger scale of ranks, we use a simple and more
fine-grained grouping strategy to create the ranks. We sort all approaches by their average ranking.
Each time that the difference between two subsequently ranked approaches is
larger than the \ac{CD}, we increase the rank. Because the rank is only
increased if the difference is larger than the \ac{CD}, we ensure that each
group only contains approaches that are statistically significantly different from the
other groups. Afterwards, we calculate the normalized rankscore as before.
Algorithm~\ref{alg:ranking} formalizes this strategy. This change in ranking
increases the sensitivity of the test and makes the results more fine-grained in
comparison to our original ranking procedure.
\begin{algorithm}
\caption{Ranking algorithm.}
\label{alg:ranking}
\label{alg:abstract_learner}
\begin{algorithmic}[1]
\State \textbf{Input:} Sorted mean ranks $R_i$ such that $\forall~i,j \in
\{1,\ldots,N\}:i<j, R_i\geq R_j$
\State \textbf{Output:} $rankscore_i$ for all ranks.
\State $rank\_tmp_1 = 1$
\State $current\_rank \leftarrow 0$
\For{i=2,\ldots, N}
\State \Comment{If difference is larger than CD increase rank}
\If{$R_{i-1}-R_i>CD$}
\State $current\_rank \leftarrow current\_rank+1$
\EndIf
\State $rank\_tmp_i \leftarrow current\_rank$
\EndFor
\State $rank\_max \leftarrow current\_rank$
\State \Comment{Determine rankscores}
\For{i=1,\ldots, N}
\State $rankscore_i \leftarrow 1-\frac{rank\_tmp_i}{rank\_max}$
\EndFor
\end{algorithmic}
\end{algorithm}
\section{Results}
\label{sec:results}
\begin{figure}
\includegraphics[width=\linewidth]{BEST_RQ1}
\caption{Mean rank score over all data sets for the metrics \emph{AUC},
\emph{F-measure}, \emph{G-measure}, and \emph{MCC}. In case multiple classifiers
were used, we list only the result achieved with the best classifier.}
\label{fig:best-rq1}
\end{figure}
\input{resultsTable.tex}
We now show the corrected results for RQ1. We will directly
compare the changes in the results with the originally published results.
Figure~\ref{fig:best-rq1} shows the mean \emph{rankscore} averaged over the four
performance metrics F-Measure, G-Measure, AUC, and MCC and the five data sets
JURECZKO, MDP, AEEEM, NETGENE, and RELINK. Table~\ref{tbl:results} shows
detailed results including the mean values and rankscores for each performance
metrics and each data set. Figure~\ref{fig:best-rq1} is the correction of
Figure~3 and Table~\ref{tbl:results} the correction of Table~8 from the original
publication. Table~\ref{tbl:results} and Figure~\ref{fig:best-rq1} only report
the results for the best classifier for each approach. In case these changed between the original results
and our correction, you will not find the exact same rows. For example, for
CamargoCruz09, we reported DT as best classifier in the original analysis, and
now NB. This is because with the problem in the statistical analysis DT was
ranked best for CamargoCruz09, but in the corrected version NB performs better.
The reasons for these, and other changes are explained in
Section~\ref{sec:reasons}
The most important finding remains the same: the
approach CamargoCruz09 still provides the best-ranking classification model with
a mean \emph{rankscore} of 0.917 for CamargoCruz09-NB. However, the
\emph{rankscore} is not a perfect 1.0 anymore. We attribute this to the more
sensitive ranking due to the correction of the Nemenyi test.
The differences to the next-ranking approaches are still rather small, though the
group of approaches that is within 10\% of the best ranking approach now only consists of CV-RF, Amasaki15, Peters15, and Ma12. The bottom of the ranking is nearly unaffected by the changes as well. The last
seven ranked approaches are still the same. Additionally, our findings regarding
the comparison of using ALL data, versus transfer learning approaches has not
changed: ALL is still in the upper mid-field of approaches. With the corrected
and more finegrained ranking, only six of the cross-project approaches actually
outperform this baseline, whereas seventeen are actually worse.
With respect to CV versus \ac{CPDP}, we still observe that \ac{CPDP} can
outperform CV in case multiple performance criteria are considered because
CV-RF is outperformed by CamargoCruz09-NB. Thus, we still note that this is
possible, but far less conclusively than before, where CV was actually only in
the mid-field of the approaches and not a close second.
Due to these small overall small differences, we change our answer to RQ1
slightly:
\begin{framed}
\noindent\textbf{Answer RQ1:}
CamargoCruz09-NB performs best among the compared \ac{CPDP} approaches and
even outperforms cross-validation. However, the differences to other
approaches are small. The baseline ALL-NB is ranked higher than seventeen of the
\ac{CPDP} approaches.
\end{framed}
\section{Reasons for changes}
\label{sec:reasons}
We checked our raw results for the reasons for all changes in rankings. The
problem with the statistical analysis actually led to two reasons for ranking
changes: first, the $z$-values already consider differences in ranks. Thus, if
the rank was very high, this could would lead to larger $z$-values, which would
negatively impact the ranking. Second, because differences were downscaled with
the $z$-values in comparison to differences in mean ranks, too many approaches
were grouped together as not statistically significantly different. For
approaches that are now better ranked than before, this means that they were
often among the best performing approaches within a group. For those that are
now ranking worse, they were at often near the bottom of their groups. For
example, CV was often among the best approaches on the middle rank. Now, it is
clearly distinguished from the others there, leading to the strong rise in the
ranking. Others that were affected the same way, though to a lesser extend are
Amasaki15, Peters15, YZhang15, and Herbold13. On the other hand Menzies11 and
Watanabe08 were often at the bottom of their groups, leading to the big losses
in rankings for both approaches.
Another change in our results is that NB is often the best
performing classifier, whereas before DT and LR were most often the best
performing classifiers. We previously already noted in our discussion that ``for
many approaches the differences between the classifiers were rather
small''~\cite{Herbold2017b}. Together with the reasons for ranking
changes explained above, theses changes are not unexpected.
Overall, all changes in the result are plausible. Moreover, our comparison
of the results of the statistical analysis with both mean values, as well as the
raw results of the benchmark did not reveal any inconsistencies of the type that
Y. Zhou reported to us. Therefore, we believe that the problem was
correctly resolved.
\section{Update of the replication kit}
\label{sec:replication}
We updated the replication kit archived at Zenodo~\cite{Herbold2017c}. The
changes two the replication kit are two-fold.
\begin{itemize}
\item We corrected the problem with the statistical analysis in the
generate\_results.R script.
\item We updated the provided CD diagrams due to the changes in the Nemenyi
test.
\end{itemize}
The changes can be reviewed in detail in the commit to the GitHub archive of the
replication kit\footnote{https://goo.gl/AbvSRj}.
|
2,869,038,156,381 | arxiv | \section{Introduction}
\IEEEPARstart{D}{iffusion} and consensus dynamics play a fundamental role in the coordination of many complex networks, from networks of autonomous vehicles~\cite{ren2008distributed}, to power grids~\cite{6345156}, to social networks~\cite{hegselmann2002opinion}, and beyond.
As such, significant research effort has been devoted to development of analytical characterizations of the performance of diffusion processes and consensus algorithms based on the network topology and the node interactions.
The vast majority of this work has considered a single, isolated network model. However, many complex networks can be more accurately represented by a set of interacting networks. For example, in vehicular ad-hoc networks, the network topology often consists of clusters of sub-networks, made up of co-located vehicles, that periodically communicate with one another~\cite{LW07}. Another example can be found in social networks, where people are often clustered into communities; interaction within communities is frequent, and interaction across communities less so. These examples motivate the \emph{Network of Networks} (NoN) model, where multiple individual networks, or \emph{subgraphs}, are connected using a few links to form a connected composite graph.
We analyze the diffusion rate of an NoN and the convergence rate of consensus algorithms in an NoN using spectral perturbation theory-based methods.
For the diffusion process in an NoN, we assume the edges between subgraphs has small weights. To formulate this setting, we study a system in which all weights between subgraphs are multiplied with a small parameter $\epsilon$. This setting captures diffusion processes in many complex network systems, for example, the social networks with weak inter-community links.
In a consensus network,
we consider a setting where the links between subgraphs may be costly to use, and so they are used sparingly in the consensus algorithm. We model this setting using a stochastic system where links that connect subgraphs are active in each iteration with some small probability $p$. This setting applies to architectures like vehicle networks and the Internet of Things, where nearby nodes can communicate using free local communication, e.g., Bluetooth, but where distant nodes must communicate using potentially costly cellular or satellite communication.
We show that the diffusion rate is directly related to the convergence rate of the expected system of the stochastic consensus network. Our results show that up to first order in $\epsilon$, the diffusion rate depends on the generalized Laplacian matrix of the connecting graph, which is determined by the number of nodes in each subgraph and the topology of the interconnecting links. The rate does not depend on the topologies of the individual subgraphs nor on which nodes are used to connect the subgraphs to one another. The second order perturbation analysis, however, shows that choosing nodes with largest information centrality~\cite{SZ89} as bridge node maximizes the diffusion rate upto second order in $\epsilon$. We also study the mean square convergence rate of the consensus network, which leads to similar results.
In addition, we conduct experiments to show that our analysis
Numerical results show that this analysis accurately captures the behavior of the studied dynamics for small values of $\epsilon$ or $p$.
\subsubsection*{Related work}
Several previous papers have studied the diffusion process in various NoN models. \cite{gomez2013diffusion} provides an upper bound for the diffusion rate of a NoN where each layer of subgraph has the same number of nodes and the inter-network links between any two adjacent layers of networks are restricted to be the same one to one map. \cite{sole2013spectral} studies the same model as \cite{gomez2013diffusion} using perturbation theory. \cite{tejedor2018diffusion} studies optimal weights for inter-layer links in the case where intra-layer network may be directed. \cite{Cencetti_2019} also studies same model as \cite{gomez2013diffusion} and derives relationships between $\lambda_2$ of the supra-Laplacian and topological properties of the subgraphs. We note that all these works are based on the homogeneous one to one inter-layer connection assumption made in \cite{gomez2013diffusion}. In addition, \cite{SHAO20175184} studies diffusion in Cartesian product of graphs as a model of NoN and gives some analysis based on numerical experiments.
As for the discrete-time consensus dynamics, there has been a significant amount of work devoted to the analysis of distributed consensus algorithms in time-varying networks and stochastic networks, e.g.,~\cite{1205192,1440896,7963624,1393134,zhou2009convergence,abaid2012consensus}. In this work, we employ a model similar to that studied in
~\cite{Kar08,fagnani2009average,PBE10}, which all study the convergence rate of the mean-square deviation from consensus in a stochastic network.
\cite{Kar08} presents bounds based on the spectrum of the expected weight matrix, whereas~\cite{fagnani2009average} and \cite{PBE10} give analytical expressions
for the convergence rate itself.
None of these previous works considered an NoN model. The NoN consensus model was introduced in \cite{gao2011robustness} and \cite{peixoto2012evolution}, where they measure network performance by analyzing its robustness against random node failures. More recent work~\cite{7963622} considers an NoN model with noisy consensus dynamics and proposes methods to identify the optimal interconnection topology. And, in~\cite{7525499}, the authors consider a similar NoN model, but with slightly different dynamics. They show that interconnection between the nodes of subgraphs with the highest degree maximizes the robustness of the NoN. While these works focus on robustness of an NoN, our work in contrast, focuses on the rate at which nodes reach consensus, and in particular, how this rate relates to the topologies of the interconnecting network and the subgraphs.
We note that a preliminary version of this work appeared in~\cite{8619565}. This conference paper presented first-order perturbation analysis only. Further, this analysis was restricted to consensus algorithms, In this paper, we study both diffusion and consensus dynamics, and more significantly, we include second-order perturbation analysis. This second-order analysis provides more insight into the role of the connecting nodes within each subgraph in determining the diffusion and convergence rate.
\subsubsection*{Outline} The rest of the paper is organized as follows. Section~\ref{modelandformulation.sec} describes our system model and the problem formulation, and it gives background on spectral perturbation analysis.
In Section~\ref{expected.sec}, we present analysis of the diffusion and convergence rate in an NoN, including its first- and second-order behaviors.
In Section~\ref{msanalysis.eq}, we present our analysis of the mean square convergence of consensus algorithms for a special case of stochastic dynamics.
Section~\ref{results.sec} gives numerical evaluations that highlight key results of our theoretical analysis, followed by the conclusion in Section~\ref{conclusion.sec}.
\section{System Model} \label{modelandformulation.sec}
\subsection{Diffusion Dynamics in Network of Networks} \label{model.sec}
We consider a system of $D$ disjoint graphs ${\mathcal G_i = (\mathcal{V}_i, \mathcal{E}_i,w_i)}$, $i = 1, \ldots, D$. Each graph $\mathcal G_i$ is weighted, undirected, and connected.
We call these graphs the \emph{subgraphs} of the NoN.
The set $\mathcal V_i$ denotes the node set of $\mathcal G_i$, with $\lvert \mathcal V_i \rvert = N_i$, and $\mathcal E_i$ is the set of links.
An edge between node $r \in \mathcal V_i$ and $s \in \mathcal V_i $ is denoted by $e(r,s)$, and $\mathcal N_i(j)$ denotes the neighbor set of node $j$ in subgraph $\mathcal G_i$.
The function $w_i : \mathcal E_i \mapsto \mathbb{R}^+$ defines a non-negative weight $w_i(r,s)$ for each edge $e(r,s) \in \mathcal E_i$.
Let $\mathbf{L}_i$ be the weighted Laplacian matrix of subgraph $\mathcal G_i$, defined as
\begin{align*}
\mathbf{L}_i(r,s) &= \left\{ \begin{array}{ll}
\sum_{k \in \mathcal N_i(r)} w_i(r,s) & \text{~for~}r=s \\
-w_i(r,s) & \text{otherwise.}
\end{array} \right.
\end{align*}
Further, we define $\Lc_{sub}$ to be the $N \times N$ block diagonal matrix with blocks $\mathbf{L}_i$, $i = 1 \ldots D$.
We construct an NoN by connecting the $D$ subgraphs with a small number of edges.
The set $\mathcal V$ is the NoN vertex set, ${\mathcal V = \bigcup_{i=1}^D \mathcal V_i}$, with $|\mathcal V| = N$. Without loss of generality, we identify the nodes in $\mathcal V$ as $1, 2, \ldots, N$.
The NoN edge set $\mathcal E$ consists of all edges in $\mathcal E_1 \cup \ldots \cup \mathcal E_D$, as well as a set of undirected \emph{connecting edges} $\mathcal E_{con} = \{e(r,s)~|~r\in \mathcal G_i, s \in \mathcal G_j, i \neq j\}$.
We call the nodes $i \in V$ that are adjacent to some edge in $\mathcal E_{con}$ \emph{connecting nodes}, and we denote the set of connecting nodes by $\mathcal V_{con}$.
We assume that there is only one connecting node $s_i$ in each subgraph $\mathcal G_i$.
The \emph{connecting graph} is defined as as $\mathcal G_{con} = (\mathcal V, \mathcal E_{con}, w_{con})$, where $w_{con}: \mathcal E_{con} \mapsto \mathbb{R}^+$ is a function that defines a non-negative weight $w_{con}(r,s)$ for each edge $e(r,s) \in \mathcal E_{con}$.
The weighted Laplacian matrix of the connecting graph is denoted by an $N\times N$ matrix $\Lc_{con}$.
For a matrix $\mathbf{Q}$, we use the symbol $\widehat{\mathbf{Q}}$ to denote the principle submatrix of $\mathbf{Q}$ whose rows and columns correspond to vertices in $\mathcal V_{con}$.
For example, $\widehat{\mathbf{L}}_{con}$ is the $D \times D$ weighted Laplacian of the graph $\widehat{\mathcal G}_{con} = (\mathcal V_{con}, \mathcal E_{con}, w_{con})$.
With these definitions, the NoN is thus formally defined as $\mathcal G = (\mathcal V, \mathcal E, w)$, where $w(r,s) = w_i(r,s)$ for $r,s \in \mathcal V_i$ and $w(r,s) = w_{con}(r,s)$ for $r\in \mathcal V_i, s\in \mathcal V_j, i\neq j$.
We further define the \emph{strength} of a node $r$ as $\Delta_r = \sum_{s\in \mathcal N(r)} w(r,s)$, where $\mathcal N(r)$ denotes the neighbor set of node $r$ in graph $\mathcal G$.
We study diffusion dynamics in this NoN where there is weak coupling between subgraphs. This weak coupling is enforced both by limiting the number of connecting nodes in each subgraph to one and by selecting a small inter-subgraph diffusion coefficient.
For each subgraph $\mathcal G_i$, every node $r \in \mathcal V_i$ has a scalar-valued state denoted by $x_{r}$.
The node dynamics are:
\begin{align*}
\dot{x}_r =\!\!\!\! \sum_{s \in \mathcal N_i(r)}\!\!\!\! w(r,s) (x_s - x_{r}) + \epsilon\!\! \sum_{e(r,u) \in \mathcal E_{con}} \!\!\!\!\!\!\!\!w(r,s)(x_u - x_r),
\end{align*}
where $\epsilon$ is the diffusion coefficient between subgraphs.
Let $\mathbf{x}_i$ denote the vector of node states for graph $\mathcal G_i$, and let $\mathbf{x}$ denote the states of all nodes in the system,
i.e., $\mathbf{x} = [\mathbf{x}_1^{T}~\mathbf{x}_2^{T}~\ldots~\mathbf{x}_D^{T}]^{T}$.
The dynamics of the entire NoN can then be written as:
\begin{align}
\label{contDynamics.eqn}
\dot{\mathbf{x}} = - (\Lc_{sub} + \epsilon \Lc_{con}) \mathbf{x}.
\end{align}
The matrix $\mathbf{L} = \Lc_{sub} + \epsilon \Lc_{con}$ is called the \emph{supra-Laplacian} of the NoN.
We investigate the smallest non-zero eigenvalue of the Laplacian matrix $\mathbf{L}$, which decides the rate of diffusion in (\ref{contDynamics.eqn}). It is also called the spectral gap of $\mathbf{L}$
\begin{definition}
\label{specGap.def}
The spectral gap of $\mathbf{L} = \Lc_{sub} + \epsilon \Lc_{con}$ is defined as the smallest non-zero eigenvalue of $\mathbf{L}$, denoted as $\alpha(\mathbf{L})$.
\end{definition}
The spectral gap determines the slowest speed that the diffusion process (\ref{contDynamics.eqn}) converges to its steady state from any initial state and therefore is also referred to as the \emph{diffusion rate}. Since $\mathbf{L}$ is positive semi-definite and has eigenvalue zero with multiplicity $1$ for any connected graph $\mathcal G$, we know that $\alpha(\mathbf{L})>0$.
In particular, we study how the spectral gap $\alpha(\mathbf{L})$ is related to matrix $ \Lc_{sub}$ and matrix $\Lc_{con}$. We recall that $\Lc_{con}$ is decided by the set of connecting nodes $\mathcal V_{con}$ and the structure of the connecting graph, characterized by $\widehat{\mathbf{L}}_{con}$.
Further, we show how are analysis can be used to select connecting nodes within the subgraphs that maximize the spectral gap.
\subsection{Connection to Consensus in Stochastic Networks}
\label{model2.subsection}
There is a close relationship between $\alpha(\mathbf{L})$ the convergence rate of discrete-time consensus dynamics in stochastic networks.
Through this relationship, we identify an alternate interpretation of $\alpha(\mathbf{L})$.
We consider a consensus network where links within each subgraph are always active, e.g., due to the proximity of agents within the subgraph to one another.
Since subgraphs may be separated spatially, communication between subgraphs may be infrequent and/or lossy.
We model this by activating the connecting edges in $\mathcal E_{con}$ each time step $\ell$ with some small probability $p$.
One can define the dynamics as a consensus network with stochastic communication links.
For a node $r\in \mathcal V_i$
\begin{align*}
x_r(\ell+1)&= x_r(\ell) - \sum\limits_{v \in \mathcal{N}_i(r) }w(r,v)\big(x_r(\ell) - x_v(\ell)\big) \\
&\quad \quad - \beta\sum_{e(r,s) \in \mathcal E_{con}} \delta_{rs}(\ell) w(r,s) (x_r(\ell) -x_s(\ell)).
\end{align*}
We assume that for all $r\in \mathcal G, r\in \mathcal G_i, \sum_{v\in \mathcal{N}_i(r)}{w(r,v)} < 1$. In addition, we assume $\beta \leq \frac{1}{2\Delta}$, where $\Delta = \max(\Delta_i)$ is the maximal node strength of $\mathcal G$.
\begin{align*}
\delta_{rs}(\ell) =
\begin{cases} 1 & \text{ with probability } p \\
0 & \text{ with probability } 1-p
\end{cases}
\end{align*}
where $\delta_{rs}(\ell)$ are Bernoulli random variables that are not necessarily mutually independent. We note that all $\delta_{rs}(\ell)$ are independent of $\mathbf{x}(\ell)$.
\subsubsection{Convergence Rate of Expected System}
Let $\mathbf{A}$ be the block diagonal matrix $\mathbf{A} = \mathbf{I} - \Lc_{sub}$.
We also define an $N \times N$ matrix ${\mathbf{B}_{rs} = \beta \cdot w(r,s)\cdot \textbf{b}_{rs} \textbf{b}_{rs}^{T}}$,
where $\textbf{b}_{rs}$ is a binary $N$-vector with the $r^{th}$ element equal to 1, the $s^{th}$ element equal to -1, and the remaining elements equal to 0.
The dynamics of the stochastic NoN can then be written as
\begin{align} \label{eq:3}
\mathbf{x}(\ell+1) = \mathbf{A}\mathbf{x}(\ell) - \sum_{e(r,s)\in \mathcal E_{con}}\delta_{rs}(\ell)\mathbf{B}_{rs}\mathbf{x}(\ell).
\end{align}
We further let $\bar{\mathbf{x}}(\ell)=\expec{\mathbf{x}(\ell)}$ and $\mathbf{B} = \sum_{e(r,s)\in\mathcal E_{con}}\mathbf{B}_{rs}$. By taking expectation of both sides of (\ref{eq:3}), we obtain
\begin{align}
\label{eq:4}
\bar{\mathbf{x}}(\ell+1) = \overline{\mathbf{A}} \bar{\mathbf{x}}(\ell)\,,
\end{align}
where $\overline{\mathbf{A}} = \mathbf{A} - p\mathbf{B}$ is the \emph{expected weight matrix}. The equality follows from the fact that $\delta_{rs}(\ell)$ is independent of $\mathbf{x}(\ell)$.
\begin{definition}
\label{rhoEss.def}
The \emph{convergence rate of the expected system} of (\ref{eq:4}), denoted $\rho_{ess}(\overline{\mathbf{A}})$, is defined as the second largest eigenvalue of $\overline{\mathbf{A}}$, also called the \emph{essential spectral radius} of $\overline{\mathbf{A}}$.
\end{definition}
Given the condition $\sum_{v\in \mathcal{N}_i(r)}{w(r,v)} < 1$, the matrix $\mathbf{A}_i:= \mathbf{I} - \mathbf{L}_i$ has $1$ as a simple eigenvalue with eigenvector $\textbf{1}$ for all subgraph $\mathcal G_i$, then matrix $\mathbf{A}$ has eigenvalue $1$ with multiplicity $D$. If $\mathcal G_{con}$ is connected, the matrix $\overline{\mathbf{A}}$ has eigenvalue $1$ with multiplicity $1$, and its corresponding eigenvector is $\textbf{1}$. Under the assumption $\beta \leq \frac{1}{2\Delta}$, the convergence rate of the expected system (\ref{eq:4}) is characterized by the second largest eigenvalue of $\overline{\mathbf{A}}$~\cite{XIAO200465}.
Next, noting that $\mathbf{A}-p\mathbf{B} = \mathbf{I} - (\mathbf{L}_{sub}+p\beta\mathbf{L}_{con})$, we state a simple relationship between $\alpha(\mathbf{L})$ and $\rho_{ess}(\overline{\mathbf{A}})$.
\begin{proposition}
\label{difftoCons.prop}
The spectral gap $\alpha(\mathbf{L})$, where $\mathbf{L} = \mathbf{L}_{sub}+\epsilon\mathbf{L}_{con}$, and the essential spectral radius $\rho_{ess}(\overline{\mathbf{A}})$, where $\overline{\mathbf{A}} = \mathbf{A}-p\mathbf{B}$, as given by Definitions~\ref{specGap.def} and~\ref{rhoEss.def}, respectively, satisfy
\begin{align}
\rho_{ess}(\mathbf{A}-p\mathbf{B}) = 1 - \alpha(\mathbf{L}_{sub}+p\beta\mathbf{L}_{con})\,.
\end{align}
\end{proposition}
\subsubsection{Mean-Square Convergence Rate}
\label{model3.subsubsec}
We also study the \emph{mean square convergence rate} of the stochastic NoN in (\ref{eq:4}).
Let $\tilde{\textbf{x}}(\ell) = \mathbf{P} \mathbf{x}(\ell)$ be the deviation from average vector,
where $\mathbf{P}$ is the projection matrix, $\mathbf{P} = (\mathbf{I}_{N} - \frac{1}{N} \textbf{1} \textbf{1}^{T})$.
If $\lim_{t \rightarrow \infty} \expec{ \| \tilde{\textbf{x}}(\ell) \|_2} = 0$, we say the system \emph{converges in mean square}.
We start by investigating the case where all edges in $\mathcal G_{con}$ are activated together with some probability $p$ in each time step $t$. We discuss the i.i.d. case in Appendix \ref{extension.sec}.
\begin{assumption}\label{activeTogether.assum}
All edges in $\mathcal G_{con}$ are online or offline with probability $p$ and $1-p$ at time step $\ell$, decided by a Bernoulli random variable $\delta(\ell)$.
\end{assumption}
We define the autocorrelation matrix of $\tilde{\textbf{x}}(\ell)$ by $\mathbf{\Sigma}(\ell) = \expec{\tilde{\textbf{x}}(\ell) \tilde{\textbf{x}}(\ell)^T}$
and note that $\mathbf{\Sigma}(\ell) = \expec{\mathbf{P} \mathbf{x}(\ell) \mathbf{x}(\ell)^{T} \mathbf{P}}$.
Using a similar method to that in \cite{patterson2010convergence}, it can be shown that $\mathbf{\Sigma}(\ell)$ satisfies the matrix recursion
\begin{align}
\mathbf{\Sigma}(\ell+1) &= (\mathbf{P}\bar{\mathbf{A}}\mathbf{P})\mathbf{\Sigma}(\ell)(\mathbf{P}\bar{\mathbf{A}}\mathbf{P}) + \sigma^2 \mathbf{B} \mathbf{\Sigma}(\ell)\mathbf{B}. \label{Mrecur.eq}
\end{align}
where the zero-mean random variable $\mu(\ell)$ is defined as $\mu(\ell) = \delta(\ell) - p$, and $\sigma^2 = \var{\mu(\ell)}$.
The variances $\mathbb{E}[\tilde{x}_r(\ell)^2]$ are given by the diagonal entries of $\mathbf{\Sigma}(\ell)$, and thus we are interested in how they evolve.
We define the matrix-valued operator,
\begin{align}
\mathcal A(X) &= (\mathbf{P}\bar{\mathbf{A}}\mathbf{P})X(\mathbf{P}\bar{\mathbf{A}}\mathbf{P}) + \sigma^2 \mathbf{B} X \mathbf{B} \label{eq:matrixOpDef}
\end{align}
and note that $\mathbf{\Sigma}(\ell+1) = \mathcal A(\mathbf{\Sigma}(\ell))$.
The rate of decay of the entries of $\mathbf{\Sigma}(\ell)$ is given by the spectral radius of $\mathcal A$, denoted $\rho(\mathcal A)$~\cite{patterson2010convergence}.
\begin{definition}
The \emph{mean square convergence rate} of the system (\ref{eq:4}), under Assumption~\ref{activeTogether.assum}, is defined as $\rho(\mathcal A)$.
\end{definition}
\section{Background on Spectral Perturbation Theory}
Our analytical approach is based on spectral perturbation analysis~\cite{baumgartel85,Ba20}, especially the analysis where repeated eigenvalues are considered~\cite{Ba20}. Here, we provide a brief overview of this material.
Let $\mathcal{M}(\epsilon, X)$ be a symmetric vector-valued (or matrix-valued operator) of a real parameter $\epsilon$ and a variable $X$ of the form
\begin{align}
\mathcal{M}(\epsilon, X) = \mathcal{M}_0(X)+ \epsilon\mathcal{M}_1(X) +\epsilon^2 \mathcal{M}_2(X) \label{eq:secOrderOperator}
\end{align}
and let $(\gamma(\epsilon), W(\epsilon))$ be an eigenvalue-eigenvector (or eigenvalue-eigenmatrix) pair of $\mathcal{M}(\epsilon, .)$, as a function of $\epsilon$
\[
\mathcal{M}(\epsilon, W(\epsilon)) = \gamma(\epsilon)W(\epsilon).
\]
According to spectral perturbation theory, the functions $\gamma$ and $W$ are well-defined and analytic for small values of ${\epsilon}$. The power series expansion of $\gamma$ is
\begin{align}
\label{perturb.eq}
\gamma(p) = \lambda(\mathcal{M}_0) + C^{(1)} \epsilon + C^{(2)} \epsilon^2 + \cdots
\end{align}
where $\lambda(\mathcal{M}_0)$ is an eigenvalue of the operator $\mathcal{M}_0$.
Let eigenvalue $\lambda(\mathcal{M}_0)$ have multiplicity $K$, and let
$\mathbf{W}_i$, $i=1 \ldots K$, be $K$ orthonormal eigenvectors (or eigenmatrices)
of $\mathcal{M}_0$ that form a basis for the eigensubspace of $\lambda(\mathcal{M}_0)$.
We form the $K \times K$ matrix $\mathbf{F} = [f_{i,j}]$, with each component given by
\begin{align} \label{eq:raleighCoeff}
f_{ij} = \frac{\langle \mathbf{W}_{i}, \mathcal{M}_1(\mathbf{W}_{j})\rangle}{\langle \mathbf{W}_{i}, \mathbf{W}_{i}\rangle}.
\end{align}
When $\mathcal{M}$ is a vector-valued operator, the inner product is the standard vector inner product
(for $\mathcal{M}$ a matrix-valued operator, the matrix inner product is $\langle \mathbf{X}, \mathbf{Y}\rangle := \tr{\mathbf{X}^*\mathbf{Y}}$).
Let $\nu_1, \nu_2, \ldots, \nu_{K}$ be the eigenvalues of $\mathbf{F}$, with repetition.
Then, the $K$ first-order perturbation constants are $C_i^{(1)} = \nu_i$, for $i=1 \ldots K$.
We also study the second order perturbation terms $C^{(2)}$.
According to \cite{baumgartel85, Ba20}, for an eigenvalue $\lambda(\mathcal{M}_0)$ with multiplicity $K > 1$, when $\mathbf{F}$ is diagonal, the second order terms
$C_i^{(2)}$, $i= 1 \ldots K$, are
\begin{align}
\label{snd_pertb.eq}
C_i^{(2)} = \sum_{\lambda_m(\mathcal{M}_0) \neq \lambda(\mathcal{M}_0)} \frac{\langle \mathbf{W}_i,\mathcal{M}_1(\mathbf{W}_m)\rangle^2}{\lambda(\mathcal{M}_0)-\lambda_m(\mathcal{M}_0)}
\end{align}
where $\mathbf{W}_i$ is the $i^{th}$ eigenvector (or eigenmatrix) of $\mathcal{M}_0$ with eigenvalue $\lambda$, for $i=1 \ldots K$, and $(\lambda_m(\mathcal{M}_0), \mathbf{W}_m)$ is an eigenpair of $\mathcal{M}_0$ with $\lambda_m(\mathcal{M}_0) \neq \lambda(\mathcal{M}_0)$.
\section{Analysis} \label{expected.sec}
In this section, we use spectral perturbation analysis to study $\alpha(\mathbf{L})$ and $\rho_{ess}(\overline{\mathbf{A}})$.
\subsection{The Spectral Gap in Diffusion Dynamics}
\label{diffAnalysis.subsec}
We first study the convergence of system (\ref{contDynamics.eqn}), assuming the diffusion coefficient $\epsilon$ between subgraphs is small.
The dynamics in \eqref{contDynamics.eqn} can be expressed using a vector-valued operator of the form given by (\ref{eq:secOrderOperator}) as $\dot{\mathbf{x}} = \mathcal{M}(\epsilon, \mathbf{x})$, where
\begin{align*}
\mathcal{M}_o(\mathbf{x}) &= \mathbf{L}_{sub} \mathbf{x} \\
\mathcal{M}_1(\mathbf{x}) &= \mathbf{L}_{con} \mathbf{x} \\
\mathcal{M}_2(\mathbf{x}) & = 0.
\end{align*}
We note that $\mathbf{L}_{sub}$ is the Laplacian matrix of a graph with $D$ connected components (the subgraphs). Thus, it has an eigenvalue of $0$ with multiplicity $D$. However, when $\widehat{G}_{con}$ is connected, $\mathbf{L}$ has an eigenvalue of $0$ with multiplicity $1$. The smallest $D-1$ nonzero eigenvalues of $\mathbf{L}$ correspond to the perturbed $0$ eigenvalue of $\mathbf{L}_{sub}$. Therefore we study the perturbations to the $0$ eigenvalue
of $\mathbf{L}_{sub}$.
We begin by defining the generalized Laplacian matrix of the connecting graph $\widehat{\mathcal G}_{con}$~\cite{rotaru2004dynamic}.
\begin{definition}
Let $\mathbf{r} = [N_1~N_2~\ldots~N_D]^T$, and let $\mathbf{R}$ be the $D \times D$ diagonal matrix with diagonal entries $\mathbf{r}$.
The \emph{generalized Laplacian}of $\widehat{\mathcal G}_{con}$ is ${\widehat{\mathbf{M}}} = \mathbf{R}^{-\frac{1}{2}} \widehat{\mathbf{L}}_{con} \mathbf{R}^{-\frac{1}{2}}$.
\end{definition}
Note that ${\widehat{\mathbf{M}}}$ is symmetric positive semidefinite. It has an eigenvalue of $0$ with eigenvector $\mathbf{r}^{1/2}$, and if $\widehat{\mathcal G}_{con}$ is connected, its second smallest eigenvalue $\lambda_2({\widehat{\mathbf{M}}})$ is greater than $0$.
We now give a relationship between this eigenvalue and the spectral gap.
\begin{theorem} \label{specGap.thm}
The spectral gap of the matrix $\mathbf{L}= \Lc_{sub} + \epsilon \Lc_{con}$, up to first order in $\epsilon$, is
\[
\alpha(\mathbf{L}) = \epsilon \lambda_2({\widehat{\mathbf{M}}})\,,
\]
in which $\lambda_2({\widehat{\mathbf{M}}})$ is the smallest nonzero eigenvalue of ${\widehat{\mathbf{M}}}$.
\end{theorem}
\begin{IEEEproof}
We determine the perturbation coefficients by forming the matrix $\mathbf{F}$ in (\ref{eq:raleighCoeff}).
To do so, we must find an orthonormal set of eigenvectors for $D$ zero eigenvalues of $\mathbf{L}_{sub}$, denoted as $\{\mathbf{v}_1, \ldots, \mathbf{v}_D\}$.
Let $\mathbf{u}_1, \ldots, \mathbf{u}_D$ be orthonormal eigenvectors of ${\widehat{\mathbf{M}}}$, and let $\lambda_1({\widehat{\mathbf{M}}}) \leq \lambda_2({\widehat{\mathbf{M}}}) \leq \ldots \leq \lambda_D({\widehat{\mathbf{M}}})$ be the corresponding eigenvalues.
We define the eigenvectors $\mathbf{v}_i$, $i=1 \ldots D$, to be ${\mathbf{v}_i = [ \theta_i^{(1)} \textbf{1}_{N_1}^{T}~ \theta_i^{(2)} \textbf{1}_{N_2}^{T}~\ldots~\theta_i^{(D)} \textbf{1}_{N_D}^{T}]^{T}}$, with
\begin{equation} \label{eq:thetaDef2}
\theta_i^{(j)} = \frac{1}{\sqrt{N_j}} u_{ij}
\end{equation}
where $u_{ij}$ denotes the $j^{th}$ component of the eigenvector $\mathbf{u}_i$. We observe that the eigenvectors $\mathbf{v}_i$, $i=1 \ldots D$, are orthonormal.
We now find the entries of the $D \times D$ matrix $\mathbf{F}$ defined by (\ref{eq:raleighCoeff}).
For $f_{ij}$, we have
\begin{align}
f_{ij} &= \langle \mathbf{v}_i, \mathbf{L}_{con} \mathbf{v}_j \rangle \nonumber \\
&= \mathbf{u}_j ^{T} \mathbf{R}^{-\frac{1}{2}} \widehat{\mathbf{L}}_{con} \mathbf{R}^{-\frac{1}{2}} \mathbf{u}_i \nonumber \\
&= \mathbf{u}_j ^{T} {\widehat{\mathbf{M}}} \mathbf{u}_i \nonumber \\
&= \lambda_i({\widehat{\mathbf{M}}}) \mathbf{u}_i^T \mathbf{u}_j. \label{gijgen.eq}
\end{align}
The equalities follow by the definition of $\mathbf{v}_i$, $\widehat{\mathbf{M}}$, and $\mathbf{u}_i$.
If $i \neq j$, then because $\mathbf{u}_i$ and $\mathbf{u}_j$ are orthonormal, $f_{ij} = 0$.
Thus $\mathbf{F}$ is a diagonal matrix, and its eigenvalues are
\begin{equation}
C_{i}^{(1)} = \lambda_i({\widehat{\mathbf{M}}}),~i = 1 \ldots D. \label{gii.eq}
\end{equation}
This completes the proof.
\end{IEEEproof}
Theorem~\ref{specGap.thm} shows that the diffusion rate, up to first order in $\epsilon$, is decided by an expression that depends on the smallest nonzero eigenvalue of ${\widehat{\mathbf{M}}}$. We note that ${\widehat{\mathbf{M}}}$ depends on the topology and edge weights of the connecting graph, as well as the number of vertices in each subgraph. However, ${\widehat{\mathbf{M}}}$ does not depend on the topology or edge weights of the subgraphs. Further, it does not depend on the choice of connecting node in each subgraph.
An intuition for this result is that the connecting link is a bottleneck in the diffusion process.
The diffusion rate within each graph is much faster than the diffusion rate across the connecting link.
The role of the connecting link is to transfer information between the two graphs, and the amount of information that needs to be exchanged is proportional to the sizes of the graphs.
It has been shown that $\lambda_i({\widehat{\mathbf{M}}})$ also determines the convergence rate of load balancing diffusion algorithms in heterogeneous systems~\cite{rotaru2004dynamic}. Following this analogy, we can view just the edges in $\mathcal G_{con}$ as executing a load balancing algorithm. The role of the connecting graph is to transfer load (i.e., node state) between the subgraphs, and the load that needs to be transferred out of each subgraph to balance the system is be proportional to the number of nodes in that subgraph.
Then we study the diffusion rate of (\ref{contDynamics.eqn}) upto second order of $\epsilon$. We note that it is decided by the spectral gap of $\mathbf{L}$.
\begin{theorem}
\label{sndPertAlpha.thm}
The spectral gap of the matrix $\mathbf{L}= \Lc_{sub} + \epsilon \Lc_{con}$, up to second order in $\epsilon$, is
\begin{align}
\label{sndPertAlpha.eqn}
\alpha(\mathbf{L}) = \epsilon \lambda_2({\widehat{\mathbf{M}}})-\epsilon^2 ((\lambda_2({\widehat{\mathbf{M}}}))^2 (\mathbf{u}_2^* \widehat{\mathcal{S}} \mathbf{u}_2)\,,
\end{align}
where $\lambda_2({\widehat{\mathbf{M}}})$ is the smallest nonzero eigenvalue of ${\widehat{\mathbf{M}}}$, and $\mathbf{u}_2$ is its corresponding eigenvector. The $D\times D$ diagonal matrix $\widehat{\mathcal{S}}$ has diagonal entries $\widehat{\mathcal{S}}(k,k):=N_k \cdot \mathbf{L}_k^{\dag}(s_k,s_k)$. $\mathbf{L}_k^{\dag}$ is the Moore-Penrose inverse of $\mathbf{L}_k$, $s_k$ is the connecting node in graph $\mathcal G_k$, and $\mathbf{L}_k^{\dag}(s_k,s_k)$ is the diagonal entry of $\mathbf{L}_k$ that corresponds to node $s_k$.
\end{theorem}
\begin{IEEEproof}
In order to study second order perturbation coefficients using (\ref{snd_pertb.eq}), we need to find all $N$ eigenvectors of the matrix $\Lc_{sub}$.
We recall that the eigenvectors of $\Lc_{sub}$ corresponding to zero eigenvalues are defined as ${\mathbf{v}_i = [ \theta_i^{(1)} \textbf{1}_{N_1}^{T}~ \theta_i^{(2)} \textbf{1}_{N_2}^{T}~\ldots~\theta_i^{(D)} \textbf{1}_{N_D}^{T}]^{T}}$, where $\theta_i^{(1)}$ is defined by (\ref{eq:thetaDef2}), for $i=1\ldots D$.
We define the remaining eigenvectors of $\mathbf{L}_{sub}$ as follows.
Consider the Laplacian matrix $\mathbf{L}_i$ for subgraph $i$, and let $\mathbf{p}_{i_\psi}$, ${\psi=1 \ldots N_i}$, be a set of $\psi$ orthonormal eigenvectors of $\mathbf{L}_i$. Since $\mathcal G_i$ is connected, its $0$ eigenvalue has multiplicity $1$. We let $\mathbf{p}_{i_\psi}$, ${\psi=2 \ldots N_i}$, be the eigenvectors associated with nonzero eigenvalues.
Then we define the remaining $\mathbf{v}_m$, $m=(D+1) \dots N$, to be ${\mathbf{v}_m = [ \mathbf{0}_{N_1}^{T}~\ldots \mathbf{0}_{N_{k-1}}^{T}~\mathbf{p}_m^{T}~\mathbf{0}_{N_{k+1}}^{T}~\ldots~\mathbf{0}_{N_{D}}^{T}]^{T}}$, where $\mathbf{p}_m \in \{\mathbf{p}_{i_\psi} : i\in[D] \textrm{ and }\psi\in\{2,\ldots, N_i\}\}$.
By applying (\ref{snd_pertb.eq}) we attain
\begin{align}
C_{i}^{(2)} & = \sum_{\lambda_m(\mathbf{L}_{sub})\neq 0}\frac{\mathbf{v}_i^*\mathbf{L}_{con}\mathbf{v}_m\mathbf{v}_m^*\mathbf{L}_{con}\mathbf{v}_i}{0-\lambda_m(\mathbf{L}_{sub})} \nonumber\\
& = \sum_{\lambda_m(\mathbf{L}_{sub})\neq 0}\frac{\widehat{\mathbf{v}}_i^*\widehat{\mathbf{L}}_{con}\widehat{\mathbf{v}}_m\widehat{\mathbf{v}}_m^*\widehat{\mathbf{L}}_{con}\widehat{\mathbf{v}}_i}{0-\lambda_m(\mathbf{L}_{sub})}\nonumber
\end{align}
We recall that $s_k$ is the vertex index of the connecting node in subgraph $\mathcal G_k$. Then
\begin{align}
& C_{i}^{(2)} = \sum_{k=1}^D\sum_{\substack{m: \\supp(\mathbf{v}_m)\subset \mathcal V_k}}\frac{\widehat{\mathbf{v}}_i^*\widehat{\mathbf{L}}_{con}(p_{m,s_k}^2\mathbf{E}_{k})\widehat{\mathbf{L}}_{con}\widehat{\mathbf{v}}_i}{-\lambda_m(\mathbf{L}_k)}\nonumber\\
& =\sum_{k=1}^D\widehat{\mathbf{v}}_i^*\widehat{\mathbf{L}}_{con}\mathbf{R}^{-\frac{1}{2}}\left(\sum_{\substack{m: \\supp(\mathbf{v}_m)\subset \mathcal V_k}}\!\!\!\!\!\!\!\!\frac{p_{m,s_k}^2\mathbf{R}^{\frac{1}{2}}\mathbf{E}_{k}\mathbf{R}^{\frac{1}{2}}}{-\lambda_m(\mathbf{L}_k)}\right)\mathbf{R}^{-\frac{1}{2}}\widehat{\mathbf{L}}_{con}\widehat{\mathbf{v}}_i \nonumber\\
& = \sum_{k=1}^D\widehat{\mathbf{v}}_i^*\widehat{\mathbf{L}}_{con}\mathbf{R}^{-\frac{1}{2}}\left(\sum_{\substack{m: \\supp(\mathbf{v}_m)\subset \mathcal V_k}}\!\!\!\!\!\!\!\!\frac{r_{kk}\cdot p_{m,s_k}^2\mathbf{E}_{k}}{-\lambda_m(\mathbf{L}_k)}\right)\mathbf{R}^{-\frac{1}{2}}\widehat{\mathbf{L}}_{con}\widehat{\mathbf{v}}_i\,,\nonumber
\end{align}
where $\mathbf{E}_k$ is a $D\times D$ matrix with only one non-zero entry $\mathbf{E}_{k,k} = 1$. $p_{m,s_k}$ is the entry of $\mathbf{p}_{m}$ associated with the connecting node $s_k$.
We can further derive
\begin{align}
C_{i}^{(2)}&= -\mathbf{u}_i^*\mathbf{R}^{-\frac{1}{2}}\widehat{\mathbf{L}}_{con}\mathbf{R}^{-\frac{1}{2}}\widehat{\mathcal{S}}\mathbf{R}^{-\frac{1}{2}}\widehat{\mathbf{L}}_{con}\mathbf{R}^{-\frac{1}{2}}\mathbf{u}_i\nonumber\\
& = -\mathbf{u}_i^* {\widehat{\mathbf{M}}} \widehat{\mathcal{S}} {\widehat{\mathbf{M}}} \mathbf{u}_i \nonumber\\
& = - (\lambda_i({\widehat{\mathbf{M}}}))^2 (\mathbf{u}_i^* \widehat{\mathcal{S}} \mathbf{u}_i)
\,,\label{expt2nd.eq}
\end{align}
where the $D\times D$ diagonal matrix $\widehat{\mathcal{S}}$ has its entries $\widehat{\mathcal{S}}(k,k):=r_{kk}\cdot \mathbf{L}_k^{\dag}(s_k,s_k)$.
From (\ref{perturb.eq}) we attain the result given in Theorem \ref{sndPertAlpha.thm}.
\end{IEEEproof}
We further obtain the following corollary for all the eigenvalues of $\mathbf{L}$ up to first order and second order in $\epsilon$.
\begin{corollary}
\label{bridgenode.thm}
For any nonzero eigenvalue $\lambda_i(\mathbf{L})$, $i= 2,\dots,D$ in the studied network of networks system (\ref{eq:3}), the first order approximation of $\lambda_i(\mathbf{L})$ is independent of the choices of connecting nodes, the second order approximation of $\lambda_i(\mathbf{L})$ is \yhy{maximized} when each connecting node is chosen as the one with maximum information centrality in each subgraph.
\end{corollary}
\begin{IEEEproof}
From Theorem \ref{specGap.thm} we know that the first order approximation of $\lambda_i(\mathbf{L})$ does not depend on the choice of the connecting nodes.
Then we take into account the second order perturbation terms given by (\ref{expt2nd.eq}). We note that once the structure and the weight function of the connecting graph are fixed, $\lambda_i({\widehat{\mathbf{M}}})$ and $\mathbf{u}_i$ are determined for all $i$. As long as the choice of connecting nodes is concerned, $C^{(2)}_{i}$ is maximized when $\widehat{\mathcal{S}}$ is minimized in the Loewner order. This is achieved when the diagonal entries $\widehat{\mathcal{S}}(k,k)$ are all minimized simultaneously. This is then achieved when each bridge node is chosen as the node with maximum information centrality~\cite{SZ89} in that subgraph, because $r_{k,k} = N_k$ is the same for any choice in that subgraph.
\end{IEEEproof}
Corollary \ref{bridgenode.thm} shows that the second-order perturbation terms are affected by the choice of connecting node in each subgraph. The second-order approximations of all eigenvalues are maximized simultaneously when each connecting node is chosen as the node with maximum information centrality in the corresponding subgraph.
\subsection{Analytical Examples}
\subsubsection{Analysis for $D=2$}
For an NoN consisting of two subgraphs $\mathcal G_1$ and $\mathcal G_2$, the backbone graph $\mathcal G_{con}$ consists of a single edge.
\begin{corollary}
\label{dif2subgraph.thm}
The spectral gap $\alpha(\mathbf{L})$ of an NoN consisting of two subgraphs $\mathcal G_1$ and $\mathcal G_2$, up to first order in $\epsilon$, is
\begin{align}
\alpha(\mathbf{L})= \epsilon \left( \frac{N}{N_1N_2}\right)
\label{alpha2.eq}
\end{align}
\end{corollary}
\begin{IEEEproof}
The generalized Laplacian matrix ${\widehat{\mathbf{M}}}$ is given by
\[
{\widehat{\mathbf{M}}} =
\begin{bmatrix}
\frac{1}{N_1} & -\frac{1}{\sqrt{N_1 N_2}}\\
-\frac{1}{\sqrt{N_1 N_2}} & \frac{1}{N_2}
\end{bmatrix}
\]
${\widehat{\mathbf{M}}}$ has two eigenvalues, $\lambda_1({\widehat{\mathbf{M}}})=0$ and $\lambda_2({\widehat{\mathbf{M}}})=\frac{1}{N_1}+ \frac{1}{N_2}$. Their corresponding eigenvectors are $\mathbf{u}_1 \!\!=\!\! \frac{1}{\sqrt{N}} [\sqrt{N_1}\, \!\sqrt{N_2}]^{T}$ and $\mathbf{u}_2=\frac{1}{\sqrt{N}} [\sqrt{N_2}\, -\sqrt{N_1}]^{T}$. Applying the definition for $F_{i}^{(1)}$ in (\ref{gii.eq}), we obtain (\ref{alpha2.eq}).
%
\end{IEEEproof}
This theorem shows that the first order approximation of $\alpha(\mathbf{L})$ depends on the number of nodes in each subgraph. The first order approximation does not depend on the structures of the subgraphs or the choice of bridge node within each subgraph, as we have observed in Theorem~\ref{specGap.thm}.
We can also observe from (\ref{alpha2.eq}) that when $N_1=N_2 = \frac{N}{2}$, the first order approximation of $\alpha(\mathbf{L})$ is minimized. In other words, when two subgraphs have the same number of nodes, the system converges rate is smallest.
\subsubsection{Analysis for $D>2$ with Equally Sized Graphs}
We next consider the case where $N_1 = N_2 = \ldots = N_D = \frac{N}{D}$, i.e., all subgraphs have the same number of nodes.
\begin{corollary} \label{equal.thm}
Consider a composite system consisting of $D$ subgraphs $\mathcal G_1, \ldots, \mathcal G_D$, each with $\frac{N}{D}$ nodes, and a backbone graph $\mathcal G_{con}$. The spectral gap $\alpha(\mathbf{L})$, up to first order in $\epsilon$, is
\begin{align*}
\alpha(\mathbf{L}) = \epsilon \left(\frac{D}{N}\right) \lambda_2(\widehat{\mathbf{L}}_{con}) \,.
\end{align*}
where $\lambda_2(\widehat{\mathbf{L}})_{con}$ is the second smallest eigenvalue of $\widehat{\mathbf{L}}_{con}$.
\end{corollary}
\begin{IEEEproof}
Given $N_1 = N_2 = \ldots = N_D = \frac{N}{D}$, we attain ${\widehat{\mathbf{M}}} = \frac{D}{N}\widehat{\mathbf{L}}_{con}$. Then we obtain the result in Corollary \ref{equal.thm} by applying Theorem \ref{specGap.thm}.
\end{IEEEproof}
As with the case where $D=2$, up to the first order approximation, the convergence factor is independent of the topology of the subgraphs, and it is independent of the choice of connecting nodes.
The diffusion rate depends on $\lambda_2(\Lc_{con})$, also called the \emph{algebraic connectivity} of the backbone graph.
If $\mathcal G_{con}$ is not connected, then $\lambda_2(\Lc_{con}) = 0$, meaning, as expected, the system does not converge.
The diffusion rate increases as the algebraic connectivity of $\mathcal G_{con}$ increases.
\subsection{Convergence Rate of the Expected Consensus Network}
Next we study the convergence rate of the the expected consensus network~(\ref{eq:4}). By using the analytic results we developed in \ref{diffAnalysis.subsec}, as well as the connection between the spectral gap of $\mathbf{L}$ and the essential spectral radius of $\overline{\mathbf{A}}$, we obtain the following corollary.
\begin{corollary} \label{expectedrho.thm}
The essential spectral radius of the expected weight matrix $\overline{\mathbf{A}}$, up to first order in $p$, is
\[
\rho_{ess}(\overline{\mathbf{A}}) = 1 - p \beta \lambda_2({\widehat{\mathbf{M}}});
\]
the essential spectral radius, upto second order in $p$, is
\[
\rho_{ess}(\overline{\mathbf{A}}) = 1 - p \beta \lambda_2({\widehat{\mathbf{M}}}) +p^2 \beta^2 ((\lambda_2({\widehat{\mathbf{M}}}))^2 (\mathbf{u}_2^* \widehat{\mathcal{S}} \mathbf{u}_2).
\]
\end{corollary}
We omit the proof of Corollary~\ref{expectedrho.thm} because the results follow straightforwardly from Proposition \ref{difftoCons.prop}, Theorem~\ref{specGap.thm}, and Theorem~\ref{sndPertAlpha.thm}.
According to Proposition \ref{difftoCons.prop} and Corollary~\ref{bridgenode.thm}, we conclude that the first order approximation of $\rho_{ess}(\overline{\mathbf{A}})$ is independent of the choices of connecting nodes; the second order approximation shows that choosing nodes with maximum information centrality as the connecting node in each subgraph leads to the fastest convergence rate for the expected consensus system (\ref{eq:4}).
\section{Analysis of Mean Square Convergence Rate} \label{msanalysis.eq}
We now use spectral perturbation analysis to study the mean square convergence rate of an NoN in which all edges in $\mathcal E_{con}$ are activated together with small probability $p$.
\subsection{Mean Square Perturbation}
We write the operator in (\ref{eq:matrixOpDef}) as a matrix-valued operator $\mathcal A(X,p)$ of both a matrix $X$ and the small probability $p$in the form (\ref{eq:secOrderOperator}), with
\begin{align}
\mathcal A_0(X) &= \tilde{\mathbf{A}} X\tilde{\mathbf{A}} \label{eq:A0} \\
\mathcal A_1(X) &= -\mathbf{B} X\tilde{\mathbf{A}} - \tilde{\mathbf{A}} X \mathbf{B} + \mathbf{B} X \mathbf{B} \label{eq:A1} \\
\mathcal A_2(X) &= \mathbf{B} X\mathbf{B} -\mathbf{B} X\mathbf{B} = \mathbf{0} \label{eq.A2}\,,
\end{align}
where $\tilde{\mathbf{A}} = \mathbf{P} \mathbf{A} \mathbf{P}$. Recall that $\mathbf{A} = \mathbf{I} - \Lc_{sub}$.
Given the assumption that for all $r\in \mathcal G, r\in \mathcal G_i, \sum_{v\in \mathcal{N}_i(r)}{w(r,v)} < 1$, then for each subgraph $\mathcal G_i$, $\mathbf{L}_i$ has a single $0$ eigenvalue. Then the matrix $\Lc_{sub}$ has eigenvalue $0$ with multiplicity $D$,
it follows that $\tilde{\mathbf{A}} = \mathbf{P} - \mathbf{L}$ has eigenvalue $1$ with multiplicity $D-1$. Therefore, the operator $\mathcal A_0$ has an eigenvalue of $1$ with multiplicity $(D-1)^2$.
When the system is perturbed by $p \mathcal A_1$, these $1$ eigenvalues are perturbed. The perturbed eigenvalue with largest magnitude is $\rho(\mathcal A)$.
For any pair of eigenvectors $\mathbf{w}_i$ and $\mathbf{w}_j$ of the matrix $\tilde{\mathbf{A}}$, $\mathbf{W}_{ij}:=\mathbf{w}_i\mathbf{w}_j^*$ is an eigenmatrix of $\mathcal A_0$ with eigenvalue $\lambda_{ij}(\mathcal A_0)=\lambda_i(\tilde{\mathbf{A}})\lambda_j(\tilde{\mathbf{A}})$. Because $\tilde{\mathbf{A}} = \mathbf{P} - \mathbf{L}$ is symmetric, its left and right eigenvectors satisfy $\mathbf{w}_i^*\mathbf{w}_i=1$ for $i\in[N]$ and $\mathbf{w}_i^*\mathbf{w}_j=0$ for any $i,j\in[N]$, $i\neq j$.
\begin{lemma} \label{rhoA.lem}
Let $\mathcal G = (\mathcal V,\mathcal E)$ be an NoN with the dynamics as defined in (\ref{eq:3}). There exists a set of vectors $\{\mathbf{w}_i :i = 2,\ldots,D\}$ and an induced set of matrices $\{{\mathbf{W}_{ij}}= \mathbf{w}_i\mathbf{w}_j^*: i,j\in \{2,\ldots,D\}\}$ such that
\begin{align}
\mathcal A_0({\mathbf{W}_{ij}}) = {\mathbf{W}_{ij}}, \qquad&\forall i,j \in \{2,\ldots,D\}\,,\label{prop1}\\
\mathbf{w}_i^*\mathbf{w}_i = 1, \qquad &\forall i, \{2,\ldots,D\}\,,\label{prop2}\\
\mathbf{w}_i^*\mathbf{w}_j = 0, \qquad &\forall i,j\in \{2,\ldots,D\}, i \neq j\,\label{prop3}\\
\mathbf{w}_i^*\textbf{1} = 0, \qquad &\forall i\in \{2,\ldots,D\}\,,\\
\mathbf{w}_i^*\mathbf{B}\mathbf{w}_j = 0, \qquad &\forall i,j\in \{2,\ldots,D\}, i \neq j \label{prop5}
\end{align}
The mean square convergence rate of system \yhy{(\ref{eq:3})} satisfying Assumption \ref{activeTogether.assum}, up to first order in $p$, is
\begin{align}
\rho(\mathcal A) = \max_{ij} \left(1 + p f^{(1)}_{ij} \right)\,,
\end{align}
in which
\begin{align}
f^{(1)}_{ij} = - \mathbf{w}_i^* \mathbf{B} \mathbf{w}_i - \mathbf{w}_j^* \mathbf{B} \mathbf{w}_j +
\left( \mathbf{w}_i^* \mathbf{B} \mathbf{w}_i \right) \left( \mathbf{w}_j^* \mathbf{B} \mathbf{w}_j \right)\,. \label{fii.eq}
\end{align}
\end{lemma}
\begin{IEEEproof}
Let $\textbf{M}_{ij} = \mathbf{m}_i \mathbf{m}_j^*$, $i,j\in \{2\ldots D\}$ be any set of (mutual) orthonormal eigenmatrices of $\mathcal A_0$ associated with eigenvalue $1$.
The vectors $\mathbf{m}_i$, $i=2 \ldots D$ are eigenvectors of $\tilde{\mathbf{A}}$ such that $\tilde{\mathbf{A}} \mathbf{m}_i = \mathbf{m}_i$; further, they are mutually orthonormal and are all orthogonal to the vector $\textbf{1}$.
We define a matrix $\mathbf{H}$ whose entries are defined as $h_{ij} = \mathbf{m}_i^*\mathbf{B}\mathbf{m}_j$. Let $\mathbf{U}$ be the matrix whose columns are $\mathbf{m}_i$, $i\in \{2\ldots D\}$. Then it is clear that $\mathbf{H} = \mathbf{U}^*\mathbf{B}\mathbf{U}$. Let $\mathbf{H} = \mathbf{S} \mathbf{\Lambda} \mathbf{S}^*$ be the spectral decomposition of $\mathbf{H}$. $\mathbf{S}$ is an unitary matrix, $\mathbf{s}_i$ is the $i$th column of $\mathbf{S}$. Therefore $\mathbf{B} = \mathbf{U}\mathbf{S}\mathbf{\Lambda}\mathbf{S}^*\mathbf{U}^*$. We define $\mathbf{w}_i := \mathbf{U}\mathbf{s}_i$, for all $i\in \{2\ldots D\}$. It is easy to verify that the vectors in $\{\mathbf{w}_i :i = 2,\ldots,D\}$ satisfy the properties (\ref{prop1})-(\ref{prop5}) stated in the lemma.
We note that by (\ref{prop2}) and (\ref{prop3}), ${{\langle \mathbf{W}_{ij}, \mathbf{W}_{ij}\rangle} = 1}$ for all $i,j\in \{2\ldots D\}$; ${{\langle \mathbf{W}_{ij}, \mathbf{W}_{pq}\rangle} = 0}$ for all $i\neq p$ or $j\neq q$. Therefore, we consider the entries of the $(D-1)^2\times (D-1)^2$ matrix $\mathbf{F}$:
\begin{align}
f_{ij,pq} &= \langle \mathbf{w}_i\mathbf{w}_j^*, \mathcal A_1(\mathbf{w}_p \mathbf{w}_q^*)\rangle \nonumber \\
&= \tr{\mathbf{w}_j\mathbf{w}_i^*\left(-\mathbf{B} \mathbf{w}_p \mathbf{w}_q^* \tilde{\mathbf{A}} - \tilde{\mathbf{A}} \mathbf{w}_p \mathbf{w}_q^* \mathbf{B} + \mathbf{B} \mathbf{w}_p \mathbf{w}_q^* \mathbf{B}\right)} \nonumber \\
&= -\tr{\mathbf{w}_j \mathbf{w}_i^*\mathbf{B} \mathbf{w}_p \mathbf{w}_q^*} - \tr{\mathbf{w}_j \mathbf{w}_i^*\mathbf{w}_p \mathbf{w}_q^* \mathbf{B}} \nonumber \\
&~~~~~+ \tr{\mathbf{w}_j \mathbf{w}_i^* \mathbf{B} \mathbf{w}_p \mathbf{w}_q^* \mathbf{B}} \label{f3.eq}
\end{align}
where the last equality holds since ${\tilde{\mathbf{A}} \mathbf{w}_p = \mathbf{w}_p}$ and similarly, ${\mathbf{w}_q^* \tilde{\mathbf{A}} = \mathbf{w}_q^*}$.
the expression can further be written as
\begin{align*}
f_{ij,pq} =& - \mathbf{w}_i^* \mathbf{B} \mathbf{w}_p \mathbf{w}_q^* \mathbf{w}_j - \mathbf{w}_i^* \mathbf{w}_p \mathbf{w}_q^* \mathbf{B} \mathbf{w}_j \nonumber\\
& +\left( \mathbf{w}_i^* \mathbf{B} \mathbf{w}_p \right) \left( \mathbf{w}_j^* \mathbf{B} \mathbf{w}_q \right).
\end{align*}
If $i=p$ and $j=q$, then noting that $\mathbf{w}_i^* \mathbf{w}_p = 1$ and $\mathbf{w}_j^*\mathbf{w}_q=1$, it follows that
\begin{align*}
f^{(1)}_{ij} := f_{ij,ij} = - \mathbf{w}_i^* \mathbf{B} \mathbf{w}_i - \mathbf{w}_j^* \mathbf{B} \mathbf{w}_j +
\left( \mathbf{w}_i^* \mathbf{B} \mathbf{w}_i \right) \left( \mathbf{w}_j^* \mathbf{B} \mathbf{w}_j \right).
\end{align*}
Furthermore, since $\mathbf{w}_i^*\mathbf{B}\mathbf{w}_j=0$ for any $i\neq j$, all off diagonal entries are zeros.
\end{IEEEproof}
We next use this lemma to characterize the convergence factor in two classes of NoNs.
\subsection{Analysis for Special Cases}
We give results for the mean square convergence rate for the two cases which we have discussed in Section~\ref{expected.sec}.
\begin{corollary}
\label{2subgraph.thm}
For an NoN consisting of two subgraphs $\mathcal G_1$ and $\mathcal G_2$, with the dynamics \yhy{(\ref{eq:3})} satisfying Assumption \ref{activeTogether.assum}, the mean square convergence rate, up to first order in $p$, is
\begin{align}
\rho(\mathcal A) = & 1- 2p\beta \left( \frac{N}{N_1N_2}\right) + p\beta^2 \left( \frac{N}{N_1N_2}\right)^2\,.
\label{rho2.eq}
\end{align}
\end{corollary}
\begin{IEEEproof}
We define the vector $\mathbf{w}_2$ as
\[
\mathbf{w}_2 =
\begin{bmatrix}
\theta^{(1)} \boldsymbol{1}_{N_1}\\
\theta^{(2)} \boldsymbol{1}_{N_2}\,.
\end{bmatrix}
\]
where $\theta^{(1)}= \sqrt{\frac{N_2}{N \cdot N_1}}$ and $\theta^{(2)}= -\sqrt{\frac{N_1}{N \cdot N_2}}$.
It is easily observed that $\mathbf{w}_2$ is an eigenvector of $\tilde{\mathbf{A}}$ with eigenvalue 1, and $\mathbf{w}_2$ is orthogonal to $\textbf{1}$.
When $D=2$, the matrix $\mathbf{F}$ consists of a single element. Applying the definition for $f^{(1)}_{22}$ in (\ref{fii.eq}), we obtain
\begin{align}\label{eq:f11}
f^{(1)}_{22} &=-2\beta(\theta^{(1)}-\theta^{(2)})^2
+ \beta^2 ( \theta^{(1)} - \theta^{(2)})^4\\
& = -2\beta\left(\frac{N}{N_1N_2}\right)
+ \beta^2\left(\frac{N}{N_1N_2} \right)^2\,.
\end{align}
This completes the proof.
\end{IEEEproof}
From (\ref{rho2.eq}) we observe that given $N$, the magnitude of $\rho(\mathcal A)$ is maximized when the graphs are of the same size, i.e., $N_1 = N_2$. It is minimized when $N_1=1$, $N_2=N-1$ or $N_2=1$, $N_1=N-1$. This means that the speed of convergence is slower between balanced subgraphs. By comparing (\ref{rho2.eq}) to (\ref{alpha2.eq}) we note that for two subgraphs, both $\rho_{ess}(\overline{\mathbf{A}})$ and $\rho(\mathcal A)$ are determined by the strength (activation probability) of the connecting edge and the number of nodes in both subgraphs.
\begin{corollary} \label{equal2.thm}
For an NoN consisting of $D$ subgraphs $\mathcal G_1, \ldots, \mathcal G_D$, each with $\frac{N}{D}$ nodes, with the system dynamics \yhy{(\ref{eq:3})} satisfying Assumption \ref{activeTogether.assum}, the mean square convergence factor, up to first order in $p$, is
\begin{align*}
\rho(\mathcal A) =& 1 -p \left(2 \beta \left(\frac{D}{N}\right) \lambda_2(\widehat{\mathbf{L}}_{con}) {-} \beta^2 \left(\frac{D}{N}\right)^2 (\lambda_2(\widehat{\mathbf{L}}_{con}))^2 \right) \,.
\end{align*}
where $\lambda_2(\Lc_{con})$ is the second smallest eigenvalue of $\Lc_{con}$.
\end{corollary}
\begin{IEEEproof}
We obtain this result by defining the $D-1$ eigenvectors of $\tilde{\mathbf{A}}$ with eigenvalue 1 as follows.
Let $\mathbf{u}_1, \ldots, \mathbf{u}_{D}$ be an orthonormal set of eigenvectors of the $D \times D$ matrix $\widehat{\mathbf{L}}_{con}$ with eigenvalues $0=\lambda_1(\widehat{\mathbf{L}}_{con}) \leq \ldots \leq \lambda_{D}(\widehat{\mathbf{L}}_{con})$.
Let ${\mathbf{u}_1 = (1/\sqrt{D}) \textbf{1}}$, and thus $\Lc_{con} \mathbf{u}_0 = 0$.
The $i^{th}$ eigenvector of $\tilde{\mathbf{A}}$, $i=2 \ldots D$, is
\[
\mathbf{w}_i = [ \theta_i^{(1)} \textbf{1}_{N_1}^{T}~~ \theta_i^{(2)} \textbf{1}_{N_2}^{T}~~~\ldots~~~\theta_i^{(D)} \textbf{1}_{N_D}^{T}]^{T}
\]
with
\begin{equation} \label{eq:thetaDef}
\theta_i^{(j)} = \frac{1}{\sqrt{N/D}} u_{ij}
\end{equation}
where $u_{ij}$ denotes the $j^{th}$ component of the eigenvector $\mathbf{u}_i$, $j=1 \ldots D$.
Therefore, the first perturbation term of the eigenvalue corresponds to eigenmatrix $\mathbf{W}_{ij}=\mathbf{w}_i\mathbf{w}_j^*$ are obtained:
\begin{align}
\label{fijequal.eq}
f_{ij}^{(1)} = &-\beta\left(\frac{D}{N}\right)\left(\lambda_i(\widehat{\mathbf{L}}_{con})+\lambda_j(\widehat{\mathbf{L}}_{con})\right) \nonumber\\
&+ \beta^2 \left(\frac{D}{N}\right)^2 \lambda_i(\widehat{\mathbf{L}}_{con})\lambda_j(\widehat{\mathbf{L}}_{con})\,.
\end{align}
By Lemma~\ref{rhoA.lem} and (\ref{fijequal.eq}), $\rho(A)$ is equal to
\begin{align}
\rho(\mathcal A)&=\max_{i,j \in \{2, \ldots, D\}} 1 -p \left(2 \beta \left(\frac{D}{N}\right) \left( \lambda_i(\widehat{\mathbf{L}}_{con}) + \lambda_j(\widehat{\mathbf{L}}_{con}) \right) \right.\nonumber\\
&~~~~~~~~~~~~~~~\left.
- \beta^2 \left(\frac{D}{N}\right) ^2 \lambda_i(\widehat{\mathbf{L}}_{con})\lambda_j(\widehat{\mathbf{L}}_{con}) \right). \label{maxequal.eq}
\end{align}
The maximum node degree of any node $v \in \mathcal V_{con}$ is $D-1$; thus, the eigenvalues of $\widehat{\mathbf{L}}_{con}$ are in the interval ${[0, 2\Delta]}$~\cite{merris1994laplacian}.
Since $\beta < \frac{1}{2\Delta}$, we have $\beta \lambda_j(\widehat{\mathbf{L}}_{con}) \in [0,1)$ for $j=2 \ldots D$. Further we attain that $2\beta(\frac{D}{N})-\beta^2(\frac{D}{N})^2\lambda_j(\widehat{\mathbf{L}}_{con})>0$ for $j=2\ldots D$. Thus, the right hand side of expression (\ref{maxequal.eq}) is maximized when $\lambda_i(\widehat{\mathbf{L}}_{con})$ is minimized. The same analysis holds for $\lambda_j(\widehat{\mathbf{L}}_{con})$. So the right hand side of expression (\ref{maxequal.eq}) is maximized when both $\lambda_i(\widehat{\mathbf{L}}_{con})$ and $\lambda_j(\widehat{\mathbf{L}}_{con})$ are equal to $\lambda_2(\widehat{\mathbf{L}}_{con})$, which proves the theorem.
\end{IEEEproof}
We observe from Corollary~\ref{equal2.thm} and Corollary~\ref{equal.thm} that for subgraphs with the same number of nodes, both $\rho_{ess}(\overline{\mathbf{A}})$ and $\rho(\mathcal A)$ are determined by the algebraic connectivity of the connecting graph as well as the number of nodes in each subgraph.
We note that the second-order perturbation analysis similar to Theorem \ref{bridgenode.thm} can also be applied to the analysis of mean-square convergence rate of (\ref{eq:4}) satisfying Assumption~\ref{activeTogether.assum}. We defer the related discussion to Appendix \ref{appendix2.sec}.
\section{Numerical Results} \label{results.sec}
In this section, we give some numerical examples to support our analytic results. Edges are weighted $1$ in these examples unless otherwise specified.
All experiments were done in MATLAB.
\begin{figure*}[htbp]
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=\linewidth]{diffusion_spt_subgraph_ER_eps_0_001.pdf}
\subcaption{$\epsilon = 0.001$}
\label{GD1_1}
\end{subfigure}%
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=\linewidth]{diffusion_spt_subgraph_ER_eps_0_010.pdf}
\caption{$\epsilon = 0.01$}
\label{GD1_4}
\end{subfigure}
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=\linewidth]{diffusion_spt_subgraph_ER_eps_0_100.pdf}
\caption{$\epsilon = 0.1$}
\label{GD1_6}
\end{subfigure}
\caption{Spectral gap of the supra-Laplacian matrix, Exact and predicated by perturbation analysis (SPA and SPA2), for composite graphs as the sizes of the individual graphs increase, for various $\epsilon$. The individual graphs are Erd\H{o}s R\'{e}nyi random graphs, where an edge exists between each pair of nodes with probability $0.6$, and the connecting graph $\mathcal G_{con}$ is a complete graph.}
\label{GD1}
\end{figure*}
\begin{figure*}[htbp]
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=\linewidth]{diffusion_spt_subgraph_path_eps_0_001.pdf}
\subcaption{$\epsilon = 0.001$}
\label{GD2_1}
\end{subfigure}%
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=\linewidth]{diffusion_spt_subgraph_path_eps_0_010.pdf}
\subcaption{$\epsilon = 0.01$}
\label{GD2_4}
\end{subfigure}
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=\linewidth]{diffusion_spt_subgraph_path_eps_0_100.pdf}
\subcaption{$\epsilon = 0.1$}
\label{GD2_6}
\end{subfigure}
\caption{Spectral gap of the supra-Laplacian matrix, Exact and evaluated by perturbation analysis (SPA and SPA2), for composite graphs as the sizes of the individual graphs increase, for various values of $\epsilon$. The individual graphs are path graphs, and the connecting graph $\mathcal G_{con}$ is a complete graph.}
\label{GD2}
\end{figure*}
First, we investigate the spectral gap of the supra-Laplacian matrix in the diffusion dynamics.
In Fig.~\ref{GD1}, we compare the spectral gap estimated by first order perturbation analysis (labeled `SPA') and second order perturbation analysis (labeled `SPA2') to the spectral gap directly computed using $\mathbf{L}$ (labeled `Exact') for various $\epsilon$. Each figure shows plots for different numbers of subgraphs, $D=2$, $D=4$, and $D=8$, as the sizes of the subgraphs increase. Each subgraph is an Erd\H{o}s R\'{e}nyi random graph with the probability of an edge existing between any two nodes equal to $0.6$. In each NoN, all subgraphs have the same number of nodes. The connecting graph $\mathcal G_{con}$ is a complete graph, and the connecting node is chosen uniformly at random in each subgraph.
As expected, the spectral gap decreases as the sizes of the individual subgraphs increase. Also, in general, we see the trend that when $\epsilon$ is held constant, with larger values of $D$, the spectral gap is higher. We explore this phenomenon further in subsequent experiments.
We observe that the spectral gap generated by first- and second-order perturbation analysis closely approximates the exact diffusion rate for $\epsilon = 0.001$ to $\epsilon=0.01$. This is in accordance with spectral perturbation theory. The result given by SPA diverges from the exact diffusion rate for a larger value $\epsilon = 0.1$. However, SPA2 still gives good approximation for the spectral gap when $\epsilon = 0.1$
In Fig.~\ref{GD2}, we show results using the same network scenarios as in Fig.~\ref{GD1}, with the exception that the connecting graphs $\mathcal G_{con}$ are path graphs. To make the experiment homogeneous, the connecting nodes are selected as end nodes of each path graph.
Again, we note the spectral gap decreases as the size of individual subgraphs increase for all $\epsilon$. The results of SPA and SPA2 closely approximate the exact spectral gap for $\epsilon=0.001$. The result of SPA2 still well approximates the spectral gap for $\epsilon=0.01$, though with less accuracy than in Fig.~\ref{GD1}.
Both SPA and SPA2 fail to closely approximate the spectral gap for $\epsilon=0.1$. Thus, we observe that the accuracy of the spectral perturbation analysis depends on the network topology. For each topology, there is some threshold for which, when $\epsilon$ is smaller than this threshold, the approximations are accurate. However, this threshold is different for different NoN topologies.
We also note that, in comparing Fig.~\ref{GD1} and Fig.~\ref{GD2}, it can be observed that the diffusion rate given by SPA coincide for networks of the same size.
This conforms with our analysis that the first-order approximation of convergence factor of the NoN obtained from spectral perturbation analysis only depends on the sizes of the subgraphs and not on their individual topologies.
In Fig. \ref{G3} we study the dependency of the spectral gap on the topology of $\mathcal G_{con}$ as the number of subgraphs varies. Each subgraph is an Erd\H{o}s R\'{e}nyi random graph with edge probability $0.6$. All subgraphs have $10$ nodes.
We let $\epsilon= 0.01$, and we compute the convergence factors when $\mathcal G_{con}$ is complete and when $\mathcal G_{con}$ is a ring.
\begin{figure}[htbp]
\centering
\includegraphics[width=.9\linewidth]{convergence_vs_D.pdf}
\caption{Spectral gap for Exact, SPA, and SPA2, with increasing NoN sizes for ring and complete $\mathcal G_{con}$ topologies. Subgraphs graphs are Erd\H{o}s R\'{e}nyi random graphs each with 10 nodes. $\epsilon$ is set to $0.01$.}
\label{G3}
\end{figure}
We observe that, when $\mathcal G_{con}$ is complete, the spectral gap of $\mathbf{L}$, in Exact, SPA , and SPA2, increases with the increase in the number of subgraphs. To better understand this phenomenon, let us assume all subgraphs are of the same size $\Phi$. For $\mathcal G_{con}$ a complete graph, $\Lc_{con}$ has one eigenvalue of $0$ and $D-1$ eigenvalues equal to $D$. For the SPA diffusion rate given in Theorem~\ref{specGap.thm}, we know that up to first order in $\epsilon$,
\begin{align}
\rho(\mathcal A) & = \epsilon \left(\frac{D}{N}\right) \lambda_2(\widehat{\mathbf{L}}_{con}) = \epsilon \left(\frac{D}{\Phi}\right).
\end{align}
Since $\Phi$ and $\epsilon$ are held constant, with the increase in $D$, the diffusion rate increases.
We also note that the diffusion rate when $\mathcal G_{con}$ is a ring graph is smaller than the diffusion rate when $\mathcal G_{con}$ is a complete graph. This can be explained in part by the fact that the algebraic connectivity of a ring graph decreases as its number of nodes increases.
In the following two examples we show that one can use the second order perturbation analysis as a heuristic for choosing connecting nodes to optimize the diffusion rate of the studied diffusion dynamics and the mean square convergence rate of the consensus dynamics.
\begin{figure}[htbp]
\centering
\includegraphics[width=.9\linewidth]{optimalLeader_beta5.pdf}
\caption{Diffusion rates for the system consists of two subgraphs connected by an edge with bridge nodes selected by different strategy. Subgraphs graphs are Erd\H{o}s R\'{e}nyi random graphs, where an edge exists between each pair of nodes with probability $0.2$. The diffusion coefficient $\epsilon$ is set to $0.1$.}
\label{G4}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=.9\linewidth]{optimalLeader_beta21.pdf}
\caption{Mean square convergence rates for the system consists of two subgraphs connected by an edge with bridge nodes selected by different strategy. Subgraphs graphs are Erd\H{o}s R\'{e}nyi random graphs, where an edge exists between each pair of nodes with probability $0.2$. The activation probability of edges in $\mathcal E_{con}$ is $p = 0.1$. $\beta$ takes the value of $\frac{1}{21}$.}
\label{G5}
\end{figure}
In Fig.~\ref{G4} and Fig.~\ref{G5} we show the exact diffusion rates (given by $\alpha(\mathbf{L})$) and the mean square convergence rates (given by $\rho(\mathcal A)$) of systems with different connecting nodes. In both examples we have two Erd\H{o}s R\'{e}nyi random subgraphs connected by a single edge. The probability that two nodes in the same subgraph are connected is set to $0.2$. And both subgraphs are connected. For the diffusion dynamics, we set $\epsilon = 0.1$. For the consensus dynamics, we let $p=0.1$ and $\beta = \frac{1}{21}$, and $w=\frac{1}{21}$ for all edges in both subgraphs. In the proposed heuristic, we choose the bridge nodes as the ones with maximum information centrality in each subgraph. We compare the results with the true optimum given by brute-force search, as well as the result of a random choice. The results show that our strategy hits optimal solutions in all occasions, and evidently outperforms the random strategy. We have shown in Theorem \ref{bridgenode.thm} that the second-order approximation of spectral radius of the supra-Laplacian is maximized when connecting nodes are chosen as the ones with largest information centrality. In Fig.~\ref{G4}, we show that by using this result we actually obtain an optimal connecting node in each subgraphs. In Fig.~\ref{G5}, we empirically show that this approach can also be used as a heuristic to find connecting nodes that lead to a good mean square convergence rate.
\section{Conclusion} \label{conclusion.sec}
We have investigated the rate of diffusion in a Network of Networks model, as well as the convergence rate in a consensus NoN with a stochastically switching connecting graph.
Using spectral perturbation analysis, we studied the diffusion rate in a NoN. We showed that the first-order perturbation term is determined by the spectral gap of the generalized Laplacian matrix of the connecting network. In addition, using second-order perturbation analysis, we showed the connection between information centrality and the optimal connecting nodes in subgraphs.
Finally, we presented numerical results to substantiate our analysis. In future work, we plan to extend our analysis to NoNs with more complex dynamics.
\bibliographystyle{IEEEtran}
|
2,869,038,156,382 | arxiv | \section{Introduction: The neuron as a system with internal and external parameters}
Experimental research on neurons, and as a consequence theoretical
analysis, is often divided into electrophysiology and molecular biology.
The first deals with membrane potentials, spikes and activations in
networks of neurons linked via synapses, the second deals with
intracellular signaling, genetic networks, release of neuromodulators,
receptors, ion channels, transporters and various processes linked to this. In both
cases we are aware of signal-induced plasticity, but an understanding of
neural plasticity concurrently on the levels of electrophysiology in
neural networks (horizontal plane) and the molecular interactions in
single neurons (vertical interactions) is missing.
The master framework linking observations on neural plasticity is
focused on associative synaptic plasticity, usually in the form of
long-term potentiation or depression \cite{MalenkaBear2004,CitriMalenka2008,KullmannLamsa2007,RemySpruston2007,LeiSetal2003,ZhongWetal2006}.
Interesting and relevant criticisms and alternative suggestions are contained in \cite{Arshavsky2021,Arshavsky2022,LangilleGallistel2020,TrettenbreinPC2016,Abrahametal2019,GallistelMatzel2013}.
Here we analyze experimental results from a primarily agnostic perspective, with
the goal of finding an adequate functional description and asking for the utility of an appropriately designed neuron model. Already, complex dynamic models of
intracellular signaling \cite{Baudotetal2008,HaneySetal2010,Crampinetal2004} and genetic read-out \cite{Ciliberti2007,LeePetal2017,Baudry2015}
exist as well as elaborate simulation models \cite{Scheler2013PLOSONE}.
However, simulation models attempt to match actual biological processes precisely,
accordingly they model only very small parts of systems in an attempt to
model them with high accuracy. But here, we are not interested in that.
Instead, we want to raise the question how the neuron is
organized as a complex system, how it processes and stores information, and how this applies to computational models.
In a first approach, we suggest to model membrane properties by a set of
external parameters, localized at the synapse, the
spine (where it exists), the soma, the dendrite or the axonal bouton. External parameters respond to signals from the outside, but they
are also guided and controlled by submembrane processes, which we describe by internal
parameters. While external parameters influence neural transmission properties which are visible to other neurons, internal parameters only have indirect
effects on neural transmission and remain hidden from other neurons.
Together external and internal parameters form the `processing layer' or the upper part of the 'vertical' system of neural plasticity. In contrast, the nucleus contains core parameters, both epigenetic (histones) and genetic (DNA) which respond to internal signaling. This provides another layer of depth for processing information and long-term memory. Core parameters
receive and send information to the internal parameters via the nuclear membrane. In this paper, they will not be analysed in detail.
Neural networks were developed to derive functional models from ideas about neural
interactions. In contrast, for internal cellular models, such as biochemical reaction systems and genetic transcription
networks, dynamical systems approaches
are often used. While neural networks are suitable for computation, but highly
inaccurate for detailed biological simulation, dynamical systems models are unwieldy for
computation, subject to dependence on small parameter
changes and only suitable for small-scale simulation models. Without
further elaboration \cite{Covert2008}
these models are also highly incompatible. They do not focus on the task of information processing that we want to analyze. Therefore we tentatively describe the systems of submembrane and cytoplasmic intracellular signaling \cite{BhallaIyengar1999} and their properties by amorphous internal parameters, to be elaborated later \cite{Jarvis2021}.
Our goal must be to isolate information processing from complex cellular
computation, since the neuronal cell performs this in the wider context of all
cell-internal mechanisms. Therefore we suggest to operate with the very
general abstraction of high-level functional parameters instead. This
could be augmented at later times with more precisely defined types of
parameters or interactions on the basis of mathematical models. Our
first goal in this paper will be to investigate what insights we can
gain from this point of view.
External parameters, roughly corresponding to membrane proteins, respond to outside signals undergo plasticity on
at least three time-scales: fast (milliseconds, desensitization), intermediate (seconds-minutes, endocytosis,
insertion), and slow (hours, new proteins or morphological features
like spines). Only the fast responses which usually involve protein conformational changes do not require the participation of the internal, vertical processing system. Both the reversible endocytosis of membrane proteins, and the long-term ubiquination of proteins or generation of new proteins require a complex system of internal interactions.
The set of internal parameters corresponding to intracellular signaling is the first line in orchestrating the plasticity of external parameters. This imposes onto the response to outside signals an internal generative model of membrane
properties \cite{ParkAetalJosselyn2020}. This is a very important point. It means that within a neural ensemble, synaptic and membrane plasticity is not only and directly dependent on the signals that are passed among the neurons. Instead the activity of the intracellular protein signaling system is necessary for selecting and responding to signals \cite{ParkAetalJosselyn2020,Alejandre-Garcia2022}. The behavior of the neuron cannot be determined from observation of the signals it receives.
Rather, adaptation is a combination of the stored model and new information.
Membrane-near control structures such as spines contain hundreds
of protein species which interact to orchestrate membrane protein
position and efficacy \cite{Donlin2021,KasaiHetal2021}.
The intracellular system contains spatially distributed parts in distinct
positions (e.g. at dendritic spines, axonal boutons) and a central 'workspace' in the
form of the cytoplasm with its organelles and protein complexes.
The membrane with its many inserted proteins, such
as ion channels, receptors, transporters etc. influences neural transmission, i.e. the horizontal
interactions between neurons. It is compartmentalized at dendrites into branches/spines/synapses, and at axons into branches and axonal boutons. Outside signals are therefore received
on a highly structured surface, such that, although neighboring
interactions exist, signals can be located by origin, and are processed
in a spatially highly distributed way. Synaptic signals received at the membrane are brief (milliseconds),
and potentially repeating. Synaptic signals
typically produce calcium transients from NMDA receptors and voltage-gated calcium channels (VGCCs), as well as calcium buffering proteins and calcium release from internal stores \cite{HumeauLuthi2007,Gulledgeetal2005}. There is evidence that calcium is a major signal for the induction of plasticity and that different shapes of the signal result in different outcomes \cite{Markram}. There are also signals, like neuromodulators, which have longer durations,
on the order of seconds, or even minutes. The short-term external parameter response is governed by outside signals, such as receptor ligands, but longer duration responses require the selection and filtering of
membrane signals by the internal system.
The innermost module is the nucleus, which has DNA as permanent
storage for protein information, and an epigenetic histone system
which codes for the accessibility of DNA \cite{JosselynTonegawa2020}. Signals are accumulated and integrated, temporally and spatially before being sent to the central core. Phosphorylations or methylations which are performed by proteins in a reversible manner code
for easy access to genes. Signals pass from the cytoplasm to the nucleus
usually in the form of proteins which are transcription factors
activating coordinating sets of genes. Thus whole genetic programs can be
started such as morphological growth of spines, axons, or dendrites.
Individual genes (e.g. AMPA receptors) may also be transcribed during periods of neural
plasticity based on histone accessibility \cite{JosselynTonegawa2020}.
One task of the processing layer is to take a spatially distributed,
temporally structured signal and produce a spatially centralized,
temporally integrated signal of transcription factors to cause DNA read-out of
individual genes or genetic programs. This is not a simple task. Signals
are selected near the membrane, they are shaped and re-structured in
time, and they need to be sorted or transferred to protein complexes in
the cytoplasm and ultimately activate proteins which enter into the
nucleus. In the nucleus, epigenetic adaptation regulates access to DNA and
also serves the function of considerable noise reduction, channeling
signals by enhancement and suppression. It is clear then that signals
from the periphery have to be processed on several levels, and with
several types of control structures before they can actually be used to
control or regulate DNA read-out, or be stored in the epigenetic layer
around it.
The other task of the internal parameter system in the membrane-near processing layer is to enact
short-term plasticity and directly shape external membrane expression on
the basis of existing information (i.e. a generative model). Signals from the environment occur
fast and they are ordered in time. When they arrive at the membrane, they are being transformed at the membrane layer
through internal parameters in feedback cycles and similar control structures. These can
suppress, lengthen, or augment a signal by engaging the
intracellular signaling network.The result is new external membrane expression as a combination of the stored generative model for external parameter response
and incoming, possibly adaptive information.
In this context, it is interesting that the intracellular system itself is devoid of memory, and cannot respond to signals by adaptive plasticity or
regulate its own plasticity \cite{Deng2011,Shinar2010, vonBertalanffy1968}. There is temporal integration for membrane
signals, and temporally orchestrated intracellular responses, such as G protein- mediated signaling by small molecules \cite{}
or the kinase/phosphatase system \cite{Hunter1995}. But the system properties can only be changed by
altering the abundances of proteins via mRNA translation, which takes hours to be reset
and recalculated. This observation strengthens the view of the internal parameter system as a processing layer, programmable by signals
from the outside and updated from inside via parameter re-sets such as local mRNA translation and the core DNA transcription system.
To summarize, the membrane layer is adaptive on
several time scales, in the long term by genetic read-out. Under certain
conditions, protein abundances in the
processing layer or at the membrane may change, adapting the neuron to a
new environment.
This shows that associative synaptic plasticity, such as long-term potentiation / depression (LTP/LTD) can be seen as a
special case of signal-induced long-term membrane plasticity.
\section{Signal Selection and Filtering}
In our conceptual model, parameters stand for proteins/molecules, sometimes for
groups of them, or variants of the same molecule. (A precise matching of
parameters to biochemistry is not intended at this point, but this is a
step that can always be added.) The values for these parameters -- both
external, at the membrane, and internal, near the membrane, or localized
somewhere in the cytosol -- change by computations or by outside
signals. Outside signals are first sensed by external parameters, they are
processed with the help of internal parameters, i.e. filtered or selected, stored or dropped.
In response to signals, external parameters may
return to their values (homeostatic regulation) or they may be re-set to new values (adaptive regulation).
Outside signals are at first pre-processed by external parameters (receptors and indirectly ion channels) at the membrane. Immediate, reversible plasticity of receptors and ligand-gated ion channels usually by protein conformational changes constitutes the first response. Calcium and small molecules, like cAMP, as well as the kinase/phosphatase system engage the internal parameter system. Control
structures and computations by internal parameters decide about the
fate of signals (cf. Fig. \ref{fig:internal-filter}).
\begin{figure}[htb]
\includegraphics[width=0.8\textwidth]{internal-filter}
\caption{Filtering and selection of signals at the membrane. External signals are created from outside signals which are interpreted by external parameters. If they pass the "filter", they are transformed by the internal parameter set into internal signals.}
\label{fig:internal-filter}
\end{figure}
The membrane has a strong tendency towards homeostasis, i.e. return to
existing values, after short-term disruption. But in some situations, the
accumulation of traces from each signaling event is predominant. Whether a signal is eliminated and ignored, or passed on to submembrane zones and
intracellular protein signaling, does not just depend on the signal,
but also on the responsivity of the neuron. From a theoretical perspective we are dealing with a problem of feature (=signal) selection and feature construction for learning and memory \cite{ElisseefGuyon2003}. The signal may have a
higher chance if it is either strong (high amplitude, co-occurring at
various sites), or repeating (at the same sites), or long-lasting (even
at a lower level of signaling). In general we may assume, (a) that even fairly weak signals can be retained,
if signals combine within a specified time frame, and if the neuron is
highly responsive and (b) even strong signals fail to register, if the
neuron has low responsivity, and if the signal is isolated. These questions have to be explored further from the perspective of utility (which feature selection processes are most useful) and experimentally (effect of responsivity, along the lines of \cite{ZhouY2009,ParkAetalJosselyn2020,DehorterN2015}).
There are several models possible for the integration of the external with the internal parameter system. For instance, we may assume that external parameters fluctuate, they follow a random walk and are reinforced by internal parameters. Such a model could emulate modern feature selection methods \cite{ElisseefGuyon2003} very effectively.
Another approach is that external parameters are corrected by values from the internal parameters, which would amount to supervised learning, or that the internal parameter system only selectively augments the external parameters by stored default values.
Calcium modulation has often been indicated as a major factor in initiating plasticity (ref). The shape of the calcium transients, where strong high-amplitude calcium transients directly permeate into intracellular signaling to begin to
make changes \cite{Dolmetsch2003,Mellstrom2008,FujiiBito2022} matters, indicating a preference for brief, strong signals. Calcium enters the cell via various ion channels and receptors, there are intracellular calcium buffers, and mechanisms for release from intracellular stores, such that the factors influencing calcium transients are an example for the integration of outside signals and internal parameters.
Another factor influencing signal selection based on external-internal integration is neuromodulation (NM) via G-protein coupled receptors (GPCRs) \cite{ShenoyLefkowitz2011}. G proteins are internally connected to ion channels which they regulate. The NM-ion channel complex determines the excitability of the neuron conditional on which (central) NM signal modulates it. NM-ion channel interactions can do reversible dendrite remodeling at intermediate time frames.
This means that the different NMs highlight the set of neurons that are modulated by them \cite{Scheler2003d}.
In this regard we may hypothesize that low-frequency neurons ('naive neurons') are dominated by a single NM, and only few, high-frequency 'hub' neurons have acquired noticeable responses to several NMs, imposing an activation scheme onto a neuronal ensemble, where hubs are preferentially modulated.
Outside signals to a neuron have the potential to activate certain proteins as immediate early genes (e.g. Arc, CREB) within minutes (5-20 minutes for Arc in hippocampal neurons \cite{Guzowski1999,Guzowski2001}), which transfer to the nucleus \cite{RieneckerKetal2022,Minatohara2016}. However, they are transferred by intracellular signaling, with a large number of control structures such as thresholds, feedback loops, feedforward loops, antagonistic signaling \cite{Scheler2013}.
For instance, negative feedback serves to broaden a signal and extend its duration, it may
also dampen a signal below threshold. Signals may be modified, such that later signals in a sequence are being
suppressed, or a sequence of signals is enhanced to reach a threshold for the processing layer \cite{Mausetal2020}.
In this sense, internal control structures serve to adapt a signal and influence whether it is passed on to the next layer.
The point is that these control structures are activated by signals but depend on internal parameters.
They may also depend on core parameters or genetic properties. An example is the co-regulation of ion channels \cite{Hudson2010c,OLeary2014,Abdelrahman2021}. Co-regulation, which requires genetic read-out, shows strong internal bias (genetic conservatism for a neuronal type) and does not require network-based regulation \cite{Golowasch2014,Drion2015}. The neuron itself "knows" its well-balanced state. This makes construction of horizontal models much easier, since each processor element monitors its own stability.
There is a notable difference between spiny and aspiny plasticity, spines being localized at projection neurons in areas of strong adult plasticity, such as pyramidal neurons in cortex, hippocampus and amygdala, medium spiny neurons in striatum and in certain dendritic regions of Purkinje cells in cerebellum - areas of high fine-tuned plasticity. Interestingly, dendritic spines (and possibly axonal boutons) act as cellular compartments with a highly reduced intracellular signaling apparatus\cite{NimchinskyEetal2002}, showing the value of mechanistic reduction in these cases. In contrast, aspiny neurons have synapses without a dedicated local system of intracellular signalling.
This can make a big difference for plasticity where spine removal and generation is an efficient
modification of existing information. Aspiny plasticity would then require
more substantial remodeling of global, rather than compartmentalised neuronal parameters.
In contrast to this exposition, in the classical associative synaptic model of plasticity the synapse signals to the core and the core signals
back to the relevant synapse. Therefore a synapse needs a tag to be identified. This "synaptic tag" hypothesis has been much discussed, but turned out not to be experimentally verifiable
\cite {Rogersonetal2014, RedondoMorris2011, LiQetal2014}. Instead we suggest that expression at the core affects the whole neuron, where localized plasticity can be handled by the combination of a global bias term from the core and local parameter setting. Since there is
local plasticity (usually fluctuating) which is direct and immediate at
each synaptic site (postsynaptic or presynaptic),
neural connections can be differentially regulated
in spite of the global bias term. The local sites can signal to
the global core. The central, global site signals back to all sites. There is spatial and temporal integration of signals happening in the intermediate layer and being present at the
core.
\section{Vertical Computation}
\label{vertical}
In this section we describe a number of adaptive processes that are being performed by vertical computation.
Our model of horizontal-vertical computation consists of spiking activity distributed over a group of connected
neuron (spatio-temporal patterns),
which are reflected as signals at a neuron or neuronal site (like a spine). In the model, the processing of the signals is determined by
external, internal and core parameters.
If a signal produces a match with the dendritic structure of the neuron, the neuron spikes. Neuronal spiking transmits a signal to all other neurons which are connected by synapses. However, transmission of
signals by itself is not sufficient to result in adaptation or parameter change.
It may lead to adaptation if calcium transients and other internal parameters fit.
What is important, is that each neuron represents a specially designed
processor, which will process information according to its own program. Such a program can be generic for a neuronal type ("naive" neuron) or it can be acquired and specific ("mature" neuron). It can be re-programmed by the complex control structures of protein signaling and in this way generate an new, adapted internal model.
Neurons have been set up by evolution with their own
genetic footprint, and it is important to realize that any neural computation is performed by a set of heterogeneous neurons which are adaptive but retain their cellular identity.
Evolution has done the `main work' in establishing
neuronal types. For instance, there are pyramidal neurons, a type of excitatory projection neuron, and then there are specifically cortical vs. hippocampal pyramidal neurons, and for cortical neurons there are surface-layer vs. deep-layer
pyramidal neurons, and as we increasingly learn, there are even a number
of genetically distinct deep-layer cortical pyramidal neurons \cite{ChangLuebke2007}, etc. For a model, the evolutionary processes do not need to be re-created directly (even though that could be very interesting).
Instead, neuronal types could be set up from a set of pre-established neuron models. Each of these neurons then has a genetic make-up and provides a type-specific
programmable unit, where programming serves to individualize the response.
There are a number of mechanisms which fulfill the idea of a programmable internal system. Internal signaling can cause activation or de-activation of a dendritic branch, it can reset presynaptic vesicle pools (e.g. by cAMP-dependence), it guides spine maturation and spine decay, and it performs spatial integration on the spine (e.g. AMPA reservoirs on the spine shaft moving to the spine synapse \cite{ArakiYHuganir2010}).
There are local buffers (cAMKII), and in the cytoplasm, there are timed delays, queues, feedback
cycles which suppress a signal or enhance it, there is spatial integration over
several compartments (\cite{ShenoyLefkowitz2011}), antagonistic signaling \cite{Scheler2013} and overflow switches \cite{Schelertransactivation}.
The intracellular signaling system passes a signal to outputs in the membrane or actuators in the nucleus after performing computation. The cytoplasm acts as a shared access system for the local compartments and protein complexes, routing inputs from many dendritic compartments to a single nuclear compartment. (The axonal system is not further described here). The output of processing guides signals from a spatially distributed system to a queued system or feeds it back to the membrane.
Here we offer a list of examples for programmable computations in neurons that have been investigated before:
\begin{enumerate}
\item
In \cite{Schelertransactivation}, we showed how a strong signal at a G-protein coupled receptor together with high protein expression at a target pathway (which is not usually involved in signaling by this receptor) results in activation of the target, a different pathway from the canonical one ('transactivation'). The result is signaling by a pathway with several new effectors which happens only under special conditions. We called this an {\it overflow switch}, as a motif in computation, emphasizing the mechanical control aspect of internal signalling.
\item
A common, ubiquitous motif in protein interactions is {\it antagonistic signaling}. This happens when an input is linked to an output by two antagonistic pathways (positive and negative) \cite{Scheler2013},
such as D1/D2-type dopaminergic signaling, or beta/alpha-adrenergic signaling. It allows re-scaling of inputs which vary over a large scale, compressing multiple orders of magnitude to a single-scale value. Also, antagonistic signaling allows for peak transients when one pathway is slightly faster than the other. These features are useful basic properties for any system that handles signals.
\item
Molecular abundances are important not just for establishing (high expression) or removing (low expression) connections between pathways as in \cite{Schelertransactivation} , they also dictate the {\it temporal execution} of the biochemical reactions underlying intracellular signaling \cite{SchelerPosterTimeF1000}. High expression means that reactions are fast and homeostatic endpoints are reached in short times. When protein expression is lower overall, internal reactions are slowed down and signals are processed sluggishly. In principle, this allows neurons to act at different time scales. This could be especially useful in situations where neural transmission (fast, high protein expression) is coupled with plasticity (slow, low protein expression).
\item
To analyze biochemical reaction networks, we employed a {\it transfer function} approximation for G protein coupled receptor (GPCR) regulation (such as for mGLU, GABA-B, monoamines, neuropeptides), in
\cite{Scheler2013PLOSONE}. Transfer functions allow to generate essentially (context-dependent) look-up tables for endpoints of internal parameters after external signaling. This is the basis for machine learning of internal parameters. It provides a suitable basis for internal computations and new values based on changes in protein expression.
\item
Using the many types of buffers available, we can achieve {\it re-sorting of signals} in time. During buffering,
spatial and temporal integration happen together.
A typical example is the protein CaMKII, which buffers calcium by
autocatalytic activity, and which releases calcium after some time --
depending on the activation status of each buffer molecule. This allows
to create a reverse calcium sequence, relative to the location received within the cytoplasm, and therefore provides a basis for a queued access to nuclear transcription.
\item
The sequence reversal motif, i.e. the reordering of signals in
time (with the help of buffers, which store a signal and release it
later, vs. later signals which pass through immediately), together with
other motifs, allows to produce {\it canonical shapes}, potentially
making it easier for other calcium-receptive pathways to process the
signal.
A sequence of synaptic inputs, which are sorted and combined from their
spatial distribution into larger components (e.g. like a bit code which is
translated into `words'), could provide a shape of signal that can be easily read.
\item There is mRNA-mediated plasticity at the spine, which
remains localized \cite{StewardO2003} or spreads {\it locally} along the dendrite. But if sufficient signals combine and the nucleus is involved,
we assume that neural remodeling affects the membrane {\it globally}, not
just an activated synapse. Instead of tags at potentiated synapses
which direct nuclear transcription products to those synapses, a simpler mechanism may be in place, a global effect or bias term that results from nuclear read-out and is combined with fast-fluctuating local plasticity. Then synaptic structure is a combination of a global (nucleus-based) and a local (spine, synapse) term.
\end{enumerate}
To summarize, outside signals are received by external parameters and activate a program on the internal system which determines the regulation for external membrane properties. The internal computational system responds to signals and produces outcomes \cite{Scheler2005}.
The system of computation, that filters and temporally re-sorts signals, that combines signals, and releases them is a programmable and re-programmable system. The system contains parameters which program its functionality. It has no memory but it can be reset by the core system. To re-program the system requires new transcription or mRNA translation.
To investigate the power of such systems, explicit programming, agent-based, dynamical systems as well as machine learning techniques are suitable.
\section{Combining Vertical and Horizontal Functions}
\label{both}
The signal that the individual neuron receives is defined by the pattern, i.e. the spatio-temporal spiking distribution
on the horizontal layer \cite{Schelerbiorxiv2019}. We can predict the signal at each neuron from the pattern, but the patterns are undergoing transformation by the neurons which carry the pattern (representational drift, \cite{MauW2020,Kappeletal2017}.
To simplify discussions of horizontal interactions, we assume neurons interact solely by spiking, which means neurons tell each other just how active they are by their spiking patterns.
Neural transmission, the
horizontal interaction between neurons, is then explicitly determined by external membrane parameters. They can be local to the synapse (e.g. AMPA, NMDA receptors), affect a dendrite (e.g. ion channels), or be restricted to neuromodulatory activation (e.g. different kinds of GPCRs), or other factors. These spiking patterns appear as spatio-temporal patterns (STPs) over a group of neurons.
For each neuron there is a set of external and internal parameters that defines the state of the neuron. External parameters adapt to and are adapted by outside signals and receive feedback through the horizontal layer. Patterns are generated by structured groups of neurons, in particular microcircuits (ensembles), which are made up of similar neural types, or neural loops which run through several different brain areas, and are composed of different neural types.
It is an interesting hypothesis rarely investigated that for the functioning of the neuron, evolution has developed neurons to be as independent as possible from outside signals. This is a question of efficiency of computation, where communication slows computation.
We also assume an internal generative model which defines a default set for the membrane parameters undergoing constant fluctuations. Some reflection of this lies in the high stochasticity of the neuron \cite{Kappeletal2017, Kasthuri2015, MongilloGLoewenstein2017}. Possibly, the neuron is mostly independent and stable as a processor, and the capacity for outside signals to create significant change - beyond the random walk fluctuations at the membrane \cite{Triesch2017} - is quite limited. It is definitely not automatic with each spike or transmission event.
In this sense then the neuron is autonomous, even though outside signals can re-program its state.
Horizontal interactions can be seen as a system with massively
parallel search for matches of signals to neurons. If there
is a match, the neuron's spiking activity changes (in most cases it increases), a subgraph of connected neurons is activated, and new patterns emerge. Since STPs rely on membrane (synaptic and non-synaptic) properties of neurons which are subject to parameter adaptation, STPs undergo constant transformation.
Thus a pattern is transformed via the signals it produces at individual neurons. While patterns are formed, new perceptual input may be suppressed.
Local neural adaptations - for instance by internal parameters - allow to adjust the
horizontal pattern. Possibly as a result the pattern better matches to existing stored templates. In this way a pattern could become adapted to existing knowledge.
We assume that internal memorization may be coordinated across a neuronal group, an ensemble. We use "strain" for an objective function which is related both to high spiking activity and to synchronization/de-synchronization. To compute strain, one could use a normalized value to measure excitation (taking peaks and temporal spacing (``bursting'') into account) and a measure for synchronization (by co-occurrence of spikes, or membrane potential fluctuations). Only under high strain may we assume vertical computation to be initiated. The objective function builds up in the
horizontal plane and gets resolved by adjusting parameters in vertical computation. Now parameter
adaptations (via vertical internalization/externalization) allow a return to a homeostatically well-adjusted horizontal spatio-temporal pattern regime of low strain. Within a horizontal structure focal areas of high activation, such as resulting from
sensory input, may be described as experiencing high "strain".
This indicates that signal information is to
be internalized, or that there is an opportunity to externalize information
to reduce strain. A neuron can
``absorb'' strain from the representation.
High strain may cause a network to store information internally and
remove it from the external membrane, or vice versa to read out
parameters so as to reduce activations.
In the short term, control structures using internal parameters in the membrane-near intracellular layer influence external membrane properties directly and immediately. Spatial resolution is kept.
This is fast adaptation of neural membrane properties.
Additionally, membrane-near control structures select strong signals, which are internalized and
maintained (e.g. with the help of calcium buffers)
without additional effects on neural transmission.
When the signal reaches cytoplasmic protein
complexes and the nucleus, spatial resolution is lost and global neural remodelling occurs. This is lasting plasticity.
\comment {Excitation in a neuron produces enhanced spikes, which is relevant for the function of the horizontal layer.
For a non-matching signal, no additional spikes result and internal suppression will stop the outside signal
from passing into the processing layer. In some cases
strong repeating (non-matching) signals may
generate internal signals and re-program a neuron.
In many neural areas, e.g. hippocampus, cortex, we see widespread electrophysiological activation of 30-50\% of neurons in response to a significant behavioral event, like a traumatic fear experience. We also often see activation of IEGs (e.g. Arc) at a similar density. While this may indicate that we have large-scale distributed plasticity, affecting a large proportion of neurons, it is also compatible with more localized plasticity, such as described in \cite{Schelerbiorxiv2019}. Distributed vs. sparse activations is a matter of brain area, or neural tissue type. First of all, most neural plasticity events will result from less significant behavioral situations than fear conditioning, and probably cause much less activation, and secondly, lasting plasticity requires more than transient activation of IEGs, which is not a reliable indicator of neural plasticity \cite{RobbinsMJetal2008,BruinsSlotLAetal2009}. }
We may distinguish between the responses of naive and mature neurons, or neurons with default control structures vs. neurons with a learned internal parameter structure. This is a concept that needs some explanation. Naive neurons exist with default configurations, possibly with random elements, and patterned exposure to signals makes them acquire a specific generative internal-external model which reflects the neuron's experience and expectation. This is a 'mature' neuron, or, rather, it is a process of maturation. Such a neuron may still learn and re-learn, and possibly even re-set to the naive state under exceptional circumstances. An outside ``raw'' signal has a different effect for a naive neuron - where it is imprinted - or a mature neuron where it is matched or non-matched. In the naive neuron there is no individualized control structure to filter the
signal. Internalization involves setting-up a copy of the external parameter set as the basis of a generative model or a "prior" for later processing \cite{Schelerbiorxiv2019}.
In a mature neuron, the signal is filtered and matched to the existing parameter set. A naive site will store or read in the next value, a mature site already contains a value and is protected against overwrite. However, maturity could be a graded property, and
a neuron could be 'chimeric' with respect to maturity. For instance spines could be either naive or mature on the same neuron,and a dendrite `bit-coded' for grade of maturity.
Neuromodulation has an interesting role in that it codes for cellular
identity in a combinatorial fashion \cite{Scheler2004Prog}. It can thus serve as labeling, both
for neurons and connections (nodes and edges of a graph
structure). Such labeling is extremely useful because
it allows for constrained inferences \cite{Scheler2003d}.
For instance, a dopamine D2 receptor may adjust
GIRK channels and subsequently the spike pattern reflects hyperpolarized
membranes at affected (labeled) neurons -- a temporary change (which requires a dopamine signal).
Overall, we anticipate a network of processors with their own individual memory, intimately connected by bit-type signals, participating in low-dimensional representational configurations, to offer a highly flexible and sophisticated framework suitable for complex computations of the kind that humans perform \cite{GallistelMatzel2013, LangilleGallistel2020}.
\section{Problems and Applications}
\subsection{Multiple Temporal Scales}
The ability of short-term signals to produce an internal response which lasts
over minutes or hours is dependent on the internal
parameters which guide the signal into the core. Within the core,
epigenetic mechanisms decide what happens with the signals
accessing the DNA.
How do we link short-term signals over milliseconds or seconds, with the long-lasting plasticity exhibited at neurons, happening over minutes or even hours?
The dynamics of plasticity is
usually simply subsumed under the concepts of either short-term or long-term
(synaptic) plasticity \cite{Deperrois2020}, without a clear understanding how the
temporal levels interact - this points to a larger problem \cite{Baudry2015}. The temporal structure of
filtering of outside signals and integration with stored information
is not well explored, neither in experiments, nor in theoretical models. The neuron receives many signals, usually with
fast durations (synaptic activations are on the order of 10-100ms, some of the longest durations have been measured for dopamine levels during feeding, for up to 10 minutes). Internal adaptations take at least
minutes and access to core parameters may take hours or even days during which
the system still adapts.
A neuron is a system with fast fluctuations of signals (milliseconds-seconds) at the periphery
where outside signals arrive and are selected or filtered, and slow
processes (hours) in the core, where long-term information (genes) is
transcribed, and genetic programs activated. From the periphery to the core, each step happens at a slower scale.
It is an unusual type of computational system which operates along those principles. Presumably,
the system is remarkable in that it can select individual short-term signals as highly relevant and transform them into long-lasting information, while at the same time filtering out very many other signals which are less relevant or redundant. Often it is assumed that a neuron collects all evidence and uses it statistically (e.g. sums it at its synapses). While such statistical screening may happen, the decisive function that the neuron is built for is to create lasting memory of individual brief events.
Dynamic processes that span many
different time scales -- from milliseconds to minutes, hours, or days --
are in general difficult to model, and difficult to understand
\cite{PigolottiVulpiani2008,TysonJJ2019,SongZ2021}.
A simple mechanism for a multiscale situation is to use a modular system, where each module
operates on its own time-scale and results are passed between the
modules at appropriate times \cite{WangZetal2022}. For instance, from a fast module an integrated term could
be passed on, whenever a computation reaches an end-point (at irregular times) and from a slow module a changing bias term could be passed to the fast
modules at regular intervals. This could be applied to the external, internal and core systems.
\subsection{Programmable elements}
A weighted neural network is a simple data structure (graph) with only one degree of freedom (weights) for recording past experience. Accordingly, it mostly stores data (or data-classification pairs) and generalizes by interpolation (or extrapolation). In contrast, a horizontal-vertical network is a network of processors with a large number of parameters sorted by function, dynamics, and structure (external, internal, core). This system can do more that adapt to outside signals and store traces of them. It can (a) select and filter, i.e. ignore many outside signals if there is no internal reinforcement and (b) it can also self-program, i.e. self-adapt its internal processing structure.
The self-programming mechanism relies essentially on a stream of outside signals (data), and feedback from the slow center to the fast periphery to handle the signals (Figure~\ref{fig:self-program}).
Here are a number of features which define the capacity for self-programming for a neuron within a network of processors:
\begin{itemize}
\item[-] re-programming the system for external and internal parameters by genetic control of protein abundances
\item[-] preset ranges for adaptation of external parameters by outside signals
\item[-] manipulating the rate of change (slow and fast) for parameters to address the stability-flexibility dichotomy
\item[-] existence of programs in the genetic core which can be enacted {\it in toto} and allow complex morphological adaptation such as spine growth, dendrite remodelling, axonal branching
\end{itemize}
Internal signaling networks, which are a type of biochemical reaction networks, guarantee a return to homeostasis
\cite{Bhattacharya2022,Scheler2013PLOSONE,vonBertalanffy1968}.
This means that
these signaling networks have no memory of their own. They react
to signals, but then return to their preset values. They are thus protected against signal-based adaptation or signal-based memory. However, there are the protein abundances which re-program the system and which are under control of another system, an outside system, namely RNA translation, or
DNA transcription.
So the system itself would be designed to not have
any memory of the actual processing it performs. This means that it acts as a processing layer sending signals into the `actuators', the membrane or the
nucleus, where signals are then absorbed, (e.g. in the histone layer) or may cause further DNA transcriptions after considerable time delay. These may re-program the system.
What can be programmed? The system has a number of control parameters available which govern neural transmission, i.e. neuronal behavior in interaction with other neurons on the horizontal plane.
These parameters code for membrane excitability, i.e. firing threshold, membrane resting potential; afterdepolarization/ afterhyperpolarization, i.e. refractory reset properties; spike latency, axonal delay, degree of myelination \cite{BonettoG2021}, etc. They set the firing rate and the capacity for synchronization. The placement of GABA-A \cite{NakamuraY2015}, and glutamate receptors influences synaptic transmission, and the capacity of the dendrites to route excitations to the soma. There are other membrane properties which also constitute programmable elements but which influence neural transmission in a more subtle way. NM receptors or GPCRs (including GABA-B, mGLU) which are regulated in response to ligand availability activate internal parameters (e.g. $G_\beta\gamma$, cAMP) from membrane receptors. These can modulate membrane properties directly by a fast feedback mechanism. They also signal to the nucleus, directly (PKA), or indirectly (MAPK, ERK) using the internal parameter system \cite{Schelertransactivation,Scheler2013PLOSONE} and in this way provide a significant part of the filtering system for outside signals.
Recently it has been argued that novel behavioral signals may affect at first
epigenetic features, i.e. histones, while only after repetition
there is read-out of DNA \cite{VanHook2021}. In this sense a naive neuron may at first have only its accessibility of DNA being affected by signals, awaiting further confirmation to actually read out DNA programs.
Programming a neuron's core identity may take several steps (Figure~\ref{fig:self-program}). In the end, a complex program guides the actions that re-define a neuron's morphology or alter its membrane composition. For instance, the core contains programs that grow or mature a spine, produce and insert neuromodulatory receptors, move new AMPA receptors to the postsynaptic density (\cite{MiskiM2022}), convert silent NMDA receptors, or balance and co-regulate ion channels \cite{Debanne2019}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{Self-program}
\end{center}
\caption{Building blocks for self-programming of neurons. Outside signals (Data) are processed by external parameters (EXT) which interact with internal parameters (INT). The EXT/INT system changes over time. The decisive feature for adaptation is the feedback between the core and internal systems. Note that data are maximally separated from the core system.}
\label{fig:self-program}
\end{figure}
To build a generative model for the external parameter set of a neuron, internal parameters can exchange or restore external parameters (e.g. they modulate receptor endocytosis followed by insertion or ubiquination).
They can protect parameters against overwrite (e.g. by placing Sk-channels next to calcium channels/AMPA/NMDA receptors
\cite{HammondRS2006}). They can also modify receptor complexes (e.g. co-localization of D1/NMDA receptors\cite{AgnatiL2003}).
The internally regulated, targeted placement of neuromodulator receptors allows to switch between several modalities on short time scales (seconds to minutes) by activation of neuromodulatory signals \cite{Scheler2003d}.The precise mechanisms by which the neuron self-programs its many features are still a matter of further research.
\subsection{Knowledge structures}
In the brain, memory items are linked together to form knowledge structures. Each neuron belongs to a constellation of neurons ('ensembles' \cite{Fuster2022,JosselynTonegawa2020}), which also may reach into different brain areas (e.g. cortical-hippocampal constellations). These can be described as sub-graphs in a 'knowledge graph' \cite{GrundemannJ2019, TanakaKZ2018}. A sub-graph can be
compressed as a single neuron \cite{Schelerbiorxiv2019}, creating a 'concept' as an abstraction of entities \cite{RutishauserU2021}.
This involves internalization of information
and adaptation of neuronal identity. The abstraction that happens for concept formation may extend from a small contribution to a stored schema to the
complete reconstruction of a spatio-temporal pattern, such as an event memory. Most importantly, so-called semantic memory \cite{ChrysanthidisLansner2022} (or factual memory) is the result of abstraction and concept formation. By belief propagation or similar techniques there may be enmeshment of the sub-graphs to more complex graphs.
This allows to build higher-order structures, for instance, by re-assigning connections in the graph or by vertical externalization to restore patterns.(Fig.)
Decisive for structure building or knowledge abstraction to work are processes of different stability, such as stable concept neurons combined with flexible graph connectivity.
For example, during a certain spatio-temporal
pattern a network state might consist of a distribution of synaptic AMPA channels, K+ and Ca-K+
channels over its neurons. Key (engram or 'concept') neurons for this pattern could employ overwrite protection, and therefore keep their parameter sets stable (without further adaptations at their sites). The flexible neurons, which do not use protection, would form a background of fluctuating, nearly random parameter sets.
In \cite{Schelerbiorxiv2019} we presented a model where a spatio-temporal spiking pattern is compressed into a couple of neurons with the highest mutual information with the pattern. It was then shown that stimulating these neurons is sufficient for re-activating the pattern. The activation of the key neurons would restore the spatio-temporal pattern independent of further learning \cite{Schelerbiorxiv2019}.
In this way, parts of the horizontal structure (the pattern) become encapsulated into single neurons and these can be stimulated so that they unfold the spatio-temporal pattern on a network with background activity.
When a network exhibits a spatio-temporal spiking pattern, its corresponding signals are matched to stored models at individual neurons. A generative model consists to a large part of the neuron's connections, its dendritic integration properties and also other short-term stable external membrane parameters. The generative neural model, defined by its parameters, is being matched to the set of signals it receives. If a neuron receives signals that match its model, it spikes and it produces a significant calcium transient as marker of the spiking event. Adaptation, after selection and filtering, is performed by updating “priors” to signal-based “posteriors”, and by optimizing the complexity of the models, which can be pruned by linking parameters.
In this way, the spatio-temporal spiking pattern starts a massively parallel search for matching neurons. For each matching neuron - typically in a sparse distribution - a sequence of new activations on the network follows. Matching neurons
activate subnetworks, or ensembles of other neurons, to complete the pattern (horizontal computing).
New patterns are created in an operation similar to
drawing inferences from data. The upshot is that horizontal computing can be likened to operations on a knowledge graph, as long as the nodes contain their own internal information and adaptation.
\comment{
What are subgraphs that are typically
activated together (entity-value constellations)? The role of lognormal
distributions of excitability? Activation of constellations and new
pattern formation as instances of inferences and the firing of a
production system?}
\section{Discussion}
From a cognitive and behavioral perspective we know that episodic memory is not a
reversible operation where a blank slate is inscribed and wiped again, but an
operation where permanent changes occur when memories are stored. A
developmental trajectory exists from neonate to juvenile to adult to
aged individual. Only a fraction of events is stored
with completeness. Much information is not kept and deemed not worth
keeping. In episodic memory, many representations of events undergo abstraction and become part
of a complex (schema) that they add to in small ways. Often, information
that is sufficiently represented by existing knowledge is merely summed
or discarded.
In a very general way, neuronal plasticity can be seen as
a form of difference learning between stored information and
new, incoming signals. This is also known as ``predictive'' or ``generative
learning''. Here we outlined a model with external parameters at the membrane, which guide transmission of information,
with internal parameters, hidden from interactions, which store and process information and are capable of
re-setting external values, and a core system that programs and re-programs the parameter system.
The difference between outside signals and internal information drives adaptation and forces a system, such as a neuron, to remain an open system with respect to its environment, i.e. a system which lacks the properties of an algorithmic or formally 'closed' system. Memory management is central in such a system, to ensure consistency, promote signal loss where appropriate \cite{Scheler2001a,Scheler2001b}, and prevent 'catastrophic forgetting' \cite{GrossbergS2020,PaikI2020}.
The main problem with the synaptic hypothesis is that adjusting weights in graphs has weak expressive power. Treating neurons as adjustable activation functions improves efficiency of implementation considerably, but does not extend the expressive power of the system \cite{Scheler2004,Kang2006,SoltizM2013}. While such systems can store data well, they don't function for more complex hierarchical knowledge and action plans, like language generation \cite{Hosseini-Asl2020}. A mathematical theory of the expressive power of various neuronal representation systems is still lacking, to my knowledge. This would involve comparing purely horizontal to vertical-horizontal models.
The synaptic theory of memory also experiences a paradox since synapses are units with high turnover \cite{MongilloGLoewenstein2017,FauthMetal2015}. This means that information cannot remain physically located at the synapses long-term without being lost or overwritten. Various schemes have been suggested to counteract this problem \cite{FauthMetal2015}. We showed that neurons as
processors operate with graded stability (from outside to inside), a principle which seems very intuitive and sound~ \ref{fig:concentric-processor}. Memorization involves transforming transient, potentially repeating information into stable information elements. Axonal/dendritic morphology must
in this case also be regarded as a stable, core component, since it
requires core-mediated programs for alteration. The peculiar nature of the neuronal cell as a processing device - concentric abstraction of temporal depth - is probably due to this unique requirement of providing short-term adaptation, long-term stability and the capacity to store information into increasingly stable formats (\ref{fig:concentric-processor}).
Types of neurons in different brain areas, defined by genetics and the product of evolutionary learning, are essentially stable, and they form part of a background structure.
Any neural system must contain elements with high stability - which provide the backbone of the knowledge structure and are protected against change - and elements which flexibly adjust to current conditions. The stability-flexibility trade-off is probably at the center of the capacity of
brains to build structured representations, subject to modifications (ART).
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{concentric-processor}
\end{center}
\caption{The neuron - here drawn as a prototypical cell - acts as a processor which receives fast signals and responds with fluctuations on different temporal planes: the membrane is fast, and the core is slow. There is an intermediate zone. This can be seen as an original solution to the flexibility-stability problem in information storage and processing. }
\label{fig:concentric-processor}
\end{figure}
Many diseases of the brain relate to processes, molecules and interactions that cannot be captured adequately with the concept of synaptic plasticity. For instance, it was recently found that ketamine drives genetic expression of the KCNQ2 channel, which increases the $I_M$ (K+) current \cite{LopezJetal2022, ShinoharaR2021,KangM2022}. In hippocampus this reduces bursting and lack of reset after spike firing, by reducing the afterdepolarization(ADP); but it also reduces spontaneous spiking at the excitatory synapse. It is possible that this latter effect transmits the long-lasting anti-depressant effect of single-dose ketamine \cite{AbdallahCGetal2015}. Here a one-time increase of the genetic expression of an ion channel provides behavioral changes for many weeks. These type of results are easy to incorporate into a model of the horizontal-vertical type but not a synaptic weight adjustment model. It is important to realize that remembering is also a biochemical event. Activation of neurons leads to specific activations via long-term connections into deep brain regions (neural 'loop' structure). This connection may mean that a cocktail of neurochemicals (monoamines, neuropeptides, hormones etc.) is released.
We notice an important advance of the proposed theoretical model - as a horizontal-vertical model - is to bridge the divide between theoretical neuroscience, artificial intelligence and disease models. While small-scale models of disease processes exist (refs), there is an unmet need for brain theories and derivative models with actual functionality (AI), which model cellular processes exhibited by the neuron, and adequately capture their disruption within a disease process.
Horizontal-vertical integration models can be considered a substantial
innovation, especially compared to associative weight
adjustment models. They use
neuron models with internal memory to build complex processor
networks, combining data and
experiments from electrophysiology and molecular biology. The proposed neuron model allows to realize self-programming of neurons and building complex models of cognition. The brain, of course, has many additional components, such as
the capillary and glymphatic networks, glia cells, and the signals that
pass the blood-brain barrier. In the future more comprehensive brain models may encompass such additional components.
\section{Conclusion}
The neuron is a highly specialized cell type, strongly compartmentalized with extensive axons and dendrites. Each compartment, dendritic or axonal, connects directly and
specifically to other neurons. For this reason, there has always been a strong focus within neuroscience on the synapse as a specialized connective element between neurons, with its postsynaptic density protein composition, and its presynaptic vesicle release mechanism. This has led to theoretical neuroscience focusing on adjusting synaptic weights as the main memory mechanism.
But by now the experimental literature on neuronal plasticity is vast, and synapses make up only a small part of it. Intrinsic
forms of plasticity via ion channels and NM receptors are well-known and have been researched in great detail (\cite{WestA2011,MahonSetal2003,Debanneetal2019, Sehgaletal2014, NatarajK2010, GillHansel2020, Scheler2013,Scheler2004,TullyPJetal2014}). The vertical, internal dimension of the neuron with its intracellular signaling network and the nuclear processes, genetic and epigenetic, has particular significance in the fields of psychopharmacology and disease modeling. But the development of a theoretical framework, linking those aspects of neural plasticity, has been missing. We have begun to fill the void by re-thinking the concept of neural plasticity from the perspective of the individual neuron. We argue that even synaptic plasticity is incompletely understood when placed outside of the context of cellular functioning.
Purely horizontal models with weight adaptation, like current neural
network models, can only be programmed by input. That is very
restrictive. Huge networks with billions of parameters are needed to solve simple problems by brute-force massive storage. Those restrictions are systematic and mathematically explainable, since the expressive power of networks with adjustable weights is weak.
A programmable memory at each neuronal site allows for more
complex and interesting operations. For instance, patterns from the horizontal plane can get stored
into a small set of dedicated neurons for a particular problem. Such models may then be used to build complex knowledge structures.
We believe that the ''third wave of AI'' will have to employ some kind of horizontal-vertical brain model to make use of the opportunities of linking intracellular intelligence with large-scale neuron modeling to achieve a truly intelligent, self-programmable system.
An immediate challenge will be to create models which run in a self-contained way,
and build up internal structures from the initial set-up and
the processing of input patterns.
\printbibliography
\end{document}
|
2,869,038,156,383 | arxiv | \section{Introduction} \label{sec1}\setcounter{equation}{0}
In this article, we are interested in the long time behavior of stochastic convective Brinkman-Forchheimer (CBF) equations with non-autonomous forcing term (deterministic term) and nonlinear diffusion term. This stochastic mathematical model describes the motion of incompressible fluid flows in a saturated porous medium (cf. \cite{PAM}). The CBF equations apply to flows when the velocities are sufficiently high and porosities are not too small, that is, when the Darcy law for a porous medium does not hold. Due to the lack of Darcy's law, it is sometimes referred as \emph{non-Darcy model} also (cf. \cite{PAM}). Given $\mathfrak{s}\in\mathbb{R}$, we consider the non-autonomous stochastic CBF equations in $\mathcal{O}\subseteq\mathbb{R}^d\ (d=2,3)$ as
\begin{equation}\label{1}
\left\{
\begin{aligned}
\frac{\partial \boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p&=\boldsymbol{f}+S(t,x,\boldsymbol{u})\circ\frac{\d \mathrm{W}}{\d t}, \\&\qquad \qquad \ \ \ \text{ in } \mathcal{O}\times(\mathfrak{s},\infty), \\ \nabla\cdot\boldsymbol{u}&=0, \qquad \ \ \text{ in } \ \mathcal{O}\times(\mathfrak{s},\infty), \\
\boldsymbol{u}(x,\mathfrak{s})&=\boldsymbol{u}_{\mathfrak{s}}(x),\ \ \ x\in \mathcal{O} \text{ and }\mathfrak{s}\in\mathbb{R},
\end{aligned}
\right.
\end{equation}
where $\boldsymbol{u}(x,t) :
\mathcal{O}\times(\mathfrak{s},\infty)\to \mathbb{R}^d$ denotes the velocity field, $p(x,t):
\mathcal{O}\times(\mathfrak{s},\infty)\to\mathbb{R}$ represents the pressure field, $\boldsymbol{f}(x,t):
\mathcal{O}\times(\mathfrak{s},\infty)\to \mathbb{R}^d$ is an external body force, $S(t,x,\boldsymbol{u})$ is a nonlinear diffusion term (see sections \ref{sec3} and \ref{sec4}), the symbol $\circ$ means that the stochastic integral should be understood in the sense of Stratonovich integral and $\mathrm{W}=\mathrm{W}(t,\omega)$ is an one-dimensional two-sided Wiener process defined on a probability space $(\Omega,\mathscr{F},\mathbb{P})$. Here $\Omega$ is given by
\begin{align*}
\Omega=\{\omega\in\mathrm{C}(\mathbb{R};\mathbb{R}):\omega(0)=0\},
\end{align*}
$\mathscr{F}$ is the Borel sigma-algebra induced by the compact-open topology of $\Omega$, and $\mathbb{P}$ is the two-sided Gaussian measure on $(\Omega,\mathscr{F})$. Consequently, $\mathrm{W}$ has a form $\mathrm{W}(t,\omega)=\omega(t)$. Moreover, $\boldsymbol{u}(\cdot,\cdot)$ satisfies $$\boldsymbol{u}=\mathbf{0}\ \text{ on }\ \partial\mathcal{O}\times(\mathfrak{s},\infty)\ \text{ or }\ \boldsymbol{u}(x,\mathfrak{s})\to 0\ \ \text{ as }\ \ |x|\to\infty,$$ when $\mathcal{O}\subset\mathbb{R}^d$ is bounded with smooth boundary or $\mathcal{O}=\mathbb{R}^d$. The positive constants $\mu,\alpha, \beta>0$ are the Brinkman (effective viscosity), Darcy (permeability of porous medium) and Forchheimer (proportional to the porosity of the material) coefficients, respectively. The absorption exponent $r\in[1,\infty)$, $r=3$ is called the critical exponent and the model \eqref{1} with $r>3$ is referred as the CBF equations with fast growing nonlinearites (\cite{KT2}). The system \eqref{1} is also known as damped Navier-Stokes equations (NSE) (cf. \cite{HZ}) because it is classical NSE for $\alpha=\beta=0$. Moreover, it is proved in \cite{HR} (see Proposition 1.1, \cite{HR}) that CBF equations \eqref{1} have the same scaling as NSE only when $r=3$ and $\alpha=0$ but no scale invariance property for other values of $\alpha$ and $r$, therefore it is sometimes called NSE modified by an absorption term (\cite{SNA}) or tamed NSE (\cite{MRXZ}).
Let us now discuss some solvability results available in the literature for the stochastic system \eqref{1} and similar models. There are a good number of works available on the solvability results for the system \eqref{1} and related models (see \cite{MRXZ1,WLMR,WL,HBAM,MTM1,LHGH1}, etc). In particular, 3D NSE with a Brinkman-Forchheimer type term subject to an anisotropic viscosity driven by multiplicative noise and stochastic tamed 3D NSE on whole domain is considered in \cite{HBAM} and \cite{MRXZ1}, respectively. An improvement of the work \cite{MRXZ1} for a slightly simplified system is done in \cite{ZBGD}. On bounded domains, the existence of martingale solutions and strong solutions (in the probabilistic sense) for stochastic CBF equations are established in \cite{LHGH1} and \cite{MTM1}, respectively. The existence of a unique pathwise strong solution to 3D stochastic NSE is a well known open problem, and the same is open for 3D stochastic CBF equations with $r\in[1,3)$. Therefore, we do not consider $d=3$ with $1\leq r<3$ for any $\mu, \beta>0$ and $d=r=3$ for $2\beta\mu<1$ (see \cite{MTM1}). In the sequel, if we write $r\in[1,3),$ then it indicates that the two dimensional case is considered.
In \cite{BCF,CDF}, authors introduced the concept of random attractors for stochastic partial differential equations (SPDEs) and applied it to some fluid flow models including 2D stochastic NSE. Later, this concept is used to establish the existence of random attractors for several fluid flow models (see \cite{FY,KM3,You} etc and references therein). In unbounded domains, due to the absence of compact Sobolev embeddings, random dynamical systems (RDS) are no more compact. In the deterministic case, the difficulty discussed above was resolved by different methods, cf. \cite{Abergel, Ghidaglia,MTM3,Rosa}, etc for the autonomous case and \cite{CLR1, CLR2,KM6}, etc for non-autonomous case. For SPDEs, the methods available in the deterministic case have also been generalized by several authors (see for example, \cite{ BLL,BLW,BCLLLR,Wang}, etc). The concept of an asymptotically compact cocycle was introduced in \cite{CLR1} and the authors proved the existence of attractors for non-autonomous 2D NSE. Later, several authors used this method to prove the existence of random attractors on unbounded domains, see for example \cite{BLL,BLW,KM,LXS,Wang} etc. Two of the methods used to prove asymptotic compactness without Sobolev embeddings are as follows: the method of energy equations (see \cite{GGW,KM,PeriodicWang} etc) and the method of uniform tail estimates (see \cite{SandN_Wang,WLW} etc).
For the existence of a unique random attractor (dynamics of almost all sample paths), it is required to define RDS or random cocycle through the solution operator of the equation \eqref{1}. To the best our knowledge, it is still an open problem to define RDS for \eqref{1}, when $S(t,x,\boldsymbol{u})$ is a general nonlinear diffusion term. In this direction, when $S(t,x,\boldsymbol{u})$ is a nonlinear diffusion term, the concept of weak pullback mean random attractors was introduced in \cite{Wang1} and this theory was applied to several physical models (cf. \cite{GU,KM4,Wang2} etc). On the other hand, using random transformations, SPDEs can generate RDS for either $S(t,x,\boldsymbol{u})=\boldsymbol{u}$ or $S(t,x,\boldsymbol{u})$ is independent of $\boldsymbol{u}$. Indeed, the existence of a unique random attractor for \eqref{1} is available for such $S(t,x,\boldsymbol{u})$ only (see \cite{KM1,KM3} for bounded domains and \cite{KM,KM6} for unbounded domains).
Wong and Zakai introduced the concept of approximating stochastic differential equations (SDEs) by deterministic differential equations in \cite{WZ,WZ1}, where they have approximated one dimensional Brownian motion. Later, this work was extended to SDEs in higher dimensions, see for example \cite{IW,DK.IM,Sussmann} etc and references therein. Due to the lack of differentiability of sample paths of Wiener process $\mathrm{W}$, the authors in \cite{UO,WU} first introduced the Ornstein-Uhlenbeck process (or colored noise) to approximate $\mathrm{W}$. The Wong-Zakai approximations (with the help of colored noise) can be used to analyze the pathwise random dynamics of \eqref{1} with nonlinear diffusion term. The approach through approximations obeys the well-known framework given by Wong and Zakai in \cite{WZ,WZ1}. Despite of analyzing the corresponding random versions of \eqref{1} acquired by random transformations, we will rather analyze the system obtained by the Wong-Zakai approximations of \eqref{1} with nonlinear diffusion term. With reference to the nonlinear diffusion term, the idea of Wong-Zakai approximations for random pullback attractors is used for a wide class of physically admissible models, see \cite{GLW,GW,LW} etc for bounded domains, \cite{GGW,WLS,WLW} etc for unbounded domains and \cite{WSLW,YSW} etc for discrete systems.
On $(\Omega,\mathscr{F},\mathbb{P})$, consider Wiener shift operator $\{\vartheta_{t}\}_{t\in\mathbb{R}}$ defined by
\begin{equation}\label{vartheta}
\vartheta_{t}\omega(\cdot)=\omega(\cdot+t) -\omega(t), \ \ \ t\in\mathbb{R}\ \text{ and }\ \omega\in\Omega.
\end{equation}
It is well known from \cite{Arnold} that the Gaussian measure $\mathbb{P}$ is ergodic and invariant for $\vartheta_{t}$. Hence, $(\Omega,\mathscr{F},\mathbb{P},\{\vartheta_{t}\}_{t\in\mathbb{R}})$ is a metric dynamical system (see \cite{Arnold}). Let us define the colored noise $\mathcal{Z}_{\delta}:\Omega\to\mathbb{R}$ with the given correlation time $\delta\in\mathbb{R}\backslash\{0\}$ such that
\begin{align*}
\mathcal{Z}_{\delta}(\omega)=\frac{\omega(\delta)}{\delta} \ \text{\ or equivalantly }\ \mathcal{Z}_{\delta}(\vartheta_{t}\omega)=\frac{1}{\delta}\left(\omega(t+\delta)-\omega(t)\right).
\end{align*}
The properties of Wiener process imply that $\mathcal{Z}_{\delta}(\vartheta_{t}\omega)$ is stationary process which follows normal distribution. It is observed that white noise can be approximated by $\mathcal{Z}_{\delta}(\vartheta_{t}\omega)$ in the sense given in Lemma \ref{WnCn} below (see Lemma 2.1, \cite{GGW} also). In \cite{LW1,SLW}, the authors used this approximation to study the chaotic behavior of RDS.
Let us now consider the following random partial differential equations as Wong-Zakai approximations of \eqref{1} for $\mathfrak{s}\in\mathbb{R}$ and $\delta\in\mathbb{R}\backslash\{0\}$:
\begin{equation}\label{2}
\left\{
\begin{aligned}
\frac{\partial \boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p&=\boldsymbol{f}+S(t,x,\boldsymbol{u})\mathcal{Z}_{\delta}(\vartheta_{t}\omega), \ \text{ in } \mathcal{O}\times(\mathfrak{s},\infty), \\ \nabla\cdot\boldsymbol{u}&=0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ in } \ \mathcal{O}\times(\mathfrak{s},\infty), \\
\boldsymbol{u}(x,\mathfrak{s})&=\boldsymbol{u}_{\mathfrak{s}}(x),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \mathcal{O} \text{ and }\mathfrak{s}\in\mathbb{R}.
\end{aligned}
\right.
\end{equation}
Also, $\boldsymbol{u}(x,t)$ satisfies $\boldsymbol{u}=\mathbf{0}$ on $\partial\mathcal{O}\times(\mathfrak{s},\infty)$ or $\boldsymbol{u}(x,\mathfrak{s})\to 0$ as $|x|\to\infty,$ when $\mathcal{O}\subset\mathbb{R}^d$ is bounded with smooth boundary or $\mathcal{O}=\mathbb{R}^d$. In bounded domains, as we are using compact Sobolev embeddings, we assume that the nonlinear diffusion term $S(t,x,\boldsymbol{u})$ satisfies Assumption \ref{NDT1}. Due to the absence of compact Sobolev embeddings in unbounded domains, we use the method of energy equations and method of uniform tail estimates, which require a stronger assumption than Assumption \ref{NDT1} on $S(t,x,\boldsymbol{u})$ (see Assumptions \ref{NDT2} and \ref{NDT3} for the method of energy equations and method of uniform tail estimates, respectively). It can be verified that Assumptions \ref{NDT2} and \ref{NDT3} are not same. In particular, both Assumptions cover different classes of nonlinear functions. For functional setting, one can refer to section \ref{sec2}.
\vskip 2mm
\noindent
\textbf{Aims and scopes of the work:} The major aims and novelties of this work are:
\begin{itemize}
\item [(i)] Existence of a unique random pullback attractor for the system \eqref{2} under Assumption \ref{NDT1} on bounded domains.
\item [(ii)] Existence of a unique random pullback attractor for the system \eqref{2} under Assumptions \ref{NDT2} and \ref{NDT3} on the whole domain.
\item [(iii)] Existence of a unique random pullback attractor for the system \eqref{1} with $S(t,x,\boldsymbol{u})=e^{\sigma t}\textbf{g}(x)$, for given $\sigma\geq0$ and $\textbf{g}\in\mathrm{D}(\mathrm{A})$ (additive noise).
\item [(iv)] For $S(t,x,\boldsymbol{u})=e^{\sigma t}\textbf{g}(x)$, for given $\sigma\geq0$ and $\textbf{g}\in\mathrm{D}(\mathrm{A})$ as well as $S(t,x,\boldsymbol{u})=\boldsymbol{u}$, convergence of solutions and upper semicontinuity of random pullback attractors for \eqref{2} towards the random pullback attractor for \eqref{1} as $\delta\to0$ (additive and multiplicative noise).
\end{itemize}
Moreover, we prove the existence of a unique random pullback attractor of the system \eqref{2} in two dimensions with $\alpha=\beta=0$ (2D NSE) under Assumption \ref{NDT4} on Poincar\'e domains.
\vskip 2mm
\noindent
\textbf{Advantages of damping term:} CBF equations are also known as NSE with damping (cf. \cite{HZ}). The damping arises from the resistance to the motion of the flow or by friction effects. Due to the presence of the damping term $\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}$, we can establish better results than which are available for NSE. In particular, in this work, linear damping $\alpha\boldsymbol{u}$ helps us to obtain results on the whole domain (for NSE, the global as well as random attractor on the whole domain is still an open problem) and the nonlinear damping $\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}$ (for $r>1$) helps us to cover a large class of nonlinearity on $S(t,x,\boldsymbol{u})$ (see subsection \ref{subsec4.3}). The authors in \cite{GGW} established the existence of a unique random attractor for 2D NSE with $S(t,x,\boldsymbol{u})=e^{\sigma t}\textbf{g}(x)$, for given $\sigma>0$ and $\textbf{g}\in\mathrm{D}(\mathrm{A})$, based on the following assumption on $\textbf{g}(\cdot)$:
\begin{assumption}[See section 3, \cite{GGW}]\label{GA}
Assume that the function \emph{$\textbf{g}(\cdot)$} satisfies the following condition: there exists a strictly positive constant $\aleph$ such that
\begin{align*}
|b(\boldsymbol{u},\emph{\textbf{g}},\boldsymbol{u})|\leq \aleph\|\boldsymbol{u}\|^2_{\mathbb{H}}, \ \ \text{ for all } \ \boldsymbol{u}\in\mathbb{H}.
\end{align*}
\end{assumption}
We also point out that the requirement of Assumption \ref{GA} is not necessary for $r>1$ ($\textbf{g}\in\mathrm{D}(\mathrm{A})$ is enough). For $r=1$, one gets linear damping, and the existence of a random attractor for \eqref{SCBF_Add} and upper semicontinuity of random attractors for \eqref{WZ_SCBF_Add} can be proved on the whole domain via same arguments as it has been done for 2D stochastic NSE on Poincar\'e domains in \cite{GGW} under Assumption \ref{GA}. For $r>1$, due to the presence of nonlinear damping term $\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}$, we provide a different treatment to avoid Assumption \ref{GA} on $\textbf{g}$ (see Lemma \ref{LemmaUe_add}).
It is well-known that passing limit in nonlinear terms is not an easy task in any analysis. In this work, while proving the pullback asymptotic compactness of solutions of the system \eqref{SCBF_Add} (Lemma \ref{Asymptotic_UB_Add}) or establishing the uniform compactness of random attractors of the system \eqref{WZ_SCBF_Add} (Lemma \ref{precompact}), it is necessary to show the convergence of the nonlinear terms appearing in the corresponding energy equations. In our model, we have two nonlinear terms given by $(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}$ and $\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}$. The required convergence is obtained by breaking down the integral containing nonlinear terms into two parts such that one part is defined in a bounded domain $\mathcal{O}_k$ with radius $k$, and another part is defined in the complement of $\mathcal{O}_k$. We then show that the nonlinear term is convergent in $\mathcal{O}_k$ and its tail on the complement of $\mathcal{O}_k$ is uniformly small when $k$ is sufficiently large, from which the required convergence follows (see Corollaries \ref{convergence_b} and \ref{convergence_c} for terms $(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}$ and $\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}$, respectively).
The remaining sections are arranged as follows. In next section, we provide the necessary function spaces required to establish the results of this work, and we define linear operator, bilinear operator and nonlinear operator to obtain an abstract formulation of the system \eqref{2}. Furthermore, we recall some well known inequalities, properties of operators, and some properties of white and colored noises. Finally, we discuss the solvability of the system \eqref{2} in the same section. In section \ref{sec3}, we prove the existence of a unique pullback random attractor for the system \eqref{2} under Assumption \ref{NDT1} on bounded domains in which we prove the asymptotic compactness using compact Sobolev embeddings (Theorem \ref{WZ_RA_B}). In section \ref{sec4}, we prove the existence of a unique pullback random attractor for the system \eqref{2} under Assumptions \ref{NDT2} and \ref{NDT3} on the whole domain in which we prove the asymptotic compactness using the idea of energy equations and uniform tail estimates, for Assumptions \ref{NDT2} and \ref{NDT3}, respectively (Theorems \ref{WZ_RA_UB} and \ref{WZ_RA_UB_GS}). Section \ref{sec6} is devoted for the stochastic CBF equations perturbed by additive white noise. Firstly, we prove the existence of a unique pullback random attractor for stochastic CBF equations driven by additive white noise (Theorem \ref{RA_add}). Next, we demonstrate the convergence of solutions and upper semicontinuity of random pullback attractors for Wong-Zakai approximations of stochastic CBF equations towards the solution and random pullback attractor of stochastic CBF equations, respectively, as the correlation time $\delta$ converges to zero, using the fundamental theory introduced in \cite{non-autoUpperWang} (Theorem \ref{Main_T_add}). Since the existence of a unique random pullback attractor for stochastic CBF equations driven by linear multiplicative noise is established in \cite{KM6}, we prove only the convergence of solutions and upper semicontinuity of random pullback attractors for its Wong-Zakai approximations towards its solution and random pullback attractor, respectively, as the correlation time $\delta\to0$ (Theorem \ref{Main_T_Multi}) in section \ref{sec7}. In Appendix \ref{sec5}, we prove the existence of a unique pullback random attractor for the system \eqref{WZ_NSE} under Assumption \ref{NDT4} on Poincar\'e domains (bounded or unbounded) (Theorem \ref{WZ_RA_UB_GS_NSE}).
\section{Mathematical Formulation}\label{sec2}\setcounter{equation}{0}
In this section, first we provide the necessary function spaces needed to obtain the results of this work. Next, we define some operators to set up an abstract formulation. Finally, we recall some properties of white noise as well as colored noise. We fix $\mathcal{O}$ as either a bounded subset of $\mathbb{R}^d$ with $\mathrm{C}^2$-boundary or whole domain $\mathbb{R}^d$.
\subsection{Function spaces}
We define the space $$\mathcal{V}:=\{\boldsymbol{u}\in\mathrm{C}_0^{\infty}(\mathcal{O};\mathbb{R}^d):\nabla\cdot\boldsymbol{u}=0\},$$ where $\mathrm{C}_0^{\infty}(\mathcal{O};\mathbb{R}^d)$ denotes the space of all infinite times differentiable functions ($\mathbb{R}^d$-valued) with compact support in $\mathbb{R}^d$. Let $\mathbb{H}$, $\mathbb{V}$ and $\widetilde\mathbb{L}^p$ denote the completion of $\mathcal{V}$ in $\mathrm{L}^2(\mathcal{O};\mathbb{R}^d)$, $\mathrm{H}^1(\mathcal{O};\mathbb{R}^d)$ and $\mathrm{L}^p(\mathcal{O};\mathbb{R}^d)$, $p\in(2,\infty)$, norms, respectively. The space $\mathbb{H}$ is endowed with the norm $\|\boldsymbol{u}\|_{\mathbb{H}}^2:=\int_{\mathcal{O}}|\boldsymbol{u}(x)|^2\d x,$ the norm on the space $\widetilde{\mathbb{L}}^{p}$ is defined by $\|\boldsymbol{u}\|_{\widetilde \mathbb{L}^p}^p:=\int_{\mathcal{O}}|\boldsymbol{u}(x)|^p\d x,$ for $p\in(2,\infty)$ and the norm on the space $\mathbb{V}$ is given by $\|\boldsymbol{u}\|^2_{\mathbb{V}}=\int_{\mathcal{O}}|\boldsymbol{u}(x)|^2\d x+\int_{\mathcal{O}}|\nabla\boldsymbol{u}(x)|^2\d x.$ The inner product in the Hilbert space $\mathbb{H}$ is denoted by $( \cdot, \cdot)$. The duality pairing between the spaces $\mathbb{V}$ and $\mathbb{V}'$, and $\widetilde{\mathbb{L}}^p$ and its dual $\widetilde{\mathbb{L}}^{\frac{p}{p-1}}$ is represented by $\langle\cdot,\cdot\rangle.$ It should be noted that $\mathbb{H}$ can be identified with its own dual $\mathbb{H}'$. We endow the space $\mathbb{V}\cap\widetilde{\mathbb{L}}^{p}$ with the norm $\|\boldsymbol{u}\|_{\mathbb{V}}+\|\boldsymbol{u}\|_{\widetilde{\mathbb{L}}^{p}},$ for $\boldsymbol{u}\in\mathbb{V}\cap\widetilde{\mathbb{L}}^p$ and its dual $\mathbb{V}'+\widetilde{\mathbb{L}}^{p'}$ with the norm $$\inf\left\{\max\left(\|\boldsymbol{v}_1\|_{\mathbb{V}'},\|\boldsymbol{v}_1\|_{\widetilde{\mathbb{L}}^{p'}}\right):\boldsymbol{v}=\boldsymbol{v}_1+\boldsymbol{v}_2, \ \boldsymbol{v}_1\in\mathbb{V}', \ \boldsymbol{v}_2\in\widetilde{\mathbb{L}}^{p'}\right\}.$$ Moreover, we have the following continuous embedding also:
$$\mathbb{V}\cap\widetilde{\mathbb{L}}^{p}\hookrightarrow\mathbb{V}\hookrightarrow\mathbb{H}\equiv\mathbb{H}'\hookrightarrow\mathbb{V}'\hookrightarrow\mathbb{V}'+\widetilde\mathbb{L}^{\frac{p}{p-1}}.$$ One can define equivalent norms on $\mathbb{V}\cap\widetilde\mathbb{L}^{p}$ and $\mathbb{V}'+\widetilde\mathbb{L}^{\frac{p}{p-1}}$ as
\begin{align*}
\|\boldsymbol{u}\|_{\mathbb{V}\cap\widetilde\mathbb{L}^{p}}=\left(\|\boldsymbol{u}\|_{\mathbb{V}}^2+\|\boldsymbol{u}\|_{\widetilde\mathbb{L}^{p}}^2\right)^{\frac{1}{2}}\ \text{ and } \ \|\boldsymbol{u}\|_{\mathbb{V}'+\widetilde\mathbb{L}^{\frac{p}{p-1}}}=\inf_{\boldsymbol{u}=\boldsymbol{v}+\boldsymbol{w}}\left(\|\boldsymbol{v}\|_{\mathbb{V}'}^2+\|\boldsymbol{w}\|_{\widetilde\mathbb{L}^{\frac{p}{p-1}}}^2\right)^{\frac{1}{2}}.
\end{align*}
\subsection{Projection operator}
Let $\mathcal{P}_p: \mathbb{L}^p(\mathcal{O}) \to\widetilde{\mathbb{L}}^p$ be the Helmholtz-Hodge (or Leray) projection (cf. \cite{DFHM,MTSS}, etc). For $p=2$, $\mathcal{P}:=\mathcal{P}_2$ becomes an orthogonal projection and for $2<p<\infty$, it is a bounded linear operator.
\vskip 2mm
\noindent
\textbf{Case I:} \textit{When $\mathcal{O}$ is a bounded subset of $\mathbb{R}^d$ with $\mathrm{C}^2$-boundary.} Since $\mathcal{O}$ has $\mathrm{C}^2$-boundary, $\mathcal{P}$ maps $\mathbb{H}^1(\mathcal{O})$ into itself (see Remark 1.6, \cite{Temam}).
\vskip 2mm
\noindent
\textbf{Case II:} \textit{When $\mathcal{O}=\mathbb{R}^d$.} The projection operator $\mathcal{P}:\mathbb{L}^2(\mathbb{R}^d) \to\mathbb{H}$ can be expressed in terms of the Riesz transform (cf. \cite{MTSS}). Moreover, $\mathcal{P}$ and $\Delta$ commutes, that is, $\mathcal{P}\Delta=\Delta\mathcal{P}$.
\subsection{Linear operator}
We define the Stokes operator
\begin{equation*}
\mathrm{A}\boldsymbol{u}:=-\mathcal{P}\Delta\boldsymbol{u},\;\boldsymbol{u}\in\mathrm{D}(\mathrm{A}):=\mathbb{V}\cap\mathbb{H}^{2}(\mathcal{O}).
\end{equation*}
\subsection{Bilinear operator}
Let us define the \emph{trilinear form} $b(\cdot,\cdot,\cdot):\mathbb{V}\times\mathbb{V}\times\mathbb{V}\to\mathbb{R}$ by $$b(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w})=\int_{\mathbb{R}^d}(\boldsymbol{u}(x)\cdot\nabla)\boldsymbol{v}(x)\cdot\boldsymbol{w}(x)\d x=\sum_{i,j=1}^d\int_{\mathbb{R}^d}\boldsymbol{u}_i(x)\frac{\partial \boldsymbol{v}_j(x)}{\partial x_i}\boldsymbol{w}_j(x)\d x.$$ If $\boldsymbol{u}, \boldsymbol{v}$ are such that the linear map $b(\boldsymbol{u}, \boldsymbol{v}, \cdot) $ is continuous on $\mathbb{V}$, the corresponding element of $\mathbb{V}'$ is denoted by $\mathrm{B}(\boldsymbol{u}, \boldsymbol{v})$. We also denote $\mathrm{B}(\boldsymbol{u}) = \mathrm{B}(\boldsymbol{u}, \boldsymbol{u})=\mathcal{P}[(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}]$.
An integration by parts gives
\begin{equation}\label{b0}
\left\{
\begin{aligned}
b(\boldsymbol{u},\boldsymbol{v},\boldsymbol{v}) &= 0,\ \text{ for all }\ \boldsymbol{u},\boldsymbol{v} \in\mathbb{V},\\
b(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}) &= -b(\boldsymbol{u},\boldsymbol{w},\boldsymbol{v}),\ \text{ for all }\ \boldsymbol{u},\boldsymbol{v},\boldsymbol{w}\in \mathbb{V}.
\end{aligned}
\right.\end{equation}
\begin{remark}
The following estimate on $b(\cdot,\cdot,\cdot)$ is helpful in the sequel (see Chapter 2, section 2.3, \cite{Temam1}). For all $\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}\in \mathbb{V}$,
\begin{align}\label{b1}
|b(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w})|&\leq C
\begin{cases}
\|\boldsymbol{u}\|^{\frac{1}{2}}_{\mathbb{H}}\|\nabla\boldsymbol{u}\|^{\frac{1}{2}}_{\mathbb{H}}\|\nabla\boldsymbol{v}\|_{\mathbb{H}}\|\boldsymbol{w}\|^{\frac{1}{2}}_{\mathbb{H}}\|\nabla\boldsymbol{w}\|^{\frac{1}{2}}_{\mathbb{H}},\ \ \ \text{ for } d=2,\\
\|\boldsymbol{u}\|^{\frac{1}{4}}_{\mathbb{H}}\|\nabla\boldsymbol{u}\|^{\frac{3}{4}}_{\mathbb{H}}\|\nabla\boldsymbol{v}\|_{\mathbb{H}}\|\boldsymbol{w}\|^{\frac{1}{4}}_{\mathbb{H}}\|\nabla\boldsymbol{w}\|^{\frac{3}{4}}_{\mathbb{H}}, \ \ \ \text{ for } d=3.
\end{cases}
\end{align}
\end{remark}
\begin{remark}
Note that $\langle\mathrm{B}(\boldsymbol{v},\boldsymbol{u}-\boldsymbol{v}),\boldsymbol{u}-\boldsymbol{v}\rangle=0$ and it implies that
\begin{align}\label{441}
\langle \mathrm{B}(\boldsymbol{u})-\mathrm{B}(\boldsymbol{v}),\boldsymbol{u}-\boldsymbol{v}\rangle &=\langle\mathrm{B}(\boldsymbol{u}-\boldsymbol{v},\boldsymbol{u}),\boldsymbol{u}-\boldsymbol{v}\rangle\nonumber\\&=-\langle\mathrm{B}(\boldsymbol{u}-\boldsymbol{v},\boldsymbol{u}-\boldsymbol{v}),\boldsymbol{u}\rangle=-\langle\mathrm{B}(\boldsymbol{u}-\boldsymbol{v},\boldsymbol{u}-\boldsymbol{v}),\boldsymbol{v}\rangle.
\end{align}
\end{remark}
\subsection{Nonlinear operator}
Consider the nonlinear operator $\mathcal{C}(\boldsymbol{u}):=\mathcal{P}(|\boldsymbol{u}|^{r-1}\boldsymbol{u})$. It is immediate that $\langle\mathcal{C}(\boldsymbol{u}),\boldsymbol{u}\rangle =\|\boldsymbol{u}\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}$ and the map $\mathcal{C}(\cdot):\mathbb{V}\cap\widetilde{\mathbb{L}}^{r+1}\to\mathbb{V}'+\widetilde{\mathbb{L}}^{\frac{r+1}{r}}$. For all $\boldsymbol{u}\in\widetilde\mathbb{L}^{r+1}$, the map is Gateaux differentiable with Gateaux derivative
\begin{align}\label{29}
\mathcal{C}'(\boldsymbol{u})\boldsymbol{v}&=\left\{\begin{array}{cl}\mathcal{P}(\boldsymbol{v}),&\text{ for }r=1,\\ \left\{\begin{array}{cc}\mathcal{P}(|\boldsymbol{u}|^{r-1}\boldsymbol{v})+(r-1)\mathcal{P}\left(\frac{\boldsymbol{u}}{|\boldsymbol{u}|^{3-r}}(\boldsymbol{u}\cdot\boldsymbol{v})\right),&\text{ if }\boldsymbol{u}\neq \mathbf{0},\\\mathbf{0},&\text{ if }\boldsymbol{u}=\mathbf{0},\end{array}\right.&\text{ for } 1<r<3,\\ \mathcal{P}(|\boldsymbol{u}|^{r-1}\boldsymbol{v})+(r-1)\mathcal{P}(\boldsymbol{u}|\boldsymbol{u}|^{r-3}(\boldsymbol{u}\cdot\boldsymbol{v})), &\text{ for }r\geq 3,\end{array}\right.
\end{align}
for all $\boldsymbol{v}\in\mathbb{V}\cap\widetilde{\mathbb{L}}^{r+1}$. Moreover, for any $r\in [1, \infty)$ and $\boldsymbol{u}_1, \boldsymbol{u}_2 \in \mathbb{V}\cap\widetilde{\mathbb{L}}^{r+1}$, we have (see subsection 2.4, \cite{MTM1})
\begin{align}\label{MO_c}
\langle\mathcal{C}(\boldsymbol{u}_1)-\mathcal{C}(\boldsymbol{u}_2),\boldsymbol{u}_1-\boldsymbol{u}_2\rangle\geq\frac{1}{2}\||\boldsymbol{u}_1|^{\frac{r-1}{2}}(\boldsymbol{u}_1-\boldsymbol{u}_2)\|_{\mathbb{H}}^2+\frac{1}{2}\||\boldsymbol{u}_2|^{\frac{r-1}{2}}(\boldsymbol{u}_1-\boldsymbol{u}_2)\|_{\mathbb{H}}^2 \geq 0
\end{align}
and
\begin{align}\label{a215}
\|\boldsymbol{u}-\boldsymbol{v}\|_{\widetilde\mathbb{L}^{r+1}}^{r+1}\leq 2^{r-2}\||\boldsymbol{u}|^{\frac{r-1}{2}}(\boldsymbol{u}-\boldsymbol{v})\|_{\mathbb{H}}^2+2^{r-2}\||\boldsymbol{v}|^{\frac{r-1}{2}}(\boldsymbol{u}-\boldsymbol{v})\|_{\mathbb{H}}^2,
\end{align}
for $r\geq 1$ (replace $2^{r-2}$ with $1,$ for $1\leq r\leq 2$).
\subsection{Inequalities}
The following inequalities are frequently used in the paper.
\begin{lemma}\label{Holder}
Assume that $\frac{1}{p}+\frac{1}{p'}=1$ with $1\leq p,p'\leq\infty$, $\boldsymbol{u}_1\in\mathbb{L}^{p}(\mathcal{O})$ and $\boldsymbol{u}_2\in\mathbb{L}^{p'}(\mathcal{O})$. Then we get
\begin{align*}
\|\boldsymbol{u}_1\boldsymbol{u}_2\|_{\mathbb{L}^1(\mathcal{O})}\leq\|\boldsymbol{u}_1\|_{\mathbb{L}^{p}(\mathcal{O})}\|\boldsymbol{u}_2\|_{\mathbb{L}^{p'}(\mathcal{O})}.
\end{align*}
\end{lemma}
\begin{lemma}\label{Interpolation}
Assume $1\leq s_1\leq s\leq s_2\leq \infty$, $\theta\in(0,1)$ such that $\frac{1}{s}=\frac{a}{s_1}+\frac{1-a}{s_2}$ and $\boldsymbol{u}\in\mathbb{L}^{s_1}(\mathcal{O})\cap\mathbb{L}^{s_2}(\mathcal{O})$, then we have
\begin{align*}
\|\boldsymbol{u}\|_{\mathbb{L}^s(\mathcal{O})}\leq\|\boldsymbol{u}\|_{\mathbb{L}^{s_1}(\mathcal{O})}^{a}\|\boldsymbol{u}\|_{\mathbb{L}^{s_2}(\mathcal{O})}^{1-a}.
\end{align*}
\end{lemma}
\begin{lemma}\label{Young}
For all $a,b,\varepsilon>0$ and for all $1<p,p'<\infty$ with $\frac{1}{p}+\frac{1}{p'}=1$, we obtain
\begin{align*}
ab\leq\frac{\varepsilon}{p}a^p+\frac{1}{p'\varepsilon^{p'/p}}b^{p'}.
\end{align*}
\end{lemma}
\subsection{White noise and colored noise}
In this subsection, we recall some properties of white noise and colored noise.
\begin{lemma}[Lemma 2.1, \cite{GGW}]\label{WnCn}
Assume that the correlation time $\delta\in(0,1]$. There exists a $\{\vartheta_t\}_{t\in\mathbb{R}}$-invariant subset $\widetilde{\Omega}\subseteq\Omega$ of full measure, such that for $\omega\in\widetilde{\Omega}$,
\begin{itemize}
\item [(i)] \begin{align}\label{N1}
\lim\limits_{t\to\pm\infty}\frac{\omega(t)}{t}=0;
\end{align}
\item[(ii)] the mapping
\begin{align}\label{N2}
(t,\omega)\mapsto\mathcal{Z}_{\delta}(\vartheta_t\omega)=-\frac{1}{\delta^2}\int_{-\infty}^{0}e^{\frac{\xi}{\delta}}\vartheta_t\omega(\xi)\d\xi
\end{align}
is a stationary solution (also called a \emph{colored noise} or an \emph{Ornstein-Uhlenbeck process}) of one-dimensional stochastic differential equation $$\d\mathcal{Z}_{\delta}+\frac{1}{\delta}\mathcal{Z}_{\delta}\d t=\frac{1}{\delta}\d\mathrm{W}$$ with continuous trajectories satisfying
\begin{align}
\lim\limits_{t\to\pm\infty}\frac{\left|\mathcal{Z}_{\delta}(\vartheta_t\omega)\right|}{t}&=0, \ \ \ \text{ for every } 0<\delta\leq1,\label{N3}\\
\lim\limits_{t\to\pm\infty}\frac{1}{t}\int_0^t\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi=\mathbb{E}[\mathcal{Z}_{\delta}]&=0, \ \ \ \text{ uniformly for } 0<\delta\leq1;\label{N4}
\end{align}
\end{itemize}
and
\begin{itemize}
\item[(iii)] for arbitrary $T>0,$
\begin{align}\label{N5*}
\lim_{\delta\to0}\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi=\omega(t) \ \text{ uniformly for } \ t\in[\mathfrak{s},\mathfrak{s}+T].
\end{align}
\end{itemize}
\end{lemma}
\begin{remark}
For convenience, we use $\Omega$ itself in place of $\widetilde{\Omega}$ throughout the work.
\end{remark}
\subsection{Abstract formulation}
In this subsection, we describe an abstract formulation and solution of \eqref{2}. Taking orthogonal projection $\mathcal{P}$ to the first equation of \eqref{2}, we obtain
\begin{equation}\label{WZ_SCBF}
\left\{
\begin{aligned}
\frac{\d \boldsymbol{u}(t)}{\d t}+\mu \mathrm{A}\boldsymbol{u}(t)+\mathrm{B}(\boldsymbol{u}(t))+\alpha\boldsymbol{u}(t)+\beta\mathcal{C}(\boldsymbol{u}(t))&=\boldsymbol{f}(t) + S(t,x,\boldsymbol{u}(t))\mathcal{Z}_{\delta}(\vartheta_t\omega), \ \ t>\mathfrak{s}, \\
\boldsymbol{u}(x,\mathfrak{s})&=\boldsymbol{u}_{\mathfrak{s}}(x), \ \ \ \ \ \ \ \ \ x\in \mathbb{R}^n \text{ and }\mathfrak{s}\in\mathbb{R}.
\end{aligned}
\right.
\end{equation}
Strictly speaking, one has to write $\mathcal{P}\boldsymbol{f}$ for $\boldsymbol{f}$ and $\mathcal{P}S(\cdot,\cdot,\cdot)$ for $S(\cdot,\cdot,\cdot)$ in \eqref{WZ_SCBF}.
\begin{definition}\label{def3.1}
Let us assume that $\mathfrak{s}\in\mathbb{R},$ $ \omega\in\Omega,$ $ \boldsymbol{u}_{\mathfrak{s}}\in \mathbb{H}$, $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R},\mathbb{V}')$, $T>0$ be any fixed time. Then, the function $\boldsymbol{u}(\cdot)$ is called a solution (in the weak sense) of the system \eqref{WZ_SCBF} on time interval $[\mathfrak{s},\mathfrak{s}+T]$, if $$\boldsymbol{u}\in \mathrm{L}^{\infty}(\mathfrak{s},\mathfrak{s}+T;\mathbb{H})\cap\mathrm{L}^2(\mathfrak{s}, \mathfrak{s}+T;\mathbb{V})\cap\mathrm{L}^{r+1}(\mathfrak{s},\mathfrak{s}+T;\widetilde{\mathbb{L}}^{r+1}),$$ with $ \partial_t\boldsymbol{u}\in \mathrm{L}^{2}(\mathfrak{s},\mathfrak{s}+T;\mathbb{V}')+\mathrm{L}^{\frac{r+1}{r}}(\mathfrak{s},\mathfrak{s}+T;\widetilde{\mathbb{L}}^{\frac{r+1}{r}})$ satisfying:
\begin{enumerate}
\item [(i)] for any $\psi\in \mathbb{V}\cap\widetilde{\mathbb{L}}^{r+1},$
\begin{align*}
(\boldsymbol{u}(t), \psi) &= (\boldsymbol{u}_{\mathfrak{s}}, \psi) - \mu \int\limits_{\mathfrak{s}}^{\mathfrak{s}+T}(\nabla\boldsymbol{u}(\xi),\nabla\psi)\d\xi-\int\limits_{\mathfrak{s}}^{\mathfrak{s}+T}b(\boldsymbol{u}(\xi),\boldsymbol{u}(\xi),\psi)\d\xi-\alpha\int\limits_{\mathfrak{s}}^{\mathfrak{s}+T}(\boldsymbol{u}(\xi),\psi)\d\xi\nonumber\\&\quad-\beta \int\limits_{\mathfrak{s}}^{\mathfrak{s}+T}\left\langle|\boldsymbol{u}(\xi)|^{r-1}\boldsymbol{u}(\xi) +\boldsymbol{f}(\xi) , \psi\right\rangle\d\xi + \int\limits_{\mathfrak{s}}^{\mathfrak{s}+T}(S(\xi,x,\boldsymbol{u}(s))\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega), \psi)\d\xi,
\end{align*}
for a.e. $t\in[\mathfrak{s},\mathfrak{s}+T]$,
\item [(ii)] the initial data is satisfied in the following sense:
$$\lim\limits_{t\downarrow \mathfrak{s}}\int_{\mathcal{O}}\boldsymbol{u}(t,x)\psi(x)\d x=\int_{\mathcal{O}}\boldsymbol{u}_{\mathfrak{s}}(x)\psi(x)\d x, $$ for all $\psi\in\mathbb{H}$,
\item [(iii)] the energy equality:
\begin{align*}
&\|\boldsymbol{u}(t)\|_{\mathbb{H}}^2+2\mu\int_{\mathfrak{s}}^t\|\boldsymbol{u}(s)\|_{\mathbb{V}}^2\d s+2\alpha\int_{\mathfrak{s}}^t\|\boldsymbol{u}(s)\|_{\mathbb{H}}^2\d s+2\beta\int_{\mathfrak{s}}^t\|\boldsymbol{u}(s)\|_{\widetilde\mathbb{L}^{r+1}}^{r+1}\d s\nonumber\\&=\|\boldsymbol{u}_{\mathfrak{s}}\|_{\mathbb{H}}^2+2\int_0^t(\boldsymbol{f}(s),\boldsymbol{u}(s))\d s+2\int_0^t(S(s,x,\boldsymbol{u}(s))\mathcal{Z}_{\delta}(\vartheta_s\omega),\boldsymbol{u}(s))\d s,
\end{align*}
for all $t\in[\mathfrak{s},T]$.
\end{enumerate}
\end{definition}
By a standard Galerkin method (see \cite{MTM,MTM1}), one can obtain that if any one of the assumptions \ref{NDT1}, \ref{NDT2} and \ref{NDT3} are fulfilled, then for all $t>\mathfrak{s}, \ \mathfrak{s}\in\mathbb{R},$ and for every $\boldsymbol{u}_{\mathfrak{s}}\in\mathbb{H}$ and $\omega\in\Omega$, \eqref{WZ_SCBF} has a unique solution in the sense of Definition \ref{def3.1}. Moreover, $\boldsymbol{u}(t,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}})$ is continuous with respect to initial data $\boldsymbol{u}_{\mathfrak{s}}(x)$ (see Lemmas \ref{Continuity}, \ref{ContinuityUB1} and \ref{ContinuityUB2}) and $(\mathscr{F},\mathscr{B}(\mathbb{H}))$-measurable in $\omega\in\Omega.$
Now, we define a cocycle $\Phi:\mathbb{R}^+\times\mathbb{R}\times\Omega\times\mathbb{H}\to\mathbb{H}$ for the system \eqref{WZ_SCBF} such that for given $t\in\mathbb{R}^+, \mathfrak{s}\in\mathbb{R}, \omega\in\Omega$ and $\boldsymbol{u}_{\mathfrak{s}}\in\mathbb{H}$, let
\begin{align}\label{Phi1}
\Phi(t,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}}) =\boldsymbol{u}(t+\mathfrak{s},\mathfrak{s},\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}}).
\end{align}
Then $\Phi$ is a continuous cocycle on $\mathbb{H}$ over $(\Omega,\mathscr{F},\mathbb{P},\{\vartheta_{t}\}_{t\in\mathbb{R}})$, where $\{\vartheta_t\}_{t\in\mathbb{R}}$ is given by \eqref{vartheta}, that is,
\begin{align}\label{Phi2}
\Phi(t+\mathfrak{s},s,\omega,\boldsymbol{u}_{\mathfrak{s}})=\Phi(t,\mathfrak{s}+s,\vartheta_{\mathfrak{s}}\omega,\Phi(\mathfrak{s},s,\omega,\boldsymbol{u}_{\mathfrak{s}})).
\end{align}
Assume that $D=\{D(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R},\omega\in\Omega\}$ is a family of non-empty subsets of $\mathbb{H}$ satisfying, for every $c>0, \mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$,
\begin{align}\label{D_1}
\lim_{t\to\infty}e^{-ct}\|D(\mathfrak{s}-t,\vartheta_{-t}\omega)\|_{\mathbb{H}}=0,
\end{align}
where $\|D\|_{\mathbb{H}}=\sup\limits_{\boldsymbol{x}\in D}\|\boldsymbol{x}\|_{\mathbb{H}}.$ Let $\mathfrak{D}$ be the collection of all tempered families of bounded non-empty subsets of $\mathbb{H}$, that is,
\begin{align}\label{D_11}
\mathfrak{D}=\big\{D=\{D(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}\text{ and }\omega\in\Omega\}:D \text{ satisfying } \eqref{D_1}\big\}.
\end{align}
\section{Random pullback attractors for Wong-Zakai approximations: bounded domains} \label{sec3}\setcounter{equation}{0}
In this section, we prove the existence of unique random $\mathfrak{D}$-pullback attractor for the system \eqref{WZ_SCBF} on bounded domains with nonlinear diffusion term. Throughout the section, we assume that $\mathcal{O}$ is bounded subset of $\mathbb{R}^d$.
\subsection{Nonlinear diffusion term}
The nonlinear diffusion term $S(t,x,\boldsymbol{u})$ appearing in \eqref{WZ_SCBF} satisfies the following assumption:
\begin{assumption}\label{NDT1}
We assume that the nonlinear diffusion term $$S(t,x,\boldsymbol{u})=e^{\sigma t}\left[\kappa\boldsymbol{u}+\mathcal{S}(\boldsymbol{u})+\boldsymbol{h}(x)\right],$$ where $\sigma\geq0, \kappa\geq0$ and $\boldsymbol{h}\in\mathbb{H}$. Also, $\mathcal{S}:\mathbb{V}\to\mathbb{H}$ is a continuous function satisfying
\begin{align}
\|\mathcal{S}(\boldsymbol{u})-\mathcal{S}(\boldsymbol{v})\|_{\mathbb{H}}&\leq s_1\|\boldsymbol{u}-\boldsymbol{v}\|_{\mathbb{V}}, \ \ \ \ \ \ \ \ \ \ \text{ for all } \boldsymbol{u},\boldsymbol{v}\in\mathbb{V},\label{S2}\\
|\left(\mathcal{S}(\boldsymbol{u})-\mathcal{S}(\boldsymbol{v}),\boldsymbol{w}\right)|&\leq s_2\|\boldsymbol{u}-\boldsymbol{v}\|_{\mathbb{H}}\|\boldsymbol{w}\|_{\mathbb{V}}, \ \ \ \text{ for all } \boldsymbol{u},\boldsymbol{v},\boldsymbol{w} \in\mathbb{V},\label{S3}\\
|\left(\mathcal{S}(\boldsymbol{u}),\boldsymbol{u}\right)|&\leq s_3 + s_4\|\boldsymbol{u}\|_{\mathbb{V}}^{1+s_5}, \ \ \ \ \ \ \text{ for all }\boldsymbol{u}\in\mathbb{V},\label{S4}
\end{align}
where $s_1$ and $s_2$ both are non-negative constants, $s_3\geq0, s_4\geq0$ and $s_5\in[0,1)$.
\end{assumption}
\begin{remark}
One can take $\boldsymbol{h}\in\widetilde{\mathbb{L}}^{\frac{r+1}{r}}$ also.
\end{remark}
\begin{example}
Let us discuss an example for such a nonlinear diffusion term which satisfies the above conditions. Let $\mathcal{S}:\mathbb{V}\to\mathbb{H}$ be a nonlinear operator defined by $\mathcal{S}(\boldsymbol{u})=\sin\boldsymbol{u}+\mathrm{B}(\boldsymbol{g}_1,\boldsymbol{u})$ for all $\boldsymbol{u}\in\mathbb{V}$, where $\boldsymbol{g}_1\in\mathrm{D}(\mathrm{A})$ is a fixed element. It is easy to verify that $\mathcal{S}$ satisfies \eqref{S2}-\eqref{S4} (see \cite{GLW}).
\end{example}
\subsection{Deterministic nonautonomous forcing term}
In the sequel, the following assumptions are needed on the external forcing term $\boldsymbol{f}$.
\begin{assumption}\label{DNFT1}
There exists a number $\gamma\in[0,\alpha)$ such that:
\begin{itemize}
\item [(i)]
\begin{align}\label{forcing1}
\int_{-\infty}^{\mathfrak{s}} e^{\gamma\xi}\|\boldsymbol{f}(\cdot,\xi)\|^2_{\mathbb{V}'}\d \xi<\infty, \ \ \text{ for all }\ \mathfrak{s}\in\mathbb{R}.
\end{align}
\item [(ii)] for every $c>0$
\begin{align}\label{forcing2}
\lim_{\tau\to-\infty}e^{c\tau}\int_{-\infty}^{0} e^{\gamma\xi}\|\boldsymbol{f}(\cdot,\xi+\tau)\|^2_{\mathbb{V}'}\d \xi=0.
\end{align}
\end{itemize}
It should be noted that \eqref{forcing2} implies \eqref{forcing1} for $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$.
\end{assumption}
\begin{lemma}\label{Continuity}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu>1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ and Assumption \ref{NDT1} is fulfilled. Then, the solution of \eqref{WZ_SCBF} is continuous in initial data $\boldsymbol{u}_{\mathfrak{s}}(x).$
\end{lemma}
\begin{proof}
Let $\boldsymbol{u}_{1}(\cdot)$ and $\boldsymbol{u}_{2}(\cdot)$ be two solutions of \eqref{WZ_SCBF}. Then $\mathfrak{X}(\cdot)=\boldsymbol{u}_{1}(\cdot)-\boldsymbol{u}_{2}(\cdot)$ with $\mathfrak{X}(\mathfrak{s})=\boldsymbol{u}_{1,\mathfrak{s}}(x)-\boldsymbol{u}_{2,\mathfrak{s}}(x)$ satisfies
\begin{align}\label{Conti1}
\frac{\d\mathfrak{X}(t)}{\d t}&=-\mu \mathrm{A}\mathfrak{X}(t)-\alpha\mathfrak{X}(t)-\left\{\mathrm{B}\big(\boldsymbol{u}_{1}(t)\big)-\mathrm{B}\big(\boldsymbol{u}_{2}(t)\big)\right\} -\beta\left\{\mathcal{C}\big(\boldsymbol{u}_1(t)\big)-\mathcal{C}\big(\boldsymbol{u}_2(t)\big)\right\}\nonumber\\&\quad+e^{\sigma t}\left[\kappa\mathfrak{X}(t)+\mathcal{S}(\boldsymbol{u}_1(t))-\mathcal{S}(\boldsymbol{u}_2(t))\right]\mathcal{Z}_{\delta}(\vartheta_t\omega),
\end{align}
in $\mathbb{V}'+\widetilde{\mathbb{L}}^{\frac{r+1}{r}}$ for a.e. $t\in[0,T]$. Taking the inner product with $\mathfrak{X}(\cdot)$ to the first equation in \eqref{Conti1}, we obtain
\begin{align}\label{Conti2}
\frac{1}{2}\frac{\d}{\d t} \|\mathfrak{X}(t)\|^2_{\mathbb{H}} &=-\mu \|\nabla\mathfrak{X}(t)\|^2_{\mathbb{H}} - \alpha\|\mathfrak{X}(t)\|^2_{\mathbb{H}} -\left\langle\mathrm{B}\big(\boldsymbol{u}_1(t)\big)-\mathrm{B}\big(\boldsymbol{u}_2(t)\big), \mathfrak{X}(t)\right\rangle \nonumber\\&\quad-\beta\left\langle\mathcal{C}\big(\boldsymbol{u}_1(t)\big)-\mathcal{C}\big(\boldsymbol{u}_2(t)\big),\mathfrak{X}(t)\right\rangle + \kappa e^{\sigma t}\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\|\mathfrak{X}(t)\|^2_{\mathbb{H}}\nonumber\\&\quad+e^{\sigma t}\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\left(\mathcal{S}(\boldsymbol{u}_1(t))-\mathcal{S}(\boldsymbol{u}_2(t)),\mathfrak{X}(t)\right) ,
\end{align}
for a.e. $t\in[\mathfrak{s},\mathfrak{s}+T] \text{ with } T>0$. By \eqref{S3}, we get
\begin{align}\label{ContiS}
\kappa & e^{\sigma t}\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\|\mathfrak{X}\|^2_{\mathbb{H}}+e^{\sigma t}\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\left(\mathcal{S}(\boldsymbol{u}_1)-\mathcal{S}(\boldsymbol{u}_2),\mathfrak{X}\right)\nonumber\\&\leq \kappa e^{\sigma t}\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|\|\mathfrak{X}\|^2_{\mathbb{H}}+s_2 e^{\sigma t}\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|\|\mathfrak{X}\|_{\mathbb{H}} \|\mathfrak{X}\|_{\mathbb{V}}\nonumber\\&\leq \frac{\alpha}{4}\|\mathfrak{X}\|^2_{\mathbb{H}}+ \frac{\min\{\mu,\alpha\}}{4}\|\mathfrak{X}\|^2_{\mathbb{V}} +Ce^{2\sigma t}\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|^2\|\mathfrak{X}\|^2_{\mathbb{H}}.
\end{align}
\vskip 2mm
\noindent
\textbf{Case I:} \textit{When $d=2$ and $r\geq1$.} Using \eqref{b1}, \eqref{441} and Lemma \ref{Young}, we obtain
\begin{align}\label{Conti3}
\left| \left\langle\mathrm{B}\big(\boldsymbol{u}_1\big)-\mathrm{B}\big(\boldsymbol{u}_2\big), \mathfrak{X}\right\rangle\right|&=\left|\left\langle\mathrm{B}\big(\mathfrak{X},\mathfrak{X} \big), \boldsymbol{u}_2\right\rangle\right|\leq\frac{\mu}{4}\|\nabla\mathfrak{X}\|^2_{\mathbb{H}}+C\|\boldsymbol{u}_2\|^4_{\widetilde{\mathbb{L}}^4}\|\mathfrak{X}\|^2_{\mathbb{H}}\nonumber\\&\leq\frac{\mu}{4}\|\nabla\mathfrak{X}\|^2_{\mathbb{H}}+C\|\boldsymbol{u}_2\|^2_{\mathbb{H}}\|\nabla\boldsymbol{u}_2\|^2_{\mathbb{H}}\|\mathfrak{X}\|^2_{\mathbb{H}},
\end{align}
and from \eqref{MO_c}, we have
\begin{align}\label{Conti4}
-\beta\left\langle\mathcal{C}\big(\boldsymbol{u}_1\big)-\mathcal{C}\big(\boldsymbol{u}_2\big),\mathfrak{X}\right\rangle\leq 0.
\end{align}
Making use of \eqref{ContiS}-\eqref{Conti4} in \eqref{Conti2}, we get
\begin{align*}
& \frac{\d}{\d t} \|\mathfrak{X}(t)\|^2_{\mathbb{H}} \leq C\bigg\{e^{2\sigma t}\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|^2+ \|\boldsymbol{u}_2(t)\|^2_{\mathbb{H}}\|\nabla\boldsymbol{u}_2(t)\|^2_{\mathbb{H}}\bigg\}\|\mathfrak{X}(t)\|^2_{\mathbb{H}},
\end{align*}
for a.e. $t\in[\mathfrak{s},\mathfrak{s}+T]$ and an application of Gronwall's inequality completes the proof.
\iffalse
gives
\begin{align*}
\|\mathfrak{X}(t)\|^2_{\mathbb{H}}&\leq e^{C\int_{\mathfrak{s}}^{t}\left\{e^{2\sigma \tau}\left|\mathcal{Z}_{\delta}(\vartheta_{\tau}\omega)\right|^2+\|\boldsymbol{u}_2(\tau)\|^2_{\mathbb{H}}\|\nabla\boldsymbol{u}_2(\tau)\|^2_{\mathbb{H}}\right\}\d \tau}\|\mathfrak{X}(\mathfrak{s})\|^2_{\mathbb{H}}\nonumber\\&\leq e^{CTe^{2\sigma (\mathfrak{s}+T)}\sup\limits_{\tau\in[\mathfrak{s},\mathfrak{s}+T]}\left|\mathcal{Z}_{\delta}(\vartheta_{\tau}\omega)\right|^2+C\sup\limits_{\tau\in[\mathfrak{s},\mathfrak{s}+T]}\|\boldsymbol{u}_2(\tau)\|^2_{\mathbb{H}}\int\limits_{\mathfrak{s}}^{\mathfrak{s}+T}\|\nabla\boldsymbol{u}_2(\tau)\|^2_{\mathbb{H}}\d \tau}\|\mathfrak{X}(\mathfrak{s})\|^2_{\mathbb{H}},
\end{align*} which completes the proof.
\fi
\vskip 2mm
\noindent
\textbf{Case II:} \textit{When $d= 3$ and $r>3$.} The nonlinear term $|(\mathrm{B}(\mathfrak{X},\mathfrak{X}),\boldsymbol{u}_2)|$ is estimated using Lemmas \ref{Holder} and \ref{Young} as
\begin{align}\label{3d-ab12}
\left|(\mathrm{B}(\mathfrak{X},\mathfrak{X}),\boldsymbol{u}_2)\right|&\leq \||\boldsymbol{u}_2||\mathfrak{X}|\|_{\mathbb{H}}\|\nabla\mathfrak{X}\|_{\mathbb{H}}\leq\frac{\mu}{4}\|\nabla\mathfrak{X}\|_{\mathbb{H}}^2+\frac{1}{\mu}\||\boldsymbol{u}_2||\mathfrak{X}|\|_{\mathbb{H}}^2\nonumber\\&\leq\frac{\mu}{4}\|\nabla\mathfrak{X}\|_{\mathbb{H}}^2+\frac{\beta}{2}\||\mathfrak{X}||\boldsymbol{u}_2|^{\frac{r-1}{2}}\|^2_{\mathbb{H}}+\eta_1\|\mathfrak{X}\|^2_{\mathbb{H}},
\end{align}
where $\eta_1= \frac{r-3}{2\mu(r-1)}\left[\frac{4}{\beta\mu (r-1)}\right]^{\frac{2}{r-3}}$ (see \cite{MTM1} for details). Using \eqref{441} and \eqref{MO_c}, we have
\begin{align}\label{Conti7}
\left|\left\langle\mathrm{B}\big(\boldsymbol{u}_1\big)-\mathrm{B}\big(\boldsymbol{u}_2\big), \mathfrak{X}\right\rangle\right|=\left|\left\langle\mathrm{B}\big(\mathfrak{X},\mathfrak{X} \big), \boldsymbol{u}_2\right\rangle\right|
\end{align}
and
\begin{align}\label{Conti8}
-\beta \left\langle\mathcal{C}\big(\boldsymbol{u}_1\big)-\mathcal{C}\big(\boldsymbol{u}_2\big),\mathfrak{X}\right\rangle\leq - \frac{\beta}{2}\||\mathfrak{X}||\boldsymbol{u}_2|^{\frac{r-1}{2}}\|^2_{\mathbb{H}},
\end{align}
respectively. Combining \eqref{Conti2}-\eqref{ContiS} and \eqref{3d-ab12}-\eqref{Conti8}, we get
\begin{align*}
& \frac{\d}{\d t} \|\mathfrak{X}(t)\|^2_{\mathbb{H}} \leq\biggl\{e^{2\sigma t}\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|^2+2\eta_1\biggr\}\|\mathfrak{X}(t)\|^2_{\mathbb{H}},
\end{align*} for a.e. $t\in[\mathfrak{s},\mathfrak{s}+T]$, which completes the proof using Gronwall's inequality.
\vskip 2mm
\noindent
\textbf{Case III:} \textit{When $d=3$ and $r=3$ with $2\beta\mu>1$.} Since $2\beta\mu>1$, there exists $0<\theta<1$ such that
\begin{align}\label{Theta}
2\beta\mu\geq\frac{1}{\theta}.
\end{align} Using \eqref{b1} and \eqref{441}, we have
\begin{align}\label{Conti5}
\left|\left\langle\mathrm{B}\big(\boldsymbol{u}_1\big)-\mathrm{B}\big(\boldsymbol{u}_2\big), \mathfrak{X}\right\rangle\right|&=\left|\left\langle\mathrm{B}\big(\mathfrak{X},\mathfrak{X} \big), \boldsymbol{u}_2\right\rangle\right|\leq\theta\mu\|\nabla\mathfrak{X}\|^2_{\mathbb{H}}+\frac{1}{4\theta\mu}\||\boldsymbol{u}_2||\mathfrak{X}|\|^2_{\mathbb{H}},
\end{align}
where $\theta$ is the same as in \eqref{Theta}. Again by \eqref{S3}, we obtain
\begin{align}\label{ContiS1}
\kappa & e^{\sigma t}\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\|\mathfrak{X}\|^2_{\mathbb{H}}+e^{\sigma t}\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\left(\mathcal{S}(\boldsymbol{u}_1)-\mathcal{S}(\boldsymbol{u}_2),\mathfrak{X}\right)\nonumber\\&\leq \frac{\alpha}{2}\|\mathfrak{X}\|^2_{\mathbb{H}}+ (1-\theta)\mu\|\mathfrak{X}\|^2_{\mathbb{V}} +Ce^{2\sigma t}\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|^2\|\mathfrak{X}\|^2_{\mathbb{H}}.
\end{align}
Using \eqref{Conti8}-\eqref{Conti5} in \eqref{Conti2}, we get
\begin{align*}
& \frac{\d}{\d t} \|\mathfrak{X}(t)\|^2_{\mathbb{H}} \leq 2(1-\theta)\mu\|\mathfrak{X}(t)\|^2_{\mathbb{H}} +Ce^{2\sigma t}\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|^2\|\mathfrak{X}(t)\|^2_{\mathbb{H}} ,
\end{align*} for a.e. $t\in[\mathfrak{s},\mathfrak{s}+T]$.
After applying Gronwall's inequality, one can conclude the proof.
\end{proof}
Next result supports us to show the existence of random $\mathfrak{D}$-pullback absorbing set for continuous cocycle $\Phi$.
\begin{lemma}\label{LemmaUe}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing1} and Assumption \ref{NDT1} is fulfilled. Then for every $0<\delta\leq1$, $\mathfrak{s}\in\mathbb{R},$ $ \omega\in \Omega$ and $D=\{D(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D},$ there exists $\mathcal{T}=\mathcal{T}(\delta, \mathfrak{s}, \omega, D)>0$ such that for all $t\geq \mathcal{T}$ and $\tau\geq \mathfrak{s}-t$, the solution $\boldsymbol{u}$ of the system \eqref{WZ_SCBF} with $\omega$ replaced by $\vartheta_{-\mathfrak{s}}\omega$ satisfies
\begin{align}\label{ue}
& \|\boldsymbol{u}(\tau,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{H}} \nonumber\\&\leq\frac{4}{\min\{\mu,\alpha\}} \int_{-\infty}^{\tau-\mathfrak{s}} e^{\int_{\tau-\mathfrak{s}}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d \xi\nonumber\\&\quad+\int_{-\infty}^{\tau-\mathfrak{s}} e^{\int_{\tau-\mathfrak{s}}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\bigg\{2s_3e^{\sigma (\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|\nonumber\\&\qquad+2s_6 e^{2\sigma (\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2+s_7\left[e^{\sigma(\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|\right]^{\frac{2}{1-s_5}}\bigg\}\d\xi,
\end{align}
where $\boldsymbol{u}_{\mathfrak{s}-t}\in D(\mathfrak{s}-t,\vartheta_{-t}\omega)$, $s_6=\frac{2}{\alpha}\|\boldsymbol{h}\|^2_{\mathbb{H}}$, $s_7=s_4(1-s_5)\left[\frac{4s_4(1+s_5)}{\min\{\mu,\alpha\}}\right]^{\frac{1+s_5}{1-s_5}}$ and, $s_3, s_4$ and $s_5$ are the constants appearing in \eqref{S4}.
\end{lemma}
\begin{proof}
From the first equation of the system \eqref{WZ_SCBF}, we obtain
\begin{align}\label{ue0}
& \frac{\d}{\d t} \|\boldsymbol{u}\|^2_{\mathbb{H}} +2\mu\|\nabla\boldsymbol{u}\|^2_{\mathbb{H}} + 2\alpha\|\boldsymbol{u}\|^2_{\mathbb{H}} + 2\beta\|\boldsymbol{u}\|^{r+1}_{\widetilde \mathbb{L}^{r+1}}\nonumber\\&= 2\left\langle\boldsymbol{f},\boldsymbol{u}\right\rangle +2e^{\sigma t}\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\left(\kappa\boldsymbol{u}+\mathcal{S}(\boldsymbol{u})+\boldsymbol{h},\boldsymbol{u}\right)\nonumber\\&\leq 2\|\boldsymbol{f}\|_{\mathbb{V}'}\|\boldsymbol{u}\|_{\mathbb{V}}+2\kappa e^{\sigma t}\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\|\boldsymbol{u}\|^2_{\mathbb{H}}+2e^{\sigma t}\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|\left(s_3+s_4\|\boldsymbol{u}\|^{1+s_5}_{\mathbb{V}}+\|\boldsymbol{h}\|_{\mathbb{H}}\|\boldsymbol{u}\|_{\mathbb{H}}\right)\nonumber\\&\leq\frac{\alpha}{2}\|\boldsymbol{u}\|^2_{\mathbb{H}}+ \frac{\min\{\mu,\alpha\}}{2}\|\boldsymbol{u}\|^2_{\mathbb{V}}+\frac{4\|\boldsymbol{f}\|^2_{\mathbb{V}'}}{\min\{\mu,\alpha\}}+2\kappa e^{\sigma t}\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\|\boldsymbol{u}\|^2_{\mathbb{H}}+2s_3e^{\sigma t}\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|\nonumber\\&\quad+s_6 e^{2\sigma t}\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|^2+s_7\left[e^{\sigma t}\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|\right]^{\frac{2}{1-s_5}},
\end{align}
where we have used \eqref{b0}, \eqref{S4}, Lemmas \ref{Holder} and \ref{Young}, and the constants $s_6=\frac{2}{\alpha}\|\boldsymbol{h}\|^2_{\mathbb{H}}$ and $s_7=s_4(1-s_5)\left[\frac{4s_4(1+s_5)}{\min\{\mu,\alpha\}}\right]^{\frac{1+s_5}{1-s_5}}$. We rewrite \eqref{ue0} as
\begin{align}\label{ue1}
&\frac{\d}{\d t} \|\boldsymbol{u}\|^2_{\mathbb{H}}+ \left(\alpha-2\kappa e^{\sigma t}\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right)\|\boldsymbol{u}\|^2_{\mathbb{H}}\nonumber\\& \leq \frac{4\|\boldsymbol{f}\|^2_{\mathbb{V}'}}{\min\{\mu,\alpha\}}+2s_3e^{\sigma t}\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|+s_6 e^{2\sigma t}\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|^2+s_7\left[e^{\sigma t}\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|\right]^{\frac{2}{1-s_5}}.
\end{align}
Making use of variation of constant formula in \eqref{ue1} and replacing $\omega$ by $\vartheta_{-\mathfrak{s}}\omega$, we get
\begin{align}\label{ue2}
& \|\boldsymbol{u}(\tau,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{H}} \nonumber\\&\leq e^{\int_{\tau}^{\mathfrak{s}-t}\left(\alpha-2\kappa e^{\sigma \zeta}\mathcal{Z}_{\delta}(\vartheta_{\zeta-\mathfrak{s}}\omega)\right)\d\zeta}\|\boldsymbol{u}_{\mathfrak{s}-t}\|^2_{\mathbb{H}}+ \frac{4}{\min\{\mu,\alpha\}}\int_{\mathfrak{s}-t}^{\tau} e^{\int_{\tau}^{\xi}\left(\alpha-2\kappa e^{\sigma \zeta}\mathcal{Z}_{\delta}(\vartheta_{\zeta-\mathfrak{s}}\omega)\right)\d\zeta} \|\boldsymbol{f}(\cdot,\xi)\|^2_{\mathbb{V}'}\d \xi\nonumber\\&\quad+\int_{\mathfrak{s}-t}^{\tau} e^{\int_{\tau}^{\xi}\left(\alpha-2\kappa e^{\sigma \zeta}\mathcal{Z}_{\delta}(\vartheta_{\zeta-\mathfrak{s}}\omega)\right)\d\zeta}\bigg\{2s_3e^{\sigma \xi}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi-\mathfrak{s}}\omega)\right|+s_6 e^{2\sigma \xi}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi-\mathfrak{s}}\omega)\right|^2\nonumber\\&\qquad+s_7\left[e^{\sigma\xi}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi-\mathfrak{s}}\omega)\right|\right]^{\frac{2}{1-s_5}}\bigg\}\d\xi\nonumber\\&\leq e^{\int_{\tau-\mathfrak{s}}^{-t}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}_{\mathfrak{s}-t}\|^2_{\mathbb{H}}\nonumber\\&\quad+ \frac{4}{\min\{\mu,\alpha\}}\int_{-t}^{\tau-\mathfrak{s}} e^{\int_{\tau-\mathfrak{s}}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d \xi\nonumber\\&\quad+\int_{-t}^{\tau-\mathfrak{s}} e^{\int_{\tau-\mathfrak{s}}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\bigg\{2s_3e^{\sigma (\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|+s_6 e^{2\sigma (\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2\nonumber\\&\qquad+s_7\left[e^{\sigma(\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|\right]^{\frac{2}{1-s_5}}\bigg\}\d\xi.
\end{align}
Now, we need to estimate each term on the right hand side of \eqref{ue2}. Depending on $\sigma$, there are two possible cases.
\vskip1mm
\noindent
\textbf{Case I:} \textit{When $\sigma=0$.} By \eqref{N4}, we have
\begin{align}\label{ue3}
\lim_{\xi\to-\infty}\frac{1}{\xi}\int_{0}^{\xi}\left(\alpha-2\kappa \mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta=\alpha-2\kappa\mathbb{E}[\mathcal{Z}_{\delta}]=\alpha.
\end{align}
Since $\gamma<\alpha$ (see Assumption \ref{DNFT1}), from \eqref{ue3}, we infer that there exists $\xi_0=\xi_0(\delta,\omega)<0$ such that for all $\xi\leq\xi_0$,
\begin{align}\label{ue4}
\int_{0}^{\xi}\left(\alpha-2\kappa \mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta<\gamma\xi.
\end{align}
Taking \eqref{forcing1} and \eqref{ue4} into account, we obtain
\begin{align}\label{ue5}
\int_{-\infty}^{\xi_0}e^{\int_{0}^{\xi}\left(\alpha-2\kappa \mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d\xi<\int_{-\infty}^{\xi_0}e^{\gamma\xi}\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d\xi<\infty,
\end{align}
and similarly from \eqref{N3} and \eqref{ue4}, we get
\begin{align}\label{ue6}
\int_{-\infty}^{\xi_0}e^{\int_{0}^{\xi}\left(\alpha-2\kappa \mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\bigg\{2s_3\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|+s_6\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2+s_7\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{2}{1-s_5}}\bigg\}\d\xi<\infty.
\end{align}
\vskip1mm
\noindent
\textbf{Case II:} \textit{When $\sigma>0$.} From \eqref{N3}, we get
\begin{align*}
\lim_{\zeta\to-\infty}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)=\alpha>\gamma,
\end{align*}
therefore, there exists $\zeta_0=\zeta_0(\mathfrak{s},\delta,\omega)<0$ such that for all $\zeta\leq\zeta_0$,
\begin{align}\label{ue7}
\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)>\gamma.
\end{align}
By \eqref{forcing1} and \eqref{ue7}, we have
\begin{align}\label{ue8}
\int_{-\infty}^{\zeta_0}e^{\int_{\zeta_0}^{\xi}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})} \mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d\xi< e^{-\gamma\zeta_0}\int_{-\infty}^{\zeta_0}e^{\gamma\xi}\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d\xi<\infty,
\end{align}
and similarly by \eqref{N3} and \eqref{ue7}, it is easy to obtain
\begin{align}\label{ue9}
& \int_{-\infty}^{\xi_0}e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})} \mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\bigg\{2s_3e^{\sigma(\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|+s_6e^{2\sigma(\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2\nonumber\\&\qquad+s_7\left[e^{\sigma(\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|\right]^{\frac{2}{1-s_5}}\bigg\}\d\xi<\infty.
\end{align}
It follows from \eqref{ue5}-\eqref{ue6} and \eqref{ue8}-\eqref{ue9} that for any $\sigma\geq0$,
\begin{align}\label{ue10}
&\int_{-\infty}^{\tau-\mathfrak{s}} e^{\int_{\tau-\mathfrak{s}}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d \xi+\int_{-\infty}^{\tau-\mathfrak{s}} e^{\int_{\tau-\mathfrak{s}}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\nonumber\\&\quad\times\bigg\{2s_3e^{\sigma (\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|+s_6 e^{2\sigma (\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2+s_7\left[e^{\sigma(\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|\right]^{\frac{2}{1-s_5}}\bigg\}\d\xi<\infty.
\end{align}
Finally, since $\boldsymbol{u}_{\mathfrak{s}-t}\in D(\mathfrak{s}-t,\vartheta_{-t}\omega)$ and $D\in\mathfrak{D}$, making use of \eqref{ue4} and \eqref{ue7}, we have for $\sigma\geq0$,
\begin{align*}
e^{\int_{0}^{-t}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}_{\mathfrak{s}-t}\|^2_{\mathbb{H}}\leq e^{\int_{0}^{-t}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|D(\mathfrak{s}-t,\vartheta_{-t}\omega)\|^2_{\mathbb{H}}\to0,
\end{align*}
as $t\to\infty$, there exists $\mathcal{T}=\mathcal{T}(\delta,\mathfrak{s},\omega,D)>0$ such that for all $t\geq\mathcal{T}$,
\begin{align}\label{ue11}
&e^{\int_{\tau-\mathfrak{s}}^{-t}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}_{\mathfrak{s}-t}\|^2_{\mathbb{H}}\nonumber\\&\leq s_6\int_{-\infty}^{\tau-\mathfrak{s}} e^{\int_{\tau-\mathfrak{s}}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} e^{2\sigma (\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2\d\xi.
\end{align}
From \eqref{ue2} along with \eqref{ue10}-\eqref{ue11}, one can conclude the proof.
\end{proof}
Further, we prove $\mathfrak{D}$-pullback asymptotic compactness using compactness of Sobolev embeddings on bounded domains.
\begin{lemma}\label{PCB}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing1} and Assumption \ref{NDT1} is fulfilled. Then for every $0<\delta\leq1, \omega\in\Omega, \mathfrak{s}\in\mathbb{R}$ and $t>\mathfrak{s}$, the solution $\boldsymbol{u}(t,\mathfrak{s},\omega,\cdot):\mathbb{H}\to\mathbb{H}$ is compact, that is, for every bounded set $B$ in $\mathbb{H}$, the image $\boldsymbol{u}(t,\mathfrak{s},\omega,B)$ is precompact in $\mathbb{H}$.
\end{lemma}
\begin{proof}
Consider the solution $\boldsymbol{u}(\tau,\mathfrak{s},\omega,\cdot)$ of \eqref{WZ_SCBF} for $\tau\in[\mathfrak{s},\mathfrak{s}+T]$, where $T>0$. Assume that the sequence $\{\boldsymbol{u}_{0,n}\}_{n\in\mathbb{N}}\subset B$. We know that (see the proof of Lemma \ref{LemmaUe})
\begin{equation}\label{PCB1}
\left\{
\begin{aligned}
\{\boldsymbol{u}(\cdot,\mathfrak{s},\omega,\boldsymbol{u}_{0,n})\}_{n\in\mathbb{N}} & \text{ is bounded in }\\ \mathrm{L}^{\infty}(\mathfrak{s},\mathfrak{s}+T;\mathbb{H})\cap\mathrm{L}^2(\mathfrak{s}, \mathfrak{s}+T;&\mathbb{V})\cap\mathrm{L}^{r+1}(\mathfrak{s},\mathfrak{s}+T;\widetilde{\mathbb{L}}^{r+1}).
\end{aligned}
\right.
\end{equation}
From \eqref{S2} along with \eqref{PCB1}, we obtain
\begin{align}\label{PCB2}
\{S(\cdot,\cdot,\boldsymbol{u}(\tau,\mathfrak{s},\omega,\boldsymbol{u}_{0,n}))\mathcal{Z}_{\delta}(\vartheta_{\tau}\omega)\}_{n\in\mathbb{N}} \text{ is bounded in } \mathrm{L}^{2}(\mathfrak{s},\mathfrak{s}+T;\mathbb{H}).
\end{align}
We also have
\begin{align}\label{PCB3}
\{\mathrm{A}(\boldsymbol{u}(\tau,\mathfrak{s},\omega,\boldsymbol{u}_{0,n}))\}_{n\in\mathbb{N}} \text{ and } \{\mathrm{B}(\boldsymbol{u}(\tau,\mathfrak{s},\omega,\boldsymbol{u}_{0,n}))\}_{n\in\mathbb{N}} \text{ are bounded in } \mathrm{L}^2(\mathfrak{s},\mathfrak{s}+T;\mathbb{V}'),
\end{align}
and
\begin{align}\label{PCB4}
\{\mathcal{C}(\boldsymbol{u}(\tau,\mathfrak{s},\omega,\boldsymbol{u}_{0,n}))\}_{n\in\mathbb{N}} \text{ is bounded in } \mathrm{L}^{\frac{r+1}{r}}(\mathfrak{s},\mathfrak{s}+T;\widetilde{\mathbb{L}}^{\frac{r+1}{r}}).
\end{align}
It follows from \eqref{PCB1}-\eqref{PCB4} and \eqref{WZ_SCBF} that
\begin{align*}
\left\{\frac{\d}{\d s}(\boldsymbol{u}(\tau,\mathfrak{s},\omega,\boldsymbol{u}_{0,n}))\right\}_{n\in\mathbb{N}} \text{ is bounded in } \mathrm{L}^2(\mathfrak{s},\mathfrak{s}+T;\mathbb{V}')+\mathrm{L}^{\frac{r+1}{r}}(\mathfrak{s},\mathfrak{s}+T;\widetilde{\mathbb{L}}^{\frac{r+1}{r}}).
\end{align*}
Since $\mathrm{L}^2(\mathfrak{s},\mathfrak{s}+T;\mathbb{V}')+\mathrm{L}^{\frac{r+1}{r}}(\mathfrak{s},\mathfrak{s}+T;\widetilde{\mathbb{L}}^{\frac{r+1}{r}})\subset\mathrm{L}^{\frac{r+1}{r}}(\mathfrak{s}+T;\mathbb{V}'+\widetilde\mathbb{L}^{\frac{r+1}{r}})$, the above sequence is bounded in $\mathrm{L}^{\frac{r+1}{r}}(\mathfrak{s}+T;\mathbb{V}'+\widetilde\mathbb{L}^{\frac{r+1}{r}})$. Note also that $\mathbb{V}\cap\widetilde\mathbb{L}^{\frac{r+1}{r}}\subset\mathbb{V}\subset\mathbb{H}\subset\mathbb{V}'\subset \mathbb{V}'+\widetilde\mathbb{L}^{\frac{r+1}{r}}$ and the embedding of $\mathbb{V}\subset\mathbb{H}$ is compact. By the \emph{Aubin-Lions compactness lemma}, there exists a subsequence (keeping as it is) and $\boldsymbol{v}\in\mathrm{L}^2(\mathfrak{s},\mathfrak{s}+T;\mathbb{H})$ such that
\begin{align}\label{PCB6}
\boldsymbol{u}(\cdot,\mathfrak{s},\omega,\boldsymbol{u}_{0,n})\to\boldsymbol{v}(\cdot) \ \text{ strongly in }\ \mathrm{L}^{2}(\mathfrak{s},\mathfrak{s}+T;\mathbb{H}).
\end{align}
Along a further subsequence (again not relabeling), we infer from \eqref{PCB6} that
\begin{align}\label{PCB7}
\boldsymbol{u}(\tau,\mathfrak{s},\omega,\boldsymbol{u}_{0,n})\to\boldsymbol{v}(\tau) \text{ in } \mathbb{H} \ \text{ for almost all }\ \tau\in(\mathfrak{s},\mathfrak{s}+T).
\end{align}
Since $\mathfrak{s}<t<T$, we obtain from \eqref{PCB7} that there exists $\tau\in(\mathfrak{s},t)$ such that \eqref{PCB7} holds true for this particular $\tau$. Then by Lemma \ref{Continuity}, we obtain
\begin{align*}
\boldsymbol{u}(t,\mathfrak{s},\omega,\boldsymbol{u}_{0,n})=\boldsymbol{u}(t,\tau,\omega,\boldsymbol{u}(\tau,\mathfrak{s},\omega,\boldsymbol{u}_{0,n}))\to \boldsymbol{u}(t,\tau,\omega,\boldsymbol{v}(\tau)),
\end{align*}
which completes the proof.
\end{proof}
In fact, Lemma \ref{PCB} helps us to prove the $\mathfrak{D}$-pullback asymptotic compactness of $\Phi$ in $\mathbb{H}$ on bounded domains.
\begin{corollary}\label{Asymptotic_B}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing1} and Assumption \ref{NDT1} is fulfilled. Then for every $0<\delta\leq1$, $\mathfrak{s}\in \mathbb{R},$ $\omega\in \Omega,$ $D=\{D(\mathfrak{s},\omega):\mathfrak{s}\in \mathbb{R},\omega\in \Omega\}\in \mathfrak{D}$ and $t_n\to \infty,$ $\boldsymbol{u}_{0,n}\in D(\mathfrak{s}-t_n, \vartheta_{-t_{n}}\omega)$, the sequence $\Phi(t_n,\mathfrak{s}-t_n,\vartheta_{-t_n}\omega,\boldsymbol{u}_{0,n})$ or $\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})$ of solutions of the system \eqref{WZ_SCBF} has a convergent subsequence in $\mathbb{H}$.
\end{corollary}
\begin{proof}
From Lemma \ref{LemmaUe} with $\tau=\mathfrak{s}-1$, we have that there exists $\mathcal{T}=\mathcal{T}(\delta,\mathfrak{s},\omega,D)>0$ such that for all $t\geq\mathcal{T}$ and $\boldsymbol{u}_{\mathfrak{s}-t}\in D(\mathfrak{s}-t, \vartheta_{-t}\omega)$,
\begin{align}\label{AB1}
\boldsymbol{u}(\mathfrak{s}-1,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\in\mathbb{H}.
\end{align}
Since $t_n\to\infty$ and $\boldsymbol{u}_{0,n}\in D(\mathfrak{s}-t_n, \vartheta_{-t_{n}}\omega)$, from \eqref{AB1}, we infer that there exists $N_1=N(\delta,\mathfrak{s},\omega,D)>0$ such that
\begin{align}\label{AB2}
\{\boldsymbol{u}(\mathfrak{s}-1,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\}_{n\geq N_1}\subset\mathbb{H}.
\end{align}
Hence, by \eqref{AB2} and Lemma \ref{PCB}, we conclude that the sequence $$\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})=\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-1,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-1,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))$$ has a convergent subsequence in $\mathbb{H}$, which completes the proof.
\end{proof}
\subsection{Existence of a unique random $\mathfrak{D}$-pullback attractor}
In this subsection, we start with the result on the existence of a $\mathfrak{D}$-pullback absorbing set in $\mathbb{H}$ for the system \eqref{WZ_SCBF}. Then, we prove the main result of this section, that is, the existence of a unique random $\mathfrak{D}$-pullback attractor for the system \eqref{WZ_SCBF}.
\begin{lemma}\label{PAS}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing2} and Assumption \ref{NDT1} is fulfilled. Then there exists a closed measurable $\mathfrak{D}$-pullback absorbing set $\mathcal{K}=\{\mathcal{K}(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D}$ for the continuous cocycle $\Phi$ associated with the system \eqref{WZ_SCBF}.
\end{lemma}
\begin{proof}
Let us denote, for given $\mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$
\begin{align}
\mathcal{K}(\mathfrak{s},\omega)=\{\boldsymbol{u}\in\mathbb{H}:\|\boldsymbol{u}\|^2_{\mathbb{H}}\leq\mathcal{L}(\mathfrak{s},\omega)\},
\end{align}
where
\begin{align}
\mathcal{L}(\mathfrak{s},\omega)&=\frac{4}{\min\{\mu,\alpha\}} \int_{-\infty}^{0} e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d \xi\nonumber\\&\quad+\int_{-\infty}^{0} e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\bigg\{2s_3e^{\sigma (\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|\nonumber\\&\qquad+2s_6 e^{2\sigma (\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2+s_7\left[e^{\sigma(\xi+\mathfrak{s})}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|\right]^{\frac{2}{1-s_5}}\bigg\}\d\xi.
\end{align}
Since $\mathcal{L}(\mathfrak{s},\cdot):\Omega\to\mathbb{R}$ is $(\mathscr{F},\mathscr{B}(\mathbb{R}))$-measurable for every $\mathfrak{s}\in\mathbb{R}$, $\mathcal{K}(\mathfrak{s},\cdot):\Omega\to2^{\mathbb{H}}$ is a measurable set-valued mapping. Furthermore, we have from Lemma \ref{LemmaUe} that for each $\mathfrak{s}\in\mathbb{R}$, $\omega\in\Omega$ and $D\in\mathfrak{D}$, there exists $\mathcal{T}=\mathcal{T}(\delta,\mathfrak{s},\omega,D)>0$ such that for all $t\geq\mathcal{T}$,
\begin{align}\label{PAS1}
\Phi(t,\mathfrak{s}-t,\vartheta_{-t}\omega,D(\mathfrak{s}-t,\vartheta_{-t}\omega))=\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,D(\mathfrak{s}-t,\vartheta_{-t}\omega))\subseteq\mathcal{K}(\mathfrak{s},\omega).
\end{align}
Now, in order to complete the proof, we only need to prove that $\mathcal{K}\in\mathfrak{D}$, that is, for every $c>0$, $\mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$ $$\lim_{t\to-\infty}e^{ct}\|\mathcal{K}(\mathfrak{s} +t,\vartheta_{t}\omega)\|^2_{\mathbb{H}}=0.$$
For every $c>0$, $\mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$,
\begin{align}\label{PAS2}
& \lim_{t\to-\infty}e^{ct}\|\mathcal{K}(\mathfrak{s} +t,\vartheta_{t}\omega)\|^2_{\mathbb{H}}\nonumber\\&=\lim_{t\to-\infty}e^{ct}\mathcal{L}(\mathfrak{s} +t,\vartheta_{t}\omega)\nonumber\\&=\lim_{t\to-\infty}\frac{4e^{ct}}{\min\{\mu,\alpha\}} \int_{-\infty}^{0} e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s}+t)}\mathcal{Z}_{\delta}(\vartheta_{\zeta+t}\omega)\right)\d\zeta} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s}+t)\|^2_{\mathbb{V}'}\d \xi\nonumber\\&\quad+\lim_{t\to-\infty}e^{ct}\int_{-\infty}^{0} e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s}+t)}\mathcal{Z}_{\delta}(\vartheta_{\zeta+t}\omega)\right)\d\zeta}\bigg\{2s_3e^{\sigma (\xi+\mathfrak{s}+t)}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|\nonumber\\&\qquad\qquad\qquad\qquad+2s_6 e^{2\sigma (\xi+\mathfrak{s}+t)}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|^2+s_7\left[e^{\sigma(\xi+\mathfrak{s}+t)}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|\right]^{\frac{2}{1-s_5}}\bigg\}\d\xi.
\end{align}
Taking into account of \eqref{N1}-\eqref{N2} and following the same steps applied in the proof of Lemma 2.6, \cite{GGW}, we obtain that there exists a $\mathcal{T}_1=\mathcal{T}_1(\omega)>0$ such that for $\xi\leq0$, $\sigma>0$, $\kappa>0$, $0<\delta\leq1$ with $\sigma\delta\neq1$ and $t\leq-\mathcal{T}_1$,
\begin{align}\label{PAS3}
-2\kappa \int_{0}^{\xi}e^{\sigma (\zeta+\mathfrak{s}+t)}\mathcal{Z}_{\delta}(\vartheta_{\zeta+t}\omega)\d\zeta\leq c_0 \left(\frac{1}{\sigma}+2-\xi-2t\right),
\end{align}
where $c_0=\min\left\{\alpha-\gamma,\frac{c}{4\kappa}\right\}$.
Let $c_1=\min\left\{\frac{c}{2},\gamma+\sigma\right\}.$ It implies from \eqref{PAS2}-\eqref{PAS3} that, $\sigma>0$, $\kappa>0$, $0<\delta\leq1$ with $\sigma\delta\neq1$ and $t\leq-\mathcal{T}_1$,
\begin{align}\label{PAS4}
&\lim_{t\to-\infty}e^{ct}\|\mathcal{K}(\mathfrak{s} +t,\vartheta_{t}\omega)\|^2_{\mathbb{H}}\nonumber\\&\leq e^{\frac{c_0(2\sigma+1)}{\sigma}}\lim_{t\to-\infty}e^{\frac{c}{2}t}\int_{-\infty}^{0} e^{\gamma\xi}\bigg\{\frac{4}{\min\{\mu,\alpha\}}\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s}+t)\|^2_{\mathbb{V}'} +2s_3e^{\sigma (\xi+\mathfrak{s}+t)}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|\nonumber\\&\qquad\qquad\qquad\qquad+2s_6 e^{2\sigma (\xi+\mathfrak{s}+t)}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|^2+s_7\left[e^{\sigma(\xi+\mathfrak{s}+t)}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|\right]^{\frac{2}{1-s_5}}\bigg\}\d\xi\nonumber\\&\leq4\frac{e^{\frac{c_0(2\sigma+1)}{\sigma}-\frac{c}{2}\mathfrak{s}}}{\min\{\mu,\alpha\}}\lim_{t\to-\infty}e^{\frac{c}{2}t}\int_{-\infty}^{0} e^{\gamma\xi}\|\boldsymbol{f}(\cdot,\xi+t)\|^2_{\mathbb{V}'}\d\xi\nonumber\\&\quad+ e^{\frac{c_0(2\sigma+1)}{\sigma}}\lim_{t\to-\infty}\int_{-\infty}^{0}e^{c_1t} \bigg\{2s_3e^{(\gamma+\sigma)\xi}e^{\sigma (\mathfrak{s}+t)}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|+2s_6 e^{(\gamma+2\sigma)\xi}e^{2\sigma (\mathfrak{s}+t)}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|^2\nonumber\\&\qquad\qquad\qquad\qquad+s_7e^{\left(\gamma+\frac{2\sigma}{1-s_5}\right)\xi}\left[e^{\sigma(\mathfrak{s}+t)}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|\right]^{\frac{2}{1-s_5}}\bigg\}\d\xi\nonumber\\&\leq4\frac{e^{\frac{c_0(2\sigma+1)}{\sigma}-\frac{c}{2}\mathfrak{s}}}{\min\{\mu,\alpha\}}\lim_{t\to-\infty}e^{\frac{c}{2}t}\int_{-\infty}^{0} e^{\gamma\xi}\|\boldsymbol{f}(\cdot,\xi+t)\|^2_{\mathbb{V}'}\d\xi\nonumber\\&\quad+ e^{\frac{c_0(2\sigma+1)}{\sigma}}\lim_{t\to-\infty}\bigg[2s_3e^{\sigma (\mathfrak{s}+t)}\int_{-\infty}^{0}e^{c_1(\xi+t)} \left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|\d\xi\nonumber\\&\quad+2s_6e^{2\sigma (\mathfrak{s}+t)}\int_{-\infty}^{0}e^{c_1(\xi+t)}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|^2\d\xi+s_7e^{\frac{2\sigma}{1-s_5} (\mathfrak{s}+t)}\int_{-\infty}^{0}e^{c_1(\xi+t)}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|^{\frac{2}{1-s_5}}\d\xi\bigg]\nonumber\\&\leq4\frac{e^{\frac{c_0(2\sigma+1)}{\sigma}-\frac{c}{2}\mathfrak{s}}}{\min\{\mu,\alpha\}}\lim_{t\to-\infty}e^{\frac{c}{2}t}\int_{-\infty}^{0} e^{\gamma\xi}\|\boldsymbol{f}(\cdot,\xi+t)\|^2_{\mathbb{V}'}\d\xi\nonumber\\&\quad+ e^{\frac{c_0(2\sigma+1)}{\sigma}}\lim_{t\to-\infty}\bigg[2s_3e^{\sigma (\mathfrak{s}+t)}\int_{-\infty}^{t}e^{c_1\xi} \left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|\d\xi+2s_6e^{2\sigma (\mathfrak{s}+t)}\int_{-\infty}^{t}e^{c_1\xi}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2\d\xi\nonumber\\&\qquad\qquad\qquad\qquad\qquad+s_7e^{\frac{2\sigma}{1-s_5} (\mathfrak{s}+t)}\int_{-\infty}^{t}e^{c_1\xi}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{2}{1-s_5}}\d\xi\bigg].
\end{align}
Using \eqref{N3}, one can easily find that
\begin{align}\label{PAS5}
&\int_{-\infty}^{0}e^{c_1\xi} \left\{\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{2}{1-s_5}}\right\}\d\xi <\infty.
\end{align}
Taking \eqref{forcing2} and \eqref{PAS5} into account, along with \eqref{PAS4}, we find that
\begin{align*}
\lim_{t\to-\infty}e^{ct}\|\mathcal{K}(\mathfrak{s} +t,\vartheta_{t}\omega)\|^2_{\mathbb{H}}=0.
\end{align*}
Also, in the case when $\sigma=0$ or $\kappa=0$ or $\sigma\delta=1$, the above convergence holds true. In fact, in these particular cases, the proof is easier.
\end{proof}
\begin{theorem}\label{WZ_RA_B}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu>1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing2} and Assumption \ref{NDT1} is fulfilled. Then there exists a unique random $\mathfrak{D}$-pullback attractor $$\mathscr{A}=\{\mathscr{A}(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D},$$ for the continuous cocycle $\Phi$ associated with the system \eqref{WZ_SCBF} in $\mathbb{H}$.
\end{theorem}
\begin{proof}
The proof follows from Corollary \ref{Asymptotic_B}, Lemma \ref{PAS} and the abstract theory given in \cite{SandN_Wang} (see Theorem 2.23 in \cite{SandN_Wang}).
\end{proof}
\section{Random pullback attractors for Wong-Zakai approximations: whole domain $\mathbb{R}^d$} \label{sec4}\setcounter{equation}{0}
In this section, we prove the existence of a unique random $\mathfrak{D}$-pullback attractor for the system \eqref{WZ_SCBF} on $\mathbb{R}^d$ with nonlinear diffusion term satisfying the assumptions given below.
\subsection{Nonlinear diffusion term}
In this work, we prove the existence of random pullback attractors for Wong-Zakai approximations of SCBF equations on $\mathbb{R}^d$ under two different assumptions on the nonlinear diffusion term $S(t,x,\boldsymbol{u})$ appearing in \eqref{WZ_SCBF}. Those two different assumptions are as follows:
\begin{assumption}\label{NDT2}
We assume that the nonlinear diffusion term $$S(t,x,\boldsymbol{u})=e^{\sigma t}\left[\kappa\boldsymbol{u}+\mathcal{S}(\boldsymbol{u})+\boldsymbol{h}(x)\right],$$ where $\sigma\geq0, \kappa\geq0$ and $\boldsymbol{h}\in\mathbb{H}$. Also, $\mathcal{S}:\mathbb{V}\to\mathbb{H}$ is a continuous function satisfying \eqref{S2}-\eqref{S3} and
\begin{align}\label{S1}
\left(\mathcal{S}(\boldsymbol{u}),\boldsymbol{u}\right)=0, \ \text{ for all }\ \boldsymbol{u}\in\mathbb{V}.
\end{align}
Note that the condition \eqref{S4} is much weaker than \eqref{S1} and for $s_3=s_4=0$ in \eqref{S4}, one can obtain \eqref{S1}.
\end{assumption}
\begin{remark}
One can take $\boldsymbol{h}\in\widetilde{\mathbb{L}}^{\frac{r+1}{r}}$ also.
\end{remark}
\begin{assumption}\label{NDT3}
Let $S:\mathbb{R}\times\mathbb{R}^d\times\mathbb{R}^d\to\mathbb{R}^d$ be a continuous function such that for all $t\in\mathbb{R}$ and $x,y\in\mathbb{R}^d$,
\begin{align}
|S(t,x,y)|&\leq \mathcal{S}_1(t,x)|y|^{q-1}+\mathcal{S}_2(t,x),\label{GS1}
\end{align}
where $1\leq q<r+1$, $\mathcal{S}_1\in\mathrm{L}^{\infty}_{\emph{loc}}(\mathbb{R};\mathbb{L}^{\frac{r+1}{r+1-q}}(\mathbb{R}^d))$ and $\mathcal{S}_2\in\mathrm{L}^{\infty}_{\emph{loc}}(\mathbb{R};\mathbb{L}^{\frac{r+1}{r}}(\mathbb{R}^d))$. Furthermore, we assume that $S(t,x,y)$ is locally Lipschitz continuous with respect to the third variable.
\end{assumption}
\begin{remark}
One can take $\mathcal{S}_2\in\mathrm{L}^{\infty}_{\emph{loc}}(\mathbb{R};\mathbb{L}^{2}(\mathbb{R}^d))$ also.
\end{remark}
\begin{example}
Let us discuss some examples for such nonlinear diffusion terms, which satisfy the above Assumptions.
\begin{itemize}
\item [(1)] Let $\mathcal{S}:\mathbb{V}\to\mathbb{H}$ be a nonlinear operator defined by $\mathcal{S}(\boldsymbol{u})=\mathrm{B}(\boldsymbol{g}_2,\boldsymbol{u})$ for all $\boldsymbol{u}\in\mathbb{V}$, where $\boldsymbol{g}_2$ is a fixed element of $\mathrm{D}(\mathrm{A})$ and $\mathcal{S}$ satisfies \eqref{S2}-\eqref{S3} and \eqref{S1}, see \cite{GGW}. Hence $S(t,x,\boldsymbol{u})=e^{\sigma t}\left[\kappa\boldsymbol{u}+\mathrm{B}(\boldsymbol{g}_2,\boldsymbol{u})+\boldsymbol{h}(x)\right]$ satisfies Assumption \ref{NDT2} but not Assumption \ref{NDT3}.
\item [(2)] Take $S(t,x,\boldsymbol{u})=\mathcal{S}_1(t,x)|\textbf{u}|$, where $\mathcal{S}_1\in\mathrm{L}^{\infty}_{\emph{loc}}(\mathbb{R};\mathbb{L}^{\frac{r+1}{r-1}}(\mathbb{R}^d))$ for $r>1$. Hence $S(t,x,\boldsymbol{u})$ satisfies Assumption \ref{NDT3} but not Assumption \ref{NDT2}.
\end{itemize}
The above examples show that Assumptions \ref{NDT2} and \ref{NDT3} cover different classes of functions.
\end{example}
\subsection{Random pullback attractors under Assumption \ref{NDT2}}
In this subsection, we prove the existence of a unique random $\mathfrak{D}$-pullback attractor under Assumptions \ref{NDT2} and \ref{DNFT1} on the nonlinear diffusion term $S(t,x,\boldsymbol{u})$ and deterministic non-autonomous forcing term $\boldsymbol{f}(x,t)$, respectively.
\begin{lemma}\label{ContinuityUB1}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu>1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ and Assumption \ref{NDT2} is fulfilled. Then, the solution of \eqref{WZ_SCBF} is continuous in initial data $\boldsymbol{u}_{\mathfrak{s}}(x).$
\end{lemma}
\begin{proof}
See Lemma \ref{Continuity}.
\end{proof}
\subsubsection{Uniform estimates and $\mathfrak{D}$-pullback asymptotic compactness of solutions}
We start by proving uniform estimates for the solutions of \eqref{WZ_SCBF}. To prove the asymptotic compactness, the method of energy equations is introduced in \cite{Ball}. Due to the lack of compact Sobolev embeddings in unbounded domains, we prove $\mathfrak{D}$-pullback asymptotic compactness of solutions of \eqref{WZ_SCBF} on unbounded domains using the idea given in \cite{Ball}. The following lemma is a particular case of Lemma \ref{LemmaUe}.
\begin{lemma}\label{LemmaUe1}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing1} and Assumption \ref{NDT2} is fulfilled. Then for every $0<\delta\leq1$, $\mathfrak{s}\in\mathbb{R},$ $ \omega\in \Omega$ and $D=\{D(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D},$ there exists $\mathcal{T}=\mathcal{T}(\delta, \mathfrak{s}, \omega, D)>0$ such that for all $t\geq \mathcal{T}$ and $\tau\geq \mathfrak{s}-t$, the solution $\boldsymbol{u}$ of the system \eqref{WZ_SCBF} with $\omega$ replaced by $\vartheta_{-\mathfrak{s}}\omega$ satisfies
\begin{align}\label{ue^1}
&\|\boldsymbol{u}(\tau,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{H}} \nonumber\\&\leq\frac{4}{\min\{\mu,\alpha\}} \int_{-\infty}^{\tau-\mathfrak{s}} e^{\int_{\tau-\mathfrak{s}}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d \xi\nonumber\\&\quad+2s_6\int_{-\infty}^{\tau-\mathfrak{s}} e^{2\sigma (\xi+\mathfrak{s})}e^{\int_{\tau-\mathfrak{s}}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2\d\xi,
\end{align}
where $\boldsymbol{u}_{\mathfrak{s}-t}\in D(\mathfrak{s}-t,\vartheta_{-t}\omega)$ and $s_6=\frac{2}{\alpha}\|\boldsymbol{h}\|^2_{\mathbb{H}}$.
\end{lemma}
\begin{proof}
When $s_3=s_4=0$, assumptions \eqref{S1} and \eqref{S4} are the same. Hence, the proof is immediate from Lemma \ref{LemmaUe} by putting $s_3=s_4=0$.
\end{proof}
The following Lemma is a direct consequence of Lemma \ref{LemmaUe1}.
\begin{lemma}\label{LemmaUe2}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing1} and Assumption \ref{NDT2} is fulfilled. Then for every $0<\delta\leq1$, $\mathfrak{s}\in\mathbb{R},$ $ \omega\in \Omega$ and $D=\{D(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D},$ there exists $\mathcal{T}=\mathcal{T}(\delta,\mathfrak{s},\omega,D)>0$ such that for every $l\geq 0$ and for all $t\geq \mathcal{T}+l$, the solution $\boldsymbol{u}$ of the system \eqref{WZ_SCBF} with $\omega$ replaced by $\vartheta_{-\mathfrak{s}}\omega$ satisfies
\begin{align}\label{ue^2}
&\|\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{H}} \nonumber\\&\leq\frac{4}{\min\{\mu,\alpha\}} \int_{-\infty}^{-l} e^{\int_{-l}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d \xi\nonumber\\&\quad+2s_6\int_{-\infty}^{-l} e^{2\sigma (\xi+\mathfrak{s})}e^{\int_{-l}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2\d\xi,
\end{align}
where $\boldsymbol{u}_{\mathfrak{s}-t}\in D(\mathfrak{s}-t,\vartheta_{-t}\omega)$ and $s_6=\frac{2}{\alpha}\|\boldsymbol{h}\|^2_{\mathbb{H}}$.
\end{lemma}
\begin{proof}
Given $\mathfrak{s}\in \mathbb{R}$ and $l\geq 0$, let $\tau=\mathfrak{s}-l$. Let $\mathcal{T}>0$ be the constant claimed in Lemma \ref{LemmaUe1}. Also, $t\geq \mathcal{T}+l$ implies $t\geq \mathcal{T}$ and $\tau\geq \mathfrak{s}-t$. Thus, the desired result is immediate from Lemma \ref{LemmaUe1}.
\end{proof}
The following Lemma is a result on weak continuity of the solutions of \eqref{WZ_SCBF}, which helps us to prove the $\mathfrak{D}$-pullback asymptotic compactness of the solutions of system \eqref{WZ_SCBF}.
\begin{lemma}\label{weak}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ and Assumption \ref{NDT2} is fulfilled. Let $0<\delta\leq1$, $\mathfrak{s}\in\mathbb{R}, \omega\in \Omega$ and $\boldsymbol{u}_{\mathfrak{s}}^0, \boldsymbol{u}_{\mathfrak{s}}^n\in \mathbb{H}$ for all $n\in\mathbb{N}.$ If $\boldsymbol{u}_{\mathfrak{s}}^n\xrightharpoonup{w}\boldsymbol{u}_{\mathfrak{s}}^0$ in $\mathbb{H}$, then the solution $\boldsymbol{u}$ of the system \eqref{WZ_SCBF} satisfies the following convergences:
\begin{itemize}
\item [(i)] $\boldsymbol{u}(\xi,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}}^n)\xrightharpoonup{w}\boldsymbol{u}(\xi,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}}^0)$ in $\mathbb{H}$ for all $\xi\geq \mathfrak{s}$.
\item [(ii)] $\boldsymbol{u}(\cdot,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}}^n)\xrightharpoonup{w}\boldsymbol{u}(\cdot,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}}^0)$ in $\mathrm{L}^2((\mathfrak{s},\mathfrak{s}+T);\mathbb{V})$ for every $T>0$.
\item [(iii)] $\boldsymbol{u}(\cdot,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}}^n)\xrightharpoonup{w}\boldsymbol{u}(\cdot,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}}^0)$ in $\mathrm{L}^{r+1}((\mathfrak{s},\mathfrak{s}+T);\widetilde{\mathbb{L}}^{r+1})$ for every $T>0$.
\end{itemize}
\end{lemma}
\begin{proof}
Using a standard method as in \cite{KM1} (see Lemmas 5.2 and 5.3 in \cite{KM1}), one can complete the proof.
\end{proof}
Now, we establish the $\mathfrak{D}$-pullback asymptotic compactness of the solutions of the system \eqref{WZ_SCBF}.
\begin{lemma}\label{Asymptotic_UB}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing1} and Assumption \ref{NDT2} is fulfilled. Then for every $0<\delta\leq1$, $\mathfrak{s}\in \mathbb{R},$ $\omega\in \Omega,$ $D=\{D(\mathfrak{s},\omega):\mathfrak{s}\in \mathbb{R},\omega\in \Omega\}\in \mathfrak{D}$ and $t_n\to \infty,$ $\boldsymbol{u}_{0,n}\in D(\mathfrak{s}-t_n, \vartheta_{-t_{n}}\omega)$, the sequence $\Phi(t_n,\mathfrak{s}-t_n,\vartheta_{-t_n}\omega,\boldsymbol{u}_{0,n})$ or $\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})$ of solutions of the system \eqref{WZ_SCBF} has a convergent subsequence in $\mathbb{H}$.
\end{lemma}
\begin{proof}
It follows from Lemma \ref{LemmaUe2} with $l=0$ that there exists $\mathcal{T}=\mathcal{T}(\delta,\mathfrak{s},\omega,D)>0$ such that for all $t\geq \mathcal{T}$,
\begin{align}\label{ac1}
\|\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{H}} &\leq\frac{4}{\min\{\mu,\alpha\}} \int_{-\infty}^{0} e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d \xi\nonumber\\&\quad+2s_6\int_{-\infty}^{0} e^{2\sigma (\xi+\mathfrak{s})}e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2\d\xi\nonumber\\&=:M(\mathfrak{s},\omega),
\end{align}
where $\boldsymbol{u}_{\mathfrak{s}-t}\in D(\mathfrak{s}-t,\vartheta_{-t}\omega).$ Since $t_n\to \infty$, there exists $N_0\in\mathbb{N}$ such that $t_n\geq \mathcal{T}$ for all $n\geq N_0$. As, it is given that $\boldsymbol{u}_{0,n}\in D(\mathfrak{s}-t_n, \vartheta_{-t_{n}}\omega)$, \eqref{ac1} implies that for all $n\geq N_0$,
\begin{align*}
\|\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\|^2_{\mathbb{H}} \leq M(\mathfrak{s},\omega),
\end{align*}
and hence $\{\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\}_{n\geq N_0}\subseteq\mathbb{H}$ is a bounded sequence, which implies that there exists $\tilde{\boldsymbol{u}}\in \mathbb{H}$ and a subsequence (keeping same label) such that
\begin{align}\label{ac2}
\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\xrightharpoonup{w}\tilde{\boldsymbol{u}}\ \text{ in }\ \mathbb{H}.
\end{align}
By the weak lower semicontinuous property of norms and \eqref{ac2}, we get
\begin{align}\label{ac3}
\|\tilde{\boldsymbol{u}}\|_{\mathbb{H}}\leq\liminf_{n\to\infty}\|\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\|_{\mathbb{H}}.
\end{align}
In order to get the desired result, we have to prove that $\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\to\tilde{\boldsymbol{u}}$ in $\mathbb{H}$ strongly, that is, we only need to show that
\begin{align}\label{ac4}
\|\tilde{\boldsymbol{u}}\|_{\mathbb{H}}\geq\limsup_{n\to\infty}\|\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\|_{\mathbb{H}}.
\end{align}
The method of energy equations introduced in \cite{Ball} will help us to prove \eqref{ac4}. For a given $l\in \mathbb{N}$ ($l\leq t_n$), we can write
\begin{align}\label{ac5}
\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})=\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})).
\end{align}
For each $l$, let $N_l$ be sufficiently large such that $t_n\geq \mathcal{T}+l$ for all $n\geq N_l$. By Lemma \ref{LemmaUe2}, we have for $l\geq N_l$,
\begin{align*}
&\|\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\|^2_{\mathbb{H}} \nonumber\\&\leq\frac{4}{\min\{\mu,\alpha\}} \int_{-\infty}^{-l} e^{\int_{-l}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d \xi\nonumber\\&\quad+2s_6\int_{-\infty}^{-l} e^{2\sigma (\xi+\mathfrak{s})}e^{\int_{-l}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2\d\xi,
\end{align*}
which infer that the sequence $\{\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\}_{n\geq N_l}$ is bounded in $\mathbb{H}$, for each $l\in \mathbb{N}$. By the diagonal process, there exists a subsequence (denoting by same symbol for convenience) and an element $\tilde{\boldsymbol{u}}_{l}\in \mathbb{H}$ for each $l\in\mathbb{N}$ such that
\begin{align}\label{ac6}
\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\xrightharpoonup{w}\tilde{\boldsymbol{u}}_{l} \ \text{ in }\ \mathbb{H}.
\end{align}
From \eqref{ac5}-\eqref{ac6} along with Lemma \ref{weak}, we obtain that for $l\in\mathbb{N}$,
\begin{align}\label{ac7}
\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\xrightharpoonup{w}\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_{l}) \ \text{ in } \ \mathbb{H},
\end{align}
\begin{align}\label{ac8}
\boldsymbol{u}(\cdot,\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))\xrightharpoonup{w}\boldsymbol{u}(\cdot,\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_{l}) \text{ in } \mathrm{L}^2((\mathfrak{s}-l,\mathfrak{s});\mathbb{V}),
\end{align}
and
\begin{align}\label{ac8'}
\boldsymbol{u}(\cdot,\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))\xrightharpoonup{w}\boldsymbol{u}(\cdot,\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_{l}) \text{ in } \mathrm{L}^{r+1}((\mathfrak{s}-l,\mathfrak{s});\widetilde{\mathbb{L}}^{r+1}).
\end{align}
Clearly, \eqref{ac2} and \eqref{ac7} imply that
\begin{align}\label{ac9}
\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_{l})=\tilde{\boldsymbol{u}}.
\end{align}
Taking into account that $\left(\mathcal{S}(\boldsymbol{u}),\boldsymbol{u}\right)=0$, from \eqref{ue0} we have
\begin{align}\label{ac10}
& \frac{\d}{\d t} \|\boldsymbol{u}\|^2_{\mathbb{H}} + \left(\alpha-2\kappa e^{\sigma t}\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right)\|\boldsymbol{u}\|^2_{\mathbb{H}}\nonumber\\&=-2\mu\|\nabla\boldsymbol{u}\|^2_{\mathbb{H}}-\alpha\|\boldsymbol{u}\|^2_{\mathbb{H}} - 2\beta\|\boldsymbol{u}\|^{r+1}_{\widetilde \mathbb{L}^{r+1}} + 2\left\langle\boldsymbol{f},\boldsymbol{u}\right\rangle +2e^{\sigma t}\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\left(\boldsymbol{h},\boldsymbol{u}\right).
\end{align}
It follows by applying variation of constant formula to \eqref{ac10}, that for each $\omega\in \Omega,$ $ \tau\in \mathbb{R}$ and $\mathfrak{s}\geq \tau$,
\begin{align}\label{ac11}
\|\boldsymbol{u}(\mathfrak{s},\tau,\omega,\boldsymbol{u}_{\tau})\|^2_{\mathbb{H}} &= e^{\int_{\mathfrak{s}}^{\tau}\left(\alpha-2\kappa e^{\sigma\zeta}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}_{\tau}\|^2_{\mathbb{H}} \nonumber\\&\quad-2\mu\int_{\tau}^{\mathfrak{s}}e^{\int_{\mathfrak{s}}^{\xi}\left(\alpha-2\kappa e^{\sigma\zeta}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\nabla\boldsymbol{u}(\xi,\tau,\omega,\boldsymbol{u}_{\tau})\|^2_{\mathbb{H}}\d\xi\nonumber\\&\quad-\alpha\int_{\tau}^{\mathfrak{s}}e^{\int_{\mathfrak{s}}^{\xi}\left(\alpha-2\kappa e^{\sigma\zeta}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}(\xi,\tau,\omega,\boldsymbol{u}_{\tau})\|^2_{\mathbb{H}}\d\xi\nonumber\\&\quad-2\beta\int_{\tau}^{\mathfrak{s}}e^{\int_{\mathfrak{s}}^{\xi}\left(\alpha-2\kappa e^{\sigma\zeta}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}(\xi,\tau,\omega,\boldsymbol{u}_{\tau})\|^{r+1}_{\widetilde \mathbb{L}^{r+1}}\d\xi\nonumber\\&\quad+2\int_{\tau}^{\mathfrak{s}}e^{\int_{\mathfrak{s}}^{\xi}\left(\alpha-2\kappa e^{\sigma\zeta}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\left\langle\boldsymbol{f}(\cdot,\xi),\boldsymbol{u}(\xi,\tau,\omega,\boldsymbol{u}_{\tau})\right\rangle\d\xi\nonumber\\&\quad+2\int_{\tau}^{\mathfrak{s}}e^{\sigma\xi}e^{\int_{\mathfrak{s}}^{\xi}\left(\alpha-2\kappa e^{\sigma\zeta}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\left(\boldsymbol{h},\boldsymbol{u}(\xi,\tau,\omega,\boldsymbol{u}_{\tau})\right)\d\xi.
\end{align}
From \eqref{ac9} and \eqref{ac11}, it is immediate that
\begin{align}\label{ac12}
\|\tilde{\boldsymbol{u}}\|^2_{\mathbb{H}}&=\|\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_{l})\|^2_{\mathbb{H}} \nonumber\\&= e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\tilde{\boldsymbol{u}}_{l}\|^2_{\mathbb{H}} \nonumber\\&\quad-2\mu\int_{-l}^{0}e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\nabla\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_{l})\|^2_{\mathbb{H}}\d\xi\nonumber\\&\quad-\alpha\int_{-l}^{0}e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_{l})\|^2_{\mathbb{H}}\d\xi\nonumber\\&\quad-2\beta\int_{-l}^{0}e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_{l})\|^{r+1}_{\widetilde \mathbb{L}^{r+1}}\d\xi\nonumber\\&\quad+2\int_{-l}^{0}e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\left\langle\boldsymbol{f}(\cdot,\xi+\mathfrak{s}),\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_{l})\right\rangle\d\xi\nonumber\\&\quad+2\int_{-l}^{0}e^{\sigma(\xi+\mathfrak{s})}e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\left(\boldsymbol{h},\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_{l})\right)\d\xi.
\end{align}
Similarly, from \eqref{ac5} and \eqref{ac11}, we obtain
\begin{align}\label{ac13}
&\|\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\|^2_{\mathbb{H}}\nonumber\\&=\|\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))\|^2_{\mathbb{H}} \nonumber\\&= e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\|^2_{\mathbb{H}} \nonumber\\&\quad-2\mu\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\nabla\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))\|^2_{\mathbb{H}}\d\xi\nonumber\\&\quad-\alpha\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))\|^2_{\mathbb{H}}\d\xi\nonumber\\&\quad-2\beta\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))\|^{r+1}_{\widetilde \mathbb{L}^{r+1}}\d\xi\nonumber\\&\quad+2\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\nonumber\\&\qquad\qquad\times\left\langle\boldsymbol{f}(\cdot,\xi+\mathfrak{s}),\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))\right\rangle\d\xi\nonumber\\&\quad+2\int_{-l}^{0}e^{\sigma(\xi+\mathfrak{s})}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\nonumber\\&\qquad\qquad\times\left(\boldsymbol{h},\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))\right)\d\xi.
\end{align}
Now, we examine the limits of each term of right-hand side of \eqref{ac13} as $n\to \infty$. By \eqref{ue2}, we examine the first term with $\tau=\mathfrak{s}-l$ and $t=t_n$ as follows
\begin{align}\label{ac14}
&e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\|^2_{\mathbb{H}} \nonumber\\&\leq e^{\int_{0}^{-t_n}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}_{0,n}\|^2_{\mathbb{H}}\nonumber\\&\quad+\frac{4}{\min\{\mu,\alpha\}}\int_{-\infty}^{-l} e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d \xi\nonumber\\&\quad+s_6\int_{-\infty}^{-l} e^{2\sigma (\xi+\mathfrak{s})} e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2\d\xi.
\end{align}
Since $\boldsymbol{u}_{0,n}\in D(\mathfrak{s}-t_n,\vartheta_{-t_n}\omega)$, we have
\begin{align}\label{ac15}
e^{\int_{0}^{-t_n}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}_{0,n}\|^2_{\mathbb{H}}\leq e^{\int_{0}^{-t_n}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|D(\mathfrak{s}-t_n,\vartheta_{-t_n}\omega)\|^2_{\mathbb{H}}\to 0,
\end{align}
as $n\to \infty$. Combining \eqref{ac15} along with \eqref{ac14}, we have
\begin{align}\label{ac16}
&\limsup_{n\to\infty}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\|^2_{\mathbb{H}}\nonumber\\& \leq\frac{4}{\min\{\mu,\alpha\}}\int_{-\infty}^{-l} e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d \xi\nonumber\\&\quad+s_6\int_{-\infty}^{-l} e^{2\sigma (\xi+\mathfrak{s})} e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2\d\xi.
\end{align}
From \eqref{ac8}, we get
\begin{align}\label{ac17}
& \lim_{n\to\infty} 2\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\nonumber\\&\qquad\qquad\times\left\langle\boldsymbol{f}(\cdot,\xi+\mathfrak{s}),\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))\right\rangle\d\xi \nonumber\\&= 2\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\left\langle\boldsymbol{f}(\cdot,\xi+\mathfrak{s}),\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_{l})\right\rangle\d\xi,
\end{align}
and
\begin{align}\label{ac18}
& \lim_{n\to\infty} 2\int_{-l}^{0}e^{\sigma(\xi+\mathfrak{s})}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\nonumber\\&\qquad\qquad\times\left(\boldsymbol{h},\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))\right)\d\xi \nonumber\\&= 2\int_{-l}^{0}e^{\sigma(\xi+\mathfrak{s})}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\left(\boldsymbol{h},\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_l)\right)\d\xi.
\end{align}
Using \eqref{ac8} and the weak lower semicontinuity property of norms, we obtain
\begin{align*}
&\liminf_{n\to\infty} \bigg\{2\mu\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\nonumber\\&\qquad\qquad\times\|\nabla\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))\|^2_{\mathbb{H}}\d\xi\bigg\} \nonumber\\&\geq 2\mu\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\nabla\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_l)\|^2_{\mathbb{H}}\d\xi,
\end{align*}
or
\begin{align}\label{ac19}
&\limsup_{n\to\infty} \bigg\{-2\mu\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\nonumber\\&\qquad\qquad\times\|\nabla\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))\|^2_{\mathbb{H}}\d\xi\bigg\} \nonumber\\&\leq -2\mu\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\nabla\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_l)\|^2_{\mathbb{H}}\d\xi.
\end{align}
Similarly, using \eqref{ac8}-\eqref{ac8'} and the weak lower semicontinuity property of norms, we get
\begin{align}\label{ac20}
&\limsup_{n\to\infty} \bigg\{-\alpha\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\nonumber\\&\qquad\qquad\times\|\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))\|^2_{\mathbb{H}}\d\xi\bigg\} \nonumber\\&\leq -\alpha\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_l)\|^2_{\mathbb{H}}\d\xi,
\end{align}
and
\begin{align}\label{ac21}
&\limsup_{n\to\infty} \bigg\{-2\beta\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\nonumber\\&\qquad\qquad\times\|\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-l,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))\|^{r+1}_{\widetilde{\mathbb{L}}^{r+1}}\d\xi\bigg\} \nonumber\\&\leq -2\beta\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_l)\|^{r+1}_{\widetilde{\mathbb{L}}^{r+1}}\d\xi.
\end{align}
Combining \eqref{ac16}-\eqref{ac21}, and using it in \eqref{ac13}, we find
\begin{align}\label{ac22}
&\limsup_{n\to\infty}\|\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\|^2_{\mathbb{H}}\nonumber\\&\leq\frac{4}{\min\{\mu,\alpha\}}\int_{-\infty}^{-l} e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d \xi\nonumber\\&\quad+s_6\int_{-\infty}^{-l} e^{2\sigma (\xi+\mathfrak{s})} e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2\d\xi\nonumber\\&\quad-2\mu\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\nabla\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_l)\|^2_{\mathbb{H}}\d\xi\nonumber\\&\quad-\alpha\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_l)\|^2_{\mathbb{H}}\d\xi\nonumber\\&\quad-2\beta\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\|\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_l)\|^{r+1}_{\widetilde{\mathbb{L}}^{r+1}}\d\xi\nonumber\\&\quad+2\int_{-l}^{0}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\left\langle\boldsymbol{f}(\cdot,\xi+\mathfrak{s}),\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_{l})\right\rangle\d\xi\nonumber\\&\quad+2\int_{-l}^{0}e^{\sigma(\xi+\mathfrak{s})}e^{\int_{0}^{-l}\left(\alpha-2\kappa e^{\sigma(\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\left(\boldsymbol{h},\boldsymbol{u}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{u}}_l)\right)\d\xi.
\end{align}
Making use of \eqref{ac12} in \eqref{ac22}, we get
\begin{align}\label{ac23}
&\limsup_{n\to\infty}\|\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\|^2_{\mathbb{H}}\nonumber\\&\leq\frac{4}{\min\{\mu,\alpha\}}\int_{-\infty}^{-l} e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d \xi\nonumber\\&\quad+s_6\int_{-\infty}^{-l} e^{2\sigma (\xi+\mathfrak{s})} e^{\int_{0}^{\xi}\left(\alpha-2\kappa e^{\sigma (\zeta+\mathfrak{s})}\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta}\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^2\d\xi+\|\tilde{\boldsymbol{u}}\|^2_{\mathbb{H}}.
\end{align}
Finally, passing the limit $l\to \infty$ in \eqref{ac23}, we arrive at \eqref{ac4}, which completes the proof.
\end{proof}
\subsubsection{Existence of random $\mathfrak{D}$-pullback attractors}
In this subsection, we start with the proof of existence of random $\mathfrak{D}$-pullback absorbing set in $\mathbb{H}$ for the system \eqref{WZ_SCBF}. Finally, we prove the main result of this section, that is, the existence of random $\mathfrak{D}$-pullback attractors for the system \eqref{WZ_SCBF}.
\begin{lemma}\label{PAS'}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing2} and Assumption \ref{NDT2} is fulfilled. Then there exists a closed measurable $\mathfrak{D}$-pullback absorbing set $\widetilde{\mathcal{K}}=\{\widetilde{\mathcal{K}}(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D}$ for the continuous cocycle $\Phi$ associated with the system \eqref{WZ_SCBF}.
\end{lemma}
\begin{proof}
When $s_3=s_4=0$, assumptions \eqref{S1} and \eqref{S4} are the same. Hence, the proof is same as the proof of Lemma \ref{PAS} by putting $s_3=s_4=0$.
\end{proof}
Now, we are able to provide the main results of this section.
\begin{theorem}\label{WZ_RA_UB}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu>1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing2} and Assumption \ref{NDT2} is fulfilled. Then there exists a unique random $\mathfrak{D}$-pullback attractor $$\widetilde{\mathscr{A}}=\{\widetilde{\mathscr{A}}(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D},$$ for the the continuous cocycle $\Phi$ associated with the system \eqref{WZ_SCBF} in $\mathbb{H}$.
\end{theorem}
\begin{proof}
The proof follows from Lemma \ref{Asymptotic_UB}, Lemma \ref{PAS'} and the abstract theory given in \cite{SandN_Wang} (Theorem 2.23 in \cite{SandN_Wang}).
\end{proof}
\begin{remark}\label{remark1}
It is remarkable to note that if we replace \eqref{S3} by
\begin{align*}
|\left(\mathcal{S}(\boldsymbol{u})-\mathcal{S}(\boldsymbol{v}),\boldsymbol{w}\right)|&\leq s_2\|\boldsymbol{u}-\boldsymbol{v}\|_{\mathbb{H}}\|\boldsymbol{w}\|_{\mathbb{H}}, \ \text{ for all }\ \boldsymbol{u},\boldsymbol{v}\in\mathbb{V} \text{ and }\boldsymbol{w} \in\mathbb{H},
\end{align*}
then we can also include the case $2\beta\mu=1$ for $d=r=3$ in Lemma \ref{Continuity} and hence our main result of section \ref{sec3} and this subsection, that is, Theorems \ref{WZ_RA_B} and \ref{WZ_RA_UB} hold true for the case $d=r=3$ with $2\beta\mu=1$ also.
\end{remark}
\subsection{Random pullback attractors under Assumption \ref{NDT3} }\label{subsec4.3}
In this subsection, we prove the existence of unique random $\mathfrak{D}$-pullback attractor under Assumption \eqref{NDT3} on nonlinear diffusion term $S(t,x,\boldsymbol{u})$. In order to prove the results of this subsection, we need the following assumption on non-autonomous forcing term $\boldsymbol{f}(\cdot,\cdot)$.
\begin{assumption}\label{DNFT3}
We assume that external forcing term $\boldsymbol{f}\in\mathrm{L}^2_{\mathrm{loc}}(\mathbb{R};\mathbb{H})$ satisfies
\begin{itemize}
\item [(i)]
\begin{align}\label{forcing3}
\int_{-\infty}^{\mathfrak{s}} e^{\alpha\xi}\|\boldsymbol{f}(\cdot,\xi)\|^2_{\mathbb{H}}\d \xi<\infty, \ \ \text{ for all }\ \mathfrak{s}\in\mathbb{R}.
\end{align}
Moreover, \eqref{forcing3} implies that
\begin{align}\label{forcing4}
\lim_{k\to\infty}\int_{-\infty}^{0}\int\limits_{|x|\geq k} e^{\alpha\xi}|\boldsymbol{f}(x,\xi+\mathfrak{s})|^2\d x\d \xi=0, \ \ \ \text{ for all }\ \mathfrak{s}\in\mathbb{R}.
\end{align}
\item [(ii)] for every $c>0$
\begin{align}\label{forcing5}
\lim_{\tau\to-\infty}e^{c\tau}\int_{-\infty}^{0} e^{\alpha\xi}\|\boldsymbol{f}(\cdot,\xi+\tau)\|^2_{\mathbb{H}}\d \xi=0,
\end{align}
where $\alpha>0$ is the Darcy coefficient.
\end{itemize}
\end{assumption}
\begin{lemma}\label{ContinuityUB2}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{H})$ and Assumption \ref{NDT3} is fulfilled. Then, the solution of \eqref{WZ_SCBF} is continuous in initial data $\boldsymbol{u}_{\mathfrak{s}}(x).$
\end{lemma}
\begin{proof}
Let $\boldsymbol{u}_{1}(t)$ and $\boldsymbol{u}_{2}(t)$ be two solutions of \eqref{WZ_SCBF}, then $\mathfrak{X}(t)=\boldsymbol{u}_{1}(t)-\boldsymbol{u}_{2}(t)$ with $\mathfrak{X}(\mathfrak{s})=\boldsymbol{u}_{1,\mathfrak{s}}(x)-\boldsymbol{u}_{2,\mathfrak{s}}(x)$ satisfies
\begin{align}\label{Conti9}
\frac{\d\mathfrak{X}(t)}{\d t}&=-\mu \mathrm{A}\mathfrak{X}(t)-\alpha\mathfrak{X}(t)-\left\{\mathrm{B}\big(\boldsymbol{u}_{1}(t)\big)-\mathrm{B}\big(\boldsymbol{u}_{2}(t)\big)\right\} -\beta\left\{\mathcal{C}\big(\boldsymbol{u}_1(t)\big)-\mathcal{C}\big(\boldsymbol{u}_2(t)\big)\right\}\nonumber\\&\quad+\left[S(t,x,\boldsymbol{u}_1(t))-S(t,x,\boldsymbol{u}_2(t))\right]\mathcal{Z}_{\delta}(\vartheta_t\omega),
\end{align}
in $\mathbb{V}'+\widetilde{\mathbb{L}}^{\frac{r+1}{r}}$. Taking the inner product with $\mathfrak{X}(\cdot)$ to the equation \eqref{Conti1}, we obtain
\begin{align}\label{Conti10}
\frac{1}{2}\frac{\d}{\d t} \|\mathfrak{X}(t)\|^2_{\mathbb{H}} &=-\mu \|\nabla\mathfrak{X}(t)\|^2_{\mathbb{H}} - \alpha\|\mathfrak{X}(t)\|^2_{\mathbb{H}} -\left\langle\mathrm{B}\big(\boldsymbol{u}_1(t)\big)-\mathrm{B}\big(\boldsymbol{u}_2(t)\big), \mathfrak{X}(t)\right\rangle \nonumber\\&\quad-\beta\left\langle\mathcal{C}\big(\boldsymbol{u}_1(t)\big)-\mathcal{C}\big(\boldsymbol{u}_2(t)\big),\mathfrak{X}(t)\right\rangle\nonumber\\&\quad + \mathcal{Z}_{\delta}(\vartheta_{t}\omega)\left\langle S(t,x,\boldsymbol{u}_1(t))-S(t,x,\boldsymbol{u}_2(t)),\mathfrak{X}(t)\right\rangle ,
\end{align}
for a.e. $t\in[\mathfrak{s},\mathfrak{s}+T] \text{ with } T>0$. By the locally Lipschitz continuity of the nonlinear diffusion term (see Assumption \ref{NDT3}), we get
\begin{align}\label{Conti11}
& \mathcal{Z}_{\delta}(\vartheta_{t}\omega)\left\langle S(t,x,\boldsymbol{u}_1)-S(t,x,\boldsymbol{u}_2),\mathfrak{X}\right\rangle\nonumber\\&\leq \left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|\|S(t,x,\boldsymbol{u}_1)-S(t,x,\boldsymbol{u}_2)\|_{\mathbb{H}}\|\mathfrak{X}\|_{\mathbb{H}}\leq C \left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|\|\mathfrak{X}\|^2_{\mathbb{H}}.
\end{align}
From \eqref{MO_c}, we have
\begin{align}\label{Conti12}
-\beta \left\langle\mathcal{C}\big(\boldsymbol{u}_1\big)-\mathcal{C}\big(\boldsymbol{u}_2\big),\mathfrak{X}\right\rangle\leq -\frac{\beta}{2}\||\mathfrak{X}||\boldsymbol{u}_1|^{\frac{r-1}{2}}\|^2_{\mathbb{H}} - \frac{\beta}{2}\||\mathfrak{X}||\boldsymbol{u}_2|^{\frac{r-1}{2}}\|^2_{\mathbb{H}}.
\end{align}
\vskip 2mm
\noindent
\textbf{Case I:} \textit{When $d=2$ and $r>1$.} Using \eqref{b1}, \eqref{441} and Lemma \ref{Young}, we obtain
\begin{align}\label{Conti13}
\left| \left\langle\mathrm{B}\big(\boldsymbol{u}_1\big)-\mathrm{B}\big(\boldsymbol{u}_2\big), \mathfrak{X}\right\rangle\right|&=\left|\left\langle\mathrm{B}\big(\mathfrak{X},\mathfrak{X} \big), \boldsymbol{u}_1\right\rangle\right|\leq\frac{\mu}{4}\|\nabla\mathfrak{X}\|^2_{\mathbb{H}}+C\|\boldsymbol{u}_1\|^4_{\widetilde{\mathbb{L}}^4}\|\mathfrak{X}\|^2_{\mathbb{H}}\nonumber\\&\leq\frac{\mu}{4}\|\nabla\mathfrak{X}\|^2_{\mathbb{H}}+C\|\boldsymbol{u}_1\|^2_{\mathbb{H}}\|\nabla\boldsymbol{u}_1\|^2_{\mathbb{H}}\|\mathfrak{X}\|^2_{\mathbb{H}}.
\end{align}
Making use of \eqref{Conti11}-\eqref{Conti13} in \eqref{Conti10}, we get
\begin{align}\label{Conti14}
& \frac{\d}{\d t} \|\mathfrak{X}(t)\|^2_{\mathbb{H}} \leq C \|\boldsymbol{u}_1(t)\|^2_{\mathbb{H}}\|\nabla\boldsymbol{u}_1(t)\|^2_{\mathbb{H}}\|\mathfrak{X}(t)\|^2_{\mathbb{H}},\text{ for a.e. } t\in[\mathfrak{s},\mathfrak{s}+T].
\end{align}
\iffalse
Consider,
\begin{align}\label{Conti15}
\int_{\mathfrak{s}}^{t}\|\boldsymbol{u}_1(\tau)\|^2_{\mathbb{H}}\|\nabla\boldsymbol{u}_1(\tau)\|^2_{\mathbb{H}}\d \tau\leq \|\boldsymbol{u}_1\|^2_{\mathrm{L}^{\infty}(\mathfrak{s},\mathfrak{s}+T;\mathbb{H})}\|\nabla\boldsymbol{u}_1\|^2_{\mathrm{L}^{2}(\mathfrak{s},\mathfrak{s}+T;\mathbb{H})}<\infty.
\end{align}
\fi
\vskip 2mm
\noindent
\textbf{Case II:} \textit{When $d= 3$ and $r\geq3$ ($r>3$ with any $\beta,\mu>0$ and $r=3$ with $2\beta\mu\geq1$).} The nonlinear term $\left|\left\langle\mathrm{B}\big(\boldsymbol{u}_1\big)-\mathrm{B}\big(\boldsymbol{u}_2\big), \mathfrak{X}\right\rangle\right|$ can be estimated using \eqref{441}, Lemmas \ref{Holder} and \ref{Young} as
\begin{align}\label{Conti16}
\left|\left\langle\mathrm{B}\big(\boldsymbol{u}_1\big)-\mathrm{B}\big(\boldsymbol{u}_2\big), \mathfrak{X}\right\rangle\right|\leq\begin{cases}
\frac{1}{2\beta}\|\nabla\mathfrak{X}\|^2_{\mathbb{H}}+\frac{\beta}{2}\||\boldsymbol{u}_1||\mathfrak{X}|\|^2_{\mathbb{H}}, &\text{ for } r=3,\\ \frac{\mu}{4}\|\nabla\mathfrak{X}\|_{\mathbb{H}}^2+\frac{\beta}{2}\||\mathfrak{X}||\boldsymbol{u}_1|^{\frac{r-1}{2}}\|^2_{\mathbb{H}}+C\|\mathfrak{X}\|^2_{\mathbb{H}} \ \ \ &\text{ for } r>3.
\end{cases}
\end{align}
Combining \eqref{Conti10}-\eqref{Conti12} and \eqref{Conti16}, we get for $r\geq3$ ($r>3$ with any $\beta,\mu>0$ and $r=3$ with $2\beta\mu\geq1$),
\begin{align}\label{Conti17}
& \frac{\d}{\d t} \|\mathfrak{X}(t)\|^2_{\mathbb{H}} \leq C\|\mathfrak{X}(t)\|^2_{\mathbb{H}},\ \ \ \text{ for a.e. }\ \ \ t\in[\mathfrak{s},\mathfrak{s}+T].
\end{align} Hence, we conclude the proof by applying Gronwall's inequality to \eqref{Conti14} and \eqref{Conti17}.
\end{proof}
Next, we prove the existence of random $\mathfrak{D}$-pullback absorbing set for continuous cocycle $\Phi$.
\begin{lemma}\label{LemmaUe3}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{H})$ satisfies \eqref{forcing3} and Assumption \ref{NDT3} is fulfilled. Then for every $0<\delta\leq1$, $\mathfrak{s}\in\mathbb{R},$ $ \omega\in \Omega$ and $D=\{D(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D},$ there exists $\mathscr{T}=\mathscr{T}(\delta, \mathfrak{s}, \omega, D)>0$ such that for all $t\geq \mathcal{T}$ and $\tau\geq \mathfrak{s}-t$, the solution $\boldsymbol{u}$ of the system \eqref{WZ_SCBF} with $\omega$ replaced by $\vartheta_{-\mathfrak{s}}\omega$ satisfies
\begin{align}\label{ue^3}
& \|\boldsymbol{u}(\tau,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{H}}+\int_{\mathfrak{s}-t}^{\tau}e^{\alpha(\xi-\tau)}\bigg[\|\boldsymbol{u}(\xi,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{H}} \nonumber\\&+\|\nabla\boldsymbol{u}(\xi,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{H}}+\|\boldsymbol{u}(\xi,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}\bigg]\d\xi \nonumber\\&\leq \widetilde{M} \int_{-\infty}^{\tau-\mathfrak{s}}e^{\alpha(\xi+\mathfrak{s}-\tau)}\left[\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{H}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r+1-q}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r}}\right]\d\xi,
\end{align}
where $\boldsymbol{u}_{\mathfrak{s}-t}\in D(\mathfrak{s}-t,\vartheta_{-t}\omega)$ and $\widetilde{M}$ is positive constant independent of $\tau, \mathfrak{s}, \omega$ and $D$.
\end{lemma}
\begin{proof}
From the first equation of the system \eqref{WZ_SCBF}, we obtain
\begin{align}\label{ue12}
& \frac{1}{2}\frac{\d}{\d t} \|\boldsymbol{u}\|^2_{\mathbb{H}} +\mu\|\nabla\boldsymbol{u}\|^2_{\mathbb{H}} + \alpha\|\boldsymbol{u}\|^2_{\mathbb{H}} + \beta\|\boldsymbol{u}\|^{r+1}_{\widetilde \mathbb{L}^{r+1}}\nonumber\\&= (\boldsymbol{f},\boldsymbol{u}) +\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\int_{\mathbb{R}^d}S(t,x,\boldsymbol{u})\boldsymbol{u}\d x\nonumber\\&\leq \|\boldsymbol{f}\|_{\mathbb{H}}\|\boldsymbol{u}\|_{\mathbb{H}}+\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|\int_{\mathbb{R}^d}\left(\mathcal{S}_1(t,x)|\boldsymbol{u}|^q+\mathcal{S}_2(t,x)|\boldsymbol{u}|\right)\d x\nonumber\\&\leq\frac{\alpha}{4}\|\boldsymbol{u}\|^2_{\mathbb{H}}+\frac{\beta}{2}\|\boldsymbol{u}\|^{r+1}_{\widetilde{\mathbb{L}}^{r+1}}+M_1\|\boldsymbol{f}\|^2_{\mathbb{H}}+M_2\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|^{\frac{r+1}{r+1-q}}\|\mathcal{S}_1(t)\|^{\frac{r+1}{r+1-q}}_{\mathbb{L}^{\frac{r+1}{r+1-q}}(\mathbb{R}^d)}\nonumber\\&\quad+M_3\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|^{\frac{r+1}{r}}\|\mathcal{S}_2(t)\|^{\frac{r+1}{r}}_{\mathbb{L}^{\frac{r+1}{r}}(\mathbb{R}^d)},
\end{align}
where we have used \eqref{b0}, \eqref{GS1}, Lemmas \ref{Holder} and \ref{Young}, and $M_1, M_2$ and $M_3$ are positive constants independent of $\tau, \mathfrak{s}, \omega$ and $D$. From \eqref{ue12}, we find
\begin{align}\label{ue13}
&\frac{\d}{\d t} \|\boldsymbol{u}\|^2_{\mathbb{H}}+ \alpha\|\boldsymbol{u}\|^2_{\mathbb{H}}+\min\left\{\frac{\alpha}{2},2\mu,\beta\right\}\left[\|\boldsymbol{u}\|^2_{\mathbb{H}}+\|\nabla\boldsymbol{u}\|^2_{\mathbb{H}} + \|\boldsymbol{u}\|^{r+1}_{\widetilde \mathbb{L}^{r+1}}\right]\nonumber\\& \leq M_1\|\boldsymbol{f}\|^2_{\mathbb{H}}+M_2\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|^{\frac{r+1}{r+1-q}}\|\mathcal{S}_1(t)\|^{\frac{r+1}{r+1-q}}_{\mathbb{L}^{\frac{r+1}{r+1-q}}(\mathbb{R}^d)}+M_3\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|^{\frac{r+1}{r}}\|\mathcal{S}_2(t)\|^{\frac{r+1}{r}}_{\mathbb{L}^{\frac{r+1}{r}}(\mathbb{R}^d)}.
\end{align}
Applying variation of constant formula to \eqref{ue13} and replacing $\omega$ by $\vartheta_{-\mathfrak{s}}\omega$, we have
\begin{align}\label{ue14}
& \|\boldsymbol{u}(\tau,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{H}}+\min\left\{\frac{\alpha}{2},2\mu,\beta\right\}\int_{\mathfrak{s}-t}^{\tau}e^{\alpha(\xi-\tau)}\bigg[\|\boldsymbol{u}(\xi,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{H}} \nonumber\\&+\|\nabla\boldsymbol{u}(\xi,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{H}}+\|\boldsymbol{u}(\xi,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}\bigg]\d\xi \nonumber\\&\leq e^{\alpha(\mathfrak{s}-t-\tau)}\|\boldsymbol{u}_{\mathfrak{s}-t}\|^2_{\mathbb{H}}+M_4 \int_{\mathfrak{s}-t}^{\tau}e^{\alpha(\xi-\tau)}\left[\|\boldsymbol{f}(\cdot,\xi)\|^2_{\mathbb{H}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi-\mathfrak{s}}\omega)\right|^{\frac{r+1}{r+1-q}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi-\mathfrak{s}}\omega)\right|^{\frac{r+1}{r}}\right]\d\xi\nonumber\\&\leq e^{\alpha(\mathfrak{s}-t-\tau)}\|\boldsymbol{u}_{\mathfrak{s}-t}\|^2_{\mathbb{H}}+M_4 \int_{-\infty}^{\tau-\mathfrak{s}}e^{\alpha(\xi+\mathfrak{s}-\tau)}\left[\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{H}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r+1-q}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r}}\right]\d\xi,
\end{align}
where $M_4$ is a positive constant independent of $\tau, \mathfrak{s}, \omega$ and $D$. Second term of the right hand side of \eqref{ue14} is finite due to \eqref{N3} and \eqref{forcing3}. Since $\boldsymbol{u}_{\mathfrak{s}-t}\in D(\mathfrak{s}-t,\vartheta_{-t}\omega)$ and $D\in\mathfrak{D}$, we have
\begin{align}\label{ue15}
e^{\alpha(\mathfrak{s}-t-\tau)}\|\boldsymbol{u}_{\mathfrak{s}-t}\|^2_{\mathbb{H}}\leq e^{\alpha(\mathfrak{s}-t-\tau)}\|D(\mathfrak{s}-t,\vartheta_{-t}\omega)\|^2_{\mathbb{H}}\to0,
\end{align}
as $t\to\infty$, there exists $\mathscr{T}=\mathscr{T}(\delta,\mathfrak{s},\omega,D)>0$ such that for all $t\geq\mathscr{T}$,
\begin{align}\label{ue16}
e^{\alpha(\mathfrak{s}-t-\tau)}\|\boldsymbol{u}_{\mathfrak{s}-t}\|^2_{\mathbb{H}}\leq \int_{-\infty}^{\tau-\mathfrak{s}}e^{\alpha(\xi+\mathfrak{s}-\tau)}\left[\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r+1-q}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r}}\right]\d\xi.
\end{align}
From \eqref{ue14} along with \eqref{ue16}, one can complete the proof.
\end{proof}
\begin{lemma}\label{PAS_GA}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{H})$ satisfies \eqref{forcing5} and Assumption \ref{NDT3} is fulfilled. Then the continuous cocycle $\Phi$ associated with the system \eqref{WZ_SCBF} possesses a closed measurable $\mathfrak{D}$-pullback absorbing set $\widehat{\mathcal{K}}=\{\widehat{\mathcal{K}}(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D}$ defined for each $\mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$
\begin{align}
\widehat{\mathcal{K}}(\mathfrak{s},\omega)=\{\boldsymbol{u}\in\mathbb{H}:\|\boldsymbol{u}\|^2_{\mathbb{H}}\leq\widehat{\mathcal{L}}(\mathfrak{s},\omega)\},
\end{align}
where
\begin{align}
\widehat{\mathcal{L}}(\mathfrak{s},\omega)=\widetilde M \int_{-\infty}^{0}e^{\alpha\xi}\left[\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{H}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r+1-q}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r}}\right]\d\xi,
\end{align}
with $\widetilde M$ appearing in \eqref{ue^3}.
\end{lemma}
\begin{proof}
Since $\widehat{\mathcal{L}}(\mathfrak{s},\cdot):\Omega\to\mathbb{R}$ is $(\mathscr{F},\mathscr{B}(\mathbb{R}))$-measurable for every $\mathfrak{s}\in\mathbb{R}$, $\widehat{\mathcal{K}}(\mathfrak{s},\cdot):\Omega\to2^{\mathbb{H}}$ is a measurable set-valued mapping. Moreover, we have from Lemma \ref{LemmaUe3} that for each $\mathfrak{s}\in\mathbb{R}$, $\omega\in\Omega$ and $D\in\mathfrak{D}$, there exists $\mathscr{T}=\mathscr{T}(\delta,\mathfrak{s},\omega,D)>0$ such that for all $t\geq\mathscr{T}$,
\begin{align}\label{PAS6}
\Phi(t,\mathfrak{s}-t,\vartheta_{-t}\omega,D(\mathfrak{s}-t,\vartheta_{-t}\omega))=\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,D(\mathfrak{s}-t,\vartheta_{-t}\omega))\subseteq\widehat{\mathcal{K}}(\mathfrak{s},\omega).
\end{align}
It only remains to show that $\widehat{\mathcal{K}}\in\mathfrak{D}$, that is, for every $c>0$, $\mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$ $$\lim_{t\to-\infty}e^{ct}\|\widehat{\mathcal{K}}(\mathfrak{s} +t,\vartheta_{t}\omega)\|^2_{\mathbb{H}}=0.$$
For every $c>0$, $\mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$,
\begin{align}\label{PAS7}
& \lim_{t\to-\infty}e^{ct}\|\widehat{\mathcal{K}}(\mathfrak{s} +t,\vartheta_{t}\omega)\|^2_{\mathbb{H}}=\lim_{t\to-\infty}e^{ct}\widehat{\mathcal{L}}(\mathfrak{s} +t,\vartheta_{t}\omega)\nonumber\\&=\widetilde{M}\lim_{t\to-\infty}e^{ct} \int_{-\infty}^{0}e^{\alpha\xi}\left[\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s}+t)\|^2_{\mathbb{H}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|^{\frac{r+1}{r+1-q}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|^{\frac{r+1}{r}}\right]\d\xi.
\end{align}
Let $c_2=\min\{c,\alpha\}.$ Consider,
\begin{align}\label{PAS8}
&\lim_{t\to-\infty}e^{ct} \int_{-\infty}^{0}e^{\alpha\xi}\left[\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|^{\frac{r+1}{r+1-q}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|^{\frac{r+1}{r}}\right]\d\xi\nonumber\\&\leq\lim_{t\to-\infty} \int_{-\infty}^{0}e^{c_2(\xi+t)}\left[\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|^{\frac{r+1}{r+1-q}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi+t}\omega)\right|^{\frac{r+1}{r}}\right]\d\xi\nonumber\\&\leq\lim_{t\to-\infty} \int_{-\infty}^{t}e^{c_2\xi}\left[\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r+1-q}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r}}\right]\d\xi=0,
\end{align}
where we have used the fact that $\int_{-\infty}^{0}e^{c_2\xi}\left[\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r+1-q}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r}}\right]\d\xi<\infty$ due to \eqref{N3}.
Hence \eqref{forcing5}, \eqref{PAS7} and \eqref{PAS8} imply
\begin{align*}
\lim_{t\to-\infty}e^{ct}\|\widehat{\mathcal{K}}(\mathfrak{s} +t,\vartheta_{t}\omega)\|^2_{\mathbb{H}}=0,
\end{align*}
as required.
\end{proof}
Next lemma plays a pivotal role to prove the $\mathfrak{D}$-pullback asymptotic compactness of $\Phi$ (see Lemma \ref{Asymptotic_UB_GS}).
\begin{lemma}\label{largeradius}
For $d=2$ with $r\geq1, d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\mathrm{loc}}(\mathbb{R};\mathbb{H})$ and satisfies \eqref{forcing3}. Then, for any $\boldsymbol{u}_{\mathfrak{s}-t}\in D(\mathfrak{s}-t,\vartheta_{-t}\omega),$ where $D=\{D(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in \Omega\}\in\mathfrak{D}$, and for any $\eta>0$, $\mathfrak{s}\in\mathbb{R}$, $\omega\in \Omega$ and $0<\delta\leq1$, there exists $\mathscr{T}^*=\mathscr{T}^*(\delta,\mathfrak{s},\omega,D,\eta)\geq 1$ and $P^*=P^*(\delta,\mathfrak{s},\omega,\eta)>0$ such that for all $t\geq \mathscr{T}^*$ and $\tau\in[\mathfrak{s}-1,\mathfrak{s}]$, the solution of \eqref{WZ_SCBF} with $\omega$ replaced by $\vartheta_{-\mathfrak{s}}\omega$ satisfy
\begin{align}\label{ep}
\int_{|x|\geq P^*}|\boldsymbol{u}(\tau,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t}) |^2\d x\leq \eta.
\end{align}
\end{lemma}
\begin{proof}
Let $\Psi$ be a smooth function such that $0\leq\Psi(s)\leq 1$ for $s\in\mathbb{R}^+$ and
\begin{align}\label{Psi}
\Psi(s)=\begin{cases*}
0,\quad \text{ for }0\leq s\leq 1,\\
1, \quad\text{ for } s\geq2 .
\end{cases*}
\end{align}
Then, there exists a positive constant $C$ such that $|\Psi'(s)|\leq C$ for all $s\in\mathbb{R}^+$. Taking the inner product of first equation of \eqref{WZ_SCBF} with $\Psi\left(\frac{|x|^2}{k^2}\right)\boldsymbol{u}$ in $\mathbb{H}$, we have
\begin{align}\label{ep1}
\frac{1}{2} \frac{\d}{\d t} \int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)|\boldsymbol{u}|^2\d x &= -\mu \int_{\mathbb{R}^d}(\mathrm{A}\boldsymbol{u}) \Psi\left(\frac{|x|^2}{k^2}\right) \boldsymbol{u} \d x-\alpha \int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)|\boldsymbol{u}|^2\d x\nonumber\\&\quad-b\left(\boldsymbol{u},\boldsymbol{u},\Psi\left(\frac{|x|^2}{k^2}\right)\boldsymbol{u}\right)-\beta \int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)|\boldsymbol{u}|^{r+1}\d x\nonumber\\&\quad+ \int_{\mathbb{R}^d}\boldsymbol{f}(x,t)\Psi\left(\frac{|x|^2}{k^2}\right)\boldsymbol{u}\d x +\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)S(t,x,\boldsymbol{u})\boldsymbol{u}\d x.
\end{align}
We estimate each term on right hand side of \eqref{ep1}. Integration by parts helps to obtain
\begin{align}\label{ep2}
-&\mu \int_{\mathbb{R}^d}(\mathrm{A}\boldsymbol{u}) \Psi\left(\frac{|x|^2}{k^2}\right) \boldsymbol{u} \d x\nonumber\\&= -\mu \int_{\mathbb{R}^d}|\nabla\boldsymbol{u}|^2 \Psi\left(\frac{|x|^2}{k^2}\right) \d x -\mu \int_{\mathbb{R}^d} \Psi'\left(\frac{|x|^2}{k^2}\right)\frac{2}{k^2}(x\cdot\nabla) \boldsymbol{u}\cdot\boldsymbol{u} \d x\nonumber\\&= -\mu \int_{\mathbb{R}^d}|\nabla\boldsymbol{u}|^2 \Psi\left(\frac{|x|^2}{k^2}\right) \d x -\mu \int\limits_{k\leq|x|\leq \sqrt{2}k}\Psi'\left(\frac{|x|^2}{k^2}\right)\frac{2}{k^2}(x\cdot\nabla) \boldsymbol{u}\cdot\boldsymbol{u} \d x\nonumber\\&\leq -\mu \int_{\mathbb{R}^d}|\nabla\boldsymbol{u}|^2 \Psi\left(\frac{|x|^2}{k^2}\right) \d x +\frac{2\sqrt{2}\mu}{k} \int\limits_{k\leq|x|\leq \sqrt{2}k}\left|\boldsymbol{u}\right| \left|\Psi'\left(\frac{|x|^2}{k^2}\right)\right|\left|\nabla \boldsymbol{u}\right| \d x\nonumber\\&\leq -\mu \int_{\mathbb{R}^d}|\nabla\boldsymbol{u}|^2 \Psi\left(\frac{|x|^2}{k^2}\right) \d x +\frac{C}{k} \int_{\mathbb{R}^d}\left|\boldsymbol{u}\right| \left|\nabla \boldsymbol{u}\right| \d x\nonumber\\&\leq -\mu \int_{\mathbb{R}^d}|\nabla\boldsymbol{u}|^2 \Psi\left(\frac{|x|^2}{k^2}\right) \d x +\frac{C}{k} \left(\|\boldsymbol{u}\|^2_{\mathbb{H}}+\|\nabla\boldsymbol{u}\|^2_{\mathbb{H}}\right),
\end{align}
and
\begin{align}\label{ep3}
-b\left(\boldsymbol{u},\boldsymbol{u},\Psi\left(\frac{|x|^2}{k^2}\right)\boldsymbol{u}\right)&=\int_{\mathbb{R}^d} \Psi'\left(\frac{|x|^2}{k^2}\right)\frac{x}{k^2}\cdot\boldsymbol{u} |\boldsymbol{u}|^2 \d x= \int\limits_{k\leq|x|\leq \sqrt{2}k} \Psi'\left(\frac{|x|^2}{k^2}\right)\frac{x}{k^2}\cdot\boldsymbol{u} |\boldsymbol{u}|^2 \d x\nonumber\\&\leq \frac{\sqrt{2}}{k} \int\limits_{k\leq|x|\leq \sqrt{2}k} \left|\Psi'\left(\frac{|x|^2}{k^2}\right)\right| |\boldsymbol{u}|^3 \d x\leq \frac{C}{k}\|\boldsymbol{u}\|^3_{\widetilde \mathbb{L}^3}.
\end{align}
Using Lemmas \ref{Interpolation} and \ref{Young} in \eqref{ep3}, we get
\begin{align}
-b\left(\boldsymbol{u},\boldsymbol{u},\Psi\left(\frac{|x|^2}{k^2}\right)\boldsymbol{u}\right)\leq\begin{cases}
\frac{C}{k}\left[\|\boldsymbol{u}\|^2_{\mathbb{H}}+\|\boldsymbol{u}\|^{4}_{\widetilde \mathbb{L}^{4}}\right],&\ \ \ \text{ for } d=2 \text{ with } r\geq1,\\ \vspace{1mm}
\frac{C}{k}\left[\|\boldsymbol{u}\|^2_{\mathbb{H}}+\|\boldsymbol{u}\|^{r+1}_{\widetilde \mathbb{L}^{r+1}}\right], &\ \ \ \text{ for } d=3 \text{ with } r\geq3.
\end{cases}
\end{align}
Finally, we estimate the last two terms of \eqref{ep1} as follows
\begin{align}
\int_{\mathbb{R}^d}\boldsymbol{f}(x,t)\Psi\left(\frac{|x|^2}{k^2}\right)\boldsymbol{u} \d x\leq \frac{\alpha}{2} \int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)|\boldsymbol{u}|^2\d x +\frac{1}{2\alpha} \int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)|\boldsymbol{f}(x,t)|^2\d x,
\end{align}
and
\begin{align}\label{ep4}
&\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)S(t,x,\boldsymbol{u})\boldsymbol{u}\d x\nonumber\\&\leq\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|\int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)\left(\mathcal{S}_1(t,x)|\boldsymbol{u}|^q+\mathcal{S}_2(t,x)|\boldsymbol{u}|\right)\d x\nonumber\\&\leq\frac{\beta}{2} \int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)|\boldsymbol{u}|^{r+1}\d x+C\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|^{\frac{r+1}{r+1-q}}\int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)\left|\mathcal{S}_1(t,x)\right|^{\frac{r+1}{r+1-q}}\d x\nonumber\\&\quad+C\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|^{\frac{r+1}{r}}\int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)\left|\mathcal{S}_2(t,x)\right|^{\frac{r+1}{r}}\d x.
\end{align}
Combining \eqref{ep1}-\eqref{ep4}, we get
\begin{align}\label{ep5}
& \frac{\d}{\d t} \int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)|\boldsymbol{u}|^2\d x \nonumber\\&\leq -\alpha \int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)|\boldsymbol{u}|^2\d x+\frac{C}{k} \left(\|\boldsymbol{u}\|^2_{\mathbb{V}}+\|\boldsymbol{u}\|^{j}_{\widetilde \mathbb{L}^{j}}\right)+\frac{1}{\alpha} \int_{|x|\geq k}|\boldsymbol{f}(x,t)|^2\d x\nonumber\\&\quad+C\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|^{\frac{r+1}{r+1-q}}\int_{|x|\geq k}\left|\mathcal{S}_1(t,x)\right|^{\frac{r+1}{r+1-q}}\d x+C\left|\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\right|^{\frac{r+1}{r}}\int_{|x|\geq k}\left|\mathcal{S}_2(t,x)\right|^{\frac{r+1}{r}}\d x,
\end{align}
where $j=4$ and $j=r+1$ for $d=2$ and $d=3$, respectively. Applying Gronwall's inequality to \eqref{ep5} on $(\mathfrak{s}-t,\tau)$ where $\tau\in[\mathfrak{s}-1,\mathfrak{s}]$ and replacing $\omega$ by $\vartheta_{-\mathfrak{s}}\omega$, we obtain that, for $\mathfrak{s}\in\mathbb{R}, t\geq 1$ and $\omega\in \Omega$,
\begin{align}\label{ep6}
&\int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)|\boldsymbol{u}(\tau,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})|^2\d x \nonumber\\&\leq e^{\alpha( \mathfrak{s}-t-\tau)}\int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)|\boldsymbol{u}_{\mathfrak{s}-t}|^2\d x\nonumber\\&\quad+\frac{C}{k}\int_{\mathfrak{s}-t}^{\tau}e^{\alpha(\xi-\tau)} \left(\|\boldsymbol{u}(\xi,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{V}}+\|u(\xi,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^{j}_{\widetilde\mathbb{L}^{j}}\right)\d\xi\nonumber\\&\quad+C\int_{-\infty}^{\tau-\mathfrak{s}}e^{\alpha (\xi+\mathfrak{s}-\tau)}\bigg[\int_{|x|\geq k}|\boldsymbol{f}(x,\xi+\mathfrak{s})|^2\d x+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r+1-q}}\int_{|x|\geq k}\left|\mathcal{S}_1(\xi+\mathfrak{s},x)\right|^{\frac{r+1}{r+1-q}}\d x\nonumber\\&\qquad+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r}}\int_{|x|\geq k}\left|\mathcal{S}_2(\xi+\mathfrak{s},x)\right|^{\frac{r+1}{r}}\d x\bigg]\d \xi,
\end{align}
where $j=4$ and $j=r+1$ for $d=2$ and $d=3$, respectively. Since $\boldsymbol{u}_{\mathfrak{s}-t}\in D(\mathfrak{s}-t,\vartheta_{-t}\omega)$ and $D\in\mathfrak{D}$, from \eqref{ue15}, we find that, for given $\eta>0,$ there exists $T_1=T_1(\delta,\mathfrak{s},\omega,D,\eta)\geq 1$ such that for all $t\geq T_1,$
\begin{align}\label{ep7}
e^{\alpha (\mathfrak{s}-t-\tau)}\int_{\mathbb{R}^d}\Psi\left(\frac{|x|^2}{k^2}\right)|\boldsymbol{u}_{\mathfrak{s}-t}|^2\d x\leq\frac{\eta}{4}.
\end{align}
Since $\boldsymbol{f}\in\mathrm{L}^2_{\mathrm{loc}}(\mathbb{R};\mathbb{H})$ satisfies \eqref{forcing4}, for a given $\eta>0,$ there exists $P_1=P_1(\delta,\mathfrak{s},\omega,\eta)>0$ such that for all $k\geq P_1$,
\begin{align}\label{ep8}
&C\int_{-\infty}^{\tau-\mathfrak{s}}e^{\alpha (\xi+\mathfrak{s}-\tau)} \int_{|x|\geq k}|\boldsymbol{f}(x,\xi+\mathfrak{s})|^2\d x\d\xi\leq Ce^{\alpha}\int_{-\infty}^{0}e^{\alpha \xi} \int_{|x|\geq k}|\boldsymbol{f}(x,\xi+\mathfrak{s})|^2\d x\d\xi\leq \frac{\eta}{4}.
\end{align}
By \eqref{N3}, we have
\begin{align*}
& C\int_{-\infty}^{\tau-\mathfrak{s}}e^{\alpha (\xi+\mathfrak{s}-\tau)}\bigg[\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r+1-q}}\int_{\mathbb{R}^d}\left|\mathcal{S}_1(\xi+\mathfrak{s},x)\right|^{\frac{r+1}{r+1-q}}\d x\nonumber\\&\qquad+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r}}\int_{\mathbb{R}^d}\left|\mathcal{S}_2(\xi+\mathfrak{s},x)\right|^{\frac{r+1}{r}}\d x\bigg]\d \xi\nonumber\\&\leq C\int_{-\infty}^{\tau-\mathfrak{s}}e^{\alpha (\xi+\mathfrak{s}-\tau)}\bigg[\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r+1-q}}+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r}}\bigg]\d \xi <\infty,
\end{align*}
from which it follows that, for given $\eta>0,$ there exists $P_2=P_2(\delta,\mathfrak{s},\omega,\eta)>0$ such that for all $k\geq P_2$,
\begin{align}\label{ep9}
& C\int_{-\infty}^{\tau-\mathfrak{s}}e^{\alpha (\xi+\mathfrak{s}-\tau)}\bigg[\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r+1-q}}\int_{|x|\geq k}\left|\mathcal{S}_1(\xi+\mathfrak{s},x)\right|^{\frac{r+1}{r+1-q}}\d x\nonumber\\&\qquad+\left|\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{r+1}{r}}\int_{|x|\geq k}\left|\mathcal{S}_2(\xi+\mathfrak{s},x)\right|^{\frac{r+1}{r}}\d x\bigg]\d \xi\leq\frac{\eta}{4}.
\end{align}
Due to \eqref{ue^3} and \eqref{ep8}-\eqref{ep9}, there exists $T_2=T_2(\delta,\mathfrak{s},\omega,D,\eta)>0$ and $P_3=P_3(\delta,\mathfrak{s},\omega,\eta)>0$ such that for all $t\geq T_2$ and $k\geq P_3$,
\begin{align}\label{ep10}
\frac{C}{k}\int_{\mathfrak{s}-t}^{\tau}e^{\alpha(\xi-\tau)} \left(\|\boldsymbol{u}(\xi,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{V}}+\|\boldsymbol{u}(\xi,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}\right)\d\xi\leq \frac{\eta}{4},
\end{align}
for $d=3$ and
\begin{align}\label{ep10*}
\frac{C}{k}\int_{\mathfrak{s}-t}^{\tau}e^{\alpha(\xi-\tau)} \left(\|\boldsymbol{u}(\xi,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{V}}+\|\boldsymbol{u}(\xi,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^{4}_{\widetilde\mathbb{L}^{4}}\right)\d\xi\leq \frac{\eta}{4},
\end{align}
where we have used $\|\boldsymbol{u}\|^4_{\widetilde\mathbb{L}^4}\leq\|\boldsymbol{u}\|^2_{\mathbb{H}}\|\nabla\boldsymbol{u}\|^2_{\mathbb{H}}$, for $d=2$.
Let $\mathscr{T}^*=\mathscr{T}^*(\delta,\mathfrak{s},\omega,D,\eta)=\max\{T_1,T_2\}, P^*=P^*(\delta,\mathfrak{s},\omega,\eta)=\max\{P_1,P_2,P_3\}$. Then it implies from \eqref{ep6}-\eqref{ep10} that, for all $t\geq \mathscr{T}^*$ and $k\geq P^*$, we obtain that for all $\tau\in[\mathfrak{s}-1,\mathfrak{s}]$,
\begin{align*}
\int\limits_{|x|\geq P^*}|\boldsymbol{u}(\tau,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})|^2\d x\leq \eta,
\end{align*}
which completes the proof.
\end{proof}
\begin{lemma}\label{PCB_U}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{H})$, $\{\boldsymbol{u}(\cdot,\mathfrak{s},\omega,\boldsymbol{u}^n)\}_{n\in\mathbb{N}}$ is a bounded sequence of solutions of \eqref{WZ_SCBF} in $\mathbb{H}$ and Assumption \ref{NDT3} is fulfilled. Then for every $0<\delta\leq1, \omega\in\Omega, \mathfrak{s}\in\mathbb{R}$ and $t>\mathfrak{s}$, there exists $\boldsymbol{u}^0\in\mathrm{L}^2(\mathfrak{s},\mathfrak{s}+T;\mathbb{H})$ with $T>0$ and a subsequence $\{\boldsymbol{u}(\cdot,\mathfrak{s},\omega,\boldsymbol{u}^{n_m})\}_{m\in\mathbb{N}}$ of $\{\boldsymbol{u}(\cdot,\mathfrak{s},\omega,\boldsymbol{u}^n)\}_{n\in\mathbb{N}}$ such that $\boldsymbol{u}(\tau,\mathfrak{s},\omega,\boldsymbol{u}^{n_m})\to\boldsymbol{u}^0(\tau)$ in $\mathbb{L}^2(\mathcal{O}_k)$ as $m\to\infty$ for every $k\in\mathbb{N}$ and for almost all $\tau\in(\mathfrak{s},\mathfrak{s}+T)$, where $$\mathcal{O}_k=\{x\in\mathbb{R}^d:|x|<k\}.$$
\end{lemma}
\begin{proof}
Since embedding $\mathbb{H}^1_0(\mathcal{O}_k)\hookrightarrow\mathbb{L}^2(\mathcal{O}_k)$ is compact, the proof can be completed analogously as in the proof of Lemma \ref{PCB}.
\end{proof}
The following theorem proves the $\mathfrak{D}$-pullback asymptotic compactness of $\Phi$.
\begin{lemma}\label{Asymptotic_UB_GS}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{H})$ satisfies \eqref{forcing4} and Assumption \ref{NDT3} is fulfilled. Then for every $0<\delta\leq1$, $\mathfrak{s}\in \mathbb{R},$ $\omega\in \Omega,$ $D=\{D(\mathfrak{s},\omega):\mathfrak{s}\in \mathbb{R},\omega\in \Omega\}\in \mathfrak{D}$ and $t_n\to \infty,$ $\boldsymbol{u}_{0,n}\in D(\mathfrak{s}-t_n, \vartheta_{-t_{n}}\omega)$, the sequence $\Phi(t_n,\mathfrak{s}-t_n,\vartheta_{-t_n}\omega,\boldsymbol{u}_{0,n})$ or $\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})$ of solutions of the system \eqref{WZ_SCBF} has a convergent subsequence in $\mathbb{H}$.
\end{lemma}
\begin{proof}
Lemma \ref{LemmaUe3} implies that there exists $\mathscr{T}=\mathscr{T}(\delta,\mathfrak{s},\omega,D)>0$ and $R(\mathfrak{s},\omega)$ such that for all $t\geq \mathscr{T}$,
\begin{align}\label{Uac1}
\|\boldsymbol{u}(\mathfrak{s}-1,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{H}} \leq R(\mathfrak{s},\omega),
\end{align}
where $\boldsymbol{u}_{\mathfrak{s}-t}\in D(\mathfrak{s}-t,\vartheta_{-t}\omega).$ Since $t_n\to \infty$, there exists $N_2\in\mathbb{N}$ such that $t_n\geq \mathscr{T}$ for all $n\geq N_2$. Since $\boldsymbol{u}_{0,n}\in D(\mathfrak{s}-t_n, \vartheta_{-t_{n}}\omega)$, \eqref{Uac1} implies that for all $n\geq N_2$,
\begin{align*}
\|\boldsymbol{u}(\mathfrak{s}-1,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\|^2_{\mathbb{H}} \leq R(\mathfrak{s},\omega),
\end{align*}
and hence
\begin{align}\label{Uac2}
\{\boldsymbol{u}(\mathfrak{s}-1,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\}_{n\geq N_2} \text{ is a bounded sequence in }\mathbb{H}.
\end{align}
It yields from \eqref{Uac2} and Lemma \ref{PCB_U} that there exists $\tau\in(\mathfrak{s}-1,\mathfrak{s})$, $\boldsymbol{u}_0$ in $\mathbb{H}$ and a subsequence (not relabeled) such that for every $k\in\mathbb{N}$ as $n\to\infty$
\begin{align}\label{Uac3}
\boldsymbol{u}(\tau,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})=\boldsymbol{u}(\tau,\mathfrak{s}-1,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\mathfrak{s}-1,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))\to \boldsymbol{u}_0 \ \text{ in }\ \mathbb{L}^2(\mathcal{O}_k).
\end{align}
Since $\boldsymbol{u}_0\in\mathbb{H}$, for given $\eta>0$, there exists $K_1=K_1(\delta,\mathfrak{s},\omega,\eta)>0$ such that for all $k\geq K_1$,
\begin{align}\label{Uac4}
\int\limits_{|x|\geq k}|\boldsymbol{u}_0|^2\d x\leq\frac{\eta}{3}.
\end{align}
Also, it follows from Lemma \ref{largeradius} that there exists $N_3=N_3(\delta,\mathfrak{s},\omega,D,\eta)\geq 1$ and $K_2=K_2(\delta,\mathfrak{s},\omega,\eta)\geq K_1$ such that for all $n\geq N_3$ and $k\geq K_2$,
\begin{align}\label{Uac5}
\int\limits_{|x|\geq k}|\boldsymbol{u}(\tau,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})|^2\d x\leq\frac{\eta}{3}.
\end{align}
From \eqref{Uac3}, we have that there exists $N_4=N_4(\delta,\mathfrak{s},\omega,D,\eta)>N_3$ such that for all $n\geq N_4$,
\begin{align}\label{Uac6}
\int\limits_{|x|< K_2}|\boldsymbol{u}(\tau,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})-\boldsymbol{u}_0|^2\d x\leq\frac{\eta}{3}.
\end{align}
Finally, Lemma \ref{ContinuityUB2} implies
\begin{align}\label{Uac7}
&\|\boldsymbol{u}(\mathfrak{s},\tau,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}(\tau,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n}))-\boldsymbol{u}(\mathfrak{s},\tau,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_0)\|^2_{\mathbb{H}}\nonumber\\&\leq C\|\boldsymbol{u}(\tau,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})-\boldsymbol{u}_0\|^2_{\mathbb{H}}\nonumber\\&\leq C\left[\int\limits_{|x|<K_2}|\boldsymbol{u}(\tau,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})-\boldsymbol{u}_0|^2\d x+\int\limits_{|x|\geq K_2}|\boldsymbol{u}(\tau,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})-\boldsymbol{u}_0|^2\d x\right]\nonumber\\&\leq C\left[\int\limits_{|x|<K_2}|\boldsymbol{u}(\tau,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})-\boldsymbol{u}_0|^2\d x+\hspace{-2mm}\int\limits_{|x|\geq K_2}(|\boldsymbol{u}(\tau,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})|^2+|\boldsymbol{u}_0|^2)\d x\right].
\end{align}
Hence, \eqref{Uac7} along with \eqref{Uac4}-\eqref{Uac6} conclude the proof.
\end{proof}
The following theorem is the main results of this section, that is, the existence of a unique random $\mathfrak{D}$-pullback attractor for $\Phi$.
\begin{theorem}\label{WZ_RA_UB_GS}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{H})$ satisfies \eqref{forcing5} and Assumption \ref{NDT2} is fulfilled. Then there exists a unique random $\mathfrak{D}$-pullback attractor $$\widehat{\mathscr{A}}=\{\widehat{\mathscr{A}}(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D},$$ for the the continuous cocycle $\Phi$ associated with the system \eqref{WZ_SCBF} in $\mathbb{H}$.
\end{theorem}
\begin{proof}
Note that Lemmas \ref{PAS_GA} and \ref{Asymptotic_UB_GS} provide the existence of a closed measurable $\mathfrak{D}$-pullback absorbing set for $\Phi$ and asymptotic compactness of $\Phi$, respectively. Hence, Lemmas \ref{PAS_GA} and \ref{Asymptotic_UB_GS} together with the abstract theory given in \cite{SandN_Wang} (Theorem 2.23, \cite{SandN_Wang}) complete the proof.
\end{proof}
\section{Convergence of attractors: Additive white noise} \label{sec6}\setcounter{equation}{0}
In this section, we examine the approximations of solutions of the following stochastic CBF equations with additive white noise. For given $\sigma\geq0$ and $\textbf{g}\in\mathrm{D}(\mathrm{A})$,
\begin{equation}\label{SCBF_Add}
\left\{
\begin{aligned}
\frac{\partial \boldsymbol{u}}{\partial t}+\mu\mathrm{A}\boldsymbol{u}+\mathrm{B}(\boldsymbol{u})+\alpha\boldsymbol{u}+\beta\mathcal{C}(\boldsymbol{u})&=\boldsymbol{f}+e^{\sigma t}\textbf{g}(x)\frac{\d \mathrm{W}}{\d t}, \ \ \ \text{ in } \mathbb{R}^n\times(\mathfrak{s},\infty), \\
\boldsymbol{u}(x,\mathfrak{s})&=\boldsymbol{u}_{\mathfrak{s}}(x),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \mathbb{R}^n \text{ and }\mathfrak{s}\in\mathbb{R}.
\end{aligned}
\right.
\end{equation}
For $\delta>0$, consider the pathwise random equations:
\begin{equation}\label{WZ_SCBF_Add}
\left\{
\begin{aligned}
\frac{\partial \boldsymbol{u}_{\delta}}{\partial t}+\mu\mathrm{A}\boldsymbol{u}_{\delta}+\mathrm{B}(\boldsymbol{u}_{\delta})+\alpha\boldsymbol{u}_{\delta}+\beta\mathcal{C}(\boldsymbol{u}_{\delta})&=\boldsymbol{f}+e^{\sigma t}\textbf{g}(x)\mathcal{Z}_{\delta}(\vartheta_{t}\omega), \ \ \text{ in } \mathbb{R}^n\times(\mathfrak{s},\infty), \\
\boldsymbol{u}_{\delta}(x,\mathfrak{s})&=\boldsymbol{u}_{\delta,\mathfrak{s}}(x),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \mathbb{R}^n \text{ and }\mathfrak{s}\in\mathbb{R}.
\end{aligned}
\right.
\end{equation}
Throughout this section, we prove results for 2D SCBF with $r>1$ and 3D SCBF with $r\geq3$ ($r>3$ for any $\beta,\mu>0$ and $r=3$ for $2\beta\mu\geq1$). Since, 2D SCBF equations are linear perturbation of 2D stochastic NSE for $r=1$, one can prove the results of this section for $d=2$ and $r=1$ by using the same arguments as it is done for 2D stochastic NSE on Poincar\'e domains in \cite{GGW} (see Section 3 in \cite{GGW}). In \cite{GGW}, authors imposed an extra condition (Assumption \ref{GA}) on $\textbf{g}(\cdot),$ which was used to prove the existence of $\mathfrak{D}$-pullback absorbing set. We observe that there is no need to impose Assumption \ref{GA} on $\textbf{g}(\cdot)$ for $r>1$.
\iffalse
Instead of using Assumption \ref{GA}, one can use obvious estimate given by
\begin{align*}
|b(\boldsymbol{u},\textbf{g},\boldsymbol{u})|=|b(\boldsymbol{u},\boldsymbol{u},\textbf{g})|\leq \|\textbf{g}\|_{\mathbb{L}^{\infty}}\|\boldsymbol{u}\|_{\mathbb{H}}\|\nabla\boldsymbol{u}\|_{\mathbb{H}}, \ \ \text{ for all } \ \boldsymbol{u}\in\mathbb{V},
\end{align*}
where
\begin{align*}
\|\textbf{g}\|_{\mathbb{L}^{\infty}}\leq C\|\textbf{g}\|^{1/2}_{\mathbb{H}}\|\textbf{g}\|^{1/2}_{\mathrm{D}(\mathrm{A})}<\infty, \ \text{ for all } \ \textbf{g}\in\mathrm{D}(\mathrm{A}),
\end{align*}
by Agmon's inequality for $d=2$.
\fi
\begin{assumption}\label{DNFT2}
For this section, we assume that external forcing term $\boldsymbol{f}$ satisfies the following two assumptions.
\begin{itemize}
\item [(i)]
\begin{align}\label{forcing6}
\int_{-\infty}^{\mathfrak{s}} e^{\alpha\xi}\|\boldsymbol{f}(\cdot,\xi)\|^2_{\mathbb{V}'}\d \xi<\infty, \ \ \text{ for all }\ \mathfrak{s}\in\mathbb{R}.
\end{align}
\item [(ii)] for every $c>0$
\begin{align}\label{forcing7}
\lim_{\tau\to-\infty}e^{c\tau}\int_{-\infty}^{0} e^{\alpha\xi}\|\boldsymbol{f}(\cdot,\xi+\tau)\|^2_{\mathbb{V}'}\d \xi=0,
\end{align}
where $\alpha>0$ is Darcy coefficient.
\end{itemize}
\end{assumption}
\subsection{Random $\mathfrak{D}$-pullback attractor for SCBF equations with additive white noise}
Consider, for some $\ell>0$
\begin{align*}
\boldsymbol{y}(\vartheta_{t}\omega) = \int_{-\infty}^{t} e^{-\ell(t-\tau)}\d \mathrm{W}(\tau), \ \ \omega\in \Omega,
\end{align*} which is the stationary solution of the one dimensional Ornstein-Uhlenbeck equation
\begin{align*}
\d\boldsymbol{y}(\vartheta_t\omega) + \ell\boldsymbol{y}(\vartheta_t\omega)\d t =\d\mathrm{W}(t).
\end{align*}
Let us recall from \cite{FAN} that there exists a $\vartheta$-invariant subset of $\Omega$ (will be denoted by $\Omega$ itself) of full measure such that $\boldsymbol{y}(\vartheta_t\omega)$ is continuous in $t$ for every $\omega\in \Omega,$ and
\begin{align}
\lim_{t\to \pm \infty} \frac{|\boldsymbol{y}(\vartheta_t\omega)|}{|t|}=0 \text{\ \ and \ \ }
\lim_{t\to \pm \infty} \frac{1}{t} \int_{0}^{t} \boldsymbol{y}(\vartheta_{\xi}\omega)\d\xi =0.\label{Y2}
\end{align}
Define
\begin{align}\label{T_add}
\boldsymbol{v}(t,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}})=\boldsymbol{u}(t,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}})-e^{\sigma t}\textbf{g}(x)\boldsymbol{y}(\vartheta_{t}\omega).
\end{align}
Then, from \eqref{SCBF_Add}, we obtain
\begin{equation}\label{CSCBF_Add}
\left\{
\begin{aligned}
\frac{\partial \boldsymbol{v}}{\partial t}+\mu\mathrm{A}\boldsymbol{v}+\mathrm{B}(\boldsymbol{v}&+e^{\sigma t}\textbf{g}\boldsymbol{y})+\alpha\boldsymbol{v}+\beta\mathcal{C}(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y})\\&=\boldsymbol{f}+\left(\ell-\sigma-\alpha\right)e^{\sigma t}\textbf{g}\boldsymbol{y}-\mu e^{\sigma t}\boldsymbol{y}\mathrm{A}\textbf{g}, \ \ \ \text{ in } \mathbb{R}^d\times(\mathfrak{s},\infty), \\
\boldsymbol{v}(x,\mathfrak{s})&=\boldsymbol{v}_{\mathfrak{s}}(x)=\boldsymbol{u}_{\mathfrak{s}}(x)-e^{\sigma\mathfrak{s}}\textbf{g}(x)\boldsymbol{y}(\omega), \quad\qquad x\in \mathbb{R}^d \text{ and }\mathfrak{s}\in\mathbb{R}.
\end{aligned}
\right.
\end{equation}
For all $\mathfrak{s}\in\mathbb{R},$ $t>\mathfrak{s},$ and for every $\boldsymbol{v}_{\mathfrak{s}}\in\mathbb{H}$ and $\omega\in\Omega$, \eqref{CSCBF_Add} has a unique solution $\boldsymbol{v}(\cdot,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}})\in \mathrm{C}([\mathfrak{s},\mathfrak{s}+T];\mathbb{H})\cap\mathrm{L}^2(\mathfrak{s}, \mathfrak{s}+T;\mathbb{V})\cap\mathrm{L}^{r+1}(\mathfrak{s},\mathfrak{s}+T;\widetilde{\mathbb{L}}^{r+1})$. Moreover, $\boldsymbol{v}(t,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}})$ is continuous with respect to initial data $\boldsymbol{v}_{\mathfrak{s}}(x)$ (Lemma \ref{Continuity_add}) and $(\mathscr{F},\mathscr{B}(\mathbb{H}))$-measurable in $\omega\in\Omega.$ Define a cocycle $\Phi_0:\mathbb{R}^+\times\mathbb{R}\times\Omega\times\mathbb{H}\to\mathbb{H}$ for the system \eqref{SCBF_Add} such that for given $t\in\mathbb{R}^+, \mathfrak{s}\in\mathbb{R}, \omega\in\Omega$ and $\boldsymbol{u}_{\mathfrak{s}}\in\mathbb{H}$,
\begin{align}\label{Phi_0}
\Phi_0(t,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}}) &=\boldsymbol{u}(t+\mathfrak{s},\mathfrak{s},\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}})=\boldsymbol{v}(t+\mathfrak{s},\mathfrak{s},\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}})+e^{\sigma(t+\mathfrak{s})}\textbf{g}\boldsymbol{y}(\vartheta_{t}\omega).
\end{align}
\begin{lemma}\label{Continuity_add}
For $d=2$ with $r>1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$. Then, the solution of \eqref{CSCBF_Add} is continuous in initial data $\boldsymbol{v}_{\mathfrak{s}}(x).$
\end{lemma}
\begin{proof}
See the proofs of Theorem 4.8 in \cite{KM} and Theorem 4.10 in \cite{KM3} for $d=2$ and $d=3$, respectively.
\end{proof}
Next, we prove the existence of $\mathfrak{D}$-pullback absorbing set of $\Phi_0.$
\begin{lemma}\label{LemmaUe_add}
For $d=2$ with $r>1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing7}. Then $\Phi_0$ possesses a closed measurable $\mathfrak{D}$-pullback absorbing set $\mathcal{K}^1_0=\{\mathcal{K}^1_0(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D}$ in $\mathbb{H}$ given by
\begin{align}\label{ue_add}
\mathcal{K}^1_0(\mathfrak{s},\omega)=\{\boldsymbol{u}\in\mathbb{H}:\|\boldsymbol{u}\|^2_{\mathbb{H}}\leq \mathcal{R}^1_0(\mathfrak{s},\omega)\},\ \ \text{ for } d=2,
\end{align}
where $\mathcal{R}^1_0(\mathfrak{s},\omega)$ is defined by
\begin{align}\label{ue_add1}
\mathcal{R}^1_0(\mathfrak{s},\omega)&=3\|\textbf{g}\|^2_{\mathbb{H}}\left|e^{\sigma\mathfrak{s}}\boldsymbol{y}(\omega)\right|^2+2R_5 \int_{-\infty}^{0}e^{\alpha\xi}\bigg[\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^2\nonumber\\&\quad+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^{r+1}+\left|e^{\sigma(\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^{\frac{2(r+1)}{r-1}}\bigg]\d\xi,
\end{align}
and a closed measurable $\mathfrak{D}$-pullback absorbing set $\mathcal{K}^2_0=\{\mathcal{K}^2_0(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D}$ in $\mathbb{H}$ given by
\begin{align}\label{ue_add2}
\mathcal{K}^2_0(\mathfrak{s},\omega)=\{\boldsymbol{u}\in\mathbb{H}:\|\boldsymbol{u}\|^2_{\mathbb{H}}\leq \mathcal{R}^2_0(\mathfrak{s},\omega)\},\ \ \text{ for } d=3,
\end{align}
where $\mathcal{R}^2_0(\mathfrak{s},\omega)$ is defined by
\begin{align}\label{ue_add3}
\mathcal{R}^2_0(\mathfrak{s},\omega)&=3\|\textbf{g}\|^2_{\mathbb{H}}\left|e^{\sigma\mathfrak{s}}\boldsymbol{y}(\omega)\right|^2+2R_8 \int_{-\infty}^{0}e^{\alpha\xi}\bigg[\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^2\nonumber\\&\quad+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^{r+1}\bigg]\d\xi.
\end{align}
Here $R_5$ and $R_8$, both are positive constants do not depend on $\mathfrak{s}$ and $\omega$.
\end{lemma}
\begin{proof}
We infer from \eqref{CSCBF_Add} that
\begin{align}\label{ue17}
\frac{1}{2}\frac{\d}{\d t}\|\boldsymbol{v}\|^2_{\mathbb{H}}=&-\mu\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}-\alpha\|\boldsymbol{v}\|^2_{\mathbb{H}}-b(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y},\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y},\boldsymbol{v})-\beta\left\langle\mathcal{C}(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}),\boldsymbol{v}\right\rangle\nonumber\\&+\left\langle\boldsymbol{f},\boldsymbol{v}\right\rangle+e^{\sigma t}\boldsymbol{y}\left(\left(\ell-\sigma-\alpha\right)\textbf{g}-\mu \mathrm{A}\textbf{g},\boldsymbol{v}\right)\nonumber\\=&-\mu\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}-\alpha\|\boldsymbol{v}\|^2_{\mathbb{H}}-\beta\|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}+b(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y},\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y},e^{\sigma t}\textbf{g}\boldsymbol{y})\nonumber\\&+\beta\left\langle\mathcal{C}(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}),e^{\sigma t}\textbf{g}\boldsymbol{y}\right\rangle+\left\langle\boldsymbol{f},\boldsymbol{v}\right\rangle+e^{\sigma t}\boldsymbol{y}\left(\left(\ell-\sigma-\alpha\right)\textbf{g}-\mu \mathrm{A}\textbf{g},\boldsymbol{v}\right).
\end{align}
Applying Lemmas \ref{Holder} and \ref{Young}, we get that there exists a constant $R_1,R_2,R_3>0$ such that
\begin{align}
\beta\left\langle\mathcal{C}(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}),e^{\sigma t}\textbf{g}\boldsymbol{y}\right\rangle&\leq\beta \left|e^{\sigma t}\boldsymbol{y}\right|\|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\|^{r}_{\widetilde\mathbb{L}^{r+1}}\|\textbf{g}\|_{\widetilde\mathbb{L}^{r+1}}\nonumber\\&\leq\frac{\beta}{4}\|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\|^{r+1}_{\widetilde\mathbb{L}^{r+1}} + R_1\left|e^{\sigma t}\boldsymbol{y}\right|^{r+1},\label{ue18}\\
\left\langle\boldsymbol{f},\boldsymbol{v}\right\rangle&\leq\|\boldsymbol{f}\|_{\mathbb{V}'}\|\boldsymbol{v}\|_{\mathbb{V}}\leq\frac{\min\{\alpha,\mu\}}{6}\|\boldsymbol{v}\|^2_{\mathbb{V}}+R_2\|\boldsymbol{f}\|^{2}_{\mathbb{V}'},\label{ue19}\\
e^{\sigma t}\boldsymbol{y}\left(\left(\ell-\sigma-\alpha\right)\textbf{g}-\mu \mathrm{A}\textbf{g},\boldsymbol{v}\right)&= \left(\ell-\sigma-\alpha\right)e^{\sigma t}\boldsymbol{y}\left(\textbf{g},\boldsymbol{v}\right)+\mu e^{\sigma t}\boldsymbol{y}\left(\nabla\textbf{g},\nabla\boldsymbol{v}\right)\nonumber\\&\leq\frac{\alpha}{6}\|\boldsymbol{v}\|^2_{\mathbb{H}}+\frac{\mu}{6}\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}+R_3\left|e^{\sigma t}\boldsymbol{y}\right|^2.\label{ue20}
\end{align}
\vskip 2mm
\noindent
\textbf{Case I:} \textit{When $d=2$ and $r>1$.}
Using \eqref{b0}, Lemmas \ref{Holder} and \ref{Young}, and Sobolev's inequality, we obtain that there exists a constant $R_4>0$ such that
\begin{align}\label{ue21}
&\left|b(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y},\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y},e^{\sigma t}\textbf{g}\boldsymbol{y})\right|\nonumber\\&= \left|b(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y},e^{\sigma t}\textbf{g}\boldsymbol{y},\boldsymbol{v})\right|\nonumber\\&\leq \left|e^{\sigma t}\boldsymbol{y}\right|\|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\|_{\widetilde\mathbb{L}^{r+1}}\|\nabla\textbf{g}\|_{\mathbb{H}}\|\boldsymbol{v}\|_{\widetilde\mathbb{L}^{\frac{2(r+1)}{r-1}}}\nonumber\\&\leq\frac{\beta}{4}\|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\|^{r+1}_{\widetilde\mathbb{L}^{r+1}} +\frac{\min\{\alpha,\mu\}}{6}\|\boldsymbol{v}\|^{2}_{\mathbb{V}}+R_4\left|e^{\sigma t}\boldsymbol{y}\right|^{\frac{2(r+1)}{r-1}}.
\end{align}
Combining \eqref{ue17}-\eqref{ue21}, we get
\begin{align}\label{ue22}
\frac{\d}{\d t}\|\boldsymbol{v}\|^2_{\mathbb{H}}+\alpha\|\boldsymbol{v}\|^2_{\mathbb{H}}\leq R_5\left[\|\boldsymbol{f}\|^{2}_{\mathbb{V}'}+\left|e^{\sigma t}\boldsymbol{y}\right|^2+\left|e^{\sigma t}\boldsymbol{y}\right|^{r+1}+\left|e^{\sigma t}\boldsymbol{y}\right|^{\frac{2(r+1)}{r-1}}\right],
\end{align}
where $R_5=\max\{2R_1,2R_2,2R_3,2R_4\}$. Applying variation of constant formula to \eqref{ue22} over $(\mathfrak{s}-t,\tau)$ with $t\geq0$, $\mathfrak{s}\in\mathbb{R}$ and $\tau\geq\mathfrak{s}-t$, and replacing $\omega$ by $\vartheta_{-\mathfrak{s}}\omega$, we obtain
\begin{align}\label{ue23}
&\|\boldsymbol{v}(\tau,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{\mathfrak{s}-t})\|^2_{\mathbb{H}} \nonumber\\&\leq e^{\alpha(\mathfrak{s}-t-\tau)}\|\boldsymbol{v}_{\mathfrak{s}-t}\|^2_{\mathbb{H}}+R_5 \int_{-\infty}^{\tau-\mathfrak{s}}e^{\alpha(\xi+\mathfrak{s}-\tau)}\bigg[\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\nonumber\\&\qquad+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^2+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^{r+1}+\left|e^{\sigma(\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^{\frac{2(r+1)}{r-1}}\bigg]\d\xi.
\end{align}
From \eqref{Phi_0}, we have
\begin{align*}
\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})=\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{\mathfrak{s}-t})+e^{\sigma\mathfrak{s}}\boldsymbol{y}(\omega)\textbf{g},
\end{align*}
with $\boldsymbol{v}_{\mathfrak{s}-t}=\boldsymbol{u}_{\mathfrak{s}-t}-e^{\sigma(\mathfrak{s}-t)}\boldsymbol{y}(\vartheta_{-t}\omega)\textbf{g}$, $\boldsymbol{u}_{\mathfrak{s}-t}\in D(\mathfrak{s}-t,\vartheta_{-t}\omega)$ and $D\in\mathfrak{D}$, which together with \eqref{ue23} gives
\begin{align}\label{ue24}
& \|\boldsymbol{u}(\tau,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{H}}\nonumber\\&\leq2\|\boldsymbol{v}(\tau,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{\mathfrak{s}-t})\|^2_{\mathbb{H}}+2\|\textbf{g}\|^2_{\mathbb{H}}\left|e^{\sigma\mathfrak{s}}\boldsymbol{y}(\omega)\right|^2 \nonumber\\&\leq 4e^{-\alpha t}\left(\|\boldsymbol{u}_{\mathfrak{s}-t}\|^2_{\mathbb{H}}+\|\textbf{g}\|^2_{\mathbb{H}}\left|e^{\sigma(\mathfrak{s}-t)}\boldsymbol{y}(\vartheta_{-t}\omega)\right|^2\right)+2\|\textbf{g}\|^2_{\mathbb{H}}\left|e^{\sigma\mathfrak{s}}\boldsymbol{y}(\omega)\right|^2+2R_5 \int_{-\infty}^{0}e^{\alpha\xi}\nonumber\\&\quad\times\bigg[\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^2+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^{r+1}+\left|e^{\sigma(\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^{\frac{2(r+1)}{r-1}}\bigg]\d\xi.
\end{align}
Using \eqref{forcing6} and \eqref{Y2}, and arguing similarly as in the proofs of Lemmas \ref{LemmaUe3} and \ref{PAS_GA}, one can conclude the proof.
\vskip 2mm
\noindent
\textbf{Case II:} \textit{When $d= 3$ and $r\geq3$ ($r>3$ with any $\beta,\mu>0$ and $r=3$ with $2\beta\mu\geq1$).}
Using \eqref{b0}, Lemmas \ref{Holder}, \ref{Interpolation} and \ref{Young}, we obtain that there exists two constants $R_6,R_7>0$ such that
\begin{align}\label{ue25}
&\left|b(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y},\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y},e^{\sigma t}\textbf{g}\boldsymbol{y})\right|\nonumber\\&=\left|b(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y},e^{\sigma t}\textbf{g}\boldsymbol{y},\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y})\right|\nonumber\\&\leq \left|e^{\sigma t}\boldsymbol{y}\right|\|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\|_{\widetilde\mathbb{L}^{r+1}}\|\nabla\textbf{g}\|_{\mathbb{H}}\|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\|_{\widetilde\mathbb{L}^{\frac{2(r+1)}{r-1}}}\nonumber\\&\leq\left|e^{\sigma t}\boldsymbol{y}\right|\|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\|^{\frac{r+1}{r-1}}_{\widetilde\mathbb{L}^{r+1}}\|\nabla\textbf{g}\|_{\mathbb{H}}\|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\|^{\frac{r-3}{r-1}}_{\mathbb{H}}\nonumber\\&\leq\frac{\beta}{4}\|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\|^{r+1}_{\widetilde\mathbb{L}^{r+1}} +\frac{\alpha}{6}\|\boldsymbol{v}\|^{2}_{\mathbb{H}}+R_6\left|e^{\sigma t}\boldsymbol{y}\right|^{2}, \ \ \text{ for } r>3,
\end{align}
and
\begin{align}\label{ue26}
&\left|b(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y},\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y},e^{\sigma t}\textbf{g}\boldsymbol{y})\right|\nonumber\\&=\left|b(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y},e^{\sigma t}\textbf{g}\boldsymbol{y},\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y})\right|\nonumber\\&\leq \left|e^{\sigma t}\boldsymbol{y}\right|\|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\|^2_{\widetilde\mathbb{L}^{4}}\|\nabla\textbf{g}\|_{\mathbb{H}}\nonumber\\&\leq\frac{\beta}{4}\|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\|^{4}_{\widetilde\mathbb{L}^{4}} +R_7\left|e^{\sigma t}\boldsymbol{y}\right|^{2}, \ \ \text{ for } r=3.
\end{align}
Combining \eqref{ue17}-\eqref{ue20} and \eqref{ue25}-\eqref{ue26}, we get
\begin{align}\label{ue27}
\frac{\d}{\d t}\|\boldsymbol{v}\|^2_{\mathbb{H}}+\alpha\|\boldsymbol{v}\|^2_{\mathbb{H}}\leq R_8\left[\|\boldsymbol{f}\|^{2}_{\mathbb{V}'}+\left|e^{\sigma t}\boldsymbol{y}\right|^2+\left|e^{\sigma t}\boldsymbol{y}\right|^{r+1}\right],
\end{align}
where $R_8=\max\{2R_1,2R_2,2(R_3+R_6),2(R_3+R_7)\}$. Applying variation of constant formula to \eqref{ue22} over $(\mathfrak{s}-t,\tau)$ with $t\geq0$, $\mathfrak{s}\in\mathbb{R}$ and $\tau\geq\mathfrak{s}-t$, and replacing $\omega$ by $\vartheta_{-\mathfrak{s}}\omega$, we obtain
\begin{align}\label{ue28}
\|\boldsymbol{v}(\tau,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{\mathfrak{s}-t})\|^2_{\mathbb{H}} &\leq e^{\alpha(\mathfrak{s}-t-\tau)}\|\boldsymbol{v}_{\mathfrak{s}-t}\|^2_{\mathbb{H}}+R_8 \int_{-\infty}^{\tau-\mathfrak{s}}e^{\alpha(\xi+\mathfrak{s}-\tau)}\bigg[\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\nonumber\\&\quad+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^2+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^{r+1}\bigg]\d\xi,
\end{align}
or
\begin{align}\label{ue29}
& \|\boldsymbol{u}(\tau,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\|^2_{\mathbb{H}}\nonumber\\&\leq2\|\boldsymbol{v}(\tau,\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{\mathfrak{s}-t})\|^2_{\mathbb{H}}+2\|\textbf{g}\|^2_{\mathbb{H}}\left|e^{\sigma\mathfrak{s}}\boldsymbol{y}(\omega)\right|^2 \nonumber\\&\leq 4e^{-\alpha t}\left(\|\boldsymbol{u}_{\mathfrak{s}-t}\|^2_{\mathbb{H}}+\|\textbf{g}\|^2_{\mathbb{H}}\left|e^{\sigma(\mathfrak{s}-t)}\boldsymbol{y}(\vartheta_{-t}\omega)\right|^2\right)+2\|\textbf{g}\|^2_{\mathbb{H}}\left|e^{\sigma\mathfrak{s}}\boldsymbol{y}(\omega)\right|^2\nonumber\\&\quad+2R_8 \int_{-\infty}^{0}e^{\alpha\xi}\bigg[\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^2+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^{r+1}\bigg]\d\xi,
\end{align}
where $\boldsymbol{u}_{\mathfrak{s}-t}\in D(\mathfrak{s}-t,\vartheta_{-t}\omega)$ and $D\in\mathfrak{D}$. Using \eqref{forcing6} and \eqref{Y2}, and arguing similarly as in the proofs of Lemmas \ref{LemmaUe3} and \ref{PAS_GA}, we conclude the proof.
\end{proof}
The next two Corollaries \ref{convergence_b} and \ref{convergence_c} help us to prove the asymptotic compactness of $\Phi_0$ in this subsection (see Lemma \ref{Asymptotic_UB_Add}) as well as the uniform compactness of random pullback attractors in next subsection (see Lemma \ref{precompact}).
\begin{corollary}\label{convergence_b}
Let $\{\boldsymbol{v}_m\}_{m\in \mathbb{N}}\subset\mathrm{L}^{\infty}(0, T; \mathbb{H})\cap\mathrm{L}^2(0, T;\mathbb{V})\cap\mathrm{L}^{r+1}(0, T;\widetilde\mathbb{L}^{r+1})$ be a bounded sequence and $\boldsymbol{v} \in \mathrm{L}^{\infty}(0, T; \mathbb{H})\cap\mathrm{L}^2(0, T;\mathbb{V})\cap\mathrm{L}^{r+1}(0, T;\widetilde\mathbb{L}^{r+1})$ such that
\begin{equation}\label{5.26}
\left\{
\begin{aligned}
\boldsymbol{v}_m&\xrightharpoonup{w}\boldsymbol{v} \ \text{ in }\ \mathrm{L}^2(0, T;\mathbb{V}),\\
\boldsymbol{v}_m&\xrightharpoonup{w}\boldsymbol{v} \ \text{ in }\ \mathrm{L}^{r+1}(0, T;\widetilde\mathbb{L}^{r+1}),\\
\boldsymbol{v}_m&\to\boldsymbol{v}\ \text{ in }\ \mathrm{L}^2(0, T; {\mathbb{L}}^2_{\emph{loc}}(\mathbb{R}^d)).
\end{aligned}
\right.
\end{equation}
Then, for $d=2$ with $r>1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$,
\begin{align}\label{Conver}
\int_{0}^{T} b(\boldsymbol{v}_m(t), \boldsymbol{v}_m(t), \uprho(t)\emph{\textbf{g}}) \d t \to \int_{0}^{T} b(\boldsymbol{v}(t), \boldsymbol{v}(t), \uprho(t)\emph{\textbf{g}}) \d t\ \ \text{ as }\ \ m\to\infty,
\end{align}
where $\uprho(t)$ is a continuous function of $t$ on $[0,T]$.
\end{corollary}
\begin{proof}
Since $\boldsymbol{v}_m, \boldsymbol{v} \in\mathrm{L}^{\infty}(0, T; \mathbb{H})\cap\mathrm{L}^2(0, T;\mathbb{V})\cap\mathrm{L}^{r+1}(0, T;\widetilde\mathbb{L}^{r+1})$ and $\uprho(t)$ is continuous, there exists a constant $\varpi>0$ such that
\begin{align*}
&\sup_{t \in [0, T]} \|\boldsymbol{v}_m(t)\|_{\mathbb{H}} +\sup_{t \in [0, T]} \|\boldsymbol{v}(t)\|_{\mathbb{H}} + \left(\int_{0}^{T}\|\boldsymbol{v}_m(t)\|^2_{\mathbb{V}} \d t\right)^{\frac{1}{2}} + \left(\int_{0}^{T}\|\boldsymbol{v}(t)\|^2_{\mathbb{V}}\d t\right)^{\frac{1}{2}}\\&+\left(\int_{0}^{T}\|\boldsymbol{v}_m(t)\|^{r+1}_{\widetilde\mathbb{L}^{r+1}} \d t\right)^{\frac{1}{r+1}} + \left(\int_{0}^{T}\|\boldsymbol{v}(t)\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}\d t\right)^{\frac{1}{r+1}} + \left(\int_{0}^{T}\left|\uprho(t)\right|^2\d t\right)^{\frac{1}{2}} \leq \varpi.
\end{align*}
Since $\textbf{g}\in\mathrm{D}(\mathrm{A})$, for given $\varepsilon>0$, we can say that there exists $k=k(\varepsilon,\textbf{g})>0$ such that
\begin{align}\label{Conver1}
\int_{\mathbb{R}^d\backslash\mathcal{O}_k}\left(|\textbf{g}(x)|^2+|\nabla\textbf{g}(x)|^2\right)\d x \leq \frac{\varepsilon^2}{256 \varpi^6}, \ \ \ \text{ for } d=2,
\end{align}
and
\begin{align}\label{Conver1*}
\int_{\mathbb{R}^d\backslash\mathcal{O}_k}\left(|\textbf{g}(x)|^2+|\nabla\textbf{g}(x)|^2\right)\d x \leq \frac{\varepsilon^2}{64 \varpi^6}, \ \ \ \text{ for } d=3,
\end{align}
where $\mathcal{O}_k=\{x\in\mathbb{R}^d:|x|<k\}$. Let us consider, for $\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}\in\mathbb{V}$,
$$b(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w})=b_1(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w})+b_2(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w})$$ where
$$b_1(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w})=\sum_{i,j=1}^d\int_{\mathcal{O}_k}\boldsymbol{u}_i\frac{\partial \boldsymbol{v}_j}{\partial x_i}\boldsymbol{w}_j\d x \ \ \text{ and } \ \ b_2(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w})=\sum_{i,j=1}^d\int_{\mathbb{R}^d\backslash\mathcal{O}_k}\boldsymbol{u}_i\frac{\partial \boldsymbol{v}_j}{\partial x_i}\boldsymbol{w}_j\d x.$$
Consider,
\begin{align}\label{Conver2}
&\int_{0}^{T} b(\boldsymbol{v}_m(t), \boldsymbol{v}_m(t), \uprho(t)\textbf{g}) \d t-\int_{0}^{T} b(\boldsymbol{v}(t), \boldsymbol{v}(t), \uprho(t)\textbf{g}) \d t \nonumber\\& =\int_{0}^{T} b(\boldsymbol{v}_m(t)-\boldsymbol{v}(t), \boldsymbol{v}_m(t), \uprho(t)\textbf{g}) \d t +\int_{0}^{T} b(\boldsymbol{v}(t), \boldsymbol{v}_m(t)-\boldsymbol{v}(t), \uprho(t)\textbf{g}) \d t\nonumber\\&=\int_{0}^{T} b_1(\boldsymbol{v}_m(t)-\boldsymbol{v}(t), \boldsymbol{v}_m(t), \uprho(t)\textbf{g}) \d t +\int_{0}^{T} b_2(\boldsymbol{v}_m(t)-\boldsymbol{v}(t), \boldsymbol{v}_m(t), \uprho(t)\textbf{g}) \d t \nonumber\\&\qquad+\int_{0}^{T} b_1(\boldsymbol{v}(t), \boldsymbol{v}_m(t)-\boldsymbol{v}(t), \uprho(t)\textbf{g}) \d t +\int_{0}^{T} b_2(\boldsymbol{v}(t), \boldsymbol{v}_m(t)-\boldsymbol{v}(t), \uprho(t)\textbf{g}) \d t .
\end{align}
In order to prove \eqref{Conver}, it is enough to show that each term on the right hand side of \eqref{Conver2} converges to $0$ as $m\to\infty$.
\vskip 2mm
\noindent
\textbf{Case I:} \textit{$d=2$ and $r>1$.}
By \eqref{b1} and \eqref{Conver1}, we obtain
\begin{align}\label{Conver3}
&\left|\int_{0}^{T} b_2( \boldsymbol{v}_m(t)-\boldsymbol{v}(t),\boldsymbol{v}_m(t), \uprho(t)\textbf{g}) \d t\right|\nonumber\\&\leq\frac{\varepsilon}{8\varpi^3}\int_{0}^{T}\left|\uprho(t)\right|\left[\|\boldsymbol{v}_m(t)\|_{\mathbb{H}}\|\nabla\boldsymbol{v}_m(t)\|_{\mathbb{H}}+\|\boldsymbol{v}(t)\|^{\frac{1}{2}}_{\mathbb{H}}\|\nabla\boldsymbol{v}(t)\|^{\frac{1}{2}}_{\mathbb{H}}\|\boldsymbol{v}_m(t)\|^{\frac{1}{2}}_{\mathbb{H}}\|\nabla\boldsymbol{v}_m(t)\|^{\frac{1}{2}}_{\mathbb{H}}\right]\d t \nonumber\\&\leq\frac{\varepsilon}{8\varpi^3}\sup_{t \in [0, T]} \|\boldsymbol{v}_m(t)\|_{\mathbb{H}}\left(\int_{0}^{T}\left|\uprho(t)\right|^2\d t\right)^{\frac{1}{2}}\left(\int_{0}^{T}\|\nabla\boldsymbol{v}_m(t)\|^2_{\mathbb{H}}\d t\right)^{\frac{1}{2}}+ \frac{\varepsilon}{8\varpi^3}\sup_{t \in [0, T]} \|\boldsymbol{v}(t)\|^{\frac{1}{2}}_{\mathbb{H}}\nonumber\\&\quad\times\sup_{t \in [0, T]} \|\boldsymbol{v}_m(t)\|^{\frac{1}{2}}_{\mathbb{H}}\left(\int_{0}^{T}\left|\uprho(t)\right|^2\d t\right)^{\frac{1}{2}}\left(\int_{0}^{T}\|\nabla\boldsymbol{v}(t)\|^2_{\mathbb{H}}\d t\right)^{\frac{1}{4}} \left(\int_{0}^{T}\|\nabla\boldsymbol{v}_m(t)\|^2_{\mathbb{H}}\d t\right)^{\frac{1}{4}}\nonumber\\&\leq \frac{\varepsilon}{4}.
\end{align}
By \eqref{b1} and continuity of $\uprho(t)$, we have
\begin{align}\label{Conver4}
&\left|\int_{0}^{T} b_1( \boldsymbol{v}_m(t)-\boldsymbol{v}(t),\boldsymbol{v}_m(t), \uprho(t)\textbf{g}) \d t\right|\nonumber\\&\leq C\|\nabla\textbf{g}\|_{\mathbb{H}}\int_{0}^{T}\left|\uprho(t)\right|\|\boldsymbol{v}_m(t)-\boldsymbol{v}(t)\|^{\frac{1}{2}}_{\mathbb{L}^2(\mathcal{O}_k)}\|\nabla\left(\boldsymbol{v}_m(t)-\boldsymbol{v}(t)\right)\|^{\frac{1}{2}}_{\mathbb{H}}\|\boldsymbol{v}_m(t)\|^{\frac{1}{2}}_{\mathbb{H}}\|\nabla\boldsymbol{v}_m(t)\|^{\frac{1}{2}}_{\mathbb{H}}\d t\nonumber\\&\leq CT^{\frac{1}{4}}\sup_{t \in [0, T]}\left[\left|\uprho(t)\right|\|\boldsymbol{v}_m(t)\|^{\frac{1}{2}}_{\mathbb{H}}\right]\left(\int_{0}^{T}\|\boldsymbol{v}_m(t)-\boldsymbol{v}(t)\|^2_{\mathbb{L}^2(\mathcal{O}_k)}\d t\right)^{\frac{1}{4}}\bigg[\left(\int_{0}^{T}\|\nabla\boldsymbol{v}_m(t)\|^2_{\mathbb{H}}\d t\right)^{\frac{1}{2}}\nonumber\\&\qquad\qquad+\left(\int_{0}^{T}\|\nabla\boldsymbol{v}(t)\|^2_{\mathbb{H}}\d t\right)^{\frac{1}{4}}\left(\int_{0}^{T}\|\nabla\boldsymbol{v}_m(t)\|^2_{\mathbb{H}}\d t\right)^{\frac{1}{4}}\bigg]\nonumber\\&\to0 \ \text{ as }\ m\to \infty.
\end{align}
\vskip 2mm
\noindent
\textbf{Case II:} \textit{$d= 3$ and $r\geq3$ ($r>3$ with any $\beta,\mu>0$ and $r=3$ with $2\beta\mu\geq1$).} By \eqref{Conver1*}, Lemmas \ref{Holder} and \ref{Interpolation}, we have
\begin{align}\label{Conver9}
&\left|\int_{0}^{T} b_2( \boldsymbol{v}_m(t)-\boldsymbol{v}(t),\boldsymbol{v}_m(t), \uprho(t)\textbf{g}) \d t\right|\nonumber\\&\leq\frac{\varepsilon}{8\varpi^3}\int_{0}^{T}\left|\uprho(t)\right|\left[\|\boldsymbol{v}_m(t)\|_{\widetilde\mathbb{L}^{r+1}}\|\boldsymbol{v}_m(t)\|_{\widetilde\mathbb{L}^{\frac{2(r+1)}{r-1}}}+\|\boldsymbol{v}(t)\|_{\widetilde\mathbb{L}^{r+1}}\|\boldsymbol{v}_m(t)\|_{\widetilde\mathbb{L}^{\frac{2(r+1)}{r-1}}}\right]\d t\nonumber\\&\leq\frac{\varepsilon}{8\varpi^3}\int_{0}^{T}\left|\uprho(t)\right|\left[\|\boldsymbol{v}_m(t)\|^{\frac{r+1}{r-1}}_{\widetilde\mathbb{L}^{r+1}}\|\boldsymbol{v}_m(t)\|^{\frac{r-3}{r-1}}_{\mathbb{H}}+\|\boldsymbol{v}(t)\|_{\widetilde\mathbb{L}^{r+1}}\|\boldsymbol{v}_m(t)\|^{\frac{2}{r-1}}_{\widetilde\mathbb{L}^{r+1}}\|\boldsymbol{v}_m(t)\|^{\frac{r-3}{r-1}}_{\mathbb{H}}\right]\d t \nonumber\\&\leq\frac{\varepsilon}{8\varpi^3}\left(\int_{0}^{T}\left|\uprho(t)\right|^2\d t\right)^{\frac{1}{2}}\left(\int_{0}^{T}\|\boldsymbol{v}_m(t)\|^2_{\mathbb{H}}\d t\right)^{\frac{r-3}{2(r-1)}}\bigg[\left(\int_{0}^{T}\|\boldsymbol{v}_m(t)\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}\d t\right)^{\frac{1}{r-1}}\nonumber\\&\quad+ \left(\int_{0}^{T}\|\boldsymbol{v}(t)\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}\d t\right)^{\frac{1}{r+1}}\left(\int_{0}^{T}\|\boldsymbol{v}_m(t)\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}\d t\right)^{\frac{2}{(r+1)(r-1)}}\bigg]\nonumber\\&\leq \frac{\varepsilon}{4},
\end{align}
for $r>3$ and
\begin{align}\label{Conver10}
&\left|\int_{0}^{T} b_2( \boldsymbol{v}_m(t)-\boldsymbol{v}(t),\boldsymbol{v}_m(t), \uprho(t)\textbf{g}) \d t\right|\nonumber\\&\leq\frac{\varepsilon}{8\varpi^3}\int_{0}^{T}\left|\uprho(t)\right|\|\boldsymbol{v}_m(t)\|^2_{\widetilde\mathbb{L}^{4}}+ \frac{\varepsilon}{8\varpi^3}\int_{0}^{T}\left|\uprho(t)\right|\|\boldsymbol{v}(t)\|_{\widetilde\mathbb{L}^{4}}\|\boldsymbol{v}_m(t)\|_{\widetilde\mathbb{L}^4}\d t\nonumber\\&\leq\frac{\varepsilon}{8\varpi^3}\left(\int_{0}^{T}\left|\uprho(t)\right|^2\d t\right)^{\frac{1}{2}}\left[\left(\int_{0}^{T}\|\boldsymbol{v}_m(t)\|^{4}_{\widetilde\mathbb{L}^{4}}\d t\right)^{\frac{1}{2}}+\left(\int_{0}^{T}\|\boldsymbol{v}(t)\|^{4}_{\widetilde\mathbb{L}^{4}}\d t\right)^{\frac{1}{4}}\left(\int_{0}^{T}\|\boldsymbol{v}_m(t)\|^{4}_{\widetilde\mathbb{L}^{4}}\d t\right)^{\frac{1}{4}}\right]\nonumber\\&\leq \frac{\varepsilon}{4},
\end{align}
for $r=3$. By \eqref{b1} and continuity of $\uprho(t)$, we have
\begin{align}\label{Conver11}
&\left|\int_{0}^{T} b_1( \boldsymbol{v}_m(t)-\boldsymbol{v}(t),\boldsymbol{v}_m(t), \uprho(t)\textbf{g}) \d t\right|\nonumber\\&\leq C\|\nabla\textbf{g}\|_{\mathbb{H}}\int_{0}^{T}\left|\uprho(t)\right|\|\boldsymbol{v}_m(t)-\boldsymbol{v}(t)\|^{\frac{1}{4}}_{\mathbb{L}^2(\mathcal{O}_k)}\|\nabla\left(\boldsymbol{v}_m(t)-\boldsymbol{v}(t)\right)\|^{\frac{3}{4}}_{\mathbb{H}}\|\boldsymbol{v}_m(t)\|^{\frac{1}{4}}_{\mathbb{H}}\|\nabla\boldsymbol{v}_m(t)\|^{\frac{3}{4}}_{\mathbb{H}}\d t\nonumber\\&\leq CT^{\frac{1}{8}}\sup_{t \in [0, T]}\left[\left|\uprho(t)\right|\cdot\|\boldsymbol{v}_m(t)\|^{\frac{1}{4}}_{\mathbb{H}}\right]\left(\int_{0}^{T}\|\boldsymbol{v}_m(t)-\boldsymbol{v}(t)\|^2_{\mathbb{L}^2(\mathcal{O}_k)}\d t\right)^{\frac{1}{8}}\bigg[\left(\int_{0}^{T}\|\nabla\boldsymbol{v}_m(t)\|^2_{\mathbb{H}}\d t\right)^{\frac{3}{4}}\nonumber\\&\quad+\left(\int_{0}^{T}\|\nabla\boldsymbol{v}(t)\|^2_{\mathbb{H}}\d t\right)^{\frac{3}{8}}\left(\int_{0}^{T}\|\nabla\boldsymbol{v}_m(t)\|^2_{\mathbb{H}}\d t\right)^{\frac{3}{8}}\bigg]\nonumber\\&\to0 \ \text{ as }\ m\to \infty.
\end{align}
Calculations similar to \eqref{Conver3} (for $d=2$ and $r>1$), \eqref{Conver9} (for $d=3$ and $r>3$) and \eqref{Conver10} (for $d=r=3$), and \eqref{Conver4} (for $d=2$ and $r>1$) and \eqref{Conver11} (for $d=3$ and $r\geq3$), we obtain
\begin{align}\label{Conver5}
&\left|\int_{0}^{T} b_2( \boldsymbol{v}(t),\boldsymbol{v}_m(t)-\boldsymbol{v}(t), \uprho(t)\textbf{g}) \d t\right|\leq \frac{\varepsilon}{4},
\end{align}
and
\begin{align}\label{Conver6}
\left|\int_{0}^{T} b_1(\boldsymbol{v}(t), \boldsymbol{v}_m(t)-\boldsymbol{v}(t), \uprho(t)\textbf{g}) \d t\right|\to0 \ \text{ as }\ m\to \infty,
\end{align}
respectively. For given $\varepsilon>0$, we infer from \eqref{Conver4} (for $d=2$ and $r>1$) and \eqref{Conver11} (for $d=3$ and $r\geq3$), and \eqref{Conver6} that there exists $M(\varepsilon)\in\mathbb{N}$ such that
\begin{align}\label{Conver7}
\left|\int_{0}^{T} b_1( \boldsymbol{v}_m(t)-\boldsymbol{v}(t),\boldsymbol{v}_m(t), \uprho(t)\textbf{g}) \d t\right| \leq \frac{\varepsilon}{4},
\end{align}
and
\begin{align}\label{Conver8}
\left|\int_{0}^{T} b_1(\boldsymbol{v}(t), \boldsymbol{v}_m(t)-\boldsymbol{v}(t), \uprho(t)\textbf{g}) \d t\right|\leq\frac{\varepsilon}{4},
\end{align}
respectively, for all $m\geq M(\varepsilon).$ Hence, \eqref{Conver2}, \eqref{Conver3} (for $d=2$ and $r>1$), \eqref{Conver9} (for $d=3$ and $r>3$), \eqref{Conver10} (for $d=r=3$), \eqref{Conver5} and \eqref{Conver7}-\eqref{Conver8} imply that, for a given $\varepsilon>0$, there exists $M(\varepsilon)\in\mathbb{N}$ such that
\begin{align*}
\left|\int_{0}^{T} b(\boldsymbol{v}_m(t), \boldsymbol{v}_m(t), \uprho(t)\textbf{g}) \d t-\int_{0}^{T} b(\boldsymbol{v}(t), \boldsymbol{v}(t), \uprho(t)\textbf{g}) \d t \right|\leq \varepsilon,
\end{align*}
for all $m\geq M(\varepsilon),$ which completes the proof.
\end{proof}
\begin{corollary}\label{convergence_c}
Let $\{\boldsymbol{v}_m\}_{m\in \mathbb{N}}\subset\mathrm{L}^{\infty}(0, T; \mathbb{H})\cap\mathrm{L}^2(0, T;\mathbb{V})\cap\mathrm{L}^{r+1}(0, T;\widetilde\mathbb{L}^{r+1})$ be a bounded sequence and $\boldsymbol{v} \in \mathrm{L}^{\infty}(0, T; \mathbb{H})\cap\mathrm{L}^2(0, T;\mathbb{V})\cap\mathrm{L}^{r+1}(0, T;\widetilde\mathbb{L}^{r+1})$ such that \eqref{5.26} be satisfied. Then, for $d=2$ with $r>1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$,
\begin{align*}
\int_{0}^{T} \big\langle\mathcal{C}(\boldsymbol{v}_m(t)) ,\uprho(t)\emph{\textbf{g}} \big\rangle \d t \to \int_{0}^{T} \big\langle\mathcal{C}(\boldsymbol{v}(t)) ,\uprho(t)\emph{\textbf{g}} \big\rangle \d t,\ \text{ as }\ m\to\infty,
\end{align*}
where $\uprho(t)$ is a continuous function of $t$ on $[0,T]$.
\end{corollary}
\begin{proof}
Since $\boldsymbol{v}_m, \boldsymbol{v} \in\mathrm{L}^{\infty}(0, T; \mathbb{H})\cap\mathrm{L}^2(0, T;\mathbb{V})\cap\mathrm{L}^{r+1}(0, T;\widetilde\mathbb{L}^{r+1})$ and $\uprho(t)$ is continuous, there exists a constant $\varpi>0$ such that
\begin{align*}
\left(\int_{0}^{T}\|\boldsymbol{v}_m(t)\|^{r+1}_{\widetilde\mathbb{L}^{r+1}} \d t\right)^{\frac{1}{r+1}} + \left(\int_{0}^{T}\|\boldsymbol{v}(t)\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}\d t\right)^{\frac{1}{r+1}} + \left(\int_{0}^{T}\left|\uprho(t)\right|^{r+1}\d t\right)^{\frac{1}{r+1}} \leq \varpi.
\end{align*}
Since $\textbf{g}\in\mathrm{D}(\mathrm{A})$, Sobolev's embeddings ($\mathbb{V}\subset\widetilde{\mathbb{L}}^{r+1}$ for $d=2$ and $\mathrm{D}(\mathrm{A})\subset\widetilde{\mathbb{L}}^{r+1}$ for $d=3$) imply that $\textbf{g}\in\widetilde{\mathbb{L}}^{r+1}$, for all $r\geq1$. Hence, for given $\varepsilon>0$, there exists $k=k(\varepsilon,\textbf{g})>0$ such that
\begin{align}\label{Conver12}
\left(\int_{\mathbb{R}^d\backslash\mathcal{O}_k}|\textbf{g}(x)|^{r+1}\d x\right)^{\frac{1}{r+1}} \leq \frac{\varepsilon}{4 \varpi^{r+1}},
\end{align}
where $\mathcal{O}_k=\{x\in\mathbb{R}^d:|x|<k\}$. Consider,
\begin{align}\label{Conver13}
&\int_{0}^{T} \big\langle\mathcal{C}(\boldsymbol{v}_m(t)) ,\uprho(t)\textbf{g} \big\rangle \d t- \int_{0}^{T} \big\langle\mathcal{C}(\boldsymbol{v}(t)) ,\uprho(t)\textbf{g} \big\rangle \d t\nonumber\\&=\int_{0}^{T}\uprho(t) \left[\int_{\mathbb{R}^d}\left(\left|\boldsymbol{v}_m(x,t)\right|^{r-1}\boldsymbol{v}_m(x,t)-\left|\boldsymbol{v}(x,t)\right|^{r-1}\boldsymbol{v}(x,t)\right)\textbf{g}(x) \d x\right] \d t\nonumber\\&=\int_{0}^{T}\uprho(t) \left[\int_{\mathcal{O}_k}\left(\left|\boldsymbol{v}_m(x,t)\right|^{r-1}\boldsymbol{v}_m(x,t)-\left|\boldsymbol{v}(x,t)\right|^{r-1}\boldsymbol{v}(x,t)\right)\textbf{g}(x) \d x\right] \d t\nonumber\\&\quad+\int_{0}^{T}\uprho(t) \left[\int_{\mathbb{R}^d\backslash\mathcal{O}_k}\left(\left|\boldsymbol{v}_m(x,t)\right|^{r-1}\boldsymbol{v}_m(x,t)-\left|\boldsymbol{v}(x,t)\right|^{r-1}\boldsymbol{v}(x,t)\right)\textbf{g}(x) \d x\right] \d t.
\end{align}
From Lemma \ref{Holder} and \eqref{Conver12}, we infer that
\begin{align}\label{Conver14}
&\left|\int_{0}^{T}\uprho(t) \left[\int_{\mathbb{R}^d\backslash\mathcal{O}_k}\left(\left|\boldsymbol{v}_m(x,t)\right|^{r-1}\boldsymbol{v}_m(x,t)-\left|\boldsymbol{v}(x,t)\right|^{r-1}\boldsymbol{v}(x,t)\right)\textbf{g}(x) \d x\right] \d t\right|\nonumber\\&\leq\frac{\varepsilon}{4\varpi^{r+1}}\int_{0}^{T}\left|\uprho(t)\right| \|\boldsymbol{v}_m(t)\|^r_{\widetilde\mathbb{L}^{r+1}} \d t +\frac{\varepsilon}{4\varpi^{r+1}}\int_{0}^{T}\left|\uprho(t)\right| \|\boldsymbol{v}(t)\|^r_{\widetilde\mathbb{L}^{r+1}} \d t\nonumber\\&\leq\frac{\varepsilon}{4\varpi^{r+1}} \left(\int_{0}^{T}\left|\uprho(t)\right|^{r+1}\d t\right)^{\frac{1}{r+1}}\left[\left(\int_{0}^{T}\|\boldsymbol{v}_m(t)\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}\d t\right)^{\frac{r}{r+1}}+\left(\int_{0}^{T}\|\boldsymbol{v}(t)\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}\d t\right)^{\frac{r}{r+1}}\right]\nonumber\\&\leq\frac{\varepsilon}{2}.
\end{align}
Using Taylor's formula, \eqref{29} and Lemma \ref{Holder}, we achieve
\begin{align*}
&\left|\int_{0}^{T}\uprho(t) \left[\int_{\mathcal{O}_k}\left(\left|\boldsymbol{v}_m(x,t)\right|^{r-1}\boldsymbol{v}_m(x,t)-\left|\boldsymbol{v}(x,t)\right|^{r-1}\boldsymbol{v}(x,t)\right)\textbf{g}(x) \d x\right] \d t\right|\nonumber\\&\leq C\int_{0}^{T}\left|\uprho(t)\right|\left[\int_{\mathcal{O}_k}\left(\left|\boldsymbol{v}_m(x,t)\right|^{r-1}+\left|\boldsymbol{v}(x,t)\right|^{r-1}\right)\left|\boldsymbol{v}_m(x,t)-\boldsymbol{v}(x,t)\right|\left|\textbf{g}(x)\right| \d x\right] \d t\nonumber\\&\leq C\int_{0}^{T}\left|\uprho(t)\right|\left(\|\boldsymbol{v}_m(t)\|^{r-1}_{\widetilde\mathbb{L}^{r+1}}+\|\boldsymbol{v}(t)\|^{r-1}_{\widetilde\mathbb{L}^{r+1}}\right)\|\boldsymbol{v}_m(t)-\boldsymbol{v}(t)\|_{\mathbb{L}^2(\mathcal{O}_k)} \|\textbf{g}\|_{\widetilde\mathbb{L}^{\frac{2(r+1)}{3-r}}}\d t\nonumber\\&\leq CT^{\frac{3-r}{2(r+1)}}\sup_{t \in [0, T]}\left|\uprho(t)\right|\left(\|\boldsymbol{v}_m\|^{r-1}_{\mathrm{L}^{r+1}(0,T;\widetilde{\mathbb{L}}^{r+1})}+\|\boldsymbol{v}\|^{r-1}_{\mathrm{L}^{r+1}(0,T;\widetilde{\mathbb{L}}^{r+1})}\right)\|\boldsymbol{v}_m-\boldsymbol{v}\|_{\mathrm{L}^2(0,T;\mathbb{L}^2(\mathcal{O}_k))}\nonumber\\&
\to 0 \ \text{ as }\ m\to \infty,
\end{align*}
for $ 1<r<3$, and
\begin{align*}
&\left|\int_{0}^{T}\uprho(t) \left[\int_{\mathcal{O}_k}\left(\left|\boldsymbol{v}_m(x,t)\right|^{r-1}\boldsymbol{v}_m(x,t)-\left|\boldsymbol{v}(x,t)\right|^{r-1}\boldsymbol{v}(x,t)\right)\textbf{g}(x) \d x\right] \d t\right|\nonumber\\&\leq C\int_{0}^{T}\left|\uprho(t)\right|\left[\int_{\mathcal{O}_k}\left(\left|\boldsymbol{v}_m(x,t)\right|^{r-1}+\left|\boldsymbol{v}(x,t)\right|^{r-1}\right)\left|\boldsymbol{v}_m(x,t)-\boldsymbol{v}(x,t)\right|\left|\textbf{g}(x)\right| \d x\right] \d t\nonumber\\&\leq C\int_{0}^{T}\left|\uprho(t)\right|\left(\|\boldsymbol{v}_m(t)\|^{r-1}_{\widetilde\mathbb{L}^{r+1}}+\|\boldsymbol{v}(t)\|^{r-1}_{\widetilde\mathbb{L}^{r+1}}\right)\|\boldsymbol{v}_m(t)-\boldsymbol{v}(t)\|^{\frac{1}{r-1}}_{\mathbb{L}^{2}(\mathcal{O}_k)}\nonumber\\&\qquad\qquad\times\|\boldsymbol{v}_m(t)-\boldsymbol{v}(t)\|^{\frac{r-2}{r-1}}_{\widetilde\mathbb{L}^{r+1}} \|\textbf{g}\|_{\widetilde\mathbb{L}^{2(r+1)}}\d t\nonumber\\&\leq CT^{\frac{1}{2(r+1)}}\sup_{t \in [0, T]}\left|\uprho(t)\right|\left(\|\boldsymbol{v}_m\|^{r-1}_{\mathrm{L}^{r+1}(0,T;\widetilde{\mathbb{L}}^{r+1})}+\|\boldsymbol{v}\|^{r-1}_{\mathrm{L}^{r+1}(0,T;\widetilde{\mathbb{L}}^{r+1})}\right)\nonumber\\&\qquad\qquad\times\|\boldsymbol{v}_m-\boldsymbol{v}\|^{\frac{1}{r-1}}_{\mathrm{L}^2(0,T;\mathbb{L}^2(\mathcal{O}_k))}\|\boldsymbol{v}_m-\boldsymbol{v}\|^{\frac{r-2}{r-1}}_{\mathrm{L}^{r+1}(0,T;\widetilde\mathbb{L}^{r+1})}\nonumber\\&
\to 0 \ \text{ as }\ m\to \infty,
\end{align*}
for $r\geq 3$, which implies that, for given $\varepsilon>0$, there exists $M(\varepsilon)\in\mathbb{N}$ such that
\begin{align}\label{Conver15}
\left|\int_{0}^{T}\uprho(t) \left[\int_{\mathcal{O}_k}\left(\left|\boldsymbol{v}_m(x,t)\right|^{r-1}\boldsymbol{v}_m(x,t)-\left|\boldsymbol{v}(x,t)\right|^{r-1}\boldsymbol{v}(x,t)\right)\textbf{g}(x) \d x\right] \d t\right|\leq\frac{\varepsilon}{2}.
\end{align}
Finally, from \eqref{Conver13}-\eqref{Conver15}, we infer that, for a given $\varepsilon>0$, there exists $M(\varepsilon)\in\mathbb{N}$ such that
\begin{align*}
\left|\int_{0}^{T} \big\langle\mathcal{C}(\boldsymbol{v}_m(t)) ,\uprho(t)\textbf{g} \big\rangle \d t- \int_{0}^{T} \big\langle\mathcal{C}(\boldsymbol{v}(t)) ,\uprho(t)\textbf{g} \big\rangle \d t\right|\leq\varepsilon,
\end{align*}
for all $m\geq M(\varepsilon),$ which completes the proof.
\end{proof}
\begin{lemma}\label{weak_add}
For $d=2$ with $r>1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$. Let $\mathfrak{s}\in\mathbb{R}, \omega\in \Omega$ and $\boldsymbol{v}_{\mathfrak{s}}^0, \boldsymbol{v}_{\mathfrak{s}}^n\in \mathbb{H}$ for all $n\in\mathbb{N}.$ If $\boldsymbol{v}_{\mathfrak{s}}^n\xrightharpoonup{w}\boldsymbol{v}_{\mathfrak{s}}^0$ in $\mathbb{H}$, then the solution $\boldsymbol{v}$ of the system \eqref{CSCBF_Add} satisfies the following convergences:
\begin{itemize}
\item [(i)] $\boldsymbol{v}(\xi,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}}^n)\xrightharpoonup{w}\boldsymbol{v}(\xi,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}}^0)$ in $\mathbb{H}$ for all $\xi\geq \mathfrak{s}$.
\item [(ii)] $\boldsymbol{v}(\cdot,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}}^n)\xrightharpoonup{w}\boldsymbol{v}(\cdot,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}}^0)$ in $\mathrm{L}^2((\mathfrak{s},\mathfrak{s}+T);\mathbb{V})$ for every $T>0$.
\item [(iii)] $\boldsymbol{v}(\cdot,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}}^n)\xrightharpoonup{w}\boldsymbol{v}(\cdot,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}}^0)$ in $\mathrm{L}^{r+1}((\mathfrak{s},\mathfrak{s}+T);\widetilde{\mathbb{L}}^{r+1})$ for every $T>0$.
\item [(iv)] $\boldsymbol{v}(\cdot,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}}^n)\to\boldsymbol{v}(\cdot,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}}^0)$ in $\mathrm{L}^2((\mathfrak{s},\mathfrak{s}+T);\mathbb{L}^2(\mathcal{O}_k))$ for every $T>0$ and $k>0$,
where $\mathcal{O}_k=\{x\in\mathbb{R}^d:|x|<k\}$.
\end{itemize}
\end{lemma}
\begin{proof}
Using the standard method as in \cite{KM1} (see Lemmas 5.2 and 5.3 in \cite{KM1}), one can complete the proof. Note that, due to compactness of Sobolev embedding in bounded domains, we get the strong convergence in $\mathrm{L}^2((\mathfrak{s},\mathfrak{s}+T);\mathbb{L}^2(\mathcal{O}_k))$.
\end{proof}
\begin{lemma}\label{Asymptotic_UB_Add}
For $d=2$ with $r>1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing6}. Then for every $\mathfrak{s}\in \mathbb{R},$ $\omega\in \Omega,$ $D=\{D(\mathfrak{s},\omega):\mathfrak{s}\in \mathbb{R},\omega\in \Omega\}\in \mathfrak{D}$ and $t_n\to \infty,$ $\boldsymbol{u}_{0,n}\in D(\mathfrak{s}-t_n, \vartheta_{-t_{n}}\omega)$, the sequence $\Phi_0(t_n,\mathfrak{s}-t_n,\vartheta_{-t_n}\omega,\boldsymbol{u}_{0,n})$ or $\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})$ of solutions of the system \eqref{SCBF_Add} has a convergent subsequence in $\mathbb{H}$.
\end{lemma}
\begin{proof}
We infer from \eqref{ue24} (for $d=2$) and \eqref{ue29} (for $d=3$) with $\tau=\mathfrak{s}$ that there exists $\mathfrak{T}=\mathfrak{T}(\mathfrak{s},\omega,D)>0$ such that for all $t\geq \mathfrak{T}$, $\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\mathfrak{s}-t})\in\mathbb{H}$ with $\boldsymbol{u}_{\mathfrak{s}-t}\in D(\mathfrak{s}-t,\vartheta_{-t}\omega).$ Since $t_n\to \infty$, there exists $N_5\in\mathbb{N}$ such that $t_n\geq \mathfrak{T}$ for all $n\geq N_5$. It is given that $\boldsymbol{u}_{0,n}\in D(\mathfrak{s}-t_n, \vartheta_{-t_{n}}\omega)$, we get $\{\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})\}_{n\geq N_5}$ is a bounded sequence in $\mathbb{H}$. Due to
\begin{align}\label{Trans}
\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})=\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{0,n}) +e^{\sigma\mathfrak{s}}\boldsymbol{y}(\omega)\textbf{g},
\end{align}
with $\boldsymbol{v}_{0,n}=\boldsymbol{u}_{0,n}-e^{\sigma(\mathfrak{s}-t)}\boldsymbol{y}(\vartheta_{-t}\omega)\textbf{g}$, we have that $\{\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{0,n})\}_{n\geq N_5}$ is also a bounded sequence in $\mathbb{H}$, which implies that there exists $\tilde{\boldsymbol{v}}\in \mathbb{H}$ and a subsequence (not relabeling) such that
\begin{align}\label{ac2add}
\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{0,n})\xrightharpoonup{w}\tilde{\boldsymbol{v}} \ \text{ in }\ \mathbb{H},
\end{align}
which gives
\begin{align}\label{ac3add}
\|\tilde{\boldsymbol{v}}\|_{\mathbb{H}}\leq\liminf_{n\to\infty}\|\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{0,n})\|_{\mathbb{H}}.
\end{align}
For our purpose, we need to show that $\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{0,n})\to\tilde{\boldsymbol{v}}$ in $\mathbb{H}$ strongly. Along with the above expression, it is enough to prove that
\begin{align}\label{ac4add}
\|\tilde{\boldsymbol{v}}\|_{\mathbb{H}}\geq\limsup_{n\to\infty}\|\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{0,n})\|_{\mathbb{H}}.
\end{align}
Now, for a given $j\in \mathbb{N}$ ($j\leq t_n$), we have
\begin{align}\label{ac5add}
\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{0,n})=\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-j,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}(\mathfrak{s}-j,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{0,n})).
\end{align}
For each $j$, let $N_j$ be sufficiently large such that $t_n\geq \mathfrak{T}+j$ for all $n\geq N_j$. From \eqref{ue23} (for $d=2$) and \eqref{ue28} (for $d=3$) with $\tau=\mathfrak{s}-j$, we have that the sequence $\{\boldsymbol{v}(\mathfrak{s}-j,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{0,n})\}_{n\geq N_j}$ is bounded in $\mathbb{H}$, for each $j\in \mathbb{N}$. By the diagonal process, there exists a subsequence (denoting by same label) and $\tilde{\boldsymbol{u}}_{j}\in \mathbb{H}$ for each $j\in\mathbb{N}$ such that
\begin{align}\label{ac6add}
\boldsymbol{v}(\mathfrak{s}-j,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{0,n})\xrightharpoonup{w}\tilde{\boldsymbol{v}}_{j}\ \text{ in } \ \mathbb{H}.
\end{align}
From \eqref{ac5add}-\eqref{ac6add} together with Lemma \ref{weak_add}, we have that for $j\in\mathbb{N}$,
\begin{align}\label{ac7add}
\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{0,n})\xrightharpoonup{w}\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-j,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{v}}_{j}) \text{ in } \mathbb{H},
\end{align}
\begin{align}\label{ac8add}
\boldsymbol{v}(\cdot,\mathfrak{s}-j,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}(\mathfrak{s}-j,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{0,n}))\xrightharpoonup{w}\boldsymbol{v}(\cdot,\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{v}}_{j}) \text{ in } \mathrm{L}^2((\mathfrak{s}-j,\mathfrak{s});\mathbb{V}),
\end{align}
\begin{align}\label{ac8'add}
\boldsymbol{v}(\cdot,\mathfrak{s}-j,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}(\mathfrak{s}-j,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{0,n}))\xrightharpoonup{w}\boldsymbol{v}(\cdot,\mathfrak{s}-j,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{v}}_{j}) \text{ in } \mathrm{L}^{r+1}((\mathfrak{s}-j,\mathfrak{s});\widetilde{\mathbb{L}}^{r+1}),
\end{align}
and
\begin{align}\label{ac1add}
\boldsymbol{v}(\cdot,\mathfrak{s}-j,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}(\mathfrak{s}-j,\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{0,n}))\to\boldsymbol{v}(\cdot,\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{v}}_{j}) \text{ in } \mathrm{L}^2((\mathfrak{s}-j,\mathfrak{s});\mathbb{L}^2(\mathcal{O}_k)),
\end{align}
where $\mathcal{O}_k=\{x\in\mathbb{R}^d:|x|<k\}$. Clearly, \eqref{ac2add} and \eqref{ac7add} imply that
\begin{align}\label{ac9add}
\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-j,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{v}}_{j})=\tilde{\boldsymbol{v}}.
\end{align}
From \eqref{ue17} we have
\begin{align}\label{ac10add}
& \frac{\d}{\d t} \|\boldsymbol{v}\|^2_{\mathbb{H}} + \alpha\|\boldsymbol{v}\|^2_{\mathbb{H}}\nonumber\\&=-2\mu\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}-\alpha\|\boldsymbol{v}\|^2_{\mathbb{H}} - 2\beta\|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\|^{r+1}_{\widetilde \mathbb{L}^{r+1}} +2\beta\left\langle\mathcal{C}(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}),e^{\sigma t}\textbf{g}\boldsymbol{y}\right\rangle\nonumber\\&\quad+2b(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y},\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y},e^{\sigma t}\textbf{g}\boldsymbol{y})+2\left\langle\boldsymbol{f},\boldsymbol{v}\right\rangle+2e^{\sigma t}\boldsymbol{y}\left(\left(\ell-\sigma-\alpha\right)\textbf{g}-\mu \mathrm{A}\textbf{g},\boldsymbol{v}\right).
\end{align}
\iffalse
which implies that for each $\omega\in \Omega,$ $ \tau\in \mathbb{R}$ and $\mathfrak{s}\geq \tau$,
\begin{align}\label{ac11add}
&\|\boldsymbol{v}(\mathfrak{s},\tau,\omega,\boldsymbol{v}_{\tau})\|^2_{\mathbb{H}} \nonumber\\&= e^{\alpha(\tau-\mathfrak{s})}\|\boldsymbol{v}_{\tau}\|^2_{\mathbb{H}} -\int_{\tau}^{\mathfrak{s}}e^{\alpha(\xi-\mathfrak{s})}\bigg[2\mu\|\nabla\boldsymbol{v}(\xi,\tau,\omega,\boldsymbol{v}_{\tau})\|^2_{\mathbb{H}}+\alpha\|\boldsymbol{v}(\xi,\tau,\omega,\boldsymbol{v}_{\tau})\|^2_{\mathbb{H}}\nonumber\\&\quad+2\beta\|\boldsymbol{v}(\xi,\tau,\omega,\boldsymbol{v}_{\tau})+e^{\sigma \xi}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega)\|^{r+1}_{\widetilde \mathbb{L}^{r+1}}\bigg]\d\xi+2\int_{\tau}^{\mathfrak{s}}e^{\alpha(\xi-\mathfrak{s})}b(\boldsymbol{v}(\xi,\tau,\omega,\boldsymbol{v}_{\tau})+e^{\sigma \xi}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega),\nonumber\\&\quad\boldsymbol{v}(\xi,\tau,\omega,\boldsymbol{v}_{\tau})+e^{\sigma \xi}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega),e^{\sigma \xi}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega))\d\xi+2\int_{\tau}^{\mathfrak{s}}e^{\alpha(\xi-\mathfrak{s})}\left\langle\boldsymbol{f}(\cdot,\xi),\boldsymbol{v}(\xi,\tau,\omega,\boldsymbol{v}_{\tau})\right\rangle\d\xi\nonumber\\&\quad+2\int_{\tau}^{\mathfrak{s}}e^{\sigma\xi}e^{\alpha(\xi-\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\left(\left(\ell-\sigma-\alpha\right)\textbf{g}-\mu \mathrm{A}\textbf{g},\boldsymbol{v}(\xi,\tau,\omega,\boldsymbol{v}_{\tau})\right)\d\xi.
\end{align}
\fi
Now, following the similar arguments (that is, the method of energy equation) as in Lemma \ref{Asymptotic_UB} (or see Lemma \ref{precompact}) together with the convergence in \eqref{ac7add}-\eqref{ac1add}, Corollary \ref{convergence_b} and Corollary \ref{convergence_c}, we obtain that $\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{0,n})\to\tilde{\boldsymbol{v}}$ strongly in $\mathbb{H}$. Hence, from \eqref{Trans}, we get that $\boldsymbol{u}(\mathfrak{s},\mathfrak{s}-t_n,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{0,n})$ has a convergent subsequence in $\mathbb{H}$, as required.
\end{proof}
The following theorem demonstrates the existence of a unique random $\mathfrak{D}$-pullback attractor for the system \eqref{SCBF_Add}.
\begin{theorem}\label{RA_add}
For $d=2$ with $r>1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing7}. Then there exists a unique random $\mathfrak{D}$-pullback attractor $$\mathscr{A}_0=\{\mathscr{A}_0(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D},$$ for the the continuous cocycle $\Phi_0$ associated with the system \eqref{SCBF_Add}, in $\mathbb{H}$.
\end{theorem}
\begin{proof}
The proof follows from Lemma \ref{Asymptotic_UB}, Lemma \ref{PAS'} and the abstract theory available in \cite{SandN_Wang} (Theorem 2.23 in \cite{SandN_Wang}).
\end{proof}
\subsection{Upper semicontinuity of random pullback attractors for SCBF equations with additive colored noise}
Consider the random equation driven by colored noise
\begin{align}\label{z1}
\frac{\d\boldsymbol{z} _{\delta}}{\d t}=-\ell \boldsymbol{z} _{\delta} + \mathcal{Z}_{\delta}(\vartheta_t\omega).
\end{align}
From \eqref{N3}, it is clear that for all $\omega\in\Omega$, the integral
\begin{align}\label{z2}
\boldsymbol{z} _{\delta}(\omega)=\int_{-\infty}^{0}e^{\ell \tau}\mathcal{Z}_{\delta}(\vartheta_{\tau}\omega)\d\tau
\end{align}
is finite, and hence $\boldsymbol{z} _{\delta}:\Omega\to\mathbb{R}$ is a well defined random variable. Let us recall some properties of $\boldsymbol{z} _{\delta}$ from \cite{GGW}.
\begin{lemma}[Lemma 3.2, \cite{GGW}]
Let $\boldsymbol{z} _{\delta}$ be the random variable given by \eqref{z2}. Then the mapping
\begin{align}\label{z3}
(t,\omega)\mapsto\boldsymbol{z} _{\delta}(\vartheta_{t}\omega)=\int_{-\infty}^{t}e^{-\ell( t-\tau)}\mathcal{Z}_{\delta}(\vartheta_{\tau}\omega)\d\tau
\end{align}
is a stationary solution of \eqref{z1} with continuous paths. Moreover, $\mathbb{E}[\boldsymbol{z} _{\delta}]=0$ and for every $\omega\in\Omega$,
\begin{align}\label{z4}
\lim_{\delta\to0}\boldsymbol{z} _{\delta}(\vartheta_{t}\omega)=\boldsymbol{y}(\vartheta_{t}\omega) \text{ uniformly on } [\mathfrak{s},\mathfrak{s}+T]\text{ with } \mathfrak{s}\in\mathbb{R}\text{ and } T>0;
\end{align}
\begin{align}\label{z5}
\lim_{t\to\pm\infty}\frac{\left|\boldsymbol{z} _{\delta}(\vartheta_{t}\omega)\right|}{|t|}=0 \text{ uniformly for } 0<\delta\leq\tilde{\ell};
\end{align}
where $\tilde{\ell}=\min\{1,\frac{1}{2\ell}\}$.
\end{lemma}
Define
\begin{align}\label{WZ_T_add}
\boldsymbol{v}_{\delta}(t,\mathfrak{s},\omega,\boldsymbol{v}_{\delta,\mathfrak{s}})=\boldsymbol{u}_{\delta}(t,\mathfrak{s},\omega,\boldsymbol{u}_{\delta,\mathfrak{s}})-e^{\sigma t}\textbf{g}(x)\boldsymbol{z} _{\delta}(\vartheta_{t}\omega).
\end{align}
Then, from \eqref{WZ_SCBF_Add}, we obtain
\begin{equation}\label{WZ_CSCBF_Add}
\left\{
\begin{aligned}
\frac{\partial \boldsymbol{v}_{\delta}}{\partial t}+\mu\mathrm{A}\boldsymbol{v}_{\delta}+\mathrm{B}(\boldsymbol{v}_{\delta}&+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta})+\alpha\boldsymbol{v}_{\delta}+\beta\mathcal{C}(\boldsymbol{v}_{\delta}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta})\\&=\boldsymbol{f}+\left(\ell-\sigma-\alpha\right)e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta}-\mu e^{\sigma t}\boldsymbol{z} _{\delta}\mathrm{A}\textbf{g}, \ \ \ \text{ in } \mathbb{R}^d\times(\mathfrak{s},\infty), \\
\boldsymbol{v}_{\delta}(x,\mathfrak{s})&=\boldsymbol{v}_{\delta,\mathfrak{s}}(x)=\boldsymbol{u}_{\delta,\mathfrak{s}}(x)-e^{\sigma\mathfrak{s}}\textbf{g}(x)\boldsymbol{z} _{\delta}(\omega), \quad\quad \ \ x\in \mathbb{R}^d \text{ and }\mathfrak{s}\in\mathbb{R}.
\end{aligned}
\right.
\end{equation}
For all $\mathfrak{s}\in\mathbb{R},$ $t>\mathfrak{s},$ and for every $\boldsymbol{v}_{\delta,\mathfrak{s}}\in\mathbb{H}$ and $\omega\in\Omega$, \eqref{WZ_CSCBF_Add} has a unique solution $\boldsymbol{v}_{\delta}(\cdot,\mathfrak{s},\omega,\boldsymbol{v}_{\delta,\mathfrak{s}})\in \mathrm{C}([\mathfrak{s},\mathfrak{s}+T];\mathbb{H})\cap\mathrm{L}^2(\mathfrak{s}, \mathfrak{s}+T;\mathbb{V})\cap\mathrm{L}^{r+1}(\mathfrak{s},\mathfrak{s}+T;\widetilde{\mathbb{L}}^{r+1})$. Moreover, $\boldsymbol{v}_{\delta}(t,\mathfrak{s},\omega,\boldsymbol{v}_{\delta,\mathfrak{s}})$ is continuous with respect to the initial data $\boldsymbol{v}_{\delta,\mathfrak{s}}(x)$ and $(\mathscr{F},\mathscr{B}(\mathbb{H}))$-measurable in $\omega\in\Omega.$ Define a cocycle $\Phi_{\delta}:\mathbb{R}^+\times\mathbb{R}\times\Omega\times\mathbb{H}\to\mathbb{H}$ for the system \eqref{WZ_SCBF_Add} such that for given $t\in\mathbb{R}^+, \mathfrak{s}\in\mathbb{R}, \omega\in\Omega$ and $\boldsymbol{u}_{\delta,\mathfrak{s}}\in\mathbb{H}$,
\begin{align}\label{Phi_d}
\Phi_{\delta}(t,\mathfrak{s},\omega,\boldsymbol{u}_{\delta,\mathfrak{s}}) &=\boldsymbol{u}_{\delta}(t+\mathfrak{s},\mathfrak{s},\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\delta,\mathfrak{s}})=\boldsymbol{v}_{\delta}(t+\mathfrak{s},\mathfrak{s},\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{\delta,\mathfrak{s}})+e^{\sigma(t+\mathfrak{s})}\textbf{g}\boldsymbol{z} _{\delta}(\vartheta_{t}\omega).
\end{align}
\begin{lemma}\label{LemmaUe_add1}
For $d=2$ with $r>1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing6}. Then $\Phi_\delta$ possesses a closed measurable $\mathfrak{D}$-pullback absorbing set $\mathcal{K}^1_\delta=\{\mathcal{K}^1_\delta(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D}$ in $\mathbb{H}$ given by
\begin{align}\label{ue_add4}
\mathcal{K}^1_\delta(\mathfrak{s},\omega)=\{\boldsymbol{u}\in\mathbb{H}:\|\boldsymbol{u}\|^2_{\mathbb{H}}\leq \mathcal{R}^1_\delta(\mathfrak{s},\omega)\},\ \ \text{ for } d=2,
\end{align}
where $\mathcal{R}^1_\delta(\mathfrak{s},\omega)$ is defined by
\begin{align}\label{ue_add5}
\mathcal{R}^1_\delta(\mathfrak{s},\omega)&=3\|\textbf{g}\|^2_{\mathbb{H}}\left|e^{\sigma\mathfrak{s}}\boldsymbol{z} _{\delta}(\omega)\right|^2+2R_{5} \int_{-\infty}^{0}e^{\alpha\xi}\bigg[\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^2\nonumber\\&\quad+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^{r+1}+\left|e^{\sigma(\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{2(r+1)}{r-1}}\bigg]\d\xi,
\end{align}
and a closed measurable $\mathfrak{D}$-pullback absorbing set $\mathcal{K}^2_\delta=\{\mathcal{K}^2_\delta(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D}$ in $\mathbb{H}$ given by
\begin{align}\label{ue_add6}
\mathcal{K}^2_\delta(\mathfrak{s},\omega)=\{\boldsymbol{u}\in\mathbb{H}:\|\boldsymbol{u}\|^2_{\mathbb{H}}\leq \mathcal{R}^2_\delta(\mathfrak{s},\omega)\},\ \ \text{ for } d=3,
\end{align}
where $\mathcal{R}^2_\delta(\mathfrak{s},\omega)$ is defined by
\begin{align}\label{ue_add7}
\mathcal{R}^2_\delta(\mathfrak{s},\omega)&=3\|\textbf{g}\|^2_{\mathbb{H}}\left|e^{\sigma\mathfrak{s}}\boldsymbol{z} _{\delta}(\omega)\right|^2+2R_{8} \int_{-\infty}^{0}e^{\alpha\xi}\bigg[\|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^2\nonumber\\&\quad+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^{r+1}\bigg]\d\xi.
\end{align}
Here $R_{5}$ and $R_{8}$ are same as in Lemma \ref{LemmaUe_add}. Furthermore, for every $\mathfrak{s}\in\mathbb{R}$, $\omega\in\Omega$ and $i\in\{1,2\}$,
\begin{align}\label{ue_add8}
\lim_{\delta\to0}\mathcal{R}^{i}_\delta(\mathfrak{s},\omega)=\mathcal{R}^{i}_0(\mathfrak{s},\omega),
\end{align}
where $\mathcal{R}^{1}_0(\mathfrak{s},\omega)$ and $\mathcal{R}^{2}_0(\mathfrak{s},\omega)$ are given by \eqref{ue_add1} and \eqref{ue_add3}, respectively.
\end{lemma}
\begin{proof}
Since, the systems \eqref{CSCBF_Add} and \eqref{WZ_CSCBF_Add} have similar terms, the existence of closed measurable $\mathfrak{D}$-pullback absorbing sets $\mathcal{K}^{i}_\delta(\mathfrak{s},\omega)$ (for every $i\in\{1,2\}$) is confirmed from the similar calculations as in Lemma \ref{LemmaUe_add}. Now, it is left to prove \eqref{ue_add8} only.
Using \eqref{z5}, we can find $\xi_0<0$ such that for all $0<\delta\leq\tilde{\ell}$,
\begin{align}\label{ue30}
\left|\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|\leq|\xi|, \ \ \ \text{ for all } \ \xi\leq\xi_0.
\end{align}
From \eqref{ue30}, we get that for all $\xi\leq\xi_0$ and $0<\delta\leq\tilde{\ell}$,
\begin{align}\label{ue31}
& e^{\alpha\xi}\left[\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^2+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^{r+1}+\left|e^{\sigma(\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{2(r+1)}{r-1}}\right]\nonumber\\&\leq e^{\alpha\xi}\left[\left|e^{\sigma (\xi+\mathfrak{s})}\xi\right|^2+\left|e^{\sigma (\xi+\mathfrak{s})}\xi\right|^{r+1}+\left|e^{\sigma(\xi+\mathfrak{s})}\xi\right|^{\frac{2(r+1)}{r-1}}\right],
\end{align}
and
\begin{align}\label{ue32}
\int_{-\infty}^{0}e^{\alpha\xi}\left[\left|e^{\sigma (\xi+\mathfrak{s})}\xi\right|^2+\left|e^{\sigma (\xi+\mathfrak{s})}\xi\right|^{r+1}+\left|e^{\sigma(\xi+\mathfrak{s})}\xi\right|^{\frac{2(r+1)}{r-1}}\right]\d\xi <\infty.
\end{align}
Consider,
\begin{align}\label{ue33}
&\int_{-\infty}^{0}e^{\alpha\xi}\left[\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^2+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^{r+1}+\left|e^{\sigma(\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{2(r+1)}{r-1}}\right]\d\xi\nonumber\\&=\int_{\xi_0}^{0}e^{\alpha\xi}\left[\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^2+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^{r+1}+\left|e^{\sigma(\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{2(r+1)}{r-1}}\right]\d\xi\nonumber\\&\quad+\int_{-\infty}^{\xi_0}e^{\alpha\xi}\left[\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^2+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^{r+1}+\left|e^{\sigma(\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{2(r+1)}{r-1}}\right]\d\xi.
\end{align}
Using \eqref{z4}, \eqref{ue31}, \eqref{ue32} and the Lebesgue Dominated Convergence Theorem, we get
\begin{align}\label{ue34}
&\lim_{\delta\to0}\int_{-\infty}^{\xi_0}e^{\alpha\xi}\left[\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^2+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^{r+1}+\left|e^{\sigma(\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{2(r+1)}{r-1}}\right]\d\xi\nonumber\\&=\int_{-\infty}^{\xi_0}e^{\alpha\xi}\left[\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^2+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^{r+1}+\left|e^{\sigma(\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^{\frac{2(r+1)}{r-1}}\right]\d\xi.
\end{align}
From \eqref{z4}, we obtain
\begin{align}\label{ue35}
&\lim_{\delta\to0}\int_{\xi_0}^{0}e^{\alpha\xi}\left[\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^2+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^{r+1}+\left|e^{\sigma(\xi+\mathfrak{s})}\boldsymbol{z} _{\delta}(\vartheta_{\xi}\omega)\right|^{\frac{2(r+1)}{r-1}}\right]\d\xi\nonumber\\&=\int_{\xi_0}^{0}e^{\alpha\xi}\left[\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^2+\left|e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^{r+1}+\left|e^{\sigma(\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\right|^{\frac{2(r+1)}{r-1}}\right]\d\xi.
\end{align}
Hence, from \eqref{z4}, \eqref{ue34} and \eqref{ue35}, we deduce the desired convergence.
\end{proof}
Next result demonstrates the convergence of solution of \eqref{WZ_SCBF_Add} to the solution of \eqref{SCBF_Add} in $\mathbb{H}$ as $\delta\to 0$.
\begin{lemma}\label{Solu_Conver}
For $d=2$ with $r>1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$. Suppose that $\{\delta_n\}_{n\in\mathbb{N}}$ be the sequence such that $\delta_n\to0$. Let $\boldsymbol{u}_{\delta_n}$ and $\boldsymbol{u}$ be the solutions of \eqref{WZ_SCBF_Add} and \eqref{SCBF_Add} with initial data $\boldsymbol{u}_{\delta_n,\mathfrak{s}}$ and $\boldsymbol{u}_{\mathfrak{s}}$, respectively. If $\|\boldsymbol{u}_{\delta_n,\mathfrak{s}}-\boldsymbol{u}_{\mathfrak{s}}\|_{\mathbb{H}}\to0$ as $n\to\infty$, then for every $\mathfrak{s}\in\mathbb{R}$, $\omega\in\Omega$ and $t>\mathfrak{s}$,
\begin{align*}
\|\boldsymbol{u}_{\delta_n}(t,\mathfrak{s},\omega,\boldsymbol{u}_{\delta_n,\mathfrak{s}})-\boldsymbol{u}(t,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}})\|_{\mathbb{H}} \to0 \ \ \text{ as }\ \ n\to\infty.
\end{align*}
\end{lemma}
\begin{proof}
Let $\mathfrak{F}=\boldsymbol{v}_{\delta_n}-\boldsymbol{v}$, where $\boldsymbol{v}_{\delta_n}$ and $\boldsymbol{v}$ are the solutions of \eqref{WZ_CSCBF_Add} and \eqref{CSCBF_Add}, respectively. Also, let $\mathfrak{z}=\boldsymbol{z} _{\delta_n}-\boldsymbol{y}$. Then we get from \eqref{WZ_CSCBF_Add} and \eqref{CSCBF_Add} that
\begin{align}\label{F1}
\frac{1}{2}\frac{\d}{\d t}\|\mathfrak{F}\|^2_{\mathbb{H}}&=-\mu\|\nabla\mathfrak{F}\|^2_{\mathbb{H}}-\alpha\|\mathfrak{F}\|^2_{\mathbb{H}} -\left\langle \mathrm{B}(\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n})-\mathrm{B}(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}), \mathfrak{F}\right\rangle\nonumber\\&\quad-\beta\left\langle\mathcal{C}(\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n})-\mathcal{C}(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}),\mathfrak{F}\right\rangle+e^{\sigma t}\mathfrak{z}\left(\left(\ell-\sigma-\alpha\right)\textbf{g}-\mu \mathrm{A}\textbf{g},\mathfrak{F}\right) \nonumber\\&=-\mu\|\nabla\mathfrak{F}\|^2_{\mathbb{H}}-\alpha\|\mathfrak{F}\|^2_{\mathbb{H}}+e^{\sigma t}\mathfrak{z}\left(\left(\ell-\sigma-\alpha\right)\textbf{g}-\mu \mathrm{A}\textbf{g},\mathfrak{F}\right)\nonumber\\&\quad-\left\langle \mathrm{B}(\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n})-\mathrm{B}(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}), (\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n})-(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y})\right\rangle \nonumber\\&\quad-\beta\left\langle\mathcal{C}(\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n})-\mathcal{C}(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}),(\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n})-(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y})\right\rangle \nonumber\\&\quad+e^{\sigma t}\mathfrak{z}b(\boldsymbol{v}_{\delta_n},\boldsymbol{v}_{\delta_n},\textbf{g})+e^{2\sigma t}\mathfrak{z}\boldsymbol{z} _{\delta_n}b(\textbf{g},\boldsymbol{v}_{\delta_n},\textbf{g})-e^{\sigma t}\mathfrak{z}b(\boldsymbol{v},\boldsymbol{v},\textbf{g})-e^{2\sigma t}\mathfrak{z}\boldsymbol{y} b(\textbf{g},\boldsymbol{v},\textbf{g})\nonumber\\&\quad+\beta e^{\sigma t}\mathfrak{z}\left\langle\mathcal{C}(\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n})-\mathcal{C}(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}),\textbf{g}\right\rangle.
\end{align}
From \eqref{MO_c}, we write
\begin{align}\label{F2}
& -\beta\left\langle\mathcal{C}(\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n})-\mathcal{C}(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}),(\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n})-(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y})\right\rangle\nonumber\\&\leq-\frac{\beta}{2}\|\left|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\right|^{\frac{r-1}{2}}\left|\mathfrak{F}+e^{\sigma t}\textbf{g}\mathfrak{z}\right|\|^2_{\mathbb{H}}\leq0.
\end{align}
Applying Lemmas \ref{Holder} and \ref{Young}, we obtain
\begin{align}\label{F3}
& \left|\beta e^{\sigma t}\mathfrak{z}\left\langle\mathcal{C}(\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n})-\mathcal{C}(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}),\textbf{g}\right\rangle\right|\nonumber\\&\leq\beta e^{\sigma t}\left|\mathfrak{z}\right|\left(\|\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n}\|^r_{\widetilde\mathbb{L}^{r+1}}+\|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\|^r_{\widetilde\mathbb{L}^{r+1}}\right)\|\textbf{g}\|_{\widetilde\mathbb{L}^{r+1}}\nonumber\\&\leq C e^{\sigma t}\left|\mathfrak{z}\right|\left(\|\boldsymbol{v}_{\delta_n}\|^r_{\widetilde\mathbb{L}^{r+1}}+\left|e^{\sigma t}\boldsymbol{z} _{\delta_n}\right|^r+\|\boldsymbol{v}\|^r_{\widetilde\mathbb{L}^{r+1}}+\left|e^{\sigma t}\boldsymbol{y}\right|^r\right)\nonumber\\&\leq C e^{\sigma t}\left|\mathfrak{z}\right|\left(1+\left|e^{\sigma t}\boldsymbol{z} _{\delta_n}\right|^r+\left|e^{\sigma t}\boldsymbol{y}\right|^r+\|\boldsymbol{v}_{\delta_n}\|^{r+1}_{\widetilde\mathbb{L}^{r+1}} +\|\boldsymbol{v}\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}\right),
\end{align}
and
\begin{align}\label{F4}
\left|e^{\sigma t}\mathfrak{z}\left(\left(\ell-\sigma-\alpha\right)\textbf{g}-\mu \mathrm{A}\textbf{g},\mathfrak{F}\right)\right|\leq\frac{\alpha}{2}\|\mathfrak{F}\|^2_{\mathbb{H}} + Ce^{2\sigma t}\left|\mathfrak{z}\right|^2.
\end{align}
\vskip 2mm
\noindent
\textbf{Case I:} \textit{$d= 2$ and $r>1$.}
Using \eqref{b1}, \eqref{441}, Lemmas \ref{Holder} and \ref{Young}, we get
\begin{align}\label{F5}
& \left|\left\langle \mathrm{B}(\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n})-\mathrm{B}(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}), (\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n})-(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y})\right\rangle\right|\nonumber\\&\leq\left|b(\mathfrak{F}+e^{\sigma t}\textbf{g}\mathfrak{z},\mathfrak{F}+e^{\sigma t}\textbf{g}\mathfrak{z},\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y})\right|\nonumber\\&\leq C\|\mathfrak{F}+e^{\sigma t}\textbf{g}\mathfrak{z}\|_{\mathbb{H}}\|\nabla(\mathfrak{F}+e^{\sigma t}\textbf{g}\mathfrak{z})\|_{\mathbb{H}}\|\nabla(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y})\|_{\mathbb{H}}\nonumber\\&\leq\frac{\mu}{2}\|\nabla\mathfrak{F}\|^2_{\mathbb{H}}+C\left(e^{2\sigma t}\left|\boldsymbol{y}\right|^2+\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}\right)\|\mathfrak{F}\|^2_{\mathbb{H}}+ Ce^{2\sigma t}\left|\mathfrak{z}\right|^2\left(1+e^{2\sigma t}\left|\boldsymbol{y}\right|^2+\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}\right).
\end{align}
Again, from \eqref{b1}, Lemmas \ref{Holder} and \ref{Young}, we have
\begin{align}
\left|e^{\sigma t}\mathfrak{z}b(\boldsymbol{v}_{\delta_n},\boldsymbol{v}_{\delta_n},\textbf{g})\right|&\leq Ce^{\sigma t}\left|\mathfrak{z}\right|\|\boldsymbol{v}_{\delta_n}\|_{\mathbb{H}}\|\nabla\boldsymbol{v}_{\delta_n}\|_{\mathbb{H}}\|\nabla\textbf{g}\|_{\mathbb{H}}\leq Ce^{\sigma t}\left|\mathfrak{z}\right|\left(1+\|\boldsymbol{v}_{\delta_n}\|^2_{\mathbb{H}}\|\nabla\boldsymbol{v}_{\delta_n}\|^2_{\mathbb{H}}\right),\label{F6}\\
\left|e^{2\sigma t}\mathfrak{z}\boldsymbol{z} _{\delta_n}b(\textbf{g},\boldsymbol{v}_{\delta_n},\textbf{g})\right|&\leq Ce^{2\sigma t}\left|\mathfrak{z}\right|\left|\boldsymbol{z} _{\delta_n}\right|\|\textbf{g}\|_{\mathbb{H}}\|\nabla\textbf{g}\|_{\mathbb{H}}\|\nabla\boldsymbol{v}_{\delta_n}\|_{\mathbb{H}}\leq Ce^{2\sigma t}\left|\mathfrak{z}\right|\left|\boldsymbol{z} _{\delta_n}\right|\left(1+\|\nabla\boldsymbol{v}_{\delta_n}\|^2_{\mathbb{H}}\right),\label{F7}\\
\left|e^{\sigma t}\mathfrak{z}b(\boldsymbol{v},\boldsymbol{v},\textbf{g})\right|&\leq Ce^{\sigma t}\left|\mathfrak{z}\right|\|\boldsymbol{v}\|_{\mathbb{H}}\|\nabla\boldsymbol{v}\|_{\mathbb{H}}\|\nabla\textbf{g}\|_{\mathbb{H}}\leq Ce^{\sigma t}\left|\mathfrak{z}\right|\left(1+\|\boldsymbol{v}\|^2_{\mathbb{H}}\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}\right),\label{F8}\\
\left|e^{2\sigma t}\mathfrak{z}\boldsymbol{y} b(\textbf{g},\boldsymbol{v},\textbf{g})\right|&\leq Ce^{2\sigma t}\left|\mathfrak{z}\right|\left|\boldsymbol{y}\right|\|\textbf{g}\|_{\mathbb{H}}\|\nabla\textbf{g}\|_{\mathbb{H}}\|\nabla\boldsymbol{v}\|_{\mathbb{H}}\leq Ce^{2\sigma t}\left|\mathfrak{z}\right|\left|\boldsymbol{y}\right|\left(1+\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}\right).\label{F9}
\end{align}
Combining \eqref{F1}-\eqref{F9}, we obtain
\begin{align}\label{F10}
\frac{\d}{\d t}\|\mathfrak{F}\|^2_{\mathbb{H}}\leq C\left[Q_1(t)\|\mathfrak{F}\|^2_{\mathbb{H}} + \left|\mathfrak{z}(\vartheta_{t}\omega)\right|Q_2(t)\right],
\end{align}
for a.e. $t\in[\mathfrak{s},\mathfrak{s}+T]$, where
\begin{align*}
Q_1(t)&=e^{2\sigma t}\left|\boldsymbol{y}(\vartheta_{t}\omega)\right|^2+\|\nabla\boldsymbol{v}(t)\|^2_{\mathbb{H}}\ \ \text{ and }\\
Q_2(t)&=e^{2\sigma t}\left|\mathfrak{z}(\vartheta_{t}\omega)\right|\bigg[1+e^{2\sigma t}\left|\boldsymbol{y}(\vartheta_{t}\omega)\right|^2+\|\nabla\boldsymbol{v}(t)\|^2_{\mathbb{H}}\bigg]+e^{\sigma t}\bigg[1+\|\boldsymbol{v}_{\delta_n}(t)\|^2_{\mathbb{H}}\|\nabla\boldsymbol{v}_{\delta_n}(t)\|^2_{\mathbb{H}}\nonumber\\&\quad+\|\boldsymbol{v}(t)\|^2_{\mathbb{H}}\|\nabla\boldsymbol{v}(t)\|^2_{\mathbb{H}}+\left|e^{\sigma t}\boldsymbol{z} _{\delta_n}(\vartheta_{t}\omega)\right|^r+\left|e^{\sigma t}\boldsymbol{y}(\vartheta_{t}\omega)\right|^r+\|\boldsymbol{v}_{\delta_n}(t)\|^{r+1}_{\widetilde\mathbb{L}^{r+1}} +\|\boldsymbol{v}(t)\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}\bigg]\nonumber\\&\quad+e^{2\sigma t}\left|\boldsymbol{z} _{\delta_n}(\vartheta_{t}\omega)\right|\bigg[1+\|\nabla\boldsymbol{v}_{\delta_n}(t)\|^2_{\mathbb{H}}\bigg]+e^{2\sigma t}\left|\boldsymbol{y}(\vartheta_{t}\omega)\right|\bigg[1+\|\nabla\boldsymbol{v}(t)\|^2_{\mathbb{H}}\bigg].
\end{align*}
Due to continuity of $e^{\sigma t},$ $\boldsymbol{y}(\vartheta_{t}\omega)$ and $\boldsymbol{z} _{\delta_n}(\vartheta_{t}\omega)$, and $\boldsymbol{v}_{\delta_n}, \boldsymbol{v}\in\mathrm{C}([\mathfrak{s},\mathfrak{s}+T];\mathbb{H})\cap\mathrm{L}^2(\mathfrak{s}, \mathfrak{s}+T;\mathbb{V})\cap\mathrm{L}^{r+1}(\mathfrak{s},\mathfrak{s}+T;\widetilde{\mathbb{L}}^{r+1})$, we infer that
\begin{align}\label{F11}
\int_{\mathfrak{s}}^{\mathfrak{s}+T}Q_1(\xi)\d\xi<\infty \ \text{ and }\ \int_{\mathfrak{s}}^{\mathfrak{s}+T}Q_2(\xi)\d\xi<\infty.
\end{align}
An application of Gronwall's inequality yields
\begin{align}\label{F12}
&\|\boldsymbol{v}_{\delta_n}(t,\mathfrak{s},\omega,\boldsymbol{v}_{\delta_n,\mathfrak{s}})-\boldsymbol{v}(t,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}})\|^2_{\mathbb{H}}\nonumber\\&\leq\left[ \|\boldsymbol{v}_{\delta_n,\mathfrak{s}}-\boldsymbol{v}_{\mathfrak{s}}\|^2_{\mathbb{H}}+C\sup_{\xi \in [\mathfrak{s},\mathfrak{s}+ T]}\left|\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega)-\boldsymbol{y}(\vartheta_{\xi}\omega)\right|\int_{\mathfrak{s}}^{\mathfrak{s}+T}Q_1(\xi)\d\xi\right]e^{C\int\limits_{\mathfrak{s}}^{\mathfrak{s}+T}Q_2(\xi)\d\xi}.
\end{align}
By \eqref{T_add}, \eqref{WZ_T_add} and \eqref{F12}, we get
\begin{align}\label{F13}
&\|\boldsymbol{u}_{\delta_n}(t,\mathfrak{s},\omega,\boldsymbol{u}_{\delta_n,\mathfrak{s}})-\boldsymbol{u}(t,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}})\|^2_{\mathbb{H}}\nonumber\\&\leq2\bigg[ 2\|\boldsymbol{u}_{\delta_n,\mathfrak{s}}-\boldsymbol{u}_{\mathfrak{s}}\|^2_{\mathbb{H}} +2e^{\sigma\mathfrak{s}}\left|\boldsymbol{z} _{\delta_n}(\omega)-\boldsymbol{y}(\omega)\right|\|\textbf{g}\|^2_{\mathbb{H}}\nonumber\\&\quad+C\sup_{\xi \in [\mathfrak{s},\mathfrak{s}+ T]}\left|\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega)-\boldsymbol{y}(\vartheta_{\xi}\omega)\right|\int\limits_{\mathfrak{s}}^{\mathfrak{s}+T}Q_1(\xi)\d\xi\bigg]e^{C\int\limits_{\mathfrak{s}}^{\mathfrak{s}+T}Q_2(\xi)\d\xi} \nonumber\\&\quad+ 2e^{\sigma t}\left|\boldsymbol{z} _{\delta_n}(\vartheta_{t}\omega)-\boldsymbol{y}(\vartheta_{t}\omega)\right|\|\textbf{g}\|^2_{\mathbb{H}},
\end{align}
for all $t\in[\mathfrak{s},\mathfrak{s}+T]$. Hence, due to \eqref{z4} and \eqref{F11}, we achieve the required convergence from \eqref{F13}.
\vskip 2mm
\noindent
\textbf{Case II:} \textit{$d= 3$ and $r\geq3$ ($r>3$ with any $\beta,\mu>0$ and $r=3$ with $2\beta\mu\geq1$).}
Using \eqref{441}, Lemmas \ref{Holder} and \ref{Young}, we get
\begin{align}\label{F14}
& \left|\left\langle \mathrm{B}(\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n})-\mathrm{B}(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}), (\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n})-(\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y})\right\rangle\right|\nonumber\\&\leq\left|b(\mathfrak{F}+e^{\sigma t}\textbf{g}\mathfrak{z},\mathfrak{F}+e^{\sigma t}\textbf{g}\mathfrak{z},\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y})\right|\nonumber\\&\leq\begin{cases}
\frac{1}{2\beta}\|\nabla\left(\mathfrak{F}+e^{\sigma t}\textbf{g}\mathfrak{z}\right)\|^2_{\mathbb{H}}+\frac{\beta}{2}\|\left|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\right|\left|\mathfrak{F}+e^{\sigma t}\textbf{g}\mathfrak{z}\right|\|^2_{\mathbb{H}}, \text{ for } r=3,\\ \frac{\mu}{4}\|\nabla\left(\mathfrak{F}+e^{\sigma t}\textbf{g}\mathfrak{z}\right)\|^2_{\mathbb{H}}+\frac{\beta}{2}\|\left|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\right|^{\frac{r-1}{2}}\left|\mathfrak{F}+e^{\sigma t}\textbf{g}\mathfrak{z}\right|\|^2_{\mathbb{H}}+C\|\mathfrak{F}+e^{\sigma t}\textbf{g}\mathfrak{z}\|^2_{\mathbb{H}}, \text{ for } r>3, \end{cases}\nonumber\\&\leq\begin{cases}
\frac{1}{2\beta}\|\nabla\mathfrak{F}\|^2_{\mathbb{H}}+C\left|e^{\sigma t}\mathfrak{z}\right|^2+C\left|e^{\sigma t}\mathfrak{z}\right|\left(1+\|\nabla\boldsymbol{v}_{\delta_n}\|^2_{\mathbb{H}}+\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}\right)+\frac{\beta}{2}\||\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}||\mathfrak{F}+e^{\sigma t}\textbf{g}\mathfrak{z}|\|^2_{\mathbb{H}},\\ \hspace{131mm} \text{ for } r=3,\\
\frac{\mu}{2}\|\nabla\mathfrak{F}\|^2_{\mathbb{H}}+C\|\mathfrak{F}\|^2_{\mathbb{H}}+ Ce^{2\sigma t}\left|\mathfrak{z}\right|^2+\frac{\beta}{2}\|\left|\boldsymbol{v}+e^{\sigma t}\textbf{g}\boldsymbol{y}\right|^{\frac{r-1}{2}}\left|\mathfrak{F}+e^{\sigma t}\textbf{g}\mathfrak{z}\right|\|^2_{\mathbb{H}}, \text{ for } r>3.
\end{cases}
\end{align}
From \eqref{b1}, Lemmas \ref{Holder} and \ref{Young}, we have
\begin{align}
\left|e^{\sigma t}\mathfrak{z}b(\boldsymbol{v}_{\delta_n},\boldsymbol{v}_{\delta_n},\textbf{g})\right|&\leq Ce^{\sigma t}\left|\mathfrak{z}\right|\|\boldsymbol{v}_{\delta_n}\|^{\frac{1}{2}}_{\mathbb{H}}\|\nabla\boldsymbol{v}_{\delta_n}\|^{\frac{3}{2}}_{\mathbb{H}}\|\nabla\textbf{g}\|_{\mathbb{H}}\leq Ce^{\sigma t}\left|\mathfrak{z}\right|\left(1+\|\boldsymbol{v}_{\delta_n}\|^{\frac{2}{3}}_{\mathbb{H}}\|\nabla\boldsymbol{v}_{\delta_n}\|^2_{\mathbb{H}}\right),\label{F16}\\
\left|e^{2\sigma t}\mathfrak{z}\boldsymbol{z} _{\delta_n}b(\textbf{g},\boldsymbol{v}_{\delta_n},\textbf{g})\right|&\leq Ce^{2\sigma t}\left|\mathfrak{z}\right|\left|\boldsymbol{z} _{\delta_n}\right|\|\textbf{g}\|^{\frac{1}{2}}_{\mathbb{H}}\|\nabla\textbf{g}\|^{\frac{3}{2}}_{\mathbb{H}}\|\nabla\boldsymbol{v}_{\delta_n}\|_{\mathbb{H}}\leq Ce^{2\sigma t}\left|\mathfrak{z}\right|\left|\boldsymbol{z} _{\delta_n}\right|\left(1+\|\nabla\boldsymbol{v}_{\delta_n}\|^2_{\mathbb{H}}\right),\label{F17}\\
\left|e^{\sigma t}\mathfrak{z}b(\boldsymbol{v},\boldsymbol{v},\textbf{g})\right|&\leq Ce^{\sigma t}\left|\mathfrak{z}\right|\|\boldsymbol{v}\|^{\frac{1}{2}}_{\mathbb{H}}\|\nabla\boldsymbol{v}\|^{\frac{3}{2}}_{\mathbb{H}}\|\nabla\textbf{g}\|_{\mathbb{H}}\leq Ce^{\sigma t}\left|\mathfrak{z}\right|\left(1+\|\boldsymbol{v}\|^{\frac{2}{3}}_{\mathbb{H}}\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}\right),\label{F18}\\
\left|e^{2\sigma t}\mathfrak{z}\boldsymbol{y} b(\textbf{g},\boldsymbol{v},\textbf{g})\right|&\leq Ce^{2\sigma t}\left|\mathfrak{z}\right|\left|\boldsymbol{y}\right|\|\textbf{g}\|^{\frac{1}{2}}_{\mathbb{H}}\|\nabla\textbf{g}\|^{\frac{3}{2}}_{\mathbb{H}}\|\nabla\boldsymbol{v}\|_{\mathbb{H}}\leq Ce^{2\sigma t}\left|\mathfrak{z}\right|\left|\boldsymbol{y}\right|\left(1+\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}\right).\label{F19}
\end{align}
Combining \eqref{F1}-\eqref{F4} and \eqref{F14}-\eqref{F19}, we obtain
\begin{align}\label{F20}
\frac{\d}{\d t}\|\mathfrak{F}\|^2_{\mathbb{H}}&\leq C\times\begin{cases}
\left|\mathfrak{z}(\vartheta_{t}\omega)\right|Q_3(t), &\text{ for } r=3 ,\\ \|\mathfrak{F}\|^2_{\mathbb{H}} + \left|\mathfrak{z}(\vartheta_{t}\omega)\right|Q_4(t), &\text{ for } r>3,
\end{cases}
\end{align}
for a.e. $t\in[\mathfrak{s},\mathfrak{s}+T]$, where
\begin{align*}
Q_3(t)&=e^{2\sigma t}\left|\mathfrak{z}(\vartheta_{t}\omega)\right|+e^{\sigma t}\bigg[1+\|\nabla\boldsymbol{v}_{\delta_n}(t)\|^2_{\mathbb{H}}+\|\nabla\boldsymbol{v}(t)\|^2_{\mathbb{H}}+\|\boldsymbol{v}_{\delta_n}(t)\|^{\frac{2}{3}}_{\mathbb{H}}\|\nabla\boldsymbol{v}_{\delta_n}(t)\|^2_{\mathbb{H}}\nonumber\\&\quad+\|\boldsymbol{v}(t)\|^{\frac{2}{3}}_{\mathbb{H}}\|\nabla\boldsymbol{v}(t)\|^2_{\mathbb{H}}+\left|e^{\sigma t}\boldsymbol{z} _{\delta_n}(\vartheta_{t}\omega)\right|^3+\left|e^{\sigma t}\boldsymbol{y}(\vartheta_{t}\omega)\right|^3+\|\boldsymbol{v}_{\delta_n}(t)\|^{4}_{\widetilde\mathbb{L}^{4}} +\|\boldsymbol{v}(t)\|^{4}_{\widetilde\mathbb{L}^{4}}\bigg]\nonumber\\&\quad+e^{2\sigma t}\left|\boldsymbol{z} _{\delta_n}(\vartheta_{t}\omega)\right|\bigg[1+\|\nabla\boldsymbol{v}_{\delta_n}(t)\|^2_{\mathbb{H}}\bigg]+e^{2\sigma t}\left|\boldsymbol{y}(\vartheta_{t}\omega)\right|\bigg[1+\|\nabla\boldsymbol{v}(t)\|^2_{\mathbb{H}}\bigg]
\end{align*}
and \begin{align*}Q_4(t)&=e^{2\sigma t}\left|\mathfrak{z}(\vartheta_{t}\omega)\right|+e^{\sigma t}\bigg[1+\|\boldsymbol{v}_{\delta_n}(t)\|^{\frac{2}{3}}_{\mathbb{H}}\|\nabla\boldsymbol{v}_{\delta_n}(t)\|^2_{\mathbb{H}}+\|\boldsymbol{v}(t)\|^{\frac{2}{3}}_{\mathbb{H}}\|\nabla\boldsymbol{v}(t)\|^2_{\mathbb{H}}+\left|e^{\sigma t}\boldsymbol{z} _{\delta_n}(\vartheta_{t}\omega)\right|^r\nonumber\\&\quad+\left|e^{\sigma t}\boldsymbol{y}(\vartheta_{t}\omega)\right|^r+\|\boldsymbol{v}_{\delta_n}(t)\|^{r+1}_{\widetilde\mathbb{L}^{r+1}} +\|\boldsymbol{v}(t)\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}\bigg]+e^{2\sigma t}\left|\boldsymbol{z} _{\delta_n}(\vartheta_{t}\omega)\right|\bigg[1+\|\nabla\boldsymbol{v}_{\delta_n}(t)\|^2_{\mathbb{H}}\bigg]\nonumber\\&\quad+e^{2\sigma t}\left|\boldsymbol{y}(\vartheta_{t}\omega)\right|\bigg[1+\|\nabla\boldsymbol{v}(t)\|^2_{\mathbb{H}}\bigg].
\end{align*}
Hence, arguing similarly as in Case I, we obtain the desired convergence.
\end{proof}
\begin{lemma}\label{Solu_Conver1}
For $d=2$ with $r>1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$. Suppose that $\{\delta_n\}_{n\in\mathbb{N}}$ be the sequence such that $\delta_n\to0$. Let $\boldsymbol{v}_{\delta_n}$ and $\boldsymbol{v}$ be the solutions of \eqref{WZ_SCBF_Add} and \eqref{SCBF_Add} with initial data $\boldsymbol{v}_{\delta_n,\mathfrak{s}}$ and $\boldsymbol{v}_{\mathfrak{s}}$, respectively. If $\boldsymbol{v}_{\delta_n,\mathfrak{s}}\xrightharpoonup{w}\boldsymbol{v}_{\mathfrak{s}}$ in $\mathbb{H}$ as $n\to\infty$, then for every $\mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$,
\begin{itemize}
\item [(i)] $\boldsymbol{v}_{\delta_n}(\xi,\mathfrak{s},\omega,\boldsymbol{v}_{\delta_n,\mathfrak{s}})\xrightharpoonup{w}\boldsymbol{v}(\xi,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}})$ in $\mathbb{H}$ for all $\xi\geq \mathfrak{s}$.
\item [(ii)] $\boldsymbol{v}_{\delta_n}(\cdot,\mathfrak{s},\omega,\boldsymbol{v}_{\delta_n,\mathfrak{s}})\xrightharpoonup{w}\boldsymbol{v}(\cdot,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}})$ in $\mathrm{L}^2((\mathfrak{s},\mathfrak{s}+T);\mathbb{V})$ for every $T>0$.
\item [(iii)] $\boldsymbol{v}_{\delta_n}(\cdot,\mathfrak{s},\omega,\boldsymbol{v}_{\delta_n,\mathfrak{s}})\xrightharpoonup{w}\boldsymbol{v}(\cdot,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}})$ in $\mathrm{L}^{r+1}((\mathfrak{s},\mathfrak{s}+T);\widetilde{\mathbb{L}}^{r+1})$ for every $T>0$.
\item [(iv)] $\boldsymbol{v}_{\delta_n}(\cdot,\mathfrak{s},\omega,\boldsymbol{v}_{\delta_n,\mathfrak{s}})\to\boldsymbol{v}(\cdot,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}})$ in $\mathrm{L}^2((\mathfrak{s},\mathfrak{s}+T);\mathbb{L}^2(\mathcal{O}_k))$ for every $T>0$ and $k>0$,
where $\mathcal{O}_k=\{x\in\mathbb{R}^d:|x|<k\}$.
\end{itemize}
\end{lemma}
\begin{proof}
Proof is similar to Lemma 3.5, \cite{GGW}, and hence we omit it here.
\end{proof}
The existence of a unique random $\mathfrak{D}$-pullback attractor (denote here by $\mathscr{A}_\delta$) for continuous cocycle $\Phi_\delta$ follows from Theorem \ref{WZ_RA_UB}. The following Lemma demonstrates the uniform compactness of family of random attractors $\mathscr{A}_{\delta}$.
\begin{lemma}\label{precompact}
For $d=2$ with $r>1, d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\mathrm{loc}}(\mathbb{R};\mathbb{V}')$ and satisfies \eqref{forcing7}. Let $\mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$ be fixed. If $\delta_n\to0$ as $n\to\infty$ and $\boldsymbol{u}_n\in\mathscr{A}_{\delta_n}(\mathfrak{s},\omega)$, then the sequence $\{\boldsymbol{u}_n\}_{n\in\mathbb{N}}$ has a convergent subsequence in $\mathbb{H}$.
\end{lemma}
\begin{proof}
Since, $\delta_n\to0$, it follows from \eqref{ue_add8} that for every $i\in\{1,2\}$ ($i=1$ and $i=2$ for $d=2$ and $d=3$, respectively), $\mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$, there exists $\mathcal{N}_1=\mathcal{N}_1(\mathfrak{s},\omega)$ such that for all $n\geq\mathcal{N}_1$,
\begin{align}\label{PC2}
\mathcal{R}^{i}_{\delta_n}(\mathfrak{s},\omega)\leq2\mathcal{R}^{i}_0(\mathfrak{s},\omega).
\end{align}
It is given that $\boldsymbol{u}_n\in\mathscr{A}_{\delta_n}(\mathfrak{s},\omega)$ and due to the property of attractors, we have that $\mathscr{A}_{\delta_n}(\mathfrak{s},\omega)$ is a subset of absorbing set, by Lemma \ref{LemmaUe_add1} and \eqref{PC2} we have, for all $n\geq\mathcal{N}_1$,
\begin{align}\label{PC3}
\|\boldsymbol{u}_n\|^2_{\mathbb{H}}\leq2\mathcal{R}^i_{0}(\mathfrak{s},\omega),
\end{align}
where, $i=1$ and $i=2$ for $d=2$ and $d=3$, respectively. It is clear that $\{\boldsymbol{u}_n:n\in\mathbb{N}\}\subset\mathbb{H}$ and therefore, there exists a subsequence (not relabeling) and $\hat{\boldsymbol{u}}\in\mathbb{H}$ such that
\begin{align}\label{PC4}
\boldsymbol{u}_n\xrightharpoonup{w}\hat{\boldsymbol{u}} \ \text{ in }\ \mathbb{H}.
\end{align}
Our next aim to prove that $\boldsymbol{u}_n\to\hat{\boldsymbol{u}}$ in $\mathbb{H}$. Since $\boldsymbol{u}_n\in\mathscr{A}_{\delta_n}(\mathfrak{s},\omega)$, by the invariance property of $\mathscr{A}_{\delta_n}(\mathfrak{s},\omega)$, for all $l\geq1$, there exists $\boldsymbol{u}_{n,l}\in\mathscr{A}_{\delta_n}(\mathfrak{s}-l,\vartheta_{-l}\omega)$ such that
\begin{align}\label{PC5}
\boldsymbol{u}_n=\Phi_{\delta_n}(l,\mathfrak{s}-l,\vartheta_{-l}\omega,\boldsymbol{u}_{n,l})=\boldsymbol{u}_{\delta_n}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{n,l}).
\end{align}
Since $\boldsymbol{u}_{n,l}\in\mathscr{A}_{\delta_n}(\mathfrak{s}-l,\vartheta_{-l}\omega)$ and $\mathscr{A}_{\delta_n}(\mathfrak{s}-l,\vartheta_{-l}\omega)\subseteq\mathcal{K}^i_{\delta_n}(\mathfrak{s}-l,\vartheta_{-l}\omega),$ by Lemma \ref{LemmaUe_add1} and \eqref{PC2} we find that for each $l\geq1$ and $n\geq\mathcal{N}_1(\mathfrak{s}-l,\vartheta_{-l}\omega)$,
\begin{align}\label{PC6}
\|\boldsymbol{u}_{n,l}\|^2_{\mathbb{H}}\leq2\mathcal{R}^i_{0}(\mathfrak{s}-l,\vartheta_{-l}\omega).
\end{align}
Due to \eqref{WZ_T_add}, we get
\begin{align}\label{PC7}
\boldsymbol{v}_{\delta_n}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{n,l})=\boldsymbol{u}_{\delta_n}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{u}_{n,l})-e^{\sigma\mathfrak{s}}\textbf{g}(x)\boldsymbol{z} _{\delta_n}(\omega),
\end{align}
where
\begin{align}\label{PC8}
\boldsymbol{v}_{n,l}=\boldsymbol{u}_{n,l}-e^{\sigma(\mathfrak{s}-l)}\textbf{g}(x)\boldsymbol{z} _{\delta_n}(\vartheta_{-l}\omega).
\end{align}
From, \eqref{PC5} and \eqref{PC7} we obtain
\begin{align}\label{PC9}
\boldsymbol{u}_n=\boldsymbol{v}_{\delta_n}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{n,l})+e^{\sigma\mathfrak{s}}\textbf{g}(x)\boldsymbol{z} _{\delta_n}(\omega).
\end{align}
Using \eqref{PC6} and \eqref{PC8}, we find that for $n\geq\mathcal{N}_1(\mathfrak{s}-l,\vartheta_{-l}\omega)$,
\begin{align}\label{PC10}
\|\boldsymbol{v}_{n,l}\|^2_{\mathbb{H}}\leq4\mathcal{R}^i_{0}(\mathfrak{s}-l,\vartheta_{-l}\omega)+2e^{2\sigma(\mathfrak{s}-l)}\left|\textbf{g}(x)\boldsymbol{z} _{\delta_n}(\vartheta_{-l}\omega)\right|^2.
\end{align}
Thanks to \eqref{z4} and \eqref{PC10}, we can find an $\mathcal{N}_2=\mathcal{N}_2(\mathfrak{s},\omega,l)\geq\mathcal{N}_1$ such that for every $l\geq1$ and $n\geq \mathcal{N}_2$,
\begin{align}\label{PC1}
\|\boldsymbol{v}_{n,l}\|^2_{\mathbb{H}}\leq4\mathcal{R}^i_{0}(\mathfrak{s}-l,\vartheta_{-l}\omega)+4e^{2\sigma(\mathfrak{s}-l)}\left|\textbf{g}(x)\right|^2(1+\left|\boldsymbol{y}(\vartheta_{-l}\omega)\right|^2).
\end{align}
By \eqref{PC4} and \eqref{PC9} along with \eqref{z4}, we get as $n\to\infty$,
\begin{align}\label{PC11}
\boldsymbol{v}_{\delta_n}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{n,l}) \xrightharpoonup{w}\hat{\boldsymbol{v}} \text{ in } \mathbb{H},\ \text{ where }\ \hat{\boldsymbol{v}}=\hat{\boldsymbol{u}}-e^{\sigma\mathfrak{s}}\textbf{g}(x)\boldsymbol{y}(\omega).
\end{align}
It follows from \eqref{PC10} that for each $l\geq1$, the sequence $\{\boldsymbol{v}_{n,l}\}_{n\in\mathbb{N}}\subset\mathbb{H}$, and hence, we get a subsequence (not relabeling) by a diagonal process such that for every $l\geq1$, there exists $\tilde{\boldsymbol{v}}_l\in\mathbb{H}$ such that
\begin{align}\label{PC12}
\boldsymbol{v}_{n,l}\xrightharpoonup{w}\tilde{\boldsymbol{v}}_l \ \text{ in }\ \mathbb{H} \ \ \text{ as }\ \ n\to\infty.
\end{align}
It yields from Lemma \ref{Solu_Conver1} and \eqref{PC12} that as $n\to\infty$,
\begin{align}
&\boldsymbol{v}_{\delta_n}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{n,l})\xrightharpoonup{w}\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{v}}_{l})\ \text{ in }\ \mathbb{H}, \label{PC13}\\
&\boldsymbol{v}_{\delta_n}(\cdot,\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{n,l})\xrightharpoonup{w}\boldsymbol{v}(\cdot,\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{v}}_{l}) \ \text{ in } \ \mathrm{L}^2(\mathfrak{s}-l,\mathfrak{s};\mathbb{V}),\label{PC14}\\
& \boldsymbol{v}_{\delta_n}(\cdot,\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{n,l})\xrightharpoonup{w}\boldsymbol{v}(\cdot,\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{v}}_{l}) \ \text{ in } \ \mathrm{L}^{r+1}(\mathfrak{s}-l,\mathfrak{s};\widetilde\mathbb{L}^{r+1}), \label{PC15}\\
& \boldsymbol{v}_{\delta_n}(\cdot,\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{n,l})\to\boldsymbol{v}(\cdot,\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{v}}_{l}) \text{ in } \mathrm{L}^{2}(\mathfrak{s}-l,\mathfrak{s};\mathbb{L}^{2}(\mathcal{O}_k)), \label{PC16}
\end{align}
where $\mathcal{O}_k=\{x\in\mathbb{R}^d:|x|<k\}$. Now, \eqref{PC11} and \eqref{PC13} imply
\begin{align}\label{PC17}
\hat{\boldsymbol{v}}=\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{v}}_{l}).
\end{align}
From \eqref{WZ_CSCBF_Add} together with \eqref{b0}, we obtain
\begin{align}\label{PC18}
&\frac{\d}{\d t}\|\boldsymbol{v}_{\delta_n}\|^2_{\mathbb{H}}+2\alpha\|\boldsymbol{v}_{\delta_n}\|^2_{\mathbb{H}}+2\mu\|\nabla\boldsymbol{v}_{\delta_n}\|^2_{\mathbb{H}}+2\beta\|\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n}\|^{r+1}_{\widetilde\mathbb{L}^{r+1}} \nonumber\\&=2b(\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n},\boldsymbol{v}_{\delta_n},e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n})+2\beta\left\langle\mathcal{C}(\boldsymbol{v}_{\delta_n}+e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n}),e^{\sigma t}\textbf{g}\boldsymbol{z} _{\delta_n}\right\rangle+2\left\langle\boldsymbol{f}(\cdot,t),\boldsymbol{v}_{\delta_n}\right\rangle\nonumber\\&\quad + 2e^{\sigma t}\boldsymbol{z} _{\delta_n}\left(\left(\ell-\sigma-\alpha\right)\textbf{g}-\mu\mathrm{A}\textbf{g},\boldsymbol{v}_{\delta_n}\right).
\end{align}
An application of variation of constant formula with $\omega$ being replaced by $\vartheta_{-\mathfrak{s}}\omega$ yields
\begin{align}\label{PC19}
&\|\boldsymbol{v}_{\delta_n}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l})\|^2_{\mathbb{H}}\nonumber\\&=I_1(n,l)+I_2(n,l)+I_3(n,l)+I_4(n,l)+I_5(n,l)+I_6(n,l)+I_7(n,l),
\end{align}
where
\begin{align*}
I_1(n,l)&=e^{-2\alpha l}\|\boldsymbol{v}_{n,l}\|^2_{\mathbb{H}},\\
I_2(n,l)&=-2\mu\int_{-l}^{0}e^{2\alpha\xi}\|\nabla\boldsymbol{v}_{\delta_n}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l})\|^2_{\mathbb{H}}\d\xi,\\
I_3(n,l)&=-2\beta\int_{-l}^{0}e^{2\alpha\xi}\|\boldsymbol{v}_{\delta_n}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l})+e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega)\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}\d\xi,\\
I_4(n,l)&= 2 \int_{-l}^{0} e^{2\alpha\xi}b(\boldsymbol{v}_{\delta_n}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l})+e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega),\nonumber\\&\qquad\qquad\qquad\qquad\boldsymbol{v}_{\delta_n}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l}),e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega))\d\xi,\\
I_5(n,l)&=2\beta \int_{-l}^{0} e^{2\alpha\xi}\left\langle\mathcal{C}(\boldsymbol{v}_{\delta_n}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l})+e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega)),e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega)\right\rangle\d\xi,\\
I_6(n,l)&=2 \int_{-l}^{0} e^{2\alpha\xi}\left\langle\boldsymbol{f}(\cdot,\xi+\mathfrak{s}),\boldsymbol{v}_{\delta_n}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l})\right\rangle\d\xi,
\end{align*}
and
\begin{align*}
I_7(n,l)&=2 \int_{-l}^{0} e^{2\alpha\xi} e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega)\big((\ell-\sigma-\alpha)\textbf{g}-\mu\mathrm{A}\textbf{g},\boldsymbol{v}_{\delta_n}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l})\big)\d\xi.
\end{align*}
Similar to \eqref{PC19}, by \eqref{CSCBF_Add} and \eqref{PC17}, it can also be obtained that
\begin{align}\label{PC19*}
\|\tilde{\boldsymbol{v}}\|^2_{\mathbb{H}}&=\|\boldsymbol{v}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{v}}_{l})\|^2_{\mathbb{H}} = e^{-2\alpha l}\|\tilde{\boldsymbol{v}}_{l}\|^2_{\mathbb{H}} -2\mu\int_{-l}^{0}e^{2\alpha \xi}\|\nabla\boldsymbol{v}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{v}}_{l})\|^2_{\mathbb{H}}\d\xi\nonumber\\&\quad-2\beta\int_{-l}^{0}e^{2\alpha \xi}\|\boldsymbol{v}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\tilde{\boldsymbol{v}}_{l})+e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega)\|^{r+1}_{\widetilde \mathbb{L}^{r+1}}\d\xi\nonumber\\&\quad+2 \int_{-l}^{0} e^{2\alpha\xi}b(\boldsymbol{v}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\tilde{\boldsymbol{v}}_l)+e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega),\nonumber\\&\qquad\qquad\qquad\qquad\boldsymbol{v}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\tilde{\boldsymbol{v}}_l),e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega))\d\xi\nonumber\\&\quad+2\beta \int_{-l}^{0} e^{2\alpha\xi}\left\langle\mathcal{C}(\boldsymbol{v}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\tilde{\boldsymbol{v}}_l)+e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega)),e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega)\right\rangle\d\xi\nonumber\\&\quad+2 \int_{-l}^{0} e^{2\alpha\xi}\left\langle\boldsymbol{f}(\cdot,\xi+\mathfrak{s}),\boldsymbol{v}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\tilde{\boldsymbol{v}}_l)\right\rangle\d\xi\nonumber\\&\quad+2 \int_{-l}^{0} e^{2\alpha\xi} e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\big((\ell-\sigma-\alpha)\textbf{g}-\mu\mathrm{A}\textbf{g},\boldsymbol{v}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\tilde{\boldsymbol{v}}_l)\big)\d\xi.
\end{align}
From \eqref{PC1}, we have
\begin{align}\label{PC20}
\limsup_{n\to\infty}I_1(n,l)\leq 4e^{-2\alpha l}\left[\mathcal{R}^i_{0}(\mathfrak{s}-l,\vartheta_{-l}\omega)+e^{2\sigma(\mathfrak{s}-l)}\left|\textbf{g}(x)\right|^2(1+\left|\boldsymbol{y}(\vartheta_{-l}\omega)\right|^2)\right].
\end{align}
Similarly, from \eqref{PC14}, we obtain
\begin{align}\label{PC21}
\limsup_{n\to\infty}I_2(n,l)\leq-2\mu\int_{-l}^{0}e^{2\alpha\xi}\|\nabla\boldsymbol{v}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\tilde{\boldsymbol{v}}_l)\|^2_{\mathbb{H}}\d\xi,
\end{align}
and
\begin{align}\label{PC22}
\lim_{n\to\infty}I_6(n,l)=2 \int_{-l}^{0} e^{2\alpha\xi}\left\langle\boldsymbol{f}(\cdot,\xi+\mathfrak{s}),\boldsymbol{v}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\tilde{\boldsymbol{v}}_l)\right\rangle\d\xi.
\end{align}
By \eqref{z4} and \eqref{PC14}, we have
\begin{align}\label{PC23}
\lim_{n\to\infty}I_7(n,l)=2 \int_{-l}^{0} e^{2\alpha\xi} e^{\sigma (\xi+\mathfrak{s})}\boldsymbol{y}(\vartheta_{\xi}\omega)\big((\ell-\sigma-\alpha)\textbf{g}-\mu\mathrm{A}\textbf{g},\boldsymbol{v}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\tilde{\boldsymbol{v}}_l)\big)\d\xi.
\end{align}
From \eqref{PC15}, we infer that
\begin{align}\label{PC24}
\limsup_{n\to\infty}I_3(n,l)\leq-2\beta\int_{-l}^{0}e^{2\alpha\xi}\|\boldsymbol{v}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\tilde{\boldsymbol{v}}_l)+e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega)\|^{r+1}_{\widetilde\mathbb{L}^{r+1}}\d\xi.
\end{align}
Using \eqref{b0}, we obtain
\begin{align*}
I_4(n,l)&= 2 \int_{-l}^{0} e^{2\alpha\xi}b(\boldsymbol{v}_{\delta_n}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l})+e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega),\nonumber\\&\qquad\qquad\boldsymbol{v}_{\delta_n}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l})+e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega),e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega))\d\xi\nonumber\\&\quad+ 2 \int_{-l}^{0} e^{2\alpha\xi}\left(\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega)-\boldsymbol{y}(\vartheta_{\xi}\omega)\right)\cdot b(\boldsymbol{v}_{\delta_n}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l})+e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega),\nonumber\\&\qquad\qquad\qquad\boldsymbol{v}_{\delta_n}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l})+e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega),e^{\sigma (\xi+\mathfrak{s})}\textbf{g})\d\xi\nonumber\\&=:I_{8}(n,l)+I_9(n,l).
\end{align*}
Making use of \eqref{z4} and Lemma \ref{convergence_b}, we get that $\lim\limits_{n\to\infty}I_9(n,l)=0$, and hence, by Lemma \ref{convergence_b}, we obtain
\begin{align}\label{PC25}
\lim_{n\to\infty}I_4(n,l)&= 2 \int_{-l}^{0} e^{2\alpha\xi}b(\boldsymbol{v}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\tilde{\boldsymbol{v}}_l)+e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega),\nonumber\\&\qquad\qquad\qquad\qquad\boldsymbol{v}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\tilde{\boldsymbol{v}}_l),e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega))\d\xi.
\end{align}
Finally, we have
\begin{align*}
I_5(n,l)=2\beta \int_{-l}^{0} e^{2\alpha\xi}\left\langle\mathcal{C}(\boldsymbol{v}_{\delta_n}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l})+e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega)),e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega)\right\rangle\d\xi\nonumber\\ \quad+2\beta \int_{-l}^{0} e^{2\alpha\xi}\left(\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega)-\boldsymbol{y}(\vartheta_{\xi}\omega)\right)\big\langle\mathcal{C}(\boldsymbol{v}_{\delta_n}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l})+e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{z} _{\delta_n}(\vartheta_{\xi}\omega)),\\varepsilon^{\sigma (\xi+\mathfrak{s})}\textbf{g}\big\rangle\d\xi=:I_{10}(n,l)+I_{11}(n,l).\qquad\qquad\qquad\qquad
\end{align*}
By \eqref{z4} and Lemma \ref{convergence_c}, we get that $\lim\limits_{n\to\infty}I_{11}(n,l)=0$, and hence, by Lemma \ref{convergence_c}, we deduce
\begin{align}\label{PC26}
\lim_{n\to\infty}I_5(n,l)&=2\beta \int_{-l}^{0} e^{2\alpha\xi}\left\langle\mathcal{C}(\boldsymbol{v}(\xi+\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\tilde{\boldsymbol{v}}_l)+e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega)),e^{\sigma (\xi+\mathfrak{s})}\textbf{g}\boldsymbol{y}(\vartheta_{\xi}\omega)\right\rangle\d\xi.
\end{align}
Combining \eqref{PC19}-\eqref{PC26}, we obtain
\begin{align}\label{PC27}
&\limsup_{n\to\infty}\|\boldsymbol{v}_{\delta_n}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l})\|^2_{\mathbb{H}}\nonumber\\&\leq 4e^{-2\alpha l}\left[\mathcal{R}^i_{0}(\mathfrak{s}-l,\vartheta_{-l}\omega)+e^{2\sigma(\mathfrak{s}-l)}\left|\textbf{g}(x)\right|^2(1+\left|\boldsymbol{y}(\vartheta_{-l}\omega)\right|^2)\right]+ \|\tilde{\boldsymbol{v}}\|^2_{\mathbb{H}}- e^{-2\alpha l}\|\tilde{\boldsymbol{v}}_{l}\|^2_{\mathbb{H}}\nonumber\\&\leq 4e^{-2\alpha l}\left[\mathcal{R}^i_{0}(\mathfrak{s}-l,\vartheta_{-l}\omega)+e^{2\sigma(\mathfrak{s}-l)}\left|\textbf{g}(x)\right|^2(1+\left|\boldsymbol{y}(\vartheta_{-l}\omega)\right|^2)\right]+ \|\tilde{\boldsymbol{v}}\|^2_{\mathbb{H}}.
\end{align}
Passing $l\to\infty$, we get
\begin{align*}
\limsup_{n\to\infty}\|\boldsymbol{v}_{\delta_n}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}},\boldsymbol{v}_{n,l})\|^2_{\mathbb{H}}\leq \|\tilde{\boldsymbol{v}}\|^2_{\mathbb{H}},
\end{align*}
which together with \eqref{PC11} gives
\begin{align}\label{PC28}
\boldsymbol{v}_{\delta_n}(\mathfrak{s},\mathfrak{s}-l,\vartheta_{-\mathfrak{s}}\omega,\boldsymbol{v}_{n,l}) \to\hat{\boldsymbol{v}} \text{ in } \mathbb{H}.
\end{align}
From \eqref{z4}, \eqref{PC9}, \eqref{PC17} and \eqref{PC28}, we get
\begin{align*}
\boldsymbol{u}_n\to \tilde{\boldsymbol{u}} \text{ in } \mathbb{H},
\end{align*}
as required, and the proof is completed.
\end{proof}
The next Theorem demonstrates the upper semicontinuity of random $\mathfrak{D}$-pullback attractors as $\delta\to0$, using the abstract theory given in \cite{non-autoUpperWang} (see Theorem 3.2, \cite{non-autoUpperWang}).
\begin{theorem}\label{Main_T_add}
For $0<\delta\leq 1, d=2$ with $r>1, d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\mathrm{loc}}(\mathbb{R};\mathbb{V}')$ and \eqref{forcing7} is satisfied. Then for every $\omega\in \Omega$ and $\mathfrak{s}\in\mathbb{R}$,
\begin{align}\label{U-SC}
\lim_{\delta\to0}\emph{dist}_{\mathbb{H}}\left(\mathscr{A}_{\delta}(\mathfrak{s},\omega),\mathscr{A}_0(\mathfrak{s},\omega)\right)=0.
\end{align}
\end{theorem}
\begin{proof}
By Lemma \ref{LemmaUe_add1}, we have for every $\mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$,
\begin{align}\label{U-SC1}
\limsup_{\delta\to0}\|\mathcal{K}^i_{\delta}(\mathfrak{s},\omega)\|^2_{\mathbb{H}}\leq\limsup_{\delta\to0}\mathcal{R}^i_{\delta}(\mathfrak{s},\omega)=\mathcal{R}^i_0(\mathfrak{s},\omega).
\end{align}
Consider a sequence $\delta\to0$ and $\boldsymbol{u}_{n,\mathfrak{s}}\to\boldsymbol{u}_{\mathfrak{s}}$ in $\mathbb{H}$. By Lemma \ref{Solu_Conver} we get that for every $t\geq0, \mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$,
\begin{align}\label{U-SC2}
\Phi_{\delta}(t,\mathfrak{s},\omega,\boldsymbol{u}_{n,\mathfrak{s}}) \to \Phi_0(t,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}}) \ \text{ in } \ \mathbb{H}.
\end{align}
Hence, by \eqref{U-SC1}, \eqref{U-SC2} and Lemma \ref{precompact} together with Theorem 3.2 in \cite{non-autoUpperWang}, one can conclude the proof.
\end{proof}
\section{Convergence of attractors: Multiplicative white noise} \label{sec7}\setcounter{equation}{0}
In this section, we examine the approximations of solutions of the following stochastic CBF equations with multiplicative white noise,
\begin{equation}\label{SCBF_Multi}
\left\{
\begin{aligned}
\frac{\partial \boldsymbol{u}}{\partial t}+\mu\mathrm{A}\boldsymbol{u}+\mathrm{B}(\boldsymbol{u})+\alpha\boldsymbol{u}+\beta\mathcal{C}(\boldsymbol{u})&=\boldsymbol{f}+\boldsymbol{u}\circ\frac{\d \mathrm{W}}{\d t}, \ \ \ \text{ in } \mathbb{R}^n\times(\mathfrak{s},\infty), \\
\boldsymbol{u}(x,\mathfrak{s})&=\boldsymbol{u}_{\mathfrak{s}}(x), \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \mathbb{R}^n \text{ and }\mathfrak{s}\in\mathbb{R}.
\end{aligned}
\right.
\end{equation}
For $\delta>0$, consider the pathwise random equations:
\begin{equation}\label{WZ_SCBF_Multi}
\left\{
\begin{aligned}
\frac{\partial \boldsymbol{u}_{\delta}}{\partial t}+\mu\mathrm{A}\boldsymbol{u}_{\delta}+\mathrm{B}(\boldsymbol{u}_{\delta})+\alpha\boldsymbol{u}_{\delta}+\beta\mathcal{C}(\boldsymbol{u}_{\delta})&=\boldsymbol{f}+\mathcal{Z}_{\delta}(\vartheta_{t}\omega)\boldsymbol{u}_{\delta}, \ \ \text{ in } \mathbb{R}^n\times(\mathfrak{s},\infty), \\
\boldsymbol{u}_{\delta}(x,\mathfrak{s})&=\boldsymbol{u}_{\delta,\mathfrak{s}}(x),\ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \mathbb{R}^n \text{ and }\mathfrak{s}\in\mathbb{R}.
\end{aligned}
\right.
\end{equation}
The existence of a unique random $\mathfrak{D}$-pullback attractor for \eqref{SCBF_Multi} and \eqref{WZ_SCBF_Multi} is established in \cite{KM6} and Section \ref{sec3}, respectively. In this section, we prove the upper semicontinuity of attractors as $\delta\to0$. Let us denote by $\widehat{\Phi}_0$ and $\widehat{\Phi}_{\delta}$, the continuous cocycles for the systems \eqref{SCBF_Multi} and \eqref{WZ_SCBF_Multi}, respectively, and $\widehat{\mathscr{A}}_0$ and $\widehat{\mathscr{A}}_{\delta}$, random $\mathfrak{D}$-pullback attractors for the systems \eqref{SCBF_Multi} and \eqref{WZ_SCBF_Multi}, respectively. Define
\begin{align}
\boldsymbol{v}(t,\mathfrak{s},\omega,\boldsymbol{v}_{\mathfrak{s}})=e^{-\omega(t)}\boldsymbol{u}(t,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}}) \ \text{ and }\ \boldsymbol{v}_{\delta}(t,\mathfrak{s},\omega,\boldsymbol{v}_{\delta,\mathfrak{s}})=e^{-\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\boldsymbol{u}_{\delta}(t,\mathfrak{s},\omega,\boldsymbol{u}_{\delta,\mathfrak{s}}).
\end{align}
Then, from \eqref{SCBF_Multi} and \eqref{WZ_SCBF_Multi} (formally), we obtain
\begin{equation}\label{CSCBF_Multi}
\left\{
\begin{aligned}
\frac{\partial \boldsymbol{v}}{\partial t}+\mu\mathrm{A}\boldsymbol{v}+e^{\omega(t)}\mathrm{B}(\boldsymbol{v})+\alpha\boldsymbol{v}+\beta e^{(r-1)\omega(t)}\mathcal{C}(\boldsymbol{v})&=e^{-\omega(t)}\boldsymbol{f}, \ \ \ \ \ \text{ in } \mathbb{R}^d\times(\mathfrak{s},\infty), \\
\boldsymbol{v}(x,\mathfrak{s})=\boldsymbol{v}_{\mathfrak{s}}(x)&=e^{-\omega(\mathfrak{s})}\boldsymbol{u}_{\mathfrak{s}}(x), \ x\in \mathbb{R}^d \text{ and }\mathfrak{s}\in\mathbb{R}.
\end{aligned}
\right.
\end{equation}
\begin{equation}\label{WZ_CSCBF_Multi}
\left\{
\begin{aligned}
\frac{\partial \boldsymbol{v}_\delta}{\partial t}+\mu\mathrm{A}\boldsymbol{v}_\delta+e^{\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\mathrm{B}(\boldsymbol{v}_\delta)+\alpha\boldsymbol{v}_\delta&+\beta e^{(r-1)\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\mathcal{C}(\boldsymbol{v}_\delta)\\&=e^{-\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\boldsymbol{f}, \ \ \ \ \ \ \text{ in } \mathbb{R}^d\times(\mathfrak{s},\infty), \\
\boldsymbol{v}_\delta(x,\mathfrak{s})=\boldsymbol{v}_{\delta,\mathfrak{s}}(x)&=e^{-\int_{0}^{\mathfrak{s}}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\boldsymbol{u}_{\delta,\mathfrak{s}}(x), \ x\in \mathbb{R}^d \text{ and }\mathfrak{s}\in\mathbb{R}.
\end{aligned}
\right.
\end{equation}
For all $\mathfrak{s}\in\mathbb{R},$ $t>\mathfrak{s},$ and for every initial data in $\mathbb{H}$ and $\omega\in\Omega$, \eqref{CSCBF_Multi} and \eqref{WZ_CSCBF_Multi} have unique solution in $\mathrm{C}([\mathfrak{s},\mathfrak{s}+T];\mathbb{H})\cap\mathrm{L}^2(\mathfrak{s}, \mathfrak{s}+T;\mathbb{V})\cap\mathrm{L}^{r+1}(\mathfrak{s},\mathfrak{s}+T;\widetilde{\mathbb{L}}^{r+1}).$ Furthermore, the solution is continuous with respect to initial data and $(\mathscr{F},\mathscr{B}(\mathbb{H}))$-measurable in $\omega\in\Omega.$
\begin{lemma}\label{LemmaUe_Multi}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing2}. Then $\widehat{\Phi}_0$ possesses a closed measurable $\mathfrak{D}$-pullback absorbing set $\mathcal{K}_0=\{\mathcal{K}_0(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D}$ in $\mathbb{H}$ given by
\begin{align}\label{ue_Multi}
\mathcal{K}_0(\mathfrak{s},\omega)=\{\boldsymbol{u}\in\mathbb{H}:\|\boldsymbol{u}\|^2_{\mathbb{H}}\leq \mathcal{R}_0(\mathfrak{s},\omega)\},
\end{align}
where $\mathcal{R}_0(\mathfrak{s},\omega)$ is defined by
\begin{align}\label{ue_Multi1}
\mathcal{R}_0(\mathfrak{s},\omega)&=\frac{4}{\min\{\mu,\alpha\}} \int_{-\infty}^{0} e^{\alpha\xi-2\omega(\xi)} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d \xi.
\end{align}
\end{lemma}
\begin{proof}
Since the existence of $\mathfrak{D}$-pullback absorbing set for $\widehat{\Phi}_0$ is proved in \cite{KM6} (see Lemma 5.6, \cite{KM6}), the rest of the proof follows in a similar way and hence we omit it here.
\end{proof}
\begin{lemma}\label{LemmaUe_Multi1}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$ satisfies \eqref{forcing2}. Then $\widehat{\Phi}_\delta$ possesses a closed measurable $\mathfrak{D}$-pullback absorbing set $\mathcal{K}_\delta=\{\mathcal{K}_\delta(\mathfrak{s},\omega):\mathfrak{s}\in\mathbb{R}, \omega\in\Omega\}\in\mathfrak{D}$ in $\mathbb{H}$ given by
\begin{align}\label{ue_Multi4}
\mathcal{K}_\delta(\mathfrak{s},\omega)=\{\boldsymbol{u}\in\mathbb{H}:\|\boldsymbol{u}\|^2_{\mathbb{H}}\leq \mathcal{R}_\delta(\mathfrak{s},\omega)\},
\end{align}
where $\mathcal{R}_\delta(\mathfrak{s},\omega)$ is defined by
\begin{align}\label{ue_Multi5}
\mathcal{R}_\delta(\mathfrak{s},\omega)=\frac{4}{\min\{\mu,\alpha\}} \int_{-\infty}^{0} e^{\int_{0}^{\xi}\left(\alpha-2\mathcal{Z}_{\delta}(\vartheta_{\zeta}\omega)\right)\d\zeta} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d \xi.
\end{align}
Furthermore, for every $\mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$,
\begin{align}\label{ue_Multi8}
\lim_{\delta\to0}\mathcal{R}_\delta(\mathfrak{s},\omega)=\frac{4}{\min\{\mu,\alpha\}} \int_{-\infty}^{0} e^{\alpha\xi-2\omega(\xi)} \|\boldsymbol{f}(\cdot,\xi+\mathfrak{s})\|^2_{\mathbb{V}'}\d \xi.
\end{align}
\end{lemma}
\begin{proof}
Since the existence of $\mathfrak{D}$-pullback absorbing set $\mathcal{K}_\delta(\mathfrak{s},\omega)$ for $\widehat{\Phi}_\delta$ is proved in Lemma \ref{PAS} and the convergence in \eqref{ue_Multi8} is proved in \cite{GLW} (see Lemma 3.7 in \cite{GLW}), hence we omit the proof here.
\end{proof}
\begin{lemma}\label{Solu_Conver2}
For $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in \mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{V}')$. Suppose that $\{\delta_n\}_{n\in\mathbb{N}}$ is a sequence such that $\delta_n\to0$. Let $\boldsymbol{u}_{\delta_n}$ and $\boldsymbol{u}$ be the solutions of \eqref{WZ_SCBF_Multi} and \eqref{SCBF_Multi} with initial data $\boldsymbol{u}_{\delta_n,\mathfrak{s}}$ and $\boldsymbol{u}_{\mathfrak{s}}$, respectively. If $\|\boldsymbol{u}_{\delta_n,\mathfrak{s}}-\boldsymbol{u}_{\mathfrak{s}}\|_{\mathbb{H}}\to0$ as $n\to\infty$, then for every $\mathfrak{s}\in\mathbb{R}$, $\omega\in\Omega$ and $t>\mathfrak{s}$,
\begin{align*}
\|\boldsymbol{u}_{\delta_n}(t,\mathfrak{s},\omega,\boldsymbol{u}_{\delta_n,\mathfrak{s}})-\boldsymbol{u}(t,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}})\|_{\mathbb{H}} \to0 \ \text{ as }\ n\to\infty.
\end{align*}
\end{lemma}
\begin{proof}
Let $\Upsilon=\boldsymbol{v}_{\delta_n}-\boldsymbol{v}$ and $\mathfrak{Z}(t)=\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi-\omega(t)$. Then from \eqref{WZ_CSCBF_Multi} and \eqref{CSCBF_Multi}, we find
\begin{align}\label{P1}
\frac{\d\Upsilon}{\d t}&=-\mu \mathrm{A}\Upsilon-\alpha\Upsilon-e^{\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\mathrm{B}\big(\boldsymbol{v}_{\delta_n}\big)+e^{\omega(t)}\mathrm{B}\big(\boldsymbol{v}\big)-\beta e^{(r-1)\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi} \mathcal{C}\big(\boldsymbol{v}_{\delta_n}\big)\nonumber\\&\quad+e^{(r-1)\omega(t)}\beta\mathcal{C}\big(\boldsymbol{v}\big) +(e^{-\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}-e^{-\omega(t)})\boldsymbol{f},
\end{align}
which gives
\begin{align}\label{P3}
&\frac{1}{2}\frac{\d}{\d t}\|\Upsilon\|^2_{\mathbb{H}}\nonumber\\&=-\mu\|\nabla\Upsilon\|^2_{\mathbb{H}}-\alpha\|\Upsilon\|^2_{\mathbb{H}}-e^{\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}b(\boldsymbol{v}_{\delta_n},\boldsymbol{v}_{\delta_n},\Upsilon) +e^{\omega(t)}b(\boldsymbol{v},\boldsymbol{v},\Upsilon) \nonumber\\&\quad-e^{(r-1)\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\left\langle\mathcal{C}(\boldsymbol{v}_{\delta_n}),\Upsilon\right\rangle+e^{(r-1)\omega(t)}\left\langle\mathcal{C}(\boldsymbol{v}),\Upsilon\right\rangle+(e^{-\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}-e^{-\omega(t)})\left\langle\boldsymbol{f},\Upsilon\right\rangle\nonumber\\&=-\mu\|\nabla\Upsilon\|^2_{\mathbb{H}}-\alpha\|\Upsilon\|^2_{\mathbb{H}}-e^{\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}b(\Upsilon,\boldsymbol{v},\Upsilon) -e^{\omega(t)}(e^{\mathfrak{Z}(t)}-1)b(\boldsymbol{v},\boldsymbol{v},\Upsilon) \nonumber\\&\quad-e^{(r-1)\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\left\langle\mathcal{C}(\boldsymbol{v}_{\delta_n})-\mathcal{C}(\boldsymbol{v}),\boldsymbol{v}_{\delta_n}-\boldsymbol{v}\right\rangle-e^{(r-1)\omega(t)}(e^{(r-1)\mathfrak{Z}(t)}-1)\left\langle\mathcal{C}(\boldsymbol{v}),\Upsilon\right\rangle\nonumber\\&\quad+e^{-\omega(t)}(e^{-\mathfrak{Z}(t)}-1)\left\langle\boldsymbol{f},\Upsilon\right\rangle.
\end{align}
From \eqref{MO_c}, we obtain
\begin{align}\label{P8}
&-e^{(r-1)\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\left\langle\mathcal{C}(\boldsymbol{v}_{\delta_n})-\mathcal{C}(\boldsymbol{v}),\boldsymbol{v}_{\delta_n}-\boldsymbol{v}\right\rangle\nonumber\\ &\leq-\frac{\beta}{2}e^{(r-1)\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\||\Upsilon||\boldsymbol{v}_{\delta_n}|^{\frac{r-1}{2}}\|^2_{\mathbb{H}}-\frac{\beta}{2}e^{(r-1)\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\||\Upsilon||\boldsymbol{v}|^{\frac{r-1}{2}}\|^2_{\mathbb{H}}\leq0.
\end{align}
\vskip 2mm
\noindent
\textbf{Case I:} \textit{$d=2$ and $r\geq1$.} Applying \eqref{b1}, Lemmas \ref{Holder} and \ref{Young}, we get
\begin{align}
\left|e^{\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}b(\Upsilon,\boldsymbol{v},\Upsilon)\right|&\leq Ce^{\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\|\Upsilon\|_{\mathbb{H}}\|\nabla\Upsilon\|_{\mathbb{H}}\|\nabla\boldsymbol{v}\|_{\mathbb{H}}\nonumber\\&\leq Ce^{2\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}\|\Upsilon\|^2_{\mathbb{H}}+\frac{\mu}{4}\|\nabla\Upsilon\|^2_{\mathbb{H}},\label{P4}\\
\left|e^{\omega(t)}(e^{\mathfrak{Z}(t)}-1)b(\boldsymbol{v},\boldsymbol{v},\Upsilon)\right|&\leq Ce^{\omega(t)}\left|e^{\mathfrak{Z}(t)}-1\right|\|\boldsymbol{v}\|^{1/2}_{\mathbb{H}}\|\nabla\boldsymbol{v}\|^{3/2}_{\mathbb{H}}\|\Upsilon\|^{1/2}_{\mathbb{H}}\|\nabla\Upsilon\|^{1/2}_{\mathbb{H}}\nonumber\\&\leq C\|\Upsilon\|^2_{\mathbb{H}}\|\nabla\Upsilon\|^2_{\mathbb{H}} + Ce^{\frac{4}{3}\omega(t)}\left|e^{\mathfrak{Z}(t)}-1\right|^{4/3} \|\boldsymbol{v}\|^{2/3}_{\mathbb{H}}\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}\nonumber\\&\leq C\|\nabla\boldsymbol{v}_{\delta_n}\|^2_{\mathbb{H}}\|\Upsilon\|^2_{\mathbb{H}}+C\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}\|\Upsilon\|^2_{\mathbb{H}} \nonumber\\&\quad+ Ce^{\frac{4}{3}\omega(t)}|e^{\mathfrak{Z}(t)}-1|^{4/3} \|\boldsymbol{v}\|^{2/3}_{\mathbb{H}}\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}},\label{P5}\\
\left|e^{(r-1)\omega(t)}(e^{(r-1)\mathfrak{Z}(t)}-1)\left\langle\mathcal{C}(\boldsymbol{v}),\Upsilon\right\rangle\right|&\leq e^{(r-1)\omega(t)}\left|e^{(r-1)\mathfrak{Z}(t)}-1\right|\|\boldsymbol{v}\|^r_{\widetilde{\mathbb{L}}^{r+1}}\|\Upsilon\|_{\widetilde{\mathbb{L}}^{r+1}}\nonumber\\&\leq C e^{(r-1)\omega(t)}\left|e^{(r-1)\mathfrak{Z}(t)}-1\right|\left[\|\boldsymbol{v}\|^{r+1}_{\widetilde{\mathbb{L}}^{r+1}}+\|\Upsilon\|^{r+1}_{\widetilde{\mathbb{L}}^{r+1}}\right],\label{P6}\\
e^{-\omega(t)}(e^{-\mathfrak{Z}(t)}-1)\left\langle\boldsymbol{f},\Upsilon\right\rangle&\leq Ce^{-2\omega(t)}\left|e^{-\mathfrak{Z}(t)}-1\right|^2\|\boldsymbol{f}\|^2_{\mathbb{V}'}+\frac{\min\{\mu,\alpha\}}{4}\|\Upsilon\|^2_{\mathbb{V}}.\label{P7}
\end{align}
Combining \eqref{P3}-\eqref{P7}, we get
\begin{align}\label{P9}
\frac{\d}{\d t}\|\Upsilon(t)\|^2_{\mathbb{H}}&\leq P_1(t)\|\Upsilon(t)\|^2_{\mathbb{H}} + P_2(t),
\end{align}
for a.e. $t\in[\mathfrak{s},\mathfrak{s}+T]$, where
\begin{align*}
P_1(t)&= C\left[(e^{2\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}+1)\|\nabla\boldsymbol{v}(t)\|^2_{\mathbb{H}}+\|\nabla\boldsymbol{v}_{\delta_n}(t)\|^2_{\mathbb{H}}\right],\\
P_2(t)&=Ce^{(r-1)\omega(t)}|e^{(r-1)\mathfrak{Z}(t)}-1|\left[\|\boldsymbol{v}(t)\|^{r+1}_{\widetilde{\mathbb{L}}^{r+1}}+\|\boldsymbol{v}_{\delta_n}(t)\|^{r+1}_{\widetilde{\mathbb{L}}^{r+1}}\right]\nonumber\\&\quad+Ce^{\frac{4}{3}\omega(t)}|e^{\mathfrak{Z}(t)}-1|^{4/3} \|\boldsymbol{v}(t)\|^{2/3}_{\mathbb{H}}\|\nabla\boldsymbol{v}(t)\|^2_{\mathbb{H}}+Ce^{-2\omega(t)}|e^{-\mathfrak{Z}(t)}-1|^2\|\boldsymbol{f}(t)\|^2_{\mathbb{V}'}.
\end{align*}
\vskip 2mm
\noindent
\textbf{Case II:} \textit{$d= 3$ and $r\geq3$ ($r>3$ with any $\beta,\mu>0$ and $r=3$ with $2\beta\mu\geq1$).} An application of Lemmas \ref{Holder} and \ref{Young} yields
\begin{align}\label{P11}
\left|e^{\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}b(\Upsilon,\boldsymbol{v},\Upsilon)\right|\leq\begin{cases}
\frac{1}{2\beta}\|\nabla\Upsilon\|_{\mathbb{H}}^2+\frac{\beta}{2}e^{2\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\||\Upsilon||\boldsymbol{v}|\|^2_{\mathbb{H}}, \text{ for } r=3,\\
\frac{\mu}{4}\|\nabla\Upsilon\|_{\mathbb{H}}^2+\frac{\beta}{4}e^{(r-1)\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\||\Upsilon||\boldsymbol{v}|^{\frac{r-1}{2}}\|^2_{\mathbb{H}}+C\|\Upsilon\|^2_{\mathbb{H}}, \text{ for } r>3,
\end{cases}
\end{align}
and
\begin{align}\label{P12}
&\left|e^{\omega(t)}(e^{\mathfrak{Z}(t)}-1)b(\boldsymbol{v},\boldsymbol{v},\Upsilon)\right|\nonumber\\&\leq\left|1-e^{-\mathfrak{Z}(t)}\right|e^{\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\|\nabla\boldsymbol{v}\|_{\mathbb{H}}\||\Upsilon||\boldsymbol{v}|\|_{\mathbb{H}}\nonumber\\&\leq\begin{cases}
\frac{1}{2\beta}\left|1-e^{-\mathfrak{Z}(t)}\right|^2\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}+\frac{\beta}{2}e^{2\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\||\Upsilon||\boldsymbol{v}|\|^2_{\mathbb{H}}, \text{ for } r=3, \\ \frac{1}{2\beta}\left|1-e^{-\mathfrak{Z}(t)}\right|^2\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}+\frac{\beta}{4}e^{(r-1)\int_{0}^{t}\mathcal{Z}_{\delta}(\vartheta_{\xi}\omega)\d\xi}\||\Upsilon||\boldsymbol{v}|^{\frac{r-1}{2}}\|^2_{\mathbb{H}}+C\|\Upsilon\|^2_{\mathbb{H}}, \text{ for } r>3.
\end{cases}
\end{align}
Combining \eqref{P3}-\eqref{P8}, \eqref{P6}-\eqref{P7} and \eqref{P11}-\eqref{P12}, we obtain
\begin{align}\label{P14}
\frac{\d}{\d t}\|\Upsilon(t)\|^2_{\mathbb{H}}&\leq C\|\Upsilon(t)\|^2_{\mathbb{H}} + P(t),
\end{align}
for a.e. $t\in[\mathfrak{s},\mathfrak{s}+T]$, where
\begin{align*}
P(t)&=Ce^{(r-1)\omega(t)}|e^{(r-1)\mathfrak{Z}(t)}-1|\left[\|\boldsymbol{v}(t)\|^{r+1}_{\widetilde{\mathbb{L}}^{r+1}}+\|\boldsymbol{v}_{\delta_n}(t)\|^{r+1}_{\widetilde{\mathbb{L}}^{r+1}}\right]\nonumber\\&\quad+\frac{1}{\beta}\left|1-e^{-\mathfrak{Z}(t)}\right|^2\|\nabla\boldsymbol{v}\|^2_{\mathbb{H}}+Ce^{-2\omega(t)}|e^{-\mathfrak{Z}(t)}-1|^2\|\boldsymbol{f}(t)\|^2_{\mathbb{V}'}.
\end{align*}
Now, applying Gronwall's inequality to \eqref{P9} and \eqref{P14}, and calculating similarly as in Lemma \ref{Solu_Conver}, one can conclude the proof with the help of the convergence \eqref{N5*}.
\end{proof}
The following result shows the uniform compactness of family of random attractors $\widehat{\mathscr{A}}_{\delta}$, which can be obtained by similar calculations of Lemma \ref{precompact}. In fact, the proof is much easier here and hence we omit here.
\begin{lemma}\label{precompact1}
For $d=2$ with $r\geq1, d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\mathrm{loc}}(\mathbb{R};\mathbb{V}')$ and satisfies \eqref{forcing2}. Let $\mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$ be fixed. If $\delta_n\to0$ as $n\to\infty$ and $\boldsymbol{u}_n\in\widehat{\mathscr{A}}_{\delta_n}(\mathfrak{s},\omega)$, then the sequence $\{\boldsymbol{u}_n\}_{n\in\mathbb{N}}$ has a convergent subsequence in $\mathbb{H}$.
\end{lemma}
Finally, we are in the position of giving main result of this section.
\begin{theorem}\label{Main_T_Multi}
For $0<\delta\leq 1, d=2$ with $r\geq1, d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$, assume that $\boldsymbol{f}\in\mathrm{L}^2_{\mathrm{loc}}(\mathbb{R};\mathbb{V}')$ and \eqref{forcing2} is satisfied. Then for every $\omega\in \Omega$ and $\mathfrak{s}\in\mathbb{R}$,
\begin{align}\label{U-SC_m}
\lim_{\delta\to0}\emph{dist}_{\mathbb{H}}\left(\widehat{\mathscr{A}}_{\delta}(\mathfrak{s},\omega),\widehat{\mathscr{A}}_0(\mathfrak{s},\omega)\right)=0.
\end{align}
\end{theorem}
\begin{proof}
By Lemma \ref{LemmaUe_Multi1}, we have for every $\mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$,
\begin{align}\label{U-SC1_m}
\limsup_{\delta\to0}\|\mathcal{K}_{\delta}(\mathfrak{s},\omega)\|^2_{\mathbb{H}}\leq\limsup_{\delta\to0}\mathcal{R}_{\delta}(\mathfrak{s},\omega)=\mathcal{R}_0(\mathfrak{s},\omega).
\end{align}
Consider a sequence $\delta\to0$ and $\boldsymbol{u}_{n,\mathfrak{s}}\to\boldsymbol{u}_{\mathfrak{s}}$ in $\mathbb{H}$. By Lemma \ref{Solu_Conver2}, we get that for every $t\geq0, \mathfrak{s}\in\mathbb{R}$ and $\omega\in\Omega$,
\begin{align}\label{U-SC2_m}
\widehat{\Phi}_{\delta}(t,\mathfrak{s},\omega,\boldsymbol{u}_{n,\mathfrak{s}}) \to \widehat{\Phi}_0(t,\mathfrak{s},\omega,\boldsymbol{u}_{\mathfrak{s}}) \ \text{ in } \ \mathbb{H}.
\end{align}
Hence, by \eqref{U-SC1_m}, \eqref{U-SC2_m} and Lemma \ref{precompact1} together with Theorem 3.2 in \cite{non-autoUpperWang}, we conclude the proof.
\end{proof}
\medskip\noindent
{\bf Acknowledgments:} The first author would like to thank the Council of Scientific $\&$ Industrial Research (CSIR), India for financial assistance (File No. 09/143(0938)/2019-EMR-I). M. T. Mohan would like to thank the Department of Science and Technology (DST), Govt of India for Innovation in Science Pursuit for Inspired Research (INSPIRE) Faculty Award (IFA17-MA110).
\begin{appendix}
\renewcommand{\thesection}{\Alph{section}}
\numberwithin{equation}{section}
\section{Random pullback attractors for Wong-Zakai approximations of 2D stochastic NSE on unbounded Poincar\'e domains} \label{sec5}\setcounter{equation}{0}
In this appendix, we discuss the existence and uniqueness of random $\mathfrak{D}$-pullback attractors for Wong-Zakai approximations of 2D stochastic NSE on Poincar\'e domains with nonlinear diffusion term. Let $\widetilde{\mathcal{O}}\subset \mathbb{R}^2$, which is open and connected. We also assume that, there exists a positive constant $\lambda_1 $ such that the following Poincar\'e inequality is satisfied:
\begin{align}\label{2.1}
\lambda_1\int_{\widetilde{\mathcal{O}}} |\phi(x)|^2 \d x \leq \int_{\widetilde{\mathcal{O}}} |\nabla \phi(x)|^2 \d x, \ \text{ for all } \ \phi \in \mathbb{H}^{1}_0 (\widetilde{\mathcal{O}}),
\end{align}
that is, $\widetilde{\mathcal{O}}$ is a Poincar\'e domain (may be bounded or unbounded). Consider the Wong-Zakai approximations of 2D NSE on $\widetilde{\mathcal{O}}$ as
\begin{equation}\label{WZ_NSE}
\left\{
\begin{aligned}
\frac{\partial \boldsymbol{u}}{\partial t}-\nu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\nabla \boldsymbol{p}&=\boldsymbol{f}(t) + S(t,x,\boldsymbol{u})\mathcal{Z}_{\delta}(\vartheta_t\omega), \ \text{ in } \ \widetilde{\mathcal{O}}\times(\mathfrak{s},\infty), \\ \nabla\cdot\boldsymbol{u}&=0, \ \text{ in } \ \widetilde{\mathcal{O}}\times(\mathfrak{s},\infty), \\
\boldsymbol{u}&=\mathbf{0}\ \ \text{ on } \ \partial\widetilde{\mathcal{O}}\times(\mathfrak{s},\infty), \\
\boldsymbol{u}(x,\mathfrak{s})&=\boldsymbol{u}_{\mathfrak{s}}(x), \ \ \ x\in \widetilde{\mathcal{O}} \text{ and }\mathfrak{s}\in\mathbb{R},
\end{aligned}
\right.
\end{equation}
where $\nu$ is the coefficient of kinematic viscosity of the fluid. In the work \cite{GGW}, authors proved the existence of unique random $\mathfrak{D}$-pullback random attractor for the system \eqref{WZ_NSE} under the Assumption \ref{NDT2}. Here, we assume that the following conditions are satisfied:
\begin{assumption}\label{NDT4}
Let $S:\mathbb{R}\times\widetilde{\mathcal{O}}\times\mathbb{R}^2\to\mathbb{R}^2$ be a continuous function such that for all $t\in\mathbb{R}$, $x\in\widetilde{\mathcal{O}}$ and $\textbf{u}\in\widetilde{\mathcal{O}}$
\begin{align*}
|S(t,x,\textbf{u})|&\leq \mathcal{S}_1(t,x)|\textbf{u}|^{q-1}+\mathcal{S}_2(t,x),
\end{align*}
where $1\leq q<2$, $\mathcal{S}_1\in\mathrm{L}^{\infty}_{\emph{loc}}(\mathbb{R},\mathbb{L}^{\frac{2}{2-q}}(\widetilde{\mathcal{O}}))$ and $\mathcal{S}_2\in\mathrm{L}^{\infty}_{\emph{loc}}(\mathbb{R},\mathbb{L}^{2}(\widetilde{\mathcal{O}}))$. Furthermore, suppose that $S(t,x,\textbf{u})$ is locally Lipschitz continuous with respect to $\boldsymbol{u}$.
\end{assumption}
\begin{remark}
Note that Assumption \ref{NDT4} is clearly different from Assumption \ref{NDT2}. Therefore, it is worth to prove the results under Assumption \ref{NDT4}.
\end{remark}
\begin{assumption}\label{DNFT4}
We assume that the external forcing term $\boldsymbol{f}\in\mathrm{L}^2_{\mathrm{loc}}(\mathbb{R},\mathbb{L}^2(\widetilde{\mathcal{O}}))$ satisfies
\begin{itemize}
\item [(i)]
\begin{align*}
\int_{-\infty}^{\mathfrak{s}} e^{\nu\lambda_1\xi}\|\boldsymbol{f}(\cdot,\xi)\|^2_{\mathbb{L}^2(\widetilde{\mathcal{O}})}\d \xi<\infty, \ \ \text{ for all } \mathfrak{s}\in\mathbb{R}.
\end{align*}
\item [(ii)] for every $c>0$
\begin{align*}
\lim_{\tau\to-\infty}e^{c\tau}\int_{-\infty}^{0} e^{\nu\lambda_1\xi}\|\boldsymbol{f}(\cdot,\xi+\tau)\|^2_{\mathbb{L}^2(\widetilde{\mathcal{O}})}\d \xi=0.
\end{align*}
\end{itemize}
\end{assumption}
Since, 2D CBF equations with $r=1$ are the linear perturbation of 2D NSE, one can prove the next theorem using the same arguments as it is done for 2D CBF equations in subsection \ref{subsec4.3}. Moreover, there are only minor changes, hence we omit the proof here.
\begin{theorem}\label{WZ_RA_UB_GS_NSE}
Assume that $\boldsymbol{f}\in\mathrm{L}^2_{\emph{loc}}(\mathbb{R};\mathbb{L}^2(\widetilde{\mathcal{O}}))$ satisfies Assumption \ref{DNFT4} and Assumption \ref{NDT4} is fulfilled. Then there exists a unique random $\mathfrak{D}$-pullback attractor for the system \eqref{WZ_NSE}, in $\mathbb{L}^2(\widetilde{\mathcal{O}})$.
\end{theorem}
\end{appendix}
|
2,869,038,156,384 | arxiv | \section{Introduction}
This report focuses on Producer-Consumer coordination, and in
particular on Single-Producer/Single-Consumer (SPSC) coordination.
The producer and the consumer are concurrent entities, i.e. processes or threads.
The first one produces items placing them in a shared structure, whereas the
the second one consumes these items by removing them from the shared structure.
Different kinds of shared data structures provide different fairness guarantees.
Here, we consider a queue data structure that provides First-In-First-Out
fairness (FIFO queue), and we assume that Producer and Consumer share a common
address space, that is, we assume threads as concurrent entities.
In the end of '70s, Leslie Lamport proved that, under Sequential Consistency
memory model \cite{LamportSC}, a Single-Producer/Single-Consumer circular buffer \footnote{A circular buffer can be used to implement a FIFO queue} can be
implemented without using explicit synchronization mechanisms between the
producer and the consumer \cite{Lamport77}. Lamport's circular buffer is a
wait-free algorithm. A wait-free algorithm is guaranteed to complete after a
finite number of steps, regardless of the timing behavior of other operations.
Differently, a lock-free algorithm guarantees only that after a finite
number of steps, \texttt{some} operation completes. Wait-freedom is a stronger
condition than lock-freedom and both conditions are strong enough to preclude
the use of blocking constructs such as locks.
With minimal modification to Lamport's wait-free
SPSC algorithm, it results correct also under
Total-Store-Order and others weaker consistency models, but it fails under weakly
ordered memory model such as those used in IBM's Power and Intel's Itanium
architectures. On such systems, expensive memory barrier (also known as memory fence)
instructions are needed in order to ensure correct load/store instructions ordering.
Maurice Herlihy in his seminal paper \cite{Herlihy91} formally proves that few
simple HW atomic instructions are enough for building any wait-free
data structure for any number of concurrent entities. The simplest and widely
used primitive is the compare-and-swap (CAS).
Over the years, many works have been proposed with focus on lock-free/wait-free
Multiple-Producer/Multiple-Consumer (MPMC) queue \cite{MNSS,mpmc1,ABA:98}.
They use CAS-like primitives in order to guarantee correct implementation.
Unfortunately, the CAS-like hardware primitives used in the implementations,
introduce non-negligible overhead on modern shared-cache architectures,
so even the best MPMC queue implementation, is not able to obtain better
performance than Lamport's circular buffer in cases with just 1 producer and
1 consumer.
FIFO queues are typically used to implement streaming networks \cite{fastflow:web, streamIt}. Streams are directional
channels of communication that behave as a FIFO queue. In many cases streams are
implemented using circular buffer instead of a pointer-based dynamic queue in order to avoid excessive memory usage. Hoverer, when
complex streaming networks have to be implemented, which have multiple nested cycles,
the use of bounded-size queues as basic data structure requires more complex and
costly communication prtocols in order to avoid deadlock situations.
Unbounded size queue are particularly interesting in these complex cases,
and in all the cases where it is extremely difficult to choose a suitable
queue size. As we shall see, it is possible to implement a wait-free unbounded
SPSC queue by using Lamport's algorithm and dynamic memory allocation.
Unfortunately, dynamic memory allocation/deallocation is costly because they use locks
to protect internal data structures, hence introduces costly memory barriers.
In this report it is presented an efficient implementation of an unbounded wait-free
SPSC FIFO queue which makes use only of a modified version of the Lamport's circular
buffer without requiring any additional memory barrier, and, at the same time,
minimizes the use of dynamic memory allocation.
The novel unbounded queue implementation presented here, is able to speed up producer-consumer
coordination, and, in turn, provides the basic mechanisms for implementing complex streaming
networks of cooperating entities.
The remainder of this paper is organized as follows. Section \ref{sec:SWSR}
reminds Lamport's algorithm and also shows the necessary modifications to
make it work efficiently on modern shared-cache multiprocessors.
Section \ref{sec:basicUnbounded} discuss the extension of the Lamport's algorithm
to the unbounded case. Section \ref{sec:uSWSR} presents the new implementations
with a proof of correctness. Section \ref{sec:exp} presents some
performance results, and Sec. \ref{sec:conclusions} concludes.
\section{Lamport's circular buffer}
\label{sec:SWSR}
\begin{figure}
\begin{Bench2}{}{}
bool push(data) {
if ((tail+1 mod N)==head )
return false; // buffer full
buffer[tail]=data;
tail= tail+1 mod N;
return true;
}
bool pop(data) {
if (head==tail)
return false; // buffer empty
data = buffer[head];
head = head+1 mod N;
return true;
}
\end{Bench2}
\caption{Lamport's circular buffer \texttt{push} and \texttt{pop} methods pseudo-code.
At the beginning \texttt{head}=\texttt{tail}=0. \label{fig:Lamport}}
\hspace{0.5ex}
\end{figure}
\begin{figure}
\begin{Bench2}{}{}
bool push(data) {
if (buffer[tail]==BOTTOM) {
buffer[tail]=data;
tail = tail+1 mod N;
return true;
}
return false; // buffer full
}
bool pop(data) {
if (buffer[head]!=BOTTOM) {
data = buffer[head];
buffer[head] = BOTTOM;
head = head+1 mod N;
return true;
}
return false; // buffer empty
}
\end{Bench2}
\caption{$P_1C_1$-buffer buffer pseudocode. Modified version of the code presented in \cite{HK97}. The buffer is initialized to BOTTOM and \texttt{head}=\texttt{tail}=0 at the beginning.\label{fig:P1C1buffer}}
\end{figure}
In Fig. \ref{fig:Lamport} the pseudocode of the
\texttt{push} and \texttt{pop} methods
of the Lamport's circular buffer algorithm, is sketched. The \texttt{buffer} is implemented
as an array of \texttt{N} entries.
Lamport proved that, under Sequential Consistency \cite{LamportSC}, no locks are
needed around \texttt{pop} and \texttt{push} methods, thus resulting
in a concurrent wait-free queue implementation.
If Sequential Consistency requirement is released, it is easy to see that Lamport's
algorithm fails. This happens for example with the
PowerPC architecture where write to write relaxation is allowed ($W \rightarrow W$ using the
same notation used in \cite{Adve95sharedmemory}),
i.e. 2 distinct writes at different memory locations may be executed
not in program order. In fact, the consumer may pop out of the
buffer a value before the data is effectively written in it, this is because
the update of the \texttt{tail} pointer (modified only by the producer) can be seen
by the consumer before the producer writes in the \texttt{tail} position of the buffer.
In this case, the test at line \lines{\ref{fig:Lamport}}{9} would be passed even though
\texttt{buffer[head]} contains stale data.
Few simple modifications to the basic Lamport's algorithm, allow the correct
execution even under weakly ordered memory consistency model.
To the best of our knowledge such modifications have been presented and
formally proved correct for the first time by Higham and Kavalsh in \cite{HK97}.
The idea mainly consists in tightly coupling control and data information into a
single buffer operation by using a know value (called BOTTOM), which cannot be used by
the application. The BOTTOM value is used to indicate whether there is an empty buffer slot,
which in turn indicates an available room in the buffer to the producer and the empty
buffer condition to the consumer.
With the circular buffer implementation sketched in Fig. \ref{fig:P1C1buffer}, the consistency
problem described for the Lamport's algorithm cannot occur provided that the generic store
\texttt{buffer[i]=data} is seen in its entirety by a processor, or not at all, i.e.
a single memory store operation is executed atomically. To the best of our knowledge,
this condition is satisfied in any modern general-purpose processor for aligned memory word stores.
As shown by Giacomoni et all. in \cite{fastforward:ppopp:08}, Lamport's circular buffer algorithm
results in cache line thrashing on shared-cache multiprocessors, as the \texttt{head} and
\texttt{tail} buffer pointers are shared between consumer and producer.
Modifications of pointers, at lines \lines{\ref{fig:Lamport}}{5} and \lines{\ref{fig:Lamport}}{12}, turn out
in cache-line invalidation (or update) traffic among processors, thus introducing unexpected overhead.
With the implementation in Fig. \ref{fig:P1C1buffer}, the head and the \texttt{tail} buffer pointers are
always in the local cache of the consumer and the producer respectively, without incurring in
cache-coherence overhead since they are not shared.
When transferring references through the buffer rather than plain data values, a memory fence is required
on processors with weakly memory consistency model, in which stores can be executed out of
program order. In fact, without a memory fence, the write of the reference in the buffer
could be visible to the consumer before the referenced data has been committed in memory.
In the code in Fig. \ref{fig:P1C1buffer}, a write-memory-barrier (WMB) must be inserted between
line \lines{\ref{fig:Lamport}}{2} and line \lines{\ref{fig:Lamport}}{3}.
\begin{figure}
\begin{Bench}{}{}
class SPSC_buffer {
private:
volatile unsigned long pread;
long padding1[longxCacheLine-1];
volatile unsigned long pwrite;
long padding2[longxCacheLine-1];
const size_t size;
void ** buf;
public:
SWSR_Ptr_Buffer(size_t n, const bool=true):
pread(0),pwrite(0),size(n),buf(0) {
}
~SWSR_Ptr_Buffer() { if (buf)::free(buf); }
bool init() {
if (buf) return false;
buf = (void **)::malloc(size*sizeof(void*));
if (!buf) return false;
bzero(buf,size*sizeof(void*));
return true;
}
bool empty() { return (buf[pread]==NULL);}
bool available() { return (buf[pwrite]==NULL);}
bool push(void * const data) {
if (available()) {
WMB();
buf[pwrite] = data;
pwrite += (pwrite+1 >= size) ? (1-size): 1;
return true;
}
return false;
}
bool pop(void ** data) {
if (empty()) return false;
*data = buf[pread];
buf[pread]=NULL;
pread += (pread+1 >= size) ? (1-size): 1;
return true;
}
};
\end{Bench}
\caption{SPSC circular buffer implementation.\label{fig:SPSC}}
\end{figure}
The complete code of the SPSC circular buffer is shown in Fig. \ref{fig:SPSC}.
\subsection{Cache optimizations}
\label{sec:cacheopt}
Avoiding cache-line thrashing due to false-sharing is a critical aspect in shared-cache multiprocessors.
Consider the case where two threads sharing a SPSC buffer are working in lock step.
The producer produces one task at a time while the consumer immediately consumes
the task in the buffer.
When a buffer entry is accessed, the system reads a portion of memory containing
the data being accessed placing it in a cache line.
The cache line containing the buffer entry is read by the consumer thread
which only consumes one single task. The producer than produces the next task pushing
the task into a subsequent entry into the buffer. Since, in general, a single cache
line contains several buffer entries (a typical cache line is 64bytes, whereas a
memory pointer on a 64bit architecture is 8 bytes) the producer's
write operation changes the cache line status invalidating the whole
contents in the line.
When the consumer tries to consume the next task the entire cache line is reloaded
again even if the consumer tries to access a different buffer location.
This way, during the entire computation the cache lines containing the buffer entries
bounce between the producer and the consumer private caches incurring in extra overhead
due to cache coherence traffic.
The problem arises because the cache coherence protocol works at cache
line granularity and because the ``distance'' between the producer and the consumer (i.e. $|pwrite-pread|$)
is less than or equal to the number of tasks which fill a cache line (on a 64bit machine with 64bytes
of cache line size the critical distance is 8).
In order to avoid false sharing between the head and tail pointers in the SPSC queue, a proper amount of padding in required to force the two pointers to reside in different cache lines (see for example Fig. \ref{fig:SPSC}).
In general, the thrashing behavior can be alleviated if the producer and the consumer are
forced to work on different cache lines, that is, augmenting the ``distance''.
The FastForward SPSC queue implementation presented in
\cite{fastforward:ppopp:08} improves
Lamport's circular buffer implementation by optimizing cache behavior and preventing
cache line thrashing.
FastForward temporally slips the producer and the consumer in such a way that
push and pop methods operate on different cache lines. The consumer,
upon receiving its first task, spins until an appropriate amount of slip
(that is the number of tasks in the queue reach a fixed value) is established.
During the computation, if necessary, the temporal slipping is maintained by the
consumer through local spinning.
FastForward obtains a performance improvement of 3.7 over Lamport's circular buffer
when temporal slipping optimization is used.
A different approach named cache line protection has been used in MCRingBuffer
\cite{MCRINGBUFFER:ipdps:10}.
The producer and consumer thread update private copies of the head and tail buffer
pointer for several iterations before updating a shared copy. Furthermore, MCRingBuffer
performs batch update of control variables thus reducing the frequency of writing the shared
control variables to main memory.
A variation of the MCRingBuffer approach is used in Liberty Queue \cite{LQ:10}. Liberty
Queue shifts most of the overhead to the consumer end of the queue. Such customization
is useful in situations where the producer is expected to be slower
than the consumer.
\begin{figure}
\begin{Bench}{}{}
bool multipush(void * const data[], int len) {
unsigned long last = pwrite + ((pwrite+ --len >= size) ? (len-size): len);
unsigned long r = len-(last+1), l=last, i;
if (buf[last]==NULL) {
if (last < pwrite) {
for(i=len;i>r;--i,--l)
buf[l] = data[i];
for(i=(size-1);i>=pwrite;--i,--r)
buf[i] = data[r];
} else
for(register int i=len;i>=0;--i)
buf[pwrite+i] = data[i];
WMB();
pwrite = (last+1 >= size) ? 0 : (last+1);
mcnt = 0; // reset mpush counter
return true;
}
return false;
}
bool flush() {
return (mcnt ? multipush(multipush_buf,mcnt) : true);
}
bool mpush(void * const data) {
if (mcnt==MULTIPUSH_BUFFER_SIZE)
return multipush(multipush_buf,MULTIPUSH_BUFFER_SIZE);
multipush_buf[mcnt++]=data;
if (mcnt==MULTIPUSH_BUFFER_SIZE)
return multipush(multipush_buf,MULTIPUSH_BUFFER_SIZE);
return true;
}
\end{Bench}
\caption{Methods added to the SPSC buffer to reduce cache trashing.\label{fig:SPSC:multipush}}
\end{figure}
\textbf{Multipush method.} Here we present a sligtly different approach
for reducing cache-line trashing which is very simple and effective,
and does not introduce any significant modification to the basic SPSC queue
implementation.
The basic idea is the following: instead of enqueuing just one item at a time
directly into the SPSC buffer, we can enqueue the items in a temporary array
and then submit the entire array of tasks in the buffer
using a proper insertion order.
We added a new method called \texttt{mpush} to the SPSC buffer implementation
(see Fig.\ref{fig:SPSC:multipush}),
which has the same interface of the push method but inserts the
data items in a temporary buffer of fixed size.
The elements in the buffer are written in the SPSC buffer
only if the local buffer is full or if the \texttt{flush} method is called.
The \texttt{multipush} method gets in input an array of items, and
writes the items into the SPSC buffer in backward order.
The backward order insertions, is particularly important to reduce
cache trashing, in fact, in this way, we enforce a distance between
the \texttt{pread} and the \texttt{pwrite} pointers thus reducing the cache invalidation ping-pong.
Furthermore, writing in backward order does not require any other
control variables or synchronisation.
This simple approach increase cache locality by reducing the
cache trashing. However, there may be two drawbacks:
\begin{enumerate}
\item we pay an extra copy for each element to push into the SPSC buffer
\item we could increase the latency of the computation if the consumer
is much faster than the producer.
\end {enumerate}
The first point, in reality, is not an issue because the cost of extra
copies are typically amortized by the better cache utilization. The
second point might represent an issue for applications exhibiting very
strict latency requirements that cannot be balanced by an increased
throughput (note however that this is a rare requirement in a streaming application).
In section \ref{sec:exp}, we try to evaluate experimentally the benefits
of the proposed approach.
\section{Unbounded List-Based Wait-Free SPSC Queue}
\label{sec:basicUnbounded}
Using the same idea of the Lamport's circular buffer algorithm,
it is possible to implement an unbounded wait-free SPSC queue
using a list-based algorithm and dynamic memory allocation/deallocation.
The implementation presented here has been inspired by the work
of Hendler and Shavit in \cite{Hendler02b}, although it is different in
several aspects.
The pseudocode is sketched in Fig. \ref{fig:listSPSC}.
\begin{figure}
\begin{Bench2}{}{}
bool push(data) {
Node * n = allocnode(data);
WMB();
tail->next = n;
tail = n;
return true;
}
bool pop(data) {
if (head->next != NULL) {
Node * n = (Node *)head;
data = (head->next)->data;
head = head->next;
deallocnode(n);
return true;
}
return false; // queue empty
}
\end{Bench2}
\caption{Unbounded list-based SPSC queue implementation.\label{fig:listSPSC}}
\hspace{0.5ex}
\end{figure}
The algorithm is very simple: the \texttt{push} method
allocates a new \texttt{Node} data structure containing the real value
to push into the queue and a pointer to the next \texttt{Node}
structure. The tail pointer is adjusted to point to the current Node.
The \texttt{pop} method gets the current head Node, sets the data value,
adjusts the head pointer and, before exiting, deallocates the head
\texttt{Node}.
In the general case, the main problem with the list-based
implementation of queues is the dynamic
memory allocation/deallocation of the \texttt{Node} structure. In fact,
dynamic memory management
operations, typically, use lock to enforce mutual exclusion to protect
internal shared data structures,
so, much of the benefits gained using lock-free implementation of the
queue are eventually
lost.
To mitigate such overhead, it is possible to use caching of list's internal
structure (e.g. \texttt{Node}) \cite{Hendler02b}. The cache is by definition
bounded in the number of
elements and so it can be efficiently implemented using a wait-free
SPSC circular buffer
presented in the previous sections.
Figure \ref{fig:SPSC_dynBuffer} shows the complete implementation of the list-based SPSC queue when
Node caching is used. In the following we will refer to this
implementation with the name dSPSC.
\begin{figure}
\begin{Bench2}{}{}
class SPSC_dynBuffer {
struct Node {
void * data;
struct Node * next;
};
volatile Node * head;
long pad1[longxCacheLine-sizeof(Node *)];
volatile Node * tail;
long pad2[longxCacheLine-sizeof(Node*)];
SPSC_Buffer cache;
private:
Node * allocnode() {
Node * n = NULL;
if (cache.pop((void **)&n)) return n;
n = (Node *)malloc(sizeof(Node));
return n;
}
public:
SPSC_dynBuffer(int size):cache(size) {
Node * n=(Node *)::malloc(sizeof(Node));
n->data = NULL; n->next = NULL;
head=tail=n; cache.init();
}
~SPSC_dynBuffer() { ... }
bool push(void * const data) {
Node * n = allocnode();
n->data = data; n->next = NULL;
WMB();
tail->next = n;
tail = n;
return true;
}
bool pop(void ** data) {
if (head->next) {
Node * n = (Node *)head;
*data = (head->next)->data;
head = head->next;
if (!cache.push(n)) free(n);
return true;
}
return false;
}
};
\end{Bench2}
\caption{Unbounded list-based SPSC queue implementation with Node(s) caching (dSPSC).\label{fig:SPSC_dynBuffer}}
\end{figure}
As we shall see in Sec. \ref{sec:exp}, caching strategies help in
improving the performance but are not sufficient
to obtain optimal figures. This is mainly due to the poor cache
locality caused by lots of memory
indirections. Note that the number of elementary
instruction per push/pop operation
is greater than the ones needed in the SPSC implementation.
\section{Unbounded Wait-Free SPSC Queue}
\label{sec:uSWSR}
We now describe an implementation of the unbounded wait-free SPSC queue
combining the ideas described in the previous sections.
We refer to the implementation with the name uSPSC.
The key idea is quite simple: the unbounded queue is based on a pool
of wait-free SPSC circular buffers (see Sec. \ref{sec:SWSR}). The pool of buffers
automatically grows and shrinks on demand. The implementation of the pool of
buffers carefully try to minimize the impact of dynamic memory allocation/deallocation
by using caching techniques like in the list-based SPSC queue.
Furthermore, the use of SPSC circular buffers as basic uSPSC data structure,
enforce cache locality hence provides better performance.
The unbounded queue uses two pointers: buf\_w that points to writer's buffer
(the same of the tail pointer in the circular buffer), and a buf\_r that points
to reader's buffer (the same of the head pointer).
Initially both buf\_w and buf\_r point to the same SPSC circular buffer.
The push method works as follow. The producer first checks whether there is an
available room in the current buffer (line \lines{\ref{fig:uSPSC}}{52}) and then push the data.
If the current buffer is full, asks the pool for a new buffer (line \lines{\ref{fig:uSPSC}}{53}),
set the buf\_w pointer and push the data into the new buffer.
\begin{figure}
\begin{Bench2}{}{}
class BufferPool {
SPSC_dynBuffer inuse;
SPSC_Buffer bufcache;
public:
BufferPool(int cachesize)
:inuse(cachesize),bufcache(cachesize) {
bufcache.init();
}
~BufferPool() {...}
SPSC_Buffer * const next_w(size_t size) {
SPSC_Buffer * buf;
if (!bufcache.pop(&buf)) {
buf = new SPSC_Buffer(size);
if (buf->init()<0) return NULL;
}
inuse.push(buf);
return buf;
}
SPSC_Buffer * const next_r() {
SPSC_Buffer * buf;
return (inuse.pop(&buf)? buf : NULL);
}
void release(SPSC_Buffer * const buf) {
buf->reset();
if (!bufcache.push(buf)) delete buf;
}
};
class uSPSC_Buffer {
SPSC_Buffer * buf_r;
long padding1[longxCacheLine-1];
SPSC_Buffer * buf_w;
long padding2[longxCacheLine-1];
size_t size;
BufferPool pool;
public:
uSPSC_Buffer(size_t n)
:buf_r(0),buf_w(0),size(size),
pool(CACHE_SIZE) {}
~uSPSC_Buffer() { ... }
bool init() {
buf_r = new SPSC_Buffer(size);
if (buf_r->init()<0) return false;
buf_w = buf_r;
return true;
}
bool empty() {return buf_r->empty();}
bool available(){return buf_w->available();}
bool push(void * const data) {
if (!available()) {
SPSC_Buffer * t = pool.next_w(size);
if (!t) return false;
buf_w = t;
}
buf_w->push(data);
return true;
}
bool pop(void ** data) {
if (buf_r->empty()) {
if (buf_r == buf_w) return false;
if (buf_r->empty()) {
SPSC_Buffer * tmp = pool.next_r();
if (tmp) {
pool.release(buf_r);
buf_r = tmp;
}
}
}
return buf_r->pop(data);
}
};
\end{Bench2}
\caption{Unbounded wait-free SPSC queue implementation.\label{fig:uSPSC}}
\end{figure}
The consumer first checks whether the current buffer is not empty and in case
pops the data. If the current buffer is empty, there are 2 possibilities:
\begin{enumerate}
\item there are no items to consume, i.e. the unbounded queue is really empty;
\item the current buffer is empty (i.e. the one pointed by buf\_r), but there may
be some items in the next buffer.
\end{enumerate}
For the consumer point of view, the queue is really empty when the current
buffer is empty and both the read and write pointers point to the same buffer.
If the read and writer queue pointers differ, the consumer have to re-check the
current queue emptiness because in the meantime (i.e. between the execution of
the instruction \lines{\ref{fig:uSPSC}}{61} and \lines{\ref{fig:uSPSC}}{62})
the producer could have written some new elements into the current buffer before
switching to a new one. This is the most subtle condition, if we switch to the
next buffer but the current one is not really empty, we definitely loose data.
If the queue is really empty for the consumer, the consumer switch to a new buffer
releasing the current one in order to be recycled by the buffer pool
(lines \lines{\ref{fig:uSPSC}}{64}--\lines{\ref{fig:uSPSC}}{67}).
\subsection{Correctness proof}
Here we provide a proof of correctness for the uSPSC queue implementation
described in the previous section.
By correctness, we mean that the consumer extracts elements from
the queue in the same order in which they were inserted by the
producer.
The proof is based on the only assumption that the SPSC circular buffer
algorithm is correct. A formal proof of the correctness of the SPSC buffer
can be found in \cite{HK97} and \cite{fastforward:ppopp:08}.
Furthermore, the previous assumption, implies that
memory word read and write are executed atomically. This is one
of the main assumption for the proof of correctness for the
SPSC wait-free circular buffer \cite{fastforward:ppopp:08}.
To the best of our knowledge, this condition is satisfied in any modern CPUs.
The proof is straightforward.
If buf\_r differs from buf\_w, the execution is correct because there is no data sharing
between producer and consumer (the push method uses only the
buf\_w pointer, whereas the pop method uses only the buf\_r pointer),
since the producer and the consumer use different SPSC buffer.
If buf\_r is equal to buf\_w (both the producer and the consumer use
the same SPSC buffer) and the buffer is neither seen full
nor empty by the producer and the consumer, the execution is correct because
of the correctness of the SPSC circular buffer.
So, we have to prove that if buf\_r is equal to buf\_w and the buffer is
seen full or empty by the producer and/or by the consumer respectively,
the execution of the \texttt{push} and \texttt{pop} methods are always correct.
The previous sentence has only one subtle condition worth proving:
buf\_r is equal to buf\_w and the producer sees the buffer full whereas
the consumer sees the buffer empty. This sound strange but it is not.
Suppose that the internal SPSC buffers used in the implementation of the uSPSC queue
has only a single slot (size=1). Suppose also that the consumer try to pop one
element out of the queue, and the queue is empty.
Before checking the condition at line \lines{\ref{fig:uSPSC}}{62},
the producer inserts an item in the queue and try to insert a second one.
At the second insert operation, the producer gets a new buffer because
the current buffer is full (line \lines{\ref{fig:uSPSC}}{53}), so, the buf\_w
pointer changes pointing to the new buffer (line \lines{\ref{fig:uSPSC}}{55}).
Since we have not assumed anything about read after write memory ordering
($R \rightarrow W$ using the same notation as in \cite{Adve95sharedmemory}),
we might suppose that the write of the buf\_w pointer is immediately visible to the consumer
end such that for the consumer results buf\_r different from buf\_w at line \lines{\ref{fig:uSPSC}}{62}.
In this case, if the consumer sees the buffer empty in the next test (line \lines{\ref{fig:uSPSC}}{63}),
the algorithm fails because the first element pushed by the produces is definitely lost.
So, depending on the memory consistency model, we could have different scenarios.
Consider a memory consistency model in which $W \rightarrow W$ program order
is respected. In this case, the emptiness check at line \lines{\ref{fig:uSPSC}}{63} could never
fail because a write in the internal SPSC buffer (line \lines{\ref{fig:SPSC}}{29}) cannot
bypass the update of the buf\_w pointer (line \lines{\ref{fig:uSPSC}}{55}).
Instead, if $W \rightarrow W$ memory ordering is relaxed, the algorithm fails if the
SPSC buffer has size 1, but it works if SPSC internal buffer has size greater than 1.
In fact, if the SPSC internal buffer has size 1 it is possible that the write in the buffer is not seen
at line \lines{\ref{fig:uSPSC}}{63} because writes can be committed out-of-order in memory, and also,
the Write Memory Barrier (WMB) at line \lines{\ref{fig:SPSC}}{28} is not sufficient, because it ensures
that only the previous writes are committed in memory.
On the other hand if the size of the SPSC buffer is at least 2
the first of the 2 writes will be visible at line \lines{\ref{fig:uSPSC}}{63} because of the WMB
instruction, thus the emptiness check could never fail.
From the above reasoning follows two theorems:
\begin{theorem}
The uSPSC queue is correct under any memory consistency
model that ensure $W \rightarrow W$ program order.
\end{theorem}
\begin{theorem}
The uSPSC queue is correct under any memory consistency
model provided that the size of the internal circular buffer is
greater than 1.
\end{theorem}
\section{Experiments}
\label{sec:exp}
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth]{andromeda_topology.pdf}
\caption{
Core's topology on the Intel Xeon E5520 workstation used for the tests.\label{fig:topology}}
\end{center}
\end{figure}
All reported experiments have been executed on an Intel
workstation with 2 quad-core Xeon E5520 Nehalem (16 HyperThreads)
@2.26GHz with 8MB L3 cache and 24 GBytes of main memory with Linux x86\_64.
The Nehalem processor uses Simultaneous MultiThreading (SMT,
a.k.a. HyperThreading) with 2 contexts per core and the Quickpath
interconnect equipped with a distributed cache coherency protocol.
SMT technology makes a single physical processor appear as two logical
processors for the operating system, but all execution resources are
shared between the two contexts.
We have 2 CPUs each one with 4 physical cores. The operating system (Linux
kernel 2.6.20) sees the two per core contexts as two distinct cores
assigning to each one a different id whose topology is sketched in
Fig. \ref{fig:topology}.
The methodology used in this paper to evaluate performance consists in
plotting the results obtained by running a simple synthetic benchmarks
and a very simple microkernel.
The first test is a 2-stage pipeline in which the first stage (P)
pushes 1 million tasks (a task is just a memory pointer) into a FIFO queue and
the second stage (C) pops tasks from the queue and checks for correct values. Neither additional memory
operations nor additional computation in the producer or consumer stage
is executed. With this simple test we are able to measure the raw performance
of a single push/pop operation by computing the average value of 100 runs and
the standard deviation.
\begin{figure}
\begin{center}
\includegraphics[width=0.32\linewidth]{noalloc_bounded_xeon_08.pdf}
\includegraphics[width=0.32\linewidth]{noalloc_bounded_xeon_02.pdf}
\includegraphics[width=0.32\linewidth]{noalloc_bounded_xeon_01.pdf}
\caption{
Average latency time and standard deviation (in nanoseconds) of a push/pop operation for the SPSC queue using different buffer size. The producer (P) and the consumer (C) are pinned: on the same core (left), on different core (middle), on different CPUs (right).\label{fig:SPSC:bounded}}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.32\linewidth]{noalloc_unbounded_xeon_08.pdf}
\includegraphics[width=0.32\linewidth]{noalloc_unbounded_xeon_02.pdf}
\includegraphics[width=0.32\linewidth]{noalloc_unbounded_xeon_01.pdf}
\caption{
Average latency time and standard deviation (in nanoseconds) of a push/pop operation for the unbounded SPSC queue (uSPSC) using different internal buffer size. The producer (P) and the consumer (C) are pinned: on the same core (left), on different core (middle), on different CPUs (right).\label{fig:SPSC:unbounded}}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.32\linewidth]{noalloc_dynamic_xeon_08.pdf}
\includegraphics[width=0.32\linewidth]{noalloc_dynamic_xeon_02.pdf}
\includegraphics[width=0.32\linewidth]{noalloc_dynamic_xeon_01.pdf}
\caption{
Average latency time and standard deviation (in nanoseconds) of a push/pop operation for the dynamic list-based SPSC queue (dSPSC) using different internal cache size. The producer (P) and the consumer (C) are pinned: on the same core (left), on different core (middle), on different CPUs (right).\label{fig:SPSC:dynamic}}
\end{center}
\end{figure}
In Fig. \ref{fig:SPSC:bounded} are reported the values obtained running the first benchmark for the
SPSC queue, varying the buffer size. We tested 3 distinct cases obtained by changing the physical mapping
of the 2 threads corresponding to the 2 stages of the pipeline:
1) the first and second stage of the pipeline are pinned on the same physical core
but on different HW contexts (P on core 0 and C on core 8), 2) are pinned on the same CPU but on different physical cores (P on core 0 and C on core 2),
and 3) are pinned on two cores of two distinct CPUs (P on core 0 and C on core 1).
In Fig. \ref{fig:SPSC:unbounded} and in Fig. \ref{fig:SPSC:dynamic} are reported the
values obtained running the same benchmark using the unbounded SPSC (uSPSC) queue and the
dynamic list-based SPSC queue (dSPSC) respectively. On top of each bar is reported the standard deviation in nanoseconds computed over 100 runs.
The SPSC queue is insensitive to buffer size in all cases. It takes on average 10--12 ns per queue
operation with standard deviations less than 1 ns when the producer and the consumer are on the same CPU,
and takes on average 11--15 ns if the producer and the consumer are on separate CPUs.
The unbounded SPSC queue (Fig. \ref{fig:SPSC:unbounded}) is more sensitive to the internal buffer size
especially if the producer and the consumer are pinned into separate CPUs. The values obtained are
extremely good if compared with the ones obtained for the dynamic list-based queue
(Fig. \ref{fig:SPSC:dynamic}), and are almost the same if compared with the bounded SPSC queue
when using an internal buffer size greater than or equal to 512 entries.
The dynamic list-based SPSC queue is sensitive to the internal cache size (implemented with
a single SPSC queue). It is almost 6 times slower than the uSPSC version if the producer and the
consumer are not pinned on the same core. In this case in fact, producer and consumer
works in lock steps as they share the same ALUs and so dynamic memory allocation is reduced
with performance improvement. Another point in this respect, is that the dynamic list-based SPSC queue uses memory
indirection to link together queue's elements thus not fully exploiting cache locality. The bigger the
internal cache the better performance is obtained. It is worth to note that caching strategies for dynamic
list-based SPSC queue implementation, significantly improve performance but are not enough
to obtain optimal figures like those obtained in the SPSC implementation.
\begin{table}[tb]
\begin{tabular*}{\linewidth}{@{\hspace{4ex}}r@{\hspace{4ex}}rrrrr}
\toprule
\ & L1 accesses & L1 misses & L2 accesses & L2 misses \\
push & 9,845,321 & 249,789 & 544,882 & 443,387 \\
mpush & 4,927,934 & 148,011 & 367,129 & 265,509 \\
\bottomrule
\end{tabular*}
\caption{push vs. mpush cache miss obtained using a SPSC of size 1024 and performing 1 million push/pop operations. \label{tab:cachemiss}}
\end{table}
We want now to evaluate the benefit of the cache optimization presented in Sec. \ref{sec:cacheopt} for the SPSC and for the uSPSC queue. We refer to mSPSC and to muSPSC the version of the SPSC queue and of the uSPSC queue which use the mpush instead of the push method.
Table \ref{tab:cachemiss} reports the L1 and L2 cache accesses and misses for the \texttt{push} and \texttt{mpush} methods using a specific
buffer size. As can be noticed, the mpush method greatly reduce cache accesses and misses.
The reduced number of misses, and accesses in general, leads to better overall performance. The average
latency of a push/pop operation, decreases from 10--11ns of the SPSC queue, to 6--9ns for the multi-push version.
The comparison of the \texttt{push} and \texttt{mpush} methods for both the SPSC and uSPSC queue, distinguishing the three mapping scenario for the producer and the consumer, are shown in Fig. \ref{fig:SPSC:multipush}. The muSPSC queue is less sensitive to
the cache optimization introduced with the \texttt{mpush} method with respect to the uSPSC queue.
\begin{figure}
\begin{center}
\includegraphics[width=0.32\linewidth]{noalloc_multipush_xeon_08.pdf}
\includegraphics[width=0.32\linewidth]{noalloc_multipush_xeon_02.pdf}
\includegraphics[width=0.32\linewidth]{noalloc_multipush_xeon_01.pdf}
\caption{
Average latency time of a multi-push/pop operation for the bounded and unbounded SPSC buffer. The multi-push internal buffer size is statically set to 16 entries. The producer (P) and the consumer (C) are pinned: on the same core (left), on different core (middle), on different CPUs (right).\label{fig:SPSC:multipush}}
\end{center}
\end{figure}
\begin{figure}
\begin{Bench}{}{}
int main() {
double x = 0.12345678, y=0.654321012;
for(int i=0;i<1000000;++i) {
x = 3.1415 * sin(x);
y += x - cos(y);
}
return 0;
}
\end{Bench}
\caption{Microbenchmark: sequential code.\label{fig:microbench}}
\end{figure}
\begin{figure}ls
\begin{Bench2}{}{}
void P() {
double x = 0.12345678;
for(int i=0;i<1000000;++i) {
x = 3.1415 * sin(x);
Q.push(x);
}
}
void C() {
double x, y=0.654321012;
for(int i=0;i<1000000;++i) {
Q.pop(&x);
y += x - cos(y);
}
}
\end{Bench2}
\caption{Microbenchmark: pipeline implementation.\label{fig:microbenchPipe}}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.32\linewidth]{computes_xeon_08.pdf}
\includegraphics[width=0.32\linewidth]{computes_xeon_02.pdf}
\includegraphics[width=0.32\linewidth]{computes_xeon_01.pdf}
\caption{Average latency time (in nanoseconds) of the pipeline microbenchmark implementation when using the mSPSC, dSPSC and muSPSC queue. P and C are pinned: on the same core (left), on different core (middle), on different CPUs (right).\label{fig:microbenchSPSC}}
\end{center}
\end{figure}
In order to test a simple but real application kernel we consider the code in Fig. \ref{fig:microbench}.
The sequential execution of such code on a single core of the tested architecture is 94.6ms. We
parallelize the code into a pipeline of 2 stages, P and C, as shown in Fig. \ref{fig:microbenchPipe}.
The 2 stages are connected by a FIFO queue. The results obtained considering for the queue the mSPSC,
dSPSC and muSPSC implementations are shown in Fig \ref{fig:microbenchSPSC}. As can be noticed
the unbounded multi-push implementation (muSPSC) obtain the best performance reaching a maximum speedup of
1.4, whereas the bounded multi-push implementation (mSPSC) reaches a maximum speedup of 1.28 and finally the
dynamic list-based queue (dSPSC) does not obtain any performance improvement reaching a maximum speedup
of 0.98. This simple test, proves the effectiveness of the uSPSC queue implementation with respect to
the list-based FIFO queue implementation when used in real case scenario.
\section{Conclusions}
\label{sec:conclusions}
In this report we reviewed several possible implementations of fast wait-free
Single-Producer/Single-Consumer (SPSC) queue for shared cache multi-core starting from
the well-known Lamport's circular buffer algorithm.
A novel implementation of unbounded wait-free SPSC queue has been introduced with a formal proof of
correctness. The new implementation is able to minimize dynamic memory allocation/deallocation
and increases cache locality thus obtaining very good performance figures on modern shared cache
multi-core. We believe that the unbounded SPSC algorithm presented here can be used as an efficient alternative to
the widely used list-based FIFO queue.
\section*{Acknoweledments}
The author gratefully acknowledge Marco Aldinucci, Marco Danelutto,
Massimiliano Meneghin and Peter Kilpatrick for their comments and suggestions.
\bibliographystyle{abbrv}
|
2,869,038,156,385 | arxiv | \section{Introduction and Outline of Results.}\label{1}
A {\bf generalized arc\/} is a continuum (i.e., a connected compact Hausdorff
space) that has exactly two non-separating points; an {\bf arc\/} is a
metrizable generalized arc. The class of generalized arcs is precisely the
class of linearly orderable continua; each generalized arc admitting exactly
two compatible linear orders. The class of (continuous
images of) generalized arcs has been extensively
studied over the years (see \cite{HY,NTT,Wil}); the most well-known results
in this area being that any two arcs are homeomorphic (to the standard closed
unit interval on the real line); and (Hahn-Mazurkiewicz)
that a Hausdorff
space is a continuous image of an arc if and only if that space is a locally
connected metrizable continuum. In this paper, a continuation of \cite{Ban3},
we study the model-theoretic topology of generalized arcs; in particular,
the ``dualized model theory'' of these spaces.
Many notions from classical first-order model theory, principally elementary
equivalence and elementary embedding, may be phrased in terms of mapping
conditions involving the ultraproduct construction. Because of the
(Keisler-Shelah) ultrapower theorem (see, e.g., \cite{CK}), two relational
structures are elementarily equivalent if and only if some ultrapower of one
is isomorphic to some ultrapower of the other; a function from one relational
structure to another is an elementary embedding if and only if there is an
ultrapower isomorphism so that the obvious square mapping diagram commutes
(see, e.g., \cite{Ban2,Ban5,Ekl}). The ultrapower construction in turn is a
direct limit
of direct products, and is hence capable of being transferred into a
purely category-theoretic setting. In this paper we focus on the category
$\bf CH$
of compact Hausdorff spaces and continuous maps, but perform the transfer
into the opposite category (thus justifying the phrase
``dualized model theory'' above).
In $\bf CH$ one then constructs ultracoproducts,
and talks of co-elementary equivalence and co-elementary maps. Co-elementary
equivalence is known \cite{Ban2,Ban5,Gur} to preserve important properties
of topological spaces, such as being infinite, being Boolean (i.e., totally
disconnected), having (Lebesgue) covering dimension $n$, and being a
decomposable continuum. If $f:X \to Y$ is a co-elementary map in $\bf CH$,
then of course $X$ and $Y$ are co-elementarily equivalent (in symbols
$X \equiv Y$). Moreover, since $f$ is a continuous surjection (see
\cite{Ban2}), additional information about $X$ is transferred to $Y$. For
instance, continuous surjections in $\bf CH$ cannot raise {\bf weight\/} (i.e.,
the smallest cardinality of a possible topological base, and for many
reasons the right cardinal invariant to replace cardiality in the dualized
model-theoretic setting), so metrizability
(i.e., being of countable weight in the compact Hausdorff context) is
preserved. Also local connectedness is preserved, since continuous surjections
in $\bf CH$ are quotient maps. Neither of these properties is an invariant
of co-elementary equivalence alone.
When attention is restricted to the full subcategory of Boolean
spaces, the dualized model theory matches perfectly with
the model theory of Boolean algebras because of Stone duality. In the
larger category there is no such match \cite{Bana,Ros}, however, and one is
forced to look for other (less direct) model-theoretic aids. Fortunately
there is a finitely axiomatizable Horn class of bounded lattices,
the so-called {\it normal disjunctive\/} lattices \cite{Ban8} (also called
{\it Wallman\/} lattices in \cite{Ban5}), comprising precisely the (isomorphic
copies of) lattices that serve as bases for the closed sets of compact
Hausdorff spaces. We go from lattices to spaces, as in the case of Stone
duality, via the {\bf maximal spectrum\/} $S(\;)$, pioneered by H. Wallman
\cite{Walm}.
$S(A)$ is the space of
maximal proper filters of $A$; a typical basic closed set in $S(A)$ is the
set of elements of $S(A)$ containing a given element of $A$. $S(\;)$
is contravariantly functorial; if $f:A \to B$ is a homomorphism of normal
disjunctive lattices and $M \in S(B)$, then $f^S(M)$ is the unique maximal
filter in $A$ containing the pre-image of $M$ under $f$. It is a fairly
straightforward task to show, then, that $S(\;)$ converts ultraproducts
to ultracoproducts, elementarily equivalent lattices to co-elementarily
equivalent compact Hausdorff spaces, and elementary embeddings to co-elementary
maps (see \cite{Ban2,Ban4,Ban5,Ban8,Gur}). An important consequence of
this is a L\"{o}wenheim-Skolem theorem for co-elementary maps: every
compact Hausdorff space maps co-elementarily onto a compact metrizable space.
(This result is used in \ref{2.4} and \ref{2.6} below.)
In \cite{Ban3} we showed that
any locally connected metrizable space co-elementarily equivalent to an arc
is already an arc; here we present the following results.
$(i)$ if $f:X \to Y$ is a co-elementary
map in $\bf CH$, and if $Y$ is locally connected (in particular, a generalized
arc), then $f$ is a monotone continuous surjection;
$(ii)$ co-elementary
images of (generalized) arcs are (generalized) arcs;
$(iii)$ any two generalized arcs are co-elementarily equivalent;
$(iv)$ if $X$ is a generalized arc and $f:X \to Y$ is
an irreducible co-elementary map in $\bf CH$, then $f$ is a homeomorphism;
$(v)$ if every locally connected co-elementary pre-image of an arc is a
generalized arc, then every locally connected compact Hausdorff space
co-elementarily equivalent to a generalized arc is also a generalized arc;
and $(vi)$ if $X$ is an arc
and $f$ is a function from $X$ to a compact Hausdorff space $Y$, then $f$ is
a co-elementary map if and only if $Y$ is an arc and $f$ is a monotone
continuous surjection.
Local connectedness is necessarily a part of $(v)$ above.
We do not know at present whether the hypothesis in $(v)$ is true; nor
do we know whether monotone surjections between
generalized arcs are always co-elementary maps.
\subsection{Remark.}\label{1.1}
By way of contrast, there is a Boolean analogue
to some of the results above. Define a {\bf generalized Cantor set\/} to be any
non-empty Boolean space with no isolated points, and a {\bf Cantor set\/} to
be a metrizable generalized Cantor set. It is well known that any two
Cantor sets are homeomorphic (to the standard Cantor middle thirds set in the
real line), and that the generalized Cantor sets are precisely the Stone
duals of the atomless Boolean algebras, constituting an elementary class
whose first-order
theory is $\aleph_0$-categorical, complete, and model complete.
In $(ii)$ and $(iii)$, one may replace ``arc'' with ``Cantor set'' uniformly;
a straightforward application of $\aleph_0$-categoricity.
The analog of $(iv)$ is
false (see Example 3.3.4$(iv)$ in \cite{Ban2}); the projective cover map to
a generalized Cantor set is always an irreducible co-elementary map between
(seldom-homeomorphic) generalized Cantor sets. As for $(v)$, it follows from
the results on dimension in \cite{Ban2} that any compact Hausdorff space
co-elementarily equivalent to a generalized Cantor set is itself a
generalized Cantor set. Finally, regarding $(vi)$, {\it all\/} continuous
surjections between
generalized Cantor sets are co-elementary maps. This is a direct consequence
of the model completeness of the theory of atomless Boolean algebras.\\
\section{Methods and Proofs.}\label{2}
We begin with a proof of $(i)$ above. Recall that a map $f:X \to Y$ is
{\bf monotone\/} (resp. {\bf strongly monotone\/}) if the inverse image
of a point (resp. a closed connected subset) of $Y$ is connected in $X$.
\subsection{Proposition.}\label{2.1}
Let $f:X \to Y$ be a co-elementary
map in $\bf CH$, with $Y$ locally connected. Then $f$ is a strongly
monotone continuous surjection.\\
\noindent
{\bf Proof.\/} Assume $f:X \to Y$ is co-elementary, $Y$ is locally
connected, and $f$ is not strongly monotone. Then there is a subcontinuum
$S$ of $Y$ such that
the inverse image $A := f^{-1}[S]$ is disconnected. Since $A$ is closed,
we can write $A = A_1 \cup A_2$ where each $A_i$ is closed non-empty,
and $A_1 \cap A_2 = \emptyset$. Let $U_i$ be an open neighborhood of $A_i$,
with $U_1 \cap U_2 = \emptyset$. If $C$ is a subcontinuum
of $X$ containing $A$, then we can pick
some $x_C \in C \setminus (U_1 \cup U_2)$. Let $B$ be the closure of
the set of all such points $x_C$, as $C$ ranges over all subcontinua
containing $A$. Since no point $x_C$ lies in $U_1 \cup U_2$, $B$
is disjoint from $A$, but intersects every subcontinuum of $X$ that
contains $A$.
Now $f[B]$ is closed in $Y$ and disjoint from $S$. Let $W$ be an open
neighborhood of $S$ whose closure misses $f[B]$.
Since $Y$ is locally connected, we have, for each $y \in S$, a connected
open neighborhood $V_y$ of $y$ such that $V_y \subseteq W$. Since $S$
is connected, so also is $V := \bigcup_{y \in S}V_y$; and the closure
$K$ of $V$ is a subcontinuum containing $S$. Since $V \subseteq W$,
and the closure of $W$ is disjoint from $f[B]$,
we know that $K$ is also disjoint from $f[B]$. We need a fact proved
elsewhere.\\
\noindent
{\bf Lemma.\/}(Lemma 2.8 in \cite{Ban5}) Let
$f:X \to Y$ be a co-elementary map in
$\bf CH$, with $K \subseteq Y$ a subcontinuum. Then
there is a subcontinuum $C \subseteq X$ such that $K = f[C]$, and
whenever $V \subseteq K$ is open in $Y$, $f^{-1}[V] \subseteq C$.\\
Using the Lemma, there exists a subcontinuum $C \subseteq X$ such that
$f[C] = K$ and $f^{-1}[V] \subseteq C$. Let $x \in A$. Then there
is a neighborhood $U$ of $x$ with $f[U] \subseteq V$. Thus
$x \in U \subseteq f^{-1}[V] \subseteq C$, hence we infer $A \subseteq
C$. Every subcontinuum of $X$ containing $A$ must intersect $B$, so
$\emptyset \neq f[B \cap C] \subseteq f[B] \cap f[C] =
f[B] \cap K = \emptyset$. This contradiction completes the proof. $\dashv$
\subsection{Remark.}\label{2.2}
The Lemma above provides only a weak consequence of co-elementarity. Indeed,
the usual projection map from the standard closed unit square in the plane
onto its first co\"{o}rdinate is not co-elementary because it does not
preserve topological dimension. Nevertheless, it does satisfy the conclusion
of the Lemma.\\
Now we are in a position to prove $(ii)$.
\subsection{Proposition.}\label{2.3}
Let $f:X \to Y$ be a co-elementary map in $\bf CH$. If $X$ is a generalized
arc, then so is $Y$.\\
\noindent
{\bf Proof.\/} Let $f:X \to Y$ be a co-elementary map in $\bf CH$, with
$X$ a generalized arc.
$Y$ is a locally connected continuum
because $X$ is locally connected and $f$ is a continuous surjection. By
\ref{2.1}, $f$ is monotone; it remains to show $Y$ has precisely two
non-separating points.
Let $a, b \in X$ be the two non-separating points of $X$. $Y$ is
non-degenerate because of co-elementarity; monotonicity then tells us that
$f(a) \neq f(b)$. If $f(a)$ were to separate $Y$, we could also separate
$X \setminus K$, where $K:= f^{-1}[\{f(a)\}]$ is a subcontinuum
(i.e., closed subinterval) containing
the endpoint $a$. This is easily seen to be impossible for
generalized arcs. Now let $y \in Y \setminus \{f(a),f(b)\}$, with
$K := f^{-1}[\{f(y)\}]$. Then $K$ is a subcontinuum of $X$ containing neither
endpoint. Thus $X \setminus K$ is disconnected; hence $y$ separates $Y$.
We therefore conclude that $Y$ is a generalized arc. $\dashv$.\\
We can very quickly settle $(iii)$.
\subsection{Proposition.}\label{2.4}
Let $X$ and $Y$ be two generalized arcs. Then $X \equiv Y$.\\
\noindent
{\bf Proof.\/} Let $X$ and $Y$ be generalized arcs. By the
L\"{o}wenheim-Skolem theorem for co-elementary maps, there exist co-elementary
maps $f:X \to X_0$ and $g:Y \to Y_0$, where $X_0$ and $Y_0$ are compact
metrizable. By \ref{2.3}, the images are generalized arcs; hence they are
arcs. Thus $X_0$ and $Y_0$ are homeomorphic, and we conclude $X \equiv Y$
because \cite{Ban2} co-elementary equivalence is an honest equivalence
relation. $\dashv$\\
To handle $(iv)$, recall that a continuous surjection $f:X \to Y$ is
{\bf irreducible\/} if $Y$ is not the image under $f$ of a proper closed
subset of $X$.
\subsection{Proposition.}\label{2.5}
Let $f:X \to Y$ be an irreducible co-elementary map in $\bf CH$. If $X$
is a generalized arc, then $f$ is a homeomorphism.\\
\noindent
{\bf Proof.\/} It suffices to show $f$ is one-one. Let $y \in Y$, with
$K := f^{-1}[\{y\}]$, a subcontinuum of $X$ by \ref{2.1}. Since $X$ is
a generalized arc, $K$ is either a singleton or a closed subinterval with
non-empty interior. The latter case easily contradicts the irreducibility
of $f$, however. $\dashv$\\
In \cite{Gur} it is shown that every infinite compact Hausdorff space is
co-elementarily equivalent to a compact Hausdorff space that is not locally
connected. (See also \cite{Ban5} for refinements.) This explains the
necessity of the local connectedness hypothesis in $(v)$.
\subsection{Proposition.}\label{2.6}
Suppose every locally connected co-elementary pre-image of an arc is a
generalized arc. Then every locally connected compact Hausdorff space
co-elementarily equivalent to a generalized arc is itself a generalized arc.\\
\noindent
{\bf Proof.\/} Suppose $X \in \bf CH$ is locally connected,
$X \equiv Y$, and $Y$ is a generalized arc. As in the proof of \ref{2.4}
above, we have co-elementary maps $f:X \to X_0$ and $g:Y \to Y_0$, where
$X_0$ and $Y_0$ are metrizable. Furthermore, we know that $X_0$ is
locally connected and that $Y_0$ is an arc (\ref{2.1} again). By the
transitivity of co-elementary equivalence, we know $X_0 \equiv Y_0$; by
the main result of \cite{Ban3}, we know $X_0$ is an arc. Our hypothesis
then tells us that $X$ is a generalized arc. $\dashv$\\
We finish with a proof of $(vi)$. If $X$ is an arc and $f:X \to Y$ is
a co-elementary map in $\bf CH$, then $Y$ is an arc and $f$ is a monotone
continuous surjection by \ref{2.1} and \ref{2.2}. So it suffices to prove
the following.
\subsection{Proposition.}\label{2.7}
Every monotone continuous surjection from
an arc to itself is a co-elementary map.\\
\noindent
{\bf Proof. } Let us take our arc to be the standard closed unit interval
${\bf I}$ with its usual order.
$f$ is either $\leq$-preserving or $\leq$-reversing, so we lose
no generality in assuming $f$ to be the former.
For any topological space $X$, we denote the closed set lattice of $X$ by
$F(X)$. $F(\;)$ converts continuous maps contravariantly into lattice
homomorphisms, and serves as a right inverse for $S(\;)$: $S(F(X))$ is
naturally homeomorphic to $X$ for any compact Hausdorff $X$.
Monotone continuous surjections from $\bf I$ to itself are strongly monotone;
hence
$f^F:F({\bf I}) \to F({\bf I})$ is a lattice embedding that takes closed
intervals (in this case the connected elements of the lattice) to closed
intervals.
However, $f^F$ will take atoms to non-atoms when $f$ is not injective.
Thus $f^F$ is not an elementary embedding without being an isomorphism.
The idea is to restrict the domain and range of $f^F$ in such a way that
the resulting lattice embedding, call it $g$, is elementary, and $g^S = f$.
Our plan is to create an elementary lattice embedding $g:{\cal A} \to
{\cal B}$, where ${\cal A}$ and ${\cal B}$ are atomless lattice bases
for ${\bf I}$ (i.e., both $\cal A$ and $\cal B$ are atomless, as well as
meet-dense in $F({\bf I})$), and $g$ agrees with the restriction
of $f^F$ to ${\cal A}$.
Since $S(\cal A)$ and $S(\cal B)$ are naturally homeomorphic to $\bf I$,
and $f$ is just $g^S$ conjugated with these homeomorphisms, $f$ is a
co-elementary map provided $g^S$ is.
For each $y \in \bf I$, let $\lambda (y) := \mbox{inf}(f^{-1}[\{y\}])$
and $\rho (y) := \mbox{sup}(f^{-1}[\{y\}])$. Then for any closed interval
$[x,y] \in F(\bf I)$, $f^F([x,y]) = [\lambda (x),\rho (y)]$. Both $\lambda$
and $\rho$ are right inverses for $f$, and are hence strictly increasing
(but not necessarily continuous). Of course $\lambda (0) = 0$ and
$\rho (1) = 1$.
Let $L,R \subseteq \bf I$, with $0 \in L$ and $1 \in R$. If ${\cal I}(L,R)$
denotes the set of all finite unions of intervals $[x,y]$ with $x \in L$
and $y \in R$, then ${\cal I}(L,R)$ is a sublattice of $F(\bf I)$, which is
atomless just in case $L \cap R = \emptyset$. If $L$ and $R$ are dense
in $\bf I$, then ${\cal I}(L,R)$ is a lattice base as well.
Now fix $L,R \subseteq \bf I$ to be disjoint countable dense subsets,
with $0 \in L$ and $1 \in R$, and set ${\cal A} := {\cal I}(L,R)$.
Then the image of $\cal A$ under $f^F$ is ${\cal I}(\lambda [L],\rho[R])$.
Clearly $\lambda [L] \cap \rho [R] = \emptyset$, $0 \in \lambda [L]$, and
$1 \in \rho [R]$.
Let $L', R' \subseteq \bf I$ be disjoint countable dense
subsets, with $\lambda [L] \subseteq L'$, $\rho [R] \subseteq R'$, and
set ${\cal B} := {\cal I}(L',R')$. Then $\cal B$ is a countable atomless
lattice base for $F(\bf I)$, and we denote by $g:{\cal A} \to \cal B$
the embedding $f^F$ with its domain and range so restricted. It remains
to show that $g$ is an elementary embedding, and for this it suffices to
show that for
each finite set $S$ in $\cal A$ and each $b \in \cal B$, there is an
automorphism on $\cal B$ that fixes $g[S]$ pointwise
and takes $b$ into $g[\cal A]$.
Let $x_1, ..., x_n$ be a listing, in increasing order, of the endpoints
of the component intervals of $g[S] \cup \{b\}$ (so each $x_i$ is in
$L' \cup R'$), with $X_i :=
f^{-1}[\{f(x_i)\}]$, $1 \leq i \leq n$. Each $X_i$ is either a singleton
or a non-degenerate closed interval, and
for $1 \leq i < j \leq n$,
either $X_i = X_j$ or each element of $X_i$ is less than each element of
$X_j$. Let $U_i$ be an open-interval neighborhood of $X_i$ such that
$U_i \cap U_j = \emptyset$ whenever $X_i \neq X_j$. Since $f$ is a
$\leq$-preserving surjection and the sets $L$ and $R$ are dense in $\bf I$,
each $U_i$ has
infinite intersection with both $\lambda [L]$ and $\rho [R]$. If
$x_i \in \lambda [L] \cup \rho [R]$, set $x_i' := x_i$. Otherwise we
know $x_i$ is an endpoint of a component interval of $b$; and we choose
$x_i' \in U_i$ in such a way that $x_i' \in \lambda [L]$ if and only if
$x_i \in L'$, and $x_i' < x_j'$ whenever $x_i < x_j$ and
$X_i = X_j$. This
procedure produces an increasing sequence $x_1',...,x_n'$ of elements
of $\lambda [L] \cup \rho [R]$; $x_i' \in \lambda [L]$ if and only if
$x_i \in L'$. For each $a \in g[S] \cup \{b\}$, let $a'$ be built up using
the endpoints $x_i'$ in the same way as $a$ is built up using the endpoints
$x_i$. Then $a' = a$ for each $a \in g[S]$, and $b' \in g[\cal A]$.
Now by a classic (Cantor) back and forth argument, there is an order
automorphism on $L' \cup R'$ that fixes $L'$ and $R'$ setwise and takes
$x_i$ to $x_i'$ for $1 \leq i \leq n$. This order automorphism gives
rise to the lattice automorphism on $\cal B$ that we require. $\dashv$\\
|
2,869,038,156,386 | arxiv |
\section{Comparing complete intersection and movable curve classes}
\label{secCI:ci}
The proof of Theorem \ref{thmCI:CIvsNef} involves comparing
intersections of pairs of nef divisor classes with movable classes. To begin this section, we give defining inequalities for the nef
and movable cones, and calculate intersections of pairs of divisors on
the varieties $X_r$ (defined below), thus providing the input data for
the algorithmic proof of Theorem \ref{thmCI:CIvsNef}. Proofs are given in Section \ref{secCI:intTh}.
\begin{defn}
\label{defnCI:kapBases}
Let $X_r$ be the composition of the blow-ups of $r$ general points in
$\mathbb{P}^3$, $1 \leq r \leq 5$, followed by the blow-ups of the
proper-transforms of the $\binom{r}{2}$ lines of $\mathbb{P}^3$ spanned by
the $r$ points.
\end{defn}
Note that $X_4 = \overline{L}_4$ and $X_5 = \overline{M}_{0,6}$. Let $H$ be the pullback of a general hyperplane, let $E_1, \ldots, E_r$ be the exceptional divisors obtained by blowing up the points, and let $E_{12}, \ldots, E_{r-1 \, r}$ be the proper transforms of the exceptional divisors obtained by blowing up the lines.
As in Section \ref{secCI:background}, the \emph{Kapranov basis} of $N^1(X_r)_\mathbb{R}$ is
\begin{equation*}
\{[H], [E_1], \ldots, [E_r], [E_{12}], \ldots, [E_{r-1 \, r}] \},
\end{equation*}
and the \emph{dual Kapranov basis} of $N_1(X_r)_\mathbb{R}$ is denoted
\begin{equation*}
\{[H]^{\vee}, [E_1]^{\vee}, \ldots, [E_r]^{\vee}, [E_{12}]^{\vee}, \ldots, [E_{r-1 \, r}]^{\vee} \}.
\end{equation*}
\noindent Again, we will abuse notation and not distinguish notationally
among the Kapranov bases from the different $X_r$.
To characterize the nef cones of the $X_r$, we adopt the notational
convention for the defining inequalities that a coefficient $d_{ij}$ is set to zero if the indices
are impossible for a given inequality. For example, if $r=1$, then all
$d_{ij}$ appearing below are taken to be 0, and if $r=2$, the final inequality below reads $d_h + d_i \geq 0$ for $i \in \{1,2\}$. We also identify
$d_{ij}$ and $d_{ji}$.
\begin{prop}
\label{lemmaCI:nefCone}
Let $[D] = d_h [H] + \sum_{i=1}^r d_i [E_i] + \sum_{1 \leq j<k \leq r} d_{jk}[E_{jk}]$ be an arbitrary divisor class in $X_r$. The cone of nef divisors is determined by the inequalities
\begin{equation*}
\begin{cases}
-d_{ij} \geq 0, &\text{for } 1 \leq i < j \leq r,\\
d_h + d_i + d_j - d_{ij} \geq 0, &\text{for } 1 \leq i < j \leq r,\\
-d_i + d_{ij} + d_{ik} \geq 0, &\text{for } 1 \leq i,j,k \leq r, i \notin\{j,k\}, \\
d_h + d_i + d_{jk} + d_{lm} \geq 0, &\text{for } \{i,j,k,l,m\} = \{1, \ldots, r\}.
\end{cases}
\end{equation*}
\end{prop}
To determine the movable cone of curves, $\overline{\mathrm{Mov}}(\overline{M}_{0,6})$, we consider intersections of one-cycles with the generators of $\overline{\mathrm{Eff}}(\overline{M}_{0,6})$, that is, with all boundary divisor classes $[\Delta_J]$ and the fifteen Keel-Vermeire divisor classes $[Q_{(ab)(cd)(e6)}]$. We apply an analogous convention used to characterize $\mathrm{Nef}(X_r)$ to the terms $c_i$ and $c_{jk}$ in the inequalities below.
\begin{prop}
\label{lemmaCI:nefCurvesCone}
Let $[C] = c_h [H]^\vee + \sum_{i=1}^r c_i [E_i]^\vee + \sum_{1 \leq j < k \leq r} c_{jk} [E_{jk}]^\vee$ be a one cycle class in $N_1(\overline{M}_{0,6})$. The cone of movable curve classes in $X_r$ is determined by the inequalities
\begin{equation*}
\begin{cases}
c_{i} \geq 0, &\text{for }i=1, \ldots, r, \\
c_{jk} \geq 0, &\text{for } 1\leq j < k \leq r, \\
c_h - c_i - c_j - c_k - c_{ij} - c_{ik} - c_{jk} \geq 0, &\text{for }1 \leq i< j < k \leq r, \\
2 c_h - \sum_{i=1}^5 c_i - c_{jl} - c_{kl} - c_{jm} - c_{km} \geq 0, &\text{for } j,k,l,m \in \{1, \ldots, r\}\text{ distinct}.
\end{cases}
\end{equation*}
\end{prop}
\noindent Note that for each inequality of the last type follows from
inequalities of the first three types when $r \leq 4$.
We will give an alternative definition of the complete intersection
cone in Lemma \ref{lemmaCI:cinef2} involving intersections of nef divisors, so we next write the remaining intersections of elements of the Kapranov basis for $N^1(\overline{M}_{0,6})_{\mathbb{R}}$ in terms of the dual basis.
\begin{prop}
\label{lemmaCI:doubleInt}
The intersections of elements of the Kapranov basis for $X_r$, in terms of the dual basis, are, for distinct $i,j,k,l \in \{1, \ldots, r\}$,
\begin{equation} \label{eq:intZero}
\; \; \qquad
0 = [H] \cdot [E_i] =
[E_i] \cdot [E_j] =
[E_i] \cdot [E_{jk}] =
[E_{ij}] \cdot [E_{kl}] =
[E_{ij}] \cdot [E_{ik}],
\end{equation}
\begin{equation} \label{eq:intDual}
\begin{split}
[H]^2& = [H]^{\vee}, \;
[H]\cdot [E_{jk}] = [E_j] \cdot [E_{jk}] = [E_k] \cdot [E_{jk}] = -[E_{jk}]^\vee,\\
[E_i]^2& = [E_i]^{\vee}, \;
[E_{jk}]^2 = 2 [E_{jk}]^\vee - [H]^\vee - [E_j]^\vee - [E_k]^\vee.
\end{split}
\end{equation}
\end{prop}
Now we turn to the algorithm used to prove Theorem \ref{thmCI:CIvsNef}. We begin with a recasting of the complete intersection cone of a projective variety $X$ with a finitely generated nef cone.
\begin{defn}
\label{defCI:Nefnm1}
For $X$ a smooth projective variety of dimension $d$ with a finitely generated nef cone, define $(\mathrm{Nef}(X))^{d-1} \subseteq N_1(X)_\mathbb{R}$ as
\begin{equation}
(\mathrm{Nef}(X))^{d-1} = \langle [N_1] \cdot \ldots \cdot [N_{d-1}]: \textrm{ each }[N_i] \textrm{ an extremal ray of }\mathrm{Nef}(X) \rangle_{\geq 0} \nonumber
\end{equation}
\end{defn}
\noindent Note that finite generation of $\mathrm{Nef}(X)$ implies that $(\mathrm{Nef}(X))^{d-1}$ is a closed cone.
\begin{lemma}
\label{lemmaCI:cinef2}
The cones $(\mathrm{Nef}(X))^{d-1}$ and $\mathcal{CI}(X) $ are equal.
\end{lemma}
\begin{proof}
To see that $(\mathrm{Nef}(X))^{d-1} \subseteq \mathcal{CI}(X)$, note first that every
nef divisor is a limit of ample divisors (see \cite{MR2095471},
Section 1.4). Since $\mathcal{CI}(X)$ is a closed cone, multilinearity and continuity of the
intersection product (\cite{MR2095471}, Section 1.1) imply the first
inclusion. The reverse inclusion is obvious.
\end{proof}
\begin{cor}
\label{corCI:algorithm}
If the nef cone of $X$ is finitely generated, the cones $\mathcal{CI}(X)$ and $\overline{\mathrm{Mov}}(X)$ coincide if and only if every extremal ray of $\overline{\mathrm{Mov}}(X)$ is a non-trivial multiple of a generator of $(\mathrm{Nef}(X))^{d-1}$.
\end{cor}
This corollary leads directly to an algorithm to test the equality of
$\mathcal{CI}(X)$ and $\overline{\mathrm{Mov}}(X)$ when $\mathrm{Nef}(X)$ and $\overline{\mathrm{Mov}}(X)$ are finitely
generated, and all necessary intersections are known. We describe the
algorithm in detail for a projective three-fold; the extension of the
algorithm to higher-dimensional varieties will be obvious.
\begin{algorithm}
Determine if $\mathcal{CI}(X) = \overline{\mathrm{Mov}}(X)$ for a smooth projective threefold.
\begin{description}
\item[Input]Extremal rays $\gamma_1, \ldots, \gamma_s$
of $\overline{\mathrm{Mov}}(X)$ with respect to the basis $\mathcal{B}_1$ of $N_1(X)_\mathbb{R}$;
extremal rays $\eta_1, \ldots, \eta_r$ of
$\mathrm{Nef}(X)$ with respect to the dual basis (with respect to the
intersection product) $\mathcal{B}^1$ of $N^1(X)_\mathbb{R}$;
intersection products of pairs from $\mathcal{B}^1$ in the basis $\mathcal{B}_1$.
\item[Output]Extremal rays $\gamma_{i_1}, \ldots, \gamma_{i_t}$ of
$\overline{\mathrm{Mov}}(X)$ not in $\mathcal{CI}(X)$ to a file \verb+NotEq+.
\end{description}
\begin{enumerate}
\item[1.a.] Read in $\gamma_1$.
\item[1.b.] For each pair $1 \leq i \leq j \leq r$, calculate $\eta_i
\cdot \eta_j$ with respect to $\mathcal{B}_1$. If $\eta_i \cdot \eta_j$ is a
non-trivial multiple of $\gamma_1$, continue to the next pair $1
\leq i' \leq j' \leq r$. Otherwise output $\gamma_1$ to \verb+NotEq+ and
continue to the next pair.
\item[2.a.] Read in $\gamma_2$.
\item[2.b.] \ldots \\
\item[$s$.a.]Read in $\gamma_s$.
\item[$s$.b.] For each pair $1 \leq i \leq j \leq r$, calculate $\eta_i
\cdot \eta_j$ with respect to $\mathcal{B}_1$. If $\eta_i \cdot \eta_j$ is a
non-trivial multiple of $\gamma_s$, continue to the next pair $1
\leq i' \leq j' \leq r$. Else output $\gamma_s$ to \verb+NotEq+ and
continue to the next pair.
\end{enumerate}
\end{algorithm}
The cones $\mathcal{CI}(X)$ and $\overline{\mathrm{Mov}}(X)$ are equal precisety when the file
\verb+NotEq+ is empty after running the algorithm. Note that the
second step for each ray $\gamma_i$ involves recalculating all generators for $(\mathrm{Nef}(X))^2$. This apparent inefficiency is in practice preferable to storing every generator $\eta_{t_1} \cdot \eta_{t_2}$ in an array due to memory requirements and the computational time required to access elements in this array.
An implementation of the algorithm as a C++ program for $X_r$, $r=2,
\ldots, 5$, is available at \href{http://www.math.hu-berlin/~larsen/papers.html}{www.math.hu-berlin/$\sim$larsen/papers.html} (for
$X_1$ the algorithm is easy to implement by hand). We obtain
enumerations of the extremal rays of $\mathrm{Nef}(X)$ and $\overline{\mathrm{Mov}}(X)$ by
inputting the inequalities from Propositions \ref{lemmaCI:nefCone} and
\ref{lemmaCI:nefCurvesCone} into software that implements
Fourier-Motzkin elimination, such as \verb+PORTA+ \cite{porta}. These \verb+PORTA+
files are also available at \href{http://www.math.hu-berlin/~larsen/papers.html}{www.math.hu-berlin/$\sim$larsen/papers.html}. Intersections of pairs of nef divisors are calculated according to Proposition \ref{lemmaCI:doubleInt}. Running these programs yields:
\begin{cor}
\label{corCI:CIvsNef}
There is a strict inclusion $\mathcal{CI}(\overline{M}_{0,6}) \subsetneq \overline{\mathrm{Mov}}(\overline{M}_{0,6})$, while $\mathcal{CI}(X_r) = \overline{\mathrm{Mov}}(X_r)$ for $r=1, \ldots, 4$.
\end{cor}
\noindent A mostly by-hand implementation of the algorithm appears in
Example \ref{exCI:toricthreefold}.
\begin{cor}
Extremal rays of $\overline{\mathrm{Mov}}(\overline{M}_{0,6})$ not contained in $\mathcal{CI}(\overline{M}_{0,6})$
intersect the canonical class of $\overline{M}_{0,6}$ negatively.
\end{cor}
\begin{proof}
The canonical class of $\overline{M}_{0,6}$ is
\begin{equation*}
[K_{\overline{M}_{0,6}}] = -4[H] + 2 \sum_{i=1}^5 [E_i] + \sum_{1 \leq j \leq k
\leq 5} [E_{jk}].
\end{equation*}
Inspection of the file \verb+NotEq+ for $r=5$ then yields the result.
\end{proof}
An obvious question to ask is whether the complete intersection and
movable cones coincide for all smooth projective toric varieties. We
next give an example of a toric blow-up of $\mathbb{P}^3$ for which the complete intersection cone is strictly contained in the movable cone.
\begin{figure}
\centering{
\toricthreefold
}
\caption{Fan of the toric variety from Example \ref{exCI:toricthreefold}}
\label{figCI:toricthreefold}
\end{figure}
\begin{example}
\label{exCI:toricthreefold}
Let $Y_2$ be the toric variety obtained by blowing up $\mathbb{P}^3$ first in
the line $V(z_1, z_3)$, followed by the blow-up of the
proper-transform of the line $V(z_1, z_2)$, where $\mathbb{C}[z_0, z_1, z_2,
z_3]$ is the homogeneous coordinate ring of $\mathbb{P}^3$. The variety $Y_2$ is smooth and projective, and its fan $\Sigma$ is depicted in Figure \ref{figCI:toricthreefold}, with the other segments indicating the two-faces of the fan. The standard facts about divisor classes and intersection theory on toric varieties reviewed and used in this example can be found in \cite{MR2810322}, Chapters 4 and 6.
The primitive generators $u_0, \ldots, u_5$ of the fan of $Y_2$ are, respectively,
\begin{equation*}
\left(\begin{array}{r} -1 \\ -1 \\ -1 \end{array} \right),
\left(\begin{array}{r} 1 \\ 0 \\ 0 \end{array} \right),
\left(\begin{array}{r} 0 \\ 1 \\ 0 \end{array} \right),
\left(\begin{array}{r} 0 \\ 0 \\ 1 \end{array} \right),
\left(\begin{array}{r} 1 \\ 0 \\ 1\end{array} \right),
\left(\begin{array}{r} 0 \\ 1 \\ 1\end{array} \right).
\end{equation*}
As usual, we label the torus-invariant divisors via the (primitive
generators of the) rays of $\Sigma(1)$, so $N^1(Y_2)_\mathbb{R}$ is generated
by the classes of the divisors $D_{0}, \ldots, D_5$. The relations among the classes of divisors $D_i$ are generated by
\begin{align*}
&[D_1] + [D_4] - [D_0] = 0, \\
&[D_2] + [D_5] - [D_0] = 0, \\
&[D_3] + [D_4] + [D_5] - [D_0] = 0.
\end{align*}
We now perform the calculations of Algorithm \ref{corCI:algorithm} to compare $\mathcal{CI}(Y_2)$ and $\overline{\mathrm{Mov}}(Y_2)$. It is clear from the relations in $N^1(Y_2)_\mathbb{R}$ that the pseudoeffective cone is
\begin{equation}
\overline{\mathrm{Eff}}(Y_2) = \langle [D_3], [D_4], [D_5] \rangle_{\geq 0}. \nonumber
\end{equation}
We therefore choose the basis for $N^1(Y_2)_\mathbb{R}$ consisting of the classes of $D_3$, $D_4$, and $D_5$, while for $N_1(Y_2)_\mathbb{R}$ we take the corresponding dual basis. It follows that the movable cone is
\begin{equation}
\overline{\mathrm{Mov}}(Y_2) = \langle [D_3]^{\vee}, [D_4]^{\vee}, [D_5]^{\vee} \rangle_{\geq 0}, \nonumber
\end{equation}
or, in coordinates, the non-negative orthant of $\mathbb{R}^3$.
As noted in Proposition \ref{propCI:pseffToric}, the closed cone of curves of $Y_2$ is generated by classes of orbit closures $V(\tau)$, $\tau \in \Sigma(2)$, and we will label them as $V(\tau) = C_{i,j}$, where $i$ and $j$ index the rays generating $\tau$.
Writing an arbitrary divisor class as $[D] = d_3 [D_3] + d_4 [D_4] + d_5 [D_5]$, the nef cone of $Y_2$ is defined by the inequalities
\begin{align*}
d_4 & =[C_{0,1}] \cdot [D] = [C_{1,2}] \cdot [D] = [C_{1,4}] \cdot [D]= [C_{2,5}] \cdot [D] \geq 0, \\
d_5 &=[ C_{0,2}] \cdot[ D] \geq 0, \\
-d_4 + d_5 & = [C_{2,4}] \cdot[ D] \geq 0, \\
-d_3 + d_4 + d_5 & =[ C_{0,3}] \cdot [D] = [C_{3,4}] \cdot[ D] = [C_{3,5}] \cdot[ D] \geq 0, \\
d_3 - d_5 & = [C_{0,5}]\cdot[ D] =[ C_{4,5}] \cdot[ D] \geq 0, \\
d_3 - d_4 & = [C_{0,4}] \cdot[ D] \geq 0.
\end{align*}
These intersections can be calculated via the geometry of the fan of $Y_2$ as in \cite{MR2810322}, Section 6.3.
For example, to obtain the fourth inequality above, we intersect $D$ with the curve $C_{0,3}$. Setting $C_{0,3} = V(\tau)$, with $\tau = \langle u_0, u_3 \rangle_{\geq 0}$, note that $\tau$ is contained precisely in the full-dimensional cones $\langle u_4, u_0, u_3 \rangle_{\geq 0}$ and $\langle u_0, u_3, u_5 \rangle_{\geq 0}$. We obtain from the coefficients of the linear dependence relation
\begin{equation*}
(1)\left(
\begin{array}{r}
1 \\
0 \\
1
\end{array} \right)
+(1)\left(
\begin{array}{r}
-1 \\
-1 \\
-1
\end{array}
\right)
+(-1)
\left(
\begin{array}{r}
0 \\
0 \\
1
\end{array}
\right)
+
(1)\left(
\begin{array}{r}
0 \\
1 \\
1
\end{array}
\right)
=
\left(
\begin{array}{r}
0 \\
0 \\
0
\end{array}
\right)
\end{equation*}
the intersection numbers $D_4 \cdot C_{0,3} = 1$, $D_0 \cdot C_{0,3} = 1$, $D_3 \cdot C_{0,3} = -1$, and $D_5 \cdot C_{0,3} = 1$, with all remaining intersection numbers equal to zero.
In particular, we obtain the coordinates for $C_{0,3}$ in the dual basis $N_1(Y_2)_\mathbb{R}$:
\begin{equation*}
[C_{0,3}] = -[D_3]^{\vee} + [D_4]^\vee + [D_5]^\vee = (-1,1,1).
\end{equation*}
Coordinates of the other generators of $N_1(Y_2)_\mathbb{R}$ with respect to
the dual basis $\{ [D_3]^{\vee}, [D_4]^{\vee}, [D_5]^{\vee} \}$ are:
\begin{align*}
&[C_{0,1}] = [C_{1,2} ] = [C_{1,4} ]= [C_{2,5} ] = (0,1,0), \\
&[C_{0,2} ] = (0,0,1) \\
& [C_{2,4} ] = (0,-1,1)\\
& [C_{0,3} ] = [C_{3,4} ] = [C_{3,5} ] =(-1,1,1), \\
& [C_{0,5}] = [C_{4,5}] = (1,0,-1), \\
& [C_{0,4}] = (1,-1,0).
\end{align*}
By a \verb+PORTA+ calculation, the nef cone is
\begin{equation*}
\mathrm{Nef}(Y_2) = \langle [D_3] + [D_5], 2[D_3] + [D_4] + [D_5], [D_3] + [D_4] + [D_5] \rangle_{\geq 0}.
\end{equation*}
We denote the three extremal rays by $\eta_1$, $\eta_2$, and $\eta_3$, respectively. To calculate all pairs of intersections $\eta_i \cdot \eta_j$, we first calculate $[D_r] \cdot [D_s]$ for $r,s = 3,4,5$. For self-intersections, we rewrite the divisor using the relations in $N^1(Y_2)_\mathbb{R}$ to make the intersection transverse. For example,
\begin{equation*}
[D_3]^2 = [D_3] \cdot([D_0] - [D_4] - [D_5]) = [C_{0,3}] - [C_{3,4}] - [C_{3,5}].
\end{equation*}
\end{example}
\noindent With respect to the dual basis $\{[D_3]^{\vee}, [D_4]^{\vee}, [D_5]^{\vee}\}$, we obtain $[D_3]^2 = (1, -1, -1)$. The other intersections are obtained analogously:
\begin{align*}
[D_4]^2 & = (1, -2, 0), \\
[D_5]^2 &= (1, -1, -1), \\
[D_3] \cdot [D_4] &= (-1, 1, 1), \\
[D_3] \cdot [D_5] &= (-1, 1, 1), \\
[D_4] \cdot [D_5] &= (1, 0, -1).
\end{align*}
Finally, we calculate the generators $\eta_i \cdot \eta_j$, $1 \leq i \leq j \leq 3$, in the dual basis $N_1(Y_2)_\mathbb{R}$ by using the above intersections among the basis elements of $N^1(Y_2)_\mathbb{R}$:
\begin{align*}
\eta_1^2 &= ([D_3] + [D_5])^2 = (0,0,0),\\
\eta_2^2 &= ( 2[D_3] + [D_4] + [D_5])^2 = (0,1,1),\\
\eta_3^2 &= ([D_3] + [D_4] + [D_5])^2 = (1,0,0),\\
\eta_1 \cdot \eta_2 &=( [D_3] + [D_5] ) \cdot ( 2[D_3] + [D_4] + [D_5] ) = (0,1,0),\\
\eta_1 \cdot \eta_3 &= ( [D_3] + [D_5] ) \cdot ([D_3] + [D_4] + [D_5]) = (0,1,0),\\
\eta_2 \cdot \eta_3 &= ( 2[D_3] + [D_4] + [D_5] ) \cdot([D_3] + [D_4] + [D_5]) = (0,1,1).
\end{align*}
Since the extremal ray $(0,0,1)$ of $\overline{\mathrm{Mov}}(Y_2)$ does not appear among the generators of $\mathcal{CI}(Y_2)$, it follows that $\mathcal{CI}(Y_2) \subsetneq \overline{\mathrm{Mov}}(Y_2)$.
To conclude this section, we use basic polyhedral geometry to give one example of how permutohedral spaces partially encode the geometry of $\overline{M}_{0,n}$. Namely, we show that extremal rays of the movable cone of $\overline{L}_{n-2}$ pull back to extremal rays of the movable cone of $\overline{M}_{0,n}$. Recalling the Kapranov blow-up construction, we define $f: \overline{M}_{0,n} \to \overline{L}_{n-2}$ to be the final (non-toric) composition of blow-ups.
\begin{prop}
\label{propCI:extRayMov}
Let $\gamma$ be an extremal ray of $\overline{\mathrm{Mov}}(\overline{L}_{n-2})$. Then $f^*(\gamma)$ is an extremal ray of $\overline{\mathrm{Mov}}(\overline{M}_{0,n})$.
\end{prop}
\begin{proof}
Note that
\begin{align*}
f^*: N_1(\overline{L}_{n-2})_\mathbb{R} &\to N_1(\overline{M}_{0,n})_\mathbb{R} \text{ and} \\
f_*: N^1(\overline{M}_{0,n})_\mathbb{R} &\to N^1(\overline{L}_{n-2})_\mathbb{R}
\end{align*}
are dual with respect
to the intersection pairing. Since $f_*$ is surjective, $f^*$ is
injective, and in particular $f^*(\gamma) \neq 0$. Moreover, by the
projection formula \ref{lemmaI:projection} with $[D] \in
\overline{\mathrm{Eff}}(\overline{M}_{0,n})$,
\begin{equation*}
[D] \cdot f^*(\gamma) = f_*([D]) \cdot \gamma \geq 0,
\end{equation*}
since $f_*([D])$ is also an effective divisor class, hence
$f^*(\gamma) \in \overline{\mathrm{Mov}}(\overline{L}_{n-2})$.
Set $\rho_{\overline{L}} = \dim N_1(\overline{L}_{n-2})_\mathbb{R}$ and
$\rho_{\overline{M}} = \dim N_1(\overline{M}_{0,n})_\mathbb{R}$. Extremality of
$\gamma$ implies that there exist $\rho_{\overline{L}} - 1$ linearly independent
defining hyperplanes of $\overline{\mathrm{Mov}}(\overline{L}_{n-2})$ intersecting $\gamma$ with
value zero, i.e. there exist linearly independent divisor classes $[D'_1],
\ldots, [D'_{\rho_{\overline{L}} - 1}]$ on $\overline{L}_{n-2}$ satisfying $[D'_i] \cdot \gamma = 0$
for all $i$. Next select $\rho_{\overline{L}} - 1$ divisor classes
$[D_i] \in N^1(\overline{M}_{0,n})_\mathbb{R}$ such that $f_*([D_i]) = [D_i]$ for all
$i$ (the $[D_i]$ are by construction linearly independent).
It is easy to see from the Kapranov blow-up construction that
\begin{equation*}
\ker f_* = \langle [E_J]: n-1 \in J\rangle,
\end{equation*}
while the projection formula implies that $[E_j] \cdot \gamma =0$ if
$n-1 \in J$.
Since the collection $\{[D_i]: i=1, \ldots, \rho_{\overline{L}}\} \cup \{[E_J]:
n-1 \in J\}$ is linearly dependent with cardinality $\rho_{\overline{M}} - 1$, it follows that $f^*(\gamma)$ is
an extremal ray of $\overline{\mathrm{Mov}}(\overline{M}_{0,n})$.
\end{proof}
By applying this proposition to $\overline{M}_{0,6}$ and varying which marked
points are chosen as poles for $\overline{L}_4$ (this choice is explained for
example in \cite{larsenThesis}, Sec. 3.3), we can obtain an enumeration of extremal rays common to $\overline{\mathrm{Mov}}(\overline{M}_{0,6})$ and $\mathcal{CI}(\overline{M}_{0,6})$. This collection, however, does not give all common extremal rays: for example, the extremal ray
\begin{equation}
\gamma = 6 [H]^\vee + 2 \sum_{i =1}^4 [E_i]^\vee + [E_{15}]^\vee + [E_{25}]^\vee + [E_{35}]^\vee, \nonumber
\end{equation}
and its symmetric analogues, is an extremal ray of both $\mathcal{CI}(\overline{M}_{0,6})$ and $\overline{\mathrm{Mov}}(\overline{M}_{0,6})$, but it is not the pull-back of an extremal ray from $\overline{\mathrm{Mov}}(\overline{L}_4)$, as can be seen by examining the \verb+PORTA+ file for $\overline{\mathrm{Mov}}(\overline{L}_4)$.
\section{Definitions and background}
\label{secCI:background}
In this section we give definitions from intersection theory on a complex projective variety before focusing on the particular examples of toric varieties and the moduli space of stable pointed rational curve, $\overline{M}_{0,n}$. We refer to \cite{MR2095471}, \cite{MR1644323}, and Appendix A of \cite{MR0463157} for the basics of intersection theory, and \cite{MR1034665} and \cite{larsenThesis} for background and examples involving $\overline{M}_{0,n}$.
Let $X$ be a smooth complex projective variety, with $\mathrm{Div}_{\mathbb{R}}(X)$ denoting the space of $\mathbb{R}$-linear formal sums of algebraic hypersurfaces on $X$ (called \emph{$\mathbb{R}$-divisors} on $X$). There is a well-defined intersection pairing between $\mathbb{R}$-divisors and $\mathbb{R}$-linear formal sums of algebraic curves on $X$ (called \emph{one-cycles}), which we denote by ``$\cdot$''.
\begin{defn}
\label{def:NS}
Two divisors $D_1, D_2 \in \mathrm{Div}_{\mathbb{R}}(X)$ are said to be \emph{numerically equivalent} if for all algebraic curves $C \subseteq X$, $D_1 \cdot C = D_2 \cdot C$.
\end{defn}
\noindent We thus obtain an equivalence relation on $\mathrm{Div}_{\mathbb{R}}(X)$, and denote the numerical equivalence class of a divisor $D$ by $[D]$.
\begin{defn}
The \emph{N\'eron-Severi} space of $X$ is defined as
\begin{equation*}
N^1(X)_{\mathbb{R}} = \{ [D] : D \in \mathrm{Div}_{\mathbb{R}}(X) \},
\end{equation*}
and the dual vector space induced by the intersection product is denoted $N_1(X)_\mathbb{R}$.
\end{defn}
\noindent We will also denote the numerical class of a one-cycle $C$ as $[C] \in N_1(X)_\mathbb{R}$. A key fact for what follows is that $N^1(X)_\mathbb{R}$ and $N_1(X)_\mathbb{R}$ are finite-dimensional $\mathbb{R}$-vector spaces.
\begin{defn}
\label{def:effDiv}
The \emph{pseudoeffective cone} of divisors, written $\overline{\mathrm{Eff}}(X)$, is the closed subcone of $N^1(X)_\mathbb{R}$ generated by classes of effective divisors. Explicitly, $\overline{\mathrm{Eff}}(X)$ is the closure in $N^1(X)_\mathbb{R}$ of
\begin{equation*}
\mathrm{Eff}(X) = \big\{ \sum d_i [D_i]: d_i \geq 0, D_i \textrm{ an effective divisor on }X \big\}.
\end{equation*}
\end{defn}
\noindent The analogous cone in $N_1(X)_\mathbb{R}$ is called the \emph{closed} (or \emph{Mori}) \emph{cone of curves}:
\begin{defn}
Define $\overline{\mathrm{NE}}(X)$ as the closed subcone of $N_1(X)_\mathbb{R}$ generated by classes of algebraic curves, that is, the closure in $N_1(X)_\mathbb{R}$ of
\begin{equation*}
\mathrm{NE}(X) = \big\{ \sum c_i [C_i]: c_i \geq 0, C_i \subseteq X \textrm{ an algebraic curve} \big\}.
\end{equation*}
\end{defn}
\begin{defn}
The cone of \emph{nef divisors} on $X$ is
\begin{equation}
\mathrm{Nef}(X) = \{[D] \in N^1(X)_\mathbb{R}: D \cdot C \geq 0 \textrm{ for all }[C] \in \overline{\mathrm{NE}}(X)\} = (\overline{\mathrm{NE}}(X))^{\vee}.\nonumber
\end{equation}
\end{defn}
We will sometimes reduce intersection properties on $\overline{M}_{0,n}$ to intersections on a more amenable variety via the \emph{projection formula}. We use this result for numerical equivalence classes on non-singular varieties, where the formula takes on a particularly simple form (see \cite{MR1644323}, Proposition 8(c), noting that rational equivalence is finer than numerical equivalence). We only give the formula for divisors and one-cycles, though the statement holds for all pairs of complementary dimensional subvarieties.
\begin{lemma}[Projection formula]
\label{lemmaI:projection}
Let $f: X \to Y$ be a proper morphism of non-singular varieties. For $\delta \in N^1(X)_\mathbb{R}$ and $\gamma \in N_1(Y)_\mathbb{R}$,
\begin{equation*}
f_*(\delta \cdot f^*\gamma) = f_*(\delta) \cdot \gamma.
\end{equation*}
\end{lemma}
We next give a short description of the moduli space $\overline{M}_{0,n}$. Set theoretically, it is defined as follows:
\begin{defn}
For $n \in \mathbb{N}$, $n \geq 3$, the elements of $\overline{M}_{0,n}$ are equivalence classes of
\begin{equation*}
\{ (C, p_1, \ldots, p_n): C \textrm{ a tree of } \mathbb{P}^1s, p_i \in C \textrm{ distinct} \}
\end{equation*}
such that
\begin{enumerate}
\item all marked points $p_i$ are distinct from the nodes of $C$,
\item each irreducible component $C$ contains at least three marked or singular points,
\item two marked curves $(C, p_1, \ldots, p_n)$ and $(C', q_1, \ldots, q_n)$ are equivalent if there is an isomorphism $\phi: C \to C'$ with $\phi(p_i) = q_i$ for all $i$.
\end{enumerate}
\end{defn}
The resulting moduli space is a smooth projective variety \cite{MR702953}, and can be realized as a sequence of blow-ups of $\mathbb{P}^{n-3}$ along linear centers \cite{MR1237834}. This blow-up construction will feature heavily in the remainder, and although it corresponds to that of \cite{MR1237834}, the ordering of the blow-ups differs slightly from that given in subsequent literature. First pick $n-2$ general points in $\mathbb{P}^{n-3}$; via a projective transformation, we may choose $x_1 = [1, 0, \ldots, 0]$, $\ldots$, $x_{n-2} = [0, \ldots, 0, 1] \in \mathbb{P}^{n-3}$. Note that these points are invariant under the action of the torus $(\mathbb{C}^*)^{n-3}$. First blow up $\mathbb{P}^{n-3}$ iteratively along $x_1, \ldots, x_{n-2}$, then along the proper transforms of the lines spanned by pairs of $x_1, \ldots, x_{n-2}$, and continue blowing up proper transforms of linear subspaces spanned by these points until all codimension two subspaces have been blown up. Since all blow-up centers were torus invariant, the result is a smooth projective toric variety.
\begin{defn}
The variety obtained from the above blow-ups of $\mathbb{P}^{n-3}$ is the \emph{permutohedral} or \emph{Losev-Manin moduli space} $\overline{L}_{n-2}$.
\end{defn}
\noindent The name \emph{permutohedral space} is given in \cite{MR1237834} since the corresponding polytope in a permutahedron, while the second variant reflects the modular interpretation given in \cite{MR1786500}. More on these varieties can be found in \cite{blume1} and \cite{pllCoxRelns}.
To obtain $\overline{M}_{0,n}$ from $\overline{L}_{n-2}$ we first blow up the point $x_{n-1} = [1, \ldots, 1]$, and then---in order of increasing dimension---all remaining proper transforms of linear subspaces spanned by $x_1, \ldots, x_{n-1}$. These blow-up constructions also give natural bases for $N_1(\overline{L}_{n-2})_\mathbb{R}$ and $N_1(\overline{M}_{0,n})_\mathbb{R}$, which we will call the \emph{Kapranov basis}:
\begin{itemize}
\item For $\overline{L}_{n-2}$, denote by $[H]$ the pull-back of the hyperplane class on $\mathbb{P}^{n-3}$, and by $[E_J]$ with $J \subseteq \{1, \ldots, n-2\}$, $1 \leq |J| \leq n-4$ the proper transforms of the exceptional divisors resulting from blowing up a linear subspace spanned by the collection of $\{x_1, \ldots, x_{n-2}\}$ indexed by $J$.
\item For $\overline{M}_{0,n}$, we abuse notation by again writing $[H]$ for the pull-back of the hyperplane class, and as above $[E_J]$ for the proper transforms of exceptional divisors, where now $J \subseteq \{1, \ldots, n-1\}$.
\end{itemize}
We will be especially concerned with two collections of divisor and curve classes on $\overline{M}_{0,n}$, namely those of \emph{boundary divisors} and \emph{F-curves}.
\begin{defn}
For $J \subseteq \{1, \ldots,n \}$ with $2 \leq |J| \leq n-2$, the \emph{boundary divisor} $\Delta_J$ is the locus of elements in $\overline{M}_{0,n}$ whose underlying curve can be decomposed into two components $C = C' \cup C''$ such that the marked points on $C'$ are indexed by $J$ and those of $C''$ are indexed by $J^c$.
\end{defn}
\noindent We will tacitly identify $\Delta_J$ and $\Delta_{J^c}$. The boundary divisors form the codimension one constituents of a stratification of $\overline{M}_{0,n}$ by dual-graph. The dimension one elements of this stratification are known as \emph{F-curves}. The numerical equivalence class of an F-curve is uniquely determine by a partition of $\{1, \ldots, n\}$ into four subsets, $\{\mu_1, \mu_2, \mu_3, \mu_4\}$.
To go from F-curves to such partitions, let $(C, p_1, \ldots, p_n) \in \overline{M}_{0,n}$ be a generic element of an F-curve. Then $C$ can be decomposed as $C = C_{spine} \cup C_1 \cup C_2 \cup C_3 \cup C_4$, where $C_{spine}$ is a $\mathbb{P}^1$ with four special points, and the marked points indexed by $\mu_i$ are located on $C_i$ for all $i$. That this partition uniquely determines the numerical class of an F-curve results from the following intersection pairings, proved in \cite{km}.
\begin{prop}
\label{propI:FcurvePartition}
Let $F_\mu$ be a one stratum, with corresponding partition $\mu = (\mu_1, \mu_2, \mu_3, \mu_4)$. For any boundary divisor $\Delta_J$,
\begin{displaymath}
F_\mu \cdot \Delta_J =
\left\{ \begin{array}{rl}
-1 & \text{if $J$ or $J^c$ equals $\mu_i$ for some $i$} \\
1 & \text{if $J = \mu_i \cup \mu_j$ for some $i \neq j$},\\
0 & \text{otherwise}.
\end{array} \right.
\end{displaymath}
\end{prop}
\noindent Since classes of boundary divisors generate $N^1(\overline{M}_{0,n})_\mathbb{R}$, these intersection numbers uniquely determine the class of $F$.
We will require one final fact about intersection theory on $\overline{M}_{0,n}$ relating boundary divisors and elements of the Kapranov basis:
\begin{equation}\label{eq:dictionary}
\begin{split}
\Delta_{J \cup \{n\}} &= E_J, \textrm { if } 1 \leq |J| \leq n-4, \\
\big[\Delta_{J \cup \{n\}}\big] & = [H] - \bigg(\sum_{J' \subsetneq J}[E_{J'}] \bigg), \textrm{ if } |J| = n-3.
\end{split}
\end{equation}
Except for small values of $n$, little is known about the pseudoeffective cone of divisors or the closed cone of curves for $\overline{M}_{0,n}$. The pseudoeffective cone of $\overline{M}_{0,n}$ (and hence, by duality, the movable cone of curve classes) is known to be finitely generated only for $n \leq 6$ \cite{MR1941624,MR2491903}, while finite-generation of the closed cone of curve classes of $\overline{M}_{0,n}$ (and hence, by duality, the cone of nef divisor classes) has been proven for $n \leq 7$ \cite{km, Larsen16112011}.
For $\overline{M}_{0,6}$, the closed cone of curve classes is generated by classes of F-curves, while the pseudoeffective cone of $\overline{M}_{0,6}$ is generated by the boundary divisors $\Delta_J$, and the \emph{Keel-Vermeire} divisors \cite{MR1882122}. In the Kapranov blow-up description of $\overline{M}_{0,6}$, Keel-Vermeire divisors are the pull-backs under the blow-up morphism $t_6: \overline{M}_{0,6} \to \mathbb{P}^3$ of the unique quadric surface containing points $p_1$, $\ldots$, $p_5$, and the lines $l_{ac}$, $l_{ad}$, $l_{bc}$ and $l_{bd}$. Taking $(a,b,c,d) = (1,2,3,4)$, the Keel-Vermeire divisor $Q_{(12)(34)(56)}$ has numerical class
\begin{equation}
\label{eqCI:KV}
[Q_{(12)(34)(56)}] = 2[H] - \sum_{i=1}^5[E_i] - [E_{13}] - [E_{14}] - [E_{23}] - [E_{24}].
\end{equation}
\noindent The remaining fourteen Keel-Vermeire divisors arise by varying the indexing product of two cycles (here meant in terms of symmetric groups, not algebraic cycles).
For toric varieties, each of the cones described above is finitely generated, and admits an explicit description (sometimes more than one) in combinatorial terms. The starting point for understanding the various cones of a toric variety $X_\Sigma$ of dimension $d$ is the Orbit-Cone correspondence (see for example \cite{MR2810322}, §3.2 and §6.3). Recalling that $\Sigma(k)$ denotes the $k$-dimensional cones of the fan $\Sigma$, and $V(\s)$ is the codimension $k$ subvariety corresponding to $\s \in \Sigma(k)$, we have the following descriptions of $\overline{\mathrm{Eff}}(X_\Sigma)$ and $\overline{\mathrm{NE}}(X_\Sigma)$:
\begin{prop} For a complete toric variety $X_\Sigma$,
\label{propCI:pseffToric}
\begin{equation}
\overline{\mathrm{Eff}}(X_\Sigma) = \langle [V(\rho)]: \rho \in \Sigma(1) \rangle_{\geq 0}, \nonumber
\end{equation}
while
\begin{equation}
\overline{\mathrm{NE}}(X_\Sigma) = \langle [V(\tau)]: \tau \in \Sigma(d-1) \rangle_{\geq 0}. \nonumber
\end{equation}
\end{prop}
\noindent The duals of these cones, $\overline{\mathrm{Mov}}(X_\Sigma)$ and $\mathrm{Nef}(X_\Sigma)$, can be calculated by the combinatorics of the defining fans. Example calculations appear in Example \ref{exCI:toricthreefold}.
\section{Introduction}
\label{secCI:introduction}
A foundational result in the geometry of projective varieties is Kleiman's theorem \cite{MR0206009}, which states the closure of the ample cone equals the nef cone. The containment of the ample cone in the nef cone is easy to prove, and since the nef cone is by definition closed, one inclusion of cones follows. The proof of the opposite inclusion is more involved; see \cite{MR0206009}, or Section 1.4.C of \cite{MR2095471}.
By duality, Kleiman's theorem is equivalent to the equality $\overline{\mathrm{NE}}(X)^{\vee} = \overline{\mathrm{Amp}}(X)$. It is natural to wonder which other cones of divisor and curve classes fit into a Kleiman-type duality. For the pseudoeffective cone of divisor classes, it is not difficult to see that dual cone $\overline{\mathrm{Eff}}(X)^{\vee}$ contains the closure of the cone of movable curve classes, where a reduced, irreducible curve $C$ is called a \emph{movable curve} if $C= C_{t_0}$ belongs to an algebraic family $(C_t)_{t \in S}$ covering $X$. To see this inclusion,
let $D$ be an effective prime divisor, and let $C$ be a movable curve. Since the support of $D$ is a codimension one subvariety, there must exist an irreducible curve $C'$ in the covering family containing $C$ such that $C'$ is not contained in the support of $D$, hence $C' \cdot D \geq 0$. Since algebraic equivalence is finer than numerical equivalence, it follows that $C \cdot D \geq 0$. The other inclusion was proved in 2004 by Boucksom, Demailly, P\v aun, and Peternell in \cite{bdpp}, where they also give an alternative characterization of the cone of movable curve classes:
\begin{defn}
\label{defCI:movable}
Let $\mu: X' \to X$ be a projective, birational morphism. A class $\gamma \in \overline{\mathrm{NE}}(X)$ is called \emph{movable} if there exists a representative one-cycle $C$ and ample divisors $A_1, \ldots, A_{\dim(X) -1}$ on $X'$ such that
\begin{equation}
\mu_*(A_1 \cdot \ldots \cdot A_{\dim(X)-1}) = C \nonumber.
\end{equation}
The closure of the cone generated by movable classes in $\overline{\mathrm{NE}}(X)$ is called the \emph{movable cone}, and is denoted $\overline{\mathrm{Mov}}(X)$.
\end{defn}
Both formulations involve non-trivial existence statements: in the first, to see that a curve $C$ is movable, we must prove the existence of a covering family to which it belongs, and in the second, we require knowledge about all projective, birational morphisms to the variety $X$. If, however, we consider only the identity morphism, we obtain a subcone of $\overline{\mathrm{Mov}}(X)$ called the \emph{complete intersection cone}:
\begin{defn}
\label{defCI:ci}
The \emph{complete intersection} cone of $X$, denoted $\mathcal{CI}(X)$, is the closed cone generated by the classes of all smooth curves obtained as an intersection of $\dim(X)-1$ ample divisors on $X$.
\end{defn}
The aim of this paper is to investigate when these cones of curve classes do and do not coincide for two natural testing grounds: moduli spaces of curves and toric varieties. Were there actual equality $\mathcal{CI}(X) = \overline{\mathrm{Mov}}(X)$, then we could characterize movable curves without having to first classify all birational morphisms to $X$. A disadvantage of working with the complete intersection cone, however, is the combinatorial complexity of $\mathcal{CI}(X)$, especially when the nef cone of $X$ has a large number of extremal rays.
\begin{example}
\label{exCI:CIvsMovSurface}
Let $X$ be a smooth projective surface. Then one-cycles and divisors coincide, so $\overline{\mathrm{Eff}}(X)^{\vee} = \mathrm{Nef}(X) = \mathcal{CI}(X)$, where the second equality follows from Kleiman's theorem, since by definition $\mathcal{CI}(X)$ is the closure of the ample cone.
\end{example}
\begin{example}
\label{exCI:CIvsMovPn}
Let $X= \mathbb{P}^n$, and let $H \subseteq \mathbb{P}^n$ be a hyperplane, and let $\ell \subseteq \mathbb{P}^n$ be a line. Then $N^1(\mathbb{P}^n)_\mathbb{R} = \langle [H]\rangle$ and $\overline{\mathrm{Eff}}(\mathbb{P}^n) = \mathrm{Nef}(\mathbb{P}^n) = \langle [H] \rangle_{\geq 0}$, while $N_1(\mathbb{P}^n)_\mathbb{R} = \langle [\ell]\rangle$, and $\overline{\mathrm{Eff}}(\mathbb{P}^n)^\vee = \overline{\mathrm{Mov}}(\mathbb{P}^n) = \langle [\ell] \rangle_{\geq 0}= \langle [H]^{n-1} \rangle_{\geq 0}$, hence $\mathcal{CI}(\mathbb{P}^n) = \overline{\mathrm{Mov}}(\mathbb{P}^n)$.
\end{example}
Peternell has calculated an example of a smooth projective threefold for which the containment of the complete intersection cone in the movable cone is strict \cite{pMov}, but one can ask if there are natural families of varieties for which these cones coincide. Two obvious testing grounds are toric varieties and moduli spaces of stable pointed rational curves, since the intersection theory on these varieties is well-understood. A connection between these two families is the Kapranov blow-up construction. In \cite{MR1237834}, $\overline{M}_{0,n}$ is constructed by a series of toric blow-ups of $\mathbb{P}^{n-3}$, culminating in the permutohedral or Losev-Manin moduli space $\overline{L}_{n-2}$, followed by (for $n \geq 5$) additional blow-ups along non-torus-invariant centers.
Example \ref{exCI:CIvsMovPn} can be taken as the base case of a progression of varieties obtained by successive Kapranov-like blow-ups. More specifically, setting $X_0 = \mathbb{P}^3$, the next variety we take to be the blow-up of $\mathbb{P}^3$ at a general point, labeling the resulting variety $X_1$. We define $X_2$ to be the blow-up of $\mathbb{P}^3$ along two general points, and the proper transform of the line spanned by the points. In general, for $1 \leq r \leq 5$, we blow-up $r$ points of $\mathbb{P}^3$ in general linear position, and then the proper transforms of the $\binom{r}{2}$ lines generated by the $r$ points. For $r \leq 4$, the centers of the blow-ups can be chosen to be torus-invariant. Then $X_4$ is the permutohedral space $\overline{L}_4$, while $X_5$ is $\overline{M}_{0,6}$. The complete intersection and movable cones of the first few varieties $X_r$ can be computed easily to show that these cones coincide, but there is little reason to expect this equality of cones to be preserved under increasing blow-ups.
The main result of this paper is the following:
\begin{thm}
\label{thmCI:CIvsNef}
There is a strict inclusion $\mathcal{CI}(\overline{M}_{0,6}) \subsetneq \overline{\mathrm{Mov}}(\overline{M}_{0,6})$, while for the toric varieties $X_r$, $1 \leq r \leq 4$, equality holds: $\mathcal{CI}(X_r) = \overline{\mathrm{Mov}}(X_r)$.
\end{thm}
\noindent In other words, the containment of these cones becomes strict when we leave the toric world in the Kapranov construction of $\overline{M}_{0,6}$.
We prove this theorem by reinterpreting the complete intersection cone in combinatorial terms (see Definition \ref{defCI:Nefnm1} and Lemma \ref{lemmaCI:cinef2}). Since the nef and pseudoeffective cones of $\overline{M}_{0,6}$ and $\overline{L}_4$ are finitely generated, it follows by this reinterpretation that equality of the moving and complete intersection cones can be tested by an algorithm that requires as input the extremal rays of the nef and effective cones of divisors, plus intersection products of divisors (see Section \ref{secCI:ci}).
That the complete intersection and movable cones coincide for the toric blow-ups of Theorem \ref{thmCI:CIvsNef} might give hope that these cones coincide for smooth projective toric varieties. It turns out, however, that even for a toric blow-up of projective space the complete intersection cone need not equal the movable cone. In Example \ref{exCI:toricthreefold}, we produce such a toric variety.
The remainder of this paper is organized as follows. We begin with some generalities on the pseudoeffective and nef cones of divisors, as well as the the closed cones of curves, for $\overline{M}_{0,n}$ and the other blow-ups $X_r$ in Section \ref{secCI:background}. In Section \ref{secCI:ci}, we establish a combinatorial definition of the complete intersection cone, and describe the algorithm used to prove Theorem \ref{thmCI:CIvsNef}. We also show that extremal movable curve classes of the toric variety $\overline{L}_{n-2}$ pull back to extremal classes in $\overline{M}_{0,n}$. Section \ref{secCI:intTh} contains proofs for intersection calculations used for our algorithm.
\textbf{Acknowledgments}: I would first like to thank Gavril Farkas for suggesting this problem, and for his help throughout. This project has benefitted greatly from conversations with Nathan Ilten, Sam Payne, and Thomas Peternell. I would also like to thank Klaus Altmann and Angela Gibney for comments on an earlier version of this paper.
\section{Calculating complete intersection and movable cones}
\label{secCI:intTh}
\begin{proof}[Proof of Proposition \ref {lemmaCI:nefCone}]
We present the proof only for $r=5$, since the other cases are standard (for details, see Chapter 4 of \cite{larsenThesis}).
These inequalities will follow by intersecting the divisor class $[D]$ with all classes of F-curves. Using the identification of the hyperplane class with the psi-class $\psi_6$ from \cite{MR1203685}, it is not hard to show that for each partition $\mu = (\mu_1, \mu_2, \mu_3, \mu_4)$,
\begin{equation*}
H \cdot F_\mu =
\begin{cases}
1 & \text{ if }\mu_i = \{6\} \text{ for some } i, \\
0 & \text{else}.
\end{cases}
\end{equation*}
To intersect F-curves with the remaining elements of the Kapranov basis, we apply Proposition \ref{propI:FcurvePartition} and the dictionary between boundary and exceptional divisor classes from Equations (\ref{eq:dictionary}). The first set of inequalities arise from F-curves with partitions $(\mu_1, \mu_2, \mu_3, \mu_4)$ satisfying $|\mu_i| = 1$ for $1 \leq i \leq 3$, with $6 \in \mu_4$, while the second set of inequalities comes from such partitions with instead $6 \notin \mu_4$. Partitions with $|\mu_1| = |\mu_2| = 1$ and $|\mu_3| = |\mu_4| = 2$ such that $i \in \mu_3 \cup \mu_4$ give the third set of inequalities, while the final set results from such partitions when $6 \in \mu_1 \cup \mu_2$.
Up to the action of the symmetric group permuting the four elements of the partition (which leaves the numerical class unchanged), the above partitions correspond to all possible partitions corresponding to F-curves in $\overline{M}_{0,6}$, so these inequalities define the nef cone.
\end{proof}
\begin{proof}[Proof of Proposition \ref{lemmaCI:nefCurvesCone}]
Since we represent $[C] \in N_1(\overline{M}_{0,6})_\mathbb{R}$ with respect to the dual Kapranov basis, by duality these inequalities can just be read off of the coordinates of the generators for $\overline{\mathrm{Eff}}(\overline{M}_{0,6})$ expressed in the Kapranov basis for $N^1(\overline{M}_{0,6})_\mathbb{R}$: the first set of inequalities are from intersecting $[C]$ with the $[E_i]$, the second from intersecting with the $[E_{ij}]$, the third from intersecting with $[\Delta_{ij}]$ where $6 \notin \{i,j\}$, and the last from intersecting with the Keel-Vermeire divisors.
\end{proof}
\begin{proof}[Proof of Proposition \ref{lemmaCI:doubleInt}]
The first equality of (\ref{eq:intZero}) holds since we can always choose a hyperplane not containing any of the points $p_i$. Since exceptional divisors corresponding to disjoint blow-up centers are also disjoint, the remaining equalities follow immediately, with the possible exception of the final one. To see that $E_{ij} \cap E_{ik} = \emptyset$ for $i,j,k$ distinct, let $\ell_{ij}, \ell_{ik} \subseteq \mathbb{P}^3$ be the corresponding lines, and $p_i$ their intersection. Since the Kapranov construction requires blowing up in order of increasing dimension, after blowing up $p_i$ the proper transforms of $\ell_{ij}$ and $\ell_{ik}$ will be disjoint, giving last equality.
To express the intersections of divisor classes $[D'], [D'']$ in (\ref{eq:intDual}) in the dual basis, we intersect an arbitrary divisor class $[D] = d_h [H] + \sum_{i=1}^r [E_i] + \sum_{j,k = 1}^r [E_{jk}]$ with $[D'] \cdot [D'']$, giving an expression in the coefficients $d_h, d_i,$ and $d_{jk}$. Since the dual bases are related by the intersection product, this expression is the one-cycle $[D'] \cdot [D'']$ in the dual Kapranov basis once we substitute $[H]^\vee$ for $d_h$, $[E_i]^\vee$ for the $d_i$, and $[E_{jk}]^\vee$ for the $d_{jk}$. The triple intersection products required are standard calculations on $\overline{M}_{0,6}$ and toric varieties. For details, see Chapter 4 of \cite{larsenThesis}.
\end{proof}
|
2,869,038,156,387 | arxiv | \section{Introduction}
Blazars are a class of radio-loud Active Galactic Nuclei (AGN) hosting a jet oriented at a small
angle with respect to the line of sight \citep{Blandford1978, Antonucci1993, Urry1995}.
The emission of these objects is non-thermal over most or the entire electromagnetic spectrum, from radio frequencies to hard $\gamma$-rays.
The observed radiation shows extreme properties, mostly due to relativistic amplification effects.
The observed Spectral Energy Distribution (SED) presents a general shape composed of two bumps, one typically located in the infrared (IR) and sometimes extending to the X-ray band and the other one in the hard X-ray to $\gamma$-rays.
If the peak frequency of the synchrotron bump ($\nu_{\rm peak}$) in $\nu$ - $\nu$F$_{\nu}$ space is larger than $10^{15}$~Hz (corresponding to $\sim$ 4~eV), a source is usually called High Synchrotron Peaked (HSP) blazars\citep{Padovani1995,Abdo2010}.
HSP blazars are also considered to be extreme sources since the Lorentz factor of the electrons radiating at the peak of the synchrotron bump $\gamma_{peak}$ are the highest within the blazar population, and likely of any other type of steady cosmic sources.
Considering a simple SSC model where $\nu_{\rm peak}=3.2 \times 10^6 \gamma^{2}_{\rm peak} B \delta $ \citep[e.g.][]{Giommi2012a}, assuming $B=0.1$ Gauss and Doppler factor $\delta =10$, HSPs characterized by $\nu_{\rm peak} $ ranging between $ 10^{15}$ and ${\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}} 10^{18}$~Hz demand $\gamma_{\rm peak} \approx 10^4-10^6$.
The typical two-bump SED of blazars and the high energies that characterize HSPs imply that these objects occupy a distinct position in the optical to X-ray spectral index ($\alpha_{\rm ox}$) versus the radio to optical spectral index ($\alpha_{\rm ro}$) colour-colour diagram \citep{Stocke1991}.
Considering the distinct spectral properties of blazars over the whole electromagnetic spectrum, selection methods based on $\alpha_{\rm ox}$ and $\alpha_{\rm ro}$ have long been used to search for new blazars.
For example, \citet{Schachter1993} discovered 10 new BL Lacs via a multi-frequency approach with radio, optical and X-ray data, and
their BL Lac nature with optical spectra.
HSP blazars play a crucial role in very high energy (VHE) astronomy.
Observations have shown that HSPs are bright and variable sources of high energy {$\gamma$-ray}\, photons (TeVCat)\footnote{http://tevcat.uchicago.edu} and that they
are likely the dominant component of the extragalactic VHE background \citep{Padovani1993,Giommi2006,DiMauro2014,Giommi2015,Ajello2015}.
In fact, most of the extragalactic objects detected so far above a few GeV are HSPs \citep[][see also TeVCat]{Giommi2009,Padovani2015a,Arsioli2015a,Fermi2fhl2015}.
However, it is known that only a few hundred HSP blazars are above the sensitivity limits of currently available $\gamma$-ray surveys.
For example, the 1WHSP catalog \citep[][hereafter Paper I]{Arsioli2015a}, which was the largest sample of HSP blazars when it was published, shows that
out of the 992 objects in the sample, 299 have an associated {$\gamma$-ray}\, counterpart in the {\it Fermi} 1/2/3FGL catalogs.
Nevertheless there is a considerable number of relatively bright HSPs which still lack a {$\gamma$-ray}\, counterpart.
These are likely faint point-like sources at or below the {\it Fermi}-LAT, detectability threshold
and were not found by the automated searches carried out so far.
Indeed, \citet{Arsioli2016} have detected $\approx\,150$ new {$\gamma$-ray}\, blazars based on a specific search around bright WHSP sources, using over 7 years of {\it Fermi}-LAT Pass 8 data.
In the most energetic part of the {$\gamma$-ray}\, band photons from high redshift sources are absorbed by the extragalactic background light (EBL) emitted by galaxies and quasars \citep{Dermer2011,Pfrommer2013,Bonnoli2015}.
Therefore, the TeV flux can drop by a very large factor compared to GeV fluxes, making distant TeV sources much more difficult to detect.
Paper I has shown that with the help of multi-wavelength analysis, HSP catalogs can provide many good candidates for VHE detection.
The currently known HSP blazars are listed in catalogs such as the 5th {\it Roma-BZCAT} \citep[][hereafter 5BZCat]{Massaro2015}, the Sedentary Survey \citep{Giommi1999,Giommi2005,Piranomonte2007}, \citet{Kapanadze2013}, and Paper I.
However, the number of known HSPs is still relatively small with less than $\approx 1000$ cataloged HSPs till now.
Significantly enlarging the number of high energy blazars is important to better understand
their role within the AGN phenomenon, and should shed light on the cosmological evolution of blazars, which is still a matter of debate.
The 5BZCat is the largest compilation of confirmed blazars, containing 3561 sources, around 500 of which are of the HSP type.
It includes blazars discovered in surveys carried out in all parts of the electromagnetic spectrum
and is also based on an extensive review of the literature and optical spectra.
The Sedentary survey comprises 150 extremely high X-ray to radio flux ratio $(\log f_{\rm x}/f_{\rm r}\geq 3\times10^{-10}~{\rm erg}~{\rm cm}^{-2}~{\rm s}^{-1}~{\rm Jy}^{-1})$ HSP BL Lacs.
The sample was obtained by cross-matching the RASS catalog of bright X-ray sources \citep{Voges1999} and the NVSS 1.4~GHz radio catalog \citep{Condon1998}.
\citet{Kapanadze2013} built a catalog of 312 HSPs with flux ratio $(f_{\rm x}/f_{\rm r}\geq 10^{-11}~{\rm erg}~{\rm cm}^{-2}~{\rm s}^{-1}~{\rm Jy}^{-1})$ selected from various X-ray catalogs, the NVSS catalog of radio sources, and the first edition of the $Roma-BZCAT$ catalog \citep{Massaro2009}.
The 1WHSP sample relied on a pre-selection based on Wide-field Infrared Survey Explorer (WISE)
IR colours, SED slope criteria, and $\nu_{\rm peak}$~$>10^{15}$ Hz. It includes 992 known, newly-identified, and candidate high galactic latitude ($b>|20^\circ|$) HSPs.
In a series of papers \cite{Massaro2011,DAbrusco2012,Massaro2012} showed that most blazars occupy a specific region of the IR colour-colour diagram, which they termed the blazar strip.
In Paper I we extended the blazar strip in the WISE colour-colour diagram to include all the Sedentary Survey blazars and called it the {\it Sedentary WISE colour domain} (SWCD).
The SWCD is wider than the WISE blazar strip since it contains some blazars whose host galaxy
is very bright, such as MKN421 (2WHSP J110427.3+381230) and MKN 501 (2WHSP J165353.2+394536).
We understood from previous work that many low-luminosity HSP blazars have the IR colours dominated by the thermal component of the host giant elliptical galaxy.
Therefore, a selection scheme adopting IR colour restrictions may work effectively for selecting cases where the non-thermal jet component dominates the IR band but is less efficient for selecting galaxy dominated sources (since they are spread over a larger area in the IR colour-colour plot).
In the present paper we extend the previous 1WHSP catalog to lower Galactic latitudes ($b>|10^\circ|$) building the larger and more complete 2WHSP catalog including over 1600 blazars expected to emit at VHE energies by means of multi-frequency data.
\section{Building the largest sample of HSP blazars}
\label{building}
\subsection{Initial data selection by spatial cross-matching}
Blazars are known to emit electromagnetic radiation over a very wide spectral range, from radio
to VHE photons. As discussed in Paper I, an effective way of building large blazar
samples is to work with multi-frequency data, especially from all-sky surveys, and apply selection
criteria based on spectral features that are known to be specific to blazar SEDs.
We followed Paper I and started by cross-matching the AllWISE whole sky infrared
catalog \citep{Cutri2013} with three radio surveys \citep[NVSS, FIRST, and SUMSS:][]
{Condon1998,White1997,Manch2003}. To take into account the positional uncertainties associated with each target, we used matching radii of 0.3~arcmin for the NVSS and the SUMSS surveys and 0.1~arcmin for the FIRST catalog.
Then we performed an internal match for all IR-radio sources to eliminate duplicate entries coming from the different radio catalogs.
Keeping only the best matches between radio and IR, we selected 2,137,505 objects.
After that, we demanded all radio-IR matching sources to have a counterpart in one of the X-ray catalogs available to us \citep[RASS BSC and FSC, 1SWXRT and deep XRT GRB, 3XMM, XMM slew, Einstein IPC, IPC slew, WGACAT, Chandra, and BMW:][]{Voges1999,Voges2000,DElia2013,Puccetti2011,Rosen2015,Saxton2008,Harris1993,Elvis1992,White2000,Evans2010,Panzera2003}. Therefore we cross-matched the IR-radio subsample with each X-ray catalog individually, taking into account their positional errors.
For instance, a radius of 0.1~arcmin was adopted for the cross-correlations (as in Paper I) unless the positional uncertainty of a source was reported to be larger than 0.1~arcmin, as e.g. in the case of many X-ray detections in the RASS survey.
In these cases, we used the 95\% uncertainty radius (or ellipse major axis) of each source as maximum distance for the cross match.
Some X-ray catalogs have a very wide range of positional uncertainties, thus we separated the data by positional errors and used different cross-matching radii for these X-ray catalogs.
The radii used for cross-matching the IR-radio subsample with each X-ray catalog are reported in Table~\ref{Xraycross}.
We also restricted the sample by Galactic latitude $|b|>10^\circ$ to avoid complications in the Galactic plane.
We combined all the IR-radio-X-ray matching sources and applied an internal cross-check keeping only single IR sources within 0.1~arcmin radius; this procedure reduced the sample to 28,376 objects.
\begin{table} [h!]
\begin{center}
\begin{tabular}{ccc}
\hline\hline
Catalog&Error position&Cross-matched\\
& & radius\\
\hline
RASS&0-36 arcsec&0.6 arcmin\\
&$>$37 arcsec&0.8 arcmin\\
Swift 1SWXRT&0-5 arcsec&0.1arcmin \\
&$>$5 arcsec&0.2 arcmin\\
Swift deep XRT GRB&all data&0.2 arcmin\\
3XMM DR4&0-5 arcsec&0.1 arcmin\\
&$>$5 arcsec& 0.2 arcmin\\
XMM Slew DR6&all data&10 arcsec\\
Einstein IPC&all data&40 arcsec\\
IPC Slew&all data&1.2 arcmin\\
WGACAT2&all data&50 arcsec\\
Chandra&all data&0.1 arcmin\\
BMW&all data&0.15 arcmin\\
\hline\hline
\end{tabular}
\caption{The cross-matching radii of the X-ray catalogs.}
\label{Xraycross}
\end{center}
\end{table}
\subsection{Further selection based on broad-band spectral slopes}
\label{slope}
Here we take advantage of the fact that HSP blazars show radio to X-ray SEDs that distinguish them from any other type of extragalactic sources by imposing two constraints on the spectral slopes, namely:
\begin{equation}
\begin{aligned}
~0.05&<\alpha_{1.4{\rm GHz}-3.4\mu{\rm m}}<0.45 \\
0.4&<\alpha_{4.6\mu{\rm m}-1{\rm keV}}< 1.1
\end{aligned}\label{eq:slope}
\end{equation}
where $\alpha_{\nu1- \nu2}=-\frac{\log(f_{\nu1}/f_{\nu2})}{\log(\nu_1/\nu_2)}$,
that is the same conditions applied to the 1WHSP catalog, with the exception that here we do not apply the criterion $-1.0< \alpha_{3.4 \mu{\rm m}-12.0\mu{\rm m}} < 0.7$.
This choice was necessary to prevent the loss of IR galaxy-dominated HSPs which could still be promising VHE candidates \citep[see][for details]{Massaro2011,Arsioli2015a}.
The parameter ranges given above are derived from the shape of the SED of HSP blazars, which is assumed to be similar to those of three well-known bright HSPs, i.e. MKN 421, MKN 501, and PKS 2155$-$304 shown in Fig.~3 of Paper I, which also fit within the limiting slopes ($\alpha_{1.4{\rm GHz}-3.4\mu{\rm m}}$ and $\alpha_{4.6\mu {\rm m}-1{\rm keV}}$) used for the selection.
By avoiding the application of the IR slope constraints used for the 1WHSP sample, we select more HSP candidates, reducing the incompleteness at low radio
luminosities where the IR flux is often dominated by the host galaxy.
\subsection{Deriving $\nu_{\rm peak}$\, and classifying the sources}
\label{DerivingPeak}
The final pre-selection led to a sample of 5,518 HSP-candidates, 922 of which are also 1WHSP sources.
Note that this initial sample includes most of the HSP blazars that had to be added to the 1WHSP sample as additional previously known sources that were missed by the
original selection procedure.
To refine and further improve the quality of the sample we used the ASDC SED builder
tool\footnote{http://tools.asdc.asi.it/SED} to examine in detail all 5,518 candidates, accepting only
those with SEDs that are consistent with that of genuine HSPs.
Finally the synchrotron component of each object that passed our screening was fitted using a third degree polynomial function so as to estimate
parameters such as $\nu_{\rm peak}$, and $\nu_{\rm peak}$f$_{\nu_{\rm peak}}$, the energy flux at the synchrotron peak.
The host galaxies of HSP blazars are typically giant ellipticals, and their optical and near IR flux sometimes dominate the SED in these bands.
In order to only fit the synchrotron component of HSP blazars, it is crucial to distinguish the non-thermal nuclear radiation from the flux coming from
the host galaxy.
To do so we used the standard giant elliptical galaxy template of the ASDC SED builder tool to judge if the optical data points were due to the host galaxy or from non-thermal synchrotron radiation.
If the source under examination had ultraviolet data (such as Swift-UVOT or GALEX measurements) it was straightforward to tell if there was non-thermal emission from the object.
In addition, to avoid selecting objects with misaligned jets, which are expected to be radio-extended, the accepted spatial extension of the radio counterparts (as reported in the original catalogues) was limited to 1 arcmin.
This procedure was carried out whenever possible, based on the 1.4~GHz radio image from NVSS, which includes the entire sky north of $\delta= -40^{\circ}$, similarly to what had been done for the 1WHSP catalog.
We could also identify radio extended sources from their SED, since radio extended objects typically display a steep radio spectrum. All cases where we could find evidence of radio (or X-ray, typically from clusters: see below) extension were eliminated from the sample.
At the end of this process we only accepted objects with $\nu_{\rm peak}$~$> 10^{15}$~Hz \citep{Padovani1995}.
Clearly, most bright sources in the current list are also included in the 1WHSP catalog. Many of the new
catalog entries are fainter sources or objects located at low Galactic latitudes ($10^{\circ} < | b |< 20^{\circ}$).
In some cases the optical data were consistent with thermal emission from the host galaxy, and the few radio, IR, or X-ray measurements that could be related to
non-thermal emission were very sparse. Clearly, more multi-frequency data are
needed for these sources.
We still have a number of unclear cases due to the lack of good multi-frequency data. We flagged them accordingly.
In addition, since the positional accuracy in X-ray surveys is usually not as precise as that of optical or radio surveys, the position of the X-ray counterparts sometimes may be 20 to 40 arcsec away from the radio and optical counterparts, introducing more uncertainty.
Many of the 2WHSP candidates have been observed by SWIFT with multiple short exposures.
To allow for a more accurate estimation of $\nu_{\rm peak}$\, and $\nu_{\rm peak}$f$_{\nu_{\rm peak}}$\, we summed all the SWIFT XRT observations that were taken within a 3 week interval.
\subsection{Avoiding X-ray contamination from cluster of galaxies}
\label{clusters}
Blazars are certainly not the only objects that emit X-rays. For instance, galaxy clusters also show X-ray emission that is, however, normally spatially extended with a spectrum that peaks at $\approx 1-3$~KeV resulting from the emission of giant clumps of hot and low density diffused gas \citep[$\approx10^8$ K and $\approx10^{-3}~\rm{atoms}/{\rm cm}^{3}$: ][]{Sarazin1988, Bohringer2010, PerezTorres2009}.
Since blazars and radio galaxies are often located in clusters of galaxies, the X-rays from the hot gas, if not correctly identified, might cause the SED of the candidate 2WHSP source
to look like that of a HSP object, introducing a source of contamination for our sample.
To avoid this problem we carried out an extensive check of bibliographic references\footnote{For the cross-check with ADS references on each source we have used the Bibliographic Tool available on
the ASDC website.} and catalogs of cluster of galaxies (e.g. ABELL, PGC, MCXC, ZW, WHL, etc), excluding cases where cluster emission could be responsible for the observed X-rays.
In addition, we used Swift XRT imaging data (which are available for $\approx 60\%$ of our sample) to distinguish between X-ray emission from blazar jets, which is point-like in the XRT count maps, and that from the clusters, which is often extended.
The same procedure was followed using XMM images, whenever these could be found in the public archive.
In addition, we cross-matched our sample with the positions of RASS extended sources and with those of the {\it Planck} catalog of Sunyaev-Zeldovich sources \citep{Planck2015}.
Finally, we visually inspected optical images and the error circle maps built with the ASDC explorer
tool\footnote{http://tools.asdc.asi.it} looking for targets that could be related to clusters of galaxies.
To illustrate how we removed objects that satisfy our multi-frequency selection criteria but where the X-ray flux is likely due to extended emission from a cluster of galaxies,
we consider the example of WHL J151056.1+054441.
This is a giant cluster of galaxies also cataloged as Abell2029. As the strong X-ray emission is clearly extended both in the Swift-XRT and XMM images (see Fig. \ref{ext1})
this source was removed from our HSP catalog.
Another example is shown in Fig. \ref{ext2}, where the candidate blazar is at the center of the cluster of galaxies LCRS B113851.7$-$115959.
Although the X-ray emission is overall extended, the region around the sources shows clumps, and there are several X-ray detections; the non-thermal emission is very clear in the SED. Apparently, there is an AGN in the center that also emits in the UV. However, based on the available data we cannot know if the X-ray is mainly from the non-thermal jet or from the cluster
and therefore we did not include this source in the catalog.
\begin{figure}[h!]
\centering
\includegraphics[width=4cm,height=4cm]{ext1_optimg.eps}
\includegraphics[width=4cm,height=4cm]{ext1_xrt2.eps}
\caption[Abell2029]{Optical (left) and X-ray (right: XRT count map) iamges of
WHL J151056.1+054441.}
\label{ext1}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=4cm,height=4cm]{ext2_optimg.eps}
\includegraphics[width=4cm,height=4cm]{ext2_xrt2.eps} \\
\includegraphics[width=0.78\linewidth,angle=90]{ext2_sed.eps}
\caption[LCRS B113851.7-115959]{Top: optical (left) and X-ray (right: XRT count map) images
of LCRS B113851.7$-$115959. Bottom: the SED of LCRS B113851.7$-$115959.}
\label{ext2}
\end{figure}
\section{Improving the sample completeness}
\label{catalog}
The procedure described above led to the selection of 734 new HSPs in addition to those
already included in the 1WHSP catalog, including previously known, newly discovered, and
candidate blazars. For each source we adopted as best coordinates those taken from the
WISE catalog.
To evaluate the efficiency of our method of selecting VHE emission blazars, we cross-matched the sample of 1,647 objects with the Second Catalog of Hard {\it Fermi}-LAT Sources (2FHL) \citep{Fermi2fhl2015} and with TeVCat.
Only 146 of the 360 sources in the 2FHL catalog (257 at $|b|>10^\circ$) are also in this preliminary sample.
To verify if there are genuine HSPs in the 2FHL catalog that were missed by our selection, we closely examined the remaining 214 2FHL sources to see if they are
cataloged as blazars.
We found 31 high Galactic latitude blazars with $\nu_{\rm peak}$~$> 10^{15}$ Hz that could be added to the catalog.
These sources were initially missed since they just did not match the optical-X-ray slope criteria (equation~\ref{eq:slope}) during the preliminary selection process.
This selection inefficiency could be due to flux variability, lack of sufficiently high quality multi-frequency data, or simply to a non-optimal choice of parameter values
in equation~\ref{eq:slope}.
Out of the 177 HSPs located at $|b|>10^\circ$ in the 2FHL catalog our selection method detected 146 objects, for an efficiency of $82.5\%$.
In addition, there are 14 HSP blazars in the 2FHL catalog that are located at latitudes $|b|<10^\circ$, the area of sky that was not considered in our work
to reduce complications connected to the Galactic plane.
Since our aim is to provide the most complete list of HSPs we added to the 2WHSP catalog the 14 low latitude objects as well as all additional HSPs found in the 2FHL catalog,
for a total of 45 sources.
Only one good HSP blazar found among the 2FHL low Galactic latitude sources had no WISE data (2WHSP J135340.2$-$663958.0).
We used the radio position instead of the IR position in this case.
We then checked catalogs of sources detected at TeV energies.
Currently, the most complete list of objects detected in this band is TeVCat, which consists of 175 sources detected by Imaging Atmospheric/Air Cherenkov Telescope/Technique (IACT).
At present there are three main IACT systems operating in the $\sim 50$~GeV to 50~TeV range: the High Energy Stereoscopic System (H.E.S.S.), MAGIC (Major Atmospheric Gamma Imaging Cherenkov Telescopes), and VERITAS (Very Energetic Radiation Imaging Telescope Array System).
There are 38 TeVCat sources that are also in the 2WHSP catalog.
We therefore checked the other high Galactic latitude sources to see if they were classified as HSP blazars, concluding that only one HSP source was missed. Note that previously we had already added 3 TeV sources to the 1WHSP catalog, since these were missed during its selection.
In total there are 39 HSPs at $|b|>10^\circ$ in TeVCat, 35 of which satisfy out selection criteria.
Our selection efficiency in this case is $89.7 \%$.
As in the case of the 2FHL catalog all missing sources have been lost because they just did not meet the slope criteria used in section~\ref{slope}.
In all cases, however the spectral parameters turned out to be very close to the limits of the selection criteria, and $\nu_{\rm peak}$\, was $\approx 10^{15}$~Hz.
The final 2WHSP catalog includes a total of 1691 sources, 288 of which are newly identified HSPs, 540 are previously known HSPs, 814 are
HSP candidates, 45 are HSP blazars taken from the 2FHL catalog, and 4 from TeVcat.
The complete list of 2WHSP sources is shown in Table~\ref{2whsptable}.
We will further discuss the incompleteness due to the inefficiency in finding sources peaking at or just above $10^{15}$ Hz in section~\ref{nupeak}.
\section{Discussion}
\label{discussion}
\subsection{The $\nu_{\rm peak}$~distribution}
\label{nupeak}
The $\nu_{\rm peak}$~distribution of the 2WHSP sources is shown in Fig.~\ref{nudist}.
The peak of the distribution is located at $\approx 10^{15.5}$~Hz and not at the threshold of $\nu_{\rm peak}$~ $= 10^{15}$~Hz used for the sample selection.
This is very likely due to incompleteness of the sample near the $\nu_{\rm peak}$\, threshold, as
our selection criteria were tuned to avoid too large an LSP contamination.
The distribution is similar to that of the 1WHSP sample and of the subsample of HSP sources in the 5BZCat
When compared with other catalogs of extreme blazars, the peak value of the $\nu_{\rm peak}$~distribution of our sample is lower. For example the Sedentary and the \citet{Kapanadze2013}(hereafter K13) catalogs have peak values $\approx 10^{16.8}$ and $\approx 10^{16.7}$~Hz, respectively.
This difference results from the criteria used and the different selected methods.
The Sedentary and \citet{Kapanadze2013} catalogs, for example, were tuned to select sources
with very large $\nu_{\rm peak}$\, values.
Note that the $\nu_{\rm peak}$\, of some sources is particularly high, with values ${\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}} 10^{18}$~Hz. We discuss these extreme sources in the next section.
Sometimes, the severe variability of HSPs may result in displacements for $\nu_{\rm peak}$ in different phases, such as MRK501 (See Fig.~\ref{extreme5}); not to mention that the intense variability will make the $\nu_{\rm peak}$f$_{\nu_{\rm peak}}$ vary 1-2 order or even worser.
In these cases, we fit the $\nu_{\rm peak}$ and $\nu_{\rm peak}$f$_{\nu_{\rm peak}}$ with the mean values in order to estimate the proper values for Synchrotron component averagely.
By doing so, we avoid having extreme values for Synchrotron peak and reduce the effects of variability.
\begin{figure}
\centering
\includegraphics[width=0.78\linewidth,angle =90]{nudist.eps}
\caption[The $\nu$ distribution]{The $\nu_{\rm peak}$\, distribution.
The black solid line, blue dotted line, and red dashed line denote well estimated $\nu_{\rm peak}$, uncertain $\nu_{\rm peak}$, and lower limits on $\nu_{\rm peak}$, respectively.
}
\label{nudist}
\end{figure}
\subsection{The highest $\nu_{\rm peak}$\, blazars}
\label{extremesec}
There are several sources in the 2WHSP sample with $\nu_{\rm peak}$\, around or above $10^{18}$Hz; these are usually called "extreme blazars".
Values of $\nu_{\rm peak}$\, ${\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}} 10^{18}$~Hz imply that the electrons responsible for the synchrotron radiation must be accelerated to extremely high energies
\citep[see the Introduction and e.g.][]{Rybicki1986,Costamante2001}.
It is hard to estimate the positions of the synchrotron peak for such extreme sources, as the available data in the X-ray band is often limited to a few keV, where most of the sensitive existing detectors operate.
For about 60 sources we could not estimate well the frequency of the synchrotron peak since the soft X-ray data show a still rising spectrum in the SED,
and no hard X-ray data exist to cover the peak of the emission. In these cases we could only estimate a lower limit to $\nu_{\rm peak}$.
For some strong X-ray variable sources with many X-ray observations we also could not obtain well-estimated $\nu_{\rm peak}$\, values with the third degree polynomial fitting in ASDC SED tool since the curvature in the X-ray spectrum (and with it $\nu_{\rm peak}$) changes with time.
However, in all these cases the available multi-frequency data imply that the synchrotron peak is within the X-ray band; in these sources we estimated an average $\nu_{\rm peak}$\, value using a second-degree polynomial in the X-ray band.
Table~\ref{extremelist} gives the list of all the extreme sources with $\nu_{\rm peak}$~$\ge 10^{17.7}$~Hz; it includes many more such objects than any previous catalog.
These extreme sources are particularly importance since they may be candidate VHE, neutrino or ultra high energy cosmic ray (UHECR) sources (section~\ref{vhecand} and \ref{neutrinocand}).
Figure~\ref{extreme1} to \ref{extreme5} illustrate five examples of SEDs of representative objects from Table~\ref{extremelist}.
\begin{itemize}
\item {\bf 2WHSP J023248.5+201717 (1ES0229+200)}.
This is an extreme source with VHE data available
\citep[the ebl-deabsorbed VHE data shown as black filled circles are from ][]{Finke2015}.
The synchrotron peak is at $\sim 10^{18}-10^{19}$ Hz and the peak flux is one of the highest among the 2WHSP sources.
In the VHE band, once one corrects the VHE fluxes for EBL absorption, the inverse Compton peak
will be at energies $>1$~TeV.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{extreme1.eps}
\caption{The SED of the extreme object 2WHSP J023248.5+201717. The dark blue points are ebl- deabsorbed data from \citet{Finke2015}. See text for details.}
\label{extreme1}
\end{figure}
\item {\bf 2WHSP J035257.4$-$683117}.
This is a known blazar with $\log$~$\nu_{\rm peak}$~$\approx 18.1$.
It has hard X-ray and $\gamma$-ray detections but no TeV detection yet.
This source might be a good target for next generation TeV telescopes.
This source is not in 5BZCat yet.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{extreme2.eps}
\caption{The SEDs of the extreme object 2WHSP J035257.4$-$683117. See text for details.}
\label{extreme2}
\end{figure}
\item {\bf 2WHSP J215305.2$-$004229 (5BZBJ2153$-$0042)}.
This source has a very hard X-ray spectrum and the SED in the X-ray band keeps increasing
up to the highest energies, implying a $\nu_{\rm peak}$\, larger that $10^{18}$~Hz.
The X-ray emission is not likely to be related to a cluster of galaxy as it is compact.
It has $\gamma$-ray data and may be a good TeV candidate source.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{extreme3.eps}
\caption{The SED of the extreme object 2WHSP J215305.2$-$004229. See text for details.}
\label{extreme3}
\end{figure}
\item {\bf 2WHSP J143342.7$-$730437}.
This is another example of a very hard X-ray SED.
It has UV data but did not have any $\gamma$-ray data yet; however, this source is in the list of
new $\gamma$-ray detections in \cite{Arsioli2016}.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{extreme4.eps}
\caption{The SEDs of the extreme object 2WHSP J143342.7$-$730437. See text for details.}
\label{extreme4}
\end{figure}
\item {\bf 2WHSP J165352.2+394536}.
This is the well-known HSP MRK501. On average
$\log \nu_{\rm peak} \sim 17.9$~Hz; however, during an X-ray flare, as shown by the BeppoSAX data (yellow points in the SED, \citet{Giommi2002}), $\nu_{\rm peak}$\, reached
$>10^{18}$~Hz.
Note that in \citet{Pian1998}, they discussed the BeppoSAX observation of MRK501 in
April, 1997 and showed that the $\nu_{\rm peak}$ of that shift at least two orders of magnitude w.r.t. previous observations of that.
The scenario is seen for the first time at that time.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{extreme5.eps}
\caption{The SEDs of the extreme object 2WHSP J165352.2+394536. See text for details.}
\label{extreme5}
\end{figure}
\end{itemize}
\begin{center}
\begin{table*}
\begin{tabular}{>{\centering\arraybackslash}p{4cm}>{\raggedleft\arraybackslash}p{1.6cm}>{\raggedleft\arraybackslash}p{1.8cm}>{\centering\arraybackslash}p{8cm}}
\hline\hline
Source&$\log$$\nu_{\rm peak}$&$\log$$\nu_{\rm peak}$f$_{\nu_{\rm peak}}$¬e\\
\hline
2WHSPJ003322.3$-$203907&17.9&-11.9&new HSP\\
2WHSPJ004013.7+405003&$>$17.5&$>$-11.5&5BZU, lower limit\\
2WHSPJ012308.5+342048&18.0&-10.8&5BZB, TeV source\\
2WHSPJ013803.7$-$215530&$>$17.5&$>$-12.0&blazar candidate lower limit\\
2WHSPJ015657.9$-$530159&18.0&-11.1&5BZB \\
2WHSPJ020412.9$-$333339&17.9&-11.7&5BZB, new $\gamma$-ray identification\\
2WHSPJ023248.5+201717&18.5&-11.0&5BZG, TeV source*\\
2WHSPJ032056.2+042447&17.9&-11.7&blazar candidate, new $\gamma$-ray identification\\
2WHSPJ032356.5$-$010833&$>$17.5&$>$-11.9&5BZB, TeV source, lower limit\\
2WHSPJ034923.1$-$115926&17.9&-11.0&5BZB, TeV source\\
2WHSPJ035257.4$-$683117&18.1&-11.0& previously known BL Lac*\\
2WHSPJ050419.5$-$095631&17.9&-11.6&new HSP, new $\gamma$-ray identification\\
2WHSPJ050709.2$-$385948&$>$17.5&$>$-12.2&blazar candidate, lower limit\\
2WHSPJ050756.0+673723&17.9&-10.7&5BZB, TeV source\\
2WHSPJ055040.5$-$321615&18.1&-10.7&5BZG, TeV source\\
2WHSPJ055716.7$-$061706&17.9&-11.5&blazar candidate, new $\gamma$-ray identification\\
2WHSPJ064710.0$-$513547&17.9&-11.2&blazar candidate\\
2WHSPJ071029.9+590820&18.1&-10.7&5BZB, TeV source\\
2WHSPJ073326.7+515354&17.9&-11.3&blazar candidate\\
2WHSPJ081917.5$-$075626&18.0&-11.5&5BZB, TeV source\\
2WHSPJ083251.4+330011&18.0&-12.0&5BZB, new $\gamma$-ray identification\\
2WHSPJ084452.2+280409&17.9&-12.3&new HSP\\
2WHSPJ092057.4$-$225720&$>$17.5&$>$-11.6&new HSP, lower limit\\
2WHSPJ094620.2+010450&17.9&-11.8&5BZB, TeV source\\
2WHSPJ095849.0+013218&17.9&-12.3&new HSP, new $\gamma$-ray identification\\
2WHSPJ102212.6+512359&18.2&-11.7&5BZG, new $\gamma$-ray identification\\
2WHSPJ104651.4$-$253544&$>$18.0&$>$-11.5&5BZB\\
2WHSPJ105606.6+025213&17.9&-11.5&5BZG\\
2WHSPJ110357.1+261117&17,9&-12.2&new HSP\\
2WHSPJ110651.7+650603&17.9&-12.7&blazar candidate\\
2WHSPJ110804.9+164820&17.9&-12.7&new HSP\\
2WHSPJ112313.2$-$090424&17.9&-12.4&blazar candidate\\
2WHSPJ113209.1$-$473853&$>$17.5&$>$-11.6&blazar candidate, lower limit\\
2WHSPJ113630.1+673704&18.1&-11.1&5BZB, TeV source\\
2WHSPJ121323.0$-$261806&17.9&-11.2&5BZB\\
2WHSPJ122044.5+690525&$>$17.5&$>$-12.0&blazar candidate, lower limit\\
2WHSPJ122208.6+030718&$>$17.5&$>$-11.8&new HSP, lower limit\\
2WHSPJ122514.2+721447&$>$17.5&$>$-11.8&lower limit, 5BZB\\
2WHSPJ125341.2$-$393159&17.9&-11.3&5BZG, new $\gamma$-ray identification\\
2WHSPJ125708.2+264924&$>$17.5&$>$-12.3&new HSP, lower limit \\
2WHSPJ132239.1+494336&$>$17.5&$>$-12.1&new HSP, lower limit\\
2WHSPJ132541.8$-$022809&17.9&-12.0&5BZB, new $\gamma$-ray identification\\
2WHSPJ140027.0$-$293936&$>$17.5&$>$-12.1&blazar candidate, lower limit\\
2WHSPJ140121.1+520928&$>$17.5&$>$-12.0&5BZB, lower limit\\
2WHSPJ142832.5+424020&18.1&-10.7&5BZB, TeV source\\
2WHSPJ143342.7$-$730437&$>$17.5&$>$-11.5&blazar candidate, lower limit, new $\gamma$-ray identification*\\
2WHSPJ151041.0+333503&$>$17.5&$>$-11.5&5BZG, lower limit, new $\gamma$-ray identification\\
2WHSPJ151618.7$-$152344&18.0&-11.7&5BZB, new $\gamma$-ray identification\\
2WHSPJ153646.6+013759&$>$18.0&$>$-11.7&5BZB\\
2WHSPJ160519.0+542058&17.9&-12.0&5BZB, new $\gamma$-ray identification\\
2WHSPJ161004.0+671026&$>$17.5&$>$-11.8&5BZB, lower limit, new $\gamma$-ray identification\\
2WHSPJ161414.0+544251&17.9&-12.6&blazar candidate\\
2WHSPJ161632.8+375603&18.0&-12.1&5BZG\\2WHSPJ161632.8+375603&18.0&-12.1&5BZG\\
2WHSPJ162330.4+085724&$>$17.5&$>$-12.1& new HSP, lower limit, new $\gamma$-ray identification\\
2WHSPJ165352.2+394536&17.9&-10.2&Variability, flaring, 5BZB, TeV source*\\
2WHSPJ171902.2+552433&17.9&-12.5&known blazar\\
2WHSPJ194333.7$-$053352&$>$17.5&$>$-11.8&blazar candidate\\
2WHSPJ194356.2+211821&18.1&-11.0&new HSP, TeV source\\
2WHSPJ205528.2$-$002116&$>$18.0&$>$-10.9&5BZB, TeV source, lower limit\\
2WHSPJ214410.0$-$195559&17.9&-12.4&blazar candidate\\
2WHSPJ215305.2$-$004229&$>$18.0&$>$-11.4&5BZB, lower limit*\\
2WHSPJ223248.7$-$202226&17.9&-11.7&blazar candidate\\
2WHSPJ225147.5$-$320611&$>$18.0&$>$-11.3&5BZU, lower limit, new $\gamma$-ray identification\\
\hline\hline \\
\end{tabular}
\caption{The extreme synchrotron peaked sources.
The sources marked with * are discussed in the text and shown in Figure \ref{extreme1} to \ref{extreme5}.}
\label{extremelist}
\end{table*}
\end{center}
\subsection{The redshift distribution}
Some 2WHSP sources lack redshift as their optical spectra are completely featureless.
As in Paper I, we estimated lower limit redshifts for these sources.
Assuming that in the optical band the host galaxy is swamped by the non-thermal emissions and leaves no imprint on the optical spectrum when the observed non-thermal flux is at least ten times larger than the host galaxy flux, we used the distance modulus (for details, see eq. 5 in Paper I) to calculate the lower limits redshifts.
For the others, we obtained the redshifts from the references listed in Table~\ref{2whsptable}.
Fig.~\ref{zdist} shows the redshift distribution, which peaks just above 0.2.
For all 2WHSP sources, $\langle z_{all} \rangle=0.371 \pm 0.005$; for firm redshift 2WHSP sources, $\langle z \rangle=0.331 \pm 0.008$.
Clearly, sources without firm redshift are on average farther away than sources with firm redshift.
High redshift sources in flux limited samples tend to have featureless optical spectra as the host galaxy contribution is overwhelmed by the synchrotron emission.
\citet{Giommi2012a} have predicted that the redshift distribution of BL Lacs without redshift in radio flux limited surveys will peak around $z_{predict} \approx 1.2$.
The results again suggest that all source with only lower limit redshift or without redshift could be much further away than objects with measured redshift.
Considering only sources with firm z values, the redshift distribution of 2WHSP sources is similar but
not identical to other HSP catalogs/subsamples.
The average redshift of the 1WHSP catalog is $\langle z_{1whsp}\rangle=0.306$, that of the
subsample of HSPs ($\nu_{\rm peak}$~$>10^{15}$~Hz) in 5BZCat is $\langle z_{bzcat}\rangle=0.294$, that
of the Sedentary sources is $\langle z_{s}\rangle=0.320$, and that of the K13 catalog is
$\langle z_{k}\rangle=0.289$.
For instance in K13, the redshifts range is $0.031<z_{k}<0.702$, while in this paper we selected a number of sources with relatively high redshift ($z >0.7$) that are not in previous catalogs.
\begin{figure}
\centering
\includegraphics[width=0.78\linewidth,angle =90]{zdist.eps}
\caption[The redshift distribution]{The redshift distribution of 2WHSP sources.
The black solid line represents the sources with firm redshifts, the red dashed line the sources with uncertain redshift, and the blue dotted line the lower limits.
}
\label{zdist}
\end{figure}
\subsection{The radio $\log$N-$\log$S of HSP blazars}
\label{lognlogssec}
The estimation of the statistical properties, such as the $\log$N-$\log$S of a population of sources, requires the availability of flux limited and complete samples.
As we demand that all 2WHSP sources have a radio, IR and X-ray counterpart, we must take into account the incompleteness resulting from
the fact that the only existing all sky X-ray survey is not sufficiently deep to ensure the detection of all radio and IR faint HSP blazars.
For the purpose of estimating the $\log$N-$\log$S we then considered the subsample of 2WHSP sources that are included in
the RASS X-ray survey, which covers the entire sky albeit with sensitivity that strongly depends on ecliptic latitude (see Sec. 4.3 of Paper I for more details).
For each source in the 2WHSP-RASS subsample we therefore calculated a contribution $``{\rm n}_{\rm i}"$ to the total density, as given by ${\rm n}_{\rm i}=1/{\rm A}_{\rm i}~{\rm deg}^{-2}$,
where the parameter ${\rm A}_{\rm i}$ is the sky area covered by RASS with sensitivity sufficient to detect the source in consideration.
We then sum the contribution of all sources in a given flux bin ${\rm N}_{\rm bin}=\sum {\rm n}_{\rm i}$ and obtain the $\log$N-$\log$S.
We use this approach to estimate the $\log$N-$\log$S of HSP blazars with respect to the radio flux density and the flux at the peak of the synchrotron component $\nu_{\rm peak}$f$_{\nu_{\rm peak}}$.
The integral radio $\log$N-$\log$S for the 2WHSP sample is shown in Fig.~\ref{rrlognlogs} where we also plot the $\log$N-$\log$S for the Sedentary HBL
\citep{Giommi1999,Giommi2005,Piranomonte2007} for comparison.
The dotted lines correspond to a fixed slope of -1.5, the expected value for a complete sample of a non-evolving population in a Euclidean Universe.
Since the radio surveys that we use have different sensitivities in the northern and southern sky, we considered only sources with $\delta > -40^{\circ}$ and radio flux density $\ge 5$ mJy.
It is clear from Fig.~\ref{rrlognlogs} that the surface density of the 2WHSP sample is approximately a
factor of ten larger than that of the Sedentary survey, which is expected since the latter includes
more extreme sources (its $\nu_{\rm peak}$\, distribution peaks at $\log \nu_{\rm peak} \sim$ 16.8,
as compared to $\log \nu_{\rm peak} \sim$ 15.5 for the 2WHSP sample). Apart from the different
normalizations the $\log$N-$\log$S of the two samples show similar trends deviating from
the Euclidean slope at radio flux densities lower than $\approx 20$~mJy. The 2WHSP flattening, however, appears
to be stronger than the one of the Sedentary survey, which suggests the onset of some degree of incompleteness
at lower radio flux densities, on top of the evolutionary effects discussed by \cite{Giommi1999}.
The 2WHSP maximum surface density corresponds
to a total of $\sim 1,900$ HSP blazars over the whole sky. Given that this number refers only to sources
with 1.4 GHz flux density $\ge 5$ mJy, and because of the incompleteness discussed above,
this has to be considered a robust lower limit.
Fig.~\ref{rrlognlogs} shows also the 5 GHz\footnote{Given that BL Lacs typically have flat
radio spectra we did not convert the 5 GHz counts to 1.4 GHz.} number counts for the Deep
X-ray Radio Blazar Survey (DXRBS) BL Lacs (red squares) and HBL only (red diamonds) from
\cite{Padovani2007}. The latter are in very good agreement with the 2WHSP number counts in the
region of overlap, which shows that our selection criteria are robust. Moreover, one can see a clear
trend going from the Sedentary survey to the 2WHSP sample and to the whole BL Lac population,
with an increase in number of a factor $\approx 10$ at every step. Given the unbiased nature of radio selection
with respect to $\nu_{\rm peak}$\, this is a direct consequence of BL Lac demographics, with HBL making up only $\sim 10\%$
of the total \citep[see also, e.g.][]{Padovani2007}.
\begin{figure}
\centering
\includegraphics[width=0.78\linewidth,angle =90]{radiologns.eps}
\caption[The integral radio 1.4 GHz LogN-LogS]{The integral radio $\log$N-$\log$S at 1.4 GHz.
The blue filled circles denote the 2WHSP catalog, the green open triangles indicate the Sedentary
one, the red open squares represent DXRBS BL Lacs of all types, while the red diamonds
are the subsample of HBLs in the DXRBS \citep[those in the HBL box: see ][for details]{Padovani2007}. The dashed lines have a
slope of $-1.5$.}
\label{rrlognlogs}
\end{figure}
\subsection{The IR Colour-Colour plot}
Figure~\ref{color} shows the WISE IR colour-colour diagram of 2WHSP sources, with signal to noise ratio (snr) in the W3 channel larger than 2, and the sources in the first WHSP sample and HSP blazars in the 5BZCat list.
As expected, all of the 1WHSP sources are within the SWCD region as this was one of the criteria of the selection.
By dropping the IR slope criterion ($-1.0< \alpha_{3.4 \mu {\rm m}-12.0\mu {\rm m}} < 0.7$) the 2WHSP sample includes more HSPs
than the 1WHSP that are located in the bottom-left region within the SWCD.
There are also 49 sources outside the SWCD region (see Table~\ref{swcdout}), six of them also in 5BZCat.
The sources at the bottom are dominated by the host galaxy in the optical and near IR bands (Class 1).
The right part of Fig.~\ref{color} is occupied by sources with problematic W3 photometry and sources
whose W3 magnitude has relatively small snr values (typically $< 4$: Class 2).
The sources located in the upper-right region have W1 fluxes similar or slightly lower than the W2 fluxes (Class 3).
The class 3 sources may be IR variable sources or could be blazars at the border between ISP and HSP objects or
might simply have poor W1 or W2 photometry.
All 49 sources were checked individually and all of them are good HSP candidates.
Thus, we suggest that the SWCD region needs to be extended to include all galaxy dominated HSPs.
\begin{figure}
\centering
\includegraphics[width=0.78\linewidth,angle =90]{color.eps}
\caption[The colour-colour diagram]{The IR colour-colour diagram. The black ones are the sources we selected in 2WHSP but not in 1WHSP, the red ones are the selected in 1WHSP, the blue crosses are the sources also in 5BZCat. The yellow line marks the SWCD region.}
\label{color}
\end{figure}
\begin{table*}
\begin{center}
\begin{tabular}{crrrr}
\hline\hline
Source&W1 mag&W2 mag&W3 mag&W3 SNR\\
\hline
Class 1: Host galaxy dominated& & & & \\
2WHSPJ180408.8+004221&12.197&12.109&10.521&12.3\\
2WHSPJ085730.1+062726&13.349&13.303&12.101&2.7\\
2WHSPJ031250.2+361519&12.137&12.089&10.432&12.5\\
2WHSPJ090802.2$-$095936&11.586&11.523&10.406&15.0\\
2WHSPJ160740.0+254113&11.401&11.443&11.057&8.7\\
2WHSPJ013626.5+302011&15.961&15.914&12.905&2.2\\
2WHSPJ023109.1$-$575505&10.546&10.515&9.039&38.3\\
2WHSPJ085958.6+294423&14.881&14.802&12.166&3.0\\
2WHSPJ094537.0$-$301332&14.864&14.850&12.317&3.1\\
2WHSPJ120850.5+452951&14.811&14.749&12.629&2.6\\
2WHSPJ130711.8+115316&12.430&12.413&11.864&4.2\\
2WHSPJ195020.9+604750&12.675&12.679&12.640&3.2\\
2WHSPJ101514.2$-$113803&12.694&12.462&12.532&2.4\\
\hline
Class 2: Mainly problematic W3& & & & \\
2WHSP J000552.9$-$284502&15.501&15.370&12.450&2.3\\
2WHSP J004501.4+051215&14.170&13.795&10.822&3.1\\
2WHSP J082030.7$-$031412&15.118&14.676&11.749&3.0\\
2WHSP J082355.6+394747&15.509&15.141&12.086&3.3\\
2WHSP J100520.4+240503&15.324&14.946&11.988&3.0\\
2WHSP J113405.8+483903&15.108&15.005&12.422&2.4\\
2WHSP J122304.9+453444&15.246&14.841&12.033&4.0\\
2WHSP J122944.5+164004&15.265&14.965&11.925&3.2\\
2WHSP J124430.7+351002&15.782&15.080&11.964&3.6\\
2WHSP J140125.3+031629&16.550&15.775&12.476&3.1\\
2WHSP J144446.0+474256&15.840&15.427&12.630&3.2\\
2WHSP J162939.4+701448&16.763&16.233&12.662&3.2\\
2WHSP J175955.2+150109&15.907&15.690&12.404&2.8\\
2WHSP J195134.7$-$154929&14.702&14.417&11.715&3.4\\
2WHSP J212233.7+192527&15.201&14.683&11.739&4.9\\
2WHSP J215355.8$-$295443&15.914&15.733&12.440&2.2\\
\hline
class 3: W2 similar to or brighter than W1& & & & \\
2WHSP J002258.9$-$244022&15.056&13.894&11.104&8.4\\
2WHSP J022941.1$-$412050&14.773&13.747&11.113&9.0\\
2WHSP J024743.3$-$481545&15.164&14.059&11.382&10.5\\
2WHSP J025057.1$-$122612&15.081&14.001&11.332&8.3\\
2WHSP J054504.3+065809&14.953&14.049&11.129&5.9\\
2WHSP J071625.6+750700&15.768&14.634&12.147&5.3\\
2WHSP J093938.5$-$031502&15.328&14.445&11.535&5.0\\
2WHSP J095518.4$-$294611&14.321&13.149&10.526&14.2\\
2WHSP J120136.0$-$060733&15.247&14.168&11.430&3.5\\
2WHSP J135043.7$-$310926&14.359&13.202&10.817&12.5\\
2WHSP J172746.3$-$754618&14.039&12.883&10.522&15.3\\
2WHSP J180158.9+610938&15.332&14.164&11.598&9.9\\
2WHSP J185550.8+805223&16.492&15.580&12.638&3.4\\
2WHSP J202803.5+720513&15.440&14.459&11.671&13.1\\
2WHSP J204734.9+793759&16.494&15.328&12.753&2.9\\
2WHSP J213533.7+314919&14.312&13.223&10.711&12.0\\
2WHSP J233207.6$-$025245&15.108&14.037&11.387&5.0\\
2WHSP J233630.4$-$635634&14.599&13.391&10.788&12.0\\
\hline\hline
\end{tabular}
\caption{Sources outside the SWCD region}
\label{swcdout}
\end{center}
\end{table*}
\subsection{Candidates for GeV and VHE $\gamma$-ray observations}
\label{vhecand}
Since HSPs are the dominant population in the extragalactic VHE sky the 2WHSP catalog
provides good candidates for the search of sources in {\it Fermi} catalogues and in the
VHE band. The Figure of Merit \citep[FOM, defined in][as the ratio between the synchrotron peak flux
$\nu_{\rm peak}$f$_{\nu_{\rm peak}}$~of a given source and that of the faintest blazar in the 1WHSP sample that has already
been detected in the TeV band]{Arsioli2015a} was introduced to provide a simple quantitative measure of
potential detectability of HSPs by TeV instruments.
The FOM parameter is reported for all 2WHSP sources and gives an objective way to assess the likelihood that a given HSP may be detectable as a TeV source.
As discussed in Paper I, relatively high FOM sources (FOM $> 0.1$) are good targets for observation with the upcoming Cherenkov Telescope Array (CTA). Another upcoming instrument, the Large High Altitude Air Shower Observatory (LHAASO), is currently designed to survey the whole northern sky for $\gamma$-ray sources above 300~GeV, with unprecedented sensitivity.
Therefore, high FOM 2WHSP sources may also provide seed-positions for searches of $\gamma$-ray signature embedded in LHAASO data \citep{Cao2010}.
For example, 2WHSP J083724.6+145820 (see Fig.~\ref{gammanew}), has $\nu_{\rm peak}$~$\sim 10^{16.7}$~Hz and $\nu_{\rm peak}$f$_{\nu_{\rm peak}}$$\sim 10^{-11}~\rm{erg}~\rm{cm}^{-2}~\rm{s}^{-1}$ (or FOM$=2$), but it had no $\gamma$-ray counterpart until recently.
The green points in Fig.~\ref{gammanew} correspond to the new $\gamma$-ray data
presented in \citet{Arsioli2016}.
Another example is 2WHSP J225147.5$-$320611, which has $\nu_{\rm peak}$~$ > 10^{18}$~Hz and
$\nu_{\rm peak}$f$_{\nu_{\rm peak}}$~$>10^{-11.3}~\rm{erg}~\rm{cm}^{-2}~\rm{s}^{-1}$ (FOM~$>1$), but also had
no $\gamma$-ray counterpart in current available $\gamma$-ray or VHE catalogs (1/2/3 FGL, 1/2 FHL, and TeVCat) until it was detected by \cite{Arsioli2016} thanks to the 2WHSP, which points to promising x-ray targets.
To better assess the percentage of detection of HSP blazars in the {$\gamma$-ray}\, band, in fact, \cite{Arsioli2016} have recently performed a dedicated $\gamma$-ray analysis of all 2WHSP sources with FOM~$\geq 0.16$, using archival {\it Fermi}-LAT observations integrated over 7.2 years of observations.
By using the position of 2WHSP sources as seeds for the data analysis, $\approx 85$ sources were identified at the $> 5\sigma~({\rm TS}>25)$ level, and another 65 at a less significant ($10<{\rm TS}<25$) level.
These results demonstrate the potential of HSP catalogs for the detection and identification of {$\gamma$-ray}\, and VHE sources.
Apart from that, the CTA flux limit/sensitivity could be as low as $3\times10^{-13}~\rm{erg}~\rm{cm}^{-2}~\rm{s}^{-1}$ \citep{Rieger2013} or $\sim 1$~mCrab at 1~TeV for 50-hour exposure.
Clearly, from Fig.~\ref{gammanew}, 2WHSP J083724.6+145820 and 2WHSP J225147.5$-$320611 may be detected by CTA in the future (since they are above the CTA sensitivity for exposure time 50 hours, the blue lines).
Therefore, with the benefit of multi-wavelength work, we provide here many candidates for future VHE observations.
\begin{figure}
\centering
\includegraphics[width=0.78\linewidth,angle=90]{gam_sed1.eps}
\includegraphics[width=0.78\linewidth,angle=90]{gam_sed2.eps}
\caption[]{VHE observations candidates. Top: 2WHSP J083724.6+145820; bottom:
2WHSP J225147.5$-$320611. The red line and blue lines are the {\it Fermi} Pass 8 and CTA sensitivities, respectively. The green circles are the data from {\it Fermi} Pass 8, and the black points are the data from other wavebands. The Pass 8 data are obtained from the {\it Fermi} tool using the 2WHSP position. These sources are not in the 3FGL catalog yet \citep[see][]{Arsioli2016}.}
\label{gammanew}
\end{figure}
\subsection{HSP blazars as neutrino and cosmic ray emitters?}
\label{neutrinocand}
Blazars have been considered as likely neutrino sources for
quite some time \citep[e.g.][]{Mannheim1995}.
\citet{Padovani2014} have suggested that blazars of the HSP type, where particles are accelerated to the highest energies,
may be good candidates for neutrino emission and presented evidence for an association between HSP blazars and neutrinos detected
by the IceCube South Pole Neutrino Observatory\footnote{http://icecube.wisc.edu}.
\citet{Petropoulou2015} further modelled the HE SED of six HSPs selected by \citet{Padovani2014}
as most probable neutrino sources and predicted their neutrino fluxes. All six predicted fluxes were consistent, within the errors, with the observed
neutrino fluxes from IceCube, especially so for two sources (MKN421 and H1914$-$194).
\citet{Padovani2016} have recently cross-matched two VHE catalogs and the 2WHSP with the most recent IceCube neutrino lists
\citep{IceCube2015}, measuring the number of neutrino events with at least one $\gamma$-ray counterpart.
In all three catalogs they observed a positive fluctuation with respect to the mean random expectation at a significance level between
0.4 and 1.3\%, with a p-value of 0.7\% for 2WHSP sources with FOM~$\ga 1$.
{\it All} HBLs considered to be the most probable counterparts of IceCube neutrinos are 2WHSP sources, which strongly suggests that
strong, VHE $\gamma$-ray HBLs are so far the most promising blazar counterparts of astrophysical neutrinos.
Finally, \cite{Resconi2016} have presented evidence of a direct connection between HSP, very high energy neutrinos, and ultra high energy cosmic rays (UHECRs) by correlating the same catalogs used by \citet{Padovani2016} with UHECRs from the Pierre Auger Observatory and the Telescope Array.
A maximal excess of 80 cosmic rays (41.9 expected) was observed for 2FHL HBL.
The chance probability for this to happen is $1.6 \times 10^{-5}$, which translates to $5.5 \times 10^{-4}$ (3.26$\sigma$) after compensation for trials.
\section{Conclusions}
\label{conclusion}
We have assembled the 2WHSP catalog, currently the largest and most complete existing catalog
of HSP blazars, by using a multi-frequency method and a detailed comparison with existing lists
of {$\gamma$-ray}\, emitting blazars. 2WHSP extends the previous 1WHSP catalog \citep{Arsioli2015a} down to
lower Galactic latitudes ($ |b|>10^{\circ} $) and to fainter IR fluxes. In addition, it includes all the
bright known HSP blazars close to the Galactic plane. The 2WHSP sample includes 1,693
confirmed or candidates HSP blazars and was also put together to provide a large list of
potential targets for VHE and multi-messenger observations.
The average $\nu_{\rm peak}$\, for our catalog is $ \langle\log\nu_{\rm peak}\rangle = 16.22 \pm 0.02$ Hz and the average redshift
is $\langle z \rangle=0.331 \pm 0.008$.
We have shown that the SWCD region needs to be extended to include HSPs in which the host galaxy is dominant.
Our radio $\log$N-$\log$S shows that the number of HSP blazars over the whole sky is $> 2,000$ and that HBL
make up $\sim 10\%$ of all BL Lacs.
Finally, we note that this catalog has already been used to provide seeds for the identification
of new {\it Fermi}-LAT objects and to look for astrophysical counterparts to neutrino and
UHECR sources \citep{Padovani2016,Resconi2016}, which proves the relevance of having a large
HSP catalog for multi-messenger astronomy.
\begin{acknowledgements}
YLC is supported by the Government of the Republic of China (Taiwan), BA is supported by the
Brazilian Scientific Program Ci$\hat{\rm{e}}$ncias sem Fronteiras - Cnpq. This work was supported by the
Agenzia Spaziale Italiana Science Data Center (ASDC) and the University La Sapienza of Rome,
Department of Physics. PP thanks the ASDC for the hospitality and partial
financial support for his visits. We made use of archival data and bibliographic information obtained
from the NASA/IPAC Extragalactic Database (NED), data and software facilities from the ASDC
managed by the Italian Space Agency (ASI). Extensive use was made of the TOPCAT software package (http://www.star.bris.ac.uk/$\sim$mbt/topcat/).
\end{acknowledgements}
\bibliographystyle{aa}
|
2,869,038,156,388 | arxiv | \section{Introduction}
The purpose of this paper is to continue the work in \cite{Nilsson:2018lof} and derive the eigenvalue spectrum of some of the operators that determine the masses and supermultiplet structure
of the entire Kaluza-Klein spectrum of the squashed seven-sphere compactification of eleven-dimensional supergravity.
The list of previously known eigenvalue spectra, containing $\Delta_0$, ${\slashed D}_{1/2}$ and $\Delta_1$, is in this paper extended by those of
$\Delta_2$, $\Delta_3$ and $\Delta_L$ while ${\slashed D}_{3/2}$ remains to be done.
Note that parts of the spectrum of the Lichnerowicz operator $\Delta_L$ are derived below but they were quoted already in \cite{Duff:1986hr}\footnote{See Ref. [198], unpublished work by Nilsson and Pope.}.
With the intention to keep this paper as brief as possible we refer the reader to the review \cite{Duff:1986hr} for a Kaluza-Klein background on the problem and
for some of the necessary details and conventions needed in the derivations below. For the full structure of irreducible isometry representations on $AdS_4$ stemming from
the squashed seven-sphere compactification we
refer to \cite{Nilsson:2018lof}. The latter paper summarises in a few pages the most relevant information from \cite{Duff:1986hr} in particular several tables
that are spread out in various chapters of \cite{Duff:1986hr}.
In this spirit we present below just the most crucial formulas that are needed to define the problem and derive the spectra. In the Appendix we collect a number of useful octonionic identities and other formulas,
as well as some for our purposes crucial Weyl tensor calculations.
The coset description of the squashed seven-sphere is
\begin{equation}
G/H=Sp_2\times Sp_1^C/Sp_1^A\times Sp_1^{B+C},
\end{equation}
where, in order to define the denominator, $Sp_2$ is split into $Sp_1^A\times Sp_1^B$ and $Sp_1^{B+C}$ is the diagonal subgroup of $Sp_1^B$ and $ Sp_1^C$. The $G$ irreps specifying the mode functions $Y$
of a general Fourier expansion on $G/H$ are denoted $(p,q;r)$ \cite{Duff:1986hr} and the entire Kaluza-Klein irrep spectrum is derived and tabulated in terms of {\it cross diagrams} in \cite{ Nilsson:2018lof}.
The present paper will fill in some gaps in our knowledge of the eigenvalue spectrum of the relevant operators on the squashed seven-sphere but some are still missing. The remaining issues that we need to address to complete the eigenvalue spectra will be elaborated upon in a forthcoming publication \cite{Nilsson:2021kn}.
The interest in deriving complete spectra in various Kaluza-Klein compactifications has recently been revived due to the discovery of new powerful versions of the embedding tensor technique.
However, these methods can be applied directly only when the vacuum is an extremum of some gauged supergravity theory in $AdS_4$ which can be lifted to ten or eleven dimensions,
see for instance \cite{Malek:2019eaz, Cesaro:2021haf} and references therein. The point we want to emphasise here is that, due to the {\it space invaders scenario} \cite{Duff:1983ajq, Duff:1986hr}, the squashed seven-sphere solution is not of this kind and it therefore
seems clear that other methods
are required to obtain the $AdS_4$ spectrum in this case.
There are also potential applications of this work in the context of the swampland program\footnote{We are grateful to M.J. Duff for raising some interesting aspects of this question that eventually led to this work.},
see for instance \cite{Vafa:2005ui, ArkaniHamed:2006dz, Ooguri:2016pdq}
and references therein. This particular connection will be addressed elsewhere.
In the next section we very briefly review the background of the problem and the method that is used in this paper to derive the eigenvalue spectrum of
operators on coset manifolds. The method is explained in many places, e.g. \cite{Salam:1981xd, Duff:1986hr},
and in Section 3 we first apply it to obtain the spectra of the spin 0 and 1 operators giving
straightforwardly the well known results cited in \cite{Duff:1986hr}.
Already when applied to the spin 1/2 Dirac operator complications arise
which get further pronounced when we subsequently turn to more and more complicated operators.
A summary of the obtained eigenvalues is provided in the Conclusions, together with comments on some of the remaining issues.
Some technical aspects needed in the derivations below are explained in the Appendix.
\section{Compactification on the seven-sphere}
The Fourier expansion technique that we will apply is, following the general strategy explained in \cite{Salam:1981xd} and summarised in \cite{Duff:1986hr},
based on the {\it coset master equation} for the $Spin(n)$ covariant derivative
on a $n$-dimensional coset space $G/H$:
\begin{equation}
\label{cma}
\nabla_a Y+\tfrac{1}{2} f^{bc}{}_a\,\Sigma_{bc} Y=-T_a Y.
\end{equation}
The mode functions $Y$ have two suppressed indices:
One corresponding to the $Spin(n)$ tangent space irrep of the field that is being Fourier expanded\footnote{Note, however, that the whole matrix of the $G$ group element is involved in this equation.} and one that specifies the mode, that is a $G$ irrep. This equation
is derived in \cite{Duff:1986hr} from group elements of the isometry group $G$. In the conventions used there the Lie algebra of the group $G$ has generators $T_A$, satisfying $[T_A, T_B]=f_{AB}{}^CT_C $,
which are divided
into $T_{\bar a}$ of the subgroup $H$ and $T_{a}$ in the complement of $H$ in $G$. Thus, if the Lie algebra of $G$ is reductive it splits as follows:
\begin{equation}
\label{reductiveliealg}
[T_{\bar a}, T_{\bar b}]=f_{\bar a \bar b}{}^{\bar c}T_{\bar c},\,\,\,[T_{\bar a}, T_{b}]=f_{\bar a b}{}^{c}T_{c},\,\,\,\,\,\,\,[T_{a}, T_{b}]=f_{a b}{}^{\bar c}T_{\bar c}+f_{a b}{}^{c}T_{c}.
\end{equation}
The indices $a,b,..$ are vector indices in the tangent space of $G/H$ and $\nabla_a$ in (\ref{cma}) is an $Spin(n)$ covariant derivative
with a torsion free spin connection while the $\Sigma_{ab}$ are the generators of $Spin(n)$ in the representation relevant for the operator equation we are solving.
Note that it is only the structure constants $f_{a b}{}^{c}$ that appear in the coset master equation (\ref{cma}). For symmetric spaces like the round seven-sphere $f_{a b}{}^{c}=0$.
Furthermore, the second algebraic relation in (\ref{reductiveliealg}) defines the imbedding of $H$ in the tangent space group $Spin(n)$.
For symmetric coset spaces the coset master equations thus reads $\nabla_a Y=-T_a Y$
and the eigenvalue spectrum for the operators on $G/H$ are rather easily derived. The complications arising in the squashed seven-sphere case therefore stem
entirely from the structure constant dependent term in (\ref{cma}). As first shown in \cite{Bais:1983wc} for the squashed seven-sphere they are given by
the structure constants of the octonions $a_{abc}$. In the normalisation used in \cite{Duff:1986hr} they read
\begin{equation}
f_{abc}=-\tfrac{1}{\sqrt{5}}a_{abc}=-\tfrac{2}{3}m\,a_{abc}.
\end{equation}
Since octonions will play a key role in the rest of this paper we have listed some useful octonionic identities in the Appendix. Furthermore, in the last expression
for these structure constants we have introduced the scale parameter $m$ arising from the ansatz $F_{\mu\nu\rho\sigma}=3m\epsilon_{\mu\nu\rho\sigma}$ in the compactification of eleven-dimensional supergravity.
This is useful since we are dealing with dimensionful quantities, like the covariant derivatives and Hodge-de Rham operators. The conventions used in \cite{Duff:1986hr} correspond to setting
\begin{equation}
m^2=\tfrac{9}{20}.
\end{equation}
We will assume $m>0$ and also use $m=\tfrac{3}{2\sqrt{5}}$ as for $f_{abc}$ above.
The squashed seven-sphere discussed in this work has the orientation that gives rise to $\mathcal N =1$ supersymmetry in $AdS_4$ after compactification. This fact will be used below
when we discuss the corresponding supermultiplets and the $SO(3,2)$ irreps entering these multiplets.
With this preparation we can readily attack the eigenvalue problems $\Delta_p Y_p = \kappa_p^2 Y_p$ where the Hodge-de Rham\footnote{This operator generalises the Laplace-Beltrami operator to $p$-forms with $p>0$.}
operator on $p$-forms is defined as $\Delta_p = \text{d}\delta + \delta \text{d}$. Here $\text{d}$ is the exterior derivative and $\delta=(-1)^p\star \text{d} \star$ its adjoint. These operators act on forms according to
\begin{equation}
(\text{d}Y)_{a_1\dots a_p} = p\nabla_{[a_1}Y_{a_2\dots a_p]},
\quad
(\delta Y)_{a_1\dots a_p} = -\nabla^{b}Y_{ba_1\dots a_p},
\end{equation}
where brackets here and in the following are weighted antisymmetrisations, $Y_{[a_1\dots a_p]} = \frac{1}{p!}(Y_{a_1\dots a_p}+\text{permutations with sign})$. The explicit form of $\Delta_p$
can for all operators considered in this paper be expressed using the Riemann tensor, $R_{abcd}$, and the d'Alembertian $\Box \equiv \nabla^a\nabla_a$.
On the squashed seven-sphere, which is an Einstein space with $R=42m^2$,
\begin{equation}\label{eq:RiemannDef}
R_{ab}{}^{cd}=W_{ab}{}^{cd}+2m^2\delta_{ab}^{cd},
\end{equation}
where $W_{abcd}$ is the Weyl tensor given in the Appendix.
The main use of the coset master equation \eqref{cma} will be to replace derivatives by group-theoretical algebraic data. In particular, for the squashed sphere, by squaring \eqref{cma}
we obtain an expression for $\Box$ which can then be used to replace $\Box$ by algebraic data plus terms linear in $\nabla_a$. This will be clear below.
\section{Eigenvalue spectra on the squashed seven-sphere}
In this section we start by briefly reviewing some of the results previously obtained with the coset space techniques mentioned above.
These results are then generalised to the more complicated operators for which we present a number of new eigenvalues.
\subsection{Review of the method applied to forms of rank 0 and 1}
The method outlined above becomes rather trivial when applied to 0-forms. In this case we want to solve the eigenvalue equation
\begin{equation}
\Delta_0 Y=\kappa_0^2 Y,
\end{equation}
where the positive semi-definite Hodge-de Rham operator on 0-forms is $\Delta_0=-\Box$. Thus, following \cite{Duff:1986hr}, the spectrum is obtained directly by squaring the coset master equation (\ref{cma}):
\begin{equation}
\Delta_0 Y=-\Box Y=-T_aT_aY=(C_G-C_H)Y=\kappa_0^2 Y,
\end{equation}
where the two second order Casimir operators for $G=Sp_2\times Sp_1^C$ and $H=Sp_1^A\times Sp_1^{B+C}$ have eigenvalues
\begin{equation}
C_G=C(p,q)+3C^C(r)=\tfrac{1}{2}(p^2+2q^2+4q+6q+2pq)+\tfrac{3}{4}r(r+2),
\end{equation}
and
\begin{equation}
C_H=2C^A(s)+\tfrac{6}{5}C^{B+C}(t)=\tfrac{1}{2}\,s(s+2)+\tfrac{3}{10}\,t(t+2).
\end{equation}
Here $(p,q)$ are Dynkin labels for $Sp_2$ irreps and $r,s,t$ those for the irreps of the various $Sp_1$ groups that occur in the squashed seven-sphere coset description.
The whole problem of deriving the eigenvalue spectrum of harmonics is reduced to determining first which $H$ irreps are contained
in the $Spin(7)$ irrep of Y (trivial for 0-forms) and then finding all $G$ irreps that when split into $H$ irreps contain any of the $H$ irreps found in the $Spin(7)$ irrep of Y.
Some partial results on the spectrum of $G$ irreps are obtained in \cite{Nilsson:1983ru, Duff:1986hr} by breaking up the spectrum on the round sphere according to $Spin(8)\rightarrow Sp_2\times Sp_1$ while the entire spectrum
of all operators is derived directly from the squashed sphere coset in \cite{Nilsson:2018lof} using the coset method just described.
For 0-forms $C_H=0$ and the spectrum therefore becomes $\kappa^2_0=C_G$.
However, as explained above, we should introduce the scale parameter $m$. In view of this we can use $m^2=\tfrac{9}{20}$ to write the
eigenvalues as follows:\footnote{We will in the rest of this paper display eigenvalues either as $\kappa_p^2$ or, equivalently, as $\Delta_p$.}
\begin{equation}
\Delta_0=\kappa^2_0=\frac{m^2}{9} 20\,C_G(p,q;r).
\end{equation}
As an example how this result is used let us consider the spectrum in the graviton sector\footnote{As in \cite{Nilsson:2018lof} we will use the notation of \cite{Breitenlohner:1982jf} which includes the parity of the field, as in, e.g., $2^+$.}: In units of $m$ the $SO(3,2)$ irrep defining energy $E_0(spin)$ is \cite{Heidenreich:1982rz, Breitenlohner:1982bm, Englert:1983rn}
\begin{equation}
E_0(2^+)=\frac{3}{2}+\frac{1}{2}\sqrt{\frac{M_2^2}{m^2}+9}=\frac{3}{2}+\frac{1}{2}\sqrt{\frac{20C_G}{9}+9}=\frac{3}{2}+\frac{1}{6}\sqrt{20C_G+81},
\end{equation}
where we have used the fact that the mass-square operator for spin 2 fields in $AdS_4$ is $M^2_2=\Delta_0$.
We will occasionally refer to results like this as being of {\it square-root form}, here a $\sqrt{81}$-form. The reason for this is that all the fields in a supermultiplet must be of the same square-root form.
We now turn to the 1-form case and review the result and the derivation in \cite{Duff:1986hr}\footnote{For a different derivation see \cite{Yamagishi:1983ri}.}. In this case the structure constant term in the coset master equation comes in and complicates the
calculation somewhat. As for the 0-form modes we write out the $\Delta_1$ eigenvalue equation explicitly:
\begin{equation}
\Delta_1:\,\,\,\,\Delta_1 Y_a = - \Box Y_a + R_a{}^bY_b=-\Box Y_a+ 6m^2 Y_a = \kappa_1^2 Y_a,
\end{equation}
We want to use the square of the coset master equation to eliminate the $\Box$ term: Inserting $f_{abc}=-\tfrac{2}{3}ma_{abc}$
into \eqref{cma} gives $\nabla_aY_b-\frac{m}{3}a_{abc}Y_c=-T_aY_b$ which when squared yields
\begin{equation}
G/H:\,\,\,\,\,\Box Y_a+\frac{2m}{3}a_{abc}\nabla_b Y_c+ \frac{m}{3}(\nabla_b a_{abc})Y_c-\frac{m^2}{9} a_{abc}a_{bcd}Y_d=T_bT_bY_a.
\end{equation}
Using the octonionic identities
$\nabla_a a_{bcd}= mc_{abcd}$ and $a_{abe}a^{cd}{}_e=2\delta_{ab}^{cd}+c_{ab}{}^{cd}$ (see the Appendix)
together with $T_aT_a=-(C_G-C_H)$, the $G/H$ equation above simplifies to
\begin{equation}
-\Box Y_a - \frac{2m}{3}a_{abc}\nabla_bY_c +6m^2Y_a =C_GY_a.
\end{equation}
Here we have also used the fact that the irrep ${\bf 7}$ of $SO(7)$ splits into $(1,1)\oplus (0,2)$ of $H=Sp_1^A\times Sp_1^{B+C}$ and that $C_H=\tfrac{12}{5}$ for both of these $H$ irreps.
So eliminating the $\Box$ term from the $\Delta_1$ and $G/H$ equations above gives
\begin{equation}
(\kappa_1^2-C_G)Y_a = \frac{2m}{3}a_{abc}\nabla _b Y_c.
\end{equation}
In order to extract the eigenvalues $\kappa_1^2$ from this equation we will have to square it. Let us define the operator
\begin{equation}
DY_a\equiv a_{abc}\nabla _b Y_c.
\end{equation}
Taking the square of $D$ then goes as follows:
\begin{equation}
D^2Y_a=a_{abc}\nabla _b a_{cde}\nabla_dY_e=a_{abc}(\nabla _b a_{cde})\nabla_dY_e+a_{abc} a_{cde}\nabla _b\nabla_dY_e.
\end{equation}
Using identities from the Appendix this equation becomes
\begin{equation}
D^2Y_a= ma_{abc}c_{bcde}\nabla_dY_e+(2\delta_{ab}^{de}+c_{ab}{}^{de})\nabla _b\nabla_dY_e= 4mDY_a+(6m^2-\Box)Y_a.
\end{equation}
Here we have also used the Ricci identity on 1-forms $[\nabla_a,\nabla_b]Y_c=R_{abc}{}^dY_d$ and the fact
that the Riemann tensor term in the computation of $D^2$ above drops due to its contraction with $c_{eabc}$. Then since
the last term in the equation for $D^2Y_a$ above is just $\Delta_1$, the equation can be written
\begin{equation}
D^2Y_a- 4mDY_a-\kappa_1^2Y_a=0.
\end{equation}
If we now use $DY_a=\frac{3}{2m}(\kappa_1^2-C_G)Y_a$ and $D^2Y_a=(\frac{3}{2m}(\kappa_1^2-C_G))^2Y_a$ we get the final result for the 1-form eigenvalues
\begin{equation}
\Delta_1=\frac{m^2}{9}(20C_G+14\pm2\sqrt{20C_G+49})=\frac{m^2}{9}(\sqrt{20C_G+49}\pm1 )^2-4m^2.
\end{equation}
The last form of the answer will be useful later.
\subsection{Spin 1/2 by squaring}
Before turning our attention to forms of rank two and three, and after that second rank symmetric tensors, we will check that our methods are able to produce the known result for the Dirac operator
acting on spin 1/2 modes. The spectrum of $\slashed D_{1/2} \equiv -i \slashed \nabla$ was derived in \cite{Nilsson:1983ru} by a different method based on the fibre bundle description of the
squashed seven-sphere. The virtue of the method in \cite{Nilsson:1983ru} is that one can follow the eigenvalues as one turns the squashing parameter from the round sphere value to
the squashed Einstein space value. This nice feature is unfortunately lacking for the methods used in this paper.
To apply our present techniques we start by squaring the Dirac operator in order to obtain a situation that is similar to the one for the Hodge-de Rham operators on $p$-forms. Using
$\Gamma^a\nabla_a\Gamma^b\nabla_b\psi=(\Box+\tfrac{1}{2}\Gamma^{ab}[\nabla_a,\nabla_b])\psi$ we find
\begin{equation}
-i\slashed \nabla \psi=\lambda\psi \Rightarrow (-\Box+\frac{R}{4})\psi=\lambda^2\psi.
\end{equation}
It is now possible to use the $G_2$ structure to split the tangent space spinor irrep into $G_2$ irreps as ${\bf 8}\rightarrow {\bf 7}\oplus {\bf 1}$, or in terms of indices $A=(a,8)$.
Then by defining two spinors $\eta$ and $ \eta_a$ we can expand a general Dirac spinor as follows
\begin{equation}
\psi=V^a\eta_a+f\,\eta,
\end{equation}
where $V^a$ is a vector field and $f$ a scalar field on the seven-sphere. While $\eta$ is the standard Killing spinor with components $\eta_B=\delta_{B}^8$, $\eta_a$ is defined by $\eta_a\equiv -i\Gamma_a \eta$
which implies the its explicit form is $(\eta_a)_B = \delta_{aB}$.
These spinors are
linearly independent and satisfy
\begin{equation}
\nabla_a\eta=\tfrac{m}{2}\eta_a, \,\,\,\,\,\nabla_a\eta_b=-\tfrac{m}{2}\delta_{ab}\eta+\tfrac{ m}{2}a_{abc}\eta_c.
\end{equation}
These equations imply
\begin{equation}
\Box\eta=-\tfrac{7m^2}{4}\eta,\,\,\,\,\Box\eta_a=-\tfrac{7m^2}{4}\eta_a,
\end{equation}
from which we obtain the equations
\begin{eqnarray}\label{BoxSpinor}
\Box(f\eta)&=&(\Box f-\tfrac{7m^2}{4}f)\eta+m(\nabla^af)\eta_a,\\
\,\,\,\,\Box(V^a\eta_a)&=&(\Box V^a-\tfrac{7m^2}{4}V^a)\eta_a+m(\nabla^aV^b)a_{abc}\eta_c.
\end{eqnarray}
Consider now the coset master equation for Dirac spinor modes. Squaring it gives
\begin{equation}
\Box \psi-\tfrac{m}{6}a_{abc}\Gamma_{ab}\nabla_c\psi-\tfrac{7m^2}{12}\psi+\tfrac{m^2}{144}c_{abcd}\Gamma_{abcd}\psi=T_aT_a\psi.
\end{equation}
So eliminating the $\Box$ term and inserting $R=42m^2$ we find that the equation we need to solve reads
\begin{equation}
\label{diracboxeq}
-\lambda^2\psi-\tfrac{m}{6}a_{abc}\Gamma_{ab}\nabla_c\psi+\tfrac{119m^2}{12}\psi+\tfrac{m^2}{144}c_{abcd}\Gamma_{abcd}\psi=T_aT_a\psi.
\end{equation}
In order to get the last term on the LHS in a nice form we introduce the projection operators for the $G_2$ split $\psi=\psi_1+\psi_7$ corresponding to ${\bf 8}\rightarrow {\bf 1}\oplus {\bf 7}$.
They are (see Appendix for more details)
\begin{equation}
P^s_1=\tfrac{1}{8}(1-\tfrac{1}{24}c_{abcd}\Gamma_{abcd}),\,\,\,\,P^s_7=1-P_1=\tfrac{1}{8}(7+\tfrac{1}{24}c_{abcd}\Gamma_{abcd}).
\end{equation}
In terms of these projectors the sum of the last two terms on the LHS above becomes
\begin{equation}
\tfrac{119}{12}\psi+\tfrac{1}{144}c_{abcd}\Gamma_{abcd}\psi=\tfrac{35}{4}P^s_1+\tfrac{121}{12}P^s_7.
\end{equation}
Then using the octonionic version of the seven-dimensional gamma matrices given in the Appendix, some algebra gives
\begin{equation}
a_{abc}\Gamma_{ab}\nabla_c\psi=(-6\nabla_aV^a-21mf)\eta+(2a^{abc}\nabla_a V_b-9mV^c+6(\nabla^cf))\eta_c.
\end{equation}
Inserting these results into (\ref{diracboxeq}) we get an equation that can be hit by the projectors $P_1^s$ and $P_7^s$ leading to the two equations
\begin{equation}
\lambda^2 f=\tfrac{20}{9}m^2 C_G f+\tfrac{35}{4}m^2f+m(\nabla_aV^a+\tfrac{7}{2}mf),
\end{equation}
and
\begin{equation}
\lambda^2 V_a=\tfrac{20}{9}m^2 (C_G -\tfrac{12}{5})V_a+(\tfrac{121}{12}+\tfrac{3}{2})m^2V_a-\tfrac{1}{3}ma_{abc}\nabla_bV_c - m\nabla_a f.
\end{equation}
To solve these equations we either have $f=0$ or $f \neq 0$. In the former case the first equation gives $\nabla^aV_a=0$ which implies that the second equation for $V_a$ can be
analysed in exactly the same way as for the 1-forms discussed previously.
Not surprisingly the result is also of $\sqrt{49}$-form and reads
\begin{equation}
\lambda^2=m^2(\tfrac{1}{6}\pm\tfrac{1}{3}\sqrt{20C_G+49})^2.
\end{equation}
Turning to the latter case, that is $f\neq 0$, we start by taking the divergence of the second equation and use the fact that $\Box f=-\frac{20m^2}{9}C_Gf$ to find
\begin{equation}
(\tfrac{20}{9}m^2C_G+\tfrac{25}{4}m^2-\lambda^2)(\nabla_aV^a)=-\tfrac{20}{9}m^3C_G\,f.
\end{equation}
Now the system of two equations for the functions $f$ and $\nabla^aV_a$ has the solutions
\begin{equation}
\lambda^2=m^2(\tfrac{1}{2}\pm\tfrac{1}{3}\sqrt{20C_G+81})^2.
\end{equation}
These two sets of eigenvalues are consistent with the known result from \cite{Nilsson:1983ru}. However, in that paper there is no sign ambiguity
since there one obtains $\lambda$ instead of $\lambda^2$. This problem is easily eliminated by applying our present result to the spin $2^+$
supermultiplet.
This completes the review. We now turn to 2-forms where the results are new.
\subsection{2-forms}
In terms of the Hodge-de Rham operator on 2-forms, the equation to be solved is
\begin{equation}
\Delta_2 Y_{ab}=-\Box Y_{ab}-2R_{acbd}Y^{cd}-2R_{[a}{}^cY_{b]c}=\kappa_2^2 Y_{ab}.
\end{equation}
Using \eqref{eq:RiemannDef} for the Riemann tensor gives the following equation for the box operator
\begin{equation}
\label{deltatwo}
\Delta_2:\,\,\,\,\Box Y_{ab}=-\kappa^2_2 Y_{ab}-2W_{acbd} Y^{cd}+10m^2 Y_{ab}.
\end{equation}
As for the previous cases the next step is to express the box operator in terms of algebraic objects on the coset manifold. The master coset formula (\ref{cma})
for 2-forms reads
\begin{equation}
\nabla_a Y_{bc}=-T_aY_{bc}-f_{a}{}^{d}{}_{[b}Y_{c]d},
\end{equation}
which can be written
\begin{equation}
\tilde\nabla_a Y_{bc}=-T_aY_{bc},
\end{equation}
by indroducing the "$G_2$-derivative" $\tilde\nabla_a$ defined on 2-forms by
\begin{equation}
\tilde\nabla_a Y_{bc}\equiv \nabla_a Y_{bc}+\frac{2m}{3}a_{a[b}{}^d Y_{c]d}.
\end{equation}
This derivative is "$G_2$" in the sense that it satisfies $\tilde\nabla_a a_{bcd}=0$ and $\tilde\nabla_a c_{bcde}=0$. This step is not crucial here but we will find more implications later of the presence of the
$G_2$ holonomy so we will have reason to return to this derivative then. (For more details about $G_2$ in this context, see for instance \cite{House:2004pm}.)
To get an algebraic expression for the box operator we can now square the master coset equation, $\tilde\nabla_a Y_{bc}=-T_aY_{bc}$, to get $\tilde\Box Y_{ab}=T_cT_cY_{ab}$. This equation
can also be written as
\begin{equation}
\label{goverhtwoforms}
G/H:\,\,\,\,\,\tilde\Box Y_{ab}=(\Box -\frac{10}{9}m^2)Y_{ab}+\frac{4}{3}ma_{cd[a}\nabla_{|c}Y_{d|b]}-\frac{2}{9}m^2c_{abcd}Y_{cd}=T_cT_cY_{ab}.
\end{equation}
Finally, to get the equation that needs to be solved to find the eigenvalues we just eliminate the box operator from (\ref{deltatwo}) and (\ref{goverhtwoforms}). Using the projectors defined in the Appendix
gives the following useful form of the resulting equation
\begin{equation}\label{algebraic2form}
(\kappa^2_2-12m^2+T_cT_c)Y_{ab}=-W_{abcd}Y^{cd}-\frac{4}{3}m^2(P_{14}Y)_{ab}+\frac{4}{3}m(a_{cd[a}\tilde\nabla_{|c}Y_{d|b]}),
\end{equation}
which, again, is not entirely algebraic due to the appearance of the operator $\tilde\nabla_a$ in the last term. This also happened in the case of the
1-form where it was easy to handle by a squaring procedure. This trick is a bit harder to apply in the case of 2-forms (as well as for the other
operators to be discussed later) as will become clear shortly. Compared to the 1-form case there is also a new feature here namely the
presence of the Weyl tensor in one of the terms which will cause additional complications. The approach used in this paper to deal with the Weyl tensor terms, in this and the other cases discussed below,
is explained in the Appendix.
Thus there are two new issues when trying to solve the 2-form equation (\ref{algebraic2form}): The Weyl tensor term and the $\tilde\nabla$ term which now tends to lead to symmetrised derivatives when squared.
To deal with the former issue we recall how $T_aT_a$ can be expressed in terms of the Casimir operators for the groups involved: $T_aT_a=-(C_G-C_H)$.
This, however, has implications for the how the spectrum is organised in terms of towers. To see this we use the tangent space decomposition
$
SO(7)\rightarrow G_2 \rightarrow Sp^A_1\times Sp^{B+C}_1
$
which makes it possible to read off the relevant decompositions directly from the McKay and Patera tables \cite{Mckay:1981mp}.
In the case of the 2-form the composition reads, see \cite{Duff:1986hr} or the summary in \cite{Nilsson:2018lof},
\begin{equation}
\bf{21}\rightarrow \bf{7}\oplus \bf{14} \rightarrow (\bf{(1,1)}\oplus \bf{(0,2)})\oplus ((0,2)\oplus (1,3) \oplus (2,0)).
\end{equation}
This means that the towers will be tabulated according to their $H$ irrep which will thus obscure the $G_2$ structure of the spectrum. As is explained in the Appendix
this problem is automatically eliminated once the Weyl tensor term is analysed and the result combined with the result from the $C_H$ term. In fact, adding these two terms gives
$\tfrac{24}{5}Y_{ab}=\tfrac{32}{3}m^2Y_{ab}$ for all the $H$ irreps in the $G_2$ irrep ${\bf 14}$. Recall that the corresponding answer for the irrep ${\bf 7}$ is $\tfrac{12}{5}=\tfrac{16}{3}m^2$.
Using this insight from the Appendix, namely that the sum $C_H+Weyl$ has a common value on all $H$ irreps in the decomposition of each $G_2$ irrep arising from the $SO(7)$ representation ${\bf 21}$,
it becomes possible to split the 2-form equation into two by acting on it with the projectors $P_7$ and $P_{14}$ and insert the respective values of $C_H+Weyl$.
We find, using the definitions $Y^{(7)}_{ab}\equiv (P_7Y)_{ab}$ and $Y_a\equiv a_{abc}Y_{bc}^{(7)}$,
\begin{equation}
{\bf 7}:\,\,\,\,\,\,\kappa_2^2Y_a=C_GY_{a}+(12+\frac{4}{3}-\frac{16}{3})m^2Y_a-\frac{4m}{3}(a_{abc}\nabla_bY_c),
\end{equation}
\begin{equation}
{\bf 14}:\,\,\,\,\,\,\kappa_2^2Y^{(14)}_{ab}=C_GY^{(14)}_{ab}+(12-\frac{4}{3}-\frac{32}{3})m^2Y^{(14)}_{ab}+\frac{2m}{3}(P_{14})_{ab}^{cd}(\nabla_c Y_{d}).
\end{equation}
Note that the last term in the second equation contains $Y_a$ and thus seems to mix with the first equation. However, the structure of the derivative terms as a 2 by 2 matrix
shows that this term will have no role in determining the eigenvalues for the 2-form as will be clear below. To proceed we write these two equations as
\begin{equation}
{\bf 7}:\,\,\,\,\,\,\kappa_2^2Y_a=\frac{m^2}{9}(20C_G+72)Y_a-\frac{4}{3}m(a_{abc}\nabla_bY_c),
\end{equation}
\begin{equation}
{\bf 14}:\,\,\,\,\,\,\kappa_2^2Y^{(14)}_{ab}=\frac{m^2}{9}\,20C_GY^{(14)}_{ab}+\frac{2}{3}m(P_{14}\nabla Y)_{ab}.
\end{equation}
Consider first the possibility to take the square of the first equation above, the one for $Y_a$. However, although this will give rise to a calculation quite similar to the one for the 1-form, there is one important difference.
While the 1-form is transverse (i.e., divergence free) this is not the case for $Y_a$ coming from the 2-form. Therefore one needs to analyse the two equations for the 2-form in stages:\\
\\
1) Either $Y_a=0$ or $Y_a\neq 0$, 2) when $Y_a\neq 0$ then either $\nabla_aY_a=0$ or $\nabla_a Y_a\neq 0$.\\
\\
So, when $Y_a=0$ the equation left to solve is the ${\bf 14}$-equation with the derivative term set to zero. This equation therefore gives the eigenvalues
\begin{equation}
\Delta_2=\frac{m^2}{9}\,20\,C_G.
\end{equation}
According to the characterisation mention above, this result leads to an energy $E_0$ of the $\sqrt{9}$-form. This fits nicely with a supermultiplet containing spins $(1^+, 1/2, 1/2, 0^+)$ with masses related to the operators
$(\Delta_2, {\slashed D}_{3/2}, {\slashed D}_{3/2}, \Delta_L)$ since we know from Ref. [198] in \cite{Duff:1986hr} that $\Delta_L$ also has an eigenvalue leading to
a $\sqrt{9}$-form. One can check that the corresponding values of $E_0$ work out as they should in relation to supersymmetry.
This eigenvalue of $\Delta_L$ leading to the $\sqrt{9}$-form will be derived in detail later in this section.
We now turn to the cases with $Y_a\neq 0$, namely $\nabla_aY_a=0$ and $\nabla_a Y_a\neq 0$. Clearly, when $\nabla_aY_a=0$ the calculation will follow the one for 1-forms very closely.
Indeed, the result is also of $\sqrt{49}$-form and reads
\begin{equation}
\Delta_2=\frac{m^2}{9}(\sqrt{20 C_G+49}\pm 2)^2-m^2.
\end{equation}
Although being of $\sqrt{49}$-form, this 2-form eigenvalue formula differs from the one for 1-forms given in \cite{Duff:1986hr}, and also derived above,
\begin{equation}
\Delta_1=\frac{m^2}{9}(20 C_G+ 14\pm 2 \sqrt{20 C_G + 49})=\frac{m^2}{9}(\sqrt{20 C_G+49}\pm 1)^2-4m^2.
\end{equation}
The differences are partly compensated for by the
different relations between the Hodge-de Rham operators and the respective $M^2$ operators \cite{Duff:1986hr}:
\begin{equation}
M^2(1^+)=\Delta_2,
\end{equation}
\begin{equation}
M^2(1^-)=\Delta_1+12m^2\pm6m\sqrt{\Delta_1+4m^2}=(\sqrt{\Delta_1+4m^2}\pm 3m)^2-m^2.
\end{equation}
If the $\Delta_1$ eigenvalues are inserted into this formula for $M^2(1^-)$ we see that the last term, that is $-4m^2$ in $ \Delta_1$, is really required to have a chance for supersymmetry to work
in a supermultiplet containing both $1^+$ and $1^-$ fields. In fact, the field content of the spin 3/2 supermultiplets has this property and requires
$E_0(1^+)=E_0(1^-)\pm 1$. To verify this relation we need the energy formula for spin 1 unitary $SO(3,2)$ irreps
\begin{equation}
E_0(1^{\pm})=\frac{3}{2}+\frac{1}{2}\sqrt{\frac{M^2}{m^2}+1}.
\end{equation}
This expression gives the energy values
\begin{equation}
E_0(1^{-(1),(2)})=\frac{3}{2}\pm\frac{3}{2}\mp\frac{1}{6}+\frac{1}{6}\sqrt{20C_G+49},
\end{equation}
where the first $\pm$ sign refers to the two towers of spin $1^{-(1),(2)}$ fields in $AdS_4$ supergravity theory.
Comparing these to
\begin{equation}
E_0(1^+)=\frac{3}{2}\pm\frac{1}{3}+\frac{1}{6}\sqrt{20C_G+49},
\end{equation}
one finds (using also the spectrum of the spin 3/2 fields in these supermultiplets) that it is possible to eliminate the sign ambiguities in the spectra of $\Delta_1$ and $\Delta_2$. For more details see \cite{Nilsson:2021kn}.
Finally, when $\nabla_a Y_a\neq 0$ we take the divergence of the ${\bf 7}$ projected part of the equation and use $a_{abc}\nabla_a\nabla_bY_c=0$ to obtain a simple equation
giving the eigenvalue
\begin{equation}
\kappa^2_2=\frac{m^2}{9}(20C_G+72)
\end{equation}
This completes the analysis of the 2-form eigenvalue spectrum. A list of the obtained eigenvalues can be found in the Conclusions.
\subsection{3-forms}
We now turn to 3-forms leaving the discussion of the Lichnerowicz modes to the next subsection. The reason for doing the analysis in this order is that
we will obtain some equations in the 3-form case that can be applied also for Lichnerowicz.
The equation for 3-forms that we wish to solve reads
\begin{equation}
\Delta_3Y_{abc}=-\Box Y_{abc}+6R^d{}_{[ab}{}^eY_{c]de}+3R_{[a}{}^dY_{bc]d}=\kappa_3^2Y_{abc}.
\end{equation}
Inserting the Riemann and Ricci tensors of the squashed seven-sphere this equation becomes
\begin{equation}
\Delta_3:\,\,\,\,\,\Box Y_{abc}=-\kappa_3^2Y_{abc}+6W^d{}_{[ab}{}^eY_{c]de}+12m^2Y_{abc}.
\end{equation}
Squaring the coset master equation for 3-forms gives
\begin{equation}
G/H:\,\,\,\,\,-(C_G-C_H)Y_{abc}=\Box Y_{abc}-2ma_{d[a}{}^e\tilde\nabla_{|d]}Y_{bc]e}+\frac{m^2}{3}(4Y_{abc}+2c_{[ab}{}^{de}Y_{c]de}),
\end{equation}
where for later convenience we have used the $G_2$ derivative
\begin{equation}
\tilde\nabla_d Y_{abc}\equiv \nabla_d Y_{abc}-ma_{d[a}{}^eY_{bc]e}.
\end{equation}
Note that this implies that the divergence
\begin{equation}
\tilde\nabla^a Y_{abc}=\nabla^a Y_{abc}-\tfrac{m}{3}(a_{aa}{}^eY_{bce}+2a_{a[b}{}^eY_{c]ae})=\tfrac{2m}{3}a_{[b}{}^{de}Y_{c]de}.
\end{equation}
Eliminating the box-operator term from the above equations gives
\begin{equation}\label{Algebraic3Form}
\kappa_3^2Y_{abc}=C_GY_{abc}-(C_HY_{abc}-6W^d{}_{[ab}{}^eY_{c]de})+\frac{m^2}{3}(40Y_{abc}+2c_{[ab}{}^{de}Y_{c]de})-2ma_{d[a}{}^e\tilde\nabla^dY_{bc]e}.
\end{equation}
To proceed we decompose the 3-form into $G_2$ irreps as ${\bf 35}={\bf 1}\oplus{\bf 7}\oplus {\bf 27}$. By defining $Y \equiv a_{abc}Y_{abc}$ and $Y_{a} \equiv c_{abcd}Y_{bcd}$ we can split the 3-form as follows
\begin{equation}
Y_{abc}=Y^{(1)}_{abc}+Y^{(7)}_{abc}+Y^{(27)}_{abc}=\frac{1}{42}a_{abc}Y-\frac{1}{24}c_{abcd}Y_d+(P_{27}Y)_{abc},
\end{equation}
where we have utilised the 3-form projectors defined in the Appendix. The term $C_HY_{abc}-6W^d{}_{[ab}{}^eY_{c]de}$ can be written (see Appendix)
\begin{equation}\label{Weyl3Form}
C_HY_{abc}-6W^d{}_{[ab}{}^eY_{c]de}=-\frac{12}{5}(\frac{1}{24}c_{abcd}Y_d)+\frac{28}{5}(P_{27}Y)_{abc}.
\end{equation}
Apart from this there are two more problematic terms, the $c_{abcd}$-term and the derivative term that will force us to square also this equation.
However, before doing that we will follow the strategy in the 2-form case and start by splitting the equation into $G_2$ pieces. The singlet term ${\bf 1}$ is obtained by contracting all three indices by $a_{abc}$ which gives
\begin{equation}
\kappa_3^2Y=C_GY+\frac{m^2}{3}(40Y+2a_{abc}c_{[ab}{}^{de}Y_{c]de})-2ma_{abc}a_{d[a}{}^e\tilde\nabla^dY_{bc]e}.
\end{equation}
Cleaning up the structure constant factors this equation becomes
\begin{equation}
{\bf 1}:\,\,\,\,\,\,\kappa_3^2Y=(C_G+16m^2)Y+2m\tilde\nabla_aY_a.
\end{equation}
Turning to the ${\bf 7}$ part, we find after some algebra
\begin{equation}
{\bf 7}:\,\,\,\,\,\,\kappa_3^2Y_a=(C_G+6m^2)Y_a-\frac{12m}{7}\nabla_aY+\frac{m}{3}a_{abc}\nabla_bY_c+2ma_{bcd}\nabla_bY^{(27)}_{cda}.
\end{equation}
When we now come to the last part, the ${\bf 27}$, it will require some new steps that will relate it to the metric and the Lichnerowicz equation.
This connection is in fact already indicated by the last term in the ${\bf 7}$ equation just discussed: $a_{bcd}\nabla_bY^{(27)}_{cda}$. This expression suggests that
the 2-index tensor $a_{acd}Y^{(27)}_{bcd}$ will be useful as we now elaborate upon (see \cite{House:2004pm} for a closely related discussion). Let us define a 2-index tensor from the ${\bf 27}$ part of
the 3-form by
\begin{equation}
\tilde Y_{ab}\equiv a_{acd}Y^{(27)}_{bcd},
\end{equation}
where we use a tilde to avoid confusing it with the antisymmetric 2-form discussed previously.
Clearly, the symmetric and traceless part of $\tilde Y_{ab}$ is also in the irrep ${\bf 27}$. However, a nice and indeed very useful fact about this definition
of $\tilde Y_{ab}$ is that it is automatically symmetric and traceless. The tracelessness follows immediately from the identity $a_{abc}c_{abcd}=0$,
or, using the 3-form projectors in the Appendix, from $P_1P_{27}=0$, while the vanishing of its
antisymmetric part follows from $a_{abc}\tilde Y_{bc}=0$ and $c_{abcd}\tilde Y_{cd}=0$\footnote{Note that these two conditions can be combined to the identity since
$\delta_{ab}^{cd}=\tfrac{1}{2}(a_{abe}a^{cde}-c_{ab}{}^{cd})$.}. These two results can be checked as follows:
\begin{equation}
a_{abc}\tilde Y_{bc}=a_{abc}a_{bde}Y^{(27)}_{cde}=-(2\delta_{ac}^{de}+c_{acde})Y^{(27)}_{cde}=-c_{acde}Y^{(27)}_{cde}=0,
\end{equation}
where we have used $P_7P_{27}=0$ in the last step, and
\begin{equation}
c_{abcd}\tilde Y_{cd}=c_{abcd}a_{cef}Y^{(27)}_{def}=-6a_{[ab}{}^{[e}\delta_{d]}^{f]}Y^{(27)}_{def}=4a_{[a}{}^{de}Y^{(27)}_{b]de}=0.
\end{equation}
Here the very last equality is a consequence of the identity
\begin{equation}
a_{abc}(P_{27})_{bcd}{}^{efg}=a_{(a}{}^{[ef}\delta_{d)}^{g]}-\frac{1}{7}\delta_{ad}a^{efg}.
\end{equation}
Clearly, also $a_{abc}(P_{27})_{abc}{}^{efg}=0$ again showing that the trace of $\tilde Y_{ab}$ vanishes.
An alternative, and perhaps more direct, way to see that $\tilde Y_{ab}$ is symmetric and traceless is to just insert the expression for the
$P_{27}$ projector which gives
\begin{equation}
a_{abc}Y^{(27)}_{bcd} = \frac{1}{2}(a_{abc}Y_{bcd}+a_{dbc}Y_{bca})-\frac{1}{7}\delta_{ad}Y.
\end{equation}
Finally, for the definition $\tilde Y_{ab}\equiv a_{acd}Y^{(27)}_{bcd}$ to be useful we need to give also the inverse
relation\footnote{Here one may use the fact that $Y^{(27)}_{abc}=(P_{27}Y^{(27)})_{abc}$ implies the identity $Y^{(27)}_{abc}=-\tfrac{3}{2}c_{de[ab}Y_{c]de}^{(27)}$.}:
\begin{equation}
Y^{(27)}_{abc}=\frac{3}{4}a_{d[ab}\tilde Y_{c]d}.
\end{equation}
The decomposition of the 3-form therefore takes the following simple form in terms of $Y, Y_a, \tilde Y_{ab}$, respectively in irreps ${\bf 1}, {\bf 7}, {\bf 27} $,
\begin{equation}
Y_{abc}=\frac{1}{42}a_{abc}Y-\frac{1}{24}c_{abcd}Y_d+\frac{3}{4}a_{d[ab}\tilde Y_{c]d}.
\end{equation}
With these preliminary results at hand it is now a rather straightforward matter to project the equation \eqref{Algebraic3Form} onto its ${\bf 27}$ component. One finds
\begin{equation}
\kappa_3^2\tilde Y_{ab}=(C_G-\frac{m^2}{3}(40-\frac{4}{3}))\tilde Y_{ab}-(C_H\delta_{(ab)}^{cd}+2W_{acbd})\tilde Y_{cd}-\frac{2}{3}ma_{cd(a}\tilde\nabla^c\tilde Y_{b)}{}^d-\frac{m}{3}(\tilde\nabla_{(a}Y_{b)}-\frac{1}{7}\delta_{ab}\tilde\nabla^cY_c).
\end{equation}
Again we can replace the second bracket by its common eigenvalue on all $H$ irreps in the $G_2$ irrep ${\bf 27}$ as explained in the Appendix, that is with $\tfrac{28}{5}=\tfrac{112m^2}{9}$.
We now insert this into the ${\bf 27}$ equation and sum up what we have found so far:
\begin{equation}
{\bf 1}:\,\,\,\,\kappa_3^2Y=(C_G+16m^2)Y+2m\tilde\nabla_aY_a,
\end{equation}
\begin{equation}
{\bf 7}:\,\,\,\,\kappa_3^2Y_a=(C_G+6m^2)Y_a-\frac{12m}{7}\nabla_aY+\frac{m}{3}a_{abc}\nabla_bY_c+2ma_{bcd}\nabla_bY^{(27)}_{cda}.
\end{equation}
\begin{equation}
{\bf 27}:\,\,\,\,\kappa_3^2\tilde Y_{ab}=(C_G+\frac{4m^2}{9})\tilde Y_{ab}-\frac{2}{3}ma_{cd(a}\tilde\nabla^c\tilde Y_{b)}{}^d-\frac{m}{3}(\tilde\nabla_{(a}Y_{b)}-\frac{1}{7}\delta_{ab}\tilde\nabla^cY_c).
\end{equation}
Note that the term $2ma_{bcd}\nabla_bY^{(27)}_{cda}$ appearing in the ${\bf 7}$ equation can be replaced by terms containing only $Y$ and $Y_a$. This relation is derived from
the gauge condition $\nabla^aY_{abc}=0$ as follows:
\begin{equation}
\nabla^a(Y^1_{abc}+Y_{abc}^7+Y_{abc}^{27})=0.
\end{equation}
Contracting this equation with $a_{dbc}$ and using $\nabla_a a_{bcd}=mc_{abcd}$ we get
\begin{equation}
\nabla^a a_{dbc}(Y^1_{abc}+Y_{abc}^7+Y_{abc}^{27})-mc_{adbc}(Y^1_{abc}+Y_{abc}^7+Y_{abc}^{27})=0.
\end{equation}
However, since $c_{abcd}$ projects onto $Y_a$ we can use the relations
\begin{equation}
a_{dbc}Y_{abc}^1=\frac{1}{42}a_{dbc}a_{abc}Y=\frac{1}{7}\delta_{da}Y,\,\,\,\,a_{dbc}Y_{abc}^7=-\frac{1}{24}a_{dbc}c_{abce}Y_e=-\frac{1}{6}a_{dab}Y_b, \,\,\,a_{dbc}Y_{abc}^{27}=\tilde Y_{da},
\end{equation}
to rewrite the above 3-form gauge condition as
\begin{equation}
\nabla^a\tilde Y_{ab}+\frac{1}{7}\nabla_bY-\frac{1}{6}a_{bcd}\nabla_cY_d+mY_b = 0.
\end{equation}
This equation will be used when we summarise the system of equations to be solved below.
Finally, we show that the last bracket in the ${\bf 27}$ equation will not play a role in the analysis of the spectrum. This can be seen by writing the three equations above in matrix form as
\begin{equation}
\kappa_3^2
\begin{pmatrix}
Y^1 \\
Y^7 \\
Y^{27}
\end{pmatrix}
=
\begin{pmatrix}
A & B & 0\\
C & D & 0\\
0 &E & F
\end{pmatrix}
\begin{pmatrix}
Y^1 \\
Y^7 \\
Y^{27}
\end{pmatrix}
\end{equation}
The eigenvalues must be the roots of the equation $\det(X-\kappa_3^2{\bf 1})=0$, where $X$ is the matrix appearing above. Since the element $E$ does not enter this equation
we can proceed and solve the three equations in two steps, first the two coupled equations for $Y$ and $Y_a$
\begin{equation}
{\bf 1}:\,\,\,\,\kappa_3^2Y=(C_G+16m^2)Y+2m\nabla_aY_a,
\end{equation}
\begin{equation}
{\bf 7}:\,\,\,\,\kappa_3^2Y_a=(C_G+4m^2)Y_a-2m\nabla_aY+\frac{2m}{3}a_{abc}\nabla_bY_c,
\end{equation}
and then the single equation involving only $\tilde Y_{ab}$
\begin{equation}
{\bf 27}:\,\,\,\,\kappa_3^2\tilde Y_{ab}=(C_G+\frac{4m^2}{9})\tilde Y_{ab}-\frac{2m}{3}a_{cd(a}\tilde\nabla^c\tilde Y_{b)}{}^d.
\end{equation}
We start by solving the first two coupled equations. There are two distinct cases: either 1) $Y=0$ or 2) $Y\neq 0$.
Case 1): Setting $Y=0$ in the ${\bf 1}$ equation gives $\nabla_aY_a=0$ which implies that the ${\bf 7}$ equation is of exactly of the same type as the equation solved
previously for 1-forms, but with different coefficients. However, one has to pay attention to the fact that the eigenvalues dealt with here are for 3-forms, not 1-forms,
so solving the present equation for $Y_a$ involves some new steps. This fact will become clear directly when squaring the operator $(DY)_a\equiv a_{abc}\nabla_bY_c$:
\begin{eqnarray}
(DDY)_a:&=&a_{abc}\nabla_ba_{cde}\nabla_dY_e=a_{abc}a_{cde}\nabla_b\nabla_dY_e+ma_{abc}c_{bcde}\nabla_dY_e=\cr
&&(2\delta_{ab}^{de}+c_{abde})\nabla_b\nabla_dY_e+4ma_{ade}\nabla_dY_e.
\end{eqnarray}
Since the $c$-term vanishes this equation simplifies to
\begin{equation}
(DDY)_a=4m(DY)_a-\Box Y_a+6m^2Y_a.
\end{equation}
At this point the $\Box Y_a$ term must be related to the 3-form eigenvalue $\kappa_3^2$. Using $Y=0$ and $\nabla^aY_{abc}=0$, we find that
\begin{eqnarray}
\Box Y_a&=&\Box c_{abcd}Y_{bcd}=c_{abcd}\Box Y_{bcd}-2m^2Y_a=\cr
&&c_{abcd}(-\kappa_3^2Y_{bcd}+6W^e{}_{[bc}{}^fY_{d]ef}+12m^2Y_{bcd})-2mY_a=-(\kappa_3^2-10m^2)Y_a,
\end{eqnarray}
where we have also used $c_{abcd}W^e{}_{[bc}{}^fY_{d]ef}=W_{adef}Y_{def}=0$ (where the first equality follows from $P_7^{(2)}W=0$). The $(DDY)_a$ equation above then reads
\begin{equation}
(DDY)_a=4m(DY)_a+(\kappa_3^2-4m^2)Y_a.
\end{equation}
Inserting the expression for $(DY)_a$ from the eigenvalue equation ${\bf 7}$ above we find
\begin{equation}
(\kappa_3^2-C_G-4m^2)^2-\frac{8m^2}{3}(\kappa_3^2-C_G-4m^2)-\frac{4m^2}{9}(\kappa_3^2-4m^2)=0,
\end{equation}
which has the following two solutions (inserting $1=\tfrac{20}{9}m^2$ in front of $C_G$)
\begin{equation}
\Delta_3=\frac{m^2}{9}(\sqrt{20C_G+49}\pm 1)^2.
\end{equation}
These eigenvalues are obtained for $\Delta_3=Q^2$ so it is gratifying to find its eigenvalues come out as a square.
Case 2): Now we turn to the second case where $Y\neq 0$. Then either $\nabla_aY_a=0$ or $\nabla_aY_a\neq 0$. The former case gives a simple
equation for the eigenvalue directly from the ${\bf 1}$ equation. However, taking the divergence of the ${\bf 7}$ equation tells us that $\Box Y=0$ so the
${\bf 1}$ equation reduces to
\begin{equation}
\kappa_3^2=16m^2,
\end{equation}
which is the eigenvalue of the singlet constant mode $Y_{abc}=a_{abc}$ as is easily verified. So let us turn to the latter case with $\nabla_aY_a\neq 0$. Taking
the divergence of the ${\bf 7}$ equation gives
\begin{equation}
\Box Y=\frac{1}{2m}(-\kappa_3^2+C_G+4m^2)\nabla_aY_a.
\end{equation}
Inserting this equation back into the ${\bf 1}$ equation, recalling that $\Box Y = -C_GY$, gives
\begin{equation}
(\kappa_3^2-C_G-16m^2)(\kappa_3^2-C_G-4m^2)=\frac{9}{5}C_G.
\end{equation}
This equation has the following solutions (inserting again $1=\tfrac{20}{9}m^2$ in front of $C_G$):
\begin{equation}
\Delta_3=\tfrac{m^2}{9}(\sqrt{20C_G+81}\pm 3)^2.
\end{equation}
Note that the single eigenvalue found above, $\kappa_3^2=16m^2$, belongs to the plus branch.
We now turn to the ${\bf 27}$ part of the 3-form equation
\begin{equation}\label{27KappaEquation}
{\bf 27}:\,\,\,\,\kappa_3^2\tilde Y_{ab}=(C_G+\frac{4m^2}{9})\tilde Y_{ab}+\frac{2m}{3}a_{c(a}{}^d\tilde\nabla^c\tilde Y_{|d|b)}.
\end{equation}
The derivative term requires as usual a squaring of the whole equation. The $G_2$ derivative on $\tilde Y_{ab}$ is given by
\begin{equation}
(D\tilde Y)_{ab}\equiv 2a_{cd(a}\tilde\nabla^c\tilde Y_{b)}{}^d=\tilde\nabla^c(a_{ca}{}^d\tilde Y_{db}+a_{cb}{}^d\tilde Y_{ad}),
\end{equation}
and computing its square, using the fact that $\tilde\nabla_a$ is zero on both $a_{abc}$ and $c_{abcd}$, gives
\begin{equation}
(DD\tilde Y)_{ab}\equiv \tilde\nabla^e\tilde\nabla^f(a_{ea}{}^ca_{fc}{}^d\tilde Y_{db}+a_{eb}{}^ca_{fc}{}^d\tilde Y_{ad}+a_{ea}{}^ca_{fb}{}^d\tilde Y_{cd}+a_{eb}{}^ca_{fa}{}^d\tilde Y_{cd}).
\end{equation}
Clearly, the last two terms in this expression contain a symmetric combination of the two $G_2$ covariant derivatives. Such terms can be a real obstacle
to carrying through the calculation but we will see below that there is a trick that can be used to eliminate this issue. Before applying this trick we simplify the other terms to get
\begin{equation}
(DD\tilde Y)_{ab}=-2\tilde\Box \tilde Y_{ab}+2\tilde\nabla^c\tilde\nabla_{(a} \tilde Y_{b)c}-2c_{cde(a}\tilde\nabla^c\tilde\nabla^d\tilde Y_{b)}{}^e+2a_{e(a}{}^ca_{|f|b)}{}^d\tilde\nabla^e\tilde\nabla^f\tilde Y_{cd}.
\end{equation}
We now address the four terms on the RHS of this equation starting with last one. The trick used to deal with the symmetric combination of derivatives in this term is to reintroduce
the 3-form via $Y^{(27)}_{abc}=\tfrac{3}{4}a_{d[ab}\tilde Y_{c]d}$ temporarily giving
\begin{equation}\label{SymmetricDerivative27}
a_{e(a}{}^ca_{|f|b)}{}^d\tilde\nabla^e\tilde\nabla^f\tilde Y_{cd}=a_{e(a}{}^c\tilde\nabla^e\tilde\nabla^fa_{|f|b)}{}^d\tilde Y_{cd}=a_{e(a}{}^c\tilde\nabla^e\tilde\nabla^f(4Y^{(27)}_{|f|b)c}-a_{b)c}{}^d\tilde Y_{fd}-a_{|cf|}{}^d\tilde Y_{b)d}).
\end{equation}
The first term in the last expression is just the $G_2$ covariant divergence of the 3-form which can be seen to satisfy
\begin{equation}
\tilde\nabla^aY_{abc}^{(27)}=\nabla^aY_{abc}^{(27)}=\nabla^aY_{abc}=0,
\end{equation}
where in the first equality we have used the fact that $\tilde Y_{ab}$ is symmetric and in the second that
$\nabla^aY_{abc}^{(1)}=\nabla^aY_{abc}^{(7)}=0$. Thus $\nabla^aY_{abc}=0$ is implies $\tilde\nabla^a\tilde Y_{ab}=0$ which gives the following much simpler form of the expression above
\begin{equation}
a_{e(a}{}^ca_{|f|b)}{}^d\tilde\nabla^e\tilde\nabla^f\tilde Y_{cd}=-a_{e(a}{}^c\tilde\nabla^e\tilde\nabla^f a_{|cf|}{}^d\tilde Y_{b)d}=-\tilde\Box \tilde Y_{ab}+\tilde\nabla^c\tilde\nabla_{(a}\tilde Y_{b)c}+
c_{(a}{}^{cde}\tilde\nabla^c\tilde\nabla^d\tilde Y_{b)}{}^e.
\end{equation}
Inserting this result into the above expression for $(DD\tilde Y)_{ab}$ we get
\begin{equation}\label{square27}
(DD\tilde Y)_{ab}=4(-\tilde\Box \tilde Y_{ab}+\tilde\nabla^c\tilde\nabla_{(a}\tilde Y_{b)c}-
c_{cde(a}\tilde\nabla^c\tilde\nabla^d\tilde Y_{b)}{}^e).
\end{equation}
Thus we have managed to eliminate the symmetric combination of covariant derivatives that has caused a bit of headache until now. The terms in the last
formula for $(DD\tilde Y)_{ab}$ can be dealt with rather easily as we now show. The first term is simply the coset master equation squared, i.e.,
\begin{equation}
\tilde\Box \tilde Y_{ab}=-(C_G-C_H)\tilde Y_{ab}.
\end{equation}
The other two terms both involve the commutator of two covariant derivatives (since $\tilde\nabla^a\tilde Y_{ab}=0$). So we need to compute
\begin{equation}
[\tilde\nabla_c,\tilde\nabla_d]\tilde Y_{ab}=\tilde\nabla_c(\nabla_d\tilde Y_{ab}-\frac{m}{3}a_{da}{}^e\tilde Y_{eb}-\frac{m}{3}a_{db}{}^e\tilde Y_{ae})-(c \leftrightarrow d)=\nonumber
\end{equation}
\begin{equation}
\nabla_c\nabla_d\tilde Y_{ab}-\frac{m}{3}a_{cd}{}^e\nabla_e\tilde Y_{ab}-\frac{m}{3}a_{ca}{}^e\nabla_d\tilde Y_{eb}-\frac{m}{3}a_{cb}{}^e\nabla_d\tilde Y_{ae}
-\frac{m}{3}a_{da}{}^e\tilde\nabla_c\tilde Y_{eb}-\frac{m}{3}a_{db}{}^e\tilde\nabla_c\tilde Y_{ae}-(c \leftrightarrow d)=\nonumber
\end{equation}
\begin{equation}
[\nabla_c,\nabla_d]\tilde Y_{ab}-\frac{2m}{3}a_{cd}{}^e\nabla_e\tilde Y_{ab
-\frac{2m^2}{9}(a_{[c|a|}{}^ea_{d]e}{}^f\tilde Y_{fb}+a_{[c|a|}{}^ea_{d]b}{}^f\tilde Y_{ef}+a_{[c|b|}{}^ea_{d]a}{}^f\tilde Y_{fe}+a_{[c|b|}{}^ea_{d]e}{}^f\tilde Y_{af}).
\end{equation}
Thus, all single derivative terms cancel except one. Noting also that the two non-derivative terms in the middle of the bracket cancel the commutator becomes
\begin{equation}
[\tilde\nabla_c,\tilde\nabla_d]\tilde Y_{ab}=[\nabla_c,\nabla_d]\tilde Y_{ab}-\frac{2m}{3}a_{cd}{}^e\nabla_e\tilde Y_{ab}
-\frac{4m^2}{9}\,a_{[c}{}^{(a|e|}a_{d]e}{}^{|f|}\tilde Y_{f}{}^{b)}.
\end{equation}
Simplifying the non-derivative terms finally gives
\begin{equation}
[\tilde\nabla_c,\tilde\nabla_d]\tilde Y_{ab}=[\nabla_c,\nabla_d]\tilde Y_{ab}-\frac{2m}{3}a_{cd}{}^e\nabla_e\tilde Y_{ab}
+\frac{4m^2}{9}(\delta_{(a}^{[c}\tilde Y_{b)}{}^{d]}-c_{cd(a}{}^e\tilde Y_{b)e}).
\end{equation}
Since we are interested in expressing the right hand side in terms of the $D$ operator defined above using $\tilde\nabla_a$ we
rewrite the last equation as
\begin{equation}
[\tilde\nabla_c,\tilde\nabla_d]\tilde Y_{ab}=[\nabla_c,\nabla_d]\tilde Y_{ab}-\frac{2m}{3}a_{cd}{}^e\tilde\nabla_e\tilde Y_{ab}
-\frac{4m^2}{9}(\delta_{(a}^{[c}\tilde Y_{b)}{}^{d]}+2c_{cd(a}{}^e\tilde Y_{b)e}).
\end{equation}
To get the final expression that will be useful here we replace the commutator on the right hand side by the Riemann tensor. This gives
\begin{equation}
[\tilde\nabla_c,\tilde\nabla_d]\tilde Y_{ab}=2W_{cd(a}{}^e\tilde Y_{b)e}-\frac{2m}{3}a_{cd}{}^e\tilde\nabla_e\tilde Y_{ab}+\frac{32m^2}{9}(\delta_{(a}^{[c}\tilde Y_{b)}{}^{d]}-\frac{1}{4}c_{cd(a}{}^e\tilde Y_{b)e}).
\end{equation}
The first term we need to compute in the $DD\tilde Y$ equation contains the contracted expression $[\tilde\nabla^b,\tilde\nabla_d]\tilde Y_{ab}$. Setting $b=c$ in the last equation above we get
\begin{equation}\label{symCommutatorContracted}
[\tilde\nabla^c,\tilde\nabla_a]\tilde Y_{bc}=W^c{}_{ab}{}^e\tilde Y_{ce}-\frac{2m}{3}a^c{}_{a}{}^e\tilde\nabla_e\tilde Y_{bc}+\frac{56m^2}{9}\tilde Y_{ab}.
\end{equation}
The second term in the $DD\tilde Y$ we need is the one with a contraction of the commutator with the $c$ symbol:
\begin{equation}\label{cCommutatorContracted}
c_{cd(a}{}^f[\tilde\nabla_c,\tilde\nabla_d]\tilde Y_{b)f}=-2W_{(a}{}^e{}_{b)}{}^f\tilde Y_{ef}+\frac{8m}{3}a_{(a}{}^{ef}\tilde\nabla^e\tilde Y_{b)}{}^f+\frac{112m^2}{9}\tilde Y_{ab}.
\end{equation}
Inserting the two results in \eqref{symCommutatorContracted} and \eqref{cCommutatorContracted} into \eqref{square27} we find
\begin{equation}
\frac{1}{4}(DD\tilde Y)_{ab}=-\tilde\Box \tilde Y_{ab}+\frac{m}{3}(D\tilde Y)_{ab}+\frac{112m^2}{9}\tilde Y_{ab}-2W_{(a}{}^e{}_{b)}{}^f\tilde Y_{ef}.
\end{equation}
Replacing the $G_2$ covariant box with Casimirs gives
\begin{equation}
\frac{1}{4}(DD\tilde Y)_{ab}=C_GY_{ab}+\frac{m}{3}(D\tilde Y)_{ab}+\frac{112m^2}{9}\tilde Y_{ab}-(C_H\tilde Y_{ab}+2W_{(a}{}^e{}_{b)}{}^f\tilde Y_{ef}).
\end{equation}
Then from the Appendix we know that the last bracket gives $\tfrac{112m^2}{9}$ which implies the amazingly simple equation
\begin{equation}\label{Squared3Form27}
(DD\tilde Y)_{ab}-\frac{4m}{3}(D\tilde Y)_{ab}-4C_G\tilde Y_{ab}=0.
\end{equation}
In view of the eigenvalue equation \eqref{27KappaEquation}, we may express the "solution" to the last equation as
\begin{equation}
D\tilde Y=\frac{2m}{3}\pm2\sqrt{C_G+\frac{m^2}{9}}=\frac{2m}{3}(1\pm\sqrt{20C_G+1}).
\end{equation}
Then by replacing $(D\tilde Y)_{ab}$ with the expression coming from the 3-form \eqref{27KappaEquation}, we find
\begin{equation}
{\bf 27}:\,\,\,\,\kappa_3^2=\frac{m^2}{9}(20C_G+2\pm 2\sqrt{20C_G+1}).
\end{equation}
Since $\kappa_3^2$ is the eigenvalue of $\Delta_3=Q^2$ this must be a square. Indeed, it can be written
\begin{equation}
{\bf 27}:\,\,\,\,\Delta_3=\frac{m^2}{9}(\sqrt{20C_G+1}\pm1)^2.
\end{equation}
\subsection{Lichnerowicz}
When we now turn to the transverse and traceless metric modes $h_{ab}$ on the squashed seven-sphere we can take advantage of
the results obtained in the previous case of the 3-form. To see how to do this let us write out the Lichnerowicz equation explicitly
\begin{equation}
\Delta_L:\,\,\,\,\,\Delta_L h_{ab}=-\Box h_{ab}-2W_{acbd}h^{cd}+14m^2h_{ab}=\kappa_L^2h_{ab}.
\end{equation}
The coset master equation reads in this case
\begin{equation}
\nabla_a h_{bc}+\frac{2m}{3}a_{ad(b}h_{c)d}=-T_a h_{bc},
\end{equation}
which when squared gives
\begin{equation}
G/H:\,\,\,\,\,\,\Box h_{ab}+\frac{4m}{3}a_{cd(a}\nabla^c h_{b)}{}^d-\frac{14m^2}{9}h_{ab}=T_cT_ch_{ab}.
\end{equation}
Eliminating the box operator from the above $\Delta_L$ and $G/H$ equations and using $T_cT_c=-(C_H-C_H)$ gives
\begin{equation}
\kappa_L^2 h_{ab}=C_Hh_{ab}-(C_Hh_{ab}+2W_{acbd}h^{cd})+(14+\frac{14}{9})m^2h_{ab}+\frac{4m}{3}a_{cd(a}\tilde\nabla^c h_{b)}{}^d,
\end{equation}
where we have used the $G_2$ covariant derivative
\begin{equation}
\tilde\nabla_a h_{bc}=\nabla_a h_{bc}-\frac{2m}{3}a_{a(b}{}^dh_{c)}{}^d.
\end{equation}
As in the previous cases we now
use the result for the
Weyl tensor term from the Appendix, i.e., that $C_H+2W$ acting on the different $H$ irreps in ${\bf 27}$ gives the universal value $\tfrac{28}{5}=\tfrac{112}{5}m^2$. This gives us the rather simple equation
\begin{equation}\label{EigenvalueLic}
\kappa_L^2 h_{ab}=\tfrac{m^2}{9}(20C_G+28)h_{ab}+\frac{4m}{3}a_{cd(a}\tilde\nabla^c h_{b)}{}^d.
\end{equation}
There is one crucial difference between the Lichnerowicz modes and the 3-form modes analysed in the last subsection: Transversality of the metric modes does not imply that the associated 3-form $Y_{abc} \equiv \frac{4}{3}a_{d[ab}h_{c]d}$ is transverse (recall the result $\tilde \nabla^a\tilde Y_{ab}=0$ derived in the previous subsection). Thus when squaring $(Dh)_{ab}\equiv 2a_{cd(a}\tilde\nabla^c h_{b)}{}^d$ we cannot simply use \eqref{Squared3Form27} since this equation was derived with the assumption of a transverse $Y^{(27)}_{abc}$. The change is however relatively small; we only need to add the term in \eqref{SymmetricDerivative27} proportional to $\nabla^a Y^{(27)}_{abc}$ to proceed. We find
\begin{equation}\label{LicSquared}
(DDh)_{ab}-\frac{4m}{3}(Dh)_{ab}-4C_G h_{ab}-8a_{cd(a}\tilde{\nabla}^c\tilde{\nabla}^eY^{(27)}_{|ed|b)} = 0.
\end{equation}
To deal with the last term we apply the $D$ operator yet another time. To simplify the computation we introduce the notation $Y_{ab}\equiv \tilde{\nabla}^e Y_{eab} = \nabla^e Y_{eab}$ where $\nabla^aY_{ab}=0$ and define $(DY)_{ab} \equiv 2a_{cd(a}\tilde{\nabla}^cY_{|d|b)} = 2a_{cd(a}\nabla^cY_{|d|b)}$. It is then immediately found that
\begin{equation}
(DDY)_{ab} = 2a_{cd(a}\tilde{\nabla}^{c}(DY)_{|d|b)} = -\frac{14m}{3}(DY)_{ab}+2a_{cd(a}\nabla^{c}(DY)_{|d|b)}
\end{equation}
Expanding out the nontrivial last term on the right side gives
\begin{equation}\label{SquaredDivergence}
2a_{cd(a}\nabla^c(DY)_{|d|b)} = 2a_{cd(a}\nabla^{c}a_{|efd}\nabla^{e}Y_{f|b)} + 2a_{cd(a}\nabla^{c}a_{|ef|b)}\nabla^{e}Y_{fd}\,.
\end{equation}
We start by analysing the first term in \eqref{SquaredDivergence}. When the first covariant derivative passes $a_{efd}$ we need to compute
\begin{equation}
\begin{aligned}
a_{cd}{}^{(a}a_{efd}\nabla_c\nabla_e Y_{f}{}^{b)}&= -2\delta^{c(a}_{ef}\nabla_{c}\nabla_e Y_{f}{}^{b)}-c_{cef}{}^{(a}\nabla_c\nabla_eY_{f}{}^{b)} \\
&= \nabla_{f}\nabla^{(a}Y_{f}{}^{b)} = 6mY^{(ab)}+R^{f(ab)g}Y_{fg}\\
&=0
\end{aligned}
\end{equation}
In the second equality we have used the antisymmetry of $Y_{ab}$ and the fact that the 2-form projector $P_7$ vanishes when acting on the Weyl tensor (which implies that $c_{ab}{}^{cd}\propto \delta_{ab}^{cd}$).
The last equality is true due to the antisymmetry of $Y_{ab}$ together with the symmetrisation $(ab)$. When the covariant derivative acts on $a_{efg}$ we find
\begin{equation}
ma_{cd}{}^{(a}c_{cefd}\nabla^{e}Y_{f}{}^{b)} = 4ma_{ef}{}^{(a}\nabla^{|e|}Y_{f}{}^{b)} = 2m(DY)_{ab}.
\end{equation}
We conclude that
\begin{equation}
2a_{cd(a}\nabla^{c}a_{|efd}\nabla^{e}Y_{f|b)} = 4m(DY)_{ab}.
\end{equation}
We then turn to the second term in \eqref{SquaredDivergence}. Here the covariant derivative can act in two different ways. First, moving the derivative past $a_{efb}$ gives the expression
\begin{equation}
a_{cd}{}^{(a}a_{ef}{}^{b)}\nabla^c \nabla^eY_{fd} = \tfrac{1}{2} a_{cd}{}^{(a}a_{ef}{}^{b)}[\nabla^c, \nabla^e]Y_{fd}\,.
\end{equation}
The action of the antisymmetrized covariant derivatives will give two terms which are essentially identical and we only show how to deal with the first one. Using the squashed sphere Riemann tensor it follows that
\begin{equation}
\begin{aligned}
a_{cd}{}^{(a}a_{ef}{}^{b)}R^{ce}{}_{fg}Y^g{}_d &= 2m^2\delta^{ce}_{fg}\,a_{cd}{}^{(a}a_{ef}{}^{b)}Y^{g}{}_d + a_{cd}{}^{(a}a_{ef}{}^{b)}W^{ce}{}_{fg}Y^{g}{}_d \\
&= m^2 a_{fd}{}^{(a}a_{gf}{}^{b)}Y^g{}_{d}\\
&=0,
\end{aligned}
\end{equation}
where we have used that $a_{efb}W^{ce}{}_{fg} = -\frac{1}{2}a_{efb}W^{cg}{}_{ef}=0$ (see the Appendix). The other term vanishes in exactly the same way. Finally, we have a term
coming from the covariant derivative hitting $a_{efb}$. After some algebra, this term becomes
\begin{equation}
2ma_{cd}{}^{(a}c_{cef}{}^{b)}\nabla^e Y_{fd} = 2ma_{cd}{}^{(a}\nabla_c Y_{d}{}^{b)} = 2m(DY)_{ab},
\end{equation}
and so $a_{cd}{}^{(a}a_{ef}{}^{b)}\nabla^c \nabla^eY_{fd} = 2m(DY)_{ab}$. Putting these results together we find that
\begin{equation}
(DDY)_{ab} = (6m-\frac{14m}{3})(DY)_{ab} = \frac{4m}{3}(DY)_{ab}.
\end{equation}
Having calculated the action of $D$ on $(DY)_{ab}$ we can apply $D$ on \eqref{LicSquared} and then subtracting the previous equation to eliminate the $(DY)_{ab}$ terms. From this procedure we find the third order equation
\begin{equation}
(D-\frac{4m}{3})(D^2h-\frac{4m}{3}Dh-4C_Gh) = 0.
\end{equation}
"Solving" for $(Dh)_{ab}$ and plugging the result back into \eqref{EigenvalueLic} gives the following three different eigenvalues:
\begin{equation}
\Delta_L = \frac{m^2}{9}(20 C_G +36),\,\,\,\, \Delta_L = \frac{m^2}{9}\left(20C_G+32 \pm 4\sqrt{20C_G+1}\right).
\end{equation}
\section{Conclusions}
Let us summarise what we know so far about the spectrum of operators on the squashed seven-sphere including the new results for $\Delta_2$ and $\Delta_3$ obtained in this paper.
The previously known eigenvalues for $\Delta_0$, $\slashed D_{1/2}$ \cite{Nilsson:1983ru}, $\Delta_1$ \cite{Yamagishi:1983ri}, and $\Delta_L$ \cite{Duff:1986hr} are
\begin{eqnarray}
\Delta_0&=&\tfrac{m^2}{9}\,20C_G,\\
{\slashed D}_{1/2}&=&-\tfrac{m}{2}\pm \tfrac{m}{3}\sqrt{20C_G+81},\\
{\slashed D}_{1/2}&=&\tfrac{m}{6}\pm \tfrac{m}{3}\sqrt{20C_G+49},\\
\Delta_1&=&\tfrac{m^2}{9}\,(20C_G+14\pm 2\sqrt{20C_G+49})=\tfrac{m^2}{9}\,(\sqrt{20C_G+49}\pm1)^2-4m^2,\\
\Delta_L&=&\tfrac{m^2}{9}\,(20C_G+36),\\
\Delta_L&=&\tfrac{m^2}{9}\,(20C_G+32\pm 4\sqrt{20C_G+1})=\tfrac{m^2}{9}\,(\sqrt{20C_G+1}\pm2)^2+3m^2,
\end{eqnarray}
while the new ones obtained in this paper are
\begin{eqnarray}
\Delta_2&=&\tfrac{m^2}{9}\,(20C_G+72),\\
\Delta_2&=&\tfrac{m^2}{9}\,(20C_G+44\pm 4\sqrt{20C_G+49})=\tfrac{m^2}{9}\,(\sqrt{20C_G+49}\pm2)^2-m^2,\\
\Delta_2&=&\tfrac{m^2}{9}\,20C_G,\\
\Delta_3&=&\tfrac{m^2}{9}\,(\sqrt{20C_G+49}\pm 1)^2,\\
\Delta_3&=&\tfrac{m^2}{9}\,(\sqrt{20C_G+81}\pm 3)^2,\\
\Delta_3&=&\tfrac{m^2}{9}\,(\sqrt{20C_G+1}\pm 1)^2.
\end{eqnarray}
Here it is appropriate to make some comments on the limitations of the obtained results. First, we have so far no results for the eigenvalues of
the spin $3/2$ operator ${\slashed D}_{3/2}$, although some can easily be extracted from supersymmetry. Secondly, for $\Delta_2$
and $\Delta_L$ we seem to lack some eigenvalues. This is indicated by the degeneracies in the cross diagrams for these two operators derived in \cite{Nilsson:2018lof},
as well as by supersymmetry. For $\Delta_3$ there may already be too many available eigenvalues but if one is supposed to pick only one sign when removing the
square (as done for ${\slashed D}_{1/2}$) one is instead lacking two eigenvalues. These problematic features are partly due to the fact that although we have extensive knowledge
of the eigenvalue spectra from the list above, the method applied here does not provide direct information how to associate these eigenvalues with the cross diagrams of \cite{Nilsson:2018lof}.
These and other issues will be elaborated upon in a forthcoming publication \cite{Nilsson:2021kn}.
The results of this paper clearly demonstrate the important role of weak $G_2$ holonomy when solving the eigenvalue equations. There are, however, deeper issues in the context of holonomy
and string/M theory that might be interesting to study in relation to the squashed seven-sphere, for instance the notion of generalised holonomy discussed in \cite{Batrachenko:2003ng}.
\section*{Acknowledgement}
One of us (BEWN) thanks M.J. Duff and C.N. Pope for a number of discussions on issues related to
the theory analysed in this paper and for collaboration at an early stage of this project.
We are also grateful to Joel Karlsson for some useful comments on the manuscript.
The work of S.E. is partially supported by the Knut and Alice Wallenberg Foundation under the grant: "Exact Results
in Gauge and String Theories", Dnr KAW 2015.0083.
|
2,869,038,156,389 | arxiv | \section{Introduction}
Bottom-up approaches give access to
systems of very reduced dimensionality with
unique physical properties. Among these systems,
{chains of a single-atom cross section} are of great interest.~\cite{wang_2011}
When the chains are formed by magnetic atoms, spin-spin
correlations can come into place leading to new
phenomena and applications.~\cite{brune_2006} Building
structures from their atomic constituents can be achieved
by the atomic-manipulation capabilities of
the scanning tunneling
microscope (STM). Magnetic atoms have been positioned one by one
at different distances and with different arrangements on a variety of
substrates.\cite{hirjibehedin_2006,khajetoorians_2012,loth_2012,holzberger_2013,yan_2014}
Spin-polarized STM~\cite{wiesendanger} or inelastic
electron tunneling spectra (IETS)~\cite{heinrich_2004,PSS}
have granted us a detailed vision of the magnetic mechanisms
at play on the atomic scale.
Recent developments in density functional
theory (DFT) have also permitted us to
attain a deep understanding
of the phenomena revealed by the above experiments.
Due to their experimental interest,
Mn chains are amongst the most studied ones.\cite{lounis_2008,rudenko_2009,lin_2011,urdaniz_2012,Tao2015,choi_2016}
Fe chains are also well known both experimentally
and theoretically.\cite{khajetoorians_2012,loth_2012,nicklas_2011,gauyacq_2013,khajetoorians_2013,spinelli_2014,spinelli_2015,Tao2015}.
The work by
Lin and Jones~\cite{hirjibehedin_2007,lin_2011}
on Fe, Co and Mn atoms
on Cu$_2$N/Cu (100)
reveals that atomic spins maintain their
nominal values on the surface ($S_{Fe}$= 2, $S_{Mn}$=5/2 and $S_{Co}$=
3/2), showing the interest of using Cu$_2$N
to preserve much of the magnetic identity of transition-metal (TM) atoms.
Studies of Fe chains~\cite{nicklas_2011}
and of Mn~\cite{rudenko_2009} reveals that close-packed
TM chains on Cu$_2$N/Cu (100) couple
antiferromagnetically due to a N-mediated superexchange
mechanism,
largely explaining experimental findings.\cite{hirjibehedin_2007,loth_2012}
In the present work, we report on a different type of magnetic atomic
chain. These chains are heterogeneous, including two types of magnetic
atoms, Fe and Mn, on a Cu$_2$N/Cu (100) substrate. An initial account of
the results has been given in a separate publication.~\cite{choi_2014}
The experimental study is based on STM and IETS results of FeMn$_x$
($x=1,\cdots, 6$) and confirms that, as in the previous cases, the magnetic
ordering along the chain is antiferromagnetic. The different anisotropy
of Fe and Mn on Cu$_2$N/Cu (100) leads to two possible orientations of
the magnetic moments (along the chain and out-of-plane). Contrary to
the case of homogeneous chains, this may indicate a non-collinear
arrangement. When $x$ is an odd number, a simple-minded evaluation of
the total spin yields 1/2, which is compatible with the appearance of a
Kondo feature at zero bias.~\cite{choi_2014} However, when $x$ is even,
the expected spin is 2 and, correspondingly, no Kondo peak is observed.
In the present work, we perform extensive calculations and compare them
with the experiments. The comparison permits us to conclude on the spin
arrangement (it is collinear along the chain) and on the behavior of
the exchange coupling between the two different atomic species of the
chain in good agreement with the experimental observations. The
antiferromagnetic coupling is confirmed and traced back to the
N-mediated superexchange mechanism as in the homogeneous chains. Finally,
the obtention of model Hamiltonians to study the magnetic structure of
these chains is discussed within the framework of
DFT calculations.~\cite{gauyacq_2013,kepenekian_2014}
\section{Experimental method}
Experiments were performed in an ultrahigh vacuum low temperature STM at
a base temperature of 0.5 K as has been partially reported
in Ref.~[\onlinecite{choi_2014}].
Differential conductance was used as a local
spectroscopic tool that gives information via inelastic
electron tunneling (IETS).\cite{heinrich_2004,PSS}
The differential conductance was directly
measured using lock-in detection with 72-$\mu$V rms modulation at ~691 Hz
to the sample bias voltage V.
The Cu (100) surface was cleaned by Ar sputtering
and then annealed up to 850 K. After having big terraces of Cu(100)
crystal, a monolayer of Cu$_2$N was formed as a decoupling layer by
nitrogen sputtering and post-annealing at 600 K. Single
Fe and Mn atoms were deposited onto the pre-cooled surface. Pt-Ir tips were
prepared by sputter-anneal cycles and coated with copper {\it in vacuo} by
soft indentations into the Cu bridges. The tip status was monitored through
STM images and controlled to manipulate the atoms.
All atomic chains were built using vertical atomic manipulation.
After identifying a given adatom by its spectroscopic features, we picked it up by voltage pulsing, dropped it off on a nitrogen site
and hop it to a copper site. We build the chain
atom by atom in
a close-packed configuration to ensure an AF coupled spin chain.
By doing so, all different kinds of FeMn$_x$ ($x$=1... 10) chains were constructed.
\begin{figure}[ht]
\includegraphics[width=0.7\columnwidth]{figure1.png}
\caption{\label{stm} {Constant current STM image of a FeMn$_3$
chain assembled on a Cu$_2$N layer and
atomic scheme (on the right of the figure) that can be inferred from
the position of the upper-right Fe atom and manipulation
with the STM tip. The STM image was scanned at 10 mV and
100 pA. The scanned area is 6 $\times$ 6 nm$^2$. The transition metal atoms
were always placed ontop of a Cu atom between two nitrogen atoms.
The topography was processed with WSXM\cite{WSXM}.}
}
\end{figure}
\section{Theoretical method}
\label{theory}
{\it Ab initio} calculations are performed within the
density-functional theory (DFT) framework as implemented in
the VASP code.\cite{kresse_efficiency_1996} We expand the wave
functions using a plane-wave basis set with a cutoff energy of
300~eV. Core electrons are treated within the projector augmented wave
method.\cite{bloechl_projector_1994, kresse_ultrasoft_1999} The PBE form
of the generalized gradient approximation is used for the exchange and
correlation functional.\cite{perdew_generalized_1996}
The system consists of a Cu(100) surface covered by a Cu$_2$N layer.
To model the surface we use a slab geometry with four Cu layers plus
the Cu$_2$N layer. Following the above experimental procedure, Fe and
Mn atoms are deposited on top Cu atoms, forming a chain in the [010]
direction. We use a unit cell {[3 $\times$ ($x$+3)] in units of the
bulk lattice parameter such that the length along the [010] direction
increases as ($x$+3)}, where $x$ is the number of Mn atoms. In this way,
we keep the distance between chain images constant for all sizes, being
of 3 lattice constants in the unrelaxed configuration. The unit cell
for FeMn$_2$ is shown in Fig.~\ref{trimer} as an example. The bottom
Cu layer was kept fixed and the remaining atoms were allowed to relax
until forces were smaller than 0.01~eV/\AA. The $k$-point sample was
varied accordingly to the unit cell, and tests were performed to assure
its convergence. The charge and magnetic moments have been calculated
using the Bader analysis.~\cite{tang_bader_2009}
A critical aspect of the calculations is the use of
a static Coulomb charging energy $U$. Lin and Jones~\cite{lin_2011}
perform a constrained DFT calculation to evaluate $U$. For
Mn they find $U$= 4.9 eV when Mn is sitting ontop a surface Cu
atom, while it is reduced to $U$= 3.9 eV ontop a N site. This is in agreement
with their previous result~\cite{hirjibehedin_2007}, where they found $U$= 5.0 eV
for Mn ontop of Cu, and $U$= 2.0 eV for Fe. To the best of our knowledge,
these are the only actual computations of $U$ for Mn on the Cu$_2$N/Cu (100)
substrate. Rudenko {\it et al.}\cite{rudenko_2009} take an
effective value $U-J$=5 eV {where $J$ is the intra-atomic
exchange coupling.} Nicklas {\it et al.}~\cite{nicklas_2011}
do not take any value of $U$ for Fe. As Lin and Jones~\cite{lin_2011}
show, the actual value of $U$ greatly affects the computed
exchange couplings, indeed they find a difference of a factor
of 2 between their calculations with $U$= 4.9 eV and without $U$,
finding DFT+U results in better agreement with the experiment.
In this work, we have used
the GGA+U method of Dudarev~\cite{dudarev_1998}
with a $U_{\texttt{eff}}=U-J$ of 1~eV for Fe and 4~eV for Mn.
The chosen values correspond to roughly subtracting $J\approx 1$ to
$U=4.9$ eV ($U=2$ eV) as computed by Lin and Jones~\cite{lin_2011} for Mn (Fe) ontop a Cu atom.
As we will see, the values of $U_{\texttt{eff}}$
become critical for the determination of the exchange couplings.
The effect of different values for $U_{\texttt{eff}}$ is discussed
below.
To rationalize our results we have fitted them to a Heisenberg Hamiltonian in the form:
\begin{equation}
\label{heisenberg}
H=-\sum_{i,j>i} J_{ij}{S}_i \cdot{} {S}_j,
\end{equation}
where $J_{ij}$ are the exchange couplings between spins ${S}_i$ and ${S}_j$.
Evaluating different magnetic configurations we have been able to extract
the $J_{ij}$ values by fitting the DFT energies to the Heisenberg Hamiltonian, Eq.~(\ref{heisenberg}).
For the shorter chains (FeMn and FeMn$_2$) we have included spin-orbit
coupling (SOC)~\cite{vasp_manual,hobbs_2000} into our scalar-relativistic
Hamiltonian. We have considered a simple anisotropic spin
Hamiltonian, in the form
\begin{equation}
\label{ani}
H=DS_z^2+E(S_x^2-S_y^2).
\end{equation}
Here $x,y$ and $z$ are such that $z$ is the computed easy axis, while $x$ and $y$
are two orthogonal directions in the plane perpendicular to $z$
(hard-plane) {in principle, different from the surface directions}. The energy of the system has been self-consistently
calculated including SOC for the different orientations $x$,
$y$, and $z$ of the magnetization axis and, from these energies,
values for $D$ and $E$ in Eq.~(\ref{ani}) have been fitted using
the evaluated magnetic moments.
\begin{figure}[ht]
\includegraphics[width=1.0\columnwidth]{figure2.png}
\caption{\label{trimer} Top (upper panel) and side (down panel) views of
the relaxed geometry of FeMn$_2$. We show Fe atoms in red; Mn atoms in
green; N atoms in blue; Cu atoms in the first layer in brown; and the
rest of Cu atoms in pink. Blue lines delimit the unit cell used in the
calculations. The vacuum side of the unit cell and the three bottom Cu layers
have been removed for pictorial reasons.}
\end{figure}
\begin{table*}[t]
\centering
\begin{ruledtabular}
\begin{tabular}{lcccccc}
& d[Fe-Mn$_1$] & d[Mn$_1$-Mn$_2$] & d[Mn$_2$-Mn$_3$] & d[Mn$_3$-Mn$_4$] & d[Mn$_4$-Mn$_5$] & d[Mn$_5$-Mn$_6$]\\
\hline
FeMn & 3.81 & & & & & \\
FeMn$_2$ & 3.77 & 3.84 & & & & \\
FeMn$_3$ & 3.75 & 3.80 & 3.82 & & & \\
FeMn$_4$ & 3.74 & 3.77 & 3.78 & 3.79 & & \\
FeMn$_5$ & 3.73 & 3.75 & 3.76 & 3.77 & 3.77 & \\
FeMn$_6$ & 3.73 & 3.74 & 3.75 & 3.74 & 3.77 & 3.74 \\
\end{tabular}
\end{ruledtabular}
\caption[] {\label{distance1} Distances between the
different TM atoms for the 6 considered chains, (in \AA{}). Note that
the unrelaxed value would be 3.65~\AA~in agreement with the
DFT lattice parameter of the Cu substrate.}
\end{table*}
\section{Results}
\subsection{Geometries and energetics}
Constant current images were obtained between atom manipulation sequences.
This permitted us to have precise knowledge of the atomic arrangement.
Unfortunately, the structure of the tip that is optimal for atom manipulation
is not necessarily good for image production and the obtained images do
not contain much information, as can be seen in
Fig.~\ref{stm}. In the inset, we show the configuration
as inferred from the atomic manipulation procedure that corresponds
to a FeMn$_3$ chain.
The calculation yields precise insight on the actual geometries. In
Fig.~\ref{trimer} we show the FeMn$_2$ geometry as an example. As we
can see in the side view, after deposition of the TM
atoms the final chain includes the N atoms in between them. These atoms
are lifted from the surface. N atoms at the edge of the chain, like
N$_1$, are moved upwards by $\approx$0.7~\AA, while N atoms in between
TM atoms, like N$_2$, are lifted by $\approx$1.6~\AA{} with respect
to their unrelaxed positions in the bare CuN$_2$ surface. Therefore,
we can assume that the final chains have the form FeMn$_x$N$_{x+2}$. Of
course, this is just a conventional choice to identify the final chain,
since other atoms in the surface are also significantly disturbed and
could be considered as parts of the chain, like Cu$_{\text{S}1}$ and Cu$_{\text{S}2}$.
It is interesting to study the evolution of the geometry of the chains
as their size increases. In Table~\ref{distance1} we show the distance
between TM atoms within the chain for all considered sizes. We observe
that these distances are bigger than the unrelaxed value (3.65~\AA\ corresponding
to the PBE lattice parameter of Cu due to the arrangement of the chain on the surface). For
example, d[Fe-Mn$_1$] for FeMn is 3.81~\AA{}, 4\% bigger than the
unrelaxed value. This distance tends to decrease as the chain size is
increased, reaching 3.73~\AA{} for FeMn$_6$.
The same behavior can be observed for other positions in the chain.
Another interesting distance to track involves the Cu atoms just
underneath Fe and Mn atoms. As can be seen in the side view of
Fig.~\ref{trimer}, these Cu atoms (Cu$_{\text{Fe}}$, Cu$_1$, Cu$_2$)
are pushed downwards by around 0.4~\AA{}.
For FeMn, the
Fe-Cu$_{\text{Fe}}$ distance is very close to the value for the isolated
atom (2.37~\AA{} vs. 2.38~\AA). However, it increases for longer chains,
with a value of 2.40~\AA{} for FeMn$_6$. The same behavior is observed
for the Mn atoms, again close to the single adsorbate for the case of FeMn
(2.50~\AA~vs. 2.47~\AA), and it increases for longer chains reaching 2.56~\AA~for
FeMn$_6$. Hence, as the number $n$ of Mn atoms increases, we find
that the intra-chain distances diminish, see Table~\ref{distance1}, while the
distance between the chain atom and the top Cu ones increases. Both
behaviors can be rationalized as stress built up as the chain increases its
size. Indeed, the elongation of these distances can be easily identify with
some destabilization of the chains as length is increased.
To check for the stability of the chains we have studied the chain's
energetics. First, we start by analyzing
the atomization energy per TM atom, $E_{at}[FeMn_x]$, defined from
\begin{eqnarray}
\label{Eat}
E_{at}[FeMn_x]&=& \\
-\frac{1}{x+1}[E[FeMn_x]&-&
E[Fe]-xE[Mn]-E_x[Cu_2N]], \nonumber
\end{eqnarray}
where E[FeMn$_n$] is the total energy of the system;
E[Fe] (E[Mn]) is the energy of a gas-phase Fe (Mn) atom; and
E$_x$[Cu$_{2}N$] is the energy of the [3 $\times$ ($x$+3)] unit cell of the
bare surface. The atomization energy per TM atom is a measure of how much
{average energy per atom} one needs to give to the adsorbed chain to separate it
in its constituent Fe and Mn gas-phase atoms and the pristine Cu$_2$N/Cu(100) substrate.
Figure \ref{fig_energy} shows
the atomization energy per TM atom as a function
of the number of Mn atoms, $x$.
The atomization energy per TM atom tends to decrease
as the chain size increases,
{ implying that long chains become energetically less favorable.}
\begin{figure}[ht]
\includegraphics[width=1.0\columnwidth]{figure3.png}
\caption{\label{fig_energy} Atomization energy, $E_{at}$ from Eq.~(\ref{Eat}), (black x's) and
the gain in energy by adding one more Mn atom the chain, $\Delta_x E_F$ from
Eq.~(\ref{ef}), (red stars). As we can see, atomization of the chain
becomes less difficult as the chain becomes longer, and correspondingly, adding one
more atom does not reduce the total energy as much for longer chains. These values reveal an increasing destabilization
of the chain as it size increases.
The dotted lines are a guide for the eye.}
\end{figure}
We can extract more information by looking at another quantity that
reflects better the experimental procedure to construct the chain. Let us
remember that the chains are constructed by STM manipulation. First,
Fe and Mn atoms are deposited on the surface, and then they are moved
with the STM tip to form the desired structure. To account for this
procedure, we can define a formation energy where the reference of energy is the
one of the
deposited TM atom, and then to study the gain in energy by adding
one more atom to the chain.
The definition that we used for the formation
energy is given by
\begin{equation}
\label{ef}
E_F[FeMn_x]=-[E_{at}[FeMn_x]-E_{at}[Fe]-xE_{at}[Mn]],
\end{equation}
where we use the atomization energies defined in equation~\ref{Eat}. From $E_F$ we can define
the gain in energy by adding one more TM atom, $\Delta_x E_F$\cite{robles_2010}, as
\begin{eqnarray}
\Delta_xE_F=E_F[FeMn_x]-E_F[FeMn_{x-1}]=\nonumber\\
-[E_{at}[FeMn_x]-E_{at}[FeMn_{x-1}]-E_{at}[Mn]].
\end{eqnarray}
This quantity, $\Delta_xE_F$, defines the energy gained by adding a Mn atom
to an existing chain starting from an Fe atom, which mimics the
experimental procedure to build the system. The calculated values
of $\Delta_x E_F$ are always positive indicating that building
the chains is an energetically favorable process. The energy gain
is modest ($\approx0.4~eV$), and decreases by 37\% from FeMn to FeMn$_6$
(Fig.~\ref{fig_energy}).
{Experimentally, the longest chains contained
9 Mn atoms.~\cite{choi_2014} Our results seem to suggest that longer chains will
be difficult to form due to the accumulated stress
imposed by the surface lattice constant and the Fe--Mn distance.}
\begin{table*}[t]
\centering
\begin{ruledtabular}
\begin{tabular}{lcccccccc}
& $\mu_T$ & $\mu$[Fe] & $\mu$[Mn$_1$] & $\mu$[Mn$_2$] & $\mu$[Mn$_3$] & $\mu$[Mn$_4$] & $\mu$[Mn$_5$]
& $\mu$[Mn$_6$]\\
\hline
Fe & 3.82 & 3.35 & & & & & & \\
Mn & 4.97 & & 4.80 & & & & & \\
FeMn & -1.30 & 3.16 & -4.65 & & & & & \\
FeMn$_2$ & 3.68 & 3.17 & -4.55 & 4.71 & & & & \\
FeMn$_3$ & -1.43 & 3.19 & -4.56 & 4.56 & -4.73 & & & \\
FeMn$_4$ & 3.78 & 3.22 & -4.56 & 4.57 & -4.57 & 4.75 & & \\
FeMn$_5$ & -1.36 & 3.23 & -4.56 & 4.58 & -4.58 & 4.58 & -4.77 & \\
FeMn$_6$ & 3.86 & 3.23 & -4.57 & 4.58 & -4.58 & 4.59 & -4.58 & 4.79 \\
\end{tabular}
\end{ruledtabular}
\caption[] {\label{magnetic} Magnetic moments (in $\mu_B$) of the TM atoms in the chains. Values for
isolated Fe and Mn atoms on the surface are included for comparison. The values
have been obtained by performing a Bader analysis.\cite{tang_bader_2009} The total
magnetic moment $\mu_T$ of the calculation cell also includes contributions from
nearby copper atoms that spin polarize, hence $\mu_T$ slighlty differs
from the sum of the magnetic moments of Fe and Mn.}
\end{table*}
\subsection{Electronic and magnetic properties and charge transfer}
The magnetic properties are one of the main motivations for the study
of these systems. They largely stem from the electronic structure of
the atomic chains on the surface. An analysis of the electronic
structure by projection onto atomic orbitals have been shown
in Ref.~[\onlinecite{choi_2014}]. The magnetic properties were
shown to be given by the population of Fe and Mn $d$ orbitals.
In the case of Fe the d$_z^2$ orbital was basically filled {(where $z$ is
the direction perpendicular to the surface \textit{in the present case})}, not
contributing to the atomic magnetic moment. The Mn atoms
show a larger spin polarization due to the complete polarization
of its $d$-shell after inclusion in the FeMn$_x$ chain. The exchange splitting
of $d$-bands is the smallest for Fe, but still it is roughly
$5$ eV, showing how robust magnetism is in these chains. The
electronic structure near the Fermi energy is largely due to
the Cu and N states and with negligible magnetic polarization.
It is interesting to inspect the total magnetic
moment of the chains shown in Table~\ref{magnetic}. We can see how the
magnetic couplings between Fe-Mn and Mn-Mn atoms are antiferromagnetic
(AF) in the most stable configuration. In addition, the magnetic moment of
atomic Fe on the surface is 3.35$\mu_B$ (formally S=2), while the value
for Mn is 4.80$\mu_B$ (formally S=$5\over2$). Therefore, formally when
adding Mn atoms to a Fe one we would get systems with S=$1\over2$ for an
odd number of Mn atoms, $x$, and S=$2$ for an even number. In the computed
results for the total magnetic moment in the cell ($\mu_T$) we observe
the expected even--odd behavior, with values of 1.30--1.43 Bohr magnetons, $\mu_B$, for
$x$ odd, and 3.68--3.86$\mu_B$ for the $x$ even.
Values of 1.30--1.43$\mu_B$ correspond well with
the simple result S=$1\over2$. The difference from 1$\mu_B$
has two different sources that are present in these calculations.
On the one hand, the TM atoms present
fractional occupancies that lead to non-integer multiples of $\mu_B$.
On the other hand, these calculations
are mean-field approximations to the difficult solutions
of correlated AF ground states. As a
consequence, the mean-field solution averages over
the possible atomic magnetic moments found in the
multiple spin configurations of the correct AF ground state.
\begin{figure}[ht]
\includegraphics[width=\columnwidth]{figure4.png}
\caption{\label{spin_density} Spin density of the FeMn$_3$ chain. Red (yellow) indicates majority
(minority) spin areas. The chosen isovalue is 0.01~eV/\AA$^3$. The color code of the atoms is the
same of Fig.~\ref{trimer}.}
\end{figure}
The AF coupling between the TM atoms is mediated
by the N atoms $via$ a superexchange mechanism as
previously shown for Mn~\cite{rudenko_2009} and Fe chains.~\cite{nicklas_2011}
To illustrate the superexchange interaction, we show
the spin-difference-density of the FeMn$_3$ chain in Fig.~\ref{spin_density}. The
spin polarization of the intercalated N atoms adopt the expected form
for superexchange, with induced spin polarization within the atom,
but net spin close to zero. Indeed, the bigger values for the induced magnetic
moments of N atoms are very small, for example 0.11$\mu_B$ for N$_1$ and 0.08$\mu_B$ for N$_2$.
Superexchange also leads to a change in the atomic angles:
the Fe-N-Mn and Mn-N-Mn angles tend to approach 180$^\circ$ maximizing the
AF interactions.~\cite{rudenko_2009,nicklas_2011}
The magnetic moments tend to increase with increasing chain size,
which is consistent with the previously mentioned progressive separation
of the chain from the surface. The magnetic moments of the Mn atoms within
a given chain are quite constant, with the exception of the final atom,
which is $\approx$4\% bigger. The reason is the lower coordination of
the edge atom, reflected in the slightly longer distance with the Cu
atom below and the longer distance with the adjacent N atoms.
The analysis of charge transfer in the system yields that each TM atom in the chain
loses around one $s$ electron to form the bond with N atoms in the
chain and the Cu atom underneath the TM atom. The $d$ charge is similar to the atomic
case, i.e., there are 6 electrons in the $d$ manifold for Fe, and 5 for Mn,
in agreement with the computed magnetic moments.
\subsection{Exchange coupling constants}
\label{U-J}
\begin{table*}[t]
\centering
\begin{ruledtabular}
\begin{tabular}{lcccc cccccc}
& \multicolumn{4}{c}{1st neighbors} & \multicolumn{3}{c}{2nd neighbors} & \multicolumn{2}{c}{3rd neighbors}
& \multicolumn{1}{c}{4th} \\
& $J_{12}$ & $J_{23}$ & $J_{34}$ & $J_{45}$ & $J_{13}$ & $J_{24}$ & $J_{35}$ & $J_{14}$ & $J_{25}$ & $J_{15}$ \\
\hline
\hline
FeMn & -10.25 & & & & & & & & & \\
FeMn$_2$ & -6.51 & -2.09 & & & 0.97 & & & & & \\
FeMn$_3$ & -7.30 & -1.97 & -1.97 & & 1.14 & 0.32 & & 0.10 & & \\
FeMn$_4$ & -7.96 & -2.69 & -2.10 & -2.16 & 0.73 & 0.05 & 0.27 & -0.36 & -0.01 & -0.02\\
\end{tabular}
\end{ruledtabular}
\caption[] {\label{jotas} Values for the exchange coupling constants (in meV) obtained by fitting Eq.~(\ref{heisenberg})
to the GGA+U total energies.}
\end{table*}
We have fitted our results to the Heisenberg Hamiltonian shown in
Eq.~(\ref{heisenberg}). In order to do so we have calculated
different spin solutions corresponding to different magnetic
ordering among the TM atoms in the chain. In all calculations, we have
used the geometry of the most stable configuration, which is always
the AF case shown in Table~\ref{magnetic}. For the atomic values of $S$, in
Eq.~(\ref{heisenberg}), we had the choice of either using the formal
value (S=2 for Fe, S=$5\over2$ for Mn), or taking the computed value
from the DFT calculation since for a single atom $S_z$ is a good quantum
number. We have opted for the later option, which
gives values for the exchange coupling constants $J$'s which are between
15\% and 20\% higher. We can justify this choice because the DFT values
reflect the local magnetic moment that interacts via other atoms (superexchange)
with the local magnetic moment of the next neighbor in the calculation.
The results are shown in Table.~\ref{jotas}. As
expected, $J$'s involving first neighbors are negative, which indicate an
AF coupling. Second neighbor $J$'s are positive, indicating a ferromagnetic
coupling which further stabilizes the global AF solution. Further order
J's are smaller, and its omission in the fit only implies a small error.
The convergence with the number of coupled atoms considered in the
Heisenberg chain, Eq.~(\ref{heisenberg}), is very fast.
We have
done a systematic study for
FeMn$_4$ where different number of neighbors are included in solving
Eq.~(\ref{heisenberg}). Considering
just first neighbors introduces a maximum error of 0.2~meV. Considering
first and second neighbors the error is less than 0.1~meV. These calculations
show that the effective interactions in these spin chains are very short
range and first-neighbors truncation is indeed an accurate approximation.
Analyzing the evaluated exchange couplings, $J$, we observe that the biggest value is obtained for
the first-neighbors interaction between Fe and Mn$_1$. Thus the AF coupling
between Fe and Mn is stronger than the one between Mn-Mn.
FeMn is a special case since it involves two edge atoms, the
resulting AF coupling is the strongest one.
Curiously, the exchange coupling between Fe and Mn$_1$ ($J_{12}$) presents
a sharp drop for FeMn$_2$ to start increasing again for FeMn$_3$ and FeMn$_4$.
The coupling between Mn$_1$ and Mn$_2$ ($J_{23}$) presents
the same behavior, it shows a minimum for FeMn$_3$.
Similar behavior is obtained for pure Mn chains
by Rudenko {\it et al.}.\cite{rudenko_2009}
These minima in the exchange couplings seem
to be independent from the atomic geometry, where
the behavior with chain length is monotonous, and with
the atomic magnetic moments. It is probably due to the
sudden appearance of an extra neighbor that symmetrizes
the interactions on the central atom of the chain.
\begin{figure}[ht]
\includegraphics[width=1.0\columnwidth]{figure5.png}
\caption{\label{uj} Evolution of J$_{12}$ for FeMn with respect to the variation of $U_{\text{eff}}$.
We have changed one of the values of U while keeping constant the other one.}
\end{figure}
As Rudenko {\it et al.}~\cite{rudenko_2009} and Lin and
Jones~\cite{lin_2011} note, there is a dependence of the values of
the interatomic exchange coupling
on the chosen value of $U_{\text{eff}}=U-J$ for the GGA+U
approximation.
Figure~\ref{uj} shows
the dependence of $J_{12}$ for FeMn on $U_{\text{eff}}$.
We find that as the value of $U_{\text{eff}}$
decreases, $|J_{12}|$ increases. For the values of the
present study~\cite{lin_2011} ($U_{\text{eff}}$ [Fe] = 1~eV and $U_{\text{eff}}$
[Mn] = 4~eV) $J_{12}$ = -10.25~meV. If $U_{\text{eff}}$[Fe] is
reduced by 1~eV this value changes to -11.60~meV. When $U_{\text{eff}}$[Mn] is
reduced by 1~eV the effect is more pronounced, obtaining a value of -16.44~meV
(60\% increase in absolute value).
These values give us an interval of exchange couplings that will be compared
with our experimental results in the following sections.
\subsection{Magnetic anisotropy}
\begin{table}[t]
\centering
\begin{ruledtabular}
\begin{tabular}{lcccc}
MAE (meV) & dir & $E_z-E_y$ & $E_x-E_y$ \\
\hline
Fe & $y$ & 1.88 & 0.74 \\
Mn & $z$ & -0.13 & -0.01 \\
FeMn & $y$ & 1.51 & 0.75 \\
FeMn$_2$ & $y$ & 1.46 & 0.48 \\
\end{tabular}
\end{ruledtabular}
\caption[] {\label{mae} Magnetic anisotropy energies (MAE) (in meV). We also show the easy axis directions, where $x$ is the direction in plane and perpendicular to the chain, $y$ along the chain, and $z$ perpendicular to the surface.}
\end{table}
We have studied the magnetic anisotropy energy (MAE) of the system by
including SOC in our calculations. We have performed self-consistent
calculations in three directions: along the chain ($y$), perpendicular
to the chain in the plane ($x$), and perpendicular to the plane ($z$). We have
also fitted the anisotropic spin Hamiltonian of Eq.~(\ref{ani}) {to
the singly adsorbed atoms using their computed magnetic moments} to obtain
values for $D$ and $E$ using the equations:
\begin{eqnarray}
D&=&\frac{2E_z-(E_x+E_y)}{S(2S-1)} \nonumber \\
E&=&\frac{E_x-E_y}{S(2S-1)},
\label{DE}
\end{eqnarray}
where $E_x$, $E_y$, $E_z$ are the magnetic anisotropy energies
when the spin, $S$, is aligned along the $x$, $y$ and $z$ directions, see
Table~\ref{mae}.
For
the supported atoms, Fe shows an easy axis along the chain, with a MAE of
1.88~meV. For Mn we get an out-of-plane easy axis, with a smaller MAE of
0.13~meV. This results qualitatively agree with the STM experimental data
of Hirjibehedin {\it et al},\cite{hirjibehedin_2007} although our values
for D and E are underestimated for Fe (D=-0.67 meV and E=0.29 meV using the DFT magnetic moment
to obtain $S$ in Eq.~(\ref{DE}))
but in good agreement for Mn. Shick
and co-workers~\cite{shick_2009} evaluated within LDA the values for D
and E of Fe and Mn on Cu$_2$N/Cu (100). Their values for Mn agree with our
calculation, probably because the MAE of Mn is so small that the value
is within the error bar of our calculations. However, our results for
Fe are closer to the experimental values of D=-1.55 meV and E=0.31
meV. {Our calculations compared better with} the values by Shick {\it et al.}~\cite{shick_2009}
{if the nominal magnetic moments are used (D=-0.44 meV and
E=0.19 meV using $S=2$ for
Fe. For Mn the values do not change because the DFT magnetic moment agrees
with the nominal one)}. Using GGA+U, Barral et al~\cite{barral_2010}
obtained similar results to Shick {\it et al.}~\cite{shick_2009}.
Recent calculations by Panda {\it et al.}~\cite{Panda2016}
yield similar values for Fe with LDA, and values
closer to the experiment using dynamical mean field theory (DMFT).
Taking into account that Mn and Fe have perpendicular easy axes,
one may wonder if SOC might induce a non-collinear alignment of the
magnetic moments in the chains. We have tested that possibility for
the shorter chains by performing an explicit non-collinear calculation
including SOC.\cite{hobbs_2000} Due to the small energies involved
in the calculation, the convergence of non-collinear solutions is very
challenging. We have been able to stabilize a non-collinear configuration
for FeMn where the magnetic moment of Fe is along the chain, while the
moment for Mn forms an angle of $27^{\circ}$ with the $y$ axis. However,
this solution is 3.79~meV higher in energy than the collinear solution.
For longer chains we have not been able to stabilize any non-collinear
solution. Therefore, we have just considered collinear configurations
for the rest of our calculations.
The larger MAE of Fe forces the full-chain magnetic axis to align
with the Fe easy axis. This is indeed seen in Table~\ref{mae}, where
the value for MAE decreases when increasing the chain length. This
reduction in MAE can be understood in terms of the addition of Mn atoms
that tend to align their magnetic easy axis perpendicular to the Fe one.
These results qualitatively agree with the results of anisotropic spin
Hamiltonians {where the total
magnetic anisotropy is approximated by adding the individual contributions of $D$ and $E$.}
This approximation does not capture the changes in geometry of the chain with
increasing lentgth. {Nevertheless, adding up the MAE contributions of each atom
leads to overestimations of MAE that are below the accuracy
of our DFT calculations.}
{The study of the magnetic anisotropy in these chains reveals their
composite structure. All FeMn$_x$ for odd-$x$ show a \textit{quasi}-spin of 1/2,
which should present zero anisotropy
if these chains were macrospins of spin 1/2, see. Eq.~(\ref{ani}).
Our DFT calculations show sizeable anisotropies that underscore
the complexity of the magnetic states of these antiferromagnetic
structures.}
\subsection{Magnetic excitation energies}
The magnetic structure of the chains can be obtained by studying the inelastic
electron tunneling spectra (IETS)~\cite{heinrich_2004} obtained with the STM.
Figure~\ref{iets} shows the differential conductance obtained for the different
FeMn$_x$ chains. The features that appear in these spectra are due to magnetic
excitations, very similarly to the ones of Mn$_x$ chains shown in Ref.~[\onlinecite{hirjibehedin_2007}].
{Nevertheless,}
there are noticeable differences regarding both the peak at zero bias
for the odd-$x$ chains and the detailed structure of the steps. Indeed,
odd-$x$ chains are singlets in the case of Mn chains, here however, the
ground states of odd-$x$ chains present
a doublet ($S\approx \frac{1}{2}$) magnetic structure.
\begin{figure}[ht]
\includegraphics[width=\columnwidth]{figure6.pdf}
\caption{\label{iets}
{
Inelastic electron tunneling spectra (IETS) of different
FeMn$_x$ chains obtained by measuring the differential conductance with
the STM tip placed on a central Mn atom.}
}
\end{figure}
We solve Eq.~(\ref{heisenberg}) and study
the different magnetic states for each chain using the
methodology of Ref.~[\onlinecite{Ternes_2015}]. The study of the ground state
is particularly important to understand the behavior of the Kondo
physics appearing in Fig.~\ref{iets}.\cite{choi_2014} Indeed, all odd-$x$ chains
display a Kondo zero-bias peak, while this peak is absent from the
chains with even $x$. Due to the Heisenberg correlations, the
experimental ground state is multi-configurational.
Solving Eq.~(\ref{heisenberg}) with the anisotropy terms, Eq.~(\ref{ani})
shows that many of these configurations are
$S={1\over2}$ states, with a weight larger than 20\% in the total state. Hence,
a spin-flip process is possible at zero-energy cost, which explains
the appearance of Kondo peaks. Likewise, even-$x$ chains do
not have degenerate ground states and Kondo physics is absent.
The absence of degenerate ground states for even-$x$ is due
to a total spin $S \approx 2$ where the large longitudinal MAE of
the Fe atom lifts the degeneracy of the ground state. The presence of
anisotropy prevents $S$ from being a good quantum number.\cite{choi_2014}
The solutions of Eq.~(\ref{heisenberg}) permit us to compare the computed
data for the chains with the experimental data. Figure~\ref{exp}, $(a)$
shows the IETS for the FeMn$_3$ chain measured of the edge Mn atom,
and $(b)$ shows two calculations. The first one (blue) performed with
{the exchange couplings computed from}
our GGA+U calculations with an effective $U$ value for
Mn of 4 eV and for Fe of 1 eV, see section~\ref{U-J},
{and adding up the experimental atomic magnetic anisotropies.} The second one
(green) scales the exchange couplings by 1.6, using the scaling found
for FeMn when the effective $U$ value for Mn was reduced to 3 eV, and
keeping the corresponding value of Fe constant. We see that the energy
thresholds are in good agreement in the second case, and the solution
of Eq.~(\ref{heisenberg}) with the third-order perturbation method
of Ref.~[\onlinecite{Ternes_2015}] largely reproduces the dynamical
phenomenology of the magnetic chain including higher-energy excitations
like the one at $\pm 10$ meV.
The asymmetry of the main inelastic thresholds
found in Fig.~\ref{exp}$(a)$ is treated
within the third-order perturbation method
by including a potential scattering term
in the Kondo scattering.\cite{Ternes_2015}
The effect of the potential scattering term is
to remove the electron-hole symmetry of the
excitation spectra of Eq.~(\ref{heisenberg}).
In order to fit the experimental spectra
we have used a $J_{\text{Kondo}} \; \rho = -0.04$ (where
$J_{\text{Kondo}} $ is the Kondo exchange coupling
with electrons from the substrate and $\rho$ is
the density of states at the Fermi energy)
and a potential scattering term ${U_{\text{Kondo}}}/{J_{\text{Kondo}}}=-0.5$.
The results indicate that larger values should be used
to reproduce the experimental data, implying the
need to go beyond third-order perturbation theory
to treat Kondo scattering in FeMn$_3$.
\begin{figure}[ht]
\includegraphics[width=0.8\columnwidth]{figure7.pdf}
\caption{\label{exp}
{
Inelastic electron tunneling spectra (IETS) of
FeMn$_3$, $(a)$ experimental, $(b)$
two computed solution using our
calculated spin-chain parameters for $(U-J)_{\text{Mn}}=4$ eV
and $(U-J)_{\text{Fe}}=1$ eV (blue) or
scaling the computed exchange coupling
by a 1.6 factor as found for FeMn with $(U-J)_{\text{Mn}}=3$ eV
and $(U-J)_{\text{Fe}}=1$ eV (green), section~\ref{U-J}.
The STM tip is placed on the edge Mn atom.}
}
\end{figure}
The magnetic behavior of
the inelastic thresholds
is correctly reproduced by a Zeeman shift.
This permits us to extract the value of
the gyromagnetic ratio $g$, Fig.~\ref{magnet}.
For the present case we find that the atomic g's {($g_{Fe}=2.1$
and $g_{Mn}=1.9$ from Ref.~[\onlinecite{hirjibehedin_2007}])} are good
approximations to obtain the correct behavior of the magnetic global
states with external B, Fig.~\ref{magnet}. To a large
extend, the atomic spin preserves its character, although very entangled
due to the sizable Heisenberg exchange interactions.
\begin{figure*}[ht]
\includegraphics[width=1.7\columnwidth]{figure8.pdf}
\caption{\label{magnet}
{
$(a)$ Inelastic
electron tunneling spectra (IETS) for FeMn$_3$ with a magnetic field
applied perpendicular to the surface increasing in 1-T steps until
9 T. $(b)$ The behavior of the different magnetic states (black dots) obtained from the
experimental figure follows the Zeeman trend expected for a Zeeman
term with the atomic $g$-factors
{($g_{Fe}=2.1$
and $g_{Mn}=1.9$ from Ref.~[\onlinecite{hirjibehedin_2007}])}.
}
}
\end{figure*}
\section{Discussion and concluding remarks}
This work is a detailed experimental and theoretical account of the
electronic and magnetic properties of a heterogeneous type of magnetic
atomic chain (FeMn$_x$ with $x=1,6$) adsorbed on Cu$_2$N/Cu (100).
The chains are assembled by atom manipulation with an STM tip, and
stable configurations are found when the TM atoms (TM = Fe and Mn) sit
ontop of Cu atoms. Experimentally, it is difficult to assemble chains
with $x > 9$ {(we rarely succeeded going beyond $x=10$
creating a straight FeMn$_x$ chain)} and theoretically we see that stress builds up as the
chain increases size due to the imposed TM-TM distance by
the underlying Cu$_2$N/Cu (100) substrate. As the chain increases its
size, the TM atoms increase their mutual distance and also their
distance to the chain, energetically this is translated into a systematic
lowering of both the atomization energy and the energy gained by the chain
every time a new Mn atom is added.
Upon adsorption, Fe and Mn lose one of their $s$ electrons in the interaction
with the substrate, mainly to form the bond with the neighboring N atoms. There
is considerable distortion and hybridization of the d-electron structure but
their occupations remain the free-atom ones, leading to magnetic moment
values close to the gas phase.\cite{hirjibehedin_2007,lin_2011}
The experimental data involve the IETS of different chains with detailed
information on the excitation energies of the chains. The lower-energy
spectral features are due to magnetic excitations of the system as was
tested by their magnetic field dependence. In parallel, the values of
the Heisenberg Hamiltonian can be obtained from DFT by evaluating the
energy of different spin arrangements of the chains. This approach gives
us a systematic insight on how the different TM atoms relate to each other
in the chain, which are mainly driven by antiferromagnetic superexchange
mediated by the non-magnetic N atoms of the surface. We find that the
Fe-Mn couplings are systematically larger than the Mn-Mn ones, and that
beyond second neighbors neglection of the magnetic coupling is a very
good approximation. Indeed, first neighbors is a sufficient approximation
to obtain exchange couplings with an error of 0.2~meV. In the present
choice of DFT+U calculations, the computed magnetic exchanges lead to
excitation energies smaller than the experimental ones.
Unfortunately, there is not a unique way of determining
the value of the Hubbard $U$ for
the calculations. Our systematic study of the values show that a change
of 1 eV in the value of the Mn $U$ leads to a $\sim$ 60\% change in the value
of the evaluated couplings yielding good agreement with the experiment.
The TM atoms are subjected to magnetic anisotropies on this surface. Our
calculations show that the MAE of the full chain is not just the sum
of the MAE's of each TM atom. Nevertheless, the smallness of Mn MAE
renders this approximation acceptable. Despite their different easy
axis, the very large MAE of the Fe dominates and the Mn spins orient
along the chain following the Fe easy axis. This leads to a collinear
solution of the initially non-collinear problem. We have not found any
spin canting or frustration although we cannot rule it out for longer
chains. {Moreover, the study of the magnetic anisotropy in these
chains reveals their composite structure. All FeMn$_x$ for odd-$x$ show
a \textit{quasi}-spin of 1/2. If these chains were spins 1/2, their
anisotropy would be stricly zero. However, our calculations show that
they have sizeable anisotropies in agreement with the complexity of the
magnetic states of these antiferromagnetic structures.}
This combined experimental and theoretical work
gives us direct insight into the
different electronic, geometric and magnetic properties
of these heterogeneous chains. In particular, we
have given an account for the
appearance of Kondo peaks and the antiferromagnetic
character of these chains, their
magnetic anisotropy that permits us to rule out a macrospin behavior,
as well as the accumulated stress that limits the length
of the chains.
\acknowledgments
DJC acknowledges the European Union for support under the
H2020-MSCA-IF-2014 Marie-Curie Individual Fellowship programme proposal
number 654469 and a previous postdoctoral fellowhship from the Alexander von Humboldt foundation. DJC and SL acknowledge Edgar Weckert and Helmut Dosch
(Deutsches Elektronen-Synchrotron, Hamburg, Germany) for providing
high-stability lab space. NL acknowledges financial support from Spanish
MINECO (Grant No. MAT2015-66888-C3-2-R).
ICN2 acknowledges support from the
Severo Ochoa Program (MINECO, Grant SEV-2013-0295).
|
2,869,038,156,390 | arxiv | \section{Introduction}
Although querying streaming data with 100$\%$ accuracy may be possible by using cutting edge servers equipped with a large memory and powerful processor(s), enabling power efficient devices such as single-board computers~(SBCs), e.g., Arduino, Raspberry Pi, Odroid, with smarter algorithms and data structures yields cheaper and energy efficient solutions. These devices are indeed cheap, are equipped with multicore processors, and portable enough to be located at the edge of a data ecosystem, which is where the data is actually generated. Furthermore, SBCs can be enhanced with various hardware such as cameras, sensors, and software such as network sniffers. Hence, exploiting their superior price/performance ratio for data streams is a promising approach. A comprehensive survey of data stream applications can be found in~\cite{muthukrishnan2005}.
Sketches can be defined as data summaries and there exist various sketches in the literature tailored for different applications. These structures help us process a query on a massive dataset with small, usually sub-linear amount of memory~\cite{alon1996,charikar2002,dobra2002,gilbert2002}. Furthermore, each data stream can be independently sketched and then these sketches can be combined to obtain the final sketch. Due to the implicit compression, there is almost always a trade-off between the accuracy of the final result and the sketch size. A complete analysis and comparison of various sketches can be found in~\cite{cormode2005}.
{\it Count-Min Sketch}~(CMS) is a probabilistic sketch that helps to estimate the frequencies, i.e., the number of occurrences, of the items in a stream~\cite{cormode2005}. The frequency information is crucial to find heavy-hitters or rare items and detecting anomalies~\cite{cormode2003,cormode2005}. A CMS stores a small counter table to keep the track of the frequencies. The accesses to the sketch are decided based on the hashes of the items and the corresponding counters are incremented. Intuitively, the frequencies of the items are not exact due to the hash collisions. An important property of a CMS is that the error is always one sided; that is, the sketch never underestimates the frequencies.
Since independent sketches can be combined, even for a single data stream, generating a sketch in parallel is considered to be a straightforward task; each processor can independently consume a different part of a stream and build a partial sketch. However, with $\tau$ threads, this straightforward approach uses $\tau$ times more memory. Although this may not a problem for a high-end server, when the cache sizes are small, using more memory can be an important burden. In this work, we focus on the frequency estimation problem on single-board multicore computers. Our contributions can be summarized as follows:
\vspace{-0.2\topsep}
\begin{enumerate}[leftmargin=*]
\item We propose a parallel algorithm to generate a CMS and evaluate its performance on a high-end server and two multicore SBCs; Raspberry Pi 3 Model B+ and Odroid-XU4. We restructure the sketch construction phase while avoiding possible race-conditions on a {\em single} CMS table. With a single table, a careful synchronization is necessary, since race-conditions not only degrade the performance but also increase the amount of error on estimation. Although we use CMS in this work, the techniques proposed in this paper can easily be extended to other table-based frequency estimation sketches such as Count-Sketch and Count Min-Min Sketch.
\item Today, many SBCs have fast and slow cores to reduce the energy consumption. However, the performance difference of these heterogenous cores differ for different devices. Under this heterogeneity, a manual optimization is required for each SBC. As our second contribution, we propose a load-balancing mechanism that distributes the work evenly to all the available cores and uses them as efficient as possible. The proposed CMS generation technique is dynamic; it is not specialized for a single device and can be employed on various devices having heterogeneous cores.
\item
As the hashing function, we use {\em tabulation hashing} which is recently proven to provide strong statistical guarantees~\cite{thorup2017} and faster than many hashing algorithms available; a recent comparison can be found in~\cite{Dahlgaard2017}. For some sketches including CMS, to reduce the estimation error, the same item is hashed multiple times with a different function from the same family. As our final contribution, we propose a cache-friendly tabulation scheme to compute multiple hashes at a time. The scheme can also be used for other applications using multiple tabulation hashes.
\end{enumerate}
\vspace{-0.2\topsep}
\section{Notation and Background}\label{sec:not}
Let $\mathcal{U} = \{1,\cdots,n\}$ be the universal set where the elements in the stream are coming from.
Let $N$ be size of the stream ${\tt s}[.]$ where ${\tt s}[i]$ denotes the $i$th element in the stream.
We will use $f_x$ to denote the frequency of an item. Hence, $$f_x = |\{x = {\tt s}[i]: 1 \leq i \leq N\}|.$$
Given two parameters $\epsilon$ and $\delta$, a Count-Min Sketch is constructed as a two-dimensional counter table with
$d = \lceil \ln(1/\delta) \rceil$ rows and $w = \lceil e/\epsilon \rceil$ columns. Initially, all the counters inside the sketch are set to $0$.
There are two fundamental operations for a CMS; the first one is {\em insert}($x$) which updates
internal sketch counters to process the items in the stream.
To insert $x \in \mathcal{U}$, the counters ${\tt cms}[i][h_i(x)]$ are incremented for $1 \leq i \leq d$, i.e., a counter from each row is incremented where the column IDs are obtained from the hash values.
Algorithm~\ref{alg:cms_construct} gives the pseudocode to sequentially process ${\tt s}[.]$ of size $N$ and construct a CMS.
The second operation for CMS is {\em query}($x$) to estimate the frequency of $x \in \mathcal{U}$ as $$f'_x = min_{1 \leq i \leq d}\{{\tt cms}[i][h_i(x)]\}.$$
\noindent With $d \times w$ memory, the sketch satisfies that
$f_x \leq f'_x$ and $\Pr\left(f'_x \geq f_x + \epsilon N\right) \leq \delta.$ Hence, the error is additive and always one-sided. Furthermore, for $\epsilon$ and $\delta$ small enough, the error is also bounded with high probability. Hence, especially for frequent items with large $f_x$, the ratio of the estimation to the actual frequency approaches to one.
\renewcommand{\baselinestretch}{0.9}
\begin{algorithm}[htbp]
\small
\caption{\textsc{CMS-Construction}}
\KwIn{ $\epsilon$: error factor, $\delta$: error probability \\
\hspace*{8ex}${\tt s}[.]$: a stream with $N$ elements from $n$ distinct elements \\
\hspace*{8ex}$h_i(.)$: pairwise independent hash functions where for \\
\hspace*{13ex}$1\leq i \leq d$, $h_i$: $\mathcal{U} \rightarrow \{1,\cdots,w\}$ and $w = \lceil e/\epsilon \rceil$\\}
\KwOut{ ${\tt cms}[.][.]$: a $d \times w$ counter sketch where $d = \lceil 1/\delta \rceil$ \\
}
\For{$i\leftarrow 1$ \KwTo $d$}{
\For{$j\leftarrow 1$ \KwTo $w$}{
${\tt cms}$[i][j] $ \leftarrow 0$
}
}
\For{$i\leftarrow 1$ \KwTo $N$}{
$x \leftarrow s[i]$
\For{$j\leftarrow 1$ \KwTo $d$}{
$col \leftarrow h_j(x)$\\
${\tt cms}$[j][$col$] $ \leftarrow {\tt cms}$[j][$col$] $ +1$
}
}
\label{alg:cms_construct}
\end{algorithm}
\renewcommand{\baselinestretch}{1}
\vspace*{-4ex}
\paragraph{Tabulation Hash:} CMS requires pairwise independent hash functions to provide the desired properties stated above. A separate hash function is used for each row of the CMS with a range equal
to the range of columns. In this work, we use tabulation hashing~\cite{zobrist1970} which has been recently analyzed by Patrascu and Thorup et al.~\cite{patrascu2012,thorup2017} and shown to provide strong statistical guarantees despite of its simplicity. Furthermore, it is even as fast as the classic multiply-mod-prime scheme, i.e., $(ax + b) \bmod p$.
Assuming each element in $\mathcal{U}$ is represented in 32 bits~(the hash function can also be used to hash 64-bit stream items~\cite{thorup2017}) and the desired output is also 32 bits, tabulation hashing works as follows: first a $4 \times 256$ table is generated and filled with random 32-bit values. Given a 32-bit input $x$, each character, i.e., 8-bit value, of $x$ is used as an index for the corresponding row. Hence, four 32-bit values, one from each row, are extracted from the table. The bitwise {\tt XOR} of these 32-bit values are returned as the hash value.
\section{Merged Tabulation with a Single Table}\label{sec:tab}
Hashing the same item with different members of a hash family is a common technique in sketching applied to reduce the error of the estimation. One can use a single row for CMS, i.e., set $d = 1$ and answer the query by reporting the value of the counter corresponding to the hash value. However, using multiple rows reduces the probability of having large estimation errors.
Although the auxiliary data used in tabulation hashing are small and can fit into a cache, the spatial locality of the accessed table elements, i.e., their distance in memory, is deteriorating since each access is performed to a different table row~(of length 256). A naive, cache-friendly rearrangement of the entries in the tables is also not possible for applications performing a single hash per item; the indices for each table row are obtained from adjacent chunks in the binary representation of the hashed item which are usually not correlated. Hence, there is no relation whatsoever among them to help us to fix the access pattern for all possible stream elements.
For many sketches, the same item is hashed more than once. When tabulation hashing is used, this yields an interesting optimization; there exist multiple hash functions and hence, more than one hash table. Although, the entries in a single table is accessed in a somehow irregular fashion, the accessed coordinates in all the tables are the same for different tables as can be observed on the left side of Figure~\ref{fig:merged_tabular_access}. Hence, the columns of the tables can be combined in an alternating fashion as shown in the right side of the figure. In this approach, when only a single thread is responsible from computing the hash values for a single item to CMS, the cache can be utilized in a better way since the memory locations accessed by that thread are adjacent. Hence, the computation will pay the penalty for a cache-miss only once for each 8-bit character of a 32-bit item. This proposed scheme is called {\em merged tabulation}.
\begin{figure}[htbp]
\begin{minipage}[c]{0.65\textwidth}
\includegraphics[width=\textwidth]{merged_tabular_access.pdf}
\end{minipage}\hfill\hfill\hfill
\begin{minipage}[c]{0.28\textwidth}
\caption{\small{Memory access patterns for naive and merged tabulation for four hashes. The hash tables are colored with different colors. The accessed locations are shown in black.}}
\label{fig:merged_tabular_access}
\end{minipage}
\end{figure}
\section{Parallel Count-Min Sketch Construction}\label{sec:par}
Since multiple CMS sketches can be combined, on a multicore hardware, each thread can process a different part of the data (with the same hash functions) to construct a partial CMS. These partial sketches can then be combined by adding the counter values in the same locations.
Although this approach has been already proposed in the literature and requires no synchronization,
the amount of the memory it requires increases with increasing number of threads. We included this {\em one sketch to one core} approach in the experiments as one of the baselines.
Constructing a single CMS sketch in parallel is not a straightforward task. One can assign an item to a single thread and let it perform all the updates (i.e., increment operations) on CMS counters. The pseudocode of this parallel CMS construction is given in Algorithm~\ref{alg:cms_construct_par_nobuf}. However, to compute the counter values correctly, this approach requires a significant synchronization overhead; when a thread processes a single data item, it accesses an arbitrary column of each CMS row. Hence, race conditions may reduce the estimation accuracy. In addition, these memory accesses are probable causes of false sharing.
To avoid the pitfalls stated above, one can allocate locks on the counters before every increment operation. However, such a synchronization mechanism is too costly to be applied in practice.
\renewcommand{\baselinestretch}{0.9}
\begin{algorithm}[htbp]
\small
\caption{\textsc{Naive-Parallel-CMS}}
\SetAlgoNoLine
\KwIn{ $\epsilon$: error factor, $\delta$: error probability \\
\hspace*{7ex}${\tt s}[.]$: a stream with $N$ elements from $n$ distinct elements \\
\hspace*{7ex}$h_i(.)$: pairwise independent hash functions where for \\
\hspace*{13ex}$1\leq i \leq d$, $h_i$: $\mathcal{U} \rightarrow \{1,\cdots,w\}$ and $w = \lceil e/\epsilon \rceil$\\
\hspace*{7ex}$\tau$: no threads\\ }
\KwOut{ ${\tt cms}[.][.]$: a $d \times w$ counter sketch where $d = \lceil 1/\delta \rceil$ \\
}
Reset all the ${\tt cms}[.][.]$ counters to 0 (as in Algorithm~\ref{alg:cms_construct}).\\
\For{$i\leftarrow 1$ \KwTo $N$ {\bf in parallel}}{
$x \leftarrow s[i]$\\
${\tt hashes}[.] \leftarrow$ {\sc MergedHash}($x$)
\For{$j\leftarrow 1$ \KwTo $d$}{
$col \leftarrow {\tt hashes}[j] $ \\
${\tt cms}$[j][$col$] $ \leftarrow {\tt cms}$[j][$col$] $ +1$ (\em{must be a critical update})
}
}
\label{alg:cms_construct_par_nobuf}
\end{algorithm}
\renewcommand{\baselinestretch}{1}
In this work, we propose a {\em buffered parallel} execution to alleviate the above mentioned issues; we (1) divide the data into batches and (2) process a single batch in parallel in two phases; a) merged-hashing and b) CMS counter updates. In the proposed approach, the threads synchronize after each batch and process the next one. For batches with $b$ elements, the first phase requires a buffer of size $b \times d$ to store the hash values, i.e., column ids, which then will be used in the second phase to update corresponding CMS counters. Such a buffer allows us to use merged tabulation effectively during the first phase. In our implementation, the counters in a row are updated by the same thread hence, there will be no race conditions and probably much less false sharing. Algorithm~\ref{alg:cms_construct_par} gives the pseudocode of the proposed buffered CMS construction approach.\looseness=-1
\renewcommand{\baselinestretch}{0.9}
\begin{algorithm}[htbp]
\small
\caption{\textsc{Buffered-Parallel-CMS}}
\KwIn{ $\epsilon$: error factor, $\delta$: error probability \\
\hspace*{7ex}${\tt s}[.]$: a stream with $N$ elements from $n$ distinct elements \\
\hspace*{7ex}$h_i(.)$: pairwise independent hash functions where for \\
\hspace*{13ex}$1\leq i \leq d$, $h_i$: $\mathcal{U} \rightarrow \{1,\cdots,w\}$ and $w = \lceil e/\epsilon \rceil$\\
\hspace*{7ex}$b$: batch size (assumption: divides $N$)\\
\hspace*{7ex}$\tau$: no threads (assumption: divides $d$)\\ }
\KwOut{ ${\tt cms}[.][.]$: a $d \times w$ counter sketch where $d = \lceil 1/\delta \rceil$ \\
}
Reset all the ${\tt cms}[.][.]$ counters to 0 (as in Algorithm~\ref{alg:cms_construct})\\[5pt]
\For{$i\leftarrow 1$ \KwTo $N/b$}{
$j_{end} \leftarrow i \times b$
$j_{start} \leftarrow j_{end} - b + 1$
\For{$j\leftarrow j_{start}$ \KwTo $j_{end}$ {\bf in parallel}}{
$x \leftarrow {\tt s}[j]$\\
$\ell_{end} \leftarrow j \times d$\\
$\ell_{start} \leftarrow \ell_{end} - d + 1$\\
${\tt buf}[\ell_{start}, \cdots, \ell_{end}] \leftarrow$ {\sc MergedHash}($x$)
}
{{\bf Synchronize} the threads, e.g., with a {\em barrier}}
\For{$t_{id}\leftarrow 1$ \KwTo $\tau$ {\bf in parallel}}{
\For{$j\leftarrow 1$ \KwTo $b$ }{
$nrows \leftarrow d / \tau$\\
$r_{end} \leftarrow t_{id} \times nrows$\\
$r_{start} \leftarrow r_{end} - nrows + 1$\\
\For{$r\leftarrow r_{start}$ \KwTo $r_{end}$ }{
$col \leftarrow {\tt buf}[((j-1) \times d) + r]$\\
${\tt cms}[r][col] \leftarrow {\tt cms}[r][col] +1$
}
}
}
}
\label{alg:cms_construct_par}
\end{algorithm}
\renewcommand{\baselinestretch}{1}
\section{Managing Heterogeneous Cores} \label{sec:load}
A recent trend on SBC design is heterogeneous multiprocessing which had been widely adopted by mobile devices. Recently, some ARM-based devices including SBCs use the {\em big.LITTLE} architecture equipped with power hungry but faster cores, as well as battery-saving but slower cores. The faster cores are suitable for compute-intensive, time-critical tasks where the slower ones perform the rest of the tasks and save more energy. In addition, tasks can be dynamically swapped between these cores on the fly. One of the SBCs we experiment in this study has an 8-core Exynos 5422 Cortex processor having four fast and four relatively slow cores.
Assume that we have $d$ rows in CMS and $d$ cores on the processor; when the cores are homogeneous, Algorithm~\ref{alg:cms_construct_par} works efficiently with static scheduling since, each thread performs the same amount of merged hashes and counter updates. When the cores are heterogeneous, the first inner loop (for merged hashing) can be dynamically scheduled: that ia a batch can be divided into smaller, independent chunks and the faster cores can hash more chunks.
However, the same technique is not applicable to the (more time consuming) second inner loop where the counter updates are performed: in the proposed buffered approach, Algorithm~\ref{alg:cms_construct_par} divides the workload among the threads by assigning each row to a different one. When the fast cores are done with the updates, the slow cores will still be working. Furthermore, faster cores cannot help to the slower ones by stealing a portion of their remaining jobs since when two threads work on the same CMS row, race conditions will increase the error.
To alleviate these problems, we propose to pair a slow core with a fast one and make them update two rows in an alternating fashion. The batch is processed in two stages as shown in Figure~\ref{fig:fastslow}; in the first stage, the items on the batch are processed in a way that the threads running on faster cores update the counters on even numbered CMS rows whereas the ones running on slower cores update the counters on odd numbered CMS rows. When the first stage is done, the thread/core pairs exchange their row ids and resume from the item their mate stopped in the first stage. In both stages, the faster threads process $fastBatchSize$ items and the slower ones process $slowBatchSize$ items where $b = fastBatchSize + slowBatchSize.$\vspace*{-2ex}
\begin{figure}[htbp]
\begin{minipage}[c]{0.65\textwidth}
\includegraphics[width=\textwidth]{fastslow.pdf}
\end{minipage}\hfill\hfill\hfill
\begin{minipage}[c]{0.32\textwidth}
\caption{\small{For a single batch, rows $i$ and $i+1$ of CMS are updated by a fast and a slow core pair in two stages. In the first stage, the fast core performs row $i$ updates and the slow core processes row $i+1$ updates. In the second stage, they exchange the rows and complete the remaining updates on the counters for the current batch.}}
\label{fig:fastslow}
\end{minipage}
\vspace*{-4ex}
\end{figure}
To avoid the overhead of dynamic scheduling and propose a generic solution, we start with $fastBatchSize = b/2$ and $slowBatchSize = b/2$ and by measuring the time spent by the cores, we dynamically adjust them to distribute the workload among all the cores as fairly as possible. Let $t_F$ and $t_S$ be the times spent by a fast and slow core, respectively, on average. Let $s_F = \frac{fastBatchSize}{t_F}$ and $s_S = \frac{slowBatchSize}{t_S}$ be the speed of these cores for the same operation, e.g., hashing, CMS update etc. We then solve the equation $\frac{fastBatchSize + x}{s_F} = \frac{slowBatchSize - x}{s_S}$ for $x$ and update the values as
\begin{align*}
fastBatchSize &= fastBatchSize + x\\
slowBatchSize &= slowBatchSize - x
\end{align*}
for the next batch. One can apply this method iteratively for a few batches and use the average values to obtain a generic and dynamic solution for such computations. To observe the relative performances, we applied this technique both for hashing and counter update phases of the proposed buffered CMS generation algorithm.
\section{Experimental Results}\label{sec:exp}
We perform experiments on the following three architectures:
\begin{itemize}[leftmargin=*]
\item {\bf Arch-1} is a server running on 64 bit CentOS 6.5 equipped with 64GB RAM and an Intel Xeon E7-4870 v2 clocked at 2.30 GHz and having 15 cores. Each core has a 32KB L1 and a 256KB L2 cache, and the size of L3 cache is 30MB.
\item {\bf Arch-2} (Raspberry Pi 3 Model B+) is a quad-core 64-bit ARM Cortex A-53 clocked at 1.4 GHz equipped with 1 GB LPDDR2-900 SDRAM. Each core has a 32KB L1 cache, and the shared L2 cache size is 512KB.
\item {\bf Arch-3} (Odroid XU4) is an octa-core heterogeneous multi-processor. There are four A15 cores running on 2Ghz and four A7 cores running on 1.4Ghz. The SBC is equipped with a 2GB LPDDR3 RAM. Each core has a 32KB L1 cache. The fast cores have a shared 2MB L2 cache and slow cores have a shared 512KB L2 cache.
\end{itemize}
For multicore parallelism, we use C++ and OpenMP. We use {\tt gcc 5.3.0} on {\bf Arch-1}. On {\bf Arch-2} and {\bf Arch-3}, the {\tt gcc} version is {\tt 6.3.0} and {\tt 7.3.0}, respectively. For all architectures, {\tt -O3} optimization flag is also enabled.
To generate the datasets for experiments, we used {\em Zipfian} distribution~\cite{Zipf1935}. Many data in real world such as number of paper citations, file transfer sizes, word frequencies etc. fit to a Zipfian distribution with the shape parameter around $\alpha = 1$. Furthermore, the distribution is a common choice for the studies in the literature to benchmark the estimation accuracy of data sketches. To cover the real-life better, we used the shape parameter $\alpha \in \{1.1, 1.5\}$. Although they seem to be unrelated at first, an interesting outcome of our experiments is that the sketch generation performance depends not only the number of items but also the frequency distribution; when
the frequent items become more dominant in the stream, some counters are touched much more than the others. This happens with increasing $\alpha$ and is expected to increase the performance since most of the times, the counters will already be in the cache. To see the other end of the spectrum, we also used {\em Uniform} distribution to measure the performance where all counters are expected to be touched the same number of times.
We use $\epsilon \in \{10^{-3}, 10^{-4}, 10^{-5}\}$ and $\delta = 0.003$ to generate small, medium and large $d \times w$ sketches where the number of columns is chosen as the first prime after $2/\epsilon$. Hence, the sketches have $w = \{2003, 20071, 200003\}$ columns and $d = \lceil \log_2(1/\delta) \rceil = 8$ rows. For the experiments on {\bf Arch-1}, we choose $N = 2^{30}$ elements from a universal set $\mathcal{U}$ of cardinality $n = 2^{25}$. For {\bf Arch-2} and {\bf Arch-3}, we use $N = 2^{25}$ and $n = 2^{20}$. For all architectures, we used $b = 1024$ as the batch size. Each data point in the tables and charts given below is obtained by averaging ten runs.
\subsection{Multi Table vs. Single Table}
Although {\em one-sketch-per-core} parallelization, i.e., using partial, multiple sketches, is straightforward, it may not be a good approach for memory/cache restricted devices such as SBCs. The memory/cache space might be required by other applications running on the same hardware and/or other types of skeches being maintained at the same time for the same or a different data stream.
Overall, this approach uses $(d \times w \times \tau)$ counters
where each counter can have a value as large as $N$; i.e., the memory consumption is
$(d \times w \times \tau \times \log N)$ bits. On the other hand, a single sketch with buffering
consumes $$(d \times ((w \times \log N) + (b \times \log w)))$$ bits since there are $(d \times b)$ entries in the
buffer and each entry is a column ID on CMS. For instance, with $\tau = 8$ threads, $\epsilon = 0.001$ and $\delta = 0.003)$, the one-sketch-per-core approach requires $(8 \times 2003 \times 8 \times 30) =$ 3.85Mbits whereas using single sketch requires
$(8 \times ((2003 \times 30) + (1024 \times 11))) =$ 0.57Mbits. Hence, in terms of memory footprint, using a single table pays off well. Figure~\ref{fig:main} shows the case for execution time.
\begin{figure}[htbp]
\centering
\begin{subfigure}[b]{0.47\textwidth}
\centering
\includegraphics[width = 6cm]{main_1_8.pdf}
\caption{{\bf Arch-1} - 8 cores}
\end{subfigure}\hspace*{1ex}
~
\begin{subfigure}[b]{0.47\textwidth}
\centering
\includegraphics[width = 5.7cm]{main_2_4.pdf}
\caption{{\bf Arch-2} - 4 cores}
\end{subfigure}
\begin{subfigure}[b]{0.47\textwidth}
\centering
\includegraphics[width = 6.1cm]{main_3_4.pdf}
\caption{{\bf Arch-3} - 4 cores}
\end{subfigure}\hspace*{1ex}
~
\begin{subfigure}[b]{0.47\textwidth}
\centering
\includegraphics[width = 5.7cm]{main_3_8.pdf}
\caption{{\bf Arch-3} - 8 cores}
\end{subfigure}
\renewcommand{\baselinestretch}{0.93}
\caption{\small{Performance comparison for multi-table~(MT) and single table~(ST) approaches. MT uses the one-sketch-per-core approach as suggested in the literature, MT+ is the MT-variant with merged tabulation. In all the figures, ST+ is the proposed scheme~(as in Algorithm~\ref{alg:cms_construct_par}), where in the last figure, ST++ is the ST+ variant using the load-balancing scheme for heterogeneous cores as described in Section~\ref{sec:load}. For all the figures, the $x$-axis shows the algorithm and $\epsilon \in \{10^{-3}, 10^{-4}, 10^{-5}\}$ pair. The $y$-axis shows the runtimes in seconds; it does not start from 0 for a better visibility of performance differences. The first bar of each group shows the case when the data is generated using uniform distribution. The second and the thirds bars show the case for Zipfian distribution with the shape parameter $\alpha = 1.1$ and $1.5$, respectively.}}
\renewcommand{\baselinestretch}{1}
\label{fig:main}
\end{figure}
In Figure~\ref{fig:main}.(a), the performance of the single-table~(ST+) and multi-table~(MT and MT+) approaches are presented on {\bf Arch-1}. Although ST+ uses much less memory, its performance is not good due to all the time spent while buffering and synchronization. The last level cache size on {\bf Arch-1} is 30MB; considering the largest sketch we have is 6.4MB~(with 4-byte counters), {\bf Arch-1} does not suffer from its cache size and MT+ indeed performs much better than ST+. However, as Fig.~\ref{fig:main}.(b) shows for {\bf Arch-2}, with a $512$KB last-level cache, the proposed technique significantly improves the performance, and while doing that, it uses significantly much less memory. As Fig.~\ref{fig:main}.(c) shows, a similar performance improvement on {\bf Arch-3} is also visible for medium~(640KB) and especially large~(6.4MB) sketches when only the fast cores with a 2MB last-level cache are used.\looseness=-1
Figure~\ref{fig:main} shows that the performance of the algorithms vary with respect to the distribution. As mentioned above, the variance on the frequencies increases with increasing $\alpha$. For uniform and Zipfian($1.1$), the execution times tend to increase with sketch sizes. Nevertheless, for $\alpha = 1.5$, sketch size does not have a huge impact on the performance, since only the {\em hot} counters of the most frequent items are frequently updated. Although each counter has the same chance to be a hot counter, the effective sketch size reduces significantly especially for large sketches. This is also why the runtimes for many configurations are less for $\alpha = 1.5$.\looseness=-1
\begin{figure*}[htbp]
\begin{subfigure}[t]{0.49\textwidth}
\includegraphics[width=\linewidth]{f2s-small.pdf}
\caption{{\bf small} on {\bf Arch-3}}
\label{fig:fs-small}
\end{subfigure}\hspace*{3ex}
\begin{subfigure}[t]{0.49\textwidth}
\includegraphics[width=\linewidth]{f2s-big.pdf}
\caption{{\bf medium} on {\bf Arch-3}}
\label{fig:fs-large}
\end{subfigure}
\caption{\small{Plots of fast-to-slow ratio $F2S = \frac{fastBatchSize}{slowBatchSize}$ of hashing and CMS update phases for consecutive batches and for small~(left) and medum~(right) sketches.}}
\label{fig:load}
\vspace*{-6ex}
\end{figure*}
\subsection{Managing Heterogeneous Cores}
To utilize the heterogeneous cores on {\bf Arch-3}, we applied the smart load distribution described in Section~\ref{sec:load}. We pair each slow core with a fast one, virtually divide each batch into two parts, and make the slow core always run on smaller part.
As mentioned before, for each batch, we dynamically adjust the load distribution based on the previous runtimes. Figure~\ref{fig:load} shows the ratio $F2S = \frac{fastBatchSize}{slowBatchSize}$ for the first $256$ batches of small and medium sketches. The best F2S changes w.r.t. the computation performed; for hashing, a 4-to-1 division of workload yields a balanced distribution. However, for CMS updates, a 1.8-to-1 division is the best. As the figure shows, the F2S ratio becomes stable after a few batches for both phases. Hence, one can stop the update process after $\thicksim$$30$ batches and use a constant F2S for the later ones. As Fig.~\ref{fig:main}.(d) shows, ST++, the single-table approach both with merged tabulation and load balancing, is always better than ST+. Furthermore, when $\tau = 8$, with the small 512KB last-level cache for slower cores, the ST++ improves MT+ much better~(e.g., when the medium sketch performance in Figs.~\ref{fig:main}.(c) and~\ref{fig:main}.(d) are compared). Overall, smart load distribution increases the efficiency by $15\%$--$30\%$ for $\tau = 8$ threads.\looseness=1
\subsection{Single Table vs. Single Table}
For completeness, we compare the performance of the proposed single-table approach. i.e., ST+ and ST++, with that of Algorithm~\ref{alg:cms_construct_par_nobuf}. However, we observed that using {\em atomic} updates drastically reduces its performance. Hence, we use the algorithm in a {\em relaxed} form, i.e., with non-atomic updates. Note that in this form, the estimations can be different than the CMS due to race conditions. As Table~\ref{tab:throughputs} shows, with a single thread, the algorithms perform almost the same except for {\bf Arch-1} for which Alg~\ref{alg:cms_construct_par_nobuf} is faster. However, when the number of threads is set to number of cores, the proposed algorithm is much better due to the negative impact of false sharing generated by concurrent updates on the same cache line. In its current form, the proposed algorithm can process approximately 60M, 4M, and 9M items on {\bf Arch-1}, {\bf Arch-2} and {\bf Arch-3}, respectively. \looseness=-1
\vspace{-2ex}
\begin{table}
\begin{center}
\scalebox{0.90}{
\begin{tabular}{r|rr|rr||r|rr|rr}
Zipfian & \multicolumn{2}{c|}{{\bf Alg 3}~(ST+ and ST++)} & \multicolumn{2}{c||}{{\bf Alg 2} - relaxed} &Zipfian & \multicolumn{2}{c|}{{\bf Alg 3}~(ST+ and ST++)} & \multicolumn{2}{c}{{\bf Alg 2} - relaxed}\\
$\alpha = 1.1$ & $\tau = 1$ & $\tau \in \{4,8\}$& $\tau = 1$ & $\tau \in \{4,8\}$ & $\alpha = 1.5$ & $\tau = 1$ & $\tau \in \{4,8\}$& $\tau = 1$ & $\tau \in \{4,8\}$ \\\hline
{\bf Arch-1} & 17.6 & 60.0 & 22.6 & 17.8 & {\bf Arch-1} & 17.9 & 57.6 & 22.6 & 12.9 \\
{\bf Arch-2} & 1.3 & 3.9 & 1.3 & 3.3 & \hspace*{2ex} {\bf Arch-2} & 1.3 & 4.1 & 1.2 & 3.2 \\
{\bf Arch-3} & 1.6 & 9.0 & 1.6 & 6.6 & {\bf Arch-3} & 1.6 & 9.0 & 1.7 & 6.1 \\
\end{tabular}
}
\caption{\small{Throughputs for sketch generation - million items per second. For each architecture, the number of threads is set to either one or the number of cores.}}
\label{tab:throughputs}
\end{center}
\vspace{-11ex}
\end{table}
\section{Related Work}\label{sec:related}
CMS is proposed by Cormode and Muthukrishnan to summarize data streams~\cite{cormode2005}. Later, they comment on its parallelization~\cite{cormode2012} and briefly mention the single-table and multi-table approaches. There are studies in the literature employing synchronization primitives such as atomic operations for frequency counting~\cite{Das2009}. However, synchronization free approaches are more popular; Cafaro et al. propose an augmented frequency sketch for time-faded heavy hitters~\cite{cafaro2018}. They divided the stream into sub-streams and generated multiple sketches instead of a single one. A similar approach using multiple sketches is also taken by Mandal~et~al.~\cite{mandal18}. CMS has also been used as an underlying structure to design advanced sketches. Recently, Roy et al. developed ASketch which filters high frequent items first and handles the remaining with a sketch such as CMS which they used for implementation~\cite{roy2016}. However, their parallelization also employs multiple filters/sketches. Another advanced sketch employing multiple CMSs for parallelization is FCM~\cite{Thomas2007}.
Although other hash functions can also be used, we employ tabular hashing which is recently shown to provide good statistical properties and reported to be fast~\cite{thorup2017,Dahlgaard2017}. When multiple hashes on the same item are required, which is the case for many sketches, our merging technique will be useful for algorithms using tabular hashing.
To the best of our knowledge, our work is the first cache-focused, synchronization-free, single-table CMS generation algorithm specifically tuned for limited-memory multicore architectures such as SBCs. Our techniques can also be employed for other table-based sketches such as Count Sketch~\cite{charikar2002} and CMS with conservative updates.
\section{Conclusion and Future Work}\label{sec:con}
In this work, we investigated the parallelization of Count-Min Sketch on SBCs. We proposed three main techniques: The first one, merged tabulation, is useful when a single is item needs to be hashed multiple times and can be used for different sketches. The second technique buffers the intermediate results to correctly synchronize the computation and regularize the memory accesses. The third one helps to utilize heterogeneous cores which is a recent trend on today's smaller devices. The experiments we performed show that the propose techniques improve the performance of CMS construction on multicore devices especially with smaller caches.
As a future work, we are planning to analyze the options on the SBCs to configure how much data/instruction cache they use, and how they handle coherency. We also want to extend the architecture spectrum with other accelerators such as FPGAs, GPUs, and more SBCs with different processor types. We believe that similar techniques we develop here can also be used for other sketches.
\renewcommand{\baselinestretch}{0.97}
\bibliographystyle{splncs04}
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2,869,038,156,391 | arxiv | \section{#1}\setcounter{equation}{0}}
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\newcommand{\noindent \mbox{{\bf Proof}: }}{\noindent \mbox{{\bf Proof}: }}
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\catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12
\def\alpha} \def\gb{\beta} \def\gg{\gamma{\alpha} \def\gb{\beta} \def\gg{\gamma}
\def\chi} \def\gd{\delta} \def\ge{\epsilon{\chi} \def\gd{\delta} \def\ge{\epsilon}
\def\theta} \def\vge{\varepsilon{\theta} \def\vge{\varepsilon}
\def\varphi} \def\gh{\eta{\varphi} \def\gh{\eta}
\def\iota} \def\gk{\kappa} \def\gl{\lambda{\iota} \def\gk{\kappa} \def\gl{\lambda}
\def\mu} \def\gn{\nu} \def\gp{\pi{\mu} \def\gn{\nu} \def\gp{\pi}
\def\varpi} \def\gr{\rho} \def\vgr{\varrho{\varpi} \def\gr{\rho} \def\vgr{\varrho}
\def\sigma} \def\vgs{\varsigma} \def\gt{\tau{\sigma} \def\vgs{\varsigma} \def\gt{\tau}
\def\upsilon} \def\gv{\vartheta} \def\gw{\omega{\upsilon} \def\gv{\vartheta} \def\gw{\omega}
\def\psi} \def\gz{\zeta{\psi} \def\gz{\zeta}
\def\Gamma} \def\Gd{\Delta} \def\Gf{\Phi{\Gamma} \def\Gd{\Delta} \def\Gf{\Phi}
\defTheta{Theta}
\def\Lambda} \def\Gs{\Sigma} \def\Gp{\Pi{\Lambda} \def\Gs{\Sigma} \def\Gp{\Pi}
\def\Omega} \def\Gx{\Xi} \def\Gy{\Psi{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}
\title{Topics in the theory of positive solutions of second-order elliptic
and parabolic partial differential equations}
\author{Yehuda Pinchover\\
{\small Department of Mathematics}\\ {\small Technion - Israel Institute of Technology}\\
{\small Haifa 32000, Israel}\\
{\small [email protected]}\\[3mm]
{\em Dedicated to Barry Simon}\\ {\em on the occasion of his 60th
birthday}}
\date{}
\maketitle
\begin{abstract}
The purpose of the paper is to review a variety of recent
developments in the theory of positive solutions of general linear
elliptic and parabolic equations of second-order on noncompact
Riemannian manifolds, and to point out a number of their
consequences.
\\[1mm]
\noindent 2000 {\em Mathematics Subject Classification.}
Primary 35J15; Secondary 35B05, 35C15, 35K10.\\[1mm]
\noindent {\em Keywords.}
Green function, ground state, heat kernel, Liouville theorem,
Martin boundary, positive solution, $p$-Laplacian.
\end{abstract}
\mysection{Introduction}\label{secint}
Positivity properties of general linear second-order elliptic and
parabolic equations have been extensively studied over the recent
decades (see for example \cite{M98,Pins95} and the references
therein). The purpose of the present paper is to review a variety
of recent developments in the theory of positive solutions of such
equations and to point out a number of their (sometimes
unexpected) consequences. The attention is focused on
generalizations of positivity properties which were studied by
Barry Simon in the special case of Schr\"odinger operators. Still,
the selection of topics in this survey is incomplete, and is
according to the author's working experience and taste. The
reference list is far from being complete and serves only this
expos\'{e}.
The outline of the paper is as follows. In
Section~\ref{secpreliminaries}, we introduce some fundamental
notions that will be studied throughout the paper. In particular,
we bring up the notions of the generalized principal eigenvalue,
criticality and subcriticality of elliptic operators, and the
Martin boundary. Section~\ref{secpert} is devoted to different
types of perturbations and their properties. In
Section~\ref{secindef}, we study the behavior of critical
operators under indefinite perturbations. In
sections~\ref{sechetk} and \ref{secup} we discuss some
relationships between criticality theory and the theory of
nonnegative solutions of the corresponding parabolic equations.
More precisely, in Section~\ref{sechetk} we deal with the large
time behavior of the heat kernel, while in Section~\ref{secup} we
discuss sufficient conditions for the nonuniqueness of the
positive Cauchy problem, and study intrinsic ultracontractivity.
In Section~\ref{seceigen}, we study the asymptotic behavior at
infinity of eigenfunctions of Schr\"odinger operators. The
phenomenon known in the mathematical physics literature as
`localization of binding', and the properties of the shuttle
operator are discussed in sections~\ref{seclocalization} and
\ref{secshttle}, respectively. The exact asymptotics of the
positive minimal Green function, and the explicit Martin integral
representation theorem for positive solutions of general
$\mathbb{Z}^d$-periodic elliptic operators on $\mathbb{R}^d$ are
reviewed in Section~\ref{secperiod}. We devote
Section~\ref{seccritliouville} to some relationships between
criticality theory and Liouville theorems. In particular, we
reveal that an old open problem of B.~Simon
(Problem~\ref{problem}) is completely solved (see
Theorem~\ref{thmDKS}). In Section~\ref{secliouville} we study
polynomially growing solutions of $\mathbb{Z}^d$-periodic
equations on $\mathbb{R}^d$. We conclude the paper in
Section~\ref{secplap} with criticality theory for the
$p$-Laplacian with a potential term.
\mysection{Principal eigenvalue, minimal growth and
classification}\label{secpreliminaries}
Consider a noncompact, connected, smooth Riemannian manifold $X$ of dimension $d$.
For any subdomain $\Omega\subseteq X$, we write $D\Subset \Omega$
if $\overline{D}$ is a compact subset of $\Omega$. The ball of
radius $r>0$ and center at $x_0$ is denoted by $B(x_0,r)$. Let
$f,g \in C(\Omega)$, we use the notation $f\asymp g$ on
$D\subseteq\Omega$ if there exists a positive constant $C$ such
that
$$C^{-1}g(x)\leq f(x) \leq Cg(x) \qquad \mbox{ for all } x\in D.$$
By ${\bf 1}$, we denote the constant function taking at any point
the value $1$.
We associate to any subdomain $\Omega\subseteq X$ {\em an
exhaustion of $\Omega$}, i.e. a sequence of smooth, relatively
compact domains $\{\Omega_{j}\}_{j=1}^{\infty}$ such that
$\Omega_1\neq \emptyset$, $\overline{\Omega}_{j}\subset
\Omega_{j+1}$ and $\cup_{j=1}^{\infty}\Omega_{j}=\Omega$. For
every $j\geq 1$, we denote $\Omega_{j}^*=\Omega\setminus
\overline{\Omega_j}$. We say that a function $f\in C(\Omega)$ {\em
vanishes at infinity of} $\Omega$ if for every $\varepsilon >0$
there exists $N\in {\mathbb N}$ such that $|f(x)|<\varepsilon$ for all $x
\in \Omega^*_{N}$.
We associate to any such exhaustion
$\{\Omega_{j}\}_{j=1}^{\infty}$ a sequence
$\{\chi_{j}(x)\}_{j=1}^{\infty}$ of smooth cutoff functions in
$\Omega$ such that $\chi_j(x)\equiv 1$ in $\Omega_j$,
$\chi_j(x)\equiv 0$ in $\Omega \setminus \Omega_{j+1}$, and $0\leq
\chi_j(x)\leq 1$ in $\Omega$. Let $0<\alpha} \def\gb{\beta} \def\gg{\gamma \leq 1$. For $W\in
C^\alpha} \def\gb{\beta} \def\gg{\gamma(\Omega)$, we denote $W_j(x)=\chi_j(x)W(x)$ and
$W^*_j(x)=W(x)-W_j(x)$.
\indent We consider a linear, second-order, elliptic operator $P$
defined in a subdomain $\Omega\subset X$. Here $P$ is an operator
with {\em real} H\"{o}lder continuous coefficients which in any
coordinate system $(U;x_{1},\ldots,x_{d})$ has the form
\begin{equation} \label{P} P(x,\partial_{x})=-\sum_{i,j=1}^{d}
a_{ij}(x)\partial_{i}\partial_{j} + \sum_{i=1}^{d}
b_{i}(x)\partial_{i}+c(x), \end{equation} where
$\partial_{i}=\partial/\partial x_{i}$. We assume that for each
$x\in \Omega$ the real quadratic form $\sum_{i,j=1}^{d}
a_{ij}(x)\xi_{i}\xi_{j}$ is positive definite on $\mathbb{R} ^d$.
We denote the cone of all positive (classical) solutions of the
elliptic equation $Pu=0$ in $\Omega$ by $\mathcal{C}_{P}(\Omega)$.
We fix a reference point $x_0 \in \Omega _1$. From time to time,
we consider the convex set
$$\mathcal{K}_{P}(\Omega):=\{u\in \mathcal{C}_{P}(\Omega)\,|\,u(x_0)=1\}$$
of all {\em normalized} positive solutions. In case that the
coefficients of $P$ are smooth enough, we denote by $P^*$ the
formal adjoint of $P$.
\begin{definition}{\em For a (real valued) function $V\in C^\alpha(\Omega)$, let
$$\lambda_0(P,\Omega,V)
:= \sup\{\lambda \in \mathbb{R} \mid \mathcal{C}_{P-\lambda V}(
\Omega)\neq \emptyset\}$$ be the {\em generalized principal
eigenvalue} of the operator $P$ with respect to the (indefinite)
weight $V$ in $\Omega$. We also denote
$$\lambda_\infty(P,\Omega, V):=
\sup_{K\Subset\Omega}\lambda_0(P,\Omega\setminus K,V).$$
For a fixed $P$ and $\Omega$, and $V=\mathbf{1}$, we simply write
$\lambda_0:=\lambda_0(P,\Omega,\mathbf{1})$ and
$\lambda_\infty:=\lambda_\infty(P,\Omega,\mathbf{1})$.
}\end{definition}
\begin{definition}{\em
Let $P$ be an elliptic operator of the form (\ref{P}) which is
defined on a smooth domain $D\Subset X$. we say that the {\em
generalized maximum principle} for the operator $P$ holds in $D$
if for any $u\in C^2(D)\cap C(\overline{D})$, the inequalities
$Pu\geq 0$ in $D$ and $u\geq 0$ on $\partial D$ imply that $u\geq
0$ in $D$.
}\end{definition}
It is well known that $\lambda_0(P,\Omega,\mathbf{1})\geq 0$ if
and only if the generalized maximum principle for the operator $P$
holds true in any smooth subdomain $D\Subset\Omega$.
The following theorem is known as the {\em Allegretto-Piepenbrink
theory}, it relates $\lambda_0$ and $\lambda_\infty$, in the
symmetric case, with fundamental spectral quantities (see for
example \cite{Agmon82,Cycon,S82} and the references therein).
\begin{theorem}\label{APthm} Suppose that $P$ is symmetric on
$C_0^\infty(\Omega)$, and that $\lambda_0>-\infty$. Then
$\lambda_0$ (resp. $\lambda_\infty$) equals to the infimum of the
spectrum (resp. essential spectrum) of the Friedrich's extension
of $P$.
\end{theorem}
Therefore, in the selfadjoint case, $\lambda_0$ can be
characterized via the classical Rayleigh-Ritz variational formula.
In the general case, a variational principle for $\lambda_0$ is
given by the Donsker-Varadhan variational formula (which is a
generalization of the Rayleigh-Ritz formula) and by some other
variational formulas (see for example \cite{NP,Pins95}).
\begin{Def} \label{defminimalg}
{\rm Let $P$ be an elliptic operator defined in a domain $\Omega
\subseteq X$. A function $u$ is said to be a {\em positive
solution of the operator $P$ of minimal growth in a neighborhood
of infinity in} $\Omega$ if $u\in \mathcal{C}_P(\Omega _j^*)$ for
some $j\geq 1$, and for any $l> j$, and $v\in C(\Omega_l^*\cup
\partial \Omega _l)\cap \mathcal{C}_P(\Omega _l^*)$, if $u\le v$
on $\partial \Omega _l$, then $u\le v$ on $\Omega _l^*$.}
\end{Def}
\begin{theorem}[\cite{Agmon82}]\label{thmmingr2}
Suppose that $\mathcal{C}_{P}(\Omega)\neq \emptyset$. Then for any
$x_0\in \Omega$ the equation $Pu=0$ has (up to a multiple
constant) a unique positive solution $v$ in
$\Omega\setminus\{x_0\}$ of minimal growth in a neighborhood of
infinity in $\Omega$.
\end{theorem}
By the well known theorem on the removability of isolated
singularity \cite{GiS}, we have:
\begin{Def} {\em
Suppose that $\mathcal{C}_{P}(\Omega)\neq \emptyset$. If the
solution $v$ of Theorem~\ref{thmmingr2} has a nonremovable
singularity at $x_0$, then $P$ is said to be a {\em subcritical
operator} in $\Omega$. If $v$ can be (uniquely) continued to a
positive solution $\tilde{v}$ of the equation $Pu=0$ in $\Omega$,
then $P$ is said to be a {\em critical operator} in $\Omega$, and
the positive global solution $\tilde{v}$ is called a {\em
ground state} of the equation $Pu=0$ in $\Omega$. The operator $P$
is said to be {\em supercritical} in $\Omega$ if
$\mathcal{C}_{P}(\Omega)=\emptyset$.
} \end{Def}
\begin{remarks}{\em
1. In \cite{S80}, B.~Simon coined the terms
`(sub)-(super)-critical operators' for Schr\"odinger operators
with short-range potentials which are defined on $\mathbb{R}^d$,
where $d\geq 3$. The definition given in \cite{S80} is in terms of
the exact (and particular) large time behavior of the heat kernel
of such operators (see \cite[p.~71]{S81} for the root of this
terminology). In \cite{M86}, M.~Murata generalized the above
classification for Schr\"odinger operators which are defined in
any subdomain of $\mathbb{R}^d$, $d\geq 1$. The definition of
subcriticality given here is due to \cite{P88}.
2. The notions of minimal growth and ground state were introduced
by S.~Agmon in \cite{Agmon82}.
3. For modified and stronger notions of subcriticality see
\cite{DS91,P88}.
}\end{remarks}
\noindent {\em Outline of the proof of Theorem~\ref{thmmingr2}.}
Assume that $\mathcal{C}_{P}(\Omega)\neq \emptyset$ and fix
$x_0\in \Omega$. Then for every $j\geq 1$, the {\em Dirichlet
Green function} $\Green{\Omega_{j}}{P}{x}{y}$ for the operator $P$
exists in $\Omega_j$. It is the integral kernel such that for any
$f\inC_0^{\infty}(\Omega)$, the function $u_j(x):=\int_{\Omega_j}
G_P^{\Omega_j}(x,y)f(y)\, \mathrm{d}y$ solves the Dirichlet
boundary value problem
$$
Pu=f \quad \mbox{ in } \Omega_j,\qquad
u=0 \quad \mbox{ on } \partial \Omega_j.
$$
It follows that $\Green{\Omega_{j}}{P}{\cdot}{x_0}\in
\mathcal{C}_{P}(\Omega_j\setminus\{x_0\})$. By the generalized
maximum principle,
$\{\Green{\Omega_{j}}{P}{x}{x_0}\}_{j=1}^{\infty}$ is an
increasing sequence which, by the Harnack inequality, converges
uniformly in any compact subdomain of $\Omega\setminus\{x_0\}$
either to $\Green{\Omega}{P}{x}{x_0}$, the positive {\em minimal
Green function} of $P$ in $\Omega$ with a pole at $x_0$ (and in
this case $P$ is subcritical in $\Omega$) or to infinity.
In the latter case, fix $x_1\in \Omega$, such that $x_1\neq x_0$.
It follows that the sequence
$\Green{\Omega_j}{P}{\cdot}{x_0}/\Green{\Omega_j}{P}{x_1}{x_0}$
converges uniformly in any compact subdomain of
$\Omega\setminus\{x_0\}$ to a ground state of the equation $Pu=0$
in $\Omega$, and in this case $P$ is critical in $\Omega$.\qed
\begin{corollary} \label{cor27}
(i) If $P$ is subcritical in $\Omega$, then for each $y\in \Omega$ the
Green function $\Green{\Omega}{P}{\cdot}{y}$ with a pole at $y$
exists, and is a positive solution of the equation $Pu=0$ of
minimal growth in a neighborhood of infinity in $\Omega$.
Moreover, $P$ is subcritical in $\Omega$ if and only if the
equation $Pu=0$ in $\Omega$ admits a positive supersolution which
is not a solution.
(ii) The operator $P$ is critical in $\Omega$ if and only if the
equation $Pu=0$ in $\Omega$ admits (up to a multiplicative
constant) a unique positive supersolution. In particular, $\dim
\mathcal{C}_{P}(\Omega)=1$.
(iii) Suppose that $P$ is symmetric on $C_0^\infty(\Omega)$ with
respect to a smooth positive density $V$, and let $\tilde{P}$ be
the (Dirichlet) selfadjoint realization of $P$ on
$L^2(\Omega,V(x)dx)$. Assume that
$\lambda\in\sigma_\mathrm{point}(\tilde{P})$ admits a nonnegative
eigenfunction $\varphi$, then $\lambda\!=\!\lambda_0$ and
$P\!-\!\lambda_0 V$ is critical in $\Omega$ (see for example
\cite{M86}).
(iv) The operator $P$ is critical (resp. subcritical) in
$\Omega$ if and only if $P^*$ is critical (resp. subcritical) in
$\Omega$.
\end{corollary}
As was mentioned, (sub)criticality is related to the large time
behavior of the heat kernel. Indeed, (sub)criticality can be also
defined in terms of the corresponding parabolic equation. Suppose
that $\lambda_0\geq 0$. For every $j\geq 1$, consider the
Dirichlet heat kernel $k_P^{\Omega_j}(x,y,t)$ of the parabolic
operator $L:=\partial_t+P$ on $\Omega_j\times (0,\infty)$. So, for
any $f\inC_0^{\infty}(\Omega)$, the function $u_j(x,t)=\int_{\Omega_j}
k_P^{\Omega_j}(x,y,t)f(y)\, \mathrm{d}y$ solves the
initial-Dirichlet boundary value problem
$$
Lu=0 \; \mbox{ in } \Omega_j\times (0,\infty),\quad
u=0 \; \mbox{ on } \partial \Omega_j\times (0,\infty),\quad
u=f \; \mbox{ on } \Omega_j\times \{0\}.
$$
By the (parabolic) generalized maximum principle, $\{k_P^{\Omega_j}(x,y,t)\}_{j=1}^{\infty}$ is
an increasing sequence which converges to $k_P^{\Omega}(x,y,t)$,
the {\em minimal heat kernel} of the parabolic operator $L$ in
$\Omega$.
\begin{lemma}\label{lemheatcrit}
Suppose that $\lambda_0\geq 0$. Let $x,y\in \Omega$, $x\neq y$.
Then
$$\int_0^\infty k_P^{\Omega}(x,y,t)\,\mathrm{d}t<\infty \qquad
\mbox{(resp. $\int_0^\infty
k_P^{\Omega}(x,y,t)\,\mathrm{d}t=\infty$),}$$ if and only if $P$
is a subcritical (resp. critical) operator in $\Omega$. Moreover,
if $P$ is subcritical operator in $\Omega$, then
\begin{equation}\label{heatgreen}G_P^{\Omega}(x,y)=\int_0^\infty
k_P^{\Omega}(x,y,t)\,\mathrm{d}t.
\end{equation}
\end{lemma}
For the proof of Lemma~\ref{lemheatcrit} see for example
\cite{Pins95}. Note that if $\lambda<\lambda_0$, then the operator
$P-\lambda$ is subcritical in $\Omega$, and that for $\lambda\leq
\lambda_0$, the heat kernel $k_{P-\lambda}^\Omega(x,y,t)$ of the
operator $P-\lambda$ is equal to $e^{\lambda t}k_P^\Omega(x,y,t)$.
\vskip 3mm
Subcriticality (criticality) can be defined also through a
probabilistic approach. If the zero-order coefficient $c$ of the
operator $P$ is equal to zero in $\Omega$, then $P$ is called a
{\em diffusion operator}. In this case, $P\mathbf{1}=0$, and
therefore, $P$ is not supercritical in $\Omega$. Moreover, for
such an operator $P$, one can associate a diffusion process
corresponding to a solution of the generalized martingale problem
for $P$ in $\Omega$. This diffusion process is either {\em
transient} or {\em recurrent} in $\Omega$. It turns out that a
diffusion operator $P$ is subcritical in $\Omega$ if and only if
the associated diffusion process is transient in $\Omega$ (for
more details see \cite{Pins95}). A Riemannian manifold $X$ is
called {\em parabolic} (resp. {\em non-parabolic}) if the Brwonian
motion, the diffusion process with respect to the Laplace-Beltrami
operator on $X$, is recurrent (resp. transient) \cite{G99}.
Suppose now that $P$ is of the form \eqref{P}, and $P$ is not
supercritical in $\Omega$. Let $\varphi\in
\mathcal{C}_{P}(\Omega)$. Then the operator $P^\varphi$ acting on
functions $u$ by
$$P^\varphi u:=\frac{1}{\varphi}P(\varphi u)$$
is a diffusion operator, and $$
k_{P^\varphi}^M(x,y,t)=\frac{1}{\varphi(x)}k_{P}^M(x,y,t)\varphi(y).$$
Therefore, $P$ is subcritical in $\Omega$ if and only if
$P^\varphi$ is transient in $\Omega$.
\vskip 3mm
We have the following general convexity results.
\begin{theorem}[\cite{P90}]\label{thmconv}
(i) Let $V\in C^\alpha} \def\gb{\beta} \def\gg{\gamma(\Omega)$, $V\neq 0$ and set
\begin{eqnarray} S_+ & = &
S_+(P,\Omega,V) =
\{\lambda\in \mathbb{R}\;|\, P-\lambda V \mbox{ is subcritical in } \Omega\},\\
S_0 & = & S_0(P,\Omega,V) = \{\lambda\in \mathbb{R}\;|\,
P-\lambda V \mbox{ is critical in } \Omega\}.
\end{eqnarray}
Then $S:=S_+\cup S_0\subseteq
\mathbb{R}$ is a closed interval and $S_0\subset \partial S$.
Moreover, if $S\neq \emptyset$, then $S$ is bounded if and only if
$V$ changes its sign in $\Omega$. \vskip 2mm
(ii) Let $W, V\in C^\alpha} \def\gb{\beta} \def\gg{\gamma(\Omega)$, then the function
$\lambda_0(\mu):=\lambda_0(P-\mu W,\Omega,V)$ is a concave
function on the interval $\{\mu\in \mathbb{R}\mid
|\lambda_0(\mu)|<\infty\}$.
\end{theorem}
\begin{proof} For $0\leq s\leq 1$, and $V_0,V_1\in C^\alpha} \def\gb{\beta} \def\gg{\gamma(\Omega)$,
let $$P_s:= P+sV_1+(1-s)V_0.$$ Assume that $u_j$ are positive
supersolutions of the equations $P_j u\geq 0$ in $\Omega$, where
$j=0,1$. It can be verified that for $0< s<1$, the function
$$u_s(x):=\left[u_0(x)\right]^{1-s}\left[u_1(x)\right]^s$$
is a positive supersolution of the equation $P_su=0$ in $\Omega$.
Moreover, for any $0< s<1$, $u_s\in \mathcal{C}_{P_s}(\Omega)$ if
and only if $V_0=V_1$, and $u_0,u_1\in \mathcal{C}_{P_0}(\Omega)$
are linearly dependent. The lemma follows easily from this
observation.
\end{proof}
\begin{corollary}[\cite{P90} and \cite{S82}]\label{cor3G}
Suppose that $P_s:= P+sV_1+(1-s)V_0$ is subcritical in $\Omega$
for $s=0,1$. Then for $0\leq s\leq 1$ we have
\begin{equation}\label{eq3G}
G_{P_s}^\Omega(x,y)\leq
\left[G_{P_0}^\Omega(x,y)\right]^{1-s}\left[G_{P_1}^\Omega(x,y)\right]^{s}.
\end{equation}
\end{corollary}
\begin{remark}{\em
The dependence of $\lambda_0$ on the higher order coefficients of
$P$ is more involved. In \cite{BNV} it was proved that in the
class of uniformly elliptic operators with bounded coefficients
which are defined on a bounded domain in $\mathbb{R}^d$,
$\lambda_0$ is locally Lipschitz continuous as a function of the
first-order coefficients of the operator $P$. A.~Ancona
\cite{An97} proved that under some assumptions, $\lambda_0$ is
Lipschitz continuous with respect to a metric
$\mathrm{dist}(P_1,P_2)$ measuring the distance between two
elliptic operators $P_1$ and $P_2$ in a certain class. Ancona's
metric depends on the difference between \emph{all} the
coefficients of the operators $P_1$ and $P_2$. }
\end{remark}
\vskip 3mm
If $P$ is subcritical in $\Omega$, then $\mathcal{C}_{P}(\Omega)$
is in general not a one-dimensional cone. Nevertheless, one can
construct the {\em Martin compactification} $\Omega_P^M$ of
$\Omega$ with respect to the operator $P$ (with a base point
$x_0$), and obtain an integral representation of any solution in
$\mathcal{C}_{P}(\Omega)$. More precisely, the {\em Martin
compactification} is the compactification of $\Omega$ such that
the function
$$K_P^\Omega(x,y):=\frac{G_P^\Omega(x,y)}{G_P^\Omega(x_0,y)}
\qquad \mbox{ on } \Omega\times \Omega \setminus\{(x_0,x_0)\}$$
has a continuous extension $K_P^\Omega(x,\eta)$
to
$\Omega\times (\Omega_P^M\setminus \{x_0\})$, and such that the
set of functions $\{K_P^\Omega(\cdot,\eta)\}_{\eta\in
\Omega_P^M}$ separates the points of $\Omega_P^M$.
The boundary of $\Omega_P^M$ is
denoted by $\partial_P^M \Omega$ and is called the {\em Martin
boundary} of $\Omega$ with respect to the operator $P$. For each
$\xi \in
\partial _P^M\Omega$, the function
$K_P^\Omega(\cdot,\xi)$ is called the {\em Martin function of the
pair $(P,\Omega)$ with a pole at $\xi$}. Note that for $\xi \in
\partial _P^M\Omega$, we have
$K_P^\Omega(\cdot,\xi)\in \mathcal{K}_{P}(\Omega)$. The set
$\partial _{m,P}^M\Omega$ of all $\xi \in
\partial _P^M\Omega$ such that $K_P^\Omega(\cdot,\xi)$ is an extreme point of
the convex set $\mathcal{K}_{P}(\Omega)$ is called the {\em
minimal Martin boundary} (for more details see \cite[and the
references therein]{M86,M98,Pins95,Taylor}).
The Martin representation theorem asserts that for any $u\in
\mathcal{K}_{P}(\Omega)$ there exists a unique probability measure
$\mu$ on $\partial_P^M \Omega$ which is supported on $\partial
_{m,P}^M\Omega$ such that
$$u(x)=\int_{\partial_P^M \Omega}K_P^\Omega(x,\xi)\,\mathrm{d}\mu(\xi).$$
There has been a great deal of work on explicit description of the Martin
compactification and representation in many concrete examples (see
for example \cite[and the references
therein]{LP,M98,MT,Pins95,Taylor}).
We present below two elementary examples of Martin
compactifications. In Section~\ref{secperiod} we discuss a recent
result on the Martin compactification of a general periodic
operator on $\mathbb{R}^d$.
\begin{example}\label{exsmoothbdd} {\em
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^d$, and
assume that the coefficients of $P$ are (up to the boundary)
smooth. Then $\partial_P^M \Omega$ is homeomorphic to $\partial
\Omega$, the euclidian boundary of $\Omega$, and for any $y\in
\partial \Omega$,
\begin{equation}\label{eqmartincalss}
K_P^\Omega(x,y):=\frac{\partial_\nu
G_P^\Omega (x,y)} {\partial_\nu G_P^\Omega(x_0,y)}\,,
\end{equation}
where $\partial_\nu$ denotes the inner normal derivative with
respect to the second variable. Note that $\partial_\nu
G_P^\Omega(\cdot,y)$ is the Poisson kernel at $y\in
\partial \Omega$. }
\end{example}
\begin{example}\label{exlaplace} {\em
Consider the equation $H_\lambda u:=(-\Delta+\lambda)u=0$ in
$\mathbb{R}^d$. Then $\mathcal{C}_{H_\lambda}(\mathbb{R}^d)\neq
\emptyset$ if and only if $\lambda\geq 0$. It is well known that
$H_0=-\Delta$ is critical on in $\mathbb{R}^d$ if and only if
$d\leq 2$. Moreover,
$$G_{H_\lambda}^{\mathbb{R}^d}(x,y)=
\begin{cases}
\dfrac{\Gamma(\nu)|x-y|^{2-d}}{4\pi^{d/2}} & \lambda=0, \text{ and } d\geq 3, \\[3mm]
(2\pi)^{-d/2}\left(\dfrac{\sqrt{\lambda}}{|x-y|}\right)^\nu K_\nu(\sqrt{\lambda}|x-y|) & \lambda>0,
\end{cases}$$
where $\nu=(d-2)/2$, and $K_\nu$ is the modified Bessel function
of order $\nu$.
Clearly,
$$\lim_{|y|\to \infty}
\dfrac{G_{-\Delta}^{\mathbb{R}^d}(x,y)}{G_{-\Delta}^{\mathbb{R}^d}(0,y)}=1.$$
Therefore, the Martin compactification of $\mathbb{R}^d$ with
respect to the Laplacian is the one-point compactification of
$\mathbb{R}^d$, and we obtained the positive Liouville theorem:
$\mathcal{K}_{-\Delta}(\mathbb{R}^d)=\{\mathbf{1}\}$.
Suppose now that $\lambda>0$. Then for any $\xi\in S^{d-1}$,
$$\lim_{\frac{y}{|y|}\to \xi,\; |y|\to \infty}\;
\dfrac{G_{H_\lambda}^{\mathbb{R}^d}(x,y)}{G_{H_\lambda}^{\mathbb{R}^d}(0,y)}=e^{\sqrt{\lambda}\,\xi\cdot
x},$$ and therefore, the Martin boundary of $\mathbb{R}^d$ with
respect to $H_\lambda$ is the sphere at infinity.
Clearly, all
Martin functions are minimal. Furthermore, $u\in
\mathcal{C}_{H_\lambda}(\mathbb{R}^d)$ if and only if there exists
a positive finite measure $\mu} \def\gn{\nu} \def\gp{\pi$ on $S^{d-1}$ such that
\[
u(x)=\int_{S^{d-1}} e^{\sqrt{\lambda}\,\xi\cdot
x}\,\mathrm{d}\mu} \def\gn{\nu} \def\gp{\pi(\xi ).
\]
}
\end{example}
\begin{remark}{\em We would like to point out that criticality
theory and Martin boundary theory are also valid for the class of
weak solutions of elliptic equations in divergence form as well as
for the class of strong solutions of strongly elliptic equations
with locally bounded coefficients. For the sake of clarity, we
prefer to concentrate on the class of classical solutions.}
\end{remark}
\mysection{Perturbations}\label{secpert}
An operator $P$ is critical in $\Omega$ if and only if any
positive supersolution of the equation $Pu=0$ in $\Omega$ is a
solution (Corollary \ref{cor27}). Therefore, if $P$ is critical in
$\Omega$ and $V\in C^\alpha} \def\gb{\beta} \def\gg{\gamma(\Omega)$ is a nonzero, nonnegative
function, then for any $\lambda>0$ the operator $P+\lambda V$ is
subcritical and $P-\lambda V$ is supercritical in $\Omega$. On the
other hand, it can be shown that subcriticality is a stable
property in the following sense: if $P$ is subcritical in $\Omega$
and $V\in C^\alpha} \def\gb{\beta} \def\gg{\gamma(\Omega)$ has a {\em compact support}, then there
exists $\epsilon>0$ such that $P-\lambda V$ is subcritical for all
$|\lambda|< \ge$, and the Martin compactifications $\Omega_{P}^M$
and $\Omega_{P-\lambda V}^M$ are homeomorphic for all $|\lambda|<
\ge$ (for a more general result see Theorem~\ref{thmssp}).
Therefore, a perturbation by a compactly supported potential (at
least with a definite sign) is well understood.
In this section, we introduce and study a few general notions of
perturbations related to positive solutions of an operator $P$ of
the form \eqref{P} by a (real valued) potential $V$. In
particular, we discuss the behavior of the generalized principal
eigenvalue, (sub)criticality, the Green function, and the Martin
boundary under such perturbations. Further aspects of perturbation
theory will be discussed in the following sections.
One facet of this study is the equivalence (or comparability) of
the corresponding Green functions.
\begin{Def} \label{equiGdef}
{\em Let $P_{j},\;j=1,2$, be two subcritical operators in
$\Omega$. We say that the Green functions $G^\Omega_{P_{1}}$ and
$G^\Omega_{P_{2}}$ are {\em equivalent} (resp. {\em
semi-equivalent}) if $G_{P_1}^\Omega \asymp G_{P_2}^\Omega$ on
$\Omega \times \Omega \setminus \{(x,x)\,|\, x \in \Omega\}$
(resp. $G_{P_1}^\Omega (\cdot,y_0)\asymp
G_{P_2}^\Omega(\cdot,y_0)$ on $\Omega \setminus \{ y_0\}$ for
some fixed $y_0\in \Omega$).
}\end{Def}
\begin{lemma}[\cite{P88}]\label{lemgreeneq} Suppose that
the Green functions $G^\Omega_{P_{1}}$ and $G^\Omega_{P_{2}}$ are
equivalent. Then there exists a homeomorphism $\Phi:\partial
_{m,P_1}^M\Omega\to
\partial _{m,P_2}^M\Omega$ such that for each minimal point $\xi \in
\partial _{m,P_1}^M\Omega$, we have $K_{P_1}^\Omega(\cdot,\xi)
\asymp K_{P_2}^\Omega(\cdot,\Phi(\xi))$ on $\Omega$. Moreover, the
cones $\mathcal{C}_{P_1}(\Omega)$ and $\mathcal{C}_{P_2}(\Omega)$
are homeomorphic.
\end{lemma}
\begin{remarks}{\em
1. It is not known whether the equivalence of $G^\Omega_{P_{1}}$
and $G^\Omega_{P_{2}}$ implies that the cones
$\mathcal{C}_{P_1}(\Omega)$ and $\mathcal{C}_{P_2}(\Omega)$ are
{\em affine} homeomorphic.
2. Many papers deal with sufficient conditions, in terms of
proximity near infinity in $\Omega$ between two given subcritical
operators $P_1$ and $P_2$, which imply that $G^\Omega_{P_{1}}$ and
$G^\Omega_{P_{2}}$ are equivalent, or even that the cones
$\mathcal{C}_{P_1}(\Omega)$ and $\mathcal{C}_{P_2}(\Omega)$ are
affine homeomorphic, see Theorem~\ref{thmssp} and \cite[and the
references therein]{Ai,An97,M86,M97,P88,P89,Sem}.
}\end{remarks}
We use the notation
$$E_+=E_+(V,P,\Omega):=\left\{\lambda\in\mathbb{R}\,|\,G^{\Omega}_{P-\lambda V} \mbox{ and }
G^{\Omega}_{P} \mbox{ are equivalent } \right \},$$
$$sE_+=sE_+ (V,P,\Omega):=\left\{\lambda\in\mathbb{R}\,|\,G^{\Omega}_{P-\lambda V}
\mbox{ and } G^{\Omega}_{P} \mbox{ are semi-equivalent }\right
\}.$$
The following notion was
introduced in \cite{P89} and is closely related to the stability
of $\mathcal{C}_P(\Omega)$ under perturbation by a potential $V$.
\begin{Def} \label{spertdef}
{\em Let $P$ be a subcritical operator in $\Omega$, and let
$V\in C^{\alpha}(\Omega)$. We say that $V$ is a {\em small
perturbation}
of $P$ in $\Omega$ if
\begin{equation} \label{sperteq} \lim_{j\rightarrow \infty}\left\{\sup_{x,y\in
\Omega_{j}^*}
\int_{\Omega_{j}^*}\frac{\Green{\Omega}{P}{x}{z}|V(z)|
\Green{\Omega}{P}{z}{y}}{\Green{\Omega}{P}{x}{y}}\,\mathrm{d}z\right\}=0.
\end{equation} }
\end{Def}
\noindent The following notions of perturbations were introduced
by M.~Murata \cite{M97}.
\begin{Def} \label{semispertdef}
{\em Let $P$ be a subcritical operator in $\Omega$, and let $V\in
C^{\alpha}(\Omega)$.
{\em (i)} We say that $V$ is a {\em semismall perturbation} of
$P$ in $\Omega$ if \begin{equation} \label{semisperteq} \lim_{j\rightarrow
\infty}\left\{\sup_{y\in \Omega_{j}^*} \int_{\Omega_{j}^*}
\frac{\Green{\Omega}{P}{x_0}{z}|V(z)|\Green{\Omega}{P}{z}{y}}
{\Green{\Omega}{P}{x_0}{y}}\,\mathrm{d}z\right\}=0. \end{equation}
{\em (ii)} We say that $V$ is a {\em $G$-bounded perturbation}
(resp. {\em $G$-semibounded perturbation}) of $P$ in $\Omega$ if
there exists a positive constant $C$ such that \begin{equation} \label{bperteq}
\int_{\Omega}\frac{\Green{\Omega}{P}{x}{z}|V(z)|\Green{\Omega}{P}{z}{y}}
{\Green{\Omega}{P}{x}{y}}\,\mathrm{d}z\leq C \end{equation} for all $x,y \in \Omega$
(resp. for some fixed $x\in \Omega$ and all $y \in \Omega
\setminus \{x\}$).
{\em (iii)} We say that $V$ is an {\em $H$-bounded perturbation}
(resp. {\em $H$-semibounded perturbation}) of $P$ in $\Omega$ if
there exists a positive constant $C$ such that \begin{equation}
\label{Hbperteq}
\int_{\Omega}\frac{\Green{\Omega}{P}{x}{z}|V(z)|u(z)}
{u(x)}\,\mathrm{d}z\leq C \end{equation} for all $x\in \Omega$ (resp. for some fixed
$x\in \Omega$) and all $u \in \mathcal{C}_P(\Omega)$.
{\em (iv)} We say that $V$ is an {\em $H$-integrable
perturbation} of $P$ in $\Omega$ if \begin{equation} \label{Iperteq}
\int_{\Omega}\Green{\Omega}{P}{x}{z}|V(z)|u(z)\,\mathrm{d}z < \infty \end{equation} for
all $x\in \Omega$ and all $u \in \mathcal{C}_P(\Omega)$.
}\end{Def}
\begin{theorem}[\cite{M97,P89,P90}]\label{thmssp}
Suppose that $P$ is subcritical in $\Omega$.
Assume that $V$ is a small (resp. semismall) perturbation of
$P^*$ in $\Omega$. Then $E_+=S_+$ (resp. $sE_+=S_+$), and
$\partial S=S_0$. In particular, $S_+$ is an open interval.
Suppose that $V$ is a semismall perturbation of $P^*$ in $\Omega$,
and $\gl\in S_0$. Let $\varphi_0$ be the corresponding ground
state. Then $\varphi_0\asymp \Green{\Omega}{P}{\cdot}{x_0}$ in
$\Omega_1^*$.
Suppose that $V$ is a semismall perturbation of $P^*$ in $\Omega$,
and $\lambda\in S_+$. Then the mapping \begin{equation} \label{sperteq7}
\Psi(u):=u(x)+\lambda\int_{\Omega}\Green{\Omega}{P-\lambda V
}{x}{z}V(z) u(z)\,\mathrm{d}z \end{equation} is an affine homeomorphism of
$\mathcal{C}_{P}(\Omega)$ onto $\mathcal{C}_{P-\lambda
V}(\Omega)$, which induces a homeomorphism between the
corresponding Martin boundaries. Moreover, in the small
perturbation case, we have $\Psi(u)\asymp u$ in $\Omega$ for all
$u\in \mathcal{C}_{P}(\Omega)$.
\end{theorem}
\begin{Rems}\label{remspert}{\em
1. Small perturbations are semismall \cite{M97}, $G$-(resp. $H$-)
bounded perturbations are $G$- (resp. $H$-) semibounded, and
$H$-semibounded perturbations are $H$-integrable. On the other
hand, if $V$ is $H$-integrable and
$\dim\mathcal{C}_P(\Omega)<\infty$, then $V$ is $H$-semibounded
\cite{M97,P88}.
There are potentials which are $H$-semibounded perturbations but
are neither $H$-bounded nor $G$-semibounded. We do not know of any
example of a semismall (resp. $G$-semibounded) perturbation which
is not a small (resp. $G$-bounded) perturbation. We are also not
aware of any example of a $H$-bounded (resp. $H$-integrable)
perturbation which is not $G$-bounded (resp. $H$-semibounded)
\cite{P99}.
2. Any small (resp. semismall) perturbation is $G$-bounded
(resp. $G$-semibounded), and any $G$-(resp. semi) bounded
perturbation is $H$-(resp. semi) bounded perturbation.
3. If $V$ is a $G$-bounded (resp. $G$-semibounded) perturbation of
$P$ (resp. $P^*$) in $\Omega$, then $G^{\Omega}_{P}$ and
$G^{\Omega}_{P-\lambda V}$ are equivalent (resp. semi-equivalent)
provided that $|\lambda |$ is small enough \cite{M97, P88, P89}.
On the other hand, if $G^{\Omega}_{P}$ and $G^{\Omega}_{P+V}$ are
equivalent (resp. semi-equivalent) and $V$ {\em has a definite
sign}, then $V$ is a $G$-bounded (resp. $G$-semibounded)
perturbation of $P$ (resp. $P^*$) in $\Omega$. In this case, by
\eqref{eq3G}, the set $E_+$ (resp. $sE_+$) is an open half line
which is contained in $S_+$ \cite[Corollary 3.6]{P90}. There are
sign-definite $G$-bounded (resp. $G$-semibounded) perturbations
such that $E_+\subsetneqq S_+$ (resp. $sE_+\subsetneqq S_+$)
\cite[Example~8.6]{P99}, \cite[Theorem~6.5]{M98}.
Note that, if $V$ is a $G$-(resp. semi-) bounded perturbation of
$P$ (resp. $P^*$) in $\Omega$ and $\Theta\in C^{\alpha} \def\gb{\beta} \def\gg{\gamma}(\Omega)$ is
any function which vanishes at infinity of $\Omega$, then clearly
the function $\Theta(x)V(x)$ is a (resp. semi-) small perturbation
of the operator $P$ (resp. $P^*$) in $\Omega$.
4. Suppose that $G^{\Omega}_{P}$ and $ G^{\Omega}_{P-|V|}$ are
equivalent (resp. semi-equivalent). Using the resolvent equation
it follows that the best equivalence (resp. semi-equivalence)
constants of $G^{\Omega}_{P}$ and $ G^{\Omega}_{P\pm |V^*_j|}$
tend to $1$ as $j \to \infty$ if and only if $V$ is a (resp.
semi-) small perturbation of $P$ (resp. $P^*$) in $\Omega$.
Therefore, zero-order perturbations of the type studied by
A.~Ancona in \cite{An97} provide us with a huge and almost optimal
class of examples of small perturbations. (see also \cite[and the
references therein]{Ai,M86,M97,P89}).
}\end{Rems}
A.~Grigor'yan and W.~Hansen \cite{GH}
have introduced the following notions of perturbations.
\begin{Def}\label{defhbig}{\em Let $P$ be a subcritical operator in
$\Omega$, and fix $h\in \mathcal{C}_P(\Omega)$. A nonnegative
function $V$ is called {\em $h$-big on $\Omega$} if any solution
$v$ of the equation $(P+V)v=0$ in $\Omega$ satisfying $0\leq v\leq
h$ is identically zero. $V$ is {\em non-$h$-big on $\Omega$} if
$V$ is not $h$-big on $\Omega$.
}\end{Def}
\begin{remark}\label{remhbig}{\em If $V$ is $H$-integrable
perturbation of $P$, then it is non-$h$-big for any $h\in
\mathcal{C}_P(\Omega)$ (see Proposition~\ref{propinteg}).
}\end{remark}
The following notion of perturbation does not involve
Green functions.
\begin{Def} \label{weakpert}
{\rm Let $P$ be a subcritical operator in $\Omega\subseteq X$. A
function $V \in C^{\alpha} \def\gb{\beta} \def\gg{\gamma}(\Omega)$ is said to be {\em a weak
perturbation} of the operator $P$ in $\Omega$ if the following
condition holds true.
\begin{description}
\item[($\ast$)] For every $\lambda \in \mathbb{R}$ there exists $N \in {\mathbb N}$ such
that the operator $P-\lambda V^*_n(x)$ is subcritical in $\Omega$
for any $n \geq N$.
\end{description}
A function $V \in C^{\alpha} \def\gb{\beta} \def\gg{\gamma}(\Omega)$ is said to be {\em a weak
perturbation} of a critical operator $P$ in $\Omega$ if there
exists a nonzero, nonnegative function $W \in C^{\alpha} \def\gb{\beta} \def\gg{\gamma}_0(\Omega)$
such that the function $V$ is a weak perturbation of the
subcritical operator $P+W$ in $\Omega$. }\end{Def}
\begin{Rems}{\em
1. If $V$ is a weak perturbation of $P$ in $\Omega$, then
$\partial S=S_0$ and $\gl_\infty(P,\Omega,\pm V)=\infty$
(\cite{P98}, see also Theorem \ref{extthm}).
2. If $V$ is a semismall perturbation of $P$ in $\Omega$, then
$|V|$ is a weak perturbation of $P$ in $\Omega$, but $G$-bounded
perturbations are not necessarily weak.
3. Let $d\geq 3$. By the Cwikel-Lieb-Rozenblum estimate, if $V\in
L^{d/2}(\mathbb{R} ^d)$, then $|V|$ is a weak perturbation of
$-\Gd$ in $\mathbb{R} ^d$. On the other hand, $(1+|x|)^{-2}$ is
not a weak perturbation of $-\Gd$ in $\mathbb{R} ^d$, while for
any $\varepsilon
>0$ the function $(1+|x|)^{-(2+\varepsilon)}$ is a small perturbation
of $-\Gd$ in $\mathbb{R} ^d, d\geq 3$ \cite{M86,P88}. }\end{Rems}
\mysection{Indefinite weight}\label{secindef}
Consider the Schr\"{o}dinger operator $H_{\lambda}:= -\Delta
-\lambda W$ in $\mathbb{R} ^d$, where $\lambda \in \mathbb{R} $ is
a spectral parameter and $W\in C_0^{\infty}(\mathbb{R} ^d),
W\not\equiv 0$. Since $-\Delta$ is subcritical in $\mathbb{R} ^d$
if and only if $d \geq 3$, it follows that for $d \geq 3$ the
Schr\"{o}dinger operator $H_{\lambda}$ has no bound states
provided that $|\lambda|$ is sufficiently small. On the other
hand, for $d=1,2,$ B.~Simon proved the following sharp result.
\begin{Thm}[\cite{S76}] \label{Thmsimon}
Suppose that $d =1,2$, and let $W\in C_0^{\infty}(\mathbb{R} ^d),
W\not\equiv 0$. Then $H_{\lambda}=-\Delta -\lambda W$ has a
negative eigenvalue for all negative $\lambda$ if and only if
$\int_{\mathbb{R} ^d} W(x) dx \leq 0.$
\end{Thm}
The following result extends Theorem~\ref{Thmsimon} to the case
of a weak perturbation of a general critical operator in $\Omega$.
\begin{Thm}[\cite{P98}] \label{mainthmindef}
Let $P$ be a critical operator in $\Omega$, and $W \in
C^{\alpha} \def\gb{\beta} \def\gg{\gamma}(\Omega)$ a weak perturbation of the operator $P$ in
$\Omega$. Denote by $\varphi _{0}$ (resp. $\varphi^*_{0}$) the
ground state of the operator $P$ (resp. $P^*$) in $\Omega$ such
that $\varphi _{0}(x_0)=1$ (resp. $\varphi^*_{0}(x_0)=1$). Assume
that $W\varphi_0 \varphi^*_0\in L^1(\Omega)$.
(i) If there exists $\lambda<0$ such that $P-\lambda W(x)$ is
subcritical in $\Omega$, then \begin{equation} \label{integralcon}
\int_{\Omega}W(x)\varphi_0(x) \varphi^*_0(x)\,\mathrm{d}x > 0. \end{equation}
(ii) Assume that for some nonnegative, nonzero function $V \in
C^\alpha} \def\gb{\beta} \def\gg{\gamma _0(\Omega)$ there exists $\tilde{\lambda}<0$ and a positive
constant $C$ such that \begin{equation} \label{cond} \Green{\Omega}{P+V-\lambda
W}{x}{x_0}\leq C\varphi_0(x) \quad \mbox{and} \quad
\Green{\Omega}{P+V-\lambda W}{x_0}{x}\leq C\varphi^*_0(x) \end{equation} for
all $x\in \Omega \setminus \Omega _1$ and $\tilde{\lambda} \leq
\lambda <0$. If the integral condition (\ref{integralcon}) holds
true, then there exists $\lambda<0$ such that $P-\lambda W(x)$ is
subcritical in $\Omega$.
(iii) Suppose that $W$ is a semismall perturbation of the
operators $P+V$ and $P^*+V$ in $\Omega$, where $V\gvertneqq 0$, $V
\in C^\alpha} \def\gb{\beta} \def\gg{\gamma _0(\Omega)$ . Then there exists $\lambda<0$ such that
$P-\lambda W(x)$ is subcritical in $\Omega$ if and only if
\eqref{integralcon} holds true.
\end{Thm}
\mysection{Large time behavior of the heat kernel}\label{sechetk}
As was already mentioned in Section~\ref{secpreliminaries}, the
large time behavior of the heat kernel is closely related to
criticality (see for example Lemma~\ref{lemheatcrit}). In the
present section we elaborate this relation further more.
Suppose that $\lambda_0(P,\Omega,\mathbf{1})\geq 0$. We consider
the parabolic operator $L$
\begin{equation}\label{eqL}
Lu=u_t+Pu \qquad \mbox{ on } \Omega\times (0,\infty).
\end{equation} \noindent
We denote by $\mathcal{H}_P(\Omega\times (a,b))$ the cone of all
nonnegative solutions of the equation $Lu=0$ in $\Omega\times
(a,b)$. Let $k_P^{\Omega}(x,y,t)$ be the heat kernel of the
parabolic operator $L$ in $\Omega$.
If $P$ is critical in $\Omega$, we denote by $\varphi_0$ the
ground state of $P$ in $\Omega$ satisfying $\varphi_0(x_0)=1$. The
corresponding ground state of $P^*$ is denoted by $\varphi^*_0$.
\begin{definition}{\em A critical operator $P$ is said to be {\em positive-critical}
in $\Omega$ if $\varphi_0\varphi_0^*\in L^1(\Omega)$, and {\em
null-critical} in $\Omega$ if $\varphi_0\varphi_0^*\not\in
L^1(\Omega)$. }
\end{definition}
\begin{theorem}[\cite{P92,P04}]\label{mainthmhk}
Suppose that $\lambda_0\geq 0$. Then for each $x,y\in \Omega$
\begin{equation*}\label{eqlimhk}
\lim_{t\to\infty} \mathrm{e}^{\lambda_0 t}k_P^{\Omega}(x,y,t)\!=\!
\begin{cases}
\dfrac{\varphi_0(x)\varphi_0^*(y)}{\int_\Omega\!
\varphi_0(z)\varphi_0^*(z)\,\mathrm{d}z} & \text{if }
P\!-\!\lambda_0 \text{ is positive-critical},
\\[5mm]
0 & \text{otherwise}.
\end{cases}
\end{equation*}
Moreover, we have the following Abelian-Tauberian type relation
\begin{equation}\label{eqgreen}
\lim_{t\to\infty} \mathrm{e}^{\lambda_0 t}k_P^{\Omega}(x,y,t)=
\lim_{\lambda\nearrow\lambda_0}(\lambda_0-\lambda)\Green{\Omega}{P-\lambda}{x}{y}.
\end{equation}
\end{theorem}
\begin{remark}{\em
The first part of Theorem~\ref{mainthmhk} has been proved by
I.~Chavel and L.~Karp \cite{CK} in the {\em selfadjoint} case.
Later, B. Simon gave a shorter proof for the selfadjoint case
using the spectral theorem and elliptic regularity \cite{S93}.
}\end{remark}
We next ask how fast $\lim_{t\to\infty} \mathrm{e}^{\lambda_0
t}k_P^{\Omega}(x,y,t)$ is approached. It is natural to conjecture
that the limit is approached equally fast for different points
$x,y\in \Omega$. Note that in the context of Markov chains, such
an {\em (individual) strong ratio limit property} is in general
not true \cite{Chu}. The following conjecture was raised by
E.~B.~Davies \cite{Dheat} in the selfadjoint case.
\begin{conjecture}\label{conjD}
Let $Lu=u_t+P(x, \partial_x)u$ be a parabolic operator which is
defined on $\Omega\subseteq X$. Fix a reference point $x_0\in
\Omega$. Then
\begin{equation}\label{eqconjD}
\lim_{t\to\infty}\frac{k_P^\Omega(x,y,t)}{k_P^\Omega(x_0,x_0,t)}=a(x,y)
\end{equation}
exists and is positive for all $x,y\in \Omega$.
\end{conjecture}
If Conjecture~\ref{conjD} holds true, then for any fixed $y\in
\Omega$ the limit function $a(\cdot,y)$ is a positive solution of
the equation $(P-\lambda_0)u = 0$ which is (up to a multiplicative
function) a parabolic Martin function in
$\mathcal{H}_P(\Omega\times \mathbb{R}_-)$ associated with any
Martin sequence of the form $(y, t_n)$ where $t_n\to-\infty$
(see \cite[and the references
therein]{Dheat,IWPT} for further partial results).
\mysection{Nonuniqueness of the positive Cauchy\\ problem and
intrinsic ultracontractivity }\label{secup}
In this section we discuss the uniqueness the Cauchy problem
\begin{equation}\label{eqL1}
\begin{cases}
Lu:=u_t+Pu=0 & \mbox{ on } \Omega\times (0,T), \\
u(x,0)=u_0(x) & \mbox{ on } \Omega,
\end{cases}
\end{equation} \noindent
in the class of nonnegative continuous solutions. So, we always
assume that $u_0 \in C(X)$, and $u_0\geq 0$.
\begin{definition}\label{defup}{\em
A {\em solution of the positive Cauchy problem} in
$\Omega_T\!:=\!\Omega\times [0,T)$ with initial data $u_0$ is a
nonnegative continuous function in $\Omega_T$ satisfying
$u(x,0)=u_0(x)$, and $Lu=0$ in $\Omega\times(0,T)$ in the
classical sense.
We say that the {\em uniqueness of the positive Cauchy problem}
(UP) for the operator $L$ in $\Omega_T$ holds, when any two
solutions of the positive Cauchy problem satisfying the same
initial condition are identically equal in $\Omega_T$.
}\end{definition}
Let $u \in {\cal C}_{P}(\Omega)$. By the parabolic generalized
maximum principle, either \begin{equation} \label{kueu} \int_\Omega\!\!
k(x,y,t)u(y)\!\,\mathrm{d}y\!=\! u(x)\;\;\mbox{for some (and hence for all)
$x \in \Omega,\, t>0$,} \end{equation} or \begin{equation} \label{kuslu} \int_\Omega\!\!
k(x,y,t)u(y)\!\,\mathrm{d}y\!<\!u(x)\;\;\mbox{for some (and hence for all)
$x \in \Omega,\, t>0,$} \end{equation} see for example \cite{EBD}. Note that
both sides of \eqref{kuslu} are solutions of the positive Cauchy
problem \eqref{eqL1} with the same initial data $u_0=u$.
Therefore, in order to show that UP does not hold for the operator
$L$ in $\Omega$, it is sufficient to show that \eqref{kuslu} holds
true for some $u \in {\cal C}_{P}(\Omega)$. It is easy to show
\cite{EBD} that
(\ref{kuslu}) holds true if and only if there exists
$\lambda <0$ such that \begin{equation} \label{Gluslu}
-\lambda\int_\Omega\Green{\Omega}{P-\lambda}{x}{y}u(y)\,\mathrm{d}y < u(x)
\end{equation}
for some (and hence for all) $x\in \Omega$. Furthermore, it
follows from \cite{M95} that (\ref{Gluslu}) is satisfied if
\begin{equation} \label{Gufnt} \int_\Omega\Green{\Omega}{P}{x}{y}u(y)\,\mathrm{d}y <
\infty \end{equation} for some (and hence for all) $x\in \Omega$. Thus, we
have:
\begin{corollary}\label{cornup}
If $\mathbf{1}$ is an $H$-integrable perturbation of a subcritical
operator $P$ in $\Omega$, then the positive Cauchy problem is not
uniquely solvable.
\end{corollary}
\begin{remarks}{\em
1. A positive solution $u \in {\cal C}_{P}(\Omega)$ which
satisfies (\ref{kueu}) is called a {\em positive invariant
solution}. If $P\mathbf{1}=0$ and (\ref{kueu}) holds for
$u=\mathbf{1}$ one says that {\em $L$ conserves probability in}
$\Omega$ (see \cite{G99}). We note that if $P$ is critical, then
the ground state $\varphi_0$ is a positive invariant solution. It
turns out that there exists a complete Riemannian manifold $X$
which does not admit any positive invariant harmonic function,
while $\lambda_0(-\Delta,X,\mathbf{1})=0$ \cite{PStroock}.
2. For necessary and sufficient conditions for UP, see
\cite{IM,M05} and the references therein.
}\end{remarks}
The following important notion was introduced by E.~B.~Davies and
B.~Simon for Schr\"odinger operators \cite{DS84,DS86a,DS86b}.
\begin{definition}{\em
Suppose that $P$ is symmetric. The Schr\"odinger semigroup
$\mathrm{e}^{-tP}$ associated with the heat kernel
$k_P^{\Omega}(x,y,t)$ is called {\em intrinsic ultracontractive}
(IU) if $P-\lambda_0$ is positive-critical in $\Omega$ with a
ground state $\varphi_0$, and for each $t>0$ there exists a
positive constant $C_t$ such that
$$C_t^{-1}\varphi_0(x)\varphi_0(y)\leq k_P^{\Omega}(x,y,t)\leq C_t\varphi_0(x)\varphi_0(y)
\qquad \forall x,y\in \Omega.$$ }\end{definition} .
\vspace{-8mm}
\begin{remarks}\label{remIU} {\em
1. If $\mathrm{e}^{-tP}$ is IU, then
\begin{equation}\label{eqlimhk1}
\lim_{t\to\infty} \mathrm{e}^{\lambda_0 t}k_P^{\Omega}(x,y,t)=
\dfrac{\varphi_0(x)\varphi_0(y)}{\int_\Omega
[\varphi_0(z)]^2\,\mathrm{d}z}
\end{equation}
{\em uniformly} in $\Omega\times \Omega$ (see for example
\cite{Ba}, cf. Theorem~\ref{mainthmhk}).
2. If $\Omega$ is a bounded uniformly H\"older domain of order
$0<\alpha<2$, then $\mathrm{e}^{-t(-\Delta)}$ is IU on $\Omega$
\cite{Ba}.
3. Let $\alpha\geq 0$. Then $\mathrm{e}^{-t(-\Delta+|x|^\alpha)}$
is IU on $\mathbb{R}^d$ if and only if $\alpha> 2$.
}\end{remarks}
Intrinsic ultracontractivity is closely related to perturbation
theory of positive solutions and hence to UP, as the following
recent result of M.~Murata and M.~Tomisaki demonstrates.
\begin{theorem}[\cite{M97,MT2006}]\label{thmiusp}
Suppose that $P$ is a subcritical symmetric operator, and that the
Schr\"odinger semigroup $\mathrm{e}^{-tP}$ is IU on $\Omega$. Then
${\bf 1}$ is a small perturbation of $P$ on $\Omega$. In
particular, UP does not hold in $\Omega$.
\end{theorem}
On the other hand, there are planner domains such that ${\bf 1}$
is a small perturbation of the Laplacian, but the semigroup
$\mathrm{e}^{-t(-\Delta)}$ is not IU (see \cite{BD} and
\cite{P99}).
\mysection{Asymptotic behavior of eigenfunctions}\label{seceigen}
In this section, we assume that $P$ is symmetric and discuss
relationships between perturbation theory, Martin boundary, and
the asymptotic behavior of weighted eigenfunctions in some general
cases (for other relationships between positivity and decay of
Schr\"odinger eigenfunctions see, \cite{Agmon84,S82,S00}).
\begin{Thm} \label{extthm}
(i) Let $V\in C^\alpha} \def\gb{\beta} \def\gg{\gamma(\Omega)$ be a positive function. Suppose that
$P$ is a symmetric, nonnegative operator on $L^2(\Omega,V(x)dx)$
with a domain $C_0^\infty(\Omega)$. Assume that $V$ is a weak
perturbation of the operator $P$ in $\Omega$. suppose that $P$
admits a (Dirichlet) selfadjoint realization $\tilde{P}$ on
$L^2(\Omega,V(x)dx)$. Then $\tilde{P}$ has a purely discrete
nonnegative spectrum (that is,
$\sigma} \def\vgs{\varsigma} \def\gt{\tau_{\mathrm{ess}}(\tilde{P})=\emptyset$). Moreover,
$$\sigma} \def\vgs{\varsigma} \def\gt{\tau(\tilde{P})=\sigma} \def\vgs{\varsigma} \def\gt{\tau_{\mathrm{discrete}}(\tilde{P})=\sigma} \def\vgs{\varsigma} \def\gt{\tau_{\mathrm{point}}(\tilde{P})=
\{\gl_n\}_{n=0}^\infty,$$ where $\lim_{n\to \infty}\gl_n=\infty$.
In particular, if $\gl_0:=\gl_0(P,\Omega,V)>0$, then the natural
embedding $E:{\cal H} \longrightarrow L^2(\Omega,V(x)dx)$ is
compact, where ${\cal H}$ is the completion of
$C_0^\infty(\Omega)$ with respect to the inner product induced by
the corresponding quadratic form.
(ii) Assume further that $P$ is subcritical and $V$ is a semismall
perturbation of the operator $P$ in $\Omega$. Let
$\{\varphi_n\}_{n=0}^\infty$ be the set of the corresponding
eigenfunctions ($P\varphi_n=\gl_nV\varphi_n$). Then for every
$n\geq 1$ there exists a positive constant $C_n$ such that
\begin{equation}\label{efest} |\varphi_n(x)|\leq C_n \varphi_0(x). \end{equation}
(iii) For every $n\geq 1$, the function $\varphi_n/\varphi_0$ has
a continuous extension $\psi_n$ up to the Martin boundary
$\partial_{P}^M\Omega$, and $\psi_n$ satisfies
$$\psi_n(\xi)\!=\!
(\psi_0(\xi))^{-1}\!\gl_n\!\!\!\int_{\Omega}\!\!K^\Omega_{P}(z,\xi)V(z)\varphi_n(z)\!\,\mathrm{d}z\!=\!
\frac{\gl_n\!\int_{\Omega}\!K^\Omega_{P}(z,\xi)V(z)\varphi_n(z)\!\,\mathrm{d}z}
{\gl_0\!\int_{\Omega}\!K^\Omega_{P}(z,\xi)V(z)\varphi_0(z)\!\,\mathrm{d}z}
$$
for every $\xi\in\partial_{P}^M\Omega$, where $\psi_0$ is the
continuous extension of $\varphi_0/\Green{\Omega}{P}{\cdot}{x_0}$
to the Martin boundary $\partial_{P}^M\Omega$.
\end{Thm}
\begin{Rems}\label{IUrem}{\em
1. By \cite{DS84}, the semigroup $\mathrm{e}^{-t\tilde{P}}$ is IU
if and only if the pointwise eigenfunction estimate (\ref{efest})
holds true with $C_n=c_t\exp (t\gl_n)\|\varphi_n\|_2$, for every
$t>0$ and $n\geq 1$. Here $c_t$ is a positive function of $t$
which may be taken as the function such that
$k_P^\Omega(x,y,t)\leq c_t\varphi_0(x)\varphi_0(y)$, where
$k_P^\Omega$ is the corresponding heat kernel. It follows that if
$\mathrm{e}^{-t\tilde{P}}$ is IU, then the pointwise eigenfunction
estimate (\ref{efest}) holds true with $C_n=\inf_{t>0}\{c_t\exp
(t\gl_n)\}\|\varphi_n\|_2$. We note that in general $\{C_n\}$ is
unbounded \cite{GD}.
Recall that if $\mathrm{e}^{-t\tilde{P}}$ is IU, then $\mathbf{1}$
is a small perturbation of $P$ (see Theorem~\ref{thmiusp}). In
particular, part (iii) of Theorem~\ref{extthm} implies that if
$\mathrm{e}^{-t\tilde{P}}$ is IU, then for any $n\geq 1$, the
quotient $\varphi_n/\varphi_0$ has a continuous extension $\psi_n$
up to the Martin boundary $\partial_{P}^M\Omega$.
2. M.~Murata \cite{Mu90} proved part (ii) of Theorem \ref{extthm}
for the special case of bounded Lipschitz domains. See also
\cite{HO} for related results on the asymptotic behavior of
eigenfunctions of Schr\"{o}dinger operators in $\mathbb{R}^d$.
}\end{Rems}
\mysection{Localization of binding}\label{seclocalization}
Let $V\in C^\alpha(\mathbb{R}^d)$ and $R\in \mathbb{R}^d$,
throughout this section we use the notation $V^R(x):=V(x-R)$. For
$j=1,2$, let $V_{j}$ be small perturbations of the Laplacian in
$\mathbb{R}^d, d\geq 3$, and assume that the operators
$P_{j}:=-\Delta + V_{j}(x)$ are nonnegative on $C_0^{\infty}(\Omega)$. We
consider the Schr\"{o}dinger operator \begin{equation} \label{dePR} P_{R} :=
-\Delta + V_{1}(x) + V_{2}^R(x) \end{equation} defined on $\mathbb{R}^d$, and
its ground state energy
$E(R):=\lambda_0(P_R,\mathbb{R}^d,\mathbf{1})$. In this section we
discuss the asymptotic behavior of $E(R)$ as $\abs{R}\to\infty$, a
problem which was studied by M.~Klaus and B.~Simon in
\cite{KS79,S80} (see also \cite{Plocal,Pins95}). The motivation
for studying the asymptotic behavior of $E(R)$ comes from a
remarkable phenomenon known as the Efimov effect for a three-body
Schr\"{o}dinger operator (for more details, see for example
\cite{T1}).
\begin{definition} \label{Kato}
Let $d\geq 3$. The space of functions \begin{equation} \label{Kndef}
K_{d}^{\infty}:=\left\{ V \in C^{\alpha}(\mathbb{R}^d) |
\lim_{M\rightarrow
\infty}\sup_{x\in\mathbb{R}^d}\int_{|z|>M}\frac{|V(z)|}
{|x-z|^{d-2}}\,\mathrm{d}z=0 \right\} \end{equation} is called the Kato class at
infinity.
\end{definition}
\begin{remark}\label{remkato}
{\em Let $d\geq 3$. If $V\in K_{d}^{\infty}$, then $V$ is a small
perturbation of the Laplacian in $\mathbb{R}^d$.
}\end{remark}
\begin{theorem}[\cite{Plocal}] \label{Thm1}
Let $d\geq 3$. For $j=1,2$, let $V_{j}(x)\in K_{d}^{\infty}$ be
two functions such that the operators $P_{j}=-\Delta + V_{j}(x)$
are subcritical in $\mathbb{R}^d$. Then there exists $r_{0}>0$
such that the operator $P_{R}$ is subcritical for any $R\in
\mathbb{R}^d \setminus B(0,r_0)$. In particular, $E(R)=0$ for all
$\abs{R}\geq r_0$.
\end{theorem}
Assume now that the operators $P_{j}=-\Delta + V_{j}(x)
,\;j=1,2$, are {\em critical} in $\mathbb{R}^d$. It turns out
that in this case, there exists $r_{0}>0$ such that $E(R)<0$ for
$\abs{R}\geq r_0$, but the asymptotic behavior of $E(R)$ depends
on the dimension $d$, as the following theorems demonstrate (cf.
\cite[the remarks in pp. 84 and 87]{KS79}).
\begin{theorem}[\cite{T1}] \label{Thmlob3}
Let $d=3$. Assume that the potentials $V_{j}, \;j=1,2$ satisfy $
|V_{j}(x)| \leq C\langle x \rangle^{-\beta}$ on $\mathbb{R}^3$,
where $\langle x \rangle:=(1+|x|^2)^{1/2}$, $\beta
>2$, and $C>0$. Suppose that
$P_j=-\Delta + V_{j}(x)$ is critical in $\mathbb{R}^3$ for
$j=1,2$.
Then there exists $r_{0}>0$ such that the operator $P_{R}$ is
supercritical for any $R\in \mathbb{R}^3 \setminus B(0,r_0)$.
Moreover, $E(R)$ satisfies
\begin{equation} \label{R2ER} \lim_{\abs{R} \rightarrow
\infty}\abs{R}^{2}E(R)=-\beta^2 <-1/4, \end{equation} where $\beta$ is the
unique root of the equation $s=\mathrm{e}^{-s}$.
\end{theorem}
\begin{theorem}[\cite{P96}] \label{Thmlob4}
Let $d=4$. Assume that for $j=1,2$ the operators $P_{j}=-\Delta +
V_{j}(x)$ are critical in $\mathbb{R}^4$, where $V_{j}\in
C^\alpha_0(\mathbb{R}^4)$.
Then there exists $r_{0}>0$ such that the operator $P_{R}$ is
supercritical for any $R\in \mathbb{R}^4 \setminus B(0,r_0)$.
Moreover, there exists a positive constant $C$ such that $E(R)$
satisfies \begin{equation} \label{ER6est} -C\abs{R}^{-2} \leq E(R) \leq
-C^{-1}|R|^{-2}(\log |R|)^{-1} \;\; \ \mbox{for all $|R|\geq
r_{0}$.} \end{equation}
\end{theorem}
\begin{theorem}[\cite{Plocal}] \label{Thmlob5}
Let $d\geq 5$. Suppose that $V_{j}, \;j=1,2$ satisfy $ |V_{j}(x)|
\leq C\langle x \rangle^{-\beta}$ in $\mathbb{R}^d$, where $\beta
> d-2$, and $C>0$. Assume that the operators
$P_{j}=-\Delta + V_{j}(x),\;j=1,2$, are critical in
$\mathbb{R}^d$.
Then there exists $r_{0}>0$ such that the operator $P_{R}$ is
supercritical for any $R\in \mathbb{R}^d \setminus B(0,r_0)$.
Moreover, there exists a positive constant $C$ such that $E(R)$
satisfies \begin{equation} \label{ER5est} -C|R|^{2-d} \leq E(R) \leq
-C^{-1}|R|^{2-d} \;\; \ \mbox{for all $|R|\geq r_{0}$.} \end{equation}
\end{theorem}
What distinguishes $d\geq 5$ from $d=3,4$, is that for a
short-range potential $V$, the ground state of a critical operator
$-\Delta + V(x)$ in $\mathbb{R}^d$ is in $L^{2}(\mathbb{R}^d)$ if
and only if $d\geq 5$ (see \cite{S81} and Theorem~\ref{thmssp}).
\mysection{The shuttle operator}\label{secshttle}
In this section we present an intrinsic criterion which
distinguishes between subcriticality, criticality and
supercriticality of the operator $P$ in $\Omega$. This criterion
depends only on the norm of a certain linear operator $S$, called
the {\em shuttle operator} which is defined on $C(\partial D)$,
where $D\Subset\Omega$.
The shuttle operator was introduced for Schr\"odinger operators on
$\mathbb{R}^d$ in \cite{C,CV,Z1,Z2}. Using Feynman-Kac-type
formulas \cite{Spath}, F.~Gesztesy and Z.~Zhao \cite{GZ2,Z2} have
studied the shuttle operator for Schr\"{o}dinger operators in
$\mathbb{R}^d$ with short-range potentials (see also \cite{GZ1}),
and its relation to the following problem posed by B.~Simon.
\begin{problem}[\cite{S81,S82}]\label{problem}
Let $V\in L^2_{\mathrm{loc}}(\mathbb{R}^2)$. Show that if the
equation $(-\Delta\!+\!V)u\!=\!0$ on $\mathbb{R}^2$ admits a
positive $L^\infty$-solution, then $-\Delta\!+\!V$ is critical.
\end{problem}
Gesztesy and Zhao used the shuttle operator and proved that for
{\em short-range} potentials on $\mathbb{R}^2$, the above
condition is a necessary and sufficient condition for criticality
(see also \cite{M84} and Theorem~\ref{thmssp} for similar results,
and Theorem~\ref{thmDKS} for the complete solution). On the other
hand, Gesztesy and Zhao showed in \cite[Example~4.6]{GZ1} that
there is a critical Schr\"odinger operator on $\mathbb{R}$ with
`almost' short-range potential such that its ground state behaves
logarithmically.
Let $P$ be an elliptic operator of the form (\ref{P}) which is
defined on $\Omega$. We assume that the following assumption {\bf
(A)} holds:
\begin{description}
\item [(A)]
There exist four smooth, relatively compact subdomains
$\Omega_j,\; 0\le j\le 3$, such that $\overline{\Omega_{j}}\subset
\Omega_{j+1},\;j=0,1,2$, and such that
$\mathcal{C}_{P}(\Omega_3)\neq \emptyset$ and
$\mathcal{C}_{P}(\Omega_{0}^*)\neq \emptyset$.
\end{description}
\begin{remarks} \label{rem(A)}
{\em 1. If assumption {\bf (A)} is not satisfied, then we shall
say that the spectral radius of the shuttle operator is infinity.
In this case, it is clear that $P$ is supercritical in $\Omega$.
2. Assumption {\bf (A)} does not imply that
$\mathcal{C}_{P}(\Omega)\neq \emptyset$. }
\end{remarks}
Fix an exhaustion $\{\Omega_{j}\}_{j=0}^{\infty}$ of $\Omega$,
such that $\Omega_j$ satisfy assumption {\bf (A)} for $0\le j\le
3$. By assumption {\bf (A)} the Dirichlet problem \begin{eqnarray} Pu=0
\;\;\;\mbox{in $\Omega_2$},\quad u=f \;\;\;&\mbox{on $\partial
\Omega_2$} \end{eqnarray} is uniquely solved in $\Omega_2$ for any $f \in
C(\partial \Omega_2)$, and we denote the corresponding operator
from $C(\partial \Omega_2)$ into $C(\Omega_2)$ by $T_{\Omega_2}$.
Moreover, for every $f \in C(\partial \Omega_1)$, one can uniquely
solve the exterior Dirichlet problem in the outer domain
$\Omega_1^*$, with `zero' boundary condition at infinity of
$\Omega$. So, we have an operator $T_{\Omega_1^*}: C(\partial
\Omega_1) \to C(\Omega_1^*)$ defined by
$$T_{\Omega_1^*}f(x):=\lim_{j\to\infty}u_{f,j}(x),$$
where $u_{f,j}$ is the solution of the Dirichlet boundary value
problem:
$$
Pu=0 \;\;\;\mbox{in $\Omega_1^*\cap \Omega_j$},\quad u=f
\;\;\;\mbox{on $\partial \Omega_1^*$},\quad u=0 \;\;\;\mbox{on
$\partial (\Omega_1^*\cap \Omega_j) \setminus
\partial \Omega_1^*$}.
$$
For any open set $D$ and $F \Subset D$, we denote by $R^D_{F}$
the restriction map $f\mapsto f\!\!\mid_F$ from $C(D)$ into
$C(F)$. The {\em shuttle operator} $S:C(\partial \Omega_1)
\longrightarrow C(\partial \Omega_1)$ is defined as follows: \begin{equation}
\label{defshuttle} S:=R^{\Omega_2}_{\partial
\Omega_1}T_{\Omega_2}R^{\Omega_1^*}_{\partial
\Omega_2}T_{\Omega_1^*}\,. \end{equation} We denote the spectral radius of
the operator $S$ by $r(S)$. We have
\begin{Thm}[\cite{Pshuttle}] \label{mainthmshuttle} The operator $P$ is
subcritical, critical, or supercritical in $\Omega$ according to
whether $r(S)<1$, $r(S)=1$, or $r(S)>1$.
\end{Thm}
The proof of Theorem~\ref{mainthmshuttle} in \cite{Pshuttle} is
purely analytic and relies on the observation that (in the
nontrivial case) $S$ is a positive compact operator defined on
the Banach space $C(\partial \Omega_1)$. Therefore, the
Krein-Rutman theorem implies that there exists a simple principal
eigenvalue $\nu_0>0$, which is equal to the norm (and also to the
spectral radius) of $S$, and that the corresponding principal
eigenfunction is strictly positive. It turns out, that the
generalized maximum principle holds in any smooth subdomain
$D\Subset \Omega$ if and only if $\nu_0 \le 1$, and that $\nu_0
<1$ if and only if $P$ admits a positive minimal Green function in
$\Omega$.
The shuttle operator can be used to prove localization of binding
for certain {\em nonselfadjoint} critical operators (see
\cite{Pshuttle}).
\mysection{Periodic operators}\label{secperiod}
In this section we restrict the form of the operator. Namely, we
assume that $P$ is defined on $\mathbb{R}^d$ and that the
coefficients of $P$ are $\mathbb{Z}^d$-periodic. For such
operators, we introduce a function $\Lambda$ that plays a crucial
role in our considerations. Its properties were studied in detail
in \cite{A1a,KS87,LP,MT,Pinsper}. Consider the function $\Lambda
:\mathbb{R}^d\rightarrow \mathbb{R}$ defined by the condition that
the equation $Pu=\Lambda (\xi )u$ on $\mathbb{R}^d$ has a positive
{\em Bloch solution} of the form
\begin{equation}\label{positiveBloch}
u_{\,\xi }(x)=e^{\xi \cdot x}\varphi_{\,\xi }(x),
\end{equation}
where $\xi\in\mathbb{R}^d$, and $\varphi_{\,\xi }$ is a positive
$\mathbb{Z}^d$-periodic function.
\begin{theorem}
\label{Lambda-lemma}
\begin{enumerate}
\item The value $\Lambda (\xi )$ is uniquely determined for any $\xi \!\in\!
\mathbb{R}^d$.
\item The function $\Lambda$ is bounded from above, strictly
concave, analytic, and has a nonzero gradient for any $\xi \in
\mathbb{R}^d$ except at its maximum point.
\item For $\xi \in
\mathbb{R}^d$, consider the operator $P(\xi ):=e^{-\xi\cdot
x}Pe^{\xi\cdot x}$ on the torus $\mathbb{T}^d$. Then $\Lambda
(\xi )$ is the principal eigenvalue of $P(\xi )$ with a positive
eigenfunction $\varphi_{\,\xi }$. Moreover, $\Lambda (\xi )$ is
algebraically simple.
\item The Hessian of $\Lambda (\xi )$ is nondegenerate at all points $\xi \in
\mathbb{R}^d$.
\end{enumerate}
\end{theorem}
Let us denote
\begin{equation}
\Lambda_{0} =\max_{\xi \in \mathbb{R}^d}\Lambda (\xi ).
\label{Lambda}
\end{equation}
It follows from \cite{A1a,LP,Pinsper} that
$\Lambda_{0}=\lambda_0$, and that $P-\Lambda_{0}$ is critical if
and only if $d=1,2$ (see also Corollary~\ref{corper}). Thus, in
the self-adjoint case, $\Lambda_{0}$ coincides with the bottom of
the spectrum of the operator $P$. Assume that $\Lambda_{0}\geq
0$. Then Theorem~\ref{Lambda-lemma} implies that the zero level
set
\begin{equation}
\Xi =\left\{ \xi \in \mathbb{R}^d|\;\Lambda (\xi )=0\right\}
\label{ksi}
\end{equation}
is either a strictly convex compact analytic surface in
$\mathbb{R}^d$ of dimension $d-1$ (this is the case if and only if
$\Lambda_{0}> 0$), or a singleton (this is the case if and only
if $\Lambda_{0}=0$).
In a recent paper \cite{MT}, M.~Murata and T.~Tsuchida have
studied the exact asymptotic behavior at infinity of the positive
minimal Green function and the Martin boundary of such periodic
elliptic operators on $\mathbb{R}^d$.
Suppose that $\Lambda_0=\Lambda(\xi_0)>0$. Then $P$ is
subcritical, and for each $s$ in the unit sphere $S^{d-1}$ there
exists a unique $\xi_s\in \Xi$ such that
$$\xi_s\cdot s=\sup_{\xi\in\Xi}\,\{\xi\cdot s\}.$$ For $s\!\in\!
S^{d-1}$ take an orthonormal basis of $\mathbb{R}^d$ of the form
$\{e_{s,1},\ldots,e_{s,d-1},s\}$. For $\xi\in \mathbb{R}^d$, let
$\varphi_\xi$ and $\varphi^*_\xi$ be periodic positive solutions
of the equation $P(\xi)u=\Lambda(\xi)u$ and
$P^*(\xi)u=\Lambda(\xi)u$ on $\mathbb{T}^d$, respectively, such
that
$$\int_{\mathbb{T}^d}\varphi_{\xi}(x)\varphi^*_{\xi}(x)\mathrm\,{d}x=1.$$
\begin{theorem}[\cite{MT}]\label{thmmuratatsuchida1} 1. Suppose that $\Lambda_0>0$.
Then the minimal Green
function $G^{\mathbb{R}^d}_P$ of $P$ on $\mathbb{R}^d$ has the
following asymptotics as $\abs{x-y}\to\infty$:
$$G^{\mathbb{R}^d}_P(x,y)\!=\!\frac{\abs{\nabla\Lambda(\xi_s)}^{(d-3)/2}
\,\mathrm{e}^{-(x-y)\cdot\xi_s}\varphi_{\xi_s}(x)\varphi^*_{\xi_s}(y)}
{(2\pi\abs{x\!-\!y})^{(d-1)/2}[\mathrm{det}(-e_{s,j}\!\cdot\!
\mathrm{Hess}
\Lambda(\xi_s)e_{s,k})]^{1/2}}\left[1\!+\!O(\abs{x\!-\!y}^{-1})\right]\!,$$
where $s:=(x-y)/\abs{x-y}$.
2. Suppose that $\Lambda_0=\Lambda(\xi_0)=0$ and $d\geq 3$. Then
the minimal Green function $G^{\mathbb{R}^d}_P$ of $P$ on
$\mathbb{R}^d$ has the following asymptotics as
$\abs{x-y}\to\infty$:
$$G^{\mathbb{R}^d}_P(x,y)\!\!=\!\!\!\frac{}{}
\frac{2^{-1}\pi^{-d/2}\Gamma(\frac{d-2}{2})\,\mathrm{e}^{-(x-y)\cdot\xi_0}\varphi_{\xi_0}(x)\varphi^*_{\xi_0}(y)}
{\{\mathrm{det}[\mathrm{Hess} \Lambda(\xi_0)]\}^{1/2}\!\abs{[-\mathrm{Hess}
\Lambda(\xi_0)]^{-1/2}(x\!-\!y)}^{d-2}}
\!\!\left[1\!+\!O(\abs{x\!-\!y}^{-1})\right]\!\!.$$
\end{theorem}
Combining the results in \cite{A1a,MT}, we have the following
Martin representation theorem.
\begin{theorem}[\cite{A1a,MT}]\label{thmperrep}
Let $\Xi$ be the set of all $\xi \in \mathbb{R}^d$ such that the
equation $Pu=0$ admits a positive Bloch solution
$u_{\,\xi}(x)=e^{\xi\cdot x}\varphi_{\,\xi }(x)$ with
$\varphi_{\,\xi }(0)=1$. Then $u$ is a positive Bloch solution if
and only if $u$ is a minimal Martin function of the equation
$Pu=0$ in $\mathbb{R}^d$. Moreover, all Martin functions are
minimal. Furthermore, $u\in \mathcal{C}_P(\mathbb{R}^d)$ if and
only if there exists a positive finite measure $\mu} \def\gn{\nu} \def\gp{\pi$ on $\Xi$ such
that
\[
u(x)=\int_\Xi u_{\,\xi }(x)\,\mathrm{d}\mu} \def\gn{\nu} \def\gp{\pi(\xi ).
\]
\end{theorem}
Theorem~\ref{thmperrep} (except the result that all Martin
functions are minimal) was extended by V.~Lin and the author to a
manifold with a group action \cite{LP}. It is assumed that $X$ is
a noncompact manifold equipped with an action of a group $G$ such
that $GV=X$ for a compact subset $V\Subset X$, and that the
operator $P$ is a $G$-invariant operator on $X$ of the form
\eqref{P}. If $G$ is finitely generated, then the set of all
normalized positive solutions of the equation $Pu=0$ in $X$ which
are also eigenfunctions of the $G$-action is a real analytic
submanifold $\Xi$ in an appropriate finite-dimensional vector
space $\mathcal{H}$. Moreover, if $\Xi$ is not a singleton, then
it is the boundary of a strictly convex body in $\mathcal{H}$. If
the group $G$ is {\em nilpotent}, then any positive solution in
${\cal C}_{P}(X)$ can be uniquely represented as an integral of
solutions over $\Xi$. In particular, $u\in {\cal C}_{P}(X)$ is a
positive minimal solution if and only if it is a positive solution
which is also an eigenfunction of the $G$-action.
\mysection{Liouville theorems for Schr\"odinger operators and
Criticality}\label{seccritliouville}
The existence and nonexistence of nontrivial bounded solutions of
the equation $Pu=0$ are closely related to criticality theory as
the following results demonstrate (see also
Section~\ref{secliouville}).
\begin{Pro}[\cite{G},{\cite[Lemma 3.4]{P99}}]\label{propinteg}
Suppose that $V$ is a nonzero, nonnegative function such that $V$
is an $H$-integrable perturbation of a subcritical operator $P$ in
$\Omega} \def\Gx{\Xi} \def\Gy{\Psi$ and let $u\in {\cal C}_{P}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$. Then for any
$\varepsilon>0$ there exists $u_\varepsilon \in {\cal
C}_{P+\varepsilon V}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ which satisfies $0<u_\varepsilon\leq u$
and the resolvent equation \begin{equation} \label{usubteq}
u_\varepsilon(x)=u(x) -
\varepsilon\int_{\Omega}\Green{\Omega}{P+\varepsilon
V}{x}{z}V(z)u(z)\,\,\mathrm{d}z. \end{equation} In particular, if $P\mathbf{1}=0$, then
for any $\varepsilon>0$ the operator $P+\varepsilon V$ admits a
nonzero bounded solution.
\end{Pro}
In \cite[Theorem~5]{DKS}, D.~Damanik, R.~Killip, and B.~Simon
proved a result which, formulated in the following new way,
reveals a complete answer to Problem~\ref{problem} posed by
B.~Simon in \cite{S81,S82} (see also \cite{GZ2,M84} and
Theorem~\ref{thmssp}). An alternative proof based on criticality
theory is presented below.
\begin{theorem}[\cite{DKS}]\label{thmDKS}
Let $d = 1$ or $2$, and $q \in L^2_{\mathrm{loc}}(\mathbb{R}^d)$.
Suppose that $H_q:=-\Delta+q$ has a bounded positive solution in
$\mathcal{C}_{H_q}(\mathbb{R}^d)$. If $V\in
L^2_{\mathrm{loc}}(\mathbb{R}^d)$ and both $H_{q\pm V}\geq 0$,
then $V=0$. In other words, $H_q$ is critical.
\end{theorem}
\begin{proof}
Theorem~\ref{thmconv} implies that we should indeed show that
$H_q$ is critical. Assume that $H_q$ is subcritical. Take a
nonzero nonnegative $W$ with a compact support. Then by
Theorem~\ref{thmssp}, there exists $\varepsilon>0$ such that
$H_{q-\varepsilon W}\geq 0$. Let $M<N$. For $d=1$ take the cutoff
function
\begin{eqnarray*} a_{M,N}(x):=
\begin{cases}
0 & |x|>N, \\
1 & |x|\leq M,\\
1-\frac{|x|-M}{N-M} & M< |x|\leq N,
\end{cases}
\end{eqnarray*}
and for $d=2$
\begin{eqnarray*} a_{M,N}(x):=
\begin{cases}
0 & |x|>N, \\
1& |x|\leq M,\\
\frac{\log N-\log |x|}{\log N-\log M} & M< |x|\leq N.
\end{cases}
\end{eqnarray*}
Let $\psi$ be a positive bounded solution of the equation $H_qu=0$
in $\mathbb{R}^d$. Then for appropriate $N, M$ with $M,N\to\infty$
(see \cite{DKS}), we have
\begin{multline*}\label{E}
0<c<\varepsilon \int_{\mathbb{R}^d} W (a_{M,N} \psi)^2\,\mathrm{d}x \leq
\int_{\mathbb{R}^d} \left[|\nabla (a_{M,N}\psi)|^2 + q (a_{M,N} \psi)^2\right]
\,\mathrm{d}x =\\
\int_{\mathbb{R}^d} |\nabla a_{M,N}|^2 \psi^2\,\mathrm{d}x\to
0,
\end{multline*}
and this is a contradiction.
\end{proof}
\begin{remarks}\label{remlioiva}{\em
1. Theorem~\ref{thmDKS} is related to Theorem~1.7 in \cite{BCN}
which claims that for $d=1,2$, if $H_q$ admits a bounded solution
that changes its sign, then $\lambda_0\!<\!0$. This claim and
Theorem~\ref{thmDKS} do not hold for $d\!\geq\! 3$ \cite{B}.
2. For other relationships between perturbation theory of positive
solutions and Liouville theorem see \cite{G99,GH}.
}\end{remarks}
After submitting the first version of the present article to the
editors, we proved the following result which generalized
Theorem~\ref{thmDKS} and the Liouville type theorems in
\cite{BCN}.
\begin{theorem}[\cite{P06}]\label{mainthmliouville} Let $\Omega\subset X$ be a domain.
Consider two Schr\"odinger operators defined on $\Omega$ of the
form
\begin{equation}\label{eqpj}
P_j:=-\nabla\cdot(A_j\nabla)+V_j\qquad j=0,1,
\end{equation}
such that $V_j\in L^{p}_{\mathrm{loc}}(\Omega;\mathbb{R})$ for
some $p>{d}/{2}$, and $A_j:\Omega \rightarrow \mathbb{R}^{d^2}$
are measurable matrix valued functions such that for any $K\Subset
\Omega$ there exists $\mu_K>1$ such that \begin{equation} \label{stell}
\mu_K^{-1}I_d\le A_j(x)\le \mu_K I_d \qquad \forall x\in K,
\end{equation}
where $I_d$ is the $d$-dimensional identity matrix.
Assume that the following assumptions hold true.
\begin{itemize}
\item[(i)] The operator $P_1$ is critical in $\Omega$. Denote
by $\varphi\in \mathcal{C}_{P_1}(\Omega)$ its ground state.
\item[(ii)] $\lambda_0(P_0,\Omega,\mathbf{1})\geq 0$, and there exists a
real function $\psi\in H^1_{\mathrm{loc}}(\Omega)$ such that
$\psi_+\neq 0$, and $P_0\psi \leq 0$ in $\Omega$, where
$u_+(x):=\max\{0, u(x)\}$.
\item[(iii)] The following matrix inequality holds
\begin{equation}\label{psialephia}
\psi^2(x) A_0(x)\leq C\varphi^2(x) A_1(x)\qquad \mbox{ a. e. in }
\Omega,
\end{equation}
where $C>0$ is a positive constant.
\end{itemize}
Then the operator $P_0$ is critical in $\Omega$, and $\psi$ is its
ground state. In particular, $\dim \mathcal{C}_{P_0}(\Omega)=1$
and $\lambda_0(P_0,\Omega,\mathbf{1})=0$.
\end{theorem}
The proof of Theorem~\ref{mainthmliouville} relies on
Theorem~\ref{mainky3}.
\begin{corollary}[\cite{Pinsper}]\label{corper}
Assume that the coefficients of the elliptic operator
$P:=-\nabla\cdot(A\nabla)+V$ are $\mathbb{Z}^d$-periodic on
$\mathbb{R}^d$. Then the operator $P-\lambda_0$ is critical in
$\mathbb{R}^d$ if and only if $d\leq 2$.
\end{corollary}
\begin{remark}\label{remcorper}{\em
One can use \cite{LP} to extend Corollary~\ref{corper} to the
case of equivariant Schr\"odinger operators on cocompact
coverings. Let $X$ be a noncompact nilpotent covering of a compact
Riemannian manifold. Suppose that $P:=-\Delta+V$ is an equivariant
operator on $X$ with respect to its {\em nilpotent} deck group
$G$. Then $P-\lambda_0$ is critical in $X$ if and only if $G$ has
a normal subgroup of finite index isomorphic to $\mathbb{Z}^d$ for
$d\leq 2$.
}\end{remark}
\mysection{Polynomially growing solutions and Liouville Theorems}\label{secliouville}
Let
$H=-\Delta +V$ be a Schr\"odinger operator on $\mathbb{R}^d$. Then
\v{S}nol's theorem asserts that, under some assumptions on the
potential $V$, if $H$ admits a polynomially growing solution of
the equation $Hu=0$ in $\mathbb{R}^d$, then $0\in \sigma(H)$.
\v{S}nol's theorem was generalized by many authors including
B.~Simon, see for example \cite{Cycon,S82} and \cite{Shubin}.
In \cite{kuchy1,kuchy2} the structure of the space of all
polynomially growing solutions of a periodic elliptic operator (or
a system) of order $m$ on an abelian cover of a compact Riemannian
manifold was studied. An important particular case of the general
results in \cite{kuchy1,kuchy2} is a real, second-order
$\mathbb{Z}^d$-periodic elliptic operator $P$ of the form
(\ref{P}) which is defined on $\mathbb{R}^d$. In this case, we can
use the information about positive solutions of such equations
described in Section~\ref{secperiod} and the results of
\cite{kuchy1} to obtain the precise structure and dimension of the
space of polynomially growing solutions.
\begin{definition} \label{defFermi} {\em 1. Let $N\geq 0$. We say that
{\em the Liouville theorem of order $N$ for the equation $Pu=0$}
holds true in $\mathbb{R}^d$, if the space $\mathrm{V}_N (P)$ of
solutions of the equation $Pu=0$ in $\mathbb{R}^d$ that satisfy
$\abs{u(x)}\leq C(1+\abs{x})^N$ for all $x\in \mathbb{R}^d$ is of
finite dimension.
2. The {\em Fermi surface} $F_P$ of the operator $P$ consists of
all vectors $\zeta\in \mathbb{C}^{d}$ such that the equation
$Pu=0$ has a nonzero {\em Bloch solution} of the form
$u(x)=e^{\mathrm{i}\zeta\cdot x}p(x)$, where
$p$ is a $\mathbb{Z}^d$-periodic function.}
\end{definition}
For a general $\mathbb{Z}^d$-periodic elliptic operator $P$ of any
order, we have:
\begin{theorem}[\cite{kuchy1}]\label{thmliouv}
\begin{enumerate}
\item If the Liouville theorem of an order $N\geq 0$ for the equation $Pu=0$ holds true,
then it holds for any order.
\item The Liouville theorem holds true if and only if the number of points in the real Fermi
surface $F_{P}\cap \mathbb{R}^d$ is finite.
\end{enumerate}
\end{theorem}
For second-order operators with real coefficients, we have:
\begin{theorem}[\cite{kuchy1}]\label{T:non-self}Let $P$ be a $\mathbb{Z}^d$-periodic operator
on $\mathbb{R}^d$ of the form (\ref{P}) such that $\Lambda_{0}
\geq 0$. Then
\begin{enumerate}
\item The Liouville theorem holds vacuously if $\Lambda(0)> 0$, i.e., the equation
$Lu=0$ does not admit any nontrivial polynomially growing
solution.
\item If $\Lambda(0)= 0$ and $\Lambda_{0}>0$, then the Liouville theorem holds for
$P$, and
$$
\dim \mathrm{V}_N
(P)=\left(\begin{array}{c}d+N-1\{\mathbb N}\end{array}\right).
$$
\item If $\Lambda(0)= 0$ and $\Lambda_{0}=0$, then the Liouville theorem holds for
$P$, and
$$
\dim \mathrm{V}_N (P)=
\left(\begin{array}{c}d+N\{\mathbb N}\end{array}\right)-\left(\begin{array}{c}d+N-2\{\mathbb N}-2\end{array}\right),
$$
which is the dimension of the space of all harmonic polynomials of
degree at most $N$ in $d$ variables.
\item Any solution $u\in \mathrm{V}_N(P)$ of the equation
$Pu=0$ can be represented as
$$
u(x)=\sum\limits_{\abs{j}\leq N} x^j p_j(x)
$$
with $\mathbb{Z}^d$-periodic functions $p_j$.
\end{enumerate}
\end{theorem}
\mysection{Criticality theory for the $p$-Laplacian with potential
term}\label{secplap} Positivity properties of quasilinear elliptic
equations defined on a domain $\Omega\subset \mathbb{R}^d$, and in
particular, those with the $p$-Laplacian term in the principal
part, have been extensively studied over the recent decades (see
for example \cite{AH1,AH2,DKN,GS,HKM,V} and the references
therein).
Let $p\in(1,\infty)$, and let $\Omega$ be a general domains in
$\mathbb{R}^d$. Denote by $\Delta_p(u):=\nabla\cdot(|\nabla
u|^{p-2}\nabla u)$ the $p$-Laplacian operator, and let $V\in
L_\mathrm{loc}^\infty(\Omega)$ be a given (real) potential.
Throughout this section we always assume that \begin{equation} \label{Q}
Q(u):=\int_\Omega \left(|\nabla u|^p+V|u|^p\right)\,\mathrm{d}x\geq 0\qquad
\forall u\in C_0^{\infty}(\Omega),\end{equation}
that is, the functional $Q$ is nonnegative on $C_0^{\infty}(\Omega)$. In
\cite{PT3}, K.~Tintarev and the author studied (sub)criticality
properties for positive weak solutions of the corresponding
Euler-Lagrange equation \begin{equation} \label{groundstate}
\frac{1}{p}Q^\prime
(v):=-\Delta_p(v)+V|v|^{p-2}v=0\quad \mbox{in } \Omega,\end{equation} along
the lines of criticality theory for second-order linear elliptic
operators that was discussed in
sections~\ref{secpreliminaries}--\ref{secindef}.
\begin{definition}{\em We say that the functional $Q$ is {\em subcritical}
in $\Omega$ (or $Q$ is {\em strictly positive} in $\Omega$) if
there is a strictly positive continuous function $W$ in $\Omega$
such that \begin{equation}\label{wsg} Q(u)\geq \int_\Omega W|u|^p\,\mathrm{d}x \qquad
\forall u\inC_0^{\infty}(\Omega).\end{equation} }\end{definition}
\begin{definition}{\em We say that a sequence
$\{u_n\}\subsetC_0^{\infty}(\Omega)$ is a {\em null sequence}, if $u_n\geq 0$
for all $n\in{\mathbb N}$, and there exists an open set $B\Subset\Omega$
such that
$\int_B|u_n|^p\,\mathrm{d}x=1$, and \begin{equation}
\lim_{n\to\infty}Q(u_n)=\lim_{n\to\infty}\int_\Omega (|\nabla
u_n|^p+V|u_n|^p)\,\mathrm{d}x=0.\end{equation}
We say that a positive function $\varphi\in
C^1_{\mathrm{loc}}(\Omega)$ is a {\em ground state} of the
functional $Q$ in $\Omega$ if $\varphi$ is an
$L^p_{\mathrm{loc}}(\Omega)$ limit of a null sequence. If $Q\geq
0$, and $Q$ admits a ground state in $\Omega$, we say that the
functional $Q$ is {\em critical} in $\Omega$. The functional $Q$
is {\em supercritical} in $\Omega$ if $Q\ngeq 0$ on $C_0^{\infty}(\Omega)$.}
\end{definition}
The following is a generalization of the Allegretto-Piepenbrink
theorem.
\begin{theorem}[see \cite{PT3}]\label{pos} Let Q be a functional of the form (\ref{Q}).
Then the following assertions are equivalent
(i) The functional $Q$ is nonnegative on $C_0^\infty(\Omega)$.
(ii) Equation (\ref{groundstate}) admits a global positive
solution.
(iii) Equation (\ref{groundstate}) admits a global positive
supersolution.
\end{theorem}
The definition of positive solutions of minimal growth in a
neighborhood of infinity in $\Omega$ in the linear case
(Definition~\ref{defminimalg}) is naturally extended to solutions
of the equation $Q'(u)=0$.
\begin{definition} \label{defminimalp}{\em
A positive solution $u$ of the equation $Q'(u)=0$ in $\Omega^*_j$
is said to be a {\em positive solution of the equation $Q'(u)=0$
of minimal growth in a neighborhood of infinity in} $\Omega$ if
for any $v\in C(\Omega_l^*\cup \partial \Omega _l)$ with $l> j$,
which is a positive solution of the equation $Q'(u)=0$ in $\Omega
_l^*$, the inequality $u\le v$ on $\partial \Omega _l$, implies
that $u\le v$ on $\Omega _l^*$.}
\end{definition}
If $1<p\leq d$, then for each $x_0\in \Omega$, any positive
solution $v$ of the equation $Q'(u)=0$ in a punctured neighborhood
of $x_0$ has either a removable singularity at $x_0$, or
\begin{equation}\label{nonremovasymp}
v(x)\sim\begin{cases}
\abs{x-x_0}^{\alpha(d,p)} & p<d, \\
-\log \abs{x-x_0} & p=d,
\end{cases} \qquad \mbox{ as } x\to x_0,
\end{equation}
where $\alpha(d,p):=(p-d)/(p-1)$, and $f\sim g$ means that $
\lim_{x\to x_0}[{f(x)}/{g(x)}]= C$ for some $C>0$ (see \cite{GiS}
for $p=2$, and \cite{Serrin1,Serrin2,V,PT3} for $1<p\leq d$).
The following result is an extension to the $p$-Laplacian of
Theorem~\ref{thmmingr2}.
\begin{theorem}[\cite{PT3}]\label{thmmingr}
Suppose that $1<p\leq d$, and $Q$ is nonnegative on $C_0^{\infty}(\Omega)$.
Then for any $x_0\in \Omega$ the equation $Q'(u)=0$ has (up to a
multiple constant) a unique positive solution $v$ in
$\Omega\setminus\{x_0\}$ of minimal growth in a neighborhood of
infinity in $\Omega$. Moreover, $v$ is either a global minimal
solution of the equation $Q'(u)=0$ in $\Omega$, or $v$ has a
nonremovable singularity at $x_0$.
\end{theorem}
The main result of this section is as follows.
\begin{theorem}[\cite{PT3}] \label{mainky3}
Let $\Omega\subseteq\mathbb{R}^d$ be a domain,
$V\in L_\mathrm{loc}^\infty(\Omega)$, and $p\in(1,\infty)$.
Suppose that the functional $Q$ is nonnegative on $C_0^{\infty}(\Omega)$. Then
\begin{itemize}
\item[(a)] The
functional $Q$ is either subcritical or critical in $\Omega$.
\item[(b)] If the functional $Q$ admits a ground state $v$, then $v$ satisfies
(\ref{groundstate}).
\item[(c)]
The functional $Q$ is critical in $\Omega$ if and only if
(\ref{groundstate}) admits a unique positive supersolution.
\item[(d)] Suppose that $1<p\leq d$. Then the functional $Q$
is critical (resp. subcritical) in $\Omega$ if and only if there
is a unique (up to a multiplicative constant) positive solution
$\varphi_0$ (resp. $G_Q^\Omega(\cdot,x_0)$) of the equation
$Q'(u)=0$ in $\Omega\setminus\{x_0\}$ which has minimal growth in
a neighborhood of infinity in $\Omega$ and has a removable (resp.
nonremovable) singularity at $x_0$.
\item[(e)] Suppose that $Q$ has a ground state $\varphi_0$. Then there exists
a positive continuous function $W$ in $\Omega$, such that for
every $\psi\in C_0^\infty(\Omega)$ satisfying $\int_\Omega \psi
\varphi_0 \,\mathrm{d}x \neq 0$ there exists a constant $C>0$ such
that the following Poincar\'e type inequality holds:
\begin{equation}\label{Poinc}
Q(u)+C\left|\int_{\Omega} \psi
u\,\mathrm{d}x\right|^p \geq C^{-1}\int_{\Omega} W|u|^p\,\mathrm{d}x \qquad
\forall u\in C_0^\infty(\Omega).\end{equation}
\end{itemize}
\end{theorem}
\begin{remarks}\label{remplapl2}{\em
1. Theorem \ref{mainky3} extends \cite[Theorem~1.5]{PT2} that
deals with the linear case $p=2$. The proof of Theorem
\ref{mainky3} relies on the {\em (generalized) Picone identity}
\cite{AH1,AH2}.
2. We call $G_Q^\Omega(\cdot,x_0)$ (after an appropriate
normalization) \emph{the positive minimal $p$-Green function of
the functional $Q$ in $\Omega$ with a pole at $x_0$}.
3. Suppose that $p=2$, and that there exists a function $\psi\in
L^2(\Omega)$ and $C\in \mathbb{R}$ such that
\begin{equation}\label{Poinc1}
Q(u)+C\left|\int_{\Omega} \psi u\,\mathrm{d}x\right|^2\geq 0 \qquad
\forall u\in C_0^\infty(\Omega),
\end{equation}
then the negative $L^2$-spectrum of $Q'$ is either empty or
consists of a single simple eigenvalue.
}\end{remarks}
We state now several positivity properties of the functional $Q$
in parallel to the criticality theory presented in
sections~\ref{secpreliminaries}--\ref{secindef}.
For $V\in L^\infty_{\mathrm{loc}}(\Omega)$, we use the notation
\begin{equation} Q_V(u):=\int_\Omega(|\nabla u|^p+V|u|^p)\,\mathrm{d}x\end{equation} to emphasize
the dependence of $Q$ on the potential $V$.
\begin{proposition}
\label{monPot} Let $V_j\in L^\infty_{\mathrm{loc}}(\Omega)$,
$j=1,2$. If $V_2\gneqq V_1$ and $Q_{V_1}\geq 0$ in $\Omega$, then
$Q_{V_2}$ is subcritical in $\Omega$.
\end{proposition}
\begin{proposition}
\label{monDom}
Let $\Omega_1\subset\Omega_2$ be domains in ${\mathbb R}^d$ such that $\Omega_2\setminus\overline{\Omega_1}\neq\emptyset$.
Let $Q_V$ be defined on $C_0^\infty(\Omega_2)$.
1. If $Q_V\geq 0$ on $C_0^\infty(\Omega_2)$, then $Q_V$ is
subcritical in $\Omega_1$.
2. If $Q_V$ is critical in $\Omega_1$, then $Q_V$ is supercritical
in $\Omega_2$.
\end{proposition}
\begin{proposition}\label{Prop2}
Let $V_0, V_1\in L^\infty_{\mathrm{loc}}(\Omega)$,
$V_0\neq V_1$. For $s\in \mathbb{R}$ we denote \begin{equation}
Q_s(u):=sQ_{V_1}(u)+(1-s)Q_{V_0}(u),\end{equation} and suppose that
$Q_{V_j}\geq 0$ on $C_0^{\infty}(\Omega)$ for $j=0,1$.
Then the functional $Q_s\geq 0$ on $C_0^{\infty}(\Omega)$ for all $s\in[0,1]$.
Moreover, if $V_0\neq V_1$, then $Q_s$ is subcritical in $\Omega$
for all $s\in(0,1)$.
\end{proposition}
\begin{proposition}\label{strictpos}
Let $Q_V$ be a subcritical in $\Omega$. Consider $V_0\in
L^\infty(\Omega)$ such that $V_0\ngeq 0$ and $\operatorname{supp}
V_0\Subset\Omega$. Then there exist $0<\tau_+<\infty$, and
$-\infty\leq \tau_-<0$ such that $Q_{V+sV_0}$ is subcritical in
$\Omega$ for $s\in(\tau_-,\tau_+)$, and $Q_{V+\tau_+ V_0}$ is
critical in $\Omega$. Moreover, $\tau_-=-\infty$ if and only if
$V_0\leq 0$.
\end{proposition}
\begin{proposition}\label{propintcond}
Let $Q_V$ be a critical functional in $\Omega$, and let
$\varphi_0$ be the corresponding ground state. Consider $V_0\in
L^\infty(\Omega)$ such that $\operatorname{supp} V_0\Subset\Omega$. Then there
exists $0<\tau_+\leq\infty$ such that $Q_{V+sV_0}$ is subcritical
in $\Omega$ for $s\in(0,\tau_+)$ if and only if
\begin{equation}\label{intcond}
\int_\Omega
V_0(x)\varphi_0(x)^p\,\mathrm{d}x>0.
\end{equation}
\end{proposition}
\begin{center}
{\bf Acknowledgments} \end{center} The author expresses his
gratitude to F.~Gesztesy and M.~Murata for their valuable remarks.
The author is also grateful to the anonymous referee for his
careful reading and useful comments. This work was partially
supported by the Israel Science Foundation founded by the Israeli
Academy of Sciences and Humanities, and the Fund for the Promotion
of Research at the Technion.
|
2,869,038,156,392 | arxiv | \section{Introduction}
\begin{figure*}[!ht]
\centering
\includegraphics[width=\textwidth]{figures/avion_both.pdf}
\caption{Plane of the ShapeNet dataset sampled with 5k points. \textbf{Left:} Point cloud produced by MongeNet. \textbf{Right:} Point cloud produced by the random uniform sampler. Note the clamping pattern across the mesh produced by the random uniform sampling approach. }
\label{fig:aviaoLupa}
\end{figure*}
Recently, computer vision researchers have demonstrated an increasing interest in developing deep learning models for 3D data understanding~\cite{Tatarchenko2019,cao2020comprehensive}. As successful applications of those models, we can mention single view object shape reconstruction~\cite{wang2018pixel2mesh,gupta2020neural}, shape and pose estimation~\cite{Kulon2020CVPR}, point cloud completion and approximation~\cite{Hanocka2020p2m}, and brain cortical surface reconstruction~\cite{SantaCruz2020deepcsr}.
A ubiquitous and important component of these 3D deep learning models is the computation of distances between the predicted and ground-truth meshes either for loss computation during training or quality metric calculations for evaluation.
In essence, the mainstream approach relies on sampling point clouds from any given meshes to be able to compute point-based distances such as the Chamfer distance~\cite{borgefors1984distance} or earth mover's distance~\cite{rubner1998metric}.
While there exist in-depth discussions about these distance metrics in the literature \cite{fan2017point,liu2020morphing}, the mesh sampling technique itself remains uncharted, where the uniform sampling approach is widely adopted by most of the 3D deep learning models \cite{wang2018pixel2mesh,mescheder2019occupancy,gupta2020neural}.
This approach is computationally efficient and is the only mesh sampling method available in existing 3D deep learning libraries like PyTorch3D~\cite{ravi2020pytorch3d} and Kaolin~\cite{kaolin2019arxiv}. However, it does not approximate the underlying surface accurately since it produces a clustering of points (clamping) along the surface resulting in large under-sampled areas and spurious artifacts for flat surfaces. For instance, Figure~\ref{fig:mainPicturePaper} shows that the Voronoi Tessellation for uniform sampled points is distributed irregularly, while Figure~\ref{fig:aviaoLupa} highlights sampling artifacts in the tail and wings of an airplane from ShapeNet.
In this paper, we revisit the mesh sampling problem and propose a novel algorithm to sample point clouds from triangular meshes. We formulate this problem as an optimal transport problem between simplexes and discrete Dirac measures to compute the optimal solution. Due to the computational challenge of this algorithm, we train a neural network, named MongeNet, to predict its solution efficiently.
As such, MongeNet can be adopted as a mesh sampler during training or testing of 3D deep learning models providing a better representation of the underlying mesh. As shown in~Figure~\ref{fig:mainPicturePaper}, MongeNet sampled points result in uniform Voronoi cell areas which better approximate the underlying mesh surfaces without producing sampling artifacts (also exemplified in Figure~\ref{fig:aviaoLupa}).
To evaluate our approach, we first compare the proposed sampling technique to existing methods, especially those within the computer graphics community. We show that our sampling scheme better approximates triangles for a reduced number of sampled points. Then, we evaluate the usefulness of MongeNet on 3D deep learning tasks as a mesh approximator to input point clouds using the state-of-the-art Point2Mesh model \cite{Hanocka2020p2m}. To conduct an in-depth analysis of our results, we detail why our method is performing better in the computation of the distance between two point clouds stemming from the same object surface.
In all tasks, the proposed approach is more robust, reliable, and better approximates the target surface when compared to the widely used uniform mesh sampling technique. We demonstrate a significant reduction in approximation error by a factor of 1.89 for results computed on the ShapeNet dataset.
\section{Related work}
\begin{figure*}[!ht]
\centering
\includegraphics[width=.24\textwidth]{figures/bench_uncut.pdf}
\includegraphics[width=.24\textwidth]{figures/3kAtquasirandom_benchDual_1.pdf}
\includegraphics[width=.24\textwidth]{figures/bench_cut.pdf}
\includegraphics[width=.24\textwidth]{figures/3kAtquasirandom_benchDual_2.pdf}
\caption{Two point clouds with 3K points drawn with the BNLD~\cite{perrier2018sequences} quasi-random sequence for two different triangulations describing the same 3D model. One would expect to have a very close distance between the point clouds, however, due to the deterministic nature of the sampler, some artefactual patterns appear.}
\label{fig:SameContinuousObject}
\end{figure*}
Sampling theory has been thoroughly studied for computer graphic applications~\cite{halton1964radical,shirley1991discrepancy,mccool1992hierarchical,kollig2002efficient}, especially for rendering and geometry processing algorithms that rely on Monte-Carlo simulation. Uniform random sampling is subject to clamping, resulting in approximation error and a poor convergence property. Several improvements over the basic uniform sampling technique have been proposed; these rely on techniques that allow the ability to enforce an even distribution of the samples. Poisson disk samplers~\cite{bridson2007fast} reject points that are too close to an existing sample, thereby enforcing a minimal distance between points. Jittering methods~\cite{cook1986stochastic,christensen2018progressive} use a divide and conquer approach to subdivide the sampling space into even-sized strata and then draw a sample in each. Quasi-random sequences~\cite{sobol1967distribution,perrier2018sequences} are deterministic sequences that result in a good distribution of the samples.
All of these samplers are very efficient, allowing the ability to sample hundreds of thousands of points within seconds, and they benefit from good asymptotic properties. However, as will be described in Section~\ref{sec:comparison}, they do not provide an optimal approximation property when the number of samples is limited. This limited number of samples is typical in numerous geometric deep learning applications where less than a hundred points are sampled per triangle.
Sampling a mesh using a point cloud has recently been proposed with elegant computational geometry algorithms~\cite{merigot2018algorithm,xin2016centroidal,wang2015intrinsic}, generalizing the euclidean Centroidal Voronoi Tessellation (CVT) for surfaces. However, the best approaches have a time complexity in $\mathcal{O}(n^2 \log n)$ which remain computationally non-affordable for computer vision applications. Training a deep learning model requires extensive use of sampling, and faster methods are needed. Indeed for a large number of applications~\cite{wang2018pixel2mesh,hanocka2020point2mesh,gupta2020neural}, this sampling is performed online and at each training step when a new mesh is predicted by the neural network.
To circumvent all the previously described impediments, we propose MongeNet, a new computationally efficient sampling method (0.32 sec for sampling 1 million points on a mesh with 250K faces), while having good approximation properties in the sense of the optimal transport distance between the generated point cloud and the associated mesh.
Related to our work, \cite{lang2020samplenet} propose a point cloud subsampling technique that selects the most relevant points of an input point cloud according to a given task, whereas \cite{liu2020morphing} propose a technique to complete sparse point clouds. Our technique is different since we propose to produce a more accurate and efficient point cloud estimate from a continuous object like a triangular mesh.
\section{Methods}
The quality of mesh sampling can be grasped visually: given a mesh, one expects that a generated point cloud should be distributed evenly over the mesh, describing its geometry accurately as illustrated in Figure~\ref{fig:aviaoLupa}, and uniform random sampling does not meet that expectation. Our proposed sampling is based on optimal transport: given a set of points, how to optimize their positions to minimize the quantity of energy needed to spread their masses on the mesh. Or less formally, given a uniform density of population, how to optimally place click-and-collect points to minimize the overall travel time. We coin our neural network MongeNet in tribute to the mathematician Gaspard Monge and his innovative work on optimal transport.
\subsection{The sampling problem}
To evaluate the distance between two objects, a natural framework is to use an optimal transport distance~\cite{villani2008optimal,santambrogio2015optimal,peyre2019computational}, while representing the mesh objects and point clouds as probability measures. Given a transportation cost $c$, the distance between two probability measures $\mu$ and $\nu$ defined on $X$ (resp. $Y$) is the solution of the following minimization problem,
\begin{equation}\label{eqn:transportOptimalDistance}
\min_{\gamma \in \Pi(\nu,\mu)} \int_{X\times Y} c(x,y) d\gamma(x,y),
\end{equation}
where $\Pi(\nu,\mu)$ is the set of couplings of $(\mu,\nu)$.
In order to consider this type of distance, one has to cast the object of interest into probability measures. For a triangular mesh $T$ of simplexes $(t_i)_{i=1\cdots m}$ one can consider the continuous measure $\mu_c$ carried by a union of simplexes which is defined for any Borel set $B \subseteq \mathbb{R}^3$ as,
\begin{equation}
\mu_c^T(B) = \frac{1}{| T |} \sum_{t_i \in T} \int_{B \cap t_i} d\mathcal{H}^2(x),
\end{equation}
with $|T|$ the total area of the mesh and $\mathcal{H}^2$ the 2-dimensional Hausdorff measure. For discrete point clouds with points $(\mathbf{x}_i)_{i=1\cdots n}$ the discrete measure $\mu_d$ reads as a sum of $n$ equi-weighted Dirac masses with mass $\frac 1 n $,
\begin{equation}
\mu_d^T = \frac{1}{n} \sum_{t_i \in T} \sum_{j=1}^{ a_i} \delta_{\mathbf{x_j}}, {\text{ with }} \mathbf{x_j}\sim U(t_i),
\end{equation}
where $a_i$ is proportional to the area of the $i$-th triangle $t_i$ with $\sum_i a_i = n$ and where $U(t_i)$ is the uniform distribution on simplex $t_i$. Note that computing an optimal transport distance between two continuous objects is practically intractable for our applications since it requires continuous probability density~\cite{benamou2000computational,papadakis2014optimal}.
To solve this problem, discretization of a continuous mesh is performed, then, with these point cloud discretizations one can compute a fast distance. More specifically, the number of sampled points per face is drawn according to a multinomial distribution parameterized by the number of queried points $n$. The points on the face are drawn using the square-root parametrization~\cite{turk1990generation,ladicky2017point}, which given the triangle $t_i = (V_i^1,V_i^2,V_i^3)$ reads as,
\begin{equation}
p = (1 - \sqrt{u_1}) V_i^1 + \sqrt{u_i}u_2 V_i^2 + (1-u_2)\sqrt{u_1}V_i^3, \label{eq:squareRootParam}
\end{equation}
where $p \in \mathbb{R}^3$ is a sampled point on the triangle $t_i$ and $u_1,u_2 \sim U([0,1])$.
Providing such a discretization of the triangular mesh allows the use of fast metrics such as the Chamfer distance (CD), earth mover's distance (EMD), and Hausdorff distance. The common issue of this technique is that one has to control the discretization error stemming from converting a continuous measure to a discrete one.
For two meshes $T^1$ and $T^2$, the distance between the two point clouds can be used as a proxy for the distance between the two corresponding meshes. Indeed, the triangle inequality for any Wasserstein metrics $W$ yields,
\begin{align}
&W(\mu_c^{T^1},\mu_c^{T^2}) \leq \nonumber\\
&\underbrace{W(\mu_c^{T^1},\mu_d^{T^1}) + W(\mu_c^{T^2},\mu_d^{T^2})}_{\text{discretization errors}} + W(\mu_d^{T^1},\mu_d^{T^2}). \label{eqn:TriangleInegality}
\end{align}
In this paper, we propose to reduce the discretization error with our new sampling technique to better represent continuous objects with point clouds. Notwithstanding this property, we want to bring the reader's attention towards the re-meshing invariance of the proposed method. From \eqref{eqn:TriangleInegality}, suppose that $T^1$ and $T^2$ are two triangulation instances from the same mesh, obtained with edge splitting or re-meshing. The distance between the two continuous objects is null and one would like the discretized form of this distance to be null as well. As illustrated in Figure~\ref{fig:SameContinuousObject}, sampled point clouds issued from deterministic quasi-random sequences~\cite{perrier2018sequences} can produce distinctive results. We observed that for very structured meshes, interrelation between points and artefactual patterns could appear, further reducing the accuracy of the estimated distance between them.
\subsection{MongeNet}
MongeNet is a model intended to replace the uniform sampling technique described in Equation~\eqref{eq:squareRootParam}: given the coordinates of a triangle (2-simplex) $t_i=(V_i^1,V_i^2,V_i^3)$ and a number of sampling points $\ell$, one generates a random sequence $S_k$ of points lying on $t_i$.
MongeNet is trained to minimize the 2-Wasserstein distance (optimal transport distance~\eqref{eqn:transportOptimalDistance} for the squared euclidean metric) between the measures carried by uniform Dirac masses $\mu(S_k)$ with positions $\mathbf{s}_k$ and the continuous measure $\mu(t_i)$ carried by the simplex $t_i$. Without loss of generality, MongeNet approximates the solution of,
\begin{equation}
\argmin_{(\mathbf{s}_k)_{k = 1 \cdots \ell} \in \mathbb{R}^{\ell\times3}} W_2^2 (\mu(t_i),\mu(S_k)), \label{eq:ViadoTransport}
\end{equation}
with $W_2$ the 2-Wasserstein distance. Such a problem falls in the semi-discrete optimal transport framework.
In this formalism, one can compute the optimal transport distance solving a convex optimization problem~\cite{merigot2011multiscale,de2012blue,levy2015numerical}. This problem relies on a powerful geometrical tool, the Laguerre Tesselation or power-diagram, corresponding to a weighted version of the Voronoi tessellation. To find a sampling with good quantization noise power~\cite{levy2015numerical,du1999centroidal} there exist iterative algorithms \cite{merigot2011multiscale}. The resulting point cloud on a triangle is a Centroidal Voronoi Tessellation (CVT) such that each Dirac mass is placed exactly on the center of mass of its Voronoi cell~\cite{aurenhammer1992minkowski}.
Solving this problem is time consuming and computationally too expensive for most applications. Instead, we propose to learn the optimal points position using a deep neural network so that a point cloud with a good approximation property can be generated within a few centiseconds. Such an optimal point sampling learned by MongeNet is displayed in Figure~\ref{fig:mainPicturePaper}~and~\ref{fig:aviaoLupa}.
\textbf{Network Training.} MongeNet is a feed-forward neural network denoted as $f_\theta$, and parameterized by its learnable parameter $\theta$. It takes as input a triangle $t$, a discrete number of output points $\ell \in \llbracket 1, 30 \rrbracket$, and a random noise $p\in \mathbb{R}$ randomized following a standard normal distribution. It outputs a random sequence of $\ell$ points $f_\theta(t,\ell,p) \in \mathbb{R}^{3\times\ell}$.
As a supervised model, MongeNet is trained to minimize the regularized empirical risk across a given training set $\mathcal{D} = \{(t_i, S_{i})_{i=1..N}\}$ consisting in a sequence of pairs of triangles and sampled points,
\[
\argmin_{\theta} \sum_{i=1}^{N} \sum_{\ell=1}^{30} \mathcal{L}(f_\theta(t_i,\ell,p_i),S_{i}),
\]
with $p_i \sim \mathcal{N}(0,1)$ and the loss function $\mathcal{L}$ defined as,
\begin{align*}
\mathcal{L}(t,\ell,p,S) &= W_2^\varepsilon(f_\theta(a,\ell,p),S) \nonumber\\
& -\alpha W_2^\varepsilon(f_\theta(a,\ell,p),f_\theta(a,\ell,p')).
\end{align*}
where $W_2^\varepsilon$ is the $\varepsilon$-regularized optimal transport~\cite{cuturi2013sinkhorn,charlier2020kernel}, and $p' \sim \mathcal{N}(0,1)$ is an adversarial Gaussian noise input that encourages entropy (different point locations) in the predicted point cloud.
We minimize this loss using the Adam Optimizer with batches of 32 triangles for approximately 15K iterations until the validation loss reaches a plateau. We set the blending parameter to $\alpha=0.01$ and the Entropic regularization $\varepsilon = 5\cdot 10^{-5}$.
\textbf{Generation of the dataset.}
The training examples consist of $19,663$ triangles sampled with $30, 50, 100, 200, 300, 500, 1000,$ and $2000$ points. The optimal point configurations are obtained using blue-noise algorithms provided in~\cite{de2019differentiation,lebrat2019optimal}. In order to obtain a continuous measure from the triangle coordinates, we discretize the measure carried by each triangle using $\mathcal{Q}^1$ quadrilateral finite elements on a grid of resolution $500\times 500$.
\textbf{Generating an arbitrary number of points per face.} The current implementation of MongeNet allows the computation of up to 30 points per-face. This is generally sufficient for most deep-learning applications. However, since this number can be arbitrarily large, we provide a local refinement technique that allows sampling any number of points per face. It amounts to splitting the largest edge of the triangle containing more than 30 points, to obtain two smaller triangles with a reduced area. This splitting operation is repeated until all of the faces are sampled with at most 30 points.
\textbf{Reduction to invariant learning-problem.} The shape space for triangles is a two-dimensional manifold~\cite{kendall2009shape}, thence up to normalization, any triangle can be parameterized with 2 unknowns. We thus map all the triangles to the unit square $[0,1]^2$, fixing the longest edge coordinates's to $(0,0)\rightarrow(0,1)$. This operation is depicted in Figure~\ref{fig:mappingUnitSquare}. All the operations are angle preserving so that the inverse transformations will conserve geometric optimality of the point cloud predicted on the unit square.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/triangles_mapping.pdf}
\caption{The query face is mapped towards the unit square using translation, rotation and reflection (isometries) and a uniform scaling (similarity).}
\label{fig:mappingUnitSquare}
\end{figure}
\textbf{Network Architecture.} Given an invariant triangle representation and a random variable sampled from a standard Gaussian distribution, MongeNet outputs 30 sets ranging from 1 to 30 random points. For computational efficiency, these output sets are tensorized as a matrix with 465 rows and 2 columns such that the $i$-th sampled point in the $j$-th output set with $j$ samples is in the $\frac{j\left( j - 1\right)}{2} + i - 1$ row of this matrix. The network architecture consists of three linear layers of 64 output neurons with a ReLU activation function and Dropout between layers. The output uses sigmoid functions to constrain the prediction to a unit square which is remapped to points in the input triangle using the area-preserving parameterization with low-distortion presented in \cite{heitz2019low}. It is then remapped to the original triangle using the inverse of the transformation described in Figure~\ref{fig:mappingUnitSquare}.
\textbf{Complexity analysis.}
The complexity of MongeNet is tantamount to the evaluation of a feedforward neural network. It is done rapidly on GPU and the computation across triangles is performed by batch. MongeNet's performance ({\bf MN}) is competitive with pytorch3D's random uniform sampler ({\bf RUS}), we compare their runtime in Table~\ref{tab:comparisonSpeed}.
\begin{table}[h]
\begin{adjustbox}{width=.95\linewidth}
\begin{tabular}{l|c|c|c|c|c|c}
\# Faces & 10k & 20k & 30k & 40k & 60k & 80k \\
\hline
{\bf RUS} & 1.14 ms & 1.50ms & 1.53ms & 1.52ms & 1.53ms & 1.53ms\\
\hline
{\bf MN} & 2.89 ms & 5.41 ms & 7.90 ms & 10.5 ms & 16.0 ms& 21.7 ms\\
\end{tabular}
\end{adjustbox}
\caption{{Runtimes for sampling 20k points on a mesh with an increasing number of faces and using an Nvidia RTX 3090.}}\label{tab:comparisonSpeed}
\end{table}
MongeNet's complexity scales linearly with the number of faces to be sampled. Non-deterministic CPU-based methods described in the next section cannot be used during training due to their longer runtime for sampling a mesh with more that 10k faces ($>1s$).
\section{Experiments}
\subsection{Comparison with pre-existing sampling methods}
\label{sec:comparison}
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth]{figures/comparison_CVPR.pdf}
\caption{Comparison of different sampling techniques for approximating a triangle with an increasing number of points.}
\label{fig:ComparisonMethods}
\end{figure}
\begin{figure*}[htb!]
\centering
\includegraphics[width=.32\textwidth]{figures/Density_Error_05.pdf}
\includegraphics[width=.32\textwidth]{figures/Density_Error_10.pdf}
\includegraphics[width=.32\textwidth]{figures/Density_Error_25.pdf}
\caption{Distribution of the approximation error for 500 triangles drawn uniformly random. From left to right: an approximating point cloud with 5 sampled points, 10 sampled points, and 25 sampled points.}
\label{fig:triangleSampledUniformly}
\end{figure*}
To evaluate the performance of our method, we compare it to three recent sampling methods: 1) Projective Blue-Noise Sampling (PNBS)~\cite{reinert2016projective} and the dart-throwing algorithm with a rule of rejection that depends on full-dimensional space and projections on lower-dimensional subspaces; 2) progressive multi-jittered sample sequence (PMJ)~\cite{christensen2018progressive}; and 3) Blue-Noise low discrepancies sequences (BNLD)~\cite{perrier2018sequences}, a quasi-random sequence with increased blue noise properties. We compare these methods against the random uniform sampling, and the sampling generated by our model MongeNet for an isosceles triangle and for an increasing number of points and 25 randomly drawn samples. All the distances between the estimated point cloud samples and the continuous object are obtained via computing the distance between the predicted points and $2000$ points sampled on the triangle using a semi-discrete optimal transport blue-noise algorithm~\cite{de2012blue}. The results are reported in Figure~\ref{fig:ComparisonMethods}.
Since our method approximates the point cloud location in the sense of optimal transport, it exhibits the best approximation error for this metric. The average 2-Wasserstein distance was 1.89 times smaller than the random uniform sampling, 1.30 times smaller than the PMJ sampling method, and 1.25 times smaller for BNLD. The MongeNet samples are different with each random sampling of the latent variables $p$, but the resulting point clouds have a similar error of approximation. Note that due to the deterministic nature of the BNLD sequence, there is no variance in the approximation error.
Of note, we are interested in the sampling patterns for a reduced number of points (typically $<100$) whereas related works ~\cite{reinert2016projective,christensen2018progressive,perrier2018sequences} have focused on sampling patterns for many more points ($>10,000$).
To show the behavior for a larger variety of triangles, we sample randomly 500 triangles uniformly into the unit square and examine the distribution of the approximation error for random uniform sampling, MongeNet, and its two forerunners PMJ and BNLD. We repeat the sampling three times using different random seeds to average the sampling performance across a given triangle. This experiment is summarized for 5, 10, and 25 sampled points in Figure~\ref{fig:triangleSampledUniformly}. Among all of the samplers, MongeNet proposes a triangle sampling with on average the best approximation quality.
\subsection{Approximation error quantification}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/mesh_approximation.pdf}
\caption{Average approximation errors between two point clouds sampled from different tessellations of the same underlying surface.}
\label{fig:MetricAcross}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/remeshing_variation.pdf}
\caption{Boxplot of the standard deviation of the approximation error across repetitions in terms of Chamfer distance and grouped by ShapeNet categories.}
\label{fig:MetricBoxPlot}
\end{figure}
We compare the approximation errors generated by sampling a similar surface from the ShapeNet dataset with MongeNet and a random uniform sampler.
Given a 3D model from a subset of 3200 models from the Shapenet dataset\footnote{The list of models used for this experiment is made available at \href{https://github.com/lebrat/MongeNet/blob/9a54c94c473d7ada5f8499232758915b9d84d067/resources/meshList.txt}{github.com/lebrat/MongeNet/resources/meshList.txt}}, we first sample 100K points using the random uniform sampler. Then we re-mesh randomly 30\% of the faces of the mesh to change its simplexes. With this new remeshing we produce a point cloud of 10K points using either the random uniform sampler or Mongenet. We report the Chamfer distance, the Hausdorff distance, and the F0.01 Score. We also report the EMD distance, but due to hardware limitations, this computation is performed using only 25K points of the original model.
In order to obtain more stable estimates, we reproduce the experiments for 10 repetitions.
In Figure~\ref{fig:MetricAcross} we display the average error produced in function of the sampler method used. Points that are sampled using MongeNet are consistently yielding a lower approximation error. In addition, the variance of the sampling is decreased, as depicted in Figure~\ref{fig:MetricBoxPlot}.
\subsection{Mesh Reconstruction}
\label{sec:Point2Mesh}
We now evaluate MongeNet on reconstructing watertight mesh surfaces to noisy point clouds which is a prerequisite for downstream applications such as rendering, collision avoidance, and human-computer interaction.
More specifically, we use the Point2Mesh \cite{Hanocka2020p2m} framework as the backbone. It consists of a MeshCNN that iteratively deforms a template mesh to tightly wrap a noisy target point cloud. At each iteration, it minimizes point cloud based distances between a point cloud sampled from the template mesh and the target point cloud. We contrast the performance of this method when configured with the standard uniform sampling and when configured with MongeNet.
Point2Mesh hyperparameters and the target noisy point clouds used in this experiment are kindly provided by the Point2Mesh authors\footnote{\href{https://github.com/ranahanocka/point2mesh}{https://github.com/ranahanocka/point2mesh}}.
Since Point2Mesh is an optimization based approach relying on random initialization, we repeat the experiment 10 times and report the average and standard deviation of the aforementioned metrics between the target point cloud and a point cloud sampled with 200K points from the produced final watertight mesh (using the same uniform sampling technique for both methods). Table~\ref{tab:point2mesh} presents the results of this experiment.
\input{point2mesh}
Point2Mesh equipped with MongeNet sampling outperforms systematically the alternative technique: it improves the Chamfer distance by a factor 1.77, the earth mover's distance by a factor 1.94, and increased on average the FS-0.01 score by 0.052.
We also noticed that larger differences could be observed for more detailed input shapes. We investigated this observation by increasing the difficulty of the reconstruction task using much more complex shapes from the Thingi10k dataset~\cite{zhou2016thingi10k}. As illustrated in Figure~\ref{fig:ComparisonTHingi}, the reconstruction errors are located either in the parts of the mesh with lots of fine details or in the areas with high curvature.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/tartaruga_details.pdf}
\caption{More difficult shape from Thingi10k reconstructed by the Point2Mesh backbone. \textbf{Left:} Point2Mesh with MongeNet sampling. \textbf{Right:} Point2Mesh with random uniform sampling.
}
\label{fig:ComparisonTHingi}
\end{figure}
\begin{figure*}[!ht]
\includegraphics[width=\textwidth]{figures/p2m_plot-crop_new.pdf}
\includegraphics[width=\textwidth]{figures/p2m_opt-crop-new.pdf}
\caption{\textbf{First row:} Loss function and metric evolution for 16 different random initializations. In blue the Point2Mesh method equipped with the MongeNet sampler and in red with the random uniform sampler, with translucent color the value for each individual run and in bold the average over all the runs.
\textbf{Second and third row:} Evolution of the optimized shape throughout the iterations for Point2Mesh equipped with MongeNet and with the random uniform sampler respectively.}
\label{fig:GalleryP2meshTraning}
\end{figure*}
We also conducted an in-depth analysis for an arduous mesh by running Point2Mesh 16 times, while monitoring all the metrics and the training loss, which are summarized in Figure~\ref{fig:GalleryP2meshTraning}. Across this experiment, all of the meta-parameters remain identical with the exception of the sampling technique. We noticed that for every scenario, because MongeNet provides a better representation for the face primitives, the distance between the target point cloud and the point cloud sample from the mesh decreased faster. As a result, for a given optimization time, the fidelity to the input point cloud was improved. At the end of the optimization, we can observe that only the MongeNet variant can recover the non-convex features of the hand.
\section{Conclusion}
In this paper, we have highlighted the limitations of the standard mesh sampling technique adopted by most of the 3D deep learning models including its susceptibility to irregular sampling and clamping, resulting in noisy distance estimation. To address this, we proposed a novel algorithm to sample point clouds from triangular meshes, formulated as an optimal transport problem between simplexes and discrete measures, for which the solution is swiftly learned by a neural network model named MongeNet. MongeNet is fast, fully differentiable, and can be adopted either for loss computation during training or for metric evaluation during testing. To demonstrate the efficacy of the proposed approach, we compared MongeNet to existing techniques widely used within the computer graphics community and evaluated the mesh approximation error using the challenging ShapeNet dataset. As a direct application, we also evaluated MongeNet on mesh approximation of noisy point clouds using the Point2Mesh backbone. In all these experiments, MongeNet outperforms existing techniques including the widely used random uniform sampling, for a modest extra computational cost.
\section*{Acknowledgement}
This work was funded in part through an Australian Department of Industry, Energy and Resources CRC-P project between CSIRO, Maxwell Plus and I-Med Radiology Network.
{\small
\bibliographystyle{ieee_fullname}
|
2,869,038,156,393 | arxiv | \section{Introduction}
It is well-known that discrete minimax problems and discrete Chebyshev problems (problems of best
$\ell_{\infty}$-approximation) can be reduced to equivalent nonlinear programming problems. Many methods for solving
minimax problems are based on application of nonlinear programming algorithms to these equivalent reformulations of
minimax problems (see such methods based on, e.g. sequential quadratic programming methods
\cite{RustemNguyen98,YuGao2002,JianQuanZhang2007,HuChenLi2009}, sequential quadratically constrained quadratic
programming methods \cite{ChaoWangLiagnHu,JianChao2010,JianMoQiu2014}, interior point methods
\cite{RustemZakovicParpas,ObasanTzalRustem}, augmented Lagrangian methods \cite{HeZhou2011,HeNie2013,HeLiuWang2016},
etc.). On the other hand, efficient, superlinearly or even quadratically convergent methods for solving minimax problems
can be also based on a convenient characterisation of an optimal solution of a minimax problem, that is, on optimality
conditions that are specific for minimax or Chebyshev problems (cf. such methods for discrete minimax problems
\cite{ConnLi92}, problems of rational $\ell_{\infty}$-approximation \cite{BarrodalePowellRoberts}, and synthesis of a
rational filter \cite{MalozemovTamasyan}). To extend such methods to the case of minimax and Chebyshev problems with
cone constraints (e.g. problems with semidefinite or semi-infinite constraints), first and second order optimality
conditions for such problems are needed.
Optimality conditions for general smooth optimisation problems with cone constraints and their particular classes were
studied in detail in multiple papers and monographs
\cite{Kawasaki,Cominetti,Shapiro97,BonComShap98,BonComShap99,BonnansShapiro,BonnansRamirez,Shapiro2009}. In the
nonsmooth case, much less attention has been paid to this subject. Optimality conditions for general nonsmooth
optimisation problems with cone constraints were studied in \cite{MordukhovichNghia}. Optimality conditions for
nonsmooth semidefinite programming problems were obtained in \cite{ZhaoGao2006,GolestaniNobakhtian2015,Tung}, while in
the case of nonsmooth semi-infinite programming problems they were analysed in
\cite{ZhenYang2007,KanziNobakhtian,Kanzi2011,CaristiFerrara,Gadhi}. However, to the best of the author's knowledge
optimality conditions for minimax problems and Chebyshev problems (problems of best $\ell_{\infty}$-approximation) with
cone constraints have not been thoroughly analysed in the literature.
In the case of unconstrained problems, optimality conditions for minimax problems can be formulated in many seemingly
non-equivalent forms some of which are not very well-known to researchers and relatively unusual in the context of
nonsmooth optimisation. In particular, optimality conditions for minimax problems can be formulated in terms of
so-called \textit{cadres} of minimax problems \cite{Descloux,ConnLi92} or in an \textit{alternance} form
\cite{MalozemovPevnyi,DaugavetMalozemov75,Daugavet,DaugavetMalozemov81,DaugavetMalozemov79,Malozemov77}, which is often
used within approximation theory \cite{Rice,Cheney}. In \cite{DemyanovMalozemov_Alternance,DemyanovMalozemov_Collect} it
was shown that the classical optimality condition $0 \in \partial f(x)$, where $\partial f(x)$ is some convex
subdifferential, can be rewritten in an alternance form. However, interconnections between various types of optimality
conditions for minimax and Chebyshev problems (particularly, sufficient optimality conditions and optimality conditions
for constrained minimax problems) have not been analysed before.
The main goal of this paper is to present a unified study of various types of optimality conditions for minimax and
Chebyshev problems with cone constraints scattered in the literature. Namely, we study six different forms of first
order necessary and sufficient optimality conditions for such problems (conditions involving a linearised problem,
Lagrange multipliers, subdifferentials and normal cones, $\ell_1$ penalty function, cadres, and alternance conditions)
and show that all these conditions are equivalent. We also demonstrate how they can be refined for particular types of
cone constraints, namely, for problems with equality and inequality constraints, problems with second order cone
constraints, as well as problems with semidefinite and semi-infinite constraints. Finally, we show how well-known
necessary and sufficient second order optimality conditions for cone constrained optimisation problems can be extended
to the case of minimax and Chebyshev problems and present several examples illustrating theoretical results.
It should be noted that although some results presented in this paper are straightforward generalisations of
corresponding results for smooth cone constrained optimisation problems to the minimax setting (e.g. optimality
conditions in terms of a linearised problem and Lagrange multipliers, Section~\ref{subsect:LagrangeMultipliers}, or
second order optimality conditions, Section~\ref{sect:SecondOrderOptCond}), many other results are completely new. In
particular, to the best of the author's knowledge interconnections between various forms of sufficient optimality
conditions for minimax problems and complete alternance (Thrms.~\ref{thrm:EquivOptCond_Subdiff} and
\ref{thrm:EquivOptCond_PenaltyFunc} and Section~\ref{subsect:Alternance_Cadre}), as well as alternance optimality
conditions for particular classes of minimax problems with cone constraints (Section~\ref{subsect:Examples}), have not
been studied before.
The paper is organised as follows. In Section~\ref{sect:FirstOrderOptCond}, we study various forms of first order
necessary and sufficient optimality conditions for cone constrained minimax problems.
Section~\ref{subsect:LagrangeMultipliers} is devoted to optimality conditions in terms of a linearised problem and
Lagrange multipliers. Optimality conditions involving subdifferentials, normal cones and a nonsmooth penalty function
are contained in Section~\ref{subsect:Subdifferentials_ExactPenaltyFunc}, while optimality conditions in terms of cadres
and in an alternance form are studied in Section~\ref{subsect:Alternance_Cadre}. A more detailed analysis of first order
optimality conditions for particular classes of cone constrained minimax problems is given in
Section~\ref{subsect:Examples}. Finally, Section~\ref{sect:SecondOrderOptCond} is devoted to second order necessary and
sufficient optimality conditions, while optimality conditions for Chebyshev (uniform approximation) problems are
discussed in Section~\ref{sect:ChebyshevProblems}.
\section{First order optimality conditions for cone constrained minimax problems}
\label{sect:FirstOrderOptCond}
Let $A \subseteq \mathbb{R}^d$ be a nonempty closed convex set, $Y$ be a Banach space, and $K \subset Y$ be a nonempty
closed convex cone. Denote by $Y^*$ the topological dual of $Y$, and by $\langle \cdot, \cdot \rangle$ either
the canonical duality pairing between $Y$ and its dual or the inner product in $\mathbb{R}^s$, $s \in \mathbb{N}$,
depending on the context.
Let $W$ be a compact Hausdorff topological space, and $f \colon \mathbb{R}^d \times W \to \mathbb{R}$ and
$G \colon \mathbb{R}^d \to Y$ be given functions. Throughout this article we suppose that the function
$f = f(x, \omega)$ is differentiable in $x$ for any $\omega \in W$, and the functions $f$ and $\nabla_x f$ are
continuous jointly in $x$ and $\omega$, while $G$ is continuously Fr\'echet differentiable. However, for the main
results below to hold true it is sufficient to suppose that $f(x, \omega)$ is continuous and continuously differentiable
in $x$ only on $\mathcal{O}(x_*) \times W$, and $G$ is continuously Fr\'echet differentiable on $\mathcal{O}(x_*)$,
where $\mathcal{O}(x_*)$ is a neighbourhood of a given point $x_*$.
Denote $F(x) = \max_{\omega \in W} f(x, \omega)$ for any $x \in \mathbb{R}^d$. Hereinafter we study the following cone
constrained minimax problem:
$$
\min F(x) \quad \text{subject to} \quad G(x) \in K, \quad x \in A. \eqno{(\mathcal{P})}
$$
Our aim is obtain several different forms of first order necessary and sufficient optimality conditions for the problem
$(\mathcal{P})$ and analyse how they relate to each other.
\subsection{Lagrange multipliers and first order growth condition}
\label{subsect:LagrangeMultipliers}
Let us start with an analysis of necessary and sufficient optimality conditions for the problem $(\mathcal{P})$
involving Lagrange multipliers. The main results of this subsection are a straightforward extension of the first order
necessary optimality conditions for cone constrained optimisation problems from \cite[Sect.~3.1]{BonnansShapiro} to the
case of cone constrained \textit{minimax} problems.
Firstly, we apply a standard linearisation procedure to the problem $(\mathcal{P})$ in order to reduce an analysis of
optimality conditions to the convex case. Then with the use of the linearised convex problem we obtain optimality
conditions involving Lagrange multipliers. To this end, we utilise the well-known \textit{Robinson's constraint
qualification} (RCQ) (see~\cite{Robinson75,Robinson76}).
Recall that RCQ is said to hold at a feasible point $x_*$ of the problem $(\mathcal{P})$, if
\begin{equation} \label{eq:RCQ}
0 \in \interior\Big\{ G(x_*) + D G(x_*)\big( A - x_* \big) - K \Big\},
\end{equation}
where $D G(x_*)$ is the Fr\'echet derivative of $G$ at $x_*$ and $\interior C$ stands for the topological interior of
a set $C$. RCQ allows one to easily compute the contingent (Bouligand tangent) cone to the feasible set of the problem
$(\mathcal{P})$.
Recal that \textit{the contingent cone} to a subset $C$ of a normed space $X$ at a point $x_* \in C$, denoted by
$T_C(x_*)$, consists of all those vectors $h \in X$ for which one can find sequences
$\{ \alpha_n \} \subset (0, + \infty)$ and $\{ h_n \} \subset X$ such that $\alpha_n \to 0$ and $h_n \to h$ as
$n \to \infty$, and $x_* + \alpha_n h_n \in C$ for all $n \in \mathbb{N}$.
Denote by $\Omega = \{ x \in A \mid G(x) \in K \}$ the feasible region of the problem $(\mathcal{P})$. The following
lemma on the contingent cone to the set $\Omega$ is well-known. Nevertheless, we present its proof for the sake of
completeness.
\begin{lemma} \label{lem:ContingConeToFeasibleSet}
Let RCQ hold true at a feasible point $x_*$ of the problem $(\mathcal{P})$. Then
\begin{equation} \label{eq:ContingConeToFeasibleSet}
T_{\Omega}(x_*) = \{ h \in T_A(x_*) \colon D G(x_*) h \in T_K(G(x_*)) \}.
\end{equation}
\end{lemma}
\begin{proof}
Introduce a function $\Phi \colon \mathbb{R}^d \to \mathbb{R}^d \times Y$ by setting $\Phi(x) = (x, G(x))$ for any
$x \in \mathbb{R}^d$. Clearly, $\Omega = \{ x \in \mathbb{R}^d \mid \Phi(x) \in A \times K \}$. By
\cite[Lemma~2.100]{BonnansShapiro} RCQ implies that
$$
0 \in \interior\big\{ \Phi(x_*) + D \Phi(x_*)\big( \mathbb{R}^d \big) - A \times K \big\}.
$$
Hence with the use of \cite[Corollary~2.91]{BonnansShapiro} one obtains that
\begin{equation} \label{eq:ContingentConeViaDerivative}
T_{\Omega}(x_*) = \big\{ h \in \mathbb{R}^d \bigm| D \Phi(x_*) h \in T_{A \times K} (\Phi(x_*)) \big\}.
\end{equation}
One can easily check that $T_{A \times K}(\Phi(x_*)) \subseteq T_A(x_*) \times T_K(G(x_*))$. On the other hand, if
$h \in T_A(x_*)$, then there exist sequences $\{ \alpha_n \} \subset (0, + \infty)$ and $\{ h_n \} \subset \mathbb{R}^d$
such that $\alpha_n \to 0$ and $h_n \to h$ as $n \to \infty$, and $x_* + \alpha_n h_n \in A$ for all $n \in \mathbb{N}$.
Consequently, for all $n \in \mathbb{N}$ one has $(x_* + \alpha_n h_n, G(x_*)) \in A \times K$ and
$(h, 0) \in T_{A \times K}(\Phi(x_*))$. Similarly, for any $w \in T_K(G(x_*))$ one has
$(0, w) \in T_{A \times K}(\Phi(x_*))$. Since $A \times K$ is a convex set, the contingent cone
$T_{A \times K}(\Phi(x_*))$ is convex. Therefore for all $h \in T_A(x_*)$ and $w \in T_K(G(x_*))$ one has
$(h, w) = (h, 0) + (w, 0) \in T_{A \times K}(\Phi(x_*))$, which implies that
$T_{A \times K}(\Phi(x_*)) = T_A(x_*) \times T_K(G(x_*))$. Hence bearing in mind
\eqref{eq:ContingentConeViaDerivative} and the fact that $D \Phi(x_*) h = (h, D G(x_*) h)$ one obtains that equality
\eqref{eq:ContingConeToFeasibleSet} holds true.
\end{proof}
Let $K^* = \{ y^* \in Y^* \mid \langle y^*, y \rangle \le 0 \: \forall y \in K \}$ \textit{the polar cone} of
$K$ and $L(x, \lambda) = F(x) + \langle \lambda, G(x) \rangle$ be the Lagrangian for the problem $(\mathcal{P})$.
Recall that a vector $\lambda_* \in Y^*$ is called \textit{a Lagrange multiplier} of $(\mathcal{P})$ at a feasible point
$x_*$, if $\lambda_* \in K^*$, $\langle \lambda_*, G(x_*) \rangle = 0$, and
$[L(\cdot, \lambda_*)]' (x_*, h) \ge 0$ for all $h \in T_A(x_*)$, where $[L(\cdot, \lambda_*)]' (x_*, h)$ is the
directional derivative of the function $L(\cdot, \lambda_*)$ at $x_*$ in the direction $h$. Finally, if $\lambda_*$ is a
Lagrange multiplier of $(\mathcal{P})$ at a feasible point $x_*$, then the pair $(x_*, \lambda_*)$ is called
\textit{a KKT-pair} of the problem $(\mathcal{P})$.
\begin{theorem} \label{thrm:NessOptCond}
Let $x_*$ be a locally optimal solution of the problem $(\mathcal{P})$ and RCQ hold at $x_*$. Then:
\begin{enumerate}
\item{$h = 0$ is a globally optimal solution of the linearised problem
\begin{equation} \label{probl:LinearisedProblem}
\min_{h \in \mathbb{R}^d} \max_{\omega \in W(x_*)} \langle \nabla_x f(x_*, \omega), h \rangle \quad
\text{subject to} \quad \enspace D G(x_*) h \in T_K\big( G(x_*) \big), \quad h \in T_A(x_*),
\end{equation}
where $W(x_*) = \{ \omega \in W \mid f(x_*, \omega) = F(x_*) \}$;
\label{stat:NessOpt_LinearisedProblem}}
\item{the set of Lagrange multipliers at $x_*$ is a nonempty, convex, bounded, and weak${}^*$ compact subset of $Y^*$.
\label{stat:NessOpt_LagrangeMult}}
\end{enumerate}
\end{theorem}
\begin{proof}
\textbf{Part~\ref{stat:NessOpt_LinearisedProblem}.} Fix an arbitrary $h \in T_{\Omega}(x_*)$. By definition there exist
sequences $\{ \alpha_n \} \subset (0, + \infty)$ and $\{ h_n \} \subset \mathbb{R}^d$ such that $\alpha_n \to 0$ and
$h_n \to h$ as $n \to \infty$, and $x_* + \alpha_n h_n \in \Omega$ for all $n \in \mathbb{N}$.
As is well-known (see, e.g. \cite[Thrm.~4.4.3]{IoffeTihomirov}), from the fact that the function $f(x, \omega)$ is
differentiable in $x$, and the gradient $\nabla_x f(x, \omega)$ is continuous jointly in $x$ and $\omega$ it follows
that the function $F(x) = \max_{\omega \in W} f(x, \omega)$ is Hadamard directionally differentiable at $x_*$ and for
any $h \in \mathbb{R}^d$ its Hadamard directional derivative at $x_*$ has the from
\begin{equation} \label{eq:DirectDerivOfMaxFunc}
F'(x_*, h) = \lim_{[h', \alpha] \to [h, +0]} \frac{F(x_* + \alpha h') - F(x_*)}{\alpha} =
\max_{\omega \in W(x_*)} \langle \nabla_x f(x_*, \omega), h \rangle.
\end{equation}
Recall that $x_*$ is a locally optimal solution of the problem $(\mathcal{P})$. Therefore, for any sufficiently large
$n \in \mathbb{N}$ one has $F(x_* + \alpha_n h_n) \ge F(x_*)$, which implies that
$$
F'(x_*, h) = \lim_{n \to \infty} \frac{F(x_* + \alpha_n h_n) - F(x_*)}{\alpha_n} \ge 0.
$$
Thus, one has
$$
F'(x_*, h) = \max_{\omega \in W(x_*)} \langle \nabla_x f(x_*, \omega), h \rangle \ge 0 \quad
\forall \, h \in T_{\Omega}(x_*),
$$
which thanks to Lemma~\ref{lem:ContingConeToFeasibleSet} implies that $h = 0$ is a globally optimal solution of the
linearised problem \eqref{probl:LinearisedProblem}.
\textbf{Part~\ref{stat:NessOpt_LagrangeMult}.} Clearly, problem \eqref{probl:LinearisedProblem} is a \textit{convex}
cone constrained optimisation problem. For any $h \in \mathbb{R}^d$ and $\lambda \in Y^*$ denote by
$L_0(h, \lambda) = F'(x_*, h) + \langle \lambda, D G(x_*) h \rangle$ the standard Lagrangian for this problem. Observe
that for all $h \in \mathbb{R}^d$ one has $L_0(h, \lambda) = [L(\cdot, \lambda)]'(x_*, h)$ .
From the facts that the sets $A$ and $K$ convex and $x_*$ is a feasible point it follows that
$A - x_* \subseteq T_A(x_*)$ and $K - G(x_*) \subseteq T_K(G(x_*))$ (choose any sequence
$\{ \alpha_n \} \subset (0, 1)$ converging to zero and for any $n \in \mathbb{N}$ define $h_n = z - x_*$ for $z \in A$
or $h_n = z - G(x_*)$ for $z \in K$). Hence RCQ (see~\eqref{eq:RCQ}) implies that
$$
0 \in \interior\Big\{ D G(x_*) \big( T_A(x_*) \big) - T_K(G(x_*)) \Big\},
$$
i.e. the standard regularity condition (Slater's condition) for problem \eqref{probl:LinearisedProblem} holds true (see,
e.g. \cite[Formula~(3.12)]{BonnansShapiro}). Consequently, by \cite[Thrm.~3.6]{BonnansShapiro} there exists
$\lambda_* \in T_K(G(x_*))^*$ such that $0 \in \argmin_{h \in T_A(x_*)} L_0(h, \lambda_*)$.
Observe that $K + G(x_*) \subseteq K$, since $K$ is a convex cone and $G(x_*) \in K$. Consequently, one has
$K \subseteq K - G(x_*) \subseteq T_K(G(x_*))$. Hence bearing in mind the fact that $\lambda_* \in T_K(G(x_*))^*$ one
gets that $\lambda_* \in K^*$, which, in particular, implies that $\langle \lambda_*, G(x_*) \rangle \le 0$. On the
other hand, since $G(x_*) \in K$ and $K$ is a cone, one has $- G(x_*) \in T_K(G(x_*))$ (choose any sequence
$\{ \alpha_n \} \subset (0, 1)$ converging to zero and put $h_n = - G(x_*)$ for all $n \in \mathbb{N}$), which yields
$\langle \lambda_*, - G(x_*) \rangle \le 0$, i.e. $\langle \lambda_*, G(x_*) \rangle = 0$. Thus, one has
$\lambda_* \in K^*$, $\langle \lambda_*, G(x_*) \rangle = 0$, and
$$
[L(\cdot, \lambda_*)]'(x_*, h) = L_0(h, \lambda_*) \ge 0 \quad \forall h \in T_A(x_*),
$$
i.e. $\lambda_*$ is a Lagrange multiplier of the problem $(\mathcal{P})$ at $x_*$.
Let us show that the set of Lagrange multipliers of the problem $(\mathcal{P})$ at $x_*$, in actuality, coincides with
the set of Lagrange multipliers of the linearised problem \eqref{probl:LinearisedProblem}. Then taking into account the
fact that the set of Lagrange multipliers of the convex problem \eqref{probl:LinearisedProblem} is a convex, bounded,
and weak${}^*$ compact subset of $Y^*$ by \cite[Thrm.~3.6]{BonnansShapiro} we arrive at the required result.
Let $\lambda_*$ be a Lagrange multiplier of the problem $(\mathcal{P})$ at $x_*$. Since
$L_0(h, \lambda) = [L(\cdot, \lambda)]'(x_*, h)$ for all $h \in \mathbb{R}^d$, by definition it is sufficient to prove
that $\lambda_* \in T_K(G(x_*))^*$. To this end, fix any $v \in T_K(G(x_*))$. By the definition of contingent cone
there exist sequences $\{ \alpha_n \} \subset (0, + \infty)$ and $\{ v_n \} \subset Y$ such that $\alpha_n \to 0$ and
$v_n \to v$ as $n \to \infty$, and $G(x_*) + \alpha_n v_n \in K$ for all $n \in \mathbb{N}$. Since $\lambda_*$ is a
Lagrange multiplier of the problem $(\mathcal{P})$ at $x_*$, one has $\langle \lambda_*, G(x_*) \rangle = 0$ and
$\lambda_* \in K^*$, which implies that
$0 \ge \langle \lambda_*, G(x_*) + \alpha _n v_n \rangle = \alpha_n \langle \lambda_*, v_n \rangle$ for all
$n \in \mathbb{N}$. Therefore $\langle \lambda_*, v \rangle \le 0$ for any $v \in T_K(G(x_*))$, i.e.
$\lambda_* \in T_K(G(x_*))^*$, and the proof is complete.
\end{proof}
Let us now turn to sufficient optimality conditions. Typically, sufficient optimality conditions ensure not only that a
given point is a locally optimal solution of an optimisation problem under consideration, but also that a certain
(usually, second order) growth condition holds at this point. Therefore it is natural to study sufficient optimality
conditions simultaneously with growth conditions.
Recall that \textit{the first order growth condition} (for the problem $(\mathcal{P})$) is said to hold true at a
feasible point $x_*$ of the problem $(\mathcal{P})$, if there exist $\rho > 0$ and a neighbourhood $\mathcal{O}(x_*)$ of
$x_*$ such that $F(x) \ge F(x_*) + \rho | x - x_* |$ for any $x \in \mathcal{O}(x_*) \cap \Omega$, where, as above,
$\Omega$ is the feasible region of $(\mathcal{P})$ and $| \cdot |$ is the Euclidean norm.
By Theorem~\ref{thrm:NessOptCond} the condition
$$
\max_{\omega \in W(x_*)} \langle \nabla_x f(x_*, \omega), h \rangle \ge 0
\quad \forall h \in T_A(x_*) \colon D G(x_*) h \in T_K\big( G(x_*) \big)
$$
is a first order necessary optimality condition for the problem $(\mathcal{P})$. Keeping this condition in mind, let us
obtain the natural ``no gap'' sufficient optimality condition that is, in fact, equivalent to the validity of the first
order growth condition.
\begin{theorem} \label{thrm:SuffOptCond}
Let $x_*$ be a feasible point of the problem $(\mathcal{P})$. If
\begin{equation} \label{eq:SuffOptCond}
\max_{\omega \in W(x_*)} \langle \nabla_x f(x_*, \omega), h \rangle > 0
\quad \forall h \in T_A(x_*) \setminus \{ 0 \} \colon D G(x_*) h \in T_K\big( G(x_*) \big),
\end{equation}
i.e. if $h = 0$ is a unique globally optimal solution of the linearised problem \eqref{probl:LinearisedProblem}, then
the first order growth condition holds at $x_*$. Conversely, if the first order growth condition and RCQ hold at $x_*$,
then inequality \eqref{eq:SuffOptCond} is valid.
\end{theorem}
\begin{proof}
Let condition \eqref{eq:SuffOptCond} hold true. Arguing by reductio ad absurdum, suppose that the first order growth
condition does not hold true at $x_*$. Then for any $n \in \mathbb{N}$ one can find $x_n \in \Omega$ such that
$F(x_n) < F(x_*) + |x_n - x_*| / n$ and $x_n \to x_*$ as $n \to \infty$.
Denote $h_n = (x_n - x_*) / |x_n - x_*|$. Without loss of generality one can suppose that the sequence $\{ h_n \}$
converges to a vector $h$ such that $|h| = 1$. From the fact that $x_n \in \Omega = \{ x \in A \mid G(x) \in K \}$ it
follows that $h \in T_A(x_*)$ and $G(x_n) = G(x_*) + |x_n - x_*| D G(x_*) h_n + o(|x_n - x_*|) \in K$ for any
$n \in \mathbb{N}$, which obviously implies that $D G(x_*) h \in T_K(G(x_*))$. Furthermore, taking into account
\eqref{eq:DirectDerivOfMaxFunc} and the definition of $x_n$ one obtains that
$$
\max_{\omega \in W(x_*)} \langle \nabla_x f(x_*, \omega), h \rangle = F'(x_*, h)
= \lim_{n \to \infty} \frac{F(x_n) - F(x_*)}{|x_n - x_*|} \le 0,
$$
which contradicts optimality condition \eqref{eq:SuffOptCond}. Thus, the first order growth condition holds at $x_*$.
Suppose now that RCQ and the first order growth condition hold at $x_*$. Then there exist a neighbourhood
$\mathcal{O}(x_*)$ of $x_*$ and $\rho > 0$ such that $F(x) \ge F(x_*) + \rho |x - x_*|$ for any
$x \in \mathcal{O}(x_*) \cap \Omega$.
Fix an arbitrary $h \in T_A(x_*) \setminus \{ 0 \}$ such that $D G(x_*) h \in T_K( G(x_*) )$.
By Lemma~\ref{lem:ContingConeToFeasibleSet} one has $h \in T_{\Omega}(x_*)$. Hence by definition there exist sequences
$\{ \alpha_n \} \subset (0, + \infty)$ and $\{ h_n \} \subset \mathbb{R}^d$ such that $\alpha_n \to 0$ and $h_n \to h$
as $n \to \infty$, and $x_* + \alpha_n h_n \in \Omega$ for all $n \in \mathbb{N}$. Clearly,
$x_* + \alpha_n h_n \in \mathcal{O}(x_*)$ for any sufficiently large $n$. Therefore
$$
F'(x_*, h) = \lim_{n \to \infty} \frac{F(x_* + \alpha_n h_n) - F(x_*)}{\alpha_n} \ge
\lim_{n \to \infty} \frac{\rho |\alpha_n h_n|}{\alpha_n} = \rho |h| > 0,
$$
i.e. \eqref{eq:SuffOptCond} holds true.
\end{proof}
\begin{remark}
From the proof of the theorem above it follows that if RCQ and the first order growth condition with constant
$\rho > 0$ hold true at a feasible point $x_*$ of the problem $(\mathcal{P})$, then the first order growth condition
with the same constant holds true at the origin for the linearised problem \eqref{probl:LinearisedProblem}, which due
to the positive homogeneity of the problem implies that
\begin{equation} \label{eq:LinProbl_FirstOrderGrowth}
\max_{\omega \in W(x_*)} \langle \nabla_x f(x_*, \omega), h \rangle \ge \rho |h|
\quad \forall h \in T_A(x_*) \colon D G(x_*) h \in T_K\big( G(x_*) \big).
\end{equation}
Conversely, if this condition holds true, then arguing in almost the same way as in the proof of the first part of
Theorem~\ref{thrm:SuffOptCond} one can check that for any $\rho' \in (0, \rho)$ the first order growth condition with
constrant $\rho'$ holds true at $x_*$. Thus, there is a direct connection between the first order growth conditions for
the problem $(\mathcal{P})$ and the linearised problem \eqref{probl:LinearisedProblem}. Moreover, note that if
\eqref{eq:SuffOptCond} holds true, then there exists $\rho > 0$ such that \eqref{eq:LinProbl_FirstOrderGrowth} is
satisfied, and the least upper bound of all such $\rho$ is equal to
$\rho_* = \min_h \max_{\omega \in W(x_*)} \langle \nabla_x f(x_*, \omega), h \rangle$, where the minimum is taken over
all those $h \in T_A(x_*)$ for which $D G(x_*) h \in T_K(G(x_*))$ and $|h| = 1$ (the set of all such $h$ is obviously
compact, which implies that the minimum in the definition of $\rho_*$ is attained and $\rho_* > 0$). \qed
\end{remark}
\begin{remark} \label{rmrk:SuffOptCond_Lagrangian}
Note that optimality condition \eqref{eq:SuffOptCond} is satisifed, provided there exists a Lagrange multiplier
$\lambda_*$ of $(\mathcal{P})$ at $x_*$ such that $[L(\cdot, \lambda_*)]'(x_*, h) > 0$ for all
$h \in T_A(x_*) \setminus \{ 0 \}$. Indeed, fix any $h \in T_A(x_*) \setminus \{ 0 \}$ such that
$D G(x_*) h \in T_K(G(x_*))$. By the definition of Lagrange multiplier one has
$\lambda_* \in K^*$ and $\langle \lambda_*, G(x_*) \rangle = 0$, which implies that
$\langle \lambda_*, y - G(x_*) \rangle \le 0$ for all $y \in K$. Since $K$ is a closed convex set, one has
$T_K(G(x_*)) = \cl[\cup_{t \ge 0} t(K - G(x_*))]$ (see, e.g. \cite[Prp.~2.55]{BonnansShapiro}). Therefore
for any $y \in T_K(G(x_*))$ one has $\langle \lambda_*, y \rangle \le 0$. Consequently, one has
$\langle \lambda_*, D G(x_*) h \rangle \le 0$ and
\begin{align*}
\max_{\omega \in W(x_*)} \langle \nabla_x f(x_*, \omega), h \rangle
&\ge \max_{\omega \in W(x_*)} \langle \nabla_x f(x_*, \omega), h \rangle + \langle \lambda_*, D G(x_*) h \rangle \\
&= [L(\cdot, \lambda_*)]'(x_*, h) > 0
\end{align*}
for any $h \in T_A(x_*) \setminus \{ 0 \}$ such that $D G(x_*) h \in T_K(G(x_*))$, i.e. optimality condition
\eqref{eq:SuffOptCond} holds true. However, note that the converse statement does not hold true in the general case.
Indeed, for any smooth problem with $A = \mathbb{R}^d$ one has $\nabla_x L(x_*, \lambda_*) = 0$ by the definition of
Lagrange multiplier, and the inequality $[L(\cdot, \lambda_*)]'(x_*, h) > 0$ for all $h \ne 0$ cannot be satisfied, but
sufficient optimality condition \eqref{eq:SuffOptCond} might hold true. Consider, for example, the problem
$$
\min\: f(x) = -x \quad \text{subject to} \quad g(x) = x \le 0.
$$
The point $x_* = 0$ is a globally optimal solution of this problem. Moreover,
$\langle \nabla f(x_*), h \rangle = - h > 0$ for any $h \ne 0$ such that
$\langle \nabla g(x_*), h \rangle = h \le 0$, i.e. optimality condition \eqref{eq:SuffOptCond} holds true. \qed
\end{remark}
Let us also note that in the convex case a necessary optimality condition becomes a sufficient condition for a
global minimum. Recall that the mapping $G$ is called \textit{convex} with respect to the cone $-K$ (or
$(-K)$-\textit{convex}), if $G(\alpha x_1 + (1 - \alpha) x_2) - \alpha G(x_1) - (1 - \alpha) G(x_2) \in K$
for any $x_1, x_2 \in \mathbb{R}^d$ and $\alpha \in [0, 1]$ (see \cite[Def.~2.103]{BonnansShapiro}).
\begin{theorem} \label{thrm:OptCond_ConvexCase}
Let for any $\omega \in W$ the function $f(\cdot, \omega)$ be convex, the mapping $G$ be $(-K)$-convex, and let $x_*$ be
a feasible point of the problem $(\mathcal{P})$. Then:
\begin{enumerate}
\item{$\lambda_* \in K^*$ is a Lagrange multiplier of $(\mathcal{P})$ at $x_*$ iff $(x_*, \lambda_*)$ is a global saddle
point of the Lagrangian $L(x, \lambda) = F(x) + \langle \lambda, G(x) \rangle$, that is,
\begin{equation} \label{eq:GlobalSaddlePoint}
L(x, \lambda_*) \ge F(x_*) \ge L(x_*, \lambda) \quad \forall x \in A, \: \lambda \in K^*;
\end{equation}
\vspace{-7mm}\label{stat:GlobalSaddlePoint}}
\item{if a Lagrange multiplier of the problem $(\mathcal{P})$ at $x_*$ exists, then $x_*$ is a globally optimal solution
of $(\mathcal{P})$; conversely, if $x_*$ is a globally optimal solution of the problem $(\mathcal{P})$ and Slater's
condition $0 \in \interior\{ G(A) - K \}$ holds true, then there exists a Lagrange multiplier of $(\mathcal{P})$
at $x_*$.
\label{stat:LagrangeMult_ConvexCase}}
\end{enumerate}
\end{theorem}
\begin{proof}
\textbf{Part~\ref{stat:GlobalSaddlePoint}.} Let $\lambda_*$ be a Lagrange multiplier of $(\mathcal{P})$ at $x_*$. Note
that $\langle \lambda, G(x_*) \rangle \le 0$ for any $\lambda \in K^*$, since $x_*$ is a feasible point
(i.e. $G(x_*) \in K$), which implies that $L(x_*, \lambda) \le F(x_*)$ for all $\lambda \in K^*$. Thus, the second
inequality in \eqref{eq:GlobalSaddlePoint} holds true.
By the definition of Lagrange multiplier $\langle \lambda_*, G(x_*) \rangle = 0$, which yields
$L(x_*, \lambda_*) = F(x_*)$. Thus, the first inequality in \eqref{eq:GlobalSaddlePoint} is satisfied iff $x_*$ is a
point of global minimum of the function $L(\cdot, \lambda_*)$ on the set $A$. Arguing by reductio ad absurdum, suppose
that this statement is false. Then there exists $x_0 \in A$ such that $L(x_0, \lambda_*) < L(x_*, \lambda_*)$.
Under our assumptions the function $F$ is convex as the maximum of a family of convex functions. Moreover, for any
$\lambda \in K^*$ the function $\langle \lambda, G(\cdot) \rangle$ is convex as well, since for any
$x_1, x_2 \in \mathbb{R}^d$ and $\alpha \in [0, 1]$
$\langle \lambda, G(\alpha x_1 + (1 - \alpha) x_2) - \alpha G(x_1) - (1 - \alpha) G(x_2) \rangle \le 0$. Thus, the
Lagrangian $L(\cdot, \lambda_*)$ is convex. Therefore, for any $\alpha \in [0, 1]$ one has
$$
L(\alpha x_0 + (1 - \alpha) x_*, \lambda_*) - L(x_*, \lambda_*)
\le \alpha \big( L(x_0, \lambda_*) - L(x_*, \lambda_*) \big).
$$
Dividing this inequality by $\alpha$ and passing to the limit as $\alpha \to +0$ one obtains that
$[L(\cdot, \lambda_*)]'(x_*, x_0 - x_*) \le L(x_0, \lambda_*) - L(x_*, \lambda_*) < 0$, which contradicts the fact that
$\lambda_*$ is a Lagrange multiplier, since $x_0 - x_* \in T_A(x_*)$ by the fact that $A$ is a convex set. Thus,
the first inequality in \eqref{eq:GlobalSaddlePoint} holds true and $(x_*, \lambda_*)$ is a global saddle point of the
Lagrangian.
Let us prove the converse statement. Suppose that $(x_*, \lambda_*)$ is a global saddle point of $L(x, \lambda)$. Then
$L(x, \lambda_*) \ge F(x_*) \ge L(x_*, \lambda_*)$ for any $x \in A$ (see~\eqref{eq:GlobalSaddlePoint}), which implies
that $x_*$ is a point of global minimum of the function $L(\cdot, \lambda_*)$ and
$\langle \lambda_*, G(x_*) \rangle = 0$, since by the definition of global saddle point
$F(x_*) = L(x_*, \lambda_*) = F(x_*) + \langle \lambda_*, G(x_*) \rangle$.
Recall that the function $F$ is Hadamard directionally differentiable by \cite[Thrm.~4.4.3]{IoffeTihomirov}.
Consequently, the function $L(\cdot, \lambda_*)$ is Hadamard directionally differentiable as well. Therefore, applying
the necessary optimality condition in terms of directional derivatives (see, e.g. \cite[Lemma~V.1.2]{DemyanovRubinov})
one obtains that $[L(\cdot, \lambda_*)]'(x_*, h) \ge 0$ for all $h \in T_A(x_*)$, i.e. $\lambda_*$ is a Lagrange
multiplier of the problem $(\mathcal{P})$ at $x_*$.
\textbf{Part~\ref{stat:LagrangeMult_ConvexCase}.} Let $\lambda_*$ be a Lagrange multiplier of $(\mathcal{P})$ at $x_*$.
Then by the first part of the theorem $L(x, \lambda_*) \ge F(x_*)$ for all $x \in A$. By the definition of Lagrange
multiplier $\lambda_* \in K^*$, which implies that $\langle \lambda_*, G(x) \rangle \le 0$ for any $x$ such that
$G(x) \in K$. Thus, for any feasible point of the problem $(\mathcal{P})$ one has
$F(x) \ge L(x, \lambda_*) \ge F(x_*)$, i.e. $x_*$ is a globally optimal solution of $(\mathcal{P})$.
It remains to note that the converse statement follows directly from Theorem~\ref{thrm:NessOptCond} and the fact that
by \cite[Prp.~2.104]{BonnansShapiro} Slater's condition $0 \in \interior\{ G(A) - K \}$ is equivalent to RCQ, provided
$G$ is $(-K)$-convex.
\end{proof}
\subsection{Subdifferentials and exact penalty functions}
\label{subsect:Subdifferentials_ExactPenaltyFunc}
Note that both necessary and sufficient optimality conditions stated in Theorems~\ref{thrm:NessOptCond} and
\ref{thrm:SuffOptCond} are very difficult to verify directly. Let us show how one can reformulate them in a more
convenient way.
Denote by $N_A(x) = \{ z \in \mathbb{R}^d \mid \langle z, v \rangle \le 0 \: \forall v \in T_A(x) \}$ \textit{the normal
cone} to the convex set $A$ at a point $x \in A$. Note that $N_A(x)$ is the polar cone of $T_A(x)$ and
$N_A(x) = \{ z \in \mathbb{R}^d \mid \langle z, v - x \rangle \: \forall v \in A \}$, since
$T_A(x) = \cl[\cup_{t \ge 0} t(A - x)]$ by virtue of the fact that the set $A$ is convex
(see, e.g. \cite[Prp.~2.55]{BonnansShapiro}). For any subspace $Y_0 \subset Y$ denote by
$Y_0^{\perp} = \{ y^* \in Y^* \mid \langle y^*, y \rangle = 0 \: \forall y \in Y_0 \}$ \textit{the annihilator} of
$Y_0$. For the sake of correctness, for any linear operator $T \colon \mathbb{R}^d \to Y$ denote by
$[T]^*$ the composition of the natural isomorphism $i$ between $(\mathbb{R}^d)^*$ and $\mathbb{R}^d$, and the adjoint
operator $T^* \colon Y^* \to (\mathbb{R}^d)^*$, i.e. $[T]^* = i \circ T^*$.
Introduce the cone
$$
\mathcal{N}(x) = [D G(x)]^* (K^* \cap \linhull(G(x))^{\perp})
= \{ i(\lambda \circ D G(x)) \mid \lambda \in K^*, \: \langle \lambda, G(x) \rangle = 0 \}.
$$
Let us verify that the convex cone $\mathcal{N}(x) \subset \mathbb{R}^d$ is, in actuality, the normal cone to the set
$\Xi = \{ z \in \mathbb{R}^d \mid G(z) \in K \}$ at the point $x$.
\begin{lemma} \label{lem:NormalCone_ConeConstr}
Let $x \in \mathbb{R}^d$ be such that $G(x) \in K$. Then
\begin{equation} \label{eq:NormalCone_ConeConstr}
\mathcal{N}(x) \subseteq \Big( \big\{ h \in \mathbb{R}^d \bigm| D G(x) h \in T_K(G(x)) \big\} \Big)^*.
\end{equation}
Furthermore, if the weakened Robinson constraint qualification $0 \in \interior\{ G(x) + D G(x)(\mathbb{R}^n) - K \}$
is satisfied at $x$, then the opposite inclusion holds true and $\mathcal{N}(x) = (T_{\Xi}(x))^* = N_{\Xi}(x)$.
\end{lemma}
\begin{proof}
Choose any $v \in \mathcal{N}(x)$. Then $v = [D G(x)]^* \lambda$ for some $\lambda \in K^*$ such that
$\langle \lambda, G(x) \rangle = 0$. By definition $\langle \lambda, y - G(x) \rangle \le 0$ for any $y \in K$. Hence
with the use of the well-known equality $T_K(G(x)) = \cl[\cup_{t \ge 0} t(K - G(x))]$ (see, e.g.
\cite[Prp.~2.55]{BonnansShapiro}) one obtains that $\langle \lambda, y \rangle \le 0$ for any $y \in T_K(G(x))$.
Consequently, for any $h \in \mathbb{R}^d$ satisfying the condition $D G(x) h \in T_K(G(x))$ one has
$\langle v, h \rangle = \langle \lambda, D G(x) h \rangle \le 0$, that is, $v$ belongs to the right-hand side of
\eqref{eq:NormalCone_ConeConstr}.
Suppose now that the weakened RCQ holds at $x_*$, and let $v$ belong to the right-hand side of
\eqref{eq:NormalCone_ConeConstr}, that is, $\langle v, h \rangle \le 0$ for any $h \in \mathbb{R}^d$ such that
$D G(x) h \in T_K(G(x))$. In other words, $h = 0$ is a point of global minimum of the conic linear problem:
\begin{equation} \label{probl:ConicLinearProblem}
\min \: \langle -v, h \rangle \quad \text{subject to} \quad D G(x) h \in T_K(G(x)).
\end{equation}
Note that the contingent cone $T_K(G(x))$ is convex, since the cone $K$ is convex. Furthermore, from the weakened RCQ
and the inclusion $(K - G(x)) \subset T_K(G(x))$ it follows that the regularity condition
$0 \in \interior\{ D G(x)(\mathbb{R}^d) - T_K(G(x)) \}$ holds true for problem \eqref{probl:ConicLinearProblem}.
Therefore by \cite[Thrm.~3.6]{BonnansShapiro} there exists a Lagrange multiplier $\lambda$ for problem
\eqref{probl:ConicLinearProblem}, i.e. $- v + [D G(x)]^* \lambda = 0$ and
$\lambda \in T_K(G(x))^*$. Bearing in mind the equality $T_K(G(x)) = \cl[\cup_{t \ge 0} t(K - G(x))]$ one obtains
that $\langle \lambda, y - G(x) \rangle \le 0$ for any $y \in K$. Putting $y = 2 G(x)$ and $y = 0$ one gets that
$\langle \lambda, G(x) \rangle = 0$, while putting $y = z + G(x) \in K$ for any $z \in K$ (recall that $K$ is a convex
cone) one gets that $\langle \lambda, z \rangle \le 0$ for any $z \in K$ or, equivalently, $\lambda \in K^*$. Thus,
$v = [D G(x)]^* \lambda$ for some $\lambda \in K^*$ such that $\langle \lambda, G(x) \rangle = 0$, i.e.
$v \in \mathcal{N}(x)$ and the inclusion opposite to \eqref{eq:NormalCone_ConeConstr} is valid.
It remains to note that $T_{\Xi}(x) = \{ h \in \mathbb{R}^d \mid D G(x) h \in T_K(G(x)) \}$, since the weakened RCQ is
satisfied at $x_*$ (see, e.g. \cite[Corollary~2.91]{BonnansShapiro}).
Thus, $\mathcal{N}(x) = T_{\Xi}(x)^* = N_{\Xi}(x)$ and the proof is complete.
\end{proof}
For any $x \in \mathbb{R}^d$ denote by $\partial F(x) = \co\{ \nabla_x f(x_*, \omega) \mid \omega \in W(x_*) \}$
\textit{the Hadamard subdifferential} of the function $F(x) = \max_{\omega \in W} f(x, \omega)$. Introduce a set-valued
mapping $\mathcal{D} \colon \Omega \rightrightarrows \mathbb{R}^d$ as follows:
$$
\mathcal{D}(x) = \partial F(x) + \mathcal{N}(x) + N_A(x).
$$
The multifunction $\mathcal{D}$ is obviously convex-valued. Our first aim is to show that optimality conditions for the
problem $(\mathcal{P})$ can be rewritten in the form of the inclusion $0 \in \mathcal{D}(x)$.
\begin{theorem} \label{thrm:EquivOptCond_Subdiff}
Let $x_*$ be a feasible point of the problem $(\mathcal{P})$. Then:
\begin{enumerate}
\item{a Lagrange multiplier of $(\mathcal{P})$ at $x_*$ exists iff $0 \in \mathcal{D}(x_*)$;
\label{stat:NessOpt_Subdiff}}
\item{sufficient optimality condition \eqref{eq:SuffOptCond} holds true at $x_*$ iff $0 \in \interior \mathcal{D}(x_*)$.
\label{stat:SuffOpt_Subdiff}}
\end{enumerate}
\end{theorem}
\begin{proof}
\textbf{Part~\ref{stat:NessOpt_Subdiff}.}~Let $\lambda_*$ be a Lagrange multiplier of $(\mathcal{P})$ at $x_*$ and
$Q(x_*) = \partial F(x_*) + [D G(x_*)]^* \lambda_*$. By the definition of Lagrange multiplier one has
\begin{equation} \label{eq:Lm_Ness_Cond}
[L(\cdot, \lambda_*)]'(x_*, h) = \max_{v \in Q(x_*)} \langle v, h \rangle \ge 0 \quad \forall h \in T_A(x_*).
\end{equation}
Let us check that this inequality implies that $0 \in Q(x_*) + N_A(x_*)$. Indeed, arguing by reductio ad absurdum,
suppose that $Q(x_*) \cap (- N_A(x_*)) = \emptyset$. Observe that $Q(x_*)$ is a compact convex set, while $N_A(x_*)$ is
a closed convex cone. Consequently, applying the separation theorem one obtains that there exists $h \ne 0$ such that
\begin{equation} \label{eq:SeparationThrm}
\langle v, h \rangle < \langle u, h \rangle \quad \forall v \in Q(x_*) \quad \forall u \in \big(- N_A(x_*) \big).
\end{equation}
Since $N_A(x_*)$ is a cone, the inequality above implies that $\langle u, h \rangle \le 0$ for all
$u \in N_A(x_*)$, i.e. $h$ belongs to the polar cone of $N_A(x_*)$. Recall that $N_A(x_*)$ is a polar cone of
$T_A(x_*)$. Therefore, $h \in T_A(x_*)^{**} = T_A(x_*)$ (see, e.g. \cite[Prp.~2.40]{BonnansShapiro}).
Taking into account inequality \eqref{eq:SeparationThrm} and the facts that $0 \in N_A(x_*)$ and $Q(x_*)$ is a compact
set one obtains that $\max_{v \in Q(x_*)} \langle v, h \rangle < 0$, which contradicts \eqref{eq:Lm_Ness_Cond}. Thus,
$0 \in Q(x_*) + N_A(x_*)$, which implies that $0 \in \mathcal{D}(x_*)$ due to the fact that by the definition of
Lagrange multiplier one has $\lambda_* \in K^*$ and $\langle \lambda_*, G(x_*) \rangle = 0$.
Let us prove the converse statement. Suppose that $0 \in \mathcal{D}(x_*)$. Then there exist $v_* \in \partial F(x_*)$
and $\lambda_* \in K^*$ such that $v_* + [D G(x_*)]^* \lambda_* \in - N_A(x_*)$ and
$\langle \lambda_*, G(x_*) \rangle = 0$. By the definition of $N_A(x_*)$ one has
$$
\max_{v \in \partial F(x_*)} \langle v, h \rangle + \langle \lambda_*, D G(x_*) h \rangle
\ge \langle v_*, h \rangle + \langle \lambda_*, D G(x_*) h \rangle \ge 0 \quad \forall h \in T_A(x_*).
$$
In other words, $[L(\cdot, \lambda_*)]'(x_*, h) \ge 0$ for all $h \in T_A(x_*)$. Thus, $\lambda_*$ is a Lagrange
multiplier of $(\mathcal{P})$ at $x_*$.
\textbf{Part~\ref{stat:SuffOpt_Subdiff}.}~Let sufficient optimality condition \eqref{eq:SuffOptCond} be satisfied. Let
us show at first that zero belongs to the relative interior $\relint \mathcal{D}(x_*)$ of $\mathcal{D}(x_*)$. Indeed,
arguing by reductio ad absurdum, suppose that $0 \notin \relint \mathcal{D}(x_*)$. Then by the separation theorem
(see, e.g. \cite[Thrm.~2.17]{BonnansShapiro}) there exists $h \ne 0$ such that $\langle v, h \rangle \le 0$ for all
$v \in \mathcal{D}(x_*)$. Hence taking into account the fact that both $\mathcal{N}(x_*)$ and $N_A(x_*)$ are convex
cones one obtains that
$$
\max_{v \in \partial F(x_*)} \langle v, h \rangle \le 0, \quad
\langle v, h \rangle \le 0 \quad \forall v \in \mathcal{N}(x_*), \quad
\langle v, h \rangle \le 0 \quad \forall v \in N_A(x_*).
$$
Therefore $h \in N_A(x_*)^* = T_A(x_*)^{**} = T_A(x_*)$ and
\begin{equation} \label{eq:FrDerInTangCone}
\langle \lambda, D G(x_*) h \rangle \le 0 \quad
\forall \lambda \in K^* \colon \langle \lambda, G(x_*) \rangle = 0.
\end{equation}
Let us verify that this inequality implies that $D G(x_*) h \in T_K(G(x_*))$. Then one obtains that we found
$h \in T_A(x_*) \setminus \{ 0 \}$ such that $D G(x_*) h \in T_K\big( G(x_*) \big)$ and
$\max_{\omega \in W(x_*)} \langle \nabla_x f(x_*, \omega), h \rangle \le 0$, which contradicts \eqref{eq:SuffOptCond}.
Arguing by reductio ad absurdum, suppose that $D G(x_*) h \notin T_K(G(x_*))$. The cone $T_K(G(x_*))$ is closed and
convex, since $K$ is a convex cone. Therefore, by the separation theorem there exists
$\lambda \in Y^* \setminus \{ 0 \}$ such that
\begin{equation} \label{eq:InclCondSepTh}
\langle \lambda, D G(x_*) h \rangle > 0, \quad \langle \lambda, y \rangle \le 0 \quad \forall y \in T_K(G(x_*)).
\end{equation}
Since $K$ is a cone and $G(x_*) \in K$, one has $G(x_*) + \alpha G(x_*) \in K$ for all $\alpha \in [-1, 1]$, which
implies that $G(x_*) \in T_K(G(x_*))$, $- G(x_*) \in T_K(G(x_*))$, and $\langle \lambda, G(x_*) \rangle = 0$.
Furthermore, as was noted above, $K \subseteq K - G(x_*) \subseteq T_K( G(x_*) )$ due to the facts that $G(x_*) \in K$
and $K$ is a convex cone. Hence with the use of \eqref{eq:InclCondSepTh} one obtains that $\lambda \in K^*$,
$\langle \lambda, G(x_*) \rangle = 0$, and $\langle \lambda, D G(x_*) h \rangle > 0$, which contradicts
\eqref{eq:FrDerInTangCone}. Thus, $D G(x_*) h \in T_K(G(x_*))$ and $0 \in \relint \mathcal{D}(x_*)$.
Let us now show that $\interior \mathcal{D}(x_*) \ne \emptyset$. Then $0 \in \interior \mathcal{D}(x_*)$ and the proof
is complete. Arguing by reductio ad absurdum, suppose that $\interior \mathcal{D}(x_*) = \emptyset$. From the facts that
$0 \in \relint \mathcal{D}(x_*)$ and $\interior \mathcal{D}(x_*) = \emptyset$ it follows
that $\linhull \mathcal{D}(x_*) \ne \mathbb{R}^d$. Therefore, there exists $h \ne 0$ such
that $\langle v, h \rangle = 0$ for all $v \in \linhull \mathcal{D}(x_*)$. Consequently, with the use of the fact that
both $\mathcal{N}(x_*)$ and $N_A(x_*)$ are convex cones one obtains that
$$
\max_{v \in \partial F(x_*)} \langle v, h \rangle = 0, \quad
\langle v, h \rangle = 0 \quad \forall v \in \mathcal{N}(x_*), \quad
\langle v, h \rangle = 0 \quad \forall v \in N_A(x_*).
$$
Hence $h \in N_A(x_*)^* = T_A(x_*)^{**} = T_A(x_*)$ and inequality \eqref{eq:FrDerInTangCone} holds true. As was
shown above, this inequality implies that $D G(x_*) h \in T_K(G(x_*))$. Thus, we found
$h \in T_A(x_*) \setminus \{ 0 \}$ such that $\max_{\omega \in W(x_*)} \langle \nabla_x f(x_*, \omega), h \rangle = 0$
and $D G(x_*) h \in T_K(G(x_*))$, which contradicts \eqref{eq:SuffOptCond}.
Therefore $0 \in \interior \mathcal{D}(x_*)$.
Let us prove the converse statement. Suppose that $0 \in \interior \mathcal{D}(x_*)$. Then there exists $\rho > 0$ such
that $\max_{v \in \mathcal{D}(x_*)} \langle v, h \rangle \ge \rho |h|$ for all $h \in \mathbb{R}^d$.
Fix any $h \in T_A(x_*)$ such that $D G(x_*) h \in T_K(G(x_*))$. By definition any $v \in \mathcal{D}(x_*)$ has
the form $v = v_1 + v_2 + v_3$, where $v_1 \in \partial F(x_*)$, $v_2 = [D G(x_*)]^* \lambda_2$ for some
$\lambda_2 \in K^* \cap \linhull(G(x_*))^{\perp}$, and $v_3 \in N_A(x_*)$.
Firstly, note that $\langle v_3, h \rangle \le 0$, since $h \in T_A(x_*)$. Secondly, recall that $K$ is a convex cone,
which implies that $T_K(G(x_*)) = \cl[\cup_{t \ge 0} t(K - G(x_*))]$ (see, e.g. \cite[Prp.~2.55]{BonnansShapiro}).
Hence taking into account the facts that $\lambda_2 \in K^*$ and $\langle \lambda_2, G(x_*) \rangle = 0$ one gets that
$\langle \lambda_2, y \rangle \le 0$ for all $y \in T_K(G(x_*))$. Consequently,
$\langle v_2, h \rangle = \langle \lambda_2, D G(x_*) h \rangle \le 0$, since $D G(x_*) h \in T_K(G(x_*))$.
Thus, for any $v \in \mathcal{D}(x_*)$ one has $\langle v, h \rangle \le \langle v_1, h \rangle$ for the corresponding
vector $v_1 \in \partial F(x_*)$, which implies that
$$
\max_{v \in \partial F(x_*)} \langle v, h \rangle \ge \max_{v \in \mathcal{D}(x_*)} \langle v, h \rangle
\ge \rho |h| \quad \forall h \in T_A(x_*) \colon D G(x_*) h \in T_K(G(x_*)),
$$
i.e. sufficient optimality condition \eqref{eq:SuffOptCond} holds true.
\end{proof}
\begin{remark} \label{rmrk:LagrangeMultViaSubdiff}
From the proof of the first part of the theorem above it follows that $\lambda_*$ is a Lagrange multiplier of the
problem $(\mathcal{P})$ at $x_*$ iff $(\partial F(x_*) + [D G(x_*)]^* \lambda_*) \cap (- N_A(x_*)) \ne \emptyset$.
In particular, if $A = \mathbb{R}^d$, then $\lambda_*$ is a Lagrange multiplier at $x_*$ iff
$0 \in \partial F(x_*) + [D G(x_*)]^* \lambda_* = \partial_x L(x_*, \lambda_*)$, where $\partial_x L(x_*, \lambda_*)$
is the Hadamard subdifferential of the function $L(\cdot, \lambda_*)$ at $x_*$. \qed
\end{remark}
The theorem above contains a reformulation of necessary and sufficient optimality conditions for the problem
$(\mathcal{P})$ in terms of the set $\mathcal{D}(x_*) = \partial F(x_*) + \mathcal{N}(x_*) + N_A(x_*)$. Note that this
convex set need not be closed, since it is the sum of a compact convex set $\partial F(x_*)$ and two closed convex
cones. In the case of necessary conditions, one can rewrite inclusion $0 \in \mathcal{D}(x_*)$ as the condition
$( \partial F(x_*) + \mathcal{N}(x_*) ) \cap (- N_A(x_*)) \ne \emptyset$ involving only closed sets; however, sufficient
optimality conditions cannot be directly rewritten in this way.
Our next goal is to show that one can replace the set $\mathcal{D}(x)$ in Theorem~\ref{thrm:EquivOptCond_Subdiff} with a
smaller \textit{closed} convex set and to simultaneously show a close connection between sufficient optimality
conditions for the problem $(\mathcal{P})$ and exact penalty functions. To this end, denote by
$\Phi_c(x) = F(x) + c \dist(G(x), K)$ a nonsmooth penalty function for the cone constraint of the problem
$(\mathcal{P})$. Here $c \ge 0$ is the penalty parameter and $\dist(y, K) = \inf\{ \| y - z \| \mid z \in K \}$ is the
distance between a point $y \in Y$ and the cone $K$. Note that the function $\Phi_c$ is nondecreasing in $c$.
Before we proceed to an analysis of optimality conditions, let us first compute a subdifferential of the penalty
function $\Phi_c$. Denote $\varphi(x) = \dist(G(x), K)$.
\begin{lemma} \label{lem:ConeConstrPenFunc_Subdiff}
Let $x$ be such that $G(x) \in K$. Then for any $c \ge 0$ the penalty function $\Phi_c$ is Hadamard subdifferentiable at
$x$ and its Hadamard subdifferential has the form $\partial \Phi_c(x) = \partial F(x) + c \partial \varphi(x)$, where
\begin{equation} \label{eq:ConeConstPenTerm_Subdiff}
\partial \varphi(x) =
\Big\{ [D G(x)]^* y^* \in \mathbb{R}^d \Bigm| y^* \in Y^*, \: \| y^* \| \le 1, \:
\langle y^*, y - G(x) \rangle \le 0 \enspace \forall y \in K \Big\}
\end{equation}
i.e. $\Phi_c$ is Hadamard directionally differentiable at $x$, for any $h \in \mathbb{R}^d$ one has
$$
\Phi'_c(x, h) = \lim_{[\alpha, h'] \to [+0, h]} \frac{\Phi_c(x + \alpha h') - \Phi_c(x)}{\alpha}
= \max_{v \in \partial \Phi_c(x)} \langle v, h \rangle,
$$
and the set $\partial \Phi_c(x)$ is convex and compact.
\end{lemma}
\begin{proof}
As was noted in the proof of Theorem~\ref{thrm:NessOptCond}, by \cite[Thrm.~4.4.3]{IoffeTihomirov} the function
$F(x)$ is Hadamard subdifferentiable. Since the sum of Hadamard subdifferentiable functions is obviously Hadamard
subdifferentiable and the Hadamard subdifferential of the sum is equal to the sum of Hadamard subdifferentials (see,
e.g. \cite[Thrm.~4.4.1]{IoffeTihomirov}), it is sufficient to prove that the penalty term $\varphi(x)$ is Hadamard
subdifferentiable and the set \eqref{eq:ConeConstPenTerm_Subdiff} is its Hadamard subdifferential.
Denote $d(y) = \dist(y, K)$. The function $d(\cdot)$ is convex due to the fact that $K$ is a convex set. By
\cite[Example~2.130]{BonnansShapiro} its subdifferential (in the sense of convex analysis) at any point $y \in K$ has
the form
$$
\partial d(y) = \Big\{ y^* \in Y^* \Bigm| \| y^* \| \le 1, \:
\langle y^*, z - y \rangle \le 0 \enspace \forall z \in K \Big\}.
$$
In turn, by \cite[Prp.~4.4.1]{IoffeTihomirov} the function $d(\cdot)$ is Hadamard subdifferentiable at $y$ and its
Hadamard subdifferential coincides with its subdifferential in the sense of convex analysis. Finally, by
\cite[Thrm.~4.4.2]{IoffeTihomirov} the function $\varphi(\cdot) = d(G(\cdot))$ is Hadamard subdifferentiable at $x$ as
well, and its Hadamard subdifferential at this point has the form $\partial \varphi(x) = [D G(x)]^* \partial d(G(x))$,
i.e. \eqref{eq:ConeConstPenTerm_Subdiff} holds true.
\end{proof}
\begin{remark} \label{remark:Subdiff_ConeConstrPenFunc}
From the equality $T_K(G(x_*)) = \cl[\cup_{t \ge 0} t(K - G(x_*))]$ (see, e.g. \cite[Prp.~2.55]{BonnansShapiro}) it
follows that
$$
\partial \varphi(x) = \Big\{ [D G(x)]^* y^* \in \mathbb{R}^d \Bigm| y^* \in (T_K(G(x)))^*, \: \| y^* \| \le 1
\Big\}.
$$
Moreover, since $\partial \varphi(x)$ is a convex set and $0 \in \partial \varphi(x)$, one has
$c \partial \varphi(x) \subseteq r \partial \varphi(x)$ for any $r \ge c \ge 0$, which implies that
$\partial \Phi_c(x) \subseteq \partial \Phi_r(x)$ for any $r \ge c \ge 0$. In addition, the inclusion
$0 \in \partial \varphi(x)$ implies that $\affine(c\partial \varphi(x)) = \linhull \partial \varphi(x)$ for any $c > 0$,
where ``$\affine$'' stands for the affine hull. As is well-known and easy to check,
$\affine( S_1 + S_2 ) = \affine S_1 + \affine S_2$ for any subsets $S_1$ and $S_2$ of a real vector space, which implies
that
$$
\affine \partial \Phi_c(x) = \affine \partial F(x) + \linhull \partial \varphi(x) = \affine \partial \Phi_r(x)
\quad \forall c, r > 0,
$$
that is, the affine hull of the subdifferential $\partial \Phi_c(x)$ does not depend on $c > 0$ and for any
$r \ge c > 0$ one has $\relint \partial \Phi_c(x) \subseteq \relint \partial \Phi_r(x)$. \qed
\end{remark}
Instead of the problem $(\mathcal{P})$ one can consider the following penalised problem:
\begin{equation} \label{probl:PenalisedProblem}
\min \: \Phi_c(x) = \max_{w \in W} f(x, \omega) + c \dist( G(x), K )
\quad \text{subject to} \quad x \in A.
\end{equation}
Recall that the penalty function $\Phi_c$ is called \textit{locally exact} at a locally optimal solution $x_*$ of the
problem $(\mathcal{P})$, if there exists $c_* \ge 0$ such that $x_*$ is a point of local minimum of the penalised
problem \eqref{probl:PenalisedProblem} for any $c \ge c_*$. We say that $\Phi_c$ satisfies \textit{the first order
growth condition} on the set $A$ at a point $x_* \in A$, if there exist a neighbourhood $\mathcal{O}(x_*)$ of $x_*$
and $\rho > 0$ such that $\Phi_c(x) \ge \Phi_c(x_*) + \rho |x - x_*|$ for all $x \in \mathcal{O}(x_*) \cap A$.
From the fact that $\Phi_c(x) = F(x)$ for any $x$ such that $G(x) \in K$ it follows that if the first order growth
condition holds true for $\Phi_c$ on $A$ at a feasible point $x_*$ of the problem $(\mathcal{P})$, then $x_*$ is a
locally optimal solution of this problem, the first order growth condition for the problem $(\mathcal{P})$ holds at
$x_*$, and $\Phi_c$ is locally exact at $x_*$.
The following theorem describes interrelations between optimality conditions for the problem $(\mathcal{P})$,
optimality conditions for the penalised problem \eqref{probl:PenalisedProblem}, the local exactness of the penalty
function $\Phi_c$, and the first order growth conditions.
\begin{theorem} \label{thrm:EquivOptCond_PenaltyFunc}
Let $x_*$ be a feasible point of the problem $(\mathcal{P})$. Then:
\begin{enumerate}
\item{a Lagrange multiplier of the problem $(\mathcal{P})$ at $x_*$ exists iff there exists $c \ge 0$ such that
$0 \in \partial \Phi_c(x_*) + N_A(x_*)$;
\label{stat:NessOpt_PenaltyFunc}}
\item{sufficient optimality condition \eqref{eq:SuffOptCond} is satisfied at $x_*$ if and only iff there exists
$c \ge 0$ such that $0 \in \interior(\partial \Phi_c(x_*) + N_A(x_*))$ iff there exists $c \ge 0$ such that $\Phi_c$
satisfies the first order growth condition on $A$ at $x_*$;
\label{stat:SuffOpt_PenaltyFunc}}
\item{if RCQ holds at $x_*$, then the penalty function $\Phi_c$ is locally exact at $x_*$; furthermore, in this case
the first order growth condition for the problem $(\mathcal{P})$ holds at $x_*$ iff $\Phi_c$ satisfies the first order
growth condition on $A$ at $x_*$.
\label{stat:Exactness_GrowthConditions}}
\end{enumerate}
\end{theorem}
\begin{proof}
\textbf{Part~\ref{stat:NessOpt_PenaltyFunc}.} Let $\lambda_*$ be a Lagrange multiplier of the problem $(\mathcal{P})$
at $x_*$. By definition $\lambda_* \in K^*$ and $\langle \lambda_*, G(x_*) \rangle = 0$, which implies that
$\langle \lambda_*, y - G(x_*) \rangle \le 0$ for all $y \in K$ and for any $c \ge \| \lambda_* \|$ one has
$[D G(x_*)]^* \lambda_* \in c \partial \varphi(x_*)$ (see \eqref{eq:ConeConstPenTerm_Subdiff}). Hence by the definition
of Lagrange multiplier and equality \eqref{eq:DirectDerivOfMaxFunc} for any $c \ge \| \lambda_* \|$ and $h \in T_A(x_*)$
one has
$$
\max_{v \in \partial \Phi_c(x_*)} \langle v, h \rangle
\ge \max_{v \in \partial F(x_*) + [D G(x_*)]^* \lambda_*} \langle v, h \rangle
= [L(\cdot, \lambda_*)]'(x_*, h) \ge 0.
$$
Then applying the separation theorem one can easily check that this inequality implies that
$0 \in \partial \Phi_c(x_*) + N_A(x_*)$ for any $c \ge \| \lambda_* \|$.
Let us now prove the converse statement. Suppose that $0 \in \partial \Phi_c(x_*) + N_A(x_*)$ for some $c \ge 0$. Recall
that by Lemma~\ref{lem:ConeConstrPenFunc_Subdiff} one has
$\partial \Phi_c(x_*) = \partial F(x_*) + c \partial \varphi(x_*)$. Therefore, there exist
$v_0 \in \partial F(x_*)$ and $y^* \in Y^*$ such that $\langle y^*, y - G(x_*) \rangle \le 0$ for any $y \in K$,
$\| y^* \| \le 1$, and $(v_0 + c [D G(x_*)]^* y^*) \in - N_A(x_*)$. Denote $\lambda_* = c y^*$. Then by the definition
of normal cone and equality \eqref{eq:DirectDerivOfMaxFunc} one has
$$
[L(\cdot, \lambda_*)]'(x_*, h)
= \max_{v \in \partial F(x_*)} \langle v, h \rangle + \langle \lambda_*, D G(x_*) h \rangle
\ge \langle v_0 + c [D G(x_*)]^* y^*, h \rangle \ge 0
$$
for all $h \in T_A(x_*)$. Furthermore, from the facts that $\langle \lambda_*, y - G(x_*) \rangle \le 0$ for any
$y \in K$, $K$ is a convex cone, and $G(x_*) \in K$ it follows that $\lambda_* \in K^*$ and
$\langle \lambda_*, G(x_*) \rangle$. Therefore $\lambda_*$ is a Lagrange multiplier of $(\mathcal{P})$ at $x_*$.
\textbf{Part~\ref{stat:SuffOpt_PenaltyFunc}.} Let sufficient optimality condition \eqref{eq:SuffOptCond} be satisfied
at $x_*$. Firstly, we show that $0 \in \relint( \partial \Phi_c(x_*) + N_A(x_*) )$ for some $c > 0$. Arguing by
reductio ad absurdum, suppose that $0 \notin \relint( \partial \Phi_c(x_*) + N_A(x_*) )$ for any $c > 0$. Then by
the separation theorem (see, e.g. \cite[Thrm.~2.17]{BonnansShapiro}) for any $n \in \mathbb{N}$ there exists $h_n \ne 0$
such that $\langle v, h_n \rangle \le 0$ for all $v \in \partial \Phi_n(x_*) + N_A(x_*)$. Replacing, if necessary,
$h_n$ by $h_n / |h_n|$ one can suppose that $|h_n| = 1$. Consequently, there exists a subsequence $\{ h_{n_k} \}$
converging to some $h_*$ with $|h_*| = 1$.
Fix any $c > 0$. As was noted in Remark~\ref{remark:Subdiff_ConeConstrPenFunc},
$\partial \Phi_c(x_*) \subseteq \partial \Phi_{n_k}(x_*)$ for any $n_k \ge c$. Therefore, for any $n_k \ge c$ and for
all $v \in \partial \Phi_c(x_*) + N_A(x_*)$ one has $\langle v, h_{n_k} \rangle \le 0$. Passing to the limit as
$k \to \infty$ one obtains that $\langle v, h_* \rangle \le 0$ for any $v \in \partial \Phi_c(x_*) + N_A(x_*)$ and
$c > 0$ or, equivalently,
\begin{equation} \label{eq:SepThrm_SubdiffConeConstrPenFunc}
\langle v_1 + v_2 + v_3, h_* \rangle \le 0 \quad
\forall v_1 \in \partial F(x_*), \: v_2 \in \bigcup_{c > 0} c \partial \varphi(x_*), \: v_3 \in N_A(x_*).
\end{equation}
Since both $\cup_{c > 0} c \partial \varphi(x_*)$ and $N_A(x_*)$ are cones (recall that $0 \in \partial \varphi(x_*)$),
one has $\langle v_2, h_* \rangle \le 0$ for all $v_2 \in \cup_{c > 0} c \partial \varphi(x_*)$, and
$\langle v_3, h_* \rangle \le 0$ for all $v_3 \in N_A(x_*)$. Consequently, by definition
$h_* \in N_A(x_*)^* = T_A(x_*)^{**} = T_A(x_*)$. Moreover, by Remark~\ref{remark:Subdiff_ConeConstrPenFunc} one
has
$$
\bigcup_{c > 0} c \partial \varphi(x_*)
= \Big\{ [D G(x)]^* y^* \in \mathbb{R}^d \Bigm| y^* \in (T_K(G(x)))^* \Big\}
$$
which implies that $\langle y^*, D G(x_*) h_* \rangle \le 0$ for all $y^* \in T_K(G(x_*))^*$, i.e. by the definition of
polar cone $D G(x_*) h_* \in [T_K(G(x_*))]^{**} = T_K(G(x_*))$. Thus, taking into account
\eqref{eq:SepThrm_SubdiffConeConstrPenFunc} one obtains that we found $h_* \in T_A(x_*) \setminus \{ 0 \}$ such that
$D G(x_*) h_* \in T_K(G(x_*))$ and $\max_{v \in \partial F(x_*)} \langle v, h_* \rangle \le 0$, which contradicts our
assumption that sufficient optimality condition \eqref{eq:SuffOptCond} holds true at $x_*$.
Therefore, $0 \in \relint( \partial \Phi_c(x_*) + N_A(x_*) )$ for some $c > 0$.
Let us verify that the topological interior of the set $\partial \Phi_c(x_*) + N_A(x_*)$ is not empty. Then one can
conclude that $0 \in \interior( \partial \Phi_c(x_*) + N_A(x_*) )$. Arguing by reductio ad absurdum, suppose that the
interior of the set $\partial \Phi_c(x_*) + N_A(x_*)$ is empty. Then taking into account the fact that
$0 \in \relint( \partial \Phi_c(x_*) + N_A(x_*) )$ one can conclude that
$$
\mathcal{E}
= \affine( \partial \Phi_c(x_*) + N_A(x_*) ) = \linhull(\partial \Phi_c(x_*) + N_A(x_*)) \ne \mathbb{R}^d.
$$
Therefore, there exists $h_* \ne 0$ such that $\langle v, h_* \rangle = 0$ for all $v \in \mathcal{E}$. Bearing in mind
the equality $\affine( \partial \Phi_c(x_*) + N_A(x_*) ) = \affine \partial \Phi_c(x_*) + \affine N_A(x_*)$
and the fact that the affine hull of $\partial \Phi_c(x_*)$ does not depend on $c > 0$ by
Remark~\ref{remark:Subdiff_ConeConstrPenFunc} one obtains that $\langle v, h_* \rangle = 0$
for all $v \in \partial \Phi_r(x_*) + N_A(x_*)$ and $r > 0$. Consequently, inequality
\eqref{eq:SepThrm_SubdiffConeConstrPenFunc} is valid, which, as was shown above, contradicts \eqref{eq:SuffOptCond}.
Thus, $0 \in \interior( \partial \Phi_c(x_*) + N_A(x_*) )$ for some $c > 0$.
Suppose now that $0 \in \interior( \partial \Phi_c(x_*) + N_A(x_*) )$ for some $c \ge 0$. Then there exists $\rho > 0$
such that
$$
\max_{v \in \partial \Phi_c(x_*) + N_A(x_*)} \langle v, h \rangle \ge \rho |h| \quad \forall h \in \mathbb{R}^d.
$$
Note that by definition for any $h \in T_A(x_*)$ one has $\langle v, h \rangle \le 0$ for all $v \in N_A(x_*)$.
Therefore
\begin{equation} \label{eq:ZeroInPenFuncSubdiffIneq}
\max_{v \in \partial \Phi_c(x_*)} \langle v, h \rangle \ge \rho |h| \quad \forall h \in T_A(x_*).
\end{equation}
Fix any $\rho' \in (0, \rho)$. Let us check that $\Phi_c(x) \ge \Phi_c(x_*) + \rho' |x - x_*|$ for any $x \in A$ lying
sufficiently close to $x_*$, i.e. $\Phi_c$ satisfies the first order growth condition on $A$ at $x_*$.
Arguing by reductio ad absurdum, suppose that there exists a sequence $\{ x_n \} \subset A$ converging to $x_*$ such
that $\Phi_c(x_n) < \Phi_c(x_*) + \rho'|x_n - x_*|$. Put $h_n = (x_n - x_*) / |x_n - x_*|$ and
$\alpha_n = |x_n - x_*|$. Without loss of generality one can suppose that the sequence $\{ h_n \}$ converges to some
vector $h_*$ with $|h_*| = 1$, which obviously belongs to $T_A(x_*)$, since $x_* + \alpha_n x_n = x_n \in A$ by
definition. Hence with the use of Lemma~\ref{lem:ConeConstrPenFunc_Subdiff} one obtains that
$$
\rho' \ge \lim_{n \to \infty} \frac{\Phi_c(x_n) - \Phi_c(x_*)}{|x_n - x_*|}
= \lim_{n \to \infty} \frac{\Phi_c(x_* + \alpha_n h_n) - \Phi_c(x_*)}{\alpha_n}
= \max_{v \in \partial \Phi_c(x_*)} \langle v, h_* \rangle,
$$
which contradicts \eqref{eq:ZeroInPenFuncSubdiffIneq}.
Suppose finally that $\Phi_c$ satisfies the first order growth condition on $A$ at $x_*$. Let us check that sufficient
optimality condition \eqref{eq:SuffOptCond} holds true at $x_*$. Indeed, by our assumption there exist $c \ge 0$,
$\rho > 0$, and a neighbourhood $\mathcal{O}(x_*)$ of the point $x_*$ such that
$\Phi_c(x) \ge \Phi_c(x_*) + \rho |x - x_*|$ for all $x \in \mathcal{O}(x_*) \cap A$.
Fix any $h \in T_A(x_*) \setminus \{ 0 \}$ such that $D G(x_*) h \in T_K(G(x_*))$. By the definition of contingent cone
there exist sequences $\{ \alpha_n \} \subset (0, + \infty)$ and $\{ h_n \} \subset \mathbb{R}^d$ such that
$\alpha_n \to 0$ and $h_n \to h$ as $n \to \infty$, and $x_* + \alpha_n h_n \in A$ for all $n \in \mathbb{N}$. Hence
for any sufficiently large $n$ one has $\Phi_c(x_* + \alpha_n h_n) - \Phi_c(x_*) \ge \rho \alpha_n |h_n|$, which
obviously implies that $\Phi_c'(x_*, h) \ge \rho |h|$.
By Remark~\ref{remark:Subdiff_ConeConstrPenFunc} for any $v \in \partial \varphi(x_*)$ one can find a vector
$y^*(v) \in (T_K(G(x_*)))^*$ such that $v = [D G(x_*)]^* y^*(v)$. Therefore for any $v \in \partial \varphi(x_*)$ one
has $\langle v, h \rangle = \langle y^*(v), D G(x_*) h \rangle \le 0$, since $D G(x_*) h \in T_K(G(x_*))$ by our
assumption. Consequently, by Lemma~\ref{lem:ConeConstrPenFunc_Subdiff} one has
$$
\max_{v \in \partial F(x_*)} \langle v, h \rangle \ge \max_{v \in \partial \Phi_c(x_*)} \langle v, h \rangle
= \Phi_c'(x_*, h) \ge \rho |h| > 0,
$$
i.e. sufficient optimality condition \eqref{eq:SuffOptCond} is satisfied at $x_*$.
\textbf{Part~\ref{stat:Exactness_GrowthConditions}.} If RCQ holds true at $x_*$, then by
\cite[Corollary~2.2]{Cominetti} there exist $a > 0$ and a neighbourhood $\mathcal{O}(x_*)$ of $x_*$ such that
$$
\varphi(x) = \dist(G(x), K) \ge a \dist(x, A \cap G^{-1}(K)) = a \dist(x, \Omega)
\quad \forall x \in \mathcal{O}(x_*) \cap A,
$$
where, as above, $\Omega$ is the feasible region of the problem $(\mathcal{P})$. Let us check that the objective
function $F$ is Lipschitz continuous near $x_*$. Then by \cite[Corollary~2.9 and Prp.~2.7]{Dolgopolik_ExPen} one can
conclude that the penalty function $\Phi_c$ is locally exact at $x_*$.
Fix any $r > 0$ and denote $B(x_*, r) = \{ x \in \mathbb{R}^d \mid |x - x_*| \le r \}$. By a nonsmooth version of the
mean value theorem (see, e.g. \cite[Prp.~2]{Dolgopolik_MCD}) for any $x_1, x_2 \in B(x_*, r)$ there exist a point
$z \in \co\{ x_1, x_2 \} \subset B(x_*, r)$ and
$v \in \partial F(z)$ such that $F(x_1) - F(x_2) = \langle v, x_1 - x_2 \rangle$.
Define $L = \max\{ |\nabla_x f(x, \omega)| \mid x \in B(x_*, r), \omega \in W \} < + \infty$. By definition
$v$ belongs to the convex hull $\co\{ \nabla_x f(z, \omega) \mid \omega \in W(z) \}$, which yields $|v| \le L$. Thus,
$|F(x_1) - F(x_2)| \le L |x_1 - x_2|$ for all $x_1, x_2 \in B(x_*, r)$, i.e. $F$ is Lipschitz continuous near $x_*$.
It remains to note that if RCQ and the first order growth condition for the problem $(\mathcal{P})$ hold at $x_*$,
then by Theorem~\ref{thrm:SuffOptCond} sufficient optimality condition \eqref{eq:SuffOptCond} holds true at $x_*$,
which by the second part of this theorem implies that $\Phi_c$ satisfies the first growth condition on $A$ at $x_*$.
The converse statement, as was noted before this theorem, holds true regardless of RCQ.
\end{proof}
\begin{remark}
{(i)~From the proof of the previous theorem it follows that $\lambda_*$ is a Lagrange multiplier of $(\mathcal{P})$ at
$x_*$ iff $0 \in \partial \Phi_c(x_*) + N_A(x_*)$ for any $c \ge \| \lambda_* \|$.
}
\noindent{(ii)~Observe that if $0 \in \partial \Phi_c(x_*) + N_A(x_*)$ for some $c \ge 0$, then
$0 \in \partial \Phi_r(x_*) + N_A(x_*)$ for any $r \ge c$, since $\partial \Phi_c(x_*) \subseteq \partial \Phi_r(x_*)$
by Remark~\ref{remark:Subdiff_ConeConstrPenFunc}. Furthermore, from this inclusion it follows that
if $0 \in \interior( \partial \Phi_c(x_*) + N_A(x_*) )$ for some $c \ge 0$, then
$0 \in \interior( \partial \Phi_r(x_*) + N_A(x_*) )$ for any $r \ge c$ as well.
}
\noindent{(iii)~Unlike the set $\mathcal{D}(x_*)$ from Theorem~\ref{thrm:EquivOptCond_Subdiff},
the set $\partial \Phi_c(x_*) + N_A(x_*)$ is always closed as the sum of a compact and a closed sets. Moreover,
the inclusion $\partial \Phi_c(x_*) + N_A(x_*) \subset \mathcal{D}(x_*)$ holds true for any $c \ge 0$. Indeed, by
Lemma~\ref{lem:ConeConstrPenFunc_Subdiff} one has $\partial \Phi_c(x_*) = \partial F(x_*) + c \partial \varphi(x_*)$.
Therefore, it is sufficient to check that $\partial \varphi(x_*) \subset \mathcal{N}(x_*)$, since $\mathcal{N}(x_*)$ is
a cone. Choose any $z^* \in \partial \varphi(x_*)$. By Lemma~\ref{lem:ConeConstrPenFunc_Subdiff}
one has $z^* = [D G(x_*)]^* y^*$ for some $y^* \in Y^*$ such that $\| y^* \| \le 1$ and
$\langle y^*, y - G(x_*) \rangle \le 0$ for all $y \in K$. Observe that $0 \in K$ and $2 G(x_*) \in K$, since $K$ is a
cone and $G(x_*) \in K$, which yields $\langle y^*, G(x_*) \rangle = 0$. Furthermore, from the fact that $K$ is a convex
cone it follows that $K + G(x_*) \subseteq K$, which implies that $\langle y^*, y \rangle \le 0$ for all $y \in K$, i.e.
$y^* \in K^*$. Thus, one can conclude that $z^* \in [D G(x_*)](K^* \cap \linhull(G(x_*))^{\perp}) = \mathcal{N}(x_*)$,
i.e. $\partial \varphi(x_*) \subset \mathcal{N}(x_*)$.
}
\noindent{(iv)~From the proof of the second part of the theorem above it follows that the inclusion
$0 \in \interior(\partial \Phi_c(x_*) + N_A(x_*))$ is a sufficient optimality condition for the penalised problem
\eqref{probl:PenalisedProblem}. Moreover, both this condition and optimality condition \eqref{eq:SuffOptCond} are
sufficient conditions for the local exactness of $\Phi_c$. Finally, note that arguing in the same way as in the proof of
the first part of Theorem~\ref{thrm:EquivOptCond_Subdiff} one can easily check that the inclusion
$0 \in \partial \Phi_c(x_*) + N_A(x_*)$ is a necessary optimality condition for problem \eqref{probl:PenalisedProblem}.
\qed
}
\end{remark}
\subsection{Alternance optimality conditions and cadres}
\label{subsect:Alternance_Cadre}
Note that the optimality condition $0 \in \mathcal{D}(x_*)$ from the previous section means that zero can be represented
as the sum of some vectors from the sets $\partial F(x_*)$, $\mathcal{N}(x_*)$, and $N_A(x_*)$. Our aim is to show that
these vectors can be chosen in such a way that they have some useful additional properties, which, in particular, allow
one to check whether the sufficient optimality condition $0 \in \interior \mathcal{D}(x_*)$ is satisfied.
Let $Z \subset \mathbb{R}^d$ be a set consisting of $d$ linearly independent vectors. Let also
$\eta(x_*) \subseteq \mathcal{N}(x_*)$ and $n_A(x_*) \subseteq N_A(x_*)$ be such that
$\mathcal{N}(x_*) = \cone \eta(x_*)$ and $N_A(x_*) = \cone n_A(x_*)$, where
$$
\cone D = \Big\{ \sum_{i = 1}^n \alpha_i x_i \Bigm|
x_i \in D, \enspace \alpha_i \ge 0, \enspace i \in \{ 1, \ldots, n \}, \enspace n \in \mathbb{N} \Big\}
$$
is the \textit{convex conic hull} of a set $D \subset \mathbb{R}^d$ (i.e. the smallest convex cone containing the set
$D$). Usually, one chooses $\eta(x_*)$ and $n_A(x_*)$ as the sets of those vectors that correspond to extreme rays
of the cones $\mathcal{N}(x_*)$ and $N_A(x_*)$ respectively.
\begin{definition} \label{def:AlternanceOptCond}
Let $p \in \{ 1, \ldots, d + 1 \}$ be fixed and $x_*$ be a feasible point of the problem $(\mathcal{P})$. One says that
\textit{a $p$-point alternance} exists at $x_*$, if there exist $k_0 \in \{ 1, \ldots, p \}$,
$i_0 \in \{ k_0 + 1, \ldots, p \}$, vectors
\begin{gather} \label{eq:AlternanceDef}
V_1, \ldots, V_{k_0} \in \Big\{ \nabla_x f(x_*, \omega) \Bigm| \omega \in W(x_*) \Big\}, \\
V_{k_0 + 1}, \ldots, V_{i_0} \in \eta(x_*), \quad
V_{i_0 + 1}, \ldots, V_p \in n_A(x_*), \label{eq:AlternanceDef2}
\end{gather}
and vectors $V_{p + 1}, \ldots, V_{d + 1} \in Z$ such that the d-th order determinants $\Delta_s$ of the matrices
composed of the columns $V_1, \ldots, V_{s - 1}, V_{s + 1}, \ldots V_{d + 1}$ satisfy the following conditions:
\begin{gather} \label{eq:DeterminantsProp}
\Delta_s \ne 0, \quad s \in \{ 1, \ldots, p \}, \quad
\sign \Delta_s = - \sign \Delta_{s + 1}, \quad s \in \{ 1, \ldots, p - 1 \}, \\
\Delta_s = 0, \quad s \in \{ p + 1, \ldots d + 1 \}. \label{eq:DeterminantsProp2}
\end{gather}
Such collection of vectors $\{ V_1, \ldots, V_p \}$ is called a $p$-point alternance at $x_*$. Any $(d + 1)$-point
alternance is called \textit{complete}.
\end{definition}
\begin{remark}
{(i)~Note that in the case of complete alternance one has
$$
\Delta_s \ne 0 \quad s \in \{ 1, \ldots, d + 1 \}, \quad
\sign \Delta_s = - \sign \Delta_{s + 1} \quad s \in \{ 1, \ldots, d \},
$$
i.e. the determinants $\Delta_s$, $s \in \{1, \ldots, d + 1 \}$ are not equal to zero and have \textit{alternating}
signs, which explains the term \textit{alternance}.
}
\noindent{(ii)~It should be mentioned that the sets $\eta(x_*)$ and $n_A(x_*)$ are introduced in order to
simplify verification of alternance optimality conditions. It is often difficult to deal with the entire cones
$\mathcal{N}(x_*)$ and $N_A(x_*)$. In turn, the introduction of the sets $\eta(x_*)$ and $n_A(x_*)$ allows
one to use only extreme rays of $\mathcal{N}(x_*)$ and $N_A(x_*)$ respectively. \qed
}
\end{remark}
Before we proceed to an analysis of optimality conditions, let us first show that the definition of $p$-point
alternance with $p \le d$ is invariant with respect to the choice of the set $Z$ and is directly connected to the notion
of \textit{cadre} (meaning \textit{frame}) of a minimax problem (see, e.g. \cite{Descloux,ConnLi92}).
\begin{proposition} \label{prp:AlternanceVsCadre}
Let $x_*$ be a feasible point of the problem $(\mathcal{P})$. Then a $p$-point alternance with
$p \in \{ 1, \ldots, d + 1 \}$ exists at $x_*$ if and only if there exist $k_0 \in \{ 1, \ldots, p \}$,
$i_0 \in \{ k_0 + 1, \ldots, p \}$, and vectors
\begin{gather} \label{eq:CadreDef}
V_1, \ldots, V_{k_0} \in \Big\{ \nabla_x f(x_*, \omega) \Bigm| \omega \in W(x_*) \Big\}, \\
V_{k_0 + 1}, \ldots, V_{i_0} \in \eta(x_*), \quad
V_{i_0 + 1}, \ldots, V_p \in n_A(x_*). \label{eq:CadreDef2}
\end{gather}
such that $\rank([V_1, \ldots, V_p]) = p - 1$ and
\begin{equation} \label{eq:CadreMultDef}
\sum_{i = 1}^p \beta_i V_i = 0
\end{equation}
for some $\beta_i > 0$, $i \in \{ 1, \ldots, p \}$. Furthermore, a collection of vectors $\{ V_1, \ldots, V_p \}$
satisfying \eqref{eq:CadreDef} and \eqref{eq:CadreDef2} is a $p$-point alternance at $x_*$ iff
$\rank([V_1, \ldots, V_p]) = p - 1$ and \eqref{eq:CadreMultDef} holds true.
\end{proposition}
\begin{proof}
Let a $p$-point alternance exist at $x_*$ and let vectors $V_i \in \mathbb{R}^d$ and indices
$k_0 \in \{ 1, \ldots, p \}$, $i_0 \in \{ k_0 + 1, \ldots, p \}$ be from the definition of $p$-point alternance.
Consider the system of linear equations $\sum_{i = 2}^{d + 1} \beta_i V_i = - V_1$ with respect to $\beta_i$. Solving
this system with the use of Cramer's rule one obtains that $\beta_i = (-1)^{i-1} \Delta_i / \Delta_1$ for all
$i \in \{ 2, \ldots, d + 1 \}$, where $\Delta_i$ are from the definition of $p$-point alternance. Taking into account
\eqref{eq:DeterminantsProp} and \eqref{eq:DeterminantsProp2} one obtains that $\beta_i > 0$ for any
$i \in \{ 2, \ldots, p \}$ and $\beta_i = 0$ for all $i \in \{ p + 1, \ldots, d + 1 \}$. Note that zero coefficients
$\beta_i$ correspond exactly to those $V_i$ that belong to $Z$.
Thus, one has $V_1 + \sum_{i = 2}^p \beta_i V_i = 0$ and $\beta_i > 0$ for all $i \in \{ 2, \ldots, p \}$.
Furthermore, from the fact that that by the definition of $p$-point alternance one has
$\Delta_1 = \determ([V_2, \ldots, V_{d + 1}]) \ne 0$ it follows that the vectors $V_2, \ldots, V_p$ are linearly
independent, which implies that $\rank([V_1, \ldots, V_p]) = p - 1$. Hence taking into account
\eqref{eq:AlternanceDef} and \eqref{eq:AlternanceDef2} one obtains that the proof of the ``only if'' part of the
proposition is complete.
Let us prove the converse statement. Suppose at first that $p = 1$. Then $V_1 = 0$ due to \eqref{eq:CadreMultDef}. Take
as $V_2, \ldots, V_{d + 1}$ all vectors from the set $Z$ in an arbitrary order. Since these vectors are linearly
independent, one has $\Delta_1 = \determ([V_2, \ldots, V_{d + 1}]) \ne 0$, and the system
$\sum_{i = 2}^{d + 1} \gamma_i V_i = - V_1$ has the unique solution $\gamma_i = 0$ for all $i$. Solving this system
with the use of Cramer's rule one obtains that $0 = \gamma_i = (-1)^{i - 1} \Delta_i / \Delta_1$ for all
$i \in \{ 2, \ldots, d + 1 \}$, where $\Delta_i = \determ([V_1, \ldots, V_{i - 1}, V_{i + 1}, \ldots V_{d + 1}])$.
Thus, $\Delta_i = 0$ for all $i \ge 2$ and the collection $\{ V_1, \ldots, V_{d + 1} \}$ satisfies the definition of
$1$-point alternance.
Suppose now that $p \ge 2$. Rewrite \eqref{eq:CadreMultDef} as follows:
$\sum_{i = 2}^p (\beta_i / \beta_1) V_i = - V_1$.
Taking into account this equality and the fact that $\rank([V_1, \ldots, V_p]) = p - 1$ one can conclude that
the vectors $V_2, \ldots, V_p$ are linearly independent. Therefore one can choose
$V_{p + 1}, \ldots, V_{d + 1} \in Z$ such that the vectors $V_2, \ldots, V_{d + 1}$ are linearly independent as well.
Consequently, $\Delta_1 = \determ([V_2, \ldots, V_{d + 1}]) \ne 0$, and the system of linear equations
$\sum_{i = 2}^{d + 1} \gamma_i V_i = - V_1$ with respect to $\gamma_i$ has the unique solution:
$\gamma_i = \beta_i / \beta_1 > 0$ for any $i \in \{ 2, \ldots, p \}$, and $\gamma_i = 0$ for all $i \ge p + 1$. On the
other hand, by Cramer's rule one has $\gamma_i = (-1)^{i - 1} \Delta_i / \Delta_1$ for all $i$, where
$\Delta_i = \determ([V_1, \ldots, V_{i - 1}, V_{i + 1}, \ldots V_{d + 1}])$. Hence conditions
\eqref{eq:DeterminantsProp} and \eqref{eq:DeterminantsProp2} hold true and the collection
$\{ V_1, \ldots, V_{d + 1} \}$ satisfies the definition of $p$-point alternance.
\end{proof}
\begin{remark}
{(i)~Any collection of vectors $V_1, \ldots, V_p$ with $p \in \{ 1, \ldots, d + 1 \}$ satisfying \eqref{eq:CadreDef},
\eqref{eq:CadreDef2} and such that
$\rank([V_1, \ldots, V_p]) = \rank([V_1, \ldots, V_{i - 1}, V_{i + 1}, \ldots, V_p]) = p - 1$ for any
$i \in \{ 1, \ldots, p \}$ is called a $p$-point \textit{cadre} for the problem $(\mathcal{P})$ at $x_*$. One can easily
verify that a collection $V_1, \ldots, V_p$ satisfying \eqref{eq:CadreDef}, \eqref{eq:CadreDef2} is a $p$-point cadre at
$x_*$ iff $\rank([V_1, \ldots, V_p]) = p - 1$ and $\sum_{i = 1}^p \beta_i V_i = 0$ for some $\beta_i \ne 0$,
$i \in \{ 1, \ldots, p \}$. Any such $\beta_i$ are called \textit{cadre multipliers}. Thus, the proposition above
can be reformulated as follows: a $p$-point alternance exists at $x_*$ iff a $p$-point cadre with positive cadre
multipliers exists at this point. Furthermore, a collection $\{ V_1, \ldots, V_p \}$ with $p \in \{ 1, \ldots, d + 1 \}$
is a $p$-point alternance at $x_*$ iff it is a $p$-point cadre with positive cadre multipliers, which implies that the
definition of $p$-point alternance is invariant with respect to the set $Z$. Note finally that optimality conditions in
terms of such cadres were utilised in \cite{ConnLi92} to design an efficient method for solving unconstrained minimax
problems, while the definition of \textit{cadre} was first given by Descloux in \cite{Descloux}.
}
\noindent{(ii)~It is worth mentioning that from the previous proposition it follows that if any $d$ vectors from the
set $\{ \nabla_x f(x_*, \omega) \mid \omega \in W(x_*) \} \cup \eta(x_*) \cup n_A(x_*)$ are linearly
independent, then only a complete alternance can exist at $x_*$. \qed
}
\end{remark}
Our next goal is demonstrate that both necessary and sufficient optimality conditions for the problem $(\mathcal{P})$
can be written in an \textit{alternance} form. To this end, we will need the following simple geometric result
illustrated by Figure~\ref{fig:ell1_interior}. This result allows one to easily prove that the origin belongs to
the interior or the relative interior of certain polytopes.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\linewidth]{Ell1_interior.mps}
\caption{The polytope $S = \co\{ x_1, x_2, -x_1 - x_2 \}$ with $x_1 = (1, 0)^T$ and $x_2 = (0, 1)^T$ contains the open
$\ell_1$ ball centered at zero with sufficiently small radius $r > 0$ that can be described as
$\{ z = \alpha_1 x_1 + \alpha_2 x_2 \in \mathbb{R}^2 \mid |\alpha_1| + |\alpha_2| < r \}$.}
\label{fig:ell1_interior}
\end{figure}
\begin{lemma} \label{lem:SimplexZeroInterior}
Let $x_1, \ldots, x_k \in \mathbb{R}^d$ be given vectors, $x = \sum_{i = 1}^k \beta_i x_i$ for some $\beta_i > 0$, and
$S = \co\{ x_1, \ldots, x_k, - x \}$. Then there exists $r > 0$ such that
\begin{equation} \label{eq:ell1_SimplexRelInt}
\Big\{ z = \sum_{i = 1}^k \alpha_i x_i \Bigm| \sum_{i = 1}^k |\alpha_i| < r \Big\} \subset S.
\end{equation}
\end{lemma}
\begin{proof}
Observe that $0 \in S$, since
$$
0 = \frac{1}{1 + \beta_1 + \ldots + \beta_k} x
+ \frac{1}{1 + \beta_1 + \ldots + \beta_k} \sum_{i = 1}^k \beta_i x_i \in S.
$$
Hence, in particular, $\co\{ 0, z \} \subset S$ for all $z \in S$.
Denote $\gamma_i = 1 + \sum_{j \ne i} \beta_j$. Then
$$
- \frac{\beta_i}{\gamma_i} x_i = \frac{1}{\gamma_i} x
+ \sum_{j \ne i} \frac{\beta_j}{\gamma_i} x_j \in S \quad \forall i \in \{ 1, \ldots, k \}.
$$
Define $r = \min\{ 1, \beta_1 / \gamma_1, \ldots, \beta_k / \gamma_k \}$. Then taking into account the fact
that $\co\{ 0, z \} \subset S$ for all $z \in S$ one obtains that $\pm r x_i \in S$ for all $i \in \{ 1, \ldots, k \}$.
Fix any $z = \sum_{i = 1}^k \alpha_i x_i$ with $\theta(z) = \sum_{i = 1}^k |\alpha_i| < r$. If $\theta(z) = 0$, then
$z = 0$ and $z \in S$. Therefore, suppose that $\theta(z) \ne 0$.
Then $\pm \theta(z) x_i \in \co \{ \pm r x_i \} \subset S$, which implies that
$$
z = \sum_{i = 1}^k \frac{|\alpha_i|}{\theta(z)} \Big( \sign(\alpha_i) \theta(z) x_i \Big) \in S
$$
(here $\sign(0) = 0$). Thus, \eqref{eq:ell1_SimplexRelInt} holds true.
\end{proof}
\begin{theorem} \label{thrm:AlternanceCond}
Let $x_*$ be a feasible point of the problem $(\mathcal{P})$. Then:
\begin{enumerate}
\item{$0 \in \mathcal{D}(x_*)$ iff for some $p \in \{1, \ldots, d + 1 \}$ a $p$-point alternance exists at~$x_*$;
\label{stat:NessOpt_Alternance}}
\item{if a complete alternance exists at $x_*$, then $0 \in \interior \mathcal{D}(x_*)$ and $\partial F(x_*) \ne \{ 0
\}$.
\label{stat:CompleteAlternance_SuffOpt}}
\end{enumerate}
\end{theorem}
\begin{proof}
\textbf{Part~\ref{stat:NessOpt_Alternance}.} ``$\implies$'' Let $0 \in \mathcal{D}(x_*)$. If
$0 \in \partial F(x_*) = \co\{ \nabla_x f(x_*, \omega) \mid \omega \in W(x_*) \}$, then by Carath\'{e}odory's theorem
(see, e.g. \cite[Corollary~17.1.1]{Rockafellar}) zero can be expressed as a convex combination
of $d + 1$ or fewer affinely independent vectors from $\{ \nabla_x f(x_*, \omega) \mid \omega \in W(x_*) \}$.
Thus, there exist $p \in \{ 1, \ldots, d + 1 \}$, $V_i \in \{ \nabla_x f(x_*, \omega) \mid \omega \in W(x_*) \}$, and
$\alpha_i > 0$, $i \in \{ 1, \ldots, p \}$, such that the vectors $V_i$ are affinely independent and
\begin{equation} \label{eq:ZeroElementOfMaxFuncSubdiff}
0 = \sum_{i = 1}^p \alpha_i V_i, \quad \sum_{i = 1}^p \alpha_i = 1.
\end{equation}
If $p = 1$, then denote by $V_2, \ldots, V_{d + 1}$ all vectors from the set $Z$. Then $\Delta_1
\ne 0$, and $\Delta_s = 0$ for all $s \in \{ 2, \ldots, d + 1 \}$, since $V_1 = 0$, that is, a $1$-point alternance
exists at $x_*$. Otherwise, note that by the definition of affine independence the vectors
$V_2 - V_1, \ldots, V_p - V_1$ are linearly independent. Hence taking into account
\eqref{eq:ZeroElementOfMaxFuncSubdiff} and the fact that
$\linhull( V_2 - V_1, \ldots, V_p - V_1) \subseteq \linhull(V_1, \ldots, V_p)$ one obtains that
$\dimens \linhull(V_1, \ldots, V_p) = p - 1$. Consequently, the collection $\{ V_1, \ldots, V_p \}$ contains exactly
$p - 1$ linearly independent vectors. Renumbering $V_i$, if necessary, one can suppose that the vectors
$V_2, \ldots, V_p$ are linearly independent. Since the set $Z$ contains $d$ linearly independent vectors, one can
choose vectors $V_{p + 1}, \ldots, V_{d + 1} \in Z$ in such a way that the vectors $V_2, \ldots, V_{d + 1}$ are linearly
independent, which yields $\Delta_1 \ne 0$.
Now, consider the system of linear equations $- V_1 = \sum_{i = 2}^{d + 1} \beta_i V_i$ with respect to $\beta_i$.
Solving this system with the use of Cramer's rule and bearing in mind equalities \eqref{eq:ZeroElementOfMaxFuncSubdiff}
one obtains that $\beta_i = (-1)^{i - 1} \Delta_i / \Delta_1 = \alpha_i / \alpha_1 > 0$ for any
$i \in \{ 2, \ldots, p \}$, and $\beta_i = (-1)^{i - 1} \Delta_i / \Delta_1 = 0$ for any $i \ge p + 1$. Thus, conditions
\eqref{eq:DeterminantsProp} and \eqref{eq:DeterminantsProp2} hold true, i.e. a $p$-point alternance exists at $x_*$.
Therefore, one can suppose that $0 \notin \partial F(x_*)$.
Since $0 \in \mathcal{D}(x_*)$ and $0 \notin \partial F(x_*)$, there exist $k, r, \ell \in \mathbb{N}$,
$\omega_i \in W(x_*)$, $\alpha_i \in (0, 1]$ , $u_j \in \eta(x_*)$, $\beta_j \ge 0$, $z_s \in n_A(x_*)$, and
$\gamma_s \ge 0$ (here $i \in \{ 1, \ldots, k \}$, $j \in \{ 1, \ldots, r \}$, and $s \in \{ 1, \ldots, \ell \}$) such
that
$$
0 = \sum_{i = 1}^k \alpha_i v_i + \sum_{j = 1}^r \beta_j u_j + \sum_{s = 1}^{\ell} \gamma_s z_s,
\quad \sum_{i = 1}^k \alpha_i = 1,
$$
where $v_i = \nabla_x f(x_*, \omega_i)$ for all $i \in \{ 1, \ldots, k \}$. Hence
$$
\sum_{i = 2}^k \frac{\alpha_i}{\alpha_1} v_i + \sum_{j = 1}^r \frac{\beta_j}{\alpha_1} u_j +
\sum_{s = 1}^{\ell} \frac{\gamma_s}{\alpha_1} z_s = - v_1,
$$
i.e. $- v_1$ belongs to $\cone(\mathcal{E})$ with
$\mathcal{E} = \{ v_2, \ldots, v_k, u_1, \ldots, u_r, z_1, \ldots, z_{\ell} \}$. Applying a simple modification of the
Carath\'eodory's theorem to the case of convex conic combinations (see, e.g. \cite[Corollary~17.1.2]{Rockafellar}) one
obtains that there exist $p \in \{ 2, \ldots, d + 1 \}$ and linearly independent vectors
$V_2, \ldots V_p \in \mathcal{E}$ such that $- v_1 = \sum_{i = 2}^p \lambda_i V_i$ for some $\lambda_i > 0$.
Clearly, one can suppose that there exist $k_0 \in \{ 1, \ldots p \}$ and $i_0 \in \{ k_0 + 1, \ldots, p \}$ such that
\eqref{eq:AlternanceDef} and \eqref{eq:AlternanceDef2} hold true.
Put $V_1 = v_1$, and choose vectors $V_{p + 1}, \ldots V_{d + 1}$ from the set $Z$ in such a way that the vectors
$V_2, \ldots, V_{d + 1}$ are linearly independent. Then one obtains that the systems
\begin{equation} \label{eq:AlternanceLinSystm}
\sum_{i = 2}^{d + 1} \beta_i V_i = - V_1.
\end{equation}
has the unique solution $\beta_i = \lambda_i$, if $2 \le i \le p$, and $\beta_i = 0$, if $p + 1 \le i \le d + 1$.
Applying Cramer's rule to system \eqref{eq:AlternanceLinSystm} one gets that
$\beta_i = (-1)^{i - 1} \Delta_i / \Delta_1$ for all $i \in \{ 2, \ldots, d + 1 \}$, where $\Delta_i$ are from
Def.~\ref{def:AlternanceOptCond}, which implies that \eqref{eq:DeterminantsProp} and \eqref{eq:DeterminantsProp2} hold
true. Thus, a $p$-point alternance exists at the point $x_*$.
\textbf{Part~\ref{stat:NessOpt_Alternance}.} ``$\impliedby$''. Let vectors $V_1, \ldots, V_{d + 1}$ be from the
definition of $p$-point alternance. Applying Cramer's rule to system \eqref{eq:AlternanceLinSystm} one obtains
that
$$
- V_1 = \sum_{i = 2}^p \beta_i V_i, \quad
\beta_i = (- 1)^{i - 1} \frac{\Delta _i}{\Delta_1} > 0 \quad \forall i \in \{ 2, \ldots, p \}.
$$
Denote $\beta_0 = 1 + \beta_2 + \ldots + \beta_{k_0} > 0$, and define $\alpha_1 = 1 / \beta_0 > 0$ and
$\alpha_i = \beta_i / \beta_0 \ge 0$ for all $i \in \{ 2, \ldots, d + 1 \}$. Then one has
\begin{equation} \label{eq:ConvConicCombOfAlterPoints}
\sum_{i = 1}^p \alpha_i V_i = 0, \quad \sum_{i = 1}^{k_0} \alpha_i = 1,
\end{equation}
i.e. $v_1 + v_2 + v_3 = 0$, where
$$
v_1 = \sum_{i = 1}^{k_0} \alpha_i V_i, \quad v_2 = \sum_{i = k_0 + 1}^{i_0} \alpha_i V_i, \quad
v_3 = \sum_{i = i_0 + 1}^{p} \alpha_i V_i
$$
(here, $v_2 = v_3 = 0$, if $k_0 = p$, and $v_3 = 0$, if $i_0 = p$). From the definition of alternance and the second
equality in \eqref{eq:ConvConicCombOfAlterPoints} it follows that $v_1 \in \partial F(x_*)$, $v_2 \in \mathcal{N}(x_*)$,
and $V_3 \in N_A(x_*)$. Thus, $0 \in \mathcal{D}(x_*)$.
\textbf{Part~\ref{stat:CompleteAlternance_SuffOpt}}. Suppose that a complete alternance $V_1, \ldots, V_{d + 1}$ exists
at $x_*$. Note that $V_1 \ne 0$, since all $\Delta_i$ are nonzero, which implies that $\partial F(x_*) \ne \{ 0 \}$.
Applying Cramer's rule to system \eqref{eq:AlternanceLinSystm} one gets that
\begin{equation} \label{eq:ReDefOfComplAlternance}
- V_1 = \sum_{i = 2}^{d + 1} \beta_i V_i, \quad
\beta_i = (- 1)^{i - 1} \frac{\Delta _i}{\Delta_1} > 0 \quad \forall i \in \{ 2, \ldots, d + 1 \}.
\end{equation}
Denote $\beta_0 = 1 + \beta_2 + \ldots + \beta_{k_0} > 0$, and define $\alpha_1 = 1 / \beta_0 > 0$ and
$\alpha_i = \beta_i / \beta_0 > 0$ for all $i \in \{ 2, \ldots, d + 1 \}$. Then \eqref{eq:ConvConicCombOfAlterPoints}
with $p = d + 1$ holds true.
Recall that by the definition of alternance $V_1, \ldots, V_{k_0} \in \partial F(x_*)$. Therefore,
$V_1, \ldots, V_{k_0} \in \mathcal{D}(x_*) = \partial F(x_*) + \mathcal{N}(x_*) + N_A(x_*)$, since
$0 \in \mathcal{N}(x_*)$ and $0 \in N_A(x_*)$. Moreover, from \eqref{eq:ConvConicCombOfAlterPoints} and the fact that
both $\mathcal{N}(x_*)$ and $N_A(x_*)$ are convex cones it follows that
\begin{align*}
V_i = 0 + V_i &= \sum_{j = 1}^{k_0} \alpha_j V_j + \sum_{j = k_0 + 1}^{i_0} \alpha_j V_j + V_i
+ \sum_{j = i_0 + 1}^{d + 1} \alpha_j V_j \\
&\in \partial F(x_*) + \mathcal{N}(x_*) + N_A(x_*) = \mathcal{D}(x_*)
\end{align*}
for any $i \in \{ k_0 + 1, \ldots, d + 1 \}$.
Therefore $S(x_*) = \co\{ V_1, \ldots, V_{d + 1} \} \subset \mathcal{D}(x)$ by virtue of the fact that
$\mathcal{D}(x_*)$ is a convex set.
Let $e_1, \ldots, e_d$ be the canonical basis of $\mathbb{R}^d$ and $\overline{e} = (- \beta_1, \ldots, - \beta_d)^T$,
where $\beta_i$ are from \eqref{eq:ReDefOfComplAlternance}. Denote $S = \co \{ e_1, \ldots, e_d, \overline{e} \}$ and
define a linear mapping $T \colon \mathbb{R}^d \to \mathbb{R}^d$ by setting $T e_i = V_{i + 1}$ for
all $i \in \{ 1, \ldots, d \}$. Then $T \overline{e} = V_1$ due to \eqref{eq:ReDefOfComplAlternance} and $T S = S(x_*)$.
Bearing in mind the fact that by the definition of complete alternance
$\Delta_1 = \determ([V_2, \ldots, V_{d + 1}]) \ne 0$, i.e. the vectors $V_2, \ldots, V_{d + 1}$ are linearly
independent, one obtains that $T$ is a linear bijection, which, in particular, implies that $T$ is an open mapping. Let
us show that $0 \in \interior S$. Then taking into account the facts that $T(\interior S)$ is an open set and
by definitions $0 \in T(\interior S) \subset S(x_*) \subset \mathcal{D}(x_*)$ one arrives at the required result.
For any $x = (x^{(1)}, \ldots, x^{(d)})^T \in \mathbb{R}^d$ denote $\| x \|_1 = |x^{(1)}| + \ldots + |x^{(d)}|$.
Applying Lemma~\ref{lem:SimplexZeroInterior} with $k = d$, $x_i = e_i$ for all $i \in \{ 1, \ldots, d \}$, and
$x = - \overline{e}$ one obtains that there exists $r > 0$ such that
$\{ x \in \mathbb{R}^d \mid \| x \|_1 < r \} \subset S$, that is, $0 \in \interior S$, and the proof is
complete.
\end{proof}
Thus, the existence of a $p$-point alternance (or, equivalently, the existence of a $p$-point cadre with positive cadre
multipliers) at a feasible point $x_*$ for some $p \in \{ 1, \ldots, d + 1 \}$ is a necessary optimality condition for
the problem $(\mathcal{P})$, while the existence of a complete alternance is a sufficient optimality condition, which
by Theorems~\ref{thrm:SuffOptCond} and \ref{thrm:EquivOptCond_Subdiff} implies that the first order growth condition
holds at $x_*$. As the following example shows, the converse statement is not true, that is, the sufficient optimality
condition $0 \in \interior \mathcal{D}(x_*)$ does not necessarily imply that a complete alternance exists at $x_*$.
\begin{example} \label{exmpl:ComplAtern_CounterExampl}
Consider the unconstrained problem
\begin{equation} \label{probl:ComplAltern_CounterExmpl}
\min_{x \in \mathbb{R}^d} \: F(x) = \| x \|_{\infty} = \max\big\{ \pm x^{(1)}, \ldots, \pm x^{(d)} \big\}.
\end{equation}
Clearly, $x_* = 0$ is a point of global minimum of this problem and the first order growth condition holds at $x_*$,
since, as is easy to see, $F(x) \ge |x| / \sqrt{n}$ for all $x \in \mathbb{R}^d$. Observe that by definition
$\partial F(0) = \co\big\{ \pm e_1, \ldots, \pm e_d \big\}$. Thus, in accordance with Thrms.~\ref{thrm:SuffOptCond}
and \ref{thrm:EquivOptCond_Subdiff} the sufficient optimality condition $0 \in \interior \partial F(0)$ is satisfied.
However, a complete alternance does not exists at $x_* = 0$.
Indeed, suppose that a $p$-point alternance for some $p \in \{ 1, \ldots, d + 1 \}$ exists at $x_*$. Then by
Proposition~\ref{prp:AlternanceVsCadre} there exist $V_1, \ldots, V_p \in \{ \pm e_1, \ldots, \pm e_d \}$ such that
$\rank([V_1, \ldots, V_p]) = p - 1$ and $\sum_{i = 1}^p \beta_i V_i = 0$ for some $\beta_i > 0$. Renumbering vectors
$V_i$, if necessary, one can suppose that the vectors $V_1, \ldots, V_{p - 1}$ are linearly independent. Hence taking
into account the fact that each $V_i$ is equal to either $e_{k_i}$ or $-e_{k_i}$ for some $k_i \in \{ 1, \ldots, d \}$
and $\sum_{i = 1}^p \beta_i V_i = 0$ for some $\beta_i > 0$ one obtains that $p = 2$. Thus, for any $d \in \mathbb{N}$
only a $2$-point alternance exists at $x_* = 0$ (note that for any $i \in \{ 1, \ldots, d \}$ the collection
$\{ e_i, -e_i \}$ satisfies the assumptions of Proposition~\ref{prp:AlternanceVsCadre}, i.e.
a $2$-point alternance does exist at $x_*$).
Note, however, that if one modifies the definition of alternance by allowing the vectors $V_1, \ldots, V_{k_0}$ to
belong to the entire subdifferential $\partial F(x_*)$ (see Definition~\ref{def:AlternanceOptCond}), then
a complete alternance exists at $x_* = 0$ in the problem under consideration. Indeed, defined $V_i = e_i$ for any
$i \in \{ 1, \ldots, d \}$ and put $V_{d + 1} = (-1/d, \ldots, -1/d)^T \in \partial F(x_*)$. Then, as is easily seen,
$\Delta_i = \determ([V_1, \ldots, V_{i - 1}, V_{i + 1}, \ldots, V_{d + 1}]) = (-1)^{d - i} (- 1/d)$ for any
$i \in \{ 1, \ldots, d \}$ and $\Delta_{d + 1} = 1$, i.e. conditions \eqref{eq:DeterminantsProp} and
\eqref{eq:DeterminantsProp2} are satisfied. \qed
\end{example}
The example above motivates us to introduce a weakened definition of alternance.
\begin{definition}
One says that \textit{a generalised $p$-point alternance} exists at $x_*$, if there exist $k_0 \in \{ 1, \ldots, p \}$,
$i_0 \in \{ k_0 + 1, \ldots, p \}$, vectors
\begin{equation} \label{eq:GenAlternanceDef}
V_1, \ldots, V_{k_0} \in \partial F(x_*), \quad
V_{k_0 + 1}, \ldots, V_{i_0} \in \mathcal{N}(x_*), \quad
V_{i_0 + 1}, \ldots, V_p \in N_A(x_*),
\end{equation}
and vectors $V_{p + 1}, \ldots, V_{d + 1} \in Z$ such that conditions \eqref{eq:DeterminantsProp} and
\eqref{eq:DeterminantsProp2} hold true. Such collection of vectors $\{ V_1, \ldots, V_p \}$ is called a
\textit{a generalised $p$-point alternance} at $x_*$. Any generalised $(d + 1)$-point alternance is called
\textit{complete}.
\end{definition}
\begin{remark} \label{rmrk:GenAlternanceVsCadre}
Almost literally repeating the proof of Proposition~\ref{prp:AlternanceVsCadre} one obtains that a generalised $p$-point
alternance with $p \in \{ 1, \ldots, d + 1 \}$ exists at $x_*$ iff there exist $k_0 \in \{ 1, \ldots, p \}$,
$i_0 \in \{ k_0 + 1, \ldots, p \}$, and vectors $V_1, \ldots, V_p$ satisfying \eqref{eq:GenAlternanceDef} such that
$\rank([V_1, \ldots, V_p]) = p - 1$ and $\sum_{i = 1}^p \beta_i V_i = 0$ for some $\beta_i > 0$,
$i \in \{ 1, \ldots, p \}$. \qed
\end{remark}
Clearly, any $p$-point alternance is a generalised $p$-point alternance as well. Therefore by
Theorem~\ref{thrm:AlternanceCond} the existence of a generalised $p$-point alternance is a necessary optimality
condition for the problem $(\mathcal{P})$ that is equivalent to the existence of a Lagrange multiplier (the fact that
the existence of a generalised $p$-point alternance implies the inclusion $0 \in \mathcal{D}(x_*)$ is proved in exactly
the same
way as the analogous statement for non-generalised $p$-point alternance).
In the general case the existence of a generalised complete alternance is not equivalent to the sufficient optimality
condition $0 \in \interior \mathcal{D}(x_*)$ (see Example~\ref{exmpl:ConstrComplAltern_CounterEx} in the following
section); however, under some additional assumptions one can prove that these conditions are indeed equivalent. To prove
this result we will need the following characterisation of relative interior points of a convex cone, which can be
viewed as an extension of a similar result for polytopes \cite[Lemma~2.9]{Zeigler} to the case of cones. Recall that
\textit{the dimension} of a convex cone $\mathcal{K} \subset \mathbb{R}^d$, denoted $\dimens \mathcal{K}$, is the
dimension of its affine hull, which obviously coincides with the linear span of $\mathcal{K}$.
\begin{lemma} \label{lem:RelIntConvexCone}
Let $\mathcal{K} \subset \mathbb{R}^d$ be a convex cone such that $k = \dimens \mathcal{K} \ge 1$. Then a point
$x \ne 0$ belongs to the relative interior $\relint \mathcal{K}$ of the cone $\mathcal{K}$ iff $x$ can be expressed as
$x = \sum_{i = 1}^k \beta_i x_i$ for some $\beta_i > 0$ and linearly independent vectors
$x_1, \ldots, x_k \in \mathcal{K}$.
\end{lemma}
\begin{proof}
Let $x \in \relint \mathcal{K}$ and $x \ne 0$. If $k = 1$, then put $x_1 = x$ and $\beta_1 = 1$. Otherwise, denote
$X_0 = \linhull \mathcal{K}$, and let $E_0 = \{ z \in X_0 \mid \langle z, x \rangle = 0 \}$ be the orthogonal complement
of $\linhull\{ x \}$ in $X_0$. As is well known, $\dimens E_0 = k - 1 \ge 1$. Let $z_1, \ldots, z_{k - 1} \in E_0$ be
any basis of $E_0$, and define $z_k = - \sum_{i = 1}^{k - 1} z_i$.
By the definition of relative interior $B(x, r) \cap X_0 \subset \mathcal{K}$ for some $r > 0$, where, as above,
$B(x, r) = \{ z \in \mathbb{R}^d \mid |z - x| \le r \}$. Let
$\delta = \max\{ |z_1|, \ldots, |z_k| \}$ and $\gamma = r / \delta$. Then for all $i \in \{ 1, \ldots, k \}$ one has
$x_i = \gamma z_i + x \in B(x, r) \cap X_0 \subset \mathcal{K}$. Furthermore,
observe that $x = \sum_{i = 1}^k (1/k) x_i$. Therefore, it remains to show that the vectors $x_1, \ldots, x_k$ are
linearly independent.
Indeed, suppose that $\sum_{i = 1}^k \alpha_i x_i = 0$ for some $\alpha_i \in \mathbb{R}$. Then by definition
$$
\sum_{i = 1}^k \alpha_i \gamma z_i = - \Big( \sum_{i = 1}^k \alpha_i \Big) x.
$$
Recall that $z_i$ belong to the orthogonal complement of $x$, i.e. $\langle z_i, x \rangle = 0$. Therefore
$\sum_{i = 1}^k \alpha_i = 0$. Hence taking into account the fact that $z_k = - \sum_{i = 1}^{k - 1} z_i$ one obtains
that $\sum_{i = 1}^{k - 1} (\alpha_i - \alpha_k) z_i = 0$, which implies that $\alpha_i = \alpha_k$ for all
$i \in \{ 1, \ldots, k - 1 \}$, since the vectors $z_1, \ldots, z_{k - 1}$ form a basis of $E_0$. Thus,
$\sum_{i = 1}^k \alpha_i = k \alpha_k = 0$, i.e. $\alpha_i = 0$ for all $i$, and one can conclude that the vectors
$x_1, \ldots, x_k$ are linearly independent.
Let us prove the converse statement. Suppose that $x = \sum_{i = 1}^k \beta_i x_i$ for some $\beta_i > 0$ and linearly
independent vectors $x_1, \ldots, x_k \in \mathcal{K}$. Denote $S(x) = \co\{ x_1, \ldots, x_k, - x \}$. Let us show that
there exists $r > 0$ such that $B(0, r) \cap X_0 \subset S(x)$, where, as above, $X_0 = \linhull \mathcal{K}$. Then
taking into account the fact that $\mathcal{K}$ is a convex cone one obtains that
\begin{equation} \label{eq:BallWithinConeSubSimples}
\big( B(x, r) \cap X_0 \big) \subset x + S(x) = \co\{ x_1 + x, \ldots, x_k + x, 0 \} \subset \mathcal{K},
\end{equation}
and the proof is complete.
Since $k = \dimens \mathcal{K}$, the collection $x_1, \ldots, x_k \in \mathcal{K}$ is a basis of the subspace
$X_0 = \linhull \mathcal{K}$. Therefore, for any $z \in X_0$ there exist unique $\alpha_i$ such that
$z = \sum_{i = 1}^k \alpha_i x_i$. For any $z \in X_0$ denote $\| z \|_{X_0} = \sum_{i = 1}^k |\alpha_i|$. One can
readily check that $\| \cdot \|_{X_0}$ is a norm on $X_0$.
With the use of Lemma~\ref{lem:SimplexZeroInterior} one obtains that
$\{ z \in X_0 \mid \| z \|_{X_0} < r \} \subset S(x)$ for some $r > 0$. Taking into account the fact that all norms on a
finite dimensional space are equivalent one gets that there exists $C > 0$ such that $\| z \|_{X_0} \le C |z|$ for all
$z \in X_0$. Therefore the inclusions
$(B(0, r / 2C) \cap X_0) \subset \{ z \in X_0 \mid \| z \|_{X_0} < r \} \subset S(x)$ hold true,
and the proof is complete.
\end{proof}
Recall that a convex cone $\mathcal{K} \subset \mathbb{R}^d$ is called \textit{pointed}, if
$\mathcal{K} \cap (-\mathcal{K}) = \{ 0 \}$.
\begin{theorem} \label{thrm:GenAlternanceCond}
Let $x_*$ be a feasible point of $(\mathcal{P})$. Then the existence of a generalised complete alternance at
$x_*$ implies that $0 \in \interior \mathcal{D}(x_*)$ and $\partial F(x_*) \ne \{ 0 \}$. Conversely, if
$0 \in \interior \mathcal{D}(x_*)$, $\partial F(x_*) \ne \{ 0 \}$, and one of the following assumptions is valid:
\begin{enumerate}
\item{$\interior \partial F(x_*) \ne \emptyset$,
}
\item{$\mathcal{N}(x_*) + N_A(x_*) \ne \mathbb{R}^d$ and either $\interior \mathcal{N}(x_*) \ne \emptyset$
or $\interior N_A(x_*) \ne \emptyset$,
}
\item{$N_A(x_*) = \{ 0 \}$ and there exists $w \in \relint \mathcal{N}(x_*) \setminus \{ 0 \}$ such that
$0 \in \partial F(x_*) + w$ (in particular, it is sufficient to suppose that $0 \notin \partial F(x_*)$ or the cone
$\mathcal{N}(x_*)$ is pointed),
}
\item{$\mathcal{N}(x_*) = \{ 0 \}$ and there exists $w \in \relint N_A(x_*) \setminus \{ 0 \}$ such that
$0 \in \partial F(x_*) + w$,
}
\end{enumerate}
then a generalised complete alternance exists at $x_*$.
\end{theorem}
\begin{proof}
If a generalised complete alternance exists at $x_*$, then literally repeating the proof of the second part of
Theorem~\ref{thrm:AlternanceCond} one obtains that $0 \in \interior \mathcal{D}(x_*)$ and $\partial F(x_*) \ne \{ 0 \}$.
Let us prove the converse statement. Consider four cases corresponding to four assumptions of the theorem.
\textbf{Case I.} Suppose that $\interior \partial F(x_*) \ne \emptyset$. If $0 \in \interior \partial F(x_*)$, then
one can find $r > 0$ such that $r e_1, \ldots, r e_d \in \partial F(x_*)$ and
$\overline{e} = (-r, \ldots, - r)^T \in \partial F(x_*)$. Note that $\rank([r e_1, \ldots, r e_d, \overline{e}]) = d$
and $\sum_{i = 1}^d r e_i + \overline{e} = 0$. Hence by Remark~\ref{rmrk:GenAlternanceVsCadre} a generalised
complete alternance exists at $x_*$.
Thus, one can suppose that $0 \notin \interior \partial F(x_*)$. Let there exists $w \in \mathcal{N}(x_*) \cup N_A(x_*)$
such that $0 \in \interior \partial F(x_*) + w$. Clearly, $w \ne 0$ and $-w \in \interior \partial F(x_*)$. If $d = 1$,
then define $V_1 = -w$, $V_2 = w$. Then $\rank([V_1, V_2]) = 1$ and $V_1 + V_2 = 0$, which due to
Remark~\ref{rmrk:GenAlternanceVsCadre} implies that a generalised complete alternance exists at $x_*$. If $d \ge 2$,
then denote by $X_0$ the orthogonal complement of the subspace $\linhull\{ w \}$. Obviously, $\dimens X_0 = d - 1$.
Let $z_1, \ldots, z_{d - 1}$ be a basis of $X_0$, and $z_d = - \sum_{i = 1}^{d - 1} z_i$.
Since $-w \in \interior \partial F(x_*)$, there exists $r > 0$ such that $V_i = -w + r z_i \in \partial F(x_*)$ for all
$i \in \{ 1, \ldots, d \}$. Denote $V_{d + 1} = w$. Observe that $\sum_{i = 1}^{d} (1/d) V_i + V_{d + 1} = 0$.
Furthermore, the vectors $V_1, \ldots, V_{d - 1}, V_{d + 1}$ are linearly independent. Indeed, suppose that
$\sum_{i = 1}^{d - 1} \alpha_i V_i + \alpha_{d + 1} V_{d + 1} = 0$ for some $\alpha_i \in \mathbb{R}$. Then
$$
r \sum_{i = 1}^{d - 1} \alpha_i z_i = \Big( \sum_{i = 1}^{d - 1} \alpha_i - \alpha_{d + 1} \Big) w.
$$
Bearing in mind the fact that $z_1, \ldots, z_{d - 1}$ is a basis of the orthogonal complement of $\linhull\{ w \}$ one
obtains that $\alpha_{d + 1} = \sum_{i = 1}^{d - 1} \alpha_i$ and $\alpha_i = 0$ for all
$i \in \{ 1, \ldots, d - 1 \}$, which implies that the vectors $V_1, \ldots, V_{d - 1}, V_{d + 1}$ are linearly
independent. Consequently, $\rank([V_1, \ldots, V_{d + 1}]) = d$ and by Remark~\ref{rmrk:GenAlternanceVsCadre}
a generalised complete alternance exists at $x_*$.
Thus, one can suppose that
\begin{equation} \label{eq:SubdiffShiftViaCones}
0 \notin \interior \partial F(x_*) + w \quad \forall w \in \mathcal{N}(x_*) \cup N_A(x_*).
\end{equation}
Note that $0 \in \interior \partial F(x_*) + w$ for some $w \in \mathcal{N}(x_*) + N_A(x_*)$. Indeed, arguing by
reductio ad absurdum, suppose that $(- \interior \partial F(x_*)) \cap (\mathcal{N}(x_*) + N_A(x_*)) = \emptyset$. Then
by the separation theorem (see, e.g. \cite[Thrm.~2.13]{BonnansShapiro}) there exists
$h \ne 0$ such that $\langle h, v \rangle \le \langle h, w \rangle$ for all $v \in - \partial F(x_*)$ and
$w \in \mathcal{N}(x_*) + N_A(x_*)$. Hence $\langle h, v \rangle \ge 0$ for all
$v \in \partial F(x_*) + \mathcal{N}(x_*) + N_A(x_*) = \mathcal{D}(x_*)$, which contradicts the assumption that
that $0 \in \interior \mathcal{D}(x_*)$.
By definition $w = w_1 + w_2$ for some $w_1 \in \mathcal{N}(x_*)$ and $w_2 \in N_A(x_*)$. The vectors $w_1$ and $w_2$
are linearly independent. Indeed, if $w_1 = \alpha w_2$ for some $\alpha \ge 0$, then
$w = (1 + \alpha) w_2 \in N_A(x_*)$, since $N_A(x_*)$ is a cone, which contradicts
\eqref{eq:SubdiffShiftViaCones}. Similarly, if $w_1 = - \alpha w_2$ for some $\alpha > 0$, then
$w = (1 - \alpha) w_2 \in N_A(x_*)$ in the case $\alpha \in (0, 1]$, and $w = (1 - 1/\alpha) w_1 \in \mathcal{N}(x_*)$
in the case $\alpha > 1$, which once again contradicts \eqref{eq:SubdiffShiftViaCones}. Thus, $w_1$ and $w_2$ are
linearly independent and $d \ge 2$.
If $d = 2$, denote $V_1 = -w \in \partial F(x_*)$, $V_2 = w_1$, and $V_3 = w_2$. Then $V_1 + V_2 + V_3 = 0$ and
$\rank([V_1, V_2, V_3]) = 2$, which implies that a generalised complete alternance exists at $x_*$ due to
Remark~\ref{rmrk:GenAlternanceVsCadre}. If $d \ge 3$, then denote by $X_0$ the orthogonal complement of
$\linhull\{ w_1, w_2 \}$. Clearly, $\dimens X_0 = d - 2$. Let $z_1, \ldots, z_{d - 2}$ be a basis of $X_0$ and
$z_{d - 1} = - \sum_{i = 1}^{d - 2} z_i$.
Since $- w \in \interior \partial F(x_*)$, there exists $r > 0$ such that $V_i = -w + r z_i \in \partial F(x_*)$ for all
$i \in \{ 1, \ldots, d - 1 \}$. Denote $V_d = w_1$ and
$V_{d + 1} = w_2$. Then $\sum_{i = 1}^{d - 1} (1/(d - 1)) V_i + V_d + V_{d + 1} = 0$. Moreover, the vectors
$V_1, \ldots, V_{d- 2}, V_d, V_{d + 1}$ are linearly independent. Indeed, if for some $\alpha_i \in \mathbb{R}$ one has
$\sum_{i = 1}^{d - 2} \alpha_i V_i + \alpha_d V_d + \alpha_{d + 1} V_{d + 1} = 0$, then
$$
r \sum_{i = 1}^{d - 2} \alpha_i z_i = \Big( \sum_{i = 1}^{d - 2} \alpha_i - \alpha_d \Big) w_1
+ \Big( \sum_{i = 1}^{d - 2} \alpha_i - \alpha_{d + 1} \Big) w_2.
$$
Taking into account the facts that $z_1, \ldots, z_{d - 2}$ is a basis of the orthogonal complement of
$\linhull\{ w_1, w_2 \}$ and the vectors $w_1$ and $w_2$ are linearly independent one can easily check that
$\alpha_i = 0$ for any $i \in \{ 1, \ldots, d - 2, d, d + 1 \}$. Thus, the vectors
$V_1, \ldots, V_{d- 2}, V_d, V_{d + 1}$ are linearly independent, which by Remark~\ref{rmrk:GenAlternanceVsCadre}
implies that a generalised complete alternance exists at $x_*$.
\textbf{Case II.} Let $\mathcal{N}(x_*) + N_A(x_*) \ne \mathbb{R}^d$ and $\interior \mathcal{N}(x_*) \ne \emptyset$ (the
case when $\interior N_A(x_*) \ne \emptyset$ is proved in the same way). Suppose that there exists
$w \in \partial F(x_*)$ such that $-w \in \interior \mathcal{N}(x_*)$. Let us show that one can assume that $w \ne 0$.
Indeed, if $w = 0$, then $0 \in \interior \mathcal{N}(x_*)$. Recall that by our assumption
$\partial F(x_*) \ne \{ 0 \}$. Choose any $v \in \partial F(x_*) \setminus \{ 0 \}$.
Since $0 \in \interior \mathcal{N}(x_*)$, there exists $\alpha \in (0, 1]$ such that
$\alpha v \in \interior \mathcal{N}(x_*)$ and $\alpha v \in \co\{ 0, v \} \subseteq \partial F(x_*)$. Thus, there
exists $w \in \partial F(x_*) \setminus \{ 0 \}$ such that $-w \in \interior \mathcal{N}(x_*)$.
Denote $V_1 = w$. Since $\interior \mathcal{N}(x_*) \ne \emptyset$, one has $\dimens \mathcal{N}(x_*) = d$. Therefore by
Lemma~\ref{lem:RelIntConvexCone} there exist linearly independent vectors $V_2, \ldots, V_{d + 1} \in \mathcal{N}(x_*)$
such that $V_1 + \sum_{i = 2}^{d + 1} \beta_i V_i = 0$ for some $\beta_i > 0$, $i \in \{2, \ldots, d + 1 \}$. Thus,
$\rank([V_1, \ldots, V_{d + 1}]) = d$, which by Remark~\ref{rmrk:GenAlternanceVsCadre} implies that a generalised
complete alternance exists at $x_*$.
Suppose now that
\begin{equation} \label{eq:SolidCone_Subdiff}
(- \partial F(x_*)) \cap \interior \mathcal{N}(x_*) = \emptyset.
\end{equation}
Then there exist $v \in \partial F(x_*)$ and $w \in N_A(x_*)$ such that $- v - w \in \interior \mathcal{N}(x_*)$.
Indeed, otherwise the sets $- (\partial F(x_*) + N_A(x_*))$ and $\interior \mathcal{N}(x_*)$ do not intersect, which by
the separation theorem implies that there exists $h \in \mathbb{R}^d \setminus \{ 0 \}$ such
that $\langle h, v \rangle \le 0$ for all $v \in - (\partial F(x_*) + N_A(x_*))$ and $\langle h, w \rangle \ge 0$ for
all $w \in \mathcal{N}(x_*)$. Hence $\langle h, v \rangle \ge 0$ for
all $v \in \partial F(x_*) + \mathcal{N}(x_*) + N_A(x_*) = \mathcal{D}(x_*)$, which contradicts the
assumption that $0 \in \interior \mathcal{D}(x_*)$.
Thus, there exist $v \in \partial F(x_*)$ and $w \in N_A(x_*)$ such that $- v - w \in \interior \mathcal{N}(x_*)$. Note
that $w \ne 0$ due to \eqref{eq:SolidCone_Subdiff}. Furthermore, one can suppose that the vectors $v$ and $w$ are
linearly independent. Indeed, if $v = \alpha w$ for some $\alpha < - 1$, then one obtains that
$- \beta v \in \interior \mathcal{N}(x_*)$, where $\beta = 1 + 1 / \alpha \in (0, 1)$. Therefore there exists
$\varepsilon > 0$ such that $- \beta v + B(0, \varepsilon) \subset \mathcal{N}(x_*)$, which implies that
$- v + B(0, \varepsilon / \beta) \subset \mathcal{N}(x_*)$ due to the fact that $\mathcal{N}(x_*)$ is a cone.
Thus, $- v \in \interior \mathcal{N}(x_*)$, which contradicts \eqref{eq:SolidCone_Subdiff}.
On the other hand, if $v = \alpha w$ for some $\alpha \ge -1$, then for $z = (1 + \alpha) w \in N_A(x_*)$ one has
$- z \in \interior \mathcal{N}(x_*)$. By definition there exists $\varepsilon > 0$ such that
$-z + B(0, \varepsilon) \subset \mathcal{N}(x_*)$. Consequently, one has
$B(0, \varepsilon) = -z + B(0, \varepsilon) + z \subset \mathcal{N}(x_*) + N_A(x_*)$. Hence with the use of the fact
that the sets $\mathcal{N}(x_*)$ and $N_A(x_*)$ are cones one obtains that $\mathcal{N}(x_*) + N_A(x_*) = \mathbb{R}^d$,
which contradicts our assumption. Thus, the vectors $v$ and $w$ are linearly independent, which implies that $d \ge 2$.
If $d = 2$, define $V_1 = v \in \partial F(x_*)$, $V_2 = - v - w \in \mathcal{N}(x_*)$, and $V_3 = w \in N_A(x_*)$. Then
$\rank([V_1, V_2, V_3]) = 2$ and $V_1 + V_2 + V_3 = 0$. Therefore by Remark~\ref{rmrk:GenAlternanceVsCadre} a
generalised complete alternance exists at $x_*$. If $d \ge 3$, denote by $X_0$ the orthogonal complement of
$\linhull\{ v, w \}$. Since $v$ and $w$ are linearly independent, one has $\dimens X_0 = d - 2$.
Let $z_1, \ldots, z_{d - 2}$ be a basis of $X_0$ and $z_{d - 1} = - \sum_{i = 1}^{d - 2} z_i$.
Since $- v - w \in \interior \mathcal{N}(x_*)$, there exists $r > 0$ such that $-v - w + r z_i \subset \mathcal{N}(x_*)$
for all $i \in \{ 1, \ldots, d - 1 \}$. Denote $V_1 = v$, $V_i = r z_{i - 1} - v - w \in \mathcal{N}(x_*)$ for all
$i \in \{ 2, \ldots, d \}$, and $V_{d + 1} = w \in N_A(x_*)$.
Then $V_1 + \sum_{i = 2}^d (1/(d - 1)) V_i + V_{d + 1} = 0$. Let us check that the vectors
$V_1, \ldots, V_{d - 1}, V_{d + 1}$ are linearly independent. Then $\rank([V_1, \ldots, V_{d + 1}]) = d$ and by
Remark~\ref{rmrk:GenAlternanceVsCadre} one concludes that a generalised complete alternance exists at $x_*$.
Let $\sum_{i = 1}^{d - 1} \alpha_i V_i + \alpha_{d + 1} V_{d + 1} = 0$ for some $\alpha_i \in \mathbb{R}$. Then
$$
r \sum_{i = 1}^{d - 2} \alpha_{i + 1} z_i = \Big( \sum_{i = 2}^{d - 1} \alpha_i - \alpha_1 \Big) v
+ \Big( \sum_{i = 2}^{d - 1} \alpha_i - \alpha_{d + 1} \Big) w.
$$
Hence bearing in mind the fact that $z_1, \ldots, z_{d - 2}$ is a basis of the orthogonal complement of
$\linhull\{ v, w \}$ one obtains that $\alpha_i = 0$ for all $i \in \{ 2, \ldots, d - 1 \}$,
$\alpha_1 = \sum_{i = 2}^{d - 1} \alpha_i = 0$, and $\alpha_{d + 1} = \sum_{i = 2}^{d - 1} \alpha_i = 0$. Thus, the
vectors $V_1, \ldots, V_{d - 1}, V_{d + 1}$ are linearly independent and the proof of Case II is complete.
\textbf{Case III.} Let $N_A(x_*) = \{ 0 \}$ and there exists $w \in \relint \mathcal{N}(x_*) \setminus \{ 0 \}$ such
that $0 \in \partial F(x_*) + w$. Let us check at first that it is sufficient to assume that $N_A(x_*) = \{ 0 \}$ and
either $0 \notin \partial F(x_*)$ or the cone $\mathcal{N}(x_*)$ is pointed.
Indeed, let $0 \notin \partial F(x_*)$. Let us verify that
$(-\partial F(x_*)) \cap \relint \mathcal{N}(x_*) \ne \emptyset$. Then taking into account the fact that
$0 \notin \partial F(x_*)$ one obtains that there exists $w \in \relint \mathcal{N}(x_*) \setminus \{ 0 \}$ such that
$0 \in \partial F(x_*) + w$.
Arguing by reductio ad absurdum, suppose that $(-\partial F(x_*)) \cap \relint \mathcal{N}(x_*) = \emptyset$. Then by
the separation theorem (see, e.g. \cite[Thrm.~11.3]{Rockafellar}) there exists $h \ne 0$ such that
$\langle v, h \rangle \le \langle w, h \rangle$ for all $v \in - \partial F(x_*)$ and $w \in \mathcal{N}(x_*)$. Hence
$\langle h, v \rangle \ge 0$ for all $v \in \partial F(x_*) + \mathcal{N}(x_*) = \mathcal{D}(x_*)$
(recall that $N_A(x_*) = \{ 0 \}$), which is impossible, since $0 \in \interior \mathcal{D}(x_*)$.
Let now the cone $\mathcal{N}(x_*)$ be pointed. If $\interior F(x_*) \ne \emptyset$, then a generalised
complete alternance exists at $x_*$ by Case~I. Therefore, we can suppose that $\interior F(x_*) = \emptyset$.
Arguing by reductio ad absurdum, suppose that $0 \notin \partial F(x_*) + w$ for any
$w \in \relint \mathcal{N}(x_*) \setminus \{ 0 \}$. As was shown above,
$(-\partial F(x_*)) \cap \relint \mathcal{N}(x_*) \ne \emptyset$, that is, there exists $w \in \relint \mathcal{N}(x_*)$
such that $0 \in \partial F(x_*) + w$. Consequently, by our assumption $0 \in \relint \mathcal{N}(x_*)$. Hence either
$\mathcal{N}(x_*) = \{ 0 \}$ or $\dimens \mathcal{N}(x_*) \ge 1$. In the former case one has
$\mathcal{D}(x_*) = \partial F(x_*)$. Therefore $0 \in \interior \partial F(x_*)$, which contradicts our assumption. In
the latter case there exists $z \in \mathcal{N}(x_*) \setminus \{ 0 \}$ and by the definition of relative interior there
exists $r > 0$ such that $\linhull \mathcal{N}(x_*) \cap B(0, r) \subset \mathcal{N}(x_*)$. Consequently,
$r z / |z| \in \mathcal{N}(x_*)$ and $-r z / |z| \in \mathcal{N}(x_*)$, which contradicts the assumption that the cone
$\mathcal{N}(x_*)$ is pointed.
Let us now turn to the proof of the main statement. Let $w_* \in \relint \mathcal{N}(x_*)$, $w_* \ne 0$, be any
vector such that $0 \in \partial F(x_*) + w_*$. By Lemma~\ref{lem:RelIntConvexCone} there exists
$k = \dimens \mathcal{N}(x_*)$ linearly independent vectors $w_1, \ldots, w_k \in \mathcal{N}(x_*)$ such that
$w_* = \sum_{i = 1}^k \beta_i w_i$ for some $\beta_i > 0$.
Note that $\linhull\{ w_1, \ldots, w_k \} = \linhull \mathcal{N}(x_*)$.
\begin{figure}[t]
\centering
\includegraphics[width=0.6\linewidth]{Reduction.mps}
\caption{In Case III we assume that $0 \in \interior (\partial F(x_*) + \mathcal{N}(x_*))$ and there exists
$w_* \in \relint \mathcal{N}(x_*) \setminus \{ 0 \}$ such that $0 \in \partial F(x_*) + w_*$. The first step of the
proof consists in showing that one can replace the cone $\mathcal{N}(x_*)$ in the condition
$0 \in \interior (\partial F(x_*) + \mathcal{N}(x_*))$ by a polyhedral cone
$\mathcal{C}_k = \cone\{ w_1, \ldots, w_k \}$ such that $w_* \in \relint \mathcal{C}_k$, where the vectors
$w_i \in \mathcal{N}(x_*)$ are linearly independent and $k = \dimens \mathcal{N}(x_*)$.}
\label{fig:reduction}
\end{figure}
Denote $\mathcal{C}_k = \cone\{ w_1, \ldots, w_k \}$. Our first goal is to check the validity of the inclusion
$0 \in \interior(\partial F(x_*) + \mathcal{C}_k)$ (see Fig.~\ref{fig:reduction} below). Indeed, let
$X_k = \linhull \mathcal{N}(x_*)$. As was shown in the proof of the ``only if'' part of Lemma~\ref{lem:RelIntConvexCone}
(see \eqref{eq:BallWithinConeSubSimples}),
$X_k \cap B(w_*, r) \subset \co\{ w_1 + w_*, \ldots, w_k + w_*, 0 \} \subset \mathcal{C}_k$ for some $r > 0$, where the
last inclusion follows from the definition of $\mathcal{C}_k$ and the fact that $w_* = \sum_{i = 1}^k \beta_i w_i$.
By our assumptions $0 \in \interior \mathcal{D}(x_*)$ and $\mathcal{D}(x_*) = \partial F(x_*) + \mathcal{N}(x_*)$.
Therefore there exists $\gamma > 0$ such that for any $i \in \{ 1, \ldots, d + 1 \}$ one can find
$v_i \in \partial F(x_*)$ and $u_i \in \mathcal{N}(x_*) \subset X_k$ for which $v_i + u_i = \gamma e_i$,
where $e_1, \ldots, e_d$ is the canonical basis of $\mathbb{R}^d$ and $e_{d + 1} = - \sum_{i = 1}^d e_i$. Clearly, there
exists $\alpha \in (0, 1)$ such that $(1 - \alpha) w_* + \alpha u_i \in X_k \cap B(w_*, r)$
for any $i \in \{ 1, \ldots, d + 1 \}$.
Let $v_* \in \partial F(x_*)$ be such that $v_* + w_* = 0$. Then for any $i \in \{ 1, \ldots, d + 1 \}$ one has
\begin{multline*}
\alpha \gamma e_i = (1 - \alpha) (v_* + w_*) + \alpha (v_i + u_i)
= \big( (1 - \alpha) v_* + \alpha v_i \big) + \big( (1 - \alpha) w_* + \alpha u_i \big) \\
\in \partial F(x_*) + \big( X_k \cap B(w_*, r) \big) \subset \partial F(x_*) + \mathcal{C}_k.
\end{multline*}
Hence taking into account the fact that the set $\partial F(x_*) + \mathcal{C}_k$ is obviously convex one gets that
$\co\{ \alpha \gamma e_1, \ldots, \alpha \gamma e_d, - \alpha \gamma \sum_{i = 1}^d e_i \}
\subset \partial F(x_*) + \mathcal{C}_k$. Consequently, with the use of Lemma~\ref{lem:SimplexZeroInterior} one obtains
that there exists $r > 0$ such that
\begin{align*}
B\left( 0, \frac{\alpha \gamma r}{2\sqrt{d}} \right)
&\subset \Big\{ x = (x^{(1)}, \ldots, x^{(d)})^T \in \mathbb{R}^d \Bigm|
\sum_{i = 1}^d |x^{(i)}| < \alpha \gamma r \Big\} \\
&\subset \co\Big\{ \alpha \gamma e_1, \ldots, \alpha \gamma e_d, - \alpha \gamma \sum_{i = 1}^d e_i \Big\}
\subset \partial F(x_*) + \mathcal{C}_k,
\end{align*}
that is, $0 \in \interior(\partial F(x_*) + \mathcal{C}_k)$.
Now we turn to the proof of the existence of generalised complete alternance. Denote $k_0 = d + 1 - k \ge 1$ and
$V_{k_0 + i} = w_i$ for any $i \in \{ 1, \ldots, k \}$. Observe that
$$
\mathbb{R}^d = \linhull\Big( \partial F(x_*) + \mathcal{C}_k \Big)
\subseteq \linhull\Big\{ \partial F(x_*), \mathcal{C}_k \Big\} \subseteq \mathbb{R}^d,
$$
where the first equality follows from the fact that and $0 \in \interior (\partial F(x_*) + \mathcal{C}_k)$. Therefore,
there exists vectors $V_2, \ldots, V_{k_0} \in \partial F(x_*)$ such that the vectors $V_2, \ldots, V_{d + 1}$ are
linearly independent.
Denote $Q(x_*) = \cone\{ V_2, \ldots, V_{d + 1} \}$ (see Fig.~\ref{fig:negative_cone}). Observe that by definition the
affine hull of $Q(x_*)$ coincides with $\mathbb{R}^d$, since $Q(x_*)$ contains $d + 1$ affinely independent vectors:
$0, V_2, \ldots, V_{d + 1}$. Therefore the relative interior of $Q(x_*)$ coincides with its topological interior, which
implies that $\interior Q(x_*) \ne \emptyset$ due to the fact that the relative interior of a convex subset of a finite
dimensional space is always nonempty.
\begin{figure}[t]
\centering
\includegraphics[width=0.6\linewidth]{Negative_cone.mps}
\caption{As soon as the condition $0 \in \interior (\partial F(x_*) + \mathcal{C}_k)$ has been checked, one can easily
find linearly independent vectors $V_2, \ldots, V_{d + 1} \in \partial F(x_*) \cup \mathcal{C}_k$. The next step is to
prove that there exists $V_1 \in \partial F(x_*)$ such that $V_1 \in - \interior \cone\{ V_2, \ldots, V_{d + 1} \}$.
Then $V_1, \ldots, V_{d + 1}$ is the desired generalised complete alternance. However, to prove the existence of
such $V_1$ one needs to properly choose the cone $\mathcal{C}_k$.}
\label{fig:negative_cone}
\end{figure}
Let us verify that $(- \interior Q(x_*)) \setminus Q(x_*) \ne \emptyset$. Indeed, arguing by reductio ad absurdum
suppose that $- \interior Q(x_*) \subset Q(x_*)$. Choose any $z \in \interior Q(x_*)$. Then
$z + B(0, \varepsilon) \subset \interior Q(x_*) \subset Q(x_*)$ for some $\varepsilon > 0$. Consequently, one has
$- z - B(0, \varepsilon) \subset - \interior Q(x_*) \subset Q(x_*)$. Hence taking into account the fact that $Q(x_*)$ is
a convex cone (which implies that $Q(x_*)$ is closed under addition) one obtains that
$$
B(0, \varepsilon) \subset \big( z + B(0, \varepsilon) \big) + \big( -z - B(0, \varepsilon) \big) \subset Q(x_*).
$$
Choose any $u \in B(0, \varepsilon)$, $u \ne 0$. Then $u \in Q(x_*)$ and $- u \in Q(x_*)$. By the definition of $Q(x_*)$
one has $u = \sum_{i = 2}^{d + 1} \alpha_i V_i$ for some $\alpha_i \ge 0$ and $- u = \sum_{i = 2}^{d + 1} \beta_i V_i$
for some $\beta_i \ge 0$. Summing up these equalities one obtains $\sum_{i = 2}^{d + 1} (\alpha_i + \beta_i) V_i = 0$,
which implies that $\alpha_i = \beta_i = 0$ for all $i \in \{ 2, \ldots, d + 1 \}$, since the vectors
$V_2, \ldots, V_{d + 1}$ are linearly independent. Consequently, $u = 0$, which contradicts our assumption that
$u \ne 0$.
Thus, there exists a nonzero vector $\xi \in (- \interior Q(x_*)) \setminus Q(x_*)$. By definition one can find
$\varepsilon > 0$ such that $- \xi + B(0, \varepsilon) \subset Q(x_*)$. Since $Q(x_*)$ is a cone,
$- \alpha \xi + B(0, \alpha \varepsilon) \subset Q(x_*)$ for any $\alpha > 0$, that is,
$\alpha \xi \in - \interior Q(x_*)$. Furthermore, $\alpha \xi \notin Q(x_*)$, since otherwise $\xi \in Q(x_*)$.
Since $0 \in \interior(\partial F(x_*) + \mathcal{C}_k)$, by choosing a sufficiently small $\alpha > 0$ we can
suppose that $\alpha \xi \in \partial F(x_*) + \mathcal{C}_k$. Therefore there exists $V_1 \in \partial F(x_*)$ and
$u \in \mathcal{C}_k \subset Q(x_*)$ such that $\alpha \xi = V_1 + u$ (the inclusion $\mathcal{C}_k \subset Q(x_*)$
follows from the fact that $\mathcal{C}_k = \cone\{ V_{k_0 + 1}, \ldots, V_{d + 1} \} \subset Q(x_*)$ by definition).
Observe that $V_1 = \alpha \xi - u \in ( - \interior Q(x_*)) - Q(x_*) = - \interior Q(x_*)$, where the last equality
follows from the fact that if $z_1 \in \interior Q(x_*)$ and $z_2 \in Q(x_*)$, then for some $\varepsilon > 0$ one has
$z_1 + B(0, \varepsilon) \subset Q(x_*)$, which implies that $z_1 + B(0, \varepsilon) + z_2 \subset Q(x_*)$, i.e.
$z_1 + z_2 \in \interior Q(x_*)$.
Note that if a vector $v \in Q(x_*)$ can be represented as a linear combination with positive coefficients of $d - 1$
or fewer vectors from the set $V_2, \ldots, V_{d + 1}$, then $v \notin \interior Q(x_*)$. Indeed, let
$v \in Q(x_*) = \cone\{ V_2, \ldots, V_{d + 1} \}$ have the form
$$
v = \beta_2 V_2 + \ldots + \beta_{i - 1} V_{i - 1} + \beta_{i + 1} V_{i + 1} + \ldots +
\beta_{d + 1} V_{d + 1},
$$
for some $\beta_j \ge 0$ and $i \in \{ 2, \ldots, d + 1 \}$. For any $\varepsilon > 0$ define
$v_{\varepsilon} = v - \varepsilon V_i$. Observe that $v_{\varepsilon} \notin Q(x_*)$, since otherwise by the
definition of $Q(x_*)$ one could find $\gamma_j \ge 0$, $j \in \{ 2, \ldots, d + 1 \}$, such that
$$
\sum_{j = 2}^{i - 1} \beta_j V_j + \sum_{j = i + 1}^{d + 1} \beta_j V_j
- \varepsilon V_i = \sum_{j = 2}^{d + 1} \gamma_j V_j,
$$
which contradicts the fact that the vectors $V_2, \ldots, V_{d + 1}$ are linearly independent. On the other hand, note
that choosing $\varepsilon > 0$ sufficiently small one can ensure that $v_{\varepsilon}$ belongs to an arbitrarily
small neighbourhood of $v$, which implies that $v \notin \interior Q(x_*)$. Thus, the vector $-V_1 \in \interior Q(x_*)$
can only be represented in the form $- V_1 = \sum_{i = 2}^{d + 1} \beta_i V_i$ for some $\beta_i > 0$,
$i \in \{ 2, \ldots, d + 1 \}$. Looking at this representation as a system of linear equations with respect to
$\beta_i$ and applying Cramer's rule one obtains that $\Delta_1 \ne 0$ and
$\beta_i = (-1)^{i - 1} \Delta_i / \Delta_1 > 0$ for any $i \in \{ 2, \ldots, d + 1 \}$, where, as in the definition of
alternance, $\Delta_i = \determ([V_1, \ldots, V_{i - 1}, V_{i + 1}, \ldots, V_{d + 1}])$.
Therefore, all determinants $\Delta_s$ are nonzero, and $\sign \Delta_s = - \sign \Delta_{s + 1}$
for all $s \in \{ 1, \ldots, d \}$, that is, a generalised complete alternance exists at $x_*$.
\textbf{Case IV.} The proof of this case repeats the proof of the previous one with $\mathcal{N}(x_*)$ replaced by
$N_A(x_*)$.
\end{proof}
\begin{remark} \label{rmrk:IsolatePoint_WeakAlternance}
{(i)~Note that the condition $\mathcal{N}(x_*) + N_A(x_*) \ne \mathbb{R}^d$ in the second assumption of the theorem
above simply means that $x_*$ is not an isolated point of the feasible region $\Omega$ of the problem $(\mathcal{P})$.
Indeed, fix any $v_1 \in \mathcal{N}(x_*)$ and $v_2 \in N_A(x_*)$. One can easily verify that, regardless of whether RCQ
holds true or not, one has $T_{\Omega}(x_*) \subseteq \{ h \in T_A(x_*) \mid D G(x_*) h \in T_K(G(x)) \}$, which by
Lemma~\ref{lem:NormalCone_ConeConstr} implies that $\langle v_1, h \rangle \le 0$ and $\langle v_2, h \rangle \le 0$ for
any $h \in T_{\Omega}(x_*)$. Therefore $\mathcal{N}(x_*) + N_A(x_*) \subset (T_{\Omega}(x_*))^* = N_{\Omega}(x_*)$.
Thus, if $\mathcal{N}(x_*) + N_A(x_*) = \mathbb{R}^d$, then $N_{\Omega}(x_*) = \mathbb{R}^d$, which with the use of
\cite[Prp.~2.40]{BonnansShapiro} implies that
$\cl \cone(T_{\Omega}(x_*)) = T_{\Omega}(x_*)^{**} = N_{\Omega}(x_*)^* = \{ 0 \}$.
On the other hand, if $x_*$ is a non-isolated point of $\Omega$, then there exists a sequence
$x_n \subset \Omega \setminus \{ x_* \}$ converging to $x_*$. Replacing $\{ x_n \}$, if necessary, with its subsequence
one can suppose that the sequence $\{ (x_n - x_*) / |x_n - x_*| \}$ converges to some $v \ne 0$, which obviously
belongs to $T_{\Omega}(x_*)$. Thus, one can conclude that the condition $\mathcal{N}(x_*) + N_A(x_*) = \mathbb{R}^d$
implies that $x_*$ is an isolated point of $\Omega$.
}
\noindent{(ii)~Let us note that by further weakening the definition of generalised alternance one can obtain sufficient
optimality conditions for the problem $(\mathcal{P})$ in an alternance form that are equivalent to the condition
$0 \in \interior \mathcal{D}(x_*)$ under less restrictive assumptions. Namely, one says that \textit{a weak $p$-point
alternance} exists at $x_*$, if there exist $k_0 \in \{ 1, \ldots, p \}$, vectors
$V_1, \ldots, V_{k_0} \in \partial F(x_*)$, $V_{k_0 + 1}, \ldots, V_p \in \mathcal{N}(x_*) + N_A(x_*)$, and
$V_{p + 1}, \ldots, V_{d + 1} \in Z$ such that conditions \eqref{eq:DeterminantsProp} and \eqref{eq:DeterminantsProp2}
hold true. Almost literally repeating the proof of the third case of the previous theorem with $\mathcal{N}(x_*)$
replaced by $\mathcal{N}(x_*) + N_A(x_*)$ one can prove that $0 \in \interior \mathcal{D}(x_*)$ and
$\partial F(x_*) \ne \{ 0 \}$, provided a weak complete alternance exists at $x_*$ and
$0 \in \partial F(x_*) + w$ for some $w \in \relint(\mathcal{N}(x_*) + N_A(x_*)) \setminus \{ 0 \}$ (in particular,
it is sufficient to assume that the necessary condition for an unconstrained local minimum $0 \in \partial F(x_*)$
is not satisfied at $x_*$). However, to obtain alternance conditions that are equivalent to the conditions
$0 \in \interior \mathcal{D}(x_*)$ and $\partial F(x_*) \ne \{ 0 \}$, in the general case one must assume that
$V_1, \ldots, V_p \in \mathcal{D}(x_*)$. Indeed, let $d = 2$ and consider the following minimax problem:
$$
\min \: F(x) = \max\{ \pm x^{(1)} \} \quad
\text{s.t.} \quad x \in A = \{ x = (x^{(1)}, x^{(2)})^T \in \mathbb{R}^2 \mid x^{(2)} = 0 \}.
$$
The point $x_* = 0$ is a globally optimal solution of this problem. Note that
$\partial F(x_*) = \co\{ (\pm 1, 0)^T \}$ and $N_A(x_*) = \{ x \in \mathbb{R}^2 \mid x^{(1)} = 0 \}$, which implies that
$\mathcal{D}(x_*) = \{ x \in \mathbb{R}^2 \mid |x^{(1)}| \le 1 \}$ and $0 \in \interior \mathcal{D}(x_*)$. However, as
is easily seen, a weak complete alternance does not exist at $x_*$ (only a $2$-point alternance exists at this point).
Note that in this example $(- \partial F(x_*)) \cap \relint N_A(x_*) = \{ 0 \}$. \qed
}
\end{remark}
Let us comment on the number $p$ in the definition of alternance (or cadre). Suppose for the sake of simplicity that
there are no constraints. From the proofs of Proposition~\ref{prp:AlternanceVsCadre} and
Theorem~\ref{thrm:AlternanceCond} it follows that a $p$-point alternance exists at $x_*$ for some
$p \in \{ 1, \ldots, d + 1 \}$ iff zero can be represented as a convex combination with nonzero coefficients of $p$
affinely independent points from the set $\{ \nabla_x f(x_*, \omega) \mid \omega \in W(x_*) \}$. Hence, in particular,
for a $p$-point alternance to exist at $x_*$ it is necessary that the cardinality of $W(x_*)$ is at least $p$ (i.e. the
maximum in the definition of $F(x_*) = \max_{\omega \in W} f(x_*, \omega)$ must be attained in at least $p$ points
$\omega$) and the set $\{ \nabla_x f(x_*, \omega) \mid \omega \in W(x_*) \}$ contains $p$ affinely independent vectors.
Thus, roughly speaking, the number $p$ in the definition of alternance (or cadre) reflects the size of the
subdifferential $\partial F(x_*)$ at a given point $x_*$ and usually corresponds to its affine dimension plus one. In
particular, in the smooth case (i.e. when $F$ is differentiable at $x_*$) only a $1$-point alternance can exist at
$x_*$. If $\partial F(x_*)$ is a line segment, then only $1$-point or $2$-point alternance can exists at $x_*$, etc. In
the constrained case, the number $p$, roughly speaking, reflects the dimension of the subdifferential $\partial F(x_*)$
and the number of active constraints at $x_*$. However, one must underline that, as
Example~\ref{exmpl:ComplAtern_CounterExampl} demonstrates, in some cases $p$ can be much smaller that the dimension of
the subdifferential.
\begin{remark}
It should be noted that in the proofs of Theorems~\ref{thrm:AlternanceCond} and \ref{thrm:GenAlternanceCond} we do not
use any particular structure of the sets $\partial F(x_*)$, $\mathcal{N}(x_*)$, and $N_A(x_*)$. Therefore, these
theorems can be restated in an abstract form. Namely, suppose that a compact convex set $P \subset \mathbb{R}^d$ and
closed convex cones $K_1, K_2 \subset \mathbb{R}^d$ are given, and let $P = \co P^0$, $K_1 = \cone K_1^0$, and
$K_2 = \cone K_2^0$ for some sets $P^0 \subseteq P$, $K_1^0 \subseteq K_1$, and $K_2^0 \subseteq K_2$. Then, for
instance, the first part of Theorem~\ref{thrm:AlternanceCond} can be reformulated as follows: $0 \in P + K_1 + K_2$ iff
there exists $p \in \{ 1, \ldots, d + 1 \}$, $k_0 \in \{ 1, \ldots, p \}$, $i_0 \in \{ k_0 + 1, \ldots, p \}$, and
vectors
$$
V_1, \ldots, V_{k_0} \in P^0, \quad
V_{k_0 + 1}, \ldots, V_{i_0} \in K_1^0, \quad
V_{i_0 + 1}, \ldots, V_p \in K_2^0
$$
such that $\rank([V_1, \ldots, V_p]) = p - 1$ and $\sum_{i = 1}^p \beta_i V_i = 0$ for some $\beta_i > 0$. Such approach
to an analysis of the condition $0 \in P$, where $P$ is a polytope, was studied in detailed by Demyanov and
Malozemov \cite{DemyanovMalozemov_Alternance,DemyanovMalozemov_Collect}. These papers, in particular, describe a
different (but equivalent) approach to the definition of alternance optimality conditions, in which instead of adding
vectors $V_{p + 1}, \ldots, V_{d + 1} \in Z$ one considers submatrices of order $p$ of the matrix $[V_1, \ldots, V_p]$.
\qed
\end{remark}
\subsection{Examples}
\label{subsect:Examples}
In this section we apply the general theory of first order optimality conditions for cone constrained minimax problems
developed in the previous sections to four particular types of such problems: problems with equality and inequality
constraints, problems with second order cone constraints, as well as problems with semidefinite and semi-infinite
constraints. We demonstrate how general conditions can be reformulated in a more convenient way for these problems and
present several examples illustrating theoretical results.
\subsubsection{Constrained minimax problems}
Let the problem $(\mathcal{P})$ be a constrained minimax problem of the form:
\begin{equation} \label{probl:ConstrainedMinimax}
\min_x \max_{\omega \in W} f(x, \omega) \quad
\text{s.t.} \quad g_i(x) \le 0, \quad i \in I, \quad g_j(x) = 0, \quad j \in J, \quad x \in A,
\end{equation}
where $g_i \colon \mathbb{R}^d \to \mathbb{R}$, $i \in I \cup J$, $I = \{ 1, \ldots, l \}$, and
$J = \{ l + 1, \ldots, l + s \}$. In this case, $Y = \mathbb{R}^{l + s}$,
$G(\cdot) = (g_1(\cdot), \ldots, g_{l + s}(\cdot))$, and $K = (- \mathbb{R}_+)^l \times 0_s$, where
$\mathbb{R}_+ = [0, + \infty)$ and $0_s$ is the zero vector from $\mathbb{R}^s$. Then one has
$K^* = \mathbb{R}_+^l \times \mathbb{R}^s$ and $L(x, \lambda) = F(x) + \sum_{i = 1}^{l + s} \lambda_i g_i(x)$.
Furthermore, as is easily seen, in the case $A = \mathbb{R}^d$, RCQ for problem \eqref{probl:ConstrainedMinimax}
coincides with the well-known Mangasarian-Fromovitz constraint qualification.
If we equip the space $Y$ with the $\ell_1$-norm, then the penalty function for problem \eqref{probl:ConstrainedMinimax}
takes the form
$$
\Phi_c(x) = \max_{\omega \in W} f(x, \omega) + c \sum_{i = 1}^l \max\{ 0, g_i(x) \}
+ c \sum_{j = l + 1}^{l + s} |g_j(x)|.
$$
Denote $I(x) = \{ i \in I \mid g_i(x) = 0 \}$. As is easy to see, one has
$$
\mathcal{N}(x) =
\Big\{ \sum_{i = 1}^{m + l} \lambda_i \nabla g_i(x) \Bigm| \lambda_i \ge 0, \: \lambda_i g_i(x) = 0 \enspace
\forall i \in I, \enspace \lambda_j \in \mathbb{R} \enspace \forall j \in J \Big\}
$$
Therefore, it is natural to choose
$$
\eta(x) = \big\{ \nabla g_i(x) \bigm| i \in I(x) \big\} \cup
\big\{ \nabla g_j(x), - \nabla g_j(x) \bigm| j \in J \big\},
$$
since this is the smallest set whose conic hull coincides with $\mathcal{N}(x)$.
Let us give several particular examples in which we demonstrate how one can verify the validity of optimality
conditions derived in the previous sections in the case of minimax problems with equality and inequality constraints. We
pay special attention to alternance optimality conditions, since these conditions along with optimality conditions in
terms of cadres are the most convenient for analytical computations and can be used to develop efficient numerical
methods (cf. \cite{ConnLi92}). To get the flavour of alternance conditions, we start with a simple nonlinear programming
problem.
\begin{example}[\cite{Bazaraa}, Exercise~4.5]
Consider the following problem:
\begin{equation} \label{probl:MathProg2}
\begin{split}
{}&\min \:
f(x) = (x^{(1)})^4 + (x^{(2)})^4 + 12 (x^{(1)})^2 + 6 (x^{(2)})^2 - x^{(1)} x^{(2)} - x^{(1)} -x^{(2)} \\
{}&\text{s.t.} \enspace x^{(1)} + x^{(2)} \ge 6, \quad 2 x^{(1)} - x^{(2)} \ge 3,
\quad x^{(1)} \ge 0, \quad x^{(2)} \ge 0.
\end{split}
\end{equation}
Define $d = 2$, $l = 2$, $J = \emptyset$, and $A = \{ x \in \mathbb{R}^2 \mid x^{(1)} \ge 0, \: x^{(2)} \ge 0 \}$. Put
also $g_1(x) = -x^{(1)} - x^{(2)} + 6$ and $g_2(x) = - 2 x^{(1)} + x^{(2)} + 3$.
Let us check that a complete alternance exists at the point $x_* = (3, 3)^T$ given in \cite[Exercise~4.5]{Bazaraa}.
Indeed, observe that $I(x_*) = I = \{ 1, 2 \}$ and $N_A(x_*) = - A$. Denote $V_1 = \nabla f(x_*) = (176, 140)^T$,
$V_2 = \nabla g_1(x_*) = (-1, -1)^T \in \eta(x_*)$, and
$V_3 = \nabla g_2(x_*) = (-2, 1)^T \in \eta(x_*)$. Then one has
$$
\Delta_1 = \begin{vmatrix} -1 & -2 \\ -1 & 1 \end{vmatrix} = -3, \quad
\Delta_2 = \begin{vmatrix} 176 & -2 \\ 140 & 1 \end{vmatrix} = 456, \quad
\Delta_3 = \begin{vmatrix} 176 & -1 \\ 140 & -1 \end{vmatrix} = -36,
$$
i.e. a complete alternance exists at $x_*$. Therefore applying
Theorems~\ref{thrm:SuffOptCond}, \ref{thrm:EquivOptCond_Subdiff}, and \ref{thrm:AlternanceCond} one obtains that $x_*$
is a strict local minimiser of problem \eqref{probl:MathProg2} at which the \textit{first} order growth condition
holds true. Note that the classical KKT optimality conditions do not allow one to verify whether the \textit{first}
order growth condition is satisfied at $x_*$. \qed
\end{example}
Let us now give a counterexample to the existence of generalised complete alternance in the general case, promised in
the previous section. In this counterexample, a generalised complete alternance does not exist at a non-isolated point
$x_*$ satisfying the sufficient optimality condition $0 \in \interior \mathcal{D}(x_*)$ and such that
$0 \notin \partial F(x_*)$.
\begin{example} \label{exmpl:ConstrComplAltern_CounterEx}
Consider the following problem:
\begin{equation} \label{probl:MathProg3_CounterEx}
\begin{split}
{}&\min \: f(x) = x^{(1)} + (x^{(2)})^2 + x^{(3)} \\
{}&\text{s.t.} \enspace x^{(2)} - |x^{(3)}| x^{(3)} \le 0, \enspace - x^{(2)} - |x^{(3)}| x^{(3)} \le 0,
\enspace x^{(1)} = 0, \enspace x^{(3)} \ge 0.
\end{split}
\end{equation}
The feasible region of this problem is depicted in Figure~\ref{fig:feasible_region}. Put $d = 3$, $l = 2$,
$J = \emptyset$, and $A = \{ x \in \mathbb{R}^d \mid x^{(1)} = 0, \: x^{(3)} \ge 0 \}$. Define
also $g_1(x) = x^{(2)} - |x^{(3)}| x^{(3)}$ and $g_2(x) = -x^{(2)} - |x^{(3)}| x^{(3)}$.
Let us check optimality conditions at the point $x_* = 0$. Firstly, note that $x_*$ is a not an isolated point of
problem \eqref{probl:MathProg3_CounterEx}, since for any $t \ge 0$ the point $x(t) = (0, 0, t)^T$ is feasible. One has
$I(x_*) = \{ 1, 2 \}$, $\nabla g_1(x_*) = (0, 1, 0)^T$, and $\nabla g_2(x_*) = (0, -1, 0)^T$, which implies that
$\mathcal{N}(x_*) = \cone\{ \nabla g_1(x_*), \nabla g_2(x_*) \} = \{ x \in \mathbb{R}^3 \mid x^{(1)} = x^{(3)} = 0 \}$.
Moreover, $N_A(x_*) = \{ x \in \mathbb{R}^3 \mid x^{(2)} = 0, \: x^{(3)} \le 0 \}$. Hence taking into account
the fact that $\nabla f(x_*) = (1, 0, 1)$ one obtains that
$$
\mathcal{D}(x_*) = \nabla f(x_*) + \mathcal{N}(x_*) + N_A(x_*) = \nabla f(x_*)
+ \{ x \in \mathbb{R}^3 \mid x^{(3)} \le 0 \}
= \{ x \in \mathbb{R}^3 \mid x^{(3)} \le 1 \}.
$$
Thus, $0 \in \interior \mathcal{D}(x_*)$ and by Theorems~\ref{thrm:SuffOptCond} and \ref{thrm:EquivOptCond_Subdiff} the
point $x_*$ is a local minimiser of problem \eqref{probl:MathProg3_CounterEx} at which the first order growth condition
holds true. Let us check that a generalised complete alternance does \textit{not} exist at~$x_*$.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\linewidth]{Feasible_region.mps}
\caption{The feasible region of problem \eqref{probl:MathProg3_CounterEx} (the shaded area).}
\label{fig:feasible_region}
\end{figure}
Note that $\interior \mathcal{N}(x_*) = \emptyset$, $\interior N_A(x_*) = \emptyset$,
$- \nabla f(x_*) \notin \mathcal{N}(x_*)$, and $- \nabla f(x_*) \in \relint N_A(x_*)$, but
$\mathcal{N}(x_*) \ne \{ 0 \}$. Thus, Theorem~\ref{thrm:GenAlternanceCond} is inapplicable. Arguing by reductio ad
absurdum, suppose that a generalised complete alternance $\{ V_1, \ldots, V_4 \}$ exists at $x_*$. Clearly,
$V_1 = \nabla f(x_*)$ and the vectors $V_2$, $V_3$, and $V_4$ are linearly independent, since
$\Delta_1 = \determ([V_2, V_3, V_4]) \ne 0$. Hence taking into account the facts that $\mathcal{N}(x_*)$ is one
dimensional and $N_A(x_*)$ is two dimensional one obtains that $V_2 \in \mathcal{N}(x_*) \setminus \{ 0 \}$ and
$V_3, V_4 \in N_A(x_*) \setminus \{ 0 \}$. However, by Remark~\ref{rmrk:GenAlternanceVsCadre} one has
$\sum_{i = 1}^4 \beta_i V_i = 0$ for some $\beta_i > 0$, which is impossible due to the fact that $V_2$ is the only
vector whose second coordinate is non-zero. Thus, a generalised complete alternance does not exist at $x_*$.
Nevertheless, observe that putting $V_1 = \nabla f(x_*)$, $V_2 = (-1, 0, 0)^T = N_A(x_*)$, and
$V_3 = (0, 0, -1)^T \in N_A(x_*)$ one has $V_1 + V_2 + V_3 = 0$ and $\rank([V_1, V_2, V_3]) = 2$, i.e.
a $3$-point alternance exists at $x_*$, which in the case $d = 3$ is not complete.
Moreover, observe that for $V_1 = \nabla f(x_*)$, $V_2 = (0, 0, -1)^T \in \mathcal{N}(x_*) + N_A(x_*)$,
$V_3 = (-0.5, 1, 0)^T \in \mathcal{N}(x_*) + N_A(x_*)$, and $V_4 = (-0.5, -1, 0)^T \in \mathcal{N}(x_*) + N_A(x_*)$ one
has $V_1 + V_2 + V_3 + V_4 = 0$ and $\rank([V_1, V_2, V_3, V_3]) = 3$. Thus, in accordance with
Remark~\ref{rmrk:IsolatePoint_WeakAlternance} a weak complete alternance exists at $x_*$.
It should be pointed out that RCQ is not satisfied at $x_*$. Therefore we pose an open problem to prove whether the
sufficient optimality condition $0 \in \interior \mathcal{D}(x_*)$ along with RCQ and the assumption that
$\partial F(x_*) \ne \{ 0 \}$ guarantee the existence of a generalised complete alternance. \qed
\end{example}
Now we give two simple examples of minimax problems.
\begin{example}[\cite{MakelaNeittaanmaki}, Problem~DEM] \label{exmpl:ProblemDEM}
Consider the following problem:
$$
\min\: F(x) = \max\{ f_1(x), f_2(x), f_3(x) \},
$$
where $f_1(x) = 5 x^{(1)} + x^{(2)}$, $f_2(x) = - 5 x^{(1)} + x^{(2)}$, and
$f_3(x) = (x^{(1)})^2 + (x^{(2)})^2 + 4 x^{(2)}$. Put $d = 2$ and $W = \{ 1, 2, 3 \}$.
Let us check optimality conditions at the point $x_* = (0, - 3)^T$. One has $W(x_*) = W$ and
$$
\partial F(x_*) = \co\{ \nabla f_1(x_*), \nabla f_2(x_*), \nabla f_3(x_*) \}
= \co\left\{ \begin{pmatrix} 5 \\ 1 \end{pmatrix}, \begin{pmatrix} -5 \\ 1 \end{pmatrix},
\begin{pmatrix} 0 \\ -2 \end{pmatrix} \right\}.
$$
\begin{figure}[t]
\centering
\includegraphics[width=0.5\linewidth]{ExampleDEM.mps}
\caption{The subdifferential $\partial F(x_*)$ (the shaded area) and the vectors $V_1, V_2, V_3 \in \partial F(x_*)$
comprising a complete alternance in Example~\ref{exmpl:ProblemDEM}.}
\label{fig:example_DEM}
\end{figure}
\noindent{}Define $V_1 = \nabla f_1(x_*)$, $V_2 = \nabla f_2(x_*)$, and $V_3 = \nabla f_3(x_*)$. Then
$$
\Delta_1 = \begin{vmatrix} -5 & 0 \\ 1 & -2 \end{vmatrix} = 10, \quad
\Delta_2 = \begin{vmatrix} 5 & 0 \\ 1 & -2 \end{vmatrix} = -10, \quad
\Delta_3 = \begin{vmatrix} 5 & -5 \\ 1 & 1 \end{vmatrix} = 10,
$$
that is, a complete alternance exists at $x_*$ (see Fig.~\ref{fig:example_DEM}). Consequently, $x_*$ is a point of
strict local minimum of the function $F$ at which the first order growth condition holds true by
Theorems~\ref{thrm:SuffOptCond}, \ref{thrm:EquivOptCond_Subdiff}, and \ref{thrm:AlternanceCond}. \qed
\end{example}
\begin{example}[\cite{MadsenSchjaer}, modified Example~4] \label{exmpl:ProblemMadsen}
Let $d = 2$ and consider the following constrained minimax problem:
\begin{equation} \label{probl:ConstrMinMaxEx}
\min\: F(x) = \max\{ f_1(x), f_2(x), f_3(x) \}
\quad \text{subject to} \quad x^{(1)} \ge 0, \quad x^{(2)} \ge 1,
\end{equation}
where $f_1(x) = (x^{(1)})^2 + (x^{(2)})^2 + x^{(1)} x^{(2)} - 1$, $f_2(x) = \sin x^{(1)}$, $f_3(x) = - \cos x^{(2)}$.
Define $W = \{ 1, 2, 3 \}$ and $A = \{ x \in \mathbb{R}^2 \mid x^{(1)} \ge 0, \: x^{(2)} \ge 1 \}$.
Let us check optimality conditions at the point $x_* = (0, 1)^T$. One has $W(x_*) = \{ 1, 2 \}$,
$N_A(x_*) = \{ x \in \mathbb{R}^2 \mid x^{(1)} \le 0, \: x^{(2)} \le 0 \}$, and
$$
\partial F(x_*) = \co\{ \nabla f_1(x_*), \nabla f_2(x_*) \}
= \co\left\{ \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \end{pmatrix} \right\}.
$$
\begin{figure}[t]
\centering
\includegraphics[width=0.4\linewidth]{Example_Madsen.mps}
\caption{The subdifferential $\partial F(x_*)$ (the vertical line segment), the normal cone $N_A(x_*)$ (the shaded
area), and the vectors $V_1, V_2 \in \partial F(x_*)$ and $V_3 \in N_A(x_*)$ comprising a generalised complete
alternance in Example~\ref{exmpl:ProblemMadsen}.}
\label{fig:Example_Madsen}
\end{figure}
\noindent{}Put $V_1 = \nabla f_1(x_*)$, $V_2 = \nabla f_2(x_*)$, and $V_3 = (-1, -1)^T \in N_A(x_*)$. Then
$$
\Delta_1 = \begin{vmatrix} 1 & -1 \\ 0 & -1 \end{vmatrix} = -1, \quad
\Delta_2 = \begin{vmatrix} 1 & -1 \\ 2 & -1 \end{vmatrix} = 1, \quad
\Delta_3 = \begin{vmatrix} 1 & 1 \\ 2 & 0 \end{vmatrix} = -2,
$$
that is, a generalised complete alternance exists at $x_*$ (see~Fig.~\ref{fig:Example_Madsen}). Consequently, by
Theorems~\ref{thrm:SuffOptCond}, \ref{thrm:EquivOptCond_Subdiff}, and \ref{thrm:GenAlternanceCond} the point $x_*$ is a
locally optimal solution of problem \eqref{probl:ConstrMinMaxEx} at which the first order growth condition holds true.
Note that it is natural to put $n_A(x_*) = \{ (-1, 0)^T, (0, - 1)^T \}$, since $N_A(x_*) = \cone n_A(x_*)$, and
analyse optimality condition in terms of non-generalised alternance. Similarly, one can consider inequality constraints
$g_1(x) = - x^{(1)} \le 0$ and $g_2(x) - x^{(2)} + 1 \le 0$, and define $A = \mathbb{R}^2$ and
$\eta(x_*) = \{ \nabla g_1(x_*), \nabla g_2(x_*) \}$. However, one can check that in both cases only a $2$-point
alternance exists at $x_*$, which in the case $d = 2$ is not complete. \qed
\end{example}
\subsubsection{Nonlinear second order cone minimax problems}
Let $(\mathcal{P})$ be a nonlinear second order cone minimax problem of the form:
\begin{equation} \label{probl:SecondOrderConeMinimax}
\min_x \max_{\omega \in W} f(x, \omega) \quad
\text{s.t.} \quad g_i(x) \in K_{l_i + 1}, \quad i \in I,
\quad b(x) = 0, \quad x \in A,
\end{equation}
where $g_i \colon \mathbb{R}^d \to \mathbb{R}^{l_i + 1}$, $I = \{ 1, \ldots, r \}$ and
$b \colon \mathbb{R}^d \to \mathbb{R}^s$ are continuously differentiable functions, and
$$
K_{l_i + 1} = \big\{ y = (y^0, \overline{y}) \in \mathbb{R} \times \mathbb{R}^{l_i} \bigm|
y^0 \ge |\overline{y}| \big\}
$$
\begin{figure}[t]
\centering
\includegraphics[width=0.4\linewidth]{Lorentz_cone.mps}
\caption{The second order (Lorentz, ice-cream) cone of dimension $3$.}
\label{fig:Lorentz_cone}
\end{figure}
\noindent{}is the second order (Lorentz, ice-cream) cone of dimension $l_i + 1$ (see Fig.~\ref{fig:Lorentz_cone}). In
this case
$$
Y = \mathbb{R}^{l_1 + 1} \times \ldots \times \mathbb{R}^{l_r + 1} \times \mathbb{R}^s, \quad
K = K_{l_1 + 1} \times \ldots \times K_{l_r + 1} \times \{ 0_s \},
$$
and $G(\cdot) = (g_1(\cdot), \ldots, g_r(\cdot), b(\cdot))$. Furthermore,
for any $\lambda = (\lambda_1, \ldots, \lambda_r, \nu) \in Y$ one has
$$
L(x, \lambda) = f(x) + \sum_{i = 1}^r \langle \lambda_i, g_i(x) \rangle + \langle \nu, g(x) \rangle,
\quad K^* = (- K_{l_1 + 1}) \times \ldots \times (- K_{l_r + 1}) \times \mathbb{R}^s.
$$
Finally, one can easily verify (cf.~\cite[Lemma~2.99]{BonnansShapiro}) that in the case $A = \mathbb{R}^d$ RCQ for
problem \eqref{probl:SecondOrderConeMinimax} is satisfied at a feasible point $x$ iff the Jacobian matrix $\nabla b(x)$
has full row rank and there exists $h \in \mathbb{R}^d$ such that $\nabla b(x) h = 0$ and
$g_i(x) + \nabla g_i(x) h \in \interior K_{l_i + 1}$ for all
$i \in I(x) = \{ i \in I \mid g_i^0(x) = |\overline{g}_i(x)| \}$,
where $g_i(x) = (g_i^0(x), \overline{g}_i(x)) \in \mathbb{R} \times \mathbb{R}^{l_i}$
(here we used the obvious equality
$\interior K_{l_i + 1} = \{ y = (y^0, \overline{y}) \in \mathbb{R} \times \mathbb{R}^{l_i} \bigm| y^0 >
|\overline{y}|$).
If we equip the space $Y$ with the norm $\| y \| = \sum_{i = 1}^r |y_i| + |z|$ for
any $y = (y_1, \ldots, y_r, z) \in Y$, then the penalty function for problem \eqref{probl:ConstrainedMinimax}
takes the form
$$
\Phi_c(x) = \max_{\omega \in W} f(x, \omega) + c \sum_{i = 1}^r \big| g_i(x) - P_{K_{l_i + 1}}(g_i(x)) \big|
+ c |b(x)|
$$
where
$$
P_{K_{l_i + 1}}(y) = \begin{cases}
\frac{\max\{ y^0 + |\overline{y}|, 0\}}{2}\left(1, \frac{\overline{y}}{|\overline{y}|}\right) &
\text{if } y^0 \le |\overline{y}|, \\
y, & \text{if } y^0 > |\overline{y}|
\end{cases}
$$
is the Euclidean projection of $y = (y^0, \overline{y}) \in \mathbb{R} \times \mathbb{R}^{l_i}$ onto the second order
cone $K_{l_i + 1}$ (see \cite[Thrm.~3.3.6]{Bauschke}; an alternative expression for the projection can be found in
\cite[Prp.~3.3]{FukushimaLuoTseng}). Note also that for any feasible point $x$ one has
\begin{multline*}
\mathcal{N}(x) = \Big\{ \sum_{i = 1}^r \nabla g_i(x)^T \lambda_i + \nabla b(x)^T \nu \Bigm|
\lambda_i \in - K_{l_i + 1}, \: \langle \lambda_i, g_i(x) \rangle = 0 \enspace \forall i \in I,
\nu \in \mathbb{R}^s \Big\} \\
= \Big\{ \sum_{i \in I_+(x)} t_i \nabla g_i(x)^T
\left( \begin{smallmatrix} - g_i^0(x) \\ \overline{g}_i(x) \end{smallmatrix} \right)
+ \sum_{i \in I_0(x)} \nabla g_i(x)^T \lambda_i
+ \nabla b(x)^T \nu \Bigm| \\
t_i \ge 0 \: \forall i \in I_+(x), \:
\lambda_i \in - K_{l_i + 1} \: \forall i \in I_0(x), \: \nu \in \mathbb{R}^s \Big\},
\end{multline*}
where $I_0(x) = \{ i \in I(x) \mid g_i(x) = 0 \}$ and $I_+(x) = I(x) \setminus I_0(x)$. Here we used the following
simple auxiliary result.
\begin{lemma}
Let $y = (y^0, \overline{y}) \in K_{l + 1} \setminus \{ 0 \}$ and
$\lambda = (\lambda^0, \overline{\lambda}) \in - K_{l + 1}$ with $l \in \mathbb{N}$ be such that
$\langle \lambda, y \rangle = 0$. Then $\lambda = 0$, if $y^0 > \overline{y}$, and $\lambda = t (- y^0, \overline{y})$
for some $t \ge 0$, if $y^0 = |\overline{y}|$.
\end{lemma}
\begin{proof}
Indeed, by definition
$\langle \lambda, y \rangle = \lambda^0 y^0 + \langle \overline{\lambda}, \overline{y} \rangle = 0$. Hence taking into
account the fact that $y^0 > 0$, since $y \in K_{l + 1} \setminus \{ 0 \}$, one obtains that
\begin{equation} \label{eq:LorentzCone_ZeroProd}
\lambda^0 = - \frac{1}{y^0} \langle \overline{\lambda}, \overline{y} \rangle \ge
- \frac{1}{y^0} |\overline{\lambda}| \cdot |\overline{y}|.
\end{equation}
Therefore, if $y^0 > |\overline{y}|$, then either (1) $\lambda = 0$ or (2) $\overline{\lambda} = 0$ and $\lambda^0 > 0$
or (3) $\lambda^0 > - |\overline{\lambda}|$. Note, however, that only the first case is possible, since
$\lambda \in - K_{l + 1}$. Thus, $\lambda = 0$, if $y^0 > \overline{y}$.
On the other hand, if $y^0 = |\overline{y}|$, then taking into account \eqref{eq:LorentzCone_ZeroProd} and the fact that
$\lambda \in - K_{l + 1}$, i.e. $\lambda^0 \le - |\overline{\lambda}|$, one obtains that
$\lambda^0 = - |\overline{\lambda}|$ and
$\langle \overline{\lambda}, \overline{y} \rangle = |\overline{\lambda}| \cdot |\overline{y}|$, that is,
$\overline{\lambda} = t \overline{y}$ for some $t \ge 0$. Thus, $\lambda = t (- y^0, \overline{y})$ for some
$t \ge 0$, if $y^0 = |\overline{y}|$.
\end{proof}
Thus, it is natural to define
\begin{align*}
\eta(x)
&= \Big\{ \nabla g_i(x)^T \left( \begin{smallmatrix} - g_i^0(x) \\ \overline{g}_i(x) \end{smallmatrix} \right)
\Bigm| i \in I_+(x) \Big\} \\
&\cup \Big\{ \nabla g_i(x)^T \left( \begin{smallmatrix} - 1 \\ |v| \end{smallmatrix} \right) \Bigm|
i \in I_0(x), \: v \in \mathbb{R}^{l_i}, \: |v| = 1 \Big\}
\cup \big\{ \nabla b_1(x), \ldots \nabla b_s(x) \big\}
\end{align*}
(here $b(\cdot) = (b_1(\cdot), \ldots, b_s(\cdot))$), since in the general case this is the smallest set such that
$\mathcal{N}(x) = \cone \eta(x)$.
Let us give an example demonstrating how one can verify alternance optimality conditions in the case of nonlinear
second order cone minimax problems.
\begin{example}
Consider the following second order cone minimax problem:
\begin{equation} \label{probl:2OrderConeMinimaxEx}
\begin{split}
&\min\: F(x) = \max\{ (x^{(1)})^2 + (x^{(2)})^2 + 4 x^{(1)} - x^{(2)}, \sin x^{(1)} - x^{(2)}, \cos x^{(2)} - 1 \} \\
&\text{s.t.} \quad g_1(x) = (- x^{(1)} + \sin x^{(2)} + 1, \sin x^{(1)} - 2 x^{(2)} - 1) \in K_2, \\
&g_2(x) = (2 (x^{(1)})^2 + 2 (x^{(2)})^2, x^{(1)} + x^{(2)}, 2 x^{(2)}) \in K_3.
\end{split}
\end{equation}
Define $d = 2$, $f_1(x) = (x^{(1)})^2 + (x^{(2)})^2 + 4 x^{(1)} - x^{(2)}$, $f_2(x) = \sin x^{(1)} - x^{(2)}$, $f_3(x) =
\cos x^{(2)} - 1$,
$W = \{ 1, 2, 3 \}$, $I = \{ 1, 2 \}$, and $A = \mathbb{R}^d$.
Let us check optimality conditions at the point $x_* = 0$. Observe that $W(x_*) = \{ 1, 2, 3 \}$ and
$$
\partial F(x_*) = \co\{ \nabla f_1(x_*), \nabla f_2(x_*), \nabla f_3(x_*) \nabla \}
= \co\left\{ \begin{pmatrix} 4 \\ -1 \end{pmatrix}, \begin{pmatrix} 1 \\ -1 \end{pmatrix},
\begin{pmatrix} 0 \\ 0 \end{pmatrix} \right\}.
$$
Note also that $g_1(x_*) = (1, - 1) \in K_2$,
$\nabla g_1(x_*)^T = \left( \begin{smallmatrix} -1 & 1 \\ 1 & -2 \end{smallmatrix} \right)$,
$g_2(x_*) = 0 \in K_3$, and
$\nabla g_2(x_*)^T = \left( \begin{smallmatrix} 0 & 1 & 0 \\ 0 & 1 & 2 \end{smallmatrix} \right)$. Therefore
$I_+(x_*) = \{ 1 \}$, $I_0(x_*) = \{ 2 \}$, and
\begin{align*}
\eta(x_*) &= \Big\{
\nabla g_1(x_*)^T \left( \begin{smallmatrix} - g_1^0(x_*) \\ \overline{g}_i(x_*) \end{smallmatrix} \right) \Big\}
\cup \Big\{ \nabla g_2(x_*)^T \left( \begin{smallmatrix} - 1 \\ |v| \end{smallmatrix} \right)
\Bigm| v \in \mathbb{R}^2 \colon |v| = 1 \Big\} \\
&= \left\{ \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right\} \cup
\left\{ \begin{pmatrix} v^{(1)} \\ v^{(1)} + 2 v^{(2)} \end{pmatrix} \Bigm|
v \in \mathbb{R}^2 \colon |v| = 1 \right\}.
\end{align*}
Let $V_1 = \nabla f_1(x_*)$, $V_2 = (0, 1)^T \in \eta(x_*)$, and
$V_3 = (v^{(1)}, v^{(1)} + 2 v^{(2)})^T \in \eta(x_*)$ with $v = (- 1 / \sqrt{2}, - 1 / \sqrt{2})^T$. Then
$$
\Delta_1 = \begin{vmatrix} 0 & -\frac{1}{\sqrt{2}} \\ 1 & -\frac{3}{\sqrt{2}} \end{vmatrix} =
\frac{1}{\sqrt{2}}, \quad
\Delta_2 = \begin{vmatrix} 4 & -\frac{1}{\sqrt{2}} \\ -1 & -\frac{3}{\sqrt{2}} \end{vmatrix} =
-\frac{13}{\sqrt{2}}, \quad
\Delta_3 = \begin{vmatrix} 4 & 0 \\ -1 & 1 \end{vmatrix} = 4,
$$
that is, a complete alternance exists at $x_*$. Therefore, by Theorems~\ref{thrm:SuffOptCond},
\ref{thrm:EquivOptCond_Subdiff}, and \ref{thrm:AlternanceCond} the point $x_*$ is a locally optimal solution of problem
\eqref{probl:2OrderConeMinimaxEx} at which the first order growth condition holds true. \qed
\end{example}
\subsubsection{Nonlinear semidefinite minimax problems}
Let now $(\mathcal{P})$ be a nonlinear semidefinite minimax problem of the form:
\begin{equation} \label{probl:NonlinearSemiDefProg}
\min_x \max_{\omega \in W} f(x, \omega) \quad
\text{subject to} \quad G_0(x) \preceq 0, \quad b(x) = 0, \quad x \in A,
\end{equation}
where $G_0 \colon \mathbb{R}^d \to \mathbb{S}^l$ and $b \colon \mathbb{R}^d \to \mathbb{R}^s$ are continuously
differentiable functions, $\mathbb{S}^l$ denotes the set of all $l \times l$ real symmetric matrices, and the relation
$G_0(x) \preceq 0$ means that the matrix $G_0(x)$ is negative semidefinite. In this case,
$Y = \mathbb{S}^l \times \mathbb{R}^s$, $G(\cdot) = (G_0(\cdot), b(\cdot))$ and
$K = \mathbb{S}^l_- \times 0_s$, where $\mathbb{S}^l_-$ is the cone of $l \times l$ negative semidefinite
matrices.
We equip $Y$ with the inner product
$\langle (B_1, z_1), (B_2, z_2) \rangle = \trace(B_1 B_2) + \langle z_1, z_2 \rangle$
for any $(B_1, z_1), (B_2, z_2) \in Y$, where $\trace(\cdot)$ is the trace of a matrix, and the corresponding norm
$\| (B, z) \|^2 = \| B \|_F^2 + |z|^2$, where $\| B \|_F = \sqrt{Tr(B^2)}$ is the Frobenius norm. Then one has
$L(x, \lambda) = F(x) + \trace(\lambda_0 \cdot G_0(x)) + \langle \nu, h(x) \rangle$ for any
$(\lambda_0, \nu) \in \mathbb{S}^l \times \mathbb{R}^s$ and $K^* = \mathbb{S}^l_+ \times \mathbb{R}^s$, where
$\mathbb{S}^l_+ = - \mathbb{S}^l_{-}$ is the cone of positive semidefinite matrices. Note also that in the case
$A = \mathbb{R}^d$ RCQ for problem \eqref{probl:NonlinearSemiDefProg} holds true at a feasible point $x$ iff the
Jacobian matrix $\nabla b(x)$ has full row rank and there exists $h \in \mathbb{R}^d$ such that $\nabla b(x) h = 0$ and
the matrix $G_0(x) + D G_0(x)h$ is negative definite (cf. \cite[Lemma~2.99]{BonnansShapiro}).
The penalty function for problem \eqref{probl:NonlinearSemiDefProg} has the form
$$
\Phi_c(x) = f(x) + c \sqrt{\| G_0(x) - P_{\mathbb{S}^l_{-}}(G_0(x)) \|_F^2 + |b(x)|^2},
$$
where $P_{\mathbb{S}^l_{-}}(G_0(x))$ is the projection of $G_0(x)$ onto the cone $\mathbb{S}^l_{-}$ of negative
semidefinite matrices. One can verify that
\begin{align*}
P_{\mathbb{S}^l_{-}}(G_0(x)) &= 0.5 (G_0(x) - \sqrt{G_0(x)^2}) \\
&= Q \diag\Big( \min\{ 0, \sigma_1(G_0(x)) \}, \ldots, \min\{ 0, \sigma_l(G_0(x)) \} \Big) Q^T,
\end{align*}
where $G_0(x) = Q \diag(\sigma_1(G_0(x)), \ldots, \sigma_l(G_0(x))) Q^T$ is a spectral decomposition of $G_0(x)$, and
$\sigma_1(G_0(x)), \ldots, \sigma_l(G_0(x))$ are the eigenvalues of $G_0(x)$ listed in the decreasing order (see, e.g.
\cite{Higham,Malick}). Consequently, one has
$$
\| G_0(x) - P_{\mathbb{S}^l_{-}}(G_0(x)) \|_F = \frac{1}{2} \| G_0(x) + \sqrt{G_0(x)^2} \|_F
= \sqrt{\sum_{i = 1}^l \max\big\{ 0, \sigma_i(G_0(x)) \big\}^2}.
$$
Observe also that for any feasible point $x$ such that $r = \rank G_0(x) < l$ one has
\begin{multline*}
\mathcal{N}(x)
= \Big\{ \Big( \langle \lambda_0, D_{x_1} G_0(x) \rangle, \ldots, \langle \lambda_0, D_{x_d} G_0(x) \rangle \Big)^T
+ \nabla b(x)^T \nu \Bigm| \\
(\lambda_0, \nu) \in \mathbb{S}^l_+ \times \mathbb{R}^s, \: \langle \lambda_0, G_0(x) \rangle = 0 \Big\}
\end{multline*}
or, equivalently,
$$
\mathcal{N}(x) = \Big\{
\Big(\langle Q_0 \Gamma Q_0^T, D_{x_1} G_0(x) \rangle, \ldots, \langle Q_0 \Gamma Q_0^T, D_{x_d} G_0(x) \rangle
\Big)^T + \nabla b(x)^T \nu \Bigm| (\Gamma, \nu) \in \mathbb{S}^{l - r}_+ \times \mathbb{R}^s \Big\},
$$
where $D_{x_i} = \partial / \partial x_i$, $r = \rank G_0(x)$, and $Q_0$ is an $l \times (l - r)$ matrix whose columns
are an orthonormal basis $q_1, \ldots, q_{l - r}$ of the null space of the matrix $G_0(x)$.
In the case $r = \rank G_0(x) = l$ one has $\mathcal{N}(x_*) = \{ \nabla b(x)^T \nu \mid \nu \in \mathbb{R}^s \}$.
Here we used the following simple auxiliary result.
\begin{lemma}
Let $\lambda_0 \in \mathbb{S}^l_+$ be a given matrix. Then the following statements are equivalent:
\begin{enumerate}
\item{$\langle \lambda_0, G_0(x) \rangle = \trace(\lambda_0 G_0(x)) = 0$;
\label{stat:ZeroFrobeniusProd}
}
\item{$\lambda_0 = Q_0 \Gamma Q_0^T$ for some $\Gamma \in \mathbb{S}^{l - r}_+$ in the case $r < l$ and
$\lambda_0 = 0$ otherwise;
\label{stat:SemidefFormNullVectors}}
\item{$\lambda_0 \in \cone\{ q q^T \mid q \in \mathbb{R}^l \colon G_0(x) q = 0 \}$.
\label{stat:ExtremeMatricesViaNullSpace}}
\end{enumerate}
\end{lemma}
\begin{proof}
Let, as above, $\sigma_1(G_0(x)), \ldots, \sigma_l(G_0(x))$ be the eigenvalues of $G_0(x)$ listed in the decreasing
order. Recall that $x$ is feasible point of problem \eqref{probl:NonlinearSemiDefProg}, i.e. $G_0(x) \preceq 0$.
Therefore
\begin{equation} \label{eq:EigenvaluesNegativeSemiDef}
\sigma_i(G_0(x)) = 0 \quad \forall i \in \{ 1, \ldots, l - r \}, \quad
\sigma_i(G_0(x)) < 0 \quad \forall i \in \{ l - r + 1, \ldots, l \}.
\end{equation}
Let also $G_0(x) = Q \diag(\sigma_1(G_0(x)), \ldots, \sigma_l(G_0(x))) Q^T$ be a spectral decomposition of $G_0(x)$ such
that the first $l - r$ columns of $Q$ coincide with $Q_0$.
\ref{stat:ZeroFrobeniusProd} ${}\implies{}$ \ref{stat:ExtremeMatricesViaNullSpace}.
Suppose that $\langle \lambda_0, G_0(x) \rangle = 0$. Bearing in mind the fact that the trace operator is invariant
under cyclic permutations one obtains that
\begin{equation} \label{eq:FrobeniusNorm_Eigenvalues}
\begin{split}
0 = \trace\big( \lambda_0 G_0(x) \big)
&= \trace\Big( Q^T \lambda_0 Q \diag(\sigma_1(G_0(x)), \ldots, \sigma_l(G_0(x))) \Big) \\
&= \sum_{i = 1}^l \sigma_i(G_0(x)) q_i^T \lambda_0 q_i,
\end{split}
\end{equation}
where $q_i$ are the columns of the matrix $Q$. Hence with the use of \eqref{eq:EigenvaluesNegativeSemiDef} and the fact
that $\lambda_0 \in \mathbb{S}^l_+$ one obtains that $q_i^T \lambda_0 q_i = 0$ for any
$i \in \{ l - r + 1, \ldots, l \}$.
Since the matrix $\lambda_0$ is positive semidefinite, there exists orthogonal vectors
$z_1, \ldots, z_k \in \mathbb{R}^l$ such that $\lambda_0 = z_1 z_1^T + \ldots + z_k z_k^T$
(see, e.g. \cite[Thrm.~7.5.2]{HornJohnson}). Consequently, one has
$$
0 = q_i^T \lambda_0 q_i = \sum_{j = 1}^k q_i^T z_j z_j^T q_i = \sum_{j = 1}^k |z_j q_i|^2
\quad \forall i \in \{ l - r + 1, \ldots, l \}.
$$
Therefore, the vectors $z_1, \ldots, z_k$ belong to the orthogonal complement of the linear span of eigenvectors $q_i$
of $G_0(x)$ corresponding to nonzero eigenvalues, which coincides with the null space of $G_0(x)$. Thus,
$G_0(x) z_i = 0$ for all $i \in \{ 1, \ldots, k \}$, that is,
$\lambda_0 \in \cone\{ q q^T \mid q \in \mathbb{R}^l \colon G_0(x) q = 0 \}$.
\ref{stat:ExtremeMatricesViaNullSpace} ${}\implies{}$ \ref{stat:SemidefFormNullVectors}. If $r = \rank G_0(x) = l$,
then $G_0(x) q = 0$ iff $q = 0$, which implies that $\lambda_0 = 0$. Thus, one can suppose that $r < l$. Then
$\lambda_0 = \sum_{i = 1}^k \alpha_i z_i z_i^T$ for some $\alpha_i \ge 0$ and $z_i \in \mathbb{R}^l$ such that
$G_0(x) z_i = 0$. Since $z_i$ belongs to the null space of $G_0(x)$ and the columns of the matrix $Q_0$ are an
orthonormal basis of this space, there exists vectors $u_i \in \mathbb{R}^{l - r}$ such that $z_i = Q_0 u_i$ for all
$i$. Therefore
$$
\lambda_0 = \sum_{i = 1}^k \alpha_i z_i z_i^T = \sum_{i = 1}^k \alpha_i Q_0 u_i u_i^T Q_0^T
= Q_0 \Big( \sum_{i = 1}^k \alpha_i u_i u_i^T \Big) Q_0^T.
$$
Define $\Gamma = \sum_{i = 1}^k \alpha_i u_i u_i^T$. Then $\lambda_0 = Q_0 \Gamma Q_0^T$ and, as is easily seen,
$\Gamma \in \mathbb{S}^{l - r}_+$.
\ref{stat:SemidefFormNullVectors} ${}\implies{}$ \ref{stat:ZeroFrobeniusProd}. Suppose now that
$\lambda_0 = Q_0 \Gamma Q_0^T$ for some $\Gamma \in \mathbb{S}^{l - r}_+$ in the case $r < l$ and $\lambda_0 = 0$
otherwise. If $\lambda_0 = 0$, then obviously $\langle \lambda_0, G_0(x) \rangle = 0$. Thus, one can suppose that
$r < l$. Observe that
$$
Q^T \lambda_0 Q = Q^T Q_0 \Gamma Q_0^T Q
= Q^T Q \begin{pmatrix} \Gamma & 0 \\ 0 & 0 \end{pmatrix} Q^T Q
= \begin{pmatrix} \Gamma & 0 \\ 0 & 0 \end{pmatrix},
$$
where $0$ are zero matrices of corresponding dimensions. Hence taking into account the fact that
$[Q^T \lambda_0 Q]_{ij} = q_i^T \lambda_0 q_j$ one obtains that $q_i^T \lambda_0 q_i = 0$ for
any $i \in \{ l - r + 1, \ldots, l \}$, which with the use of the last two equalities in
\eqref{eq:FrobeniusNorm_Eigenvalues} implies that $\langle \lambda_0, G_0(x) \rangle = 0$.
\end{proof}
Taking into account the equality $\trace(q q^T D_{x_i} G_0(x)) = q^T D_{x_i} G_0(x) q$ and the previous lemma one can
define
\begin{multline*}
\eta(x) = \big\{ \nabla b_1(x), \ldots \nabla b_s(x) \big\} \\
\cup \Big\{ \Big( q^T D_{x_1} G_0(x) q, \ldots, q^T D_{x_d} G_0(x) q \rangle \Big)^T \in \mathbb{R}^d \Bigm|
q \in \mathbb{R}^l \colon |q| = 1, \: G_0(x) q = 0 \Big\}
\end{multline*}
in the case $\rank G_0(x) < l$, and $\eta(x) = \{ \nabla b_1(x), \ldots \nabla b_s(x) \}$, if $\rank G_0(x) = l$. Let us
give a simple example illustrating alternance optimality conditions in the case of nonlinear semidefinite minimax
problems.
\begin{example}
Let $d = 3$, $W = \{ 1, 2, 3 \}$, $l = 3$, and $A = \mathbb{R}^d$. Consider the following nonlinear semidefinite minimax
problem:
\begin{equation} \label{probl:SemiDefMinimaxEx}
\min \: F(x) = \max\big\{ f_1(x), f_2(x), f_3(x) \big\} \quad \text{subject to} \quad
G_0(x) \preceq 0,
\end{equation}
where $f_1(x) = - 3 x^{(1)} - 3 x^{(2)} - 2 \sin x^{(3)}$,
$f_2(x) = - x^{(1)} + (x^{(2)})^2 + (x^{(3)})^2 - 1$, and $f_3(x) = (x^{(1)} - 1)^2 + 2 x^{(3)}$, and
$$
G_0(x) =
\begin{pmatrix} x^{(1)} - (x^{(2)})^2 & \sin x^{(3)} & x^{(1)} + x^{(2)} + x^{(3)} \\
\sin x^{(3)} & x^{(2)} & x^{(1)} x^{(2)} + (x^{(3)} + 1)^2 \\
x^{(1)} + x^{(2)} + x^{(3)} & x^{(1)} x^{(2)} + (x^{(3)} + 1)^2 & (x^{(1)})^2 + (x^{(2)})^2 - x^{(3)} - 2
\end{pmatrix}.
$$
Let us check optimality conditions at the point $x_* = (1, -1, 0)^T$. One has $W(x_*) = \{ 1, 3 \}$ and
$$
\partial F(x_*) = \co\{ \nabla f_1(x_*), \nabla f_3(x_*) \}
= \co\left\{ \left( \begin{smallmatrix} -3 \\ -3 \\ -2 \end{smallmatrix} \right),
\left( \begin{smallmatrix} 0 \\ 0 \\ 2 \end{smallmatrix} \right) \right\}, \quad
G_0(x_*) = \left( \begin{smallmatrix} 0 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{smallmatrix} \right).
$$
Consequently, $G_0(x_*) \preceq 0$ and $\rank G_0(x_*) = 1$, which implies that $x_*$ is a feasible point of problem
\eqref{probl:SemiDefMinimaxEx} and by definition one has
$$
\eta(x)
= \Big\{ \Big( q^T D_{x_1} G_0(x) q, q^T D_{x_2} G_0(x) q, q^T D_{x_3} G_0(x) q \Big)^T
\in \mathbb{R}^d \Bigm|
q \in \mathbb{R}^3 \colon |q| = 1, \: G_0(x) q = 0 \Big\}.
$$
Let $V_1 = \nabla f_1(x_*)$ and $V_2 = \nabla f_3(x_*)$. For $q_1 = (1, 0, 0)^T$ and
$q_2 = (0, 0, 1)^T$ one has $G_0(x_*) q_1 = 0$, $G_0(x_*) q_2 = 0$, and
$$
V_3 = \begin{pmatrix} q_1^T D_{x_1} G_0(x_*) q_1 \\ q_1^T D_{x_2} G_0(x_*) q_1 \\
q_1^T D_{x_3} G_0(x_*) q_1 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}, \quad
V_4 = \begin{pmatrix} q_2^T D_{x_1} G_0(x_*) q_2 \\ q_2^T D_{x_2} G_0(x_*) q_2 \\
q_2^T D_{x_3} G_0(x_*) q_2 \end{pmatrix} = \begin{pmatrix} 2 \\ -2 \\ -1 \end{pmatrix}.
$$
By definition $V_3, V_4 \in \eta(x_*)$. For the chosen vectors $V_1, V_2, V_3$, and $V_4$ one has
\begin{align*}
\Delta_1 &= \left|\begin{smallmatrix} 0 & 1 & 2 \\ 0 & 2 & -2 \\ 2 & 0 & -1 \end{smallmatrix}\right| = -12, \quad
\Delta_2 = \left|\begin{smallmatrix} -3 & 1 & 2 \\ -3 & 2 & -2 \\ -2 & 0 & -1 \end{smallmatrix}\right| = 15, \\
\Delta_3 &= \left|\begin{smallmatrix} -3 & 0 & 2 \\ -3 & 0 & -2 \\ -2 & 2 & -1 \end{smallmatrix}\right| = -24, \quad
\Delta_4 = \left|\begin{smallmatrix} -3 & 0 & 1 \\ -3 & 0 & 2 \\ -2 & 2 & 0 \end{smallmatrix}\right| = 6.
\end{align*}
Thus, a complete alternance exists at $x_*$, which by Theorems~\ref{thrm:SuffOptCond},
\ref{thrm:EquivOptCond_Subdiff}, and \ref{thrm:AlternanceCond} implies that the point $x_*$ is a locally optimal
solution of problem \eqref{probl:SemiDefMinimaxEx} at which the first order growth condition holds true. \qed
\end{example}
\subsubsection{Semi-infinite minimax problems}
Let finally $(\mathcal{P})$ be a nonlinear semi-infinite minimax problem of the form:
\begin{equation} \label{probl:SemiInfProg}
\min_x \max_{\omega \in W} f(x, \omega) \quad
\text{s.t.} \quad g_i(x, t) \le 0, \quad t \in T, \quad i \in I, \quad b(x) = 0, \quad x \in A,
\end{equation}
where the mapping $b \colon \mathbb{R}^d \to \mathbb{R}^s$ is continuously differentiable, $T$ is a compact metric
space, and the functions $g_i \colon \mathbb{R}^d \times T \to \mathbb{R}$, $g_i = g_i(x, t)$, are continuous jointly in
$x$ and $t$, differentiable in $x$ for any $t \in T$, and the functions $\nabla_x g_i$ are continuous,
$i \in I = \{ 1, \ldots, l \}$.
Let $C(T)$ be the space of all real-valued continuous functions defined on $T$ equipped with the uniform norm
$\| \cdot \|_{\infty}$, and $C_{-}(T)$ be the closed convex cone consisting of all nonpositive functions from $C(T)$. As
is well-known (see, e.g. \cite[Thrm.~IV.6.3]{DunfordSchwartz}), the topological dual space of $C(T)$ is isometrically
isomorphic to the space of signed (i.e. real-valued) regular Borel measures on $T$, denoted by $rca(T)$, while the set
of regular Borel measures (which constitute a closed convex cone in $rca(T)$) is denoted by $rca_+(T)$. Define
$Y = (C(T))^l \times \mathbb{R}$, $K = (C_{-}(T))^l \times \{ 0_s \}$, and introduce the mapping
$G \colon \mathbb{R}^d \to Y$ by setting $G(x) = (g_1(x, \cdot), \ldots, g_l(x, \cdot), b(x))$. Then problem
\eqref{probl:SemiInfProg} is equivalent to problem $(\mathcal{P})$. We endow the space $Y$ with the norm
$\| y \| = \sum_{i = 1}^n \| y_i \|_{\infty} + |z|$ for all $y = (y_1, \ldots, y_l, z) \in Y$.
Observe that the dual space $Y^*$ is isometrically isomorphic (and thus can be identified with)
$rca(T)^l \times \mathbb{R}^s$, while the polar cone $K^*$ can be identified with the cone
$(rca_+(T))^l \times \mathbb{R}^s$. Then for any $\lambda = (\mu_1, \ldots, \mu_l, \nu) \in Y^*$ one has
$L(x, \lambda) = F(x) + \sum_{i = 1}^l \int_T g(x, t) d \mu_i(t) + \langle \nu, b(x) \rangle$. Note also that in the
case $A = \mathbb{R}^d$ RCQ for problem \eqref{probl:SemiInfProg} is satisfied at a feasible point $x$ iff the
Jacobian matrix $\nabla b(x)$ has full row rank and there exists $h \in \mathbb{R}^d$ such that $\nabla b(x) h = 0$ and
$\langle \nabla_x g_i(x, t), h \rangle < 0$ for all $t \in T$ and $i \in I$ such that $g_i(x, t) = 0$.
The penalty function for problem \eqref{probl:NonlinearSemiDefProg} has the form
$$
\Phi_c(x) = f(x) + c \Big( \sum_{i = 1}^l \max_{t \in T} \{ g_i(x, t), 0 \} + |h(x)| \Big).
$$
For any feasible point $x$ one has
\begin{multline*}
\mathcal{N}(x) = \Big\{ \sum_{i = 1}^l \int_T \nabla_x g_i(x, t) d \mu_i(t) + \nabla b(x)^T \nu \Bigm|
\mu_i \in rca_+(T), \\
\support(\mu_i) \subseteq \{ t \in T \mid g_i(x, t) = 0 \} \enspace \forall i \in I,
\nu \in \mathbb{R}^s \Big\},
\end{multline*}
where $\support(\mu)$ is the support of a measure $\mu$. We define $\eta(x) = \mathcal{N}(x)$, since it does
not seem possible to somehow reduce the set $\mathcal{N}(x)$ due to the infinite dimensional nature of the problem.
When it comes to numerical methods, it is very difficult to deal with measures $\mu_i \in rca_+(T)$ directly (especially
in the case when the sets $\{ t \in T \mid g_i(x, t) = 0 \}$ have infinite cardinality). Apparently, the general theory
of optimality conditions for cone constrained minimax problems developed in the previous sections cannot overcome this
obstacle for semi-infinite minimax problems. That is why such problems require a special treatment. Our aim is to show
that necessary optimality conditions for semi-infinite minimax problems, including such conditions in terms of cadre and
alternance, can be completely rewritten in terms of discrete measures whose supports consist of at most $d + 1$ points,
which allows one to avoid the use of Radon measures.
To this end, suppose that $x_*$ is a feasible point of problem \eqref{probl:SemiInfProg}, and let there exist
a Lagrange multiplier $\lambda = (\mu_1, \ldots, \mu_l, \nu) \in K^*$ of problem \eqref{probl:SemiInfProg} at $x_*$. We
say that $\lambda$ is \textit{a discrete Lagrange multiplier}, if for any $i \in I$ the measure $\mu_i$ is discrete and
its support consists of at most $d + 1$ points, i.e. $\mu_i = \sum_{j = 1}^{m_i} \lambda_{ij} \delta(t_{ij})$ for some
$t_{ij} \in T$, $\lambda_{ij} \ge 0$, and $m_i \le d + 1$. Here $\delta(t)$ is the Dirac measure of mass one at the
point $t \in T$. If $\lambda$ is a discrete Lagrange multiplier, then one has
$L(x, \lambda) = F(x) + \sum_{i = 1}^l \sum_{j = 1}^{m_i} \lambda_{ij} g_i(x, t_{ij})
+ \langle \nu, b(x) \rangle$ for all $x \in \mathbb{R}^d$.
Let us check that necessary optimality conditions for problem \eqref{probl:SemiInfProg} can be expressed in terms of
discrete Lagrange multipliers. Denote $I(x) = \{ i \in I \mid \max_{t \in T} g_i(x, t) = 0 \}$, and let
$T_i(x) = \{ t \in T \mid g_i(x, t) = 0 \}$.
\begin{theorem} \label{thrm:DiscreteLagrangeMultipliers}
Let $x_*$ be a feasible point of problem \eqref{probl:SemiInfProg}. Then the following statements hold true:
\begin{enumerate}
\item{if $x_*$ is a locally optimal solution of problem \eqref{probl:SemiInfProg} at which RCQ holds true, then
there exists a discrete Lagrange multiplier of this problem at $x_*$;
\label{stat:ExistenceDiscreteLagrMult}}
\item{a discrete Lagrange multiplier exists at $x_*$ if and only if for any $i \in I(x_*)$ one can find
$m_i \in \{ 1, \ldots, d + 1 \}$ and $t_{ij} \in T_i(x_*)$, $j \in \{ 1, \ldots, m_i \}$, such that
there exists at $x_*$ a Lagrange multiplier of the discretised problem
\begin{equation} \label{probl:DiscretisedSemiInfProb}
\begin{split}
&\min_x \: \max_{\omega \in W} f(x, \omega) \\
&\text{s.t.} \enspace g_i(x, t_{ij}) \le 0, \enspace j \in \{ 1, \ldots, m_i \}, \enspace i \in I(x_*),
\enspace b(x) = 0, \enspace x \in A;
\end{split}
\end{equation}
\label{stat:LagrMultViaDiscreteProb}}
\vspace{-5mm}
\item{if $b(\cdot) \equiv 0$, the function $f(\cdot, \omega)$ is convex for any $\omega \in W$, the functions
$g_i(\cdot, t)$ are convex for any $t \in T$ and $i \in I$, and there exists $x_0 \in A$ such that $g_i(x_0, t) < 0$ for
all $t \in T$ and $i \in I$, then a discrete Lagrange multiplier exists at $x_*$ iff $x_*$ is a globally optimal
solution of problem \eqref{probl:SemiInfProg} iff for any $i \in I(x_*)$ there exist $m_i \in \{ 1, \ldots, d + 1 \}$
and $t_{ij} \in T_i(x_*)$, $j \in \{ 1, \ldots, m_i \}$, such that $x_*$ is a globally optimal solution of problem
\eqref{probl:DiscretisedSemiInfProb}.
\label{stat:DiscreteLagrMult_ConvexCase}}
\end{enumerate}
\end{theorem}
\begin{proof}
\textbf{Part~\ref{stat:ExistenceDiscreteLagrMult}.} Introduce the function
$$
z(x) = \max\big\{ F(x) - F(x_*), \max_{t \in T} g_1(x, t), \ldots, \max_{t \in T} g_l(x, t) \}.
$$
Observe that $z(x_*) = 0$, and if $z(x) < 0$ for some $x \in A$ such that $b(x) = 0$, then $x$ is a feasible point of
problem \eqref{probl:SemiInfProg} for which $F(x) < F(x_*)$. Hence taking into account the fact that $x_*$ is a locally
optimal solution of problem \eqref{probl:SemiInfProg} one obtains that $x_*$ is a locally optimal solution of
the problem
\begin{equation} \label{probl:ReducedSemiInfProblem}
\min\: z(x) \quad \text{subject to} \quad b(x) = 0, \quad x \in A
\end{equation}
as well. Note that this is a constrained minimax problem, since the function $z$ can be written as
$z(x) = \max_{\omega \in \widetilde{W}} \widetilde{f}(x, \omega)$, where
$\widetilde{W} = W \cup (T \times \{ 1 \}) \cup \ldots \cup (T \times \{ l \})$,
$\widetilde{f}(x, \omega) = f(x, \omega) - F(x_*)$, if $\omega \in W$, and $\widetilde{f}(x, \omega) = g_i(x, t)$, if
$\omega = (t, i) \in T \times \{ i \}$ for some $i \in I$.
Recall that by our assumption Robinson's constraint qualification for problem \eqref{probl:SemiInfProg} holds true at
$x_*$, i.e. $0 \in \interior\{ G(x_*) + D G(x_*)( A - x_* ) - K \}$ or, equivalently,
\begin{equation} \label{eq:RCQ_SemiInfProblems}
0 \in \interior\left\{ \begin{pmatrix} g(x_*, \cdot) + \nabla_x g(x_*, \cdot )h \\ \nabla b(x_*) h \end{pmatrix}
+ \begin{pmatrix} (C_+(T))^l \\ 0_s \end{pmatrix} \Biggm| h \in A - x_* \right\}
\end{equation}
where $g = (g_1, \ldots, g_l)^T$ and $C_+(T) = - C_{-}(T)$ is the cone of nonnegative continuous functions defined
on $T$. Hence, in particular, one gets that $0 \in \interior\{ \nabla b(x_*) (A - x_*) \}$, that is, RCQ for problem
\eqref{probl:ReducedSemiInfProblem} is satisfied at $x_*$. Consequently, by Theorem~\ref{thrm:NessOptCond} there exists
a Lagrange multiplier of problem \eqref{probl:ReducedSemiInfProblem} at $x_*$, which by
Remark~\ref{rmrk:LagrangeMultViaSubdiff} implies that
$(\partial z(x_*) + \nabla b(x_*)^T \nu) \cap (- N_A(x_*)) \ne \emptyset$ for some $\nu \in \mathbb{R}^s$, where
$$
\partial z(x_*) = \co\Big\{\nabla_x f(x_*, \omega), \nabla_x g_i(x_*, t) \Bigm|
\omega \in W(x_*), t \in T_i(x_*), i \in I(x_*) \Big\}.
$$
Hence there exist $v_1 \in \partial F(x_*)$,
$v_2 \in \co\{ \nabla_x g_i(x_*, t) \mid t \in T_i(x_*), i \in I(x_*) \}$, and $\alpha \in [0, 1]$ such that
$\alpha v_1 + (1 - \alpha) v_2 + \nabla b(x_*)^T \nu \in - N_A(x_*)$. By Carath\'{e}odory's theorem for any
$i \in I(x_*)$ there exist $m_i \le d + 1$, $t_{ij} \in T_i(x_*)$, and $\alpha_{ij} \ge 0$,
$j \in \{ 1, \ldots, m_i \}$, such that
$$
v_2 = \sum_{i \in I(x_*)} \sum_{j = 1}^{m_i} \alpha_{ij} \nabla_x g_i(x_*, t_{ij}), \quad
\sum_{i \in I(x_*)} \sum_{j = 1}^m \alpha_{ij} = 1.
$$
Let us check that $\alpha \ne 0$. Then putting
$\mu_i = \sum_{j = 1}^{m_i} (1 - \alpha) (\alpha_{ij} / \alpha) \delta(t_{ij})$ for all $i \in I(x_*)$, $\mu_i = 0$
for $i \in I \setminus I(x_*)$, and $\lambda = (\mu_1, \ldots, \mu_l, \nu / \alpha) \in K^*$ one obtains that
$\langle \lambda, G(x_*) \rangle = 0$,
$$
\frac{1 - \alpha}{\alpha} v_2 + \frac{1}{\alpha} \nabla b(x_*)^T \nu =
\sum_{i = 1}^l \int_T \nabla_x g_i(x, t) d \mu_i(t) + \frac{1}{\alpha} \nabla b(x_*)^T \nu = [D G(x_*)]^* \lambda,
$$
and $(\partial F(x_*) + [D G(x_*)]^* \lambda) \cap (- N_A(x_*)) \ne \emptyset$, which by
Remark~\ref{rmrk:LagrangeMultViaSubdiff} implies that $\lambda$ is a discrete Lagrange multiplier at $x_*$.
Thus, it remains to check that $\alpha \ne 0$. Arguing by reductio ad absrudum suppose that $\alpha = 0$. Then
$v_2 + \nabla b(x_*)^T \nu \in - N_A(x_*)$. Note that from \eqref{eq:RCQ_SemiInfProblems} it follows that
there exists $h \in A - x_* \subset T_A(x_*)$ such that $\nabla b(x_*) h = 0$ and
$\langle \nabla_x g_i(x_*, t), h \rangle < 0$ for all $t \in T_i(x_*)$ and $i \in I(x_*)$. Hence by the definition of
$v_2$ one has $\langle v_2 + \nabla b(x_*)^T \nu, h \rangle < 0$, which is impossible, since by our assumption
$v_2 + \nabla b(x_*)^T \nu \in - N_A(x_*)$. Thus, $\alpha \ne 0$ and the proof of the first part of the theorem is
complete.
\textbf{Part~\ref{stat:LagrMultViaDiscreteProb}.} The validity of this statement follows directly from the definitions
of a discrete Lagrange multiplier and a Lagrange multiplier for problem \eqref{probl:DiscretisedSemiInfProb}.
\textbf{Part~\ref{stat:DiscreteLagrMult_ConvexCase}.} Observe that the assumptions on the functions $b(\cdot)$ and
$g_i(\cdot, t)$ imply that the mapping $G(\cdot)$ is $(-K)$-convex, while the existence of $x_0 \in A$ such that
$g_i(x_0, t) < 0$ for all $t \in T$ and $i \in I$ is equivalent to Slater's condition
$0 \in \interior\{ G(A) - K \}$ and implies the validity of Slater's conditions for the discritised problem
\eqref{probl:DiscretisedSemiInfProb}.
Suppose that there exists a discrete Lagrange multiplier at $x_*$. Then by the second part of the theorem for any
$i \in I(x_*)$ one can find $m_i \in \{ 1, \ldots, d + 1 \}$ and $t_{ij} \in T_i(x_*)$, $j \in \{ 1, \ldots, m_i \}$,
such that there exists a Lagrange multiplier of the discretised problem \eqref{probl:DiscretisedSemiInfProb}. Hence
by Theorem~\ref{thrm:OptCond_ConvexCase} the point $x_*$ is a globally optimal solution of problem
\eqref{probl:DiscretisedSemiInfProb}.
If $x_*$ is a globally optimal solution of the discretised problem \eqref{probl:DiscretisedSemiInfProb}, then $x_*$ is
obviously a globally optimal solution of problem \eqref{probl:SemiInfProg} as well, since the feasible region of
problem \eqref{probl:SemiInfProg} is contained in the feasible region of problem \eqref{probl:DiscretisedSemiInfProb}.
Finally, if $x_*$ is a globally optimal solution of problem \eqref{probl:SemiInfProg}, then taking into account the
fact that in the convex case by \cite[Prp.~2.104]{BonnansShapiro} Slater's condition $0 \in \interior\{ G(A) - K \}$ is
equivalent to RCQ and applying the first part of the theorem one obtains that there exists a discrete Lagrange
multiplier at $x_*$.
\end{proof}
\begin{remark}
Note that from the proof of the theorem above it follows that in the definition of discrete Lagrange multiplier one can
suppose that \textit{the union} of the supports of all measures $\mu_i$ consists of at most $d + 1$ points. Furthermore,
dividing the inclusion $\alpha v_1 + (1 - \alpha) v_2 + \nabla b(x_*)^T \nu \in - N_A(x_*)$ by $\alpha$ one obtains
that
$$
\Big( v_1 + \cone\big\{ \nabla_x g_i(x_*, t) \bigm| t \in T_i(x_*), i \in I(x_*) \big\}
+ \frac{1}{\alpha} \nabla b(x_*)^T \nu \Big) \cap (- N_A(x_*)) \ne \emptyset.
$$
Hence taking into account the fact that any point from the convex conic hull can be expressed as a non-negative linear
combination of $d$ or fewer linearly independent vectors (see, e.g. \cite[Corollary~17.1.2]{Rockafellar}) one can check
that in the definition of discrete Lagrange multiplier it is sufficient to suppose that the union of the supports of
the measures $\mu_i$ consists of at most $d$ points. \qed
\end{remark}
With the use of the theorem above one can easily obtain convenient necessary optimality conditions for problem
\eqref{probl:SemiInfProg} in terms of cadre and alternance. Let $Z \subset \mathbb{R}^d$ be a set consisting of $d$
linearly independent vectors and let $n_A(x)$ be a nonempty set such that $N_A(x) = \cone n_A(x)$ for any
$x \in \mathbb{R}^d$.
\begin{definition}
Let $x_*$ be a feasible point of problem \eqref{probl:SemiInfProg} and $p \in \{ 1, \ldots, d + 1 \}$ be fixed. One
says that \textit{a discrete $p$-point alternance} exists at $x_*$, if there exist $k_0 \in \{ 1, \ldots, p \}$,
$i_0 \in \{ k_0 + 1, \ldots, p \}$, vectors
\begin{gather} \label{eq:DiscreteAlternanceDef}
V_1, \ldots, V_{k_0} \in \Big\{ \nabla_x f(x_*, \omega) \Bigm| \omega \in W(x_*) \Big\}, \\
V_{k_0 + 1}, \ldots, V_{i_0} \in \Big\{ \nabla_x g_i(x_*, t) \Bigm| i \in I(x_*), t \in T_i(x_*) \Big\}, \:
V_{i_0 + 1}, \ldots, V_p \in n_A(x_*), \label{eq:DiscreteAlternanceDef2}
\end{gather}
and vectors $V_{p + 1}, \ldots, V_{d + 1} \in Z$ such that the $d$th-order determinants $\Delta_s$ of the matrices
composed of the columns $V_1, \ldots, V_{s - 1}, V_{s + 1}, \ldots V_{d + 1}$ satisfy the following conditions:
\begin{gather*}
\Delta_s \ne 0, \quad s \in \{ 1, \ldots, p \}, \quad
\sign \Delta_s = - \sign \Delta_{s + 1}, \quad s \in \{ 1, \ldots, p - 1 \}, \\
\Delta_s = 0, \quad s \in \{ p + 1, \ldots d + 1 \}.
\end{gather*}
Any such collection of vectors $\{ V_1, \ldots, V_p \}$ is called \textit{a discrete $p$-point alternance} at $x_*$. Any
discrete $(d + 1)$-point alternance is called \textit{complete}
\end{definition}
Bearing in mind Theorem~\ref{thrm:DiscreteLagrangeMultipliers} and applying Proposition~\ref{prp:AlternanceVsCadre} and
Theorem~\ref{thrm:AlternanceCond} to the discretised problem \eqref{probl:DiscretisedSemiInfProb} one obtains that the
following result holds true.
\begin{corollary}
Let $x_*$ be a feasible point of problem \eqref{probl:SemiInfProg}. Then the following statements are equivalent:
\begin{enumerate}
\item{a discrete Lagrange multiplier exists at $x_*$;}
\item{a discrete $p$-point alternance exists at $x_*$ for some $p \in \{ 1, \ldots, d + 1 \}$;}
\item{a discrete $p$-point cadre with positive cadre multipliers exists at $x_*$ for some
$p \in \{ 1, \ldots, d + 1 \}$, that is, there exist $k_0 \in \{ 1, \ldots, p \}$, $i_0 \in \{ k_0 + 1, \ldots, p \}$,
and vectors satisfying \eqref{eq:DiscreteAlternanceDef} and \eqref{eq:DiscreteAlternanceDef2}
such that $\rank([V_1, \ldots, V_p]) = p - 1$ and $\sum_{i = 1}^p \beta_i V_i = 0$ for some $\beta_i > 0$.
}
\end{enumerate}
Furhtermore, if a complete discrete alternance exists at $x_*$, then $x_*$ is a local minimiser of problem
\eqref{probl:SemiInfProg} at which the first order growth condition holds true.
\end{corollary}
\begin{remark}
It should be noted that it is unclear whether first order sufficient optimality condition for problem
\eqref{probl:SemiInfProg} can be rewritten in an equivalent form involving discrete Lagrange multipliers. One can
consider sufficient optimality conditions for the discretised problem \eqref{probl:DiscretisedSemiInfProb}. These
conditions are obviously sufficient optimality conditions for problem \eqref{probl:SemiInfProg}, since the feasible
region of this problem is contained in the feasible region of problem \eqref{probl:DiscretisedSemiInfProb}. However, it
seems that ``abstract'' sufficient optimality conditions for problem $(\mathcal{P})$ rewritten in terms of the
semi-infinite minimax problem are not equivalent to such conditions for the discretised problem. \qed
\end{remark}
\section{Second order optimality conditions for cone constrained minimax problems}
\label{sect:SecondOrderOptCond}
First order information is often insufficient to identify whether a given point is a locally optimal solution of a
minimax problem. For instance, in the case of unconstrained problems first order sufficient optimality conditions
cannot be satisfied, if the set $W(x_*) = \{ \omega \in W \mid F(x_*) = f(x_*, \omega) \}$ consists of less than $d + 1$
points. In such cases one obviously has to use \textit{second} order optimality conditions, whose analysis is the main
goal of this section. To simplify this analysis, we will mainly utilise a standard reformulation of cone constrained
minimax problems as equivalent smooth cone constrained problems and apply well-known second order optimality conditions
for such problems from \cite{Kawasaki,Cominetti,BonComShap98,BonComShap99,BonnansShapiro} to obtain optimality
conditions for minimax problems.
Let us introduce some auxiliary definitions first. Let $(x_*, \lambda_*)$ be a KKT-pair of the problem $(\mathcal{P})$,
that is, $x_*$ is a feasible point of this problem and $\lambda_*$ is a Lagrange multiplier at $x_*$. Then
$(\partial F(x_*) + [D G(x_*)]^* \lambda_*) \cap (- N_A(x_*)) \ne \emptyset$ by
Remark~\ref{rmrk:LagrangeMultViaSubdiff}, which implies that there exists $v \in \partial F(x_*)$ such that
$\langle v, h \rangle + \langle \lambda_*, D G(x_*) h \rangle \ge 0$ for all $h \in T_A(x_*)$. By definition there exist
$k \in \mathbb{N}$, $\omega_i \in W(x_*)$, and $\alpha_i \ge 0$, $i \in \{ 1, \ldots, k \}$, such
that $v = \sum_{i = 1}^k \alpha_i \nabla_x f(x, \omega_i)$ and $\sum_{i = 1}^k \alpha_i = 1$.
Let $\alpha = \sum_{i = 1}^k \alpha_i \delta(\omega_i)$ be the discrete Radon measure on $W$ corresponding to $\alpha_i$
and $\omega_i$. Then
$$
\left\langle \int_W \nabla_x f(x, \omega) d \alpha(\omega), h \right\rangle +
\langle \lambda_*, D G(x_*) h \rangle \ge 0 \quad \forall h \in T_A(x_*),
\quad \alpha(W) = 1.
$$
Denote by $\alpha(x_*, \lambda_*)$ the set of all Radon measures $\alpha \in rca_+(W)$ satisfying the conditions above
and such that $\support(\alpha) \subset W(x_*)$. It is easily seen that this set is convex, bounded and weak${}^*$
closed, i.e. $\alpha(x_*, \lambda_*)$ is a weak${}^*$ compact set. Any measure $\alpha \in \alpha(x_*, \lambda_*)$ is
called \textit{a Danskin-Demyanov multiplier} corresponding to the KKT-pair $(x_*, \lambda_*)$ (see, e.g.
\cite[Sect.~2.1.1]{Polak}). Note that in the case of discrete minimax problems, i.e. when $W = \{ 1, \ldots, m \}$ (or
in the case when the set $W(x_*)$ consists of a finite number of points), the set of Danskin-Demyanov multipliers
$\alpha(x_*, \lambda_*)$ is simply a closed convex subset of the standard (probability) simplex in $\mathbb{R}^m$.
Denote by $\mathcal{L}(x, \lambda, \alpha) = \int_W f(x, \omega) d \alpha(\omega) + \langle \lambda, G(x) \rangle$
\textit{the integral Lagrangian} for the problem $(\mathcal{P})$, where $x \in \mathbb{R}^d$, $\lambda \in Y^*$, and
$\alpha \in rca_+(W)$. Note that $\alpha_*$ is a Danskin-Demyanov multiplier corresponding to $(x_*, \lambda_*)$ iff
$\langle \nabla_x \mathcal{L}(x_*, \lambda_*, \alpha_*), h \rangle \ge 0$ for all $h \in T_A(x_*)$,
$\support(\alpha) \subset W(x_*)$, and $\alpha(W) = 1$.
Let $S \subset Y$ be a given set, and $y \in S$ and $h \in Y$ be fixed. Recall that \textit{the outer second order
tangent set} to the set $S$ at the point $y$ in the direction $h$, denoted by $T_S^2(x, h)$, consists of all those
vectors $w \in Y$ for which one can find a sequence $\{ t_n \} \subset (0, + \infty)$ such that $\lim t_n = 0$ and
$\dist(x + t_n h + 0.5 t_n^2 w, S) = o(t_n^2)$. See \cite[Sect.~3.2.1]{BonnansShapiro} for a detailed treatment of
second-order tangent sets. Here we only note that the second order tangent set $T_S^2(x, h)$ might be nonconvex even in
the case when the set $S$ is convex.
For any $\lambda \in Y^*$ denote by $\sigma(\lambda, S) = \sup_{y \in S} \langle \lambda, y \rangle$ the support
function of the set $S$. Also, for any feasible point $x_*$ of the problem $(\mathcal{P})$ denote by
$\Lambda(x_*)$ the set of all Lagrange multipliers of $(\mathcal{P})$ at $x_*$. Finally, for any feasible point
$x_*$ of the problem $(\mathcal{P})$ denote by
$$
C(x_*) = \Big\{ h \in T_A(x_*) \Bigm| D G(x_*) h \in T_K(G(x_*)), \enspace F'(x_*, h) \le 0 \Big\}
$$
\textit{the critical cone} at the point $x_*$. Observe that if $\Lambda(x_*) \ne \emptyset$, then by definition for any
$\lambda \in \Lambda(x_*)$ one has
$[L(\cdot, \lambda_*)]'(x_*, h) = F'(x_*, h) + \langle \lambda_*, D G(x_*) h \rangle \ge 0$ for any
$h \in T_A(x_*)$, which implies that
$$
C(x_*) = \Big\{ h \in T_A(x_*) \Bigm| D G(x_*) h \in T_K(G(x_*)), \enspace F'(x_*, h) = 0 \Big\},
$$
since $\langle\lambda_*, D G(x_*) h \rangle \le 0$ for any $h$ such that $D G(x_*) h \in T_K(G(x_*))$
(see Remark~\ref{rmrk:SuffOptCond_Lagrangian}). Moreover, one also has
\begin{equation} \label{eq:CriticalConeViaLagrangian}
\begin{split}
C(x_*) = \Big\{ h \in T_A(x_*) \Bigm| &D G(x_*) h \in T_K(G(x_*)), \\
&\langle \lambda_*, D G(x_*) h \rangle = 0, \enspace [L(\cdot, \lambda_*)]'(x_*, h) = 0 \Big\}
\end{split}
\end{equation}
for any $\lambda_* \in \Lambda(x_*)$.
For the sake of simplicity, we derive second order necessary optimality conditions only in the case when
$x_* \in \interior A$ and the set $W(x_*)$ is discrete. Arguing in the same way one can derive second order conditions
in the general case. However, it should be noted that in the general case these conditions are very cumbersome, since
they involve complicated expressions depending on the second order tangent sets to $A$ and $C_{-}(W)$.
In this section we suppose that the mapping $G$ is twice continuously Fr\'{e}chet differentiable in a neighbourhood of
a given point $x_*$, the function $f(x, \omega)$ is twice differentiable in $x$ in a neighbourhood $\mathcal{O}(x_*)$ of
$x_*$ for any $\omega \in W$, and the function $\nabla^2_{xx} f(\cdot)$ is continuous on $\mathcal{O}(x_*) \times W$.
\begin{theorem} \label{thrm:2Order_NessOptCond}
Let $W = \{ 1, \ldots, m \}$, $f(x, i) = f_i(x)$ for any $i \in W$, and $x_* \in \interior A$ be a locally optimal
solution of the problem $(\mathcal{P})$ such that RCQ holds true at $x_*$. Then for any $h \in C(x_*)$ and for any
convex set $\mathcal{T}(h) \subseteq T_K^2(G(x_*), D G(x_*) h)$ one has
$$
\sup_{\lambda \in \Lambda(x_*)} \Big\{
\sup_{\alpha \in \alpha(x_*, \lambda)} \big\langle h, \nabla^2_{xx} \mathcal{L}(x_*, \lambda, \alpha) h \big\rangle
- \sigma(\lambda, \mathcal{T}(h)) \Big\} \ge 0.
$$
\end{theorem}
\begin{proof}
From the facts that $x_*$ is a locally optimal solution of $(\mathcal{P})$ and $x_* \in \interior A$ it
follows that $(x_*, F(x_*))$ is a locally optimal solution of the problem
$$
\min_{(x, z)} z \quad \text{subject to} \quad
f(x, \omega) - z \le 0 \quad \omega \in W, \quad G(x) \in K.
$$
This problem can be rewritten as the cone constrained problem
\begin{equation} \label{prob:MinMaxViaConeConstr}
\min \: \widehat{f}(x, z) \quad \text{subject to} \quad \widehat{G}(x, z) \in \widehat{K},
\end{equation}
where $\widehat{f}(x, z) = z$, $\widehat{Y} = K \times \mathbb{R}^m$,
$\widehat{G}(x, z) = (G(x), f_1(x) - z, \ldots, f_m(x) - z)$, and $\widehat{K} = K \times \mathbb{R}_{-}^m$, where
$\mathbb{R}_{-} = (- \infty, 0]$. Our aim is to prove the theorem by reformulating second order optimality conditions
for problem \eqref{prob:MinMaxViaConeConstr} in terms of the problem $(\mathcal{P})$.
For any $x \in \mathbb{R}^d$, $z \in \mathbb{R}$, $\lambda \in Y^*$ and $\alpha \in \mathbb{R}^m$ denote by
$$
\mathcal{L}_0(x, z, \lambda, \alpha) = \widehat{f}(x, z) + \langle (\lambda, \alpha), \widehat{G}(x, z) \rangle
= z + \sum_{i = 1}^m \alpha^{(i)} \big( f_i(x) - z \big) + \langle \lambda, G(x) \rangle
$$
the Lagrangian for cone constrained problem \eqref{prob:MinMaxViaConeConstr}. Observe that
$\mathcal{L}_0(x, z, \lambda, \alpha) = \mathcal{L}(x, \lambda, \alpha) + (1 - \sum_{i = 1}^m \alpha^{(i)}) z$. One can
easily see that $(\lambda_*, \alpha_*)$ is a Lagrange multiplier of problem \eqref{prob:MinMaxViaConeConstr} at $(x_*,
F(x_*))$ iff $\lambda_*$ is a Lagrange multiplier of the problem $(\mathcal{P})$ at $x_*$ and $\alpha_*$ is a
Danskin-Demyanov multiplier corresponding to $(x_*, \lambda_*)$.
Let $z_* = F(x_*)$. Observe that
\begin{multline*}
\widehat{G}(x_*, z_*) + D \widehat{G}(x_*, z_*)\big( \mathbb{R}^d \times \mathbb{R} \big) - \widehat{K} \\
= \left\{ \begin{pmatrix} G(x_*) \\ f(x_*) - z_* \mathbf{1}_m \end{pmatrix}
+ \begin{pmatrix} D G(x_*) h_x \\ \nabla f(x_*) h_x - h_z \mathbf{1}_m \end{pmatrix}
- \begin{pmatrix} K \\ \mathbb{R}_{-}^m \end{pmatrix} \biggm| (h_x, h_z) \in \mathbb{R}^d \times \mathbb{R}
\right\}.
\end{multline*}
where $f(\cdot) = (f_1(\cdot), \ldots, f_m(\cdot))^T \in \mathbb{R}^m$ and
$\mathbf{1}_m = (1, \ldots, 1)^T \in \mathbb{R}^m$. Taking into account the fact that RCQ for the problem
$(\mathcal{P})$ is satisfied at $x_*$ one can easily check that RCQ for problem \eqref{prob:MinMaxViaConeConstr} is
satisfied at $(x_*, z_*)$. Therefore, by \cite[Thrm.~3.45]{BonnansShapiro} the second order necessary optimality
conditions for problem \eqref{prob:MinMaxViaConeConstr} are satisfied at $(x_*, 0)$, that is,
for every $\widehat{h} = (h_x, h_z) \in C(x_*, z_*)$, where
\begin{multline*}
C(x_*, z_*) = \Big\{ (h_x, h_z) \in \mathbb{R}^d \times \mathbb{R} \Bigm|
D G(x_*) h_x \in T_K(G(x_*)), \\
\nabla f(x_*) h_x - h_z \mathbf{1}_m \in T_{\mathbb{R}^m_{-}}(f(x_*) - z_* \mathbf{1}_m),
\enspace h_z = 0 \Big\}
\end{multline*}
and any convex set
$\mathcal{T}(\widehat{h}) \subseteq T^2_{\widehat{K}}(\widehat{G}(x_*, z_*), D \widehat{G}(x_*, z_*) \widehat{h})$ one
has
$$
\sup\Big\{ \langle h_x, \nabla^2_{xx} \mathcal{L}(x_*, \lambda, \alpha) h_x \rangle
- \sigma\big( (\lambda, \alpha), \mathcal{T}(\widehat{h}) \big) \Big\}
$$
where the supremum is taken over all Lagrange multipliers $(\lambda, \alpha)$ of problem
\eqref{prob:MinMaxViaConeConstr} at $(x_*, z_*)$. As was noted above, for any such $(\lambda, \alpha)$ one has
$\lambda \in \Lambda(x_*)$ and $\alpha \in \alpha(x_*, \lambda)$. Furthermore, note that
\begin{align*}
C(x_*, z_*) &= \Big\{ (h, 0) \in \mathbb{R}^d \times \mathbb{R} \Bigm| D G(x_*) h \in T_K(G(x_*)), \enspace
\langle \nabla f_i(x_*), h \rangle \le 0 \quad \forall i \in W(x_*) \Big\} \\
&= C(x_*) \times \{ 0 \}.
\end{align*}
Therefore for every $h \in C(x_*)$ and for any convex subset $\mathcal{T}(h, 0)$ of the second order tangent set
$T^2_{\widehat{K}}(\widehat{G}(x_*, z_*), D \widehat{G}(x_*, z_*)(h, 0))$ one has
$$
\sup_{\lambda \in \Lambda(x_*)} \Big\{
\sup_{\alpha \in \alpha(x_*, \lambda)} \big\langle h, \nabla^2_{xx} \mathcal{L}(x_*, \lambda, \alpha) h \big\rangle
- \sigma\big( (\lambda, \alpha), \mathcal{T}(h, 0) \big) \Big\} \ge 0.
$$
It remains to note that for every $h \in C(x_*)$ and for any convex set
$\mathcal{T}(h) \subseteq T_K^2(G(x_*), D G(x_*) h)$ one has
$\mathcal{T}(h) \times \{ 0 \} \subseteq T^2_{\widehat{K}}(\widehat{G}(x_*, z_*), D \widehat{G}(x_*, z_*)(h, 0))$, since
for all $w \in \mathcal{T}(h)$ and for any sequence $\{ t_n \} \subset (0, + \infty)$ such that $\lim t_n = 0$ and
$\dist(G(x_*) + t_n D G(x_*) h + 0.5 t_n^2 w, K) = o(t_n^2)$ (note that at least one such sequence exists due to the
fact that $\mathcal{T}(h) \subseteq T_K^2(G(x_*), D G(x_*) h)$) one has
\begin{multline*}
\dist\Big( \widehat{G}(x_*, z_*) + t_n D \widehat{G}(x_*)(h, 0) + \frac{1}{2} t_n^2 (w, 0), \widehat{K} \Big) \\
\le \dist\Big( G(x_*) + t_n D G(x_*) h + \frac{1}{2} t_n^2 w, K \Big)
+ \dist\big( f(x_*) - z_* \mathbf{1}_m + t_n \nabla f(x_*) h, \mathbb{R}^m_{-} \big) = o(t_n^2).
\end{multline*}
Here we used the fact that $\dist\big( f(x_*) - z_* \mathbf{1}_m + t_n \nabla f(x_*) h, \mathbb{R}^m_{-} \big) = 0$ for
any sufficiently large $n$, since $h \in C(x_*)$ and by the definition of critical cone one has
$\langle \nabla f_i(x_*), h \rangle \le 0$ for any $i \in W(x_*)$.
\end{proof}
Almost literally repeating the proof of \cite[Prp.~3.46]{BonnansShapiro} one can prove the following useful corollary to
the theorem above. For the sake of completeness, we outline its proof.
\begin{corollary}
Let all assumptions of the previous theorem be valid and suppose that there exists a unique Lagrange
multiplier at $x_*$, i.e. $\Lambda(x_*) = \{ \lambda_* \}$ for some $\lambda_* \in K^*$. Then for any
$h \in C(x_*)$ one has
$$
\sup_{\alpha \in \alpha(x_*, \lambda_*)} \big\langle h, \nabla^2_{xx} \mathcal{L}(x_*, \lambda_*, \alpha) \big\rangle
- \sigma\big( \lambda_*, T^2_K(G(x_*), D G(x_*) h) \big) \ge 0
$$
\end{corollary}
\begin{proof}
Let $\Sigma$ be the set consisting of all sequences $\sigma = \{ t_n \} \subset (0, + \infty)$ such that
$\lim t_n = 0$. For any $\sigma \in \Sigma$ and $h \in C(x_*)$ denote by $\mathcal{T}_{\sigma}(h)$ the set of all those
vectors $w \in Y$ for which $\dist(G(x_*) + t_n D G(x_*) h + 0.5 t_n^2 w, K) = o(t_n^2)$. Observe that the set
$\mathcal{T}_{\sigma}(h)$ is convex, since for any $n \in \mathbb{N}$ the
function $w \mapsto \dist(G(x_*) + t_n D G(x_*) h + 0.5 t_n^2 w, K)$ is convex. Furthermore, one has
$\mathcal{T}_{\sigma}(h) \subseteq T_K^2(G(x_*), D G(x_*) h)$. Hence by Theorem~\ref{thrm:2Order_NessOptCond} for any
$h \in C(x_*)$ the following inequality holds true:
$$
\inf_{\sigma \in \Sigma} \Big\{ \sup_{\alpha \in \alpha(x_*, \lambda_*)}
\big\langle h, \nabla^2_{xx} \mathcal{L}(x_*, \lambda_*, \alpha) h \big\rangle
- \sigma(\lambda_*, \mathcal{T}_{\sigma}(h)) \Big\} \ge 0.
$$
It remains to note that
$$
\inf_{\sigma \in \Sigma} \big( - \sigma(\lambda_*, \mathcal{T}_{\sigma}(h)) \big)
= - \sup_{\sigma \in \Sigma} \sup_{w \in \mathcal{T}_{\sigma}(h)} \langle \lambda_*, w \rangle
= - \sigma\big( \lambda_*, T^2_K(G(x_*), D G(x_*) h) \big),
$$
since $T^2_K(G(x_*), D G(x_*) h) = \bigcup_{\sigma \in \Sigma} \mathcal{T}_{\sigma}(h)$ by definition.
\end{proof}
Let us briefly discuss optimality conditions from Theorem~\ref{thrm:2Order_NessOptCond}. Firstly, note that they mainly
differ from classical optimality conditions by the presence of the sigma term $\sigma(\lambda, \mathcal{T}(h))$, which,
in a sense, represents a contribution of the curvature of the cone $K$ at the point $G(x_*)$ to optimality conditions.
This term is a specific feature of second order optimality conditions for cone constrained optimisation
problems \cite{Kawasaki,Cominetti,BonComShap98,BonComShap99,BonnansShapiro}. See
\cite{BonnansShapiro,BonnansRamirez,Shapiro2009} for explicit expressions for the critical cone $C(x_*)$, the second
order tangent set $T^2_K(G(x_*), D G(x_*)h)$, and the sigma term $\sigma(\lambda, \mathcal{T}(h))$ in various particular
cases.
Secondly, it should be pointed out that $\sigma(\lambda, \mathcal{T}(h)) \le 0$ for all $\lambda \in \Lambda(x_*)$ and
$h \in C(x_*)$. Furthermore, if $0 \in T^2_K(G(x_*), D G(x_*)h)$ (in particular, if the cone $K$ is polyhedral), then
$\sigma(\lambda, \mathcal{T}(h)) = 0$ for all $\lambda \in \Lambda(x_*)$ and $h \in C(x_*)$
(see~\cite[pp.~177--178]{BonnansShapiro}). In this case, the optimality conditions from
Theorem~\ref{thrm:2Order_NessOptCond} take the more traditional form:
$$
\sup_{\lambda \in \Lambda(x_*)} \sup_{\alpha \in \alpha(x_*, \lambda)}
\big\langle h, \nabla^2_{xx} \mathcal{L}(x_*, \lambda, \alpha) h \big\rangle \ge 0
\quad \forall h \in C(x_*).
$$
As was noted in the proof of Theorem~\ref{thrm:2Order_NessOptCond}, the set
$\{ (\lambda, \alpha) \mid \lambda \in \Lambda(x_*), \alpha \in \alpha(x_*, \lambda_*) \}$
coincides with the set of Lagrange multipliers of problem \eqref{prob:MinMaxViaConeConstr} at the point $(x_*, F(x_*))$.
Consequently, this set is convex and weak${}^*$ compact, since RCQ for problem \eqref{prob:MinMaxViaConeConstr} holds
at $(x_*, F(x_*))$. It is easily seen that the function
$(\lambda, \alpha) \mapsto \langle h, \nabla^2_{xx} \mathcal{L}(x_*, \lambda, \alpha) h \rangle$ is weak${}^*$
continuous. Therefore, if $0 \in T^2_K(G(x_*), D G(x_*)h)$ (in particular, if the cone $K$ is polyhedral), then under
the assumptions of Theorem~\ref{thrm:2Order_NessOptCond} for any $h \in C(x_*)$ one can find $\lambda \in \Lambda(x_*)$
and $\alpha \in \alpha(x_*, \lambda)$ such that
$\langle h, \nabla^2_{xx} \mathcal{L}(x_*, \lambda, \alpha) h \rangle \ge 0$.
Now we turn to second order sufficient optimality conditions. Similar to the case of first order optimality conditions,
we study second order sufficient optimality conditions in the context of second order growth condition. Recall that
\textit{the second order growth condition} (for the problem $(\mathcal{P})$) is said to be satisfied at a
feasible point $x_*$ of $(\mathcal{P})$, if there exist $\rho > 0$ and a neighbourhood $\mathcal{O}(x_*)$ of
$x_*$ such that $F(x) \ge F(x_*) + \rho | x - x_* |^2$ for any $x \in \mathcal{O}(x_*) \cap \Omega$, where $\Omega$ is
the feasible region of $(\mathcal{P})$.
We start with simple sufficient conditions that do not involve the sigma term.
\begin{theorem}
Let $x_* \in \interior A$ be a feasible point of the problem $(\mathcal{P})$ such that $\Lambda(x_*) \ne \emptyset$ and
for any $h \in C(x_*) \setminus \{ 0 \}$ one can find $\lambda \in \Lambda(x_*)$ and $\alpha \in \alpha(x_*, \lambda)$
such that $\langle h, \nabla^2_{xx} \mathcal{L}(x_*, \lambda, \alpha) h \rangle > 0$. Then $x_*$ is a locally optimal
solution of the problem $(\mathcal{P})$ at which the second order growth condition holds true.
\end{theorem}
\begin{proof}
Consider the following smooth cone constrained optimisation problem:
\begin{equation} \label{prob:MinMaxViaConeConstr_Contin}
\min_{(x, z)} z \quad \text{subject to} \quad f(x, \omega) - z \le 0 \quad \omega \in W,
\quad G(x) \in K, \quad z \in \mathbb{R}.
\end{equation}
Let us check that sufficient optimality condition for this problem hold true at the point $(x_*, F(x_*))$. Indeed,
the Lagrangian for problem \eqref{prob:MinMaxViaConeConstr_Contin} has the form
$$
\mathcal{L}_0(x, z, \lambda, \alpha) = z + \int_W \big( f(x, \omega) - z \big) \, d \alpha(\omega)
+ \langle \lambda, G(x) \rangle
$$
for any $\lambda \in K^*$ and $\alpha \in rca_+(W)$. As was noted in the proof of Theorem~\ref{thrm:2Order_NessOptCond},
the critical cone for problem \eqref{prob:MinMaxViaConeConstr_Contin} at $(x_*, F(x_*))$ has the form
$C(x_*, F(x_*)) = C(x_*) \times \{ 0 \}$. Therefore by our assumptions for any
$\widehat{h} = (h, 0) \in C(x_*, F(x_*))$, $\widehat{h} \ne 0$, one can find $\lambda \in \Lambda(x_*)$ and
$\alpha \in \alpha(x_*, \lambda_*)$ such that
$$
\big\langle \widehat{h}, \nabla^2_{(x, z)(x, z)} \mathcal{L}_0(x_*, F(x_*), \lambda, \alpha) \widehat{h} \big\rangle
= \big\langle h, \nabla^2_{xx} \mathcal{L}(x_*, \lambda, \alpha) h \big\rangle > 0
$$
As was pointed out in the proof of Theorem~\ref{thrm:2Order_NessOptCond}, the pair $(\lambda, \alpha)$ is a Lagrange
multiplier of problem \eqref{prob:MinMaxViaConeConstr_Contin} at $(x_*, F(x_*))$. Thus, one can conclude that the second
order sufficient optimality condition for problem \eqref{prob:MinMaxViaConeConstr_Contin} holds true at $x_*$, which by
\cite[Thrm.~3.63]{BonnansShapiro} implies that $(x_*, F(x_*))$ is a locally optimal solution of
\eqref{prob:MinMaxViaConeConstr_Contin} at which the second order growth condition holds true. Thus, by definition there
exist $\rho > 0$ and $\varepsilon > 0$ such that $z \ge F(x_*) + \rho (|x - x_*|^2 + |z - F(x_*)|^2)$ for all
$x \in B(x_*, \varepsilon)$ and $z \in \mathbb{R}$ such that $|z - F(x_*)| < \varepsilon$, $F(x) \le z$, and
$G(x) \in K$. Note that the function $F(\cdot) = \max_{\omega \in W} f(\cdot, \omega)$ is continuous, since by our
assumptions the space $W$ is compact and the function $f$ is continuous. Consequently, there exists
$r \in (0, \varepsilon)$ such that $|F(x) - F(x_*)| < \varepsilon$ for all $x \in B(x_*, r)$. Therefore, putting
$z = F(x)$ one obtains that $F(x) \ge F(x_*) + \rho |x - x_*|^2$ for all $x \in B(x_*, r)$ such that $G(x) \in K$, that
is, $x_*$ is a locally optimal solution of the problem $(\mathcal{P})$ at which the second order growth condition holds
true.
\end{proof}
In the case when the space $Y$ is finite dimensional and the cone $K$ is second order regular one can strengthen
the previous theorem and obtain simple sufficient optimality conditions involving the sigma term. Recall that the cone
$K$ is said to be \textit{second order regular} at a point $y \in K$, if the following two conditions are satisfied:
\begin{enumerate}
\item{for any $h \in T_K(y)$ and any sequence $\{ y_n \} \subset K$ of the form $y_n = y + t_n h + 0.5 t_n^2 w_n$ where
$t_n > 0$ for all $n \in \mathbb{N}$, $\lim t_n = 0$, and $\lim t_n w_n = 0$ one has
$\lim \dist(w_n, T_K^2(y, h)) = 0$;}
\item{$T^2_K(y, h) = \{ w \in Y \mid \dist(x + th + 0.5 t^2 w, K) = o(t^2), t \ge 0 \}$ for any $h \in Y$.}
\end{enumerate}
We say that the cone $K$ is second order regular, if it is second order regular at every point $y \in K$.
For more details on second order regular sets see \cite{BonComShap98,BonComShap99} and
\cite[Sect.~3.3.3]{BonnansShapiro}. Here we only mention that the cone $\mathbb{S}^l_{-}$ of negative semidefinite
matrices is second order regular (see~\cite[p.~474]{BonnansShapiro}) and the second order cone is second order regular
by \cite[Prp.~3.136]{BonnansShapiro} and \cite[Lemma~15]{BonnansRamirez}.
Below we do not assume that $x_* \in \interior A$, but avoid the usage of the second order tangent set to the set $A$
for the sake of simplicity and due to the fact that we are mainly interested in the case when the set $A$ is polyhedral.
\begin{theorem}
Let $Y$ be a finite dimensional Hilbert space, the cone $K$ be second order regular, and $(x_*, \lambda_*)$ be a
KKT-pair of the problem $(\mathcal{P})$ such that the restriction of the function
$\sigma(\lambda_*, T^2_K(G(x_*), \cdot))$ to its effective domain is upper semicontinuous. Suppose also that
\begin{equation} \label{eq:2OrderSuffCond_SigmaTerm}
\sup_{\alpha \in \alpha(x_*, \lambda_*)}
\big\langle h, \nabla^2_{xx} \mathcal{L}(x_*, \lambda_*, \alpha) h \big\rangle
- \sigma\big( \lambda_*, T^2_K(G(x_*), D G(x_*) h) \big) > 0
\end{equation}
for all $h \in C(x_*) \setminus \{ 0 \}$. Then $x_*$ is a locally optimal solution of the problem $(\mathcal{P})$ at
which the second order growth condition holds true.
\end{theorem}
\begin{proof}
Introduce the Rockafellar-Wets augmented Lagrangian
\begin{equation} \label{eq:RockWetsAugmLagr}
\mathscr{L}(x, \lambda, c) = F(x) + \Phi(G(x), \lambda, c), \quad
\Phi(y, \lambda, c) = \inf_{z \in K - y} \big\{ - \langle \lambda, z \rangle + c \| z \|^2 \big\}
\end{equation}
for the problem $(\mathcal{P})$ (see~\cite{RockafellarWets,ShapiroSun,Dolgopolik_AugmLagr}), where
$\langle \cdot, \cdot \rangle$ is the inner product in $Y$ and $c \ge 0$ is the penalty parameter. It is easily seen
that
\begin{equation} \label{eq:AugmLagrangian}
\Phi(y, \lambda, c) = c \big( \dist(y + (2c)^{-1} \lambda, K) \big)^2 - \frac{1}{4c} \| \lambda \|^2.
\end{equation}
Let us compute a second order expansion of the function $x \mapsto \mathscr{L}(x, \lambda, c)$.
Denote $\delta(y) = \dist(y, K)^2$. By a generalisation of the Danskin-Demyanov theorem
\cite[Thrm.~4.13]{BonnansShapiro} the function $\delta(\cdot)$ is continuously Fr\'{e}chet differentiable and
$D \delta(y) = 2(y - P_K(y))$, where $P_K$ is the projection of $y$ onto $K$ (note that the projection exists,
since $Y$ is finite dimensional). Hence by the chain rule the function $x \mapsto \Phi(G(x), \lambda, c)$ is
continuously Fr\'{e}chet differentiable and
$$
D_x \Phi(G(x), \lambda, c) h
= 2c \Big\langle G(x) + (2c)^{-1} \lambda - P_K(G(x) + (2c)^{-1} \lambda), D G(x) h \Big\rangle
$$
for all $h \in \mathbb{R}^d$. To simplify this expression in the case $x = x_*$ and $\lambda = \lambda_*$ note that
$$
\left\langle z_* - G(x_*) - \frac{1}{2c} \lambda_*, z - z_* \right\rangle \ge 0 \quad \forall z \in K,
$$
if $z_* = G(x_*)$ (recall that $\langle \lambda_*, G(x_*) \rangle = 0$ and $\lambda_* \in K^*$ by the definition of
KKT-point). Thus, the point $z = G(x_*)$ satisfies the necessary and sufficient optimality conditions for the convex
problem
$$
\min\: \| z - G(x_*) - (2c)^{-1} \lambda_* \|^2 \quad \text{subject to} \quad z \in K,
$$
that is, $P_K(G(x_*) + (2c)^{-1} \lambda_*) = G(x_*)$. Consequently, for any $c > 0$ one has
$D_x \Phi(G(x_*), \lambda_*, c) = [D G(x_*)]^* \lambda_*$.
Recall that the cone $K$ is second order regular and the space $Y$ is finite dimensional. Therefore by
\cite[Thrm.~4.133]{BonnansShapiro} (see also \cite[Thrm.~3.1]{Shapiro2016}) for all $y, v \in Y$ there
exists the second-order Hadamard directional derivative
$$
\delta''(y; v) := \lim_{[v', t] \to [v, +0]}
\frac{\delta(y + t v') - \delta(y) - t D \delta(y) v'}{\frac{1}{2} t^2}
$$
and it has the form
\begin{equation} \label{eq:SecondOrderHadamardDeriv}
\delta''(y; v) = \min_{z \in \mathscr{C}(y)}
\Big[ 2 \| v - z \|^2 - 2 \sigma(y - P_K(y), T^2_K(P_K(y), z) \Big],
\end{equation}
where $\mathscr{C}(y) = \{ z \in T_K(P_K(y)) \mid \langle y - P_K(y), z \rangle = 0 \}$. Bearing in mind the definition
of the second-order Hadamard directional derivative one can easily check that the function $\delta''(y, \cdot)$ is
continuous and positively homogeneous of degree two. Hence taking into account the definition of this derivative one
can easily check that for any linear operator $T \colon \mathbb{R}^d \to Y$ one has
$$
\delta(y + T h + o(|h|)) = \delta(y) + D \delta(y) \Big( T h + o(|h|) \Big)
+ \frac{1}{2} \delta''(y; T h) + o(|h|^2 ).
$$
Consequently, putting $y = G(x_*) + (2c)^{-1} \lambda_*$ and $T h = D G(x_*) h$, taking into account the fact that
$D \delta(y) = c^{-1} \lambda_*$, and utilising the second order expansion
$$
G(x_* + h) = G(x_*) + D G(x_*) h + \frac{1}{2} D^2 G(x_*)(h, h) + o(|h|^2)
$$
one obtains that
\begin{align*}
\Phi(G(x_* + h), \lambda_*, c) &= \Phi(G(x_*), \lambda_*, c) + \langle \lambda_*, D G(x_*) h \rangle
+ \frac{1}{2} \langle \lambda_*, D^2 G(x_*)(h, h) \rangle \\
&+ \frac{c}{2} \delta''\Big(G(x_*) + \frac{1}{2c} \lambda_*; D G(x_*) h \Big) + o(|h|^2).
\end{align*}
Hence with the use of the well-known second-order expansion for the max-function of the form
$$
F(x_* + h) - F(x_*) = \max_{\omega \in W} \Big( f(x_*, \omega) - F(x_*)
+ \langle \nabla_x f(x_*, \omega), h \rangle
+ \frac{1}{2} \langle h, \nabla_{xx}^{2} f(x_*, \omega) h \rangle \Big) + o(|h|^2)
$$
one finally gets that for any $c \ge 0$ there exists $r_c > 0$ such that for all $h \in B(0, r_c)$ one has
\begin{multline} \label{eq:AugmLagr_2OrderExpansion}
\Big| \mathscr{L}(x_* + h, \lambda_*, c) - \mathscr{L}(x_*, \lambda_*, c)
- \max_{\omega \in W} \Big( f(x_*, \omega) - F(x_*) + \langle \nabla_x f(x_*, \omega), h \rangle
+ \frac{1}{2} \langle h, \nabla_{xx}^{2} f(x_*, \omega) h \rangle \Big) \\
- \langle \lambda_*, D G(x_*) h \rangle
- \frac{1}{2} \langle \lambda_*, D^2 G(x_*)(h, h) \rangle - \frac{1}{2} \omega_c(h) \Big|
\le \frac{1}{c} |h|^2,
\end{multline}
where
\begin{align*}
\omega_c(h) &= c \delta''\Big( G(x_*) + (2c)^{-1} \lambda_*), D G(x_*) h) \Big) \\
&= \min_{z \in C_0(x_*, \lambda_*)}
\Big[ 2 c \| D G(x_*) h - z \|^2 - \sigma\big( \lambda_*, T^2_K(G(x_*), z) \big) \Big]
\end{align*}
and $C_0(x_*, \lambda_*) = \{ z \in T_K(G(x_*)) \mid \langle \lambda_*, z \rangle = 0 \}$
(see \eqref{eq:SecondOrderHadamardDeriv}). By \cite[formula~(3.63)]{BonnansShapiro} one has
$T^2_K(G(x_*), z) \subseteq T_{T_K(G(x_*))}(z)$ for all $z \in Y$. Note also that the cone $T_K(G(x_*))$ is convex,
since $K$ is a convex cone. Therefore
$$
T_K(G(x_*)) = \cl \Big[ \bigcup_{t \ge 0} t\big( K - G(x_*) \big) \Big], \enspace
T_{T_K(G(x_*)}(z) = \cl \Big[ \bigcup_{t \ge 0} t\big( T_K(G(x_*)) - z \big) \Big]
$$
(see, e.g. \cite[Prp.~2.55]{BonnansShapiro}). Hence bearing in mind the facts that $\lambda_* \in K^*$ and
$\langle \lambda_*, G(x_*) \rangle = 0$ one obtains that $\langle \lambda_*, y \rangle \le 0$ for all
$y \in T_K(G(x_*))$, which implies that $\langle \lambda_*, y \rangle \le 0$ for any
$y \in T^2_K(G(x_*), z) \subseteq T_{T_K(G(x_*)}(z)$ and all $z \in C_0(x_*, \lambda_*)$. Consequently,
$\sigma(\lambda_*, T^2_K(G(x_*), z)) \le 0$ for all $z \in C_0(x_*, \lambda_*)$. Recall also that
the restriction of the function $z \mapsto \sigma(\lambda_* , T^2_K(G(x_*), z))$ to its effective domain is upper
semicontinuous by our assumption. Therefore $\lim_{c \to + \infty} \omega_c(h) = + \infty$, if
$D G(x_*) h \notin T_K(G(x_*))$ or $\langle \lambda_*, D G(x_*) h \rangle \ne 0$, and
\begin{equation} \label{eq:SecondOrderTermLimit}
\lim_{c \to + \infty} \omega_c(h) \ge - \sigma\big( \lambda_* , T^2_K(G(x_*), D G(x_*)) \big)
\end{equation}
otherwise. Utilising this fact and the second order expansion for the augmented Lagrangian we can easily prove the
statement of the theorem.
Indeed, let us show that there exist $\rho, c > 0$, and a neighbourhood $\mathcal{O}(x_*)$ of $x_*$ such that
$\mathscr{L}(x, \lambda_*, c) \ge \mathscr{L}(x_*, \lambda_*, c) + \rho |x - x_*|^2$ for any
$x \in A \cap \mathcal{O}(x_*)$. Then taking into account the facts that $\Phi(y, \lambda_*, c) \le 0$ for any
$y \in K$ thanks to \eqref{eq:AugmLagrangian} and $\Phi(G(x_*), \lambda_*, c) = 0$ due to the fact that
$P_K(G(x_*) + (2c)^{-1} \lambda_*) = G(x_*)$ one obtains that
$$
F(x) \ge \mathscr{L}(x, \lambda_*, c) \ge \mathscr{L}(x_*, \lambda_*, c) + \rho |x - x_*|^2
= F(x_*) + \rho |x - x_*|^2
$$
for all $x \in A \cap \mathcal{O}(x_*)$ such that $G(x) \in K$, and the proof is complete.
Arguing by reductio ad absurdum suppose that for any $n \in \mathbb{N}$ there exists
$x_n \in A$ such that $\mathscr{L}(x_n, \lambda_*, n) < \mathscr{L}(x_*, \lambda_*, n) + n^{-1} |x_n - x_*|^2$
and $x_n \in B(x_*, \min\{ \frac{1}{n}, r_n \})$. With the use of \eqref{eq:AugmLagr_2OrderExpansion}
for any $n \in \mathbb{N}$ one has
\begin{multline*}
0 > \mathscr{L}(x_n, \lambda_*, n) - \mathscr{L}(x_*, \lambda_*, n) - \frac{1}{n} |x_n - x_*|^2 \\
\ge \max_{\omega \in W} \Big( f(x_*, \omega) - F(x_*) + \langle \nabla_x f(x_*, \omega), u_n \rangle
+ \frac{1}{2} \langle u_n, \nabla_{xx}^{2} f(x_*, \omega) u_n \rangle \Big) \\
+ \Big\langle \lambda_*, D G(x_*) u_n + \frac{1}{2} D^2 G(x_*)(u_n, u_n) \Big\rangle + \frac{1}{2} \omega_n(u_n)
- \frac{2}{n} |u_n|^2,
\end{multline*}
where $u_n = x_n - x_*$. Consequently, for any $\alpha \in \alpha(x_*, \lambda_*)$ one has
\begin{multline} \label{eq:AugmLagr_2OrderExp_DanskinDemyanov}
0 > \mathscr{L}(x_n, \lambda_*, n) - \mathscr{L}(x_*, \lambda_*, n) - \frac{1}{n} |x_n - x_*|^2
\ge \big\langle \nabla_x \mathcal{L}(x_*, \lambda_*, \alpha), u_n \big\rangle \\
+ \frac{1}{2} \big\langle u_n, \nabla^2_{xx} \mathcal{L}(x_*, \lambda_*, \alpha) u_n \big\rangle
+ \frac{1}{2} \omega_n(u_n) - \frac{2}{n} |u_n|^2,
\end{multline}
since $\alpha \in rca_+(W)$, $\support(\alpha) \subseteq W(x_*)$, and $\alpha(W) = 1$ by the definition of
Danskin-Demyanov multipliers.
Define $h_n = u_n / |u_n|$. Without loss of generality one can suppose that the sequence $\{ h_n \}$ converges to some
$h_* \in \mathbb{R}^d$ with $|h_*| = 1$. Moreover, $h_* \in T_A(x_*)$ by virtue of the facts that the set $A$ is convex
and $\{ x_n \} \subset A$. Let us show that $[L(\cdot, \lambda_*)]'(x_*, h_*) = 0$.
Indeed, suppose that $[L(\cdot, \lambda_*)]'(x_*, h_*) \ne 0$. Note that by the definition of Lagrange multiplier one
has $[L(\cdot, \lambda_*)]'(x_*, h_*) \ge 0$. Thus, $[L(\cdot, \lambda_*)]'(x_*, h_*) > 0$, which thanks to the
equality $D_x \Phi(G(x_*), \lambda_*, c) = [D G(x_*)]^* \lambda_*$ implies that
\begin{align*}
\lim_{n \to \infty}
\frac{\mathscr{L}(x_* + \beta_n h_n, \lambda_*, c) - \mathscr{L}(x_*,\lambda_*, c) - \beta_n^2}{\beta_n}
&= F'(x_*, h_*) + \langle \lambda_*, D G(x_*) h_* \rangle \\
&= [L(\cdot, \lambda_*)]'(x_*, h_*) > 0,
\end{align*}
for any $c > 0$, where $\beta_n = |u_n| = |x_n - x_*|$ (note that $x_* + \beta_n h_n = x_n$). Consequently, there
exists $n_0 \in \mathbb{N}$ such that $\mathscr{L}(x_n, \lambda_*, 1) > \mathscr{L}(x_*, \lambda_*, 1) + |x_n - x_*|^2$
for all $n \ge n_0$. As was noted above, $\Phi(G(x_*), \lambda_*, c) = 0$ for any $c > 0$. Consequently,
$\mathscr{L}(x_*, \lambda_*, 1) = \mathscr{L}(x_*, \lambda_*, c) = F(x_*)$ for any $c > 0$. Hence bearing in mind the
fact that the function $c \mapsto \mathscr{L}(x, \lambda, c)$ is obviously non-decreasing (see
\eqref{eq:RockWetsAugmLagr}) one obtains that
$$
\mathscr{L}(x_n, \lambda_*, c) \ge \mathscr{L}(x_n, \lambda_*, 1)
> \mathscr{L}(x_*, \lambda_*, 1) + |x_n - x_*|^2 = \mathscr{L}(x_*, \lambda_*, c) + |x_n - x_*|^2
$$
for all $c \ge 1$, which contradicts the definition of $x_n$. Thus, $[L(\cdot, \lambda_*)]'(x_*, h_*) = 0$.
Note that $\langle \nabla_x \mathcal{L}(x_*, \lambda_*, \alpha_*), u_n \big\rangle \ge 0$ for all $n \in \mathbb{N}$ due
to the definition of Danskin-Demyanov multiplier and the fact that $u_n = x_n - x_* \in T_A(x_*)$, since $A$ is a convex
set. Hence with the use of \eqref{eq:AugmLagr_2OrderExp_DanskinDemyanov} one obtains that
$$
0 > \frac{1}{2} \big\langle u_n, \nabla^2_{xx} \mathcal{L}(x_*, \lambda_*, \alpha_*) u_n \big\rangle
+ \frac{1}{2} \omega_n(u_n) - \frac{2}{n} |u_n|^2
$$
for any $n \in \mathbb{N}$. Dividing this inequality by $|u_n|^2$ (recall that $\omega_c(\cdot)$ is positively
homogeneous of degree two), passing to the limit as $n \to \infty$ with the use of \eqref{eq:SecondOrderTermLimit}, and
taking the supremum over all $\alpha \in \alpha(x_*, \lambda_*)$ one finally gets that
$$
0
\ge \sup_{\alpha \in \alpha(x_*, \lambda_*)} \big\langle h_*, \nabla^2_{xx} \mathcal{L}(x_*, \lambda, \alpha) h_*
\big\rangle
- \sigma\big( \lambda_*, T^2_K(G(x_*), D G(x_*) h_*) \big),
$$
and $D G(x_*) h_* \in T_K(G(x_*))$, $\langle \lambda_*, D G(x_*) h_* \rangle = 0$, and
$[L(\cdot, \lambda_*)]'(x_*, h_*) = 0$, that is, $h_* \in C(x_*)$ (see \eqref{eq:CriticalConeViaLagrangian}), which
contradicts \eqref{eq:2OrderSuffCond_SigmaTerm}.
\end{proof}
\begin{remark}
{(i)~Note that the restriction of the function $\sigma(\lambda_*, T^2_K(G(x_*), \cdot))$ to its effective domain is
continuous in the case when $K$ is the second order cone (see~\cite[Formula~(42) and Thrm.~29]{BonnansRamirez})
or the cone $\mathbb{S}^l_{-}$ (see~\cite[Sect.~5.3.5]{BonnansShapiro}).
}
\noindent{(ii)~It should be noted that one can obtain second order sufficient optimality conditions for
the problem $(\mathcal{P})$ involving the sigma term that are equivalent to the second order growth condition without
the additional assumption that the space $Y$ is finite dimensional. However, this condition is much more cumbersome than
the one stated in the theorem above, since it involves the second order tangent sets to $A$ and $C_{-}(W)$. That is why
we leave the derivation of such second order conditions to the interested reader (see \cite[Sect.~3.3.3]{BonnansShapiro}
for more details in the smooth case). \qed
}
\end{remark}
\section{Optimality conditions for Chebyshev problems with cone constraints}
\label{sect:ChebyshevProblems}
In this section we study optimality conditions for cone constrained Chebyshev problems of the form:
$$
\min_x\: \max_{\omega \in W} \big| f(x, \omega) - \psi(\omega) \big| \quad
\text{subject to} \quad G(x) \in K, \quad x \in A.
\eqno{(\mathcal{C})}
$$
Here $\psi \colon W \to \mathbb{R}$ is a continuous function. This problem is a particular case of the problem
$(\mathcal{P})$. Indeed, define $\widehat{W} = W \times \{ 1, -1 \}$,
$\widehat{f}(x, \omega, 1) = f(x, \omega) - \psi(\omega)$ and
$\widehat{f}(x, \omega, -1) = - f(x, \omega) + \psi(\omega)$ for any $\omega \in W$. Then the problem $(\mathcal{C})$
can be rewritten as the problem $(\mathcal{P})$ of the form:
\begin{equation} \label{prob:EquivChebyshevProblem}
\min_x \: \max_{\widehat{\omega} \in \widehat{W}} \widehat{f}(x, \widehat{\omega}) \quad
\text{subject to} \quad G(x) \in K, \quad x \in A.
\end{equation}
Therefore, optimality conditions for the problem $(\mathcal{C})$ can be easily obtained as a direct corollary to
optimality conditions for the problem $(\mathcal{P})$. Nevertheless, it is worth explicitly formulating these
conditions. Furthermore, the following sections can be viewed as a convenient and concise summary of the main results
obtained in this article.
\subsection{First order optimality conditions}
Define $F(x) = \max_{\omega \in W} |f(x, \omega) - \psi(\omega)|$, and let
$W(x) = \{ \omega \in W \mid F(x) = |f(x, \omega) - \psi(\omega)| \}$ the set of points of maximal deviation. Under our
assumptions on $f$, the function $F$ is Hadamard directionally differentiable and its Hadamard directional
derivative has the form
$$
F'(x, h) = \max_{v \in \partial F(x)} \langle v, h \rangle
= \max_{\omega \in W(x)}
\Big( \sign(f(x, \omega) - \psi(\omega)) \big\langle \nabla_x f(x, \omega), h \big\rangle \Big)
$$
for any $h \in \mathbb{R}^d$, where
$\partial F(x) = \co\{ \sign(f(x, \omega) - \psi(\omega)) \nabla_x f(x, \omega) \mid \omega \in W(x) \}$ is the
Hadamard subdifferential of the function $F$ at the point $x$. In this section we suppose that $\sign(0) = \{-1, 1 \}$.
For any $\lambda \in Y^*$ denote by $L(x, \lambda) = F(x) + \langle \lambda, G(x) \rangle$ the Lagrangian for
the problem $(\mathcal{C})$. A vector $\lambda_* \in Y^*$ is called \textit{a Lagrange multiplier} of the problem
$(\mathcal{C})$ at a feasible point $x_*$, if $\lambda_* \in K^*$, $\langle \lambda_*, G(x_*) \rangle = 0$, and
$[L(\cdot, \lambda_*)]'(x_*, h) \ge 0$ for all $h \in T_A(x_*)$. In this case, the pair $(x_*, \lambda_*)$ is called
\textit{a KKT-pair} of the problem $(\mathcal{C})$.
Applying Theorems~\ref{thrm:NessOptCond}--\ref{thrm:OptCond_ConvexCase} to problem
\eqref{prob:EquivChebyshevProblem} one obtains that the following results hold true.
\begin{theorem} \label{thrm:NessOptCond_Chebyshev}
Let $x_*$ be a locally optimal solution of the problem $(\mathcal{C})$ such that RCQ holds at $x_*$. Then:
\begin{enumerate}
\item{$h = 0$ is a globally optimal solution of the linearised problem
$$
\min_{h \in \mathbb{R}^d} \max_{v \in \partial F(x_*)} \langle v, h \rangle \quad
\text{s.t.} \quad D G(x_*) h \in T_K\big( G(x_*) \big), \quad h \in T_A(x_*);
$$
}
\vspace{-5mm}
\item{the set of Lagrange multipliers at $x_*$ is a nonempty, convex, bounded, and weak${}^*$ compact subset of $Y^*$.}
\end{enumerate}
\end{theorem}
\begin{theorem} \label{Thrm:ConvexCase_Chebyshev}
Let there exist continuous functions $\phi \colon W \to \mathbb{R}^d$ and $\phi_0 \colon W \to \mathbb{R}$ such that
$f(x, \omega) = \langle \phi(\omega), x \rangle + \phi_0(\omega)$ for all $x$ and $\omega$. Suppose also that the
mapping
$G$ is $(-K)$-convex and $x_*$ is a feasible point of the problem $(\mathcal{C})$. Then:
\begin{enumerate}
\item{$\lambda_*$ is a Lagrange multiplier of $(\mathcal{C})$ at $x_*$ iff $(x_*, \lambda_*)$ is a global saddle point
of the Lagrangian $L(x, \lambda) = F(x) + \langle \lambda, G(x) \rangle$, that is, for all $x \in A$ and
$\lambda \in K^*$ one has $L(x, \lambda_*) \ge F(x_*) \ge L(x_*, \lambda)$;
}
\item{if a Lagrange multiplier of the problem $(\mathcal{C})$ at $x_*$ exists, then $x_*$ is a globally optimal solution
of $(\mathcal{C})$; conversely, if $x_*$ is a globally optimal solution of the problem $(\mathcal{C})$ and Slater's
condition $0 \in \interior\{ G(A) - K \}$ holds true, then there exists a Lagrange multiplier of $(\mathcal{C})$
at $x_*$.
}
\end{enumerate}
\end{theorem}
\begin{theorem} \label{thrm:SuffOptCond_Chebyshev}
Let $x_*$ be a feasible point of the problem $(\mathcal{C})$. If
\begin{equation} \label{eq:SuffOptCond_Chebyshev}
\max_{v \in \partial F(x_*)} \langle v, h \rangle > 0
\quad \forall h \in T_A(x_*) \setminus \{ 0 \} \colon D G(x_*) h \in T_K\big( G(x_*) \big),
\end{equation}
then the first order growth condition holds at $x_*$. Conversely, if the first order growth condition and RCQ hold at
$x_*$, then inequality \eqref{eq:SuffOptCond_Chebyshev} is valid.
\end{theorem}
Next we present several equivalent reformulations of necessary and sufficient optimality conditions for the problem
$(\mathcal{C})$ from Theorems~\ref{thrm:NessOptCond_Chebyshev} and \ref{thrm:SuffOptCond_Chebyshev}. Recall that
$$
\mathcal{N}(x) = [D G(x)]^* (K^* \cap \linhull(G(x))^{\perp})
= \{ i(\lambda \circ D G(x)) \mid \lambda \in K^*, \: \langle \lambda, G(x) \rangle = 0 \},
$$
where $i$ is the natural isomorphism between $(\mathbb{R}^d)^*$ and $\mathbb{R}^d$. For any $c \ge 0$ a penalty function
for the problem $(\mathcal{C})$ is denoted by $\Phi_c(x) = F(x) + c \dist(G(x), K)$. By
Lemma~\ref{lem:ConeConstrPenFunc_Subdiff} this function is Hadamard subdifferentiable and for any $x$ such that
$G(x) \in K$ its Hadamard subdifferential has the form
$$
\partial \Phi_c(x) = \partial F(x)
+ c \Big\{ [D G(x)]^* y^* \in \mathbb{R}^d \Bigm| y^* \in Y^*, \: \| y^* \| \le 1, \:
\langle y^*, y - G(x) \rangle \le 0 \enspace \forall y \in K \Big\}.
$$
Let us reformulate alternance optimality conditions in terms of the problem $(\mathcal{C})$. Let, as earlier,
$Z \subset \mathbb{R}^d$ be any collection of $d$ linearly independent vectors, and
$\eta(x)$ and $n_A(x)$ be any sets such that $\mathcal{N}(x) = \cone \eta(x)$ and
$N_A(x) = \cone n_A(x)$.
\begin{definition}
Let $p \in \{ 1, \ldots, d + 1 \}$ be given and $x_*$ be a feasible point of the problem $(\mathcal{C})$. One says that
\textit{a $p$-point alternance} exists at $x_*$, if there exist $k_0 \in \{ 1, \ldots, p \}$,
$i_0 \in \{ k_0 + 1, \ldots, p \}$, vectors
\begin{gather} \label{eq:AlternanceDef_Ch}
V_1, \ldots, V_{k_0} \in
\Big\{ \sign\big(f(x_*, \omega) - \psi(\omega) \big)\nabla_x f(x_*, \omega) \Bigm| \omega \in W(x_*) \Big\}, \\
V_{k_0 + 1}, \ldots, V_{i_0} \in \eta(x_*), \quad
V_{i_0 + 1}, \ldots, V_p \in n_A(x_*), \label{eq:AlternanceDef_Ch_2}
\end{gather}
and vectors $V_{p + 1}, \ldots, V_{d + 1} \in Z$ such that the $d$th-order determinants $\Delta_s$ of the matrices
composed of the columns $V_1, \ldots, V_{s - 1}, V_{s + 1}, \ldots V_{d + 1}$ satisfy the following conditions:
\begin{gather*}
\Delta_s \ne 0, \quad s \in \{ 1, \ldots, p \}, \quad
\sign \Delta_s = - \sign \Delta_{s + 1}, \quad s \in \{ 1, \ldots, p - 1 \}, \\
\Delta_s = 0, \quad s \in \{ p + 1, \ldots d + 1 \}.
\end{gather*}
Such collection of vectors $\{ V_1, \ldots, V_p \}$ is called a $p$-point alternance at $x_*$. Any $(d + 1)$-point
alternance is called \textit{complete}. If the set in the right-hand side of \eqref{eq:AlternanceDef_Ch} is replaced by
$\partial F(x_*)$ and the sets $\eta(x_*)$ and $n_A(x_*)$ in \eqref{eq:AlternanceDef_Ch_2} are replaced by
$\mathcal{N}(x_*)$ and $N_A(x_*)$ respectively, then one says that \textit{a generalised $p$-point alternance} exists at
$x_*$, and the corresponding collection of vectors $\{ V_1, \ldots, V_p \}$ is called \textit{a generalised $p$-point
alternance} at $x_*$.
\end{definition}
Finally, if $x_*$ is a feasible point of $(\mathcal{C})$, then any collection of vectors $V_1, \ldots, V_p$
with $p \in \{ 1, \ldots, d + 1 \}$ satisfying \eqref{eq:AlternanceDef_Ch}, \eqref{eq:AlternanceDef_Ch_2}, and such
that $\rank([V_1, \ldots, V_p]) = \rank([V_1, \ldots, V_{i - 1}, V_{i + 1}, \ldots, V_p]) = p - 1$ for any
$i \in \{ 1, \ldots, p \}$ is called a $p$-point \textit{cadre} for the problem $(\mathcal{C})$ at $x_*$. It is easily
seen that a collection $V_1, \ldots, V_p$ satisfying \eqref{eq:AlternanceDef_Ch}, \eqref{eq:AlternanceDef_Ch_2} is
a $p$-point cadre at $x_*$ iff $\rank([V_1, \ldots, V_p]) = p - 1$ and
$\sum_{i = 1}^p \beta_i V_i = 0$ for some $\beta_i \ne 0$, $i \in \{ 1, \ldots, p \}$. Any such $\{ \beta_i \}$ are
called \textit{cadre multipliers}.
Applying the main results of Sections~\ref{subsect:Subdifferentials_ExactPenaltyFunc} and
\ref{subsect:Alternance_Cadre} to problem \eqref{prob:EquivChebyshevProblem} one obtains the following six equivalent
reformulations of necessary/sufficient optimality conditions for the cone constrained Chebyshev problem
$(\mathcal{C})$.
\begin{theorem} \label{thrm:EquivNeccOptCond_Chebyshev}
Let $x_*$ be a feasible point of the problem $(\mathcal{C})$. Then the following statements are equivalent:
\begin{enumerate}
\item{there exists a Lagrange multiplier of $(\mathcal{C})$ at $x_*$;}
\item{there exists $v \in \partial F(x_*)$ and $\lambda_* \in K^*$ such that $\langle \lambda_*, G(x_*) \rangle = 0$
and $\langle v, h \rangle + \langle \lambda_*, D G(x_*) h \rangle \ge 0$ for all $h \in T_A(x_*)$;}
\item{$0 \in \partial F(x_*) + \mathcal{N}(x_*) + N_A(x_*)$;}
\item{$0 \in \partial \Phi_c(x_*) + N_A(x_*)$ for some $c > 0$;}
\item{a $p$-point alternance exists at $x_*$ for some $p \in \{1, \ldots, d + 1 \}$;}
\item{a $p$-point cadre with positive cadre multipliers exists at $x_*$ for some $p \in \{ 1, \ldots, d + 1 \}$.}
\end{enumerate}
\end{theorem}
\begin{theorem} \label{thrm:EquivSuffOptCond_Chebyshev}
Let $x_*$ be a feasible point of the problem $(\mathcal{C})$. Then the following statements are equivalent:
\begin{enumerate}
\item{sufficient optimality condition \eqref{eq:SuffOptCond_Chebyshev} holds true at $x_*$;}
\item{$0 \in \interior(\partial F(x_*) + \mathcal{N}(x_*) + N_A(x_*))$;}
\item{$0 \in \interior(\partial \Phi_c(x_*) + N_A(x_*))$ for some $c > 0$;}
\item{$\Phi_c$ satisfies the first order growth condition on $A$ at $x_*$ for some $c \ge 0$.}
\end{enumerate}
Moreover, all these conditions are satisfied, if a complete alternance exists at $x_*$. In addition, if one of
the following assumptions is valid
\begin{enumerate}
\item{$\interior \partial F(x_*) \ne \emptyset$,}
\item{$\mathcal{N}(x_*) + N_A(x_*) \ne \mathbb{R}^d$ and either $\interior \mathcal{N}(x_*) \ne \emptyset$
or $\interior N_A(x_*) \ne \emptyset$,}
\item{$N_A(x_*) = \{ 0 \}$ and there exists $w \in \relint \mathcal{N}(x_*) \setminus \{ 0 \}$ such that
$0 \in \partial F(x_*) + w$ (in particular, it is sufficient to suppose that $0 \notin \partial F(x_*)$ or the cone
$\mathcal{N}(x_*)$ is pointed),}
\item{$\mathcal{N}(x_*) = \{ 0 \}$ and there exists $w \in \relint N_A(x_*) \setminus \{ 0 \}$ such that
$0 \in \partial F(x_*) + w$,}
\end{enumerate}
then the four equivalent sufficient optimality conditions stated in this theorem are satisfied iff a generalised
complete alternance exists at $x_*$.
\end{theorem}
\begin{remark} \label{rmrk:ChebyshevSquared}
It should be noted that the Chebyshev problem $(\mathcal{C})$ can be reduced to the minimax problem $(\mathcal{P})$ in
a different way. Namely, define $F_2(\cdot) = \max_{\omega \in W} 0.5 (f(\cdot, \omega) - \psi(\omega))^2$ and consider
the following cone constrained problem:
\begin{equation} \label{prob:ChebyshevSquared}
\min \: F_2(x) \quad \text{subject to} \quad G(x) \in K, \quad x \in A.
\end{equation}
Note that $W(x) = \{ \omega \in W \mid F_2(x) = 0.5 (f(x, \omega) - \psi(\omega))^2 \}$. Furthermore, one has $
F(x_1) \ge F(x_2)$ for some $x_1$ and $x_2$ if and only if
$$
|f(x_1, \omega_*) - \psi(\omega_*)| \ge |f(x_2, \omega) - \psi(\omega)|
\quad \forall \omega_* \in W(x_1), \quad \forall \omega \in W,
$$
while this inequality is satisfied if and only if
$$
\frac{1}{2} \big( f(x_1, \omega_*) - \psi(\omega_*) \big)^2 \ge \frac{1}{2} \big( f(x_2, \omega) - \psi(\omega) \big)
\quad \forall \omega_* \in W(x_1), \quad \forall \omega \in W,
$$
or, equivalently, if and only if $F_2(x_1) \ge F_2(x_2)$. Therefore, $x_*$ is a locally/globally optimal solution of the
problem $(\mathcal{C})$ iff $x_*$ is a locally/globally optimal solution of problem \eqref{prob:ChebyshevSquared}.
Moreover, it is easily seen that the function $F_2$ is Hadamard subdifferentiable, $F_2'(x, h) = \max_{v \in \partial
F_2(x)} \langle v, h \rangle$ for all $h \in \mathbb{R}^d$, where
$$
\partial F_2(x) = \big\{ (f(x, \omega) - \psi(\omega)) \nabla_x f(x, \omega) \bigm| \omega \in W(x) \big\},
$$
that is, $F_2'(x, \cdot) = F(x) F'(x, \cdot)$ and $\partial F_2(x) = F(x) \partial F(x)$ for all $x$. Consequently,
$\lambda_*$ is a Lagrange multiplier of the problem $(\mathcal{C})$ at a feasible point $x_*$ such that $F(x_*) \ne 0$
iff $F(x_*) \lambda_*$ is a Lagrange multiplier of problem \eqref{prob:ChebyshevSquared} at $x_*$. Therefore, replacing
$\partial F(x_*)$ with $F(x_*) \partial F(x_*)$ in
Theorems~\ref{thrm:NessOptCond_Chebyshev}--\ref{thrm:EquivSuffOptCond_Chebyshev} one obtains equivalent
necessary/sufficient optimality conditions for the cone constrained Chebyshev problem $(\mathcal{C})$. \qed
\end{remark}
\subsection{Second order optimality conditions}
Let us finally formulate second order optimality conditions for the problem $(\mathcal{C})$. To this end, suppose that
the mapping $G$ is twice continuously Fr\'{e}chet differentiable in a neighbourhood of a given point $x_*$, the function
$f(x, \omega)$ is twice differentiable in $x$ in a neighbourhood $\mathcal{O}(x_*)$ of $x_*$ for any $\omega \in W$, and
the function $\nabla^2_{xx} f(\cdot)$ is continuous on $\mathcal{O}(x_*) \times W$.
Firstly, note that if for a feasible point $x_*$ one has $F(x_*) = 0$, then $x_*$ is a globally optimal solution of the
problem $(\mathcal{C})$, since this function is nonnegative. Therefore, below we suppose that the optimal value of
the problem $(\mathcal{C})$ is strictly positive.
Let $(x_*, \lambda_*)$ be a KKT-pair of the problem $(\mathcal{C})$. Then by the second part of
Theorem~\ref{thrm:EquivNeccOptCond_Chebyshev} there exist $v \in \partial F(x_*)$ and
$\lambda_* \in K^*$ such that $\langle \lambda_*, G(x_*) \rangle = 0$ and
$\langle v, h \rangle + \langle \lambda_*, D G(x_*) h \rangle \ge 0$ for all $h \in T_A(x_*)$. Then by the definition of
$\partial F(x_*)$ there exist $k \in \mathbb{N}$, $\omega_i \in W(x_*)$, and $\alpha_i \ge 0$,
$i \in \{ 1, \ldots, k \}$, such that
$$
v = \sum_{i = 1}^k \alpha_i \sign(f(x, \omega_i) - \psi(\omega_i)) \nabla_x f(x, \omega_i), \quad
\sum_{i = 1}^k \alpha_i = 1.
$$
Let $\alpha = \sum_{i = 1}^k \sign(f(x, \omega_i) - \psi(\omega_i)) \alpha_i \delta(\omega_i)$ be the discrete Radon
measure on $W$ corresponding to $\alpha_i$ and $\omega_i$. Then
$$
\left\langle \int_W \nabla_x f(x, \omega) d \alpha(\omega), h \right\rangle +
\langle \lambda_*, D G(x_*) h \rangle \ge 0 \quad \forall h \in T_A(x_*),
\quad |\alpha|(W) = 1,
$$
where $|\alpha| = \alpha^+ + \alpha^-$ is the total variation of the measure $\alpha$, while $\alpha^+$ and $\alpha^-$
are positive and negative variations of $\alpha$ respectively (see, e.g. \cite{Folland}). Denote by
$\alpha(x_*, \lambda_*)$ the set of all Radon measures $\alpha \in rca(W)$ satisfying the conditions above and the
inclusions
\begin{align*}
\support(\alpha^+) \subseteq W_+(x_*) &:= \{ \omega \in W(x_*) \mid f(x_*, \omega) - \psi(\omega) > 0 \}, \\
\support(\alpha^-) \subseteq W_-(x_*) &:= \{ \omega \in W(x_*) \mid f(x_*, \omega) - \psi(\omega) < 0 \}.
\end{align*}
One can easily verify that $\alpha(x_*, \lambda_*)$ is a convex, bounded and weak${}^*$ closed (and, therefore,
weak${}^*$ compact) set. Any measure $\alpha \in \alpha(x_*, \lambda_*)$ is called \textit{a Danskin-Demyanov
multiplier} corresponding to the KKT-pair $(x_*, \lambda_*)$.
For any $x \in \mathbb{R}^d$, $\lambda \in Y^*$, and $\alpha \in rca(W)$ denote by
$$
\mathcal{L}(x, \lambda, \alpha) = \int_W f(x, \omega) d \alpha(\omega) + \langle \lambda, G(x) \rangle
$$
the integral Lagrangian for the problem $(\mathcal{C})$. It is easily seen that $\alpha_*$ is a Danskin-Demyanov
multiplier corresponding to $(x_*, \lambda_*)$ if and only if $|\alpha_*|(W) = 1$,
$\support(\alpha_*^{\pm}) \subseteq W_{\pm}(x_*)$,
and $\langle \nabla_x \mathcal{L}(x_*, \lambda_*, \alpha_*), h \rangle \ge 0$ for all $h \in T_A(x_*)$.
Applying the main results of Section~\ref{sect:SecondOrderOptCond} to problem~\eqref{prob:EquivChebyshevProblem} one
gets the following necessary/sufficient second order optimality conditions for the problem $(\mathcal{C})$.
\begin{theorem}
Let $W = \{ 1, \ldots, m \}$, $f(x, i) = f_i(x)$ for any $i \in W$, and $x_* \in \interior A$ be a locally optimal
solution of the problem $(\mathcal{C})$ such that RCQ holds true at $x_*$. Then for any vector $h$ from the critical
cone
$$
C(x_*) = \Big\{ h \in T_A(x_*) \Bigm| D G(x_*) h \in T_K(G(x_*)), \enspace F'(x_*, h) \le 0 \Big\}
$$
and for any convex set $\mathcal{T}(h) \subseteq T_K^2(G(x_*), D G(x_*) h)$ one has
$$
\sup_{\lambda \in \Lambda(x_*)} \Big\{
\sup_{\alpha \in \alpha(x_*, \lambda)} \big\langle h, \nabla^2_{xx} \mathcal{L}(x_*, \lambda, \alpha) h \big\rangle
- \sigma(\lambda, \mathcal{T}(h)) \Big\} \ge 0.
$$
Furthermore, if $\Lambda(x_*) = \{ \lambda_* \}$, then for any $h \in C(x_*)$ one has
$$
\sup_{\alpha \in \alpha(x_*, \lambda_*)} \big\langle h, \nabla^2_{xx} \mathcal{L}(x_*, \lambda_*, \alpha) \big\rangle
- \sigma\big( \lambda_*, T^2_K(G(x_*), D G(x_*) h) \big) \ge 0.
$$
\end{theorem}
\begin{theorem}
Let $x_* \in \interior A$ be a feasible point of the problem $(\mathcal{C})$ such that $\Lambda(x_*) \ne \emptyset$ and
for any $h \in C(x_*) \setminus \{ 0 \}$ one can find $\lambda \in \Lambda(x_*)$ and $\alpha \in \alpha(x_*, \lambda)$
such that $\langle h, \nabla^2_{xx} \mathcal{L}(x_*, \lambda, \alpha) h \rangle > 0$. Then $x_*$ is a locally optimal
solution of the problem $(\mathcal{C})$ at which the second order growth condition holds true.
\end{theorem}
\begin{theorem}
Let $Y$ be a finite dimensional Hilbert space, the cone $K$ be second order regular, and $(x_*, \lambda_*)$ be a
KKT-pair of the problem $(\mathcal{C})$ such that the restriction of the function
$\sigma(\lambda_*, T^2_K(G(x_*), \cdot))$ to its effective domain is upper semicontinuous. Suppose also that
$$
\sup_{\alpha \in \alpha(x_*, \lambda_*)}
\big\langle h, \nabla^2_{xx} \mathcal{L}(x_*, \lambda_*, \alpha) h \big\rangle
- \sigma\big( \lambda_*, T^2_K(G(x_*), D G(x_*) h) \big) > 0
$$
for all $h \in C(x_*) \setminus \{ 0 \}$. Then $x_*$ is a locally optimal solution of the problem $(\mathcal{C})$ at
which the second order growth condition holds true.
\end{theorem}
\section{Conclusions}
In this article we presented a unified theory of first and second order necessary and sufficient optimality
conditions for minimax and Chebyshev optimisation problems with cone constraints, including such problems with equality
and inequality constraints, problems with second order cone constraints, problems with semidefinite constraints, as well
as problems with semi-infinite constraints. We analysed different, but equivalent forms of first order optimality
conditions and demonstrated how they can be reformulated in a more convenient way for particular classes of
cone constrained minimax problems. These results can be utilised to develop new methods for solving cone constrained
minimax and Chebyshev problems based on structural properties of optimal solutions (cf. such methods for discrete
minimax problems \cite{ConnLi92}, problems of rational $\ell_{\infty}$-approximation \cite{BarrodalePowellRoberts},
and synthesis of a rational filter \cite{MalozemovTamasyan}). A development of such methods is an interesting topic of
future research.
\section*{Acknowledgements}
The author wishes to express his sincere gratitude to prof. V.N.~Malozemov and the late prof. V.F.~Demyanov. Their
research on minimax problems and alternance optimality conditions, as well as inspiring lectures, were the main source
of inspiration for writing this article. In particular, the main results of Section~\ref{subsect:Alternance_Cadre} are
a natural continuation of their research on alternance optimality conditions
\cite{DemyanovMalozemov_Alternance,DemyanovMalozemov_Collect}.
\bibliographystyle{abbrv}
|
2,869,038,156,394 | arxiv | \section{Introduction}
LMC X-4 is a 13.5s pulsar
orbiting a 20$M_{\odot}$ O7 III-V companion every 1.4 days
(Kelley {\it et~al.}~1983; Ilovaisky {\it et~al.}~1984).
The system exhibits a long term cycle
with a roughly 30-day period
(Lang {\it et~al.}~1981; Ilovaisky
{\it et~al.}~1984) similar to the 35-day period of
Her~X-1.
Although the high-mass companion of LMC~X-4 implies the presence
of a wind, LMC X-4 has many properties in common with the disk-fed
system Her~X-1.
In both systems the long term periods are roughly 20 times
the orbital periods.
Both systems are fully eclipsing
(Tananbaum {\it et~al.}~1972; Li {\it et~al.}~1978); show
a power-law spectrum with a soft X-ray ``excess"
in the 0.5-10 keV band (Dennerl 1989);
optical light curves
with a power spectrum that
shows power at the sum of the orbital and long term
frequencies, but not in the difference,
implying that the long-term variations are due to precession
of an accretion disk (Ilovaisky {\it et~al.}~1984);
and correlation
between times of low hard X-ray luminosity and episodes of spindown
(Dennerl 1991; Wilson {\it et~al.}~1994).
The optical lightcurves of
LMC X-4 show changes in peak-to-peak amplitude of up to 40\% relative
to the mean over the 30-day
period whereas the optical lightcurves of Her X-1 show less than 20\%
change in peak-to-peak amplitude over the 35-day cycle.
For LMC X-4 the X-ray flux varies by a factor of 60 between the high
and low states of the 30-day period; however, the flux of scattered
X-rays, measured during X-ray eclipses, remains unchanged throughout
the 30 day cycle (Woo {\it et~al.}~1995), implying that the low state is
not caused by a decrease in X-ray luminosity but by attenuation
in intervening matter.
LMC X-4 also displays
flares which typically occur once a day
(Dennerl 1989; Levine {\it et~al.}~1991; Woo {\it et~al.}~1996).
The X-ray luminosity during the flares rises
to $\sim$5 times the Eddington
luminosity and the spectrum changes from a power law to a very hot
thermal spectrum.
There are several mechanisms that could cause the UV continuum or lines to
vary with the 13.5 second neutron star rotation period.
In Vela~X-1 it has been shown
that the Si\,IV and N\,V P~Cygni lines of Vela~X-1 vary
with the 283 second pulsar period, presumably as a result of
time-dependent
photoionization of the stellar wind by the X-rays
(Boroson {\it et~al.}~1996a). The UV
continuum could pulsate as a result of X-ray heating of either the normal
star or the accretion disk, as is the case in Her~X-1 (Boroson
{\it et~al.}~1996b).
Here we present coordinated ultraviolet and X-ray observations of
LMC X-4 taken with the GHRS on HST and with the GIS and SIS on ASCA.
The high temporal resolution of the GHRS allowed us to search for
a manifestation of the pulsations in the UV and the high spectral
resolution allows us to study
the geometry of the system as reflected in the line profiles.
Some results from an ASCA observation covering a full binary orbit
taken an year earlier are also presented.
We compare the HST data as well as archival IUE data with models
predicting UV continuum emission from the X-ray heated disk and star
that has been successfullly applied to the Her X-1 system.
We interpret the dramatic changes with orbital phase observed in
the ultraviolet spectra in terms of
the effects of X-ray photonionization on the stellar wind
of the normal companion. The observations and analysis are described
in section 2 and our interpretation
is discussed in section 3.
\section{Observations and Analysis}
Figure 1 shows the location of our observation in comparison with
the one-day averaged lightcurves obtained from the All Sky Monitor on
XTE
and the same data folded with the long term ephemeris and period of
Dennerl {\it et~al.}~(1992).
This indicates that our simultaneous HST and ASCA observations
occurred during the
high state of the 30-day cycle,
corresponding to a phase coverage of $\phi_{30-day}$=0.20-0.22.
The ASCA lightcurves and HST coverage are plotted on Figure 2.
HST took eight seperate observations, but as the target was
nearly in HST's Continuous
Viewing Zone the gaps between the observations are small.
A journal of the observations is given in Table 1.
HST captured an eclipse
egress that did not have simultaneous X-ray coverage.
The HST observations were taken with the GHRS G160M grating
centered alternately at 1240 \AA~(N V) and 1550 \AA~(C IV) in the RAPID
mode with a time resolution of 0.5s.
Our use of the GHRS
RAPID mode prevented us from over-sampling the spectrum, checking for bad
counts, and correcting for the Doppler shift due to the spacecraft orbit.
The observed flux in the N\thinspace V $\lambda 1240$ line measured with IUE
(van der Klis {\it et~al.}~1982) ranges from
$F \approx 1 \times 10^{-12}$ergs cm$^{-2}$
$s^{-1}$
at minimum ($\phi_{orb} = \approx 0.5$) to
$F \approx 3 \times 10^{-12}$ ergs cm$^{-2}$ s$^{-1}$
at maximum ($\phi_{orb} = \approx 0.9)$.
The fluxes measured from our observations are listed in Table 1.
The 1996 ASCA observations caught an eclipse ingress and
two pre-eclipse dips that
did not have simultaneous UV coverage.
The 1994 ASCA observations covered more than a full binary orbit including
an eclipse.
Detailed studies of the ASCA observations will appear elsewhere
(Boroson {\it et~al.}~1997b).
No flares occurred during either the 1994 or 1996 ASCA observations
although each covered more than a
binary orbit and flares are reported to occur roughly once a day.
It is possible that flares (which last roughly 30 minutes) occurred during
the data gaps (roughly 40-50 minutes) due to earth occultation.
\subsection{Continuum Fits}
A model involving X-ray heating of the disk and star as previously applied
to Her X-1 (Cheng, Vrtilek, \& Raymond~1995) was adapted for use with LMC X-4.
Changes to the model from that described in Cheng, Vrtilek, \& Raymond
are inclusion of both gravity darkening and limb darkening effects
(important for the massive companion in LMC X-4) as well as a different
reddening curve towards the LMC (Nandy~{\it et~al.}~1981).
We use a distance to the source of 50 kpc
and an E$_{B-V}$ of 0.05.
The companion star is modelled as a 20 $M_{\odot}$ O7III star with an
effective temperature of 35,000K
and IUE spectra of stars of varying temperature are used to determine
the spectral shape at different points on the star and disk surface.
Predictions of the model from the ultraviolet
and optical B band are shown in Figure 3. We are using archival
IUE data for comparison, hence
we restrict ourselves to the
region near C~IV since the presence of strong geocoronal Lyman Alpha
prevents accurate determination of the IUE flux near N~V.
In addition our model, which is constructed to predict continuum emission,
consistently overpredicts the flux near N~V in this system; this is
likely due to the presence
of a stellar wind---not included in the model---which results in
strong absorption at N~V.
Because the 30.25d period has a large uncertainty the long term phase of the
IUE observations cannot be determined.
The ultraviolet data including
the current observations and all archival IUE observations fall within extremes
given by a lower limit to $\dot M$ of
3.2~$\times~10^{-9}M_{\odot}$ yr$^{-1}$ and an upper limit of
1.0~$\times~10^{-7}M_{\odot}$ yr$^{-1}$.
The best fit curve to the HST continuum data (shown as filled triangles)
corresponds
to an $\dot M$ of 4.0~$\times~$10$^{-8}M_{\odot}$ yr$^{-1}$.
There is uncertainty in the flux level, both for the model and the data.
Our HST continuum
determination is based on a very narrow wavelength band
since the observations were centered on lines and the total
available bandpass was
only 37$\AA\/$ for the spectral resolution we required;
the HST data are
absolute flux calibrated to only 10\%; there is a difference in
resolution between IUE and GHRS. As for the model, the uncertainties in the
reddening and stellar temperature can shift the flux. The fluxes
near C IV and N V show a similar flux vs. orbit trend. The relative
errors in flux are likely to be much smaller than the overall flux
normalization.
The change in peak-to-peak amplitude of the optical flux from the 30-day
low to the 30 day
high
predicted by the model is
consistent with that observed (Ilovaisky {\it et~al.}~1984).
Figure 3b shows the extremes over the 30.25 day cycle for B magnitude
variations and Figure 3c shows
the model predictions at the X-ray on and off states for
a given $\dot M$ reproducing the variations shown by
Ilovaisky~{\it et~al.}~1984 in their Figure 1.
During binary eclipse the total UV and optical flux is due to the
companion star. During the time of maximum UV flux (binary phase 0.75)
for the highest $\dot M$ the contribution from the disk is 8\% and from
the heated star 10\%.
A comprehensive presentation of the fits to individual IUE spectra and
archival optical data will appear in a later paper
(Preciado {\it et~al.}~1997).
The X-ray spectra obtained by ASCA are consistent with a power law
of photon index $\alpha$ = 0.63$\pm$0.02, a 0.180$\pm$.006 keV blackbody,
and an iron line at
6.54$\pm$0.03 keV with an equivalent width of 99$\pm$25eV, typical
for this source over the energy range observed.
The 2-10 keV flux is
2.9~$\times 10^{-10}$ergs~s$^{-1}$~cm$^{-2}$ with about 10\% changes from
the mean during the period of observations.
The luminosity obtained from the $\dot M$ measured with the UV continuum
fits, using L$_x$~=~0.5GM$\dot M$/r, is consistent with the luminosity implied
by this flux.
\subsection{Spectral Features}
Figure 4 shows the N~V and C~IV line profiles at four orbital phases each;
these are different for the two lines since they could not be observed
simultaneously at this spectral resolution.
These two lines
are the strongest in the UV spectrum and
are known from IUE observations
to be strongly and weakly phase dependent, respectively.
The C~IV profiles show little evolution from binary phase 0.15-0.49.
At binary phase 0.11, the observation closest to the X-ray eclipse
(binary phase 0.0), we see
broad absorption in the N~V doublets. The maximum velocity at this
phase (1150 km/s) is reasonable for a terminal velocity for the wind
however it is likely that by phase 0.11 some ionization has already
taken place and terminal velocities of up to 1300 km/s are possible
(Boroson {\it et~al.}~1997a). As we progress in phase
towards seeing more of the X-ray source we find that the broad absorption
disappears and what remains are relatively narrow absorption profiles.
The width of the residual absorption left at phase 0.41 in N~V is
consistent with the width of the optical lines which Hutchings {\it et~al.}~
attributed to the photosphere.
Comparison with low-resolution IUE archival data suggests that the
variations are phase dependent effects, presumably arising in the
geometry-dependent interaction between the X-ray source and stellar wind.
Interstellar absorption lines from S\,II$~\lambda\lambda1250.578,1253.805$
are visible in our exposures in the wavelength region near
N\,V$~\lambda\lambda1238,1242$. The velocities of these absorption
lines (heliocentric redshifts are consistent with zero within the errors
of our measurement) suggest that they arise in gas within the Galaxy, and
not in the
LMC (where we would expect velocities of the order 280-320km~s$^{-1}$;
Bomans {\it et~al.}~1996).
The narrow C\,IV absorption lines are superimposed on broader absorption
features, which we interpret as arising in the photosphere of the O~star in
the LMC~X-4 system. As a result of the overlap, the velocity and
especially the equivalent widths of the interstellar C\,IV lines are
uncertain. The heliocentric redshifts of
the C\,IV lines are also consistent with zero within the uncertainties of
our measurement implying that the
C\,IV absorber is also within the Galaxy.
There is no evidence for a C\,IV absorber associated with LMC~X-4, which
might be expected if the X-ray source photoionizes surrounding
interstellar gas (McCray, Wright, and Hatchett 1977). The N\,V line at
$\phi=0.41$ shows what appears to be a narrow feature at the base of the
absorption due to the blue doublet component. However, observations at
higher signal-to-noise ratio are needed to establish that this results from an
interstellar feature and not merely a fluctuation in the photospheric line
profile.
\subsection{Search for Pulsations}
The pulse periods that we measured from ASCA data of 1994
(13.5069$\pm$0.0002s) and 1996
(13.5090$\pm$0.0002s)
are consistent with spin-down over the past 10 years (Figure 5).
While earlier observations are scant they indicate that
sometime between 10 and 20 years ago LMC X-4 went through a period
of spin-up.
Such rapid, erratic, changes from spin-up to spin-down are expected
from equilibrium rotators (Chakrabarty {\it et~al.}~1997), systems in which
the spin period equals the Kepler period near the inner boundary of
the accretion disk.
We searched for pulsations in the UV spectrum at the 13.5 second
pulsar period using
an analysis of variance (ANOVA) method
(Davies 1990, 1991). This approach, which involves binning the data
over trial periods, is well-suited to data with gaps or with non-uniform
readout times. To detect pulsations from the accretion disk, we need to
subtract the changes in light-travel time from the orbiting pulsar to the
orbiting HST from the
uniform readout times of the GHRS. Thus we report only the ANOVA method
and not a power-spectral search to find pulsations from material
moving with the neutron star.
From the 4 HST orbits we find a 5$\sigma$
limit of 1.8\% for the fractional peak-to-peak pulse amplitude in the continuum
centered at 1240\AA~
surrounding the N\,V line. For the continuum surrounding C\,IV,
the limit is 2.7\%.
Our limits for pulsation
in the P~Cygni absorption troughs are 12.4\% in the N\,V line and
7\% in the C\,IV line.
\section{Discussion and Conclusions}
A model that incorporates X-ray heating of the companion star
and the accretion disk
provides good fits to the continuum UV emission from LMC X-4.
The value of $\dot M$ derived from GHRS observations is consistent
with that from X-ray flux measured during
simultaneous observations.
Owing to the size and temperature of the companion the major
contribution to the ultraviolet and
optical flux is the unheated companion star.
At maximum light (binary phases 0.25 and 0.75) the
contribution of the unheated star is 82\%;
heating of the primary contributes 10\% and the
disk 8\%.
Although we captured no flares during our observations
the X-ray heating model can accomodate flares by using unsteady accretion
from the stellar wind. During flares the intensity can increase
by factors of up to 20 for times
ranging from $\sim$20s to 45 minutes, resulting in super Eddington
luminosities (Dennerl 1989; Woo {\it et~al.}~1995). The necessary $\dot M$ is
3.2~$\times~10^{-7}~M_{\odot}$~yr$^{-1}$
and implies a range of a factor of 100 in $\dot M$ for LMC X-4.
By contrast the flaring states of Z-source LMXBs such as Sco X-1 and Cyg X-2,
where the
flaring state is also associated with super-Eddington accretion,
are produced with changes of only a factor of 2-4 in $\dot M$; this is because
Sco X-1 and Cyg X-2 are disk fed systems normally accreting
at just under the Eddington limit.
It is possible that the greater range and overall greater $\dot M$ required for
LMC X-4 is due to some intrinsic difference between
X-ray binaries in the Magellanic Clouds
compared with systems within our own
Galaxy. The mean luminosity of those sources in the Clouds which have massive
OB-type companions similar to LMC X-4 is
50 times that of the counterparts
in our Galaxy (van Paradijs \& McClintock 1995). It has been
suggested that this is due to the lower abundances of metals in the
Clouds. One linking mechanism
is the effect of X-ray heating of gas as it
falls toward a compact object: this depends strongly on the atomic
number Z via the photoelectric cross section ($\sigma \propto Z^4$).
For spherical accretion, such heating can seriously impede the
accretion flow and thereby reduce the limiting luminosity to a value
far below the Eddington limit; so the LMC sources may be more luminous because
their low-Z accretion flow is less impeded by heating.
A second metallicity dependent effect
takes place in wind-fed systems. The accretion rate depends sensitively
on the velocity of the stellar wind; all available evidence indicates that
the terminal wind velocity decreases with Z. Such behavior is predicted
by successful theories of radiation-driven stellar winds (Kudrizki \&
Hummer 1990; Kudritzki~{\it et~al.}~1991).
In this case the lower metallicity in the LMC means a lower terminal velocity,
a higher $\dot M$ and a more luminous X-ray source.
The dramatic orbital variations shown by the N~V profiles can be
interpreted in terms of X-ray photoionization of the stellar wind
from the companion.
The narrow absorption lines shown in Figure~4d
can be attributed to the
surface of the companion star
and the broad lines to
the wind. In this scenario,
the X-ray source
ionizes nearly the entire stellar wind that is not in the shadow of
the companion star, so that
when the neutron star is in front of the normal star,
the wind absorption disappears and mainly the photospheric absorption lines
are visible.
It is only near phose 0.0, i.e. during X-ray eclipse, that the broad
lines that reveal the high wind velocities become visible.
Detailed fits to the line profiles with a model that takes into account
the structure of the wind
and its influence on spectral features will be presented in a later paper
(Boroson {\it et~al.}~1997a).
The pulse amplitude in the X-rays is lower in LMC X-4
(the pulsed fraction of LMC X-4 is 10\% in quiescence)
than in Her~X-1 and Vela~X-1.
In Her~X-1 the X-ray pulsed fraction is close
to 50\% and
UV~continuum pulsations have an amplitude of $\sim0.5$\% (Boroson {\it et~al.}~1996).
If LMC X-4 resembles Her X-1 than we would expect UV~continuum pulsations
with an amplitude of $\sim0.1$\%, which is below our detection limit.
The narrow wavelength regions surrounding the
lines provided too low a count rate to detect
pulsations similar to those seen in other systems.
In Vela X-1 the pulsed fraction
for 2-10 keV X-ray is 0.3, and, while no continuum pulsations were detected in
the UV using FOS observations, 3\% pulsations in the P-Cygni lines were
detected.
P~Cygni line pulsations,
seen in Vela~X-1, may be smeared by light travel time in LMC~X-4,
owing to a more extended
X-ray photoionized region.
The STIS, recently installed on HST, extends in several significant
ways the
ultraviolet capabilities that became available with the GHRS:
with the echelle grating it is possible to sample continuously a broad
region (600$\AA\/$) of the spectrum at greater spectral
resolution than with the GHRS. Observations with the STIS that cover binary
phase 0.0 can confirm our interpretation of the line profiles, provide
improved measures of the continuum,
and enable a more sensitive search for UV manifestation of the
X-ray pulses.
Until recently, long-term approximately-periodic variability attributed
to disk precession
has been
known for only three systems: Her X-1, LMC X-4, and SS433. XTE
observations suggest similar long-term behavior in SMC X-1 and
Cyg X-2 (Levine {\it et~al.}~1996; Wijnands, Kuulkers, \& Smale 1996).
The interpretation of these periods in terms of disk precession has been
questioned
by several authors including, e.g., Kondo {\it et~al.}~1983, who showed that
precession of a disk controlled by the gravitational fields of the
neutron and companion stars is untenable: any induced precession disppears
rapidly because of differential precession within the disk.
Recently Iping and Petterson (1990) proposed that the
behavior attributed to precession is maintained by
the influence of the X-ray emission on the structure of the disk.
Accordingly, in these systems, the tendency of disks to undergo periodic
changes in their orbital
orientations is more appropriately termed radiation-driven warping.
Iping and Petterson's primary result, that strong central illumination can
maintain disk warping, has been obtained
analytically by Pringle (1996) and Maloney, Begelman, \& Pringle (1996).
The success of our X-ray heated star plus warped disk model in
fitting the optical, UV, and X-ray lightcurves for both the orbital
and long-term periods of Her X-1 and LMC X-4
supports an interpretation in terms of radiation-driven warping.
\vskip 0.2in
Based on observations with the NASA/ESA {\it Hubble Space Telescope},
obtained at the Space Telescope Science Institute, which is operated
by the Association of Universities for Research in Astronomy, Inc.,
under NASA contract GO-05874.01-94A.
We are grateful for quick-look results provided by the ASM/RXTE team.
SDV and BB were supported in part by NASA
(NAG5-2532, NAGW-2685), and NSF
(DGE-9350074).
\newpage
\normalsize
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\newpage
\centerline{\bf Figure Captions}
\vskip 0.1in
{\bf Figure 1.} (a) One day averages of the flux observed from
LMC X-4 with the All Sky Monitor
on board the ROSSI X-ray Timing
Explorer. (Quick-look results were provided by the ASM/RXTE team.)
The arrow indicates the time of the simultaneous HST/ASCA observations,
and the horizontal bar represents one 30.25 day interval.
(b) The data from (a) (light lines)
with the lightcurve from (c) (dark lines) superposed.
(c) The data from (a) folded with the ephemeris for the long-term
period provided by Dennerl {\it et~al.}~1992 ( P$_{30}$ = 30.25+/-.03d and
$\phi^0_{30}$ = JD 2,448,226.0). Arrow indicates the phase
of the
simultaneous HST/ASCA observations.
The errors are 1$\sigma$ from counting statistics.
\vskip 0.1in
{\bf Figure 2.}
(a) The exposure times
of the HST observations which covered the binary phases
0.08-0.49 (start time = JD 2,450,228.44). The first observation
is N~V and is alternated with C~IV.
(b) ASCA lightcurves (0.5-10 keV): observations
covered the binary phases 0.75-1.84 (start time =
JD 2,450,227.92) determined from the ephemeris of
Woo {\it et~al.}~(1996) with P$_{orb}$ = 1.40840249$\pm6.0\times10^{-7}$d and
$\phi^0_{orb}$ = JD 2,446,729.84878$\pm$.0041d.
\vskip 0.1in
{\bf Figure 3.} (a) Average UV continuum flux near C~IV {\it vs.} binary phase.
The solid squares represent the GHRS observations. The open triangles
are archival IUE data. The dashed line represents a 30.25 day phase when
the X-ray flux is in the low state; the solid line a 30.25d phase during
an X-ray high state; and the
dotted line represents the best
fit continuum for the GHRS observations.
(b) The B magnitude predictions of the model for the three states depicted
in 3a. (c) The B magnitude
predictions for a given $\dot M$ during the
X-ray on and off states. (b) and (c) should be compared with
Figure 1 in Ilovaisky~{\it et~al.}~1984.
\vskip 0.1in
{\bf Figure 4.} The GHRS observations of N~V and C~IV.
\vskip 0.1in
{\bf Figure 5.} The pulse period history of LMC X-4.
\newpage
\centerline{\bf Table 1: Log of GHRS Observations}
\small
\begin{center}
\begin{tabular}{cccccc}\hline
HST&&Orbital&&Wavelength&Continuum\\
Observation & Start time & phase$^1$ & Duration & range&Flux\\
Number&(JD-2,450,000)&at midpoint&(s)&(\AA)&
(ergs~cm$^{-2}~$s$^{-1}~\AA\/^{-1}$)\\
\hline
\hline
Z3AA0104T & 228.4428 & 0.111 & 4352 & 1222.5-1258.6 &
4.364$\times 10^{-13}\pm$0.008$^{2}$\\
Z3AA0106T & 228.5083 & 0.159 & 4704 & 1532.3-1567.5&
2.673$\times 10^{-13}\pm$0.004$^{3}$\\
Z3AA0108T & 228.5798 & 0.209 & 4484 & 1225.5-1258.6&
4.564$\times 10^{-13}\pm$0.008$^{2}$\\
Z3AA010AT & 228.6511 & 0.259 & 4424 & 1532.3-1567.5&
2.761$\times 10^{-13}\pm$0.004$^{3}$\\
Z3AA010CT & 228.7223 & 0.312 & 4423 & 1225.5-1258.6&
4.457$\times 10^{-13}\pm$0.008$^{2}$\\
Z3AA010ET & 228.7926 & 0.360 & 4543 & 1532.3-1567.5&
2.600$\times 10^{-13}\pm$0.004$^{3}$\\
Z3AA010GT & 228.8619 & 0.410 & 4721 & 1225.5-1258.6&
4.177$\times 10^{-13}\pm$0.007$^{2}$\\
Z3AA010IT & 228.9285 & 0.465 & 6628 & 1532.3-1567.5&
2.411$\times 10^{-13}\pm$0.004$^{3}$\\
\hline
\end{tabular}
\end{center}
$^1$Using the ephemeris of
Woo {\it et~al.}~(1996) with the nth eclipse in JD given by
a0~+~a1*n~+~a2*n$^2$
where a0~=~JD~2,446,729.84878$\pm$0.0041,
a1~=~1.40840249$\pm~6.0~\times~10^{-7}$ days, and
a2~=~-1.45E-9. $\phi^0_{orb}$~=~a0 and $P_{orb}$~=~a1~+~a2*n.\\
$^2$Flux near N~V ([1226-1234$\AA\/$]+[1255-1259$\AA\/$]).\\
$^3$Flux near C~IV ([1532-1544$\AA\/$]+[1556-1567$\AA\/$]).\\
\vskip 0.2in
\vfill\eject
\newpage
\section{FIGURES}
\centerline{\psfig{figure=figure1.eps}}
\newpage
\centerline{\psfig{figure=figure2.eps}}
\newpage
\centerline{\psfig{figure=figure3.eps}}
\newpage
\centerline{\psfig{figure=figure4.eps}}
\newpage
\centerline{\psfig{figure=figure5.eps}}
\end{document}
|
2,869,038,156,395 | arxiv | \section{Introduction}
Networks are present in human life in multiple forms from social networks to communication networks (such as the Internet), and they have been widely studied as complex networks of nodes and relationships, describing their structure, relations, etc. Nowadays, studies go deep into network knowledge, and using standard network metrics such as degree distribution and diameter one can determine how robust and/or resilient a network is.
Even though both terms (robustness and resilience) have been used with the same meaning, we will consider \textit{robustness} as the network inner capacity to resist failures, and \textit{resilience} as the ability of a network to resist and recover after such failures. However, both terms have in common the methodology for testing robustness and/or resilience. Usually, they consist on planned attacks against nodes failures or disconnections, from a random set of failures to more elaborated strategies using well known network metrics.
We consider that an ``adversary'' should plan a greedy strategy aiming to maximize damage with the minimum number of strikes. We present a new network impact metric called the \textit{Hayastan Shakarian} index (HS), which reflects the size of the larger connected component after removing a network edge. Then, we plan a greedy strategy called the \textit{Hasyastan Shakarian} cut (HSc), which is to remove the edge with higher HS value, aiming to partition the network in same-sized connected components, and recalculating the network \textit{HS}s after each cut. We discuss the performance of attacks based on \textit{the Hayastan Shakarian cut} and the edge betweenness centrality metric \cite{bersano2012metrics}, compared by the robustness index ($R$-index). Our main conclusion is that the first strikes of the strategy \textit{the Hayastan Shakarian} cut strategy causes more ``\textit{damage}'' than the classical betweenness centrality metric in terms of network disconnection ($R$-index).
The article is organized as follows, next section presents related work, followed by the definition of \textit{the Hayastan Shakarian cut}, its attacking strategy, and its simulation results in Section \ref{miuz}. Conclusions are presented in Section \ref{conclusions}.
\section{Related Work}
\label{related}
Over the last decade, there has been a huge interest in the analysis of complex networks and their connectivity properties~\cite{Albert2000}. During the last years, networks and in particular social networks have gained significant popularity. An in-depth understanding of the graph structure is key to convert data into information. To do so, complex networks tools have emerged~\cite{Albert2002} to classify networks~\cite{Watts1998}, detect communities~\cite{Leskovec2008}, determine important features and measure them~\cite{bersano2012metrics}.
Concerning robustness metrics, betweenness centrality deserves special attention. Betweenness centrality is a metric that determines the importance of an edge by looking at the paths between all of the pairs of remaining nodes. Betweenness has been studied as a resilience metric for the routing layer~\cite{smith2011network} and also as a robustness metric for complex networks \cite{iyer2013attack} and for internet autonomous systems networks~\cite{mahadevan2006internet} among others. Betweenness centrality has been widely studied and standardized as a comparison base for robustness metrics, thus in this study it will be used for performance comparison.
Another interesting edge-based metrics are: the increase of distances in a network caused by the deletion of vertices and edges \cite{krishnamoorthy1990incremental}; the mean connectivity \cite{tainiter1975statistical}, that is, the probability that a network is disconnected by the random deletion of edges; the average connected distance (the average length of the shortest paths between connected pairs of nodes in the network) and the fragmentation (the decay of a network in terms of the size of its connected components after random edge disconnections) \cite{Albert2000}; the balance-cut resilience, that is, the capacity of a minimum cut such that the two resulting vertex sets contain approximately the same number (similar to the HS-index but aiming to divide the network only in halves \cite{tangmunarunkit2002network}, not in equal sized connected components); the effective diameter (``\textit{the smallest $h$ such that the number of pairs within a $h$-neighborhood is at least $r$ times the total number of reachable pairs}'' \cite{palmer2001connectivity}); and the Dynamic Network Robustness (DYNER) \cite{singer2006dynamic}, where a backup path between nodes after a node disconnection and its length is used as a robustness metric in communication networks.
The idea of planning a ``network attack'' using centrality measures has captured the attention of researchers and practitioners nowadays. For instance, Sterbenz et al.~\cite{sterbenz2011modelling} used bet\-ween\-ness-centrality (\textit{bcen}) for planning a network attack, calculating the \textit{bcen} value for all nodes, ordering nodes from higher to lower \textit{bcen}, and then attacking (discarding) those nodes in that order. They have shown that disconnecting only two of the top \textit{bcen}-ranked nodes, their packet-delivery ratio is reduced to $60\%$, which corresponds to~$20\%$ more damage than other attacks such as random links or nodes disconnections, tracked by link-centrality and by node degrees.
A similar approach and results were presented by {\c{C}}etin\-kaya et al.~\cite{ccetinkaya2013modelling}. They show that after disconnecting only $10$ nodes in a network of $100+$ nodes the packet-delivery ratio is reduced to $0\%$. Another approach, presented as an improved network attack \cite{rak2010survivability, sydney2010characterising}, is to recalculate the betweenness-centrality after the removal of each node \cite{holme2002attack,molisz2006end}. They show a similar impact of non-recalculating strategies but discarding sometimes only half of the equivalent nodes.
In the study of resilience after edge removing, Rosenkratz et al. \cite{rosenkrantz2009resilience} study backup communication paths for network services defining that a network is ``\textit{k-edge-failure-resilient if no matter which subset of k or fewer edges fails, each resulting subnetwork is self-sufficient}'' given that ``\textit{the edge resilience of a network is the largest integer k such that the network is k-edge-failure-resilient}''.
For a better understanding of network attacks and strategies, see \cite{holme2002attack,molisz2006end,rak2010survivability, sydney2010characterising}.
\section{The Hayastan Shakarian Cut}
\label{miuz}
Given a network $\mathcal{N}$ of size $N$, we denote by $\mathcal{C}(\mathcal{N} \setminus e)$ the set of connected components in $\mathcal{N}$ after disconnecting edge~$e$. The \textit{Hayastan Shakarian} index (HS-index) for an edge $e$ in $\mathcal{N}$ is defined as follows:
\begin{equation}
{\scriptsize
\textit{HS}_\mathcal{N}(e) =
\left\{
\begin{array}{l}
\frac{\sum_{\textit{c} \in \mathcal{C} (\mathcal{N}\setminus e)}{\|\textit{c}}\|}{max_{\textit{c} \in \mathcal{C} (\mathcal{N}\setminus e)}\|\textit{c}\|} - 1, \textit{if } \|\mathcal{C} (\mathcal{N}\setminus e)\| \neq \|\mathcal{C} (\mathcal{N})\| \\
\\
0 \hfill \textit{otherwise}
\end{array}
\right.
}
\end{equation}
\noindent where $\|c\|$, with $\textit{c} \in \mathcal{C} (\mathcal{N}\setminus e)$, is the size of the connected component $c$ of the network $\mathcal{N}$ after disconnecting edge $e$. Notice that $\textit{HS}_\mathcal{N}$($e$) reflects the partition of a network in several sub-networks after the disconnection of edge $e$ and how these sub-networks remain interconnected. Strictly speaking, it compares in size the core network (the largest connected component) with the other remaining sub-networks. $\textit{HS}_{\mathcal{N}}$ takes values between $0$ (the whole network remains connected) to $N-1$ (the whole network is disconnected).
Then, we define that the \textit{Hayastan Shakarian cut} as the one that disconnects the node with the highest \textit{HS-index} value (when there is more than one, we choose it at random). That is:
\[
\textit{HSc} = \mathcal{N}\setminus \hat{e}, \textit{where}~\hat{e} = \textit{max} (\textit{HS}_\mathcal{N}(e)) ~\forall ~e \in \mathcal{N}
\]
\begin{figure}[htp]
\begin{center}
\subfigure[Original network]{\includegraphics[width=.55\linewidth]{original_graph.png}}
\subfigure[HS removing] {\includegraphics[width=.45\linewidth]{sin_max_miuz.png}} \hfill
\subfigure[BC removing]{\includegraphics[width=.45\linewidth]{sin_max_bet.png}}
\end{center}
\caption{Effects of removing edges by (b) its maximal HS, and (c) its maximal betweenness centrality value.}
\label{fig:figure1}
\end{figure}%
\subsection{Network Attack Plan}
If we plan a network attack by disconnecting nodes with a given strategy (e.g. Figure \ref{fig:figure1}), it is widely accepted to compare it against the use of betweenness centrality metric, because the latter reflects the importance of an edge in the network \cite{iyer2013attack}. These attack strategies are compared by means of the \textit{Unique Robustness Measure} ($R$-index)~\cite{schneider2011mitigation}, defined as:
\begin{equation}
R = \frac{1}{N}\sum_{Q=1}^{N} {s(Q)},
\end{equation}
where $N$ is the number of nodes in the network and $s(Q)$ is the fraction of nodes in the largest connected component after disconnecting (remove the edges among connected components) $Q$ nodes using a given strategy. Therefore, the higher the $R$-index, the better in terms of robustness. Our strategy is:\\
In each step, (re)compute the \textit{HS} index for all the edges, and perform the \textit{Hayastan Shakarian} cut.\\
\subsection{Simulations}
In an attempt to study the behavior of the internet backbone, we test our strategy in simulated power-law degree distributions with exponents $\alpha \in \{1.90,2.00,2.05,2.10,2.20\}$ using Gephi. A power-law distribution is one that can be approximated by the function $f(x) =k*x^{-\alpha}$, where $k$ and $\alpha$ are constants. For each $\alpha$, we simulate $50$ networks of size $1000$ (i.e., with $1000$ nodes).
Then, we tested and compared strategies ranked in terms of the $R$-index.
Instead of just comparing the robustness, after the removal of all of the nodes, we studied the behavior of the attacks after only a few strikes. To do so, we define a variant of the $R$-index which takes into account only the first $n$ strikes of an attack. Thus, for a simultaneous attack (where the nodes are ranked by a metric only once at the beginning), the $R_n$-index is defined as:
\begin{equation}
R_n = \frac{1}{n}\sum_{Q=1}^{n} {s(Q)}.
\end{equation}
For a sequential attack, the order of node disconnection is recomputed after each disconnection. Similar to the $R$-index, notice that the lower the $R_n$-index, the more effective the attack is, since that gives us a higher reduction of robustness.
Results are shown in Figure~\ref{fig:figureX}. We tested sequential attacks: At each strike, the next node to disconnect was the one with the highest metric (whether it be \textit{HSc} or betweenness centrality) in the current network. The figure shows the behavior of the $R_n$-index in 50 scale-free networks (generated as before) with exponent 1.9 to 2.2. The \textit{Hayastan Shakarian} cut proves to be very effective in attacks with only a few disconnections. Moreover, it shows that the effectiveness persists over the number of strikes in scale-free networks with lower exponent.
It is interesting to note that, no matter the metric used, the damage decreases in the long term with the number of strikes. Revisiting the definition of network robustness, that is, \textit{``a measure of the decrease of network functionality under a sinister attack''}\cite{holme2002attack}, it is important to notice that $R$-index is a metric for a general view of robustness, and it gives no information of how fast the network is disconnected. We suggest that weighted versions of the $R$-index could be a good compromise. This is the case of the $R_n$-index proposed above which takes into account the removal of only a percentage of the best ranked nodes (or until the network reaches its percolation threshold \cite{holme2002attack}) could be a more useful metric for robustness and/or resilience in worst-case scenarios.
We start analyzing what would be the worst ``attack''. In the worst case, a ``\textit{malicious adversary}'' will try to perform the maximum damage with the minimum number of strikes, that is, with the minimum number of node disconnections (made by the attacker).
In terms of decreasing the size of the largest connected component, the \textit{Hayastan Shakarian cut} is the best attack strategy during the first strikes. It achieves the disconnection of more than a quarter of the network only after $200$ strikes for $\alpha=2.10$ and after $150$ strikes for $\alpha=2.05$ (see Figure \ref{fig:figureX}). It is important to notice that the greater the value of $\alpha$ the lower the number of strikes needed by the \textit{Hayastan Shakarian cut} for reducing the original network to a connected component of $0.75$ of the original size.
\begin{figure*}[thb!]
\begin{center}
\subfigure[$\alpha=1.9$]{\includegraphics[width=.195\linewidth]{hs19.png}}
\subfigure[$\alpha=2.0$]{\includegraphics[width=.195\linewidth]{hs20.png}}
\subfigure[$\alpha=2.05$]{\includegraphics[width=.195\linewidth]{hs205.png}}
\subfigure[$\alpha=2.1$]{\includegraphics[width=.195\linewidth]{hs21.png}}
\subfigure[$\alpha=2.2$]{\includegraphics[width=.195\linewidth]{hs22.png}}
\end{center}
\caption{Size of the largest fractional connected component (Y-axis) vs. number of disconnected edges (X-axis) using as strategies the Hayastan Shakarian cut (blue) and edge betweenness centrality (red).}
\label{fig:figureX}
\end{figure*}
\subsection {Revisiting the Hayastan Shakarian cut}
Concerning the attacking strategies, it is important to notice that the \textit{Hayastan Shakarian cut} was designed for disconnecting the network from the first strikes, aiming to get connected components with similar sizes. This is the main reason of its better performance compared to the betweenness centrality metric in a worst-attack scenario. For instance, in Figure \ref{fig:figureX} we present examples where the \textit{HS} cut performs better attacks than an strategy using the betweenness centrality metric for different values of $\alpha$. Notice that the \textit{Hayastan Shakarian cut} performs better than the edge betweenness centrality strategy for values $\alpha < 2.1$, disconnecting the former more than the half of the network in less strikes than the latter.
Given that for a high coefficient (of the power law) the network would have a densely connected core with a few nodes, and many nodes of degree $1$. In this case a \textit{Hayastan Shakarian cut} will not be as effective as an edge betweenness attack since the \textit{HS}-index of the nodes in the core is probably $0$ (because is highly connected), performing a \textit{Hayastan Shakarian cut} over those edges attached to nodes of degree $1$.
On the other hand, for a low coefficient (of the power law) the proportion of high degree nodes and low degree nodes is more similar than those of a high coefficient, this means the network is well connected even if it is not densely connected. In this case the \textit{Hayastan Shakarian cut} will not cause that much damage (because the network is well connected) but it still causes more damage than edge betweenness, because edge betweenness chooses the edges with high traffic, and since the network is well connected it has many paths between any two nodes. Instead the \textit{Hayastan Shakarian cut} will keep choosing the edges that causes more nodes to be disconnected.
Betweenness ``generates'' weak points by removing the edges with high traffic until there are no more alternative paths between 2 nodes. On the other hand the \textit{Hayastan Shakarian cut} finds weak points that are already on the network (before removing a single edge, see Figure: \ref{fig:figure1}). Therefore, the \textit{Hayastan Shakarian cut} will not produce a network necessarily weaker, but a more fragmented and with stronger fragments.
It is important to notice that there is a breaking point where \textit{Hayastan Shakarian cut} is no longer the best attacking strategy. As shown in Figure \ref{fig:figureX}, this breaking point appears after the largest connected component is less than $1/4$ of the original network. While an edge betweenness attack starts attacking the edges in the core with high traffic, this will not produce loss of nodes in the beginning but after a number of edge removals, making the network weak and vulnerable to the point where a single additional edge removal may greatly reduce the fractional size of the largest component, in our examples, of around $0.1$. After the breaking point and for the same number of edges removed by the \textit{Hayastan Shakarian cut} the fractional size of the largest component will be higher than the one produced by a edge betweenness attack.
Another interesting property of $\textit{HS}$-index occurs when $\textrm{max} (\textit{HS}_\mathcal{N}(e)) = 0$. In other words, when the \textit{Hayastan Shakarian} index is 0 for any node. In that case, the full network will remain connected no matter which edge is disconnected. Therefore, we suggest a resilience metric as the number (or percentage) of disconnected nodes until $\textrm{max} (\textit{HS}_\mathcal{N}(e)) > 0$, a pre-defined threshold, or the percolation threshold \cite{holme2002attack}.
\section{Conclusions and Future Work}
\label{conclusions}
In this article we have presented the \textit{Hayastan Shakarian cut} (HSc), a robustness index for complex networks defined as the inverse of the size of the remaining largest connected component divided by the sum of the sizes of the remaining connected components.
We tested our index as a measure to quantify the impact of node removal in terms of the network robustness metric $R$-index. We compared \textit{HSc} with other attacks based on well-known centrality measures (betweenness centrality) for node removal selection. The attack strategy used was sequential targeted attack, where every index is recalculated after each removal, and the highest one is selected for the next extraction.
Preliminary results show that the \textit{Hayastan Shakarian cut} performs better compared to a betweenness centrality-based attack in terms of decreasing the network robustness in the worst-attack scenario (disconnecting more than the half of the network) for values of $\alpha < 2.1$. We suggest that the \textit{Hayastan Shakarian cut}, as well as other measures based on the size of the largest connected component, provides a good addition to other robustness metrics for complex networks.
As future work, we can study improving through new connections an already existing network to make it more robust to attacks, and we can study networks in which the topological space is correlated with the cost of nodes disconnections, nodes which are nearby have a lower cost to be disconnected compared with nodes which are far apart.
\bibliographystyle{abbrv}
|
2,869,038,156,396 | arxiv | \section{Introduction}
History seems to have separated much of chemistry \cite{brush1976kind,garber1995maxwell} from the classical theory of fields \cite{simpson1998maxwell,zangwill2013modern,chree1908mathematical}. Chemical reactions are found throughout the ionic solutions of biology and chemistry but they are usually described in a language apparently disjoint from that of classical field theory even though the reactants and products of chemical reactions are almost always charged and carry significant electrical current. The reactants, catalysts and enzymes of chemistry and biology depend on charge interactions for much of their function.
Field theory has much to offer chemistry, particularly in the study of charged systems
as admirably reviewed in \cite{RN46052}.
The Maxwell equations are as universal and precise as any theory and apply to chemical reactions in the plasmas of gases and ionic solutions, and liquids in general. Indeed, the Maxwell equations can be written without any adjustable parameters, implying the universal conservation of total current \cite{eisenberg2019kirchhoff,eisenberg2020electrodynamics}. A theory of the electromechanical response of charge to the electric
field is needed in that case to make a complete description of the charged system, i.e. an electromechanical theory of polarization
phenomena \cite{wang2021variational}.
But some properties of the electromagnetic field (e.g., conservation
of total current) are true entirely independent of the electromechanical response. The special systems that are almost completely described by conservation of current (without specification of charges) include the electronic circuits of our computers and many properties of the action potential of nerve and muscle fibers. Both systems are almost entirely specified by Kirchhoff's current law. The question then arises how do we fit chemical reactions into the framework of Kirchhoff's law and conservation of total current that is so remarkably simpler to implement than a full accounting of all the charges in a chemical system?
Here we show one way to describe chemical reactions in ionic solutions with an extension of classical field theory that does not violate the traditions of either chemistry or electrodynamic field theory. In this work, we take advantage of the energy variation method \cite{xu2018osmosis,shen2020energy,shen2022energy} that treats ionic solutions as complex fluids, with interactions, internal stored energy, flow, and dissipation like other complex fluids
\cite{eisenberg2010energy,chen2020differential,liu2020molecular,li2020generalized,giga2017variational}. It begins with defining two functionals for the total energy and dissipation of the system, and introducing the kinematic equations based on physical laws of conservation. The specific forms of the flux and stress functions in the kinematic equations are obtained by taking the time derivative of the total energy functional and comparing with the defined dissipation functional. More details of this method can be found in \cite{shen2020energy}.
We use energy variation methods to link the electric field and reaction dynamics as Wang, et al, \cite{wang2020field,RN46052} have for reactions that do not involve charge or electrodynamics.
The generalization to charged systems allows us to study systems of some importance. We study active transporters of biological membranes. In a sense, we extend the electrical treatment of the membrane proteins called channels to deal with transporters. Conservation of current (in the form of Kirchhoff's law) provides the crucial coupling between the properties of disjoint sodium and potassium channel proteins that act independently in the atomic and molecular sense because they are well screened. Channels are many Debye lengths apart, shielded from each other by the ions, water dipoles (and quadrupoles), that also form the ionic atmosphere of proteins and lipid bilayer. The atomic scale function of these proteins is crucial to their biological function. Their coupling is just as important to their function, but the coupling of these channel proteins is not chemical. The coupling is provided by the cable equation \cite{eisenberg1975electrophysiology}
as biophysicists have called the telegrapher equation version of the Maxwell equations (and Kirchhoff's law) used by Kelvin to design the Atlantic cable
\cite{thompson1855theory}
well before Maxwell wrote his equations.
Active transport is one of the most important processes in life. It maintains the concentration gradients and membrane potentials that allow biological cells to function. Indeed, without active transport animal cells swell, burst, and die. Active transport powers the generation of ATP in both animals (in mitochondria, by oxidative phosphorylation) and plants (in chloroplasts, by photosynthesis). ATP is the currency of chemical energy in all plants and animals, storing in a small organic compound the energy from photosynthesis or oxidation. When hydrolyzed to ADP, the chemical energy is available for the myriad of dissipative processes essential to life. Nearly all of them use ATP as their energy source. Life is complex with many facets. The energy source of life is not.
The coupling of flows in transporters in the inner membrane of mitochondria allows one substance to move uphill (against its gradient of electrochemical potential) using the energy from the downhill movement of another substance. Coupling is inherently about the relationship of flows, yet most analysis and simulations of active transporters do not explicitly include a variable for flow. Most do not use the electrodynamic equations for flow, whether conservation of total current or Kirchhoff’s law or a nonlinear version of Ohm’s law. Indeed, most analysis and simulations use methods that are derived assuming zero flow and do not include changes in potential or free energy associated with flow. The problems with this approach become apparent if one tries to analyze electron flow through a resistor or semiconductor diode or rectifier in that tradition.
We analyze an active transporter by studying the currents through it, as a specific example of our general field theory for ionic flows with chemical reactions. We combine Kirchhoff’s law and chemical reaction energetics with diffusion, migration, and convection of ions and water to make an 'electro-osmotic' model. The name is chosen to emphasize the important role of electricity in this system, implying the need to deal with electrical flows by methods used to deal with electrical flows in other systems, like electrical and electronic circuits. This approach seems sensible for the systems like oxidative phosphorylation and photosynthesis where electron flows are involved. The analysis of electron flows has been well established in physics and engineering for more than a century. The analysis of ionic flows has been well established in membrane biophysics for some seventy years. We combine them here with chemical reactions hoping to construct a useful electro-osmotic theory of cytochrome $c$ oxidase.
We do not have to deal with the myriad of charges involved in the transport of current or the incalculable number of interactions of those charges. Analysis of current flow is all we use in this conservative coupling approach, following the practice of circuit analysis. We do not have to assume equilibrium or zero flow. We do not have to deal explicitly with the charges involved in the transport. Analysis of current flow is all we use. In particular, we do not assume equilibrium or near equilibrium flows, as has been done in previous analysis. Note that near equilibrium analysis (using the Green-Kubo formalism, for example) is inappropriate for devices that function with large flows. These devices are not near equilibrium. The electronic devices of our digital technology function far from equilibrium and they are not analyzed by assuming nearly zero flow. Indeed, they usually use power supplies to maintain spatially nonuniform potentials, often described by inhomogeneous Dirichlet boundary conditions. Traditionally, electronic devices are analyzed by studying small changes around a nonequilibrium operating point, which we hasten to add is maintained by large---not small--- flows from the power supplies. But even that linearization is not necessary nowadays because the full flow nonequilibrium problems can be solved conveniently with readily available software.
We are certainly not the first to exploit the simplification provided by the analysis of current instead of charge. Circuit designers have used this approach 'forever' \cite{bush1929operational,RN45733}.
Charges are hardly mentioned when circuits are designed as a glance at textbooks shows \cite{RN45934,RN25358,RN21598,RN45662,RN28108,RN26593,RN45935}, perhaps most eloquently in symbolic circuit design \cite{RN45999}. Analysis of current---not charge---characterizes the study of ion channels since they were discovered as conductances by Hodgkin and Huxley some seventy years ago \cite{RN267,RN309,RN28957,RN12551,RN28654,RN291,}.
Analysis of current is not a prominent feature of the study of active transporters, however, even in the work of Hodgkin and collaborators that started the physiological analysis of active transport in cell membranes (not whole epithelia or biochemical preparations), at much the same time as their work defining channels \cite{RN45995,RN9363,RN10470}.
Most models of active transport include conformational changes of the protein that require a model to compute the spatial distribution of mass in the protein, as it changes during active transport \cite{RN29619,RN29611,}.
The conformational changes usually provide \textbf{alternating} access to an occluded state that is not connected (or accessible) to either side of the protein. The occluded state blocks the conduction path (and incidentally often traps ions in the 'middle' of the transporter) and thus prevents backflux. Alternating access mechanisms create flux across the transporter protein without allowing backflux that would seriously degrade the efficiency of the transporter. Transporters were first thought to use quite different mechanisms from channels \cite{RN434,RN155}.
However, recent work \cite{RN23} shows otherwise. Alternating access is apparently created by correlated motions of gates that account for activation and inactivation in classical voltage activated sodium channels \cite{RN29619,RN29607}. The physical basis of the gates is not discussed in the classical literature. Later in the paper, we speculate that the switches that provide alternating access might be like the switches of bipolar transistors. .
In this work, we use an electro-osmotic approach to describe cytochrome $c$ oxidase (or Complex IV). We choose cytochrome $c$ oxidase because experimental work and simulations of the highest quality have shown that “cytochrome $c$ oxidase is a remarkable energy transducer [i.e., coupled transporter of electrons and protons] that seems to work almost purely by Coulombic principles without the need for significant protein conformational changes” \cite{RN45750}. Cytochrome $c$ oxidase depends on an "occluded state" containing the reaction center(s) to prevent backflow as do other active transporters, but it uses some type of the 'water-gate' mechanism proposed in \cite{RN4995,RN45750}. The alternating access in cytochrome $c$ oxidase occurs without conformation change (of the spatial distribution of mass), in marked contrast to the usual alternating access models of transporters. Perhaps the gate in the oxidase is like the switch in a semiconductor (diode) rectifier. The switches might be rectifiers produced by spatial distributions of permanent charge, of opposite signs, as rectification (and switching) is produced in PN diodes, and bipolar transistors. Diode rectifiers depend on changes in shape (i.e., conformation) of the electric field, not changes in the distribution (i.e., conformation) of mass. This idea is outlined in the Discussion Section following \cite{RN24,RN15949,RN6246}.
The rest of the paper is organized as follows. In section 2, we derive the general three dimensional field equations for an ionic system with reaction and use it to create a general framework of an electro-osmotic model. In section 3, we propose a specific, simplified model for cytochrome $c$ oxidase. In section 4, we carry out computational studies of our cytochrome $c$ oxidase model and explore the effects of various conditions on the transport of protons across the mitochondria membrane. In section 5, we conclude our paper with a discussion of our cytochrome $c$ oxidase model and future directions.
\section{Derivation of Electro-osmotic Model }
We mainly focus on a mathematical model of elementary reactions
\begin{equation}\label{reacton_de}
\alpha_1 C_1^{z_1}+\alpha_2 C_2^{z_2}+\alpha_3 C_3^{z_3}\overset{k_f}{\underset{k_r}{\rightleftharpoons}} \alpha_4 C_4^{z_4},
\end{equation}
where $k_f$ and $k_r$ are two constants for forward and reverse directions, $[C_i]$ is the concentration of $i^{th}$ species, respectively. Here $\alpha_i$ is the stoichiometric coefficient, $z_i$ is the valence of $i^{th}$ species and together they satisfy
\begin{equation}\label{zrelation}
\sum_{i=1}^3 \alpha_iz_i = \alpha_4z_4.
\end{equation}
In particular, we have in mind a case where an active transporter ('pump') uses the energy supplied by a chemical reaction to pump molecules. Later, we will focus on the reaction for cytochrome $c$ oxidase, i.e., for Complex IV of the respiratory chain
\begin{equation}
2 H^{+}+\frac 1 2 O_2+2e^{-}\overset{k_f}{\underset{k_r}{\rightleftharpoons}} H_2O.
\end{equation}
According to the conservation laws, we have the following conservation of chemical elements (like sodium, potassium and chloride). Note that this conservation is in addition to the conservation of mass, because nuclear reactions that change one element into another are prohibited in our treatment, as in laboratories and most of life.
\begin{subequations}\label{conservation of element}
\begin{align}
&\frac{d}{dt} (\alpha_4 [C_1] +\alpha_1 [C_4])=0,\\
&\frac{d}{dt} (\alpha_4 [C_2] +\alpha_2 [C_4])=0,\\
&\frac{d}{dt} (\alpha_4 [C_3] +\alpha_3 [C_4])=0.
\end{align}
\end{subequations}
In order to derive a thermal dynamical consistent model, the Energy Variation Method \cite{shen2020energy} is used.
Based on the laws of conservation of elements and Maxwell equations, we have the following kinematic system
\begin{equation}\label{assumption_de}
\left\{
\begin{array}{l}
\frac{d [C_1]}{d t} =-\nabla\cdot \boldsymbol{j}_1 -\nabla\cdot \boldsymbol{j}_p-\alpha_1 \mathcal{R},\\
\frac{d [C_2]}{d t} = -\nabla\cdot \boldsymbol{j}_2 -\alpha_2\mathcal{R},\\
\frac{d [C_3]}{d t} =-\nabla\cdot \boldsymbol{j}_3 - \alpha_3 \mathcal{R},\\
\frac{d [C_4]}{d t} =-\nabla\cdot \boldsymbol{j}_4 + \alpha_4 \mathcal{R},\\
\nabla\cdot(\bD) = \sum_{i=1}^4z_i[C_i]F,\\
\nabla\times \bE = \boldsymbol{0},
\end{array}
\right.
\end{equation}
where $\boldsymbol{j}_l, l=1,2,3,4$ are the passive fluxes and $\boldsymbol{j}_p$ is the pump flux, $\mathcal{R}$ is reaction rate function. All these variables are unknown and will be derived by using the Energy Variational method.
$\bj_{ex}$ is the flux of electrons supplied from an external source. In mitochondiral membranes this will include special pathways linking one enzyme and one complex to another by the movement of lipid soluble or water soluble electron donors and water soluble electron donors or acceptors like the quinones.
$\bD$ is Maxwell's electrical displacement field and $\bD = \varepsilon_0\varepsilon_r \bE$ with electric field $\bE$, dielectric constant $\varepsilon_0$ and relative dielectric constant $\varepsilon_r$.
The equation $ \nabla\times \bE = \boldsymbol{0}$
implies that there exists a $\phi$ such that $\bE = -\nabla \phi$.
We consider a system with structure and boundary conditions defined on that structure.
The structures are given to us by structural biologists. The structures are decorated with molecules (proteins and lipids for the most part) that use particular atomic arrangements to channel physical forces into physiological function. We describe a constant flux for one species, say $C_3$, that serve as the input of the system. In cytochrome $c$ oxidase the input is electrons carried on the heme groups of cytochrome oxidase.
\begin{equation}\label{bd_flux}
\left\{
\begin{array}{ll}
\boldsymbol{j}_i\cdot\boldsymbol{n} = j_{i,extra}, i=1\cdots, 4, & \mbox{on~} \partial\Omega,\\
\boldsymbol{D}\cdot\boldsymbol{n} = 0, & \mbox{on~} \partial\Omega.
\end{array}\right.
\end{equation}
\begin{rmk}
By multiplying $z_i e$ on both sides of the first three equations and $-e$ on both sides of the fourth equation, we have
\begin{eqnarray}
\frac{d}{dt}(\nabla \cdot\bD) &=&\sum_{i=1}^4z_iF\frac{d[C_i]}{dt} \\
&=& -\sum_{i=1}^4\nabla\cdot(z_iF\bj_i)-z_iF\nabla\cdot\bj_p- (z_1\alpha_1+z_2\alpha_2+z_3\alpha_3-z_4\alpha_4)F\mathcal{R}\\
&=& -\sum_{i=1}^4 \nabla\cdot(z_iF\bj_i)-z_iF\nabla\cdot\bj_p,
\end{eqnarray}
which is consistent with the electrostatic Maxwell equations. Treatment of transient problems, involving displacement currents is needed to deal with some important experimental work \cite{RN30714,RN30642,RN6359,RN30008,RN30276,RN45724}.
\end{rmk}
The total energetic functional is defined as the summation of entropies of mixing, \cite{RN46053}
internal energy and electrical static energy.
\begin{eqnarray}\label{totalenergy}
E_{tot} &=& E_{ent}+E_{int}+E_{ele}\nonumber\\
&= &\sum_{i=1}^4 \int_{\Omega}RT\left\{ [C_i]\left(\ln{\left(\frac{[C_i]}{c_0}\right)}-1 \right)\right\} dx+\int_{\Omega} \sum_{i=1}^4[C_i]U_idx
+
\int_{\Omega} \frac{\boldsymbol{D}\cdot\boldsymbol{E}}{2}dx.
\end{eqnarray}
Then the chemical potentials can be calculated from the variation of total energy
\begin{equation}
\tilde\mu_l=\frac{\delta E_{tot}}{\delta [C_i]} = RT\ln\frac{[C_i]}{c_0} +U_i +z_l\phi e, l=1,\cdots, 4.
\end{equation}
It is assumed in the present work that dissipation of the system energy is due to passive diffusion, chemical reaction and the pump. Accordingly, the total dissipation functional $\Delta$ is defined as follows
\begin{eqnarray}
\Delta = \int_{\Omega}\left\{\sum_{j=1}^4|\bj_i|^2+ RT\mathcal{R}\ln \left(\frac{\mathcal{R}}{k_{r}\left(\frac{[C_4]}{c_0}\right)^{\alpha_4}}+1\right)\right\} dx -\int_{\Omega} f_p dx,
\end{eqnarray}
where $f_p = f_p(\mathcal{R}, \mu,x)\ge 0 $ is the term from the pump.
Open systems in which some fluxes flow in or out, entering or leaving the system altogether, have distinctive energy dissipation laws that differ from those of closed systems. The natural mitochondrion is an UNclamped system, in which the electrical potential assumes whatever value satisfies the field equations. The sum of all currents across the membrane of the natural mitochondrion is zero (including the capacitive displacement current) as it is in small biological cells. Many experiments are done in voltage clamped systems. In these the sum of the currents does not equal zero just as the sum of currents in the classical Hodgkin Huxley experiments was not zero. Of course, the ratio of fluxes will be different in the clamped and unclamped cases, as we document at length later in this paper.
In the natural Unclamped mitochondrion, we have the following generalized energy dissipation law
\begin{equation}
\frac{dE_{tot}}{dt} = J_{E,\partial\Omega}-\Delta.
\end{equation}
Here $J_{E,\partial\Omega}$ is the rate of boundary energy absorption or release that measures the energy of flows that enter or leave the system altogether through the boundary.
Recall that the chemical potential of a species is the energy that can be absorbed or released due to a change of the number of particles of the given species and $J_i\cdot n $ is the total number of $i^{th}$ particles passing through the boundary, per area per unit time. We define $J_{E,\partial\Omega}$ as follows
\begin{equation}\label{J_Edefinition}
J_{E,\partial\Omega}=\int_{\partial\Omega} \sum_{i=1}^4 \tilde{\mu}_i \bj_i\cdot\boldsymbol{n} dS.
\end{equation}
In general, different types of boundary conditions can be written in the following general format
\begin{equation}
\bj_i\cdot\boldsymbol{n} = g_i(f([C_i])-f([C_i]_{extra})),
\end{equation} where $g_i$ is the conductance of $i^{th}$ species on the boundary, $[C_i]_{extra}$ is the fix reservoir's concentration of $i^{th}$ species and $f$ is some specific function. Then the rate of boundary energy absorption or release is
\begin{equation}
J_{E,\partial\Omega}= \int_{\partial\Omega} \sum_{i=1}^4g_i\tilde{\mu}_i(f([C_i])-f([C_i]_{extra})).
\end{equation}
In this case energy can change both because of the flux across the boundary and also because of the change in dissipation.
and
\begin{equation}
\frac{dE}{dt}-J_{E,\partial\Omega} = -\Delta.
\end{equation}
\begin{rmk} Boundary Conditions, Structure, Evolution, and Engineers
These boundary conditions serve as the link
between general field equations and structures
that serve as devices. Structures are chosen
and devices designed (by evolution or engineers) so these boundary conditions
are satisfied. The boundary conditions are chosen so devices have almost the same properties no matter where they are placed in a network. The structures and boundary conditions on those structures are not automatic properties of nature. The structures are decorated with (i.e., include) specific substructures (like power transistors) that exploit arrangements of atoms (like doping charges) to create properties that are useful. The properties are summarized by boundary conditions located on the structures provided by evolution and engineers. These boundary conditions help make the idea of a component useful. They help ensure that
a component in one part of a system does what it
does in another part of the system and so can be
described by a 'transfer function' independent of
its location in the system.
It is clear that channels
and transporters in biological systems behave as components.
Indeed, most of classical physiology and biophysics is devoted
to identifying such components, on a wide variety of length scales from
atoms to organisms, and studying how they interact in the hierarchy
of structures that make animals and plants \cite{RN7109,RN45615,RN28910,RN26033,RN45909,RN45950,RN134,RN295}.
\end{rmk}
\begin{rmk}
\begin{enumerate}
\item A closed system allows no flux across the boundaries. It has the following no-flux boundary conditions
\begin{equation}\label{bd_noflux}
\left\{
\begin{array}{ll}
\boldsymbol{j}_i\cdot\boldsymbol{n} = 0~ i =1,2,3,4, & \mbox{on~} \partial\Omega,\\
\boldsymbol{D}\cdot\boldsymbol{n} = 0, & \mbox{on~} \partial\Omega.
\end{array}\right.
\end{equation}
In a closed system, $J_{E,\partial\Omega} = 0$ and the energy dissipation law is
$$\frac{dE_{tot}}{dt} = -\Delta.$$
In a closed system, the energy changes into dissipation. That is the only way energy can change in a closed system.
\item An open system has flow across the boundaries. An open system might have constant inflow/outflow
$\bj_i\cdot\boldsymbol{n} = J_{i,extra} $.
In that case,
$g_i = \frac{J_{i,extra}}{f([C_i])-f([C_i]_{extra})}$, and
$$J_{E,\partial\Omega} =\int_{\partial\Omega}\sum_{i}^4 \tilde{\mu}_i J_{i,extra} dS .$$
\item For the Dirichlet boundary condition $[C_i]=[C_i]_{extra}$ on $\partial\Omega$, the flux $\bj_i\cdots\boldsymbol{n}$ is unknown and part of the solution. In this case, $J_{E,\partial\Omega}$ is an unknown flux needed to ensure that the Dirichlet condition $[C_i]=[C_i]_{extra}$ is obeyed on $\partial\Omega$.
It is very important to understand this requirement. In reality, i.e., in experiments and their models, supplying the unknown flux requires specialized instrumentation, for example, a patch clamp amplifier in a voltage clamp setup. Almost always, that flux is supplied at one location in space. In that way a classical voltage clamp can be established. However, if one wishes to "clamp" a field, one must control the potential at many locations. Each location requires a different flux and thus a different amplifier and different electrodes to supply that flux. Without such complicated apparatus, it is almost impossible to maintain a constant field in space \cite{han1993superconducting}.
Indeed, it is nearly impossible to maintain any pre-specified field because it is practically impossible to apply different fluxes at different locations. If one assumes a constant field in a theory, without such apparatus in an experiment, one is in effect introducing flux into the calculation and model that is not present in the experimental setup. One is introducing an artifactual flux likely to produce artifactual conclusions that are not relevant to the original experiment. \cite{eisenberg2010computing,eisenberg1998ionic}.
\end{enumerate}
\end{rmk}.
By taking the time derivative of total energy function \eqref{totalenergy}, we have
\begin{eqnarray}
\frac{dE_{tot}}{dt} &=&\int_{\Omega}\sum_{i=1}^4\left\{\mu_i \frac {d[C_i]}{dt}\right\}dx +\int_{\Omega} \bE\cdot\frac{d\bD}{dt}dx\nonumber\\
&=&\int_{\Omega}\sum_{i=1}^4\left\{\mu_i \frac {d[C_i]}{dt}\right\}dx -\int_{\Omega}\nabla \phi\cdot\frac{d\bD}{dt}dx\nonumber\\
&=& \int_{\Omega}\sum_{i=1}^4\left\{\mu_i \frac {d[C_i]}{dt}\right\}dx+\int_{\Omega} \phi\nabla\cdot\left(\frac{d\bD}{dt}\right)dx\nonumber\\
&=& \int_{\Omega}\sum_{i=1}^4\left\{\mu_i \frac {d[C_i]}{dt}\right\}dx +\int_{\Omega} \phi F\sum_{i=1}^4\left\{z_i \frac {d[C_i]}{dt}\right\} dx\nonumber\\
&=& \int_{\Omega} \sum_{i=1}^4\left\{\tilde{\mu}_i \frac {d[C_i]}{dt}\right\}dx\nonumber\\
&=&-\int_{\Omega}\sum_{i=1}^4\left\{\tilde{\mu}_i\nabla\cdot \boldsymbol{j}_i\right\}dx -\int_{\Omega}\tilde{\mu}_1 \nabla\cdot \bj_p dx
- \int_{\Omega}\mathcal{R}(\sum_{i=1}^3\alpha_i \tilde{\mu}_i-\alpha_4\tilde{\mu}_4 )dx\nonumber \nonumber\\
&=&\int_{\Omega}\sum_{i=1}^4\left\{\nabla\tilde{\mu}_i\cdot \boldsymbol{j}_i\right\}dx+\int_{\Omega}\nabla\tilde{\mu}_1 \cdot \bj_p dx -\int_{\Omega}\mathcal{R}(\sum_{i=1}^3\alpha_i \mu_i -\alpha_4 \mu_4)dx+\int_{\partial\Omega} \sum_{i=1}^4 \mu_i \bj_i\cdot\boldsymbol{n} dS,
\end{eqnarray}
where $\mu_i = RT\ln\frac{[C_i]}{c_0} +U_i$ and Eq. \eqref{zrelation} is used.
By comparing with the dissipation functional, we have
\begin{subequations}
\begin{align}
&\bj_i = -\frac{D_i}{RT}[C_i]\nabla\tilde\mu_i,~ i=1,2,3, \label{fluxi} \\
&RT\ln\left(\frac{\mathcal{R}}{k_{r}\left(\frac{[C_4]}{c_0}\right)^{\alpha_4}}+1\right) = \sum_{i=1}^3\alpha_i \mu _i -\alpha_4 \mu_4\label{reactionrateele}.
\end{align}
\end{subequations}
And the corresponding energy influx rate is
\begin{equation}
J_E = \sum_{i=1}^4\int_{\partial\Omega} \tilde{\mu}_i j_{i,extra}.
\end{equation}
For the pump flux, if we assume flux is only along the $z$ direction, then,
\begin{equation}
\bj_p = (0,0,\frac{f_p}{\partial_z\mu_1}). \label{pumpmu1}
\end{equation}
At equilibrium, we have
\begin{equation}
\left\{\begin{array}{l}
\bj_i = \nabla [C_i]_{eq}+\frac{z_iF}{RT}[C_i]_{eq}\nabla\phi_{eq} =0,\nonumber\\
\sum_{i=1}^3\alpha_i \mu_i([C_i]_{eq} ) -\alpha_4\ \mu_4([C_4]_{eq} ) = 0,
\end{array}
\right.
\end{equation}
The last equation means
\begin{eqnarray}
0 = RT \ln \left(\frac{\Pi^3_{i=1} (\frac{[C_i]_{eq}}{c_0})^{\alpha_i}}{(\frac{[C_4]_{eq}}{c_0})^{\alpha_4}}\right) +(\sum_{i=1}^3\alpha_i U_i -\alpha_4 U_4) ,\label{equilibriumcondition}
\end{eqnarray}
According to the definition of equilibrium constant $k_{eq}$,
\begin{equation}
k_{eq} = \frac{\Pi^3_{i=1} (\frac{[C_i]_{eq}}{c_0})^{\alpha_i}}{(\frac{[C_4]_{eq}}{c_0})^{\alpha_4}}
\end{equation}
Eq. \eqref{equilibriumcondition} yields
\begin{equation}\label{keqele}
\ln{k_{eq}} = e^{-\frac{\Delta U}{RT}},
\end{equation}
with $\Delta U = \sum_{i=1}^3\alpha_i U_i -\alpha_4 U_4$ .
Then combining Eqs. \eqref{reactionrateele} and \eqref{keqele} yields
\begin{eqnarray}
\ln\left(\frac{\mathcal{R}}{k_{r}\left(\frac{[C_4]}{c_0}\right)^{\alpha_4}}+1\right) &=& \ln \left(\frac{\Pi_{i=1} \left(\frac{[C_i]}{c_0}\right)^{\alpha_i}}{\left(\frac{[C_4]}{c_0}\right)^{\alpha_4}}\right),
\end{eqnarray}
which implies
\begin{equation}
\mathcal{R} = k_f\left(\frac{[C_1]}{c_0}\right)^{\alpha_1}\left(\frac{[C_2]}{c_0}\right)^{\alpha_2}\left(\frac{[C_3]}{c_0}\right)^{\alpha_3}-k_{r}\left(\frac{[C_4]}{c_0}\right)^{\alpha_4},\nonumber
\end{equation}
where $k_f = \frac{k_r}{k_{eq}}$ \cite{wang2020field}.
\begin{rmk}
Here $k_{eq}$ is dimensionless. $k_r$ and $k_f$ are with unit $s^{-1}$ \cite{ozcan2022equilibrium}.
\end{rmk}
Then the whole system is as follows
\begin{equation}\label{model_ele}
\left\{
\begin{array}{l}
\frac{d [C_1]}{d t} =\nabla\cdot (D_1 \nabla [C_1]+D_1\frac{z_1F}{RT}[C_1]\nabla\phi) -\partial_z j_p-\alpha_1 \mathcal{R},\\
\frac{d [C_2]}{d t} =\nabla\cdot (D_2 \nabla [C_2]+D_2\frac{z_2F}{RT}[C_2]\nabla\phi)-\alpha_2 \mathcal{R},\\
\frac{d [C_3]}{d t} =\nabla\cdot (D_3 \nabla [C_3]+D_3\frac{z_3F}{RT}[C_3]\nabla\phi)-\alpha_3\mathcal{R},\\
\frac{d [C_4]}{d t} =\nabla\cdot (D_4 \nabla [C_4]+D_4\frac{z_4F}{RT}[C_4]\nabla\phi)+\alpha_4 \mathcal{R},\\
-\nabla\cdot(\varepsilon_0\varepsilon_r \nabla \phi) = \sum_{i=1}^4z_iF[C_i],
\end{array}
\right.
\end{equation}
with
\begin{eqnarray}
\mathcal{R} = k_f\left(\frac{[C_1]}{c_0}\right)^{\alpha_1}\left(\frac{[C_2]}{c_0}\right)^{\alpha_2}\left(\frac{[C_3]}{c_0}\right)^{\alpha_3}-k_{r}\left(\frac{[C_4]}{c_0}\right)^{\alpha_4}.
\end{eqnarray}
and boundary conditions
\begin{equation}\label{bd_flux}
\left\{
\begin{array}{ll}
\boldsymbol{j}_i\cdot\boldsymbol{n} = j_{extra}, i=1\cdots 4, & \mbox{on~} \partial\Omega,\\
\boldsymbol{D}\cdot\boldsymbol{n} = 0, & \mbox{on~} \partial\Omega.
\end{array}\right.
\end{equation}
\begin{rmk}
If we assume that one of the reactants is an electron, for instance $C_3$,
and supplied by an thin electrode along z direction, the density of electron $[C_3]=\rho_e = \rho(z)\delta(x_0, y_0)$. Then the model is changed to
\begin{equation}\label{model_eletron}
\left\{
\begin{array}{l}
\frac{d [C_1]}{d t} =\nabla\cdot (D_1 \nabla [C_1]+D_1\frac{z_1e}{RT}[C_1]\nabla\phi) -\partial_{z}(j_p) -\alpha_1 \mathcal{R}\delta(x_0, y_0),\\
\frac{d [C_2]}{d t} =\nabla\cdot (D_2 \nabla [C_2]+D_2\frac{z_2e}{RT}[C_2]\nabla\phi)-\alpha_2 \mathcal{R}\delta(x_0, y_0),\\
\frac{d [C_4]}{d t} =\nabla\cdot (D_4 \nabla [C_4]+D_4\frac{z_4e}{RT}[C_4]\nabla\phi)+\alpha_4 \mathcal{R}\delta(x_0, y_0),\\
\frac{d\rho(z)}{dt} = -\partial_z \bj_e -\alpha_3\mathcal{R}\delta(x_0, y_0),\\
-\nabla\cdot(\varepsilon_0\varepsilon_r \nabla \phi) = \sum_{i=1,2,4}z_ie[C_i] -F\rho(z)\delta(x_0, y_0).
\end{array}
\right.
\end{equation}
\end{rmk}
\begin{rmk} When the reaction and ions are in an electrolyte, the fluid effect needs to be taken into consideration. In this case, the energy functional is changed to be
\begin{eqnarray}
E_{tot} &=&E_{kin}+ E_{ent}+E_{int}+E_{ele}\nonumber\\
&= &\int_{\Omega}\frac{\rho |\mathbf{u}|^2}2 dx +\sum_{i=1}^4 \int_{\Omega}RT\left\{ [C_i]\left(\ln{\left(\frac{[C_i]}{c_0}\right)}-1 \right)\right\} dx+\int_{\Omega} \sum_{i=1}^4[C_i]U_idx
\end{eqnarray}
and the dissipation functional is changed to
\begin{eqnarray}
\Delta = \int_{\Omega}2\eta|\boldsymbol{D}_{\eta}|^2 dx + \int_{\Omega}\left\{\sum_{j=1}^4|\bj_i|^2 +RT\mathcal{R}\ln \left(\frac{\mathcal{R}}{k_{r}\left(\frac{[C_4]}{c_0}\right)^{\alpha_4}}+1\right)\right\} dx -\int_{\Omega} f_p dx,
\end{eqnarray}
where $\boldsymbol{D}_{\eta} = \frac{\nabla\boldsymbol{u} + (\nabla\boldsymbol{u})^T}{2}$ and $\boldsymbol{u}$ is the velocity.
We can use the Energy variation method to get the diffusion-reaction-convection model as follows
\begin{equation}\label{model_ele_full}
\left\{
\begin{array}{l}
\frac{\partial [C_1]}{\partial t} +\nabla\cdot([C_1]\boldsymbol{u}) =\nabla\cdot (D_1 \nabla [C_1]+D_1\frac{z_1F}{RT}[C_1]\nabla\phi) -\partial_z j_p-\alpha_1 \mathcal{R},\\
\frac{\partial [C_2]}{\partial t} +\nabla\cdot([C_2]\boldsymbol{u}) =\nabla\cdot (D_2 \nabla [C_2]+D_2\frac{z_2F}{RT}[C_2]\nabla\phi)-\alpha_2 \mathcal{R},\\
\frac{\partial [C_3]}{\partial t} +\nabla\cdot([C_3]\boldsymbol{u}) =\nabla\cdot (D_3 \nabla [C_3]+D_3\frac{z_3F}{RT}[C_3]\nabla\phi)-\alpha_3\mathcal{R},\\
\frac{\partial [C_4]}{\partial t} +\nabla\cdot([C_4]\boldsymbol{u})=\nabla\cdot (D_4 \nabla [C_4]+D_4\frac{z_4F}{RT}[C_4]\nabla\phi)+\alpha_4 \mathcal{R},\\
-\nabla\cdot(\varepsilon_0\varepsilon_r \nabla \phi) = \sum_{i=1}^4z_iF[C_i], \\
\rho (\frac{\partial\boldsymbol{u} }{\partial t} +(\boldsymbol{u}\nabla)\cdot \boldsymbol{u} +\nabla p= \nabla(\eta (\nabla\boldsymbol{u}+ (\nabla\boldsymbol{u})^T)) -(\sum_{i=1}^4 z_iF[C_i])\nabla\phi\\
\nabla\cdot \boldsymbol{u} = 0.
\end{array}
\right.
\end{equation}
\end{rmk}
Note we are not here considering transient problems in which charge is stored in polarization fields. These will be studied separately so we can deal with the important experiments reported in \cite{RN30714,RN30642,RN6359,RN30008,RN30276,RN45724}. Transient problems are obviously important if reactions are studied on the atomic scale of distance and time (angstroms and femtoseconds) because the polarization currents are large. Dealing with those currents requires use of a universal form of the Maxwell equations combined with an appropriate model of the stress strain relation of charge in a viscoelastic structure, commonly called polarization. Speaking loosely, the transient problems can be dealt with in circuits by a generalization of Kirchhoff’s law
\cite{RN45998,eisenberg2019kirchhoff} to describe the actual transient currents that flow through an ideal resistor \cite{eisenberg2018current}.
It is important to realize that currents (and fluxes) cannot be computed by methods that assume the
currents and fluxes are zero. Electrostatics cannot compute currents because currents and fluxes involve time and electrostatics does not \cite{sugitani2008theoretical}. Electrostatics does not include Maxwell's Ampere law that is the universal coupler of current to electric and magentic fields. In the context of cytochrome $c$ oxidase these issues come to the fore. Models without electron or proton current as variables do not describe the 'transfer function' of the transporter being studied. Models cannot calculate Ohm's law (for system with large and small currents and electrical potentials) if the models assume currents are zero.
In fact, using a formulation of electrodynamics that explicitly
involves current is straightforward, as engineers have known for a very long time, going back to Heaviside \cite{belevitch1962summary,darlington1984history} and are worked out in practical detail in \cite{RN45998}.
Kirchhoff's current law allows analysis of systems of great importance, without dealing with charges explicitly. That is why analysis of electronic circuits does not need to use distributions of charges but rather uses Kirchhoff's current law or its generalization, conservation of total current.
Kirchhoff's current law is an exact corollary of the Maxwell equations themselves, if current includes the displacement current
\cite{RN45998,eisenberg2018current,eisenberg2019kirchhoff}.
It might seem that another corollary of the Maxwell equations, the continuity equation, can be used instead of Kirchhoff's law for total current. And it is certainly true that the continuity equation of electrodynamics contains
the same information as conservation of (total) current, all conjoined with
the Maxwell equations. But that information is not useful when enormous numbers of charges are involved as in cytochrome $c$ oxidase or in other macroscopic scale systems like the electronic circuits of our computers. The information implicit in the flux of charges is only usable when written as total current that is conserved perfectly whenever the Maxwell equations are valid. This formulation using conservation of total current
does not require explicit treatment of charges. The continuity equation does require the explicit treatment of charges, and their significant interactions, whether involving two charge interactions, three charge, .... or the interactions of an entire cluster expansion. The significant interactions of charge are difficult to understand or even enumerate and more difficult to compute \cite{RN46056,RN46055}. Kirchhoff's current law is easy to understand and trivial to compute.
\section{An Electro-osmotic Model of cytochrome $c$ oxidase}
Here we propose a specific model of cytochrome $c$ oxidase (or Complex IV) as an example so our approach can be seen in action.
The schematic structure of cytochrome $c$ is shown in Fig.\ref{fig:Schematic} a, where both two channels from mitochondria matrix (inside), D and K, are taken into consideration. Here, $E$ denotes the end of $D$ channel. And the end of K channel is asumped to be the binuclear center (BNC) denoted by $B$ where the chemical reaction \eqref{HOreaction} occurs. The protons accumulated in E are transported to the BNC and the proton loading site (PLS), denoted by X. A pump is located between E and PLS. The pump provides energy that comes from concentration gradients, namely gradients of chemical potential at BNC. Then finally, the proton is pushed out from PLS to the inter membrane space, outside the mitochondrion.
It is clear that
this model is incomplete at best, and in some sense wrong, at worst. We depend on our experimental colleagues to help us correct and improve the model, for example, by including mechanisms we have over simplified. Enormous detail of the
chemical reactions is described in the literature, with more intermediates being reported frequently. We do not include these intermediates.
Let $\mathcal{\rho}_e = \rho_0\delta(x_0,y_0,z_0, t)$.
Integrating the diffusion-reaction equation
\begin{equation}
\frac{d [C]_i}{dt} = -\nabla\cdot \bj_i -\alpha_i \mathcal{R},
\end{equation}
in the complex IV compartment yields
\begin{equation}
\eta\frac {d\bar{C}_i}{dt} = J_i^{in} - J_i^{out}-\alpha_i \mathcal{R},
\end{equation}
where $\eta_{mat}$, $
\eta_{ims}$ and $\eta$ are the volumes of mitochondrial compartment, inter membrane space and reaction compartment, respectively.
The chemical reaction in the cytochrome $c$ oxidase Complex IV is
\begin{equation}\label{HOreaction}
2 H^{+}+\frac 1 2 O_2+2e^{-}\overset{k_f}{\underset{k_r}{\rightleftharpoons}} H_2O.
\end{equation}
\begin{figure}[!ht]
\centering
\begin{subfigure}[]
{\includegraphics[width=3.in]{Figure/schematicnstructure.png}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=0.45\textwidth]{Figure/schematicn2e.png}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=0.45\textwidth]{Figure/schematicx2p.png}}
\end{subfigure}
\caption{Schematic. (a) Complex IV structure; (b)
Circuit diagram of model \eqref{current_model} - \eqref{curentdef}; (c) Circuit diagram when the rectifier is located between the protein loading site PLS and P side. The drawing of (a) is taken from \cite{belevich2010initiation} according to the policies of the Proceedings of the National Academy of Sciences \text{https://www.pnas.org/author-center/publication-charges\#author-rights-and-permissions} examined on June 18, 2022. We thank the authors for providing all of us such a helpful figure.
} \label{fig:Schematic}
\end{figure}
For simplicity, we follow the Hodgkin Huxley tradition and fix the proton concentrations in the mitochondrial matrix (inside) and the inter membrane space outside the mitochondria so they do not vary with time or flow. More general treatments in which concentrations are changed with time by flow are possible as have been done in even more complex structures, Such analysis has been done in a bi-domain model of the lens of the eye and a tri-domain model of the optic nerve and glia \cite{RN28691,RN30228,RN30228,RN30601,RN30649}.
Here for simplicity, we assume that the concentration of oxygen at the B site is a constant.
If the
oxygen varies with time, an additional equation can be used to describe the dynamics of oxygen. The properties of that term will be
determined either directly by experimentation or by a
higher resolution model as in \cite{yamashita2012insights}. We do not expect the extra term to introduce significant mathematical, numerical, or computational difficulties.
Many variables are needed to keep track of all the potentials and concentrations in the variouis regions of our model. The concentrations and potentials at E242; at BNC; and at the proton loading site (PLS), are different inside and outside the mitochondria, as is the electron concentration (see Fig.\eqref{fig:Schematic} (a)). They are described by the variables
$[H]_E$, $[H]_B$, $[H]_x$,$\phi_E$, $\phi_B$, $\phi_X$, $\phi_N$, $\phi_P$ and $\rho_e$, respectively.
\begin{subequations}\label{current_model}
\begin{align}
&\frac{d[H]_E} {dt} =\frac {S_v} F(I_{N2 E}-I_{E2 X}-I_{E2 B}),\label{HEequation} \\
&\frac{d[H]_B} {dt}= \frac {S_v} F( I_{E2 B}+I_{N2 B})-2\mathcal{R},\label{HBquation} \\
&\frac{d[H]_X} {dt}= \frac {S_v} F( I_{E2 X}-I_{X2 P} ),\label{HXquation} \\
&\frac{d\rho_e}{dt} = \frac {-S_v} F I_e-2\mathcal{R}, \label{rhoequation}\\
&C_E \frac{d(\phi_E-\phi_N)}{dt} = (I_{N2 E}-I_{E2 X}-I_{E2 B}), \\
&C_B \frac{d(\phi_B-\phi_N)}{dt} = I_{E2 B} +I_{N2 B}+I_e, \\
&C_X \frac{d(\phi_X-\phi_P)}{dt} = ( I_{E2 X}-I_{X2 P}),\\
& C_m\frac{d(\phi_N-\phi_P)}{dt} +I_{leak} +I_{X2 p}+I_e = 0,
\end{align}
\end{subequations}
with currents
\begin{subequations}\label{curentdef}
\begin{align}
& I_{N2 B} = g_K(\phi_N-\phi_B-\frac{RT}{F}\ln\frac{[H]_B}{[H]_N})=\frac{g_K}{F}(\mu_N-\mu_B),\\
&I_{E2 B} = g_B(\phi_E-\phi_B-\frac{RT}{F}\ln\frac{[H]_B}{[H]_E})=\frac{g_B}{F}(\mu_E-\mu_B), \\
&I_{E2 X} =I_{pump} +I_{xleak},\\
&I_{leak} = g_m (\phi_N-\phi_P -E_{other} ), \\
&I_e = -FJ_e,\\
&I_{xleak} = -g_E(\mu_X-\mu_E),\\
& I_{pump} = \left\{
\begin{array}{cc}
g_{pump}max(R_c,0) (\mu_X-\mu_E), \mu_X-\mu_E< \delta_{th}, \\
g_{pump}max(R_c,0)\delta_{th} \exp{\left(- \frac{(\mu_X-\mu_E)}{\varepsilon}\right)}, \mu_X-\mu_E\ge \delta_{th}, \end{array}
\right.\label{pumpcurrent}\\
&\mathcal{R} = k_f[H^+]^{2}[O_2]^{1/2}\rho_e^2-k_{r}[H_2O].\label{reaction}
\end{align}
\end{subequations}
We follow the review of Wikström \cite{RN45750} and implement switching functions without invoking conformation changes of the distribution of mass. We treat cytochrome c oxidase as a Coulombic system and use rectifiers to implement the switching functions that provide alternating access of an occluded state.
Here we discuss two cases. In one case, a rectifier between $N$ and $E$ blocks the proton flows. In the other case, a rectifier between $X$ and $P$ blocks the backward proton flows.
Then, the currents $I_{N2 E}$ and $I_{X2 P}$ are modelled in the follows two cases.
\begin{itemize}
\item Case 1: the rectifier is between $N$ and $E$ as shown in Fig. \ref{fig:Schematic}b
\begin{subequations}\label{rect_n2e}
\begin{align}
& I_{N2 E} = max\left(g_D \left(\phi_N-\phi_E-\frac{RT}{F}\ln\frac{[H]_E}{[H]_D}\right), -SW_{0}\right)=max\left(\frac{g_D}{F}(\mu_N-\mu_E),-SW_{0}\right)\label{swith},\\
&I_{X2 P} = g_X(\phi_X-\phi_P-\frac{RT}{F}\ln\frac{[H]_P}{[H]_X})=\frac{g_X}{F}(\mu_X-\mu_P),
\end{align}
\end{subequations}
\item Case 2: the rectifier is between $X$ and outside as shown in Fig. \ref{fig:Schematic}c
\begin{subequations}\label{rect_x2p}
\begin{align}
& I_{N2 E} = g_D \left(\phi_N-\phi_E-\frac{RT}{F}\ln\frac{[H]_E}{[H]_D}\right) \label{swith2},\\
&I_{X2 P} =max\left( g_X(\phi_X-\phi_P-\frac{RT}{F}\ln\frac{[H]_P}{[H]_X}),-SW_{0}\right)=max\left(\frac{g_X}{F}(\mu_X-\mu_P),-SW_{0}\right),
\end{align}
\end{subequations}
\end{itemize}
where $SW_{0}$ is the threshold for the turn-off of the rectifier and $SW_{0}=0$ stands for perfect rectifier. We reiterate that pn junctions are used to rectify the movement of the pseudo-ions holes and electrons throughout our digital circuitry. Analogous distributions of permanent charge provided by acid and base side chains of proteins produce rectification of charge movement in ionic systems.
We use Kirchhoff's law and the conductance formulation of Hodgkin and Huxley. Complex properties are hidden by a nonlinear time dependent version of Ohm's law and modelled by a conductance, as did Hodgkin and Huxley \cite{RN267,RN309,RN28957,RN12551,RN28654,RN291,}.
Alternating access (with its implied occluded state) is described by a switching function for the D channel using equation \eqref{swith}. This is a classical rectifier function and (when $SW_0 =0$), allows current only to flow from D to E: no backward flow is allowed.
Many properties of the model depend on the pump current between E242 and the Proton Loading Site PLS. We assume that in the ordinary situation the pump strength depends on the reaction rate and the chemical reaction difference between two sites. However, in the less ordinary situation, when the difference is too large, the pump may not be able to overcome the barrier. A turn-off threshold is assigned to the pump for that reason. We assume that when the difference in chemical potential $\mu_x - \mu_E$ is greater than the threshold, the pump current decreases exponentially to zero as it turns off. More realistic, and complex properties of the pump will undoubtedly be needed to explain some functions of cytochrome c oxidase. They can easily be incorporated into our model, as these properties are measured and modelled.
Our `electro-osmotic' model is a `master equation' approach building on the work of Hummer and
Kim, \cite{kim2012proton,kim2007kinetic,kim2009kinetic}
but showing how to exploit conservation of current in the form of Kirchhoff's current law. This approach is used throughout electrical and electronic engineering to design semiconductor devices, as textbooks document ($op. cit.$) perhaps most eloquently in the modern automated circuit design literature built on Kirchhoff's law \cite{RN45999}. Currents are sufficient for such automated design. Charges are not needed except in switched-capacitor networks (p. 64 of \cite{RN45999}) .
We extend the classical use of Kirchhoff's law that forms the foundation of circuit design to include chemical reactions. We must include chemical reactions to drive currents of electrons, protons and other ions because that is how cytochrome $c$ oxidase functions. The essential function of cytochrome $c$ oxidase is to convert a flow of electrons to a flow of protons from inside the mitochondrion to outside it. The electrons that are inputs to the cytochrome $c$ oxidase are presented to the enzyme attached to the heme group of cytochrome $c$ itself.
The existing literature analyzes these systems without explicitly dealing with currents, making the task either impossible (by using a theory that assumes a zero value for the fluxes being studied) or very difficult (by involving a staggering number of charges). Using currents instead of charges avoids these difficulties and has the added advantage of automatically satisfying Maxwell's equations, if total current is used in Kirchhoff's law.
This approach is incomplete because it does not deal with all the charge in the circuit formed by cytochrome $c$ oxidase But \textbf{those details of charge are not needed in the design of electronic circuits.} That simple fact can be verified by examining textbooks of circuit design (as already cited).
In circuit analysis of this type, some questions about charges need not be asked: for example, the atomic mechanism of current flow (particularly electron flow) can be ignored. The function of the circuit is independent of the details of the components of current in wires, for example, with only a few exceptions \cite{RN45998}. Thus, we de-emphasize the atomic details of the various pathways that provide electron flow to the main reaction centers. For us, these pathways are wires. The atomic and chemical details of electron flow in these wires are known in breathtaking detail and we are sorry that we do not seem to need to use these magnificent results, but, at this resolution, we do not.
A key biological result is that some of the coupling so important to understand the electro-osmotic properties of a mitochondrion depends
on the macroscopic conservation of current, i.e., Kirchhoff's law applied to the entire mitochondrion. The application of Kirchhoff's law to mitochondrial transport, and active transport in general, is not common in the literature. But Kirchhoff's law has been used in another branch of biophysics for a long time, for some eighty years. Kirchhoff's law is the keystone of the analysis of ion channels. Kirchhoff's law is the keystone that supports the structure of the Hodgkin approach to the action potential by balancing the various components of current, summing them to zero
in the appropriate (finite) geometries, like those of mitochondria, the way the keystone of an arch sums mechanical forces.
Conservation of current provides the coupling in other biophysical applications, e.g., generation and
propagation of the action potential, linking atomic scale properties of ion channels of one type
to properties (e.g., opening) of another type.
In a classical action potential, the opening of sodium channels is coupled to the opening of other sodium channels, and to the closing of potassium channels by the electric field, not by anything else. There is no steric or chemical interaction between the channels. The coupling is essential to the function of the nerve cell, but that coupling is described by a version of the Maxwell equations (called the cable or telegrapher's equation) not by equations of chemical kinetics. The ion channels of the action potential act independently in the chemical sense because they are so far apart, without opportunity for short range or chemical interactions. The ion channels are not independent, in the physiological or physical sense, however. Rather they are coupled by the electric field. The electrical field is that which satisfies the Maxwell equations, or their equivalent, Kirchhoff's current law.
What we propose here is in the tradition of Hodgkin's treatment of ionic channels, but we
include the chemical reactions that are the essence of oxidative phosphorylation and the life and function of mitochondria.
We are more than aware that a detailed analysis of alternating access, the occluded states, and the switching function
is needed to understand cytochrome $c$ oxidase. That analysis needs the currents flowing to be analyzed along with the atomic detail of the water-gate switch \cite{RN4995,RN45750}, in our view. The switches act on currents and of course satisfy conservation of current.
An analysis of charges cannot easily guarantee conservation of current, and classical chemical analysis precludes large currents because assuming equilibrium or near equilibrium conditions is clearly inappropriate for a system like Reaction Center IV designed for the efficient handling of large flows of electrons and protons.
Here we describe alternating access with the classical equation of a rectifier to highlight the possibility that occluded states
and alternating access are the biochemical names for what engineers call rectification.
It is important to realize that rectification is an automatic unavoidable consequence of the distribution of doping in semiconductors, for example in the classical PN diode. This rectification occurs with no change in the spatial distribution of mass (i.e., with no change in what is usually called conformation)
and so it is compatible with the view cited above that cytochrome c oxidase functions without changes in the spatial distribution of mass, i.e., without what is classically called conformation change. In the rectification mechanism, the switching (rectification) occurs because of a radical change in the distribution of electrical potential, which in turn allows current flow in one direction and not another. The distribution of potential depends on the distribution of mobile electrons which have almost no mass. The conformation of the potential profile and thus the electric field creates a barrier for current flow in one direction but not in another, because of the effects of doping (permanent charge) and mobile charge combined in the Maxwell Gauss law, or the Poisson equation. This system is rather complex, although completely understood and used in literally billions of places in each of our computers, The system involves diffusive and electrical movement of electrons (and holes) driven by the gradients of chemical potential (e.g., concentration) and electrical potential. As the electrical potential changes sign, diffusive and electrical flow changes. As concentrations change, diffusive and electrical flow change in other ways. All interact through the changing fields of electrical and chemical potential. A few pages (not just a few words or sentences) are needed to explain how each kind of movement (diffusive, electrical, holes and electrons) contribute to rectification. See textbooks of semiconductor circuit design, e.g, \cite{colinge2005physics,pierret1996semiconductor}. It is also important to consult
research articles \cite{laux1999revisiting,haggag2000analytical} to understand the oversimplifications of the textbook discussions and to validate them.
More elaborate patterns of doping, starting with the PNPN designs (thyristors, Silicon Controlled Rectifiers = SCR) are used in power transistors. Analogous spatial distributions of permanent charge (and acid and base side chains) might be used to implement switches in cytochrome c oxidase.
We note rectification arising from the distribution of permanent charge in a protein was proposed by Mauro a very long time ago \cite{RN23400,RN23397}. Such rectifiers of ionic current have even more complex properties than semiconductor rectifiers because concentrations of current carriers in biological solutions can be changed independently of electrical potential, which is not often
the case in analogous semiconductor systems. Ionic rectifiers were built a long time ago
using a biological protein as a template
\cite{miedema2007biological} and are now used routinely in
the ionic channels of nanotechnology \cite{RN45776}
even in a hybrid chip that can enable a scalable integrated ionic circuit platform for micro-total-analytical systems \cite{RN45777}.
The switch of Reaction Complex IV is likely to involve both the distribution of permanent charge (mostly acid and base side chains),
and chemical interactions as described in the water-gate model \cite{RN4995,RN45750}, perhaps also involving spatial distribution of dielectric properties as well \cite{RN30417}.
It seems premature to attack this problem here, as important as it is for the function of cytochrome $c$ oxidase, and all alternating access transporters, for that matter. Here, we simply describe the rectification without further analysis of how it arises from the distribution of permanent charge and other properties of the transporter structure.
Of course, other possibilities exist.
Alternating access might arise, for example, from bubbles in the conduction pathway, as we are studying in other work \cite{RN46054}.
Two bubbles might act as coupled activation and inactivation gates, correlated to provide alternating access to an occluded state, for example.
\section{Computational Studies}
In this section, we carry out several computational studies to explore the effects of various conditions on proton transport efficiency.
The initial values and default parameters are listed in Table 1-2.
\begin{table}[]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Variable & Notations & Values (with Unit) \\
\hline
$E_{242}$ site $H^+$ concentration & $[H]_{E}$ & 0.01196 $\mu M$ \\
\hline
BNC site $H^+$ concentration & $[H]_{B}$ &0.01682 $\mu M$ \\
\hline
PLS site $H^+$ concentration & $[H]_{X}$ & 0.01441 $\mu M$ \\
\hline
BNC site electric density & $\rho_{e}$& 0.01166 $\mu M$ \\
\hline
$E_{242}$ site electric potential & $\phi_{E}$ & -5 $mV$ \\
\hline
BNC site electric potential & $\phi_{B}$ & -14.1562 $mv$ \\
\hline
PLS site electric potential & $\phi_{X}$& 200 $mv$ \\
\hline
N site electric potential & $\phi_{N}$& 0 $mv$ \\
\hline
P site electric potential & $\phi_{P}$& 160 $mv$ \\
\hline
\end{tabular}
\caption{Default Initial Values}
\end{center}
\end{table}
\begin{table}[]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Variable & Notations & Values (with Unit) \\
\hline
$E_{242}$ site effective capacitance & $C_D$& 1E-1 $fA ms/mV /(\mu m)^2 $\\
\hline
BNC site effective capacitance & $C_B$& 1E-1 $fA ms/mV /(\mu m)^2 $\\
\hline
PLS site effective capacitance & $C_X$&1E-1 $fA ms/mV /(\mu m)^2 $ \\
\hline
Membrane capacitance & $C_X$&7.5E-2 $fA ms/mV /(\mu m)^2 $ \\
\hline
D channel conductance for $H^+$ & $g_D$& 3.75E-3$ pS/(mum)^2$ \\
\hline
K channel conductance for $H^+$ & $g_K$& 1E-3 $ pS/(\mu m)^2$\\
\hline
E2B channel conductance for $H^+$ & $g_B$& 5E-2 $ pS/(\mu m)^2$\\
\hline
E2X channel conductance for $H^+$ & $g_E$& 1E-3 $ pS/(\mu m)^2$\\
\hline
E2X Pump rate for $H^+$ & $g_P$& 369 $pS ms/(\mu m)^2 \mu M$\\
\hline
X2P channel conductance for $H^+$ & $g_X$& 9.8E-4 $ pS/(\mu m)^2$\\
\hline
Membrane conductance for leak & $g_m$& 1E-5$ pS/(\mu m)^2$\\
\hline
Mito. matrix $H^+$ concentration & $[H]_{mat}$ & 0.01 $\mu M$ \\
\hline
Mito. inner membrane space $H^+$ concentration & $[H]_{ims}$ & 0.063 $\mu M$ \\
\hline
Nernst Potential due to other Ions & $E_{Other}$ & $-160mV$ \\
\hline
Reaction site $[O_2]$ concentration & $[O_{2}]$ & 0.0028 $\mu M$ \\
\hline
Reaction site $[H_2O]$ concentration & $[H_2O]$ & 0 $\mu M$ \\
\hline
Electron current &$I_e$ & -5.24 $fA$\\
\hline
Forward reaction rate coefficient & $k_f$ & 1333
\\
\hline
Backward reaction rate coefficient & $k_r$ & 0.005 \\
\hline
surface volume ratio &
$S_v$ &1000\\
\hline
Potential Threshold & $\delta_{th}$ & 210 mv\\
\hline
Decay rate & $\varepsilon$& 1 $(ms)^{-1}$ \\
\hline
\end{tabular}
\caption{Parameters}
\end{center}
\end{table}
\subsection{Effect of electron current: Input to Output Relations}
We first check Figs.\ref{fig:ratio_Ie_eq} the effect of electron current $I_e$ on the efficiency of Complex IV. The case 1 (inside to E rectifier) results are represented by blue circle and the case 2 (X to outside rectifier) results are represented by red square.
The ratios between currents and the supplied electron current are measures of the transfer function or 'gain' of cytochrome c oxidase. According to the previous study \cite{RN45750}, the ratios are $\frac{I_{X2P}}{I_e} = -1$, $\frac{I_{E2X}}{I_e} = -1$ and $\frac{I_{E2P}}{I_{N2E}+I_{N2B}} = -0.5$ at the normal state. These ratios mean that (nominally) each input electron will bring 2 protons from the N side. One of the protons is used for the chemical reaction and the other one is pumped to the P side becoming an output in that way.
Fig. \ref{fig:ratio_Ie_eq} (d)-(h) confirm that when the electron supply is sufficient ($|I_e|\ge 5.24 fA$) these ratios can be maintained.
However, if the input electron current decreases (in magnitude), the reaction rate decreases linearly (see Fig. \ref{fig:ratio_Ie_eq} (a)) since $\mathcal{R} = \frac{-S_v}{2F}I_e$ at equilibrium according to Eq. \eqref{rhoequation}.
The pump strength depends on reaction rate (Eq.\eqref{pumpcurrent}), so the pump current $I_{Pump}$ decreases hand in hand with reaction rate.
Beyond a threshold, the total current between E242 and PLS $I_{E2x}$ becomes negative, $provided$ the rectifier is located $between$ the inside and E site (blue lines with circles). The the protons leak back form PLS to E242. This `leak back' can be seen in Fig. \ref{fig:ratio_Ie_eq} (d), where the positive ratio means means that the $I_{E2X}$ is negative (because $I_e$ is negative, with our sign conventions). Proton back flow from outside to the proton loading site PLS occurs in this case. The accumulated protons in E242 increase the chemical potential $\mu_E$ to be greater than $\mu_N$ and $\mu_E$, which leads to more current from E242 to the reaction cite B (see Fig. \ref{fig:ratio_Ie_eq} (f)) and back flow from reaction cite to N side (see Fig.\ref{fig:ratio_Ie_eq} (h)). The rectifier blocks the direct back flow from E242 to the N side. The urrent $I_{N2E}$ becomes zeros (see Fig. \ref{fig:ratio_Ie_eq} (g)). In this case, the proton flow pattern is shown in Fig. \ref{fig:protonflow} (b).
The location of the rectifier is important. Behavior is different when the rectifier is moved. When the rectifier is between the protein loading site PLS and outside (red lines with squares), the backward flow from outside to PLS is blocked to be zero. Then the current $I_{E2X}$ is also zero at the equilibrium, according to Eq. \eqref{HXquation}, which means $\mu_E = \mu_X$ as shown in Fig. \ref{fig:ratio_Ie_eq} (b). The protons are still transported from inside to E242 then to BNC. $\frac{I_{N2E}}{I_e}=-0.68$ through D channel and directly to BNC with ratio $\frac{I_{N2B}}{I_e}=-0.32$ through the K channel. Fig. \ref{fig:protonflow}(c) shows the proton flow pattern in this situation.
We suspect that the rectifier between the protein loading site PLS and outside is closer to the real biology setup, because it blocks backward flow. For that reason, we mainly present the results with the rectifier between PLS and outside. Our approach can of course handle almost any location or properties of the rectifier/switch once they are specified by experiment or higher resolution models.
\begin{figure}[!ht]
\centering
\begin{subfigure}[]{
\includegraphics[width=2.5in]{Figure/diffIE_retifier/reaction.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=2.5in]{Figure/diffIE_retifier/muXmuE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=2.5in]{Figure/diffIE_retifier/Ipump.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=2.5in]{Figure/diffIE_retifier/IE2XIE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=2.5in]{Figure/diffIE_retifier/IX2PIE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=2.5in]{Figure/diffIE_retifier/IE2BIE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=2.5in]{Figure/diffIE_retifier/IN2EIE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=2.5in]{Figure/diffIE_retifier/IN2BIE.eps}}
\end{subfigure}
\caption{Pump efficiency at equilibrium states with different electron current and different threshold. (a) Reaction rate; (b) $\mu_X -\mu_E$; (c) $I_{Pump}$;(d) $I_{E2X}/I_e$; (e) $I_{X2P}/I_e$; (f) $I_{E2B}/I_e$; (g) $I_{N2E}/I_e$; (h) $I_{N2B}/I_e$. Red line: Switch between X and outside; Blue dash line: Switch between inside and E.}
\label{fig:ratio_Ie_eq}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentnormal.png}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentbackw.png}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentbackw3.png}}
\end{subfigure}
\caption{ Schematic for the proton flow pattern. Equation numbers that define arrows are shown, e.g., eq. (38b) for $I_{X 2 P}$. (a) Normal state; (b) Backward flow state with Perfect switch between N and E242; (c) Flow state with perfect switch between the proton Loading Site PLS and outside. }
\label{fig:protonflow}
\end{figure}
\subsection{Effect of Proton Concentration }
In this section, we study the effect of proton concentrations in the intermembrane space (outside) by increasing from the default value $0.06 \mu M$ to $0.15 \mu M$.
Figs. \ref{fig:Concentration_p_eq} and \ref{fig:Pump_p_eq} show the equilibrium states of concentrations and pump efficiency at different proton concentrations with difference leak conductance $g_m$.
First, Fig. \ref{fig:Pump_p_eq} a. illustrates the reaction rate with different $[H]_P$ keeps a constant since $\mathcal{R} = \frac{-S_v}{2F}I_e$ at equilibrium.
When the leak conductance is zero, the complex IV efficiency does not change, i.e. $\frac{I_{X2P}}{I_e} =1$ and the flow pattern is shown in Fig.\ref{fig:protonflow}a. When the shunt conductance $g_m$ is larger than zero, the pump resistance increases with the outside proton concentration. This produces a decrease of the complex IV efficiency all the way to zero after the threshold. Then the proton pattern is the same as in Fig.\ref{fig:protonflow}c, where all the protons pumped from inside through D and K channels are consumed by the reaction at BNC.
The concentrations of electrons and protons at $E$ and $B$ sites are small perturbations in all cases. The concentration in the PLS is almost a constant with different $[H]_P$ when the leak conductance is large. However, it increases tremendously when the leak conductance is small. Large leak conductance simulates voltage clamp conditions, which do not describe the normal functional state of the mitochondrion. Small leak conductance presumably corresponds to the natural state in which the sum of all currents across the mitochondrion is `clamped' to zero, by Kirchhoff's current law, because there is nowhere else for the current to flow.
\begin{figure}[!ht]
\centering
\begin{subfigure}[]{
\includegraphics[width=2.5in]{Figure/diffHP_dy/RC.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=2.5in]{Figure/diffHP_dy/IN2EIE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=2.5in]{Figure/diffHP_dy/IN2BIE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=2.5in]{Figure/diffHP_dy/IE2XIE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=2.5in]{Figure/diffHP_dy/IE2BIE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=2.5in]{Figure/diffHP_dy/IX2PIE.eps}}
\end{subfigure}
\caption{Pump efficiency at equilibrium states with different proton concentration and different leak conductance. (a) Reaction rate; (b) $I_{N2E}/I_e$; (c) $I_{N2B}/I_e$; (d) $I_{E2X}/I_e$; (e) $I_{E2B}/I_e$; (f) $I_{X2P}/I_e$. Black dash line: $g_m =0$; Red line with circle: $g_m=10^{-5}$; Red line with square: $g_m = 10^{-3}$. }
\label{fig:Pump_p_eq}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/diffHP_dy/rhoE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/diffHP_dy/HE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/diffHP_dy/HB.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/diffHP_dy/HX.eps}}
\end{subfigure}
\caption{Concentration at equilibrium states with different proton concentration and different leak conductance. (a) Electron concentration $\rho_e$; (b) $[H]_E$; (c) $[H]_B$;(d) $[H]_X$. Black dash line: $g_m =0$; Red line with circle: $g_m=10^{-5}$; Red line with square: $g_m = 10^{-3}$. }
\label{fig:Concentration_p_eq}
\end{figure}
\subsection{Kirchhoff clamp }
Most of this paper describes cytochrome c oxidase embedded in a mitochondrion approximating a preparation without other members of the respiratory chain, but with otherwise normal properties. The mitochondrion is a small cell, as it were, in which the interior potential is unlikely to vary substantially with macroscopic location, on the micron scale, because the cell is much smaller than the length constant of cable theory. In such a system, Kirchhoff’s current law ensures that the sum of all currents across the mitochondrial membrane is zero. The currents are necessarily coupled by electrodynamics, whether or not they are coupled by chemistry. If one current increases, the sum of the others must decrease. A graph of one current against another will give a definite ratio, a coupling ratio, if you will, when other variables are held constant.
The various currents are coupled by the electric potential. The electrical potential contributes to the forces that drive the currents. Chemical reactions may contribute as well. But even without chemical reactions, coupling can occur through the electric field. While this may seem strange in context of classical transport biophysics, it is well precedent ed and understood in channel biophysics, that presumably follows the same laws of physics. The coupling of sodium and potassium conductances that allow the action potential to propagate are an example of coupling by the electric field, without chemical interaction of the underlying protein molecules, as we have discussed previously in this paper.
Coupling occurs because the electric field adopts the values that conserve current. That is easy to prove from the Maxwell equations \cite{RN28898,RN30138}.
In fact, the conservation of current is a form of Kirchhoff’s law, so currents are clamped to one another (i.e., coupled) in a “Kirchhoff clamp” if we want to coin a phrase for what is really just the UNclamped, natural situation.
If the electrical potential is controlled, so it is not free to adopt the value that conserves current, a different situation occurs altogether. This situation is called a voltage clamp in electrophysiology, and was invented by Cole and used by Hodgkin and Huxley to understand the mechanism of the action potential. The voltage clamp loosens the Kirchhoff clamp because it has an amplifier (that is outside the biological system) to supply current and energy. Indeed, the Kirchhoff clamp of the natural mitochondrion is entirely removed by the currents supplied by the voltage clamp amplifier.
In the voltage clamp, one current is not coupled to another current by the voltage. They cannot be, because the voltage does not vary with the current. The result is that coupling and flux ratios reflect chemical coupling, not voltage coupling, in the voltage clamp setup. The result is that nearly every experimental result is different in a voltage clamp and the natural UNclamped situation.
The voltage clamp was invented to gain experimental control of currents so they can be studied as Cole made abundantly clear, followed by Hodgkin and Huxley. But \textbf{the voltage clamp is not natural}. It removes a natural form of flux coupling. Flux coupling by the electric field is absent. Flux coupling by the electric field is natural, just as natural in the mitochondrion as in the nerve, just as natural in the generation of ATP as it is in the generation of the nerve signal.
It is difficult to voltage clamp mitochondria, and preparations reconstituted into bilayers (that can be voltage clamped) have other difficulties that experimentalists often wish to avoid. Other methods are used to simulate a voltage clamp, quite well, as it turns out.
In work on mitochondria, voltage clamp is usually produced indirectly, by artificially increasing the leak conductance. An effective carrier of potassium current like valinomycin is often added to solutions. When valinomycin is present in large enough concentrations, it partitions into the mitochnondrial membrane, and the leak conductance dominates. The potential across the mitochondrial membrane is set by the equilibrium potential of the leak conductance. If valinomycin is used to increase the leak conductance, the potential is in fact close to the potassium equilibrium potential, independent of current because valinomycin is remarkably selective for potassium ions. Valinomycin clamps the potential to the potassium equilibrium potential.
Figs. \ref{fig:Concentration_Eother_gm_eq}-\ref{fig:ratio_Eother_gm_eq} illustrate the effect of Nernst potential $E_{other}$ on concentrations, electric potentials and currents under different leak conductance. $E_{other}$ is an approximation to the potassium equilibrium potential The red lines with circles, squares and triangles are denotes the different leak conductance $g_m = 10^{-6}, 10^{-5}, 10^{-3}$, correspondingly. The black dash lines are the results by setting zero leakage, i.e. $g_m =0$. And the blue dash lines denotes voltage clamp results where the electric potential at inside $\phi_N$ is set to be zero and electric potential at outside $\phi_P = -E_{other} $ in system \eqref{current_model}.
First, consider the natural case, when shunt conductance $g_m=0$. Cytochrome $c$ oxidase is not affected. The efficiency of complex IV is not changed by the Nernst potential because $I_{leak}$ is always zero in this case.
When $g_m>0$, as the Nernst potential become more negative, the resistance for proton pumping increases which leads to the decrease of proton pump efficiency. Of course, when the leak conductance is large enough, the system is nearly voltage clamped to the equilibrium potential for the leak. The Kirchhoff clamp (red lines with triangles) is unlocked, removed by the large leak conductance. The results are the same as the results of the voltage clamp (blue dash lines).
Fig. \ref{fig:Concentration_Eother_gm_eq}(d), Fig. \ref{fig:phi_Eother_gm_eq}(c)-(d), and Fig. \eqref{fig:ratio_Eother_gm_eq} show the difference in various quantities between voltage and UNclamped natural situations.
The dramatic effect of antibiotics on membrane properties was studied by \cite{RN938}.
Valinomycin has been studied in many other papers, which include \cite{RN46023,RN45770,RN27722,RN46059,RN46060,RN46061}.
\begin{figure}[!ht]
\centering
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentvsvotclamp/rhoE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentvsvotclamp/HE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentvsvotclamp/HB.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentvsvotclamp/HX.eps}}
\end{subfigure}
\caption{Concentration at equilibrium states with different $E_{other} $and $g_m$. (a) Electron concentration $\rho_e$; (b) $[H]_E$; (c) $[H]_B$;(d) $[H]_X$. Black dash line: $g_m=0$; Red line with circles: $g_m =10^{-6}$; Red line with squares: $g_m =10^{-5}$; Red line with triangles: $g_m =10^{-3}$; Blue dash lines: Voltage clamp.}
\label{fig:Concentration_Eother_gm_eq}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentvsvotclamp/phiE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentvsvotclamp/phiB.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentvsvotclamp/phiX.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentvsvotclamp/phiP.eps}}
\end{subfigure}
\caption{Electric potential at equilibrium states with different $E_{other} $and $g_m$. (a) $\phi_E$; (b) $\phi_B$; (c) $\phi_X$;(d) $\phi_P$. Red line with squares: $g_m =10^{-5}$; Red line with triangles: $g_m =10^{-3}$; Blue dash lines: Voltage clamp.}
\label{fig:phi_Eother_gm_eq}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentvsvotclamp/IN2EIE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentvsvotclamp/IN2BIE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentvsvotclamp/IE2BIE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentvsvotclamp/IE2XIE.eps}}
\end{subfigure}
\begin{subfigure}[]{
\includegraphics[width=3.in]{Figure/currentvsvotclamp/IX2PIE.eps}}
\end{subfigure}
\caption{Evolution of current at equilibrium states with different $E_{other} $and $g_m$. (a) $I_{N2E}/I_e$; (b) $I_{N2B}/I_e$; (c) $I_{E2B}/I_e$; (d) $I_{E2X}/I_e$. (e) $I_{X2P}/I_e$. Red line with squares: $g_m =10^{-5}$; Red line with triangles: $g_m =10^{-3}$; Blue dash lines: Voltage clamp.}
\label{fig:ratio_Eother_gm_eq}
\end{figure}
\newpage
\section{Discussion and Conclusion}
As history separated chemistry and field theory, so it separated chemical theory from devices. Almost all of chemical theory devalues the significance of boundary conditions and flows. Almost none of chemical theory allows flows from boundary to boundary. These bald statements are easy to confirm. Most monographs and texts of chemical theory barely mention flow from boundaries, and most deal with equilibrium zero flow systems, using those results to discuss what happens when flow is not zero with certain inherent difficulties (given the obvious inconsistencies involved). References are hard to find that discuss spatially nonuniform boundary conditions or flows from boundary to boundary driven by external sources. Power supplies are essential for most engineering devices and they require different locations on boundaries to have different potential. That is to say, they require spatially nonuniform boundary conditions for potential with nonzero flows on the boundary.
Devices are important. Our entire electronic technology is built from devices that function more or less the same way no matter where they are located (within reasonable limits, it goes without saying. Nothing in engineering or technology is true in general. Everything exists and functions only within reasonable limits). It would obviously be useful if chemical systems could be easily and routinely shaped into devices.
Devices depend on spatially complex boundary conditions. A device has inputs and outputs with different locations and different boundary equations. If inputs and outputs are at the same location, and have the same properties, there is no device! Most devices have power supplies as well as inputs and outputs. These supply flows of energy that allow the device to have well defined input output relations that are robust, quite independent of what is connected to the input or output of the device. A transfer function relates input and output when they are related by a constant coefficient causal ordinary differential equation in time.
Devices maintain these properties almost entirely by using electricity and energy from power supplies. They are fundamentally nonequilibrium systems with spatially nonuniform Dirichlet boundary conditions for the electrical potential.
Electricity is used to make engineering devices for good reason. Electrical potentials, currents, and electrical energy are described by the Maxwell equations with greater precision over a wider range of conditions than almost any other physical phenomenon.
The generality of the Maxwell equations in classical form is obscured because they embody an outdated and seriously inadequate representation of dielectrics and polarization. We say this with no disrespect for the enormous contributions of Maxwell and Heaviside, et al. But a representation that described dielectric measurements slower than some 0.05 sec (in the 1890's) cannot be expected to describe systems studied in the 2020's that function on atomic time scales of femtoseconds, let alone the much faster time scales of visible and ultraviolet light ($3x10^{12}$ - $7x10^{12}$) to $10^{16}Hz$ and even higher energy radiation like x-rays ($10^{16}$ - $10^{20}Hz$) and gamma rays($>10^{20}Hz$).
The classical Maxwell equations use a single real number to describe how charge moves when an electric field is changed. Charge moves (and is said to polarize) on an enormous range of times scales, when electric fields are applied to matter, that cannot usefully be described by the single dielectric constant. The classical formulation of the Maxwell equations embody---as well as depend on--- the classical, but crude representation of dielectrics. They use a dielectric constant in the very definition of key variables. Those variables in fact depend on an out of date constitutive model. If time dependent, complicated charge movements are present, as is always the case in liquids, and in solids over the time scale of modern technology, the Maxwell equations need to be rewritten to isolate the movements of material charge (with mass) from other types of current (e.g., from the displacement current found throughout space $\varepsilon_0 \partial E/\partial t$). The rewritten ``Core Maxwell Equations'' must then be joined to a description of polarization showing how charge moves when electric and magnetic forces are applied \cite{wang2021variational}. That description is not very different from the stress strain description of how mass moves when a force is applied. Mathematicians and physicists have dealt with such stress strain relations in solids and complex fluids with many types of flow (migration, convection, diffusion) and those methods (chiefly of the theory of complex fluids, in its energetic variational flavor) can be applied to the polarization phenomena of charge.
The Core Maxwell Equations have a special property not found in many other field equations because of the displacement current term $\varepsilon_0 \partial E/\partial t$ that is universal, present everywhere including inside atoms and in empty space, wherever the Maxwell equations are valid.
The Core Maxwell Equations are of use even when the spatial and temporal location of charge is not known because the displacement current term $\varepsilon_0 \partial E/\partial t$ is universal, It is present everywhere, inside atoms and between stars. For this reason, the Core Maxwell Equations (surprisingly) are very useful for any description of polarization and for any type of charge movement, whether created or driven by electrodynamics, diffusion, convection, sunlight (in solar cells), or even heat flow even though they themselves do not describe polarization phenomena.
The Core Maxwell Equations have this special property because of Maxwell’s Ampere law. This law has no counterpart in the field theories of mechanics. This law allows electric and magnetic fields to flow in perfect vacuum so they can create propagated waves we call `the light of the sun', even though the vacuum contains no charge (with mass).
The Maxwell Ampere law has a corollary: the divergence of total current is always zero, everywhere to the same accuracy and in the same domain that the Maxwell equations are valid. The Maxwell Ampere law implies a unique definition of total current as the entire source (i.e., right hand side) of\textbf{ curl B} in Maxwell’s Ampere law {\textbf{no matter how material charge moves.}} In fact, Maxwell’s Ampere law defines total current everywhere, including in a perfect vacuum where there is no mass or charge with mass or movement of charge with mass whatsoever. In a total vacuum, total current is $\varepsilon_0 \partial E/\partial t$). In the presence of matter total current is $J_{total- current} = \ J+ \ \varepsilon_0 \partial E/\partial t$) where $J$ described the movement of charge with mass no matter how fast, transient, or small it is. This fundamental property of electricity was well known to Maxwell and his followers, as is made clear on p.155 and p. 511-512 of \cite{RN28696}.
and is central to the discussion of current flow in
the classic text of \cite{RN45998}
and the work of Landauer \cite{RN45927,RN26340}.
The idea of total current is defined to focus attention on these issues and is discussed in many papers cited in \cite{eisenberg2021maxwell}.
If total current is confined to a single one dimensional path, conservation of total current becomes equality of total current everywhere (see Fig. 2 of \cite{RN25303}
for an extensive physical discussion with examples). If that current is confined to a circuit, conservation of total current becomes Kirchhoff’s current law \cite{RN46001}. In a particular system, it is easy to verify whether the one dimensional approximation for current flow is accurate (enough). Just measure the currents and see if they are equal (in unbranched systems) or whether they sum to zero as Kirchhoff’s law requires at nodes in circuits in general as Kirchhoff’s law requires. It would be interesting to learn to specify both the necessary and sufficient conditions in abstract mathematical terms (that can be evaluated before a particular system is specified) under which Kirchhoff's law is accurate (enough).
The Kirchhoff’s law just described is not quite the Kirchhoff’s law of textbooks of electrodynamics or engineering. The textbook law is derived and presented as valid for long times many orders of magnitude longer than the time scales of electronic devices, let alone atomic motion. There should be no misunderstanding of this crucial point. The literature of circuit design shows that Kirchhoff's law is used as an essential design tool or analog and digital circuits and integrated circuits \cite{RN25358,RN4927,RN45662,RN4927,RN26358,RN28108,RN25161,RN26575} that can even function on the $10^{-12}$ second time scale \cite{RN46002}.
It seems clear to us that the generalization of Kirchhoff’s law to total current either solves this problem, or makes it moot, as you wish \cite{eisenberg2019kirchhoff,eisenberg2018current}, following the practice of Maxwell himself, and his followers, according to p.155 and p. 511-512 of \cite{RN28696}.
This paper combines Kirchhoff’s law for total current with a quite general description of chemical reactions with a general description of ion and water flow, using the EnVarA (energy variational) approach in the tradition of the theory of complex fluids. We use this electro-osmotic framework to analyze a coarse grained description of cytochrome $c$ oxidase in the tradition of `master equations'. Our model is built on the carefully constructed and well analyzed models of many others, but here we analyze the master equations using currents defined as in Kirchhoff's law for total current.
Our analysis is in the tradition of engineering. It does not depend on details of electron current flow. It does not compute properties of charges and their interactions, except implicitly as defined by the Maxwell equations, particularly Maxwell's Ampere's law $curl \ B=\mu_0 \ J_{total-current} =\mu_0 \ J+ \ \mu_0\varepsilon_0 \partial E/\partial t$
It seems obvious to us that each of the systems of oxidative phosphorylation and photosynthesis require a circuit analysis embedded in the theory of complex fluids. Evolution has built structures that conduct electron currents, as in our electronic technology and so should be analyzed by the extraordinarily successful methods of electronic circuit analysis. Evolution has used ion current flow and chemical reactions in addition to electron flow so the circuit approach is embedded here using the theory of complex fluids. Complex fluid theory is designed to combine a wide variety of flows and the forces and energies that drive them in a mathematically consistent way. The structures evolution uses to control these flows form the geometry---or anatomy or histology, depending on the length scale---of the system. The channels and transporters (and electron transport pathways) built by evolution form boundary conditions that decorate (i.e., are located on) these structures and thus describe how they work.
Cytochrome $c$ oxidase is called Complex IV for a reason. It is embedded in a lipid bilayer, connected to electron pathways we approximate as wires (in the engineering tradition), and is surrounded by electrolytes that store energy in their electric and chemical potential fields that form a complex fluid. Complex IV includes the cytochrome $c$ oxidase enzyme, electron pathways, channels for protons and potassium ions, pathways for oxygen diffusion, and the membrane that encloses it and ions that surround it. Each subsystem stores energy and responds to energy gradients with different types of flows. Complex fluid mathematics is designed to handle systems of this complexity, although biological applications involve more preset structural complexity than in many physical systems of fluids. Complex fluid theory treats all fields, flows and boundary conditions---including spatially nonuniform conditions that power the system as they power electronic devices---consistently. The electro-osmotic extension of the theory hopefully joins the forces, flows and energies of chemical reactions into this formulation, while preserving the mathematical consistency of the original theory, without violating the traditions of chemistry.
Our work is significantly incomplete and limited. We over-approximate several important biophysical mechanisms, including the water-gate switch, and the oxygen reduction mechanism. We are more than aware of the need for higher resolution in later work, with specific atomic scale models that compute the electric field and flows from underlying structures and chemical reactions on time scales of displacement (capacitive) currents that have been so well resolved in experiments of great difficulty. These currents are important in understanding the switches and mechanisms by which cytochrome c oxidase couples electron flow, oxidative chemical reactions, and proton flow to make oxidative phosphorylation possible in mitochondria.
\section*{Acknowledgement}
This work was partially supported by the
National Natural Science Foundation of China no. 12071190 and Natural Sciences and Engineering Research Council of Canada (NSERC).
We also would like to thank American Institute of Mathematics where this project initiated.
\newpage
\printbibliography
\newpage
|
2,869,038,156,397 | arxiv | \section{Introduction}
After both stars in a binary system have ended their main sequence phase, a common result is a
white dwarf (WD) binary, and a large fraction of these binaries are expected to merge in less than a
Hubble time \citep{nel01}, due to various causes like gravitational wave emission or magnetic braking
and so on. Numerical simulations indicate that the post-merger system consists of a fast-rotating central
core surrounded by a Keplerian disk \citep{ji13}, which have inherited the orbital angular momentum of
the progenitor WD binary. A hot corona above the disk can form as a result of the development of the
magnetorotational instability within the disk. This corona is highly magnetized, with field strengths
of order $10^{10}-10^{11}\,\rm G$ within a radius of $R\sim 10^9\,\rm cm$, with strong outflows emerging from this central region (see Figure 2 in \citet[][]{ji13}). The ejecta velocity of this outflow is of order $10^9\,\rm cm\,s^{-1}$ and the total ejected mass is about $10^{-3}~M_\odot\sim10^{30}\,\rm g$, creating an expanding fireball. \citet{bel14} argued that internal shocks may develop in this outflowing fireball, increasing its radiative output and leading to a predicted bright optical transient of luminosity $10^{41}-10^{42}\,\rm erg\,s^{-1}$.
Here we examine possible energy dissipation scenarios in such merger events which could lead to cosmic-ray (CR) acceleration and high-energy neutrino emission.
The origin of the diffuse high-energy (TeV to PeV) neutrino flux discovered by IceCube
\citep{aar13a,aar13b,aar15,IC3+15tevnu,aar15c,aar16} is currently under intense debate.
Among various possible astrophysical sources which can contribute to this flux \citep[e.g., for reviews, see][]{wax13,mes14,anc14,Ahlers+15nurev}, the most commonly discussed are CR reservoirs including star-forming galaxies (SFGs) and starbursts galaxies (SBGs) \citep{loe06}, and galaxy clusters and groups \citep{mur08,kot09},
in which confined CRs can produce neutrinos via $pp$ interactions.
In SFGs and SBGs, CRs are accelerated in supernova and hypernova remnants~\citep{mur13,sov15,nick15, cha15,xia16} as well as fast outflows and possible jets from active galactic nuclei~\citep{mur13, tam14, mur14,wl16}.
The IceCube neutrino flux can be accounted for in the SBG scenario without violating the extragalactic gamma-ray background \citep{mur13,cha14,xia16} or even the simultaneous explanation of neutrinos, gamma-rays and CRs is possible \citep{mw16}.
However, the neutrino data point around $30\,\rm TeV$ remains unsolved, and the fact that a large fraction of the isotropic extragalactic gamma-ray background can be explained by other sources such as blazars serves as motivation for investigating ``hidden" (i.e. $\gamma$-ray dim) neutrino sources \citep{mur16a}.
In this work, we argue that WD merger events belong to such kinds of hidden CR accelerators, as often discussed in the context of low-power gamma-ray bursts and choked jets~\citep{mw01,mi13,xia14,xia15,senno16,ta16}.
We calculate neutrino spectra of individual mergers and show that nearby events are detectable if CRs are accelerated efficiently by magnetic dissipation. We also discuss the diffuse neutrino flux, and find that WD mergers could also provide an interesting fraction of the IceCube diffuse neutrino background.
The paper is organized as follows. We introduce the model and method of calculation in Section 2. Then we consider the possibility of detection of individual WD merger sources in Section 3.
Section 4 presents our results of the diffuse neutrino flux from WD mergers, compared with the
IceCube data. Lastly, we provide a summary and final discussion in Section 5.
\section{Model and Calculation}
\subsection{Outflow from the merger region}
\label{sec:outflow}
The merger of a white dwarf binary is expected to generate strong magnetic fields ($10^{10}-10^{11}$ G), as demonstrated by recent numerical simulations \citep[e.g.][]{ji13, zhu15}. For various types of WD, the birth and merger rate is slightly different \citep{bog09}. Here in our work we consider typical carbon-oxygen WD binary of equal mass $\sim0.6\,M_\odot$.
Consistent with these simulations, to within an order of magnitude, the outflow from the central core and disrupted disk plus corona region can be characterized by an initial radius $R_0\sim 10^9 R_9~\,\rm cm$, magnetic field $B_0\sim 10^{10}B_{10}\,\rm G$, temperature $T_0\sim 10^8 T_8\,\rm K$, and mass outflow rate ${\dot M} \sim 2\times 10^{26}\,\rm g\,s^{-1}$. Considering a residence time for the baryons inside this region of order or somewhat longer than the sound crossing or virial time, $t_c \sim 100~\rm s$, the mass density at the boundary is $\rho_0\sim {\dot M}t_c/(4\pi R_0^3) \sim 1~{\dot M}_{26} (t_c/100 ~\rm s)$ g cm$^{-3}$, and the outflow velocity is $v_0\sim 10^9 \,\rm cm\,s^{-1}$, \citep[e.g.,][]{ji13,bel14}. This magnetic field is characteristic of the corona above the disk, and the total instantaneous coronal magnetic energy is
$\mathcal{E}_{B,0}\sim 10^{48}\,\rm erg$. One can verify that for these conditions, the initial magnetic energy density $B_0^2/8\pi\sim 4\times 10^{18}~B_{10}^2~\rm erg~cm^{-3}$, initial kinetic energy density $(1/2)\rho_0 v_0^2\sim 0.5\times 10^{18}~\rho_0 v_9^2~\rm erg~cm^{-3}$ and initial radiation energy density $a_R T_0^4\sim 0.75\times 10^{18}~T_8^4~\rm erg~cm^{-3}$ are in approximate equipartition. However, it is possible to have parameters that are not covered by the present numerical simulations.
The magnetic luminosity of the outflow can be estimated as $L_B\sim \mathcal{E}_B/t_c
\sim 4\pi R_0^2 v_0(B_0^2/8\pi) \sim 10^{46}~\rm erg/s$, which may be an optimistic maximum luminosity.
A more conservative estimate \citep{bel14} is $L_B\sim 10^{44}~\rm erg/s$.
The outflow is expected to last for a viscous time $t_{\rm visc}$ characterizing the draining of the
disrupted disk, which in terms of an $\alpha_B$ magnetic viscosity prescription is $t_{\rm visc}\sim
\alpha_B^{-1}(H/r)^{-2}\Omega^{-1}\simeq 3\times10^3\,\rm s$, where we assume $\alpha_B\sim 10^{-3}$ \citep[e.g.][]{bel13, hos13} and the scale height at the tidal disruption radius $H/r\sim 1$ \citep{kas15}. As a result of the merger, the total amount
of magnetic energy ejected by the remnant is $\mathcal{E}_B\sim L_B t_{\rm visc}\sim 10^{48}-10^{50}~\rm erg$.
The toroidal component of the magnetic field in the outflow would decrease as $B_t\propto R^{-1}$, as
in a pulsar striped wind or magnetar wind, while a poloidal component decreases as $B_p\propto R^{-2}$. In general, we can expect the total magnetic field to scale as a power-law in radius, and the magnetic
luminosity to scale as a power-law in radius, $L_B\sim 4\pi R^2 v (B^2/8\pi) \propto R^{\alpha}$.
For random equipartition magnetic fields, acting as a relativistic gas, purely adiabatic expansion
would imply $L_B\propto R^{-2/3}$, i.e. $\alpha=-2/3$. Alternatively, for a strong ordered magnetic field,
one would have $B\propto R^{-1}$ and $L_B=\rm const$, i.e. $\alpha=0$. This latter case is of
greater interest for our purposes here.
The evolution of other quantities in the outflow are calculated as follows \citep[e.g.,][]{mur16b}. Since the outflow ceases at $t_{\rm visc}\sim3\times10^3\,\rm s$, at time $t<t_{\rm visc}$, we get $\rho\propto R^{-2}$ and after that $\rho\propto R^{-3}$.
We can define a critical radius $R_{\rm cr}\equiv v_0t_{\rm visc}=3\times10^{12}\,\rm cm$, so
\begin{equation}
\rho(R)=
\begin{cases}
\rho_0(R/R_0)^{-2} &\mbox{if $R_0\leq R<R_{\rm cr}$,}\\
\rho_0(R_{\rm cr}/R_0)^{-2}(R/R_{\rm cr})^{-3} &\mbox{if \,\,\,\,\,\,$R\geq R_{\rm cr}$.}
\end{cases}
\label{eq:rho}
\end{equation}
The blackbody temperature evolution follows $T\propto \rho^{1/3}$ for adiabatic index $4/3$, thus
\begin{equation}
T(R)=
\begin{cases}
T_0(R/R_0)^{-2/3} &\mbox{if $R_0\leq R<R_{\rm cr},$}\\
T_0(R_{\rm cr}/R_0)^{-2/3}(R/R_{\rm cr})^{-1} &\mbox{if \,\,\,\,\,\,$R\geq R_{\rm cr}$.}
\end{cases}
\label{eq:temp}
\end{equation}
\subsection{Diffusion radius and magnetic energy dissipation}
\label{sec:diss}
Initially the outflow is optical thick $\tau_{T,0}\gg 1$, and the radiation is thermalized and trapped. As the
expansion proceeds, the optical depth decreases and when the characteristic diffusion length is equal to the characteristic dimension of the outflow (e.g. the radius of the leading gas particles), the radiation
begins to diffuse out faster then the gas expands. The diffusion timescale is $t_{\rm diff}\sim \tau_TR/c$, where the Thomson depth evolves as
\begin{equation}
\tau_T(R)=
\begin{cases}
\tau_{T,0}(R/R_0)^{-1} &\mbox{if $R_0\leq R<R_{\rm cr}$,}\\
\tau_{T,0}(R_{\rm cr}/R_0)^{-1}(R/R_{\rm cr})^{-2} &\mbox{if \,\,\,\,\,\,$R\geq R_{\rm cr}$.}
\end{cases}
\label{eq:tauT}
\end{equation}
The expansion time is $t_{\rm exp}\sim R/v_0$, and by equaling $t_{\rm diff}=t_{\rm exp}$ we can get the diffusion radius of $R_D=6.3\times10^{13}\,\rm cm$, which naturally falls into $R>R_{\rm cr}$ regime.
For radii less than the diffusion radius, $R \leq R_D$, it is likely that any magnetic reconnection
process is suppressed, due to the high photon drag. In this regime, radiation pressure works against
the development of turbulence and against regions of opposite magnetic polarity approaching. Beyond
$R_D$, however, radiation pressure start to drop, and reconnection may start to occur, although the
transition threshold from one regime to the other is not well-known \citep[][and references therein]{Uzdensky11magrec}.
The Thomson optical depth at the diffusion radius $\tau_T(R_D) \sim c/v$ remains above unity for at least two orders of magnitude in radius beyond $R_D$, and the dependence of the magnetic reconnection rate on the flow parameters in this still optically thick regime is speculative. For simplicity, here we adopt a simple power-law dependence for the magnetic energy dissipation rate, $\dot{\mathcal{E}}_B\propto t^{-q}$, where $q$ is a phenomenological parameter. Thus, beyond the dissipation radius, for $v=$ constant we can adopt a power-law dependence of the dissipation rate with radius of $d\mathcal{E}_B/dR=\mathcal{A}R^{-q}$,
where $\mathcal{A}= \frac{\mathcal{E}_B}{\int_{R_{\rm begin}}^{R_{\rm end}}R^{-q}dr}$ is a normalization factor.
As the photons begin to escape the outflow and reconnection begins, we could expect CR acceleration
to occur \citep[e.g.,][]{Giannios10crmag,Kagan+15recon}, which is also facilitated by the decreasing
chance of scattering against photons. The cooling mechanisms for accelerated CRs include mainly
synchrotron and inverse-Compton (IC) losses, inelastic $pp$ scattering, Bethe-Heitler pair-production and photomeson production processes.
\subsection{Cooling timescales of protons}
\label{sec:cooling}
We consider the cooling at the diffusion radius, where we assume CR acceleration begins ($R_{\rm begin} \sim R_D$). The density there can be expressed as $\rho_D=\rho_0(R_{\rm cr}/R_0)^{-2}(R_D/R_{\rm cr})^{-3}$, where $\rho_0=1\,\mathrm{g\,cm^{-3}}, R_0=10^9~R_9\,\rm cm$ is the density and radius at the initial stage of the hot coronal outflow. Also, the magnetic field $B_D=B_0(R_D/R_0)^{-1}$ and blackbody temperature $T_D=T_0(R_{\rm cr}/R_0)^{-2/3}(R_D/R_{\rm cr})^{-1}$, and we choose nominal values $T_0=10^8~T_8\,\mathrm{K},\, B_0=10^{10}~B_{10}\,\rm G$ (see Section \ref{sec:outflow}).
In this work, we calculate cooling rates of high-energy protons, using the numerical code developed in \citet{mur07} and \citet{mur08ph}.
\par
High-energy protons lose their energies through adiabatic, radiative and hadronic processes. The adiabatic cooling timescale $t_{\rm ad}$ is comparable to the dynamical timescale.
Radiative cooling includes synchrotron and IC scattering, and the synchrotron cooling timescale is
\begin{equation}
t_{\rm syn}=\frac{6\pi m_p^4c^3}{\sigma_Tm_e^2B^2\epsilon_p}
\label{eq:sync}
\end{equation}
and the IC cooling timescale is
\begin{equation}
t_{\rm IC}^{-1}=\frac{c}{2\gamma_p^2}\left(\frac{m_e^2}{m_p^2}\right)\pi r_e^2m_p^2c^4\int_0^\infty{\epsilon^{-2}\frac{dn}{d\epsilon}\frac{F(\epsilon,\gamma_p)}{\beta_p(\gamma_p-1)}d\epsilon},
\label{eq:IC}
\end{equation}
where $\sigma_T$ is the
Thomson cross section, $\gamma_p$ is the Lorentz factor of protons and the expression of function $F(\epsilon,\gamma_p)$ can be found in \citet{mur07}. For a blackbody spectrum of scattering photons, the average photon energy is
$\epsilon=2.7kT$ and the average number density is $n\simeq19.232\pi\times\frac{1}{(hc)^3}\times(kT)^3$. Throughout the paper, we label the energies in the local frame as $\epsilon$ (e.g. $\epsilon_p,\,\epsilon_\nu$) and as $E$ in the observer's frame.
Hadronic cooling mechanisms mainly contain inelastic $pp$ collisions, the Bethe-Heitler and photomeson production processes in the following ways respectively,
$$p+p\longrightarrow p/n+N\pi$$
$$p+\gamma\longrightarrow p+e^{\pm},$$
$$p+\gamma\longrightarrow p/n+N\pi$$
We can expect muon neutrinos to be produced in $pp$ and $p\gamma$ processes, and electron neutrinos especially from muon decay.
The cooling timescale of inelastic $pp$ scattering is
\begin{equation}
t_{pp}=\frac{1}{c\sigma_{pp}n_p\kappa_{pp}}.
\label{eq:pp}
\end{equation}
We estimate the proton number density as $n_p=\rho_DY_p/m_p\simeq 7.2\times 10^{11}{\,\rm cm}^{-3}$, where
the assumed $Y_p=0.1$ is proton mass fraction of the ejected material since there will be heavy nuclei ejected from a carbon-oxygen white dwarf \citep{sch12, dan14}. The neutrino emission from these nuclei is expected in a low-energy range (see the Appendix) and is out of our interest.
Assuming in each collision a fraction 50\% of the proton energy is lost and using the energy-dependent $pp$ cross section given by \citet{kel06}, we can get the $pp$ cooling timescale.
The photomeson production dominates the cooling at sufficiently high energies, and the timescale can be expressed as
\citep[e.g.,][]{ste68}
\begin{equation}
t_{p\gamma}^{-1}=\frac{c}{2\gamma_p^2}\int_{\bar{\epsilon}_{\rm th}}^\infty{d\bar{\epsilon}\sigma_{p\gamma}(\bar{\epsilon})\kappa_{p\gamma}(\bar{\epsilon})\bar{\epsilon}\int_{\bar{\epsilon}/2\gamma_p}^\infty{\epsilon^{-2}\frac{dn}{d\epsilon}d\epsilon}},
\label{eq:pgamma}
\end{equation}
where $\bar{\epsilon}$ is the photon energy in the rest frame of proton, $\kappa_{p\gamma}$ is the inelasticity and $\bar{\epsilon}_{\rm th}$ is the threshold photon energy for the photomeson production process.
At relative higher energy, the protons start to cool through the BH pair-production process.
The same formula is applied with the replacement of the cross section and threshold energy~\citep{cho92}.
In particular, the high-energy BH cross section is $\sigma_{\rm BH}\approx(28/9)\alpha r_e^2\ln[(2\epsilon_p\epsilon)/(m_pm_ec^4)-106/9]$.
The timescale of the magnetic reconnection acceleration is assumed to be comparable to that
of diffusive shock acceleration, $t_{\rm acc}=\epsilon_p/(e\beta_{\mathrm{rec}}Bc)$, where
$\beta_{\rm rec}$ is the reconnection speed, $\beta_{\rm rec}\sim0.1-0.2$ \citep[e.g.,][]{Giannios10crmag}.
We now plot the inverse of all these timescales as functions of proton energy at $R_D$ in Figure 1. The maximum proton energy can be found by equaling total energy loss time with acceleration time. Since we know that only the $pp$ reaction and photomeson production process will produce neutrinos, the other interactions would provide a strong suppression on the final neutrino spectrum. This suppression factor due to proton cooling can be written as
$\zeta_{\rm CRsup}$~\citep{mur08ph,wd09},
\begin{equation}
\zeta_{\rm CRsup}(\epsilon_\nu)=\frac{t_{pp}^{-1}+t_{p\gamma}^{-1}}{t_{\rm syn}^{-1}+t_{\rm IC}^{-1}+t_{\rm ad}^{-1}+t_{pp}^{-1}+t_{\rm BH}^{-1}+t_{p\gamma}^{-1}}.
\label{eq:suppr}
\end{equation}
\begin{figure}
\plotone{fig1.eps}
\caption{The inverse of cooling timescales for protons at the diffusion radius: blue dashed-synchrotron, blue dotdashed-IC, blue dotted--adiabatic cooling, green dashed--inelastic $pp$ scattering, green dotdashed--BH process, green dotted--photomeson production, black solid--total. Also shown is the reconnection acceleration timescale--red solid.
\label{fig1}}
\end{figure}
Further on, the cooling of the mesons also need to be considered. It is similar to the proton radiative and hadronic cooling times are
\begin{equation}
t_{\rm syn}=\frac{6\pi m_\pi^4c^3}{\sigma_Tm_e^2B_r^2\epsilon_\pi},
\label{eq:pionsync}
\end{equation}
\begin{equation}
t_{\rm had}=1/(c\sigma_{\pi p} n_p \kappa_{\pi p}).
\label{eq:pionhad}
\end{equation}
The pion-proton scattering cross section is $\sigma_{\pi p}\approx 5\times10^{-26}\,\rm cm^2$ at the energies of interest, and the inelasticity is $\kappa_{\pi p}=0.8$ \citep{Oli14}. For the emission region, we use the magnetic field $B_r$ that remains after reconnection events with the energy fraction $\epsilon_B=0.01$ \citep[e.g][]{med99}. For our outflow parameters, the meson goes from decay dominated to radiation cooling dominated. We can define the break energy for neutrinos, $\epsilon_{\nu,\rm brk}$ satisfies $t_{\rm dec}\equiv\gamma_\pi\tau_\pi\sim t_{\pi,\rm cool}$, and thus the suppression factor due to meson cooling is expressed to be
\begin{equation}
\zeta_{\pi \rm sup}(\epsilon_\nu)=\frac{t_{\rm dec}^{-1}}{t_{\rm dec}^{-1}+t_{\rm syn}^{-1}+t_{\rm had}^{-1}}
\label{eq:pionsup}
\end{equation}
Since the mean lifetime of muons is much longer than that of pions, the break energy due to muon
cooling is much lower than $\epsilon_{\nu,\rm brk}$, and all flavors from muon decay can contribute only
at lower energies with no effect on the high-energy spectrum, so for simplicity we ignore the muon
cooling effect.
\section{Neutrino Emission from Nearby Mergers}
\label{sec:indiv}
The neutrino spectrum should follow the initial proton spectrum if there are no
energy-dependent suppression factors, such as Bethe-Heitler or radiative cooling losses.
We shall assume the initial accelerated proton spectrum to be a power-law
$dN_p/d\epsilon_p\propto\epsilon_p^{-s}$ with index $s=2$, and we consider two suppression factors
which modify the neutrino spectrum, see eqs.(\ref{eq:suppr},\ref{eq:pionsup}).
The neutrino luminosity is then
\begin{equation}
\epsilon_\nu L_{\epsilon_\nu}\propto \eta L_B\zeta_{\rm CRsup}(\epsilon_\nu)\zeta_{\pi \rm sup}(\epsilon_\nu).
\label{eq:nulum}
\end{equation}
For one single merger event with nominal parameters, we plot the neutrino spectrum in Figure 2.
At the low energy end, the neutrino spectrum is flat, $\epsilon_\nu L_{\epsilon_\nu}\sim\rm const$.
However, the neutrino flux has been suppressed by synchrotron, IC, adiabatic and BH cooling of protons. For higher energies, the radiative cooling of pions become severe, leading to a strong suppression
of the neutrino spectrum, and a sharp drop feature appears.
\begin{figure}
\plotone{fig2.eps}
\caption{The neutrino spectra of a single merger event. The total dissipated magnetic energy is assumed to be $\mathcal{E}_B=10^{50}\,\rm erg$.
\label{fig2}}
\end{figure}
Based on the neutrino spectrum above, it is of interest to discuss the prospects for the
detection of individual nearby merger events, since the rate of these mergers is relatively high.
Let us consider a merger event at distance $D_L=10\,\rm Mpc$.
We assume that the CR acceleration begins at $R_D$ and ends at roughly $R_{\rm end}\sim 100R_D$, and the CR efficiency is $\eta=0.1$. Therefore for our optimistic case of $\mathcal{E}_B=10^{50}\,\rm erg$, the total local injection power into CRs by white dwarf mergers is $Q_{\rm inj}=\eta\mathcal{R}\mathcal{E}_B\sim10^{45}\,\rm erg\,Mpc^{-3}\,yr^{-1}$, where the rate of white dwarf mergers is set to $\mathcal{R}\sim10^{-4}\,\rm Mpc^{-3}\,yr^{-1}$ \citep[e.g][]{bad12}, comparable to the rate of type Ia supernovae.
The neutrino fluence for single merger event can be expressed as
\begin{equation}
\begin{split}
E_\nu^2\mathcal{F}_\nu(E_\nu)=&\int_{R_D}^{R_{\rm end}}dR\frac{K}{4(1+K)}\times \\
&\frac{d\mathcal{E}_{\rm CR}/dR}{4\pi D_L^2\ln{(E_{p,\rm max}/E_{p,\rm min})}}
\zeta_{\rm CRsup}(E_\nu)\zeta_{\pi \rm sup}(E_\nu),
\label{eq:fluence}
\end{split}
\end{equation}
where $K$ denotes the average ratio of charged to neutral pions, with $K\approx1$ for $p\gamma$ and
$K\approx2$ for $pp$ interactions \citep{mur16a}, the differential CR power $d\mathcal{E}_{\rm CR}/dR=\eta\mathcal{A}R^{-q}$ and
we take $q=2$ in this section. The integration on $R$ gives $\mathcal{E}_{\mathrm {CR}}
\equiv\int_{R_D}^{R_{\mathrm {end}}}{\eta\mathcal{A}R^{-q}}=\eta \mathcal{E}_B\sim10^{49}\,\rm erg$
for the optimistic case.
Using the latest IceCube effective area $A(E_\nu)$ given by \citet{aar15c}, we can now estimate
the number of neutrino events in the IceCube detector. The number of muon neutrinos above 1~TeV is
\begin{equation}
N(>1\mathrm{TeV})=\int_{1\mathrm{TeV}}^{E_{\nu,\max}}{dE_\nu A(E_\nu)\mathcal{F}_\nu(E_\nu)}.
\label{eq:num}
\end{equation}
For our parameters, $N(>1\mathrm{TeV})\sim0.08$. Note that the fluence depends on the inverse of
the distance square, therefore, for a possible IceCube observation, this source should be within
$D_{L}^\prime=(0.08/1)^{1/2}\times10\,\rm Mpc\sim3\,Mpc$ from the earth. This implies that on average
we would have to wait for $((4/3)\pi D_{L}^{\prime3}\times\mathcal{R})^{-1}\sim 10^2\,
\mathcal{R}_{-4}^{-1}\eta_{-1}^{-1}~ \rm yr$. Poisson fluctuation effects or a higher rate or efficiency
(or including related mergers) could conceivably reduce this wait time.
If such merger events occur very close to us, one may ask whether {\it Fermi-LAT} can observe the
GeV $\gamma$-ray signal from the source. Importantly, this WD merger scenario is optically
thick, and they may be considered as hidden sources \citep{mur16a}.
The Thomson optical depth is $\tau_T=n_p\sigma_T R=\tau_T(R_D)(R/R_D)^{-2}$, where $\tau_T(R_D)=n_p\sigma_T R_D\sim30$ is the depth at the diffusion radius. Correspondingly, the photosphere radius is $R_{\rm ph}\sim6R_D$, so we can expect that most of the accompanying high-energy gamma-rays may be absorbed inside the fireball. More accurately, to verify this argument, we need to consider the $\gamma\gamma$ annihilation, for which the cross section is
\begin{equation}
\sigma(\epsilon_\gamma, \epsilon)=
\frac{\pi r_e^2}{2}(1-\beta^2)[2\beta(\beta^2-2)+(3-\beta^4)\ln(\frac{1+\beta}{1-\beta})],
\label{eq:sigmagg}
\end{equation}
where $\beta=(1-\frac{m_e^2c^4}{\epsilon_\gamma\epsilon})^{1/2}$, classical electron radius $r_e=2.818\times10^{-13}\,\rm cm$, and the thermal photon energy $\epsilon=2.7kT=\epsilon(R)$, density $n=n(R)$, so the $\gamma\gamma$ optical depth is $\tau_{\gamma\gamma}=\tau_{\gamma\gamma}(R)=n\sigma(\epsilon_\gamma, \epsilon)R$.
For photons of energy below 1\,GeV, $\tau_{\gamma\gamma}<1$ so they may still escape from the source and could possibly trigger {\it Fermi-LAT}. At a distance $D_L=10\,\rm Mpc$, the accompanying
fluence of GeV photons is of order $\mathcal{F}_\gamma\sim 8\times10^{-3}\,\rm GeV\cdot cm^{-2}$.
The differential sensitivity of {\it Fermi-LAT} at 1\,GeV is about $\mathcal{F}_{\mathrm {sens}}\sim6.4\times10^{-11}
\,\rm erg\cdot cm^{-2}\cdot s^{-1}$. For a merger event of duration $\mathcal{T}\sim 10^4\,\rm s$, at that distance it would be likely to be observed. From the non-detection of such merger events
by LAT we can derive a constraint, namely $\frac{4}{3}\pi d_{\max}^3\times \mathcal{R}\times
\mathcal{T}_{\rm obs}<1$, where the {\it Fermi-LAT} operation time $\mathcal{T}_{\rm obs}\sim 8\,\rm yr$, the WD merger
event rate is $\mathcal{R}=10^{-4}\,\rm Mpc^{-3}\cdot yr^{-1}$ and the maximum distance
$d_{\max}$ can be related to the CR energy $\mathcal{E}_{\rm CR}$ as
$\frac{\mathcal{E}_{\rm CR}}{d_{\max}^2\cdot \mathcal{F}_{\rm sens}}=\frac{10^{49}\rm erg}
{(10\mathrm {Mpc})^2\cdot \mathcal{F}_\gamma}$. This leads to an upper limit of $\mathcal{E}_{\rm CR}
\lesssim2.2\times10^{47}\,\rm erg$. Finally, substituting $\eta=0.1$ we can get a rough constraint
on the dissipated magnetic energy $\mathcal{E}_B\lesssim 2\times10^{48}\,\rm erg$.
However, we neglected the matter attenuation effect in the above estimate, e.g., even GeV photons may undergo BH pair-production interactions and be strongly attenuated \citep{mur15}. We adopt the attenuation coefficients in \citet{mur15} and find that the optical depth is $\tau_M(R_D)\sim 7.7$ for $1\,\rm GeV$ photons.
That means that the flux of GeV photons will fade away sufficiently (by a factor of $e^{-\tau_M}$)
and would not trigger any detection.
If the ejecta is clumpy, some GeV gamma-rays may escape, which may lead to a detectable signal for {\it Fermi-LAT}. However, one should keep in mind that in practice the chance of being observed should be less due to the finite field of view and the response time of the instrument, or other accidental reasons.
Most of the high-energy photons are absorbed and the re-radiation of electron-positron pairs is more likely to be concentrated in the soft X-ray band. Eventually, a significant fraction of the energy would be radiated in the ultraviolet or optical band. Using the diffusion time $t_{\rm diff}\approx \tau_T R/c\sim1.3\times{10}^6$~s, the luminosity of this transient is estimated to be $L_{\gamma}\approx{\mathcal E}_B/t_{\rm diff}\sim10^{42}-{10}^{44}\,\rm erg\,s^{-1}$ or $L_\gamma=4\pi R_D^2ca_RT_D^4\sim3\times10^{42}\,\rm erg\,s^{-1}$, which is
an order of magnitude lower than the peak luminosity of nearby event SN 2011fe ($a \,few\times10^{43}\,\rm erg\,s^{-1}$) but detectable. Thus, the non-detection of bright thermal transients in the optical survey would also enable us to put constraints on the WD merger model especially if the non-thermal injection is large.
\section{Diffuse neutrino flux}
\label{sec:difnuflux}
For a flat CR energy spectrum, the local neutrino energy budget is estimated to be
\begin{equation}
\begin{split}
\epsilon_\nu Q_{\epsilon_{\nu_i}}=&\int_{R_D}^{R_{\rm end}}\frac{K}{4(1+K)}\times\\
&\frac{\eta\mathcal{R}\mathcal{A}R^{-q}dR}{\ln{(\epsilon_{p,\rm max}/\epsilon_{p,\rm min})}}\zeta_{\rm CRsup}(\epsilon_\nu)\zeta_{\pi\rm sup}(\epsilon_\nu).
\label{eq:localnu}
\end{split}
\end{equation}
Defining $Q_\nu$ in the comoving volume, the diffuse neutrino flux per flavor is given by \citep[e.g.,][]{mur16a}
\begin{equation}
E_{\nu}^2\Phi_{\nu_i}=\frac{c}{4\pi H_0}\int_0^{z_{\rm max}}\frac{\epsilon_\nu Q_{\epsilon_{\nu_i}}S(z)}{(1+z)^2\sqrt{\Omega_M(1+z)^3+\Omega_\Lambda}}dz,
\label{eq:diffnu}
\end{equation}
where we assume that the source evolution traces the cosmological star formation history, which can be
expressed as $S(z)=[(1+z)^{-34}+(\frac{1+z}{5000})^3+(\frac{1+z}{9})^{35}]^{-0.1}$ \citep{hop06,yuk08},
$z_{\rm max}=4$, and the cosmology parameters are $H_0=67.8\,{\rm km\,s^{-1}\,Mpc^{-1}}, \Omega_M=0.308$
\citep{pla15}.
Figure 3 shows the diffuse neutrino flux of our model. The black solid line is for the fiducial values
$Q_{\rm inj}=\eta\mathcal{R}\mathcal{E}_B\sim10^{43}\,\rm erg\,Mpc^{-3}\,yr^{-1}$ based on
$\mathcal{E}_B=10^{48}~\rm erg$, while the dashed line is for the optimistic case $Q_{\rm inj}=10^{45}\,
\rm erg\,Mpc^{-3}\,yr^{-1}$ corresponding to $\mathcal{E}_B=10^{50}~\rm erg$.
In the fiducial CR injection scenario, the WD mergers does not contribute much to the diffuse neutrino flux, but the CR injection rate of the WD mergers is highly uncertain. We can see that, for our optimistic injection scenario, this kind of WD merger events is a potentially interesting source to account for the IceCube neutrino data. However, a very large injection power $Q_{\rm inj}\sim 8\times10^{45}\,\rm erg\,Mpc^{-3}\,yr^{-1}$ (dotted line in Figure 3) is needed to reach the flux level at $\sim30\,\rm TeV$, which is unlikely from WD-WD mergers alone, although such numbers might be obtainable if one adds the combined effect of similar merger events such as WD-NS and NS-NS mergers \citep[e.g][]{met12}.
\begin{figure}[t]
\plotone{fig3.eps}
\caption{Diffuse neutrino flux of the WD merger scenario. The solid line represent a nominal CR injection power $Q_{\rm inj}=\eta\mathcal{R}\mathcal{E}_B=(0.1)\times(10^{-4}\,\rm Mpc^{-3}\,yr^{-1})\times(10^{48}\,erg)=10^{43}\,\rm erg\,Mpc^{-3}\,yr^{-1}$ and the dashed line shows an optimistic case of 100 times higher ($\mathcal{E}_B=10^{50}\,\rm erg$). For illustration, we also plot a case in which the injection power needed to reach the IceCube data around $30\,\rm TeV$ is $Q_{\rm inj}\sim8\times10^{45}\,\rm erg\,Mpc^{-3}\,yr^{-1}$ (dotted line). The IceCube data is indicated by blue points \citep{aar15}. Here we take $q=2$ as an example.
\label{fig3}}
\vspace{-1.\baselineskip}
\end{figure}
In Figure 4 we investigate the effects of varying the parameter $q$ on the magnetic reconnection rate
and the final spectrum, taking the optimistic case $\mathcal{E}_B=10^{50}\,\rm erg$ as an example.
We see that the diffuse neutrino flux depends strongly on the magnetic dissipation rate. For larger $q$, the neutrino flux is higher. This is easy to see if we take into account the radial dependence that $t_{pp}^{-1}\propto n_p\propto R^{-3},\,t_{p\gamma}^{-1}\propto n\propto R^{-3}$, and $t_{\rm sync}^{-1}\propto R^{-2},\,t_{\rm ad}^{-1}\propto R^{-1}$. From eq.(\ref{eq:suppr}) we can derive that $\zeta_{\rm CRsup}$ decreases with $R$. If the magnetic energy dissipates more quickly (larger $q$), the shape of the spectrum will be closer to the shape at $R_D$, where $\zeta_{\rm CRsup}$ is larger and thus leads to higher neutrino flux.
\begin{figure}
\plotone{fig4.eps}
\caption{Same as Figure 3 but with different $q$. Green solid: $q=10$; green dashed: $q=5$; green dotted: $q=3$; green dotdashed: $q=2.5$; black solid: $q=2.0$; black dashed: $q=1.5$; black dotted: $q=1.0$. The injection power is set to $Q_{\rm inj}=(0.1)\times(10^{-4}\,\rm Mpc^{-3}\,yr^{-1})\times(10^{50}\,erg)=10^{45}\,\rm erg\,Mpc^{-3}\,yr^{-1}$.
\label{fig4}}
\end{figure}
The neutrino and gamma-ray energy generation rates are conservatively related as
$\epsilon_{\gamma}^2\Phi_{\gamma}=\frac{4}{K}\epsilon_{\nu}^2\Phi_{\nu}|_{\epsilon_\nu=0.5\epsilon_\gamma}$.
The diffuse gamma-ray flux is shown by the red lines in Figure 5 (Only the component attenuated by
$\gamma\gamma$ absorption is relevant, since cascades occur in the synchrotron-dominated regime).
We can clearly see that even for the very optimistic case of an injection power $8\times10^{45}\,\rm erg\,Mpc^{-3}\,yr^{-1}$, the diffuse gamma-ray flux from these sources is below the extragalactic gamma-ray background measured by {\it Fermi-LAT}.
\begin{figure}
\plotone{fig5.eps}
\caption{The ``optimistic'' diffuse gamma-ray flux of the WD merger scenario, which shows that the predicted gamma-ray flux should be far below the extragalactic gamma-ray background measured by {\it Fermi-LAT} (red data points) \citep{ack15}. The cyan area shows the allowed region for the non-blazar gamma-ray flux in \citet{dim16}. Only the $\gamma\gamma$ absorption effect is included in this figure, although the Bethe-Heitler pair production in the ejecta is also likely to be important. The thin red solid line is for the optimistic injection with $Q_{\rm inj}=10^{45}\,\rm erg\,Mpc^{-3}\,yr^{-1}$ and thin red dashed line is for the even more optimistic case of 8 times higher. The thick black lines are the neutrino fluxes, correspondingly. We take $q=2$ as an example.
\label{fig5}}
\end{figure}
\section{Discussions and Conclusions}
\label{sec:disc}
In this work we discussed the high-energy neutrino emission from WD mergers, and showed that they may be potentially interesting sources for IceCube observations.
Since the total CR injection power is highly uncertain, we considered both a nominal case of $\mathcal{E}_B=10^{48}\,\rm erg$ and an optimistic case of $\mathcal{E}_B=10^{50}\,\rm erg$.
This kind of merger events at cosmological distances are essentially hidden sources and would not contribute significantly to the high energy gamma-ray background, thus eliminating the tension that exists with {\it Fermi-LAT} for both the (optically thin) hadronuclear and photohadronic scenarios. Note that the $p\gamma$ efficiency is larger if $\mathcal{E}_B$ is sufficiently larger than $10^{48}\,\rm erg$, since the synchrotron photons due to reconnections may become dominant as target photons, which is beyond the scope of this work but interesting to investigate in future.
Besides, we found that the diffuse neutrino flux depends strongly on the dissipation rate of magnetic energy.
The faster the magnetic energy dissipates, the more high-energy neutrino flux is expected.
Even if these mergers are not responsible for IceCube's diffuse neutrino flux, searches for
high-energy neutrino and gamma-ray signals from a single merger event are useful and interesting.
The neutrino spectra of a single merger event is characterized by a global suppression which
is caused by the other cooling mechanisms competing with $pp$ and $p\gamma$ neutrino production process, and a sharp drop at energies $\gtrsim100\,\rm TeV$ due to the radiative cooling suppression of pions.
For individual WD mergers at Mpc scale distances, the attenuation of the gamma-rays by $\gamma\gamma$ absorption below $1\,\rm GeV$ is negligible, allowing a constraint to be placed on such mergers by the fact that they have not been observed by {\it Fermi-LAT}.
Based on a simple estimate, this constraint implies a total CR energy production per event of $\lesssim10^{47}$ erg, which in turn constrains the contribution that such sources can make to the diffuse neutrino background.
However, this constraint should be relieved if we take into account the matter attenuation effect,
which is likely to occur in a non-clumpy and spherical setup.
It is possible that not all WD mergers result in the type of remnants discussed here. Some WD merger
events may be able to ignite a thermonuclear explosion promptly after merger \citep{1984ApJ...277..355W,1984ApJS...54..335I,dan12}, and these will be identified as
type Ia supernovae (e.g., if mergers are violent) \citep{2010Natur.463...61P,2012ApJ...747L..10P,liu16}. There are also other possible outcomes of WD mergers, such as a massive fast-rotating WD \citep[e.g.,][]{seg97, gar12} or an accretion-induced collapse into a NS \citep[e.g.,][]{sai85, sai04}. In this work we have mainly focused on the merger events that do not explode on a dynamical time \citep{ras09}, yet no such events have been identified so far. Nevertheless, with accumulating {\it Fermi-LAT} and IceCube operation hours, this kind of merger events may finally be detected, allowing a test of our model in the near future.
\acknowledgements
We acknowledge support by the National Basic Research Program of China (973 Program grant 2014CB845800 and the National Natural Science Foundation of China grant 11573014 (D.X. and D.Z.G.), by the program for studying abroad supported by China Scholarship Council (D.X.), by Pennsylvania State University (K.M.) and by NASA NNX13AH50G (P.M.). The work of K. M. is also supported by NSF Grant No. PHY-1620777.
\clearpage
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2,869,038,156,398 | arxiv | \section{Copyright}
\end{document}
\section{Introduction}
Studies have found that while many people think positively about
Battery Electric Vehicles (BEV) and would be interested in buying one in the future, they have many uncertainties and unknowns that need to be addressed before buying one for themselves (e.g., \cite{preston2020}). There are a variety of concerns and beliefs about BEV ownership that contribute to this uncertainty, such as ownership cost, limited range, availability of charging, and social and safety factors. These concerns and beliefs can be due to misperceptions about BEVs or not being aware of facts. Some people who live in a house may believe that charging is inconvenient, and they may not know that BEVs less expensive to maintain than internal combustion engine (ICE) vehicles. Because people have a variety of concerns, and there are many possible interventions to address the range of concerns, interventions will vary in their suitability and effectiveness for a particular individual. When there are many interventions that could be presented, it is important to present the most relevant interventions to a person, rather than showing a fixed set of interventions which may not address their concerns.
In this paper, we examine methods for identifying persuasive interventions for each individual given their background. We use demographic information about an individual to provide a background context. We gather this information through a survey/questionnaire. One of the difficulties in predictions based on survey data from human subjects is the limited data set size for training and testing a model.
We investigate two primary approaches. The first integrates human knowledge with supervised machine learning for predicting barriers to reduce the number of subjects needed for training a model. Interventions to address the identified barriers can then be presented to a subject. For this approach, the experiments are performed on a dataset collected from the survey.
In the second approach, reinforcement learning (RL) is used to directly learn which interventions are most effective. These experiments examine the case when no demographic information is available and a limited set of demographic information is available.
\section{Related Work}
The related work falls into several areas: (1) behavior change (2) studies on consumer preferences towards BEVs (3) prediction models with small training sets.
\subsubsection{Behavior Change}
There has been an increasing acknowledgement of the important role that behavioral change will play for combating environmental challenges \cite{bujold2020}. Techniques and methods which have been developed for improving physical health, mental well-being, educational achievements or productivity levels are currently adapted and applied to resource conservation and climate action \citep{williamson2018climate}. While there are multiple approaches to behavior changes, in this paper we focus on attitudes, which have been considered by many as the core component which behavior change interventions should target \citep{petty2018attitudes}. A popular distinction between different types of interventions is between affective and cognitive interventions \citep{heimlich2008understanding}. Here we are focusing on cognitive interventions alone.
\subsubsection{Consumer Preferences towards BEVs}
BEVs are seen as a major part of the global effort toward carbon neutrality. When combined with clean energy sources, BEVs will lead to a significant reduction in greenhouse emissions. Various legislative programs in the EU, USA, UK and other countries will be limiting or stopping the sales of non-BEVs in the next decade. The popularity of BEVs, however, is still low, and although consumers are generally interested in BEVs, they still perceive multiple challenges and barriers on the path towards owning one \citep{giansoldati2020barriers}.
\subsubsection{Small Training Sets} With human subject survey data, it can be challenging to collect a large data set for training a prediction model based on some of the deep learning models. Although a number of techniques, such as transfer learning (e.g., \cite{zhuang2020xferLearning}), fine-tuning (e.g., \cite{tajbakhsh2016fineTuning}), and zero-shot or few-shot learning \cite{snell2017prototypical}, have been developed to address creating models for with new datasets with very few, if any, labels, these techniques make use of an existing large labeled dataset where the input is similar to the input for the target task, e.g., images of objects represented as pixel values. In contrast, human subject data varies widely depending on the question being studied.
We chose to instead examine models which have been shown to be more efficient to train than deep learning models.
\section{Predicting Barriers to Adoption}
The effect of an intervention in shifting a person’s preferences, such as towards BEVs or PHEVs, can be assessed by either asking a person directly about their preference or showing them a task to indirectly assess it, such as building a car on a website and observing whether they choose to build a BEV, PHEV, hybrid or ICE vehicle. Each presented intervention can affect a person’s preferences, but the effect of an intervention tends to decrease with each presented intervention. Thus it can be difficult to run enough subjects to assess each intervention for the different backgrounds that can influence the effect of an intervention.
An alternative, indirect approach to predicting personalized interventions that might be most effective for a given individual is to instead use elicited factors. Many factors can be assessed per person, thus reducing the number of subjects needed. We hypothesize that factors such as barriers and motivators towards a vehicle engine preference can inform the prediction of interventions.
Examples of barriers are shown in Figure \ref{fig:barrierPlot}. Since such factors are elicited from a person, each person can provide information about their set of barriers, enabling collection of information about the best type of intervention for each person. This contrasts with the constraint of asking a person about the effect of only one intervention or a small number of interventions.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{barrier_plot.png}
\caption{The raw counts of each time a barrier was mentioned by a participant. Note participants could list more than one barrier.}
\label{fig:barrierPlot}
\end{figure}
\subsection{Barriers Dataset}
We ran a study asking 500 drivers in the United States for demographic information in a multiple-choice format and for their top barriers in a free-response format. The top barriers were then coded into a set of barriers, a subset of which are shown in Figure \ref{fig:barrierPlot}. We filtered out all barriers that were mentioned fewer than 10 times, except for `power\_outage', which was kept because it was the only barrier mentioned by one person. This produced the list of barriers in Figure \ref{fig:barrierPlot} that we used in our experiments.
The answers by each subject to set of 14 demographic questions were collected as part of the study. The answers were then considered as features for predicting barriers. Many of the features can be intuitively related to barriers. For example, income bracket may influence how likely `cost\_upfront' (e.g., the price of the vehicle) is to be a barrier, and whether a person lives in a house or apartment or whether a person lives in an urban or rural location may influence whether `charging' is a barrier.
The demographic features we used included whether they live in a house/townhouse or apartment, rent, have charging available, live in an urban or rural area, age bracket, education level, members in household, are employed, income bracket, political leaning, gender, prefer gas or BEVs, and whether their driving includes commutes, long distances, or off road.
\subsection{Prediction and Ranking of Barriers}
Since the intent is to present one intervention at a time to a subject, the interventions should be ranked to inform the presentation order, rather than simply classified as to inform whether to present. We envision that the rank of an intervention roughly corresponds to the importance of the barrier that the intervention addresses. Thus, the barriers should also be predicted and ranked for each person, where the predictions are based on a subject's demographics. There were a total of 18 barriers to be predicted in a multi-label task, since subjects were allowed to list more than one barrier.
We examined two models: (1) simple Multi-Layer Perceptron (MLP) and (2) Support Vector Machine (SVM).
\subsubsection{MLP}
We used 3 layers with a logistic activation function and 10 nodes. The output layer uses a softmax nonlinearity, where each of 18 nodes corresponded to a barrier. The MLP performed multi-label classification using a mean squared error loss.
\subsubsection{SVM} With the small dataset size, we also examined the use of an SVM. To handle the multi-label classification task, a one-vs-rest model was used for each of the 18 barriers. To further reduce the effect of the small dataset, we used cross-validation with stratification of the folds for training and evaluating the model. This enabled us to use 95\% of the data for training when there are at least 20 exemplars for a class. When there were fewer than 20 exemplars, the number of partitions was set to the number of exemplars. Both an RBF and a polynomial kernel of order 3 were examined; we report the results for the polynomial kernel, which had slightly better performance. Since the target barriers were highly imbalanced, the weighting of the target and background class was balanced so that the weights were inversely proportional to their frequencies.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figs/PR_barriers_SVM.png}
\caption{Precision-Recall curves for 18 barriers.}
\label{fig:PRcurves_barrier}
\end{figure}
\subsection{Barrier Prediction Evaluation}
In the survey to collect our dataset, the subjects listed their top barriers but did not rank them. Thus, the evaluation task is similar to an information retrieval task of identifying and ranking relevant documents for presenting to a user, where documents are only labeled as relevant or not. We used Precision and Recall for evaluation, a metric commonly used to evaluate information retrieval tasks.
\subsubsection{Ranking Performance}
Figure~\ref{fig:PRcurves_barrier} shows the precision vs recall curves for the SVM model for each of the 18 barriers, as well as the macro-average over all barriers. The overall macro precision-recall (PR) area under the curve is 0.2968. Note that the barriers that were mentioned more frequently tend to have higher PR values, possibly related to the effect when the number of targets available for training is small.
\begin{figure}
\centering
\subfloat[\centering]{\includegraphics[width=.5\linewidth]{figs/accuracy_barriers_2models_stderr_v2.png}}
\subfloat[\centering]{\includegraphics[width=.5\linewidth]{figs/precision_barriers_2models_stderr_v2.png}}
\caption{(a) Accuracy and (b) precision (error bars indicate one standard error) of the SVM model and a baseline of always predicting each of the top 5 most frequent barriers.}
\label{fig:classification_results}
\end{figure}
The MLP model predicted the same ranking of barriers with the less than 1\% difference in probability values for all individuals. The top-ranked barriers corresponded to the most frequently mentioned barriers. Thus, for a baseline model, we used the most frequently mentioned barriers.
\subsubsection{Classification Performance}
Since the probability of each barrier varies less than 1\% over the users, we evaluated the performance in a classification task.
The classification performance of always predicting the top-5 barriers compared to the classification performance of the SVM models for the same barriers is shown in Figure~\ref{fig:classification_results}. Precision and accuracy were used. (We did not consider recall or hit rate and false alarm rate since the baseline always predicts each of the top-5, so each of the measures is always 1.) These measures are shown for the SVM and the baseline models in Figure~\ref{fig:classification_results}. The mean accuracy and precision of the SVM model for each of the classes is higher then presenting the top 5.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figs/barriers2interventions.png}
\caption{Intervention selection based on predicted barriers.}
\label{fig:barriers2interventions}
\end{figure}
\subsection{Use of Predicted Barriers for Intervention Selection}
From the predicted barriers, the interventions that best address those barriers can then be identified for presentation to a user, as shown in Figure~\ref{fig:barriers2interventions}.
The interventions are manually defined by experts in Behavioral Science and for rich modalities, also by experts in Human Computer Interaction. If there are multiple interventions for a barrier, a study to assess the effectiveness of each intervention (as measured by preference shift) by presenting one or a few interventions per subject can be conducted, and the effectiveness of the intervention (see the MTurk study described in the next section) used when deciding which intervention to present for a predicted barrier.
\begin{figure*}[h!]
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{figs/nondemographic_deterministic_learningcurve.png} &
\includegraphics[width=0.3\textwidth]{figs/nondemographic_stochastic_learningcurve.png} &
\includegraphics[width=0.3\textwidth]{figs/demographic_deterministic_learningcurve.png} \\
\textbf{(a)} & \textbf{(b)} & \textbf{(c)} \\
\end{tabular}
\caption{(a) Learning curve for the non-demographic, deterministic human model. (b) Learning curve for the non-demographic, stochastic human model. (c) Learning curve for the demographic-aware, deterministic human model. In each case, interactions are bundled in episodes of 100 interventions each, and the curve represents the mean of those 100 interventions. Shaded areas around the average represent standard deviations.}
\label{fig:learningcurves}
\end{figure*}
\section{Recommending Interventions with Reinforcement Learning}\label{sec:rl_interventions_and_survey}
To recommend interventions, we would like to use a reinforcement learning model to learn the best intervention for each user. Since reinforcement learning approaches are data-hungry and participant data is expensive to gather, we decided to run an initial data-gathering survey which would inform simulation models. These simulation models could then be used to pre-train reinforcement learning models, and allow us to select the model that is most likely to perform well in deployment. We experimented with a model that takes account demographics and one that does not.
To gather data for our simulation models,
4136 MTurk subjects participated in a survey to measure intervention effectiveness. At the beginning of the survey, the subjects answered the demographic questions and indicated their initial preferences for BEVs. Then they were randomly exposed to one of the 35 interventions we designed and then they answered the BEV preference question again. A preference changes score was computed as the difference between pre- and post-intervention answers. For each intervention, we computed a mean intervention effectiveness. The mean intervention effectiveness ranged from $0.275$ to $13.10$, with an
average of $5.52$, and a standard deviation of $2.9$.
\subsection{Non-Demographic Model}
We developed a reinforcement learning model that learns to select the intervention that is most effective, on average, regardless of demographic information. To train this model, we developed two models to simulate human responses to interventions. The first is a deterministic model, in which the preference shift is always equal to the mean preference shift observed for each intervention. The second is a stochastic model, in which the preference shift is modeled by a Gaussian distribution with mean and variance equal to those observed in the data.
The reinforcement learning agent uses an $\epsilon$-soft policy with a SARSA update. We anneal the exploration parameter $\epsilon$ according to the following schedule, where $t$ is a time variable incremented at every step at which the agent learns.
\begin{equation}
\small \epsilon(t) = \frac{0.4}{1+(1\times 10^{-5}) t}
\end{equation}
The learning rate is similarly annealed according to the following schedule.
\begin{equation}
\small \alpha(t) = \frac{10}{1+(1\times 10^{-2}) t}
\end{equation}
Figure \ref{fig:learningcurves}(a) shows the learning curve for the non-demographic scenario in which the simulation model of the human deterministically shifts its preferences with the mean observed preference shift for each intervention. We have bundled the interventions into ``episodes'' of 100 interventions each, even though the human model is stateless, so that previous interventions do not have an effect on future interactions. The solid blue curve represents the mean reward in each episode, and the shaded area represents the standard deviation of reward during the episode.
Figure \ref{fig:learningcurves}(b) shows the learning curve for the non-demographic scenario with the stochastic human model. We observe that the model converges more slowly than for the deterministic case, and also that its performance has much more variance because the human model itself is stochastic.
\subsection{Demographic-Aware Model}
The models without demographics inform a baseline on the effectiveness of interventions that are conducted independent of demographics. Next, we consider a simulation of a survey in which different participants with different demographics are presented. We selected two genders and two age groups that were well-represented in the data, and created a simulation model for a survey that presented randomly selected individuals from those four possible demographic combinations.
Figure \ref{fig:learningcurves}(c) shows the learning curve for the survey experiment. We note that the learning curve has higher variance than that of Figure \ref{fig:learningcurves}(a), but also that the intervention effectiveness is converging to a higher value. This reflects the effectiveness of targeting interventions with the help of demographic information.
In these experiments, we used two types of demographics and hypothesize performance may improve with additional demographics. However, the improvement would trade off with a requirement for more training episodes to learn the larger space, which we leave for future work.
\section{Discussion and Future Work}
We proposed two approaches to handle the limited amount of human subject data for suggesting interventions to increase electric vehicle preference. We have demonstrated the use of machine learning models to predict barriers to the adoption of electric vehicles from demographic information, as well as to suggest interventions. Our preliminary results demonstrate that better intervention effectiveness can be attained by tailoring the choice of intervention to the demographics of the specific survey participant. In future work, we will validate our trained models on new survey participants, and compare the effectiveness of the interventions suggested by the RL model against interventions recommended by a human expert to address barriers that are predicted for the participant.
|
2,869,038,156,399 | arxiv | \section{Introduction}
\label{sec:intro}
Spoken language understanding (SLU) systems have traditionally been a cascade of an automatic speech recognition (ASR) system converting speech into text followed by a natural language understanding (NLU) system that interprets the meaning, or intent, of the text. In contrast, an end-to-end (E2E) SLU system~\cite{serdyuk2018towards,qian2017exploring,chen2018spoken,ghannay2018end,lugosch2019speech,Haghani2018,caubriere2019curriculum} processes speech input directly into intent without going through an intermediate text transcript.
There are many advantages of end-to-end SLU systems \cite{lugosch2019speech}, the most significant of which is that E2E systems can directly optimize the end goal of intent recognition, without having to perform intermediate tasks like ASR. %
Compared to end-to-end SLU systems, cascaded systems are modular and each component can be optimized separately or jointly (also with end-to-end criteria~\cite{Goel2005,yaman2008integrative,Haghani2018}). One key advantage of modular components is that each component can be trained on data that may be more abundant. For example, there is a lot of transcribed speech data that can be used to train an ASR model. In comparison, there is a paucity of speech data with intent labels, and intent labels, unlike words, are not standardized and may be inconsistent from task to task. Another advantage of modularity is that components can be re-used and adapted for other purposes, e.g. an ASR service used as a component for call center analytics, closed captioning, spoken foreign language translation, etc.
While end-to-end SLU is an active area of research, currently the most promising results under-perform or just barely outperform traditional cascaded systems~\cite{caubriere2019curriculum,qian2018speech}. One reason is that deep learning models require a large amount of appropriate training data. To train an end-to-end speech-to-intent model, we need intent-labeled speech data, and such data is usually scarce. \cite{caubriere2019curriculum,tomashenko2019investigating} address this problem using a curriculum and transfer learning approach whereby the model is gradually trained on increasingly relevant data until it is fine-tuned on the actual domain data. Similarly,~\cite{lugosch2019speech,bhosale2019end} advocate pre-training an ASR model on a large amount of transcribed speech data to initialize a speech-to-intent model that is then trained on a much smaller training set with both transcripts and intent labels.
Training data for end-to-end SLU is much scarcer than training data for ASR (speech and transcripts) or NLU (text and semantic annotations). In fact, there are many relevant NLU text resources and models (e.g. named entity extraction) and information in the world is mostly organized in text format, without corresponding speech. As SLU becomes more sophisticated, it is important to be able to leverage such text resources in end-to-end SLU models. Pre-training on ASR resources is straightforward, but it is less clear how to use NLU resources. This problem has not been adequately addressed in the literature that we are aware of.
In this paper, we pose an interesting question: for an end-to-end S2I model, how can we take advantage of text-to-intent training data without speech? There are many possible approaches, but we focus on two methods in this paper. In the first, we jointly train the speech-to-intent model and a text-to-intent model, encouraging the acoustic embedding from the speech model to be close to a fine-tuned BERT-based text embedding, and using a shared intent classification layer. The second method involves data augmentation,
where we convert the additional text-to-intent data into synthetic speech-to-intent data using a multi-speaker text-to-speech (TTS) system.
To evaluate our methods, we performed carefully controlled experiments. First we built
strong baselines with conventional cascaded systems, where the acoustic, language, and intent classification models are adapted on in-domain data. We also built a strong speech-to-intent model using pre-trained acoustic models for initialization and multi-task training to optimize ASR and intent classification objectives, producing competitive results compared with the cascaded system. We evaluated these models on varying amounts of speech and text training data. Through these experiments, we seek to answer two questions. First, can we improve S2I models with additional text-to-intent data? Second, how does that compare to having actual speech-to-intent data?
\section{Training Speech-to-Intent Systems}
\label{sec:s2i}
End-to-end speech-to-intent systems directly extract the intent label associated with a spoken utterance without explicitly transcribing the utterance. However, it is still useful to derive an intermediate ASR embedding that summarizes the message component of the signal for intent classification. An effective approach to achieve this goal is to train the S2I classifier starting from a pre-trained ASR system.
ASR pre-training is also beneficial since intent labels are not required in this step; hence, we can use ASR speech data instead of specific in-domain intent data, which is usually limited.
\begin{figure}[t]
\begin{center}
\includegraphics[width=6cm]{asr_intent.jpg}
\end{center}
\vspace{-4mm}
\caption{A S2I system with pre-trained ASR}
\label{fig:asr_pretrain}
%
\end{figure}
In our work, we use a phone-based connectionist temporal classification (CTC)~\cite{graves2006connectionist} acoustic model (AM) trained on general speech data as the base ASR system. First, we initialize the S2I model with this model and adapted it to the in-domain data. Once the adapted ASR system is trained, it is modified for intent classification using speech that was transcribed and also annotated with intent labels. As shown in Fig.~\ref{fig:asr_pretrain}, to construct the intent recognition system we modify the model by adding a classification layer that predicts intent targets. Unlike phone targets which are predicted by the ASR system at the frame level, intent targets span larger contexts. In our case, we assume that each training utterance corresponds to a {\em single}\/ intent, although it might be composed of several words. To better capture intents at the utterance level, we derive an acoustic embedding (AE) corresponding to each training utterance. This embedding is computed by time averaging all the hidden states of the final LSTM layer to summarize the utterance into one compact vector that is used to predict the final intent. The final fully connected layer introduced in this step to process the acoustic embeddings can be viewed as an intent classifier. While training the network to predict intent, given that transcripts for the utterances are also available, we continue to refine the network to predict ASR targets as well. With this multi-task objective, the network adapts its layers to the channel and speakers of the in-domain data.
During test time, only the outputs of the intent classification layer are used, while the ASR branch is discarded. To improve robustness of the underlying speech model, we also employ data augmentation techniques for end-to-end acoustic model training, namely speed and tempo perturbation, in all our experiments.
\section{Improvements using Text-to-Intent data}
\label{sec:limited_data}
In practice, we expect that end-to-end S2I classifiers will be trained in conditions where there is a limited amount of transcribed S2I data and significantly more text-to-intent (T2I) data. To investigate this setting, we develop two approaches to use additional text data for building S2I systems.
\subsection{Leveraging pre-trained text embedding}
\label{sec:bert}
Leveraging text embedding (TE) from models pre-trained on large amounts of data, such as BERT~\cite{devlin2018bert} and GPT-2~\cite{radford2019language}, has recently improved performance in a number of NLU tasks. In this paper, we use BERT-based text embeddings to transfer knowledge from text data into a speech-to-intent system. The text embeddings are used to ``guide'' acoustic embeddings which are trained with a limited amount of S2I data, in the same spirit as learning a shared representation between modalities~\cite{ngiam2011multimodal,andrew2013deep,wang2015deep,harwath2018jointly}.
We employ the following steps to train the final model. \\
{\bf (A) T2I model pre-training:} As in the standard process outlined in the original BERT paper~\cite{devlin2018bert}, we first fine-tune BERT on the available text-to-intent data using a masked language model (LM) task as the intermediate task. The model is further fine-tuned with intent labels as the target classification task before the representation of the special token \texttt{[CLS]} is used as the text embedding of an utterance. \\
{\bf (B) ASR pre-training for S2I model:} As described in Section~\ref{sec:s2i}, the base ASR model is trained on non-parallel ASR data and subsequently adapted using an augmented S2I data set. This step adapts the model to the acoustic conditions of the intent data to extract better acoustic embeddings. Next, multi-task training further fine-tunes the acoustic model to generate embeddings that are better for intent classification. We now have an initial S2I system that has seen only limited novel intent content. Note that we add a fully connected layer before the classifier to make sure the dimensions of the acoustic embedding and text embedding match for joint-training in the next step. \\
{\bf (C) Full S2I model training:} The final S2I classifier is then assembled by combining the fine-tuned T2I classifier (step A) and the pre-trained S2I system (step B). We jointly train the fine-tuned BERT model with the pre-trained ASR acoustic model in an attempt to leverage the knowledge extracted from larger amounts of text data to improve the quality of the acoustic embedding for intent classification. The training framework is shown in Figure~\ref{fig:joint-train}. We extract text embeddings from the fine tuned BERT model with the reference text as input. Acoustic embeddings are also extracted in parallel from a corresponding acoustic signal. These two embeddings are used to train a shared intent classification layer which has been initialized from the text-only classification task described above. Given that the text embedding comes from a well trained extractor, the acoustic embeddings are explicitly forced to match the better text embeddings. We hypothesize that this matching will also allow the shared classifier layer to train better. During test time, we only use the acoustic branch for intent inference.
To achieve these goals, a training procedure that optimizes two separate loss terms is employed. %
The first loss term corresponds to a composite cross-entropy intent classification loss derived by using the text embeddings, ${L}_{CE}(TE)$, and the acoustic embeddings, ${L}_{CE}(AE)$, separately to predict intent labels using the shared classifier layer. In the combined classification loss, the text-embedding classification loss is scaled by a weight parameter $\alpha$.
The second loss is the mean squared error (MSE) loss between the text embedding and acoustic embedding ${L}_{MSE}(AE,TE)$. It is important to note that while the gradients from the combined classification loss are propagated back to both the text and acoustic embedding networks, the MSE loss is only back-propagated to the acoustic side because we presume that the acoustic embeddings should correspond closely to the BERT embeddings, which have been trained on massive quantities of text and perform better on intent classification. On the speech branch the minimized loss is \mbox{${L}_{MSE}(AE,TE)+{L}_{CE}(AE)+\alpha{L}_{CE}(TE)$}, while the loss on the text branch is \mbox{${L}_{CE}(AE)+\alpha{L}_{CE}(TE)$}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=7.5cm]{joint_training.jpg}
\end{center}
\vspace{-3mm}
\caption{Joint-training of the S2I system with text embeddings}
\label{fig:joint-train}
\vspace{-5mm}
\end{figure}
\subsection{Using text data for speech data augmentation}
Instead of using available text data for pre-training the T2I system, we also try converting the text data to speech using a multi-speaker TTS system. %
Like in Section~\ref{sec:s2i}, the S2I classifier in this case is trained in two steps. \\
{\bf (A) ASR pre-training:} The base ASR model (trained on non-parallel ASR data) is first adapted using the speech portion of the S2I data set. \\
{\bf (B) Training with synthesized data:} The TTS-synthesized data is used along with the limited amount of S2I data, without augmentation, for training.
Section~\ref{sec:tts} describes the generation of the TTS data set in detail.
Compared with tying text and acoustic embeddings, S2I data augmentation might be more effective at training the ``embedding'' to intent classifier layers of the neural network, because novel (utterance, intent) pairs are used in the final training phase rather than just in pre-training.
\section{Experiments and Results}
\label{sec:expt}
\subsection{Experimental setup}
Experiments were performed on a corpus consisting of call center recordings of open-ended first
utterances by customers describing the reasons for their calls, which is described in~\cite{Goel2005}. The
8kHz telephony speech data was manually transcribed and labeled with
correct intents. The corpus contains real customer spontaneous utterances,
not crowd-sourced data of people reading from a script, and includes a variety
of ways customers naturally described their intent. For example,
the intent ``BILLING'' includes short sentences such as ``{\em billing}''
and longer ones such as ``{\em i need to ask some questions on uh to get
credit on an invoice adjustment}.''
The training data consists of 19.5 hours of speech that was first divided
into a training set of 17.5 hours and a held-out set of 2 hours. The held-out set
was used during training to track the objective function and
tune certain parameters like the initial learning rate. In addition to
the 17.5-hour training set (which we call {\em 20hTrainset}, containing 21849
sentences, 145K words), we also extracted a 10\% subset (1.7h) for
low-resource experiments (which we call {\em 2hTrainset}, containing 2184
sentences, 14K words). We augmented the training data via speed and tempo perturbation (0.9x and 1.1x), so {\em 2hTrainset}\/ finally contains about 8.7 hours and {\em 20hTrainset}\/ about 88 hours of speech. The {\em devset}\/ consists of 3182 sentences (2.8 hours) and was used for hyperparameter tuning, e.g., tuning the acoustic weight to optimize the word error rate (WER). A
separate data set containing 5592 sentences (5h, 40K words) was used
as the final {\em testset}.
In the training set, each sentence had a single intent, and there were
29 intent classes. The testset contains additional unseen
intent classes and multiple intents per sentence, as naturally happens
in real life. For simplicity, in this paper we do not address these
cases and always count such sentences as errors when calculating
intent accuracy; they account for about 1\% of the utterances. The {\it testset}\/ has an
average of 7 words per utterance, with the longest sentence being over 100
words long. 70\% of the sentences are unique (not repetitions).
\subsection{Pre-trained models}
\subsubsection{ASR CTC model}
\label{sec:asr_sys}
We pre-train an ASR system on the 300-hour Switchboard English conversational speech corpus. The AM is a 6-layer bidirectional LSTM network with every layer containing 640 LSTM units per direction. The AM is trained using CTC loss over 44 phones and the blank symbol. Our AM training recipe follows our prior work. We first perform speed and tempo perturbation (0.9x and 1.1x) resulting in a 1500-hour audio data set. We train the AM for 20 epochs using CTC loss, followed by 20 epochs of soft forgetting training~\cite{audhkhasi2019forget}, followed by 20 epochs of guided training~\cite{kurata2019guiding}. We use sequence noise injection~\cite{saon2019sequence} and SpecAugment~\cite{park2019specaugment} throughout the training to provide on-the-fly data augmentation, and we also use a dropout probability of 0.5 on the LSTM output at each layer. %
\subsubsection{BERT based T2I model}
We start with pre-trained BERT\textsubscript{base} model of~\cite{devlin2018bert}. Using the implementation introduced in \cite{wolf2019transformers}, we first pre-train using a masked LM target with learning rate $3\text{e-}5$ for 10 epochs, followed by 3 epochs of fine-tuning on the intent classification task with learning rate $2\text{e-}5$. This text-to-intent BERT based classifier trained on {\em 20hTrainset}\/ gives 92.0\% accuracy on human-generated reference transcripts.
\subsubsection{TTS system}
\label{sec:tts}
The TTS system architecture is similar to the single speaker system described in \cite{Kons2019}. It is a modular system based on three neural-net models: one to infer prosody, one to infer acoustic features, and an LPCNet~\cite{Valin2019LPCNet} vocoder.
The main difference between the single and multi-speaker systems is that both the prosody and the acoustic networks are converted to multi-speaker models by conditioning them on a speaker embedding vector. Each of the three models was independently trained on 124 hours of 16KHz speech from 163 English speakers. The speaker set is composed of 4 high quality proprietary voices with more than 10 hours of speech, 21 VCTK~\cite{VCTK2017} voices, and 138 LibriTTS~\cite{Zen2019LibriTTSAC} voices. During synthesis, for each sentence we select a random speaker out of the known speakers set. We then synthesize the sentence with this voice. Finally, the samples are downsampled to 8KHz to match the S2I audio sampling rate.
\subsection{Results}
We first establish the strongest possible baseline results for the conventional cascaded system where the ASR and T2I models are trained separately. %
Using the ASR system described above and a BERT T2I model trained on the domain data {\it 20hTrainset}, we obtained 21.9\% WER and 73.4\% intent classification accuracy (IntAcc) for {\it testset}, as shown in Table~\ref{tab:baseline}. %
The same T2I model has 92.0\% intent accuracy on human reference transcripts; thus, there is a significant degradation in accuracy with speech input due to ASR errors. %
Both the AM and LM of this system are then adapted using the domain data {\it 20hTrainset}. Table~\ref{tab:baseline} shows that such adaptation
dramatically improved both WER and intent accuracy, which increased from 73.4\% to 89.7\%. There is now only about a 2\% accuracy gap between using human transcripts (92.0\%) and using ASR outputs (89.7\%). In the low-resource scenario, adapting the AM and LM on {\it 2hTrainset} and also training T2I on only {\it 2hTrainset} results in intent accuracy of 82.8\%.
\begin{table}%
%
\centering
\begin{tabular}{@{}lllcc@{}} \toprule
\multicolumn{1}{c}{\bf AM} & \multicolumn{1}{c}{\bf LM} & \multicolumn{1}{c}{\bf T2I} & {\bf WER} & \bf{IntAcc} \\ \cmidrule(lr){1-3}
\cmidrule(lr){4-5}
unadapted & unadapted & 20hTrainset & 21.9\% & 73.4\% \\
20hTrainset & 20hTrainset & 20hTrainset & 10.5\% & 89.7\% \\
2hTrainset & 2hTrainset & 2hTrainset & 11.6\% & 82.8\% \\ \bottomrule
\end{tabular}
\caption{WER and IntAcc (intent accuracy) of baseline cascaded systems (ASR followed by T2I) for speech-to-intent recognition}
\label{tab:baseline}
\end{table}
As shown in Table~\ref{tab:E2E}, when paired speech-to-intent data is used our proposed E2E S2I approach gives comparable accuracy as the cascaded approach, both with full training data ({\it 20hTrainset}) or in the low-resource setting ({\it 2hTrainset}).
In the low-resource scenario where only a limited amount of speech is available ({\em 2hTrainset}), frequently one may have extra text data with intent labels but no corresponding speech data. For the cascaded system, it is straightforward to train different components (AM, LM, T2I) with whatever appropriate data is available. %
Table~\ref{tab:extraText} shows how the intent accuracy varies when the full {\it 20hTrainset} (but not the speech) is available as text-to-intent data. By training the LM and T2I on this data, the intent accuracy increased to 89.6\%, basically matching the best accuracy of 89.7\%, where the AM is also adapted on {\it 20hTrainset}. The third row in Table~\ref{tab:extraText} shows that if only the T2I model is trained on {\it 20hTrainset}, the accuracy is still quite high: 88.9\%. Comparing results from the first and third rows, we observe that the accuracy difference between training the T2I model on {\it 2hTrainset}\/ versus {\it 20hTrainset}\/ is about 6\%, accounting for most of the gap between the full resource and limited resource performance. In other words, the T2I model is the weakest link, the component that is most starved for additional data. For the AM and LM, because they were pre-trained on plenty of general data, even just 2h of adaptation is almost enough, but the intent classifier needs much more data. This makes sense because the intent classification task can be quite domain specific.
\begin{table}%
\centering
\begin{tabular}{@{}lcc@{}} \toprule
& \multicolumn{1}{c}{\bf 20hTrainset} & \multicolumn{1}{c}{\bf 2hTrainset} \\ \midrule
Cascaded(ASR+T2I) & 89.7\% & 82.8\% \\
E2E CTC & 89.8\% & 82.2\% \\
\bottomrule
\end{tabular}
\caption{End-to-end speech-to-intent classification accuracy.}
\label{tab:E2E}
\end{table}
\begin{table}%
\centering
\begin{tabular}{@{}lllcc@{}}
\toprule
\multicolumn{1}{c}{\bf AM} & \multicolumn{1}{c}{\bf LM} & \multicolumn{1}{c}{\bf T2I} & {\bf WER} & \bf{IntAcc} \\ \cmidrule(lr){1-3}
\cmidrule(lr){4-5}
2hTrainset & 2hTrainset & 2hTrainset & 11.6\% & 82.8\% \\
2hTrainset & 20hTrainset & 20hTrainset & 10.3\% & 89.6\% \\
2hTrainset & 2hTrainset & 20hTrainset & 11.6\% & 88.9\% \\
\bottomrule
\end{tabular}
\caption{Limited speech resources with extra text-to-intent data.}
\label{tab:extraText}
\end{table}
For the end-to-end speech-to-intent system, leveraging the text-to-intent data is not straightforward. If it were unable to take advantage of such data, it would be at a significant 6-7\% accuracy disadvantage compared to the cascaded system in this scenario.
\begin{table}%
%
\centering
\begin{tabular}{@{}lc@{}}
\toprule
\multicolumn{1}{c}{\bf Method} & \multicolumn{1}{c}{\bf IntAcc} \\ \midrule
E2E S2I system trained on 2hTrainset & 82.2\% \\ \midrule
Joint training tying speech/text embeddings & 84.7\% \\
Adding synthetic multi-speaker TTS speech & 87.8\% \\
Joint training + adding synthetic speech & 88.3\% \\ \midrule
E2E S2I system trained on 20hTrainset & 89.8\% \\
\bottomrule
\end{tabular}
\caption{End-to-End models using extra text-to-intent data to recover accuracy lost by switching from {\em 20hTrainset}\/ to {\em 2hTrainset}.}
\label{tab:extraTextE2E}
\end{table}
In the next set of experiments, in Table~\ref{tab:extraTextE2E}, we show results from leveraging extra text data to reduce the gap caused by having less S2I training data.
Our first approach described in Section~\ref{sec:bert} and illustrated in Figure~\ref{fig:joint-train} ties the speech and text embeddings and trains the intent classifier on speech and text embeddings. By joint training end-to-end S2I CTC model with BERT fine tuned on full text-to-intent data, we observe accuracy improvement from 82.2\% to 84.7\%.
In our second approach, we took the extra text-to-intent data ({\em 20hTrainset}) and generated synthetic speech with multi-speaker TTS to create new speech-to-intent data. Adding this synthetic data to {\em 2hTrainset} to train the E2E model resulted in an intent accuracy of 87.8\%, a significant improvement from 82.2\%. If we generated the synthetic data using single speaker TTS, the accuracy was roughly the same: 87.3\%. These results were quite surprising. %
We hypothesize that this improvement is due to the S2I model learning new semantic information (``embedding''-to-intent) from the new synthetic data rather than adapting to the acoustics. Therefore it was not necessary to generate a lot of variability in the speech (e.g. speakers, etc.) with the TTS data. Running ASR on the TTS speech, the WER was very low, around 4\%, so there was little mismatch between the TTS speech and the underlying ASR model. One can imagine that the speech encoder portion of the model removes speaker variability, etc. to produce an embedding that depends largely on just the word sequence; hence, any representative TTS speech would be sufficient because the weakest link was the intent classifier. Finally, if we combine the two methods, joint training text and speech embeddings with synthetic TTS speech data, we obtain modest gains, resulting in an intent classification accuracy of 88.3\%.
\section{Conclusion}
\label{sec:conclusion}
End-to-end spoken language understanding systems require paired speech and semantic annotation data, which is typically quite scarce compared to NLU resources (semantically annotated text without speech). We have made progress on this important but neglected issue by showing that an end-to-end speech-to-intent model can learn from annotated text data without paired speech. By leveraging pre-trained text embeddings and data augmentation using speech synthesis, we are able to improve the intent classification error rate by over 60\% and achieve over 80\% of the improvement from paired speech-to-intent data.
\section{Acknowledgements}
We thank Ellen Kislal for her initial studies on E2E S2I.
\vfill\pagebreak
\bibliographystyle{IEEEbib}
|
2,869,038,156,400 | arxiv | \section{Introduction}
\label{intro}
Any continuous physical model is empirically equivalent to a certain finite model.
This is widely used in practice:
solutions of differential equations by the finite difference method or by using truncated series are typical examples.
It is often believed that continuous models are ``more fundamental'' than discrete or finite models.
However, there are many indications that nature is fundamentally discrete at small (Planck) scales, and is possibly finite.%
\footnote{The total number of binary degrees of freedom in the Universe is about \Math{~ 10^{122}} as estimated via the holographic principle and the Bekenstein--Hawking formula.}
Moreover, description of physical systems by, e.g., differential equations can not be fundamental in principle, since it is based on approximations of the form
\Math{f\vect{x}\approx{}f\vect{x_0}+\nabla{f\vect{x_0}}\Delta{}x}.
In this paper we consider some approaches to constructing discrete combinatorial models of quantum evolution.
\par
The classical description of a \emph{reversible} dynamical system looks schematically as follows.
There are a set \Math{W} of states%
\footnote{The set \Math{W} often has the structure of a set of functions: \Math{W=\Sigma^X}, where \Math{X} is a \emph{space}, and \Math{\Sigma} is a set of \emph{local} states.}
and a group \Math{G_\mathrm{cl}\leq\SymG{W}} of transformations (bijections) of \Math{W}.
Evolutions of \Math{W} are described by sequences of group elements \Math{g_t\in{}G_\mathrm{cl}} parameterized by the {continuous} time \Math{t\in\Time=\ordset{t_a,t_b}\subseteq\R}.
The observables are functions \Math{h: W\rightarrow\R}.
\par
An arbitrary set \Math{W} can be ``quantized'' by assigning numbers from a number system \Math{\mathcal{F}} to the elements \Math{w\in{}W}, i.e., by interpreting \Math{W} as a basis of the module \Math{\mathcal{F}^{\otimes{W}}}.
The quantum description of a dynamical system assumes that the module spanned by the set of classical states \Math{W} is a Hilbert space \Math{\Hspace_W} over the field of complex numbers, i.e., \Math{\mathcal{F}=\C}.
The transformations \Math{g_t} and the observables \Math{h} are replaced by {unitary}, \Math{U_t\in\Aut{\Hspace_W}}, and {Hermitian}, \Math{H}, operators on \Math{\Hspace_W}, respectively.
A constructive version of quantum description is reduced to the following:
\begin{itemize}
\item
time is \emph{discrete} and can be represented as a sequence of integers, typically \Math{\Time = \ordset{0,1,\ldots,T}};
\item
the set \Math{W} is finite and, respectively, the space \Math{\Hspace_W} is finite-dimensional;
\item
the general unitary group \Math{\Aut{\Hspace_W}} is replaced by a finite group \Math{G};
\item
the field \Math{\C} is replaced by \Math{\period}th cyclotomic field \Math{\Q_\period}, where \Math{\period} depends on the structure of \Math{G};
\item
the evolution operators \Math{U_t} belong to a unitary representation of \Math{G} in the Hilbert space \Math{\Hspace_W} over \Math{\Q_\period}.
\end{itemize}
\par
It is clear that a single unitary evolution is \emph{not sufficient} for describing the physical reality.
Such evolution is nothing more than a physically trivial change of coordinates (a symmetry transformation).
This means that observable values or relations, being invariant functions of states, do not change with time.
As an example, consider a unitary evolution of a pair of state vectors: \Math{\barket{\varphi_1}=U\barket{\varphi_0},} \Math{\barket{\psi_1}=U\barket{\psi_0}}.
For the scalar product we have \Math{\inner{\varphi_1}{\psi_1}=\brabar{\varphi_0}U^{-1}U\barket{\psi_0}\equiv\inner{\varphi_0}{\psi_0}}.
There are two ways to obtain observable effects in the scenario of unitary evolution:
(a) in quantum mechanics measurements are described by non-unitary operators --- projections into subspaces of the Hilbert space;
(b) in gauge theories collections of evolutions are considered, and comparing results of different evolutions can lead to observable effects (in the case of a non-trivial gauge holonomy).
\par
The role of observations in quantum mechanics is very important --- it is sometimes said that ``observation creates reality''.%
\footnote{The phrase is often attributed to John Archibald Wheeler.}
We pay special attention to the explicit inclusion of observations in the models of evolution.
While the states of a system are fixed in the moments of observation, there is no objective way to trace the identity of the states between observations.
In fact, all identifications --- i.e., parallel transports provided by the gauge group which describes symmetries of the states --- are possible.
This leads to a kind of fundamental indeterminism.
To handle this indeterminism we need a way to describe statistically collections of parallel transports.
Then we can formulate the problem of finding trajectories with maximum probability that pass through a given sequence of states fixed by observations.
In a properly formulated model, the principle of selection of the most probable trajectories should reproduce in the continuum limit the principle of least action.
\section{Constructive description of quantum behavior}
\label{sec-quantum}
The transition from a continuous quantum problem to its constructive counterpart can be done by replacing a unitary group of evolution operators with some finite group.
To justify such a replacement \cite{KornyakPEPAN} one can use the fact from the theory of quantum computing that any unitary group contains a dense finitely generated subgroup.
This \emph{residually finite} \cite{Magnus} group has infinitely many finite homomorphic images.
The infinite set of non-trivial homomorphisms allows to find a finite group that is empirically equivalent to the original unitary group in any particular problem.
\subsection{Permutations and natural quantum amplitudes}
\label{quantperm
As it is well known, any representation of a finite group is a subrepresentation of some permutation representation.
Namely, a representation \Math{\repq} of \Math{\mathsf{G}} % Whole symmetry group: \mathcal{W}\mathrm{G} in a \Math{\adimH}-dimensional Hilbert space \Math{\Hspace_{\adimH}} can be embedded into a permutation representation \Math{\regrep} of \Math{\mathsf{G}} % Whole symmetry group: \mathcal{W}\mathrm{G} in an \Math{\mathsf{N}} % Size of whole set of states:{N_\wS}-dimensional Hilbert space \Math{\Hspace_{\mathsf{N}} % Size of whole set of states:{N_\wS}}, where \Math{\mathsf{N}} % Size of whole set of states:{N_\wS\geq\adimH}.
The representation \Math{\regrep} is equivalent to an action of \Math{\mathsf{G}} % Whole symmetry group: \mathcal{W}\mathrm{G} on a set of things \Math{\Omega} % Whole set of states:\mathrm{S=\set{\omega} % Element of whole set of states:\mathrm{s_1,\ldots,\omega} % Element of whole set of states:\mathrm{s_\mathsf{N}} % Size of whole set of states:{N_\wS}} by permutations.
If \Math{\adimH=\mathsf{N}} % Size of whole set of states:{N_\wS} then \Math{\repq\cong\regrep}. Otherwise, if \Math{\adimH<\mathsf{N}} % Size of whole set of states:{N_\wS}, the embedding has the structure
\MathEq{
\transmatr^{-1}\regrep\transmatr
=\Vtwo{
\left.
\begin{aligned}
\!\IrrRep{1}&\\[-2pt]
&\hspace{8pt}\mathrm{V}
\end{aligned}
\right\}\Hspace_{\mathsf{N}} % Size of whole set of states:{N_\wS-\adimH}
}{
\left.
\hspace{29pt}
{\repq}
\right\}\Hspace_{\adimH}
},\hspace{10pt}
\Hspace_{\mathsf{N}} % Size of whole set of states:{N_\wS} = \Hspace_{\mathsf{N}} % Size of whole set of states:{N_\wS-\adimH}\oplus\Hspace_{\adimH}.
}
Here \Math{\IrrRep{1}} is the trivial one-dimensional representation. It is a mandatory subrepresentation of any permutation representation. \Math{\mathrm{V}} is an optional subrepresentation.
We can treat the unitary evolutions of data in the spaces \Math{\Hspace_{\adimH}} and \Math{\Hspace_{\mathsf{N}} % Size of whole set of states:{N_\wS-\adimH}} {independently}, since both spaces are invariant subspaces of \Math{\Hspace_{\mathsf{N}} % Size of whole set of states:{N_\wS}}.
\par
The embedding into permutations provides a simple explanation of the presence of complex numbers and complex amplitudes in the formalism of quantum mechanics.
We interpret complex quantum amplitudes as
projections onto invariant subspaces of vectors with natural components for a suitable permutation representation \cite{KornyakPEPAN,Kornyak12,Kornyak13a}.
It is natural to assign natural numbers --- multiplicities --- to elements of the set \Math{\Omega} % Whole set of states:\mathrm{S} on which the group \Math{\mathsf{G}} % Whole symmetry group: \mathcal{W}\mathrm{G} acts by permutations.
The vector of multiplicities,
\MathEq{\barket{n} = \Vthree{n_1}{\vdots}{n_{\mathsf{N}} % Size of whole set of states:{N_\wS}},}
is an element of the \emph{module} \Math{\natmod_\mathsf{N}} % Size of whole set of states:{N_\wS = \mathbb{N}^\mathsf{N}} % Size of whole set of states:{N_\wS}, where \Math{\mathbb{N}=\set{0,1,2,\ldots}} is the \emph{semiring} of natural numbers.
The permutation action defines the \emph{permutation representation} of \Math{\mathsf{G}} % Whole symmetry group: \mathcal{W}\mathrm{G} in the module \Math{\natmod_\mathsf{N}} % Size of whole set of states:{N_\wS}.
Using the fact that all eigenvalues of any linear representation of a finite group are \emph{roots of unity}, we can turn the module \Math{\natmod_\mathsf{N}} % Size of whole set of states:{N_\wS} into a Hilbert space \Math{\Hspace_{\mathsf{N}} % Size of whole set of states:{N_\wS}}.
We denote by \Math{\mathbb{N}_\period} the \emph{semiring} formed by linear combinations of \Math{\period}th roots of unity with natural coefficients.
The so-called \emph{conductor} \Math{\period} is a divisor of the \emph{exponent}%
\footnote{The exponent of a group is defined as the least common multiple of the orders of its elements.}
of \Math{\mathsf{G}} % Whole symmetry group: \mathcal{W}\mathrm{G}.
In the case \Math{\period>1} the semiring \Math{\mathbb{N}_\period} becomes a \emph{ring of cyclotomic integers}.
The introduction of the \emph{cyclotomic field} \Math{\Q_\period} as the field of fractions of the ring \Math{\mathbb{N}_\period} completes the conversion of the module \Math{\natmod_\mathsf{N}} % Size of whole set of states:{N_\wS} into the Hilbert space \Math{\Hspace_{\mathsf{N}} % Size of whole set of states:{N_\wS}}.
If \Math{\period>2}, then \Math{\Q_\period} is empirically equivalent to the field of complex numbers \Math{\C} in the sense that \Math{\Q_\period} is a dense subfield of \Math{\C}.
\subsection{Measurements and the Born rule}\label{Born
A quantum measurement is, in fact, a selection among all the possible state vectors that belong to a given subspace of a Hilbert space.
This subspace is specified by the experimental setup.
The probability to find a state vector in the subspace is described by the Born rule.
There have been many attempts to derive the Born rule from other physical assumptions --- the Schr\"{o}dinger equation, Bohmian mechanics, many-worlds interpretation, etc.
However, the Gleason theorem \cite{Gleason} shows that the Born rule is a logical consequence of the very definition of a Hilbert space and has nothing to do with the laws of evolution of the physical systems.
\par
The Born rule expresses the probability to register a particle described by the amplitude \Math{\barket{\psi}} by an apparatus configured to select the amplitude \Math{\barket{\phi}} by the formula (in the case of pure states):
\MathEq{\ProbBorn{\phi}{\psi} = \frac{\cabs{\inner{\phi}{\psi}}^2}{\inner{\phi}{\phi}\inner{\psi}{\psi}}\equiv\frac{\brabar{\psi}\projector{\phi}\barket{\psi}}{\inner{\psi}{\psi}}\equiv\tr\vect{\projector{\phi}\projector{\psi}}\,,}
where \Math{\displaystyle\projector{a}=\frac{\barket{a}\!\brabar{a}}{\inner{a}{a}}} is the {projector} onto subspace spanned by \Math{\barket{a}}.\\
\textbf{Remark.}
In the ``finite'' background the only reasonable interpretation of probability is the \emph{frequency interpretation}:
probability is the ratio of the number of ``favorable'' combinations to the total number of combinations. So we expect that \Math{\ProbBorn{\phi}{\psi}} must be a \emph{rational number} if everything is arranged correctly.
Thus, in our approach the usual \alert{non-constructive} contraposition
--- \alert{complex numbers} as intermediate values vs. \alert{real numbers} as observable values
--- is replaced by the \alert{constructive} one --- \alert{irrationalities} vs. \alert{rationals}.
From the constructive point of view, there is no fundamental difference between irrationalities and constructive complex numbers: both are elements of algebraic extensions.
\subsection{Illustration: constructive view of the Mach--Zehnder interferometer}
\label{MZI
The Mach--Zehnder interferometer is a simple but
important example of a two-level quantum system.
The device consists of a single-photon light source, beam splitters, mirrors and photon detectors (see Figure~\ref{MachZehnder}).
\begin{figure}[h]
\centering
\sidecaption
\includegraphics[width=0.4\textwidth]{MachZehnder}
\caption{Mach--Zehnder interferometer. Balanced setup: both beam splitters are \Math{50/50} and there is no phase shift between upper and lower paths.}
\label{MachZehnder}
\end{figure}
Consider a two-dimensional Hilbert space spanned by the two orthonormal basis vectors \Math{\barket{\nearrow}} --- ``right upward beams'', and \Math{\barket{\searrow}} --- ``right downward beams''.
Then the \Math{50/50} beam splitter (i.e., a photon has equal probability of being reflected and transmitted) is described by the matrix
\MathEqLab{S=\frac{1}{\sqrt{2}}\Mtwo{1}{\im}{\im}{1}\,.}{Smatrix}
The mirror matrix is \Math{~M=\Mtwo{0}{\im}{\im}{0}}\,.
Notice that \Math{M=S^2}, and, on the other hand, \Math{S} can be expressed via \Math{M} as an element of the group algebra: \Math{S=\frac{1}{\sqrt{2}}\vect{\mathbb{I}} % Unit matrix:\mathcal{I}\mathcal{I}\mathrm{I}\mathbb{I}\mathbf{I}\mathrm{\mathbf{I}+M}}, where \Math{\idma
} is the identity matrix.
The scheme in the figure implements the unitary evolution \Math{S\!MS\barket{\nearrow}=S^4\barket{\nearrow}=-\barket{\nearrow}},
which means that only the upper detector will register photons, the lower detector will always be inactive.
\par
This device is able to demonstrate many interesting features of the quantum behavior.
Consider, for example, the scheme of quantum \emph{interaction-free measurement} proposed by Elitzur and Vaidman \cite{ElitzurVaidman}.
The Penrose version of this example is called the \emph{bomb-testing problem}.
Suppose we have a collection of bombs, of which some are defective.
The detonator of a good bomb causes explosion after absorbing a single photon.
The detonators of defective bombs reflect photons without any consequences.
Classically, the only way to verify that a bomb is good is to touch the detonator.
However, as shown in Figure~\ref{Bomb}, the quantum interference makes it possible to select \Math{25\%}
of good bombs without exploding them:
the signal of the lower detector ensures that the unexploded bomb is good.
\def0.46{0.46}
\def0.6{0.6}
\begin{figure}[h]
\centering
\begin{minipage}{0.46\textwidth}
\centering
\includegraphics[width=0.6\textwidth]{MachZehnderBombDud}\\
{\small\Math{\barket{\nearrow}\xrightarrow{S\!MS}-\barket{\nearrow}~~\Prob=1}\\{}testing \alert{defective} bomb}
\end{minipage}
\begin{minipage}{0.46\textwidth}
\centering
\includegraphics[width=0.6\textwidth]{MachZehnderBombExplodes}\\
{\small\Math{\barket{\nearrow}\xrightarrow{\projector{\searrow}S}\frac{\im}{\sqrt{2}}\barket{\searrow}~~\Prob=\frac{1}{2}}\\good bomb \alert{exploded}}
\end{minipage}
\\[5pt]
\begin{minipage}{0.46\textwidth}
\centering
\includegraphics[width=0.6\textwidth]{MachZehnderBombUntested}\\
{\small\Math{\barket{\nearrow}\xrightarrow{\projector{\nearrow}S\,M\,\projector{\nearrow}S}-\frac{1}{2}\barket{\nearrow}~~\Prob=\frac{1}{4}}\\bomb remains \alert{untested}}
\end{minipage}
\begin{minipage}{0.46\textwidth}
\centering
\includegraphics[width=0.6\textwidth]{MachZehnderBombGood}\\
{\small\Math{\barket{\nearrow}\xrightarrow{\projector{\searrow}S\,M\,\projector{\nearrow}S}\frac{\im}{2}\barket{\searrow}~~\Prob=\frac{1}{4}}\\bomb is \alert{good and intact}}
\end{minipage}
\caption
Penrose bomb tester. \Math{\Prob} is the probability of a branch of evolution. \Math{\projector{a}} denotes the projector onto \Math{\barket{a}}.}
\label{Bomb}
\end{figure}
\par
A slight modification of the scheme shown in Figure~\ref{MachZehnder} allows us to implement any unitary operator \Math{U\in\UG{2}} by the Mach--Zehnder interferometer.
This is easily verified by direct calculation.
Since \Math{\dim\UG{2}=4}, we should add four parameters in a proper way.
For example, we can change the transparency of the beam splitter. Mathematically this means replacing the matrix \eqref{Smatrix} by another one of the form \Math{\alpha\mathbb{I}} % Unit matrix:\mathcal{I}\mathcal{I}\mathrm{I}\mathbb{I}\mathbf{I}\mathrm{\mathbf{I}+\beta{\,}M}, where \Math{\cabs{\alpha}^2+\cabs{\beta}^2=1}.
Another possibility is to introduce \emph{phase shifters}. The phase shifter matrix related, e.g., to a ``right upward beam'' has the form \Math{\Mtwo{\e^{\im\omega}}{0}{0}{1}}.
Moreover, combining many Mach--Zehnder interferometers \cite{Zeilinger}, one can realize elements of any unitary group \Math{\UG{n}}.
\par
Since a ``mirror'' is the square of a ``beam splitter'', any unitary evolution in a sequence of balanced Mach--Zehnder interferometers can be described by degrees of \Math{S}.
The operator \Math{S} generates the cyclic group \Math{\CyclG{8}}.
The smallest degree faithful action of \Math{\CyclG{8}} is realized by permutations of \Math{8} objects.
Any of the four permutations, that generate \Math{\CyclG{8}} as a group of permutations, can be put in correspondence with the beam splitter, e.g., \Math{S\longleftrightarrow{g}=\vect{1,2,3,4,5,6,7,8}}.
The generator \Math{g} can be represented by matrix \Math{P_{\!g}} acting in the module \Math{\mathbb{N}^8} that consists of the vectors with natural components: \MathEq{N=\vect{n_1,n_2,n_3,n_4,n_5,n_6,n_7,n_8}^T\in\mathbb{N}^8.}
To ``extract'' the beam splitter from the matrix \Math{P_{\!g}} we should extend the natural numbers by \Math{8}th roots of unity --- the conductor \Math{\period=8} in this case.
Any \Math{8}th root of unity can be represented as a power of any of the four \emph{primitive} roots defined by the \emph{cyclotomic polynomial} \Math{\Phi_8\vect{\runi}=\runi^4+1}.
Let us denote by \Math{\mathbb{N}_8} the set of linear combinations of \Math{8}th roots of unity with {natural} coefficients. This is a ring since \Math{\period=8>1}.
The ring \Math{\mathbb{N}_8} is isomorphic to the ring of \Math{8}th cyclotomic integers.
In principle, due to the projective nature of the quantum states, we could perform all calculations using only natural numbers and roots of unity.
But it is convenient to use also the \Math{8}th cyclotomic field, which we will denote by \Math{\Q_8}.
The field \Math{\Q_8} is the fraction field of the ring \Math{\mathbb{N}_8}.
\par
The matrix \Math{P_{\!g}} by a transformation \Math{T} over the field \Math{\Q_8} can be reduced to the form
\MathEq{S_g=T^{-1}P_{\!g}T=\Mtwo{A}{0}{0}{S_{\!\runi}},}
where \Math{A=\diag\vect{1,-1,\runi^2,-\runi^2,\runi^3,-\runi}}, \Math{\runi} is a primitive \Math{8}th root of unity, and
\MathEqLab{\displaystyle{}S_{\!\runi}=\frac{1}{2}\Mtwo{\runi-\runi^3}{\runi+\runi^3}{\runi+\runi^3}{\runi-\runi^3}}{Srmatrix}
is the beam splitter matrix \Math{S} expressed in terms of the cyclotomic numbers.
Quantum amplitude of the Mach--Zehnder interferometer can be approximated by the projection of the natural vector \Math{N} into the ``splitter'' subspace:
\MathEqLab{\barket{\psi}=\Vtwo{\psi_1}{\psi_2}=\frac{1}{8}\Vtwo{-\runi{}^3\vect{n_1+n_3-n_5-n_7}+\vect{1-\runi{}^2}\vect{n_2-n_6}}{\runi{}\vect{n_1-n_3-n_5+n_7}+\vect{1+\runi{}^2}\vect{-n_4+n_8}}.}{N-to-psi}
It can be shown that expression \eqref{N-to-psi} can approximate with arbitrary precision any point on the Bloch sphere --- a standard representation of the complex projective line \Math{\C{}P^1}.
\section{Combinatorial models of evolution}
\label{models
Let us begin with some general
considerations concerning the evolution of a probabilistic system subject to observations.
The evolution of such a system can be described as follows.
We have a fundamental (``Planck'') time which is the sequence of integers:
\MathEqLab{\Time = \ordset{0,1,\ldots,T}.}{Tfund}
There is also a sequence of ``times of observations''.
For simplicity, we assume that the observation time is a subsequence of the fundamental time
\MathEqLab{\mathcal{T}=\ordset{t_0=0,\ldots,t_{i-1},t_{i},\ldots,t_{N}=T}}{Tobserv}
(otherwise we could assume that the times of observations are not determined exactly, e.g., they could be random variables with probability distributions localized within subintervals of the fundamental time).
Let \Math{W_{t_i}} denote the state of a system observed at the time \Math{t_i}, and
\MathEqLab{W_{t_0}\rightarrow\cdots{\rightarrow}W_{t_{i-1}}{\rightarrow}W_{t_i}\rightarrow\cdots{\rightarrow}W_{t_N}}{traj}
denote a trajectory of the system.
Whereas the states \Math{W_{t_{i-1}}} and \Math{W_{t_i}} are fixed by observation, the transition between them can be described only probabilistically.
\par
The selection of the most probable trajectories is the main problem in the study of the evolution.
If we can specify \Math{\Prob_{W_{t_{i-1}}{\rightarrow}W_{t_{i}}}} --- the \emph{one-step transition probability} ---
then the probability of trajectory \eqref{traj} can be calculated as the product
\MathEqLab{\Prob_{W_{t_0}\rightarrow\cdots{\rightarrow}W_{t_{N}}}=\prod\limits_{i=1}^N\Prob_{W_{t_{i-1}}{\rightarrow}W_{t_{i}}}.}{Probtraj}
The inconvenience of dealing with the product of large number of multipliers can be eliminated by introducing the \emph{entropy}, which is defined as the logarithm of probability.
The transition to logarithms allows us to replace the products by sums.
On the other hand, taking the logarithm does not change the positions of the extrema of a function due to the monotonicity of the logarithm.
Thus, for searching the most likely trajectories we introduce the \emph{one-step entropy}
\MathEqLab{\Entropy_{W_{t_{i-1}}\rightarrow{}W_{t_{i}}}=\log\Prob_{W_{t_{i-1}}\rightarrow{}W_{t_{i}}}}{Entropy-one-step}
and use instead of \eqref{Probtraj} the \emph{entropy of trajectory}:
\MathEqLab{\Entropy_{W_{t_{0}}\rightarrow\cdots\rightarrow{}W_{t_{N}}}=\sum\limits_{i=1}^N\Entropy_{W_{t_{i-1}}\rightarrow{}W_{t_{i}}}.}{Entropytraj}
\par
The formulation of any dynamical model usually begins with postulating a Lagrangian.
However, it would be desirable to derive Lagrangians from more fundamental principles.
One can see that continuum approximations of \eqref{Entropy-one-step} and \eqref{Entropytraj} lead to the concepts of Lagrangian and action, respectively.
The reasoning is schematically the following.
The states \Math{W_{t_i}} are specified by sets of numerical parameters (coordinates) \Math{\mathbf{X}_{t_i}=\vect{X_{1,{t_i}}, X_{2,{t_i}},\ldots,X_{K,{t_i}}}}.
For a specific model one-step entropy \eqref{Entropy-one-step} can be calculated as a function of the coordinates:
\Math{\Entropy_{W_{t_{i-1}}\rightarrow{}W_{t_{i}}}=S\!\vect{\mathbf{X}_{t_{i}},\Delta\mathbf{X}_{t_i}}}, where \Math{\Delta\mathbf{X}_{t_i}=\mathbf{X}_{t_{i}}-\mathbf{X}_{t_{i-1}}}.
Assuming that \Math{\displaystyle{}N\rightarrow\infty,} \Math{\displaystyle{}t_i\!-\!t_{i-1}\rightarrow0} and embedding the sequence \Math{\mathbf{X}_{t_{i}}} into the continuous function \Math{\mathbf{X}\!\vect{t}},
we can represent the one-step entropy in the form \Math{S\!\vect{\mathbf{X}\!\vect{t_i}, \Delta\mathbf{X}\!\vect{t_i}}.}
The second order Taylor approximation of this function has the form \Math{\displaystyle{}S\approx{}A+b_{kk'}\vect{\Delta{X}_k\vect{t_i}-\Delta{X}_k^*\vect{t_i}}\vect{\Delta{X}_{k'}\vect{t_i}-\Delta{X}_{k'}^*\vect{t_i}}},
where \Math{\Delta\mathbf{X}^*\!\vect{t_i}} is the solution of the system of equations \Math{\displaystyle\frac{\partial{S}}{\partial\Delta\mathbf{X}\!\vect{t_i}}=0.}
Since the discrete time is a dimensionless counter, the differences can be approximated in the continuum limit by introducing derivatives, and we come to the Lagrangian
\MathEq{\displaystyle\mathcal{L}=A+B_{kk'}\vect{\frac{dX_k}{dt}-a_k}\vect{\frac{dX_{k'}}{dt}-a_{k'}},} where \Math{B_{kk'}} is a \emph{negative definite} quadratic form;~
\Math{B_{kk'},~A} and \Math{a_{k}} depend on \Math{X_1\!\vect{t},} \Math{X_2\!\vect{t},\ldots,} \Math{X_K\!\vect{t}.}
The action \MathEq{\displaystyle\mathcal{S}=\int\!\mathcal{L}dt} is a continuum approximation of the entropy of trajectory \eqref{Entropytraj},
so the principle of least action can be treated as a continuous remnant of the principle of selection of the most likely trajectories.
\subsection{Example: extracting Lagrangian from combinatorics}
\label{Random-walk
As an illustration of the above let us consider the one-dimensional random walk.
This model studies the statistics of sequences of positive (\Math{+1}) and negative (\Math{-1}) unit steps on the integer line \Math{\mathbb{Z}}.
Any statistical description is based on the concepts of \emph{microstates} and \emph{macrostates} --- the last can naturally be treated as equivalence classes of microstates \cite{Kornyak15}.
In this model, microstates are individual sequences of steps. The probability of a microstate consisting of \Math{k_+} positive and \Math{k_-} negative steps is equal to \Math{\alpha_+^{k_+}\alpha_-^{k_-}},
where \Math{\alpha_+} and \Math{\alpha_-} denote probabilities of single steps (\Math{\alpha_++\alpha_-=1}).
The macrostates are defined by the equivalence relation: two sequences \Math{u} and \Math{v} are equivalent if \Math{k_+^u+k_-^u=k_+^v+k_-^v=t} and \Math{k_+^u-k_-^u=k_+^v-k_-^v=x},
i.e., both sequences have the same length \Math{t} and define the same point \Math{x} on \Math{\mathbb{Z}}.
The probability of an arbitrary microstate to belong to a given macrostate is described by the \emph{binomial distribution}, which in terms of the variables \Math{x} and \Math{t} takes the form
\MathEqLab{P\vect{x,t}=\frac{t!}{\vect{\frac{t+x}{2}}!\vect{\frac{t-x}{2}}!}\vect{\frac{1+v}{2}}^{\frac{t+x}{2}}\vect{\frac{1-v}{2}}^{\frac{t-x}{2}},}{binom}
where \Math{v=\alpha_+-\alpha_-} is the ``drift velocity''
\footnote{It has been shown \cite{Knuth} that the velocity, defined in a similar way, i.e., as the difference of probabilities of steps in opposite directions, satisfies the relativistic velocity addition rule: \Math{w=\vect{u+v}/\vect{1+uv}}.}
Obviously, \Math{-1\leq{v}\leq1}.
\par
Let \Math{\ordset{x_0,\ldots,x_{i-1},x_{i},\ldots,x_N}} be a sequence of points (observed values) corresponding to the sequence of times of observations \eqref{Tobserv}.
We assume that the time differences \Math{\Delta{t}_i=t_i-t_{i-1}} are much larger than the unit of fundamental time \eqref{Tfund} but much less than the total time: \Math{1\ll\Delta{t}_i\ll{T}}.
Applying formula \eqref{binom} to \Math{i}th time interval we can write the one-step entropy:
\MathEq{\Entropy_{x_{i-1}\rightarrow{}x_i}=\ln\Delta{t}_{i}!-\ln\vect{\frac{\Delta{t}_{i}+\Delta{}x_i}{2}}!-\ln\vect{\frac{\Delta{t}_{i}-\Delta{}x_i}{2}}!
+\frac{\Delta{t}_{i}+\Delta{}x_i}{2}\ln\vect{\frac{1+v_i}{2}}+\frac{\Delta{t}_{i}-\Delta{}x_i}{2}\ln\vect{\frac{1-v_i}{2}},}
where \Math{\Delta{}x_i=x_i-x_{i-1}}, and \Math{v_i} denotes the drift velocity in the \Math{i}th interval.\\
Applying the Stirling approximation, \Math{\ln{}n!\approx{}n\ln{n}-n}, we have
\MathEqLab{\Entropy_{x_{i-1}\rightarrow{}x_i}\approx{}S_i=\Delta{t}_{i}\ln\Delta{t}_{i}-\frac{\Delta{t}_{i}+\Delta{}x_i}{2}\ln\vect{\frac{\Delta{t}_{i}+\Delta{}x_i}{1+v_i}}-\frac{\Delta{t}_{i}-\Delta{}x_i}{2}\ln\vect{\frac{\Delta{t}_{i}-\Delta{}x_i}{1-v_i}}.}{EStirling}
Solving the equation \Math{\partial{S_i}/\partial{\Delta{}x_i}=0} we obtain the stationary point: \Math{\Delta{}x_i^*=v_i\Delta{t}_{i}}.
Replacing the sequences \Math{x_i,} \Math{v_i} by continuous functions \Math{x\vect{t},} \Math{v\vect{t}} and introducing the approximation \Math{\Delta{}x_i\approx\dot{x}\vect{t}\Delta{t}_{i}}
in the second order Taylor expansion of \eqref{EStirling} around the point \Math{\Delta{}x_i^*} we have finally
\MathEq{\displaystyle\Entropy_{x_{i-1}\rightarrow{}x_i}\approx-\frac{1}{2}\vect{\frac{\dot{x}\vect{t}-v}{\sqrt{1-v^2}}}^2\!\Delta{t}_{i}\,.}
Thus we come to the Lagrangian \Math{\displaystyle\mathcal{L}=\vect{\frac{\dot{x}\vect{t}-v}{\sqrt{1-v^2}}}^2}
with the corresponding Euler-Lagrange equation \MathEq{\displaystyle\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial{\dot{x}}}-\frac{\partial\mathcal{L}}{\partial{x}}=0~\Longrightarrow~\ddot{x}\vect{1-v^2}+2\dot{x}v\frac{\partial{v}}{\partial{t}}-\vect{1+v^2}\frac{\partial{v}}{\partial{t}}=0\,.}
\subsection{Scheme for constructing models of quantum evolution}
\label{subsec-Qmodel
The trajectory of a quantum system is a sequence of observations with unitary evolutions between them.
We propose a scheme to construct quantum models that combine unitary evolutions with observations.
The scheme assumes that transitions between observations are described by bunches of properly weighted unitary parallel transports.
The standard scheme of quantum mechanics with single unitary evolutions can be reproduced in our scheme by a special choice of weights.
But in our scheme such unique evolutions are assumed to be obtained as statistically dominant elements of the bunches.
\par
We use the following notations
\begin{itemize}
\item
\Math{\Hspace}: a Hilbert space;
\item
\Math{\projector{\psi_{t_0}},\ldots,\projector{\psi_{t_i}},\ldots,\projector{\psi_{t_N}}}: a sequence of observations,\\
where \Math{\projector{\psi_{t_i}}=\barket{\psi_{t_i}}\!\brabar{\psi_{t_i}}} is the projector that fixes \Math{\psi_{t_i}\in\Hspace} as the result of observation at the time \Math{t_i};
\item
\Math{\Delta{t}_i=t_i-t_{i-1}}: the length of \Math{i}th time interval;
\item
\Math{\mathsf{G}} % Whole symmetry group: \mathcal{W}\mathrm{G=\set{\mathsf{g}} % Element of whole symmetry group: \mathrm{g}\mathsf{w}\mathfrak{w_1,\ldots,\mathsf{g}} % Element of whole symmetry group: \mathrm{g}\mathsf{w}\mathfrak{w_\mathsf{M}} % Size of whole group: {N_\wG}}: a finite \emph{gauge group};
\item
\Math{\repq}: a unitary representation of \Math{\mathsf{G}} % Whole symmetry group: \mathcal{W}\mathrm{G} in the space \Math{\Hspace};
\item
\Math{\gamma=g_{1},\ldots,g_{\Delta{t}_i}}: a sequence of the length \Math{\Delta{t}_i} of elements from \Math{\mathsf{G}} % Whole symmetry group: \mathcal{W}\mathrm{G};
\item
\Math{\Value\!\vect{\gamma}=\prod_{j=1}^{\Delta{t}_i}{g_{j}}\in\mathsf{G}} % Whole symmetry group: \mathcal{W}\mathrm{G}: the (group) \emph{value} of the sequence \Math{\gamma} --- the parallel transport;
\item
\Math{\Gamma_i=\set{\gamma_1,\ldots,\gamma_k,\ldots,\gamma_{\mathsf{K}_i}}}: an (arbitrary) \emph{enumeration} of the set of all sequences \Math{\gamma},\\
where \Math{\mathsf{K}_i\equiv\cabs{\Gamma_i}=\mathsf{M}} % Size of whole group: {N_\wG^{\Delta{t}_i}} is the total number of the sequences;
\item
\Math{w_{ki}}: a \emph{non-negative weight} of \Math{k}th sequence (in \Math{i}th time interval).
\end{itemize}
With these notations we come to the scheme shown in Figure~\ref{QModel}.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\textwidth]{QModelMMCP}
\caption{Scheme of quantum evolution with observations}
\label{QModel}
\end{figure}
The probability of transition from \Math{\psi_{t_{i-1}}} to \Math{\psi_{t_i}} is given by the formula
\MathEq{\Prob_{\psi_{t_{i-1}}\rightarrow\psi_{t_{i}}}=\sum\limits_{k=1}^{\mathsf{K}_i}\!\!w_{ki}\brabar{\varphi_{ki}}\projector{\psi_{t_i}}\barket{\varphi_{ki}},
\text{\color{black}~~where~~} \varphi_{ki}=\repq\!\vect{\Value\!\vect{\gamma_k}}\psi_{t_{i-1}}\,.}
The case of standard quantum mechanics with a single unitary evolution between observations is obtained in our scheme by selecting a sequence \Math{\gamma} formed by an element \Math{g\in\mathsf{G}} % Whole symmetry group: \mathcal{W}\mathrm{G} repeated \Math{\Delta{t}_i} times.
The weight of the sequence \Math{\gamma} is set to \Math{1}, and the weights of all other sequences are equated to \Math{0}.
In other words, the set of weights is the Kronecker delta on the set of sequences: \Math{w_{ki}=\delta_{\gamma,\gamma_k},~\gamma_k\in\Gamma_i}.
Introducing the Hamiltonian \Math{H=\im\ln\repq\!\vect{g}}, we can write the evolution in the usual form
\MathEq{U\equiv\repq\vect{g^{\Delta{t}_i}}=\e^{-\im{}H\vect{t_i-t_{i-1}}}.}
Since the notion of Hamiltonian stems from the principle of least action, it is natural to assume the existence of some mechanism of selecting sequences of the form \Math{g,g,\ldots,g} as dominant elements in the set of all sequences.
This requires a detailed analysis of the combinatorics of steps in fundamental time \eqref{Tfund} for particular models.
\subsection{Dynamics of observed quantum system. Quantum Zeno effect and finite groups}
\label{subsec-Zeno
Consider the issue concerning the connection between the quantum dynamics and the group properties of unitary evolution operators.
Namely, we consider the quantum Zeno effect for operators that belong to representations of finite groups.
\par
The ``quantum Zeno effect'
\footnote{This effect is also known under the name ``the Turing paradox''.}
(see the review \cite{ZenoRev})
is a feature of the quantum dynamics, which is manifested in the fact that frequent measurements can stop (or slow down) the evolution of a system --- for example, inhibit decay of an unstable particle --- or force it to evolve in a prescribed way.
In the latter case, the phenomenon is called the ``anti-Zeno effect''.
\par
Consider a quantum system that evolves from the initial (at \Math{t=0}) normalized pure state \Math{\barket{\psi_0}} under the action of the unitary operator \Math{U=\e^{-\im{H}t}}, where \Math{H} is the Hamiltonian.
The probability to find the system in the initial state at time \Math{t} is the following
\MathEqLab{{}p_H\vect{t}=\cabs{\left\langle{\psi_0}\cabs{\,\e^{-\im{H}t}}{\psi_0}\right\rangle}^2.}{ZenoProb}
The most important characteristics of any dynamical process are its temporal parameters.
For the quantum Zeno effect such a parameter is called the ``Zeno time'', denoted \Math{\tau_Z}.
It is determined from the short-time expansion of \eqref{ZenoProb}:
\MathEqLab{p_H\vect{t}=1-t^2/\tau_Z^2+\Obig\vect{t^4}.}{ZenoExp}
Calculation of \eqref{ZenoExp} shows that~~
\Math{\displaystyle\tau_Z^{-2}=\left\langle{\psi_0}\cabs{H^2}{\psi_0}\right\rangle-{\left\langle{\psi_0}\cabs{H^{\phantom{1}\!\!}}{\psi_0}\right\rangle}^2.}
\par
Let us present the so-called \emph{Zeno dynamics} in the framework of scheme proposed in Section~\ref{subsec-Qmodel}.
We have here the sequence of observations
\Math{\projector{\psi_{t_0}},\projector{\psi_{t_1}},\ldots,\projector{\psi_{t_N}}},
each of which selects the same state \Math{\psi_0}, i.e., \Math{\psi_{t_0}=\psi_{t_1}=\cdots=\psi_{t_N}\equiv\psi_0}.
Assuming that \Math{t_0=0,~ t_N=T} and the times of observations are equidistant: \Math{t_i-t_{i-1}=T/N}, we can write, using \eqref{ZenoExp}, the approximation for the one-step transition probability
\MathEq{\Prob_{\psi_{t_{i-1}}\rightarrow\psi_{t_{i}}}\approx1-\frac{1}{N^2}\vect{\frac{T}{\tau_Z}}^2}
with the corresponding approximation for the one-step entropy
\MathEq{\Entropy_{\psi_{t_{i-1}}\rightarrow{}\psi_{t_{i}}}\approx-\frac{1}{N^2}\vect{\frac{T}{\tau_Z}}^2.}
For the entropy of the trajectory we have
\MathEq{\Entropy_{\psi_{t_{0}}\rightarrow\cdots\rightarrow{}\psi_{t_{N}}}=\sum\limits_{i=1}^N\Entropy_{\psi_{t_{i-1}}\rightarrow{}\psi_{t_{i}}}\approx-\frac{1}{N}\vect{\frac{T}{\tau_Z}}^2\xrightarrow{N~\rightarrow~\infty}0}
and, respectively, for the probability of trajectory:\hspace{15pt}
\Math{\displaystyle\Prob_{\psi_{t_{0}}\rightarrow\cdots\rightarrow{}\psi_{t_{N}}
\xrightarrow{N~\rightarrow~\infty}\e^0=1.}\\
This is precisely the essence of the Zeno effect.
\par
Now assume that the evolution operator \Math{U} belongs to a representation of a finite group \Math{\mathsf{G}} % Whole symmetry group: \mathcal{W}\mathrm{G}, i.e., \Math{U=\repq\vect{\mathsf{g}} % Element of whole symmetry group: \mathrm{g}\mathsf{w}\mathfrak{w},~\mathsf{g}} % Element of whole symmetry group: \mathrm{g}\mathsf{w}\mathfrak{w\in\mathsf{G}} % Whole symmetry group: \mathcal{W}\mathrm{G}, and the time is the sequence of natural numbers: \Math{t={0,1,2,\ldots}}~.
A natural way to define the Zeno time in this case follows from the observation that the leading part of expansion \eqref{ZenoExp} vanishes at \Math{t=\tau_Z}.
By analogy we can define the \emph{natural Zeno time} \Math{\text{\Large\Math{\tau}}_{\!Z}} as the first \Math{t\in\ordset{0,1,2,\ldots}} that provides minimum of the expression
\MathEqLab{p_U\vect{t}=\cabs{\left\langle{\psi_0}\cabs{U^{t}}{\psi_0}\right\rangle}^2.}{nZeno}
Obviously expression \eqref{nZeno} is either constant (namely, \Math{p_U\vect{t}=1}) or periodic.
In the latter case its period is a divisor of the order of \Math{U}.
The \emph{order} of an element \Math{a} of a group is the smallest natural number \Math{n>0} such that \Math{a^n=e}, where \Math{e} denotes the identity element of the group.
The order of \Math{a} will be denoted \Math{\ord\vect{a}}. For the faithful representation we have \Math{\ord\vect{U}\equiv\ord\vect{\repq\vect{\mathsf{g}} % Element of whole symmetry group: \mathrm{g}\mathsf{w}\mathfrak{w}}=\ord\vect{\mathsf{g}} % Element of whole symmetry group: \mathrm{g}\mathsf{w}\mathfrak{w}}.
\par
Consider, for example, the ``Max-Zehnder'' representation \Math{\repq_{MZ}} of the group \Math{\CyclG{8}}, i.e., the ``beam splitter'' matrix \eqref{Smatrix} is taken as a generator of \Math{\CyclG{8}}.
Table~\ref{tabZ8} presents the Zeno times for all operators from the representation \Math{\repq_{MZ}}.
We adopt the convention (motivated by formula \eqref{ZenoExp}) that \Math{\text{\Large\Math{\tau}}_{\!Z}=\infty} if probability \eqref{nZeno} is constant.
\begin{table}[h]
\centering
\caption{Zeno times for all operators from \Math{\repq_{MZ}\vect{\CyclG{8}}}}
\label{tabZ8}
\begin{tabular}{c|c|c|c}
\Math{U=\repq_{MZ}\vect{\mathsf{g}} % Element of whole symmetry group: \mathrm{g}\mathsf{w}\mathfrak{w}} & \Math{\ord\!\vect{\mathsf{g}} % Element of whole symmetry group: \mathrm{g}\mathsf{w}\mathfrak{w}} & \Math{\Period\!\vect{p_U\vect{t}}} & \Math{\text{\Large\Math{\tau}}_{\!Z}} \\\hline
\Math{S^0=\mathbb{I}} % Unit matrix:\mathcal{I}\mathcal{I}\mathrm{I}\mathbb{I}\mathbf{I}\mathrm{\mathbf{I}} & \Math{1} & \Math{p_U\vect{t}=1} & \Math{\infty} \\\hline
\Math{S^4} & \Math{2} & \Math{p_U\vect{t}=1} & \Math{\infty} \\\hline
\Math{S^2=M, S^6} & \Math{4} & \Math{2} & \Math{1} \\\hline
\Math{S, S^3, S^5, S^7} & \Math{8} & \Math{4} & \Math{2}
\end{tabular}
\end{table}
\par
The two-dimensional ``Max-Zehnder'' representation \Math{\repq_{MZ}} can be generalized to an arbitrary cyclic group \Math{\CyclG{N}} by replacing the ``beam splitter'' matrix of the form \eqref{Srmatrix} with the unitary matrix
\MathEq{S_{\!N}=\frac{1}{2}\Mtwo{\runi+\runi^{N-1}}{\runi-\runi^{N-1}}{\runi-\runi^{N-1}}{\runi+\runi^{N-1}}\,,}
where \Math{\runi} is an \Math{N}th primitive root of unity.
Figure~\ref{ZenoZ100} shows the evolution of the probability to observe the initial state for the evolution operator \Math{S_{\!100}} in the time interval \Math{0\leq{t}\leq100}.
The quadratic short-time behavior, described by the formula \eqref{ZenoExp}, is clearly visible in the figure. The Zeno time in this example is \Math{\text{\Large\Math{\tau}}_{\!Z}=25}.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth,height=0.31\textwidth]{ZenoZ100}
\caption{Probability \Math{p_U\vect{t}} vs. time \Math{t} for the operator \Math{U=S_{\!\!100}\in\repq_{MZ}\vect{\CyclG{100}}}.}
\label{ZenoZ100}
\end{figure}
\par
As a non-commutative example, consider the icosahedral group \Math{\AltG{5}} --- the smallest (\Math{\cabs{\AltG{5}}=60}) non-commutative simple group.
It has applications for model building in the particle physics, especially in issues beyond the standard model, such as the flavor physics \cite{EverettStuart}.
The non-trivial elements of \Math{\AltG{5}} have orders \Math{2}, \Math{3} and \Math{5}.
The irreducible representations of \Math{\AltG{5}} are: one trivial singlet, \Math{\IrrRep{1}}, two triplets, \Math{\IrrRep{3}} and \Math{\IrrRep{3'}}, one quartet, \Math{\IrrRep{4}}, and one quintet, \Math{\IrrRep{5}}.
Figure \ref{ZenoA5} shows the evolution of ``Zeno probabilities'' for the following matrices of orders \Math{2}, \Math{3} and \Math{5}, respectively,
\MathEqLab{U=\frac{1}{2}\Mthree{-\phi}{\Frac{1}{\phi}}{1}{\Frac{1}{\phi}}{-1}{\phi}{1}{\phi}{\Frac{1}{\phi}},~V=\Mthree{0}{0}{1}{1}{0}{0}{0}{1}{0},~W=\frac{1}{2}\Mthree{-\phi}{-\Frac{1}{\phi}}{1}{\Frac{1}{\phi}}{1}{\phi}{-1}{\phi}{-\Frac{1}{\phi}}\,,}{operators}
where \Math{\phi=\frac{1+\sqrt{5}}{2}} is the ``golden ratio''.
To write these matrices, we added an
element of order \Math{3} (the simplest among randomly selected) to the generators of orders \Math{2} and \Math{5} proposed in \cite{Shirai} for the representation \Math{\IrrRep{3'}}.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth,height=0.31\textwidth]{ZenoA5rep3p}
\caption{Zeno dynamics in the representation \Math{\IrrRep{3'}} of the group \Math{\AltG{5}} for unitary operators \eqref{operators}.}
\label{ZenoA5}
\end{figure}
\section{Summary}
\label{Summary}
\begin{enumerate}
\item
We adhere to the idea of empirical universality of discrete, more specifically, finite models for describing physical reality.
In other words, any continuous model can be replaced by a finite model that fit the same observable behavior.
\item
This idea, in application to quantum problems, means that unitary groups of evolution operators can be replaced by unitary representations of finite groups.
\item
The mathematical fact that any representation of a finite group can be embedded in a permutation representation allows to approximate, with arbitrary precision, quantum amplitudes by projections of vectors with natural components.
The complex components of these projections are combinations of natural numbers and roots of unity.
\item
To illustrate the content of the article, we have used the Mach-Zehnder interferometer --- a simple but important example of a two-level quantum system with rich behavior.
\item
We propose a scheme for constructing quantum models.
Taking into account that a single unitary evolution, being a simple change of coordinates, is not sufficient to describe physical phenomena,
the scheme involves sequences of observations with bunches of unitary parallel transports between the observations.
\item
The principle of selection of the most probable trajectories in such models via the large numbers approximation leads in the continuum limit to the principle of least action with appropriate Lagrangians and deterministic evolution equations.
\item
To look at the connection between quantum dynamics and the group properties of unitary evolution operators, we have considered the quantum Zeno effect in the context of our approach.
\end{enumerate}
\begin{acknowledgement}
The work is supported in part by the Ministry of Education and Science of the Russian Federation (grant 3003.2014.2) and the Russian Foundation for Basic Research (grant 13-01-00668).
\end{acknowledgement}
|
2,869,038,156,401 | arxiv | \section{Acknowledgments}
\begin{acknowledgments}
We thank C.~A.~Watson for experimental assistance, W.~F.~Kindel and K.~W.~Lehnert for the parametric amplifier, and A.~F.~Kockum and M.~Dukalski for helpful discussions. We acknowledge funding from the Dutch Organization for Fundamental Research on Matter (FOM), the Netherlands Organization for Scientific Research (NWO, VIDI scheme), and the EU FP7 integrated projects SOLID and SCALEQIT.
\end{acknowledgments}
\section{Device parameters}
We used the same device as in Ref.~\onlinecite{Riste13b}. The experimental parameters differ slightly due to a different choice of cavity couplings and aging of the qubit junctions during the thermal cyclings. Throughout the experiment, only the single-junction transmon is used and the double-junction qubit is parked at the $7.689~\mathrm{GHz}$ sweet spot.
The copper cavity has fundamental mode $f_\mathrm{r}=6.5433~\mathrm{GHz}$ and coupling limited linewidth $\kappa/2\pi=1.4~\mathrm{MHz}$, with asymmetric coupling $\kappa_\mathrm{out}/\kappa_\mathrm{in} \approx 30$. The single-junction transmon has transition frequency $\omega_{Q}/2\pi=5.430~\mathrm{GHz}$, relaxation time $T_{1}=27~\mu\mathrm{s}$, Ramsey time $T_{2}^{\ast}=5~\mu\mathrm{s}$, and echo time $T_{2,\mathrm{echo}}=8~\mu\mathrm{s}$. The residual excitation of $\sim1\%$ is neglected in the analysis.
\section{Analog feedback for phase cancellation}
The analog feedback loop consists of a FPGA-based controller, a voltage-controlled delayed trigger, and an arbitrary waveform generator (Tektronix AWG520). Another arbitrary waveform generator (Tektronix AWG5014) provides all the deterministic qubit control and measurement pulses and synchronizes the feedback-loop components. Measurement pulses, with carrier frequency $f_\mathrm{m}$ and envelope $\epsilon_\mathrm{m}(t)$, are applied to the cavity input. The output signal is amplified and demodulated to produce the homodyne voltage $V_Q$, constituting the input to the feedback loop.
The feedback controller digitizes $V_Q$ at $100~\mathrm{MSamples/s}$ and $8$-bit resolution. The digitized signal is weighed in real time by a sequence $w$ of $7-$bit signed integers, generating a running integral over $1.25~\mu\mathrm{s}$. The resulting weighted integral $V_\mathrm{int}$ is multiplied by an integer $c_\mathrm{fb}$, setting the analog feedback strength.
Following digital-to-analog conversion, $c_\mathrm{fb}V_\mathrm{int}$ provides the input to the delayed trigger. Upon activation by a marker bit from the AWG5014, this device starts ramping an internal voltage linearly (see Fig.~S5). A trigger is generated when the running voltage crosses $c_\mathrm{fb}V_\mathrm{int}$. The trigger delay determines the phase correction for the measurement-induced phase shift and is here compiled into the tomographic pulse.
This correction is applied by time-shifting the envelope of the tomographic pulse modulating the qubit drive tone. The use of single-sideband modulation translates this delay into a difference $\delta \varphi$ in the rotation axis in $R_{\vec{n},{\varphi}}$ [Fig.~2(b)]. The modulation frequency of $30~\mathrm{MHz}$ achieves a phase resolution of $\sim10$~degrees, set by the $1~\mathrm{ns}$ clock of the AWG520. This discretization corresponds to an error of $\sim0.1\%$ in $r_\mathrm{cl}$.
\section{Pump leakage suppression}
To ensure that the qubit does not suffer unintended measurement-induced dephasing when $\epsilon_\mathrm{m}=0$, the cavity needs to be empty at steady state.
This requires cancelling the leakage of the JPA pump towards the cavity. The three circulators between cavity and JPA provide $\sim70~\mathrm{dB}$ suppression, but additional $20~\mathrm{dB}$ are desirable to prevent unwanted dephasing.
To suppress the residual leakage, we supply a continuous-wave tone at $f_\mathrm{m}$, by fine-tuning the DC offsets at the corresponding mixer. To calibrate these offsets, we integrate the homodyne voltage before and after applying a $\pi$ pulse to the qubit. If there are no photons in the cavity, the signal remains unchanged. In the presence of a residual photon population, instead, the transient of the intra-cavity field from the one corresponding to qubit in $\ket{0}$ to the one for $\ket{1}$ produces a variation in the homodyne signal. We optimize the amplitude and phase of the input offset by minimizing the variance of the homodyne voltage over this interval. From the magnitude of the applied offsets, we estimate a pump leakage of $\sim10^{-2}$ intra-cavity photons without nulling. Although complete cancellation from the cavity input port cannot be achieved, due to the asymmetry in input and output coupling rates, we estimate from Fig.~S3(c) that the pump leakage is suppressed to better than $10^{-4}$ photons. This is at least two orders of magnitude lower than the steady-state population at $\epsilon_\mathrm{m}$ values used in the experiment.
\section{Weight function optimization}
We obtain the weight function $w_\mathrm{opt}$ by the following numerical optimization procedure. The results of $200,000$ pairs of experimental homodyne records $V_Q$ and tomographic measurements $M_I$ for each tomographic pre-rotation [Fig.~2(d)] are stored and processed to calculate $r_\mathrm{con}$ for each $w$. Fig.~S7 depicts the optimization procedure starting from $w=0$. The optimization routine randomly selects one of $25$ blocks $w_i$ (each $50~\mathrm{ns}$ long) of $w$. Every $w_i$ is stepped across $5$ values, while keeping the remaining blocks fixed. A quadratic fit of $r_\mathrm{con}$ selects the optimum $w_i$ at each iteration. The optimization over the whole integration window is repeated with increasingly smaller steps for $w_i$.
To speed up the optimization procedure for the $w_\mathrm{opt}$ used in Figs.~2-4 and S2, the optimization starts from $w = \mathrm{Re} \left[\alpha_0\right]$, the expected optimum for a detector with infinite bandwidth~\cite{FriskKockum12} and is repeated three times.
The final shape is obtained after final linear interpolation between adjacent $w_i$ and smoothing by averaging each value $w[n]$ with its nearest neighbors (repeated twice).
\section{Mode-matching theory for the optimal weight function}
In order to obtain the maximum correlation between measurement record and qubit state, we here apply the method described in Ref.~\onlinecite{EichlerPhD13}
to derive the state-dependent field propagating in the coax line from the cavity to the JPA, where it is reflected and amplified.
Applying a measurement pulse to the cavity, with the qubit in a superposition state, entangles qubit and cavity due to their dispersive interaction [Eq.~(1)].
The intra-cavity field mainly exits the output port at a rate $\kappa_\mathrm{out}\approx \kappa$. The qubit-cavity interaction is complete when the cavity has returned to the vacuum state, i.e., several cavity decay times $1/\kappa$ after the measurement pulse is turned off. The entanglement is now between the qubit and the outgoing field $a_\mathrm{out}$ in the coax line.
Input-output theory~\cite{Gardiner04} connects this field to the incoming field $a_\mathrm{in}$ and the field $a$ inside the cavity:
\begin{equation}
\label{eq:iotheory}
a_\mathrm{out} = \sqrt{\kappa}a -a_\mathrm{in}.
\end{equation}
Ideally, the incoming field is in the vacuum state $\ket{\mathrm{vac}}$. The field $a_\mathrm{out}$ is the input to the JPA ($b_\mathrm{in} = a_\mathrm{out}$) and is transformed to the outgoing field $b_\mathrm{out}$.
A time-independent field can be defined by integrating the time-dependent $b_\mathrm{out}(t)$:
\begin{equation*}
B = \int w (t) b_\mathrm{out}(t) \mathrm{d}t,
\end{equation*}
where $w (t)$
is normalized
to preserve the commutation relation $\left[B,B^\dagger\right]=1$.
The joint qubit-field state is now $\ket{\Psi} = \left(\ket{0}\ket{\beta_0, G_{\phi}e^{i \phi}} + \ket{1}\ket{\beta_1,G_{\phi}e^{i \phi}}\right)/\sqrt{2}$,
where the squeezed state $\ket{\beta_i,G_{\phi}e^{i \phi}} = D(\beta_i)S(G_{\phi}e^{i \phi})\ket{\mathrm{vac}}$ is defined by the displacement operator $D(\beta_i) = \exp(\beta_i B^\dagger - \beta_i^\ast B)$ and the squeezing operator $S(G_{\phi}e^{i\phi}) = \exp(G_{\phi}e^{-i\phi}B^2- G_{\phi}e^{i\phi}B^{\dagger2})$~\cite{Gardiner04}.
Here, $\beta_i = \avg{B}_i $ for qubit in $\ket{i}$ and $G_{\phi}e^{i\phi}$ is the complex-valued amplitude gain of the JPA, for the quadrature with phase $\phi$.
In our experiment $\phi = \pi/2$, resulting in amplification of the $Q$-quadrature.
The entanglement between the field $B$ and the qubit is maximized when the distance $\abs{\beta_0 - \beta_1}$ is largest. This condition is matched for $w(t)=w_\mathrm{mm}(t) = N \langle \bd_\mathrm{out}(t)Z\rangle$~\cite{EichlerPhD13}. When the qubit starts in a maximal superposition state, this gives
\begin{equation*}
\abs{\beta_0 - \beta_1}_{\mathrm{max}}/2 = N \int \abs{\langle b_\mathrm{out}(t)\rangle_0-\langle b_\mathrm{out}(t) \rangle_1 }^2\mathrm{d}t,
\end{equation*}
with the normalization constant
\begin{equation*}
N=\frac{1}{\sqrt{\int|\langle \bd_\mathrm{out}(t)\rangle_0-\langle \bd_\mathrm{out}(t) \rangle_1|^2\mathrm{d}t}}.
\end{equation*}
To determine $w_\mathrm{mm} (t)$ for finite JPA bandwidth we move to the frequency domain, where
\begin{equation}
\label{eq:bout}
b_\mathrm{out,\Delta} = G_{\mathrm{s},\Delta} b_\mathrm{in,\Delta} + G_{\mathrm{i},\Delta} \bd_\mathrm{in,-\Delta},
\end{equation}
with $G_{\mathrm{s},\Delta}$ and $G_{\mathrm{i},\Delta}$ complex gain factors.
Throughout the text, we refer to $\abs{G_{\mathrm{s},0}}$ as the JPA voltage gain $G$.
In the small-signal approximation~\cite{EichlerPhD13} and with the pump resonant with the JPA,
\begin{align}
\label{eq:complexgain}
G_{\mathrm{s},\Delta} &=& -1 + \frac{\kappa_\mathrm{e}\left({\kappa_\mathrm{e}}/{2}-i\Delta\right)}{\left(i\Delta - \lambda_- \right)\left(i\Delta - \lambda_+ \right)}\\
G_{\mathrm{i},\Delta} &=& \frac{\mathcal{G}-1}{\mathcal{G}} \frac{\kappa_\mathrm{e}^2/{2}e^{-i2\phi}}{\left(i\Delta - \lambda_- \right)\left(i\Delta - \lambda_+ \right)},
\end{align}
with $\kappa_\mathrm{e}$ the extrinsic loss rate of the JPA and $\lambda_{\pm} = \kappa_\mathrm{e}/2 \left[1 \pm (\mathcal{G}-1)/\mathcal{G}\right]$, where $\mathcal{G}$ is set by the pump power.
For the amplified $Q$-quadrature (for $\phi = \pi/2$) the peak gain is $G_{Q}=\abs{G_{\mathrm{s},0}-G_{\mathrm{i},0}} = 2(\mathcal{G}-1/2)$. For the deamplified $I$-quadrature, $G_{I}=\abs{G_{\mathrm{s},0}+G_{\mathrm{i},0}} = \left[2(\mathcal{G}-1/2)\right]^{-1}$.
Increasing $\mathcal{G}$ reduces the detection bandwidth as $\kappa_{\mathrm{JPA}} \approx \kappa_\mathrm{e}/\mathcal{G}$ (Fig.~S6).
Combining Eqs.~\eqref{eq:iotheory} and ~\eqref{eq:bout}, we obtain the expectation values
\begin{align*}
&\langle{\bd_\mathrm{out,\Delta}}\rangle_0 = \sqrt{\kappa}\left[G^*_{\mathrm{s},\Delta} \avgacg{\Delta}+G^*_{\mathrm{i},\Delta} \avgag{-\Delta} \right] \\
&\langle{\bd_\mathrm{out,\Delta}}\rangle_1 = \sqrt{\kappa}\left[G^*_{\mathrm{s},\Delta} \avgace{\Delta}+G^*_{\mathrm{i},\Delta} \avgae{-\Delta} \right].
\end{align*}
Using $\avgag{\Delta} = -\avgace{-\Delta}$, valid for our choice of $f_\mathrm{m}$, we arrive to
\begin{align*}
\langle{\bd_\mathrm{out,\Delta} Z}\rangle &= (\langle{\bd_\mathrm{out,\Delta}}\rangle_0 - \langle{\bd_\mathrm{out,\Delta}}\rangle_1)/2\\
&=\sqrt{\kappa}\left(G^*_{\mathrm{s},\Delta} +G^*_{\mathrm{i},\Delta}\right)\left[\avgacg{\Delta}-\avgace{\Delta}\right]/2.
\end{align*}
and $w_\mathrm{mm}(t) = N \mathcal{F}^{-1} \left[\langle{\bd_\mathrm{out,\Delta} Z}\rangle\right]$, with $\mathcal{F}$ the Fourier transform. Thus, $w_\mathrm{mm}$ is proportional to the average deamplified $I$-quadrature, $\mathrm{Re} \left[\avg{b_\mathrm{out}(t)}_0 \right]$. In the limit $G \gg 1$ and for $\phi = \pi/2$, $ G^*_{\mathrm{s},\Delta} +G^*_{\mathrm{i},\Delta} \approx \left(2G_{\mathrm{s},\Delta}\right)^{-1}$.
For $G=1$, $w_\mathrm{mm}(t) \propto \avgag{t} - \avgae{t} = 2\mathrm{Re} \left[\avgag{t}\right]$, corresponding to the $\avg{V_I}$ that would be measured in the absence of the JPA, reproducing the result in Ref.~\onlinecite{FriskKockum12}.
\begin{figure*}
\includegraphics[width=1.3\columnwidth]{DeLangeRiste_131121_FigS1}
\caption{Detailed schematic of the experimental setup. The components of the analog feedback loop are highlighted in purple. The homemade feedback controller is based on an FPGA board (Altera Cyclone IV), programmed with the weight function $w$. The integrated, rescaled homodyne signal $c_\mathrm{fb}V_\mathrm{int}$ is fed to a voltage-controlled delayed trigger. The signal-dependent delay translates into a phase shift in the tomographic pulse, generated by a Tektronix AWG520 modulating the qubit drive generator (see text). Qubit rotations (both deterministic and conditional) are Gaussian DRAG pulses~\cite{Motzoi09} ($\sigma=6~\mathrm{ns}$, $24~\mathrm{ns}$ total duration), with $30~\mathrm{MHz}$ single-sideband modulation. The homodyne signal is offset-subtracted by a bias tee and amplified in multiple stages (including a home-built amplifier with $40~\mathrm{dB}$ gain, $100~\mathrm{MHz}$ bandwidth) before entering the FPGA to span most of the fixed ADC input range ($-1$ to $1~\mathrm{V}$).
Other system components (black) are described in Ref.~\onlinecite{Riste12}.}
\end{figure*}
\begin{figure}
\includegraphics[width=\columnwidth]{DeLangeRiste_131121_FigS3}
\caption{Linearity of homodyne voltage. Averaged homodyne response ($100,000$ repetitions) for various measurement drive amplitudes $\epsilon_\mathrm{m}$ and homodyne detection phases $\phi=\{0,\pi/2\}$. The qubit is prepared in either $\ket{0}$ (a,c) or $\ket{1}$ (b,d) and the applied measurement pulse is phase-shifted by either $\phi=\pi/2$ (a,b) or $\phi=0$ (c,d) relative to the pump. The excellent overlap between all curves, rescaled by $\tilde{\epsilon}_\mathrm{m}/0.2~\mathrm{V}$, demonstrate the linearity of the JPA in the operating regime and evidence near-perfect agreement with the model.}
\end{figure}
\begin{figure*}
\includegraphics[width=2\columnwidth]{DeLangeRiste_131121_FigS2}
\caption{Conditional state tomography for various measurement configurations. (a) $\phi=0$, giving maximum discrimination between qubit in $\ket{0}$ and in $\ket{1}$. (b) $\phi=\pi/2$, replicating Fig.~2(d). (c) No measurement pulse. The observed independence of the qubit state on $V_\mathrm{int}$ shows that any residual pump leakage into the cavity is negligible. In the three cases, $V_\mathrm{int}$ is calculated using the numerically optimized $w_\mathrm{opt}$ for (b). Panels (a-b) connect with the experimental results first shown in Ref.~\onlinecite{Murch13}.}
\end{figure*}
\begin{figure*}
\includegraphics[width=1.7\columnwidth]{DeLangeRiste_131121_FigS4}
\caption{Timings in the feedback scheme. Time $t=0$ corresponds to the end of the $\pi$ pulse in the echo sequence. From top to bottom, first row: intra-cavity quadratures $\avg{I}$ and $\avg{Q}$ upon microwave excitation at $f_\mathrm{m}$ with pulse envelope $\epsilon_\mathrm{m}$, followed by conditional tomographic pulse and final readout. Second row: FPGA integration window, with weight function $w_\mathrm{opt}$ and output voltage $c_\mathrm{fb}V_\mathrm{int}$. Third row: tomographic pulses generated by an AWG520, and delayed by a voltage-controlled trigger (fourth row, Fig.~S5). The feedback latency, defined as the time between the end of the FPGA integration and the earliest tomographic pulse reaching the cavity, is $260~\mathrm{ns}$.}
\end{figure*}
\begin{figure*}
\includegraphics[width=1.3\columnwidth]{DeLangeRiste_131121_FigS5}
\caption{Voltage-controlled delayed trigger. (a) Simplified diagram of the homemade circuit.
A current source, switched by a marker bit from the AWG5014, ramps a voltage $V_C$ linearly from $-2$ to $2~\mathrm{V}$ in $200~\mathrm{ns}$. This voltage is compared to the signal from the FPGA, $c_\mathrm{fb}V_\mathrm{int}$. (b) The output trigger is generated when $V_c$ crosses $c_\mathrm{fb}V_\mathrm{int}$.}
\end{figure*}
\begin{figure}
\includegraphics[width=0.75\columnwidth]{DeLangeRiste_131121_FigS6}
\caption{JPA small-signal amplitude gain at different bias points.
The values of $G$ are obtained by fitting $\abs{G_{\mathrm{s},\Delta}}$ to each curve using Eq.~\eqref{eq:complexgain}. The other fit parameter is $\kappa_\mathrm{e}/2\pi = 83, 91, 92$ and $91\pm1~\mathrm{MHz}$ for $G=2.5, 16, 23$ and $36$, respectively. As expected, $\kappa_\mathrm{e} \approx \mathcal{G}\kappa_{\mathrm{JPA}}$ is approximately constant for $G\gg1$. }
\end{figure}
\begin{figure*}
\includegraphics[width=1.8\columnwidth]{DeLangeRiste_131121_FigS7}
\caption{Weight function optimization. (a) Five consecutive iterations of the optimization procedure for $\kappa/2\pi=5.7~\mathrm{MHz}$. For each iteration, every $w_i$ is stepped across seven values, with step size decreasing with the iteration number as $\left[1\colon1\colon0.8\colon0.8\colon0.5\colon0.2\right]$.
To illustrate the convergence, the whole procedure is repeated eight times, each time starting from $w=0$. The results of each iteration are superimposed.
In every case, the result of the optimization converges to $w_\mathrm{opt}$ used in Figs.~2, 3, and S2 after smoothing (red curve).
(b) $r_\mathrm{con}$ after each optimization step in $w_i$, with 25 steps per iteration (dashed lines), shown for one of the optimization runs shown in (a). (c) Conditioned $r$ on $V_\mathrm{int}$ for successive iterations of the optimization run shown in (b). The horizontal axis is rescaled by the standard deviation $\sigma$ of $V_\mathrm{int}$.}
\end{figure*}
|
2,869,038,156,402 | arxiv | \section{Introduction}
Asymptotically anti-de Sitter (AAdS) spaces are naturally endowed with a
conformal structure at their boundary
\cite{FG,Imbimbo:1999bj,Schwimmer:2008yh}, as evidenced from the divergent
conformal factor in the Fefferman-Graham (FG) expansion of the boundary metric
\cite{FG}. This divergence leads to an infinite volume, which in the context
of Einstein-AdS gravity corresponds to the infrared divergence of the
Einstein-Hilbert (EH) part of the action, being proportional to the volume of
the space when evaluated on-shell. In particular, for a $2n-$dimensional AAdS
manifold $M_{2n}$, we have that
\begin{equation}
I_{EH}\left[ M_{2n}\right] =-\frac{\left( 2n-1\right) }{8\pi G{\ell}^{2
}Vol\left[ M_{2n}\right] , \label{RelationBare
\end{equation}
where ${\ell}$ is the AdS radius of the manifold and $Vol\left[
M_{2n}\right] $ is its bare (divergent) volume. Therefore, it is reasonable
to expect that the problem of volume renormalization should be intimately
related to the renormalization of the Einstein-AdS action.
Volume renormalization has been thoroughly studied in the mathematical
literature. In the context of asymptotically hyperbolic (AH)
manifolds~\cite{2000math.....11051A,Yang2008,Alexakis2010,ALBIN2009140,Graham:1999pm,2016arXiv161209026F
, its resolution is known to provide relevant information about the boundary
geometry, in the form of conformal invariant quantities. More recently and by
means of tractor calculus techniques~\cite{Gover2003,bailey1994}, the
renormalized volume problem has been extended to more general setups
in~\cite{Gover:2016xwy,Gover:2016hqd}, where hidden algebraic structures are
enhanced by considering the bulk itself as a conformal manifold.
For the particular case of AAdS Einstein spaces, as discussed by Albin in
Ref.\cite{ALBIN2009140}, the renormalized volume can be expressed in terms of
the integral of some (unspecified) polynomial in contractions of the Weyl
curvature tensor, and the Euler characteristic of the manifold. Concrete
realizations of Albin's prescription exist in the cases of AAdS Einstein
manifolds in four \cite{2000math.....11051A} and six \cite{Yang2008}
dimensions. In the four-dimensional case, as discussed by Anderson in
Ref.\cite{2000math.....11051A}, the renormalized volume $Vol_{ren}\left[
M_{4}\right] $ of the four-dimensional manifold $M_{4}$ obeys a relation
given b
\begin{equation}
\frac{1}{32\pi^{2}
{\displaystyle\int\limits_{M_{4}}}
d^{4}x\sqrt{G}W^{\alpha\beta\mu\nu}W_{\alpha\beta\mu\nu}=\chi\left[
M_{4}\right] -\frac{3}{4\pi^{2}}Vol_{ren}\left[ M_{4}\right] ,
\end{equation}
where $W$ is its Weyl tensor and $\chi\left[ M_{4}\right] $ is its Euler characteristic.
The renormalization of the Einstein-AdS gravity action has been achieved
through the Holographic Renormalization procedure
\cite{Emparan:1999pm,Kraus:1999di,deHaro:2000vlm,Balasubramanian:1999re,Henningson:1998gx,Papadimitriou:2004ap,Papadimitriou:2005ii
, where one considers the addition of the Gibbons-Hawking-York term
\cite{York:1972sj,Gibbons:1976ue} and of intrinsic counterterms at the AdS
boundary. These boundary terms are fixed by requiring the cancellation of
divergences in the FG expansion of the action at the AdS boundary, while
maintaining consistency with a well-posed variational principle for the
conformal structure. However, as all the added counterterms are boundary
terms, it is difficult to see their relation to the renormalized volume, which
is defined in terms of a bulk integral. It is convenient, therefore, to
consider an alternative but equivalent renormalization scheme, named the
Kounterterms procedure
\cite{Olea:2005gb,Olea:2006vd,Kofinas:2007ns,Miskovic:2014zja,Miskovic:2009bm
, which is coordinate independent and considers the addition of topological
terms in order to achieve the renormalization of the action. In particular,
for the $2n-$dimensional case, the procedure considers the addition of the
$n-th$ Chern form, which corresponds to the boundary term arising from the
Euler theorem in $2n-$ dimensions. As shown in Ref.\cite{Miskovic:2014zja},
the renormalized Einstein-AdS action $I_{EH}^{ren}$, can be written precisely
in terms of the volume integral of a polynomial of antisymmetric contractions
of the Weyl tensor, hinting at its relation with the renormalized volume.
In this paper, we show that the relation between the EH part of the action and
the bare volume given by Eq.(\ref{RelationBare}), still holds between the
renormalized action $I_{EH}^{ren}\left[ M_{2n}\right] $ and the renormalized
volume $Vol_{ren}\left[ M_{2n}\right] $, for the cases where a definite
expression for the renormalized volume exists (the cases of AAdS manifolds in
four \cite{2000math.....11051A} and six \cite{Yang2008} dimensions). We also
conjecture that the relation holds for all even-dimensional AAdS Einstein
manifolds, and that therefore, the polynomial structure of the renormalized
action provides a concrete realization of Albin's prescription for the
renormalized volume.
In the context of the gauge/gravity duality
\cite{Maldacena:1997re,Gubser:1998bc,Witten:1998qj}, the volume
renormalization here discussed is also related to the renormalization
of\ holographic entanglement entropy (EE)
\cite{Ryu:2006bv,Rangamani:2016dms,Lewkowycz:2013nqa,Taylor:2016aoi} and
entanglement R\'{e}nyi entropy (ERE)
\cite{Dong:2016fnf,Headrick:2010zt,2011arXiv1102.2098B,Hung:2011nu}. This is
understood from the fact that both the EE and the ERE can be expressed in
terms of the areas of certain codimension-2 surfaces that are embedded within
an AAdS bulk manifold and which extend to the AdS boundary. Their
corresponding areas are infinite due to the same divergent conformal factor
that affects the bulk volume, and thus, they can be renormalized in a similar
way. Regarding the renormalization of EE, in Refs.
\cite{Anastasiou:2017xjr,Anastasiou:2018rla}, we developed a topological
scheme, which is applicable to holographic odd-dimensional conformal field
theories (CFTs) with even-dimensional AAdS gravity duals. The resulting
renormalized EE $\left( S_{EE}^{ren}\right) $ can be written as a polynomial
in contractions of the AdS curvature tensor $\mathcal{F}_{AdS}$ of a
codimension-2 surface $\Sigma$ in the AAdS bulk, and a purely topological term
that depends on its Euler characteristic $\chi\left[ \Sigma\right] $. The
$\mathcal{F}_{AdS}$ tensor \cite{Mora:2006ka} is defined a
\begin{equation}
\left. \mathcal{F}_{AdS}\right. _{b_{1}b_{2}}^{a_{1}a_{2}}=\mathcal{R
_{b_{1}b_{2}}^{a_{1}a_{2}}+\frac{1}{{\ell}^{2}}\delta_{\left[ b_{1
b_{2}\right] }^{\left[ a_{1}a_{2}\right] }, \label{FAdS
\end{equation}
where $\mathcal{R}_{b_{1}b_{2}}^{a_{1}a_{2}}$ is the intrinsic Riemann tensor
of $\Sigma$, ${\ell}$ is the bulk AdS radius and $\delta_{\left[ b_{1
b_{2}\right] }^{\left[ a_{1}a_{2}\right] }$ is the antisymmetric
generalization of the Kronecker delta. Also, considering the standard
Ryu-Takayanagi (RT) minimal area construction \cite{Ryu:2006bv}, the bulk
surface $\Sigma$ is the minimum of area (codimension-2 volume) which is
conformally cobordant with the entangling region in the CFT. As we show in
this paper, the interpretation of the renormalized EE as renormalized area is
then natural, as it has the same polynomial structure mentioned above, but
considering the AdS curvature of $\Sigma$ instead of the Weyl tensor of
$M_{2n}$. We therefore consider tha
\begin{equation}
S_{EE}^{ren}=\frac{Vol_{ren}\left[ \Sigma\right] }{4G}\text{,}
\label{SEERen
\end{equation}
in analogy with the standard RT formula.
Beyond the renormalization of EE, it is of interest to study the
renormalization of ERE, as the latter is also expressible in terms of areas of
codimension-2 surfaces. As shown by Xi Dong, in \cite{Dong:2016fnf}, the
$m-th$ ERE $S_{m}$ can be written in terms of the integral of a quantity
called the modular entropy $\widetilde{S}_{m}$ \cite{Nishioka:2018khk}, which
in turn is obtained as the area of certain cosmic brane. In particular,
$\widetilde{S}_{m}$ is given by the area of a minimal cosmic brane $\Sigma
_{T}$ with tension $T\left( m\right) =\frac{\left( m-1\right) }{4mG}$,
such that it is conformally cobordant with the entangling region in the CFT.
The cosmic brane is minimal in the sense that it extremizes the total action
consisting on the Einstein-AdS action for the bulk manifold plus the
Nambu-Goto (NG) action for the brane, thus accounting for the backreaction of
the brane on the ambient geometry. $\widetilde{S}_{m}$ obeys an analog of the
RT area law, such tha
\begin{equation}
\widetilde{S}_{m}=\frac{Vol\left[ \Sigma_{T}\right] }{4G}, \label{ModEnt
\end{equation}
and the $m-th$ R\'{e}nyi entropy $S_{m}$ can be computed a
\begin{equation}
S_{m}=\frac{m}{m-1
{\displaystyle\int\limits_{1}^{m}}
\frac{dm^{\prime}}{m^{\prime2}}\widetilde{S}_{m^{\prime}}, \label{RenyiEnt
\end{equation}
where the brane tension provides a natural analytic continuation of the
integer $m$ into the real numbers, which is required for performing the
integral. As we show in this paper, by considering the same volume
renormalization prescription discussed above, it is possible to obtain the
renormalized version of $\widetilde{S}_{m}$, given in terms of $Vol_{ren
\left[ \Sigma_{T}\right] $, from which the renormalized ERE is computed in
the same way as in Eq.(\ref{RenyiEnt}). Also, a renormalized version of the
total action can derived by evaluating the renormalized Einstein-AdS action on
the conically singular bulk manifold that considers the brane as a singular
source in the Riemann curvature. Then, the contribution due to the cosmic
brane can be rewritten as a renormalized version of the NG action.
The organization of the paper is as follows: In Section \ref{Section II}, we
explicitate the relation between renormalized volume and renormalized
Einstein-AdS action, making contact with the mathematical literature in the
four \cite{2000math.....11051A} and six \cite{Yang2008} dimensional cases. We
also give our conjecture for the general relation in the $2n-$dimensional
case, relating it to Albin's prescription \cite{ALBIN2009140}. In Section
\ref{Section III}, we exhibit the emergence of the renormalized total action,
in agreement with Dong's cosmic brane prescription \cite{Dong:2016fnf}, from
evaluating the $I_{EH}^{ren}$ action on the conically singular manifold
considering the brane as a source. We also comment that the inclusion of the
renormalized NG action for the brane can be considered as a one-parameter
family of deformations in the definition of the renormalized bulk volume. In
Section \ref{Section IV}, we obtain the renormalized modular entropy
$\widetilde{S}_{m}^{ren}$ and the renormalized ERE $S_{m}^{ren}$ starting from
the renormalized total action, and we compare the resulting modular entropy
with the existing literature for renormalized areas of minimal surfaces
\cite{Alexakis2010}. In Section \ref{Section V}, we consider the computation
of $S_{m}^{ren}$ for the particular case of a ball-like entangling region at
the CFT$_{2n-1}$, and we check that the usual result for $S_{EE}^{ren}$ is
recovered in the $m\rightarrow1$ limit. Finally, in Section \ref{Section VI},
we comment on the physical applications of our conjectured renormalized volume
formula, on our topological procedure for computing renormalized EREs and on
future generalizations thereof.
\bigskip
\section{Renormalized Einstein-AdS action is renormalized
volume\label{Section II}}
\bigskip
The standard EH action, when evaluated on an AAdS Einstein manifold, is
proportional to the volume of the manifold, which is divergent. We propose
that for $2n-$dimensional spacetimes, the renormalized Einstein-AdS action
$I_{EH}^{ren}$ is also proportional to the corresponding renormalized volume
of the bulk manifold. To motivate this conjecture, we first introduce
$I_{EH}^{ren}$, and we then compare it with known formulas for the
renormalized volume of AAdS Einstein manifolds in four and six-dimensions.
Finally, we give a concrete formula for the renormalized volume in the general
$2n-$dimensional case, and we comment on its properties.
There are different but equivalent prescriptions for renormalizing the action.
For example, the standard Holographic Renormalization scheme
\cite{Emparan:1999pm,Kraus:1999di,deHaro:2000vlm,Balasubramanian:1999re,Henningson:1998gx,Papadimitriou:2004ap,Papadimitriou:2005ii}
and the Kounterterms procedure
\cite{Olea:2005gb,Olea:2006vd,Kofinas:2007ns,Miskovic:2014zja,Miskovic:2009bm
. The equivalence between the two renormalization schemes, for the case of
Einstein-AdS gravity, was explicitly discussed in
Refs.\cite{Miskovic:2014zja,Miskovic:2009bm}, so using either one or the other
is a matter of convenience. However, as discussed in the introduction, we
consider the Kounterterms-renormalized action as it can be readily compared
with the existing renormalized volume formulas.
We consider the $2n-$ dimensional Einstein-AdS action $I_{EH}^{ren}$ as
derived using the Kounterterms prescription \cite{Olea:2005gb}, which is given
by
\begin{equation}
I_{EH}^{ren}\left[ M_{2n}\right] =\frac{1}{16\pi G}\displaystyle\int
\limits_{M_{2n}}d^{2n}x\sqrt{G}\left( R-2\Lambda\right) +\frac{c_{2n}}{16\pi
G}\displaystyle\int\limits_{\partial M_{2n}}B_{2n-1}, \label{IRenChern
\end{equation}
where $c_{2n}$ is defined a
\begin{equation}
c_{2n}=\frac{\left( -1\right) ^{n}{\ell}^{2n-2}}{n\left( 2n-2\right) !}
\label{c_2n
\end{equation}
and $B_{2n-1}$ is the $n-th$ Chern form, which in Gauss normal coordinates
(considering a radial foliation) corresponds t
\begin{align}
B_{2n-1}\overset{\text{def}}{=}-2n\displaystyle\int\limits_{0}^{1}dtd^{2n-1}x
& \sqrt{h}\delta_{\left[ i_{1}\cdots i_{2n-1}\right] }^{\left[ j_{1}\cdots
j_{2n-1}\right] }K_{j_{1}}^{i_{1}}\left( \frac{1}{2}\mathcal{R}_{j_{2}j_{3
}^{i_{2}i_{3}}-t^{2}K_{j_{2}}^{i_{2}}K_{j_{3}}^{i_{3}}\right) \times
\nonumber\\
& \cdots\times\left( \frac{1}{2}\mathcal{R}_{j_{2n-2}j_{2n-1}
^{i_{2n-2}i_{2n-1}}-t^{2}K_{j_{2n-2}}^{i_{2n-2}}K_{j_{2n-1}}^{i_{2n-1
}\right) . \label{Chern
\end{align}
In Eq.(\ref{Chern}), $\mathcal{R}_{j_{1}j_{2}}^{i_{1}i_{2}}$ is the intrinsic
Riemann tensor at the AdS boundary and $K_{j}^{i}$ is its extrinsic curvature
with respect to the radial foliation. We emphasize that the addition of the
$B_{2n-1}$ term, which has an explicit dependence on the extrinsic curvature
$K$, is consistent with a well-posed variational principle with Dirichlet
boundary conditions for the CFT metric $g_{\left( 0\right) }$, instead of
the usual Dirichlet condition for the induced metric $h$. In
particular, considering the FG expansion, variations of $K$ are proportional to
variations of $g_{\left( 0\right) }$.
As shown in Ref.\cite{Miskovic:2014zja}, the renormalized action $I_{EH
^{ren}\left[ M_{2n}\right] $ can also be written a
\begin{equation}
I_{EH}^{ren}\left[ M_{2n}\right] =\frac{1}{16\pi G}\displaystyle\int
\limits_{M_{2n}}d^{2n}x\sqrt{G}\left( {\ell}^{2\left( n-1\right)
P_{2n}\left[ W_{\left( E\right) }\right] \right) -\frac{c_{2n}}{16\pi
G}\left( 4\pi\right) ^{n}n!\chi\left[ M_{2n}\right] , \label{IRenPoly
\end{equation}
where $P_{2n}\left[ X\right] $ is a polynomial of a rank $\binom{2}{2
-$tensor $X$, given b
\begin{equation}
P_{2n}\left[ X\right] =\frac{1}{2^{n}n\left( 2n-2\right) !
{\displaystyle\sum\limits_{k=1}^{n}}
\frac{\left( -1\right) ^{k}\left[ 2\left( n-k\right) \right] !2^{\left(
n-k\right) }}{{\ell}^{2\left( n-k\right) }}\binom{n}{k}\delta_{\left[
\mu_{1}...\mu_{2k}\right] }^{\left[ \nu_{1}...\nu_{2k}\right] }X_{\nu
_{1}\nu_{2}}^{\mu_{1}\mu_{2}}\cdots X_{\nu_{2k-1}\nu_{2k}}^{\mu_{2k-1}\mu
_{2k}}, \label{P2n
\end{equation}
$\delta_{\left[ \nu_{1}...\nu_{k}\right] }^{\left[ \mu_{1}...\mu
_{k}\right] }$ is the totally antisymmetric generalization of the Kronecker
delta defined as $\delta_{\left[ \nu_{1}...\nu_{k}\right] }^{\left[ \mu
_{1}...\mu_{k}\right] }=\det\left[ \delta_{\nu_{1}}^{\mu_{1}}\cdots
\delta_{\nu_{k}}^{\mu_{k}}\right] $ and $W_{\left( E\right) }$ is the Weyl
tensor of an AAdS Einstein manifold, which can be written a
\begin{equation}
\left. W_{\left( E\right) }\right. _{\nu_{1}\nu_{2}}^{\mu_{1}\mu_{2
}=R_{\nu_{1}\nu_{2}}^{\mu_{1}\mu_{2}}+\frac{1}{{\ell}^{2}}\delta_{\left[
\nu_{1}\nu_{2}\right] }^{\left[ \mu_{1}\mu_{2}\right] }. \label{Weyl(E)
\end{equation}
In order to show the equivalence between Eq.(\ref{IRenChern}) and
Eq.(\ref{IRenPoly}), one starts by considering the $2n-$dimensional Euler
theorem, which relates the $n-th$ boundary Chern form with the $2n-
dimensional Euler density $\mathcal{E}_{2n}$. In particular, one has tha
\begin{equation}
\displaystyle\int\limits_{M_{2n}}\mathcal{E}_{2n}=\left( 4\pi\right)
^{n}n!\chi\left[ M_{2n}\right] +\displaystyle\int\limits_{\partial M_{2n
}B_{2n-1}, \label{EulerTheo
\end{equation}
where $\mathcal{E}_{2n}$ is given in terms of the bulk Riemann tensor b
\begin{equation}
\mathcal{E}_{2n}=\frac{1}{2^{n}}d^{2n}x\sqrt{G}\delta_{\left[ \mu_{1
...\mu_{2n}\right] }^{\left[ \nu_{1}...\nu_{2n}\right] }R_{\nu_{1}\nu_{2
}^{\mu_{1}\mu_{2}}\cdots R_{\nu_{2n-1}\nu_{2n}}^{\mu_{2n-1}\mu_{2n}},
\label{EulerDensity
\end{equation}
and $\chi\left[ M_{2n}\right] $ is the Euler characteristic of the bulk
manifold. One can then re-write the $n-th$ Chern form appearing in
Eq.(\ref{IRenChern}) in terms of $\mathcal{E}_{2n}$ and considering that for
AAdS Einstein manifold
\begin{equation}
R_{\nu_{1}\nu_{2}}^{\mu_{1}\mu_{2}}=\left. W_{\left( E\right) }\right.
_{\nu_{1}\nu_{2}}^{\mu_{1}\mu_{2}}-\frac{1}{{\ell}^{2}}\delta_{\left[ \nu
_{1}\nu_{2}\right] }^{\left[ \mu_{1}\mu_{2}\right] },
\end{equation}
it is possible to show tha
\begin{equation}
\displaystyle\int\limits_{M_{2n}}d^{2n}x\sqrt{G}{\ell}^{2\left( n-1\right)
}P_{2n}\left[ W_{\left( E\right) }\right] =\displaystyle\int
\limits_{M_{2n}}d^{2n}x\sqrt{G}\left( R-2\Lambda\right) +c_{2n
\displaystyle\int\limits_{M_{2n}}\mathcal{E}_{2n}, \label{PtoE
\end{equation}
wit
\begin{equation}
\Lambda=-\frac{\left( 2n-1\right) \left( 2n-2\right) }{2{\ell}^{2
},~R=-\frac{2n\left( 2n-1\right) }{{\ell}^{2}},\ \ \ \ \ \label{LandR
\end{equation}
$c_{2n}$ as given in Eq.(\ref{c_2n}) and $P_{2n}\left[ X\right] $ as given
in Eq.(\ref{P2n}), making use of the properties of the generalized Kronecker
delta functions. Then, the equivalence between Eq.(\ref{IRenChern}) and
Eq.(\ref{IRenPoly}) follows trivially. We note that $I_{EH}^{ren}\left[
M_{2n}\right] $ as given in Eq.(\ref{IRenPoly}), when considering that
$tr[W_{\left( E\right) }]=0$ for Einstein manifolds, only differs from the
renormalized action given in Ref.\cite{Miskovic:2014zja} by the constant term
proportional to $\chi\left[ M_{2n}\right] $. However, as discussed in
Ref.\cite{Olea:2005gb}, this term does not alter the dynamics of the equations
of motion, nor changes the overall thermodynamic properties of the solutions,
and it only amounts to a trivial shift in the value of the horizon entropy for
black-hole solutions.
Considering the form of $I_{EH}^{ren}\left[ M_{2n}\right] $ given in
Eq.(\ref{IRenPoly}), we now show that, as mentioned in the introduction, the
renormalized volume of AAdS Einstein manifolds in four and six dimensions is
indeed proportional to the renormalized Einstein-AdS action, with the same
proportionality factor considered in Eq.(\ref{RelationBare}). These cases
serve as examples for our conjectured relation in the general $2n-$dimensional
case, which we discuss afterwards.
\subsection{The 4D case}
In the case of four-dimensional AAdS manifolds, as seen from setting $n=2$ in
Eq.(\ref{IRenPoly}), the renormalized Einstein-AdS action is given b
\begin{equation}
I_{EH}^{ren}\left[ M_{4}\right] =\frac{{\ell}^{2}}{256\pi G
{\displaystyle\int\limits_{M_{4}}}
d^{4}x\sqrt{G}\delta_{\left[ \mu_{1}\mu_{2}\mu_{3}\mu_{4}\right] }^{\left[
\nu_{1}\nu_{2}\nu_{3}\nu_{4}\right] }\left. W_{\left( E\right) }\right.
_{\nu_{1}\nu_{2}}^{\mu_{1}\mu_{2}}\left. W_{\left( E\right) }\right.
_{\nu_{3}\nu_{4}}^{\mu_{3}\mu_{4}}-\frac{\pi{\ell}^{2}}{2G}\chi\left[
M_{4}\right] . \label{I4D
\end{equation}
Now, considering the definition of $\left\vert W_{\left( E\right)
}\right\vert ^{2}$ a
\begin{equation}
\left\vert W_{\left( E\right) }\right\vert ^{2}\overset{\text{def}
{=}\left. W_{\left( E\right) }\right. ^{\alpha\beta\mu\nu}\left.
W_{\left( E\right) }\right. _{\alpha\beta\mu\nu},
\end{equation}
and tha
\begin{equation}
\frac{1}{16
{\displaystyle\int\limits_{M_{4}}}
d^{4}x\sqrt{G}\delta_{\left[ \mu_{1}\mu_{2}\mu_{3}\mu_{4}\right] }^{\left[
\nu_{1}\nu_{2}\nu_{3}\nu_{4}\right] }\left. W_{\left( E\right) }\right.
_{\nu_{1}\nu_{2}}^{\mu_{1}\mu_{2}}\left. W_{\left( E\right) }\right.
_{\nu_{3}\nu_{4}}^{\mu_{3}\mu_{4}}=\frac{1}{4
{\displaystyle\int\limits_{M_{4}}}
d^{4}x\sqrt{G}\left\vert W_{\left( E\right) }\right\vert ^{2},
\end{equation}
$I_{EH}^{ren}\left[ M_{4}\right] $ can be re-written a
\begin{equation}
I_{EH}^{ren}\left[ M_{4}\right] =\frac{{\ell}^{2}}{64\pi G
{\displaystyle\int\limits_{M_{4}}}
d^{4}x\sqrt{G}\left\vert W_{\left( E\right) }\right\vert ^{2}-\frac{\pi
{\ell}^{2}}{2G}\chi\left[ M_{4}\right] .
\end{equation}
Finally, our proposal for the renormalized volume is given b
\begin{equation}
Vol_{ren}\left[ M_{4}\right] =-\frac{8\pi G{\ell}^{2}}{3}I_{EH}^{ren}\left[
M_{4}\right] , \label{RenVol4
\end{equation}
and so we have tha
\begin{equation}
\frac{1}{32\pi^{2}
{\displaystyle\int\limits_{M_{4}}}
d^{4}x\sqrt{G}\left\vert W_{\left( E\right) }\right\vert ^{2}=\chi\left[
M_{4}\right] -\frac{3}{4\pi^{2}{\ell}^{4}}Vol_{ren}\left[ M_{4}\right] ,
\end{equation}
in agreement with Anderson's formula \cite{2000math.....11051A}. By
considering Eq.(\ref{RenVol4}), we then conclude that in four dimensions the
renormalized Einstein-AdS action is indeed proportional to the renormalized
volume, and it is trivial to check that the proportionality factor is the same
as the one between the EH part of the action and the unrenormalized volume.
\subsection{The 6D case\label{2.2}}
In the case of six-dimensional AAdS manifolds, as seen from setting $n=3$ in
Eq.(\ref{IRenPoly}), the renormalized Einstein-AdS action is given b
\begin{equation}
I_{EH}^{ren}\left[ M_{6}\right] =\frac{1}{16\pi G
{\displaystyle\int\limits_{M_{6}}}
d^{6}x\sqrt{G}{\ell}^{4}P_{6}\left[ W_{\left( E\right) }\right] +\frac
{\pi^{2}{\ell}^{4}}{3G}\chi\left[ M_{6}\right] , \label{I6D
\end{equation}
where the $P_{6}$ polynomial in contractions of the Weyl tensor is given b
\begin{align}
P_{6}\left[ W_{\left( E\right) }\right] & =\frac{1}{2\left( 4!\right)
{\ell}^{2}}\delta_{\left[ \mu_{1}\mu_{2}\mu_{3}\mu_{4}\right] }^{\left[
\nu_{1}\nu_{2}\nu_{3}\nu_{4}\right] }\left. W_{\left( E\right) }\right.
_{\nu_{1}\nu_{2}}^{\mu_{1}\mu_{2}}\left. W_{\left( E\right) }\right.
_{\nu_{3}\nu_{4}}^{\mu_{3}\mu_{4}}\nonumber\\
& -\frac{1}{\left( 4!\right) ^{2}}\delta_{\left[ \mu_{1}\mu_{2}\mu_{3
\mu_{4}\mu_{5}\mu_{6}\right] }^{\left[ \nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu
_{5}\nu_{6}\right] }\left. W_{\left( E\right) }\right. _{\nu_{1}\nu_{2
}^{\mu_{1}\mu_{2}}\left. W_{\left( E\right) }\right. _{\nu_{3}\nu_{4
}^{\mu_{3}\mu_{4}}\left. W_{\left( E\right) }\right. _{\nu_{5}\nu_{6
}^{\mu_{5}\mu_{6}}. \label{P6
\end{align}
Therefore, the Euler characteristic of the bulk manifold $M_{2n}$ can be
written a
\begin{equation}
\chi\left[ M_{6}\right] =\frac{3G}{\pi^{2}{\ell}^{4}}I_{EH}^{ren}\left[
M_{6}\right] -\frac{3}{16\pi^{3}
{\displaystyle\int\limits_{M_{6}}}
d^{6}x\sqrt{G}P_{6}\left[ W_{\left( E\right) }\right] .
\end{equation}
\qquad
Now, we can rewrite $P_{6}\left[ W_{\left( E\right) }\right] $ in terms of
$\left\vert W_{\left( E\right) }\right\vert ^{2}$ and the Weyl invariants in
six dimensions (setting ${\ell=1}$ for simplicity). In order to do this, we
consider that the first two Weyl invariants, $I_{1}$ and $I_{2}$, are given b
\begin{align}
I_{1}\left[ W\right] & =W_{\alpha\beta\mu\nu}W^{\alpha\rho\lambda\nu
}W_{\rho\text{\ \ ~~~}\lambda}^{\text{ \ }\beta\mu},\nonumber\\
I_{2}\left[ W\right] & =W_{\mu\nu}^{\alpha\beta}W_{\alpha\beta
^{\rho\lambda}W_{\rho\lambda}^{\mu\nu},\ \ \ \ \label{WeylInvs
\end{align}
where $W$ denotes the Weyl tensor of a manifold that need not be Einstein.
Then, considering tha
\begin{align}
\delta_{\left[ \nu_{1}\nu_{2}\nu_{3}\nu_{4}\right] }^{\left[ \mu_{1}\mu
_{2}\mu_{3}\mu_{4}\right] }\left. W_{\left( E\right) }\right. _{\mu
_{1}\mu_{2}}^{\nu_{1}\nu_{2}}\left. W_{\left( E\right) }\right. _{\mu
_{3}\mu_{4}}^{\nu_{3}\nu_{4}} & =4\left\vert W_{\left( E\right)
}\right\vert ^{2},\nonumber\\
\delta_{\left[ \nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}\nu_{6}\right] }^{\left[
\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}\mu_{6}\right] }\left. W_{\left(
E\right) }\right. _{\mu_{1}\mu_{2}}^{\nu_{1}\nu_{2}}\left. W_{\left(
E\right) }\right. _{\mu_{3}\mu_{4}}^{\nu_{3}\nu_{4}}\left. W_{\left(
E\right) }\right. _{\mu_{5}\mu_{6}}^{\nu_{5}\nu_{6}} & =16\left(
2I_{2}\left[ W_{\left( E\right) }\right] +4I_{1}\left[ W_{\left(
E\right) }\right] \right) ,
\end{align}
we obtain
\begin{equation}
-4!P_{6}\left[ W_{\left( E\right) }\right] =-2\left\vert W_{\left(
E\right) }\right\vert ^{2}+\frac{7}{3}I_{2}\left[ W_{\left( E\right)
}\right] -\frac{4}{3}I_{1}\left[ W_{\left( E\right) }\right] +\left(
4I_{1}\left[ W_{\left( E\right) }\right] -I_{2}\left[ W_{\left(
E\right) }\right] \right) .
\end{equation}
Now, we consider an identity given by Osborn and Stergiou in
Ref.\cite{Osborn:2015rna}, which for Einstein manifolds in six dimensions
states tha
\begin{equation}
4I_{1}\left[ W_{\left( E\right) }\right] -I_{2}\left[ W_{\left(
E\right) }\right] =\left. W_{\left( E\right) }\right. ^{\rho\mu
\nu\lambda}\square\left. W_{\left( E\right) }\right. _{\rho\mu\nu\lambda
}+10\left\vert W_{\left( E\right) }\right\vert ^{2}, \label{IdentityOsborn
\end{equation}
where $\square\overset{\text{def}}{=}\nabla_{\mu}\nabla^{\mu}$ is the
covariant Laplacian operator (for more details on the identity, see Appendix
\ref{Appendix A}). Then, integrating by parts, we have
\begin{equation}
4I_{1}\left[ W_{\left( E\right) }\right] -I_{2}\left[ W_{\left(
E\right) }\right] =-\left\vert \nabla W_{\left( E\right) }\right\vert
^{2}+10\left\vert W_{\left( E\right) }\right\vert ^{2}+\text{b.t.},
\label{BTEq
\end{equation}
where b.t. is a boundary term that in our case plays no role, as it vanishes
asymptotically near the AdS boundary, and is therefore neglected (see Appendix
\ref{Appendix B}). Then, we obtain
\begin{equation}
-4!P_{6}=-\left\vert \nabla W_{\left( E\right) }\right\vert ^{2}+8\left\vert
W_{\left( E\right) }\right\vert ^{2}+\frac{7}{3}I_{2}\left[ W_{\left(
E\right) }\right] -\frac{4}{3}I_{1}\left[ W_{\left( E\right) }\right] .
\end{equation}
Finally, we consider the definition of the conformal invariant $J$, given by
Chang, Quing and Yang in Ref.\cite{Yang2008} a
\begin{equation}
J\left[ W\right] =-\left\vert \nabla W\right\vert ^{2}+8\left\vert
W\right\vert ^{2}+\frac{7}{3}W_{\mu\nu}^{\text{ \ \ }\alpha\beta
W_{\alpha\beta}^{\text{ \ \ \ }\lambda\rho}W_{\lambda\rho}^{\text{ \ \ \
\mu\nu}+\frac{4}{3}W_{\mu\nu\rho\lambda}W^{\mu\alpha\rho\beta}W_{\text{
~}\alpha\text{ \ ~~}\beta}^{\nu~~\lambda},
\end{equation}
and we see tha
\begin{equation}
P_{6}\left[ W_{\left( E\right) }\right] =-\frac{1}{4!}J\left[ W_{\left(
E\right) }\right] .
\end{equation}
Therefore, the Euler characteristic of the $M_{6}$ bulk manifold can be
written a
\begin{equation}
\chi\left[ M_{6}\right] =\frac{3G}{\pi^{2}{\ell}^{4}}I_{EH}^{ren}\left[
M_{6}\right] -\frac{1}{128\pi^{3}
{\displaystyle\int\limits_{M_{6}}}
d^{6}x\sqrt{G}J\left[ W_{\left( E\right) }\right] ,
\end{equation}
and by considering that, according to our proposal for renormalized volume
\begin{equation}
I_{EH}^{ren}\left[ M_{6}\right] =-\frac{5}{8\pi G{\ell}^{2}}Vol_{ren}\left[
M_{6}\right] , \label{RenVol6
\end{equation}
we have tha
\begin{equation}
\chi\left[ M_{6}\right] =-\frac{15}{8\pi^{3}{\ell}^{6}}Vol_{ren}\left[
M_{6}\right] +\frac{1}{128\pi^{3}
{\displaystyle\int\limits_{M_{6}}}
d^{6}x\sqrt{G}J\left[ W_{\left( E\right) }\right] ,
\end{equation}
in agreement with the renormalized volume formula proposed by Chang, Qing and
Yang \cite{Yang2008}. Inspection of Eq.(\ref{RenVol6}) allows us to conclude
that, in the six-dimensional case, the renormalized action is indeed
proportional to the renormalized volume, where the proportionality factor is
again the same as between the EH part of the action and the unrenormalized volume.
\subsection{The general even-dimensional case}
In agreement with the four-dimensional and six-dimensional cases, we propose
that the renormalized volume of $2n-$dimensional AAdS Einstein manifolds is
proportional to the renormalized Einstein-AdS action, such tha
\begin{equation}
I_{EH}^{ren}\left[ M_{2n}\right] =-\frac{\left( 2n-1\right) }{8\pi G{\ell
}^{2}}Vol_{ren}\left[ M_{2n}\right] . \label{I_ren/Vol_ren
\end{equation}
Considering this proposal and the polynomial form of the renormalized action
as presented in Eq.(\ref{IRenPoly}), we conjecture that the renormalized
volume of $M_{2n}$ is given b
\begin{equation}
Vol_{ren}\left[ M_{2n}\right] =-\frac{{\ell}^{2}}{2\left( 2n-1\right)
}\left(
{\displaystyle\int\limits_{M_{2n}}}
d^{2n}x\sqrt{G}{\ell}^{2\left( n-1\right) }P_{2n}\left[ W_{\left(
E\right) }\right] -c_{2n}\left( 4\pi\right) ^{n}n!\chi\left[
M_{2n}\right] \right) , \label{VolRenM2n
\end{equation}
where $c_{2n}$ and $W_{\left( E\right) }$ are defined in Eq.(\ref{c_2n}) and
Eq.(\ref{Weyl(E)}) respectively.
We note that our expression for $Vol_{ren}\left[ M_{2n}\right] $ corresponds
to a concrete realization of Albin's prescription, given in
Ref.\cite{ALBIN2009140}, which considers that for even-dimensional AAdS
Einstein spaces, the renormalized volume can be expressed in terms of the
integral of a polynomial in contractions of the Weyl curvature tensor, and the
Euler characteristic of the manifold. Also, the obtained expression for the
renormalized volume has the same form as the expression for the renormalized
EE of holographic CFTs obtained in Ref.\cite{Anastasiou:2018rla}, multiplied
by $4G$. Therefore, both instances of renormalized volumes (for both the bulk
manifold $M_{2n}$ and the minimal surface $\Sigma$ of the RT construction) can
be put in the same footing. We also note that the renormalized volume
expression is trivial for constant curvature AdS manifolds. In particular, in
the constant curvature case, the Weyl tensor $W_{\left( E\right) }$ of
$M_{2n}$ vanishes identically, and so the only remaining contribution to the
renormalized volume is the purely topological constant, which is proportional
to the Euler characteristic of the manifold. The renormalized volume can
therefore be understood as a measure of the deviation of a manifold with
respect to the constant curvature case, which corresponds to the maximally
symmetric vacuum of the Einstein-AdS gravity theory (usually global AdS).
We will see in the following sections that the renormalized volume formula is
also applicable to codimension-2 surfaces that minimize a certain total action
which corresponds to the renormalized version of Dong's proposed action for
the bulk and a cosmic brane with a certain tension \cite{Dong:2016fnf}, which
is applicable in the computation of holographic R\'{e}nyi entropies.
\section{Action on replica orbifold and cosmic branes\label{Section III}}
In the computation of holographic R\'{e}nyi entropies, it is useful to
consider the replica trick in order to construct a suitable $2n-$dimensional
bulk replica orbifold ${M}_{2n}^{\left( \alpha\right) }$, which is a
squashed cone (conically singular manifold without U$\left( 1\right) $
isometry \cite{Fursaev:2013fta,atiyah_lebrun_2013}) having a conical angular
parameter $\alpha$, such that $2\pi\left( 1-\alpha\right) $ is its angular
deficit. Then, using the AdS/CFT correspondence in the semi-classical limit,
the $m-th$ modular entropy $\widetilde{S}_{m}$
\cite{Dong:2016fnf,Nishioka:2018khk} which, as mentioned in the introduction,
is used in the computation of EREs, can be computed a
\begin{equation}
\widetilde{S}_{m}=\left. -\partial_{\alpha}I_{E}\left[ {M}_{2n}^{\left(
\alpha\right) }\right] \right\vert _{\alpha=\frac{1}{m}}, \label{SModular
\end{equation}
where $I_{E}\left[ {M}_{2n}^{\left( \alpha\right) }\right] $ is the
Euclidean bulk gravitational action evaluated on ${M}_{2n}^{\left(
\alpha\right) }$. As shown by Lewkowycz and Maldacena
\cite{Lewkowycz:2013nqa}, when one considers the Einstein-AdS action and the
limit of $\alpha\rightarrow1$, this prescription recovers the well-known RT
area formula \cite{Ryu:2006bv}, considering that the locus of the conical
sigularity (which is the fixed-point set of the replica symmetry) defines a
codimension-2 surface which coincides with $\Sigma$. In the case of
$\alpha=\frac{1}{m}$, with $m\in\mathbb{N}$ and $m>1$, the $m-th$ modular
entropy can be used to construct the $m-th$ R\'{e}nyi entropy $S_{m}$
according to Eq.(\ref{RenyiEnt}). In particular, as shown by Dong
\cite{Dong:2016fnf}, the locus of the conical singularity will correspond to
that of a cosmic brane with constant tension given by $T=\frac{\left(
1-\alpha\right) }{4G}$, whose coupling to the bulk metric is implemented by
the Nambu-Goto (NG) action for the induced metric $\gamma$ on the brane.
Therefore, the conical defect of the replica orbifold ${M}_{2n}^{\left(
\alpha\right) }$ is sourced by the cosmic brane and its location is
determined by minimizing the full action, which considers the contributions of
both the bulk Einstein-AdS action and the NG action of the cosmic brane. This
idea is further implemented by Nishioka \cite{Nishioka:2018khk}, where it is
explained that the resulting total action (which is refered to as the
bulk-per-replica action), contains the contribution at the conical
singularity, which precisely gives the usual area formula of RT and its
correct generalization beyond the tensionless limit, which is needed to
compute the modular entropy, and from it, the R\'{e}nyi entropy.
We first show how the evaluation of the standard Einstein-AdS action in the
replica orbifold ${M}_{2n}^{\left( \alpha\right) }$ directly leads to the
total action considered by Dong. Then, we consider its renormalized version,
in light of the volume renormalization procedure developed in Section
\ref{Section II}.
From the computation of curvature invariants defined on conically singular
manifolds
\cite{Fursaev:1995ef,Fursaev:2013fta,Mann:1996bi,Dahia:1998md,atiyah_lebrun_2013
, and in particular for the case of squashed-cone manifolds as studied by
Fursaev, Patrushev and Solodukhin in Ref.\cite{Fursaev:2013fta}, we have tha
\begin{equation}
R^{\left( \alpha\right) }=R+4\pi\left( 1-\alpha\right) \delta_{\Sigma_{T
}, \label{RicciConical
\end{equation}
where $R^{\left( \alpha\right) }$ is the Ricci scalar of the orbifold
${M}_{2n}^{\left( \alpha\right) }$, $R$ is its regular part, $2\pi\left(
1-\alpha\right) $ is the angular defect of the squashed cone and
$\delta_{\Sigma_{T}}$ is a $\left( 2n-2\right) -$dimensional $\delta$
function which has support only at the location of the conical singularity
(which coincides with the on-shell position of the cosmic brane). Therefore,
by using the definition of the $\delta_{\Sigma_{T}}$, which is such tha
\begin{equation
{\displaystyle\int\limits_{{M}_{2n}^{\left( \alpha\right) }}}
d^{2n}x\sqrt{G}\delta_{\Sigma_{T}}
{\displaystyle\int\limits_{\Sigma_{T}}}
d^{2n-2}y\sqrt{\gamma},
\end{equation}
where $\Sigma_{T}$ is the codimension-2 geometric locus of the conical
singularity and $\sqrt{\gamma}$ is the induced metric at the $\Sigma_{T}$
surface, we have tha
\begin{equation
{\displaystyle\int\limits_{{M}_{2n}^{\left( \alpha\right) }}}
d^{2n}x\sqrt{G}R^{\left( \alpha\right) }
{\displaystyle\int\limits_{M_{2n}^{\left( \alpha\right) }\setminus\Sigma
_{T}}}
d^{2n}x\sqrt{G}R+4\pi\left( 1-\alpha\right)
{\displaystyle\int\limits_{\Sigma_{T}}}
d^{2n-2}y\sqrt{\gamma}.\label{RicciScalarConical
\end{equation}
Finally, considering that the NG action of a codimension-2 brane $\Sigma_{T}$
with tension T is given b
\begin{equation}
I_{NG}\left[ \Sigma_{T}\right] =
{\displaystyle\int\limits_{\Sigma_{T}}}
d^{2n-2}y\sqrt{\gamma},
\end{equation}
we have that the Einstein-AdS action evaluated on ${M}_{2n}^{\left(
\alpha\right) }$ give
\begin{equation}
I_{EH}\left[ {M}_{2n}^{\left( \alpha\right) }\right] =\frac{1}{16\pi G
{\displaystyle\int\limits_{M_{2n}^{\left( \alpha\right) }\setminus\Sigma
_{T}}}
d^{2n}x\sqrt{G}\left( R-2\Lambda\right) +\frac{\left( 1-\alpha\right)
}{4G
{\displaystyle\int\limits_{\Sigma_{T}}}
d^{2n-2}y\sqrt{\gamma},
\end{equation}
and therefore
\begin{align}
I_{EH}\left[ {M}_{2n}^{\left( \alpha\right) }\right] & =I_{EH}\left[
M_{2n}^{\left( \alpha\right) }\setminus\Sigma_{T}\right] +I_{NG}\left[
\Sigma_{T}\right] =I_{tot},\nonumber\\
T\left( \alpha\right) & =\frac{\left( 1-\alpha\right) }{4G},
\end{align}
which corresponds to the total action $I_{tot}$ considered by Dong. Hence, the
NG action arises as the conical contribution to the EH action evaluated on the
replica orbifold. The $m-th$ modular entropy $\widetilde{S}_{m}$ can then be
computed, according to Eq.(\ref{SModular}), thus obtaining Eq.(\ref{ModEnt}).
Finally, the holographic ERE can be computed according to Eq.(\ref{RenyiEnt}).
We now proceed to obtain the renormalized version of the total action
$I_{tot}^{ren}$, by considering the evaluation of the renormalized
Einstein-AdS action on the orbifold ${M}_{2n}^{\left( \alpha\right) }$.
\subsection{Renormalized NG action from the conical singularity}
We evaluate the renormalized Einstein-AdS action on the replica orbifold, obtainin
\begin{equation}
I_{EH}^{ren}\left[ {M}_{2n}^{\left( \alpha\right) }\right] =\frac{1}{16\pi
G
{\displaystyle\int\limits_{{M}_{2n}^{\left( \alpha\right) }}}
d^{2n}x\sqrt{G}{\ell}^{2\left( n-1\right) }P_{2n}\left[ W_{\left(
E\right) }^{\left( \alpha\right) }\right] -\frac{c_{2n}}{16\pi G}\left(
4\pi\right) ^{n}n!\chi\left[ {M}_{2n}^{\left( \alpha\right) }\right] ,
\end{equation}
where the conically-singular Einstein Weyl is defined a
\begin{equation}
\left. W_{\left( E\right) }^{\left( \alpha\right) }\right. _{\nu_{1
\nu_{2}}^{\mu_{1}\mu_{2}}\overset{\text{def}}{=}\left. R^{\left(
\alpha\right) }\right. _{\nu_{1}\nu_{2}}^{\mu_{1}\mu_{2}}+\frac{1}{{\ell
}^{2}}\delta_{\left[ \nu_{1}\nu_{2}\right] }^{\left[ \mu_{1}\mu_{2}\right]
},
\end{equation}
and $c_{2n}$ is given in Eq.(\ref{c_2n}). Now, using tha
\begin{equation
{\displaystyle\int\limits_{{M}_{2n}^{\left( \alpha\right) }}}
d^{2n}x\sqrt{G}{\ell}^{2\left( n-1\right) }P_{2n}\left[ W_{\left(
E\right) }^{\left( \alpha\right) }\right]
{\displaystyle\int\limits_{{M}_{2n}^{\left( \alpha\right) }}}
d^{2n}x\sqrt{G}\left( R^{\left( \alpha\right) }-2\Lambda\right) +c_{2n
{\displaystyle\int\limits_{{M}_{2n}^{\left( \alpha\right) }}}
\mathcal{E}_{2n}^{\left( \alpha\right) },
\end{equation}
where $P_{2n}\left[ X\right] $ is given in Eq.(\ref{P2n}) and $\varepsilon
_{2n}$ is defined in Eq.(\ref{EulerDensity}), we hav
\begin{equation}
I_{EH}^{ren}\left[ {M}_{2n}^{\left( \alpha\right) }\right] =\frac{1}{16\pi
G
{\displaystyle\int\limits_{{M}_{2n}^{\left( \alpha\right) }}}
d^{2n}x\sqrt{G}\left( R^{\left( \alpha\right) }-2\Lambda\right)
+\frac{c_{2n}}{16\pi G
{\displaystyle\int\limits_{{M}_{2n}^{\left( \alpha\right) }}}
\mathcal{E}_{2n}^{\left( \alpha\right) }-\frac{c_{2n}}{16\pi G}\left(
4\pi\right) ^{n}n!\chi\left[ {M}_{2n}^{\left( \alpha\right) }\right] .
\end{equation}
Also, from the properties of the Euler density for squashed cones as
conjectured in Ref.\cite{Anastasiou:2018rla}, we have tha
\begin{align
{\displaystyle\int\limits_{{M}_{2n}^{\left( \alpha\right) }}}
\mathcal{E}_{2n}^{\left( \alpha\right) } &
{\displaystyle\int\limits_{M_{2n}^{\left( \alpha\right) }\setminus\Sigma
_{T}}}
\mathcal{E}_{2n}+4\pi n\left( 1-\alpha\right)
{\displaystyle\int\limits_{\Sigma_{T}}}
\varepsilon_{2n-2}~+O\left( \left( 1-\alpha\right) ^{2}\right)
,\nonumber\\
\chi\left[ {M}_{2n}^{\left( \alpha\right) }\right] & =\chi\left[
M_{2n}^{\left( \alpha\right) }\setminus\Sigma_{T}\right] +\left(
1-\alpha\right) \chi\left[ \Sigma_{T}\right] +O\left( \left(
1-\alpha\right) ^{2}\right) ,
\end{align}
and by considering Eq.(\ref{RicciScalarConical}), we can write $I_{EH
^{ren}\left[ {M}_{2n}^{\left( \alpha\right) }\right] $ a
\begin{align}
I_{EH}^{ren}\left[ {M}_{2n}^{\left( \alpha\right) }\right] & =\frac
{1}{16\pi G}\left( \displaystyle\int\limits_{M_{2n}^{\left( \alpha\right)
}\setminus\Sigma_{T}}d^{2n}x\sqrt{G}{\ell}^{2\left( n-1\right)
P_{2n}\left[ W_{\left( E\right) }\right] -c_{2n}\left( 4\pi\right)
^{n}n!\chi\left[ M_{2n}^{\left( \alpha\right) }\setminus\Sigma_{T}\right]
\right) \nonumber\\
+ & \frac{\left( 1-\alpha\right) }{4G}\left( \displaystyle\int
\limits_{\Sigma_{T}}d^{2n-2}y\sqrt{\gamma}+nc_{2n}\displaystyle\int
\limits_{\Sigma_{T}}\varepsilon_{2n-2}-nc_{2n}\left( 4\pi\right)
^{n-1}\left( n-1\right) !\chi\left( \Sigma_{T}\right) \right) \nonumber\\
& +O\left( \left( 1-\alpha\right) ^{2}\right) .
\end{align}
Finally using that (as shown in Ref.\cite{Anastasiou:2018rla}
\begin{equation
{\displaystyle\int\limits_{\Sigma_{T}}}
d^{2n-2}y\sqrt{\gamma}+nc_{2n
{\displaystyle\int\limits_{\Sigma_{T}}}
\varepsilon_{2n-2}=-\frac{{\ell}^{2}}{2\left( 2n-3\right)
{\displaystyle\int\limits_{\Sigma_{T}}}
d^{2n-2}y\sqrt{\gamma}{\ell}^{2\left( n-2\right) }P_{2n-2}\left[
\mathcal{F}_{AdS}\right] , \label{SigmaVolRenorm
\end{equation}
where $\mathcal{F}_{AdS}$ for $\Sigma_{T}$ is defined in Eq.(\ref{FAdS}), and tha
\begin{equation}
c_{2n-2}=-\frac{2\left( 2n-3\right) }{{\ell}^{2}}nc_{2n},
\end{equation}
we obtai
\begin{align}
I_{EH}^{ren}\left[ {M}_{2n}^{\left( \alpha\right) }\right] & =\frac
{1}{16\pi G}\displaystyle\int\limits_{M_{2n}^{\left( \alpha\right)
}\setminus\Sigma_{T}}d^{2n}x\sqrt{G}{\ell}^{2\left( n-1\right)
P_{2n}\left[ W_{\left( E\right) }\right] -\frac{c_{2n}}{16\pi G}\left(
4\pi\right) ^{n}n!\chi\left[ M_{2n}^{\left( \alpha\right) }\setminus
\Sigma_{T}\right] \nonumber\\
& +\frac{\left( 1-\alpha\right) }{4G}\left( -\frac{{\ell}^{2}}{2\left(
2n-3\right) }\right) \Bigg(\displaystyle\int\limits_{\Sigma_{T}
d^{2n-2}y\sqrt{\gamma}{\ell}^{2\left( n-2\right) }P_{2n-2}\left[
\mathcal{F}_{AdS}\right] \nonumber\\
& -c_{2n-2}\left( 4\pi\right) ^{n-1}\left( n-1\right) !\chi\left[
\Sigma_{T}\right] \Bigg)+O\left( \left( 1-\alpha\right) ^{2}\right) .
\end{align}
Therefore
\begin{equation}
I_{EH}^{ren}\left[ {M}_{2n}^{\left( \alpha\right) }\right] =I_{EH
^{ren}\left[ M_{2n}^{\left( \alpha\right) }\setminus\Sigma_{T}\right]
+I_{NG}^{ren}\left[ \Sigma_{T}\right] +O\left( \left( 1-\alpha\right)
^{2}\right) ,
\end{equation}
where $I_{EH}^{ren}\left[ M_{2n}^{\left( \alpha\right) }\setminus\Sigma
_{T}\right] $ is given in Eq.(\ref{IRenPoly}), and $I_{NG}^{ren}\left[
\Sigma_{T}\right] $ is defined a
\begin{equation}
I_{NG}^{ren}\left[ \Sigma_{T}\right] =\frac{\left( 1-\alpha\right)
{4G}Vol_{ren}\left[ \Sigma_{T}\right] , \label{IRenNG
\end{equation}
for $Vol_{ren}\left[ \Sigma_{T}\right] $ given b
\begin{equation}
Vol_{ren}\left[ \Sigma\right] =-\frac{{\ell}^{2}}{2\left( 2n-3\right)
}\Bigg(
{\displaystyle\int\limits_{\Sigma}}
d^{2n-2}y\sqrt{\gamma}{\ell}^{2\left( n-2\right) }P_{2n-2}\left[
\mathcal{F}_{AdS}\right] -c_{2n-2}\left( 4\pi\right) ^{n-1}\left(
n-1\right) !\chi\left[ \Sigma\right] \Bigg) , \label{VolRenSigma
\end{equation}
in complete analogy with the formula for the renormalized volume of the bulk
manifold. By definin
\begin{equation}
I_{tot}^{ren}\overset{\text{def}}{=}I_{EH}^{ren}\left[ M_{2n}^{\left(
\alpha\right) }\setminus\Sigma_{T}\right] +I_{NG}^{ren}\left[ \Sigma
_{T}\right] \text{,}\label{ITotalRenorm
\end{equation}
we note that our expression for $I_{tot}^{ren}$ is consistent with Dong's
proposal for $I_{tot}$, as it corresponds to the renormalized version of it.
Therefore, our renormalized total action is understood as the sum of the
renormalized Einstein-AdS action for the regular part of the bulk plus the
renormalized NG action of the cosmic brane. We emphasize that the dynamics
obtained by extremizing $I_{tot}^{ren}$ is the same as that obtained by
extremizing $I_{tot}$ as given in Ref.\cite{Dong:2016fnf}, because both
actions only differ by topological bulk terms that are equivalent to boundary
terms through use of the Euler theorem, and therefore, they lead to the same
bulk Euler-Lagrange equations of motion. Furthermore, the boundary terms
generated on the AdS boundary and on $\partial\Sigma_{T}$, are precisely the
ones that cancel the bulk divergences, while at the same time being consistent
with Dirichlet boundary conditions for the CFT metric $g_{\left( 0\right) }$
and for the induced metric $\sigma_{\left( 0\right) }$ on the entanglement
surface in the CFT, in the notation of
Ref.\cite{Anastasiou:2017xjr,Anastasiou:2018rla}.
We note that, $I_{EH}^{ren}\left[ {M}_{2n}^{\left( \alpha\right) }\right]
$ appears to differ from $I_{tot}^{ren}$ by an unspecified $O\left( \left(
1-\alpha\right) ^{2}\right) $ part. However, we conjecture that there should
not be extra terms of higher order in $\left( 1-\alpha\right) $ and, that
therefore, we should hav
\begin{align
{\displaystyle\int\limits_{{M}_{2n}^{\left( \alpha\right) }}}
\mathcal{E}_{2n}^{\left( \alpha\right) } &
{\displaystyle\int\limits_{M_{2n}^{\left( \alpha\right) }\setminus\Sigma
_{T}}}
\mathcal{E}_{2n}+4\pi n\left( 1-\alpha\right)
{\displaystyle\int\limits_{\Sigma_{T}}}
\varepsilon_{2n-2},\nonumber\\
\chi\left[ {M}_{2n}^{\left( \alpha\right) }\right] & =\chi\left[
M_{2n}^{\left( \alpha\right) }\setminus\Sigma_{T}\right] +\left(
1-\alpha\right) \chi\left[ \Sigma_{T}\right] ,\ \ \ \ \label{NewProposal
\end{align}
with no additional $O\left( \left( 1-\alpha\right) ^{2}\right) $ pieces,
instead. The reasoning for this is that the location of the conical
singularity or, equivalently, the position of the brane, should not be changed
by the renormalization procedure. In turn, the bulk dynamics should not be
affected and, consequently, the extra bulk terms added to the NG action of
$\Sigma_{T}$ in order to achieve the renormalization should be equivalent to a
boundary term at $\partial\Sigma_{T}$, which is fixed by the boundary
conditions. This is precisely our case, as can be seen in
Eq.(\ref{SigmaVolRenorm}) for the $O\left( \left( 1-\alpha\right) \right)
$ part, as the $\varepsilon_{2n-2}$ is equivalent to the $\left( n-1\right)
-$th Chern form $B_{2n-3}$ evaluated at $\partial\Sigma_{T}$ by virtue of the
Euler theorem in $\left( 2n-2\right) $ dimensions. Hence, any extra
contribution located at $\delta_{\Sigma_{T}}$ of higher order in $\left(
1-\alpha\right) $ would necessarily have to be an extra contribution
proportional to $\varepsilon_{2n-2}$, or to some other bulk topological term
dynamically equivalent to a boundary term at $\partial\Sigma_{T}$. In the
former case, the effect would be to change the form of the tension $T$ as a
function of $\alpha$, which is not allowed as it would change the physics.
Finally, the latter case seems very contrived and unlikely as other
topological terms are not always defined for $\left( 2n-2\right)
-$dimensional manifolds. Therefore, our conjecture of Eq.(\ref{NewProposal})
follows, on physical grounds, for the particular case of conical singularities
induced by cosmic branes with tension whose position is fixed by requiring the
on-shell condition. Finally, we then have tha
\begin{equation}
I_{EH}^{ren}\left[ {M}_{2n}^{\left( \alpha\right) }\right] =I_{tot}^{ren},
\end{equation}
with $I_{tot}^{ren}$ as given in Eq.(\ref{ITotalRenorm}), in analogy with
Dong's total action for the unrenormalized case.
\subsection{Action on conical orbifold as deformed volume}
The renormalized volume of the conically singular manifold $M_{2n}^{(\alpha)}$
can be thought of as a one-parameter family of deformations with respect to
the renormalized volume of the non--singular manifold $M_{2n}^{\left(
\alpha\right) }\setminus\Sigma_{T}$, that is
\begin{equation}
Vol_{ren}\big[M_{2n}^{(\alpha)}\big]=Vol_{ren}\big[M_{2n}^{\left(
\alpha\right) }\setminus\Sigma_{T}\big]+q_{\alpha}~,\label{Vshift
\end{equation}
where $q_{\alpha}$ is a finite deformation that vanishes in the tensionless
limit of $\alpha\rightarrow1$. From the bulk perspective, Eq.\eqref{Vshift}
amounts to reinterpreting the problem of computing the renormalized modular
entropy $\widetilde{S}_{m}^{ren}$, which in turns yields a renormalized
R\'{e}nyi entropy, as the problem of extremizing a bulk hypersurface while
fixing the renormalized $q$--deformed volume bounded by it.
Starting from $Vol_{ren}\left[ M_{2n}^{(\alpha)}\right] $ as given in
Eq.(\ref{VolRenM2n}) and considering that, as discussed above, $I_{EH
^{ren}\left[ {M}_{2n}^{\left( \alpha\right) }\right] =I_{tot}^{ren}$, for
$I_{tot}^{ren}$ given in Eq.(\ref{ITotalRenorm}), we can write the
renormalized volume of the replica orbifold as
\begin{equation}
Vol_{ren}\big[M_{2n}^{(\alpha)}\big]=Vol_{ren}\big[M_{2n}^{(\alpha)
\setminus\Sigma_{T}\big]-\left( 1-\alpha\right) \frac{2\pi{\ell}^{2
}{\left( 2n-1\right) }Vol_{ren}\left[ \Sigma_{T}\right]
.\label{Shifted_Bulk_Volume
\end{equation}
Therefore, the volume deformation $q_{\alpha}$ is given b
\begin{equation}
q_{\alpha}=-\left( 1-\alpha\right) \frac{2\pi{\ell}^{2}}{\left(
2n-1\right) }Vol_{ren}\left[ \Sigma_{T}\right] , \label{Shift
\end{equation}
and it is labelled by the angular parameter $\alpha$, or equivalently by the
tension $T\left( \alpha\right) =\frac{\left( 1-\alpha\right) }{4G}$. This
deformation has support only at the codimension-2 surface $\Sigma_{T}$.
As an example, we consider the $n=2$ case for an AAdS$_{4}$ bulk, where the
renormalized volume evaluated on the orbifold become
\begin{align}
Vol_{ren}\left[ M_{4}^{\left( \alpha\right) }\right] & =-\frac{{\ell
}^{4}}{96
{\displaystyle\int\limits_{M_{4}}}
d^{4}x\sqrt{G}\delta_{\left[ \mu_{1}\mu_{2}\mu_{3}\mu_{4}\right] }^{\left[
\nu_{1}\nu_{2}\nu_{3}\nu_{4}\right] }\left. W_{\left( E\right) }\right.
_{\nu_{1}\nu_{2}}^{\mu_{1}\mu_{2}}\left. W_{\left( E\right) }\right.
_{\nu_{3}\nu_{4}}^{\mu_{3}\mu_{4}}+\frac{4\pi^{2}{\ell}^{4}}{3}\chi\left[
M_{4}\right] \nonumber\\
& -\left( 1-\alpha\right) \left( \frac{\pi{\ell}^{4}}{6
{\displaystyle\int\limits_{\Sigma_{T}}}
d^{2}y\sqrt{\gamma}\delta_{\left[ a_{1}a_{2}\right] }^{\left[ b_{1
b_{2}\right] }\left. \mathcal{F}_{AdS}\right. _{b_{1}b_{2}}^{a_{1}a_{2
}-\frac{4\pi^{2}{\ell}^{4}}{3}\chi\left[ \Sigma_{T}\right] \right)
\end{align}
and therefore, in this case, the deformation in the bulk volume is given b
\begin{equation}
q_{\alpha}=-\left( 1-\alpha\right) \left( \frac{\pi{\ell}^{4}}{6
{\displaystyle\int\limits_{\Sigma_{T}}}
d^{2}y\sqrt{\gamma}\delta_{\left[ a_{1}a_{2}\right] }^{\left[ b_{1
b_{2}\right] }\left. \mathcal{F}_{AdS}\right. _{b_{1}b_{2}}^{a_{1}a_{2
}-\frac{4\pi^{2}{\ell}^{4}}{3}\chi\left[ \Sigma_{T}\right] \right) .
\end{equation}
\section{Renormalized R\'{e}nyi entropy from renormalized area of cosmic
branes\label{Section IV}}
Considering $I_{tot}^{ren}$ as defined in Eq.(\ref{ITotalRenorm}), the
renormalized modular entropy can be trivially computed a
\begin{equation}
\widetilde{S}_{m}^{ren}=-\partial_{\alpha}I_{tot}^{ren}=-\partial_{\alpha
}I_{NG}^{ren},
\end{equation}
in accordance with the discussion given in the introduction. Therefore, using
the form of $I_{NG}^{ren}$ defined in Eq.(\ref{IRenNG}), we obtai
\begin{equation}
\widetilde{S}_{m}^{ren}=\frac{Vol_{ren}\left[ \Sigma_{T}\right] }{4G},
\end{equation}
where $T\left( m\right) =\frac{\left( m-1\right) }{4mG}$, in agreement
with Eq.(\ref{ModEnt}). Now, the renormalized ERE can be computed from
$\widetilde{S}_{m}^{ren}$ using Eq.(\ref{RenyiEnt}), such tha
\begin{equation}
S_{m}^{ren}=\frac{m}{m-1
{\displaystyle\int\limits_{1}^{m}}
\frac{dm^{\prime}}{{m^{\prime}}^{2}}\widetilde{S}_{{m}^{\prime}}^{ren}.
\end{equation}
We remark that in computing $\partial_{\alpha}I_{tot}^{ren}$, the $\alpha$
dependence of the location of $\Sigma_{T}$ should no be taken into account,
because first one requires its location to be determined by the extremization
of $I_{tot}^{ren}$, and then, this location is taken as a given.
Now, for illustrative purposes, we present the form adopted by the modular
entropy for the cases of AAdS$_{4}$ and AAdS$_{6}$ bulk manifolds.
\subsection{Modular entropy in AdS$_{4}$/CFT$_{3}$ and in AdS$_{6}$/CFT$_{5}$}
For an AAdS$_{4}$ bulk manifold and an embedded codimension-2 surface
$\Sigma_{T}$ whose position is determined by minimizing $I_{tot}^{ren}$ for a
cosmic brane with tension $T\left( m\right) $, such that $\partial\Sigma
_{T}$ lies at the AdS boundary, we have that the modular entropy in the
CFT$_{3}$ is given b
\begin{equation}
\widetilde{S}_{m}^{ren}=-\partial_{\alpha}I_{tot}^{ren}=\frac{{\ell}^{2}}{16G
{\displaystyle\int\limits_{\Sigma_{T}}}
d^{2}y\sqrt{\gamma}\delta_{\left[ a_{1}a_{2}\right] }^{\left[ b_{1
b_{2}\right] }\left. \mathcal{F}_{AdS}\right. _{b_{1}b_{2}}^{a_{1}a_{2
}-\frac{\pi{\ell}^{2}}{2G}\chi\left[ \Sigma_{T}\right] , \label{SModRenn2
\end{equation}
where $\mathcal{F}_{AdS}$ is defined in Eq.(\ref{FAdS}), and $\chi\left[
\Sigma_{T}\right] $ is the Euler characteristic of $\Sigma_{T}$. We note that
in the tensionless limit, we recover our result for $S_{EE}^{ren}$ given in
Ref.\cite{Anastasiou:2017xjr}. Also, the resulting expression for
$\widetilde{S}_{m}^{ren}$, when multiplied by $4G$, matches the formula given
by Alexakis and Mazzeo for the renormalized area of a codimension-2 minimal
surface embedded in a four-dimensional AAdS Einstein manifold, given in
Ref.\cite{Alexakis2010}.
Analogously, for an AAdS$_{6}$ bulk manifold, the corresponding renormalized
modular entropy of the CFT$_{5}$ is given b
\begin{align}
\widetilde{S}_{m}^{ren} & =-\frac{{\ell}^{2}}{48G}\displaystyle\int
\limits_{\Sigma_{T}}d^{4}y\sqrt{\gamma}\left( \frac{{\ell}^{2}}{8
\delta_{\left[ a_{1}a_{2}a_{3}a_{4}\right] }^{\left[ b_{1}b_{2}b_{3
b_{4}\right] }\left. \mathcal{F}_{AdS}\right. _{b_{1}b_{2}}^{a_{1}a_{2
}\left. \mathcal{F}_{AdS}\right. _{b_{3}b_{4}}^{a_{3}a_{4}}\right.
\nonumber\\
& -\left. \delta_{\left[ a_{1}a_{2}\right] }^{\left[ b_{1}b_{2}\right]
}\left. \mathcal{F}_{AdS}\right. _{b_{1}b_{2}}^{a_{1}a_{2}}\right)
+\frac{\pi^{2}{\ell}^{4}}{3G}\chi\left[ \Sigma_{T}\right] .
\label{SModRenn3
\end{align}
We note that in the tensionless limit, we recover our result for $S_{EE
^{ren}$ in the AdS$_{6}$/CFT$_{5}$ case, as given in
Ref.\cite{Anastasiou:2018rla}. We also note that, when multiplying by $4G$,
the resulting expression is equal to the renormalized volume of a
four-dimensional AAdS manifold, having the same structure as the renormalized
Einstein-AdS$_{4}$ action, but with an extra $\delta_{\left[ a_{1
a_{2}\right] }^{\left[ b_{1}b_{2}\right] }\left. \mathcal{F}_{AdS}\right.
_{b_{1}b_{2}}^{a_{1}a_{2}}$ term, which is equal to $2tr\left[ \mathcal{F
_{AdS}\right] $. For Einstein manifolds, the Weyl tensor $W_{\left(
E\right) }$ is equal to $\mathcal{F}_{AdS}$, and because $tr\left[
W_{\left( E\right) }\right] =0$, this term vanishes. This shows that
although the minimal surface $\Sigma_{T}$ is not an Einstein manifold, its
renormalized volume has the same form. The reason for this is that our
renormalized volume formula given in Eq.(\ref{VolRenM2n}), evaluated using
$\mathcal{F}_{AdS}$, is valid for AAdS Einstein manifolds and also for minimal
submanifolds of codimension-2 embedded therein, where the meaning of minimal
is that the submanifold extremizes the $I_{tot}^{ren}$ action.
\section{Example: Renormalized R\'{e}nyi entropies for ball-shaped entangling
regions in odd-dimensional CFTs\label{Section V}}
We now compute the renormalized ERE in the case of a ball-shaped entangling
region on the CFT. This case is of interest as it can be computed exactly, and
in the limit of $m\rightarrow1$ (tensionless limit), one can directly check
that the result obtained for $S_{EE}^{ren}$ in Ref.\cite{Anastasiou:2018rla}
is recovered.
The first computation of EREs for ball-shaped regions in holographic CFTs was
performed by Hung, Myers, Smolkin and Yale in Ref.\cite{Hung:2011nu}, by using
the Casini-Huerta-Myers (CHM) map \cite{Casini:2011kv} to relate the
computation of the modular entropy (used to obtain the ERE) to that of the
Bekenstein-Hawking entropy of a certain topological BH. In particular, by
considering a conformal transformation, the CFT was put in a hyperbolic
background geometry, such that the reduced density matrix for the entangling
region in the vacuum state was unitarily mapped into the thermal density
matrix of a Gibbs state. By considering the usual holographic embedding of the
CFT into global AdS, the conformal transformation in the CFT was seen to
correspond to a coordinate transformation in the bulk, mapping two different
foliations of AdS space. Also, by the standard AdS/CFT dictionary, the thermal
state in the CFT was identified as the holographic dual of the topological
black hole, whose (Euclidean) metric is given b
\begin{equation}
ds^{2}=\frac{{\ell}^{2}}{R^{2}}\left[ f\left( r\right) d\tau^{2
+\frac{dr^{2}}{f\left( r\right) }+r^{2}\left( du^{2}+\sinh^{2
u~d\Omega_{d-2}^{2}\right) \right] ,
\end{equation}
wher
\begin{equation}
f\left( r\right) =\frac{r^{2}}{R^{2}}-1-\frac{r_{H}^{d-2}}{r^{d-2}}\left(
\frac{r_{H}}{R^{2}}-1\right) ,
\end{equation}
$d\Omega_{d-2}^{2}$ is the line element in a unit $\left( d-2\right) -$
sphere, $R$ is the radius of the original ball-shaped entangling region,
$r_{H}$ is the horizon radius where $f\left( r\right) $ vanishes and
$d=2n-1$ is the dimension of the CFT. Following Ref.\cite{Nishioka:2018khk},
from the metric we see that the topological BH has a temperature given b
\begin{equation}
T\left( x\right) =\frac{1}{4\pi R}\left[ dx-\frac{d-2}{x}\right] ,
\label{Tgeom
\end{equation}
where $x=\frac{r_{H}}{R}$ is the dimensionless horizon radius. Now, by
considering the form of the density matrix in the Gibbs state, the temperature
should also be equal t
\begin{equation}
T_{m}=\frac{1}{2\pi Rm}, \label{TGibbs
\end{equation}
where $m$ is the replica index (such that we are computing the $m-th$ modular
entropy). By equating the expressions given in Eqs.(\ref{Tgeom}) and
(\ref{TGibbs}), we can solve for the dimensionless horizon radius $x_{m}$,
obtaining tha
\begin{equation}
x_{m}=\frac{1+\sqrt{dm^{2}\left( d-2\right) +1}}{dm}. \label{x_m
\end{equation}
Therefore, the Bekenstein-Hawking entropy of the BH, and consequently (through
the CHM map) the $m-th$ modular entropy in the CFT$_{d}$, is given b
\begin{equation}
\widetilde{S}_{m}=\frac{Vol\left[ \mathcal{H}^{d-1}\right] {\ell}^{d-1}
{4G}x_{m}^{d-1}, \label{ModSph
\end{equation}
where $\mathcal{H}^{d-1}$ denotes a constant curvature $\left( d-1\right)
-$dimensional hyperbolic space with unit AdS radius. Then, finally, using
Eq.(\ref{RenyiEnt}), the $m-th$ R\'{e}nyi entropy is given b
\begin{equation}
S_{m}=\frac{m}{m-1
{\displaystyle\int\limits_{1}^{m}}
\frac{d{m}^{\prime}}{{m^{\prime}}^{2}}\widetilde{S}_{{m}^{\prime}
=\frac{Vol\left[ \mathcal{H}^{d-1}\right] {\ell}^{d-1}}{4G}\frac{m}{2\left(
m-1\right) }\left( 2-x_{m}^{d-2}-x_{m}^{d}\right) . \label{RenyiSph
\end{equation}
We note that $Vol\left[ \mathcal{H}^{d-1}\right] $ appearing in both the
modular entropy given in Eq.(\ref{ModSph}) and in the ERE given in
Eq.(\ref{RenyiSph}) is infinite, and therefore, the corresponding entropies
are divergent. We now proceed to renormalize them considering that
$\mathcal{H}^{d-1}$ is a $\left( 2n-2\right) -$dimensional AAdS Einstein
manifold (in particular, it is global AdS), and therefore, the correctly
renormalized $Vol_{ren}\left[ \mathcal{H}^{d-1}\right] $ can be computed
using our formula as given in Eq.(\ref{VolRenSigma}). Considering that
$\mathcal{H}^{d-1}$ has constant Riemannian curvature, and therefore
$\mathcal{F}_{\left. AdS\right\vert _{\mathcal{H}^{d-1}}}=0$, and also that
its Euler characteristic $\chi\left[ \mathcal{H}^{d-1}\right] =1$, we obtain tha
\begin{equation}
Vol_{ren}\left[ \mathcal{H}^{d-1}\right] =\frac{c_{d-1}\left( 4\pi\right)
^{\frac{d-1}{2}}\left( \frac{d-1}{2}\right) !}{2\left( d-2\right) },
\end{equation}
and using tha
\begin{equation}
c_{d-1}=\frac{2\left( -1\right) ^{\frac{d-1}{2}}}{\left( d-1\right)
\left( d-3\right) !},
\end{equation}
we have tha
\begin{equation}
Vol_{ren}\left[ \mathcal{H}^{d-1}\right] =\frac{\left( -1\right)
^{\frac{d-1}{2}}\left( 4\pi\right) ^{\frac{d-1}{2}}\left( \frac{d-1
{2}\right) !}{\left( d-1\right) !}.
\end{equation}
We therefore find that the renormalized modular entropy and renormalized ERE
are given b
\begin{equation}
\widetilde{S}_{m}^{ren}=\frac{\left( -1\right) ^{\frac{d-1}{2}}\left(
4\pi\right) ^{\frac{d-1}{2}}\left( \frac{d-1}{2}\right) !{\ell}^{d-1
}{4G\left( d-1\right) !}x_{m}^{d-1
\end{equation}
an
\begin{equation}
S_{m}^{ren}=\frac{\left( -1\right) ^{\frac{d-1}{2}}\left( 4\pi\right)
^{\frac{d-1}{2}}\left( \frac{d-1}{2}\right) !{\ell}^{d-1}}{4G\left(
d-1\right) !}\frac{m}{2\left( m-1\right) }\left( 2-x_{m}^{d-2}-x_{m
^{d}\right) .
\end{equation}
We now check that in the $m\rightarrow1$ limit, we recover the standard result
for the universal part of the entanglement entropy obtained by Kawano,
Nakaguchi and Nishioka in Ref.\cite{Kawano:2014moa}, which is equal to
$S_{EE}^{ren}$ as discussed in Ref.\cite{Anastasiou:2018rla}. Considering
Eq.(\ref{x_m}), we evaluat
\begin{equation}
\lim\limits_{m\rightarrow1}x_{m}=1
\end{equation}
an
\begin{equation}
\lim\limits_{m\rightarrow1}\frac{m}{2\left( m-1\right) }\left(
2-x_{m}^{d-2}-x_{m}^{d}\right) =1,
\end{equation}
and therefore we obtain tha
\begin{equation}
\lim\limits_{m\rightarrow1}\widetilde{S}_{m}^{ren}=\lim\limits_{m\rightarrow
1}S_{m}^{ren}=\frac{\left( -1\right) ^{\frac{d-1}{2}}\left( 4\pi\right)
^{\frac{d-1}{2}}\left( \frac{d-1}{2}\right) !{\ell}^{d-1}}{4G\left(
d-1\right) !},
\end{equation}
which precisely corresponds to $S_{EE}^{ren}$ as it should.
We comment more on the utility of computing the renormalized ERE, and on the
physical applications of our renormalized volume formula, in the next section.
\section{Outlook\label{Section VI}}
We conjectured the proportionality between renormalized Einstein-AdS action
and renormalized bulk volume for even-dimensional AAdS Einstein manifolds (see
Section \ref{Section II}). We also compared our proposal for the renormalized
volume with existing expressions given in the conformal geometry literature
for the four-dimensional \cite{2000math.....11051A} and six-dimensional
\cite{Yang2008} cases. Our resulting formula for the renormalized volume,
which applies for $2n-$dimensional AAdS Einstein manifolds, is given in
Eq.(\ref{VolRenM2n}) and has the form of a polynomial in full contractions of
powers of the Weyl tensor plus a topological constant that depends on the
Euler characteristic of the manifold. Therefore, our formula corresponds to a
concrete realization of Albin's prescription given in Ref.\cite{ALBIN2009140}.
We also reinterpreted modular entropies (which are used in the computation of
EREs) in terms of the renormalized areas of codimension-2 cosmic branes
$\Sigma_{T}$ with tension (see Section \ref{Section IV}), whose border
$\partial\Sigma_{T}$ is situated at the AdS boundary (being conformally
cobordant with the entangling region in the CFT), and whose precise location
is determined by requiring the total configuration to be a minimum of the
$I_{tot}^{ren}$ action as given in Eq.(\ref{ITotalRenorm}), which in turn is
the renormalized version of Dong's total action prescription
\cite{Dong:2016fnf} including the NG action of the brane. The obtained formula
for the renormalized volume of $\Sigma_{T}$ given in Eq.(\ref{VolRenSigma}),
has the same form as the renormalized volume for the bulk manifold, but in
codimension-2, and therefore, the same factors in the polynomial expansion are
recovered by replacing $n$ for $\left( n-1\right) $. However, there is one
important difference: in the case of $Vol_{ren}\left[ \Sigma_{T}\right] $,
the polynomial is evaluated on $\mathcal{F}_{AdS}$ (as given in Eq.(\ref{FAdS
)) instead of on the Weyl tensor. Of course, in the case of Einstein
manifolds, both tensors are the same, but because the minimal codimension-2
surfaces $\Sigma_{T}$ (minimal in the sense of minimizing $I_{tot}^{ren}$)
need not be an Einstein manifold, the renormalized volume formula is seen to
be more general. Thus, we propose that the renormalized volume formula of
Eq.(\ref{VolRenM2n}) is valid for both AAdS Einstein manifolds and
codimension-2 minimal submanifolds embedded therein, such that $D=2n$ is the
dimension of the corresponding manifold or submanifold, where in general, the
polynomial is evaluated on $\mathcal{F}_{AdS}$. Our expression for the
renormalized volume can be understood as a measure of the deviation of a
manifold from the maximally symmetric constant curvature case, for which it is
only given by a constant proportional to the Euler characteristic of the
manifold. The obtained formula for the renormalized area of $\Sigma_{T}$
agrees with the formula given by Alexakis and Mazzeo in
Ref.\cite{Alexakis2010}, for the case when the brane is a 2-dimensional
extremal surface embedded in AAdS$_{4}$.
We also explicitated the relation between the geometrical interpretation of
the Einstein-AdS action evaluated on the conically singular replica orbifold,
as a one-parameter family of deformations to the renormalized bulk volume, and
Dong's codimension-2 cosmic brane construction of the total action
\cite{Dong:2016fnf} which includes the NG action of the brane, but in
renormalized form. We showed (in Section \ref{Section III}) that both
approaches are equivalent, and that the contribution to the bulk action at the
codimension-2 locus of the conical singularity is precisely the renormalized
NG action of the cosmic brane with a tension given by $T=\frac{\left(
m-1\right) }{4mG}$, where $m$ is the replica index.
For the case of ball-shaped entangling regions (see Section \ref{Section V}),
we computed the renormalized ERE in $\left( 2n-1\right) -$dimensional
holographic CFTs, following the computation performed by Hung, Myers, Smolkin
and Yale \cite{Hung:2011nu} using the CHM map \cite{Casini:2011kv}, and
renormalizing the horizon area of the corresponding topological BH. We have
also explicitly checked that in the tensionless limit, the known results for
the renormalized EE \cite{Anastasiou:2018rla} are correctly reproduced. This
case is of interest because, as discussed in Ref.\cite{Anastasiou:2018rla}, in
the tensionless limit, the renormalized EE is directly related to the $a_{d
-$charge \cite{Myers:2010xs}, which is a $C-$function candidate quantity that
decreases along renormalization group (RG) flows between conformal fixed
points and it counts the number of degrees of freedom of the CFT, providing a
generalization of the $c-$theorem \cite{Zamolodchikov:1986gt}.
The equivalence between the Kounterterms-renormalized Einstein-AdS action and
the renormalized volume is of interest as it recasts the problem of action
renormalization in gravity into the framework of volume renormalization in
conformal geometry. It therefore constitutes a mathematical validation of the
Kounterterms scheme, and it also emphasizes the topological nature of the
renormalized action, which has been systematically overlooked by the standard
Holographic Renormalization framework. For the recent discussions about
holographic complexity
\cite{Alishahiha:2015rta,Stanford:2014jda,Brown:2015bva,Abt:2017pmf,Banerjee:2017qti
, this result is interesting as it suggests a relation between the
\textit{complexity equals action} (CA) \cite{Brown:2015bva} and the
\textit{complexity equals volume} (CV) \cite{Stanford:2014jda} proposals. We
note, however, that our result does not imply that both proposals are directly
equivalent, as the volume considered in the CV proposal is an extremal
codimension-1 volume at constant boundary time, while the renormalized action
considered in the CA proposal is integrated over a region of the full
spacetime manifold. In trying to relate the two proposals, there may also be
more subtleties regarding the different domains of integration considered in
both (e.g., the Wheeler-de Witt patch in the CA case and the extremal spatial
slice crossing the Einstein-Rosen bridge in the CV case, when computing the
complexity of the thermofield-double state), and the differences between
Lorentzian and Euclidean gravity, which nonetheless are beyond the scope of
this paper.
The new computational scheme for obtaining renormalized EREs from the
renormalized volumes of codimension-2 minimal cosmic branes is interesting,
because as discussed by Headrick in Ref.\cite{Headrick:2010zt}, the EREs
encode the information of the full eigenvalue spectrum of the reduced density
matrix for the entangling region in the CFT, which has potential applications
for state reconstruction or, proceeding in reverse, for bulk geometry
reconstruction starting from the CFT. Furthermore, as discussed by Hung,
Myers, Smolkin and Yale in Ref.\cite{Hung:2011nu}, the EREs are, in general,
non-linear functions of the central charges and other CFT parameters, and
therefore, they are useful for characterizing CFTs and their behavior, for
example under RG flows, providing extra tools for a more detailed analysis
than what is possible from the renormalized EE only.
As future work, we will examine the significance of the equivalence between
renormalized action and renormalized volume for the study of holographic
complexity and its corresponding renormalization, revisiting the hinted
equivalence between the CA and CV proposals for the case of Einstein-AdS
gravity. We will also analyze the issue of volume renormalization for
odd-dimensional AAdS Einstein manifolds, attempting to relate it to the
Kounterterms-renormalized Einstein-AdS$_{2n+1}$ action presented in
Ref.\cite{Olea:2006vd}. Finally, we will investigate possible additions to the
holographic dictionary by considering a bulk configuration with a series of
embedded branes of different codimension, such that the full configuration is
required to be the minimum of an extended total action (in the spirit of
$I_{tot}^{ren}$ as given in Eq.(\ref{ITotalRenorm})), including the
corresponding codimension-$k$ renormalized NG actions for the new objects.
This renormalized action, constructed as a sum over different renormalized
volumes of objects with different codimension, should correspond to a
generalized notion of complexity \cite{Carmi:2017ezk} which seems worthy of
further enquiry.
\begin{acknowledgments}
The authors thank D.E. D\'{i}az, P. Sundell and A. Waldron for interesting discussions. C.A. is a Universidad Andres Bello (UNAB) Ph.D. Scholarship holder, and
his work is supported by Direcci\'{o}n General de Investigaci\'{o}n
(DGI-UNAB). This work is funded in part by FONDECYT Grants No. 1170765 ``\textit{Boundary dynamics in anti-de Sitter gravity and gauge/gravity duality}'' and No.
3180620 ``\textit{Entanglement Entropy and AdS gravity}'', and CONICYT Grant DPI 20140115.
\end{acknowledgments} |
2,869,038,156,403 | arxiv | \section{Introduction}\label{intro}
The hybrid Monte Carlo (HMC) method is a widely used molecular simulation technique for computing the equilibrium properties of condensed matter systems \cite{Duane87, Mehlig92,Allen13}. The basic HMC algorithm for sampling from the canonical (NVT) ensemble consists of three steps \cite{Allen13}:
\begin{enumerate}[label=\textbf{Step \arabic*.}, leftmargin=4\parindent]
\item Draw a complete set of initial momenta $\textbf{p} \equiv \left\lbrace\textbf{p}_i\right\rbrace_{i=1}^N$ from the distribution $P(\textbf{p})$. \label{step1}
\item Propagate a microcanonical molecular dynamics (MD) trajectory using a time-reversible and volume-preserving integrator (e.g., the velocity-Verlet algorithm) to take the system from state $\textbf{x}$ to $\textbf{x}'$, where $\textbf{x} \equiv \left\lbrace\textbf{x}_i\right\rbrace_{i=1}^N$ is a complete set of particle coordinates for the system. The length of the trajectory $\Delta t = n_{\text{steps}} \times \delta t$ is specified by the number of integration steps $n_{\text{steps}}$ and time step $\delta t$. \label{step2}
\item Accept or reject the new configuration $\textbf{x}'$ according to the Metropolis-Hastings criterion, \label{step3}
\begin{equation} \label{metropeq}
P_{\text{acc}}(\left\lbrace \textbf{x},\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x}',\textbf{p}' \right\rbrace) = \text{min} \left(1, \frac{e^{-\beta U(\textbf{x}')} P(\textbf{p}') }{e^{-\beta U(\textbf{x})}P(\textbf{p})}\right),
\end{equation}
where $U(\textbf{x})$ is the potential energy of the system.
\end{enumerate}
\par The most common choice for $P(\textbf{p})$ is the Maxwell-Boltzmann (MB) distribution \cite{Duane87, Mehlig92,Allen13},
\begin{equation} \label{pmb}
P_{\text{MB}}^{\text{std}}(\textbf{p}) \propto Ce^{-\beta K(\textbf{p})},
\end{equation}
where $\beta = (k_{B}T)^{-1}$ and $C$ is a temperature specific normalization constant. The kinetic energy $K(\textbf{p})$ is defined using the standard expression from classical mechanics. For rigid bodies, for example, $\textbf{p} \equiv \left\lbrace\textbf{p}_i^\text{com}, \bm{\Omega}_i\right\rbrace_{i=1}^N$, where $\textbf{p}_i^{\text{com}} \equiv m_i \textbf{v}_i^{\text{com}}$, $\textbf{v}_i^{\text{com}}$ is the linear center of mass (`com') velocity for molecule $i$, and $\bm{\omega}_i$ and $\bm{\Omega}_i$ denote molecule $i$'s angular velocity and angular momentum, respectively. The $j^\text{th}$ component of molecule $i$'s angular momentum ($j = 1,2,3$) is $\Omega_j = {I_{ j,k} \omega_{i,k}}$ $(k=1,2,3)$, where $I_{ j,k}$ denotes the generic molecule's ($i$ in this case) inertia tensor. The kinetic energy for rigid bodies is thus given by $K(\textbf{p}) \equiv \frac{1}{2} \sum_{i=1}^{N} \left[ \textbf{p}_i^{\text{com}} \cdot \textbf{p}_i^{\text{com}}/m_i + \bm{\omega}_i \cdot \bm{\Omega}_i\right]$.
Insertion of $P_{\text{MB}}(\textbf{p})$ into Eq.\ \ref{metropeq} yields:
\begin{equation}
P_{\text{acc}}^{\text{std}}(\left\lbrace \textbf{x},\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x}',\textbf{p}' \right\rbrace) = \text{min}\left(1,e^{-\beta \left[ H(\textbf{x}',\textbf{p}') - H(\textbf{x},\textbf{p}) \right]} \right), \label{pacc}
\end{equation}
where $H(\textbf{x}',\textbf{p}') - H(\textbf{x},\textbf{p})\equiv U(\textbf{x}') -U(\textbf{x}) + K(\textbf{p}') - K(\textbf{p})$ is the difference between the classical Hamiltonians $H(\textbf{x},\textbf{p})$ and $H(\textbf{x}',\textbf{p}')$.
\par The superscript `std' in Eqs. \ref{pmb} and \ref{pacc} denotes that these are the standard choices for $P(\textbf{p})$ and $P_{\text{acc}}$, which are used in the overwhelming majority of HMC simulation studies of condensed matter systems. When used together, they ensure that the HMC algorithm will satisfy detailed balance and thus asymptotically sample configurations from the Boltzmann distribution $P_{\text{eq}}(\textbf{x}) \propto e^{-\beta U(\textbf{x})}$. Sampling can also be performed using other initial momentum distributions that are even functions of $\textbf{p}$ (i.e., $P(\textbf{p}) = P(-\textbf{p})$) \cite{Mehlig92,Allen13}. As we demonstrate below, however, it is generally necessary to modify the acceptance criterion as prescribed by Eq.\ \ref{metropeq} to preserve detailed balance. Hence if $P(\textbf{p}) \ne P_{\text{MB}}(\textbf{p})$, then using the standard acceptance criterion in Eq. \ref{pacc} will generally result in detailed balance violations and concomitant sampling errors.
\section{Detailed Balance}\label{bg}
\par The detailed balance condition is given by:
\begin{equation}
P_{\text{eq}}(\textbf{x}) \Gamma(\textbf{x} \rightarrow \textbf{x}') = P_{\text{eq}}(\textbf{x}') \Gamma(\textbf{x}' \rightarrow \textbf{x}),\label{bal}
\end{equation}
where $P_{\text{eq}}(\textbf{x})$ is the equilibrium distribution and $\Gamma(\textbf{x} \rightarrow \textbf{x}')$ is the Markov transition probability from state $\textbf{x}$ to $\textbf{x}'$. This condition is sufficient to ensure that $P_{\text{eq}}(\textbf{x})$ will be a stationary state for the Markov processes generated by an MC sampling algorithm. Here, we examine the constraints Eq.\ \ref{bal} imposes on the choice of $P(\textbf{p})$ in the HMC algorithm under the following conditions:
\begin{enumerate}[label=\textbf{(\roman*)}]
\item The MD trajectories are propagated using a time-reversible and volume-preserving integration scheme. (The latter properties ensures that $d\textbf{x}d\textbf{p} = d\textbf{x}'d\textbf{p}'$; non-volume-preserving integrators may be used in HMC, but a Jacobian factor must be incorporated into the acceptance criterion (Eq.\ \ref{metropeq}) to correct for the resulting compression of phase space \cite{Matubayasi99}). \label{c1}
\item $P_{\text{eq}}(\textbf{x})$ is given by the Boltzmann distribution ($P_{\text{eq}}(\textbf{x}) \propto e^{-\beta U(\textbf{x})}$) \label{c2}
\item The trial trajectories are accepted using the standard criterion $P_{\text{acc}}^{\text{std}}$ (Eq.\ \ref{pacc}). \label{c3}
\item The Hamiltonian $H(\textbf{x},\textbf{p})$ is an even function of $\textbf{p}$ (i.e., $H(\textbf{x},-\textbf{p}) = H(\textbf{x},\textbf{p})$) and defined using the standard expression for kinetic energy from classical mechanics. \label{c4}
\item The function $P(\textbf{p})$ is unspecified, but has the following properties: \label{c5}
\begin{enumerate}[label=\textbf{(\alph*)}]
\item \subitem $P(\textbf{p}) = P(-\textbf{p})$
\item \subitem $P(\textbf{p})$ is stationary (i.e., the definition of $P(\textbf{p})$ does not change with time). The initial $\textbf{p}$ and final $\textbf{p}'$ momenta are thus both treated as variates from this distribution. Note that this does not imply $P(\textbf{p}) = P(\textbf{p}')$, which would suggest ``uniformity''.
\end{enumerate}
\end{enumerate}
For HMC, the forward transition probability from $\textbf{x}$ to $\textbf{x}'$ is given by \cite{Matubayasi99}:
\begin{eqnarray}
\Gamma(\textbf{x} \rightarrow \textbf{x}') = \iint d\textbf{p} d\textbf{p}' P(\textbf{p})g(\left\lbrace \textbf{x},\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x}',\textbf{p}' \right\rbrace) P_{\text{acc}}^{\text{std}}(\left\lbrace \textbf{x},\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x}',\textbf{p}' \right\rbrace) \label{forward0} \\
=\iint d\textbf{p} d\textbf{p}' P(\textbf{p})g(\left\lbrace \textbf{x},\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x}',\textbf{p}' \right\rbrace) \text{min}\left(1,e^{-\beta \left[ H(\textbf{x}',\textbf{p}') - H(\textbf{x},\textbf{p}) \right]} \right), \label{forward1}
\end{eqnarray}
where $g(\left\lbrace \textbf{x},\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x}',\textbf{p}' \right\rbrace) = \delta(\textbf{x}' - \textbf{x}(\Delta t))\delta(\textbf{p}' - \textbf{p}(\Delta t))\delta(\textbf{x} - \textbf{x}(0))\delta(\textbf{p} - \textbf{p}(0))$ and $\delta$ is the Kronecker delta function. The function $g(\left\lbrace \textbf{x},\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x}',\textbf{p}' \right\rbrace)$ gives the probability that an MD trajectory originating from $\lbrace \textbf{x}(0),\textbf{p}(0) \rbrace = \lbrace \textbf{x},\textbf{p} \rbrace$ will end up in state $\lbrace \textbf{x}(\Delta t),\textbf{p}(\Delta t) \rbrace = \lbrace \textbf{x}',\textbf{p}' \rbrace$ after an elapsed time $\Delta t$.
\par Similarly, the reverse transition probability $\Gamma(\textbf{x}' \rightarrow \textbf{x})$ can be written as \cite{Matubayasi99}:
\begin{eqnarray}
\Gamma(\textbf{x}' \rightarrow \textbf{x}) = \iint d\textbf{p} d\textbf{p}' P(\textbf{p})g(\left\lbrace \textbf{x}',\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x},\textbf{p}' \right\rbrace) P_{\text{acc}}^{\text{std}}(\left\lbrace \textbf{x}',\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x},\textbf{p}' \right\rbrace) \label{reverse0} \\
=\iint d\textbf{p} d\textbf{p}' P(\textbf{p}) g(\left\lbrace \textbf{x}',\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x},\textbf{p}' \right\rbrace) \text{min}\left(1,e^{-\beta \left[ H(\textbf{x},\textbf{p}') - H(\textbf{x}',\textbf{p}) \right]} \right). \label{reverse1}
\end{eqnarray}
In Eqs.\ \ref{reverse0} and \ref{reverse1} we follow the notation convention used in Ref. \cite{Matubayasi99}, whereby $\textbf{p}$ denotes the generic set of momenta over which integration is performed, and hence we can use $\textbf{p}$ in conjunction with $\textbf{x}'$. The notation is arbitrary, provided that one defines and uses $\textbf{p}$ and $\textbf{p}'$ consistently and does not confuse these variables in manipulating the equations or making variable substitutions. In particular, because $P$ refers to a generic momentum distribution associated with the initial state of the transition under consideration, one can write, for a $\textbf{x}' \rightarrow \textbf{x}$ transition (Eqs.\ \ref{reverse0} or \ref{reverse1}) $P(\textbf{p})g(\left\lbrace \textbf{x}',\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x},\textbf{p}' \right\rbrace) P_{\text{acc}}^{\text{std}}(\left\lbrace \textbf{x}',\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x},\textbf{p}' \right\rbrace)$ or $P(\textbf{p}')g(\left\lbrace \textbf{x}',\textbf{p}' \right\rbrace \rightarrow \left\lbrace \textbf{x},\textbf{p} \right\rbrace) P_{\text{acc}}^{\text{std}}(\left\lbrace \textbf{x}',\textbf{p}' \right\rbrace \rightarrow \left\lbrace \textbf{x},\textbf{p} \right\rbrace)$. Changing (redefining) integration variables \cite{Tuckerman10} $\textbf{p} \rightarrow -\textbf{p}'$ and $\textbf{p}' \rightarrow -\textbf{p}$ and noting that $d\textbf{p} d\textbf{p}' = d\left[ -\textbf{p}' \right]d\left[-\textbf{p}\right]$ yields:
\begin{eqnarray}
\Gamma(\textbf{x}' \rightarrow \textbf{x}) = \iint d\textbf{p} d\textbf{p}' P(-\textbf{p}') g(\left\lbrace \textbf{x}',-\textbf{p}' \right\rbrace \rightarrow \left\lbrace \textbf{x},-\textbf{p} \right\rbrace) \text{min}\left(1,e^{-\beta \left[ H(\textbf{x},-\textbf{p}) - H(\textbf{x}',-\textbf{p}') \right]} \right). \label{reverse2}
\end{eqnarray}
For a time-reversible and volume-preserving integrator $ g(\left\lbrace \textbf{x}',-\textbf{p}' \right\rbrace \rightarrow \left\lbrace \textbf{x},-\textbf{p} \right\rbrace ) = g(\left\lbrace \textbf{x},\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x}',\textbf{p}' \right\rbrace)$.
Additionally, assuming $H(\textbf{x},-\textbf{p}) = H(\textbf{x},\textbf{p})$ and $P(-\textbf{p}) = P(\textbf{p})$, we obtain:
\begin{eqnarray}
\Gamma(\textbf{x}' \rightarrow \textbf{x}) = \iint d\textbf{p} d\textbf{p}' P(\textbf{p}') g(\left\lbrace \textbf{x},\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x}',\textbf{p}' \right\rbrace) \text{min}\left(1,e^{-\beta \left[ H(\textbf{x},\textbf{p}) - H(\textbf{x}',\textbf{p}') \right]} \right). \label{reverse3}
\end{eqnarray}
Invoking the identity $\text{min}\left(1,e^{-\beta \left[ H(\textbf{x},\textbf{p}) - H(\textbf{x}',\textbf{p}') \right]} \right) = \frac{e^{-\beta H(\textbf{p},\textbf{x})}}{e^{-\beta H(\textbf{p}',\textbf{x}')}}\text{min}\left(1,e^{-\beta \left[ H(\textbf{x}',\textbf{p}') - H(\textbf{x},\textbf{p}) \right]} \right)$:
\begin{eqnarray}
\Gamma(\textbf{x}' \rightarrow \textbf{x}) = \iint d\textbf{p} d\textbf{p}' P(\textbf{p}') g(\left\lbrace \textbf{x},\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x}',\textbf{p}' \right\rbrace) \frac{e^{-\beta H(\textbf{p},\textbf{x})}}{e^{-\beta H(\textbf{p}',\textbf{x}')}}\text{min}\left(1,e^{-\beta \left[ H(\textbf{x}',\textbf{p}') - H(\textbf{x},\textbf{p}) \right]} \right) \label{reverse4}\\
=\iint d\textbf{p} d\textbf{p}' P(\textbf{p}) g(\left\lbrace \textbf{x},\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x}',\textbf{p}' \right\rbrace) \frac{P(\textbf{p}')}{P(\textbf{p})} \frac{e^{-\beta U(\textbf{x})}e^{-\beta K(\textbf{p})}}{e^{-\beta U(\textbf{x}')}e^{-\beta K(\textbf{p}')}}\text{min}\left(1,e^{-\beta \left[ H(\textbf{x}',\textbf{p}') - H(\textbf{x},\textbf{p}) \right]} \right) \label{reverse5}\\
= \frac{e^{-\beta U(\textbf{x})}}{e^{-\beta U(\textbf{x}')}} \iint d\textbf{p} d\textbf{p}' P(\textbf{p}) g(\left\lbrace \textbf{x},\textbf{p} \right\rbrace \rightarrow \left\lbrace \textbf{x}',\textbf{p}' \right\rbrace) \boxed{\frac{P(\textbf{p}')e^{-\beta K(\textbf{p})}}{P(\textbf{p})e^{-\beta K(\textbf{p}')}}}\text{min}\left(1,e^{-\beta \left[ H(\textbf{x}',\textbf{p}') - H(\textbf{x},\textbf{p}) \right]} \right).\label{reverse6}
\end{eqnarray}
Comparison with Eq. \ref{forward1} reveals that the integrands differ by the boxed term in Eq. \ref{reverse6}. To satisfy detailed balance (Eq.\ \ref{bal}), the choice of $P(\textbf{p})$ must therefore either ensure that the boxed term in Eq. \ref{reverse6} is always unity or has no effect on the value of the integral when the integration over $\textbf{p}$ and $\textbf{p}'$ is carried out. It is clear from our derivation that neither of these criteria will be automatically satisfied by an arbitrary distribution $P(\textbf{p})$, even if it has the properties listed in point \ref{c5} above. The former criterion is satisfied, however, by the Maxwell-Boltzmann distribution $P(\textbf{p}) = P^{\text{std}}_{\text{MB}}(\textbf{p})$ (Eq.\ \ref{pmb}). Inserting the MB distribution into Eqs.\ \ref{forward1} and \ref{reverse6}, we find that the boxed term in Eq.\ \ref{reverse6} is identically equal to 1. Subsequent inspection of Eqs.\ \ref{forward1} and \ref{reverse6} reveals that $\Gamma(\textbf{x} \rightarrow \textbf{x}') = \frac{e^{-\beta U(\textbf{x}')}}{e^{-\beta U(\textbf{x})}} \Gamma(\textbf{x}' \rightarrow \textbf{x}) $, which is the detailed balance condition (Eq.\ \ref{bal}). By contrast, identifying a distribution that unambiguously satisfies only the latter criterion is extremely challenging, as the integrals in Eq. \ref{forward1} and \ref{reverse6} cannot usually be evaluated analytically. Other choices for $P(\textbf{p})$ are certainly possible, but HMC acceptance criterion in Eqs.\ \ref{forward1} and \ref{reverse6} must be redefined as prescribed by Eq.\ \ref{metropeq} to avoid violating detailed balance.
\section{Detailed Balance for the One-dimensional Harmonic Oscillator}\label{MBproof}
\par The analysis above demonstrates that choosing $P(\textbf{p})=P^{\text{std}}_{\text{MB}}(\textbf{p})$ is generally required to satisfy detailed balance when HMC sampling is performed using the standard acceptance criterion $P_{\text{acc}}^{\text{std}}$ (Eq.\ \ref{pacc}). \colorred{Here, we show analytically for a simple model system and velocity Verlet integrator that detailed balance can \emph{only} be satisfied by $P(\textbf{p})=P^{\text{std}}_{\text{MB}}(\textbf{p})$ when $P_{\text{acc}}=P_{\text{acc}}^{\text{std}}$. For this case, the condition that $P(\textbf{p})=P^{\text{std}}_{\text{MB}}(\textbf{p})$ is thus not only sufficient but also necessary.} Consider the simple case of applying the HMC algorithm to sample the configurations of a particle of mass $m$ in a one-dimensional harmonic well described by the potential energy function $U(x) = \frac{1}{2} k x^2$. Further, suppose that we define the kinetic energy using the standard expression $K(p) \equiv \frac{p^2}{2m}$ and consider HMC moves that consist of only a \emph{single} molecular dynamics integration step propagated using the time reversible and volume-preserving velocity Verlet scheme. Under these conditions, $\left\lbrace x', p'\right\rbrace$ is related to $\left\lbrace x, p\right\rbrace$ via:
\begin{eqnarray}
x' = x + \frac{p}{m}\delta t + \frac{f(x)}{2m}\delta t^2 \\
p' = p + \frac{1}{2} \left[ f(x') + f(x) \right] \delta t,
\end{eqnarray}
where $\delta t$ is the integration time step and $f(x) = -\frac{dU(x)}{dx} = -kx $ is the force acting on the particle. The detailed balance condition (Eq.\ \ref{bal}) must be satisfied for all $x$ and $x'$. Thus we are free to choose $x$ and $x'$ arbitrarily to simplify our analysis. Letting $x=0$ and $x' \ne 0$, and noting that $f(0)=0$ and $f(x') = -kx'$, we obtain:
\begin{eqnarray}
x' = \frac{p}{m}\delta t \\
p' = p - \frac{1}{2} kx' \delta t = p(1-\frac{1}{2m}k\delta t^2).
\end{eqnarray}
These equations show that for a given choice of the sampling parameter $\delta t$, only $p = \frac{x'm}{\delta t}$ will lead to an attempted HMC move from $\left\lbrace x,p \right\rbrace \rightarrow \left\lbrace x',p' \right\rbrace$. Similarly, they also reveal that $p' = \frac{x'm}{\delta t} - \frac{1}{2} kx' \delta t$ is unique for a given $\delta t$.
Therefore $g$ will be given by:
\begin{eqnarray} \label{gho}
g(\left\lbrace x, p\right\rbrace \rightarrow \left\lbrace x', p'\right\rbrace)=\delta\left(p-\frac{x'm}{\delta t}\right)
\delta\left(p'-\frac{x'm}{\delta t}+\frac12kx'\delta t\right).
\end{eqnarray}
Additionally, we find that:
\begin{eqnarray}
H(x,p) = \frac{x'^2m}{2\delta t^2} \label{hho0} \\
H(x',p') = \frac{x'^2m}{2\delta t^2} + \frac{1}{8m}k^2x'^2 \delta t^2 \label{hho1}
\end{eqnarray}
Using these expressions, we can now analyze the conditions under which the detailed balance (Eq.\ \ref{bal}) will be satisfied. For the selected $x$ and $x'$, the detailed balance condition (Eq.\ \ref{bal}) reduces to:
\begin{eqnarray}
\Gamma(x \rightarrow x') = \frac{e^{-\beta U(x')}}{e^{-\beta U(x)}} \Gamma(x' \rightarrow x) \\
= {e^{-\frac{\beta}{2}kx'^2 }} \Gamma(x' \rightarrow x) \label{balho},
\end{eqnarray}
where we have substituted $U(x) = 0 $ and $U(x') = \frac{1}{2}kx'^2$. Inserting Eqs.\ \ref{gho}, \ref{hho0}, and \ref{hho1} into the expression for the forward transition probability given by Eq.\ \ref{forward1}, we obtain:
\begin{eqnarray}
\Gamma(x \rightarrow x') =\iint dp dp' P(p)g(\left\lbrace x, p \right\rbrace \rightarrow \left\lbrace x',p' \right\rbrace) \text{min}\left(1,e^{-\beta \left[ H(x',p') - H(x,p) \right]} \right) \\
=\iint dp dp' P(p)\delta\left(p-\frac{x'm}{\delta t}\right) \delta\left(p'-\frac{x'm}{\delta t}+\frac12kx'\delta t\right) \text{min}\left(1,e^{-\beta \left[ \frac{x'^2m}{2\delta t^2} + \frac{1}{8m}k^2x'^2 \delta t^2 - \frac{x'^2m}{2\delta t^2} \right]} \right) \\
= P\left(\frac{x'm}{\delta t}\right) \text{min}\left(1,e^{ -\frac{\beta}{8m}k^2x'^2 \delta t^2 } \right) \\
= P\left(\frac{x'm}{\delta t}\right) e^{ -\frac{\beta}{8m}k^2x'^2 \delta t^2 }, \label{forwardho}
\end{eqnarray}
where the last equality holds because $\frac{\beta}{8m}k^2x'^2 \delta t^2 \ge 0$. The reverse transition probability can also be evaluated in a similar fashion from Eq.\ \ref{reverse4}:
\begin{eqnarray}
\Gamma(x' \rightarrow x) = \iint dp dp' P(p') g(\left\lbrace x,p \right\rbrace \rightarrow \left\lbrace x',p' \right\rbrace) \frac{e^{-\beta H(p,x)}}{e^{-\beta H(p',x')}}\text{min}\left(1,e^{-\beta \left[ H(x',p') - H(x,p) \right]} \right) \\
= \iint dp dp' P(p') \delta\left(p-\frac{x'm}{\delta t}\right) \delta\left(p'-\frac{x'm}{\delta t}+\frac12kx'\delta t\right) \frac{e^{-\beta H(p,x)}}{e^{-\beta H(p',x')}}\text{min}\left(1,e^{-\beta \left[ H(x',p') - H(x,p) \right]} \right) \\
= P\left(\frac{x'm}{\delta t}-\frac12kx'\delta t \right) e^{\frac{\beta}{8m}k^2x'^2 \delta t^2 }\text{min}\left(1,e^{-\frac{\beta}{8m}k^2x'^2 \delta t^2} \right)\\
= P\left(\frac{x'm}{\delta t}-\frac12kx'\delta t \right) e^{\frac{\beta}{8m}k^2x'^2 \delta t^2 }e^{-\frac{\beta}{8m}k^2x'^2 \delta t^2} \\
= P\left(\frac{x'm}{\delta t}-\frac12kx'\delta t \right) \label{reverseho}
\end{eqnarray}
Substituting Eqs.\ \ref{forwardho} and \ref{reverseho} into Eq.\ \ref{balho}, we find that for detailed balance to hold, we need to have:
\begin{eqnarray}
P\left(\frac{x'm}{\delta t}\right)e^{-\frac{\beta}{8m}k^2x'^2\delta t^2}&=&
e^{-\frac{\beta}2kx'^2}P\left(\frac{x'm}{\delta t}-\frac12kx'\delta t\right)
\end{eqnarray}
which is clearly not satisfied for an arbitrary $P(\cdot)$. Letting $\phi(p):=\ln P(p)+\beta p^2/2m$, we obtain:
\begin{eqnarray}
\ln P\left(\frac{x'm}{\delta t}\right) - \frac{\beta}{8m}k^2x'^2\delta t^2 &=&\ln P\left(\frac{x'm}{\delta t}-\frac12kx'\delta t\right) - \frac{\beta}2kx'^2\notag\\
\phi\left(\frac{x'm}{\delta t}\right)-\frac{\beta m x'^2}{2\delta t^2} - \frac{\beta}{8m}k^2x'^2\delta t^2&=& \phi\left(\frac{x'm}{\delta t}-\frac12kx'\delta t\right) -\frac{\beta}{2m}\left(\frac{x'm}{\delta t}-\frac12kx'\delta t\right)^2- \frac{\beta}2kx'^2 \notag\\
\phi\left(\frac{x'm}{\delta t}\right)-\cancel{\frac{\beta m x'^2}{2\delta t^2}} - \cancel{\frac{\beta}{8m}k^2x'^2\delta t^2} &=& \phi\left(\frac{x'm}{\delta t}-\frac12kx'\delta t\right) -\cancel{\frac{\beta m x'^2}{2\delta t^2}} + \cancel{\frac{\beta}2kx'^2}-\cancel{\frac{\beta}{8m}k^2x'^2\delta t^2} - \cancel{\frac{\beta}2kx'^2}\notag\\
\phi\left(\frac{x'm}{\delta t}\right) &=& \phi\left(\frac{x'm}{\delta t}-\frac12kx'\delta t\right)\label{eqphi}
\end{eqnarray}
which can be mathematically stated as $\phi(bx')=\phi(ax')$ with $b=m/\delta t, a=m/\delta t-k\delta t/2$. Because $\phi(bx')=\phi(ax')$ for all $x'$,
\begin{eqnarray}
\phi(b^2x') = \phi\left[ b(bx') \right] = \phi\left[ a(bx') \right] = \phi\left[ b(ax') \right] =\phi\left[ a(ax') \right] = \phi( a^2x'),
\end{eqnarray}
or more generally,
\begin{eqnarray}
\phi(b^nx') = \phi(a^nx')
\end{eqnarray}
for $n\in\mathbb{Z}$.
Similarly,
\begin{eqnarray}\label{phin}
\phi(x') = \phi\left[b^n \frac{x'}{b^n} \right] = \phi\left[a^n \frac{x'}{b^n} \right] =\phi\left[\left(\frac{a}{b}\right)^nx'\right].
\end{eqnarray}
Without loss of generality, suppose $a<b$. Because Eq.\ \ref{phin} is valid for any $n\in\mathbb{Z}$, it must also hold in the limit that $n\rightarrow\infty$. Taking this limit, we find that:
$$\phi(x') = \phi\left[\left(\frac{a}{b}\right)^nx'\right] = \lim_{n\rightarrow\infty} \phi\left[\left(\frac{a}{b}\right)^nx'\right] = \phi(0)=\text{const.}\notag$$
Thus, in order for Eq.\ \ref{eqphi} to hold for \emph{any} $x'$, $\phi(p)$ needs to be constant:
\begin{eqnarray}
\phi(p)=\ln P(p)+\beta p^2/2m = \text{const.},
\end{eqnarray}
or equivalently,
\begin{eqnarray}
P(p) \propto e^{-\beta \frac{p^2}{2m}} = e^{-\beta K(p)},
\end{eqnarray}
which is the Maxwell-Boltzmann distribution. Hence, this derivation proves for the 1-D harmonic oscillator model system that for detailed balance to hold when the standard HMC acceptance criterion (Eq.\ \ref{pacc}) is used in conjunction with $K(p) \equiv \frac{p^2}{2m}$, $P(p)$ can \emph{only} have the Maxwell-Boltzmann distribution.
\section{A Stationary and even momentum distribution is not sufficient for detailed balance}\label{LJ}
\par The derivations outlined in Secs.\ \ref{bg} and \ref{MBproof} assume that $P(\textbf{p})$ is even (i.e., $P(\textbf{p})=P(-\textbf{p})$) and stationary (i.e., the initial momenta $\textbf{p}$ and final momenta $\textbf{p}'$ are treated as variates from the same distribution, $P(\textbf{p})$). As demonstrated by our analysis, however, these properties are not sufficient to satisfy detailed balance. This fact can also be shown numerically by performing HMC simulations of the Lennard-Jones model for argon ($\sigma_{\text{Ar}} = 0.3405$ nm, $\epsilon_{\text{Ar}}/k_B = 119.8 $ K), in which the potential is truncated at $ 3 \times \sigma_{\text{Ar}}$ and standard long-range tail corrections are applied. We consider three HMC schemes in which the target sampling temperature is specified by $\beta_1 = (k_BT_1)^{-1}$:
\begin{itemize}
\item \textbf{Scheme I:} HMC using $P(\textbf{p}) \propto e^{-\beta_2K(\textbf{p})}$ and $P^{\text{std}}_{\text{acc}} = \text{min}\left(1, \frac{e^{-\beta_1 U(\textbf{x}')}e^{-\beta_1 K(\textbf{p}')}}{e^{-\beta_1 U(\textbf{x})}e^{-\beta_1 K(\textbf{p})}} \right)$, where $\beta_2 = \beta_1$. This scheme corresponds to the standard HMC algorithm, where the initial momenta are drawn from the Maxwell-Boltzmann distribution at the target sampling temperature $\beta_1 = \beta_2 = (k_BT_1)^{-1}$. As proved in Sec.\ \ref{bg}, this scheme satisfies detailed balance.
\item \textbf{Scheme II:} HMC using $P(\textbf{p}) \propto e^{-\beta_2K(\textbf{p})}$ and $P^{\text{mod}}_{\text{acc}} = \text{min}\left(1, \frac{e^{-\beta_1 U(\textbf{x}')}e^{-\beta_2 K(\textbf{p}')}}{e^{-\beta_1 U(\textbf{x})}e^{-\beta_2 K(\textbf{p})}} \right)$, where $\beta_2 < \beta_1$. In this scheme, the initial momenta are drawn from a Maxwell-Boltzmann distribution corresponding to an artificially high temperature $\beta_2 = (k_BT_2)^{-1}$, but the acceptance criterion has been modified such that $P^{\text{mod}} _{\text{acc}}= \text{min} \left(1, \frac{e^{-\beta_1 U(\textbf{x}')} P(\textbf{p}') }{e^{-\beta_1U(\textbf{x})}P(\textbf{p})}\right)$, as prescribed by Eq.\ \ref{metropeq}. This modified acceptance criterion is sufficient to ensure that detailed balance is satisfied for all choices of $\beta_1$ and $\beta_2$.
\item \textbf{Scheme III:} HMC using $P(\textbf{p}) \propto e^{-\beta_2K(\textbf{p})}$ and $P^{\text{std}}_{\text{acc}} = \text{min}\left(1, \frac{e^{-\beta_1 U(\textbf{x}')}e^{-\beta_1 K(\textbf{p}')}}{e^{-\beta_1 U(\textbf{x})}e^{-\beta_1 K(\textbf{p})}} \right)$, where $\beta_2 < \beta_1$. In this scheme, the initial momenta are drawn from a Maxwell-Boltzmann distribution corresponding to an artificially high temperature $\beta_2 = (k_BT_2)^{-1}$, but no modifications are made to the standard acceptance criterion. \textbf{As proved in Sec.\ \ref{MBproof}, this sampling scheme, does not satisfy detailed balance, even though $P(\textbf{p})$ is an even function and stationary (Fig.\ \ref{LJvel})}
\end{itemize}
\par We simulate $N=500$ particles at a fixed density of 1.3778 g/cm$^3$ and choose $T_1 =107.82 K$ as our target sampling temperature. In terms of Lennard-Jones units, this corresponds to $\rho^* = 0.820$ and $T^* = 0.9$. In each case, sampling is performed for $\sim 5 \times 10^5$ HMC moves, where each move consists of 10 molecular dynamics integration steps using a 30 fs time step.
\par As expected, Scheme I correctly predicts the average potential energy, as illustrated by the excellent agreement with benchmark data from the National Institute for Science and Technology for the Lennard-Jones fluid (Table \ref{LJeos}). As expected, it also satisfies the normalization condition $\langle e^{-\beta \Delta H} \rangle = 1$ (Table \ref{LJeos}). This rigorous statistical mechanical relationship must hold for all valid HMC sampling schemes. Thus it provides a convenient consistency check for detecting sampling errors associated with detailed balance violations. In the absence of sampling errors, we expect that $\left| \langle e^{-\beta \Delta H} \rangle -1 \right| \times \sigma^{-1} \lesssim 1 $, where $\sigma$ is the estimated uncertainty. Accordingly, we find that Scheme I satisfies this expectation. Similarly, Scheme II correctly predicts the average potential energy and obeys the normalization condition. Even though the initial velocities in Scheme II are drawn from the Maxwell-Boltzmann distribution at an artificially high temperature $T_2 = 117.82$ K, the modified HMC acceptance ensures that detailed balance is satisfied and that the correct result is recovered. By contrast, Scheme III does not satisfy detailed balance and hence fails to obey the normalization condition and correctly predict the equation of state (Table \ref{LJeos}). The sampling errors produced by Scheme III become more pronounced as the velocity generation temperature $T_2$ increases from 117.82 to 200.82 K.
\begin{table}[h]
\caption{\label{LJeos} Equation of state data for Lennard-Jones argon at a target sampling temperature $T_1 = 107.82$ K}
\begin{tabular}{lcc}
\hline
& $\langle U \rangle/N$ (kJ mol$^{-1}$) & $ \left| \langle e^{-\beta \Delta H} \rangle -1 \right| \times \sigma^{-1} $ \\
\hline
NIST Ref. Data\footnote{Benchmark data from the \href{https://mmlapps.nist.gov/srs/LJ_PURE/mc.htm}{National Institute of Standards and Technology \url{https://mmlapps.nist.gov/srs/LJ_PURE/mc.htm}}, which has been converted from Lennard-Jones units} & -5.7230(7)\footnote{Number in parentheses denotes uncertainty in the last significant digit} & -- \\
Scheme I ($T_1 = 107.82$ K) & -5.7231(1) & 0.68 \\
Scheme II ($T_1 = 107.82$ K; $T_2 = 117.82$ K) & -5.7231(2) & 0.47 \\
Scheme III ($T_1 = 107.82$ K; $T_2 = 117.82$ K) & -5.6444(1) & 3.5 \\
Scheme III ($T_1 = 107.82$ K; $T_2 = 150.82$ K) & -5.3998(2) & 21.4 \\
Scheme III ($T_1 = 107.82$ K; $T_2 = 200.82$ K) & -5.0623(2) & 12.5 \\
\hline
\end{tabular}
\end{table}
\begin{figure}[h]
\hspace{20px}
\includegraphics [width =0.25 \linewidth ]{a.png}
\includegraphics [width =0.25\linewidth ]{b.png}
\hspace{20px}
\includegraphics [width =0.25 \linewidth ]{c.png}
\includegraphics [ width =0.25 \linewidth ]{d.png}
\includegraphics [width =0.25 \linewidth ]{e.png}
\caption {\label{LJvel} The proposed (red) and accepted (blue) velocity distributions are statistically indistinguishable for all sampling schemes and choices of $T_2$. Thus, the velocity distribution is stationary regardless of whether detailed balance is satisfied by the algorithm. The green lines show the Maxwell-Boltzmann distribution at the indicated temperature $T_2$. Scheme III does not satisfy detailed balance and hence does not yield the correct equation of state for the Lennard-Jones fluid (Table \ref{LJeos}), even though the velocity distribution is an even function and stationary. These two criteria are therefore \emph{not sufficient} to ensure detailed balance}
\end{figure}
\par In each sampling scheme, $P(\textbf{p})$ is an even function of $\textbf{p}$ by construction. Additionally, $P(\textbf{p})$ is stationary under each sampling scheme, regardless of whether detailed balance is satisfied by the algorithm (Fig.\ \ref{LJvel}). Thus, in agreement with our proof in Sec.\ \ref{MBproof}, these results demonstrate that stationarity and evenness of $P(\textbf{p})$ are \emph{not sufficient} conditions for detailed balance. If one wishes to use the standard HMC acceptance criterion and definition of kinetic energy given in Sec.\ \ref{intro}, \colorred{initial velocities should be drawn from the Maxwell-Boltzmann distribution at the target sampling temperature to satisfy detailed balance (see Sec.\ \ref{MBproof})}. As Scheme II illustrates, detailed balance can be satisfied using other choices for $P(\textbf{p})$, if the HMC acceptance criterion is modified appropriately.
\section{HMC using partial momentum updates}\label{corrp}
\par The HMC algorithm outlined in Sec.\ \ref{intro} assumes that a completely new set of initial momenta are drawn from $P(\textbf{p})$ in \ref{step1} It also possible to perform HMC using correlated samples from $P(\textbf{p})$ that are generated by partially refreshing the momenta from the end of the previous MC step. Wagoner and Pande \cite{Wagoner12} have rigorously proved, however, that the sign of the momenta must be changed (negated) either upon acceptance or rejection of each HMC move for this scheme to satisfy balance (i.e., leave $P_{\text{eq}}(\textbf{x}) \propto e^{-\beta U(\textbf{x})}$ stationary) and thus sample from the correct equilibrium distribution. Indeed, numerical studies have shown that omitting this important step results in detectable sampling errors \cite{Akhmatskaya09}.
\vspace{0.5in}
|
2,869,038,156,404 | arxiv | \section{Introduction}
Born-Oppenheimer approximation \cite{Born-Opp} is the basic tool of molecular physics \cite{landau, bethe-jackiw,weinbergQM}, it is also fundamental in solids, for example in modelling coupling of lattice vibrations to electronic degrees of freedom (for a rather detailed coverage of various aspects of this field, one may look for example at \cite{mahan}, there are also classics on the subject such as \cite{born-huang, ziman}). In the case of molecular physics, where all particles interact via Coulomb forces, the Born-Oppenheimer vibrational energy levels go with $(m/M)^{1/2}$, where $m/M$ refers to the light mass to heavy mass ratio, rotational energy levels as well as anharmonic corrections go with $(m/M)$, therefore they are at a higher order, the relevant expansion parameter being considered as $(m/M)^{1/4}$. Our main concern here is essentially the static Born-Oppenheimer approximation in which one is interested in stationary levels of the system. There is a large literature on this subject, we will not be able to cover all of it, we only mention some works aiming at a rigorous approach to the Born-Oppenheimer approximation, which is somewhat related to our desire to get some control over the error terms in our toy model. We are not discussing the time-dependent Born-Oppenheimer approximation, this is a very interesting and closely related subject, the reader can consult the review articles \cite{Spohn, Hagedorn-Joye, Jecko-5} for more information.
An interesting toy model worked out by R. Seiler in \cite{seiler-2}, where two heavy and one light particle all interact via harmonic oscillator potentials. In this work it is verified that the assumptions of the Born-Oppenheimer approach hold. Following this, some rigorous aspects of Born-Oppeheimer approximation is presented in \cite{combes-seiler}. It is not at all clear that the eigenvalues of the light degrees of freedom, that one computes, under the influence of potentials when the heavy centers are clamped, actually define well-behaved nonintersecting surfaces when one considers the heavy degrees of freedoms as parameters. This difficult problem is solved by Hunziker in \cite{hunziker}, where even for Coulomb type potentials energy eigenfunctions are shown to be essentially analytic functions of the heavy coordinates. These problems further investigated in a series of papers by Hagedorn \cite{hagedorn4,hagedorn5,hagedorn6}. An attempt to include higher order corrections to Born-Oppenheimer approximation is given by Weingert and Littlejohn \cite{weigert} as an example of their diagonalization technique in the deformation quantization approach. They discover derivative terms in the corrections and it is consistent with what we find here as well. Higher order corrections also were rigorously investigated by Hagedorn in a series of papers \cite{hagedorn1,hagedorn2}, they are essentially angular momentum and nonlinear oscillation terms, as predicted originally by Born-Oppenheimer. Further investigations along similar lines are presented in \cite{seiler-klein}. The reader can find a large collection references and mathematically precise statements on Born-Oppenheimer approximation in a recent review by Jecko \cite{Jecko-5}
In a similar spirit to \cite{seiler-2}, a slightly simpler model for its pedagocial value was proposed by G. Gangopadhyay and B. Dutta-Roy in \cite{dutta-roy} where the authors consider a light particle coupled to a heavy particle via a delta function potential, which makes it slightly singular, and the whole system is confined to a box in one dimension for which analytical treatment is possible. Our toy model investigates Born-Openheimer approximation in a very similar singular system, albeit leading to finite results again thanks to being formulated in one dimension, but it is physically more interesting. The simplicity of the model allows us to test various aspects and higher order corrections to Born-Oppenheimer approximation. We consider two heavy particles interacting with a light particle through an attractive delta function potential. As a result of the attraction the heavy particles form a kind of molecule, but a singular one, since there is no repulsion they collapse onto one another if we consider them as classical particles on the ground state energy surface. When the two heavy particles are separated by a small distance there is an effective linear attractive potential acting on them. This leads to a linear oscillator that one can solve exactly.
In our problem, we note that the relevant expansion parameter is $(m/M)^{1/3}$, different from the usual molecular systems. The consistency of our approximations are verified in the first Appendix, by computing the order of each neglected term, using the proposed solution for the estimates. Moreover, we note that the higher order corrections can be introduced in a second quantized language, using an approach suggested by Rajeev\cite{rajeev-asymptotic}, which contain some higher order derivative terms, not so simple to identify as in the case of rotational degrees of freedom of molecules. These are some of the novel aspects of this problem. The relativistic dispersion relation for the light particle could be important to gain some insight into the heavy quark systems, in which the gluons, being massles always to be treated relativistically leading to a linear attractive potential between the heavy quarks. Here we study the one dimensional version of the model studied previsously for two dimensions in \cite{caglar-teo} to understand a proper finite formulation via a nonperturbative renormalization process. Again we find a linear effective potential as a result of the interactions with the light particle, the derivation of which is given in Appendix-II. What would be more interesting is to study the two dimensional version of the present problem where there are renormalization issues to be taken care of before one employs the Born-Oppenheimer type approximations\cite{akbas-teo}.
\section{Born-Oppenheimer for Delta Function Potentials}
In this section we apply the conventional Born-Oppenheimer approximation to a very simple model in one-dimension. Let us consider two heavy particles, each one of which is interacting with a light particle through an attractive delta function potential.
The Hamiltonian of the system can be written as,
\begin{equation}
\Big[-\frac{\hbar^2}{2M}\sum_{i}{\nabla_i}^2-\frac{\hbar^2}{2m}\nabla^2-\lambda\delta(x-x_1)-\lambda\delta(x-x_2)]\Big]\Psi(x;x_1, x_2)=E\Psi(x;x_1,x_2).
\end{equation}
Here $x_1,x_2$ refer to the heavy particles coordinates and $x$ refers to the light one. The choices of the masses also reflect this difference.
Let us assume that the Born-Oppenheimer approximation can be applied to this system, that is we introduce a decomposition of the wave function into fast and
slow degrees of freedom:
\begin{equation}
\Psi(x;x_1, x_2)=\phi(x|x_1, x_2)\psi(x_1,x_2).
\end{equation}
We assume that this decomposition respects the translational invariance of the system and we will make use of this when we estimate the error terms. We can always assume that the wave function has such a decomposition. Note that the conventional assumption would be that the wave function can be expanded into a series of complete eigenfunctions of the light-degree of freedom with coefficients depending on the heavy degrees of freedom.
After completing our calculations, we will see that the derivative of the wave function $ \phi $ with respect to the large mass coordinates $ x_i $ will give us smaller terms, which will entail us to decouple the light degree of freedom from the heavy ones. We substitute the proposed solution into the Schr\"odinger equation,
\begin{eqnarray}
&\ & \Big[-\frac{\hbar^2}{2M}\sum_{i}{\nabla_i}^2\psi(x_1,x_2)\Big]\phi(x|x_1,x_2)+ \Big[-\frac{\hbar^2}{2M}\sum_{i}{\nabla_i}^2\phi(x|x_1,x_2)\Big]\psi(x_1,x_2) \nonumber\\
&\ & -\frac{\hbar^2}{2M}\sum_{i} {\partial \phi\over \partial x_i}{\partial \psi\over \partial x_i}
+\Big[\big(-\frac{\hbar^2}{2m}\nabla_x^2-\lambda\delta(x-x_1)-\lambda\delta(x-x_2)\big)\phi(x |x_1,x_2)\Big]\psi(x_1, x_2)\nonumber\\
&\ & \quad \quad \quad \quad\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \quad =E\phi(x|x_1,x_2)\psi(x_1,x_2).
\end{eqnarray}
Let us assume that we find the solution to the equation below,
\begin{equation}
-\frac{\hbar^2}{2m}\nabla_x^2 \phi(x|x_1, x_2)-\lambda[\delta(x-x_1)+\delta(x-x_2)]\phi(x|x_1, x_2) =E(x_1,x_2)\phi(x|x_1, x_2)
.\end{equation}
This has a simple interpretation, we assume that the heavy particles act like fixed centers and the light particle moves in this background.
As a result we find,
\begin{eqnarray}
&\ & \Big[-\frac{\hbar^2}{2M}\sum_{i}{\nabla_i}^2+E(x_1,x_2)\Big]\psi(x_1,x_2)\phi(x|x_1,x_2)+ \Big[-\frac{\hbar^2}{2M}\sum_{i}{\nabla_i}^2\phi(x|x_1,x_2)\Big]\psi(x_1,x_2) \nonumber\\
&\ & -\frac{\hbar^2}{2M}\sum_{i} {\partial \phi\over \partial x_i}{\partial \psi\over \partial x_i} =E\phi(x|x_1,x_2)\psi(x_1,x_2).
\end{eqnarray}
Therefore, if we can neglect the last two terms on the lefthand side, we end up with the Born-Oppenheimer result,
\begin{equation}
\Big[-\frac{\hbar^2}{2M}\sum_{i}{\nabla_i}^2+E(x_1,x_2)\Big]\psi(x_1,x_2)=E\psi(x_1,x_2)
.\end{equation}
Thus the solution of equation (4) for the light degrees of freedom generates an effective potential for the heavy degrees of freedom, which we can then solve in principle.
Let us recall the solution to this fixed center problem, the heavy degrees of freedom are frozen at their locations, thus a particle is moving under the influence of two delta potentials.
This is a standart quantum mechanics problem, which we can easily solve by the following approach, inspired from the two dimensional version of delta function potentials \cite{fatih-teo}.
Let us make the following ansatz for the ground state wave-function
\begin{equation}
\phi(x|x_1, x_2) = \sum_{i=1}^2\int_0^\infty {dt\over \hbar} K_t(x,x_i)e^{-\frac{\nu^2}{\hbar}t},
\end{equation}
here $K_t(x,y)$ refers to the heat kernel on $\bf{R}$, which is simply a Gaussian. The heat kernel satisfies the well-known heat equation,
\begin{equation}
-\frac{\hbar^2}{2m}\nabla^2K_t(x,y) +\hbar\frac{\partial K_t(x,y)}{\partial t}= 0
,\end{equation}
the solution of which is unique under the conditions,
\begin{equation}
K_t(x,y)=K_t(y,x),\quad{\rm and} \qquad \lim_{t\to 0^+} K_t(x,y)=\delta(x-y)
,\end{equation}
moreover we impose $K_t(x,y)\geq 0$, which is required for the solutions in more general contexts.
One can see that the solution in flat space is given by a Gaussian,
\begin{equation}
K_t(x,y) =\sqrt{\frac{2m}{4\pi\hbar t}}e^{-\frac{2m}{4\hbar t}|x-y|^2}.
\end{equation}
The proposed wave function is positive and symmetric as required by the properties of the ground state wave function and the symmetry of the problem due to equal strength delta functions. The antisymmetric combination is of a higher energy level.
We verify that this ansatz indeed solves the equation with a negative eigenvalue,
if and only if $E(x_1,x_2)=-\nu^2$ satisfies
\begin{equation}
\frac{1}{\lambda} -\frac{1}{\hbar}\int_0^\infty dt K_t(x_2,x_2)e^{-\frac{\nu^2}{\hbar} t}=\frac{1}{\hbar}\int_0^\infty{dt}K_t(x_1,x_2)e^{-\frac{\nu^2 t}{\hbar}}
\end{equation}
this claim can be verified by the following integration by parts trick,
\begin{eqnarray}
\int_0^\infty {dt\over \hbar} \Big(-{\hbar^2\over 2m}\Big){\partial^2\over \partial x^2} K_t(x,x_i)e^{-\frac{\nu^2}{\hbar} t}& =&-\int_0^\infty {dt\over \hbar} \hbar{\partial\over \partial t} K_t(x,x_i)e^{-\frac{\nu^2}{\hbar} t}\nonumber\\
& =&-\int_0^\infty dt \Big({\partial\over \partial t} (K_t(x,x_i)e^{-\frac{\nu^2}{\hbar} t})-K_t(x,x_i)e^{-\frac{\nu^2}{\hbar} t}\Big)\nonumber\\
&=& \delta(x-x_i)-(-\nu^2) \int_0^\infty {dt\over \hbar} K_t(x,x_i)e^{-\frac{\nu^2}{\hbar} t} \nonumber
,\end{eqnarray}
where we used the inital condition on the heat kernel and its boundedness as $t\to \infty$.
One can see that for any given positive value of $\lambda$ there is always a solution for $\nu$.
Moreover, one can verify that the above solution agrees with the usual solution one would find by the Fourier transform method. Nevertheless, this representation of the solution is more useful for our calculations, and it can be generalized to the case of curves embedded in a Riemannian manifold.
When we place the two delta functions onto the same location, that is $z=|x_1-x_2|=0$, we find
\begin{equation}
\frac{1}{2\lambda} = \frac{1}{\hbar}\int_0^\infty dt K_t(x_1,x_1)e^{-\frac{\nu_0^2}{\hbar}t}=\sqrt{\frac{m}{2\hbar^2\nu_0^2}}.
\end{equation}
As a result we find for the zeroth order energy,
\begin{equation}
\nu_0^2= \frac{{2m}\lambda^2}{\hbar^2}.
\end{equation}
Normally, we expect a nonzero distance between the two centers, in this case the soltion is found from the full expression. One would see immediately that the energy achieves minimum when the distance between the centers is zero, that corresponds to the equlibrium configuration for the heavy system. When we displace them slightly from this equlibrium configuration, we may calculate the resulting energy change, which would act like an effective potential for the slow degrees of freedom in the Born-Oppenheimer approach. So under the assumption that small values of $z$ make the main contribution to the dynamics, we get a small correction to the energy, $E(x_1,x_2)=-\nu_0^2 +\Delta E(z)$, when we insert this back again into the equations,
\begin{eqnarray}
\frac{1}{\lambda} -\sqrt{\frac{m}{2\pi\hbar}}\int_0^\infty \frac{dt }{\hbar}\frac{e^{-\frac{\nu^2-\Delta E}{\hbar}t}}{\sqrt{t}}&=&\sqrt{\frac{m}{2\pi\hbar}}\int_0^\infty\frac{dt}{\hbar}\frac{e^{-\frac{(\nu^2-\Delta E_1)}{\hbar}t-\frac{mz^2}{2\hbar t}}}{\sqrt{t}}\nonumber\\
&=& \sqrt{\frac{m}{2\hbar^2(\nu_0^2-\Delta E)}}e^{\frac{|z|}{\hbar}\sqrt{2m(\nu_0^2-\Delta E)}}
\end{eqnarray}
We could neglect $\Delta E|z|/\nu_0 $ terms since both $\Delta E$ and $z$ are assumed small, as we will verify, we then find,
\begin{equation}
\frac1{\lambda}-\sqrt{\frac{m}{2\hbar^2(\nu_0^2-\Delta E)}} =\sqrt{\frac{m}{2\hbar^2(\nu_0^2-\Delta E)}}e^{-\frac{\nu_0 |z|\sqrt{2m}}{\hbar}}
\end{equation}
and expanding everything in the same order leads to the following relation between
$\Delta E $ and $z$:
\begin{equation}
\Delta E= \lambda^3|z|\Big(\frac{ 2m}{\hbar^2}\Big)^2
.\end{equation}
We are now ready for the Born-Oppenheimer approach, we introduce this energy as an effective potential acting between the heavy particles, since it depends on the separation between them. This gives us the following Schr\"odinger equation:
\begin{equation}
-\frac{\hbar^2}{2\mu}{\partial^2\over \partial z^2}\psi +|z|\lambda^3\left(\frac{2m}{\hbar^2}\right)^2\psi =\delta E\psi,
\end{equation}
here we represent the excitation energies, corresponding to $E+\nu_0^2$, by $\delta E$, also we use the reduced mass $\mu=M_1M_2/(M_1+M_2)=M/2$ since only the relative coordinate appears in the equation.
Note that this is a particle under the influence of a linear potential, the solutions of which are well-known, an especially beautiful presentation can be found in \cite{schwinger}.
Let us define the variables below,
\begin{eqnarray}
\beta^3 &=& (2\mu)\left(\frac{2m}{\hbar^3}\right)^2\lambda^3,\quad {\rm or}\quad \beta=\Big({\mu\over m}\Big)^{1/3}{\nu_0^2\over \lambda} \quad{\rm also,}\nonumber\\
u &=&|z| -\left(\frac{\hbar^2}{2m}\right)^2\frac{\delta E}{\lambda^3}\ \ {\rm and}\ \
\sigma = \beta u
.\end{eqnarray}
Note that $\lambda$ has dimensions energy-length.
As a result we find the solutions given by the Airy functions, being separated into even or odd ones,
\begin{equation}
\psi_{\pm}(z) =C\Big(\begin{matrix}{\rm sgn}(z)\\ 1\end{matrix}\Big) Ai\left((2\mu)^\frac1{3}\frac{(2m)^\frac{2}{3}}{\hbar^2}\lambda\left(|z| - \left(\frac{\hbar^2}{2m}\right)^2\frac{\delta E}{\lambda^3}\right)\right)
.\end{equation}
If we now impose the continuity of the wave functions and their derivates at
$z$= 0, we find, for the odd and even respectively,
either the zeros of $Ai(\sigma)$ or $Ai'(\sigma)$, which we collectively denote by $-\sigma_n$ with $n$ odd refering to the odd and $n$ even refering to the even solutions \cite{schwinger}. This leads to the following quantization conditions for the eigenvalues of the linear oscillator,
\begin{equation}
\delta E_n=(-\sigma_n) {2 m\lambda^2\over \hbar^2}\Big({m\over \mu}\Big)^{1/3}=(-\sigma_n)\nu_0^2\Big({m\over \mu}\Big)^{1/3},
\end{equation}
which shows explicitly that $\delta E_n << \nu_0^2$.
We can now accomplish the normalization of each wave function,
which can be turned into the following integral,
\begin{equation}
2{ C_n}^2\int_{\sigma_n}^\infty d\sigma| Ai(\sigma)|^2 = 1
.\end{equation}
This integral can be evaluated explicitly, giving us,
\begin{equation}
\frac{1}{2{ C_n}^2} = \left(\frac{\partial Ai(\sigma)}{\partial\sigma}\right)^2|_{\sigma=\sigma_n}-\sigma_n Ai(\sigma_n)^2
.\end{equation}
As a result, for $n$ is even, we get
\begin{equation}
\frac{1}{2{ C_n}^2} = -\sigma_n Ai(\sigma_n)^2
\end{equation}
and similarly, for $n$ is odd, we have,
\begin{equation}
\frac{1}{2{ C_n}^2}= \left(\frac{\partial Ai(\sigma)}{\partial\sigma}\right)^2|_{\sigma=\sigma_n}
.\end{equation}
As a result we find the wave function,
\begin{equation}
\psi_n(z) = C_n({\mu\over m})^{1/6} {\nu_0\over \lambda^{1/2}}\Big(\begin{matrix}{\rm sgn}(z)\\ 1\end{matrix}\Big) Ai\left((2\mu)^\frac1{3}\frac{(2m)^\frac{2}{3}}{\hbar^2}\lambda\left(|z| - \left(\frac{\hbar^2}{2m}\right)^2\frac{\delta E_n}{\lambda^3}\right)\right)
,\end{equation}
indeed here $({\mu\over m})^{1/6} {\nu_0\over \lambda^{1/2}}=(2\mu)^{1/6}{(2 m)^{1/3}\lambda^{1/2}\over \hbar}$.
If we assume that we have two identical bosonic heavy particles, we only need to consider the even wave functions which are symmetric under the interchange of these two particles, which corresponds to $z\mapsto-z$ here.
This essentially completes the discussion on Born-Oppenheimer approximation, apart from checking the consistency of our approximations. For this, we will calculate the expectation value of the variable $|z|$, it must be of the same order as the kinetic energy, and we find that
\begin{eqnarray}
(2{C_n}^2)\int_{\sigma_n}^\infty d\sigma \left( \frac{\sigma\hbar^2}{(2\mu)^\frac1{3}(2m)^\frac{2}{3}\lambda} + \left(\frac{\hbar^2}{2m}\right)^2\frac{\delta E}{\lambda^3}\right)|Ai(\sigma)|^2 &=& -\frac{2}{3}\frac{{\sigma_n}}{\lambda }\frac{\hbar^2}{(2\mu)^\frac{1}{3}(2m)^\frac{2}{3}}\nonumber\\
&=&\Big[-\frac{2}{3}\sigma_n\Big] \frac{\lambda}{\nu_0^2}\Big(\frac{m}{\mu}\Big)^{1/3}.\nonumber
\end{eqnarray}
Next, to ensure that higher order terms in the potential are negligible, we will calculate the spread of the wave function, by evaluating the expectation value of $z^2$.
This can be done, based on the fomulae given in \cite{Airy}, we calculate $< z^2>$ as follows,
\begin{equation}
< z^2> = 2{C_n}^2\int_{\sigma_n}^\infty d\sigma |Ai(\sigma)|^2\left( \frac{\sigma\hbar^2}{(2\mu)^\frac1{3}(2m)^\frac{2}{3}\lambda} + \left(\frac{\hbar^2}{2m}\right)^2\frac{\delta E}{\lambda^3}\right)^2
\end{equation}
we can calculate this integral term by term, as an example we have,
\begin{equation}
\frac{\hbar^4}{(2\mu)^\frac{2}{3}(2m)^\frac{4}{3}\lambda^2}(2{C_n}^2)\int_{\sigma_n}^\infty d\sigma {\sigma}^2|Ai(\sigma)|^2 =\frac1{5}\frac{{\sigma_n}^2}{\lambda^2}\frac{\hbar^4}{(2\mu)^\frac{2}{3}(2m)^\frac{4}{3}}\left(1-\frac1{{\sigma_n}^3}\right)
,\end{equation}
and we only quote the result for $n$ even, since we are mainly interested in the identical particle case,
which leads to the following result,
$<z^2>$ ($n$ even case) :
\begin{equation}
<z^2> =\frac{8}{15}\frac{{\sigma_n}^2}{\lambda^2}\frac{\hbar^4}{(2\mu)^\frac{2}{3}(2m)^\frac{4}{3}} -\frac1{5}\frac{\hbar^4}{(2\mu)^\frac{2}{3}(2m)^\frac{4}{3}\lambda^2 \sigma_n}=A_n \Big({m\over \mu}\Big)^{2/3}\Big( {\lambda\over \nu_0^2}\Big)^2
,\end{equation}
where $A_n$ is a numerical factor.
Having found this solution, we may go back and check the consistency of these approximations. This is presented in the Appendix-I, where we show for the proposed solution, that indeed the terms we neglect, lead to smaller order corrections.
\section{A Many Body View}
We will approach the same problem from the many-body perspective, construct the principal operator for a set of interacting particles, the Hamiltonian of which is given as follows:
\begin{equation}
H=\int dx \phi^\dagger (x) [-{\nabla^2\over 2m}] \phi(x)+\int dx \psi^\dagger(x) [-{\nabla^2\over 2M}] \psi(x)-\lambda \int dx \phi^\dagger (x) \psi^\dagger(x) \phi(x)\psi(x)
\end{equation}
For the extension, we will use the algebra of orthofermions as suggested by Rajeev in\cite{rajeev-asymptotic} (named as angels in Rajeev's work).
We therefore modify the problem using an extended Fock space construction. This is accomplished by an algebra defined through a set of operation rules,
\begin{eqnarray} \chi(x)\chi^{\dagger}(y) &=& \delta(x-y) \Pi_0,\cr\nonumber
\chi_(x)\chi(y)=0&=&
\chi^{\dag}(x)\chi^{\dag} (y) ,
\end{eqnarray}
where
\begin{equation} \Pi_1 = \int dx \,
\chi^{\dag}(x) \chi(x) , \;\;\;\Pi_0 = 1- \Pi_1 \;
\end{equation}
are the projection operators onto the one-angel and no-angel states,
respectively. This algebra has a realization on ${\bf C}\oplus {\cal L}^2({\bf R})$.
Thus we extend the Hilbert space of the theory from ${\cal F}_M\otimes {\cal F}_m$ to
${\cal F}_M\otimes {\cal F}_m\oplus {\cal F}_M\otimes {\cal F}_m\otimes L^2({\bf R})$.
Define a new Hamiltonian on this extended space in matrix form as follows
\begin{eqnarray}
&\ &\hat H-E\Pi_0=\left (
\begin{array}{cc}
(H_0-E) \Pi_0 & \int dx \psi^\dag(x)\phi^\dag (x) \chi(x) \\
\int dy \psi(y) \phi(y) \chi^\dag(y) & {1\over \lambda}\Pi_1 \\
\end{array} \right )\equiv \left (
\begin{array}{cc}
a & b^\dagger \\
b & d \\
\end{array} \right )
\end{eqnarray}
The resolvent or the Green's function of this extended system is defined as
\begin{equation}
(\hat H-E \Pi_0)^{-1} \equiv \left (
\begin{array}{cc}
\alpha & \beta^\dagger \\
\beta & \delta \\
\end{array} \right )
\end{equation}
The projection of this matrix Green's function on to the no-angel (no-orthofermion) subspace can be fromally written in two alternative ways:
\begin{eqnarray}
\nonumber \alpha = (a-b^\dagger d^{-1} b)^{-1} = (H-E)^{-1}
=a^{-1}+a^{-1}b^\dagger \Phi^{-1}ba^{-1},
\end{eqnarray}
where we introduce the principal matrix $\Phi$, given by
\begin{equation}
\Phi \equiv d-ba^{-1}b^\dagger
.\end{equation}
The second expression becomes more useful in our calculations.
The first relation for $\alpha$, by the properties of the orthofermion operators, shows that the projection of the resolvent of the new operator onto the no-angel subspace reproduces the Green's function of the original Hamiltonian. In the same way therefore the second representation also reproduces the Green's function.
In our case we have the explicit expression,
\begin{eqnarray}
\nonumber \Phi = {1\over \lambda} \Pi_1 -\int dx \phi(x)\psi(x)\chi^\dag(x) {1\over H_0-E}\int dy \psi^\dag (y) \phi^\dag (y) \chi(y)\nonumber
\end{eqnarray}
Let us normal order this operator, the result, can be written in the momentum representation. We warn the reader that {\it we use the following notational convention}, $[dp]={dp\over 2\pi}$ and
delta function, written in momentum space as $\delta[p-q]$ refers to $2\pi\delta(p-q)$, having established that, the normal ordered operator becomes,
\begin{eqnarray}
\Phi&=&{1\over \lambda} \Pi_1-\int [dpdq] \chi^\dagger(p+q){1\over H_0+\nu_0^2-\delta' E+p^2/2M+q^2/2m}\chi(p+q)\nonumber\\
&-&\int [dpdqdr] \chi^\dag(p+q)\psi^\dag(r){1\over H_0+\nu_0^2
-\delta' E+p^2/2M+r^2/2M+q^2/2m}\psi(p) \chi(r+q),\nonumber
\end{eqnarray}
where we dropped a term which contains normally ordered light particles since we are assuming that there is only a single light particle. Moreover, we use again the splitting of the energy as $-\nu_0^2+\delta' E$. In this formalism two heavy and a single light is replaced with an orthofermion and a single heavy particle.
In the first term change to the center of momentum and relative momentum,
$P=p+q$ and $\eta=\mu[p/M-q/m]$, where $\mu$ is the reduced mass, the first term becomes:
\begin{equation}
\int [dPd\eta] \chi^\dagger(P){1\over H_0+\nu_0^2-\delta' E+P^2/2(M+m)+\eta^2/2\mu}\chi(P)
\end{equation}
Integral over the relative momenta can be executed;
\begin{equation}
{1\over 2\hbar}\sqrt{{2mM\over m+M}}\int [dP] \chi^\dagger(P){1\over \sqrt{H_0+\nu_0^2-\delta' E+P^2/2(M+m)}}\chi(P)
.\end{equation}
The key idea, due to Rajeev is this, the bound state solutions can only come from the zero eigenvalues of the $\Phi(E)$ operator.
If we are interested in the bound states of this system we will look for an eigenfunction, $|\omega>$,
\begin{equation}
\Phi(E) |\omega>=0
,\end{equation}
this normalizable solution $|\omega>$ can be used to obtain the actual bound state solution easily.
In $H_0$ we have both the light and heavy particle free Hamiltonians, but recall that we have no light particle when we switch to the principal operator in this sector, that means $H_0=H_0(\psi)$. We thus assume that $H_0-\delta' E$ in the kinetic term, and $p^2/2M+r^2/2M-\delta' E$ in the potential term, are small relative to
$\nu_0^2$ type terms. Everything can be expanded relative to the large $\nu_0^2$ term or $\nu_0^2+q^2/2m$ term respectively, depending on the kinetic or potential terms, and we drop the extra term ${m/M}$ in the kinetic energy as well as in the reduced mass. The reason behind this complication can be understood as follows, we cannot assume that $\delta' E$ is small, because there is also the total kinetic energy of center of mass motion, which could be large, but it is also included in the other kinetic energy terms, as we will see in a moment, hence only their combined sum can be small. It turns out that to get everything consistent we need to do a second order expansion, as we will see.
\begin{eqnarray}
\Phi&=& \frac1{\lambda}\Pi_1-\frac{\sqrt{2m}}{2\hbar\nu_0}\int [dP]\chi^\dagger(P)\left[1-\frac1{2}\frac{(H_0-\delta' E+\frac{P^2}{2M})}{\nu_0^2}+\frac{3}{8}\frac{(H_0-\delta' E+\frac{P^2}{2M})^2}{\nu_0^4}+...\right]\chi(P)\nonumber\\
&-&\!\!\!\!\int [dpdqdr]\chi^\dagger(p+q)\psi^\dagger(r)\left[\frac1{(\nu_0^2+\frac{q^2}{2m})}-\frac{(\frac{p^2}{2M}+\frac{r^2}{2M}-\delta' E)}{(\nu_0^2+\frac{q^2}{2m})^2}+\frac{(\frac{p^2}{2M}+\frac{r^2}{2M}-\delta' E)^2}{(\nu_0^2+\frac{q^2}{2m})^3}+...\right]\nonumber\\
&\ & \qquad \qquad \qquad \qquad \qquad \qquad \times \psi(p)\chi(q+r)\nonumber
\end{eqnarray}
Using this expansion for $\Phi$, we make the following ansatz for the zero eigenvalue solution of $\Phi$ operator.
\begin{equation}
|\omega> =\int [d\xi dQ]f(\xi)e^{-iQX/\hbar}\chi^\dagger(\frac{Q}{2}+\xi)\psi^\dagger(\frac{Q}{2}-\xi)|\Omega>,
\end{equation}
where, $|\Omega>$ denotes the vacuum for the combined Fock space of particles and orthofermion. Here we may assume that the relative wave function in position space is real, this implies a symmetry for the Fourier transform, $f(\xi)=f^*(-\xi)$. Note that in principle we can improve this ansatz by taking into account the fact that the orthofermion is a composite of light and heavy, hence its mass is actually $M+m$, but this is a smaller order improvement, hence we may ignore it. This approximation actually implies an approximate symmetry for the wave function, up to order $m/M$, we may assume that due to the bosonic nature of the heavy particles, the wave function $f(z)$ is ivariant under inversion, i. e. $f(z)=f(-z)$ (this symmetry should be violated by terms of order $m/M$). This means that Fourier transform satisfies $f(\xi)=f(-\xi)$.
Let us now compute the action of $\Phi$ on our ansatz. The first part of $\Phi$ operator produces,
\begin{eqnarray}
&\ &\Big(\frac1{\lambda}-\frac1{2}\frac{\sqrt{2m}}{\hbar\nu_0}\Big)\Pi_1|\omega>+\frac1{4}\frac{\sqrt{2m}}{\hbar\nu^3}\int [dP] \chi^\dagger(P)\big(H_0-\delta' E+\frac{P^2}{2M}\big)\chi(P)|\omega>\nonumber\\
&\ &=\Big(\frac1{\lambda}-\frac1{2}\frac{\sqrt{2m}}{\hbar\nu_0}\Big)\Pi_1|\omega>+\frac{1}{4}\frac{\sqrt{2m}}{\hbar\nu^3}\int [dQd\xi] f(\xi)e^{-i{QX\over \hbar}}\big(\frac1{2}\frac{Q^2}{2M}+2\frac{\xi^2}{2M}-\delta' E\big)\nonumber\\
&\ & \qquad\qquad\qquad \qquad\qquad\qquad \qquad \times \chi^\dagger(\frac{Q}{2}+\xi)\psi^\dagger(\frac{Q}{2}-\xi)|\Omega>
,\end{eqnarray}
using the convolution to express the particle-orthofermion Fock state in relative coordinate space and then stripping off the Fock state vector, we have the expression,
\begin{eqnarray}
\left(\frac1{\lambda}-\frac1{2}\frac{\sqrt{2m}}{\hbar\nu_0}\right)f(z) +\frac1{4}\frac{\sqrt{2m}}{\hbar\nu^3}\left[\frac1{2}\frac{Q^2}{2M}-\frac{\hbar^2}{2\mu}\nabla^2_z-\delta' E\right]f(z)
.\end{eqnarray}
Similarly we work on the "potential" part of the $\Phi$ operator, the first term of which is given by
\begin{eqnarray}
&\ &\int [dpdqdr]\chi^\dagger(p+q)\psi^\dagger(r)\frac1{\nu_0^2+\frac{q^2}{2m}}\psi(p)\chi(q+r)|\omega>\nonumber\\
&\ & =\int[ dqdQd\xi] f(\xi)e^{-iQX/\hbar}\frac1{\nu_0^2+\frac{q^2}{2m}}\chi^\dagger(\frac{Q}{2}-\xi +q)\psi^\dagger(\frac{Q}{2}+\xi -q)|\Omega>.
\end{eqnarray}
Here, we redefine $\xi \rightarrow -\xi $ and $ \xi \rightarrow \xi -q $
\begin{equation}
\int [dqdQd\xi] f(q-\xi)e^{-iQX/\hbar}\frac1{\nu_0^2+\frac{q^2}{2m}}\chi^\dagger(\frac{Q}{2}+\xi )\psi^\dagger(\frac{Q}{2}-\xi )|\Omega>
\end{equation}
and assuming the ground state wave function $f(\xi)$ is symetric so $ f(q-\xi) \rightarrow f(\xi-q)$
the final result can be turned into an expression for $f(z)$ only;
\begin{equation}
\int [dqdQd\xi] f(q-\xi)e^{-iQX/\hbar}\frac1{\nu_0^2+\frac{q^2}{2m}}\chi^\dagger(\frac{Q}{2}+\xi )\psi^\dagger(\frac{Q}{2}-\xi )|\Omega> \mapsto {1\over 2}\frac{\sqrt{2m}}{\hbar\nu_0}e^{-\frac{\sqrt{2m}}{\hbar}\nu_0 |z|}f(z)
.\end{equation}
Thus, combining the first result with this one, we have,
\begin{equation}
\left(\frac1{\lambda}-\frac1{2}\frac{\sqrt{2m}}{\hbar\nu_0}\right)f(z)+\frac1{4}\frac{\sqrt{2m}}{\hbar\nu_0^3}\left[\frac1{2}\frac{Q^2}{2M}-\frac{\hbar^2}{2\mu}\nabla^2_z -\delta' E\right]f(z)-\frac1{2}\frac{\sqrt{2m}}{\hbar\nu_0}e^{-\frac{\sqrt{2m}}{\hbar}\nu_0 |z|}f(z)
.\end{equation}
There is one more term coming from the potential part, this last term acting on $|\omega>$ gives us,
\begin{eqnarray}
&\ &\!\!\!\!\!\!\!\!\!\!\!\int[ dpdqdr]\chi^\dagger(p+q)\psi^\dagger(r)\frac{(\frac{p^2}{2M}+\frac{r^2}{2M}-\delta' E)}{(\nu_0^2+\frac{q^2}{2m})^2}\psi(p)\chi(q+r)|\omega>\nonumber\\
&\ &=\int [dqdQd\xi] f(\xi)e^{-iQX/\hbar}\left(\frac{\frac{(\frac{Q}{2}-\xi)^2}{2M}+\frac{(\frac{Q}{2}+\xi-q)^2}{2M}-\delta' E}{(\nu_0^2+\frac{q^2}{2m})^2}\right)\chi^\dagger(\frac{Q}{2}-\xi+q)\psi^\dagger(\frac{Q}{2}+\xi-q)|\Omega>\nonumber\\
&\ &=\int [dqdQd\xi ]f(\xi-q)e^{-iQX/\hbar}\left(\frac{\frac{(\frac{Q}{2}+\xi-q)^2}{2M}+\frac{(\frac{Q}{2}-\xi)^2}{2M}-\delta' E}{(\nu_0^2+\frac{q^2}{2m})^2}\right)\chi^\dagger(\frac{Q}{2}+\xi)\psi^\dagger(\frac{Q}{2}-\xi)|\Omega>
.\end{eqnarray}
Note that if we consider the expression below,
\begin{equation}
\frac{({Q\over 2} +\xi-q)^2}{2M}=\frac{({Q\over 2}+\xi)^2}{2M}\underbrace{-\frac{2m}{2M}\frac{2q({Q\over 2}+\xi)}{2m}+\left(\frac{2m}{2M}\right)\frac{q^2}{2m}}_{(*)}
,\end{equation}
inside the above kernel, the second and the last terms, denoted by $(*)$ collectively here, are negligible corrections, if we ignore these two terms, we restore the symmetric structure of this kernel.
Let us remark that this is the same order of magnitude approximation as assuming the bosonic inversion symmetry with respect to the relative coordinate of the orthofermion and the heavy particle in our ansatz and they are clearly related, the equations essentially signal that the assumed symmetry cannot be exact. Let us briefly digress on this issue,
an exact calculation of this term leads to the following expression,
\begin{eqnarray}
(*)&\mapsto& -{2m\over M}\left(\frac1{4}\frac{\left(\frac{Q}{2}-i\hbar\frac{\partial}{\partial z}\right)}{2m}\frac{\hbar}{i}\frac{\partial}{\partial z}+\frac1{8}\frac{\hbar^2}{2m}\frac{\partial^2}{\partial z^2}\right)\frac{\sqrt{2m}}{\hbar\nu_0^3}e^{-\frac{\sqrt{2m}}{\hbar}\nu_0 |z|}\left[1+\frac{\sqrt{2m}}{\hbar}\nu_0 |z|\right]f(z)\nonumber
\end{eqnarray}
which explicitly shows that it is of order $m/M$, and it contains terms which will break the reflection symmetry of $f(z)$ via mixing with the center of mass momentum $Q$. Note that, $Q$ is left undetermined, assuming the system moves with a well-defined value, but in reality this should also be smeared out with some function $g(Q)$, and every $Q$-term leads to $-i\hbar {\partial \over \partial X}$ acting on the wave function $g(X)$ in the coupled set of equations. A more complete treatment should take this into account. An estimate of this term can be made by using the first order solution we have in the previous section and it produces,
\begin{eqnarray}
{\nu_0^2\over 8}\Big(\frac{\sqrt{2m}}{\hbar\nu_0^3}\Big)\left[\frac{iQ}{\sqrt{2m}\nu_0}\Bigg(O\left(\frac{m}{\mu}\right)^\frac{4}{3}+O\left(\frac{m}{\mu}\right)^\frac{5}{3}+...\Bigg)+\Bigg(O\left(\frac{m}{\mu}\right)+O\left(\frac{m}{\mu}\right)^\frac{4}{3}+O\left(\frac{m}{\mu}\right)^\frac{5}{3}+....\Bigg)\right]\nonumber
.\end{eqnarray}
Here, we deliberately kept the coefficients in front of it in an unsimplified form, since in further calculations, these coefficients show up in all the terms. Note that $Q$ term above actually brings a contribution at a higher order. In principle, we ignore this term and then check the consistency after we find the first order solution, which will agree with the above estimate.
Hence, after ignoring these terms, the final result of this part becomes,
\begin{eqnarray}
&\ & \int [dqdQd\xi] f(\xi-q)e^{-iQX/\hbar}\left(\frac{\frac{(\frac{Q}{2}+\xi)^2}{2M}+\frac{(\frac{Q}{2}-\xi)^2}{2M}-\delta' E}{(\nu_0^2+\frac{q^2}{2m})^2}\right)\chi^\dagger(\frac{Q}{2}+\xi)\psi^\dagger(\frac{Q}{2}-\xi)|\Omega>\mapsto\nonumber\\
&\ & \qquad \qquad \qquad \mapsto\frac1{4}\frac{\sqrt{2m}}{\hbar\nu_0^3}\left[\frac1{2}\frac{Q^2}{2M}-\frac{\hbar^2}{2\mu}\nabla^2_z -\delta' E \right]e^{-\frac{\sqrt{2m}}{\hbar}\nu_0 |z|}\Big[1+\frac{\sqrt{2m}}{\hbar}\nu_0 |z|\Big]f(z)
\nonumber
\end{eqnarray}
We remark that {\it there is another valid choice for this term}, that will be obtained by adding an extra $q$ into it, giving us,
\begin{eqnarray}
&\ &\int [dqdQd\xi] f(\xi)e^{-iQX/\hbar}\left(\frac{\frac{(\frac{Q}{2}-\xi)^2}{2M}+\frac{(\frac{Q}{2}+\xi-q)^2}{2M}-\delta' E}{(\nu_0^2+\frac{q^2}{2m})^2}\right)\chi^\dagger(\frac{Q}{2}-\xi+q)\psi^\dagger(\frac{Q}{2}+\xi-q)|\Omega>\nonumber\\
&\ &=\int [dqdQd\xi ]f(\xi-q)e^{-iQX/\hbar}\left(\frac{\frac{(\frac{Q}{2}+\xi-q)^2}{2M}+\frac{(\frac{Q}{2}-\xi+q)^2}{2M}-\delta' E}{(\nu_0^2+\frac{q^2}{2m})^2}\right)\chi^\dagger(\frac{Q}{2}+\xi)\psi^\dagger(\frac{Q}{2}-\xi)|\Omega>\nonumber
.\end{eqnarray}
The correct expression is then found by taking the half sum of these terms, since that means the integral expression is actually a symmetric kernel, that is the usual ordering prescription for noncommuting operators, essentially leading to expressions like,
\begin{equation}
(-\frac{\hbar^2}{2\mu}\nabla^2_z) V(z)+V(z)(-\frac{\hbar^2}{2\mu}\nabla^2_z )
.\end{equation}
The corrections coming from this non-commuting expressions are of order $m/M$, as we will verify shortly.
As a result we combine all the first order in the kinetic energy terms, which implies that we find the general solution for the eigenvector $|\omega>$, to this order (yet we keep the symmetric structure), from the equation,
\begin{eqnarray}
&\ &\left(\frac1{\lambda}-\frac1{2}\frac{\sqrt{2m}}{\hbar\nu_0}\right)f(z)+\frac1{4}\frac{\sqrt{2m}}{\hbar\nu_0^3}\left[\frac1{2}\frac{Q^2}{2M}-\frac{\hbar^2}{2\mu}\nabla^2_z -\delta' E\right]f(z)-\frac1{2}\frac{\sqrt{2m}}{\hbar\nu_0}e^{-\frac{\sqrt{2m}}{\hbar}\nu_0 |z|}f(z)\nonumber\\
&\ &\qquad\qquad +\frac{1}{8}\frac{\sqrt{2m}}{\hbar\nu_0^3}\left[\Big[\frac{1}{2}\frac{Q^2}{2M}-\frac{\hbar^2}{2\mu}\nabla^2_z -\delta' E\Big],e^{-\frac{\sqrt{2m}}{\hbar}\nu_0 |z|}\Big[1+\frac{\sqrt{2m}}{\hbar}\nu_0 |z|\Big]\right]_+f(z)=0\nonumber
.\end{eqnarray}
Note that the anti-commutator is not important to this order, it is kept to emphasize the hermitian nature of the problem.
Now, {\it we assume that higher orders in $z$ are also small}, the consistency of which can be verified after the solution, so we expand this equation in $z$ to find the first order solution.
So the final equation one considers for this system becomes
\begin{equation}
\left(\frac1{\lambda}-\frac{\sqrt{2m}}{\hbar\nu_0} \right)f(z) +\frac1{2}\frac{\sqrt{2m}}{\hbar\nu_0^3}\left[\frac1{2}\frac{Q^2}{2M}-\frac{\hbar^2}{2\mu}\nabla^2_z -\delta' E \right]f(z) +\frac1{2}\frac{2m}{\hbar^2}|z|f(z) =0
\end{equation}
If we solve this system order by order, the zeroth order term gives us
\begin{equation}
\nu_0^2= {2m\lambda^2\over \hbar^2}
,\end{equation}
which then can be inserted into the remaining part to give us the equation;
$$
\left(-\frac{\hbar^2}{2\mu}\nabla^2_z -\Big[\delta' E-\frac1{2}\frac{Q^2}{2M}\Big] \right)f(z) + \frac{\sqrt{2m}}{\hbar}\nu_0^3 |z| f(z) = 0,
$$
or after introducing the small energy difference $\delta E=\delta' E-{1\over 2} {Q^2\over 2M}$, can be equivalently rewritten as,
$$
\left(-\frac{\hbar^2}{2\mu}\nabla^2_z -\delta E \right)f(z) + \left(\frac{2m}{\hbar^2}\right)^2\lambda^3 |z| f(z) = 0,
$$
which is exactly the equation found by the Born-Oppenheimer approximation in the previous section.
Having found the solution, an immediate computation of the actual symmetrized term reveals that, the ordering ambiguity brings the following contribution,
\begin{eqnarray}
&\ &\!\!\!\!\!\!\!\!\!\!\!\frac1{8}\frac{\sqrt{2m}}{\hbar\nu_0^3}(2C_n^2)\int_{\sigma_n}^\infty d\sigma Ai(\sigma)\left(-\frac{\hbar^2}{2\mu}\nabla^2_z e^{-\frac{\sqrt{2m}}{\hbar}\nu_0| z|}\Big[\frac{\sqrt{2m}}{\hbar}\nu_0 |z|+1\Big]\right)Ai(\sigma)\nonumber\\
&= & -\frac{\sqrt{2m}}{8\hbar\nu_0^3}\frac{\hbar^2}{2\mu}(2C_n^2)\int_{\sigma_n}^\infty d\sigma Ai(\sigma)\left(\nabla^2_z\Big[-\frac1{2}\frac{2m}{\hbar^2}\nu_0^2 z^2+\frac1{3}\frac{(2m)^\frac{3}{2}}{\hbar^3}\nu_0^3|z|^3-\frac1{8}\frac{(2m)^2}{\hbar^4}\nu_0^4z^4+...\Big]\right) Ai(\sigma)\nonumber\\
&=&-\frac{\sqrt{2m}}{8\hbar\nu_0^3}\frac{\hbar^2}{2\mu}(2C_n^2)\int_{\sigma_n}^\infty d\sigma Ai(\sigma)\left(-\frac{2m}{\hbar^2}\nu_0^2+2\frac{(2m)^\frac{3}{2}}{\hbar^3}\nu_0^3|z|-\frac{3}{2}\frac{(2m)^2}{\hbar^4}\nu_0^4z^2\right)Ai(\sigma)\nonumber\\
& =&{\sqrt{2m}\over \hbar\nu_0^3}\left[O\left(\nu_0^2\left(\frac{m}{\mu}\right)\right)+O\left(\nu_0^2\left(\frac{m}{\mu}\right)^\frac{4}{3}\right)+O\left(\nu_0^2\left(\frac{m}{\mu}\right)^\frac{5}{3}\right)+...\right]\nonumber
,\end{eqnarray}
where $\sigma$ is the shifted coordinate and $\sigma_n<0$ is the root of the Airy function corresponding to the energy level, $C_n$ is the normalization constant as defined in the previous section. Hence, we verify that this ordering ambiguity has no effect at this order as claimed.
This raises the possibility of obtaining higher order corrections, which we know to be of order $({m\over M})^{2/3}$ from our previous estimates. Note that to find these terms, we may still neglect terms of order ${m\over M}$, so the reflection symmetry of $f(z)$ can be kept. To find the next order correction we resort to perturbation theory. We indicate the general approach and leave some of the details to the reader.
We assume that $\delta E$ can be expanded as $\delta_1 E+\delta_2 E+...$ where the terms represent contributions of order $({m\over M})^{1/3}, ({m\over M})^{2/3}...$ respectively. Moreover $\Phi$ itself has such an expansion too, that we should consider.
Let us then expand the operator $\Phi$ further,
\begin{eqnarray}
\Phi&=&\frac1{\lambda}\Pi_1 -\frac{\sqrt{2m}}{\hbar\nu_0}\int [dP]\chi^\dagger(P)\left[1-\frac1{2}\frac{(H_0-\delta_1 E -\delta_2 E+\frac{P^2}{2M})}{\nu_0^2}+\frac{3}{8}\frac{(H_0-\delta_1 E+\frac{P^2}{2M})^2}{\nu_0^4}+...\right]\chi(P)
\nonumber\\
&-&\int [dpdqdr]\chi^\dagger(p+q)\psi^\dagger(r)\left[\frac1{\nu_0^2+\frac{q^2}{2m}}-\frac{(\frac{p^2}{2M}+\frac{r^2}{2M}-\delta_1 E -\delta_2 E)}{(\nu_0^2+\frac{q^2}{2m})^2}+\frac{(\frac{p^2}{2M}+\frac{r^2}{2M}-\delta_1 E)^2}{(\nu_0^2+\frac{q^2}{2m})^3}+..\right]\nonumber \\
&\ & \qquad \qquad\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \times \psi(p)\chi(q+r)\nonumber
\end{eqnarray}
The first order term $\delta_1 E$ is found exactly as above, the eigenvector to this order is called $|\omega_1>$. Note that the zeroth order does not determine the eigenvector, it is a scalar identity. The next order term can be found by using {\it first order perturbation theory} on the eigenvalue equation. This comes from our defining equation $\Phi(E)|\omega>=\omega(E)|\omega>$. Here the equation $\omega(\nu_0+\delta_1E+\delta_2E)+\delta_2\omega(\nu_0+\delta_1 E)=0$ should be solved.
The variation of the eigenvalue can be found from first order perturbation theory, the change of the first order part is found by the Feynmann-Helmann theorem.
Note that, we should now {\it expand the potential terms to higher orders to improve our solution to first order equation, so that it includes the next order corrections as well}. This implies $f(z)$, hence its eigenvalue, should further be corrected in the first order solution.
Thus, we should compute,
\begin{eqnarray}
<\omega_1|\delta_2 \Phi|\omega_1>\!\!&+&\!\!<\omega_1|{\partial\Phi\over \partial E}|\omega_1>\delta_2E =-\frac1{4}\frac{\sqrt{2m}}{\hbar\nu_0^3}\delta_2 E-\frac1{4}\frac{(2m)^\frac{3}{2}}{\hbar^3}\nu_0<z^2>\nonumber\\
&\ & -\frac{3}{16}\frac{\sqrt{2m}}{\hbar\nu_0^5}\int [dP]<\omega_1|\chi^\dagger(P)
\left(H_0-\delta_1 E +\frac{P^2}{2M}\right)^2\chi(P)|\omega_1>\nonumber\\
&\ &-\int [dpdqdr] <\omega_1|\chi^\dagger(p+q)\psi^\dagger(r)\frac{(\delta_2 E)}{(\nu_0^2+\frac{q^2}{2m})^2}\psi(p)\chi(q+r)|\omega_1>\nonumber\\
& \ &-\int [dpdqdr] <\omega_1|\chi^\dagger(p+q)\psi^\dagger(r)\frac{(\frac{p^2}{2M}+\frac{r^2}{2M}-\delta_1 E)^2}{(\nu_0^2+\frac{q^2}{2m})^3}\psi(p)\chi(q+r)|\omega_1>\nonumber
.\end{eqnarray}
Note that in the third and the last terms we did not keep $\delta_2 E$. Setting the whole expression equal to zero will determine the unknown $\delta_2 E$. Here $<z^2>$ again refers to the expectation with respect to the Airy functions we have found in the previous section, as a result,
the second term can be easily computed.
The kinetic energy correction can be found as,
\begin{eqnarray}
(*)&=&-\frac{3}{16}\frac{\sqrt{2m}}{\hbar\nu_0^5}\int [dP]<\omega_1|\chi^\dagger(P)\left(H_0-\delta_1 E +\frac{P^2}{2M}\right)^2\chi(P)|\omega_1>\nonumber\\
& =& -\frac{3}{16}\frac{\sqrt{2m}}{\hbar\nu_0^5}(2 C^2_n)\int_{\sigma_n}^\infty d\sigma Ai(\sigma)\left(\frac{\hbar^4}{(2\mu)^2}\nabla^4_z+2\delta_1 E\frac{\hbar^2}{2\mu}\nabla^2_z+\delta_1 E^2\right)Ai(\sigma)\nonumber
,\end{eqnarray}
where $\delta_1 E$ could be at its $n$th level, as we discussed previously, $\sigma_n<0$ refers to the corresponding root of the Airy function again.
Let us recall that $\sigma$ was defined through,
\begin{equation}
\frac{\sigma\hbar^2}{(2m)^\frac{2}{3}(2\mu)^\frac1{3}\lambda}+\frac{\delta_1 E}{\lambda^3}\left(\frac{\hbar^2}{2m}\right)^2 =z
,\end{equation}
and this root in turn determines the energy level,
\begin{equation}
-\sigma_n =(2\mu)^\frac1{3}\frac{\hbar^2}{(2m)^\frac{4}{3}\lambda^2}\delta_1 E^{(n)}
.\end{equation}
We suppress this dependence of $\delta_1 E$ on $n$ for simplicity. This computation can be done and the result comes out to be,
\begin{equation}
(*)=-a_n\left(\frac{m}{\mu}\right)^\frac{2}{3}\frac1{\lambda} =-a_n\frac{\sqrt{2m}}{\hbar\nu_0}\left(\frac{m}{\mu}\right)^\frac{2}{3}
\end{equation}
where $a_n$ is a known numerical factor.
The expectation value of $z^2$ has already been computed,
\begin{equation}
-\frac{1}{4}\frac{(2m)^\frac{3}{2}}{\hbar^3}\nu_0<z^2> =-b_n\left(\frac{m}{\mu}\right)^\frac{2}{3}\frac1{\lambda}=-b_n\frac{\sqrt{2m}}{\hbar\nu_0}\left(\frac{m}{\mu}\right)^\frac{2}{3}
.\end{equation}
The next term has already been computed, we only need to add $\delta_2 E$ to it,
\begin{equation}
(**)=-\int[ dpdqdr] <\omega_1|\chi^\dagger(p+q)\psi^\dagger(r)\frac{\delta_2 E}{(\nu_0^2+\frac{q^2}{2m})^2}\psi(p)\chi(q+r)|\omega_1>
\end{equation}
which leads to,
\begin{equation}
(**)=-\frac{1}{4}\frac{\sqrt{2m}}{\hbar\nu_0^3}\delta_2 E
\end{equation}
We may now compute the last term,
\begin{equation}
(***)=-\int [dpdqdr]<\omega_1|\chi^\dagger(p+q)\psi^\dagger(r)\frac{\left(\frac{p^2}{2M}+\frac{r^2}{2M}-\delta_1 E\right)^2}{(\nu_0^2+\frac{q^2}{2m})^3}\psi(p)\chi(q+r)|\omega_1>
.\end{equation}
To this order of accuracy, this term can also be symmetrized, to be fully consistent one may choose a Weyl ordering, that will lead to terms of the form,
\begin{equation}
({-\hbar^2\over 2\mu} \nabla^2_z )^2 V(z)+{-\hbar^2\over 2\mu} \nabla^2_z V(z){-\hbar^2\over 2\mu} \nabla^2_z +V(z)({-\hbar^2\over 2\mu} \nabla^2_z )^2
,\end{equation}
by doing a calculation similar to our consistency check, one can see that the ordering ambiguities will bring terms of order $m/M$ hence can be neglected at this order. Derivatives acting on the Airy functions will give us contributions at the desired order, they can all be computed.
As a result, one finds,
$$
(***)=-d_n\frac{\sqrt{2m}}{\hbar\nu_0}\left(\frac{m}{\mu}\right)^\frac{2}{3},
$$
where $d_n$ is an explicitly computable constant. Adding all the contributions and then setting them to zero, we see that
one can find the second order energy expression,
$$
\delta_2 E^{(n)}=-\alpha_n\nu_0^2\left(\frac{m}{\mu}\right)^\frac{2}{3}
$$
where $\alpha_n$ is a known purely numerical constant.
This completes our discussion on the nonrelativistic case.
\section{Relativistic Treatment of the Light Particle}
Here we will introduce a slightly modified version of the previous problem, where the light particle is treated with a relativistic dispersion relation. This is in a sense similar to the heavy quark problem, where gauge fields binding the heavy quarks should be considered relativistic, yet the resulting effective heavy quark system is nonrelativistic.
First let us define the Hamiltonian of the system, where the light particle binding is so strong that its binding energy is comparable to its mass, hence we should treat it relativistically. Yet the coupling is not so strong to cause pair creation. On the other hand the heavy particles are so much heavier compared to the light one that the effect of the point interactions on them can be treated nonrelativistically. This can be modeled by a non-local many-body Hamiltonian,
\begin{equation}
H={1\over 2}\int dx :\pi^2+\phi (-\nabla^2+m^2)\phi :+\int dx \Psi^\dagger(x) \Big( -{\nabla^2\over 2M}\Big)\Psi(x)-\lambda \int \phi^{(-)}(x)\Psi^\dagger(x)\Psi(x) \phi^{(+)}(x) dx\nonumber
\end{equation}
We follow the usual convention of setting $\hbar=1$ and $c=1$ in this section.
The light particle can be assumed to have no charge so it is simply represented by a real field. Quantization of this real field is done in the usual way, by first considering the full solution to the equations of motion and then imposing the real valuedness, as well as assuming the canonical commutation relations.
As a result of this, we have the representation in terms of creation and annihilation operators,
\begin{equation}
\phi(x,t)=\int {[dk]\over \sqrt{2\omega(k)}} \left( a^\dagger(k)e^{-ikx+i\omega(k) t}+a(k)e^{ikx-i\omega(k)t}\right)
,\end{equation}
where
\begin{equation}
[a(k), a^\dagger(p)]=\delta[k-p]\quad {\rm and} \quad \omega(k)=\sqrt{k^2+m^2}
.\end{equation}
We have the nonlocal splitting into positive and negative frequency (energy) parts,
\begin{equation}
\phi^{(+)}(x,t)=\int {[dk]\over \sqrt{2\omega(k)}} a(k)e^{ikx-i\omega(k)t}
,\end{equation}
similarly for the negative frequency part. The free Hamiltonian part of the relativistic field itself does not
evolve in time, as can be verified. Our choice of the interaction forbids the particle creation-annihilation process, it is not truely a delta function, it is a truncated version of it, in the limit where the particle creation-annihilation is negligible.
As a first approximation to this still complicated problem, we will invoke a kind of Born-Oppenheimer approximation and pretend that the heavy particles are actually static, therefore are completely localized at $x_1$ and $x_2$. This approximation can be formulated in the following way,
\begin{equation}
H^{(0)}=\int [dk]\sqrt{k^2+m^2} a^\dagger(k)a(k)-\lambda \sum_{i=1,2}\phi^{(-)}(x_i)\phi^{(+)}(x_i)
.\end{equation}
A nice way to attack this problem is to utilize a discrete version of the orthofermion algebra as it was done in \cite{caglar-teo}, as a result one finds the following $\Phi_{ij}(\mu)$ operator, where we wrote
$-m<\mu(|x_1-x_2|)<m$ for the binding energy, while emphasizing its dependence on the distance between the two centers:
\begin{equation}
\Phi(\mu(|z|))=\begin{cases} {1\over \lambda}-{1\over 2}\int_{-\infty}^\infty [dp]\frac{1}{(\sqrt{p^2+m^2})(\sqrt{p^2+m^2}-\mu(|z|))}& \mbox{if } i=j\\ -{1\over 2}\int_{-\infty}^\infty [dp] \frac{e^{ipz}}{\sqrt{p^2+m^2}(\sqrt{p^2+m^2}-\mu(|z|))}& \mbox{if } i\neq j \end{cases}
,\end{equation}
where we set $z=x_1-x_2$. To find the ground state energy we need to study the solutions of the $zero$ eigenvalues of the matrix $\Phi_{ij}$, and similar to the nonrelativistic case, one finds that the symmetric solution corresponds to lower energy, which, in turn corresponds to the solution of the following equation:
\begin{equation}
\frac1{\lambda} -\int_{-\infty}^\infty {[dp]\over 2} \frac{1}{\sqrt{p^2+m^2}(\sqrt{p^2+m^2}-\mu(|z|))} =\int_{-\infty}^\infty {[dp]\over 2} \frac{e^{ipz}}{\sqrt{p^2+m^2}(\sqrt{p^2+m^2}-\mu(|z|))}
.\end{equation}
As we will see shortly, the binding energy decreases as we increase the distance in this case, hence the most strongly bound case, that is, the true ground state enery is found by setting $z=0$.
This corresponds to the binding of the two heavy particles, a kind of molecule formation. It is possible to get this zeroth order solution as,
\begin{equation}
\frac{1}{\lambda}-\int_{-\infty}^\infty {[dp]\over 2}\frac{1}{(\sqrt{p^2+m^2})(\sqrt{p^2+m^2}-\mu_0)}=\int_{-\infty}^\infty{[ dp]\over 2}\frac{1}{(\sqrt{p^2+m^2})(\sqrt{p^2+m^2}-\mu_0)}
.\end{equation}
Let us cast this expression into something that we can work with. To this purpose, we use a
Feynman parametrization,
\begin{equation}
\int_{-\infty}^\infty [dp]\frac{1}{(\sqrt{p^2+m^2})(\sqrt{p^2+m^2}-\mu_0)} =\int_0^1 du\int_{-\infty}^\infty [dp]\frac{1}{(\sqrt{p^2+m^2}-\mu_0 u)^2}
,\end{equation}
then we employ an exponentiation trick,
\begin{equation}
\frac{1}{(\sqrt{p^2+m^2}-\mu_0 u)^2} =\int_0^\infty dt\, t\, e^{-t(\sqrt{p^2+m^2}-\mu_0 u)}
,\end{equation}
and finally,
to calculate the resulting integral, we resort to the subordination identity, given by
\begin{equation}
e^{-t\sqrt{p^2+m^2}}=\frac{1}{2\sqrt{\pi}}\int_0^\infty ds \frac{t}{s^{\frac3{2}}}e^{-s(p^2+m^2)-\frac{t^2}{4s}}
.\end{equation}
Collecting all these manipulations, we find,
\begin{equation}
\int_{-\infty}^\infty [dp] \frac{1}{\sqrt{p^2+m^2}(\sqrt{p^2+m^2}-\mu_0 )}= \frac1{2\sqrt{\pi}} \int_0^1 du\int_0^\infty dt t^2e^{ut\mu_0}\int_0^\infty ds\frac1{s^{\frac3{2}}}e^{-m^2s-\frac{t^2}{4s}}\int_{-\infty}^\infty [dp] e^{-sp^2},
\end{equation}
after performing the momentum integral we end up with,
\begin{equation}
\int_{-\infty}^\infty [dp] \frac{1}{\sqrt{p^2+m^2}(\sqrt{p^2+m^2}-\mu_0 )} =\frac{1}{4\pi}\int_0^1 du \int_0^\infty dt t^2e^{ut\mu_0}\int_0^\infty ds \frac{1}{s^2}e^{-sm^2-\frac{t^2}{4s}}.
\end{equation}
Now we recall the well-known formula for the modified Bessel functions (for most of the standart functions and integrals thereof we use \cite{gradsh}),
\begin{equation}
K_{\nu}(w) =\frac1{2}\left(\frac{w}{2}\right)^\nu\int_0^\infty ds \frac{e^{-s-\frac{w^2}{4s}}}{s^{\nu+1}}
.\end{equation}
Using this we simplify our expression into,
\begin{equation}
\int_{-\infty}^\infty [dp] \frac{1}{\sqrt{p^2+m^2}(\sqrt{p^2+m^2}-\mu_0 )} ={m\over \pi}\int_0^1 du\int_0^\infty dt t e^{ut\mu_0}K_1(mt)= \frac{m}{\pi \mu_0}\int_0^\infty dt(e^{\mu_0 t}-1)K_1(mt).
\end{equation}
We employ the identity,
\begin{equation}
K_1(mt) =-\frac1{m}\frac{\partial K_0(mt)}{\partial t}
\end{equation}
to rewrite this expression as,
\begin{equation}
\int_0^\infty dt K_1(mt)(e^{\mu_0 t}-1) =\frac1{m} (1-e^{\mu_0 t})K_0(mt)|_0^\infty +\frac{\mu_0}{m}\int_0^\infty dt e^{\mu_0 t}K_0(mt).
\end{equation}
Note that,
\begin{equation}
(e^{\mu_0 t}-1)K_0(mt)|_0^\infty = 0
,\end{equation}
as long as $\mu_0<m$,
so we can calculate the remaining integral, and find,
\begin{equation}
\int_0^\infty dte^{\mu_0 t}K_0(mt) =\frac{ \arccos(-\frac{\mu_0}{m})}{\sqrt{m^2-\mu_0^2}}
.\end{equation}
So finally, we have,
\begin{equation}
\int_{-\infty}^\infty [dp] \frac1{\sqrt{p^2+m^2}(\sqrt{p^2+m^2}-\mu_0 )} = {1\over \pi} \frac{ \arccos(-\frac{\mu_0}{m})}{\sqrt{m^2-\mu_0^2}},
\end{equation}
or inserting this expression back into the eigenvalue equation, we have for $\mu_0$ the equation,
\begin{equation}
\frac1{\lambda} ={1\over \pi}\frac{ \arccos(-\frac{\mu_0}{m})}{\sqrt{m^2-\mu_0^2}}
.\end{equation}
This equation will always have a solution for any given $\lambda>0$, up to $\lambda\mapsto (\pi m)^{-}$ which actually leads to $\mu_0\mapsto -m^+$. We may consider $\lambda=2m$ as the critical value, since $\mu_0\mapsto 0^+$ corresponds to this critical value.
Let us now obtain the effective potential generated by a small separation of the heavy particles, that is, we assume that
$ z\ne 0$, yet it is small, in a sense to be made more precise later on.
This corresponds to the idea that the light degrees of freedom is averaged out assuming some arbitrary (yet small) $z$ in the background. Hence we should write an effective Hamiltonian for this case again as in the nonrelativistic version, or equivalently an effective Schr\"odinger equation,
\begin{equation}
\Big[-\frac{1}{2M}\sum_{i}{\nabla_i}^2+\mu(|x_1-x_2|)\Big]\psi(x_1,x_2)=E\psi(x_1,x_2)
.\end{equation}
To find the dynamics of the heavy system we need the effective potential term in between.
Assuming in our solution $|z|$ remains small, so we also get a small correction to $\mu_0$, we will expand this term. In order to do this, we write the general case,
\begin{equation}
\frac{1}{\lambda} -\int_{-\infty}^\infty {[dp]\over 2} \frac1{\sqrt{p^2+m^2}(\sqrt{p^2+m^2}-(\mu_0+\delta\mu))} =\int_{-\infty}^\infty {[dp]\over 2} \frac{e^{ipz}}{\sqrt{p^2+m^2}(\sqrt{p^2+m^2}-(\mu_0+\delta\mu(|z|)))},
\end{equation}
or we can write it as,
\begin{equation}
\frac{1}{\lambda}-\frac{1}{2\pi}\frac{\arccos(-\frac{(\mu_0+\delta\mu)}{m})}{\sqrt{m^2-(\mu_0+\delta\mu)^2}} =\int_{-\infty}^\infty {[dp]\over 2} \frac{e^{ipz}}{\sqrt{p^2+m^2}(\sqrt{p^2+m^2}-(\mu_0+\delta\mu))}
.\end{equation}
By going through similar steps as before for the righthand side, we end up with,
\begin{eqnarray}
\int_{-\infty}^\infty [dp] \frac{e^{ipz}}{\sqrt{p^2+m^2}(\sqrt{p^2+m^2}-(\mu_0+\delta\mu))}&=&
\frac{1}{4\pi}\int_0^1 du\int_0^\infty dt t^2e^{ut(\mu_0+\delta\mu)}\int_0^\infty ds\frac{e^{-m^2s-\frac{(t^2+z^2)}{4s}}}{s^2} \nonumber\\
&=&
\frac{m}{(\mu_0+\delta\mu)\pi}\int_0^\infty dtt \frac{(e^{t(\mu_0+\delta\mu)}-1)K_1(m\sqrt{t^2+z^2})}{\sqrt{t^2+z^2}}\nonumber
\end{eqnarray}
We now use the relationship between $K_1(Z)$ and $K_0(Z) $, we notice that,
$$
\frac{tK_1(m\sqrt{t^2+z^2})}{\sqrt{t^2+z^2}}=-\frac1{m}\frac{\partial K_0(m\sqrt{t^2+z^2})}{\partial t}
$$
hence,
\begin{equation}
\int_{-\infty}^\infty [dp]\frac{e^{ipz}}{\sqrt{p^2+m^2}(\sqrt{p^2+m^2}-(\mu_0+\delta\mu))}={1\over \pi}\int_0^\infty dte^{t(\mu_0+\delta\mu)}K_0(m\sqrt{t^2+z^2}).
\end{equation}
This finally leads to the result
\begin{equation}
\frac1{\lambda} -{1\over 2\pi}\frac{\arccos(-\frac{(\mu_0+\delta\mu)}{m})}{\sqrt{m^2-(\mu_0+\delta\mu)^2}}={1\over2\pi}\int_0^\infty dte^{t(\mu_0+\delta\mu)}K_0(m\sqrt{t^2+z^2})
\end{equation}
By expanding the energy equation in $\delta \mu$,
\begin{equation}
\frac{1}{\lambda}-\frac1{2\pi}\frac{\arccos\left(-\frac{\mu_0}{m}\right)}{\sqrt{m^2-\mu_0^2}}-\frac{\delta\mu}{2\pi}\left(\mu_0\frac{\arccos\left(-\frac{\mu_0}{m}\right)}{(m^2-\mu_0^2)^\frac{3}{2}}+\frac1{m^2-\mu_0^2}\right)=\frac1{2\pi}\int_0^\infty dt e^{t(\mu_0+\delta\mu)}K_0(m\sqrt{t^2+z^2})
,\end{equation}
or equivalently,
\begin{equation}
\frac{1}{\lambda}-\frac{1}{\pi}\frac{\arccos\left(-\frac{\mu_0}{m}\right)}{\sqrt{m^2-\mu_0^2}}-\frac{\delta\mu}{m^2-\mu_0^2}\left({\mu_0\over 2\lambda}+{1\over 2\pi}\right)=\frac1{2\pi}\int_0^\infty dt [e^{t(\mu_0+\delta\mu)}K_0(m\sqrt{t^2+z^2})-e^{t\mu_0}K_0(mt)]
.\end{equation}
By also expanding the $\delta\mu$ term on the right side, we have,
\begin{equation}
\delta\mu\int_0^\infty dt te^{t\mu_0}K_0(m\sqrt{t^2+z^2}) \rightarrow \delta\mu \int_0^\infty dtte^{t\mu_0}K_0(mt)=\delta\mu\frac{\partial}{\partial\mu_0}\int_0^\infty dt e^{t\mu_0}K_0(mt).
\end{equation}
And we notice that
\begin{equation}
\delta\mu\frac{\partial}{\partial\mu_0}\int_0^\infty dt e^{t\mu_0}K_0(mt)=\delta\mu\left[\mu_0\frac{\arccos\left(-\frac{\mu_0}{m}\right)}{(m^2-\mu_0^2)^\frac{3}{2}}+\frac1{m^2-\mu_0^2}\right]
.\end{equation}
As a result we end up with,
\begin{equation}
\frac{1}{\lambda}-\frac{1}{\pi}\frac{\arccos\left(-\frac{\mu_0}{m}\right)}{\sqrt{m^2-\mu_0^2}}-\frac{\delta\mu}{m^2-\mu_0^2}\left({\mu_0\over \lambda}+{1\over \pi}\right)=\frac1{2\pi}\int_0^\infty dt e^{t\mu_0}[K_0(m\sqrt{t^2+z^2})-K_0(mt)]
.\end{equation}
We recognize that the first two terms correpond to the equation defining $\mu_0$, moreover we may compute the left side by expanding in $z$, assuming the wave function corresponding to this potential keeps $z$ small, we find,
\begin{equation}
-\frac{\delta\mu}{m^2-\mu_0^2}\left({\mu_0\over \lambda}+{1\over \pi}\right)=-{1\over 4}|z|
\end{equation}
or we find the first order effective potential term, to be used in the Schr\"odinger equation, as,
\begin{equation}
\delta\mu(z)=\frac{|z|}{4}\frac{(m^2-\mu_0^2)\lambda}{\left(\mu_0+\frac{1}{\pi}\lambda\right)}
.\end{equation}
We relegate the derivation of this term in Appendix-II not to distract the reader from the main line of argument.
Thus, we arrive the Schr\"odinger equation, separating the center of mass motion again and using the wave function for the relative coordinate, reduced mass being equal to $M/2$, gives us,
\begin{equation}
\Big[-\frac{1}{M}{\partial^2\over \partial z^2}+\frac{|z|}{4}\frac{(m^2-\mu_0^2)\lambda}{\left(\mu_0+\frac{1}{\pi}\lambda\right)}\Big]\psi(z)=(E-{Q^2\over 4M})\psi(z)=\delta E \psi(z)
.\end{equation}
The rest of the argument goes similar to the nonrelativistic case.
\section{Once Again A Many Body View }
In this section, we will approach the same problem directly from the many-body formulation, hoping that this will give us some more insight on the Born-Oppenheimer approximation. Let us write down the resulting principal operator $\Phi$ in this case, again introducing the orthofermion Fock space as in the nonrelativistic situation. For the sake of brevity, we skip details. We use momentum space description and use the fact that there is a single relativistic particle and two heavy particles, hence after normal ordering $\Phi$ acts on no-light particle, one orthofermion plus one heavy particle Fock state. This means we have,
\begin{eqnarray}
\Phi &=& \frac1{\lambda}\Pi_1 -\int [dp dq]\chi^{\dagger}(p+q)\frac1{2\sqrt{q^2+m^2}[\sqrt{q^2+m^2}+H_0+\frac{p^2}{2M}-(\mu_0+\delta' E)]}\chi(p+q)
\nonumber\\
&\ &-\int [dpdqdr]\chi^{\dagger}(p)\Psi^{\dagger}(r-q)\frac1{2\sqrt{q^2+m^2}[{r^2\over 2M}+{p^2\over 2M}+\sqrt{q^2+m^2}-(\mu_0+\delta' E)]}\Psi(p-q)\chi(r)\nonumber
,\end{eqnarray}
where we wrote for the energy, $E$, an expansion $E=\mu_0+\delta' E$ anticipating higher order terms, and for the potential part we dropped $H_0$, here $H_0$ only refering to the heavy particle free Hamiltonian, since there are no heavy particles left after normal ordering of the potential part, acting on the special sector that we are interested in.
Let us first understand the operator which only contains orthofermion creation and annihilation operators, by using Feynmann parametrization, exponentiation and subordination consecutively,
\begin{equation}
\int [dpdq]\chi^{\dagger}(p+q)\frac1{\sqrt{q^2+m^2}[\sqrt{q^2+m^2}+H_0+\frac{p^2}{2M}-(\mu_0+\delta' E)]}\chi(p+q)=\nonumber
\end{equation}
$$
\frac1{2\sqrt{\pi}}\int [dpdq]\chi^{\dagger}(p+q)\int_0^1 du\int_0^\infty dt t^2e^{-ut(H_0+\frac{p^2}{2M}-(\mu_0+\delta' E))}\int_0^\infty ds \frac{e^{-s(m^2+q^2)-\frac{t^2}{4s}}}{s^\frac3{2}}\chi(p+q)
.$$
Let us define new coordinates as
$$ p+q= R$$and $$p-\alpha q=Q$$ with some $\alpha$ and require that we can write the sum of the energy expressions as a sum of two squares,
\begin{equation}
\frac{tup^2}{2M}+sq^2 =A(p+q)^2+B(p-\alpha q)^2
.\end{equation}
This allows us to calculate these unkown constants $\alpha, A$ and $B$ as,
\begin{equation}
\alpha =\frac{2Ms}{tu}
,\quad
A= \frac{stu}{2M}\left(\frac1{\frac{tu}{2M}+s}\right)\quad {\rm and} \quad
B=\left(\frac{tu}{2M}\right)^2\left(\frac1{\frac{tu}{2M}+s}\right)
.\end{equation}
As a result the Jacobian of this transformation becomes,
\begin{equation}
[dpdq]=\left(\frac{tu}{2M}\right)\left(\frac1{\frac{tu}{2M}+s}\right)[dRdQ]
.\end{equation}
Subsequently, we can rewrite this part of the operator $\Phi$ again as,
\begin{eqnarray}
&\ &\!\!\!\!\!\!\!\!\int [dpdq]\chi^{\dagger}(p+q)
\frac1{\sqrt{q^2+m^2}[\sqrt{q^2+m^2}+H_0 +\frac{p^2}{2M} -(\mu_0+\delta' E)]}\chi(p+q)\nonumber\\
&\ &\quad =\frac1{2\sqrt{\pi}}\int [dR]\chi^{\dagger}(R)e^{-AR^2}\int_0^1 du \int_0^\infty dt\, t^2 e^{-ut[H_0-(\mu_0+\delta' E)]}\nonumber\\
&\ &\quad \quad \quad \quad \quad \quad \quad \quad \times\int_0^\infty ds\frac{e^{-sm^2-\frac{t^2}{4s}}}{s^{\frac3{2}}}\int [dQ]\left(\frac{tu}{2M}\right)\frac{e^{-BQ^2}}{\left(\frac{tu}{2M}+s\right)}\chi(R)
.\end{eqnarray}
One can evaluate $dQ$-integral and this expression takes the form,
\begin{equation}\label{key1}
\frac{1}{4\pi}\int [dR]\chi^{\dagger}(R)e^{-AR^2}\int_0^1 du \int_0^\infty dt t^2 e^{-ut(H_0-(\mu_0+\delta' E))}\int_0^\infty ds \frac{e^{-sm^2-\frac{t^2}{4s}}}{s^{\frac3{2}}\sqrt{\frac{tu}{2M}+s}}\chi(R).
\end{equation}
Here $M$ is very large compared to $m$. Therefore we expect that some of the terms here are of lower order hence can be neglected. To be precise, $tu/M$ terms can be neglected in comparison to $s$ terms to leading order. To make sense of this claim we need to scale $s$ by $1/m^2$ to get dimensionless variables, and also $t$ by $1/m$. This argument is given in Appendix-III, after neglecting this term, the result becomes,
\begin{eqnarray}
&\ &\frac1{4\pi}\int [dR]\chi^{\dagger}(R)\int_0^1 du \int_0^\infty dt t^2e^{-ut(H_0+\frac{R^2}{2M}-(\mu_0+\delta'E))}\int_0^\infty ds\frac{e^{-sm^2-\frac{t^2}{4s}}}{s^2}\chi(R) \nonumber\\
&\ &\quad\quad \quad =-{m\over\pi} \int [dR]\chi^{\dagger}(R)\int_0^\infty dt\frac{ (e^{-t(H_0+\frac{R^2}{2M}-(\mu_0+\delta' E))}-1)}{(H_0+\frac{R^2}{2M}-(\mu_0+\delta' E))}K_1(mt)\chi(R)\nonumber\\
&\ &\quad \quad \quad ={1\over \pi}\int [dR]\chi^{\dagger}(R)\frac{\arccos(\frac{H_0+\frac{R^2}{2M}-\delta' E-\mu_0}{m})}{\sqrt{m^2-(H_0+\frac{R^2}{2M}-\delta' E-\mu_0)^2}}\chi(R)
\end{eqnarray}
Here, we changed the first exponential term due to the replacement,
\begin{equation}
A =\left(\frac{stu}{2M}\right)\left(\frac1{\frac{tu}{2M}+s}\right)\rightarrow \frac{tu}{2M}
.\end{equation}
So finally, the operator $\Phi$ takes the form,
\begin{eqnarray}
\Phi=\frac1{\lambda}\Pi_1&-&\!\!\!\!\!\frac1{2\pi}\int [dR]\chi^\dagger(R)\frac{\arccos\left(\frac{\frac{R^2}{2M}+H_0-\mu_0-\delta' E}{m}\right)}{\sqrt{m^2-\left(\frac{R^2}{2M}+H_0-\mu_0-\delta' E\right)^2}}\chi(R)\nonumber\\
&-&\!\!\!\!\int [dpdqdr]\chi^\dagger(p)\Psi^\dagger(r-q)\frac1{2\sqrt{q^2+m^2}(\sqrt{q^2+m^2}-\mu_0 +{r^2\over 2M}+{p^2\over 2M}-\delta' E)}\Psi(p-q)\chi(r)\nonumber
\end{eqnarray}
We now note that the "kinetic" part of $\Phi$ operator can be expanded, assuming all the kinetic energy contributions of the heavy degrees of freedom are small compared to the binding energy $m-\mu_0$, thus,
\begin{eqnarray}
&\ &\frac{\arccos\left(\frac{\frac{R^2}{2M}+H_0-\mu_0-\delta\mu}{m}\right)}{\sqrt{m^2-\left(\frac{R^2}{2M}+H_0-\mu_0-\delta' E\right)^2}}=\left(\arccos\left(-\frac{\mu_0}{m}\right)-\frac{H_0+\frac{R^2}{2M}-\delta' E}{\sqrt{m^2-\mu_0^2}}\right)\nonumber\\
&\ & \quad \quad\qquad \qquad \qquad\qquad\qquad \qquad \times\frac{1}{\sqrt{m^2-\mu_0^2}}\left(1-\mu_0\frac{(H_0+\frac{R^2}{2M}-\delta' E)}{m^2-\mu_0^2}\right)+...\nonumber
\end{eqnarray}
We may now further expand the small part of the potential term,
\begin{eqnarray}
&\ &\int [dpdqdr]\chi^\dagger(p)\Psi^\dagger(r-q)\frac1{2\sqrt{q^2+m^2}(\sqrt{q^2+m^2}-\mu_0 +{r^2\over 2M}+{p^2\over 2M}-\delta' E)}\Psi(p-q)\chi(r)\nonumber\\
&\ &=\int [dpdqdr]\chi^\dagger(p)\Psi^\dagger(r-q)\frac1{2\sqrt{q^2+m^2}(\sqrt{q^2+m^2}-\mu_0 )}\Psi(p-q)\chi(r)\nonumber\\
&\ &-\int [dpdqdr]\chi^\dagger(p)\Psi^\dagger(r-q)\frac{({r^2\over 2M}+{p^2\over 2M}-\delta' E)}{2\sqrt{q^2+m^2}(\sqrt{q^2+m^2}-\mu_0 )^2}\Psi(p-q)\chi(r)+...
\end{eqnarray}
This leads to an expansion of $\Phi(E)$,
\begin{eqnarray}
\Phi&=&\!\!\!\left[\frac1{\lambda}-\frac1{2\pi}\frac{\arccos\left(-\frac{\mu_0}{m}\right)}{\sqrt{m^2-\mu_0^2}}\right]\Pi_1\nonumber\\
&+&\!\!\!\frac1{2\pi}\left(\mu_0\frac{\arccos\left(-\frac{\mu_0}{m}\right)}{(m^2-\mu_0^2)^\frac{3}{2}}+\frac1{m^2-\mu_0^2}\right)\int [dR]\chi^\dagger(R)\left(H_0+\frac{R^2}{2M}-\delta'E\right)\chi(R)\nonumber\\
&-&\!\!\!{1\over 2}\int [dpdqdr]\chi^\dagger(p)\Psi^\dagger(r-q)\frac1{\sqrt{q^2+m^2}(\sqrt{q^2+m^2}-\mu_0)}\Psi(p-q)\chi(r)\nonumber\\
&+&\!\!\!{1\over 2}\int [dpdqdr]\chi^\dagger(p)\Psi^\dagger(r-q)\frac{({r^2\over 2M}+{p^2\over 2M}-\delta' E)}{\sqrt{q^2+m^2}(\sqrt{q^2+m^2}-\mu_0 )^2}\Psi(p-q)\chi(r)+...\nonumber
.\end{eqnarray}
Let's define a particular wave function as our ansatz for the solution,
$$
|\omega>=\int [dQd\xi]e^{-iQX}f(\xi)\chi^\dagger(Q/2+\xi)\Psi^\dagger(Q/2-\xi)|\Omega>
,$$
and demand as before
\begin{equation}
\Phi(\mu_0+\delta' E) |\omega>=0
.\end{equation}
When we solve this equation order by order, we will find $f$ and the true energy of the system. Going through the same discussion as in the nonrelativistic case this problem can be reduced to an eigenfunction and eigenvalue equation for $f$. Again we assume $f$ is symmetric, and the last part of the potential term again can be turned into a convolution in two different ways, which corresponds to the ordering ambiguity. We will spare the details since they are very similar to the previous case. The only thing we should find is the corrected potential corresponding to the inverse Fourier transform of
\begin{equation}
\frac{1}{\sqrt{q^2+m^2}(\sqrt{q^2+m^2}-\mu_0 )^2}
.\end{equation}
Going through exactly the same steps as before, using a Feynmann parametrization and an exponentiation, recognizing the modified Bessel function inside,
we obtain the inverse Fourier transform as,
\begin{eqnarray}
\int_{-\infty}^\infty [dp] \frac{e^{ipz}}{\sqrt{p^2+m^2}(\sqrt{p^2+m^2}-\mu_0)^2}&=&
\frac{1}{4\pi}\int_0^1 u du\int_0^\infty dt t^3e^{ut\mu_0}\int_0^\infty ds\frac{e^{-m^2s-\frac{(t^2+z^2)}{4s}}}{s^2} \nonumber\\
&=&
\frac{m}{\pi\mu_0}\int_0^1 u du\int_0^\infty dtt^3 \frac{e^{ut\mu_0}K_1(m\sqrt{t^2+z^2})}{\sqrt{t^2+z^2}}\nonumber\\
&=&\frac{m}{\pi \mu_0}{\partial \over \partial \mu_0}\int_0^1 du\int_0^\infty dtt^2 \frac{e^{ut\mu_0}K_1(m\sqrt{t^2+z^2})}{\sqrt{t^2+z^2}}\nonumber\\
&=&{1 \over \pi}{\partial\over \partial \mu_0}\int_0^\infty dt e^{\mu_0 t} K_0(m\sqrt{t^2+z^2})\nonumber
\end{eqnarray}
Inserting all these expressions back, we end up with an eigenvalue equation,
\begin{eqnarray}
&\ & \left(\frac1{\lambda}-\frac1{2\pi}\frac{\arccos\left(-\frac{\mu_0}{m}\right)}{\sqrt{m^2-\mu_0^2}}\right)f(z)+\frac1{2\pi}\left(\mu_0\frac{\arccos\left(-\frac{\mu_0}{m}\right)}{(m^2-\mu_0^2)^\frac{3}{2}}+\frac1{m^2-\mu_0^2}\right)\left(-\frac{\nabla^2_z}{M}+\frac1{2}\frac{Q^2}{2M}-\delta' E\right)f(z)\nonumber\\
&\ & -{1\over 2\pi}\left(\int_0^\infty dt e^{\mu_0 t} K_0(m\sqrt{t^2+z^2} )\right) f(z)\nonumber\\
&\ & +{1\over 2}\left[-\frac{\nabla^2_z}{M}+\frac1{2}\frac{Q^2}{2M}-\delta' E \, , {1\over 2\pi}{\partial\over \partial \mu_0}\int_0^\infty dt e^{\mu_0 t} K_0(m\sqrt{t^2+z^2})\right]_+ f(z)=0\nonumber
\end{eqnarray}
We now introduce the small parameter $\delta E=\delta' E-{Q^2\over 4M}$, and expanding the third potential term up to $|z|$, and keeping only the constant contribution from the last anticommutator, dropping all the derivative corrections to this order, we have,
\begin{eqnarray}
&\ & \left(\frac1{\lambda}-\frac{1}{\pi}\frac{\arccos\left(-\frac{\mu_0}{m}\right)}{\sqrt{m^2-\mu_0^2}}\right)f(z)\nonumber\\
&\ & \ \ \ +\frac{1}{\pi}\left(\mu_0\frac{\arccos\left(-\frac{\mu_0}{m}\right)}{(m^2-\mu_0^2)^\frac{3}{2}}+\frac1{m^2-\mu_0^2}\right)\left(-\frac{\nabla^2_z}{M}-\delta E\right)f(z)
+{1\over 4} |z|f(z)=0\nonumber
.\end{eqnarray}
If we now look at the zeroth order solution, we find as expected the equation for $\mu_0$ as before, but that does not determine $f$.
Assuming $\mu_0$ solves the zeroth order equation, thus removing the constant multiple of $f$ in the equation, the next order equation becomes,
$$
\left(-\frac{\nabla^2_z}{M}+\frac1{2}\frac{Q^2}{2M}-\delta E\right)f(z)+\frac1{4}\frac{(m^2-\mu_0^2)}{(\mu_0+\frac{1}{\pi}\lambda)}\lambda|z|f(z)=0
,$$
which is exactly as before.
Nevertheless, we remark that this picture again allows us to go one step further and obtain the second order corrections to the energy.
We leave this more cumbersome calculation to the reader, which can be found exactly following the previous part.
\section{Conclusions}
We show that for a simple model which consists of two heavy particles interacting with a light particle through attractive delta function potentials in one dimension, we can apply the Born-Oppenheimer approximation to understand the spectrum. The simplicity of the model allows us to test the validity of the Born-Oppenheimer approximation self-consistently. A novel approach, originally proposed by Rajeev for interacting bosons in two dimensions can also be applied to this problem not only to recover the previous results but also obtain in a more systematic way higher order corrections. It turns out that for this system the relevant expansion paramater becomes $(m/M)^{1/3}$. A modification of this problem where the light particle is very small yet the binding is not so strong to cause pair creation is also proposed. Along similar lines this problem is discussed. The many body perspective can be adapted to this problem as well and we can again recover the Born-Oppenheimer result from this point of view.
\section{Appendix-I}
In this appendix we present the error estimates for the Born-Oppenheimer Approximation.
In order to do this we first write down the
the wave function normalization for the light degree of freedom.
Let us recall that the wave function we found is given by,
\begin{equation}
\phi(x|x_1,x_2) =\frac{N}{\hbar}\left[\int_0^\infty dt K_t(x,x_1)e^{-\frac{\nu^2}{\hbar}t}+\int_0^\infty dt K_t(x,x_2)e^{-\frac{\nu^2}{\hbar}t}\right]
.\end{equation}
We demand the normalization,
$$
\int_{-\infty}^\infty dx |\phi(x|x_1,x_2)|^2= 1
$$
The convolution property of the heat kernel,
\begin{equation}
\int dx K_{t_1}(x,x_1)K_{t_2}(x,x_2)=K_{t_1+t_2}(x_1,x_2)
\end{equation}
simplifies the calculations, giving us the condition,
\begin{equation}
1=2\frac{N^2}{\hbar^2}\left[\int dt_1 dt_2 K_{t_1+t_2}(x_1,x_1)e^{-\frac{\nu^2}{\hbar} (t_1+t_2)}+\int dt_1 dt_2K_{t_1+t_2}(x_1,x_2)e^{-\frac{\nu^2}{\hbar}(t_1+t_2)} \right]
.\end{equation}
Defining new parameters,
\begin{equation}
t=t_1+t_2, \quad
{\rm and} \quad
s=t_1-t_2
,\end{equation}
and executing the $s$-integrals,
$$
1=\frac{N^2}{\hbar^2}\left[\int_0^\infty dt\, t [K_t(x_1,x_1)+K_t(x_2,x_2)]e^{-\frac{\nu^2}{\hbar}t}+2\int_0^\infty dt\, t K_t(x_1,x_2)e^{-\frac{\nu^2}{\hbar}t}\right].
$$
The integrals can be calculated giving us the condition,
$$
1=\frac{N^2}{2}\frac{\sqrt{2m}}{\hbar\nu^3}\left[1+\left[1+\frac{\sqrt{2m}}{\hbar}\nu|x_1-x_2|\right]e^{-\frac{\sqrt{2m}}{\hbar}\nu|x_1-x_2|}\right].
$$
As a result the normalized light particle wave function becomes,
\begin{equation}
\phi(x|x_1,x_2) =\frac{\sqrt{\nu}}{\sqrt{2}} \frac{(2m)^\frac1{4}}{\sqrt{\hbar}}\frac1{\sqrt{\left[1+\left[1+\frac{\sqrt{2m}}{\hbar}\nu|x_1-x_2|\right]e^{-\frac{\sqrt{2m}}{\hbar}\nu|x_1-x_2|}\right]}}\left[e^{-\frac{\sqrt{2m}}{\hbar}\nu|x-x_1|}+e^{-\frac{\sqrt{2m}}{\hbar}\nu|x-x_2|}\right]
.\end{equation}
Since the whole system is translationally invariant, it is more natural to define new coordinates,
\begin{equation}
X=\frac{x_1+x_2}{2}\quad
{\rm and }\quad
z=x_1-x_2
\end{equation}
and express the wave function in terms of these coordinates, moreover, to isolate the relevant contributions, we shift all the coordinates by $X$, which will not change $z$, yet will modify $x$ to $x+X$, this gives us,
\begin{equation}
\phi(x,z)=\frac1{\sqrt{2}} \frac{(2m)^\frac1{4}}{\sqrt{\hbar}}\sqrt{\nu}\frac1{\sqrt{\left[1+\left[1+\frac{\sqrt{2m}}{\hbar}\nu|z|\right]e^{-\frac{\sqrt{2m}}{\hbar}\nu|z|}\right]}}\left[e^{-\frac{\sqrt{2m}}{\hbar}\nu|x-\frac{z}{2}|}+e^{-\frac{\sqrt{2m}}{\hbar}\nu|x+\frac{z}{2}|}\right]
.\end{equation}
In general, the derivatives with respect to $X$ coordinates may give us a large contribution, but that is the total translational kinetic energy of the system, which is not interesting from our perspective. Therefore, this shift of coordinates
will be natural in order to remove such derivative contributions from our estimates. In principle, they can also be estimated but the results becomes more combersome.
Note that $\nu$ itself is a function of $z$, as we determined thorugh the eigenvalue equation previously,
\begin{equation}
{1\over \lambda} =\sqrt{m\over 2\hbar^2 \nu^2(z)}\Big(e^{-\sqrt{2m}\nu(z)/\hbar}+1\Big)
.\end{equation}
To simplify our estimates we rewrite the wave function in the following decomposed form,
\begin{equation}
\phi(z;x)= A(z)(\epsilon_+(x,z)+\epsilon_-(x,z)),
\end{equation}
where,
$$
\epsilon_+= e^{-\frac{\sqrt{2m}}{\hbar}\nu|x+\frac{z}{2}|}\quad{\rm and} \quad
\epsilon_-= e^{-\frac{\sqrt{2m}}{\hbar}\nu|x-\frac{z}{2}|}
$$
and $A(z)$ refers to the common multiplicative part. Let us recall that within the approximations we use, $\nu(z)$ can be factored as, $\nu^2(z)=\nu^2_0+\delta E_1(z)$, with
$$
\nu_0=\frac{\sqrt{2m}}{\hbar}\lambda \quad {\rm and} \quad
\delta E_1=-{|z|}\lambda^3 \Big(\frac{2m}{\hbar^2}\Big)^2
.$$
Let us expand the derivatives acting on the product wave function as,
\begin{equation}
\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right)\phi(x,z)\psi(z)=\psi(z)\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right)\phi(x,z)+\phi(x,z)\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right)\psi(z)
\end{equation}
$$
+2\frac{\partial \psi}{\partial x_1}\frac{\partial\phi}{\partial x_1}+2\frac{\partial \psi}{\partial x_2}\frac{\partial\phi}{\partial x_2}
.$$
We use only the following terms
$$
\psi(z)\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right)\phi(x,z)+2\frac{\partial \psi}{\partial x_1}\frac{\partial\phi}{\partial x_1}+2\frac{\partial \psi}{\partial x_2}\frac{\partial\phi}{\partial x_2}
$$
since the derivatives acting on the wave function $\psi(z)$ is precisely the heavy particle part that we use in our approximations. Let us note that,
$$
\frac{\partial}{\partial x_1}=\frac1{2}\frac{\partial}{\partial X}+\frac{\partial}{\partial z}\quad {\rm and} \quad
\frac{\partial}{\partial x_2}=\frac1{2}\frac{\partial}{\partial X}-\frac{\partial}{\partial z}
,$$
thus we need to estimate these two expressions,
\begin{equation}
2\psi(z)\frac{\partial^2}{\partial z^2}\phi(x,z)\quad {\rm and} \quad 4\frac{\partial\psi}{\partial z}\frac{\partial\phi}{\partial z}.
\end{equation}
We note that
$$
\frac{\partial \epsilon_+}{\partial z}=\frac{1}{2}\frac{\partial\epsilon_+}{\partial x}-\frac{\sqrt{2m}}{\hbar}{\partial \nu\over \partial z} |x+\frac{z}{2}|\epsilon_+,
$$
$$
\frac{\partial\epsilon_-}{\partial z}=-\frac{1}{2}\frac{\partial\epsilon_-}{\partial x}-\frac{\sqrt{2m}}{\hbar}{\partial \nu\over \partial z} |x-\frac{z}{2}|\epsilon_-.
$$
We show the consistency of our approximations by evaluating the average of all these terms with the presumed solutions of the heavy particles wave functions $\psi$.
We start with the second term,
$$
4\frac{\partial \phi}{\partial z}\frac{\partial\psi}{\partial z}=4\frac{\partial\psi}{\partial z}\left[\frac{\partial A}{\partial z}(\epsilon_++\epsilon_-)+A\left(\frac{1}{2}[\frac{\partial\epsilon_+}{\partial x}-\frac{\partial\epsilon_-}{\partial x}]-\frac{\sqrt{2m}}{\hbar}{\partial \nu\over \partial z}( |x-\frac{z}{2}|\epsilon_-+|x+\frac{z}{2}|\epsilon_+)\right)\right]
.$$
By taking the average of this term we estimate the contribution of this term we to the first order energy calculation we already have. We replace this derivative term in the Schr\"odinger equation by the expectation value,
\begin{eqnarray}
-\frac{4\hbar^2}{2\mu}\int dx dz \phi \psi \frac{\partial \phi}{\partial z}\frac{\partial\psi}{\partial z}&=&\!\!
-\frac{\hbar^2}{\mu}\Bigg[\int dz\psi\frac{\partial\psi}{\partial z}\int dx \left[2\frac{\partial A}{\partial z}(\epsilon_++\epsilon_-)+A\left(\frac{\partial\epsilon_+}{\partial x}-\frac{\partial\epsilon_-}{\partial x}\right)\right]A(\epsilon_++\epsilon_-)\nonumber\\
&-&\frac{\sqrt{2m}}{\hbar}{\partial \nu\over \partial z}( |x-\frac{z}{2}|\epsilon_-+|x+\frac{z}{2}|\epsilon_+) A^2(\epsilon_-+\epsilon_+)\Bigg]\nonumber\\
&=&-\frac{\hbar^2}{2\mu}\int dz\psi\frac{\partial \psi}{\partial z}\int dx \Bigg[4A\frac{\partial A}{\partial z}(\epsilon_++\epsilon_-)^2+2A^2\left(\epsilon_+\frac{\partial\epsilon_+}{\partial x}-\epsilon_-\frac{\partial\epsilon_-}{\partial x}\right)\nonumber\\
&\ &+2A^2\epsilon_+\epsilon_-\left(\frac1{\epsilon_+}\frac{\partial\epsilon_+}{\partial x}-\frac1{\epsilon_-}\frac{\partial \epsilon_-}{\partial x}\right)\nonumber\\
&\ &-2 A^2\frac{\sqrt{2m}}{\hbar}{\partial \nu\over \partial z}( |x-\frac{z}{2}|\epsilon_-+|x+\frac{z}{2}|\epsilon_+) (\epsilon_-+\epsilon_+)\Bigg]\nonumber
.\end{eqnarray}
To this purpose, we can use the first order result for the derivative of energy,
\begin{equation}
{\partial \nu\over \partial z}\approx -{1\over 2}{\rm sgn} (z) \Big({2m\over \hbar^2} \Big)^{3/2} \lambda^2
\end{equation}
Let us now look at the first term on the right side inside the integral, after $x$ integration it becomes,
\begin{equation}
(1)=-\frac{\hbar^2}{2\mu}4\int dz \psi \frac{\partial \psi}{\partial z}\frac{\partial}{\partial z}\ln A
.\end{equation}
The second term is an exact differential in $x$ and hence gives zero upon integration. The third term can be written as,
$$
(3)=-2\frac{\hbar^2}{2\mu}\int dxdz \psi\frac{\partial\psi}{\partial z}A^2\epsilon_+\epsilon_-\frac{\partial}{\partial x}\ln\left(\frac{\epsilon_+}{\epsilon_-}\right)
.$$
Last part contains various cross terms which we will estimate later.
Note that for the first term it suffices to use the leading nonzero term of $\ln A$'s derivative,
$$
\frac{\partial}{\partial z}\ln A \approx \frac{\partial }{\partial z}\ln\sqrt{\nu}\approx -\frac1{4}\lambda\frac{2m}{\hbar^2}{\rm sgn}(z).
$$
As a result it has an upper bound,
$$\Bigg|\frac{\hbar^2}{2\mu}\nu_0\frac{\sqrt{2m}}{\hbar}\int dz {\rm sgn}(z)\psi \frac{\partial\psi}{\partial z}\Bigg|\le \nu_0\sqrt{\frac{m}{\mu}}\left|\int dz \left|\frac{\hbar}{\sqrt{2\mu}}\frac{\partial\psi}{\partial z}\right|^2\right|^\frac1{2}\left|\int dz |{\rm sgn}(z)\psi|^2\right|^\frac{1}{2}\le \nu_0^2\left(\frac{m}{\mu}\right)^\frac{2}{3}
,$$
where we used the fact that the kinetic energy is less than the total energy, since the potential, being proportional to $|z|$, is positive everywhere.
Let us come to the second nonzero term we found,
$$
2\Bigg|\frac{\hbar^2}{2\mu}\frac{\sqrt{2m}}{\hbar}\nu_0\int dz \psi\frac{\partial\psi}{\partial z} A^2\int_{-\frac{|z|}{2}}^\frac{|z|}{2} dx e^{-\frac{\sqrt{2m}}{\hbar}\nu(|x+\frac{z}{2}|+|x-\frac{z}{2}|)}\frac{\partial}{\partial x}\left(|x+\frac{z}{2}|-|x-\frac{z}{2}|\right)\Bigg|
$$
$$
\le 16\frac{2m}{\hbar\sqrt{2\mu}}\nu_0^2\left|\int dz \left|\frac{\hbar}{\sqrt{2\mu}}\frac{\partial\psi}{\partial z}\right|^2\right|^\frac1{2}\left|\int dz z^2|\psi|^2\right|^\frac{1}{2}\approx \frac{2m}{\hbar\sqrt{2\mu}}\nu_0^3\left(\frac{m}{\mu}\right)^\frac1{6}\sqrt{<z^2>}\approx C\frac{m}{\mu}\nu_0^2
$$
here we restrict the range of $x$ integration to $[-{|z|\over 2},{|z|\over 2}]$ because outside of this interval the expression inside the derivative has no dependence on $x$ hence gives zero upon differentiation. Since we are in a finite interval the exponential term is replaced with its upper bound $1$, for this the absolute value of the derivative term ${\partial \psi\over \partial z}$ should be taken. Moreover, $A^2$ is replaced with its leading order constant value, since there is already $|z|$ term multiplying the whole expression as well as ${\partial \psi\over \partial z}$ term.
This leaves us with the last term, which contains various combinations like
\begin{equation}
\int dx\, \epsilon_-\epsilon_+|x-{z\over 2}|
.\end{equation}
We note that the absolute value of all such combinations are smaller than the following integral
\begin{equation}
2\int dx e^{-{\sqrt{2m} \over \hbar}\nu|x|} (|x|+|z|)
\end{equation}
as can be seen by using the $x\to -x$ transformation for the negative part of the real axis, replacing one of the exponentials by its upper limit $1$, afterwards shifting the integration variable, and
then extending again the integration region to the full real axis.
This can be used to estimate the full expression,
\begin{equation}
(4)\leq 8 {\hbar^2\over \mu} {\sqrt{2m}\over \hbar} \int dz A^2 \Bigg| {\partial \nu\over \partial z}{\partial \psi\over \partial z}\Bigg| \psi \int dx e^{-{\sqrt{2m} \over \hbar}\nu|x|} (|x|+|z|)
.\end{equation}
We will only estimate the $|x|$ term since it is easy to see that the $|z|$ term is even smaller.
Note that the $|x|$ part satisfies, to leading order,
\begin{equation}
(4')\leq 8 {\hbar\over \sqrt{\mu}} \Big({\sqrt{2m}\over \hbar}\Big)^{5}\nu_0 \Big({\sqrt{2m}\over \hbar}\nu_0\Big)^{-2}\lambda^2 \Big[ \int dz ({\rm sgn}(z)\psi)^2 \Big]^{1/2} \Big[ {\hbar^2\over 2\mu}\int dz \Big|{\partial \psi\over \partial z}\Big|^2\Big]^{1/2}
.\end{equation}
Here we replaced $A^2$ by its constant value to leading order, that means $\nu$ is replaced by $\nu_0$ to leading order everywhere.
Using $ {2m\over \hbar^2} \lambda^2=\nu_0^2$, we can reorganize these terms to find,
\begin{equation}
(4')\leq C_4 \Big({m\over \mu}\Big)^{2/3} \nu_0^2
\end{equation}
as desired.
Let us now evaluate
the average of the second order derivatives, $ [2\psi\frac{\partial^2}{\partial z^2}\phi]$, we divide this into a group of terms, since it is a long expression, we have the first group of terms, most conveniently written as averages,
$$
(1)={\hbar^2\over \mu} \int dx dz \psi^2 A (\epsilon_++\epsilon_-)\left[\frac{\partial^2A}{\partial z^2} (\epsilon_++\epsilon_-)+4\frac{\partial A}{\partial z}\left(\frac{\partial\epsilon_+}{\partial x}-\frac{\partial\epsilon_-}{\partial x}\right)+4A\left(\frac{\partial^2\epsilon_+}{\partial x^2}+\frac{\partial^2\epsilon_-}{\partial x^2}\right)\right]
.$$
Then we have a second group of terms after taking into account the repetitions, we also face with second order derivatives. These are hard to estimate when we have only first order expansions given by simple expressions. To overcome this before we do any approximations, we first integrate by parts, to reduce these more complicated terms into products of derivatives. As a result, there are some terms {\it coming from the integration by parts inside the average}, all of it put together amount to the following,
\begin{eqnarray}
(2)={\hbar^2\over \mu } \int dx dz \psi^2 A(\epsilon_++\epsilon_-){\sqrt{2m}\over \hbar} {\partial A\over \partial z} {\partial \nu\over \partial z} \Big[\epsilon_+|x+{z\over 2} |+\epsilon_-|x-{z\over 2}|\Big]
\end{eqnarray}
Similarly, we use {\it an integration by parts trick} to write an average of a group of terms
\begin{eqnarray}
(3)&=&{\hbar^2\over \mu } \int dx dz \psi^2 A^2\Big[({\partial \epsilon_+\over \partial x}-{\partial \epsilon_-\over \partial x})-{\sqrt{2m}\over \hbar} {\partial \nu\over \partial z}(\epsilon_+|x+{z\over 2} |+\epsilon_-|x-{z\over 2}|)\Big]\nonumber\\
&\ & \quad \quad \quad \quad \quad\quad \quad \quad \quad \quad \ \ \ \ \ \ \ \ \ \ \ \times {\sqrt{2m}\over \hbar} {\partial \nu\over \partial z} \Big[\epsilon_+|x+{z\over 2} |+\epsilon_-|x-{z\over 2}|\Big]
.\nonumber\end{eqnarray}
Integration by parts in the average produces one more term of the form
\begin{equation}
\int dx dz A^2 (\epsilon_++\epsilon_-) \psi {\partial \psi\over \partial z} {\sqrt{2m}\over \hbar} {\partial \nu\over \partial z} \Big[\epsilon_+|x+{z\over 2} |+\epsilon_-|x-{z\over 2}|\Big]
,\end{equation}
however this is exactly one of the terms we estimated on the cross-derivative terms above.
Let us now estimate the averages of the first group of terms,
$$
(1^a)=-2\frac{\hbar^2}{2\mu}\int dz |\psi|^2\frac1{A}\frac{\partial^2}{\partial z^2}A
$$
$$
(1^b)=-4\frac{\hbar^2}{2\mu}\int dxdz|\psi|^2\Big( \frac{\partial}{\partial z}A^2\Big)\Big(\epsilon_+\epsilon_-\frac{\partial}{\partial x}\ln\left(\frac{\epsilon_+}{\epsilon_-}\right)+{1\over 2}{\partial \over \partial x}( \epsilon_+^2+\epsilon_-^2 )
\Big)
$$
$$
(1^c)=-8\frac{\hbar^2}{2\mu}\int dxdz |\psi|^2 A^2\left(\frac{\partial^2\epsilon_+}{\partial x^2}+\frac{\partial^2\epsilon_-}{\partial x^2}\right)(\epsilon_++\epsilon_-)
.$$
Since it is hard to estimate the second derivative terms for normalization part, we note the identity,
$$
\frac1{A}\frac{\partial^2}{\partial z^2}A =\frac{\partial^2}{\partial z^2}\ln A+\left(\frac{\partial}{\partial z}\ln A\right)^2
$$
Let us now consider
the first integral expression $(1^a)$,
$$
-2\frac{\hbar^2}{2\mu}\int dz|\psi|^2\frac{\partial^2}{\partial z^2}\ln A=4\frac{\hbar^2}{2\mu}\int dz \psi\frac{\partial\psi}{\partial z}\frac{\partial\ln A}{\partial z}
$$
$$
\le\sqrt{\frac{m}{\mu}}\nu_0\left|\int dz\left|\frac{\hbar}{\sqrt{2\mu}}\frac{\partial\psi}{\partial z}\right|^2\right|^\frac1{2} \left|\int dz |\psi|^2\right|^\frac1{2}\approx \left(\frac{m}{\mu}\right)^\frac{2}{3}\nu_0^2
.$$
Moreover, the second term becomes,
$$
-2\frac{\hbar^2}{2\mu}\int dz |\psi|^2\left(\frac{\partial}{\partial z}\ln A\right)^2\approx\frac{m}{\mu}\nu_0^2.
$$
The second part $(1^b)$ has the following combination,
\begin{eqnarray}
4\frac{\hbar^2}{2\mu}\Bigg|\int dz|\psi|^2\Big( \frac{\partial}{\partial z}A^2\Big)\int^{|z|/2}_{-|z|/2} dx \epsilon_+\epsilon_-\frac{\partial}{\partial x}\ln\left(\frac{\epsilon_+}{\epsilon_-}\right)\Bigg|\le 8\frac{\hbar^2}{2\mu}\frac{(2m)^\frac{3}{2}}{\hbar^3}\nu_0^3\left[\int dz |z| |\psi|^2\right]\nonumber
,\end{eqnarray}
using $<|z|>=C {\lambda\over\nu_0^2}({m\over \mu})^{1/3}$, the above expression becomes,
$$
\le 8\frac{\hbar^2}{2\mu}\frac{(2m)^\frac{3}{2}}{\hbar^3}C {\lambda\over\nu_0^2}({m\over \mu})^{1/3}\nu_0^3 \approx \left(\frac{m}{\mu}\right)^\frac{4}{3}\nu_0^2.
$$
The expression for $(1^b)$ contains one more term of the form,
\begin{equation}
\int dx \Big( \epsilon_+{\partial \epsilon_+\over \partial x}+ \epsilon_-{\partial \epsilon_-\over \partial x}\Big)={1\over 2}\int dx {\partial \over \partial x}( \epsilon_+^2+\epsilon_-^2 ),
\end{equation}
which is a total derivative and integrates out to zero.
Let us now consider
the last term,
$$
(1^c)=-8\frac{\hbar^2}{2\mu}\int dxdz |\psi|^2 A^2\left(\frac{\partial^2\epsilon_+}{\partial x^2}+\frac{\partial^2\epsilon_-}{\partial x^2}\right)(\epsilon_++\epsilon_-)\approx C_8 \frac{m}{\mu}\nu_0^2,
$$
since
this integral is the average kinetic energy of the light particle multiplied with $\frac{m}{\mu}$.
The potential energy can be computed using the known wave function and shown to be less than
some multiple of $\nu^2$ easily, this gives the estimate on the kinetic energy.
Due to the factor in front we can replace it with its leading constant value.
Let us discuss the terms in $(2)$,
we have combinations of the form,
\begin{equation}
\int dx \epsilon_{\pm} \epsilon_\pm |x\pm {z\over 2}|
\end{equation}
using our previous estimate on these terms, combining $A{\partial A\over \partial z}$ into ${1\over 2} {\partial A^2\over \partial z}$, and using the leading order term for this derivative, which is given by ${\sqrt{2m}\over \hbar}{\partial \nu\over \partial z}$, and as we indicated before, the leading term for ${\partial \nu\over \partial z}\approx -{1\over 2}$sgn$(z)({\sqrt{2m}\over \hbar})^3\lambda^2$, combining all these estimates and identities, we find,
\begin{equation}
| (2)| \leq C'_9 {\hbar^2\over \mu} \int dz \psi^2\Big( {\sqrt{2m}\over \hbar}\Big)^{2} \Big[ \Big({\sqrt{2m}\over \hbar}\Big)^3\lambda^2\Big]^2 \Big({\sqrt{2m}\over \hbar}\nu\Big)^{-2}\leq C_9 {m\over \mu} \nu_0^2
.\end{equation}
Let us consider the first group of terms in our combination labeled as $(3)$, integration by parts in $x$ leads to
\begin{eqnarray}
\int dx dz \psi^2 A^2 (\epsilon_+-\epsilon_-){\sqrt{2m}\over \hbar}{\partial \nu\over \partial z} {\partial \over \partial x} (\epsilon_+|x+{z\over 2} |+\epsilon_-|x-{z\over 2}|)
.\end{eqnarray}
The last term in fact can be turned into $\epsilon_+|x+{z\over 2}|$ by using $z\to-z$ symmetry, and after that we can estimate the absolute value of each term.
Using, $|\epsilon_+-\epsilon_-|<1$, and evaluating the derivative ${\partial \over \partial x}$ acting on $\epsilon_+|x+{z\over 2}|$, and using a similar estimate as before for the funciton inside of the $x$-integral,
\begin{equation}
|(3^a)|\leq C'_{10} {\hbar^2 \over \mu} \int dz \Big({\sqrt{2m}\over \hbar}\Big)^2\nu \Big|{\partial \nu \over \partial z}\Big|({\sqrt{2m}\over \hbar} \nu) \int dx e^{-{\sqrt{2m}\over\hbar}\nu |x|} |x|\leq C_{10} {m\over \mu} \nu_0^2
.\end{equation}
Let us now look at the rest,
which contains integrals like
\begin{equation}
\int dx \epsilon_\pm \epsilon_\pm |x\pm {z\over 2}|^2
,\end{equation}
by similar arguments they are less than combinations
\begin{equation}
\int dx e^{-{\sqrt{2m}\over \hbar} }[ |x|^2+2|x||z|+|z|^2]
,\end{equation}
the leading term of which comes from $|x|^2$ integration.
Thus, to the leading order, we have an upper bound,
\begin{equation}
|(3^b )| \leq C_{11}' {\hbar^2\over \mu}\int dz \psi^2 \Big({\sqrt{2m}\over \hbar}\nu\Big)\Big({\sqrt{2m}\over \hbar} \Big)^2\Big|{\partial \nu\over \partial z}\Big|^2 \Big({\sqrt{2m}\over \hbar}\nu \Big)^{-3}
\leq C_{11} {m\over \mu} \nu_0^2
.\end{equation}
Estimation of each one of these terms are therefore shown to be smaller than the leading term within this approximation, as claimed.
\section{Appendix-II}
Here, we show that the short distance behavior of the effective potential for the relativistic particle indeed goes as $-{1\over 2}|z|$ as claimed in the main text. To do this we need to expand the original term as follows,
\begin{equation}
\int _0^\infty dt e^{t\mu_0}K_0(m\sqrt{t^2+z^2})=\frac1{2}\sum_{k=0}^\infty\frac{\mu_0^k}{k!}\int_0^\infty dt t^k\int_0^\infty \frac{du}{u}e^{-u(t^2+z^2)-\frac{m^2}{4u}}
.\end{equation}
Using the fact that for any finite value of $t$ the exponential is unformly convergent and there is Gaussian suppression for very large values of $t$, we get,
\begin{eqnarray}
\int _0^\infty dt e^{t\mu_0}K_0(m\sqrt{t^2+z^2})&=&\frac1{2}\sum_{k=0}^\infty \frac{\mu_0^k}{k!}\int_0^\infty dt t^ke^{-ut^2}\int_0^\infty \frac{du}{u}e^{-uz^2-\frac{m^2}{4u}}\nonumber\\
&=&\frac1{2}\sum_{k=0}^\infty \frac{\mu_0^k}{k!}\int_0^\infty dt t^ke^{-t^2}\int_0^\infty \frac{du}{u^{\frac{k+1}{2}+1}}e^{-uz^2-\frac{m^2}{4u}}\nonumber\\
&=&\frac1{4}\sum_{k=0}^\infty \Gamma\left(\frac{k+1}{2}\right)\frac{\mu_0^k}{k!}|z|^{k+1}\int_0^\infty \frac{du}{u^{\frac{k+1}{2}+1}}e^{-u-\frac{m^2z^2}{4u}}\nonumber
\end{eqnarray}
We then recognize that the last integrals correspond to modified Bessel functions, we get,
$$
\int _0^\infty dt e^{t\mu_0}K_0(m\sqrt{t^2+z^2})
=\frac1{2}\sum_{k=0}^\infty \Gamma\left(\frac{k+1}{2}\right)\frac{\mu_0^k}{k!}2^\frac{k+1}{2}\left(\frac{|z|}{m}\right)^\frac{k+1}{2}K_{\frac{k+1}{2}}(m|z|)
.$$
We recall the duplication formula for the gamma function,
\begin{equation}
\Gamma(x)\Gamma\left(x+\frac1{2}\right)=2^{1-2x}\sqrt{\pi}\Gamma(2z)
,\end{equation}
Using this identity the integral expression becomes,
$$
(*)=\int _0^\infty dt e^{t\mu_0}K_0(m\sqrt{t^2+z^2})=\sqrt{\pi}\sum_{k=0}^\infty \frac{\mu_0^k}{2^\frac{k+1}{2}\Gamma\left(\frac{k}{2}+1\right)}\left(\frac{|z|}{m}\right)^\frac{k+1}{2}K_{\frac{k+1}{2}}(m|z|)
$$
To identify the proper $z$ behaviour we need to write the sum on the right side over integer and half-integer terms separately,
$$
(*)=\sqrt{\pi}\sum_{n=0}^\infty \frac{\mu_0^{2n}}{2^{n+\frac1{2}}\Gamma(n+1)}\left(\frac{|z|}{m}\right)^{n+\frac1{2}}K_{n+\frac1{2}}(m|z|)+\sqrt{\pi}\sum_{n=1}^\infty \frac{\mu_0^{2n-1}}{2^n\Gamma\left(n+\frac1{2}\right)}\left(\frac{|z|}{m}\right)^nK_n(m|z|).
$$
We now remind the reader
the series expantion of modified Bessel functions with integer and half-integer order (see \cite{gradsh}) respectively,
\begin{eqnarray}
K_{n+\frac1{2}}(x)&=&\sqrt{\frac{\pi}{2x}}e^{-x}\sum_{k=0}^n \frac{\Gamma(n+k+1)}{k!\Gamma(n-k+1)(2x)^k},\nonumber\\
K_n(x)&=&\frac1{2}\sum_{k=0}^{n-1}(-1)^k\frac{(n-k-1)!}{k!\left(\frac{x}{2}\right)^{n-2k}}\nonumber\\
&+&(-1)^{n+1}\sum_{k=0}^\infty \frac{\left(\frac{x}{2}\right)^{n+2k}}{k!(n+k)!}\left[\ln\left(\frac{x}{2}\right)-\frac1{2}\psi(k+1)-\frac1{2}\psi(k+n+1)\right]\nonumber
.\end{eqnarray}
When we use these expansions in the sum, the first part of the sum takes the form,
\begin{eqnarray}
\Sigma_1&=&\sqrt{\pi}\sum_{n=0}^\infty \frac{\mu_0^{2n}}{2^{n+\frac1{2}}\Gamma(n+1)}\left(\frac{|z|}{m}\right)^{n+\frac1{2}}K_{n+\frac1{2}}(m|z|)\nonumber\\
&=&\pi\sum_{n=0}^\infty\sum_{k=0}^n\frac{\mu_0^{2n}}{2^{n+k+1}k!}\frac{\Gamma(n+k+1)}{\Gamma(n+1)\Gamma(n-k+1)}\frac{|z|^{n-k}}{m^{n+k+1}}e^{-m|z|}\nonumber\\
&=&\pi\sum_{n=0}^\infty\sum_{k=0}^n\frac{\mu_0^{2n}}{2^{n+k+1}k!}\frac{\Gamma(n+k+1)}{\Gamma(n+1)\Gamma(n-k+1)}\frac{|z|^{n-k}}{m^{n+k+1}}(1-m|z|+\frac1{2}m^2z^2+...).\nonumber
\end{eqnarray}
The constant term and $|z|$ term can be found by setting $k=n$ and $k=n-1$, and then identifying each contribution.
For $k=n$ the first term gives,
$$
\pi\sum_{n=0}^\infty \frac{\mu_0^{2n}}{2^{2n+1}(\Gamma(n+1))^2}\frac{\Gamma(2n+1)}{m^{2n+1}}(1-m|z|+\frac1{2}m^2z^2+...)
,$$
which contains a constant term as well as a $|z|$ term, which we write as,
$$
C_1=\pi\sum_{n=0}^\infty \frac{\mu_0^{2n}}{2^{2n+1}(\Gamma(n+1))^2}\frac{\Gamma(2n+1)}{m^{2n+1}}
$$
$$
C_2|z|=-\pi\sum_{n=0}^\infty \left(\frac{\mu_0}{m}\right)^{2n}\frac{\Gamma(2n+1)}{2^{2n+1}(\Gamma(n+1))^2}|z|
.$$
If we set $k=n-1$, in the expression for $\Sigma_1$ above, we find,
$$
=\pi\sum_{n=1}^\infty \left(\frac{\mu_0}{m}\right)^{2n}\frac{\Gamma(2n)}{2^{2n}\Gamma(n)\Gamma(n+1)}|z|
.$$
Note that {\it here the sum should begin} from $n=1$.
In these two expressions we use now the duplication formula, and this leads to the cancelations except the $n=0$ term in the first expression.
As a result, the leading orders of the first summation $\Sigma_1$ becomes,
$$
\pi\sum_{n=0}^\infty\frac{\mu_0^{2n}}{2^{2n+1}(\Gamma(n+1))^2}\frac{\Gamma(2n+1)}{m^{2n+1}}-\frac{\pi}{2}|z|
.$$
Let us now focus on the second summation, $\Sigma_2$,
$$
\Sigma_2=\sqrt{\pi}\sum_{n=1}^\infty \frac{\mu_0^{2n-1}}{2^n\Gamma\left(n+\frac1{2}\right)}\left(\frac{|z|}{m}\right)^nK_n(m|z|)=\sqrt{\pi}\frac1{2}\sum_{n=1}^\infty\sum_{k=0}^{n-1}(-1)^k\frac{\mu_0^{2n-1}}{2^{2k}\Gamma\left(n+\frac1{2}\right)}\frac{(n-k-1)!}{k!}\frac{|z|^{2k}}{m^{2n-2k}}
$$
$$
+\sqrt{\pi}\sum_{n=1}^\infty\sum_{k=0}^\infty(-1)^{n+1}\frac{\mu_0^{2n-1}}{2^{2k+2n}\Gamma\left(n+\frac1{2}\right)}\frac{m^{2k}|z|^{2n+2k}}{k!(k+n)!}\left[\ln\left(\frac{m|z|}{2}\right)-\frac1{2}\psi(k+1)-\frac1{2}\psi(k+n+1)\right]
.$$
The second expression is of higher order than $O(z)$ so we can neglect it, thus we consider only the first expression. The first term also contributes higher order terms except $k=0$ for each $n$, thus we replace it, to order $z$, as follows,
$$
\sqrt{\pi}\frac1{2}\sum_{n=1}^\infty\sum_{k=0}^{n-1}(-1)^k\frac{\mu_0^{2n-1}}{2^{2k}\Gamma\left(n+\frac1{2}\right)}\frac{(n-k-1)!}{k!}\frac{|z|^{2k}}{m^{2n-2k}}\rightarrow\sqrt{\pi} \frac1{2}\sum_{n=1}^\infty \frac{\mu_0^{2n-1}}{\Gamma\left(n+\frac1{2}\right)}\frac{\Gamma(n)}{m^{2n}}
$$
As a result the total constant contribution becomes,
$$
\pi\sum_{n=0}^\infty \frac{\mu_0^{2n}}{2^{2n+1}(\Gamma(n+1))^2}\frac{\Gamma(2n+1)}{m^{2n+1}}+\frac{\sqrt{\pi}}{2}\sum_{n=1}^\infty \frac{\mu_0^{2n-1}}{\Gamma\left(n+\frac1{2}\right)}\frac{\Gamma(n)}{m^{2n}}
$$
$$
=\frac{\sqrt{\pi}}{2}\sum_{n=0}^\infty\frac{\mu_0^{2n}}{m^{2n+1}}\frac{\Gamma\left(n+\frac1{2}\right)}{\Gamma(n+1)}+\frac{\sqrt{\pi}}{2}\sum_{n=0}^\infty \frac{\mu_0^{2n+1}}{m^{2n+2}}\frac{\Gamma(n+1)}{\Gamma\left(n+\frac{3}{2}\right)}
$$
$$
=\frac1{m}\frac{\sqrt{\pi}}{2}\sum_{n=0}^\infty\left(\frac{\mu_0}{m}\right)^{2n}\left(\frac{\Gamma\left(n+\frac1{2}\right)}{\Gamma(n+1)}+\frac{\mu_0}{m}\frac{\Gamma(n+1)}{\Gamma\left(n+\frac{3}{2}\right)}\right)=\frac{\arccos\left(-\frac{\mu_0}{m}\right)}{\sqrt{m^2-\mu_0^2}}
,$$
as can be verified by expanding the integral term for $z=0$ (or by setting $z=0$ and identifying this integral as before).
So, as claimed, we find for the integral expression,
\begin{equation}
{1\over \pi}\int_0^\infty dt e^{t\mu_0}[K_0(m\sqrt{t^2+z^2})-K_0(mt)]=-\frac1{2}|z|+O(z^2)
\end{equation}
\section{Appendix-III}
In this short appendix we will demostrate that indeed the main contribution of the integral given in (\ref{key1}) comes from the region where $tu/M$ terms in comparison to $s$ terms are neglected.
To make sense of this claim we rewrite the integral in terms of sclaed out variables, $s\mapsto s/m^2$ and $t\mapsto t/m$. This leads to
\begin{eqnarray}
&\ &\frac{1}{4\pi m}\int [dR]\chi^{\dagger}(R)\exp\left[-{stu\over 2}\Big({m\over M}\Big){1\over tu \frac{m}{2M}+s}\left({R^2\over m^2}\right)\right]\nonumber\\
&\ &\times\int_0^1 du \int_0^\infty dt t^2 e^{-ut[H_0-(\mu_0+\delta' E)]/m}\int_0^\infty ds \frac{e^{-s-\frac{t^2}{4s}}}{s^{\frac3{2}}\sqrt{tu\frac{m}{2M}+s}}\chi(R).
\end{eqnarray}
Note that the term in question is important when $u$ is not too small, if $u<<1$ we can neglect these terms. However for the time being let us not put a restriction on $u$. If $s$ is large the term in question is negligible, unless $t$ is very large, but then there are exponential $t$ terms to suppress the integrals. Then the main contributions may come when $t$ and $s$ are both small.
Note that $t$ should be much smaller in this case since otherwise exponential term $e^{-t^2/4s}$ leads to a high suppression. If $s$ is small, we need at most $t\sim s^{1/2}$. So the terms we want to drop off may not be negligible if we have $s^{1/2} m/M\sim s$, that means $s\sim (m/M)^2$.
Let us look at the integral within this interval, note that for $0<t<s^{1/2}$ and $0<s<(m/M)^2$,
the second exponential term is of order one plus lower order corrections in $m/M$. So we replace it with $1$ to estimate. The first one can be written as,
\begin{equation}
\exp\left[-{stu\over 2}\Big({m\over M}\Big){1\over tu \frac{m}{2M}+s}\left({R^2\over m^2}\right)\right]=\exp\left[-s{tu\over s}\Big({m\over 2M}\Big){1\over \frac{tu}{s} \frac{m}{2M}+1}\left({R^2\over m^2}\right)\right],
\end{equation}
which is of the form $\exp [-\alpha\frac{x}{1+x}]$ and this negative exponential is maximized when the function of $x$ is minimized, hence we set $x=0$.
That means the exponential could be replaced by $1$. Hence the integral has an upper bound in the interval of interest,
\begin{eqnarray}
\frac{1}{4\pi m}\int [dR]\chi^{\dagger}(R)\chi(R)
\int_0^1 du \int_0^{(m/M)^2} ds\int_0^{s^{1/2}} t^2 dt \frac{e^{-s-\frac{t^2}{4s}}}{s^{\frac3{2}}\sqrt{tu\frac{m}{2M}+s}}.\nonumber
\end{eqnarray}
By replacing the upper limit of the $t$ integral with $m/M$ and using Cauchy-Schwartz inequality, replacing $e^{-s}$ by $1$, we estimate that the integral expression without the orthofermion part (which gives a projection) is smaller than,
\begin{eqnarray}
\int_0^1 du \int_0^{m/M} t^2 dt\left[ \int_0^{(m/M)^2} ds\frac{e^{-\frac{t^2}{2s}}}{s^{3}}\right]^{1/2}\left[\int_0^{(m/M)^2} {ds\over{tu\frac{m}{2M}+s}}\right]^{1/2}.\nonumber
\end{eqnarray}
Now we estimate as follows,
\begin{eqnarray}
&\ & \int_0^1 du \int_0^{m/M} t^2 dt\left[ \int_0^{(m/M)^2} ds\frac{e^{-\frac{t^2}{2s}}}{s^{3}}\right]^{1/2}\left[\int_0^{(m/M)^2} {ds\over{tu\frac{m}{2M}+s}}\right]^{1/2}\nonumber\\
&\ & =\int_0^1 du \int_0^{m/M} t^2 dt\left[ {1\over t^4}\int_0^{(m/M)^2/t^2} ds\frac{e^{-\frac{1}{2s}}}{s^{3}}\right]^{1/2}\left[\ln\Big({tu\frac{m}{2M}+(m/M)^2\over tu\frac{m}{2M}}\Big)\right]^{1/2}\nonumber\\
&\ &\leq \int_0^1 du \int_0^{m/M} dt \left[ \int_0^{\infty} ds\frac{e^{-\frac{1}{2s}}}{s^{3}}\right]^{1/2}\left[\ln\Big({tu\frac{M}{2m}+1\over tu\frac{M}{2m} }\Big)\right]^{1/2}\nonumber\\
&\ & ={m\over M}\left[ \int_0^{\infty} ds\frac{e^{-\frac{1}{2s}}}{s^{3}}\right]^{1/2}\int_0^1 du \int_0^{1} dv \ln^{1/2}\Big({{uv/2}+1\over uv/2}\Big)\nonumber\\
&\ &\leq C{m\over M},
\end{eqnarray}
hence unimportant at this level as claimed.
\section{Acknowledgment}
O. T. Turgut would like to express his deep gratitude to Jens Hoppe for discussions and the kind invitation to KTH, Stockholm, where parts of this work are completed. O. T. Turgut also would like to thank F. Erman and L Akant for discussions.
|
2,869,038,156,405 | arxiv | \section{Introduction}
\label{intro}
In the first part of this article we study the Lax-Oleinik semi-group $\mathcal L_t$ defined by a Tonelli Lagrangian on
a graph and prove that for any continuous function $u$, $\mathcal L_tu+c t$
converges as $t\to\infty$ where $c$ is the critical value of the Lagrangian.
For Lagrangians on compact manifolds, Fathi \cite{F} proved the convergence
using the Euler-Lagrange flow and conservation of energy.
In our case we do not have these tools but
we can follow ideas of Roquejoffre \cite{R} and Davini-Siconolfi \cite{DS}.
Camilli and collaborators \cite{ACCT,CM,CS} have studied viscosity
solutions of the Hamilton-Jacobi equation, and given sufficient conditions
for a set to be a uniqueness set and a representation formula.
In the second part of this article we prove that, under the assumption that the
Lagrangian is symmetric at the vertices, the sets of weak KAM and viscosity
solutions of the Hamilton-Jacobi equation defined coincide.
We consider a graph $G$ without boundary
consisting of finite sets of unoriented edges $\mathcal I=\{I_j\}$ and
vertices $\mathcal V=\{e_l\}$. The interior of $I_j$ is $I_j-\mathcal V$.
Parametrizing each edge by arc length
$\sigma_j:I_j\to[0,s_j]$ we can write its tangent bundle as
$TI_j=I_j\times\mathbb R$ and
\[TG=\bigcup_i \{j\}\times TI_j/\sim\]
where $(i,x,v)\sim(j,y,w)\iff (i,x,v)=(j,y,w)$ or
$x=y\in I_i\cap I_j, v=w=0$.
Thus, a function $L: TG\to\mathbb R$ is given by a collection of functions
$L_j: TI_j\to\mathbb R$ such that $L_i (e_l,0) = L_j(e_l,0)$ for $e_l\in I_i\cap I_j$.
A Lagrangian in $G$ is a function $L: TG\to\mathbb R$
such that each $L_j$ is $C^k$, $k\ge 2$, and $L_j(x,\cdot)$ is
strictly convex and super-linear for any $x\in I_j$.
We will say that a Lagrangian is {\it symmetric at the vertices} if
at each vertix $e_l$ there is a function $\lambda_l:\{u\in\mathbb R:u\ge 0\}\to\mathbb R$
such that $L_j(e_l,z)=\lambda_l(|z|)$ if $e_l\in I_j$. As an
example consider the mechanical Lagrangian given by $L_j(x,v)=\frac 12v^2-U_j(x)$.
For $x\in I_j\setminus\mathcal V$, we say that $(x,v)$ points towards
$\sigma(s_j)$ if $v>0$ and points towards $\sigma(0)$ if $v<0$.
We say that $(\sigma_j^{-1}(0),v)$ is an $I_j$- incoming or outgoing
vector according to whether $v>0$ or $v<0$, and we say that $(\sigma_j^{-1}(s_j),v)$
is an $I_j$- incoming or outgoing vector according to wether $v<0$ or $v>0$.
We let $T^+_{e_l}I_j$ ($T^-_{e_l}I_j$) to be the set of $I_j$- outgoing
(incoming or zero) vectors in $T_{e_l}I_j$.
\section{Basic properties of the action}
\label{sec:basic}
\subsection{A distance on a graph}
\label{sec:distance-graph}
We start defining a distance in the most natural way.
We say a continuous path $\alpha:[a,b]\to G$ is a
{\em unit speed geodesic} (u.s.g.) if there is a partition
$a=t_0<\ldots<t_m=b$ such that for each $1\le i\le m$ there is $j(i)$
such that
\[\alpha([t_0,t_1])\subset I_{j(1)}, \alpha([t_1,t_2])=I_{j(2)},\ldots,
\alpha([t_{m-2},t_{m-1}])=I_{j(m-1)}, \alpha([t_{m-1},t_m])\subset I_{j(m)},\]
$\sigma_{j(i)}\circ\alpha|_{[t_{i-1},t_i]}$ is differentiable and either
$(\sigma_{j(i)}\circ\alpha)'\equiv 1$ or $(\sigma_{j(i)}\circ\alpha)'\equiv-1$.
We set the length of a u.s.g. to be
\[\ell(\alpha)=|\sigma_{j(1)}(\alpha(t_1))-\sigma_{j(1)}(\alpha(a))|+\sum_{i=2}^{m-1}s_{j(i)}+
|\sigma_{j(m)}(\alpha(b))-\sigma_{j(m)}(\alpha(t_{m-1}))|\]
and define a distance on $G$ by
\[d(x,y)=\min\{\ell(\alpha): \alpha:[a,b]\to G \mbox{ is a u.s.g.},
\alpha(a)=x, \alpha(b)=y\}\]
\subsection{Absolute continuity}
\label{sec:absolute-contiouity}
We say a path $\gamma:[a,b]\to G$ is {\em absolutely continuous} if for any
$\varepsilon>0$ there is $\delta>0$ such that for any finite collection
of disjoint intervals $\{[c_i,d_i]\}$ with $\sum_i(d_i-c_i)<\delta$ we have
$\sum_id(\gamma(c_i),\gamma(d_i))<\varepsilon$.
If $\gamma:[a,b]\to G$, $\gamma(t)\in\mathcal V$, we define $\dot\gamma(t)=0$ if for
any $\varepsilon>0$ there is $\delta>0$ such that $d(\gamma(s),\gamma(t))<\varepsilon|s-t|$
when $|s-t|<\delta$.
Let $\gamma:[a,b]\to G$ be absolutely continuous and consider
the closed set $V=\gamma^{-1}(\mathcal V)$
so that $(a,b)\setminus V=\bigcup_i(a_i,b_i)$ where the intervals $(a_i,b_i)$
are disjoint and $\gamma([a_i,b_i])\subset I_{j(i)}$. It is clear that each
$\sigma_{j(i)}\circ\gamma:[a_i,b_i]\to[0,s_{j(i)}]$ is absolutely continuous.
We set $\dot\gamma(t)=(\sigma_{j(i)}\circ\gamma)'(t)$
whenever is defined.
Next Proposition will allow us to define the action of an
absolutely continuous curve.
\begin{proposition}\label{abs-cont}
Let $\gamma:[a,b]\to G$ be absolutely continuous and $V=\gamma^{-1}(\mathcal V)$
\begin{enumerate}[(a)]
\item $\dot\gamma=0$ Lebesgue almost everywhere in $V$.
\item $\dot\gamma$ is integrable and for
any $[c,d]\subset[a,b]$ we have
$d(\gamma(c),\gamma(d))\le\int\limits_c^d|\dot\gamma|$.
\end{enumerate}
\begin{proof}
Write $(a,b)\setminus V=\bigcup_i(a_i,b_i)$ as above with the intervals $(a_i,b_i)$
disjoint.
Since $\dot\gamma=0$ on the interior of $V$ and
$\cup\{a_n,b_n\}$ is numerable to stablish item (a) it remains to prove
that $\dot\gamma=0$ Lebesgue almost everywhere in $\partial V\setminus\cup\{a_n,b_n\}$.
Let $s=\min_j s_j$ and take $\delta>0$ such that $d(\gamma(s),\gamma(t))<s$ if $|s-t|<\delta$
There is $N$ such that $b_i-a_i<\delta$ for $i>N$. Since $\gamma(a_i), \gamma(b_i)\in\mathcal V$
we have that $\gamma(a_i)=\gamma(b_i)$ for $i>N$.
We change the labeling of the first $N$ terms
to have $a_1<b_1\le a_2<\cdots<b_m$ and $\gamma(a_i)=\gamma(b_i)$ for $i>m$.
Letting $J_0=[a,a_1]$, $J_i=[b_i,a_{i+1}]$,
$1\le i<m$, $J_m=[b_m,b]$, and $V_i=V\cap J_i$ we have $\gamma(V_i)=e_{l_i}$,
$0\le i\le m$. We can forget about the cases $b_i=a_{i+1}$.
Define the function $f_i:J_i\to\mathbb R$ by $f_i(t)=d(e_{l_i},\gamma(t))$.
For $t,s\in J_i$ we have $|f_i(t)-f_i(s)|\le d(\gamma(t),\gamma(s))$, so $f_i$ is absolutely
continuous and then $f_i'$ exists Lebesgue almost everywhere in $J_i$.
Let $t\in\partial V_i\setminus\bigcup_n\{a_n,b_n\}$ be a point where $f_i'$ exists.
There is a sequence $n_k\to\infty$ such that $a_{n_k}\to t$
and $\gamma(a_{n_k})=e_{l_i}$. Thus $f_i'(t)=0$, which means that $\dot\gamma(t)=0$.
If $(a_j,b_j)\subset J_i$ then $|\dot\gamma|=|f_i'|$ Lebesgue
almost everywhere in $(a_j,b_j)$, so that
\[\int_{J_i}|\dot\gamma|=\int_{J_i}|f_i'|,\]
and then
\[\int_a^b|\dot\gamma|=\sum_{i=0}^m\int_{J_i}|\dot\gamma|+\sum_{i=1}^m\int_{a_i}^{b_i}|\dot\gamma|<\infty.\]
It is also easy to see that for $t,s\in J_i$ we have $d(\gamma(t),\gamma(s))\le\int\limits_s^t|f_i'|$,
and using the partition $a\le a_1<b_1\le a_2<\cdots<b_m\le b$ to make a
partition of $[c,d]$ we get $d(\gamma(c),\gamma(d))\le\int\limits_c^d|\dot\gamma|$.
\end{proof}
\end{proposition}
\subsection{Lower semicontinuity and apriori bounds}
\label{sec:lower-semicontunuity}
In this crucial part of the paper we prove that in the
framework of graphs we have the the lower semicontinuity of
the action and apriori bounds for the Lipschitz norm of minimizers.
The proofs have the same spirit as in euclidean space, paying attention
to what happens at the vertices.
Denote by $\mathcal C^{ac}([a,b])$ the set of absolutely
continuous functions $\gamma:[a,b]\to G$ provided with the topology of
uniform convergence.
We define the action of $\gamma\in \mathcal C^{ac}([a,b])$ as
\[A(\gamma)=\int_a^bL(\gamma(t),\dot\gamma(t))dt\]
A minimizer is a $\gamma\in \mathcal C^{ac}([a,b])$ such that for any
$\alpha\in\mathcal C^{ac}([a,b])$ with $\alpha(a)=\gamma(a)$, $\alpha(b)=\gamma(b)$ we have
\[A(\gamma)\le A(\alpha)\]
The following two properties of the Lagrangian are important to
achieve our goal and follow from its strict convexity and super-linearity.
\begin{proposition}\label{Cle}
If $C\ge 0$, $\varepsilon>0$, there is $\eta>0$ such that for $x,y\in I_j$,
$d(x,y)<\eta$ and $v,w\in\mathbb R$, $|v|\le C$, we have
\[L(y,w)\ge L(x,v)+L_v(x,v)(w-v)-\varepsilon.\]
\end{proposition}
\begin{proposition}\label{Cle2}
If $L_{vv}\ge\theta>0$, $C\ge 0$, $\varepsilon>0$, there is $\eta>0$ such that for
$x,y\in I_j$, $d(x,y)<\eta$ and $v,w\in\mathbb R$, $|v|\le C$, we have
\[L(y,w)\ge L(x,v)+L_v(x,v)(w-v)+\frac{3\theta}4|w - v|^2-\varepsilon.\]
\end{proposition}
\begin{lemma}\label{lower}
Let $L$ be a Lagrangian on $G$. If a sequence $\gamma_n\in\mathcal C^{ac}([a,b])$
converges uniformly to the curve $\gamma:[a,b]\to G$ and
\[\liminf_{n\to\infty} A(\gamma_n)<\infty\]
then the curve $\gamma$ is absolutely continuous and
\[A(\gamma)\le\liminf_{n\to\infty} A(\gamma_n).\]
\end{lemma}
\begin{proof}
By the super-linearity we may assume that $L\ge 0$.
Let $c= \liminf\limits_{n\to\infty} A(\gamma_n)$. Passing to a subsequence we
can assume that
\[ A(\gamma_n)<c+1,\quad \forall n\in\mathbb N\]
Fix $\varepsilon>0$ and
take $B> 2(c+1)/\varepsilon$. Again by super-linearity there is a positive number $C(B)$
such that
\[L(x,v)\ge B|v|-C(B),\quad x\in G\setminus\mathcal V, v\in \mathbb R\]
From Proposition \ref{abs-cont} and $L\ge 0$, for $E\subset[a,b]$ measurable we have
\[-C(B)\hbox{Leb}(E)+B\int_E|\dot\gamma_n|
\le\int_E L(\gamma_n,\dot\gamma_n)+\int_{[a,b]\setminus E}L(\gamma_n,\dot\gamma_n)
\le c+1.\]
Thus
\[\int_E|\dot\gamma_n|\le \frac 1B (c+1+C(B)\hbox{Leb}(E))\le
\frac\varepsilon 2+\frac{C(B)\hbox{Leb}(E)}B.\]
Choosing $0<\delta<\dfrac{\varepsilon B}{2C(B)} $ we have that
\[\hbox{Leb}(E)<\delta\Rightarrow\forall n\in\mathbb N\, \int_E|\dot\gamma_n|<\varepsilon.\]
Since the sequence $\dot\gamma_n$ is uniformly integrable, we have that $\gamma$ is
absolutely continuous and $\dot\gamma_n$
converges to $\dot\gamma$ in the $\sigma(L^1,L^\infty)$ weak topology.
Set $V=\gamma^{-1}(\mathcal V)$. Let $\varepsilon>0$ and
$E_k=\bigl\{t:|\dot\gamma(t)|\le k, d(t,V)\ge\dfrac 1k\bigr\}$.
By Propositions \ref{Cle}, \ref{abs-cont}, for $n$ large,
\begin{align*}
\int_{\gamma^{-1}(e_l)}\big[L(e_l,0)+L_v(e_l,0)\dot\gamma_n(t)-\varepsilon\big]&\le
\int_{\gamma^{-1}(e_l)} L(\gamma_n,\dot\gamma_n) ,\\
\int_{E_k} \big[L(\gamma,\dot\gamma)+L_v(\gamma,\dot\gamma)(\dot\gamma_n-\dot\gamma)-\varepsilon\big]
&\le\int_{E_k} L(\gamma_n,\dot\gamma_n), \\
\int_{E_k\cup V} \big[L(\gamma,\dot\gamma)+L_v(\gamma,\dot\gamma)(\dot\gamma_n-\dot\gamma)-\varepsilon\big]
& \le\int_{E_k\cup V} L(\gamma_n,\dot\gamma_n)
\le A(\gamma_n) .
\end{align*}
Letting $n\to+\infty$ we have that
\[ \int_{E_k\cup V}L(\gamma,\dot\gamma)\le c+\varepsilon\;(b-a)\]
Since $E_k\uparrow [a,b]\setminus V$ when $k\to+\infty$ and $L\ge 0$, we have
\[ A(\gamma)=\lim_{k\to+\infty}\int_{E_k\cup V}L(\gamma,\dot\gamma)\le c+\varepsilon\;(b-a)\]
Now let $\varepsilon\to 0$.
\end{proof}
Lemma \ref{lower} implies
\begin{theorem}\label{semicontinua}
Let $L$ be a Lagrangian on $G$.
The action $A:\mathcal C^{ac}([a,b])\to\mathbb R\cup\{\infty\}$ is lower semicontinuous.
\end{theorem}
\begin{proposition}\label{tonelli}
Let $L$ be a Lagrangian on $G$.
The set $\{\gamma\in\mathcal C^{ab}([a,b]): A(\gamma)\le K\}$
is compact with the topology of uniform convergence.
\end{proposition}
Let $C_t=\sup\{L(x,v):x\in G, |v|\le\frac{\hbox{diam}(G)}t\}$,
then for any minimizer $\gamma:[a,b]\to G$ with $b-a\ge t$ we have
\[A(\gamma)\le C(b-a).\]
\begin{proposition}\label{addendum}
Suppose $\gamma_n:[a,b]\to G$ converge uniformly to $\gamma:[a,b]\to G$ and
$A(\gamma_n)$ converges to $A(\gamma)$, then $\dot\gamma_n$ converges to
$\dot\gamma$ in $L^1[a,b]$
\end{proposition}
\begin{proof}
Let $F\subset[a,b]$ be a finite union of intervals.
From Lemma \ref{lower} we have
\[\int_FL(\gamma,\dot\gamma)\le\liminf_n\int_F L(\gamma,\dot\gamma) \hbox{ and }
\int_{[a,b]\setminus F}L(\gamma,\dot\gamma)\le\liminf_n\int_{[a,b]\setminus F}L(\gamma,\dot\gamma)\]
Since
\[\lim_n\int_FL(\gamma_n,\dot\gamma_n)+\int_{[a,b]\setminus F}L(\gamma_n,\dot\gamma_n)
=\lim_nA(\gamma_n)=A(\gamma)\]
we have
\begin{equation}
\label{eq:I}
\lim_n\int_{F}L(\gamma_n,\dot\gamma_n)=\int_{F}L(\gamma,\dot\gamma).
\end{equation}
If $V=\gamma^{-1}(\mathcal V)\subset F$ then
\[\limsup_n\int_{ V}L(\gamma_n,\dot\gamma_n)\le\lim_n\int_{F}L(\gamma_n,\dot\gamma_n)
=\int_{F}L(\gamma,\dot\gamma).\]
Thus
\begin{equation}
\label{eq:0}
\limsup_n\int_{ V}L(\gamma_n,\dot\gamma_n)\le \int_{ V}L(\gamma,\dot\gamma).
\end{equation}
As in Lemma \ref{lower}, the $\dot\gamma_n$ are uniformly integrable, so they
converge to $\dot\gamma$ in the $\sigma(L_1,L_\infty)$ weak topology and then,
for any Borel set $B$ where $\dot\gamma$ is bounded,
\begin{equation}
\label{eq:II}
\lim_n\int_{B}L_v(\gamma,\dot\gamma)(\dot\gamma_n-\dot\gamma)=0.
\end{equation}
Given $\varepsilon>0$, from
Proposition \ref{Cle2} we have that for $n$ large enough
\[ \frac{3\theta}4\int_{\gamma^{-1}(e_l)}|\dot\gamma_n|^2\le\int_{\gamma^{-1}(e_l)}
\big[L(\gamma_n,\dot\gamma_n)-L(e_l,0)-L_v(e_l,0)\dot\gamma_n+\varepsilon\big].
\]
Which together with Proposition \ref{abs-cont} and equations \eqref{eq:0}, \eqref{eq:II} give
$\limsup\limits_n\int\limits_{ V}|\dot\gamma_n|^2\le\varepsilon$ for any $\varepsilon>0$. So
\begin{equation}
\label{eq:a}
\lim_n\int_{ V}|\dot\gamma_n|^2=0.
\end{equation}
For $k>0$ let
$D_k:=\{t\in[a,b]:|\dot\gamma(t)|>k\}$, $B_k:=\{t\in[a,b]: d(t,V)>\frac 1k\}$.
Then $\lim\limits_{k\to\infty}$Leb$(D_k)=\lim\limits_{k\to\infty}$Leb$([a,b]\setminus V\setminus B_k)=0$.
Let $F_k$ be a finite union of intervals such that
$D_k\cap B_k\subset F_k\subset B_k$ and Leb$(F_k\setminus(D_k\cap B_k))<\frac 1k.$
Then $\lim\limits_{k\to\infty}$ Leb$(F_k)=0$.
Given $\varepsilon>0$, from
Proposition \ref{Cle2} we have that for $n$ large enough
\[ \frac{3\theta}4\int_{B_k\setminus F_k}|\dot\gamma_n-\dot\gamma|^2\le\int_{B_k\setminus F_k}
\big[L(\gamma_n,\dot\gamma_n)-L(\gamma,\dot\gamma)-L_v(\gamma,\dot\gamma)(\dot\gamma_n-\dot\gamma)+\varepsilon\big].
\]
From \eqref{eq:I}, \eqref{eq:II} we get that
$\limsup\limits_n\int\limits_{B_k\setminus F_k}|\dot\gamma_n-\dot\gamma|^2\le\varepsilon$
for any $\varepsilon>0$. So
\begin{equation}
\label{eq:b}
\lim_n\int_{B_k\setminus F_k}|\dot\gamma_n-\dot\gamma|^2=0.
\end{equation}
Since $\{\dot\gamma_n\}$ is uniformly integrable, given $\varepsilon>0$, for $k$
sufficiently large we have
\begin{equation}\label{eq:c}
\int_{F_k\cup[a,b]\setminus V\setminus B_k}|\dot\gamma_n-\dot\gamma|\le
\int_{F_k\cup[a,b]\setminus V\setminus B_k}|\dot\gamma_n|+|\dot\gamma|<\varepsilon,
\end{equation}
From \eqref{eq:a}, \eqref{eq:b}, \eqref{eq:c}
and Cauchy-Schwartz inequality, we have that for any $\varepsilon>0$
\[ \limsup_n\int_a^b|\dot\gamma_n-\dot\gamma|\le
\lim_n\underset{V}{\int}|\dot\gamma_n|+
\limsup_n\underset{F_k\cup[a,b]\setminus V\setminus B_k}{\int|\dot\gamma_n-\dot\gamma|}+
\lim_n\underset{B_k\setminus F_k}{\int}|\dot\gamma_n-\dot\gamma|\le \varepsilon\]
\end{proof}
\begin{lemma}\label{aprioriac}
Let $L$ be a Lagrangian in $G$.
For $\varepsilon>0$ there exists $K_{\varepsilon}$ that is a Lipschitz
constant for any minimizer $\gamma:[a,b]\to G$ with
$b-a\ge \varepsilon$.
\end{lemma}
\begin{proof}
Note that if $\gamma$ is a minimizer and $\gamma(c,d)\subset I_j$ then
$\gamma|_{(c,d)}$ is a solution of the Euler Lagrange equation for $L_j$.
Suppose the Lemma is not true, then by Proposition \ref{abs-cont},
for any $i\in\mathbb N$ there are a minimizer
$\gamma_i: [s_i,t_i]\to G$ with $t_i-s_i\ge\varepsilon$ and a set
$E_i\subset [s_i,t_i]\setminus\gamma_i^{-1}(\mathcal V)$ with
Leb$(E_i)>0$ such that $|\dot\gamma|>i$ on $E_i$. Let $c_i\in E_i$.
Translating $[s_i,t_i]$ we can assume that $c_i=c$ for all $i$ and taking
a subsequence that there is $a\in\mathbb R$ such that $\gamma_i$ is defined in
$[a,a+\frac{\varepsilon}2], c\in[a,a+\frac{\varepsilon}2]$.
As $A(\gamma_i|[a,a+\frac{\varepsilon}2])$ is bounded, by Proposition \ref{tonelli} there
is subsequence $\gamma_i|[a,a+\frac{\varepsilon}2]$ which converges uniformly to
$\gamma:[a,a+\frac{\varepsilon}2]\to G$. Since $\gamma$ is limit of minimizers, it is a
minimizer and $A(\gamma)\le\liminf A(\gamma_i|[a,a+\frac{\varepsilon}2])$.
We can not have that $A(\gamma)<\limsup A(\gamma_i|[a,a+\frac{\varepsilon}2])$
because that would contradict that the $\gamma_i$ are minimizers. Thus
$A(\gamma)=\lim A(\gamma_i|[a,a+\frac{\varepsilon}2])$.
If $\gamma(c)\in I_j\setminus\mathcal V$, there is $\delta>0$ such that
$\gamma([c-\delta,c+\delta])\subset I_j$.
If $\gamma(c)=e_l$ we have 2 possibilities (not mutually exclusive)
a) There is an edge $I_j$ with $e_l\in I_j$ and infinitely many $i$'s
such that $\gamma_i(c)\in I_j$ and $\dot\gamma_i(c)$ points towards $e_l$.
b) There is an edge $I_j$ with $e_l\in I_j$ and infinitely many $i$'s such that
$\gamma_i(c)\in I_j$ and $\dot\gamma_i(c)$ points towards the other vertex.
In case a) there is $\delta>0$ such that $\gamma([c-\delta,c])\subset I_j$.
In case b) there is $\delta>0$ such that $\gamma([c,c+\delta])\subset I_j$.
We have that $\gamma$ is a solution of the Euler-Lagrange equation for
$L_j$ either on $[c-\delta,c]$ or on $[c,c+\delta]$ and then $|\dot\gamma(t)|\le K$ on
$[c-\delta,c]$ or $[c,c+\delta]$.
For some $0<\delta_1<\delta$ we have that $\gamma_i$ are solutions
of the Euler-Lagrange equation for $L_j$ on $[c-\delta_1,c]$ or on $[c,c+\delta_1]$.
For $i$ suficiently large, we have that $|\dot\gamma_i|>2 K$ either
on $[c-\delta_1,c]$ or on $[c,c+\delta_1]$. This would contradict
Proposition \ref{addendum}.
\end{proof}
\section{Weak KAM theory on graphs}
\label{wkam-graph}
The content of this section is similar to that for Lagrangians
on compact manifolds. We only give the proofs that are different from
those in the compact manifold case, which can be found in \cite{F}.
\subsection{The Peierls barrier}
\label{sec:barrier}
Given $x, y \in G$ let $\mathcal C^{ac}(x,y,t)$ be the set of curves
$\alpha\in\mathcal C^{ac}([0,t])$ such that $\alpha (0)=x$ and $\alpha (t) = y$.
For a given real number $k$ define
$$h_t(x,y)=\min_{\alpha \in \mathcal C^{ac}(x,y,t)} A(\alpha) $$
and
$$h^k(x,y)= \liminf_{t\rightarrow\infty}h_t(x,y)+kt $$
\begin{lemma}\label{hLip}
For $\varepsilon>0$ the function $F:[\varepsilon,\infty)\times G\times G\to\mathbb R$
defined by $F(t,x,y)= h_t(x,y)$ is Lipschitz.
\end{lemma}
\begin{lemma}\label{critical}
There exists a real $c$ independent of $x$ and $y$ such that
\begin{enumerate}
\item For all $k>c$ we have $h^k(x,y) =\infty $.
\item For all $k<c$ we have $h^k(x,y) =-\infty $
\item $h^c(x,y) $ is finite. The function $h:=h^c$ is called the \emph{Peierls barrier}.
\end{enumerate}
\end{lemma}
\begin{lemma}\label{c-cerradas}
The value $c$ is the infimum of $k$ such that $\int\limits_\gamma L+k\ge 0$
for all closed curves $\gamma$.
\end{lemma}
\begin{definition}\quad
The \emph{Ma\~n\'e potencial} $\Phi:G\times G\to\mathbb R$ is defined by
\[\Phi(x,y)=\inf_{t>0} h_t(x,y)+ct.\]
Clearly we have $\Phi(x,y)\le h(x,y)$ for any $x,y\in G$.
\end{definition}
\begin{proposition}\label{propiedades-h}
Functions $h$ and $\Phi$ have the following properties.
\begin{enumerate}
\item \label{3fi} $\Phi(x,z)\le \Phi(x,y)+\Phi(y,z)$.
\item \label{3h} $h(x,z)\le h(x,y)+\Phi(y,z)$, $h(x,z)\le\Phi(x,y)+h(y,z)$.
\item \label{lip} $h$ and $\Phi$ are Lipschitz
\item \label{unif} If $\gamma_n:[0,t_n]\to G$ is a sequence of absolutely continuous curves
with $t_n\to\infty$ and $\gamma_n(0)\to x$, $\gamma_n(t_n)\to y$, then
\begin{equation}
\label{ineq-unif}
h(x,y)\le\liminf_{n\to\infty}A(\gamma_n)+ct_n.
\end{equation}
\end{enumerate}
\end{proposition}
\begin{definition}
A curve $\gamma:J\to G$ is called
\begin{itemize}
\item {\em semi-static} if
\[\Phi(\gamma(t),\gamma(s)=\int_t^s L(\gamma,\dot\gamma)+c(s-t)\]
for any $t,s\in J$, $t\le s$.
\item {\em static} if
\[\int_t^s L(\gamma,\dot\gamma)+c(s-t)=-\Phi(\gamma(s),\gamma(t))\]
for any $t,s\in J$, $t\le s$.
\item
The \emph{Aubry set} $\mathcal A$ is the set of points $x\in G$ such that
$h(x,x)=0$.
\end{itemize}
\end{definition}
Notice that by item \eqref{3h} in Proposition \ref{propiedades-h},
$h(x,z)=\Phi(x,z)$ if $x\in\mathcal A$ or $z\in\mathcal A$.
\begin{proposition}
If $\eta:\mathbb R\to G$ is static then $\eta(s)\in\mathcal A$ for any $s\in\mathbb R$.
\end{proposition}
Although we do not conservation of energy we can prove
that semi-static curves have energy $c(L)$.
\begin{proposition}\label{energy}
Let $\eta:J\to G$ be semi-static. For almost every $t\in J$
\[L_v(\eta(t),\dot\eta(t))\dot\eta(t) = L(\eta(t),\dot\eta(t))+c\]
\end{proposition}
\begin{proof}
For $\lambda>0$, let $\eta_\lambda(t):=\eta(\lambda t)$ so that
$\dot\eta_\lambda(t)=\lambda\dot\eta(\lambda t)$ almost everywhere.
For $r,s\in J$ let
$$\mathcal A_{rs}(\lambda):=\int_{r/\lambda}^{s/\lambda}[L(\eta_\lambda(t),\dot\eta_\lambda(t))+c]\, dt
=\int_r^s[L(\eta(s),\lambda\dot\eta(s))+c]\,\frac{ds}{\lambda}. $$
Since $\eta$ is a free-time minimizer, differentiating
$\mathcal A_{rs}(\lambda)$ at $\lambda=1$, we have that
$$ 0=\mathcal A_{rs}'(1)=
\int_0^T[L_v(\eta(s),\dot\eta(s))\dot\eta(s)-L(\dot\eta(s),\dot\eta(s)-c]\,ds.
$$
Since this holds for any $r,s\in J$ we have
\[L_v(\eta(t),\dot\eta(t))\dot\eta(t) = L(\eta(t),\dot\eta(t))+c\]
for almost every $t\in J$.
\end{proof}
\subsection{Weak KAM solutions}\label{wkam-sol}
Following Fathi \cite{F}, we define weak KAM solutions and give some of
their properties
\begin{definition}
Let $c$ be given by Lemma \ref{critical}.
\begin{itemize}
\item A function $u:G\to\mathbb R$ is \emph{dominated} if for any $x,y\in G$, we have
\[u(y)-u(x)\le h_t(x,y)+ct\quad \forall t>0 ,\]
or equivalently
\[u(y)-u(x)\le\Phi(x,y).\]
\item
$\gamma:I\to G$ \emph{calibrates} a dominated function $u:G\to\mathbb R$ if
\[u(\gamma(s))-u(\gamma(t))=\int_t^sL(\gamma,\dot\gamma)+c(s-t)\quad\forall s,t\in I\]
\item
A continuous function $u:G\to\mathbb R$ is a
\emph{backward (forward) weak KAM solution} if
it is dominated and for any $x\in G$ there is $\gamma:(-\infty,0]\to G$
($\gamma:[0,\infty)\to G$) that calibrates $u$ and $\gamma(0)=x$
\end{itemize}
\end{definition}
\begin{corollary}\label{est-calibra}
Any static curve $\gamma:J\to G$ calibrates any dominated function
$u:G\to\mathbb R$
\end{corollary}
\begin{proposition}\label{h-wkam}
For any $x\in G$, $h(x,\cdot)$ is a backward weak KAM solution and
$-h(\cdot,x)$ is a forward weak KAM solution.
\end{proposition}
\begin{proof}
By item \eqref{3h} of Proposition \ref{propiedades-h}, $h(x,\cdot)$ is dominated.
The standard construction of calibrating curves for compact manifolds
involves the Euler Lagange flow that we do not have, so
we use a diagonal trick.
Let $\gamma_n:[-t_n,0]\to G$ be a sequence of minimizing curves
connecting $x$ to $y$ such that
\[h(x,y)=\lim_{n\to\infty}A(\gamma_n)+ct_n\]
By Lemma \ref{aprioriac}, $\{\gamma_n\}$ is uniformly Lipschitz and then
equicontinuous.
It follows from the Arzela Ascoli Theorem that there is a sequence
$n^1_j\to\infty$ such that $\gamma_{n^1_j}$ converges uniformly on $[-1, 0]$.
Again, by the Arzela Ascoli Theorem, there is a subsequence $(n^2_j)_j$ of
the sequence $(n^1_j)_j$ such that $\gamma_{n^1_j}$ converges uniformly on $[-2,0]$.
By induction, this procedure gives for each $k\in\mathbb N$ a sequence $(n^k_j)_j$ that
is a subsequence of the sequence $(n^{k-1}_j)_j$ and such that $\gamma_{n^k_j}$
converges uniformly on $[-k,0]$ as $j\to\infty$. Letting $m_k=n^k_k$, the sequence
$\gamma_{m_k}$ converges uniformly on each $[-l,0]$. For $s<0$ define
$\gamma(s)=\lim\limits_{k\to\infty}\gamma_{m_k}(s)$.
Fix $t<0$, for $k$ large $t+t_{m_k}\ge 0$ and
\begin{equation}\label{diag}
A(\gamma_{m_k})+ct_{m_k}=
\int\limits_{-t_{m_k}}^tL(\gamma_{m_k},\dot\gamma_{m_k})+c(t+t_{m_k})+
\int_t^0L(\gamma_{m_k},\dot\gamma_{m_k})-ct.
\end{equation}
Since $\gamma_{m_k}$ converges to $\gamma$ uniformly on $[t,0]$, we have
\[\liminf_{k\to\infty}\int_t^0L(\gamma_{m_k},\dot\gamma_{m_k})\ge \int_t^0L(\gamma,\dot\gamma).\]
From item \eqref{unif} of Proposition \ref{propiedades-h} we have
\[h(x,\gamma(t))\le\liminf_{k\to\infty}
\int_{-t_{m_k}}^tL(\gamma_{m_k},\dot\gamma_{m_k})+c(t+t_{m_k}).\]
Taking $\liminf\limits_{k\to\infty}$ in \eqref{diag} we get
\[h(x,y)\ge h(x,\gamma(t))+\int_t^0L(\gamma,\dot\gamma)-ct.\]
So $\gamma$ calibrates $h(x,\cdot)$.
\end{proof}
From Proposition \ref{h-wkam} we have
\begin{corollary}\label{aubry-calibra}
If $x\in\mathcal A$ there exists a curve $\gamma:\mathbb R\to G$ such that $\gamma(0)=x$ and for
all $t\ge 0$
\begin{align*}
h(\gamma(t),x)&=-\int_0^tL(\gamma,\dot\gamma)-ct\\
h(x,\gamma(-t))&=-\int_{-t}^0L(\gamma,\dot\gamma)-ct.\\
\end{align*}
In particular the curve $\gamma$ is static and calibrates any dominated function
$u:G\to\mathbb R$.
\end{corollary}
\begin{theorem}\label{aubry-potential-sol}
The function $\Phi(x,\cdot)$ is a backward weak KAM solution if and
only if $x\in\mathcal A$.
\end{theorem}
\begin{corollary}\label{kam}
Let $C\subset G$ and $w_0:C\to\mathbb R$ be bounded from below. Let
\[w(x)=\inf_{z\in C}w_0(z)+\Phi(z,x)\]
\begin{enumerate}
\item\label{max-dom}
$w$ is the maximal dominated function not exceeding $w_0$ on $C$.
\item\label{aubry-kam}
If $C\subset\mathcal A$, $w$ is a backward weak KAM solution.
\item\label{dom-coinc} If for all $x,y\in C$
\[w_0(y)-w_0(x)\le\Phi(x,y),\]
then $w$ coincides with $w_0$ on $C$.
\end{enumerate}
\end{corollary}
For $u:G\to\mathbb R$ let $I(u)$ be the set of points $x\in G$ for which exists
$\gamma:\mathbb R\to G$ such that $\gamma(0)=x$ and $\gamma$ calibrates $u$.
\begin{corollary}\label{calibra-todo}
\[\mathcal A=\bigcap\limits_{u \emph{ dominated}}I(u)\]
\end{corollary}
\begin{proposition}
For each $x,y\in G$ with $x\ne y$ we can find $\varepsilon>0$ and a curve
$\gamma:[-\varepsilon,0]\to G$ such that $\gamma(0)=y$ and for all $t\in[0,\varepsilon]$
\[\Phi(x,\gamma(0))-\Phi(x,\gamma(-t))=\int_{-t}^0L(\gamma,\dot\gamma)+ct.\]
In particular, for each $x\in G$ the function $G\setminus\{x\}\to\mathbb R$;
$y\mapsto\Phi(x,y)$ is a backward weak KAM solution.
\end{proposition}
\begin{theorem}\label{gonzalo}
$\mathcal A$ is nonempty and
if $u:G\to\mathbb R$ is a backward weak KAM solution then
\begin{equation}\label{RF}
u(x)=\min_{q\in\mathcal A}u(q)+h(q,x)
\end{equation}
\end{theorem}
\begin{corollary}\label{gonza}
\[h(x,y)=\min_{q\in\mathcal A}h(x,q)+h(q,y)=\min_{q\in\mathcal A}\Phi(x,q)+\Phi(q,x)\]
\end{corollary}
\section{The Lax semigroup and its convergence}
\subsection{The Lax semigroup}
Let $\mathcal F$ be the set of real functions on $G$, bounded from below.
The backward Lax semigroup $\mathcal L_t:\mathcal F\to\mathcal F$, $t>0$ is defined by
\[\mathcal L_t f (x)= \inf_{y\in G}f(y)+h_t(y,x).\]
It is clear that $f\in \mathcal F$ is dominated if and only if $f\le\mathcal L_tf+ct$
for any $t>0$.
It follows at once that $\mathcal L_t\circ\mathcal L_s=\mathcal L_{t+s}$ and
\begin{equation}
\label{eq:cd}
\|\mathcal L_t f-\mathcal L_t g\|_\infty\le\|f-g\|_\infty
\end{equation}
The proof of the following Lemma is the same as in the compact
manifold case.
\begin{lemma}
Given $\varepsilon>0$ there is $K_\varepsilon>0$ such that for each $u:G\to\mathbb R$ continuous,
$t\ge\varepsilon$, we have $\mathcal L_tu:G\to\mathbb R$ is a Lipschitz with constant $K_\varepsilon$.
\end{lemma}
\begin{theorem}\label{fixed=weak}
A continuous function $u:G\to\mathbb R$ is a fixed point of the semigroup $\mathcal L_t+ct$
if and only if it is a backward weak KAM solution
\end{theorem}
\begin{proof}
Suppose $u:G\to\mathbb R$ is a fixed point of the semigroup $\mathcal L_t+ct$.
For each $T\ge 2$ there is a curve $\alpha_T: [-T, 0]\to G$ such
that $\alpha_T(0)=x$ and
\[u (x)-u (\alpha_T (-T))=A (\alpha_T) + cT. \]
By Lemma \ref{aprioriac} $\{\alpha_T\}$ is uniformly Lipschitz.
As in Propostion \ref{h-wkam}
one obtains a sequence $t_k\to\infty $ and
$\gamma:(-\infty,0]\to G$ such that $\alpha_{t_k}$ converges to
$\gamma$, uniformly on each $[-n, 0]$.
By Lemma \ref {semicontinua}
\begin{align*}
\int_{-n}^0L(\gamma,\dot\gamma)+nc &\le\liminf_{k\to\infty} \int_{-n}^0L(\alpha_{t_k},\dot\alpha_{t_k})+nc\\
&=\liminf_{k\to\infty} u(x)-u(\alpha_{t_k}(-n))\\
&=u(x)-u(\gamma(-n))
\end{align*}
Suppose now that $u:G\to\mathbb R$ is a backward weak KAM solution.
Since $u$ is dominated, $u\le\mathcal L_t u+ct$.
For $x\in G$ let $\gamma:(-\infty,0]\to G$ be such that $\gamma(0)=x$ and for all $t>0$
\[u(x)-u(\gamma(-t))=\int_{-t}^0L(\gamma,\dot\gamma)+ct.\]
Thus
\[u(x)\ge u(\gamma(-t))+h_t(\gamma(t),x)+ct\ge\mathcal L_tu(x)+ct.\]
\end {proof}
From Proposition \ref{h-wkam} and Theorem \ref{fixed=weak} one obtains
\begin{corollary}
The semigroup $\mathcal L_t+ct$ has fixed points.
\end{corollary}
\subsection{Convergence of the Lax semigroup}
\label{sec:lax-converge}
Without loss of generality assume $c=0$. For $u\in C(G)$ define
\begin{equation}
\label{eq:limite}
v(x):=\min_{z\in G}u(z)+h(z,x).
\end{equation}
\begin{proposition}\label{ge}
Let $\psi=\lim\limits_{n\to\infty}\mathcal L_{t_n}u$ for some $t_n\to\infty$, then
\begin{equation}\label{arriba}
\psi\ge v.
\end{equation}
\end{proposition}
\begin{proof}
For $x\in G$ let $\gamma_n:[0,t_n]\to G$ be such that $\gamma_n(t_n)=x$ and
\begin{equation}\label{other}
\mathcal L_{t_n}u(x)=u(\gamma_n(0))+A(\gamma_n).
\end{equation}
Passing to a subsequence if necessary we may assume that
$\gamma_n(0)$ converges to $y\in G$. Taking $\liminf$ in \eqref{other},
we have from item \eqref{unif} of Proposition \ref{propiedades-h}
\[\psi(x)=u(y)+\liminf_{n\to\infty}A(\gamma_n)\ge u(y)+h(y,x). \]
\end{proof}
\begin{proposition}\label{igual}
If $\mathcal L_tu$ converges as $t\to\infty$, then the limit is
function $v$ defined in \eqref{eq:limite}.
\end{proposition}
\begin{proof}
For $x\in G$ let $z\in G$ be such that $v(z)=u(z)+h(z,x)$.
Since $\mathcal L_t u(x)\le u(z)+h_t(z,x)$, we have
\[\lim_{t\to\infty}\mathcal L_t u(x)\le \liminf_{t\to\infty}u(z)+h_t(z,x)= v(z)\]
which together with Proposition \eqref{ge} gives $\lim\limits_{t\to\infty}\mathcal L_t u=v$.
\end{proof}
Thus, given $u\in C(G)$ our goal is to prove that
$\mathcal L_tu$ converges to $v$ defined in \eqref{eq:limite}.
\begin{remark}
Using Corollary \ref{gonza} we can write \eqref{eq:limite} as
\begin{align}\label{eq:limite1}
v(x)&=\min_{y\in\mathcal A}\Phi(y,x)+w(y)\\
\label{eq:w}
w(y)&:= \inf_{z\in G}u(z)+\Phi(z,y)
\end{align}
Item \eqref{max-dom} of Corollary \ref{kam} states that $w$
is the maximal dominated function not exceeding $u$.
Items \eqref{aubry-kam}, \eqref{dom-coinc} of the same Corollary imply that
$v$ is the unique backward weak KAM solution that coincides with $w$
on $\mathcal A$.
\end{remark}
\begin{proposition}\label{dom-conv}
Suppose that $u$ is dominated, then $\mathcal L_tu$ converges uniformly as
$t\to\infty$ to the function $v$ given by \eqref{eq:limite}.
\end{proposition}
\begin{proof}
Since $u$ is dominated, the function $t\mapsto\mathcal L_tu$ nondecreasing.
As well, in this case, $w$ given by \eqref{eq:w} coincides with $u$.
Items \eqref{max-dom} and \eqref{dom-coinc} of
Corollary \ref{kam} imply that $v$ is the maximal dominated function
that coincides with $u$ on $\mathcal A$ and then $u\le v$ on $G$.
Since the semigroup $\mathcal L_t$ is monotone and $v$ is a backward weak KAM solution
\[\mathcal L_tu\le\mathcal L_tv=v\hbox{ for any } t>0.\]
Thus the uniform limit $\lim\limits_{t\to\infty}u$ exists.
\end{proof}
We now address the convergence of $\mathcal L_t$ following the lines in
\cite{DS} which coincide in part with those in \cite{R}.
For $u\in C(G)$ let
\[\omega_\mathcal L(u):=\{\psi\in C(G):\exists t_n\to\infty \hbox{ such that }
\psi=\lim_{n\to\infty}\mathcal L_{t_n}u\}.\]
\begin{align}\label{uu}
\underline{u}(x)&:=\sup\{\psi(x):\psi\in\omega_\mathcal L(u)\}\\
\overline{u}(x)&:=\inf\{\psi(x):\psi\in\omega_\mathcal L(u)\}\label{ou}
\end{align}
From these and Proposition \ref{ge}
\begin{proposition}\label{v<us}
Let $u\in C(G)$, $v$ be the function given by \eqref{eq:limite},
$\underline{u}, \overline{u}$ defined in \eqref{uu} and \eqref{ou}. Then
\begin{equation}\label{eq:v<u}
v\le \overline{u}\le\underline{u}
\end{equation}
\end{proposition}
\begin{proposition}
For $u\in C(G)$, function $\underline{u}$
given by \eqref{uu} is dominated.
\end{proposition}
\begin{proof}
Let $x,y\in G$. Given $\varepsilon>0$ there is $\psi=\lim\limits_{n\to\infty}\mathcal L_{t_n}u$
such that $\underline{u}(x)-\varepsilon<\psi(x).$ For $n>N(\varepsilon)$ and $a>0$
\[\underline{u}(x)-2\varepsilon<\psi(x)-\varepsilon\le \mathcal L_{t_n}u(x)=\mathcal L_a(\mathcal L_{t_n-a}u)(x)\le
\mathcal L_{t_n-a}u(y)+h_a(y,x).\]
Choose a divergent sequence $n_j$ such that $(\mathcal L_{t_{n_j}-a}u)_j$
converges uniformly. For $j>\bar N(\varepsilon)$, $\mathcal L_{t_{n_j}-a}u(y)<\underline{u}(y)+\varepsilon$,
and then
\[\underline{u}(x)-3\varepsilon<\mathcal L_{t_{n_j}-a}u(y)+h_a(y,x)-\varepsilon<\underline{u}(y) +h_a(y,x).\]
\end{proof}
Denote by $\cK$ the family of static curves $\eta:\mathbb R \to G$,
and for $y\in\mathcal A$ denote by $\cK(y)$ the set of curves $\eta\in\cK$
with $\eta(0)=y$.
\begin{proposition}
$\cK$ is a compact metric space with respect to the uniform convergence on
compact intervals.
\end{proposition}
\begin{proof}
Let $\{\eta_n\}$ be a sequence in $\cK$. By Lemma \ref{aprioriac}, $\{\eta_n\}$ is uniformly Lipschitz.
As in Proposition \ref{h-wkam} we obtain a sequence $n_k\to\infty$ such that
$\eta_{n_k}$ converges to $\eta:\mathbb R\to G$ uniformly on each $[a,b]$ and
then $\eta$ is static.
\end{proof}
\begin{proposition}\label{Acoincide}
Two dominated functions that coincide on
$\mathcal M=\bigcup\limits_{\eta\in\cK}\omega(\eta)$ also coincide on $\mathcal A$.
\end{proposition}
\begin{proof}
Let $\varphi_1$, $\varphi_2$ be two dominated functions coinciding on
$\mathcal M$. Let $y\in\mathcal A$ and $\eta\in\cK(y)$. Let $(t_n)_n$ be a
diverging sequence such that $\lim_n\eta(t_n)= x\in\mathcal M$.
By Corollary \ref{est-calibra}
\[\varphi_i(y) = \varphi_i(\eta(0)) - \Phi(y, \eta(0)) = \varphi_i(\eta(t_n)) -
\Phi(y, \eta(t_n))\]
for every $n\in N, i=1,2$. Sending $n$ to $\infty$, we get
\begin{align*}
\varphi_1(y)&= \lim_{n\to\infty}\varphi_1(\eta(t_n)) -\Phi(y, \eta(t_n))
=\varphi_1(x) - \Phi(y, x) =\varphi_2(x) - \Phi(y, x) \\
&= \lim_{n\to\infty}\varphi_2(\eta(t_n)) -\Phi(y, \eta(t_n))
=\varphi_2(y).
\end{align*}
\end{proof}
\begin{proposition}\label{nonincreasing}
Let $\eta\in\cK$, $\psi\in C(G)$ and $\varphi$ be a dominated function.
Then the function
$t\mapsto(\mathcal L_t\psi)(\eta(t))-\varphi(\eta(t))$ is nonincreasing on $\mathbb R_+$.
\end{proposition}
\begin{proof} From Corollary \ref{est-calibra}, for $t<s$ we have
\[(\mathcal L_s\psi)(\eta(s))-(\mathcal L_t\psi)(\eta(t))\le\int_t^s L(\eta(\tau),\dot\eta(\tau))d\tau
=\varphi(\eta(s))-\varphi(\eta(t))\]
\end{proof}
\begin{lemma}\label{perturbacion}
There is a $M>0$ such that, if $\eta$ is any curve in $\cK$ and $\lambda$
is sufficiently close to 1, we have
\begin{equation}\label{ineq}
\int_{t_1}^{t_2}L(\eta_\lambda,\dot{\eta}_\lambda)\le
\Phi(\eta_\lambda(t_1),\eta_\lambda(t_2))+M(t_2-t_1)(\lambda-1)^2
\end{equation}
for any $t_2>t_1$, where $\eta_\lambda(t)=\eta(\lambda t)$.
\end{lemma}
\begin{proof}
Let $K>0$ be a Lipschitz constant for any minimizer $\gamma:[a,b]\to G$ with $b-a>1$,
$2R=\sup\{|L_{vv}(x,v)|:|v|\le K\}$.
For $\lambda\in(1-\delta,1+\delta)$ fixed, using Proposition \ref{energy}
\begin{align*}
\int_{t_1}^{t_2}L(\eta_\lambda(t),\dot\eta_\lambda(t))dt
&=\int_{t_1}^{t_2}[L(\eta(\lambda t),\dot\eta(\lambda t))+(\lambda-1)
L_v(\eta(\lambda t),\dot\eta(\lambda t)) \dot\eta(\lambda t)\\
&+\frac 12(\lambda-1) ^2L_{vv}(\eta(\lambda t),\mu\dot\eta(\lambda t))(\dot\eta(\lambda t))^2]\, dt\\
&\le\lambda \int_{t_1}^{t_2}L(\eta(\lambda t),\dot\eta(\lambda t))\,dt+(t_2-t_1)RK^2(\lambda-1)^2\\
&=\Phi(\eta(\lambda t_1),\eta(\lambda t_2))+(t_2-t_1)RK^2(\lambda-1)^2
\end{align*}
\end{proof}
\begin{proposition}\label{superdiff}
Let $\eta\in\cK$, $\psi\in C(G)$ and $\varphi$ be a dominated function.
Assume that $D^+((\psi-\varphi)\circ\eta)(0)\setminus\{0\}\ne\emptyset$
where $D^+$ denote the super-differential.
Then for all $t>0$ we have
\begin{equation}\label{eq:2}
(\mathcal L_t\psi)(\eta(t))-\varphi(\eta(t))<\psi(\eta(0))-\varphi(\eta(0))
\end{equation}
\end{proposition}
\begin{proof}
Fix $t>0$. By Corollary \ref{est-calibra} it is enough to prove
\eqref{eq:2} for $\varphi=-\Phi(\cdot,\eta(t))$. Since $\mathcal L_t(\psi+a)=\mathcal L_t\psi+a$
we can assume that $\psi(\eta(0))=\varphi(\eta(0))$.
\[(\mathcal L_t\psi)(\eta(t))-\varphi(\eta(t))=(\mathcal L_t\psi)(\eta(t))\le
\int_{(1/\lambda-1)t}^{t/\lambda}L(\eta_\lambda,\dot{\eta}_\lambda)+\psi(\eta((1-\lambda)t)),\]
thus, by Lemma \ref{perturbacion}
\[(\mathcal L_t\psi)(\eta(t))-\varphi(\eta(t))\le
\psi(\eta((1-\lambda)t))-\varphi(\eta((1-\lambda)t))+Mt(\lambda-1)^2.\]
If $m\in D^+((\psi-\varphi)\circ\eta)(0)\setminus\{0\}$, we have
\[(\mathcal L_t\psi)(\eta(t))-\varphi(\eta(t))\le m((1-\lambda)t)+o((1-\lambda)t))+Mt(\lambda-1)^2,\]
where $\lim\limits_{\lambda\to 1}\dfrac{o((1-\lambda)t)}{1-\lambda}=0$.
Choosing appropriately $\lambda$
close to $1$, we get
\[(\mathcal L_t\psi)(\eta(t))-\varphi(\eta(t))<0.\]
\end{proof}
\begin{proposition}\label{superincreasing}
Suppose $\varphi$ is dominated and $\psi\in\omega_\mathcal L(u)$. For any
$y\in\mathcal M$ there exists $\gamma\in\cK(y)$ such that the function
$t\mapsto\psi(\gamma(t))-\varphi(\gamma(t))$ is constant.
\end{proposition}
\begin{proof}
Let $(s_k)_k$ and $(t_k)_k$ be diverging sequences, $\eta$ be a
curve in $\cK$ such that $y=\lim\limits_k \eta(s_k)$, and $\psi$ is
the uniform limit of $\mathcal L_{t_k}u$. As in Proposition \ref{h-wkam},
we can assume that the sequence of functions $t\mapsto\eta(s_k+t )$ converges
uniformly on compact intervals to $\gamma:\mathbb R\to G$, and so $\gamma\in\cK$.
We may assume moreover that $t_k-s_k\to\infty$, as $k\to\infty$, and
that $\mathcal L_{t_k-s_k}u$ converges uniformly to $\psi_1\in\omega_\mathcal L(u)$.
By the semi-group property and \eqref{eq:cd}
\[\|\mathcal L_{t_k}u-\mathcal L_{s_k}\psi_1\|_\infty\le\|\mathcal L_{t_k-s_k}u -\psi_1\|_\infty\]
which implies that $\mathcal L_{s_k}\psi_1$ converges uniformly to $\psi$.
From Proposition \ref{nonincreasing}, we have that for any $\tau\in\mathbb R$
$s\mapsto(\mathcal L_s\psi_1)(\eta(\tau+s))-\varphi(\eta(\tau+s))$ is a nonincreasing
function in $\mathbb R^+$, and hence it has a limit $l(\tau)$ as $s\to\infty$,
which is finite since $l(\tau)\ge-\|\overline u-\varphi\|_\infty$.
Given $t>0$, we have
\[l(\tau) = \lim_{k\to\infty} (\mathcal L_{s_k+t}\psi_1) (\eta(s_k +\tau+t))-\varphi(\eta(s_k+\tau+t)) = (\mathcal L_t\psi) (\gamma(\tau+t))-\varphi(\gamma(\tau+t))\]
The function $t\mapsto(\mathcal L_t\psi)(\gamma(\tau+t))-\varphi(\gamma(\tau+t))$ is therefore constant on $\mathbb R^+$.
Applying Proposition \ref{superdiff} to
the curve $\gamma(\tau +\cdot)\in\cK$, we have
$D^+((\psi-\varphi)\circ\gamma)(\tau)\setminus\{0\} =\emptyset$ for any
$\tau\in\mathbb R$. This implies that $\psi-\varphi$ is constant on $\gamma$.
\end{proof}
\begin{proposition}\label{acercanse}
Let $\eta\in\cK$, $\psi\in\omega_\mathcal L(u)$ and $v$ be defined by \eqref{eq:limite}.
For any $\varepsilon>0$ there exists $\tau\in\mathbb R$ such that
\[\psi(\eta(\tau))-v(\eta(\tau))<\varepsilon.\]
\end{proposition}
\begin{proof}
Since the curve $\eta $ is contained in $\mathcal A$, we have
\[v(\eta(0)) = \min_{z\in G} u(z) + \Phi(z,\eta(0)),\]
and hence $v(\eta(0)) = u(z_0) + \Phi(z_0, \eta(0))$, for some
$z_0\in G$. Take a curve $\gamma:[0,T]\to G$ such that
\[v(\eta(0))+\frac\varepsilon 2 = u(z_0)+\Phi(z_0,\eta(0))+\frac\varepsilon 2
> u(z_0)+\int_0^TL(\gamma,\dot\gamma)\ge\mathcal L_Tu(\eta(0)).\]
Choosing a divergent sequence $(t_n)_n$ such that $\mathcal L_{t_n}u$
converges uniformly to $\psi$ we have for $n$ sufficiently large
\[\|\mathcal L_{t_n}u-\psi\|_\infty<\frac\varepsilon 2, \quad t_n-T>0.\]
Take $\tau=t_n-T$
\begin{align*}
\psi(\eta(\tau))-\frac\varepsilon 2&<\mathcal L_{t_n}u(\eta(\tau)=\mathcal L_{\tau}\mathcal L_Tu\\
&\mathcal L_Tu(\eta(0)+\int_0^\tau L(\eta,\dot\eta)\\
&\frac\varepsilon 2+v(\eta(0))+\int_0^\tau L(\eta,\dot\eta)=\frac\varepsilon 2+v(\eta(\tau))
\end{align*}
\end{proof}
From Propositions \ref{superincreasing} and \ref{acercanse} we obtain
\begin{theorem}\label{Mcoincide}
Let $\psi\in\omega_\mathcal L(u)$ and $v$ be defined by \eqref{eq:limite}. Then $\psi=v$
on $\mathcal M$.
\end{theorem}
\begin{theorem}
Let $u\in C(G)$, then $\mathcal L_tu$ converges uniformly as $t\to\infty$ to $v$
given by \eqref{eq:limite}.
\end{theorem}
\begin{proof}
The function $\underline{u}$
is dominated and coincides with $v$ on $\mathcal M$ by Theorem
\ref{Mcoincide}. Proposition \ref{Acoincide} implies that
$\underline{u}$ coincide with $v$ on $\mathcal A$ and so does with $w$.
By item \eqref{max-dom} of Corollary \ref{kam} we have $\underline{u}\le v$.
\end{proof}
\section{Viscosity solutions of the Hamilton - Jacobi equation}
\label{sec:visco-sol}
In this section we compare weak KAM and viscosity solutions.
\begin{definition}\quad
\begin{itemize}
\item A real function $\varphi$ defined on the neighborhood of $e_l$ is $C^1$ if for
every $j$ with $e_l\in I_j$, $\varphi|I_j$ is $C^1$.
\item A real function $\varphi$ defined on the neighborhood of $(e_l,t) $ is $C^1$
if for every $j$ with $e_l\in I_j$, $\varphi|I_j\times(t-\delta,t+\delta)$ is $C^1$.
\end{itemize}
\end{definition}
Note that if $\alpha: [0,\delta]\to I_j$ is differentiable and $\alpha(0)=e_l$, then
$\alpha_+'(0)\in T^-_{e_l}I_j$ and
we have
\[D^j\varphi(e_l)z=(\varphi\circ \alpha)_+'(0).\]
We consider the Hamiltonian consisting in
functions $H_j:I_j\times\mathbb R\to\mathbb R$ given by
\[H_j(x,p)=\max\left\{-pz-L_j(x,z) :
\begin{array}{ll}z\in T^-_xI_j, & x\in\mathcal V \\
z\in T_xI_j, & x\in I_j\setminus\mathcal V
\end{array}\right\}
\]
and the Hamilton Jacobi equations
\begin{equation}
\label{eq:HJ}
H(x,Du(x))=c,
\end{equation}
\begin{equation}\label{eq:hjt}
u_t(x,t)+H(x,D_xu(x,t))=0.
\end{equation}
Note that if $L$ is symmetric at the vertices, then for any vertix $e_l$ there
is a function $h_a$ such that $H_j(e_l,p)=h_a(|p|)$ for any $j$ with
$e_l\in I_j$. This kind of Hamiltonians are called of eikonal type \cite{CS}.
The following definition appeared in \cite{CS} and \cite{CM}.
\begin{definition}
A function $u:G\to\mathbb R$ is a
\begin{itemize}
\item {\em viscosity subsolution} of \eqref{eq:HJ} if
satisfies the usual definition in $G\setminus\mathcal V$ and for any $C^1$
function $\varphi$ on the neighborhood of any $e_l$ s.t. $u-\varphi$
has a maximum at $e_l$ we have
\[\max\{H_j(e_l,D^j\varphi(e_l)):e_l\in I_j\}\le c.\]
\item {\em viscosity supersolution} of \eqref{eq:HJ} if
satisfies the usual definition in $G\setminus\mathcal V$ and for any $C^1$
function $\varphi$ on the neighborhood of any $e_l$ s.t. $u-\varphi$
has a minimum at $e_l$ we have
\[\max\{H_j(e_l,D^j\varphi(e_l)):e_l\in I_j\}\ge c\]
\item {\em viscosity solution} if it is both, a subsolution and a
supersolution.
\end{itemize}
A function $u:G\times[0,\infty)\to\mathbb R$ is a
\begin{itemize}
\item {\em viscosity subsolution} of \eqref{eq:hjt} if
satisfies the usual definition in $G\setminus\mathcal V\times[0,\infty)$ and for any $C^1$
function $\varphi$ on the neighborhood of any $(e_l,t)$ s.t. $u-\varphi$
has a maximum at $(e_l,t)$ we have
\[\varphi_t(e_l,t)+\max\{H_j(e_l,D^j\varphi(e_l,t)):e_l\in I_j\}\le c.\]
\item {\em viscosity supersolution} of \eqref{eq:HJ} if
satisfies the usual definition in $G\setminus\mathcal V\times[0,\infty)$ and for any $C^1$
function $\varphi$ on the neighborhood of any $(e_l,t)$ s.t. $u-\varphi$
has a minimum at $(e_l,t)$ we have
\[\varphi_t(e_l,t)+\max\{H_j(e_l,D^j\varphi(e_l)):e_l\in I_j\}\ge c\]
\item {\em viscosity solution} if it is both, a subsolution and a supersolution.
\end{itemize}
\end{definition}
\begin{proposition}\label{KAM=viscosity}
If $u:G\to\mathbb R$ is dominated then then it is a viscosity subsolution
of \eqref{eq:HJ}. If $u$ is a backward weak KAM solution then it is a
viscosity solution.
\end{proposition}
\begin{proof} Suppose $u:G\to\mathbb R$ is dominated.
Let $\varphi$ be a $C^1$ function on the neighborhood of $e_l$ s.t. $u-\varphi$
has a maximum at $e_l$, $j$ s.t. $e_l\in I_j$, $\alpha:[0,\delta]\to I_j$ differentiable with
$\alpha(0)=e_l$, $z=\alpha'(0)$. Define $\gamma:[-\delta,0]\to I_j$ by $\gamma(s)=\alpha(-s)$.
\begin{align*}
\varphi(e_l)-\varphi(\gamma(s))&\le u(e_l)-u(\gamma(s))\le\int_s^0L_j(\gamma,\dot\gamma)-cs\\
\frac{\varphi(e_l)-\varphi(\alpha(t)))}t&\le\frac 1t\int_{-t}^0L_j(\gamma,\dot\gamma)+c\\
-D^j\varphi(e_l)z&\le L_j(e_l,z)+c.
\end{align*}
So $u$ is a subsolution.
Let $\varphi$ be a $C^1$ function on the neighborhood of $e_l$ s.t. $u-\varphi$ has a minimum
at $e_l$. Let $\gamma:(-\infty,0]\to G$ be such that $\gamma(0)=e_l$ and for $t<0$
\[u(e_l)-u(\gamma(t))=\int_t^0L_j(\gamma,\dot\gamma)-ct\]
Let $\delta>0, j$ be such that $\gamma([-\delta,0])\subset I_j$.
\[\varphi(e_l)-\varphi(\gamma(s))\ge\int_s^0L_j(\gamma,\dot\gamma)-cs\]
Define $\alpha:[0,\delta]\to I_j$ by $\alpha(t)=\gamma(-t)$, $z=\alpha'(0)$,
\begin{align*}
\frac{\varphi(e_l)-\varphi(\alpha(t))}t&\ge\frac 1t\int_{-t}^0L_j(\gamma,\dot\gamma)+c\\
-D^j\varphi(e_l)z&\ge L_j(e_l,z)+c.
\end{align*}
So $u$ is a supersolution.
\end{proof}
\begin{proposition}\label{laxsemi}
Let $f:G\to\mathbb R$ be continuous and define $u:G\times[0,\infty)\to\mathbb R$ by
$u(x,t)=\mathcal L_tf(x)$, then $u$ is a viscosity solution of \eqref{eq:hjt}
\end{proposition}
\begin{proof}
Since $\mathcal L_tf=\mathcal L_{t-s}(\mathcal L_sf)$ if $0\le s<t$, for any $\gamma:[s,t]\to G$
\begin{equation}
\label{domina}
u(\gamma(t),t)-u(\gamma(s),s)\le\int_s^tL(\gamma,\dot\gamma)
\end{equation}
and for any $x\in G$ there is $\gamma:[s,t]\to G$ with $\gamma(t)=x$ such
that equality in \eqref{domina} holds.
Let $\varphi$ be a $C^1$ function on the neighborhood of $(e_l,t)$ s.t. $u-\varphi$
has a maximum at $(e_l,t)$, $j$ s.t. $e_l\in I_j$, $\alpha:[0,\delta]\to I_j$
differentiable with
$\alpha(0)=e_l$, $z=\alpha'(0)$. Define $\gamma:[t-\delta,t]\to I_j$ by
$\gamma(s)=\alpha(t-s)$.
\begin{align*}
\varphi(e_l,t)-\varphi(\gamma(s),s)&\le
u(e_l,t)-u(\gamma(s),s)\le\int_s^tL_j(\gamma,\dot\gamma)\\
\frac{\varphi(e_l,t)-\varphi(\alpha(t-s),s))}{t-s}&
\le\frac 1{t-s}\int_{s}^tL_j(\gamma,\dot\gamma)\\
\varphi_t(e_l,t)-D^j_x\varphi(e_l,t)z &\le L_j(e_l,z).
\end{align*}
So $u$ is subsolution.
Let $\varphi$ be a $C^1$ function on the neighborhood of $(e_l,t)$ s.t. $u-\varphi$
has a minimum at $(e_l,t)$. Let $\gamma:[t-1,t]\to G$ be such that $\gamma(t)=e_l$ and
\[u(e_l,t)-u(\gamma(t-1),t-1)=\int_{t-1}^tL(\gamma,\dot\gamma)\]
Let $\delta>0, j$ be such that $\gamma([t-\delta,t])\subset I_j$. For $s\in[t-\delta,t]$
\[\varphi(e_l,t)-\varphi(\gamma(s),s)\ge\int_s^tL_j(\gamma,\dot\gamma)\]
Define $\alpha:[0,\delta]\to I_j$ by $\alpha(s)=\gamma(t-s)$, $z=\alpha'(0)$,
\begin{align*}
\frac{\varphi(e_l,t)-\varphi(\alpha(t-s),s)}{t-s}&\ge\frac 1{t-s}\int_s^tL_j(\gamma,\dot\gamma)\\
\varphi_t(e_l,t)-D^j_x\varphi(e_l,t)z&\ge L_j(e_l,z).
\end{align*}
So $u$ is supersolution.
\end{proof}
\begin{proposition}\label{comparacion}
Suppose the Lagrangian is symmetric at the vertices.
Let $u,v:G\times[0,T]\to\mathbb R$ be respectively a Lipschitz viscosity sub, supersolution of
\eqref{eq:hjt} such that $u(x,0)\le v(x,0)$, for any $x\in G$. Then $u\le v$.
\end{proposition}
\begin{proof}
Suppose that there are $x^*,t^*$ such that $\delta=u(x^*,t^*)- v(x^*,t^*)>0$.
Let $0<\rho\le\dfrac\delta{4t^*}$ and define $\Phi:G^2\times[0,T]^2$ by
\[\Phi(x,y,t,s)=u(x,t)-v(y,s)-\frac{d(x,y)^2+|t-s|^2}{2\varepsilon}-\rho(t+s),\]
where $d(x,y)$ is the shortest lenght of a path in $G$ connecting $x$
and $y$, and so $d(x,y)=d(y,x)$
From the previous definitions we have
\begin{equation}
\label{eq:3}
\frac\delta 2\le\delta-2\rho t^*=\Phi(x^*,x^*,t^*,t^*)\le\sup_{G^2\times[0,T]^2}\Phi=
\Phi(x_\varepsilon,y_\varepsilon,t_\varepsilon,s_\varepsilon).
\end{equation}
It follows from
$\Phi(x_\varepsilon,x_\varepsilon,t_\varepsilon,t_\varepsilon)+\Phi(y_\varepsilon,y_\varepsilon,s_\varepsilon,s_\varepsilon)\le 2\Phi(x_\varepsilon,y_\varepsilon,t_\varepsilon,s_\varepsilon)$
that
\begin{align*}
\frac{d(x_\varepsilon,y_\varepsilon)^2+|t_\varepsilon-s_\varepsilon|^2}{2\varepsilon}&\le
u(x_\varepsilon,t_\varepsilon)-u(y_\varepsilon,s_\varepsilon)+v(x_\varepsilon,t_\varepsilon)-v(y_\varepsilon,s_\varepsilon)\\
&\le C(d(x_\varepsilon,y_\varepsilon)^2+|t_\varepsilon-s_\varepsilon|^2)^{1/2}
\end{align*}
Thus, there is a sequence $\varepsilon\to 0$ such that $x_\varepsilon,y_\varepsilon$
converge to $\bar{x}\in G$ and $t_\varepsilon,s_\varepsilon$ converge to
$\bar{t}\in[0,T]$ and \eqref{eq:3} gives
\[\frac\delta 2\le\Phi(\bar{x},\bar{x},\bar{t},\bar{t})\le u(\bar{x},\bar{t})-v(\bar{x},\bar{t}),\]
and so $\bar{t}\ne 0$.
Define the test functions
\begin{align*}
\varphi(x,t)&=v(y_\varepsilon,s_\varepsilon)+\frac{d(x,y_\varepsilon)^2+|t-s_\varepsilon|^2}{2\varepsilon}+\rho(t+s_\varepsilon)\\
\psi(y,s)&=u(x_\varepsilon,t_\varepsilon) -\frac{d(x_\varepsilon,y)^2+|t_\varepsilon-s|^2}{2\varepsilon}-\rho(t_\varepsilon+s).
\end{align*}
\[\varphi_t (x_\varepsilon,t_\varepsilon)=\dfrac{t_\varepsilon-s_\varepsilon}\varepsilon+\rho,
\quad\psi_s(y_\varepsilon,s_\varepsilon)=\dfrac{t_\varepsilon-s_\varepsilon}\varepsilon-\rho\]
Since $u-\varphi$ has maximum at $(x_\varepsilon,t_\varepsilon)$, $v-\psi$ has minimum
at $(y_\varepsilon,s_\varepsilon)$, $u$ is subsolution and $v$ is supersolution,
\begin{align}\nonumber
2\rho=\varphi_t (x_\varepsilon,t_\varepsilon)-\psi_s(y_\varepsilon,s_\varepsilon) \le
& \max\{H_j\bigl(y_\varepsilon,-D^j_y\Bigl(\frac{d(x_\varepsilon,y)^2}{2\varepsilon}\Bigr)(y_\varepsilon)\bigr):
x\in I_j\}\\
- & \max\{H_j\bigl(x_\varepsilon,D^j_x\Bigl(\frac{d(x,y_\varepsilon)^2}{2\varepsilon}\Bigr)(x_\varepsilon)\bigr):
x\in I_j\}
\label{fundamental}
\end{align}
Since $\rho>0$ we can not have $x_\varepsilon=y_\varepsilon$.
If $\bar{x}$ is not a vertix, $\bar{x}\in I_j$, for $\varepsilon>0$ small we have
\[D^j_x\Bigl(\frac{d(x,y_\varepsilon)^2}{2\varepsilon}\Bigr)(x_\varepsilon)=
\pm\frac{d(x_\varepsilon,y_\varepsilon)}{\varepsilon}=-D^j_y\Bigl(\frac{d(x_\varepsilon,y)^2}{2\varepsilon}\Bigr)(y_\varepsilon).\]
If we denote by $a(x_\varepsilon,y_\varepsilon)$ this common value, then \eqref{fundamental} becomes
\[2\rho\le H_j(y_\varepsilon,a(x_\varepsilon,y_\varepsilon))
- H_j\bigl(x_\varepsilon,a(x_\varepsilon,y_\varepsilon))\]
with $a(x_\varepsilon,y_\varepsilon)$ bounded as $\varepsilon\to 0$,
giving a contradiction.
Suppose now that $\bar{x}=e_l$. For $\varepsilon>0$ small we
distinguish the following cases
1. Neither $x_\varepsilon$ nor $ y_\varepsilon$ is a vertix.
If $x_\varepsilon, y_\varepsilon\in I_j$, $d(x_\varepsilon,y_\varepsilon)=|\sigma_j(x_\varepsilon)-\sigma_j(y_\varepsilon)|$.
If $x_\varepsilon\in I_i$, $ y_\varepsilon\in I_j$, and
$e_l\in I_i\cap I_j$, then $d(x_\varepsilon,y_\varepsilon)=d(x_\varepsilon,e_l)+d(e_l,y_\varepsilon)$.
In both subcases
\begin{align*}
|D^i_x\Bigl(\frac{d(x,y_\varepsilon)^2}{2\varepsilon}\Bigr)(x_\varepsilon)|&=
\frac{d(x_\varepsilon,y_\varepsilon)}{\varepsilon}\\
|D^j_y\Bigl(\frac{d(x_\varepsilon,y)^2}{2\varepsilon}\Bigr)(y_\varepsilon)|&=
\frac{d(x_\varepsilon,y_\varepsilon)}{\varepsilon}
\end{align*}
Then \eqref{fundamental} becomes
\[2\rho\le H_j\bigl(y_\varepsilon,\pm\frac{d(x_\varepsilon,y_\varepsilon)}{\varepsilon}\bigr)
- H_i\bigl(x_\varepsilon,\pm\frac{d(x_\varepsilon,y_\varepsilon)}{\varepsilon}\bigr).\]
2. Suppose $x_\varepsilon=e_l$, $y_\varepsilon\in I_j\setminus\mathcal V$.
\begin{align*}
|D^j_x\Bigl(\frac{d(x,y_\varepsilon)^2}{2\varepsilon}\Bigr)(e_l)|&=
\pm\frac{d(e_l,y_\varepsilon)}{\varepsilon}\\
|D^j_y\Bigl(\frac{d(e_l,y)^2}{2\varepsilon}\Bigr)(y_\varepsilon)|&=
\pm\frac{d(e_l,y_\varepsilon)}{\varepsilon}
\end{align*}
Since
\[H_j\bigl(e_l,\pm\frac{d(e_l,y_\varepsilon)}{\varepsilon}\bigr)
=h_a\bigl(e_l,\frac{d(e_l,y_\varepsilon)}{\varepsilon}\bigr),\]
we have that \eqref{fundamental} becomes
\[2\rho\le H_j\bigl(y_\varepsilon,\pm\frac{d(x_\varepsilon,y_\varepsilon)}{\varepsilon}\bigr)
-H_j\bigl(x_\varepsilon,\frac{d(x_\varepsilon,y_\varepsilon)}{\varepsilon}\bigr).\]
3. If $y_\varepsilon=e_l$, $x_\varepsilon\in I_j\setminus\mathcal V$ we get in the same way that
\eqref{fundamental} becomes
\[2\rho\le H_j\bigl(y_\varepsilon,\frac{d(x_\varepsilon,y_\varepsilon)}{\varepsilon}\bigr)
-H_j\bigl(x_\varepsilon,\pm\frac{d(x_\varepsilon,y_\varepsilon)}{\varepsilon}\bigr).\]
Since $\dfrac{d(x_\varepsilon,y_\varepsilon)}{\varepsilon}$ remains bounded as $\varepsilon\to 0$
in all cases, we get a contradiction.
\end{proof}
\begin{corollary}\label{unicidad}
Suppose the Lagrangian is symmetric at the vertices.
Let $u,v:G\times[0,T]\to\mathbb R$ be viscosity solutions of \eqref{eq:hjt}
such that $u(x,0)= v(x,0)$ for any $x\in G$. Then $u=v$.
\end{corollary}
\begin{corollary}\label{viscosity=fixed}
Suppose the Lagrangian is symmetric at the vertices.
Let $f:G\to\mathbb R$ be a viscosity solution of \eqref{eq:HJ}, then
$f$ is a fixed point of the Lax semigroup $\mathcal L_t+ct$.
\end{corollary}
\begin{proof}
We next show that $u(x,t)=f(x)-ct$ is a viscosity solution of \eqref{eq:hjt}.
Proposition \ref{laxsemi} and Corollary \ref{unicidad} then imply that
$f-ct=\mathcal L_tf$.
Let $\varphi$ be a $C^1$ function on the neighborhood of $(e_l,t)$ s.t. $u-\varphi$
has a maximum at $(e_l,t)$. Then $s\to -cs-\varphi(e_l,s)$ has a maximum at
$t$ and so $\varphi_t(e_l,t)=-c$. Since $f-\varphi(\cdot,t)$ has a maximum
at $e_l$ we have
\[\sup\{H_j(x,D^j\varphi(x)):x\in I_j\}\le c=-\varphi_t(e_l,t),\]
so $u$ is a subsolution of \eqref{eq:hjt}. Similarly $u$ is a
supersolution of \eqref{eq:hjt}.
\end{proof}
\begin{corollary}\label{con-unic}
Suppose the Lagrangian is symmetric at the vertices.
Let $u:G\to\mathbb R$ be a viscosity solution of \eqref{eq:HJ} then
the representation formula \eqref{RF} holds.
\end{corollary}
\begin{proof}
By Proposition \ref{KAM=viscosity} and Corollary \ref{viscosity=fixed},
$u$ is a backward weak KAM solution and by Theorem \eqref{gonzalo},
formula \eqref{RF} holds.
\end{proof}
|
2,869,038,156,406 | arxiv | \section{Introduction}\label{sec:prelims}
This paper is dedicated to the memory of Don Burkholder, a great probabilist and a kind man.
Recent work in random spatial networks \citep{AldousGanesan-2013,Aldous-2012} has focussed on specification and analysis of an intriguing class of random networks known as
\emph{scale-invariant random spatial networks} (SIRSN).
Motivated by the success of Google Maps and Bing Maps, \cite{Aldous-2012} shows how a natural collection of desirable properties (statistical invariance under translation, rotation and scale-change,
and some integrability conditions) define a class of models with a useful structure theory.
\begin{defn}[Definition of a SIRSN, \citealp{Aldous-2012}]\label{def:SIRSN}
Consider a \(d\)-dimensional random mechanism, which provides random routes connecting any two points \(x_1,x_2\in\mathbb{R}^d\).
We say that this is a \emph{SIRSN} if the following properties hold:
\begin{enumerate}
\item\label{def:SIRSN-item-route}
Between any specified two points \(x_1,x_2\in\mathbb{R}^d\), almost surely the random mechanism provides just one connecting random route \(\mathcal{R}(x_1,x_2)=\mathcal{R}(x_2,x_1)\), which is a finite-length path connecting \(x_1\) to \(x_2\).
\item\label{def:SIRSN-item-invariance}
For a finite set of points \(x_1,\ldots,x_k\in\mathbb{R}^d\), consider the random network \(\mathcal{N}(x_1,\ldots,x_k)\) formed by the random routes provided by the structure to connect all \(x_i\) and \(x_j\).
Then \(\mathcal{N}(x_1,\ldots,x_k)\) is statistically invariant (strictly speaking, equivariant) under translation, rotation, and re-scaling:
if \(\mathfrak{S}\) is a Euclidean similarity of \(\mathbb{R}^d\) then the networks \(\mathfrak{S}\mathcal{N}(x_1,\ldots,x_k)\) and \(\mathcal{N}(\mathfrak{S}x_1,\ldots,\mathfrak{S}x_k)\)
have the same distribution.
\item\label{def:SIRSN-item-finite-length}
Let \(D_1\) be the length of the route between two points separated by unit Euclidean distance. Then \(\Expect{D_1}<\infty\).
\item\label{def:SIRSN-item-locally-finite}
Suppose that \(\Xi_\lambda\) is a Poisson point process in \(\mathbb{R}^d\), of intensity \(\lambda>0\) and independent of the random mechanism in question.
Then \(\mathcal{N}(\Xi_\lambda)\), the union of all the networks \(\mathcal{N}(x_1,\ldots,x_k)\) for \(x_1,\ldots,x_k\in\Xi_\lambda\), is a locally finite fibre process in \(\mathbb{R}^d\).
That is to say, for any compact set \(K\) the total length of \(\mathcal{N}(\Xi_\lambda)\cap K\) is almost surely finite.
\item\label{def:SIRSN-item-finite-intensity}
The length intensity \(\ell\) of \(\mathcal{N}(\Xi_1)\) (the mean length per unit area) is finite.
\item\label{def:SIRSN-item-SIRSN}
Suppose the Poisson point processes \(\{\Xi_\lambda:\lambda>0\}\) are coupled so that \(\Xi_{\lambda_1}\supseteq\Xi_{\lambda_2}\) if \(\lambda_1<\lambda_2\). The fibre process
\[
\bigcup_{\lambda>0}\bigcup_{x_1,x_2\in\Xi_\lambda} \left( \mathcal{R}(x_1,x_2)\setminus(\operatorname{\mathcal B}(x_1,1)\cup\operatorname{\mathcal B}(x_2,1)) \right)
\]
has length intensity bounded above by a finite constant \(p(1)\).
\end{enumerate}
If only properties \ref{def:SIRSN-item-route}-\ref{def:SIRSN-item-finite-intensity} are satisfied, then the random mechanism is called a \emph{weak SIRSN}.
If only properties \ref{def:SIRSN-item-route}-\ref{def:SIRSN-item-locally-finite} are satisfied, then the random mechanism is called a \emph{pre-SIRSN}.
\end{defn}
\cite{AldousGanesan-2013} describe the \emph{binary hierarchy model},
a structure for providing planar routes, based on minimum-time paths using a dyadic grid furnished with speeds and uniformly randomized in orientation and position.
\cite{Aldous-2012} proves that this is a full planar SIRSN satisfying all the requirements of Definition \ref{def:SIRSN}.
\cite{AldousGanesan-2013} also propose two other candidates for planar SIRSNs which do not involve the somewhat unnatural randomization required for the binary hierarchy model:
one is based on route-provision \emph{via} a scale-invariant improper Poisson line process marked with random speeds
(the \emph{Poisson line process model}); and the other uses a dynamic proximity graph related to the Gabriel graph.
The purpose of the present paper is to explore the Poisson line process model: we will show that it is at least a pre-SIRSN if \(d=2\), and moreover we will show that even in dimension \(d>2\)
the construction provides a random metric space on \(\mathbb{R}^d\) (in particular it satisfies at least properties \ref{def:SIRSN-item-route}-\ref{def:SIRSN-item-invariance}
of Definition \ref{def:SIRSN}, with the possible exception of uniqueness of route).
This therefore establishes the significance of the Poisson line process model as a scale-invariant random spatial network, while leaving open the question of whether it is a weak SIRSN or
even a full SIRSN, not just a pre-SIRSN.
The chief difficulty in analyzing any of these random mechanisms lies in the fact that it is hard to work with explicit minimum-time paths, whose explicit construction would involve solving a non-local minimization problem to determine geodesics.
\citet{AldousKendall-2007} and \citet{Kendall-2011b,Kendall-2014a} use approximations known as ``near-geodesics'', constructed using a kind of greedy algorithm.
\citet{BaccelliTchoumatchenkoZuyev-2000} and \citet{BroutinDevillerHemsley-2014} study Delaunay tessellation paths that are determined using either their relationship to appropriate Euclidean straight lines or the so-called ``cone walk''.
\citet{LaGatta-2011} studies geodesics determined by random \emph{smooth} Riemannian structures, for which conventional calculus methods are available.
In the following, we argue for existence of minimum-time paths by exploiting properties of a
Sobolev space of paths, and then by using indirect arguments.
The structure of the paper is as follows.
The rest of this introduction (Section \ref{sec:prelims}) is concerned with basic notions of stochastic geometry (Subsection \ref{sec:notation})
and with the definition of the underlying improper Poisson line process \(\Pi\) marked with speeds (Subsection \ref{sec:improper}).
This improper Poisson line process \(\Pi\) is defined by an intensity measure \((\gamma-1) v^{-\gamma}\d{v}\,\mu_d(\d{\ell})\) (for speed \(v>0\), parameter \(\gamma>1\), and invariant measure \(\mu_d\) on line-space)
and supplies a measurable orientation field marked by speeds:
Section \ref{sec:lipschitz-paths} then explores the way in which the measurable orientation field can be integrated to provide Lipschitz paths based on the marked line process, namely \(\Pi\)-paths.
Sobolev space and comparison arguments can then be used to establish \emph{a priori} bounds on Lipschitz constants for finite-time \(\Pi\)-paths (Theorem \ref{thm:a-priori-bound}),
hence closure, weak closure, and finally weak compactness (Corollary \ref{cor:compactness}) of finite-time \(\Pi\)-paths. All these results require \(\gamma\geq d\).
Note that dimension \(d>1\) if line-process theory is to be non-vacuous.
Section \ref{sec:pi-paths and points} shows that, given \(\gamma>d\) and fixed points \(x_1,x_2\in\mathbb{R}^d\), it is almost surely possible to connect \(x_1\) to \(x_2\) in finite time with \(\Pi\)-paths
(Theorem \ref{thm:connection}), and indeed with probability \(1\) it is possible to connect \emph{all} pairs of points in this way (Theorem \ref{thm:metric-space}).
Combined with Corollary \ref{cor:compactness}, this implies the existence of minimum-time \(\Pi\)-paths, namely \emph{\(\Pi\)-geodesics} (Definition \ref{def:geodesic}, Corollary \ref{cor:geodesics}).
In dimension \(d>2\) this is a rather unexpected result, since almost surely none of the lines of \(\Pi\) will then intersect.
Nevertheless, \(\Pi\) then furnishes \(\mathbb{R}^d\) with the structure of a random geodesic metric space.
In these higher dimensions it is difficult to imagine what a \(\Pi\)-geodesic might look like (Figure \ref{fig:dumbbell} illustrates the easier \(d=2\) case):
however Definition \ref{def:near-sequential-pi-path}, Theorem \ref{thm:pi-path-approximation} and Corollary \ref{cor:pi-path-approximation} describe a class of ``\(\varepsilon\)-near-sequential-\(\Pi\)-paths''
which can be used to approximate (and to simulate) \(\Pi\)-geodesics (Theorem \ref{thm:near-sequential-compactness}).
In particular these results imply measurability of the random time taken to pass from one point to another using a \(\Pi\)-geodesic (Corollary \ref{cor:measurable-time}).
The remainder of the paper is restricted to the planar case of \(d=2\), since the arguments now make essential use of point-line duality.
Consider the extent to which networks formed by \(\Pi\)-geodesics fulfil the requirements of Definition \ref{def:SIRSN}.
The statistical invariance property \ref{def:SIRSN-item-invariance} follows immediately from similar invariance of the underlying intensity measure of the improper Poisson line process
(whether planar or not).
Property \ref{def:SIRSN-item-route} requires almost sure uniqueness of network routes:
Section \ref{sec:pi-geodesics-uniqueness} establishes this for \(\gamma>d=2\) (Theorem \ref{thm:uniqueness}),
using a careful analysis of the nature of planar \(\Pi\)-geodesics (Theorem \ref{thm:encounters}) which falls just short of establishing that planar \(\Pi\)-geodesics can be made up of consecutive sequences of line segments.
While \(\Pi\)-geodesics between pairs of points are minimum-time paths, the fact that they have finite mean length is not immediately apparent;
this is established in Section \ref{sec:pi-geodesics-finite-mean-length}, first for restricted planar \(\Pi\)-geodesics (Lemma \ref{lem:finite-mean-length-in-ball}),
then for general planar \(\Pi\)-geodesics (Theorem \ref{thm:finiteness-of-mean}).
Thus the finite-mean-length property \ref{def:SIRSN-item-finite-length} of Definition \ref{def:SIRSN} is verified for \(d=2\).
Finally the pre-SIRSN property \ref{def:SIRSN-item-SIRSN} is established for the planar case in Theorem \ref{thm:pre-SIRSN} of Section \ref{sec:properties};
here also is established the uniqueness of planar \(\Pi\)-geodesics reaching out to infinity (Theorem \ref{thm:unique-to-infinity}) and,
for any specified point \(x\in\mathbb{R}^2\),
the fact that all \(\Pi\)-geodesics emanating from \(x\) must coincide for initial periods (Theorem \ref{thm:coalescence-of-geodesics}).
These results are established using an essentially soft argument concerning the existence of certain structures in \(\Pi\) (Lemma \ref{lem:pre-SIRSN-structure});
the concluding Section \ref{sec:conclusion} notes that more quantitative arguments would be required to decide whether the weak SIRSN or full SIRSN properties hold.
Section \ref{sec:conclusion} also notes some other interesting open questions.
\subsection[Notation and basic results]{Notation and basic results for random line processes}\label{sec:notation}
Random line processes (random patterns of lines) play a fundamental r\^ole in this study.
Here we review notation and basic results for un-sensed random line processes in Euclidean space, as described in \citet[Chapter 8]{ChiuStoyanKendallMecke-2013}.
(By an ``un-sensed line'', we mean a line without preferred direction.)
The corresponding theory for sensed lines follows from the observation that the space of sensed lines forms a double cover of the space of un-sensed lines.
Consider \emph{line-space}, the space \(\mathcal{L}^d\) of all un-sensed lines in \(\mathbb{R}^d\), for dimension \(d\geq2\).
In the planar case \(d=2\) there is a natural geometric representation of \(\mathcal{L}^2\) as a punctured projective plane,
since there is a \(3\)-space construction of the family of planar lines as the family of intersections of \(2\)-subspaces with a reference plane (say \(x_3=1\)).
More visually, but less naturally, \(\mathcal{L}^2\) can be viewed as a M\"obius band of infinite width.
Similar but less graphic geometric descriptions of \(\mathcal{L}^d\) (and its sensed counterpart) can be given in higher dimensional cases (\(d>2\)):
for example, the space of \emph{sensed} lines in \(\mathbb{R}^d\) can be represented using the standard immersion of the tangent bundle \(TS^{d-1}\) of the \((d-1)\)-sphere in \(\mathbb{R}^d\).
It is convenient to introduce notation for hitting events and hitting sets. For a line \(\ell\in\mathcal{L}^d\) and for \(K\) a compact subset of \(\mathbb{R}^d\), we write
\begin{equation}\label{eqn:def-hit}
\ell \Uparrow K
\end{equation}
for the statement that \(\ell\) intersects \(K\).
We also introduce the \emph{hitting set} of \(K\) (the set of lines that hit \(K\)):
\begin{equation}\label{eqn:hitting-set}
\hittingset{K} \quad=\quad
\left\{\ell\in\mathcal{L}^d\;:\; \ell\Uparrow K\right\}\,.
\end{equation}
General arguments show that there exists a measure on \(\mathcal{L}^d\) that is invariant under Euclidean isometries and unique up to a scaling factor.
Line-space \(\mathcal{L}^d\) can be constructed as the quotient space of the group of \(d\)-dimensional rigid motions by the subgroup that leaves a specified line invariant.
The existence of invariant measure on line-space
follows from the study of quotient measures for locally compact topological groups; a conceptual and general treatment of existence and uniqueness
is given by \citet[Section 2.3]{AbbaspourMoskowitz-2007} (see also \citealp[pp.~130-133]{Loomis-1953}), and follows here from unimodularity of the two groups in question.
\citet[Chapter 10]{Santalo-1976} and \citet{Ambartzumian-1990} describe alternative approaches that are direct but are computational rather than conceptual.
\begin{defn}\label{def:invariant-line-measure}
\emph{Invariant line measure} \(\mu_d(\d{\ell})\) is the unique measure on \(\mathcal{L}^d\) that is invariant under Euclidean isometries and is normalized by
the following requirement:
for all compact convex sets \(K\subset\mathbb{R}^d\) of non-empty interior (``convex bodies''),
the \(\mu_d\)-measure of the hitting set \(\hittingset{K}\) is half the Hausdorff \((d-1)\)-dimensional measure of the boundary of \(K\):
\begin{equation}\label{eqn:normalized-measure}
\mu_d(\hittingset{K})\quad=\quad\frac12 m_{d-1}(\partial K)\,.
\end{equation}
\end{defn}%
Here and in the following, \(m_{d-1}\) denotes Hausdorff measure of dimension \(d-1\).
The purpose of the normalization factor \(\tfrac12\) is to ensure that the \(\mu_d\)-measure of the hitting set of a fragment \(A\) of a flat hyper-surface is equal to
its hyper-surface area \(m_{d-1}(A)\).
In the important special case of \(d=2\),
we can parametrize an un-sensed line \(\ell\in\mathcal{L}^2\) by (a) the angle \(\theta=\theta(\ell)\in[0,\pi)\) that it makes with a reference line (say, the \(x\)-axis),
and (b) the \emph{signed} distance \(r=r(\ell)\) between the line \(\ell\) and a reference point
(conventionally taken to belong to the reference line; say, the origin \(\text{\textbf{o}}=(0,0)\)).
Equation \eqref{eqn:normalized-measure} then takes a more explicit form:
\begin{equation}\label{eqn:planar-normalized-measure1}
\mu_2(\d{\ell})\quad=\quad\frac12 \;\d{r}\,\d{\theta}\,.
\end{equation}
More generally, the line measure \(\mu_d(\d{\ell})\) can be disintegrated using \((d-1)\)-dimensional Hausdorff measure on the hyperplane perpendicular to \(\ell\).
Let \(\varpi\) be the un-sensed direction of \(\ell\) and let \(y\) be its point of intersection on the perpendicular hyperplane.
Let \(\kappa_{s}=\tfrac{s/2}{\Gamma(1+s/2)}\) denote the \(s\)-dimensional volume of the unit ball in \(\mathbb{R}^s\), and for later convenience let \(\omega_{s-1}=s\kappa_s\)
denote the hyper-surface area of its boundary. Then
\begin{equation}\label{eqn:disintegration1}
\mu_d(\d\ell) \quad=\quad \frac{1}{\kappa_{d-1}} \; m_{d-1}(\d y)\, m_{S_+^{d-1}}(\d\varpi)\,,
\end{equation}
where the measure \(m_{S_+^{d-1}}\) is defined on the space of un-sensed line directions and can be thought of as \((d-1)\)-dimensional Hausdorff measure on the unit hemisphere \(S_+^{d-1}\)
in \(\mathbb{R}^d\). Proper interpretation of the representation \eqref{eqn:disintegration1} requires the space of un-sensed directions to
be considered as a further projective space,
and the product measure to be twisted to take account of the fact that \(m_{d-1}\) here is defined on the hyperplane normal to the un-sensed direction of the line in question.
However the resulting discrepancies are confined to a null-set which can be ignored when considering invariant Poisson line processes.
An alternative representation, useful for certain calculations,
describes \(\mu_d\) in terms of the intersection of \(\ell\) with a fixed reference hyperplane. In two dimensions we obtain
\begin{equation}\label{eqn:planar-normalized-measure2}
\mu_2(\d{\ell})\quad=\quad\frac12 \, \sin\theta \; \d{p}\, \d{\theta}\,,
\end{equation}
where \(p=p(\ell)\) is the signed distance from the reference point \(\text{\textbf{o}}\) to the intersection of \(\ell\) with the reference line.
This alternative representation is defective: if \(\theta=0\) then there is no intersection and so \(p\) is ill-defined.
However once again the resulting discrepancies are confined to a null-set which can be ignored when considering invariant Poisson line processes.
In higher dimensions the corresponding representation is
\begin{equation}\label{eqn:disintegration2}
\mu_d(\d\ell) \quad=\quad \frac{\sin\theta}{\kappa_{{d-1}}} \;m_{d-1}(\d z)\, m_{S_+^{d-1}}(\d\varpi) \,,
\end{equation}
where \(\theta\) is the angle made by the un-sensed direction \(\varpi\) of the line \(\ell\) with the fixed reference hyperplane, and \(z\) locates the intersection of \(\ell\) with the reference hyperplane.
Note finally that the arguments of this paper depend only on the general forms of Equations (\ref{eqn:planar-normalized-measure1}, \ref{eqn:disintegration1}, \ref{eqn:planar-normalized-measure2}, \ref{eqn:disintegration2});
the exact constants involved
are not crucial.
\subsection{Improper Poisson line processes}\label{sec:improper}
Our constructions use \emph{Poisson} line processes. A unit-intensity Poisson line process in \(\mathbb{R}^d\) is obtained simply
by generating a Poisson point process on the corresponding representing space \(\mathcal{L}^d\) using the invariant measure
\(\mu_d\). It is a geometric consequence of the \(\sigma\)-finiteness of \(\mu_d\) that the resulting random line pattern is locally finite: only finitely many lines hit any given compact set.
However our constructions will use \emph{improper} Poisson line processes, which can be viewed as superpositions of infinitely many independent Poisson line processes,
different line processes being thought of as representing highways with speed limits lying in different ranges.
If we augment the representation space by a mark space \((0,\infty)\) of speed-limits, then the improper Poisson
line process can be represented as a Poisson point process on \(\mathcal{L}^d\times(0,\infty)\), with a \(\sigma\)-finite intensity measure on \(\mathcal{L}^d\times(0,\infty)\) which is invariant under rigid motions but which
does not project down onto a \(\sigma\)-finite intensity measure on \(\mathcal{L}^d\).
Thus the main actors in this account are invariant {improper} un-sensed Poisson line processes,
with each line \(\ell\) being marked by a different positive speed-limit \(v=v(\ell)>0\).
Scaling arguments \citep{Aldous-2012,AldousGanesan-2013} lead to a natural family of intensity measures
for such a marked line process, based on
a positive parameter \(\gamma>1\):
\begin{equation}\label{eqn:improper}
(\gamma-1) \, v^{-\gamma} \,\d{v} \;\mu_d(\d{\ell})\,.
\end{equation}
The factor \(\gamma-1\) ensures that for all \(\gamma>1\) the sub-process of lines with marks \(v>1\) forms a unit-intensity Poisson line process which is of \emph{unit intensity}, in the sense that its mean intensity is the invariant measure given in \eqref{eqn:normalized-measure}, so that the mean number of lines hitting a flat fragment of hyper-surface is equal to its hyper-surface area.
In case \(d=2\) we may write this intensity measure as \(\frac12 (\gamma-1) v^{-\gamma}\,\d{v}\,\d{r}\,\d{\theta}\).
Fixing a general dimension \(d\) and parameter \(\gamma>1\), let \(\Pi=\Pi^{(d,\gamma)}\) denote the resulting random process of marked lines \((\ell, v(\ell))\).
In the following, the dependence on \(d\) and \(\gamma\) will be clear from the context,
and consequently will be suppressed. Figure \ref{fig:dumbbell} illustrates the formation of minimum-time routes between two fixed collections of nodes, for varying values of the parameter \(\gamma>2\).
Note that spatial networks formed in this way will automatically satisfy
property \ref{def:SIRSN-item-invariance} of Definition \ref{def:SIRSN}, because of the invariance properties of the intensity measure \eqref{eqn:improper}.
\begin{Figure}
\centering
\includegraphics[width=1.5in]{dumbbell-image-n128-g2-1-s2.jpg}\hfil
\includegraphics[width=1.5in]{dumbbell-image-n128-g4-0-s2.jpg}\hfil
\includegraphics[width=1.5in]{dumbbell-image-n128-g8-0-s2.jpg}\hfil
\includegraphics[width=1.5in]{dumbbell-image-n128-g16-0-s2.jpg}
\caption{\label{fig:dumbbell}
Minimum-time routes between two separated collections of nodes for networks built from the improper Poisson line process described in Section \ref{sec:improper},
for parameter \(\gamma\) taking values \(2.1,4.0,8.0,16.0\). Lighter segments have lower speed-limits.
Note that as \(\gamma\) increases so the routes become more direct, but also there is less route-sharing.
}
\end{Figure}
The intensity measure gives infinite mass to the set of lines intersecting any convex body, and therefore the union
of all lines from \(\Pi\) is \emph{not}
a random closed set.
Consequently it is not possible to make direct application of the classic theory of random closed sets (as surveyed, for example, in \citealp[Chapter 6]{ChiuStoyanKendallMecke-2013}).
Indeed the union of all lines from \(\Pi\)
is almost surely everywhere dense, and the theory for such sets is obscure (see for example \citealp{AldousBarlow-1981,Kendall-2000b}).
We therefore focus on sub-patterns of lines subject to a positive lower bound on their speeds.
Consider the set of lines hitting a convex body \(K\) and having speed-limits no slower than \(v_0>0\):
\[
\left\{ (\ell, v)\;:\; \ell\in\hittingset{K} \text{ and } v\geq v_0\right\}\,.
\]
This has finite intensity measure, since \(\gamma>1\).
It follows that the full set of marked lines \(\left\{ (\ell,v) : \ell\in\hittingset{K}\right\}\) can be expressed as
a countable union of random closed sets (indeed, locally finite unions of lines) when broken up according to different ranges of speed-limit.
Hence the union of all these lines does have a natural representation as a random \(F_\sigma\).
Indeed it can be related to random closed set theory as follows.
\begin{defn}\label{def:silhouette}
For given \(d\geq2\) and \(\gamma>1\), and fixed \(v_0>0\),
let \(\Pi_{v_0}\) denote the \emph{proper} marked Poisson line process of all lines with speed-limits no slower than \(v_0\):
\[
\Pi_{v_0}\quad=\quad \left\{(\ell,v)\in\Pi\;:\; v\geq v_0\right\}\,.
\]
The \emph{silhouette} \(\mathcal{S}_{v_0}\) of \(\Pi_{v_0}\) is the random closed set which is the union of all lines in \(\Pi_{v_0}\):
\begin{equation}\label{eqn:silhouette}
\mathcal{S}_{v_0}\quad=\quad
\bigcup\left\{\ell\;:\;(\ell, v)\in\Pi_{v_0}\right\}
\quad=\quad\bigcup\left\{\ell\;:\;(\ell,v)\in\Pi \text{ and } v\geq v_0\right\}\,.
\end{equation}
So \(\mathcal{S}_{0+}=\bigcup_{v_0>0}\mathcal{S}_{v_0}\) can be viewed as a random \(F_\sigma\).
\end{defn}
Note that almost surely the \emph{unmarked} line process \(\{\ell:(\ell,v)\in\Pi_{v_0}\}\) can be recovered from the silhouette \(\mathcal{S}_{v_0}\).
Moreover we can recover \(\Pi\) in entirety from
the details of the changes in \(\mathcal{S}_v\) as \(v\) varies, since \(\mathcal{S}_v\setminus\mathcal{S}_{v+}\) is the locally finite union of
lines whose speed-limits are exactly equal to \(v\).
Indeed \(\Pi_v =\Pi_{v_-}=\bigcap_{v<v_0}\Pi_v\) for \(v_0>0\), and \(\Pi_v\) changes only at countably
many values of \(v>0\), and, almost surely, for all \(v>0\) if \(\mathcal{S}_v\setminus\mathcal{S}_{v+}\) is non-empty then it is composed of just one line.
Thus
\[
\Pi\quad=\quad
\left\{(v,\ell)\;:\;\ell= \mathcal{S}_v\setminus\mathcal{S}_{v_+} \text{ for some } v>0\right\}\,.
\]
For notational convenience we introduce
the \emph{maximum speed-limit function} \(V\) holding everywhere on \(\mathbb{R}^d\) and imposed by \(\Pi\).
This is a random upper-semicontinuous function
\(V:\mathbb{R}^d\to[0,\infty)\) defined in terms of its upper level sets:
\begin{equation}\label{eqn:speed-limit}
\left\{x\;:\; V(x)\geq v_0\right\} \quad=\quad \mathcal{S}_{v_0}
\quad=\quad\bigcup\left\{\ell\;:\;(\ell,v)\in\Pi_{v_0}\right\}\,.
\end{equation}
As with random dense line patterns, there is no satisfactory theory for general random upper-semicontinuous functions: we
use \(V\) merely as a convenient mathematical short-hand for the filtration of random closed sets \(\{\mathcal{S}_v:v>0\}\).
\section[Lipschitz paths]{Lipschitz paths and networks}\label{sec:lipschitz-paths}
This section introduces the notion of \(\Pi\)-paths; locally Lipschitz paths traversing a system of ``roads'' (supplied by \(\Pi\)) furnished with speed-limits \emph{via} the maximum speed-limit function \(V\).
We will formulate this notion carefully and prove results yielding a variational context within which to study the minimum-time \(\Pi\)-paths (``\(\Pi\)=geodesics'') between specified points.
Care is needed, because we cannot assume that such paths are built up using consecutive sequences of intervals spent on different roads (and indeed this absolutely cannot be the case for dimension \(d>2\)).
Instead the \(\Pi\)-paths are best viewed using such intervals arranged in a tree pattern rather than ordered sequentially (compare the use of trees to represent bounded variation paths in \citealp{HamblyLyons-2010}).
From henceforth we shall fix a dimension \(d\geq2\) (since the case \(d=1\) is trivial) and a parameter \(\gamma>1\)
(note however that the discussion
of this section will lead to imposition of progressively more severe
constraints on \(\gamma\)).
Recall (for example, from \citealp[ch.5]{Evans-1998}) that a Lipschitz curve \(\xi:[0,T)\to\mathbb{R}^d\) satisfies \(|\xi(s)-\xi(t)|\leq A|s-t|\) when \(0\leq s<t<T\),
for some constant \(A>0\).
The least such \(A\) is the \emph{Lipschitz constant} \(\text{Lip}(\xi)\).
The Lipschitz property for \(\xi\) using constant \(A\) holds if and only if \(\xi\) is absolutely continuous with almost-everywhere defined weak derivative \(\xi'\), with \(\operatorname{ess\ sup}_t|\xi'(t)|\leq A\).
We first discuss two preparatory results; a simple
lemma relating the velocity of a general Lipschitz path (defined for almost all time) to the directions of lines which it visits,
and a corollary concerning the way in which such a Lipschitz path behaves
at the intersections formed by a pattern of lines. Intuitively speaking, if the path velocity has non-zero component normal to a given line
then it must move away immediately, so for almost all time either the path
runs on the line and has velocity parallel to the line, or the path does not lie on the line at all.
\begin{lemma}\label{lem:direction}
Let \(\xi:[0,T)\to\mathbb{R}^d\) be a locally Lipschitz path and let \(\ell\) be a line, both lying in \(d\)-dimensional space \(\mathbb{R}^d\).
Suppose that \(\underline{e}\) is a unit vector parallel to the direction of \(\ell\).
Then the time-set
\[
\left\{t\in[0,T)\;:\; \xi(t)\in\ell \text{ and }\xi^\prime(t)\neq \langle\xi^\prime(t),\underline{e}\rangle\underline{e}\;\right\}
\]
is a Lebesgue-null subset of \([0,T)\).
\end{lemma}
\begin{proof}
Let \(P\) denote projection onto the hyperplane normal to \(\ell\).
The line \(\ell\) projects under \(P\) to a singleton point which we denote by \(P\ell\).
Restricting to compact subsets of \([0,T)\) if necessary, it suffices to treat the case in which \(\xi\) is globally Lipschitz over \([0,T)\); let \(\alpha=\text{Lip}(\xi)\) be the corresponding Lipschitz constant, so that
\[
|\xi^\prime(t)| \quad\leq\quad \alpha \qquad \text{for almost all }t\in[0,T)\,.
\]
The set \(\{t\in[0,T):P\xi(t)=P\ell \text{ and }P\xi^\prime(t)\neq0\}\) is the countable union of time-sets
\[
A^\pm_{j,n} \quad=\quad \left\{t\in[0,T):P\xi(t)=P\ell \text{ and }\pm\langle P(\xi^\prime)(t), \underline{e}_j\;\rangle>\tfrac1n\right\}\,,
\quad\text{for } j=1, \ldots, d-1\,,
\]
where \(\underline{e}_1, \ldots, \underline{e}_{d-1}\) are orthogonal unit vectors perpendicular to \(\ell\),
so it suffices to show each \(A^\pm_{j,n}\) is Lebesgue-null (note that \(P\xi^\prime\) is not continuous, so that \(A^\pm_{j,n}\) may not be open).
Without loss of generality, fix attention on \(A^+_{1,n}\); we will complete the proof by showing that this is Lebesgue-null.
Fix arbitrary \(\varepsilon>0\) and cover \(A^+_{1,n}\) by a disjoint countable union of closed bounded intervals \(F=\bigcup_i [a_i,b_i]\), such that
\begin{equation}\label{eq:residue}
\operatorname{Leb}(F\setminus A^+_{1,n})\quad<\quad \frac{\varepsilon}{\alpha n}\,.
\end{equation}
Since \(\xi\) is continuous, we may shrink each interval \([a_i,b_i]\) so as to ensure that \(P\xi=P\ell\) at \(a_i\) and \(b_i\), while still preserving \eqref{eq:residue} and the
covering property \(A^+_{1,n}\subseteq F\). Since \(P\xi=P\ell\) on the end-points of each \([a_i,b_i]\),
\[
\int_{F\setminus A^+_{1,n}} P\xi^\prime(t)\d{t} + \int_{A^+_{1,n}} P\xi^\prime(t)\d{t}
\quad=\quad \int_F P\xi^\prime(t)\d{t}\quad=\quad \sum_i (P\xi(b_i)-P\xi(a_i))\quad=\quad 0\,.
\]
However we can apply the Lipschitz property of \(\xi\) to show that
\begin{equation*}
\alpha \operatorname{Leb}(F\setminus A^+_{1,n})
\quad\geq\quad \left|\int_{F\setminus A^+_{1,n}} P\xi^\prime(t)\d{t}\right|
\quad\geq\quad \left|\int_{A^+_{1,n}} \langle P\xi^\prime(t),\underline{e}_1\rangle\d{t}\right|\\
\quad\geq\quad \frac{1}{n}\operatorname{Leb}(A^+_{1,n})
\end{equation*}
and therefore
\[
\operatorname{Leb}(A^+_{1,n}) \quad\leq\quad \alpha \, n \operatorname{Leb}(F\setminus A^+_{1,n})\quad\leq\quad \varepsilon\,.
\]
Since \(\varepsilon>0\) is arbitrary, the result follows.
\end{proof}
As a direct consequence of Lemma \ref{lem:direction}, the only way in which a Lipschitz path can spend positive time on the intersection
of two distinct lines is by being at rest.
\begin{cor}\label{cor:network-intersections}
Consider a network in \(\mathbb{R}^d\) formed by the union of a countable number of distinct lines \(\ell_1\), \(\ell_2\), \ldots, and form the
\emph{intersection point pattern}
\[
\mathcal{I} \quad=\quad \bigcup_{1\leq i < j < \infty} \ell_i\cap\ell_j\,.
\]
If \(\xi:[0,T)\to\mathbb{R}^d\) is a locally Lipschitz curve in \(\mathbb{R}^d\) then
the time-set
\begin{equation}\label{eqn:nullity}
\left\{t\in[0,T)\;:\; \xi(t)\in\mathcal{I} \text{ and } |\xi^\prime(t)|>0\right\}
\end{equation}
must be a Lebesgue-null subset of \([0,T)\).
\end{cor}
\begin{proof}
Since \(\mathcal{I}\) is a countable union of points, it suffices to consider two distinct lines \(\ell_1\) and \(\ell_2\)
with non-empty intersection.
Let \(\underline{e}_i\) be a unit vector parallel to the direction of \(\ell_i\). Note that, since the \(\ell_i\) are distinct and meet (and therefore cannot be parallel), it follows that
the unit vectors \(\underline{e}_1\) and \(\underline{e}_2\) must be linearly independent.
By Lemma \ref{lem:direction}
the following two time-sets are both Lebesgue-null:
\begin{eqnarray*}
&\left\{t\in[0,T)\;:\; \xi(t)\in\ell_1 \text{ and } \xi^\prime(t) \neq \langle\xi^\prime(t),\underline{e}_1\rangle\underline{e}_1\;\right\}\,,\\
&\left\{t\in[0,T)\;:\; \xi(t)\in\ell_2 \text{ and } \xi^\prime(t) \neq \langle\xi^\prime(t),\underline{e}_2\rangle\underline{e}_2\;\right\}\,.
\end{eqnarray*}
If \(\xi^\prime\) is
simultaneously parallel to the two
linearly independent \(\underline{e}_i\) then it must vanish. Consequently
the time-set defined by Equation \eqref{eqn:nullity} is a subset of the union of these two sets, and is therefore Lebesgue-null.
\end{proof}
We now define the fundamental notion explored in this paper.
\begin{defn}\label{def:pi-path}
A \emph{\(\Pi\)-path} is a locally Lipschitz path in \(\mathbb{R}^d\) which for almost all time obeys the speed-limits imposed
by \(\Pi\) \emph{via} the random upper semicontinuous function \(V:\mathbb{R}_d\to[0,\infty)\).
Expressed more precisely, the \(\Pi\)-path is given by an \(\mathbb{R}^d\)-valued function
\[
\xi:[0,T)\to\mathbb{R}^d\,
\]
defined up to a (possibly infinite) \emph{terminal time} \(T>0\), and satisfying the condition
that, for all \(v>0\), the time-set
\[
\left\{t\in[0,T)\;:\; |\xi^\prime(t)| > v \text{ and } \xi(t)\not\in\mathcal{S}_v\right\}
\]
is a Lebesgue-null subset of \([0,\infty)\).
Let \(\mathcal{A}_T\) be the space of all \(\Pi\)-paths defined up to time \(T>0\), and let \(\mathcal{A}=\bigcup_{T>0}\mathcal{A}_T\) be the space of all \(\Pi\)-paths whatsoever.
\end{defn}
\begin{rem}\label{rem:pi-path}
Direct arguments using Lemma \ref{lem:direction} show that
the Lebesgue-null condition in Definition \ref{def:pi-path} can be replaced by any one of three equivalent conditions:
\begin{enumerate}
\item\label{def:pi-path-condition1} for almost all \(t\in[0,T)\) for which \(\xi^\prime(t)\neq0\),
it is the case that \(\xi(t)\in\mathcal{S}_{|\xi^\prime(t)|}\);
\item\label{def:pi-path-condition2} for almost all \(t\in[0,T)\) for which \(\xi^\prime(t)\neq0\),
it is the case that the line \(\xi(t)+\xi'(t)\mathbb{R}\) belongs to \(\Pi_{|\xi'(t)|}\);
\item\label{def:pi-path-condition3} for almost all \(t\in[0,T)\),
it is the case that \(|\xi'(t)|\leq V(\xi(t))\).
\end{enumerate}
\end{rem}
Here
\(\xi(t)+\xi'(t)\mathbb{R}\) denotes the line through \(\xi(t)\) with orientation \(\xi'(t)\).
Condition \ref{def:pi-path-condition1} implies that
for almost all \(t\in[0,T)\), if \(\xi(t)\not\in\bigcup_{v>0}\mathcal{S}_v\)
then \(\xi'(t)=0\).
Crucially, \(\Pi\)-paths integrate the measurable orientation field provided by \(\Pi\) and obey the speed-limits imposed by \(\Pi\).
\begin{lem}\label{lem:pi-path-and-intersections}
Suppose that \(\xi\) is a \(\Pi\)-path. For any \((\ell,v)\in\Pi\), let \(\underline{e}\) be a unit vector parallel to the direction of \(\ell\). Then
the following time-sets are both Lebesgue-null:
\begin{align}
&\left\{t\in[0,T)\;:\; \xi(t)\in\ell, \xi'(t)\neq\langle\xi'(t),\underline{e}\rangle\underline{e}\;\right\}\,,\label{eq:pi-path-and-intersections1}\\
&\left\{t\in[0,T)\;:\; \text{ for some }(\ell,v)\in\Pi, \xi(t)\in\ell, |\xi'(t)|>v\right\}\,.\label{eq:pi-path-and-intersections2}
\end{align}
\end{lem}
\begin{proof}
It is a direct consequence of Lemma \ref{lem:direction} that the time-set at \eqref{eq:pi-path-and-intersections1} is Lebesgue-null.
As for the time-set at \eqref{eq:pi-path-and-intersections2}, note that, for some Lebesgue null-set \(\mathcal{N}_1\),
\begin{multline*}
\left\{t\in[0,T)\;:\; |\xi'(t)|>v \text{ and } \xi(t)\in\ell \text{ for some }(\ell,v)\in\Pi\right\} \quad\subseteq\quad\\
\left\{t\in[0,T)\;:\; \xi(t)\in\ell \text{ and }\xi(t)\in\widetilde\ell \text{ for some } (\ell,v), (\widetilde\ell,w)\in\Pi\text{ with }w>v\right\}\cup \mathcal{N}_1
\end{multline*}
(use the equivalent \(\Pi\)-path definition \ref{def:pi-path}). But this implies that
\begin{multline*}
\left\{t\in[0,T)\;:\; |\xi'(t)|>v \text{ and } \xi(t)\in\ell \text{ for some }(\ell,v)\in\Pi\right\} \\
\quad\subseteq\quad
\left\{t\in[0,T)\;:\; |\xi'(t)|>v \text{ and } \xi(t)\in\mathcal{I}\right\}\,,
\end{multline*}
where \(\mathcal{I}\) is the intersection point pattern derived from \(\Pi\) as in the statement of Corollary \ref{cor:network-intersections}.
By Corollary \ref{cor:network-intersections}, the time-set on the right-hand side
is Lebesgue-null.
\end{proof}
As noted at the start of this section, there are two related reasons for taking
this rather abstract approach to \(\Pi\)-paths, as opposed to working only with paths built up sequentially from segments of lines in \(\Pi\). Firstly, in dimension \(d>2\) there are no
non-trivial sequential \(\Pi\)-paths, since
almost surely no two lines of \(\Pi\) will intersect.
Secondly, even in the planar case of \(d=2\) we must consider
non-sequential \(\Pi\)-paths as possible
limits of simple \(\Pi\)-paths, for example when establishing the existence of minimizers of \(\Pi\)-path functionals (specifically,
the functional yielding accumulated elapsed time).
Here and in the following, we establish a number of statements about \(\mathcal{A}_T\) and \(\mathcal{A}\), all of which should be qualified as holding ``almost-surely'', since they depend on the random construction of
the marked line process \(\Pi\). Similarly ``constants'' are actually random variables measurable with respect to the line process \(\Pi\).
Borrowing the terminology of Random Walks in Random Environments, it might be said that we are concerned with the quenched law based on the random environment provided by \(\Pi\).
It is immediate from the definition that
\(\Pi\)-paths are individually fully Lipschitz over time intervals in which the \(\Pi\)-path in question belongs to a specified compact set.
As a consequence, we can establish \emph{a priori} bounds on the length of any \(\Pi\)-path beginning in a specified compact set \(K\) and with terminal time at most \(T<\infty\)
(hence finite), so
long as \(\gamma\geq d\). These bounds will depend on \(T<\infty\), \(K\), \(\gamma\), and also on the random marked line pattern \(\Pi\), and
will allow us to deduce a uniform Lipschitz property for
all \(\Pi\)-paths from \(\mathcal{A}_T\) which start in \(K\).
Moreover, if \(\gamma\geq d\) then the set \(\mathcal{A}_T\) is weakly closed, and therefore the family of
all \(\Pi\)-paths from \(\mathcal{A}_T\) started in \(K\) is weakly compact. This allows us to make sense of the notion of \emph{\(\Pi\)-geodesics}, measuring path-length
not as geometric length but using total time of passage along the path.
We first establish an \emph{a priori} upper bound for the Lipschitz constants of \(\Pi\)-paths begun near the origin and defined up to a finite time.
\begin{thm}[An \emph{a priori} bound for path space]\label{thm:a-priori-bound}
Suppose that \(\gamma\geq d\geq2\). Fix \(T<\infty\) and \(r_o>0\), and consider a \(\Pi\)-path \(\xi\) with \(\xi(0)\in\operatorname{\mathcal B}(\text{\textbf{o}},r_0)\).
Then there is \(C=C(\gamma, T, r_0, \Pi)<\infty\) (which is a constant when conditioned on the particular realization of the marked line process \(\Pi\)) such that almost surely
\[
|\xi(t)| \quad\leq\quad C \qquad \text{for all }t\in[0,T)\,.
\]
\end{thm}
\begin{proof}
We consider a given realization of the speed-limit function \(V\) (equivalently, of \(\Pi\)). It suffices to obtain a lower bound on the time at
which \(\xi\) first exits \(\operatorname{\mathcal B}(\text{\textbf{o}},r)\) for a given \(r>r_0\), and to show that this bound tends to infinity as \(r\to\infty\).
We achieve this by constructing a comparison with a one-dimensional system (controlled by the speed-limit function \(V\)), thus delivering an upper bound on \(|\xi|\),
and then by showing that with probability \(1\) the underlying configuration of \(\Pi\) is such that
the comparison system takes infinite time to diverge to infinity.
Our comparison system \(y:[0,\infty)\to[0,\infty)\) satisfies
\begin{align}\label{eq:comparison-system}
y'(t) \;&=\; \overline{V}(y(t)) \;=\; \sup\{V(x)\;:\; |x|\leq y(t)\}
\;=\; \sup\{v:(\ell,v)\in\Pi, \ell\Uparrow\operatorname{\mathcal B}(\text{\textbf{o}},r)\}\,,\\
y(0) \;&=\; r_0\,.\nonumber
\end{align}
So \(y\) could be thought of as the distance from \(\text{\textbf{o}}\) of an idealized path which always travels radially at the maximum speed \(\overline{V}\)
available from \(\Pi\) within that distance. Standard comparison techniques show that if \(\xi\) is a \(\Pi\)-path then
\[
y(t) \quad\geq\quad |\xi(t)| \qquad \text{for almost all \(t\) whenever }|\xi(0)|\leq r_0\,.
\]
The next step is to control the growth of the maximum speed-limit \(\overline{V}(y)\) as a function of \(y\).
We introduce a nonlinear projection from marked line-space to the quadrant \([0,\infty)^2\):
\[
(\ell, v) \quad\mapsto\quad \left(|r|^{d-1}, v^{-(\gamma-1)}\right)\quad=\quad (p, s)\,.
\]
We think of \(p=|r|^{d-1}\) as ``generalized distance'', and \(s=v^{-(\gamma-1)}\) as ``meta-slowness''. The fibres of the projection have zero invariant line measure.
Bearing in mind that \(\gamma\geq d\geq2\), and using the expression for the intensity measure at \eqref{eqn:improper}, also for \(\mu_d\) at \eqref{eqn:disintegration1}, the image of \(\Pi\)
under the projection is a stationary Poisson point process on the quadrant \([0,\infty)^2\), with intensity measure
\[
(\gamma-1)\frac{\omega_{d-1}}{2} \times(-v^{-\gamma}\d{v}) \times\d(|r|^{d-1})
\quad=\quad
\frac{\omega_{d-1}}{2} \,\d(v^{-(\gamma-1)}) \,\d(|r|^{d-1})
\quad=\quad
\frac{\omega_{d-1}}{2} \,\d{p}\,\d{s}
\]
(recall that \(\omega_{d-1}\) is the hyper-surface area of the unit ball in \(\mathbb{R}^d\)).
This projection to a planar point process
delivers a Poisson point process \(\widetilde\Pi\) of constant intensity \(\tfrac12{\omega_{d-1}}\)
in the quadrant. Note that
\[
\left(\overline{V}(r)\right)^{-(\gamma-1)}\quad=\quad\inf\left\{s : (p,s) \in \widetilde\Pi \text{ and } p\leq r^{d-1}\right\}\,,
\qquad \text{ for } r>0\,.
\]
Accordingly we can use \eqref{eq:comparison-system} to obtain information about the sequence of generalized distances at which the meta-slowness of the comparison system \(y\) changes,
deriving a recursive expression (compare the use of perpetuities in \citealp{Kendall-2011b,Kendall-2014a}).
Initially the random meta-slowness of \(y\) at
generalized distance \(P_0=r_0^{d-1}\) is
\[
S_0 \text{ of distribution } \text{Exponential}\left(\tfrac12{\omega_{d-1}} \, P_0\right)\,.
\]
Let \(P_0<P_1<P_2<\ldots\) be the sequence of generalized distances at which the meta-slowness changes through values \(S_0>S_1>S_2>\ldots\).
Poisson process arguments show
\begin{align}\nonumber
P_n-P_{n-1} \quad&=\quad
\frac{1}{S_{n-1}} T_n \qquad \text{ for } T_n
\text{ of distribution } \text{Exponential}\left(\tfrac12{\omega_{d-1}}\right)\,,\\
S_n \quad&=\quad
U_n S_{n-1} \qquad \text{ for } U_n
\text{ of distribution } \text{Uniform}(0,1)\,.
\label{eqn:meta-slowness-recursion}
\end{align}
Moreover the \(T_i\)'s and \(U_i\)'s are all independent.
It is useful to re-organize this recursion
so as to recognize the product \(X_n=S_n P_n\)
as a Markov chain (in fact a perpetuity):
\begin{equation}\label{eq:perpetuity}
X_n=S_n P_n \quad=\quad U_n (T_n + S_{n-1} P_{n-1})\quad=\quad U_n(T_n+X_{n-1})\,.
\end{equation}
Iteration shows that this perpetuity has a limiting distribution, expressible as an infinite
almost-surely convergent sum
\[
U_1 T_1 + U_1 U_2 T_2 + U_1U_2U_3T_3 + \ldots
\]
(stronger results on perpetuities can be found in
the foundational paper of \citealp{Vervaat-1979}). Indeed, Markov chain arguments show that \(X_n\) converges
to this distribution in total variation with geometric ergodicity \citep[Ch.~15 especially Theorem 15.0.1]{MeynTweedie-1993}. This follows by noting that the Markov chain
\(\{X_n:n\geq0\}\) is Lebesgue-irreducible and satisfies a geometric Foster-Lyapunov condition based on (for example)
\begin{multline*}
\Expect{1+X_n \;|\; X_{n-1}}\quad=\quad
1 + \frac12\left(\frac{2}{\omega_{d-1}}+X_{n-1}\right)\\
\quad\leq\quad
\begin{cases}
\frac{2}{3} (1+X_{n-1}) & \text{ if } 1+X_{n-1} > 3\left(1+ \frac{2}{\omega_{d-1}}\right)\,,\\
\frac{2}{3} (1+X_{n-1})+\frac12 +\frac{1}{\omega_{d-1}} & \text{ otherwise. }
\end{cases}
\end{multline*}
The interval \([0, 3(1 + \tfrac{2}{\omega_{d-1}}))\) is a small set for the Markov chain \(X\)
(since \(U_n\) and \(T_n\) are independent and
have continuous densities which are strictly positive over their ranges of \([0,1]\) and \([0,\infty)\)),
so this is indeed a geometric Foster-Lyapunov condition, and establishes geometric ergodicity for the Markov chain \(X\).
Consider the elapsed actual time between generalized positions \(P_{n-1}\) and \(P_n\), namely
\[
(S_{n-1})^{\frac{1}{\gamma-1}} \times \left(P_n^{\frac{1}{d-1}}-P_{n-1}^{\frac{1}{d-1}}\right)\,.
\]
Summing over \(n\), and using the requirement that \(\gamma\geq d\), the total time till \(y\) reaches infinity is given by the sum
\begin{multline*}
\sum_n (S_{n-1})^{\tfrac{1}{\gamma-1}} \times \left(P_n^{\frac{1}{d-1}}-P_{n-1}^{\frac{1}{d-1}}\right)
\;=\;
\sum_n (S_{n-1})^{\tfrac{1}{\gamma-1}-\tfrac{1}{d-1}} \times \left((S_{n-1}P_n)^{\frac{1}{d-1}}-(S_{n-1}P_{n-1})^{\frac{1}{d-1}}\right)\\
\quad=\quad
\sum_n (S_{n-1})^{\tfrac{1}{\gamma-1}-\tfrac{1}{d-1}} \times \left((S_{n-1}P_{n-1}+T_n)^{\frac{1}{d-1}}-(S_{n-1}P_{n-1})^{\frac{1}{d-1}}\right)\,.
\end{multline*}
If \(\gamma=d\) then \((S_{n-1})^{\tfrac{1}{\gamma-1}-\tfrac{1}{d-1}}=1\).
If on the other hand \(\gamma>d\) then the exponent \(\tfrac{1}{\gamma-1}-\tfrac{1}{d-1}\) is negative, while from the recurrence it follows that \(S_n\to0\) almost surely.
In either case, the above sum diverges if the same can be said for
\begin{equation}\label{eqn:infinite-sum}
\sum_n \left((S_{n-1}P_{n-1}+T_n)^{\frac{1}{d-1}}-(S_{n-1}P_{n-1})^{\frac{1}{d-1}}\right)\,.
\end{equation}
However geometric ergodicity of \(X_n=S_nP_n\), together with independence of the \(T_i\)'s and \(U_i\)'s, and the exponential distribution of \(T_n\), allows us to deduce that
almost surely infinitely many of these summands must satisfy
\[
(S_{n-1}P_{n-1}+T_n)^{\frac{1}{d-1}}-(S_{n-1}P_{n-1})^{\frac{1}{d-1}} \quad>\quad 1\,.
\]
Hence it follows that the infinite sum \eqref{eqn:infinite-sum} almost surely diverges, and therefore \(y\)
will almost surely takes infinite time to reach infinity.
This suffices to prove the
theorem.
\end{proof}
\begin{rem}\label{rem:equivalence}
It can be shown that in case \(\gamma<d\) then there are indeed \(\Pi\)-paths which reach infinity in finite time, and even in finite expected time.
\end{rem}
\begin{rem}
We will see (Theorem \ref{thm:connection}) that if we require \(\gamma>d\) then in addition we can connect specified pairs of points by \(\Pi\)-paths in finite time.
\end{rem}
We can now deduce the existence of an \emph{a priori} bound on the global Lipschitz constant of \(\Pi\)-paths of finite length begun in a fixed compact set.
For any constant \(C\), the lines from \(\Pi\) meeting \(\operatorname{\mathcal B}(\text{\textbf{o}},C)\)
have speed-limits bounded above by \(\overline{V}(C)\), a random value depending on the random environment \(\Pi\).
Fixing \(T<\infty\) and \(r_0>0\), and taking \(C\) to be the random value depending on \(\Pi\) in the statement of Theorem \ref{thm:a-priori-bound}, then
Remark \ref{rem:pi-path} Condition \ref{def:pi-path-condition3} implies that
the \(\Pi\)-paths in \(\mathcal{A}_T\) beginning in \(\operatorname{\mathcal B}(\text{\textbf{o}},r_0)\) satisfy a uniform Lipschitz property with random Lipschitz constant depending on \(\Pi\), \(r_0\), and \(T\).
Note that \(r_0>0\) can be chosen arbitrarily.
A closure result for \(\mathcal{A}_T\) follows immediately.
Recall (again using results expounded in \citealp[ch.5]{Evans-1998})
that the Sobolev space \(W^{1,2}([0,T)\to\mathbb{R}^d)\) can be viewed as the space of absolutely continuous curves \(\xi:[0,T)\to\mathbb{R}^d\)
whose first derivatives \(\xi'\) satisfy \(\int_0^T|\xi'|^2\,\d{t}<\infty\).
Thus \(W^{1,2}([0,T)\to\mathbb{R}^d)\) forms a separable Hilbert space when furnished with an inner-product norm given by
\(\sqrt{\int_0^T|\xi|^2\,\d{t}+\int_0^T|\xi'|^2\,\d{t}}\).
Recall also the Hilbert-space fact that bounded and weakly-closed subsets of \(W^{1,2}([0,T)\to\mathbb{R}^d)\) are weakly compact.
\begin{lem}[Closure of path space]\label{lem:closure}
Suppose \(\gamma\geq d\). For finite \(T\), the path space \(\mathcal{A}_T\) is closed in the Sobolev space \(W^{1,2}([0,T)\to\mathbb{R}^d)\).
\end{lem}
\begin{proof}
Consider any sequence \(\xi_1\), \(\xi_2\), \ldots of paths drawn from \(\mathcal{A}_T\).
Suppose \(\xi_n\to \xi\) when considered as elements of \( W^{1,2}([0,T)\to\mathbb{R}^d)\). By Sobolev space arguments,
taking a convergent subsequence if necessary, we may suppose that
\begin{align*}
\xi_n(t) \quad&\to\quad \xi(t) \qquad \text{for all }t\in[0,T)\,;\\
\xi_n'(t)\quad&\to\quad \xi'(t)\qquad \text{for almost all }t\in[0,T)\,.
\end{align*}
Consider the following time-set, which can be seen to be Lebesgue-null by
combining the Lipschitz property of \(\xi\), the choice of the convergent sequence, and
using the equivalent definition of \(\Pi\)-paths given by Remark \ref{rem:pi-path} Condition \ref{def:pi-path-condition2}:
\begin{align*}
\mathcal{N}\quad=\quad\Big\{t\;:\; &\text{ either } \xi'(t) \text{ does not exist, or } \xi'_n(t)\not\to \xi'(t),\\
&\text{ or, for some \(n\), } \xi'_n(t)\neq0 \text{ but }\xi_n(t)+\xi'_n(t)\mathbb{R}\not\in \mathcal{S}_{|\xi'_n(t)|}\Big\}\,.
\end{align*}
Fix attention on \(t\not\in\mathcal{N}\) for which \(|\xi'(t)|>u>0\). Then \(\xi_n'(t)\to\xi'(t)\), so
\[
\xi_n(t)+\xi'_n(t)\mathbb{R} \;\in\; \Pi_u \qquad\text{ for all large enough }n\,.
\]
Furthermore,
since \(\Pi_u\) yields a locally finite line process,
\begin{align*}
\text{either }
& \xi_n(t)+\xi'_n(t)\mathbb{R} \text{ eventually equals } \ell_t \text{ for some fixed } \ell_t\text{ from }\Pi_u \\
& \qquad\qquad\qquad\text{ and moreover } \xi(t)+\xi'(t)\mathbb{R}=\ell_t \\
& \text{or }
\xi_n(t) \quad\to\quad \text{ an intersection point of the line process }\Pi_u\,.
\end{align*}
In the first case, since \(\ell_t\) is a line from \(\Pi_{|\xi'_n(t)|}\) for all large enough \(n\),
it follows from the convergence \(\xi_n'(t)\to\xi'(t)\) that \(\ell_t\) is a line from \(\Pi_{|\xi'(t)|}\).
In the second case, Corollary \ref{cor:network-intersections} implies that
\[
\left\{t\;:\; \xi(t) \text{ is an intersection point of \(\Pi\) and } \xi'(t)\neq0\right\}
\]
must be a null time-set.
Either way,
up to a Lebesgue-null time-set,
\begin{itemize}
\item[] either \(\xi'(t)=0\),
\item[] or \(\xi(t)+\xi'(t)\mathbb{R}\in \Pi_{|\xi'(t)|}\).
\end{itemize}
Hence \(\xi\) is a \(\Pi\)-path (using Remark \ref{rem:pi-path}, specifically the equivalent defining Condition \ref{def:pi-path-condition2} for \(\Pi\)-paths), hence belongs to \(\mathcal{A}_T\).
\end{proof}
We can improve on this lemma to show that \(\mathcal{A}_t\) is \emph{weakly closed}, from which there follows a useful compactness result.
\begin{thm}[Weak closure of path space]\label{thm:weak-closure}
Suppose \(\gamma\geq d\). For finite \(T\), the path space \(\mathcal{A}_T\) is weakly closed in \(W^{1,2}([0,T)\to\mathbb{R}^d)\).
\end{thm}
\begin{proof}
Consider any sequence \(\xi_1\), \(\xi_2\), \ldots in \(\mathcal{A}_T\). Suppose \(\xi_n\to \xi\) \emph{weakly} in \(W^{1,2}([0,T)\to\mathbb{R}^d)\).
We will invoke the equivalent defining Condition
\ref{def:pi-path-condition3} from Remark \ref{rem:pi-path}, namely, if we can show that \(|\xi'(t)|\leq V(\xi(t))\) for almost all \(t\in[0,T)\) then \(\xi\) is a \(\Pi\)-path.
As an immediate consequence, this will imply that \(\mathcal{A}_T\) is weakly closed.
If \(T<\infty\) then weak convergence in \(W^{1,2}([0,T)\to\mathbb{R}^d)\) implies uniform convergence;
in particular this implies \(\xi\) is continuous.
Fix \(v>0\), and consider a non-empty time interval \([r,s]\subseteq[0,T)\), such that \([r,s]\) is contained in a single connected component of the open time-set
\(\{t\in[0,T):\xi(t)\not\in\mathcal{S}_{v}\}\). Thus the compact set \(\{\xi(t):r\leq t\leq s\}\) does not intersect the closed set \(\mathcal{S}_{v}\)
and indeed
will not intersect the closed set \(\mathcal{S}_{v-\varepsilon}\) for some (perhaps small) \(\varepsilon\in(0,v)\).
By the uniform convergence of \(\xi_n\) to \(\xi\),
we know that for all large enough \(n\)
the compact set \(\{\xi_n(t):r\leq t\leq s\}\) does not intersect \(\mathcal{S}_{v-\varepsilon}\); since \(\xi_n\) is a \(\Pi\)-path, this implies
that \(|\xi_n'(t)|<v-\varepsilon\) for almost all \(t\in[r,s]\), and hence that \(\xi_n\) is Lipschitz with Lipschitz constant \(v-\varepsilon\) over the time interval \([r,s]\).
But the uniform convergence of \(\xi_n\) to \(\xi\) implies that \(\xi\) itself is Lipschitz with Lipschitz constant \(v-\varepsilon\) over the time interval \([r,s]\),
therefore that \(|\xi'(t)|\leq v-\varepsilon<v\) for almost all \(t\in[r,s]\).
Expressing the open time-set
\(\{t\in[0,T):\xi(t)\not\in\mathcal{S}_{v-\varepsilon}\}\) as a countable union of closed intervals,
and letting \(\varepsilon\to0\),
it follows that for almost all \(t\in[0,T)\) if \(\xi(t)\not\in\mathcal{S}_{v}\)
then \(|\xi'(t)|<v\). Consequently \(|\xi'(t)|\leq V(\xi(t))\) for almost all \(t\in[0,T)\) as required.
\end{proof}
It follows immediately that bounded subsets of \(\mathcal{A}_T\) are weakly precompact, and this is crucial for the discussion of minimum-time \(\Pi\)-paths given in the
next section.
\begin{cor}[Weak compactness and path space]\label{cor:compactness}
Suppose \(\gamma\geq d\). Fix \(T>0\) and consider the set of paths in \(\mathcal{A}_T\) which begin in a compact set \(K\).
This set is weakly compact as a subset of \(W^{1,2}([0,T)\to\mathbb{R}^d)\).
\end{cor}
\begin{proof}
Immediate from Theorems \ref{thm:a-priori-bound} and \ref{thm:weak-closure} together with weak compactness results for
the Hilbert space \(W^{1,2}([0,T)\to\mathbb{R}^d)\).
\end{proof}
\section[\texorpdfstring{$\Pi$}{Π}-paths between point pairs]{\(\Pi\)-paths between arbitrary points}\label{sec:pi-paths and points}
We have shown that the condition \(\gamma\geq d\) ensures that almost surely \(\Pi\)-paths
do not diverge to infinity in finite time (as noted in Remark \ref{rem:equivalence}, this condition is also necessary).
However it is not yet clear whether \(\Pi\)-paths can move from one point to another in finite time.
In dimension \(d=2\) the question is essentially one of whether one can reach a specified point in finite time by travelling along paths of progressively slower and slower speed.
Dimension \(d\geq3\) appears intransigent at first glance, since one knows that lines of a Poisson line process do not intersect each other in dimension \(3\) and higher.
Nevertheless, almost surely there are \(\Pi\)-paths connecting \emph{all} pairs of points in dimension \(2\) \emph{and} higher, so long as we strengthen the condition on \(\gamma\) to \(\gamma>d\).
We begin by showing how to connect specified pairs of points.
\begin{thm}[\(\Pi\)-paths connect given pairs of points in finite time]\label{thm:connection}
Suppose \(\gamma>d\).
Then specified \(x_1\) and \(x_2\) in \(\mathbb{R}^d\)
can almost surely be connected in finite time \(T\) by a \(\Pi\)-path \(\xi\).
\end{thm}
\begin{proof}
The construction connects segments of lines from \(\Pi\) together in a tree-like fashion, rather than sequentially.
The basic idea is as follows:
for a sufficiently large constant \(\alpha\) (indeed, \(\alpha>2^{(\gamma-1)/(\gamma-d)}\) suffices),
construct disjoint balls of radius \(|x_1-x_2|/\alpha\) around \(x_1\) and \(x_2\).
Choose the fastest line \(\ell\) in \(\Pi\)
hitting both balls, corresponding to the root of a binary tree representation of a path connecting \(x_1\) to \(x_2\). Then create two daughter nodes, repeating the construction
based on (a) \(x_1\) and the closest point to \(x_1\) on \(\ell\), and (b) \(x_2\) and the closest point to \(x_2\) on \(\ell\).
Extend this recursively to generate a binary tree-indexed collection of line segments.
Figure \ref{fig:binary-tree} illustrates the first two stages of the \(d=2\) case.
\begin{Figure}
\includegraphics[width=5in]{binary-tree}
\centering
\caption{\label{fig:binary-tree}
First two stages of a recursive construction of a \(\Pi\)-path from \(x_1\) to \(x_2\) in two dimensions,
using fastest available lines taken from an improper Poisson line process marked by speeds.
Note that in the case of higher dimensions the lines will almost surely not intersect.
}
\end{Figure}
The path \(\xi\) formed from this binary tree is evidently a \(\Pi\)-path.
The issue is to show that it makes the connection from \(x_1\) to \(x_2\) in finite time.
Firstly, we need a stochastic lower bound for the speed-limit of the fastest line connecting two balls, namely the speed of the fastest line of \(\Pi\) in
\[
\hittingset{\operatorname{\mathcal B}(x_1,\alpha^{-1}r)}\cap\hittingset{\operatorname{\mathcal B}(x_2,\alpha^{-1}r)}\,.
\]
Here we write \(r=|x_1-x_2|\) for the Euclidean distance between \(x_1\) and \(x_2\).
We obtain a stochastic lower bound for the speed distribution in two steps:
\begin{itemize}
\item[(a)] Shrink \(\operatorname{\mathcal B}(x_1,\alpha^{-1}r)\) to \(\operatorname{\mathcal B}(x_1,\alpha^{-1}r/2)\) and then replace \(\operatorname{\mathcal B}(x_1,\alpha^{-1}r/2)\)
by the hyper-disk \(D(x_1,\alpha^{-1}r/2)\) obtained by intersecting \(\operatorname{\mathcal B}(x_1,\alpha^{-1}r/2)\) with the hyperplane through \(x_1\) which is normal to the vector \(x_2-x_1\);
\item[(b)] Consider the bundle of lines in \(\hittingset{D(x_1,\alpha^{-1}r/2)}\cap\hittingset{\operatorname{\mathcal B}(x_2,\alpha^{-1}r)}\) which run through a given point
\(z\in D(x_1,\alpha^{-1}r/2)\). For each such \(z\), reduce the bundle size by restricting attention to lines which additionally intersect
\(\operatorname{\mathcal B}(z+x_2-x_1,\alpha^{-1}r/2)\subset \operatorname{\mathcal B}(x_2,\alpha^{-1}r)\).
\end{itemize}
This geometric construction is illustrated in Figure \ref{fig:reduction}.
\begin{Figure}
\includegraphics[width=3.5in]{reduction}
\centering
\caption{\label{fig:reduction}
Reduction of \(\hittingset{\operatorname{\mathcal B}(x_1,\alpha^{-1}r)}\cap\hittingset{\operatorname{\mathcal B}(x_2,\alpha^{-1}r)}\) to a smaller hitting set for
which the line measure is more easily computable yet still provides a useful lower bound.
}
\end{Figure}
Using inclusion of hitting sets, this produces an easily computable lower bound for the line measure:
\begin{multline*}
\mu_d\left(\hittingset{\operatorname{\mathcal B}(x_1,\alpha^{-1}r)}\cap\hittingset{\operatorname{\mathcal B}(x_2,\alpha^{-1}r)}\right)
\quad\geq\quad
\mu_d\left(\hittingset{D(x_1,\alpha^{-1}r/2)}\cap\hittingset{\operatorname{\mathcal B}(x_2,\alpha^{-1}r)}\right)\\
\quad\geq\quad
m_{d-1}\left(D(x_1,\alpha^{-1}r/2)\right)\times
\mu^{(\text{\textbf{o}})}_{d-1} \left(\hittingset{\operatorname{\mathcal B}(x_2-x_1,\alpha^{-1}r/2)}\right)\,.
\end{multline*}
Here \(\mu^{(\text{\textbf{o}})}_{d-1}=\tfrac{\sin\theta}{\kappa_{d-1}}m_{S^{d-1}_+}\) is derived from the disintegration of \(\mu_d\) by Lebesgue measure on the hyperplane through \(x_1\) which is normal to \(x_2-x_1\),
using \eqref{eqn:disintegration2}. Thus \(\mu^{(\text{\textbf{o}})}_{d-1}\) is a weighted version of
the invariant (hyper-surface area) measure on the hemisphere of un-sensed lines \(\ell\) passing through the origin \(\text{\textbf{o}}\), weighted by \(\sin\theta\) where
\(\theta\) is the angle between \(\ell\) and the hyperplane, normalized to have unit total measure.
Recall that \(\kappa_{d-1}\) denotes the \((\d-1)\)-volume of the unit \((d-2)\)-ball. Thus
\[
m_{d-1}\left(D(\xi_1,\alpha^{-1}r/2)\right) \quad=\quad \kappa_{d-1} \left(\frac{r}{2\alpha}\right)^{d-1}\,,
\]
On the other hand, from \eqref{eqn:disintegration2}
the relevant computation of weighted hyper-surface area for the visibility hemisphere is
\[
\frac{\omega_{d-2}}{\kappa_{d-1}} \int^{\pi_2}_{\pi/2-\theta_0} \sin\theta \,\cos^{d-2}\theta\,\d{\theta} \quad=\quad \frac{\omega_{d-2}}{(d-1)\kappa_{d-1}} \,\cos^{d-1}\theta_0
\quad=\quad \cos^{d-1}\theta_0\,,
\]
where \(\cos\theta_0=1/(2\alpha)\) (and noting that \(\omega_{d-2}=(d-1)\kappa_{d-1}\)); hence
\[
\mu^{(\text{\textbf{o}})}_{d-1} \left(\hittingset{\operatorname{\mathcal B}(x_2-x_1,\alpha^{-1}r/2)}\right)\quad=\quad \left(\frac{1}{2\alpha}\right)^{d-1}\,.
\]
These considerations yield the lower bound
\[
\mu_d\left(\hittingset{\operatorname{\mathcal B}(\xi_1,\alpha^{-1}r)}\cap\hittingset{\operatorname{\mathcal B}(\xi_2,\alpha^{-1}r)}\right)
\quad\geq\quad
\kappa_{d-1}\left(\frac{r}{4\alpha^2}\right)^{d-1}\,.
\]
This reasoning can be applied to the recursive construction indicated above. Let \(r_h\) be the distance between points at node \(h\) on the binary tree representing the path, then
(omitting some implicit conditioning on \(r_h\))
\[
\Prob{\text{ fastest line speed at } h \leq v_h}
\quad\leq\quad\exp\left(
-\frac{\kappa_{d-1}}{(4\alpha^2)^{d-1}}\,\frac{r_h^{d-1}}{v_h^{\gamma-1}}
\right)\,.
\]
By construction, if node \(h\) is at level \(n\) of the tree then \(r_n\leq \alpha^{-n}r_0\), where \(r_0\) is the Euclidean distance between the original
start and finish points. Fixing \(\varepsilon>0\), we set
\[
v_h \quad=\quad \frac{r_h^{(d-1)/(\gamma-1)}}{(n \zeta)^{\frac{1}{\gamma-1}}} \qquad \text{where }\zeta=\frac{(4\alpha^2)^{d-1}}{\kappa_{d-1}}(1+\varepsilon)\log 2
\]
Then \(r_h^{d-1}/v_h^{\gamma-1}=n \zeta\). Use the first Borel-Cantelli lemma, and the convergence of
\begin{multline*}
\sum_h \exp\left(
-\frac{\kappa_{d-1}}{(d-1)(4\alpha^2)^{d-1}}\,\frac{r_h^{d-1}}{v_h^{\gamma-1}}
\right)
\;=\; \sum_n 2^n \exp\left(-(1+\varepsilon) n \log 2\right)
\;=\; \sum_n 2^{-\varepsilon n}\,<\, \infty\,,
\end{multline*}
to deduce that it is
almost surely the case that the speed limits of all but finitely many segments \(h\) in the binary tree representation
will exceed
\[
v_h \quad=\quad \frac{r_h^{(d-1)/(\gamma-1)}}{(n \zeta)^{\frac{1}{\gamma-1}}}\,.
\]
By the triangle inequality the relevant path length for node \(h\) is no greater than \((1+\tfrac{2}{\alpha})r_h\).
So the total time spent traversing
the path is finite when
\[
\sum_h \left(1+\frac{2}{\alpha}\right) \frac{r_h}{v_h}
\quad=\quad
\sum_n 2^n \times \left(1+\frac{2}{\alpha}\right) (n \zeta)^{\frac{1}{\gamma-1}} \left(\frac{r_0}{\alpha^n}\right)^{(\gamma-d)/(\gamma-1)}
\quad<\quad\infty\,.
\]
But this sum converges when \(\alpha> 2^{(\gamma-1)/(\gamma-d)}\):
thus in this case the construction gives a finite-time \(\Pi\)-path
between \(x_1\) and \(x_2\).
\end{proof}
\begin{rem}\label{rem:connection-by-geodesics}
It can be shown that \(\gamma>d\) is also a necessary condition for connection of \(x_1\) to \(x_2\) by a \(\Pi\)-path in \(\mathbb{R}^d\).
For if \(\gamma=d\) then all \(\Pi\)-paths leaving the origin are subject to an upper bound using the comparison process of Theorem \ref{thm:a-priori-bound},
and a direct calculation shows that this comparison process takes infinite expected time to leave the origin.
\end{rem}
Note that, for \(d>2\), the finite-time path has a curious fractal-like property: whenever the \(\Pi\)-path changes from one line of positive speed-limit to another,
then it must shift gears right down to zero speed then right up again to the new speed. (And the same applies to each change of speed-limit whilst shifting gears down, and so on \emph{ad infinitum}, as in the case of the fleas and poets of \citealp[On Poetry: a Rhapsody]{Swift-1733}.)
Since there are only countably many lines in \(\Pi\), a simple modification of the above construction shows that almost surely all lines in \(\Pi\) are connected.
\begin{cor}\label{cor:connect-all-lines}
If \(\gamma>d\) then almost surely all lines of \(\Pi\) are joined by finite-time \(\Pi\)-paths.
\end{cor}
If two points \(x_1\) and \(x_2\) are joined by \(\Pi\)-paths taking finite time, then it is reasonable to ask whether there is a minimum-time \(\Pi\)-path. As a consequence of
Corollary \ref{cor:compactness}, we know that this occurs, since \(\gamma>d\) and hence \emph{a fortiori} the compactness condition \(\gamma\geq d\) holds.
We summarize this conclusion by means of a definition and a further corollary.
We note in passing that \(\Pi\)-geodesics inherit all the properties of minimal geodesics in metric spaces; for example a minimal \(\Pi\)-geodesic cannot intersect itself.
\begin{defn}[\(\Pi\)-geodesic]\label{def:geodesic}
The \(\Pi\)-path \(\xi:[0,T]\to\mathbb{R}^d\) is said to be a \emph{\(\Pi\)-geodesic} (or
\emph{minimum-time geodesic}) from \(\xi(0)=x_1\) to \(\xi(T)=x_2\) if there are no \(\Pi\)-paths connecting \(x_1\)
and \(x_2\) in \(\mathcal{A}_S\) for \(S<T\).
\end{defn}
\begin{cor}[Existence of \(\Pi\)-geodesics]\label{cor:geodesics}
Suppose \(\gamma>d\). Consider the set of paths \(\xi\) in \(\mathcal{A}\) which begin at fixed location \(\xi(0)=x_1\) and end at fixed location \(\xi(T)=x_2\)
(here \(T\) depends on \(\xi\)).
Almost surely there exist \(\Pi\)-geodesics in \(\mathcal{A}\) from \(x_1\) to \(x_2\).
\end{cor}
\begin{proof}
By Theorem \ref{thm:connection}, almost surely there are connecting \(\Pi\)-paths in \(\mathcal{A}_T\) for large enough \(T<\infty\). Consider a sequence of such paths
\(\xi_1\), \(\xi_2\), \ldots,
starting at \(x_1\) and ending at \(x_2\), such that \(\xi_n(t)=x_2\) for all \(t\in[T_n,T)\), and such that \(T_n\) tends to \(T_\infty\) the infimum of all connection times for \(\Pi\)-paths
between \(x_1\) and \(x_2\). By Corollary \ref{cor:compactness} we may extract a weakly convergent subsequence, and the limit \(\xi\in\mathcal{A}_T\)
will satisfy \(\xi(t)=x_2\) for all \(t\in[T_\infty,T)\) and hence realize the infimum. The
resulting \(\Pi\)-path \(\xi\) will be a \(\Pi\)-geodesic between \(x_1\) and \(x_2\).
\end{proof}
In the next section we will examine the extent to which \(\Pi\)-geodesics are uniquely determined by their end-points.
Before turning to this matter,
we improve on Theorem \ref{thm:connection} by showing that if \(\gamma>d\) then almost surely \emph{all} pairs of points in \(\mathbb{R}^d\)
are connected by finite-time \(\Pi\)-paths.
That is to say, almost surely
there are no infinite singularities in the metric space induced by the time taken by fastest \(\Pi\)-path transit.
The proof closely follows that of Theorem \ref{thm:connection}, but splits paths apart in a hierarchical way so as to access entire regions rather than single points.
\begin{thm}\label{thm:metric-space}
Suppose \(\gamma>d\) and \(d\geq2\). With probability \(1\), the network formed by \(\Pi\) connects up all pairs of points in \(\mathbb{R}^d\) using
finite-time \(\Pi\)-paths.
\end{thm}
\begin{proof}
It suffices to establish that, almost surely, finite-time \(\Pi\)-paths can be used to connect a specified point to all the points of a single hypercube of positive area.
Consider then the construction of a path \(\xi\) from \(x_1\) to a hypercube centred on \(x_2\), where \(|x_1-x_2|=r_0\) and the hypercube is of
side-length \(\alpha^{-1}r_0\) for some sufficiently large integer \(\alpha\) (indeed, \(\alpha>(1+\tfrac{\sqrt{d}}{2})2^{(\gamma-1)/(\gamma-d)}\) suffices).
The construction commences as in Theorem \ref{thm:connection},
choosing the fastest line \(\ell\) of \(\Pi\) in
\(\hittingset{\operatorname{\mathcal B}(x_1,\alpha^{-1}r_0)}\cap\hittingset{\operatorname{\mathcal B}(x_2,\alpha^{-1}r_0)}\), and this corresponds to the root of a tree now representing a whole family of paths.
Repeat the construction, adding a further line from \(\Pi\) which nearly connects \(x_1\) to the point on \(\ell\) closest to \(x_1\), as in Theorem \ref{thm:connection}.
However on the other side we generate \(\alpha^d\) separate paths, using the fastest possible line to connect
the ball of radius \(\alpha^{-1}r_0\) centred on the point on \(\ell\) closest to \(x_2\), to each of a total of \(\alpha^d\)
balls of radius \(\alpha^{-1}r_0\) centred on
centroids of cells arising from
a dissection of the original hypercube of side-length \(\alpha^{-1}r_0\) into \(\alpha^d\) sub-hypercubes each of side-length \(\alpha^{-2}r_0\).
This
is illustrated in Figure \ref{fig:multiple}.
\begin{Figure}
\includegraphics[width=3.5in]{multiple}
\centering
\caption{\label{fig:multiple}
Initial stage of connecting \(x_1\) to points in a specified hypercube (case of \(d=2\)).
}
\end{Figure}
In the first case the new distance is at most \(\alpha^{-1} r_0\). In the second case we may use Pythagoras to show that the new distance is at most \((1+\frac12\sqrt{d})\alpha^{-1}r_0\).
This construction step generates \(1+\alpha^d\) new segments at the second level of the tree, the first one being like the segments generated in Theorem \ref{thm:connection},
while the remaining \(\alpha^d\) nearly connect a given point to \(\alpha^d\) centroids of sub-hypercubes.
Repeating the construction down to level \(n\), we generate \(h_n\) segments of the first kind and \(k_n\) segments of the second kind, where
\begin{align}\label{eqn:binary-recursion}
h_n \quad&=\quad 2 h_{n-1} + k_{n-1}\,, & \qquad & h_1 \quad=\quad 1\,,\\
k_n \quad&=\quad \alpha^d k_{n-1}\,, & \qquad & k_1 \quad=\quad \alpha^d\,.\nonumber
\end{align}
The total number of segments which have been built at level \(n\) is therefore
\[
h_n+k_n \quad=\quad 2 (h_{n-1} + k_{n-1}) + (\alpha^d - 1)k_{n-1}\,.
\]
The bound on \(\alpha\) imposed at the start of the proof, together with \(\gamma>d\geq2\), shows that \(\alpha^d>8\),
and so we can use the recursion \eqref{eqn:binary-recursion} and the fact that \(h_1+k_1=1+\alpha^d\) to deduce
\begin{multline}
h_n+k_n \quad\leq\quad 2 (h_{n-1} + k_{n-1}) + (\alpha^d - 1)k_{n-1}\\
\quad=\quad 2^2 (h_{n-2} + k_{n-2}) + (\alpha^d - 1) (k_{n-1} + 2 k_{n-2})\\
\quad=\ldots=\quad 2^{n-1} (h_{1} + k_{1}) + (\alpha^d - 1) (k_{n-1} + 2 k_{n-2} + \ldots + 2^{n-2} k_{1})
\quad\leq\quad
\text{constant}\times \alpha^{nd}\,.
\end{multline}
Now consider the speed-limit of the line forming node \(h\) at level \(n\). Suppose the relevant distance between target points is \(r_h\). Then (as in Theorem \ref{thm:connection})
\[
\Prob{\text{ fastest line speed-limit at } h \leq v_h}
\quad\leq\quad\exp\left(
-\frac{\kappa_{d-1}}{(4\alpha^2)^{d-1}}\,\frac{r_h^{d-1}}{v_h^{\gamma-1}}
\right)\
\]
We know
\[
r_h \quad\leq\quad \left(\frac{1+\sqrt{d}/2}{\alpha}\right)^n r_0\,.
\]
Choose
\[
v_h \quad=\quad \frac{r_h^{(d-1)/(\gamma-1)}}{(n \zeta)^{\frac{1}{\gamma-1}}} \qquad \text{as before, but with }\zeta=\frac{(4\alpha^2)^{d-1}}{\kappa_{d-1}}(1+\varepsilon)d\log \alpha
\]
Using the first Borel-Cantelli lemma once again, all but finitely many of the segments in this construction have speed-limit exceeding the respective \(v_h\), since
\[
\sum_n \alpha^{dn} \exp\left(-(1+\varepsilon)nd\log \alpha)\right) \quad=\quad \sum_n \alpha^{-\varepsilon n d}\quad<\quad\infty\,.
\]
Thus each one of the paths can be traversed in finite time if the following sum converges:
\begin{multline*}
\sum_{h \text{ in specified path}} \frac{r_h}{v_h} \quad\leq\quad \sum_n 2^n (n \zeta )^{\frac{1}{\gamma-1}} \left(\frac{1+\sqrt{d}/2}{\alpha}\right)^{n(\gamma-d)/(\gamma-1)}\\
\quad\leq\quad \zeta^{\frac{1}{\gamma-1}}\sum_n n^{\frac{1}{\gamma-1}} \left(2 \left(\frac{1+\sqrt{d}/2}{\alpha}\right)^{(\gamma-d)/(\gamma-1)}\right)^n\,.
\end{multline*}
Recalling the stipulation that \(\gamma>d\), this converges if we choose
\[
\alpha \quad>\quad \left(1+\frac{\sqrt{d}}{2}\right) 2^{(\gamma-1)(\gamma-d)}\,.
\]
The family of \(\Pi\)-paths used here is weakly compact (Corollary \ref{cor:compactness}).
It follows therefore that this construction almost surely delivers \(\Pi\)-paths which within finite time connect a specified point \(x_1\) to all points in a non-empty hypercube with centroid \(x_2\) and side-length \(\alpha^{-1}|x_2-x_1|\).
Using this fact together with judicious
concatenation of \(\Pi\)-paths, it follows
that almost surely all pairs of points in \(\mathbb{R}^d\) are connected by \(\Pi\)-geodesics.
\end{proof}
In the case \(d=2\), both Theorems \ref{thm:connection} and \ref{thm:metric-space} can be proved more directly,
exploiting the fact that non-parallel lines in \(\mathbb{R}^2\) always meet.
The resulting \(\Pi\)-paths are then formed from consecutive sequences of line segments drawn from \(\Pi\).
However our interest is in \emph{\(\Pi\)-geodesics}, and even in case \(d=2\) it is not yet known whether \(\Pi\)-geodesics can be constructed as
consecutive sequences of line segments.
While we define \(\Pi\)-geodesics as minimum-\emph{time} paths, we retain an interest in the actual lengths of \(\Pi\)-geodesics.
It is a consequence of Theorem \ref{thm:a-priori-bound} that a \(\Pi\)-geodesic between two points is almost surely of finite length:
this follows because if the \(\Pi\)-geodesic has finite duration \(T\)
then it must be contained in a sufficiently large ball, and therefore its maximum speed is bounded, which in turn bounds the length.
There is a more subtle question, namely whether the \emph{mean} length of the \(\Pi\)-geodesic is finite.
We shall answer this question in the affirmative in Section \ref{sec:pi-geodesics-finite-mean-length}, but only for the case of dimension \(d=2\).
The above results establish the existence of \(\Pi\)-geodesics, but only non-constructively. The
principal difficulty in taking a
constructive approach lies in the implicit tree-like way in which \(\Pi\)-paths are constructed
in Theorems \ref{thm:connection} and \ref{thm:metric-space}. In the remainder of this section we
show how to approximate \(\Pi\)-paths by sequentially-defined Lipschitz paths which are almost \(\Pi\)-paths.
The major benefit of this result is that it implies the measurability of the random time which a \(\Pi\)-geodesic would take to move from one specified point \(x\) to another specified point \(y\).
As is commonly the case for measurability arguments, the details are a little tedious; however the result does provide theoretical justification
for some simulation constructions of \(\Pi\)-geodesics (for example, the construction in Figure \ref{fig:dumbbell}).
The essence of the matter is to work
with Lipschitz paths which would be \(\Pi\)-paths if the upper-semicontinuous speed-limit \(V\) were
replaced by \(\max\{\varepsilon,V\}\) for some small \(\varepsilon>0\).
The methods of proof of the following results also justify the simulation algorithm used to produce the realizations of networks in Figure \ref{fig:dumbbell}.
\begin{defn}[\(\varepsilon\)-near-sequential-\(\Pi\)-path]
\label{def:near-sequential-pi-path}
For given \(\varepsilon>0\), a continuous path \(\widetilde{\xi}:[0,T]\to\mathbb{R}^d\) is an
\emph{\(\varepsilon\)-near-sequential-\(\Pi\)-path} if there is a finite dissection of the interval \([0,T]\) as
\[
0=b_0\leq a_1\leq b_1\leq a_2\leq b_2\leq \ldots\leq a_m\leq b_m\leq a_{m+1}=T\,,
\]
associated with a finite sequence of marked lines from \(\Pi\) (possibly with repeats),
\[
(\widetilde{\ell}_1,\widetilde{v}_1), (\widetilde{\ell}_2,\widetilde{v}_2), \ldots, (\widetilde{\ell}_m,\widetilde{v}_m)\,,
\]
such that
\begin{itemize}
\item[(a)] \(\widetilde{\xi}(t)\in\widetilde{\ell}_r\) when \(a_r\leq t\leq b_r\), and
\(|\widetilde{\xi}'(t)|\leq \widetilde{v}_r\) for almost all \(t\in(a_r,b_r)\);
\item[(b)] \(|\widetilde{\xi}'(t)|< \varepsilon\) for almost all \(t\in\bigcup_{r=0}^m[b_r,a_{r+1}]\)
and \(\widetilde{\xi}'\) is constant on each \([b_r,a_{r+1}]\);
\item[(c)] \(\sum_{r=0}^m |a_{r+1}-b_r|< \varepsilon\).
\end{itemize}
\end{defn}
The next result shows that \(\Pi\)-paths can be approximated by \(\varepsilon\)-near-sequential-\(\Pi\)-paths for small \(\varepsilon>0\), simply by
using the principal marked lines involved in \(\xi\) to generate the finite marked line sequence
\((\widetilde{\ell}_1,\widetilde{v}_1)\), \((\widetilde{\ell}_2,\widetilde{v}_2)\), \ldots, \((\widetilde{\ell}_m,\widetilde{v}_m)\).
\begin{thm}\label{thm:pi-path-approximation}
Suppose only that \(\gamma> 1\), so that the line processes \(\Pi_v\) are locally finite for each \(v>0\).
Consider a \(\Pi\)-path \(\xi:[0,T]\to\mathbb{R}^d\), defined up to some fixed finite time \(T\) and running from \(x_1\) to \(x_2\).
For each \(\varepsilon>0\) there can be found an \(\varepsilon\)-near-sequential-\(\Pi\)-path \(\widetilde{\xi}:[0,T]\to\mathbb{R}^d\) such that
\begin{itemize}
\item[] \(\widetilde{\xi}(0)=x_1\), \(\widetilde{\xi}(T)=x_2\);
\item[] \(\sup\{|\xi(t)-\widetilde{\xi}(t)|:t\in[0,T]\}<\varepsilon\).
\end{itemize}
\end{thm}
Before proving this, we state the following important corollary, whose proof is an immediate consequence.
\begin{cor}\label{cor:pi-path-approximation}
Suppose only that \(\gamma> 1\). Every \(\Pi\)-path defined up to finite time \(T\) can be uniformly
approximated by a sequence of \(\varepsilon\)-near-sequential-\(\Pi\)-paths such that \(\varepsilon\downarrow0\) along the sequence.
\end{cor}
\begin{proof}[Proof of Theorem \ref{thm:pi-path-approximation}]
It suffices to prove the result for a fixed positive \(\varepsilon<1\).
The Poisson line process \(\Pi\) has no triple intersections, and therefore a
given \(\Pi\)-path \(\xi\) can only ever lie on at most two lines simultaneously. Hence by countable exhaustion (based on ordering by \(\operatorname{Leb}\{t\in[0,T]:\xi(t)\in\ell\}\)) it follows that there can be only countably many marked lines \((\ell,v)\in\Pi\) such that \(\operatorname{Leb}\{t\in[0,T]:\xi(t)\in\ell\}>0\) (so that \(\ell\) and \(\xi\) intersect in a time-set of positive measure).
Since \(\xi\) is continuous, we know that
the image \(\operatorname{Im}(\xi)\) of \(\xi\) is compact, and therefore (since \(\Pi_v\) is locally finite for any positive \(v\)) the lines intersecting \(\xi\) in time-sets of positive measure can be sequentially ordered by speed in decreasing order:
\begin{align}
& (\ell_1, v_1), \quad (\ell_2, v_2), \quad (\ell_3, v_3), \quad \ldots \label{eq:positive-decreasing-order}\\
\text{ and } \quad & v_1 \quad\geq\quad v_2 \quad\geq\quad v_3 \quad\geq\quad \ldots \nonumber \,.
\end{align}
Note that we do \emph{not} presume that the \(\ell_i\) are visited sequentially in order.
It is an immediate consequence of the above that \(\xi\) is \(\operatorname{Lip}(v_1)\). Moreover a consequence of the countable exhaustion
construction is that
\[
\operatorname{Leb}\{t\in[0,T]: \xi(t)\not\in \ell_1\cup\ell_2\cup\ldots\cup\ell_n\} \to 0 \text{ as }n\to\infty\,.
\]
Therefore for all \(\varepsilon>0\), for all sufficiently large \(n\),
\begin{align*}
&\operatorname{Leb}\{t\in[0,T]: \xi(t)\not\in \ell_1\cup\ell_2\cup\ldots\cup\ell_n\} \quad<\quad\varepsilon\,,
\\
\text{ and } \quad&\qquad v_{n+1} \quad<\quad \varepsilon\,.
\end{align*}
We use the finite sequence \((\ell_1,v_1)\), \((\ell_2,v_2)\), \ldots \((\ell_n,v_n)\)
to generate an \(\varepsilon\)-near-sequential-\(\Pi\)-path \(\widetilde{\xi}\) which approximates \(\xi\) in uniform norm.
Note that this finite sequence is \emph{not} the same as the finite sequence
\((\widetilde{\ell}_1,\widetilde{v}_1)\), \((\widetilde{\ell}_2,\widetilde{v}_2)\), \ldots, \((\widetilde{\ell}_m,\widetilde{v}_m)\)
from Definition \ref{def:near-sequential-pi-path}, but is used to construct it iteratively.
We begin by setting \(\widetilde{\xi}(0)=\xi(0)\) and \(\widetilde{\xi}(T)=\xi(T)\).
Consider first the time set \(\{t\in[0,T]:\xi(t)\in\ell_1\}\). The \(\Pi\)-path \(\xi\) makes a countable number of excursions
away from \(\ell\), and the excursion intervals form the connected components of the (relatively) open set
\([0,T]\setminus\{t\in[0,T]:\xi(t)\in\ell_1\}\).
There are at most two incomplete excursions in the interval \([0,T]\), namely the beginning excursion,
anchored to \(x_1\) at time \(0\) on the left, and the end excursion, anchored to \(x_2\) at time \(T\) on the right. In addition there can be at most finitely
many complete excursions for which \(\operatorname{dist}(\xi,\ell)\) reaches the level \(\varepsilon\) (in fact a calculation using the \(\operatorname{Lip}(v_1)\)
property of \(\xi\) gives an upper bound on the number of such excursions, namely \(\frac12 v_1 T/\varepsilon\)). We set
\begin{align*}
& U_0 \quad=\quad [0,T]\,,\\
& U_1 \quad=\quad
\\
&\{t\in U_0:\xi(t)\not\in\ell_1\}
\setminus \bigcup\{(a,b)\subset U_0: \xi(a), \xi(b)\in\ell_1, \; 0<\operatorname{dist}(\xi(t),\ell_1)<\varepsilon \text{ for }a<t<b\}\,.
\end{align*}
So \(U_1\) is a finite union of intervals (relatively open in \(U_0\)).
Define \(\widetilde{\xi}\) on \(U_0\setminus U_1\) as the orthogonal projection of \(\xi\) on \(\ell_1\). By the properties of
orthogonal projection and the \(\operatorname{Lip}(v_1)\) property of \(\xi\) on \(U_0\), it follows that \(|\widetilde{\xi}'(t)|\leq v_1\)
for almost all \(t\in U_ 0\setminus U_1\). Moreover, by construction,
\begin{align*}
|\widetilde{\xi}(t)-\xi(t)| \quad&=\quad 0 & \text{ if } \xi(t)\in \ell_1 \text{ and } t\in U_0\setminus U_1 \,,\\
|\widetilde{\xi}(t)-\xi(t)| \quad&<\quad \varepsilon & \text{ for other } t \in U_0\setminus U_1\,.
\end{align*}
Hence \(|\widetilde{\xi}(t)-\xi(t)|<\varepsilon\) for all \(t \in U_0\setminus U_1\). Finally, note that
\(\{t\in[0,T]:\xi(t)\in\ell_1\}\subseteq [0,T]\setminus U_1\).
Now define
\begin{align*}
& U_2 \quad=\quad \\
& \{t\in U_1:\xi(t)\not\in\ell_2\}
\setminus \bigcup\{(a,b)\subset U_1: \xi(a), \xi(b)\in\ell_2, \; 0<\operatorname{dist}(\xi(t),\ell_2)<\varepsilon \text{ for }a<t<b\}\,,
\end{align*}
and note that by construction \(\ell_1\) cannot intersect \(\xi\) in a time-set of positive measure in \(U_1\), so that \(\xi\) is
\(\operatorname{Lip}(v_2)\) in the time-set \(U_1\). We can argue as before that \(U_2\) is a finite union of relatively open intervals.
We can extend the definition of \(\widetilde{\xi}\) to \(U_1\setminus U_2\) by using orthogonal projection of \(\xi\) onto \(\ell_2\):
we have \(|\widetilde{\xi}'(t)|\leq v_2\)
for almost all \(t\in U_1\setminus U_2\), and \(|\widetilde{\xi}(t)-\xi(t)|<\varepsilon\) for all \(t \in U_1\setminus U_2\).
Moreover, \(\{t\in[0,T]:\xi(t)\in\ell_1\cup\ell_2\}\subseteq [0,T]\setminus U_2\).
The construction is illustrated in Figure \ref{fig:near-pi}
\begin{Figure}
\includegraphics[width=3in]{near-pi}
\centering
\caption{\label{fig:near-pi}
First two stages of the iterative construction of an \(\varepsilon\)-near-sequential-\(\Pi\)-path. The solid curve
represents the trajectory of the \(\Pi\)-path \(\xi\). The dotted segments represent the partially-defined trajectory of \(\widetilde{\xi}\)
after these first two stages (later stages successively fill in the gaps).
Note that \(\widetilde{\xi}\) is defined (a) when \(\xi\) runs along one of the lines \(\ell_1\), \(\ell_2\) and (b) when
\(\xi\) makes small excursions from one of these lines.
}
\end{Figure}
Iterating this construction, we end up defining \(\widetilde{\xi}\) on a time-set \([0,T]\setminus U_n\)
containing \(\{t\in[0,T]:\xi(t)\in \ell_1\cup\ldots\cup\ell_n\}\), such that
\(\widetilde{\xi}(t)\in\ell_1\cup\ell_2\cup\ldots\cup\ell_n\) for \(t\in[0,T]\setminus U_n\),
with
\(|\widetilde{\xi}'(t)|\leq v_r\)
for almost all \(t\) such that \(\widetilde{\xi}(t)\in \ell_r\), for \(r=1\), \(2\), \ldots \(n\), and finally
\(|\widetilde{\xi}(t)-\xi(t)|<\varepsilon\) if \(t\in[0,T]\setminus U_n\). Since
\(\{t\in[0,T]:\xi(t)\in \ell_1\cup\ldots\cup\ell_n\}\subseteq[0,T]\setminus U_n\), it follows from the countable exhaustion construction
that \(\operatorname{Leb}([0,T]\setminus U_n)<\varepsilon\).
We next complete the construction on the finite family of excursion intervals which are the connected components of the relatively
open set \(U_n\).
Note that by construction \(\widetilde{\xi}\) agrees with \(\xi\) on the end-points of these excursion intervals.
None of the lines \(\ell_1\), \(\ell_2\), \ldots, \(\ell_n\) intersect \(U_n\) in a time-set of positive measure:
therefore \(\xi\) satisfies a \(\operatorname{Lip}(v_{n+1})\) property on \(U_n\). Hence for each of these excursion intervals, if \(a\) and \(b\) are the
end-points then \(|\xi(b)-\xi(a)|\leq (b-a)v_{n+1}<(b-a)\varepsilon\).
Accordingly we can define \(\widetilde{\xi}\)
by linear interpolation over the excursion interval
(so that \(\widetilde{\xi}'\) is indeed constant over this excursion interval),
with the result that \(|\widetilde{\xi}'(t)|<\varepsilon\) for almost all \(t\in[a,b]\). Finally
the \(\operatorname{Lip}(v_{n+1})\) property implies that \(|\xi(t)-\xi(a)|\) and \(|\xi(t)-\xi(b)|\) are both strictly bounded above
by \(|b-a|\varepsilon \leq \varepsilon^2 \leq \varepsilon\) when \(a<t>b\) (use \(|b-a|\leq \operatorname{Leb}([0,T]\setminus U_n)<\varepsilon\) and \(\varepsilon\leq1\)); it follows by convexity that the same bound holds
if \(a\) and \(b\) are replaced by the piecewise interpolant \(\widetilde{\xi}(t)\):
\[
|\xi(t)-\widetilde{\xi}(t)|\quad<\quad \varepsilon\,.
\]
It follows that \(\widetilde{\xi}\) is the required \(\varepsilon\)-near-sequential-\(\Pi\)-path approximating \(\xi\) to within \(\varepsilon\)
in uniform norm. The sequence
\((\widetilde{\ell}_1,\widetilde{v}_1)\), \((\widetilde{\ell}_2,\widetilde{v}_2)\), \ldots, \((\widetilde{\ell}_m,\widetilde{v}_m)\)
is obtained from the successive visits (with repetitions) of \(\widetilde{\xi}\) to the finite sequence of lines
\((\ell_1,v_1)\), \((\ell_2,v_2)\), \ldots \((\ell_n,v_n)\).
\end{proof}
\begin{rem}\label{rem:geodesic-simplification}
If \(\xi\) is a \(\Pi\)-geodesic then the above construction can be simplified. Using the notation of the proof, suppose
that \((a,b)\) is a connected component of \(U_n\). The maximum speed of \(\xi\)
in \(U_n\) is \(v_{n+1}\): consequently if \(\xi(s)\) and \(\xi(t)\) belong to \(\ell_{n+1}\) for \(s<t\), both
belonging to \((a,b)\),
then the fastest route from \(\xi(s)\) to \(\xi(t)\) must run along \(\ell_{n+1}\) at maximum permitted speed \(v_n\). Consequently \(\{t\in(a,b):\xi\in\ell_{n+1}\}\)
must already be a relatively closed interval in \((a,b)\), so that there is no need to use the excursion construction in the proof of the theorem.
\end{rem}
We have seen that, under the weak condition \(\gamma>1\),
every \(\Pi\)-path can be uniformly approximated by a sequence of \(\varepsilon\)-near-sequential-\(\Pi\)-paths with \(\varepsilon\downarrow0\).
Conversely, if we strengthen the condition on \(\gamma\) to \(\gamma\geq d\) (so that the \emph{a priori} bound
of Theorem \ref{thm:a-priori-bound} is available) then
there is a kind of compactness result for \(\varepsilon\)-near-sequential-\(\Pi\)-paths.
\begin{thm}\label{thm:near-sequential-compactness}
Suppose that \(\gamma\geq d\geq2\) and \(T<\infty\).
For \(n=1\), \(2\), \ldots, let \(\widetilde{\xi}_n:[0,T]\to\mathbb{R}^d\) be an
\(\varepsilon_n\)-near-sequential-\(\Pi\)-path from \(x\) to \(y\), with \(\varepsilon_n\downarrow0\).
Then there are subsequences \(\{\widetilde{\xi}_{n_k}: k=1, 2, \ldots\}\) which converge uniformly
to \(\Pi\)-path limits.
\end{thm}
\begin{proof}
Since \(\varepsilon_n\) is decreasing in \(n\),
each \(\varepsilon_n\)-near-sequential-\(\Pi\)-path \(\widetilde{\xi}_n\) obeys the single modified speed-limit \(\max\{\varepsilon_1,V\}\).
Hence the comparison argument of Theorem \ref{thm:a-priori-bound} can be adapted to show that all the
\(\varepsilon_n\)-near-sequential-\(\Pi\)-paths \(\widetilde{\xi}_1\), \(\widetilde{\xi}_2\), \ldots lie in a single ball \(B\)
of radius \(R\) depending on \(V\) and \(\varepsilon_1\).
Consequently the \(\widetilde{\xi}_1\), \(\widetilde{\xi}_2\), \ldots obey a uniform Lipschitz condition (with Lipschitz
constant given by the speed of the fastest line to hit the ball \(B\)); therefore by the Arzela-Ascoli theorem we can extract a uniformly convergent
subsequence \(\{\widetilde{\xi}_{n_k}: k=1, 2, \ldots\}\) whose limit \(\widetilde{\xi}_\infty\) is also a Lipschitz path
with the same Lipschitz constant.
The persistence of Lipschitz constants in the limit also holds locally.
For fixed \(\lambda>0\), consider the open set \(\mathcal{S}_v^c=\{x:V(x)<v\}\).
Fix \(0<s<t<T\) such that \(\operatorname{Im}(\widetilde{\xi}_\infty|_{[s,t]})\subset\mathcal{S}_v^c\).
But \(\widetilde{\xi}_\infty\) is continuous, so \(\operatorname{Im}(\widetilde{\xi}_\infty|_{[s,t]})\) is compact; therefore the uniform convergence
of \(\widetilde{\xi}_{n_k}\to\widetilde{\xi}_\infty\) implies that for all \(k\geq k_\lambda\) we have
\[
\operatorname{Im}(\widetilde{\xi}_n|_{[s,t]})\quad\subset\quad \mathcal{S}_v^c\,.
\]
It follows that if \(k\geq k_\lambda\) then \(\widetilde{\xi}_{n_k}\) satisfies a \(\operatorname{Lip}(\max\{\varepsilon_{n_k}, v\})\) condition
over the time set \([s,t]\).
Bearing in mind that \(\varepsilon_n\downarrow0\),
we deduce that \(\widetilde{\xi}_\infty\) satisfies
a \(\operatorname{Lip}(v)\) condition whenever \(\widetilde{\xi}_\infty\) belongs to \(\mathcal{S}_v^c\).
This implies that the following is a Lebesgue-null subset of \([0,T]\):
\[
\{t\in[0,T]: |\widetilde{\xi}_\infty'(t)|> v \text{ and } \widetilde{\xi}_\infty(t)\not\in\mathcal{S}_v\}\,.
\]
Thus the subsequential limit \(\widetilde{\xi}_\infty\) is a \(\Pi\)-path (Definition \ref{def:pi-path}).
\end{proof}
This allows us to deduce the measurability of the random variable which is given by the time taken for a \(\Pi\)-geodesic to move between
specified end-points \(x_1\) and \(x_2\).
\begin{cor}\label{cor:measurable-time}
Suppose that \(\gamma>d\). Fix \(x_1\) and \(x_2\) in \(\mathbb{R}^d\), and let \(T\) be the least time such that there is a \(\Pi\)-path running from \(x_1\) to \(x_2\) in time \(T\), equivalently,
such that the (possibly non-unique) \(\Pi\)-geodesic from \(x_1\) to \(x_2\) has duration \(T\).
Then \(T\) is a function of the marked line process \(\Pi\): it is in fact measurable and hence a random variable.
\end{cor}
\begin{proof}
Consider the event \(E_{\varepsilon,\tau}\) that there is an \(\varepsilon\)-near-sequential-\(\Pi\)-path from \(x_1\) to \(x_2\) of duration at most \(\tau\).
This event is measurable,
because we may restrict attention to a countable sub-family of \(\varepsilon\)-near-sequential-\(\Pi\)-paths, determined for example
so that constituent line segments are bounded by the intersection point process of \(\Pi\).
But it is a consequence of the above results that
\[
\bigcap_\varepsilon E_{\varepsilon,\tau} \quad=\quad [\text{ duration of \(\Pi\)-geodesic from \(x_1\) to \(x_2\) is no more than \(\tau\) }]\,.
\]
For Theorem \ref{thm:pi-path-approximation} shows that the existence of such a \(\Pi\)-geodesic leads to the construction
of \(\varepsilon\)-near-sequential-\(\Pi\)-paths from \(x_1\) to \(x_2\) of the same duration as the \(\Pi\)-geodesic. On the other hand
it is an immediate consequence of
Theorem \ref{thm:near-sequential-compactness} that if \(E_{\varepsilon_n,\tau}\) is non-empty for a sequence \(\varepsilon_n\downarrow0\)
then there must exist a \(\Pi\)-path from \(x_1\) to \(x_2\) of duration \(\tau\). (Note that
we may prolong the duration of any \(\varepsilon\)-near-sequential-\(\Pi\)-path simply by holding it at its destination.)
Finally, the events \(E_{\varepsilon,\tau}\) are decreasing in \(n\), so the intersection \(\bigcap_\varepsilon E_{\varepsilon,\tau}\)
can be reduced to a countable intersection. It follows that
\[
\left[\text{ duration of \(\Pi\)-geodesic from \(x_1\) to \(x_2\) is no more than \(\tau\) }\right]
\]
is a measurable event.
\end{proof}
We note that simple selection criteria can be used to establish measurable maps which yield intervening \(\Pi\)-geodesics for each
pair of end-points \(x_1\) and \(x_2\).
\section[\texorpdfstring{$\Pi$}{Π}-geodesics: a.s. uniqueness in d=2]{\(\Pi\)-geodesics: almost-sure uniqueness in dimension \(2\)}\label{sec:pi-geodesics-uniqueness}
In the simplest non-trivial case, namely \(d=2\), it can be shown that the \(\Pi\)-geodesic between two specified points
is almost surely unique.
The method of proof makes essential use of the point-line duality of the plane, so
will not extend to the case \(d\geq3\).
\begin{rem}\label{rem:non-uniqueness}
The assertion of almost sure uniqueness between
of \(\Pi\)-geodesic connections between two specified points does \emph{not} imply that almost surely all pairs of points are connected by
unique \(\Pi\)-geodesics: a simple counterexample can be constructed by considering the possibility that three lines, of speeds only just below unit speed,
form an approximate equilateral triangle \(\Delta\)
of near unit-side length. Let \(\rho\) be the perimeter of \(\Delta\), and suppose all other lines within \(\rho/4\) of \(\Delta\) are of less than speed \(1/2\), while all lines hitting
the interior of
\(\Delta\) are of speed substantially less than \(1/2\). (How much less depends on the approximation to equilateral shape.) Both these events have positive probability.
Then consider the two \(\Pi\)-paths of length \(\rho/2\) running either way round \(\Delta\) from a given reference point on the boundary of \(\Delta\). These form two distinct \(\Pi\)-geodesics
between the same end-points. (The construction is illustrated in Figure \ref{fig:non-unique}.)
\end{rem}
Note that structures similar to this counterexample will exist at all scales in the random metric space produced from \(\mathbb{R}^2\) furnished with a random metric derived from \(\Pi=\Pi^{(2,\gamma)}\) for \(\gamma>2\).
So these random metric spaces are far from hyperbolic in the sense of Gromov (defined for example in \citealp[\S8.4]{BuragoBuragoIvanov-2001}).
\begin{Figure}
\includegraphics[width=2.5in]{non-unique}
\centering
\caption{\label{fig:non-unique}
Within the approximate equilateral triangle delineated by three fast lines, speeds are slow enough to prevent short-cuts.
Outside the triangle, up to a distance of one quarter of the perimeter, speeds are slow enough that there is no possibility of \(\Pi\)-geodesics
looping outside this region and then returning to the triangle.
}
\end{Figure}
We begin with a structural result about \(\Pi\)-geodesics in dimension \(2\), namely that if \(\ell\) is a line from \(\Pi\) (hence of positive speed-limit)
which contributes a segment to a \(\Pi\)-geodesic \(\xi\) then \(\xi\) joins and leaves \(\ell\) using simple intersections of \(\ell\) with other lines in \(\Pi\).
\begin{defn}[Proper encounter of a line by a \(\Pi\)-path]\label{def:proper-encounter}
Suppose \(d=2\) and \(\gamma>2\). We say that a \(\pi\)-path \(\xi\) \emph{encounters a line \(\ell\) of \(\Pi\) properly}
at \(\xi(t)\in\ell\) if there is \(\varepsilon>0\) such that within \(\operatorname{\mathcal B}(\xi(t),\varepsilon)\) the \(\Pi\)-path \(\xi\) is not contained solely in \(\ell\),
but lies in the union of \(\ell\) and a further line \(\widetilde\ell\)
from \(\Pi\).
\end{defn}
The notion of a proper encounter
is vacuous for \(\Pi\)-paths in the case \(d>2\), because then almost surely lines of the Poisson line process \(\Pi\) do not intersect each other.
\begin{thm}\label{thm:encounters}
Suppose \(d=2\) and \(\gamma>2\).
With probability \(1\), for each line \(\ell\in\Pi\) and each \(\Pi\)-geodesic \(\xi\), the intersections of \(\xi\)
with \(\ell\) form a finite disjoint union of intervals, such that the non-singleton intervals are delimited by proper encounters of \(\xi\) with \(\ell\).
\end{thm}
\begin{proof}
First note that with probability \(1\) there are no triple intersections of lines \(\ell_1\), \(\ell_2\), \(\ell_3\) from \(\Pi\).
Given this, the remainder of the argument is purely geometric, and therefore applies to all \(\Pi\)-geodesics simultaneously.
Consider the set of all lines \(\ell\) in \(\Pi\) intersecting \(\xi\). As noted in Remark \ref{rem:geodesic-simplification},
the \(\Pi\)-geodesic property implies that
the intersection of \(\xi\) with the fastest such line must be a single (possibly trivial) interval.
This is because the fastest route between first and last intersection with this fastest line must lie along the line.
For the second fastest line, the intersection must be formed as the union of at most two
intervals, which must be encountered respectively before and after the encounter with the fastest line. Continuing this argument, the
intersection of \(\xi\) with the \(k^\text{th}\) fastest line must be the union of at most \(2^{k-1}\) intervals.
Thus for \emph{any} line \(\ell\) from \(\Pi\),
if \(\xi\) intersects \(\ell\) at all then the intersection set must be the union of a finite number of intervals, some of which may be trivial.
For a given \(\Pi\)-geodesic \(\xi\), fix a given line \(\ell_1\) from \(\Pi\) which intersects \(\xi\), and consider the start \(\xi(t)\) of a
non-singleton intersection interval \([\xi(t),\xi(s)]\) between \(\xi\) and \(\ell_1\).
(Reversing time, the following argument will apply to the departure point \(\xi(s)\) as well as to the arrival point \(\xi(t)\).)
For sufficiently small \(\varepsilon\), either \(\ell_1\) will be the fastest line in \(\operatorname{\mathcal B}(\xi(t),\varepsilon)\), or it will be the second fastest, and the fastest line \(\ell_2\)
will intersect \(\ell_1\) at \(\xi(t)\). Choose \(u<t\) to be as small as possible subject to the requirement that \(\xi|_{(u,t]}\subset \operatorname{\mathcal B}(\xi(t),\varepsilon)\). It follows from the \(\Pi\)-geodesic property that if \(\xi|_{(u,t)}\) hits \(\ell_2\) at time \(v\in(u,t)\) (assuming \(\ell_2\) exists) then \(\xi|_{[v,t]}\) must run along \(\ell_2\), so that \(\xi\) makes a proper encounter with \(\ell_1\) using the line \(\ell_2\). On the other
hand, if \(\xi|_{(u,t)}\) does not hit \(\ell_2\) then it cannot hit \(\ell_1\), since if it did so at time \(v\in(u,t)\) then \(\xi|_{[v,t]}\) would have to run along \(\ell_1\), contradicting the maximality of \([t,s]\). Thus we may restrict attention to the case when \(\xi|_{(u,t)}\) hits neither \(\ell_1\) nor \(\ell_2\). This and following features of the construction are illustrated in Figure \ref{fig:encounter}.
\begin{Figure}
\includegraphics[width=2.5in]{encounter}
\centering
\caption{\label{fig:encounter}
Illustration of the construction used in the proof of Theorem \ref{thm:encounters} to demonstrate that a geodesic \(\xi\) must make a proper encounter on
a line \(\ell_1\in\Pi\). Here \(\ell_2\) is a possible faster line; by making the enclosing ball small
enough we can exclude the possibility that such a faster \(\ell_2\) hits \(\ell_1\) in any place other than the
point \(\xi(t)\) where \(\xi\) hits \(\ell_1\) for the first time. Dotted lines are lines of low cost, where the notion of cost is described in the
construction.}
\end{Figure}
We introduce the notion of \emph{cost}, based on comparison between the motion defined by \(\xi\) over \((u,t)\) (say)
and the motion defined by a comparison particle \(\widetilde{\xi}\), which begins at time \(u\) at the location which is the
projection of \(\xi(u)\) on \(\ell_1\), and which continues at maximum speed (\(w\), say) along \(\ell_1\) in the direction from \(\xi(t)\) to \(\xi(s)\).
We compute the cost of following \(\xi\) rather than \(\widetilde{\xi}\)
in terms of the time by which \(\widetilde{\xi}\) leads \(\xi\) when \(\xi\) hits \(\ell_1\) (namely, at time \(t\)). This is given by the integral
\[
\frac{1}{w}\int_{u}^t(w - |\xi'(a)|\cos\theta(a))\d{a}\quad=\quad \frac{1}{w}\int_{u}^t(w - v(a)\cos\theta(a))\d{a}\,,
\]
where \(\theta(a)\) is the angle that \(\xi'(a)\) makes with \(\ell\),
and setting
\(v(a)=|\xi'(a)|\). (Note that \(v(a)=|\xi'(a)|=V(\xi(a))\), because \(\xi\) is a \(\Pi\)-geodesic).
Let \(H\) be the perpendicular distance between \(\xi(u)\) and \(\ell_1\). We re-parametrize in terms of the perpendicular distance between \(\xi(a)\) and \(\ell_1\),
removing those parts of the integral for which \(\xi'(a)\) is not directed towards \(\ell_1\) (in which case the contribution to the integral is certainly positive, since \(\ell_1\) is faster than any other line
used by \(\xi\) over \((u,t)\)).
Setting
\[
\overline{a}(h)
\quad=\quad
\inf\left\{a:\text{ perpendicular distance of }\xi(a)\text{ from }\ell_1\text{ is }h\right\}\,,
\]
and \(\overline{v}(h)=v(\overline{a}(h))\),
and \(\overline{\theta}(h)=\theta(\overline{a}(h))\), we find
\begin{multline*}
\frac{1}{w}\int_{u}^t(w - |\xi'(a)|\cos\theta(a))\d{a}\quad\geq\quad \frac{1}{w}\int_{0}^H(w - \overline{v}(h)\cos\overline{\theta}(h))\frac{\d{h}}{\overline{v}(h)\sin\overline{\theta}(h)}\\
\quad=\quad
\int_{0}^H\left(
\frac{\csc\overline{\theta}}{\overline{v}}
- \frac{\cot\overline{\theta}}{w}
\right)\d{h}
\,.
\end{multline*}
Accordingly, define the relative \emph{cost index} of a given line \(\ell\) from \(\Pi\)
(compared with \(\ell_1\)) by
\begin{equation}\label{eqn:cost-index}
c(\ell)\quad=\quad \frac{\csc{\theta}}{{v}}
- \frac{\cot{\theta}}{w}\,,
\end{equation}
where \(v\) is the speed-limit of \(\ell\), and \(\theta\) is the angle it makes with \(\ell_1\). Evidently
the time by which \(\widetilde{\xi}\) leads \(\xi\) when \(\xi\) hits \(\ell_1\) can be controlled
in terms of an integral of cost indices of lines along which \(\xi\) travels when directed towards \(\ell_1\).
The cost index of \(\ell_1\) is not defined, though a limiting argument gives the value \(0\). Note too that, for any line \(\ell\) of speed \(v\),
the cost index of \(\ell\) turns out to be positive if \(v< w\).
Consider line-space parametrized using \(\xi(t)\) and \(\ell_1\) as reference point and reference line,
restricting attention to lines with speed-limit less than \(w\) (the speed-limit for \(\ell_1\)). The intensity measure \(\tfrac{\gamma-1}{2} \,v^{-\gamma}\d{v}\,\d{r}\, \d{\theta}\)
may be re-expressed in terms of \(c\), \(r\), and \(\theta\): since
\[
\frac{\d{c}}{\d{v}}\quad=\quad - \frac{\csc\theta}{v^2}\,,
\]
it follows that in the new coordinates the intensity measure is
\begin{equation}\label{eq:cost-measure}
\frac{\gamma-1}{2} \,\sin\theta\left(c \sin\theta + \frac{\cos\theta}{w}\right)^{\gamma-2} \,\d{c}\,\d{r}\,\d{\theta}\,.
\end{equation}
Now \(v^{-1}=c \sin\theta +\tfrac{\cos\theta}{w}>0\), so the measure determined by \eqref{eq:cost-measure}
is non-negative.
Consider the line pattern of lines with cost smaller than a specified threshold \(c_0\).
From the form of \eqref{eq:cost-measure}, this pattern is locally finite. On the other hand, the line pattern of lines
with cost exceeding a specified threshold must be locally dense even when constrained by
requiring
angle \(\theta\) to lie within a small interval.
We pick \(\widetilde\ell\) to be the lowest cost line separating \(\xi(s)\) from \(\xi_{(u,t)}\)
(see Figure \ref{fig:encounter} for a possible configuration, notice that this line may or may not be \(\ell_2\)),
and we
determine the minimal \(v\in[u,t]\) such that all the lines involved in \(\xi|_{(v,t)}\)
are more expensive than \(\widetilde\ell\).
If \(v=t\) then consider the \(\liminf\) as \(\delta\to0\) of the costs of lines involved in \(\xi|_{(t-\delta,t)}\).
This must be finite, for otherwise a low-cost line would produce a path faster than the \(\Pi\)-geodesic.
Since there are only finitely many low-cost lines near \(\xi(t)\),
this means there must be at least one low-cost line which is repeatedly visited by \(\xi\)
in every interval \((t-\delta,t)\); so this line must pass through \(\xi(t)\) and therefore must either be \(\ell_1\)
(which we have excluded) or \(\ell_2\) (which is a case already disposed of). Hence we may suppose
\(v<t\).
If \(v<t\), pick the line \(\ell^*\) of least cost which is hit by \(\xi|_{(v,t)}\).
Suppose (a) this hits the component of \(\ell_1\setminus\xi(s)\) not containing \(\xi(t)\). Then a combination of \(\ell^*\) and \(\widetilde\ell\) and \(\ell_1\) provides a faster way to get to \(\xi(s)\) than is provided by \(\xi\), again violating the \(\Pi\)-geodesic property of \(\xi\).
Otherwise (b) this line \(\ell^*\) does not hit the component of \(\ell_1\setminus\xi(s)\) not containing \(\xi(t)\).
If \(\xi\) does not meet \(\ell_1\) at \(\xi(t)\) using \(\ell^*\), then \(\ell^*\)
followed by \(\ell_1\) provides a faster way to get to \(\xi(s)\) than is provided by \(\xi\), violating the \(\Pi\)-geodesic property of \(\xi\).
It follows that if \(\xi\) does not meet \(\ell_1\) at \(\xi(t)\) using \(\ell_2\) then
it must meet \(\ell_1\) at \(\xi(t)\) using \(\ell^*\), proving the result.
As noted above, a time-reversal argument deals with the departure time \(s\). Consequently all the countably many non-singleton intersections of \(\xi\) with lines of positive speed-limit must be proper.
\end{proof}
Note in passing that in higher dimension \(d>2\) the quantity analogous to the cost \eqref{eqn:cost-index} varies along each line.
We can now prove almost sure uniqueness of \(\Pi\)-geodesics between specified pairs of points in the planar case.
\begin{thm}\label{thm:uniqueness}
Suppose \(\gamma>d=2\). Consider two points \(x\) and \(y\) in the plane \(\mathbb{R}^2\). Almost surely there is just one \(\Pi\)-geodesic connecting
\(x\) and \(y\).
\end{thm}
\begin{proof}
First, note the following consequence of Theorem \ref{thm:a-priori-bound}: as \(R\to\infty\), so
\[
\Prob{\text{ all geodesics from \(x\) to \(y\) are contained in }\operatorname{\mathcal B}(\text{\textbf{o}},R) }\quad\to\quad 1\,.
\]
Furthermore, Theorem \ref{thm:encounters} implies that the following assertion holds almost surely:
all \(\Pi\)-geodesics join or leave any line \(\ell\) in \(\Pi\) at only countably many possible places, namely the intersection
points of \(\ell\) with the rest of \(\Pi\). Moreover any particular \(\Pi\)-geodesic joins or leaves any particular line at only finitely many of these places.
Finally, note that if two \(\Pi\)-geodesics from \(x\) to \(y\) intersect at \(z\) then they must do so at the same relative time as measured from \(x\).
Bearing these facts in mind, we now develop the proof.
For fixed \(v>0\), pick a line \(\ell_0\) uniformly at random from \(\Pi_{v_0}\) such that \(\ell_0\Uparrow\operatorname{\mathcal B}(\text{\textbf{o}},R)\).
Note that the speed \(V\) of \(\ell_0\) has a Pareto distribution, with density \((\gamma-1) (v/v_0)^{-\gamma}\) for \(v>v_0\).
Note too that \(V\) is independent of the physical location of \(\ell_0\) and (by Slivnyak's theorem) is independent of
\(\Pi\setminus\{\ell_0\}\) which itself is a statistical copy of \(\Pi\).
Then (almost surely) for all sufficiently large \(R>0\) we know all \(\Pi\)-geodesics from \(x\) to \(y\) belong to the path-set
\(\mathcal{P}^{\ell_0}(x,y)=\bigcup_{u>0}\mathcal{P}^{\ell_0}_{u}(x,y)\), where \(\mathcal{P}^{\ell_0}_{u}(x,y)\) is the set of \(\Pi\)-paths from \(x\) to \(y\) lying in \(\operatorname{\mathcal B}(\text{\textbf{o}},R)\) for which:
\begin{itemize}
\item the \(\Pi\)-path is always run at maximal permissible speed;
\item excursions of the \(\Pi\)-path away from \(\ell_0\) are \(\Pi\)-geodesics;
\item the intersections of the \(\Pi\)-path with \(\ell_0\) form a finite disjoint union of intervals \([a,b]\);
\item and from these intervals the non-singleton intervals are delineated by proper encounters of the \(\Pi\)-path and \(\ell_0\),
moreover these end-points lie in \(\ell_0\cap\bigcup\{\ell:\ell\in\Pi_u\}\).
\end{itemize}
We further decompose \( \mathcal{P}^{\ell_0}_{u}(x,y)=\bigcup_\sigma \mathcal{P}^{\ell_0}_{u}(x,y;\sigma)\), where \(\sigma\) ranges over the family of finite sequences of pairs of (signed) integers, such that the closed intervals delineated by the pairs of integers are disjoint. To define \(\mathcal{P}^{\ell_0}_{u}(x,y;\sigma)\), consider the doubly-infinite point sequence \(\ell\cap\bigcup\{\ell:\ell\in\Pi_u\}\), and index the points by \(\mathbb{Z}\) once and for all, using a fixed sense of direction along \(\ell_0\) and arranging for the interval determined by the points indexed by \(0\) and \(1\) to be the (almost surely unique) interval nearest to \(\text{\textbf{o}}\). Then \(\mathcal{P}^{\ell_0}_{u}(x,y;\sigma)\)
is composed of those \(\Pi\)-paths in \(\mathcal{P}_{u}(x,y)\) for which the union of disjoint non-singleton intervals of intersection with \(\ell_0\) equals the union of the intervals bounded by pairs of points indexed by the pairs of \(\sigma\), moreover \(\sigma\) lists these intervals in the order in which they are visited and according to the direction in which they are travelled.
It is a consequence of the defining properties of \(\mathcal{P}^{\ell_0}(x,y)\) \emph{etc}, and the property that intersecting
\(\Pi\)-geodesics from \(x\) to \(y\) must visit their intersections at the same relative times, that all \(\Pi\)-paths in \(\mathcal{P}^{\ell_0}_{u}(x,y;\sigma)\)
spend the same amount of time \(S_\sigma\) outside \(\ell_0\).
Moreover, consider the lengths \(L_{\sigma_{1}}\), \(L_{\sigma_{2}}\), \ldots, \(L_{\sigma_{k}}\) corresponding to the intervals bounded by pairs of points indexed by the elements of \(\sigma=(\sigma_1, \sigma_2, \ldots, \sigma_k)\). These are
sums of independent Gamma random variables of the same rate, and all \(\Pi\)-paths in \(\mathcal{P}^{\ell_0}_{u}(x,y;\sigma)\)
spend the same amount of time \(L_\sigma/V=(L_{\sigma_{1}}+L_{\sigma_{2}}+\ldots+L_{\sigma_{k}})/V\) on \(\ell_0\).
Moreover the \(S_\sigma\) and the \(L_{\sigma_i}\) random variables are statistically independent of the speed \(V\) of \(\ell_0\).
If \(\sigma\neq\widetilde{\sigma}\) then the sums \(L_\sigma=L_{\sigma_{1}}+L_{\sigma_{2}}+\ldots+L_{\sigma_{k}}\), \(L_{\widetilde{\sigma}}=L_{\widetilde{\sigma}_{1}}+L_{\widetilde{\sigma}_{2}}+\ldots+L_{\widetilde{\sigma}_{\widetilde{k}}}\) can be decomposed into summands over shared or distinct Gamma random variables to reveal that \(L_\sigma-L_{\widetilde{\sigma}}\) has a non-degenerate probability density whenever \(\sigma\neq\widetilde{\sigma}\), and therefore \(\Prob{L_\sigma=L_{\widetilde{\sigma}}}=0\). Hence \(\Pi\)-paths from \(\mathcal{P}^{\ell_0}_{u}(x,y;\sigma)\) have a common duration of \(S_\sigma+L_\sigma/V\), and
\(\Pi\)-paths from \(\mathcal{P}^{\ell_0}_{u}(x,y;\widetilde{\sigma})\) have a common duration of \(S_{\widetilde{\sigma}}+L_{\widetilde{\sigma}}/V\), and the Pareto distribution and independence of \(V\) implies that
\[
\Prob{S_\sigma+L_\sigma/V=S_{\widetilde{\sigma}}+L_{\widetilde{\sigma}}/V}=0 \text{ if }\sigma\neq\widetilde{\sigma}\,.
\]
Thus almost surely, for all the countably many different \(\sigma\neq\widetilde{\sigma}\), the common durations of \(\Pi\)-paths from \(\mathcal{P}^{\ell_0}_{u}(x,y;\sigma)\) and \(\mathcal{P}^{\ell_0}_{u}(x,y;\widetilde{\sigma})\) are different.
It follows that almost surely all the \(\Pi\)-geodesics between \(x\) and \(y\) must traverse the same set of non-singleton intervals in the same direction along \(\ell_0\), since any two such \(\Pi\)-geodesics will have to belong to the same \(\mathcal{P}^{\ell_0}_{u}(x,y;\sigma)\) for sufficiently small \(u>0\). But this must then hold for all \(\ell\in\Pi\), and therefore (since \(\Pi\)-geodesics
must intersect at the same time as measured from \(x\)) almost surely all \(\Pi\)-geodesics between \(x\) and \(y\) must agree.
\end{proof}
The argument here is delicate: for example it is \emph{not} the case that the set of lengths along lines between intersections is linearly independent
if we consider the ensemble of lengths
of a unit Poisson line process. Indeed the tessellation produced by a Poisson line process will be rigid; consideration of various triangles shows that
the length of any single segment will be determined
by the lengths of all the other segments, so long as the incidence geometry of the segments is known.
The almost-sure uniqueness of planar \(\Pi\)-geodesics implies that planar
spatial networks formed from the Poisson line process model satisfy
property \ref{def:SIRSN-item-route} of Definition \ref{def:SIRSN}.
\section[\texorpdfstring{$\Pi$}{Π}-geodesics: finite mean-length in d=2]{\(\Pi\)-geodesics: finiteness of mean-length in dimension \(2\)}\label{sec:pi-geodesics-finite-mean-length}
One might conceive that a \(\Pi\)-geodesic between two fixed points might be of finite length but not of finite mean length.
However this is not the case, at least in dimension \(d=2\). We begin to show this by first establishing the finite mean length of constrained \(\Pi\)-geodesics, {restricted} to lie within specified
(two-dimensional) balls.
\begin{lemma}\label{lem:finite-mean-length-in-ball}
Suppose \(d=2\) and \(\gamma>2\). Consider \(x_1\), \(x_2\in\mathbb{R}^2\), fix \(r_0>|x_2-x_1|\), and consider
the least time by which it is possible to connect \(x_1\) to \(x_2\) by a \(\Pi\)-path which remains entirely within \(\operatorname{\mathcal B}(\tfrac{x_1+x_2}{2},r_0)\):
\begin{multline*}
T_{r_0}^* \quad=\quad \inf\Big\{T \;:\; \text{ there is }\xi\in\mathcal{A}_T \text{ such that } \\
\xi(0)=x_1, \; \xi(T)=x_2, \; \text{ and }\xi(t)\in\operatorname{\mathcal B}(\tfrac{x_1+x_2}{2},r_0)\text{ for all }t\in[0,T] \Big\}\,.
\end{multline*}
Then \(\Expect{T_{r_0}^*\;|\: V_0}<\infty\), where \(V_0\) is the speed-limit of the fastest line hitting \(\operatorname{\mathcal B}(\tfrac{x_1+x_2}{2},r_0)\);
moreover the following finite expectation provides an upper bound on the mean length of a \(\Pi\)-path connecting \(x_1\) to \(x_2\) within \(\operatorname{\mathcal B}(\tfrac{x_1+x_2}{2},r_0)\):
\[
\Expect{V_0 T_{r_0}^*}\quad<\quad \infty\,.
\]
\end{lemma}
\begin{proof}
Because we work only in dimension \(2\),
and seek an upper bound on \(\Pi\)-geodesic length, we are able to concentrate on \(\Pi\)-paths defined by joining together sequences of line segments;
there is no need to
negotiate the complexities of the tree construction described in Theorems \ref{thm:connection} and \ref{thm:metric-space}.
The time taken by such a \(\Pi\)-path, constrained to lie within \(\operatorname{\mathcal B}(\tfrac{x_1+x_2}{2},r_0)\) and connecting \(x_1\) to \(x_2\), necessarily provides an upper bound
on the \(\operatorname{\mathcal B}(\tfrac{x_1+x_2}{2},r_0)\)-constrained \(\Pi\)-geodesic connecting \(x_1\) to \(x_2\). Thus the finiteness of \(\Expect{V_0 T_{r_0}^*}\), together with the fact that \(V_0\)
is the maximum speed attainable within \(\operatorname{\mathcal B}(\tfrac{x_1+x_2}{2},r_0)\), provides an upper bound on the mean length of the
constrained \(\Pi\)-geodesic connecting \(x_1\) to \(x_2\) which is restricted to lie within \(\operatorname{\mathcal B}(\tfrac{x_1+x_2}{2},r_0)\).
Suppose that \(|x_2-x_1|=r_1<\frac12 r_0\).
Without loss of generality, we suppose that \(x_1+x_2=\text{\textbf{o}}=(0,0)\), and \(x_1=(-\frac12 r_1,0)\), \(x_2=(\frac12 r_1,0)\).
We shall join \(x_1\) and \(x_2\) together by working towards the two points by two paths commencing on the line segment \(\sigma_1=\{(0,h):0\leq h\leq r_1\}\);
we will then be able to join the two \(\Pi\)-paths together by prolonging the first line segment used in the construction of one of the \(\Pi\)-paths.
The constructions of the two \(\Pi\)-paths are entirely similar, so we shall focus on the \(\Pi\)-path leading to \(x_1\).
Suppose that the fastest line intersecting \(\operatorname{\mathcal B}(\text{\textbf{o}},r_0)\) has speed-limit \(V_0\). Exploiting the notion of meta-slowness described above
in the proof of Theorem \ref{thm:a-priori-bound}, we know that
if \(S_0=V_0^{-(\gamma-1)}\) is the meta-slowness of this line then
\begin{equation}\label{eq:fastest-line}
S_0 \quad=\quad \frac{1}{\pi r_0} E_0 \qquad \text{ where \(E_0\) is distributed as }\text{Exponential}(1)\,.
\end{equation}
The following formulae are simplified if
our \(\Pi\)-path constructions are required to avoid using this line. The first line used in the \(\Pi\)-path running to \(x_1\) will be the fastest line \(\ell_1\) with speed less than \(V_0\)
and intersecting both the line segment \(\sigma_1\) and the line segment from \(x_1\) to \(\tfrac{3}{2}x_1\).
Suppose that the speed-limit of this line is \(V_1\), so the meta-slowness is \(S_1=V_1^{-(\gamma-1)}\).
We use standard integral geometry of lines and Pythagoras to show that the line measure of all such lines is
\[
\frac12\left(|x_1-(0,r_1)|+|\tfrac{3}{2}x_1|-|\tfrac{3}{2}x_1-(0,r_1)|-|x_1\|\right)
\quad=\quad
\frac{\sqrt{5}-2}{4} \; r_1\,.
\]
Consequently we may deduce
\[
S_1 \quad=\quad S_0 + \frac{4}{(\sqrt{5}-2) r_1} E_1 \qquad \text{ where \(E_1\) is distributed as }\text{Exponential}(1)\,,
\]
and \(E_1\) is independent of \(S_0\) (equivalently \(V_0\)) and the geometry of the line \(\ell_1\).
\begin{Figure}
\includegraphics[width=2in]{mean-length}
\centering
\caption{\label{fig:mean-length}
Successive construction of one of two components of a \(\Pi\)-path from \(x_1\) to \(x_2\).
}
\end{Figure}
Let \(y_1\) be the point on \(\ell_1\) closest to \(x_1\), and note that the distance from \(y_1\) to \(\sigma_1\) along \(\ell_1\)
is bounded above by the distance from \(\tfrac{3}{2}x_1\) to \((0,r_1)\), namely
\[
\sqrt{\frac{9}{16}r_1^2 + r_1^2} \quad=\quad \frac{5}{4}\, r_1\,.
\]
The construction is illustrated in Figure \ref{fig:mean-length}.
This construction is continued recursively, for example replacing the origin by the point \(y_1\) on \(\ell_1\) closest to \(x_1\),
\(r_0\) by \(r_1\),
and replacing the segment \(\sigma_1\) by a segment \(\sigma_2\) begun at \(y_1\), directed along \(\ell_1\) towards the start of the \(\Pi\)-path,
of length \(r_2=2|y_1-x_1|\). Simple geometric arguments show that both \(|y_1-x_1|\) and \(|y_1-\tfrac{3}{2}x_1|\) are bounded above by \(\tfrac{1}{4}r_1\), so
the distance that \(\sigma_2\) extends
from \(\tfrac32 x_1\) cannot exceed \(\tfrac{3}{4}r_1\), while the distance between \(\tfrac32 x_1\) and \(\sigma_1\) is \(\tfrac34 r_1\).
This construction can be used to generate a new line \(\ell_2\), of meta-slowness \(S_2=V_2^{-(\gamma-1)}\)
which is required to be strictly greater than \(S_1\),
and a new closest distance \(r_2/2\) from \(x_1\) to \(\ell_2\). The calculations show that \(r_2\leq r_1/2\).
In general the \(n^\text{th}\) line \(\ell_n\) of the construction has meta-slowness \(S_n=V_n^{-(\gamma-1)}\)
with
\begin{equation}\label{eqn:recursion}
S_n \quad=\quad S_{n-1} + \frac{4}{(\sqrt{5}-2) r_n} E_n \qquad \text{ where \(E_n\) is distributed as }\text{Exponential}(1)\,,
\end{equation}
and \(T_1\) is independent of \(S_0\), \ldots, \(S_{n-1}\) (equivalently \(V_0\), \ldots, \(V_{n-1}\)) and the geometry of the lines \(\ell_1\), \ldots, \(\ell_{n-1}\).
Here \(r_n\) is the closest distance from \(x_1\) to \(\ell_n\), and \(r_n<r_{n-1}/2\); the length of the new segment
(running from \(\sigma_n\) to \(y_n\) along \(\ell_n\)) is bounded above by \(\tfrac{5}{4}r_n\).
Evidently we have constructed a \(\Pi\)-path from \(\sigma_1\) to \(x_1\), built as a sequence of line segments.
Total time of travel is bounded above by
\begin{multline*}\label{eqn:time-of-travel}
\sum_{n=1}^\infty S_n^{\frac{1}{\gamma-1}} \times \frac{5}{4} r_n
\quad=\quad
\frac{5}{4}
\sum_{n=1}^\infty \left(
S_0+ \frac{4}{\sqrt{5}-2}\left(\frac{E_1}{r_1}+\ldots+ \frac{E_n}{r_n}\right)\right)^{\frac{1}{\gamma-1}} r_n
\quad\leq\quad
\\
\frac{5r_0^{\frac{\gamma-2}{\gamma-1}}}{4}
\sum_{n=1}^\infty \left(
2^{-(n-1)}r_0S_0+
\frac{4}{\sqrt{5}-2}\left(
2^{-(n-1)}{E_1}+ 2^{-(n-2)}{E_2}+\ldots+ {E_n}\right)\right)^{\frac{1}{\gamma-1}}
\left(2^{\frac{\gamma-2}{\gamma-1}}\right)^{-n}
\,,
\end{multline*}
where the second step uses \(r_n<r_{n-1}/2\) and \(r_1<r_0\). We can use the conditional Jensen's inequality for the concave function
\(u\mapsto u^{\frac{1}{\gamma-1}}\) (concave because \(\gamma>2\)) to deduce that the mean total time of travel, conditional on \(V_0\)
(equivalently \(S_0\)), is bounded above by
\begin{equation}\label{eqn:time-of-travel}
\frac{5r_0^{\frac{\gamma-2}{\gamma-1}}}{4}
\sum_{n=1}^\infty \left(
2^{-(n-1)}r_0S_0+
\frac{8}{\sqrt{5}-2}\right)^{\frac{1}{\gamma-1}}
\left(2^{\frac{\gamma-2}{\gamma-1}}\right)^{-n}\,.
\end{equation}
Comparison with a geometric sum shows that this sum is finite, since \(\gamma>2\).
We deduce the finiteness of the conditional mean time from \(x_1\) to \(x_2\), since the path can be completed by extending either \(\ell_1\) or its counterpart
in the \(x_2\) path construction; the extra length required is bounded above by \(r_1<r_0\), and the extra time required is therefore bounded above by \(V_0^{-1} r_0\).
Finally, finiteness of mean length follows by multiplying the conditional mean time by \(V_0=S_0^{-\tfrac{1}{\gamma-1}}\) and then taking the expectation. The decisive calculation concerns
what happens to the conditional bound \eqref{eqn:time-of-travel} when multiplying through by \(S_0^{-\tfrac{1}{\gamma-1}}\)
and taking the expectation; we obtain a mean length upper bound of
\begin{multline}\label{eqn:length-of-travel}
\Expect{ \left(\frac{1}{S_0}\right)^{\frac{1}{\gamma-1}}
\frac{5r_0^{\frac{\gamma-2}{\gamma-1}}}{4}
\sum_{n=1}^\infty \left(
2^{-(n-1)}r_0 S_0+
\frac{8}{\sqrt{5}-2}\right)^{\frac{1}{\gamma-1}}
\left(2^{\frac{\gamma-2}{\gamma-1}}\right)^{-n}
}
\quad=\quad\\
\quad=\quad
\frac{5r_0^{\frac{\gamma-2}{\gamma-1}}}{4}
\sum_{n=1}^\infty
\Expect{ \left(\frac{1}{S_0}\right)^{\frac{1}{\gamma-1}}
\left(
2^{-(n-1)}r_0 S_0+
\frac{8}{\sqrt{5}-2}\right)^{\frac{1}{\gamma-1}}
}
\left(2^{\frac{\gamma-2}{\gamma-1}}\right)^{-n}
\quad\leq\quad\\
\quad\leq\quad
\frac{5r_0^{\frac{\gamma-2}{\gamma-1}}}{4}
\sum_{n=1}^\infty
\Expect{
\left(
2^{-(n-1)}r_0 S_0+
\frac{8}{\sqrt{5}-2}\right)^{\frac{1}{\gamma-1}} \;;\; S_0\geq1
}
\left(2^{\frac{\gamma-2}{\gamma-1}}\right)^{-n}
+\\
+
\frac{5r_0^{\frac{\gamma-2}{\gamma-1}}}{4}
\Expect{ \left(\frac{1}{S_0}\right)^{\frac{1}{\gamma-1}}\;;\;S_0<1}
\sum_{n=1}^\infty
\left(
2^{-(n-1)}r_0 +
\frac{8}{\sqrt{5}-2}\right)^{\frac{1}{\gamma-1}}
\left(2^{\frac{\gamma-2}{\gamma-1}}\right)^{-n}
\,.
\end{multline}
Finiteness of the first summand follows by using the conditional Jensen's inequality as before (noting that \(\gamma>2\)).
Finiteness of the second summand follows by noting, as \(\gamma>2\),
\[
\Expect{ \left(\frac{1}{S_0}\right)^{\frac{1}{\gamma-1}}\;;\;S_0<1}\quad<\quad\infty\,.
\]
\end{proof}
We can now prove the full result: the \(\Pi\)-geodesics between specified points are of finite mean length if \(d=2\).
\begin{thm}\label{thm:finiteness-of-mean}
Suppose \(d=2\) and \(\gamma>2\).
Consider a \(\Pi\)-geodesic \(\xi\) connecting two points \(x_1\) and \(x_2\). The mean length of \(\xi\) is finite.
\end{thm}
\begin{proof}
Consider two points \(x_1\), \(x_2\). Without loss of generality, set \(r_0>\tfrac{3}{\sqrt2}r_1=\tfrac{3}{\sqrt2}|x_2-x_1|\), \(\frac12(x_1+x_2)=\text{\textbf{o}}\), and \(x_1\), \(x_2\in\operatorname{\mathcal B}(\text{\textbf{o}}, r_0)\).
Note that we can pick \(r_0\) as large as we please.
We wish to show that the \(\Pi\)-geodesic \(\xi\) from \(x_1\) to \(x_2\) is of finite length.
It is immediate from the \(\Pi\)-geodesic property that the time spent by \(\xi\) in \(\operatorname{\mathcal B}(\text{\textbf{o}}, r_0)\) cannot exceed the time spent travelling from \(x_1\) to \(x_2\)
using the path described in Lemma \ref{lem:finite-mean-length-in-ball}.
Following the arguments of Lemma \ref{lem:finite-mean-length-in-ball}, we deduce finiteness of mean for the length of the portion of \(\xi\)
lying in \(\operatorname{\mathcal B}(\text{\textbf{o}}, r_0)\).
Let \(V_0=S_0^{-\frac{1}{\gamma-1}}\) be the fastest line hitting \(\operatorname{\mathcal B}(\text{\textbf{o}}, r_0)\). Recall from Equation \eqref{eq:fastest-line} of Lemma \ref{lem:finite-mean-length-in-ball}
that
\[
S_0 \quad=\quad \frac{1}{\pi r_0} E_0 \qquad \text{ where \(E_0\) is distributed as }\text{Exponential}(1)\,.
\]
\begin{Figure}
\includegraphics[width=2in]{racetrack}
\centering
\caption{\label{fig:racetrack}
Illustration of the racetrack construction: four \(r_1\times 3r_1\) rectangles placed to surround a central \(r_1\times r_1\) square,
which is centred inside a disc of radius \(r_0> \tfrac{{3}}{\sqrt2}r_1\). The racetrack is formed by four lines connecting the short sides of each rectangle,
chosen to be the fastest such lines which are strictly slower than the fastest line hitting the disc.
}
\end{Figure}
On the other hand, consider the ``racetrack'' around \(\text{\textbf{o}}\) formed by the fastest lines slower than \(V_0\) and connecting the short sides of rectangles
of sides \(r_1\) and \(3r_1\), placed to surround a central \(r_1\times r_1\) square
(see Figure \ref{fig:racetrack}).
By our choice of \(r_0\), the rectangles are all contained in \(\operatorname{\mathcal B}(\text{\textbf{o}},r_0)\).
Each of these lines intersects the \(3r_1\times3r_1\) square in a segment of length at most \(\sqrt{10}r_1\). Moreover the invariant line measure of the set of lines
joining the short sides of a rectangle
of sides \(r_1\) and \(3r_1\) is given by
\[
\frac12\left(
2\times \sqrt{10}r_1 - 2\times 3r_1
\right)\quad=\quad (\sqrt{10}-3)r_1\,.
\]
Therefore the speed-limits \(V'_i=(S'_i)^{-\frac{1}{\gamma-1}}\) (\(i=1,2,3,4\)) of these lines have distributions given by
\[
S'_i \quad=\quad S_0 + \frac{1}{(\sqrt{10}-3) r_1} E'_i \qquad \text{ where \(E'_i\) is distributed as }\text{Exponential}(1)\,.
\]
Here the \(E'_1\), \(E'_2\), \(E'_3\), \(E'_4\) are independent of each other and of \(S_0\);
this can be argued based on the facts that they are based on line-sets which are disjoint and conditioned on being slower than \(S_0\).
The racetrack establishes a path of length at most \(4\sqrt{10}r_1\), which can be traversed in time at most
\begin{multline}\label{eqn:max-time}
T_* \;=\; \sqrt{10}r_1 \sum_{i=1}^4 \left(S_0 + \frac{1}{(\sqrt{10}-3) r_1} E'_i\right)^{\frac{1}{\gamma-1}}
\;=\;
\sqrt{10}r_1^{\frac{\gamma-2}{\gamma-1}} \sum_{i=1}^4 \left(r_1 S_0 + \frac{1}{\sqrt{10}-3} E'_i\right)^{\frac{1}{\gamma-1}}
\\
\quad\leq\quad
\sqrt{10}r_1^{\frac{\gamma-2}{\gamma-1}} \left(4 r_1^{\frac{1}{\gamma-1}} S_0^{\frac{1}{\gamma-1}} + \sum_{i=1}^4 \left(\frac{1}{\sqrt{10}-3 } E'_i\right)^{\frac{1}{\gamma-1}}\right)
\\
\quad=\quad
4\sqrt{10}r_1 S_0^{\frac{1}{\gamma-1}}
+
\sqrt{10}r_1^{\frac{\gamma-2}{\gamma-1}}\sum_{i=1}^4 \left(\frac{1}{\sqrt{10}-3 } E'_i\right)^{\frac{1}{\gamma-1}}
\,,
\end{multline}
where the inequality follows from the Minkowski inequality (note that \(\gamma>2\)).
It follows that \(\xi\) cannot spend more than \(T_*\) of time outside of \(\operatorname{\mathcal B}(\text{\textbf{o}}, r_0)\), since otherwise
it would be possible to take a short-cut involving only some of the racetrack and two portions of \(\xi\) lying within \(\operatorname{\mathcal B}(\text{\textbf{o}}, r_0)\), thus travelling from
\(x_1\) to \(x_2\) in less time overall.
We now apply the comparison technique used in the proof of Theorem \ref{thm:a-priori-bound}, using a scalar comparison process \(y\). We suppose that \(\xi\) starts at
\(\partial\operatorname{\mathcal B}(\text{\textbf{o}}, r_0)\), so \(\operatorname{dist}(\xi(0),\text{\textbf{o}})=r_0\). Then \(|\xi|<y\), where \(y(0)=0\) and \(y'(t)=\overline{V}(y(t))\). Note that the fastest line hitting \(\operatorname{\mathcal B}(\text{\textbf{o}}, r_0)\)
has speed-limit \(V_0\), so \(\overline{V}(y(0))=V_0\). Moreover \(\overline{V}(r)\) for \(r>r_0\) is based entirely on lines with speeds faster than \(V_0\),
and is therefore independent of \(E'_1\), \(E'_2\), \(E'_3\), \(E'_4\).
In dimension \(d=2\), generalized distance is simply ordinary distance. So the recursive formulation \eqref{eqn:meta-slowness-recursion} becomes
\begin{align}\label{eqn:meta-slowness-recursion2}
R_n-R_{n-1} \quad&=\quad \frac{1}{S_{n-1}}\;\text{Exponential}\left(\pi\right)\,,\\
S_n \quad&=\quad S_{n-1} U_n \,,
\end{align}
for independent Uniform\((0,1)\) random variables \(U_i\),
with
distribution of
\(S_0\)
as above.
The times between successive changes of speed are given by
\[
S_{n-1}^{\frac{1}{\gamma-1}}(R_n-R_{n-1})
\quad=\quad
{S_{n-1}^{-(\gamma-2)/(\gamma-1)}}\text{Exponential}\left(\pi\right)\,.
\]
We know that \(S_n\) decreases as \(n\to\infty\).
Accordingly, a coupling argument shows that the number \(N_{T_*}\) of changes of speed by time \(T_*\) will not exceed \(\widetilde{N}\), where
\(\widetilde{N}\) has distribution \(\text{Poisson}\left({\pi S_0^{\frac{\gamma-2}{\gamma-1}} T_*}\right)\) when conditioned on \(S_0\) and \(T_*\),
and is independent of the actual changes of speed (though not of \(T^*\) or \(S_0\)).
Thus the final speed is no more than
\[
S_0^{-\frac{1}{\gamma-1}} \prod_{n=1}^{\widetilde{N}} U_n^{-\frac{1}{\gamma-1}}\,,
\]
and the distance travelled by the \(\Pi\)-geodesic outside \(\operatorname{\mathcal B}(\text{\textbf{o}}, r_0)\) cannot exceed
\[
S_0^{-\frac{1}{\gamma-1}} T_* \prod_{n=1}^{\widetilde{N}} U_n^{-\frac{1}{\gamma-1}}\,.
\]
Conditioning on \(E'_1\), \(E'_2\), \(E'_3\), \(E'_4\), and \(S_0\), we can
integrate out first the \(U_i\)'s and then the Poissonian variation \(\widetilde{N}\) from the resulting bound on mean distance travelled, and then use
the upper bound on \(T_*\) specified by \eqref{eqn:max-time}.
We thus obtain the following bound on mean distance travelled, using \(S_0=\tfrac{1}{\pi r_0} E_0\) and exchangeability of the \(E'_i\):
\begin{multline*}
\Expect{S_0^{-\frac{1}{\gamma-1}} T_* \times \prod_{n=1}^{\widetilde{N}} U_n^{-\frac{1}{\gamma-1}}}=
\Expect{S_0^{-\frac{1}{\gamma-1}} T_* \times \left(\frac{\gamma-1}{\gamma-2}\right)^{\widetilde{N}} }
=
\Expect{S_0^{-\frac{1}{\gamma-1}} T_* \times \exp\left( \frac{\pi S_0^{\frac{\gamma-2}{\gamma-1}} }{\gamma-2}T_*\right)}\\
\quad\leq\quad
4\sqrt{10}\;\mathbb{E}\Big[
\left(r_1
+
S_0^{-\frac{1}{\gamma-1}}r_1^{\frac{\gamma-2}{\gamma-1}}\left(\frac{1}{\sqrt{10}-3 } E'_1\right)^{\frac{1}{\gamma-1}}
\right)
\times
\exp\left( \frac{4\sqrt{10} \pi r_1 S_0}{\gamma-2}\right)\times\\
\times
\exp\left( \frac{\sqrt{10} \pi (r_1 S_0)^{\frac{\gamma-2}{\gamma-1}} }{\gamma-2}\sum_{i=1}^4 \left(\frac{1}{\sqrt{10}-3 } E'_i\right)^{\frac{1}{\gamma-1}}\right)
\Big]\\
\quad\leq\quad
4\sqrt{10}\; r_1\;\mathbb{E}\Big[
\left(1
+
\left(\frac{\pi}{\sqrt{10}-3 } \right)^{\frac{1}{\gamma-1}} \left(\frac{r_0}{r_1}\right)^{\frac{1}{\gamma-1}} \left(\frac{E'_1}{E_0}\right)^{\frac{1}{\gamma-1}}
\right)
\times
\exp\left( \frac{4\sqrt{10}}{\gamma-2}\; \frac{r_1}{r_0} E_0\right)\times\\
\times
\exp\left(\frac{\sqrt{10}}{\gamma-2} \frac{\pi^{\frac{1}{\gamma-1}} }{(\sqrt{10}-3)^{\frac{1}{\gamma-1}}}
\left(\frac{r_1}{r_0}\right)^{\frac{\gamma-2}{\gamma-1}}
\sum_{i=1}^4 E_0^{\frac{\gamma-2}{\gamma-1}} (E'_i)^{\frac{1}{\gamma-1}}
\right)
\Big]\,.
\end{multline*}
But now we can apply the simple inequality
\[
(E_0)^{\frac{\gamma-2}{\gamma-1}}(E'_i)^{\frac{1}{\gamma-1}}\quad=\quad
(E_0)^{1-\frac{1}{\gamma-1}}(E'_i)^{\frac{1}{\gamma-1}} \quad\leq\quad
E_0 + E'_i\qquad\text{(for \(E_0>0\), \(E'_i>0\))}\,,
\]
to deduce that
\begin{multline}\label{eqn:controlling-inequality}
\Expect{S_0^{-\frac{1}{\gamma-1}} T_* \times \prod_{n=1}^{N_{T_*}} U_n^{-\frac{1}{\gamma-1}}}\quad\leq\quad
4\sqrt{10}\; r_1\;\mathbb{E}\Big[
\left(1
+
\left(\frac{\pi}{\sqrt{10}-3 } \right)^{\frac{1}{\gamma-1}} \left(\frac{r_0}{r_1}\right)^{\frac{1}{\gamma-1}} \left(\frac{E'_1}{E_0}\right)^{\frac{1}{\gamma-1}}
\right)
\times
\\
\times
\exp\left( \frac{4\sqrt{10}}{\gamma-2}\; \frac{r_1}{r_0} E_0\right)\times
\exp\left(\frac{\sqrt{10}}{\gamma-2} \frac{\pi^{\frac{1}{\gamma-1}} }{(\sqrt{10}-3)^{\frac{1}{\gamma-1}}}
\left(\frac{r_1}{r_0}\right)^{\frac{\gamma-2}{\gamma-1}}
\left(4 E_0 + E'_1 + E'_2 + E'_3 + E'_4\right)
\right)
\Big]\,.
\end{multline}
Now the expectation can be bounded above by an expression involving finite Gamma integrals of the forms
\[
\int_0^\infty \exp(-\beta u)\d{u}\,,\qquad \int_0^\infty u^{\frac{1}{\gamma-1}} \exp(-\beta u)\d{u}\,,\qquad \int_0^\infty u^{-\frac{1}{\gamma-1}} \exp(-\beta u)\d{u}\,,
\]
for \(\beta>0\) (once \(r_0\) is chosen sufficiently large) and \(\gamma>2\).
Consequently the mean length of the \(\Pi\)-geodesic outside of \(\operatorname{\mathcal B}(\text{\textbf{o}},r_0)\) must also be finite, proving
the theorem.
\end{proof}
This work shows that that planar
spatial networks formed from the Poisson line process model satisfy
property \ref{def:SIRSN-item-finite-length} of Definition \ref{def:SIRSN}.
\section[Further properties of \texorpdfstring{$\Pi$}{Π}-geodesics in d=2]{Further properties of \(\Pi\)-geodesics in \(d=2\)}\label{sec:properties}
Finally we show that in dimension \(d=2\) any specified point \(x\) almost surely possesses just one \(\Pi\)-geodesic to \(\infty\);
moreover that for any three distinct points \(x,y,z\in\mathbb{R}^2\) almost surely the \(\Pi\)-geodesics from \(x\) to \(y\) and from \(x\) to \(z\) coincide for a non-trivial initial segment;
and also that if \(\Xi\) is an independent Poisson point process in \(\mathbb{R}^2\)
then almost surely the totality of all \(\Pi\)-geodesics between points of \(\Xi\) forms a fibre process \citep[\S8.3]{ChiuStoyanKendallMecke-2013}
which places finite total length in any given compact subset of \(\mathbb{R}^2\).
This last result shows that the network generated by \(\Pi\) possesses a very weak variant of Aldous' \emph{SIRSN property} \citep{AldousGanesan-2013,Aldous-2012}; a SIRSN
(scale-invariant random spatial network) would have the property that the \emph{mean} total length per unit area was finite (weak SIRSN property)
and moreover the property that the mean total length of connecting routes of distance at least \(1\) from start and source would remain bounded as
the intensity of \(\Xi\) increased to infinity. It is conjectured that the network generated by \(\Pi\) is a true SIRSN,
but at present all we can prove is the above ``pre-SIRSN'' property.
All three of these results depend on the same construction, based on \citet[Figure 6]{Aldous-2012}:
consider the behaviour of \(\Pi\)-geodesics starting from points in a \(2\times2\) square centred on the origin and ending outside a \(10\times10\) square centered at \((0,2)\).
Condition on the sides of the two squares and the \(y\)-axis all being subsets of lines from \(\Pi\), with speeds as follows:
the sides of the \(2\times2\) square have speed \(a\),
the \(y\)-axis has speed \(b\),
and the sides of the \(10\times10\) square have speed \(c\).
Suppose further that no other lines of \(\Pi_1\) (speed exceeding \(1\)) hit the \(10\times10\) square.
Figure \ref{fig:construction} illustrates the construction.
\begin{Figure}
\includegraphics[width=2in]{construction}
\centering
\caption{\label{fig:construction}
Construction forcing certain \(\Pi\)-geodesics to pass through the points \(A\) and \(B\).
}
\end{Figure}
\begin{lemma}\label{lem:pre-SIRSN-structure}
In the above situation, suppose that \(c > 10b > 59 a/3 > 354 / 3\).
Then any \(\Pi\)-geodesic connecting the interior of the \(2\times2\) square and the exterior of the \(10\times10\) square must pass through the points \(A=(0, -1)\)
and \(B=(0,-3)\).
\end{lemma}
\begin{proof}
To simplify exposition, we can and shall confine our attention to \(\Pi\)-geodesics constrained to lie in or on the \(10\times10\) square.
First note from the figure that the construction can be divided into rectangles of dimensions \(2\times2\), \(1\times2\), \(1\times6\), \(2\times4\), and \(6\times4\).
For each of these rectangles the sides have speed at least \(c\), while any other lines intersecting the rectangles have speed not exceeding \(1\).
Geometric comparisons show that, for any of these squares, \(\Pi\)-geodesics between pairs of points on the perimeter cannot intersect the interior.
This is a ``no short-cut'' condition for \(\Pi\)-geodesics.
In particular, a (constrained) \(\Pi\)-geodesic from the \(2\times2\) square to point \(C\) must be confined to the union of the \(2\times2\) square and the other square boundaries.
Consequently, such a \(\Pi\)-geodesic must have a final segment which is one of
\begin{enumerate}
\item \(A\to B\to C\) (the \(B\to C\) part using the perimeter of the \(10\times10\) square);
\item \(D\to C\);
\item \(E\to E^2\to C\) (last part using perimeter);
\item \(E\to E^1\to C\) (last part using perimeter);
\item \(F\to F^2\to C\) (last part using perimeter);
\item \(F\to F^1\to C\) (last part using perimeter);
\item or one of four cases which are mirror images of cases \(3-6\).
\end{enumerate}
We can now compare times taken by these alternative routes:
under the condition \(c > 10b > 59 a/3 > 354 / 3\) it transpires that the quickest route always passes through the points \(A\) and \(B\) as required.
\end{proof}
This lemma enables soft proofs of the three theorems of this section.
\begin{thm}\label{thm:unique-to-infinity}
Suppose \(\gamma>d=2\). With probability \(1\), for any point \(x\) there is one and only one \(\Pi\)-geodesic from \(x\) to \(\infty\); moreover all such infinite \(\Pi\)-geodesics eventually coalesce when sufficiently far away from the origin.
\end{thm}
\begin{proof}
Small perturbations of the structure described in Lemma \ref{lem:pre-SIRSN-structure} will have the same property (\(\Pi\)-geodesics from within small squares to exteriors of large squares all pass through specified points),
and so there is a positive probability \(\varepsilon>0\) that \(\Pi\) will generate a structure ensuring that all \(\Pi\) geodesics from the \(2\times2\) square reaching out further than the \(10\times10\) square
will have to pass through a specified pair of points near to \(A\) and \(B\).
Moreover we can use scale-invariance to generate further structures at larger scales, such that whether or not a corresponding perturbation of each structure is realized is independent of whether or not the other structures are realized.
An appeal to the second Borel-Cantelli lemma then shows that there must be an infinite sequence of planar points \((0, -3a_1)\), \((0, -3a_2)\), \ldots \(\to\infty\), such that if
\(x\in[-a_n,a_n]^2\) and \(y\not\in[-5a_n,5a_n]\times[-3a_n,7a_n]\)
then any \(\Pi\)-geodesic from \(y\) to \(x\) must pass through \((0, -3a_n)\). Moreover the section of this \(\Pi\)-geodesic from \(x\) to \((0,-3a_n)\) is uniquely determined, since for almost all \(y\) the \(\Pi\)-geodesic from \(y\) to \(x\) will be unique (Theorem \ref{thm:uniqueness}). Successive sections of similar \(\Pi\)-geodesics therefore build up a unique \(\Pi\)-geodesic from \(x\) to \(\infty\). Moreover the nature of the structure described in Lemma \ref{lem:pre-SIRSN-structure} ensures that, for any other planar point \(y\), this other point will be included in the smaller of the two rectangles of structures at sufficiently large scales: eventual coalescence of all infinite \(\Pi\)-geodesics thus follows.
\end{proof}
We can now establish a result similar to that of \cite{Bettinelli-2014} for the planar Brownian map: almost surely \(\Pi\)-geodesics emanating from a given point must initially coalesce.
(Evidently this cannot hold for \emph{all} points: consider points actually lying on a \(\Pi\)-geodesic!)
\begin{thm}\label{thm:coalescence-of-geodesics}
Suppose \(\gamma>d=2\). Almost surely for any distinct points \(x\), \(y\), \(z\), the \(\Pi\)-geodesics from \(x\) to \(y\) and from \(x\) to \(z\) coincide for a non-trivial initial segment.
\end{thm}
\begin{proof}
The argument follows that of Theorem \ref{thm:unique-to-infinity}, except that structures are now generated at increasingly smaller scales, all surrounding \(x\).
\end{proof}
\begin{thm}\label{thm:pre-SIRSN}
Suppose \(\gamma>d=2\). The network generated by \(\Pi\) has the \emph{pre-SIRSN property}, in the sense that if \(\Xi\) is an independent Poisson point process in \(\mathbb{R}^2\)
then almost surely the totality of all \(\Pi\)-geodesics between points of \(\Xi\) intersected with a compact set has finite total length.
\end{thm}
\begin{proof}
By scaling and monotonicity, we may suppose that the compact set in question is the \(1\times1\) square centred at the origin. Arguing as in Theorem \ref{thm:coalescence-of-geodesics},
at a suitably large scale there will be a structure which forces all \(\Pi\)-geodesics between points in the \((1\times1)\) square and points in the exterior of a \(L\times L\) square to pass through a specified point \(H\)
near the boundary of the \(L\times L\) square. Here \(L\) is random but depends only on \(\Pi\), not \(\Xi\).
Let \(N\) be the random number of points placed by \(\Xi\) in the \(L\times L\) square.
Then at most \(\binom{N+1}{2}\) \(\Pi\)-geodesics can intersect the \(1\times1\) square (based on these points and on \(H\)).
Each of these \(\Pi\)-geodesics has finite length (Theorem \ref{thm:finiteness-of-mean}), so the result follows.
\end{proof}
This proves property \ref{def:SIRSN-item-locally-finite} of Definition \ref{def:SIRSN} for the planar case.
\section{Conclusion}\label{sec:conclusion}
This paper has established:
\begin{enumerate}
\item Basic metric space properties of the Poisson line process model,
including existence of minimum-time paths;
\item Extension of the metric space properties of the Poisson line process model to higher dimensions;
\item Approximation results for minimum-time paths (``\(\Pi\)-geodesics'');
\item Almost-sure uniqueness and finite mean length of \(\Pi\)-geodesics in the planar case;
\item Local finiteness of resulting networks in the planar case.
\end{enumerate}
As a result, it follows that the planar Poisson line process model produces a pre-SIRSN (Definition \ref{def:SIRSN}).
The major outstanding question is, whether in fact the weak SIRSN or even full SIRSN properties hold for the planar Poisson line process model.
Extending the method of Theorem \ref{thm:pre-SIRSN} would require a much more quantitative approach;
it would be necessary to estimate the scale at which
there would exist structures forcing large \(\Pi\)-geodesics to pass through specified points.
A linked question concerns the nature of \(\Pi\)-geodesics in the planar case: can they be represented using sequences of line segments from the Poisson line process,
or do they necessarily involve the tree-like representations described in the proofs of Theorems \ref{thm:connection} and \ref{thm:metric-space}?
The methods of Section \ref{sec:pi-geodesics-uniqueness} are suggestive that the answer is yes,
but do not entirely exclude the possibility of slow-down at points not lying on \(\Pi\). One must show that \(\Pi\)-geodesics
between pairs of points can \emph{never} slow down to zero speed \emph{en route}.
This is conceptually linked to the notion of network transit points \citep{BastFunkeSandersSchultes-2007}, as discussed in \cite{Aldous-2012}.
A further question concerns whether the pre-SIRSN property extends to higher dimensions.
Point-line duality is pervasive in the arguments of the second half of this paper, so presently it is not clear how to proceed with this.
Possibly a quantitative form of \(\Pi\)-geodesic coalescence, as described in Theorem \ref{thm:coalescence-of-geodesics}, might allow headway to be made.
The paper has focussed throughout on results relating directly to the pre-SIRSN property of the Poisson line process model.
We have noted in passing and without proof some computations which establish sharpness of conditions on \(\gamma\) in our results
(Remarks \ref{rem:equivalence}, \ref{rem:connection-by-geodesics});
similar calculations show that the topology of Euclidean space \(\mathbb{R}^d\) viewed as a \(\Pi\)-geodesic metric space (for \(\gamma>d\)) is the normal Euclidean topology.
Finally, we note that an alternative motivation for the above work is given by recent developments in the study of Brownian maps;
the random metric space given here can be compared to the Brownian map (note for example that both situations exhibit coalescence of geodesics)
and promises by its constructive nature to be (relatively) more amenable to rigorous mathematical investigation, as well as providing higher dimensional constructions.
It would be of great interest to clarify the extent to which the two theories can be linked.
Note in particular the intriguing prospect of mimicking the Brownian map theory by the construction of ``Liouville Brownian motions'', perhaps using Dirichlet form theory
(compare \citealp{Berestycki-2013,GarbaRhodesVargas-2013}).
\bigskip
\noindent
\textbf{Acknowledgements:} My thanks to David Aldous, who challenged me to try to understand the Poisson line process model for a scale-invariant network.
\bibliographystyle{chicago}
|
2,869,038,156,407 | arxiv | \section{Introduction}
This paper is devoted to the nonexistence of positive solution to the Lane-Emden system,
\begin{align}\label{laneEmdenSystem}
\left\{\begin{array}{ll}
-\Delta u = v^p , \\
-\Delta v = u^q ,
\end{array}
\right.
\quad \text{in } \mathbb{R}^n,
\end{align}
where $\ u,v \geq 0$, $0<p, q < +\infty$.
The hyperbola \cite{Mitidieri93,Mitidieri96}
\begin{align*}
\frac{1}{p+1}+\frac{1}{q+1}=\frac{n-2}{n}
\end{align*}
is called critical curve because it is known that on or above it, i.e.
\begin{align*}
\frac{1}{p+1}+\frac{1}{q+1} \leq \frac{n-2}{n},
\end{align*}
which is called critical and supercritical respectively,
the system \eqref{laneEmdenSystem} admits (radial) non-trivial solutions, cf. Serrin and Zou \cite{SZ98}, Liu, Guo and Zhang \cite{LGZ06} and Li \cite{Li13}.
However, for subcritical cases, i.e. $(p,q)$ satisfying,
\begin{align}\label{subcriticalRegion}
\frac{1}{p+1}+\frac{1}{q+1}>\frac{n-2}{n},
\end{align}
people guess that the following statement holds and call it the Lane-Emden conjecture:
\begin{conjecture*}
$u=v\equiv0$ is the unique nonnegative solution for system \eqref{laneEmdenSystem}.
\end{conjecture*}
The full Lane-Emden conjecture is still open. Only partial results are known, and many researchers have made contribution in pushing the progress forward. We shall briefly present some important recent developments of the Lane-Emden conjecture.
Denote the scaling exponents of system \eqref{laneEmdenSystem} by
\begin{align}\label{scalingComponent}
\alpha = \dfrac{2(p+1)}{pq-1}, \ \ \beta = \dfrac{2(q+1)}{pq-1},\quad \text{for }pq>1.
\end{align}
Then subcritical condition \eqref{subcriticalRegion} is equivalent to
\begin{align}\label{subcriticalRegion2}
\alpha + \beta > n-2,\quad \text{for } pq>1.
\end{align}
For $p,q$ in the following region
\begin{align}
pq\leq 1, \ \text{or } pq>1 \ \text{and } \max\{\alpha,\beta\}\geq n-2,
\end{align}
\eqref{laneEmdenSystem} admits no positive entire \emph{supersolution}, cf. Serrin and Zou \cite{SZ96}. This implies the conjecture for $n=1,2$.
Also, the conjecture is true for
\begin{align}\label{BMRegion}
\min\{\alpha,\beta\}\geq \frac{n-2}{2}, \ \text{with } (\alpha,\beta) \neq (\frac{n-2}{2},\frac{n-2}{2}),
\end{align}
cf. Busca and Man\'{a}sevich \cite{BM02}. Note that \eqref{BMRegion} covers the case that both $(p,q)$ are subcritical, i.e. $\max\{p,q\}\leq \frac{n+2}{n-2}$, with $(p,q)\neq (\frac{n+2}{n-2},\frac{n+2}{n-2})$, which is treated earlier, cf. de Figueiredo and Felmer \cite{FF94} and Reichel and Zou \cite{RZ00}.
Also, Mitidieri \cite{Mitidieri96} has proved that the system admits no radial positive solution.
Chen and Li \cite{CL09a} have proved that any solution with finite energy must be radial, therefore combined with Mitidieri \cite{Mitidieri96}, no finite-energy non-trivial solution exists.
For $n=3$, the conjecture is solved by two papers. First, Serrin and Zou \cite{SZ96} proved that there is no positive solution with polynomial growth at infinity. Then Pol\'{a}\v{c}ik, Quittner and Souplet \cite{PQS07} removed the growth condition. In fact, they proved that no bounded positive solution implies no positive solution. This result has two important consequences. One is that combining with Serrin and Zou's result, one can prove the conjecture for $n=3$. The other is that proving the Lane-Emden conjecture is equivalent to proving nonexistence of bounded positive solution.
Thus, we always assume that $(u,v)$ are bounded in this paper.
For $n=4$, the conjecture is recently solved by Souplet \cite{Souplet09}. In \cite{SZ96}, Serrin and Zou used the integral estimates to derive the nonexistence results. Souplet further developed the approach of integral estimates and solved the conjecture for $n=4$ along the case $n=3$. In higher dimensions, this approach provides a new subregion where the conjecture holds, but the problem of full range in high dimensional space still seems stubborn.
Souplet has proved that if
\begin{align}\label{soupletRegion}
\max\{\alpha,\beta\}>n-3,
\end{align}
then \eqref{laneEmdenSystem} with $(p,q)$ satisfying \eqref{subcriticalRegion} has no positive solution. Notice that \eqref{soupletRegion} covers \eqref{subcriticalRegion} only when $n\leq 4$, and when $n\geq 5$ \eqref{soupletRegion} covers a subregion of \eqref{subcriticalRegion}.
The approach developed by Souplet in \cite{Souplet09} is also effective on non-existence of positive solution to Hardy-H\'{e}non type equations and systems (cf. \cite{Fazly2014, FG2014, Phan12, PS12}):
\begin{equation*}
\begin{cases}-\Delta u=|x|^a v^p,\\
-\Delta v=|x|^b u^q,
\end{cases}\quad \text{ in }\mathbb R^n.
\end{equation*}
This approach can also be applied to more general elliptic systems, for further details, we refer to \cite{Souplet12} and \cite{QS12}. Moreover, a natural extension and application of this tool is the high order Lane-Emden system which was done by Arthur, Yan and Zhao \cite{AYZ14}.
In this paper, we point out that the key to the Lane-Emden conjecture is obtaining a certain type of energy estimate. This estimate is in fact a necessary and sufficient condition to the conjecture. Connecting the estimate and the conjecture is a laborious work and needs to combine many types of estimates. We believe that with the result here people can refocus on proving the crucial estimate and thus solve the conjecture.
\begin{theorem}\label{liouvilleThm}
Let $n\geq 3$ and $(u,v)$ be a non-negative bounded solution to \eqref{laneEmdenSystem}.
Assume
there exists an $s>0$ satisfying $n-s\beta<1$ such that
\begin{align}
\int_{B_R} v^s \leq CR^{n-s\beta}, \label{vEstimateBetter}
\end{align}
then $u, v \equiv 0$ provided $0<p,q<+\infty$ and $\frac 1{p+1}+\frac 1{q+1}>1-\frac 2n$.
\end{theorem}
\begin{remark}\label{remark2}
\begin{enumerate}
\item[(a)] Energy estimate \eqref{vEstimateBetter} is a necessary condition to the Lane-Emden conjecture. One just needs to notice that when $u,v\equiv 0$, \eqref{vEstimateBetter} is obviously satisfied. The key to the proof of Theorem \ref{liouvilleThm} is to show \eqref{vEstimateBetter} is sufficient.
\item[(b)] If $p\geq q$, the assumption on $v$ is weaker than the corresponding assumption on $u$ due to a comparison principle between $u$ and $v$ (i.e. Lemma \ref{comparisonPrinciple}).
In other words, if $p\geq q$, and we assume for some $r>0$, such that $n-r\alpha <1$,
\begin{align}
\int_{B_R} u^r \leq CR^{n-r\alpha}. \label{uEstimateBetter}
\end{align}
Then \eqref{uEstimateBetter} implies \eqref{vEstimateBetter} by Lemma \ref{comparisonPrinciple}.
\item[(c)] By taking $s=p$ Theorem \ref{liouvilleThm} recovers the result in \cite{Souplet09}.
\item[(d)] A technical issue is that the standard $W^{2,p}$-estimate used in \cite{Souplet09} is not enough to establish Theorem \ref{liouvilleThm} (see the footnote of Proposition \ref{estimateOnBR}). To overcome this difficulty, a mixed type $W^{2,p}$-estimate is introduced in Lemma \ref{w2pEstimate}.
\end{enumerate}
\end{remark}
\begin{remark}
\begin{enumerate}
\item[(a)] It is worthy to point out an interesting role that the coefficient ``1" of the nonlinear source term plays in the Lane-Emden system. Consider the following system
\begin{equation}\label{laneEmdenSystem2}
\begin{cases}
-\Delta u=c_1(x)v^p,\\
-\Delta v=c_2(x)u^q,
\end{cases}\quad\text{in}\quad \mathbb R^n,
\end{equation}
where $0<a\leq c_1(x),c_2(x)\leq b<\infty$ and $x\cdot\nabla c_1(x),x\cdot\nabla c_2(x)\geq 0$ for some positive constants $a,b>0$. We can also have the following Rellich-Poho\v{z}aev type identity for some constants $d_1,d_2$ such that $d_1+d_2=n-2$,
\begin{equation}
\label{pohozaevId2}
\begin{split}
&\int_{B_R}(\frac{nc_1}{p+1}-d_1c_1+\frac{x\cdot \nabla c_1(x)}{p+1})v^{p+1}+(\frac{nc_2}{q+1}-d_2c_2+\frac{x\cdot \nabla c_2(x)}{q+1})u^{q+1}dx \\
&= R^n \int_{\mathbb S^{n-1}} \frac{c_1(R)v^{p+1}(R)}{p+1}+\frac{c_2(R)u^{q+1}(R)}{q+1}d\sigma \\
& +R^{n-1} \int_{\mathbb S^{n-1}} d_1v'u+d_2u'v d\sigma +
R^n\int_{\mathbb S^{n-1}}(v'u'-R^{-2}\nabla_{\theta}u\cdot\nabla_{\theta}v)d\sigma.
\end{split}
\end{equation}
By the constrains on $c_1(x),c_2(x)$, we can have the left terms (LT) in \eqref{pohozaevId2} as
\begin{equation}
LT\geq \delta_0\int_{B_R}v^{p+1}+u^{q+1}dx, \quad \text{for some}\quad \delta_0>0.
\end{equation}
The argument in \cite{Souplet09} is also valid for this case, and we still can prove nonexistence for $n\leq 4$ and for $\max(\alpha,\beta)>n-3, n\geq 5$.
On the other hand, for $c_1(x),c_2(x)$ such that $x\cdot \nabla c_1(x), x\cdot \nabla c_2(x)<0$, there exist non-zero solutions of \eqref{laneEmdenSystem2} in some subcritical cases (see Lei and Li \cite{LL13} for detail).
\item[(b)] Theorem \ref{liouvilleThm} is still true if we consider \eqref{laneEmdenSystem2} with $0<a\leq c_1(x),c_2(x)\leq b<\infty$ and $x\cdot \nabla c_1(x), x\cdot \nabla c_2(x)\geq 0$. And the proof is highly similar to the case $c_1=c_2=1$. So in this paper, we only prove for $c_1=c_2=1$.
\end{enumerate}
\end{remark}
The complete solution of the Lane-Emden conjecture may be a longstanding work. Hence, it will be interesting to consider the Lane-Emden conjecture under some conditions weaker than \eqref{vEstimateBetter}.
\\
\textbf{Open problem 1.} Can we prove the Lane-Emden conjecture under the following pointwise asymptotic:
\begin{equation*}
|v(x)|\leq C|x|^{-\gamma},\quad \text{for some}\quad 0<\gamma<\beta.
\end{equation*}
\\
\textbf{Open problem 2.} Can we prove the Lane-Emden conjecture under the following integral asymptotic:
\begin{equation*}
\int_{B_R}v^s\leq CR^\delta, \quad \text{for some} \quad s>0,\quad 0<\delta<1.
\end{equation*}
Clearly, if problem 2 is solved, problem 1 directly follows by choosing sufficiently large $s$.
The paper is organized as follows. In Section 2, we provide a few preliminary results. Some simplified proofs are given for the completeness and convenience of readers. One of the difficulty in the proof of Theorem \ref{liouvilleThm} was to control the embedding index, and we derived a varied form of $W^{2,p}$-estimate (see Lemma \ref{sphereEstimateW2p}) to solve this problem. In Section 3, we give the proof of Theorem \ref{liouvilleThm}. Our proof by classifying the argument into two cases hopefully can deliver the idea and the structure of the proof to readers in a clearer way.
\section{Preliminaries}
Throughout this paper, the standard Sobolev embedding on $\mathbb S^{n-1}$ is frequently used. Here we make some conventions about the notations.
Let $D$ denote the gradient with respect to standard metric on manifold. Let $n\geq 2$, $j\geq 1$ be integers and $1\leq z_1<\lambda\leq +\infty$, $z_2\neq (n-1)/j$. For $u=u(\theta)\in W^{j,z_1}(\mathbb S^{n-1})$, we have
\begin{align}\label{sobolevEmbedding}
\|u\|_{L^{z_2}(\mathbb S^{n-1})} \leq C\left( \|D_{\theta}^j u\|_{L^{z_1}(\mathbb S^{n-1})} + \|u\|_{L^1(\mathbb S^{n-1})}\right) ,
\end{align}
where
\begin{align*}
\left\lbrace
\begin{array}{ll}
\frac{1}{z_2} = \frac{1}{z_1} - \frac{j}{n-1}, \ &\text{if } z_1<(n-1)/j, \\
z_2 = \infty, \ &\text{if } z_1>(n-1)/j,
\end{array}
\right.
\end{align*}
and $C=C(j,z_1,n)>0$. Although $C$ may be different from line to line, we always denote the universal constant by $C$. For simplicity, in what follows, for a function $f(r,\theta)$, we define
\begin{align}\label{sphereNorm}
\|f\|_{p}(r)=\|f(r,\cdot)\|_{L^p(\mathbb S^{n-1})},
\end{align}
if no risk of confusion arises. Also let $s,p,q$ be defined as in Theorem \ref{liouvilleThm} and
$$l=s/p, \quad k=\frac{p+1}{p},\quad m=\frac{q+1}q.$$
By Remark \ref{remark2} (b) and Lemma \ref{comparisonPrinciple}, throughout the paper, we always assume $p\geq q$.
At last, we set
$$F(R)=\int_{B_R}u^{q+1}dx.$$
\subsection{Basic Inequalities}
Let us start with a basic yet important fact.
Considering $L^t$-norm on $B_{2R}$, we can write
\begin{align*}
\|f\|_{L^t(B_{2R})}^t = \int_0^{2R} \|f(r)\|_{L^t(\mathbb S^{n-1})}^t r^{n-1} dr,
\end{align*}
then by a standard measurement argument (cf. \cite{SZ96}, \cite{Souplet09}) one can prove that:
\begin{lemma}\label{sphereEstimate}
Let $f_i\in L^{p_i}_{loc}(\mathbb{R}^n)$, and $i=1,\ldots,N$, then for any $R>0$, there exists $\tilde{R}\in[R,2R]$ such that
\begin{align*}
\|f_i\|_{L^{p_i}(\mathbb S^{n-1})}(\tilde{R}) \leq (N+1) R^{-\frac{n}{p_i}}\|f_i\|_{L^{p_i}(B_{2R})}, \ \text{for each } i=1,\ldots,N.
\end{align*}
\end{lemma}
The following lemma is a varied $W^{2,p}$-estimate which seems not to appear in any literature, so we give a simple proof.
\begin{lemma}\label{w2pEstimate}
Let $1<\gamma<+\infty$ and $R>0$. For $u\in W^{2,\gamma}(B_{2R})$, we have
\begin{align*}
\|D^2 u\|_{L^\gamma(B_R)} \leq C\left( \|\Delta u\|_{L^\gamma(B_{2R})} + R^{\frac{n}{\gamma}-(n+2)}\|u\|_{L^1(B_{2R})}\right)
\end{align*}
where $C=C(\gamma,n)>0$.
\end{lemma}
Proof. First we deal with functions in $C^2(B_2)\cap C^0(\overline{B_2})$. By standard elliptic $W^{2,p}$-estimate, we have
\begin{align}\label{w2pEstimateStd}
\|D^2 u\|_{L^\gamma(B_1)} &\leq C( \|\Delta u\|_{L^\gamma(B_{\frac{3}{2}})}+\|u\|_{L^\gamma(B_{\frac{3}{2}})}).
\end{align}
By Lemma \ref{sphereEstimate}, $\exists \tilde{R}\in[\frac{7}{4},2]$ such that on $B_{\tilde{R}}$,
$u$ can be written as $u=w_1+w_2$, where respectively $w_1$ and $w_2$ are solutions to
\begin{align*}
\left\lbrace
\begin{array}{ll}
\Delta w_1 = \Delta u, \ &\text{in } B_{\tilde{R}}, \\
w_1=0, \ &\text{on } \partial B_{\tilde{R}},
\end{array}
\right.
\end{align*}
and
\begin{align*}
\left\lbrace
\begin{array}{ll}
\Delta w_2 = 0, \ &\text{in } B_{\tilde{R}}, \\
w_2=u, \ &\text{on } \partial B_{\tilde{R}},
\end{array}
\right.
\end{align*}
and additionally,
\begin{align}\label{sphereEstimateW2p}
\int_{\partial B_{\tilde R}}ud\sigma \leq C\|u\|_{L^1(B_2)}.
\end{align}
By standard $W^{2,p}$-estimate with homogeneous boundary condition, we have
\begin{align*}
\|w_1\|_{L^\gamma(B_{\frac{3}{2}})} \leq \| w_1\|_{W^{2,\gamma}(B_{\frac 32})}
\leq C\|\Delta w_1\|_{L^\gamma(B_{\tilde{R}})}.
\end{align*}
Since $w_2$ can be solved explicitly by Poisson formula on $B_{\tilde{R}}$, we see that by \eqref{sphereEstimateW2p} for any $x\in B_{\frac{3}{2}}\subsetneq B_{\tilde{R}}$, $w_2(x)$ can be bounded pointwisely by
\begin{align*}
|w_2(x)| \leq C \int_{\partial B_{\tilde{R}}} |u| \leq C\|u\|_{L^1(B_2)}.
\end{align*}
So,
\begin{align*}
\|w_2\|_{L^\gamma(B_{\frac{3}{2}})} \leq C \|u\|_{L^1(B_2)}.
\end{align*}
Hence,
\begin{align*}
\|u\|_{L^\gamma(B_{\frac{3}{2}})} &\leq \|w_1\|_{L^\gamma(B_{\frac{3}{2}})}+\|w_2\|_{L^\gamma(B_{\frac{3}{2}})} \\
&\leq C(\|\Delta u\|_{L^\gamma(B_{\tilde{R}})}+\|u\|_{L^1(B_2)}).
\end{align*}
Therefore, \eqref{w2pEstimateStd} becomes
\begin{align*}
\|D^2 u\|_{L^\gamma(B_1)} &\leq C( \|\Delta u\|_{L^\gamma(B_2)}+\|u\|_{L^1(B_2)}).
\end{align*}
Then the lemma follows from a dilation and approximation argument.
$\Box$
\begin{lemma}[Interpolation inequality on $B_R$]\label{interpolation}
Let $1\leq \gamma<+\infty$ and $R>0$. For $u\in W^{2,\gamma}(B_R)$, we have
\begin{align*}
\|D_x u\|_{L^1(B_R)} \leq C\left( R^{n(1-\frac{1}{\gamma})+1} \|D_x^2 u\|_{L^\gamma(B_R)} + R^{-1} \|u\|_{L^1(B_R)}\right),
\end{align*}
where $C=C(\gamma,n)>0$.
\end{lemma}
\subsection{Poho\v{z}aev Identity, Comparison Principle and Energy Estimates}
For system \eqref{laneEmdenSystem} we have a Rellich-Poho\v{z}aev identity, which is the starting point of the proof of Theorem \ref{liouvilleThm},
\begin{lemma}\label{pohozaevId}
Let $d_1,d_2\geq0$ and $d_1+d_2=n-2$, then
\begin{align*}
&\int_{B_R}(\frac{n}{p+1}-d_1)v^{p+1}+(\frac{n}{q+1}-d_2)u^{q+1}dx \\
&= R^n \int_{\mathbb S^{n-1}} \frac{v^{p+1}(R)}{p+1}+\frac{u^{q+1}(R)}{q+1}d\sigma
+R^{n-1} \int_{\mathbb S^{n-1}} d_1v'u+d_2u'v d\sigma +
R^n\int_{\mathbb S^{n-1}}(v'u'-R^{-2}\nabla_{\theta}u\cdot\nabla_{\theta}v)d\sigma.
\end{align*}
\end{lemma}
\begin{lemma}[Comparison Principle]\label{comparisonPrinciple}
Let $p\geq q >0,pq>1$ and $(u,v)$ be a positive bounded solution of \eqref{laneEmdenSystem}. Then we have the following comparison principle,
\begin{align*}
v^{p+1}(x)\leq \frac{p+1}{q+1} u^{q+1}(x), \ x\in \mathbb{R}^n.
\end{align*}
\end{lemma}
Proof. Let $l=(\frac{p+1}{q+1})^{\frac{1}{p+1}}$, $\sigma=\frac{q+1}{p+1}$. So $l^{p+1}\sigma=1$, and $\sigma\leq 1$. Denote
\begin{align*}
\omega = v- lu^{\sigma}.
\end{align*}
We will show that $\omega \leq 0$.
\begin{align*}
\Delta\omega &= \Delta v - l \nabla\cdot(\sigma u^{\sigma-1}\nabla u) \\
&= \Delta v - l\sigma(\sigma-1)|\nabla u|^2 -l\sigma u^{\sigma-1}\Delta u\\
&\geq -u^q+l\sigma u^{\sigma-1}v^p \\
&= u^{\sigma-1}((\frac{v}{l})^p - u^{q+1-\sigma}) \\
&= u^{\sigma-1}((\frac{v}{l})^p - u^{\sigma p}).
\end{align*}
So, $\Delta\omega>0$ if $w>0$.
Now, suppose $w>0$ for some $x\in\mathbb{R}^n$, and there are two cases:
Case 1: $\exists x_0\in \mathbb{R}^n$, such that $\omega(x_0)=\displaystyle \max_{\mathbb{R}^n} \omega(x)>0$, and $\Delta \omega(x_0)\leq 0$. However, when $w>0$, $\Delta\omega>0$, a contradiction.
Case 2: There exists a sequence $\{x_m\}$ with $|x_m|\rightarrow+\infty$, such that $\displaystyle \lim_{m\rightarrow+\infty} \omega(x_m) = \displaystyle \max_{\mathbb{R}^n} \omega(x) >c_0>0$ for some constant $c_0$.
Let $\omega_R(x)=\phi(\frac{x}{R})\omega(x)$, where $\phi(x)\in C_0^{\infty}(B_1)$ is a cutoff function and $\phi(x)\equiv1$ in $B_{\frac{1}{2}}$. Since $\omega_R(x)=0$ on $\partial B_R$, there exists an $x_R\in B_R$ such that $\omega_R(x_R)=\displaystyle \max_{B_R}\omega_R(x)$ and $\displaystyle \lim_{R\rightarrow+\infty} \omega(x_R) = \displaystyle \max_{\mathbb{R}^n} \omega(x) >0$. Also,
\begin{align*}
0=\nabla \omega_R(x_R) = \phi(\frac{x_R}{R})\nabla\omega(x_R) + \frac{1}{R}\nabla\phi(\frac{x_R}{R})\omega(x_R).
\end{align*}
As $\phi(\frac{x_R}{R})\geq c_1>0$ for some constant $c_1$ (in fact, $\phi(\frac{x_R}{R})\rightarrow 1$) and $\omega(x_R)$ is bounded since $u,v$ are bounded in $\mathbb{R}^n$, we see that $\nabla\omega(x_R)\rightarrow 0$ as $R\rightarrow +\infty$.
So,
\begin{align*}
0 &\geq \Delta\omega_R(x_R)=\frac{1}{R^2}\Delta\phi(\frac{x_R}{R})\omega(x_R)+\frac{2}{R}\nabla\phi(\frac{x_R}{R})\cdot\nabla\omega(x_R)+\phi(\frac{x_R}{R})\Delta\omega(x_R) \\
\Rightarrow 0 &\geq \Delta\omega(x_R) + O(\frac{1}{R^2})
\end{align*}
Since $\omega(x_R)>c_0/2$ for sufficiently large $R$, $\Delta\omega(x_R)>c_2>0$ for some constant $c_2$, a contradiction.
$\Box$
\begin{remark}
For general Lane-Emden type system \eqref{laneEmdenSystem2}, we can choose
$$w=v-Clu^{\sigma},\quad \text{where}\quad C^{p+1}=\sup_{x\in\mathbb R^n}\frac{c_2(x)}{c_1(x)}.$$
By the same arguments, we can also get the desired comparison principle.
\end{remark}
Next we prove a group of energy estimates which are crucial to the entire argument in this paper. As Theorem \ref{liouvilleThm} points out, better energy estimates are the key to the Lane-Emden conjecture. Unfortunately, efforts have been made so far only provide the following inequalities, which are first obtained by Serrin and Zou \cite{SZ96} (1996). Here we give a simpler proof than the original one for the convenience of readers.
\begin{lemma}\label{energyEstimates}
Let $p,q>0$ with $pq>1$. For any positive solution $(u,v)$ of \eqref{laneEmdenSystem}
\begin{align}\label{uvEstimate1}
\int_{B_R} u \leq C R^{n-\alpha}, \ \text{and } \int_{B_R} v \leq C R^{n-\beta},
\end{align}
\begin{align}\label{uvEsitmatePQ}
\int_{B_R} u^q \leq C R^{n-q\alpha}, \ \text{and } \int_{B_R} v^p \leq C R^{n-p\beta}.
\end{align}
\end{lemma}
Proof. Without loss of generality, we can assume that $p\geq q$.
Let $\phi\in C^{\infty}(B_R(0))$ be the first eigenfunction of $-\Delta$ in $B_R$ and $\lambda$ be the eigenvalue. By definition and rescaling, it is easy to see that $\phi\mid_{\partial B_R} =0$ and $\lambda\sim \frac{1}{R^2}$. By normalizing, one gets $\phi\geq c_0>0$ on $B_{R/2}$ for some constant $c_0$ independent of $R$, $\phi(0)=\|\phi\|_{\infty}=1$. So, multiplying \eqref{laneEmdenSystem} by $\phi$ then integrating by parts on $B_R$ we have,
\begin{align*}
\int_{B_R} \phi u^q = -\int_{B_R} \phi \Delta v &= \int_{\partial B_R} v \frac{\partial \phi}{\partial n} d\sigma + \lambda \int_{B_R} \phi v.
\end{align*}
By Hopf's Lemma we know that $\frac{\partial \phi}{\partial n}<0$ on $\partial B_R$, so
\begin{align}\label{uqBoundedByV}
\int_{B_R} \phi u^q \leq \lambda \int_{B_R} \phi v.
\end{align}
Similarly, we have
\begin{align}\label{vpBoundedByU}
\int_{B_R} \phi v^p \leq \lambda \int_{B_R} \phi u.
\end{align}
Applying Lemma \ref{comparisonPrinciple} to \eqref{uqBoundedByV}, we have
\begin{align*}
\frac{1}{R^2} \int_{B_R} \phi v\geq C \int_{B_R} \phi v^{\frac{q(p+1)}{q+1}}.
\end{align*}
Notice that $\frac{q(p+1)}{q+1}>1$ as $pq>1$, so by H\"{o}lder inequality
\begin{align*}
\int_{B_R} \phi v^{\frac{q(p+1)}{q+1}} &\geq (\int_{B_R} \phi v)^{\frac{q(p+1)}{q+1}}(\int_{B_R} \phi)^{-(\frac{q(p+1)}{q+1}-1)} \\
&\geq C(\int_{B_R} \phi v)^{\frac{q(p+1)}{q+1}} R^{-n\frac{qp-1}{q+1}}.
\end{align*}
So,
\begin{align*}
&\frac{1}{R^2} \int_{B_R} \phi v\geq C (\int_{B_R} \phi v)^{\frac{q(p+1)}{q+1}} R^{-n\frac{qp-1}{q+1}} \\
\Rightarrow & \int_{B_R} \phi v \leq C R^{n-\beta}.
\end{align*}
Therefore, by \eqref{uqBoundedByV}
\begin{align*}
\int_{B_R} \phi u^q \leq C R^{n-\beta-2} =CR^{n-q\alpha}.
\end{align*}
Now, \textbf{Case 1:} If $q>1$, then by H\"{o}lder inequality
\begin{align*}
\int_{B_R} \phi u \leq (\int_{B_R} \phi u^q)^{\frac{1}{q}}(\int_{B_R} \phi )^{\frac{1}{q'}}
\leq CR^{\frac{n}{q}-\alpha} R^{\frac{n}{q'}}
= CR^{n-\alpha}, \quad \frac 1q+\frac 1{q'}=1.
\end{align*}
Mean while, by \eqref{vpBoundedByU}
\begin{align*}
\int_{B_R} \phi v^p \leq CR^{n-\alpha-2} = CR^{n-p\beta}.
\end{align*}
This finishes the proof for Case 1.
\textbf{Case 2:} Assume that $q\leq 1$. To prove this trickier case, we begin with a lemma of energy-type estimate,
\begin{lemma}
If $\Delta u\leq 0$, then for $\gamma\in(0,1)$, $\eta\in C^\infty_0(\mathbb{R}^n)$,
\begin{align}\label{energy}
\int_{\mathbb{R}^n} \frac{4}{\gamma^2}|D(u^{\frac{\gamma}{2}})|^2\eta^2 = \int \eta^2 |Du|^2 u^{\gamma-2}
\leq C \int |D\eta|^2u^{\gamma}.
\end{align}
\end{lemma}
Proof. Multiply $\eta^2 u^{\gamma-1}$ to $\Delta u\leq 0$ then integrate over the whole space. $\Box$
We rewrite \eqref{energy} as
\begin{equation}\label{energy2}
\int_{B_R}|D u|^2u^{\gamma-2}\leq \frac{C_\gamma}{R^2}\int_{B_{2R}}u^{\gamma}
\end{equation}
where $C_\gamma\rightarrow+\infty$ as $\gamma\rightarrow 1$.
From Poincar\'e's Inequality, we have
\begin{align}\label{embedding}
|f|_{\frac{na}{n-a},\Omega_R} \leq C(n,a,\Omega)\left(|Df|_{a,\Omega_R} + |R|^{\frac{n-a}{a}}|f_{\Omega_R}|\right),
\end{align}
where
\begin{align*}
f_{\Omega_R}=\Xint-_{\Omega_R} f=\frac{1}{|\Omega_R|}\int_{\Omega_R} f,\quad \Omega_R=\{Rx|x\in\Omega\}.
\end{align*}
\par
Next we prove a variation of embedding inequality,
\begin{lemma}
For any $l\geq 1$,
\begin{align}\label{variationEmbedding}
|f^l|_{\frac{an}{n-a},\Omega_R} \leq C(n,a,\Omega)\left( |D(f^l)|_{a,\Omega_R} + |R|^{\frac{n-a}{a}} |f_{\Omega_R}|^l\right)
\end{align}
\end{lemma}
Proof. By \eqref{embedding},
\begin{align*}
|f^l|_{\frac{an}{n-a},\Omega_R} &\leq C(n,a,\Omega)\left( |D(f^l)|_{a,\Omega_R} + |R|^{\frac{n-a}{a}} |(f^l)_{\Omega_R}|\right) \\
&\leq C(n,a,\Omega)\left( |D(f^l)|_{a,\Omega_R} + |R|^{\frac{n-a}{a}-n} \int_{\Omega_R} f^l dx \right) \\
&\leq C(n,a,\Omega)\left\lbrace |D(f^l)|_{a,\Omega_R} + |R|^{\frac{n-a}{a}-n} (\int_{\Omega_R} f dx)^{\theta l} (\int_{\Omega_R} f^{l\frac{an}{n-a}}dx)^{(1-\theta)l \frac{n-a}{lan}}\right\rbrace,\quad\theta=\frac{1-\frac{n-a}{na}}{l-\frac{n-a}{na}} \\
&\leq C(n,a,\Omega) |D(f^l)|_{a,\Omega_R} + \frac 1 2 |f^l|_{\frac{an}{n-a},\Omega_R} + C(n,a,\Omega) |R|^{\frac{n-a}{a}} |f_{\Omega_R}|^l.
\end{align*}
In getting the last inequality, we have used the Young's inequality. So we get \eqref{variationEmbedding}.
$\Box$
Let $l\geq 1$, $\theta\leq 2q<2$, $\gamma=l\theta<1$, $f=u^{\frac\theta2}$, $a=2$. Then
\begin{equation}
\begin{split}
|f^l|_{\frac{2n}{n-2},B_R}&\leq C\left(|Df^l|_{2,B_R}+R^{\frac{n-2}2}|f_{B_R}|^l\right)\\
&\leq C\left(|D(u^{\frac{l\theta}{2}})|_{2,B_R}+R^{\frac{n-2}{2}}|(u^{\frac\theta2})_{B_R}|^l\right)\\
&\leq \frac CR\left(\int_{B_{2R}}u^{l\theta}\right)^{\frac 12}+R^{\frac{n-2}2}\left(\Xint-_{B_R}u^{\frac\theta2}\right)^l.
\end{split}
\end{equation}
The last term on the right can be estimate by H\"{o}lder and the fact that $\Xint-_{B_R} u^q\leq CR^{-q\alpha}$ since $\frac{\theta}{2}<q$. This yields that
\begin{equation}
\int_{B_R}u^{\frac n{n-2}\theta l}\leq C\left(R^{-\frac{2n}{n-2}}\left(\int_{B_{2R}}u^{l\theta}\right)^{\frac n{n-2}}+R^{n-\frac n{n-2}l\theta\alpha}\right).
\end{equation}
This means if $\Xint-_{B_R}u^{l\theta}\leq CR^{-l\theta\alpha}$, we have $\Xint-_{B_R}u^{\frac n{n-2}l\theta}\leq CR^{-\frac{n}{n-2}l\theta\alpha}$ provided $l\theta<1$. By $\Xint-_{B_R} u^q\leq CR^{-q\alpha}$, one gets
\begin{equation}
\Xint-_{B_R}u^s\leq C(s)R^{-s\alpha},\quad \text{for}\quad s<\frac n{n-2}
\end{equation}
where $C(s)\rightarrow+\infty$ as $s\rightarrow \frac{n}{n-2}$.
By taking $s=1$, the above inequality immediately leads to $$\int_{B_R} u \leq C R^{n-\alpha}.$$
Since $pq>1$ and we assume that $p\geq q$, $q$ must be greater than 1, then by H\"{o}lder and \eqref{vpBoundedByU} we get
$$\int_{B_R} v^p \leq C R^{n-p\beta}.$$
This finishes the proof of Lemma \ref{energyEstimates}. $\Box$
\subsection{Key Estimates on $\mathbb{S}^{n-1}$}
Now that we have energy inequalities \eqref{uvEsitmatePQ}, in our assumption \eqref{vEstimateBetter} we can always assume $s\geq p$.
Since $l=\frac{s}{p}$, we have $l\geq 1$. The following estimates for quantities on sphere $\mathbb S^{n-1}$ are necessary to the proof.
\begin{proposition}\label{estimateOnSphere}
For $R\geq 1$, there exists $\tilde{R}\in[R,2R]$ such that for $l=\frac s p\geq 1$, $k=\frac{p+1}{p}$ and $m=\frac{q+1}{q}$, we have
\begin{align*}
\|u\|_1(\tilde{R}) \leq CR^{-\alpha}, & \
\|v\|_1(\tilde{R}) \leq CR^{-\beta}, \\
\|D^2_x u\|_l(\tilde{R}) \leq CR^{-\frac{lp\beta}{l+\varepsilon}}, &\
\|D^2_x v\|_{1+\varepsilon}(\tilde{R}) \leq CR^{-\frac{q\alpha}{1+\varepsilon}}, \\
\|D_x u\|_1(\tilde{R}) \leq CR^{1-\frac{\alpha+2}{1+\varepsilon}}, &\
\|D_x v\|_1(\tilde{R}) \leq CR^{1-\frac{\beta+2}{1+\varepsilon}}, \\
\|D^2_x u\|_k(\tilde{R}) \leq C(R^{-n}F(2R))^{\frac{1}{k}},&\
\|D^2_x v\|_m(\tilde{R}) \leq C(R^{-n}F(2R))^{\frac{1}{m}}.
\end{align*}
\end{proposition}
In view of Lemma \ref{sphereEstimate}, to prove Proposition \ref{estimateOnSphere}, we shall give the corresponding estimates on $B_{2R}$ first. We use the varied $W^{2,p}$-estimate (i.e. Lemma \ref{w2pEstimate}) to achieve this.
\begin{proposition}\label{estimateOnBR}
For $R\geq 1$, we have
\begin{align}\label{1stEstimate}
\left\{\begin{array}{ll}
\|u\|_{L^1(B_R)} &\leq CR^{n-\beta}, \\
\|v\|_{L^1(B_R)} &\leq CR^{n-\alpha},
\end{array}
\right.
\end{align}
\begin{align}\label{2ndEstimate}
\left\{\begin{array}{ll}
\|D_x^2 u\|^{l+\varepsilon}_{L^{l+\varepsilon}(B_R)} \leq CR^{n-lp\beta}, \ \text{with } l=\frac s p\geq 1, \\
\|D_x^2 v\|^{1+\varepsilon}_{L^{1+\varepsilon}(B_R)} \leq CR^{n-q\alpha},
\end{array}
\right.
\end{align}
\begin{align}\label{3rdEstimate}
\left\{\begin{array}{ll}
\|D_x u\|_{L^1(B_R)} &\leq C R^{n+1-\frac{\alpha+2}{1+\varepsilon}},\\
\|D_x v\|_{L^1(B_R)} &\leq C R^{n+1-\frac{\beta+2}{1+\varepsilon}},
\end{array}
\right.
\end{align}
and let $k=\frac{p+1}{p}$, $m=\frac{q+1}{q}$,
\begin{align}\label{4thEstimate}
\left\{\begin{array}{ll}
\|D_x^2 u\|_{L^k(B_R)}^k \leq CF(2R), \\
\|D_x^2 v\|_{L^m(B_R)}^m \leq CF(2R).
\end{array}
\right.
\end{align}
\end{proposition}
Proof.
Some frequently used facts include, $q\alpha=\beta+2$, $p\beta=\alpha+2$ and hence $n-kp\beta<0$ (due to \eqref{subcriticalRegion2}) and therefore $l<k$ (since $n-lp\beta\geq 0$).
Estimates \eqref{1stEstimate} directly follow from \eqref{uvEstimate1} in Lemma \ref{energyEstimates}.
For the first estimate of \eqref{2ndEstimate}, after applying Lemma \ref{w2pEstimate}, the mixed type $W^{2,p}$-estimate\footnote{Notice that with the standard $W^{2,p}$-estimate, we end up with a term of $\|u\|_{l+\epsilon}$ which cannot be estimated by any energy bound.}, we get
\begin{align*}
\|D_x^2 u\|^{l+\varepsilon}_{L^{l+\varepsilon}(B_R)} &\leq C\left( \|\Delta u\|^{l+\varepsilon}_{L^{l+\varepsilon}(B_{2R})} + R^{n-(l+\varepsilon)(n+2)}\|u\|^{l+\varepsilon}_{L^1(B_{2R})}\right).
\end{align*}
Then we use the assumed estimate \eqref{vEstimateBetter} and Lemma \ref{energyEstimates} to get
\begin{align*}
\|D_x^2 u\|^{l+\varepsilon}_{L^{l+\varepsilon}(B_R)}
&\leq C\left( \int_{B_{2R}} v^{p(l+\varepsilon)}dx + R^{n-(l+\varepsilon)(n+2)}R^{(l+\varepsilon)(n-\alpha)}\right) \\
&\leq C\left( R^{n-pl\beta}+R^{n-(l+\varepsilon)(2+\alpha)}\right) \\
&\leq C R^{n-pl\beta},
\end{align*}
where the last inequality is due to $\alpha+2=p\beta$.
For the second estimate of \eqref{2ndEstimate},
\begin{align*}
\|D_x^2 v\|^{1+\varepsilon}_{L^{1+\varepsilon}(B_R)} &\leq C\left( \|\Delta v\|^{1+\varepsilon}_{L^{1+\varepsilon}(B_{2R})} + R^{n-(1+\varepsilon)(n+2)}\|v\|^{1+\varepsilon}_{L^1(B_{2R})} \right) \\
&\leq C\left( \int_{B_{2R}} u^{q(1+\varepsilon)}dx + R^{n-(1+\varepsilon)(n+2)} R^{(1+\varepsilon)(n-\beta)} \right) \\
&\leq C\left( R^{n-q\alpha} + R^{n-(1+\varepsilon)(\beta+2)} \right) \\
&\leq C R^{n-q\alpha}.
\end{align*}
For the first estimate of \eqref{3rdEstimate}, by Lemma \ref{interpolation},
\begin{align*}
\|D_x u\|_{L^1(B_R)} &\leq C\left( R^{n(1-\frac{1}{1+\varepsilon})+1} \|D_x^2 u\|_{L^{1+\varepsilon}(B_R)} + R^{-1} \|u\|_{L^1(B_R)}\right) \\
&\leq C\left( R^{n(1-\frac{1}{1+\varepsilon})+1} R^{\frac{n-p\beta}{1+\varepsilon}} +R^{-1}R^{n-\alpha}\right) \\
&\leq C R^{n+1-\frac{\alpha+2}{1+\varepsilon}},
\end{align*}
The second estimate in \eqref{3rdEstimate} can be obtained by a similar process. Last, the fact that $n-(p+1)\beta<0$ gives
\begin{align*}
\|D_x^2 u\|_{L^k(B_R)}^k &\leq C \left( \int_{B_{2R}} |\Delta u|^kdx + R^{n-k(n+2)}(\int_{B_{2R}} |u|dx)^k\right) \\
&\leq C\left( \int_{B_{2R}} v^{p+1}dx + R^{n-k(n+2)} R^{k(n-\alpha)}\right) \\
&\leq C\left( F(2R) + R^{n-(p+1)\beta}\right) \\
&\leq C F(2R),
\end{align*}
and hence the first estimate in \eqref{4thEstimate} follows, and similarly we get the second estimate.
$\Box$
\textbf{Proof of Proposition \ref{estimateOnSphere}:} By Lemma \ref{sphereEstimate}, $\exists \tilde{R}\in[R,2R]$, Proposition \ref{estimateOnSphere} follows from Proposition \ref{estimateOnBR} immediately. $\Box$
\begin{lemma}\label{F4R}
There exists $M>0$ such that $\exists \{R_j\}\rightarrow +\infty$,
\begin{align*}
F(4R_j) \leq M F(R_j).
\end{align*}
\end{lemma}
Proof. Suppose not, then for any $M>0$ and any $\{R_j\}\rightarrow+\infty$, we have
\begin{align*}
F(4R_j) > M F(R_j).
\end{align*}
Take $M>5^n$ and $R_{j+1}= 4R_j$ with $R_0>1$. Therefore,
\begin{align*}
F(R_j) > M^j F(R_0),
\end{align*}
which leads to a contradiction with $F(R_j)\leq CR_j^n \leq C(4^jR_0)^n$. $\Box$
\section{Proof of Liouville Theorem}
From now on, without loss of generality, we may assume $p\geq q.$ By Lemma \ref{comparisonPrinciple}, $\|v\|^{p+1}_{L^{p+1}(B_R)}\leq \|u\|^{q+1}_{L^{q+1}(B_R)}$. By the Rellich-Poho\v{z}aev type identity in Lemma \ref{pohozaevId}, we can denote
\begin{align}\label{FR}
F(R):=\int_{B_R}u^{q+1}\leq CG_1(R)+CG_2(R),
\end{align}
where
\begin{align}
& G_1(R)=R^n\int_{\mathbb S^{n-1}}u^{q+1}(R)d\sigma, \label{g1} \\
& G_2(R)=R^n\int_{\mathbb S^{n-1}}(|D_x u(R)|+R^{-1}u(R))(|D_x v(R)|+R^{-1}v(R))d\sigma. \label{g2}
\end{align}
Heuristically, we are aiming for estimate as
\begin{align}\label{goalEstimate}
G_i(R)\leq CR^{-a_i}F^{1-\delta_i}(4R), \quad i=1,2.
\end{align}
Then by Lemma \ref{F4R} there exists a sequence $\{R_j\}\rightarrow+\infty$ such that
\begin{align*}
G_i(R_j)\leq CR^{-a_i}F^{1-\delta_i}(R_j), \quad i=1,2.
\end{align*}
Suppose there are infinitely many $R_j$'s such that $G_1(R_j)\geq G_2(R_j)$, then take that subsequence of $\{R_j\}$ and still denote as $\{R_j\}$. We do the same if there are infinitely many $R_j$'s such that $G_1(R_j)\leq G_2(R_j)$. So, there are only two cases we shall deal with: there exists a sequence $\{R_j\}\rightarrow+\infty$ such that
\begin{enumerate}
\item[Case] 1: $G_1(R_j)\geq G_2(R_j)$. Then we prove $a_1>0$, $\delta_1>0$. So, $F^{\delta_1}(R_j)\leq CR_j^{-a_1}\rightarrow 0$,
\item[Case] 2: $G_1(R_j)\leq G_2(R_j)$. Then we prove $a_2>0$, $\delta_2>0$. So, $F^{\delta_2}(R_j)\leq CR_j^{-a_2}\rightarrow 0$.
\end{enumerate}
Then we conclude that $F(R)\equiv 0$.
\begin{comment}
Let $\bar{a}=\min\{a,\tilde{a}\}$ and $\bar{b}=\max\{b,\tilde{b}\}$. Suppose we have
\begin{align}\label{aBarbBar}
\bar{a}>0, \ \bar{b}<1,
\end{align}
and we show that
\begin{align*}
F(R)\leq F(\tilde{R}) \leq CR^{-\bar{a}}F^{\bar{b}}(4R), \quad R\geq 1,
\end{align*}
implies $F(R)\equiv 0$.
First, we claim that there exists $M>0$ such that $\exists \{R_j\}$ with $R_j\rightarrow +\infty$,
\begin{align*}
F(4R_j) \leq M F(R_j).
\end{align*}
Suppose not, then for any $M>0$ and any $\{R_j\}\rightarrow+\infty$ we have
\begin{align*}
F(4R_j) > M F(R_j).
\end{align*}
Take $M>5^n$ and $R_{j+1}= 4R_j$ with $R_0>1$. Therefore,
\begin{align*}
F(R_j) > M^j F(R_0),
\end{align*}
which leads to a contradiction with $F(R_j)\leq CR_j^n \leq C(4^jR_0)^n$.
\end{comment}
Surprisingly, for both cases $a_i\approx(\alpha+\beta+2-n)\delta_i$. Indeed, we have
\begin{theorem}\label{liouvilleThm2}
For $F(R)$ defined as \eqref{FR} and $\alpha,\beta$ defined as \eqref{scalingComponent}, there exists a sequence $\{R_j\}\rightarrow+\infty$ such that
\begin{align*}
F(R_j) \leq C R_j^{-(\alpha+\beta+2-n)+o(1)}.
\end{align*}
\end{theorem}
Hence, Theorem \ref{liouvilleThm} is a direct consequence of Theorem \ref{liouvilleThm2}, and we only need to prove Theorem \ref{liouvilleThm2} for case 1 and 2.
\subsection{Case 1: Estimate for $G_1(R)$}
According to previous discussion in the introduction, we assume that
\begin{align*}
p\geq q >0, \quad pq>1, \quad \beta\leq \alpha<n-2, \quad
n\geq 3,
\end{align*}
hence in particular
\begin{align}\label{pLowerBound}
p>\frac{n}{n-2}.
\end{align}
\begin{remark}
For systems \eqref{laneEmdenSystem2} with double bounded coefficients, \eqref{pLowerBound} is a necessary
condition for existence of positive solution, see \cite{LL13}.
\end{remark}
In addition to our assumption that $n-s\beta<1$,
since we have energy inequalities \eqref{uvEsitmatePQ}, we can assume $s\geq p$.
Also, if $n-s\beta < 0$, \eqref{vEstimateBetter} implies $v\equiv 0$ and hence $u\equiv 0$. So, we assume $n-s\beta \geq 0$.
Let $l=\frac{s}{p}$, then
\begin{align}\label{rangeL}
l\geq 1, \ \text{and } \frac{n-1}{p\beta} < l \leq \frac{n}{p\beta}.
\end{align}
It is worthy to point out that, what the proof of Lane-Emden conjecture really needs is a ``breakthrough" on the energy estimate \eqref{uvEsitmatePQ}. $s$ in \eqref{vEstimateBetter} needs not be very large but enough to satisfy $n-s\beta<1$. In other words, $s$ can be very close to $\frac{n-1}{\beta}$, and it is sufficient to prove Theorem \ref{liouvilleThm}.
The strategies of attacking $G_1$ and $G_2$ are the same. Basically, first by H\"{o}lder inequality we split the quantities on sphere $\mathbb S^{n-1}$ into two parts. One has a lower (than original) index after embedding, and the other has a higher one. Then we estimate the latter part by $F(R)$, and thus we get a feedback estimate as \eqref{goalEstimate}.
Let
\begin{align*}
k=\frac{p+1}{p}.
\end{align*}
Since $p\beta=\alpha+2$, $n-(p+1)\beta=n-2-(\alpha+\beta)<0$ by \eqref{subcriticalRegion2}. Thus, $n-kp\beta<0$ as $n-lp\beta\geq 0$, it follows that $l<k$.
{\bf{Subcase 1.1}} $\frac{1}{l} \geq \frac{2}{n-1} + \frac{1}{q+1}$.
Note that in this subcase, since $l\geq 1$, we must have $n\geq 4$ (i.e., $n\neq 3$).
By \eqref{pLowerBound} we see that $k=1+\frac 1 p<1+\frac{n-2}{n} = \frac 2 n (n-1)\leq \frac{n-1}{2}$. Take
\begin{align*}
\frac{1}{\mu} = \frac{1}{k} - \frac{2}{n-1}.
\end{align*}
So, $W^{2,k}(\mathbb S^{n-1})\hookrightarrow L^{\mu}(\mathbb S^{n-1})$.
Take
\begin{align*}
\frac{1}{\lambda} = \frac{1}{l} - \frac{2}{n-1} \geq \frac{1}{q+1}.
\end{align*}
Then $W^{2,l+\varepsilon}(\mathbb S^{n-1})\hookrightarrow L^{\lambda}(\mathbb S^{n-1})$.
Direct verification shows that $ \frac{1}{\mu}= \frac{1}{k} - \frac{2}{n-1} \leq \frac{1}{q+1}$ which is due to \eqref{subcriticalRegion},
so we have
\begin{align*}
\frac{1}{\mu}\leq \frac{1}{q+1}\leq \frac{1}{\lambda}.
\end{align*}
Then by H\"{o}lder inequality and Sobolev embedding \eqref{sobolevEmbedding}, we have (with notation \eqref{sphereNorm})
\begin{align}
\|u\|_{q+1}(R) &\leq \|u\|_{\lambda}^{\theta}\|u\|_{\mu}^{1-\theta}(R) \label{uHolderInequality} \\
&\leq C(R^2\|D_x^2 u\|_{l+\varepsilon}(R)+\|u\|_1(R))^{\theta}(R^2\|D_x^2 u\|_k(R)+\|u\|_1(R))^{1-\theta}, \label{uInequality}
\end{align}
where $\theta\in [0,1]$ and
\begin{align} \label{theta}
\frac{1}{q+1}=\frac{\theta}{\lambda}+\frac{1-\theta}{\mu}.
\end{align}
Since $l$ can be 1 (then $W^{2,p}$-estimate fails for $\|D_x^2 u\|_1(R)$), we add an $\varepsilon$ to $l$ for later use of $W^{2,p}$-estimate. $\varepsilon$ can be any real positive number and later will be chosen sufficiently small.
To get desired estimate, we have requirements in form of inequalities involving parameters, such as $\alpha,\beta,\varepsilon$ and etc. To verify those requirements very often we just verify strict inequalities with $\varepsilon=0$ because such inequalities continuously depend on $\varepsilon$.
So, by \eqref{g1} and \eqref{uInequality}
\begin{align}\label{g1Estimate1}
G_1(R) \leq CR^n \left( (R^2\|D_x^2 u\|_{l+\varepsilon}(R)+\|u\|_1(R))^{\theta}(R^2\|D_x^2 u\|_k(R)+\|u\|_1(R))^{1-\theta}\right) ^{q+1}.
\end{align}
Then by Proposition \ref{estimateOnSphere}, there exists $\tilde{R}\in[R,2R]$ such that
\begin{align*}
G_1(\tilde{R})
&\leq CR^n\left(( R^2R^{-\frac{lp\beta}{l+\varepsilon}}+R^{-2-\alpha})^{\theta}(R^2(R^{-n}F(4R))^{\frac{1}{k}} +R^{-\alpha})^{1-\theta} \right) ^{q+1}\\
&\leq CR^n\left( R^{2-\frac{lp\beta\theta}{l+\varepsilon}-\frac{n(1-\theta)}{k}}F^{\frac{1-\theta}{k}}(4R)\right) ^{q+1} \\
&\leq R^{-a_1}F^{1-\delta_1}(4R),
\end{align*}
where the last inequality is due to $R^{-\frac{n}{k}}>R^{-\alpha-2}$ and
\begin{align}\label{abG1}
a_1 &= a_1^{\varepsilon}=(q+1)(\frac{lp\beta\theta}{l+\varepsilon}+\frac{np(1-\theta)}{p+1}-2-\frac{n}{1+q}),\\
1-\delta_1 &= \frac{(1-\theta)p(q+1)}{p+1}.
\end{align}
Since for sufficiently small $\varepsilon$, $a_1^{\varepsilon}>0$ and $\delta_1>0$ are just a perturbation of
\begin{align}\label{a1delta1}
a_1^0>0, \ \text{and } \delta_1>0,
\end{align}
we only need to prove \eqref{a1delta1} is true.
Since $lp=s$, $p\beta=\alpha+2$ and $q\alpha=\beta+2$,
\begin{align*}
a_1^0 &= p\beta\theta(q+1) + (1-\delta_1)n - 2(q+1)-n \\
&= (q+1)(p\beta\theta-2) -\delta_1n \\
&= (q+1)(p\beta(\theta-1)+p\beta -2) -\delta_1n \\
&= (q+1)(-\alpha (1-\delta_1)+\alpha)-\delta_1n \\
&= \delta_1((q+1)\alpha-n) \\
&= (\alpha + \beta +2 -n)\delta_1.
\end{align*}
So we just need to prove $\delta_1>0$. By \eqref{abG1} and \eqref{theta} we have
\begin{align*}
&(1-\theta)p(q+1) < p+1 \\
\Leftrightarrow& \frac{\frac{1}{l}-\frac{2}{n-1}-\frac{1}{1+q}}{\frac{1}{l}-\frac{1}{k}}(q+1)<k \\
\Leftrightarrow& (\frac{1}{l}-\frac{2}{n-1})(q+1) -1< \frac{k}{l}-1 \\
\Leftrightarrow& \frac{1}{l}(q+1-1-\frac{1}{p}) < \frac{2}{n-1}(q+1) \\
\Leftrightarrow& \frac{pq-1}{s} < \frac{2(q+1)}{n-1} \\
\Leftrightarrow& n-1 < s\beta,
\end{align*}
and the last inequality is included in our assumption. So, we have proved subcase 1.1.
{\bf{Subcase 1.2}} $\frac{1}{l} < \frac{2}{n-1} + \frac{1}{q+1}$.
As discussed in the beginning of subcase 1.1, $k<\frac{n-1}{2}$ if $n>3$. Since $l<k$, $\frac{1}{l} > \frac{2}{n-1}$ for $n>3$. When $n=3$, since $l\geq 1$ by \eqref{rangeL}, $\frac{1}{l} \leq 1 =\frac{2}{n-1}$.
Therefore, for $n>3$, take
\begin{align*}
\frac{1}{\lambda} = \frac{1}{l} - \frac{2}{n-1} < \frac{1}{q+1},
\end{align*}
and for $n=3$, take
\begin{align*}
\lambda = \infty,
\end{align*}
so we have
\begin{align*}
W^{2,l+\varepsilon}(\mathbb S^{n-1})\hookrightarrow L^{\lambda}(\mathbb S^{n-1}), \ n\geq 3.
\end{align*}
So,
\begin{align*}
\|u\|_{q+1}(R) \leq C\|u\|_{\lambda}(R)\leq C(R^2\|D_x^2 u\|_{l+\varepsilon}(R)+\|u\|_1(R) ).
\end{align*}
Therefore, by Proposition \ref{estimateOnSphere} there exists $\tilde{R}\in[R,2R]$ such that
\begin{align}
G_1(\tilde{R}) &\leq CR^n (R^2\|D_x^2 u\|_{l+\varepsilon}(R)+\|u\|_1(R))^{q+1} \\
&\leq CR^{n}(R^2 R^{-\frac{lp\beta}{l+\varepsilon}}+R^{-\alpha})^{q+1} \\
&\leq CR^{n+(2-\frac{lp\beta}{l+\varepsilon})(q+1) }.
\end{align}
So,
\begin{align*}
F(\tilde{R}) &\leq C R^{n+(2-\frac{lp\beta}{l+\varepsilon})(q+1) } \\
&\leq CR^{n+(2-p\beta)(q+1)+\frac{\varepsilon p\beta}{l+\varepsilon}(q+1)} \\
&\leq CR^{-(\alpha+\beta+2-n) + \frac{\varepsilon p\beta}{l+\varepsilon}(q+1)}.
\end{align*}
Since $\varepsilon$ can be arbitrarily small,
\begin{align*}
F(\tilde{R}) \leq CR^{-(\alpha+\beta+2-n) + o(1)}.
\end{align*}
Thus, we have proved Case 1.
\subsection{Case 2: Estimate for $G_2(R)$.}
Let
\begin{align*}
m=\frac{q+1}{q}.
\end{align*}
{\bf{Subcase 2.1}} $m<n-1$.
With $z,z'>0$ and $\frac 1 z + \frac{1}{z'}=1$, \eqref{g2} becomes,
\begin{align}\label{G2holderInequality}
\left.\begin{array}{ll}
G_2(R) &\leq CR^n\||D_x u| + R^{-1}u\|_z\||D_x v| + R^{-1}v\|_{z'}(R) \\
&\leq CR^n(\|D_x u\|_z(R)+R^{-1}\|u\|_z(R))(\|D_x v\|_{z'}(R) + R^{-1}\|v\|_z(R)) \\
&\leq CR^n(\|D_x u\|_z(R)+R^{-1}\|u\|_1(R))(\|D_x v\|_{z'}(R) + R^{-1}\|v\|_1(R)),
\end{array}
\right.
\end{align}
where the last inequality is due to
\begin{align*}
\|u\|_z(R) \leq C(R\|D_x u\|_z(R) + \|u\|_1(R)), \ \text{and } \|v\|_{z'}(R) \leq C(R\|D_x v\|_{z'}(R) + \|v\|_1(R)).
\end{align*}
Assume there exists $z$ (we shall check the existence later) such that by Sobolev Embedding \eqref{sobolevEmbedding},
\begin{align}
&\left.\begin{array}{ll}\label{g2UEmbedding}
\|D_x u\|_z(R) &\leq \|D_x u\|_{\rho_1}^{\tau_1}(R)\|D_x u\|_{\gamma_1}^{1-\tau_1}(R) \\
&\leq C(R\|D^2_x u\|_{l+\varepsilon}(R) + \|D_x u\|_1(R))^{\tau_1}(R\|D^2_x u\|_k(R)+ \|D_x u\|_1(R))^{1-\tau_1},
\end{array}
\right. \\
& \left.\begin{array}{ll}\label{g2VEmbedding}
\|D_x v\|_{z'}(R) &\leq \|D_x v\|_{\rho_2}^{\tau_2}(R)\|D_x v\|_{\gamma_2}^{1-\tau_2}(R) \\
&\leq C(R\|D^2_x v\|_{1+\varepsilon}(R) + \|D_x v\|_1(R))^{\tau_2}(R\|D^2_x v\|_m(R)+ \|D_x v\|_1(R))^{1-\tau_2},
\end{array}
\right.
\end{align}
where $\tau_1,\tau_2\in[0,1]$ and
\begin{align}
\frac{1}{z} = \frac{\tau_1}{\rho_1} + \frac{1-\tau_1}{\gamma_1}, \label{zDef} \\
\frac{1}{z'} = \frac{\tau_2}{\rho_2} + \frac{1-\tau_2}{\gamma_2}, \label{zDefPrime}
\end{align}
and since $l<k\leq m<n-1$,
define
\begin{align}
\frac{1}{\rho_1} &= \frac{1}{l} - \frac{1}{n-1}, \ \ \frac{1}{\gamma_1} = \frac{1}{k} - \frac{1}{n-1}, \\
\frac{1}{\rho_2} &= 1 - \frac{1}{n-1}, \ \ \frac{1}{\gamma_2} = \frac{1}{m} - \frac{1}{n-1}. \label{rhoGammaDef}
\end{align}
So, we have
\begin{align*}
W^{1,l+\varepsilon}(\mathbb{S}^{n-1})\hookrightarrow L^{\rho_1}(\mathbb{S}^{n-1}), \ W^{1,k}(\mathbb{S}^{n-1})\hookrightarrow L^{\gamma_1}(\mathbb{S}^{n-1}), \\
W^{1,1+\varepsilon}(\mathbb{S}^{n-1})\hookrightarrow L^{\rho_2}(\mathbb{S}^{n-1}), \ W^{1,m}(\mathbb{S}^{n-1})\hookrightarrow L^{\gamma_2}(\mathbb{S}^{n-1}).
\end{align*}
To verify the existence of such $z$, by \eqref{zDef}-\eqref{rhoGammaDef}, we expect that
\begin{align}\label{rangeZ}
\max\left\lbrace \frac{1}{k} - \frac{1}{n-1}, \frac{1}{n-1}\right\rbrace \leq \frac{1}{z} \leq \min \left\lbrace \frac{1}{l}-\frac{1}{n-1}, \frac{1}{q+1}+\frac{1}{n-1}\right\rbrace.
\end{align}
Thus, we need to verify, (i) $\frac{1}{k} - \frac{1}{n-1}\leq\frac{1}{l}-\frac{1}{n-1}$, (ii) $\frac{1}{n-1}\leq\frac{1}{l}-\frac{1}{n-1}$, (iii) $\frac{1}{n-1}\leq\frac{1}{q+1}+\frac{1}{n-1}$, (iv) $\frac{1}{k}-\frac{1}{n-1}\leq\frac{1}{q+1}+\frac{1}{n-1}$.
Since $l<k$, (i) is true. (ii) holds for $n> 3$ as discussed at the beginning of subcase 1.2 $\frac 1 l>\frac 1 k>\frac{2}{n-1}$; for $n=3$, take $s=p$ and then $l=1$, so (ii) still holds. (iii) is obvious. (iv) is equivalent to $\frac{1}{p+1}+\frac{1}{q+1}\geq 1-\frac{2}{n-1}$, which is guaranteed by \eqref{subcriticalRegion}.
\begin{comment}
\begin{align*}
\frac{1}{k} - \frac{1}{n-1}<\frac{1}{l}-\frac{1}{n-1}.
\end{align*}
Meanwhile,
\begin{align*}
\frac{1}{k}-\frac{1}{n-1}<\frac{1}{q+1}+\frac{1}{n-1}
\Leftrightarrow \frac{1}{p+1}+\frac{1}{q+1}> 1-\frac{2}{n-1},
\end{align*}
which is guaranteed by \eqref{subcriticalRegion}.
Last,
\begin{align*}
\frac{1}{n-1} \leq \frac{1}{l} - \frac{1}{n-1}
\end{align*}
holds for $n\geq 3$ as discussed at the beginning of Step 2.
\end{comment}
So, we put \eqref{g2UEmbedding} and \eqref{g2VEmbedding} in \eqref{G2holderInequality} and get
\begin{align}\label{g2Estimate}
\left. \begin{array}{ll}
G_2(R)\leq & C R^{n+2}(\|D_x^2 u\|_{l+\varepsilon}(R) + R^{-1}\|D_x u\|_1(R) + R^{-2}\|u\|_1(R))^{\tau_1} \\
& \times(\|D_x^2 u\|_k(R) + R^{-1}\|D_x u\|_1(R) + R^{-2}\|u\|_1(R))^{1-\tau_1} \\
& \times(\|D_x^2 v\|_{1+\varepsilon}(R) + R^{-1}\|D_x v\|_1(R) + R^{-2}\|v\|_1(R))^{\tau_2} \\
& \times(\|D_x^2 v\|_m(R) + R^{-1}\|D_x v\|_1(R) + R^{-2}\|v\|_1(R))^{1-\tau_2}.
\end{array}\right.
\end{align}
Then by Proposition \ref{estimateOnSphere}, there exists $\tilde{R}\in[R,2R]$ such that
\begin{align*}
G_2(\tilde{R}) &\leq C R^{n+2} R^\frac{-p\beta\tau_1}{1+\varepsilon/l} \left( (R^{-n}F(4R))^{\frac{1}{k}}+ R^{-\frac{\alpha+2}{1+\varepsilon}} +R^{-\alpha-2}\right) ^{1-\tau_1} \\
&\times R^{-\frac{q\alpha\tau_2}{1+\varepsilon/l}} \left( (R^{-n}F(4R))^{\frac{1}{m}} +R^{-\frac{\beta+2}{1+\varepsilon}}+R^{-\beta-2}\right)^{1-\tau_2} \\
&\leq C R^{-a_2^{\varepsilon}}F^{1-\delta_2}(4R),
\end{align*}
where the last inequality is due to $R^{-\frac{n}{k}}>R^{-\alpha-2}$ and $R^{-\frac{n}{m}}>R^{-\beta-2}$. Meanwhile,
\begin{align}
a_2&= a_2^{\varepsilon} = -n-2 +\frac{p\beta\tau_1}{1+\varepsilon/l} + \frac{q\alpha\tau_2}{1+\varepsilon/l} + n\frac{1-\tau_1}{k} + n\frac{1-\tau_2}{m}, \\
1-\delta_2 &= \frac{1-\tau_1}{k} + \frac{1-\tau_2}{m}. \label{delta2}
\end{align}
Similar to subcase 1.1, we only need to prove
\begin{align*}
a_2^0 >0, \ \delta_2>0.
\end{align*}
Surprisingly, similar to $a_1\approx (\alpha+\beta+2-n)\delta_1$, we have $a_2\approx (\alpha+\beta+2-n)\delta_2$ since we can prove $a_2^0 =(\alpha+\beta+2-n)\delta_2$. Indeed,
\begin{align*}
a^0_2 &= -n-2 +p\beta(\tau_1-1)+p\beta+q\alpha(\tau_2-1)+q\alpha+n(1-\delta_2) \\
&= -n-2 - p\beta k \frac{1-\tau_1}{k} - q\alpha m\frac{1-\tau_2}{m} + \alpha+\beta+4 +n(1-\delta_2) \\
&= \alpha + \beta +2-n - (\alpha+\beta+2) (1-\delta_2) + n(1-\delta_2)\\
&= (\alpha+\beta+2-n)\delta_2,
\end{align*}
where the third equality above is due to $p\beta k=(p+1)\beta=(q+1)\alpha=q\alpha m$ and $(p+1)\beta=\alpha+\beta+2$.
So, we only need to prove $\delta_2 >0$ or equivalently by \eqref{zDef}, \eqref{zDefPrime} and \eqref{delta2},
\begin{align}
(m-\frac{k}{l})\frac{1}{z}+(\frac{k}{n-1}+(m-1)(k-1))\frac{1}{l}+\frac{m-2}{n-1}-(m-1)>0, \label{zAndl}
\end{align}
To achieve this, we take the upper bound of $\frac{1}{z}$ in \eqref{rangeZ} and see whether \eqref{zAndl} holds.
{\bf{Case 2.1.1}} If $\frac{1}{l}-\frac{1}{n-1} \geq \frac{1}{q+1}+\frac{1}{n-1}$, then let $\frac{1}{z}=\frac{1}{q+1}+\frac{1}{n-1}$, and \eqref{zAndl} becomes,
\begin{align*}
&(\frac{1}{pq}-\frac{p+1}{p(q+1)})\frac{1}{l}+\frac{q+1}{q}(\frac{1}{n-1}+\frac{1}{q+1})+\frac{1-q}{(n-1)q}-\frac{1}{q}>0 \\
&\Leftrightarrow (\frac{1}{pq}-\frac{p+1}{p(q+1)})\frac{1}{l}+ \frac{2}{q(n-1)} > 0 \\
&\Leftrightarrow -\frac{2}{\beta s} + \frac{2}{n-1} > 0 \\
&\Leftrightarrow s\beta > n-1.
\end{align*}
{\bf{Case 2.1.2}} If $\frac{1}{l}-\frac{1}{n-1} < \frac{1}{q+1}+\frac{1}{n-1}$, then let $\frac{1}{z}=\frac{1}{l}-\frac{1}{n-1}$, and \eqref{zAndl} becomes,
\begin{align*}
&(m-\frac{k}{l})(\frac{1}{l}-\frac{1}{n-1})+(\frac{k}{n-1}+(m-1)(k-1))\frac{1}{l}+\frac{m-2}{n-1}-(m-1)>0 \\
&\Leftrightarrow -\frac{k}{l^2}+(m+\frac{k}{n-1}+\frac{k}{n-1}+(m-1)(k-1))\frac{1}{l}+\frac{m-2}{n-1}-(m-1)>0 \\
&\Leftrightarrow -\frac{k}{l^2}+(\frac{p}{p+1}+\frac{1}{q}+\frac{2}{n-1})\frac{k}{l}>\frac{2}{n-1}+\frac{1}{q} \\
&\Leftrightarrow - \frac{k}{l^2} + (1+k(\frac{2}{n-1}+\frac{1}{q}))\frac{1}{l} - (\frac{2}{n-1}+\frac{1}{q}) >0 \\
&\Leftrightarrow (\frac{k}{l} -1)(\frac{1}{l} -(\frac{2}{n-1}+\frac{1}{q})) < 0 \\
&\Leftrightarrow \frac{1}{k} < \frac{1}{l} < \frac{2}{n-1}+\frac{1}{q}.
\end{align*}
Notice that $\frac{1}{l} < \frac{2}{n-1}+\frac{1}{q}$ holds under the assumption of case 2.1.2, and $\frac{1}{k} < \frac{1}{l}$ since $l<k$.
In all, \eqref{zAndl} always holds under our assumption $n-s\beta<1$.
{\bf{Subcase 2.2}} $m\geq n-1$.
First, we have for any $\gamma\in[1,\infty)$,
\begin{align*}
W^{1,m}(\mathbb{S}^{n-1}) \hookrightarrow L^{\gamma}(\mathbb{S}^{n-1}).
\end{align*}
Then we claim $\frac{1}{l}>\frac{1}{n-1}$. Suppose $\frac{1}{l}\leq \frac{1}{n-1}$, then $k>l\geq n-1$, hence $p \leq \frac{1}{n-2}$, which is not possible due to \eqref{pLowerBound}.
Take $\frac{1}{z}=\frac{1}{l}-\frac{1}{n-1}$
then
\begin{align*}
W^{1,l+\varepsilon}(\mathbb{S}^{n-1}) \hookrightarrow L^{z}(\mathbb{S}^{n-1}).
\end{align*}
Therefore, by Sobolev embedding and \eqref{G2holderInequality}
\begin{align*}
G_2(R) &\leq CR^n(\|D_x u\|_z(R)+R^{-1}\|u\|_1(R))(\|D_x v\|_{z'}(R) + R^{-1}\|v\|_1(R)) \\
&\leq C R^{n+2}(\|D_x^2 u\|_{l+\varepsilon} + R^{-1}\|D_x u\|_1 + R^{-2}\|u\|_1)(\|D_x^2 v\|_m + R^{-1}\|D_x v\|_1 + R^{-2}\|v\|_1).
\end{align*}
Similarly to previous work, there exists a $\tilde{R}\in[R,2R]$ such that
\begin{align*}
G_2(\tilde{R}) &\leq C R^{n+2} R^\frac{-p\beta}{1+\varepsilon/l}
\left( (R^{-n}F(4R))^{\frac{1}{m}} +R^{-\frac{\beta+2}{1+\varepsilon}}+R^{-\beta-2}\right) \\
&\leq C R^{-a_2^{\varepsilon}}F^{1-\delta_2}(4R),
\end{align*}
where
\begin{align}\label{abG2}
a_2 &= a_2^{\varepsilon} = -n-2 +\frac{p\beta}{1+\varepsilon/l} + \frac{n}{m}, \\
1-\delta_2 &= \frac{1}{m}.
\end{align}
Direct verification shows that
\begin{align*}
a_2^0 = (\alpha+\beta+2-n)\delta_2,
\end{align*}
and obviously $\delta_2>0$ so $\alpha_2^0>0$.
Thus, we have proved Case 2.
\newpage
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2,869,038,156,408 | arxiv | \section{Introduction}
Image classification and recognition has been an active research spot for many years with explosive growth of image data from daily life and internet. The far-reaching study can be employed in practical applications, including face recognition, species categorization, object detection, for a more intellective world. However, there exist threefold challenges in this research. The first is to extract numerical feature representation from images since original image cannot be exploited directly. Extensive studies have explored this area\cite{smeulders2000content}. Classical methods for feature extraction include Haar-like feature\cite{viola2001rapid,viola2004robust}, Scale-invariant feature transform(SIFT)\cite{lowe2004distinctive}, Histograms of Oriented Gradients(HOG)\cite{dalal2005histograms}, Local binary pattern(LBP)\cite{ahonen2004face}, Speeded up robust feature(SURF)\cite{bay2006surf}, etc. The above methods can be classified into two kinds: single feature vector representation(HAAR, HOG, LBP) and bag-of-words representation(SIFT, SURF, patches of LBP and HOG). The difference between the two kinds features is that a image is represented by a single vector or a bag of instances, leading to two areas in machine learning, standard classification and multi-instance learning. As that the contents in a image are not distributed uniformly or regularly,
bag-of-words representation has the advantage that every word expresses a image's key feature independently, without the negative impact of unfixed locations of these key features in a single vector.
The second challenge is to integrate the information from multiple sources or multiple feature sets. In fact, the extracted features from different methods are complementary to each other and can be combined to improve the performance
of image recognition, which is in the scope of multi-view learning. Multi-view learning has been developed from co-training, to multiple kernel learning and subspace learning\cite{xu2013survey}. Extensive experiments have verified that information form multiple views can boost the performance of methods in machine learning.
The third key challenge is to design an efficient data-dependent distance function to show the image relationships and distributions in feature space. The researchers in distance metric learning has proposed many algorithms to improve the performance of distance related methods based on the idea that a desired metric should shrink the distance between similar points and expand the distance between dissimilar points as much as possible\cite{kulis2012metric,moutafis2016overview,bellet2013survey}.
The distance between images has been seldom studied when multi-instance features are extracted from images.
However, in practical, the features of image can be extracted from multiple views, each view consists of multi-instance features. It is worthy and important to unify the information from multiple views, investigating the relationship between different views and different bags in the same view.
It is much more interesting to explore multiple data-dependent metrics in multi-instance task with multiple views.
In this paper, we propose a new approach to improve the performance of image classification, named MVML. For each image, multi-instance features are extracted due to the merits of bag-of-words representation and different feature extraction methods are implemented to constructed multiple views. To combine the features from multi-view effectively, we first define a new distance function for bags and then seek for a instance-dependent metric by maximizing the average conditional probability that every image is similar with its nearest image. The distance between images is computed by the weighted sum of the distance between bags from each single view. So the metrics and weights are both needed to be optimized.
The tricks of gradient descent and alternate optimization are used to solve our approach. The efficiency of our novel method in making image classification, compared with single view multi-instance learning, has been demonstrated in the numerical experiments.
In summary, the main contributions of our work are as follows:
\begin{itemize}
\item A new approach for multi-instance classification, based on multi-view learning with the technique of metric learning is introduced. The combination of multi-instance, multi-view and metric learning has never been studied before. Multi-view learning have been explored to improve the performance of classification.
Metric learning can devise a view-dependent metric for every view to further boost the performance of multi-view learning.
\item The proposed method is a conditional probability model, maximizing the average probability that every image is similar with its nearest image. In multi-view condition, the distance between images is the weighted sum of the distance between bags from each single view. Our method can be solved by gradient descent and alternate optimization iteratively.
\item We have designed a new distance function between bags, integrating the distances between instances of bags skillfully. Compared with the previous two distances, experiments on $k$NN classification have validated the efficiency and advantage of our new distance function.
\end{itemize}
The followings of the paper are organized in this way. In Section \ref{sec:related}, we will introduce previous related works about metric learning, multi-view learning and multi-instance classification. Our model, including multi-instance problem with multi-view, distance function for bags and the probability framework will be provided in Section \ref{sec:model}. Then we will optimize our approach in this Section. In Section \ref{sec:exp}, numerical experiments will be made to demonstrate that our model can deal with multi-instance classification effectively and efficiently. The conclusions will be summarized in Section \ref{sec:conclu}.
\section{Related Works}
\label{sec:related}
\subsection{Metric learning}
Metric learning aims to learn a distance function to improve the performance of distance-related methods, including $k$-nearest neighbors($k$NN), $k$-means, which are very classical and important methods in classification, clustering, etc. For a dataset with $ c $ classes
\begin{equation}\label{tr01}
T = \{(x_1 ,y_1), \cdots ,(x_m ,y_m )\},
\end{equation}
where $(x_i,y_i) \in R^n \times \{ 1, 2, \cdots, c\}, i=1, \cdots, m$ and $ m $ is the total number of samples, $ n $ is the number of features. Two sets are defined as following
\begin{eqnarray}
S &=& \{(x_i,x_j) | y_i=y_j \} \label{simpair} \\
D &=& \{(x_i,x_l) | y_i \neq y_l \} \label{difpair}
\end{eqnarray}
The points in each pair of $ S $ are from the same class and $ D $ contains pairs of dissimilar points. Metric learning seeks for a metric to recompute the distance between two different points as
\begin{equation}\label{dist}
d_M(x_i,x_j)= (x_i -x_j)^\top M (x_i-x_j)
\end{equation}
to make similar points closer and dissimilar points farther. In the equation (\ref{dist}), an effective metric $ M $ should satisfies the conditions\cite{royden1988real,wang2014survey}:
\begin{enumerate}[(1)]
\item distinguishability: $ d_M(x_i,x_i)=0 $;
\item non-negativity: $ d_M(x_i,x_j) \ge 0 $;
\item symmetry: $ d_M(x_i,x_j)=d_M(x_j,x_i) $;
\item triangular inequality: $ d_M(x_i,x_j)+d_M(x_i,x_k) \ge d_M(x_j,x_k)$;
\end{enumerate}
A great many methods of metric learning have been proposed to show the strong ability of data-dependent distance in adjusting the original structure of feature space, resulting in the formations of more advantageous neighborhoods. The label information of pairwise relationship in (\ref{simpair})-(\ref{difpair}) has been exploited in most of these previous works. One of the earliest efforts in pursuing ideal metric is the method MLSI(Metric Learning with Side Information), presented by Xing et.al\cite{xing2002distance}. It uses similarity side-information to improve $k$NN performance based on the idea that similar points should be as near as possible and the distance between dissimilar points should be larger than a threshold. Goldberger et.al proposes NCA(neighborhood component analysis)\cite{goldberger2004neighbourhood} which directly maximizes leave-one-out accuracy by learning a low-rank quadratic metric. Then LMNN(large margin nearest neighbor)\cite{weinberger2009distance} is presented on the basis of NCA to minimize the distance between any two similar and close points, subject to the constraints that any point should be pushed away from the neighborhood of its different labeled points by a large margin.
Due to the limitations of LMNN, several extensions are introduced to improve LMNN, including solving LMNN more efficiently\cite{park2011efficiently,weinberger2008fast}, introducing kernel into LMNN\cite{torresani2006large}, multi-task version of LMNN\cite{parameswaran2010large}, etc. Golberson has constructed a convex optimization problem\cite{globerson2005metric} to learn a quadratic Gaussian metric, trying to, though impractical, collapse all the similar inputs to a single point. From the view of information theory, ITML(information-theoretic metric learning)\cite{davis2007information} minimizes the relative entropy between two multivariate Gaussian distribution, leading to a Bregman optimization problem.
Metric learning for special task has also been studied extensively. SSM-DML\cite{yu2012semisupervised} pays attention to multi-view metric learning to improve the performance of cartoon synthesis. Jin et.al\cite{jin2009learning} propose an iterative metric learning algorithm for multi-instance and multi-label problem to improve the quality of associations between instances and class labels. MildML\cite{guillaumin2010multiple} learns a data-dependent metric of bags from multi-instance task based on the information that two bags in positive pair share at least one label, negative otherwise. MIMEL\cite{xu2011multi} aims to maximize inter-class bag distance and minimize intra-class bag distance, constructing a minimization problem of KL divergence between two multivariate Gaussians.
\subsection{Multi-view learning}
To identify a person, the information of face, fingerprint or DNA can take effect independently. These three kinds of information are from different sources. The features of color, shape or texture can be used to classify pictures individually. They can be seen as different feature subsets of the images. What's more, we can also combine features from different sources or feature subsets to identify objects with higher accuracy since diverse characters are synthesized. It is the basic idea of multi-view learning. Each source or subset is called a view.
In contrast to single view learning, multi-view learning can improve the learning performance by optimizing multiple functions, each one is used to model a single view, to extract the information from different views of the same data inputs. Multi-view learning has been a hot research spot recently and the related studies can be divided into three directions: co-training, multiple-kernel learning and subspace learning\cite{xu2013survey,sun2013survey}. The first co-training algorithm was proposed for semi-supervised learning\cite{blum1998combining}, which combines the information from two views, labeled and unlabeled, to boost the performance of the algorithms with only few labeled data points. It had been analyzed and extended further\cite{nigam2000analyzing,muslea2002active+,yu2011bayesian}. Kernel function is an important and useful tool in machine learning, which maps original example into higher dimensional Hilbert space for easier learning with better performance. Multiple kernel learning(MKL) can be applied into multi-view learning task, each kernel corresponds to a particular view. Multiple kernels will be combined linearly or non-linearly to model multi-view data with more proper way. MKL had been studied in diverse formations, including semi-definite program\cite{lanckriet2004learning,sonnenburg2005general}, second order cone program\cite{bach2004multiple}, 2-norm regularization program\cite{rakotomamonjy2007more}, etc and explored widely\cite{ying2009generalization,kloft2011local}.
Canonical correlation analysis(CCA)\cite{hotelling1936relations} and its kernel version\cite{akaho2006kernel} are two early works in subspace learning. They seek basis vectors for two sets of variables, each set can be seemed as the data corresponds to a single view. They have been extended to clustering\cite{chaudhuri2009multi} and regression\cite{diethe2008multiview} and analyzed for consistency\cite{cai2011convergence}.
\subsection{Multi-instance learning}
Different from standard supervised learning, in which the input is often described by a single feature vector,
every input in multi-instance learning(MIL) is a set of labeled instances, called bag. The label of a bag is determined by one or several instances. The multi-instance problem was first introduced by Dietterich\cite{dietterich1997solving} in the study of drug activity prediction. Drug contains many molecules, valid and invalid. If the molecules in a drag are all invalid, the drug is labeled as negative, other wise positive. Classification or pattern recognition of multi-instance task can be considered on bag level and/or instance level. APR(Axis-Parallel Rectangle)\cite{dietterich1997solving} is an early work in dealing with MIL, starting from a initial point and expanding a rectangle to find the smallest rectangle that covers at least one instance of each positive bag and zero instance of any negative bag. A probabilistic framework called Diverse Density(DD)\cite{maron1998framework} was proposed to learn a concept by maximizing a defined likelihood function. Zhang et.al\cite{zhang2001dd} proposes an improved version of DD which combines Expectation-Maximization and diverse density. Citation-$k$NN\cite{wang2000solving} is introduced to decide the label of a bag not only by its neighbors but also citers. Support vector machine had been adopted in MIL with modified constraints that some instances are unlabeled but should be constrained\cite{andrews2002support,tao2004svm}.
However, there are few research on multi-instance problem with multiple views, let alone the technique of metric learning embedded. MVML is a probability framework, defining a distance function for multi-instance problem and integrating the information from multiple views by learning multiple metrics. The experiments verifies that our approach is effective in dealing with multi-instance task.
\section{Model and optimization}
\label{sec:model}
\subsection{Multi-instance classification task with multi-view }
In multi-instance task, the features of every image are extracted in the form of bags, each of which contains multiple instances. Under the condition of multi-view, every image can be described by $ v $ different but complementary views. We can classify these images based on a single view or multiple views. Generally, multi-view data can provide more information.
To make classification on the batch of images, the training set with $ v $ views is constructed as following:
The $k$-th($ k=1, 2, \cdots, v$) view of the training set is
\begin{equation}\label{eq:prbk}
T^k=\{(X_1^k,y_1), (X_2^k,y_2), \cdots, (X_{m_k}^k,y_{m_k})\}
\end{equation}
where $ X_i^k=\{x_{i1}^k, x_{i2}^k, \cdots, x_{i m_i^k}^k \}, y_{j_k} \in \{1,2, \cdots, c\} $. $ T^k $ is composed of $ m_k $ bags with their corresponding labels. The bag $ X_i^k $ contains $ m_i^k $ instances.
It should be noted that the above mentioned problem is not the same as the traditional multi-instance task, in which the label of a bag is determined by whether the bag contains positive instance.
In our problem, the classes $ c $ is equal with or larger than 2 and all the instances in a bag may be helpful in deciding the label.
The task of the paper is to find a prediction function $ f $, with respect to multiple distance metrics $ M_1, \cdots, M_v $, each corresponds to an unique view: given an image with the information from $ v $ views, $ X^1, X^2, \cdots, X^v $, the label of the image can be predicted by $ y=f(X^1, X^2, \cdots, X^v; M_1, M_2, \cdots, M_k) $.
\subsection{Distance between Bags}
\label{sec:dist}
In the traditional distance metric learning, the distance is measured between two feature vectors $ x_i. x_j $, that is,
\begin{equation}
d_M(x_i, x_j)=(x_i-x_j)^\top M(x_i-x_j)
\end{equation}
where the metric $ M $ should satisfies the property of distinguishability, non-negativity, symmetry, triangular inequality. However, in the feature extraction of image, the feature is often in the form of bag, which contains multiple vectors. It is not suitable to concatenate these vectors into a longer vector. So it is hard but very important to measure the distance of different images, each of which is represented by a bag of features.
In metric learning, there are two distance functions for bags proposed before, both of which are designed to measure the distance between bags as accurately as possible. The first one is to measure the distance by the average distance of pairwise examples from different bags\cite{xu2011multi}.
The second one calculates the minimum distance of pairwise instances\cite{jin2009learning}, also called \textit{minimal Hausdorff distance}. For two bags $ X_i=\{x_{i1}, x_{i2}, \cdots, x_{im_{i}}\}, X_j=\{x_{j1}, x_{j2}, \cdots, x_{jm_{j}}\} $, the distances defined by the above two functions are
\begin{equation}
D_{ave}(X_i,X_j;M)=\frac{1}{m_i m_j} \sum\limits_{k=1}^{m_i} \sum\limits_{l=1}^{m_j} d_M(x_{ik}, x_{jl})
\end{equation}
and
\begin{equation}
D_{min}(X_i,X_j;M)= \min\limits_{1 \le k \le m_i, 1 \le l \le m_j} d_M(x_{ik}, x_{jl})
\end{equation}
respectively.
However, the two functions can be improved in virtue of their own drawbacks. For the function $ D_{ave} $, calculating the distance of pairwise instances can bring the trouble of much redundant information. The distance between two same bags is not zero in $ D_{ave} $. For the function $ D_{min} $, it determines the distance of bags only by the minimum distance of pairwise instances, which ignores too much useful information. If two different bags contain similar instances, $ D_{min} $ will make improper judge. So we argue that both $ D_{ave}$ and $D_{min} $ can not measure the bag distance properly.
In the paper, a new distance function is defined as follows:
for each instance $ x $ in each bag, the nearest instance in the other bag is found and the distance is recorded. The distance is the minimum for $ x $ in searching the other bag. The new defined distance function calculates the Average of these Minimums, named as $ D_{am} $.
We define the distance between two bags $ X_i, X_j $ as
\begin{equation}
D_{am} (X_i, X_j; M) =\frac{1}{m_i} \sum\limits_{p=1}^{m_i} \min\limits_{x_{jl} \in X_j} d_M(x_{ip}, x_{jl}) + \frac{1}{m_j} \sum\limits_{q=1}^{m_j} \min\limits_{x_{ih} \in X_i} d_M(x_{jq}, x_{ih})
\end{equation}
\begin{figure}
\centering
\subfigure[]{\includegraphics[width=3.5in]{distconcept.pdf}}
\subfigure[]{\includegraphics[width=2.5in]{dist2.pdf}}
\caption{A simple task}\label{fig:distconcept}
\end{figure}
Suppose that there is a simple task in Figure \ref{fig:distconcept}: Give a picture \textbf{a} belonged to the class of \textit{car}, find the picture with the same label from \textbf{b} and \textbf{c}. There are 7, 4, 3 key points in the picture \textbf{a, b, c} respectively. The blue squares are the key points indicating the label. On the instance level, the key points b1,b2 in b are similar with a3, a4 and c2, c3 are similar with a1, a2. In the function $ D_{min} $, the distances between pictures are $ D_{min}(a,b)=d(a3,b1) $ and $ D_{min}(a,c)=d(a1,c2) $(The black lines in Figure \ref{fig:distconcept}.(b) are pairwise distances calculated by $D_{min}$), both are relative small. We will make wrong judge if the latter is a slight smaller than the former one. So similar instances from different classes can weaken the quality of $ D_{min} $. In the function $ D_{ave} $, all the distances of pairwise instances will be computed and averaged, bringing negative information, such as $ d(a3,b3), d(a4,b4), d(a3,b4) $, etc, which will not be calculated in $ D_{am} $(In Figure \ref{fig:distconcept}.(b), the red lines are redundant distance information in computing bag distance). The experiments in the Section \ref{sec:exp} will verify the advantages of our proposed distance function.
In multi-view condition, every image $ I $ is extracted as $ v $ bags $ X_i^1, X_i^2, \cdots, X_i^v $. The distance between two images $ I, J $ is defined as
\begin{equation}
D_M(I, J)=\sum\limits_{k=1}^v \alpha_k D_{am}(X_i^k, X_j^k)
\end{equation}
where $ \alpha_i, i=1,\cdots, v $ are the weights to be learned.
\subsection{Metric learning in probability framework}
Although the distance between bags is designed, the relationship between instances is also very important since it affects bag distance significantly. Fortunately, metric learning can be applied to establish favorable relationships for instances form bag.
Inspired by the ideas of $k$NN and NCA, for each image, we just aim to maximize the probability of that its nearest image has the same label with it.
In multi-view situation, we consider optimizing the joint conditional probability distribution of the image $ I $ with $ v $ views, but not simply maximizing the sum of the marginal probability distribution,
\begin{equation}\label{mlp01}
p(y_i|X_i^1, X_i^2, \cdots, X_i^v;\mathscr{M}) =\frac{\exp(-f(X_i,y_i))}{ \sum\limits_{y=1}^c \exp(-f(X_i,y))}
\end{equation}
where
\begin{equation}\label{mlp02}
f(X_i, y) =\min\limits_{y_j=y} D_M(I,J)= \min\limits_{y_j=y} \sum\limits_{k=1}^v \alpha_k D_{am} (X_i^k, X_j^k)
\end{equation}
and $ \mathscr{M}=\{M_1, \cdots, M_v\} $.
On the whole training set, the following regularized likelihood function is constructed to learn the desired distance metrics
\begin{eqnarray}
&& \max\limits_{\mathscr{M},\alpha} E(\mathscr{M},\alpha) \nonumber \\
&=& \frac{1}{m} \sum\limits_{i=1}^m \ln p(y_i|X_i^1, X_i^2, \cdots, X_i^v) \nonumber - \frac{\lambda}{2} \sum\limits_{k=1}^v \|M_k\|_F^2 -\frac{\mu}{2} \|\alpha\|^2 \\
&=& -\frac{1}{m} \sum\limits_{i=1}^m f(X_i,y_i)- \frac{1}{m} \sum\limits_{i=1}^m \ln \sum\limits_{y=1}^c \exp(- f(X_i,y)) \nonumber \\
&& - \frac{\lambda}{2} \sum\limits_{k=1}^v \|M_k\|_F^2 -\frac{\mu}{2} \|\alpha\|^2 \label{model01}
\end{eqnarray}
To ensure the basic property of metric, $E(M,\alpha)$ should be subject to $M_k \succeq 0, k=1,\cdots,v$(positive semi-definite).
Let $ v=1 $, the primary model degenerates into a single view version and the objective function (\ref{model01}) becomes
\begin{equation}
\max\limits_{M,\alpha} E_s(M,\alpha)
= \frac{1}{m} \sum\limits_{i=1}^m \ln p(y_i|X_i) \nonumber - \frac{\lambda}{2} \|M\|_F^2 \label{model02}
\end{equation}
which can be used to learn metric in multi-instance classification with single view.
\subsection{Optimization}
\label{sec:opt}
Although the model is constructed with the constraints that all the metrics should be positive semi-definite,
we can first optimize the model in an unconstrained condition and get $ M_1^*, M_2^*,\cdots, M_v^* $ and then project the metrics into positive semi-definite space. The procedure proceeds iteratively until convergence.
Our algorithm can be solved by gradient ascent and projection alternately.
In single view version, only one metric need to be optimized. The model can be solved in the Algorithm \ref{alg:svmiml}. Given an unknown image \textit{L} with bag $ X_l $, its label can be decided by
\begin{equation}
y=\arg \max\limits_{y=1,\cdots,c} p(y_l|X_l; M^*)
\end{equation}
\begin{algorithm}
\caption{Single view metric learning for multi-instance task (SVML)}\label{alg:svmiml}
\textbf{Input:} The training set $ T^1$; The penalty parameters $ \lambda$, gradient step-size $ \eta $, maximum of iterations $ R $.\\
\textbf{Output:} The target metrics $ M^*$;\\
\textbf{Procedure:}\\
1. Let $ r=1$ and initialize $ M$ as identity matrix;\\
2. Update $ M $ alternately using gradient ascent method by
\begin{equation}
M^{\text{new}}=M^{\text{old}}+\eta \frac{\partial E_s}{\partial M}
\end{equation}
and make projections by
\begin{equation}
M^{\text{new}}=\text{PSD}(M^{\text{new}})
\end{equation}
where PSD denotes the projection operator of positive semi-definite space.\\
3. Let $ r=r+1 $, if $ r > R $, stop iteration and obtain the output, otherwise go to step 2.\\
\end{algorithm}
First, fix $\alpha_k, k=1,\cdots,v$, the gradient of $ E(M,\alpha) $ with respect to $ M_k $ is
\begin{eqnarray}
\frac{\partial E}{\partial M_k} &=& -\frac{1}{m} \sum\limits_{i=1}^m \frac{\partial f(X_i,y_i)}{\partial M_k} - \lambda M_k \nonumber \\
&& + \frac{1}{m} \sum\limits_{i=1}^m \frac{\sum\limits_{y=1}^c \exp(-f(X_i,y)) \frac{\partial f(X_i,y)}{\partial M_k}}{\sum\limits_{y=1}^c \exp(-f(X_i,y))}
\end{eqnarray}
To obtain $\frac{\partial f(X_i,y)}{\partial M_k}$, we decompose it as
\begin{eqnarray}
\frac{\partial f(X_i,y)}{\partial M_k} &=& \frac{\partial }{\partial M_k} \alpha_k^* D_{am} (X_i^k, X_{j^*}^k) \\
&=& \frac{\alpha_k^*}{m_i^k} \sum\limits_{p=1}^{m_i^k} \frac{\partial }{\partial M_k} \min\limits_{x_{jl} \in X_j} d_M(x_{ip}, x_{jl}) \nonumber \\
&& + \frac{\alpha_k^*}{m_j^k} \sum\limits_{q=1}^{m_j^k} \frac{\partial }{\partial M_k} \min\limits_{x_{ih} \in X_i} d_M(x_{jq}, x_{ih})\\
&=& \frac{\alpha_k^*}{m_i^k} \sum\limits_{p=1}^{m_i^k} (x_{ip}-x_{jl^*})(x_{ip}-x_{jl^*})^\top \nonumber \\
&& +\frac{\alpha_k^*}{m_j^k} \sum\limits_{q=1}^{m_j^k} (x_{jq}-x_{ih^*})(x_{jq}-x_{ih^*})^\top \label{eq:fpar}
\end{eqnarray}
where
\begin{eqnarray}
(\alpha_k^*, j^*)=\arg\min\limits_{\alpha,j} \sum\limits_{k=1}^v \alpha_k D_{am} (X_i^k, X_j^k)
\end{eqnarray}
For each $ p=1,\cdots, m_i^k $,
\begin{equation}
l^*=\arg \min\limits_{x_{jl} \in X_j} d_M(x_{ip}, x_{jl})
\end{equation}
and for each $ q=1,\cdots, m_j^k $,
\begin{equation}
h^*=\arg \min\limits_{x_{ih} \in X_i} d_M(x_{jq}, x_{ih})
\end{equation}
Second, fix $ M_k, k=1,\cdots,v $, update $ \alpha $ by
$ \alpha= \alpha+ \eta_2 \frac{\partial E}{\partial \alpha} $, where
\begin{eqnarray}
\frac{\partial E}{\partial \alpha} &=& -\frac{1}{m} \sum\limits_{i=1}^m F(X_i,y_i) -\mu \alpha \nonumber \\
&& + \frac{1}{m} \sum\limits_{i=1}^m \frac{\sum\limits_{y=1}^c \exp(- f(X_i,y)) F(X_i,y)}{\sum\limits_{y=1}^c \exp(-f(X_i,y))}
\end{eqnarray}
where
\begin{equation}
F(X_i,y)=(D(X_i^1,X_j^1), \cdots, D(X_i^v,X_j^v))^\top
\end{equation}
and
\begin{equation}
X_j=\arg \min\limits_{y_j=y} \sum\limits_{k=1}^v \alpha_k D_M (X_i^k, X_j^k)
\end{equation} .
It should be noted that the equation (\ref{eq:fpar}) is computed on the basis of the metrics and weights, which are unknown before the equation. So iterative optimization will be implemented in our method. The metrics and weights for different views will be first initialized and then $ M_k, \alpha_k, k=1,\cdots,v $ can be updated by gradient ascent and projection to the space of positive semi-definite iteratively until convergence.
The detailed procedure of our method is shown in the Algorithm \ref{alg:mvmiml}.
Given an unknown image \textit{L} with $ v $ bags $ X_l^1, \cdots, X_l^v $, its label can be decided by
\begin{equation}
y=\arg \max\limits_{y=1,\cdots,c} p(y_l|X_l^1, \cdots, X_l^v; \mathscr{M}^*)
\end{equation}
where $ \mathscr{M}^*=\{M_1^*, \cdots, M_v^*\} $.
\begin{algorithm}\label{algo:mvml}
\caption{Multi-view metric learning for multi-instance task (MVML)}\label{alg:mvmiml}
\textbf{Input:} The training set $ T^1, T^2, \cdots, T^v $; The penalty parameters $ \lambda, \mu $, gradient step-size $ \eta $, maximum of iterations $\tau, R $.\\
\textbf{Output:} The target metrics $ M_1^*, M_2^*, \cdots, M_v^* $;\\
\textbf{Procedure:}\\
1. Let $ t=1, r=1 $ and Initialize $ M_1, M_2, \cdots, M_v $ as identity matrix;\\
2. Fix $ \alpha_k, k=1, 2, \cdots, v $, update $ M_k, k=1,\cdots,v $ alternately using gradient ascent method by
\begin{equation}
M_k^{\text{new}}=M_k^{\text{old}}+\eta_1 \frac{\partial E}{\partial M_k}
\end{equation}
and make projections to positive semi-definite space
\begin{equation}
M_k^{\text{new}}=\text{PSD}(M_k^{\text{new}})
\end{equation}
3. Let $ t=t+1 $, if $ t > \tau $, stop iteration, otherwise go to step 2;\\
4. Update the weight $ \alpha $,
\begin{equation}
\alpha^{\text{new}}= \alpha^{\text{old}}+\eta_2 \frac{\partial E}{\partial \alpha}
\end{equation}
5. Let $t=1, r=r+1 $, if $ r > R $, stop iteration and obtain the output, otherwise go to step 2.\\
\end{algorithm}
\begin{figure*}
\centering
\includegraphics[width=7in]{workflow.pdf}
\caption{Workflow}\label{fig:work}
\end{figure*}
\subsection{Computational cost}
Our approach is solved iteratively with alternate optimization. The computation of gradient descent is the main part of the computational cost. It contains two parts: updating metrics and weights. In updating the metrics, the computational cost in every iteration is $O(V^2 K^2 mn(m+n)) $, where $ V $ is the number of views, $ K $ is the number of images, $ m $ is the average number of instances in each bag and $n$ is the average length of each instance. And the cost is $O(V K^2 mn(m+n)) $ in updating weights. In consideration of the iteration number $ R, \tau $. The total computational cost of our model is $ O(R\tau V^2 K^2 mn(m+n))$. The cost is relatively high, but tractable in experiments, which can be improved by parallel computing or GPU acceleration in the future.
\section{Experiments}
\label{sec:exp}
In this section, numerical experiments on image datasets are made to demonstrate the efficacy of our new method. Six datasets Corel, Caltech, GRAZ02(people, cars, bikes, background), Butterfly, Galaxy Zoo, FERET are selected and converted into standard format(Each dataset contains three parts: features, bag ids and labels).
All the experiments are made on Matlab 2015a(PC, 8GB RAM).
\subsection{Datasets}
We will first describe our datasets since they are different in bag numbers, class numbers, image contents, etc.
The selected datasets can be divided into four categories: (1)Object detection. Detect particular object in a image by the unique features of the object. The object often appears with complex background that could affect feature extraction. (2) Species recognition. A species may contains several classes. The task is to recognize the class of a species. The difficulty is that there exist exiguous difference between different classes of the same species. (3) Galaxy discrimination. Discriminate the shape of galaxy in a image. (4) Face identification. Identify the gender of a given face.
The detailed information of the datasets is introduced as follows.
\begin{itemize}
\item Object detection: \textbf{Corel}\cite{duygulu2002object} is a famous natural scene image database which provides ten categories images, including architecture, bus, dinosaur, elephant, face, flower, food, horse, sky, snowberg. It was originally used for image annotation since every image contains several word annotations. But in our paper, we just use one main word for each image and make classification on the dataset.
\textbf{Caltech} is another well-known and widely studied\cite{bosch2007image,zhang2006svm} image dataset for pattern recognition. The dataset is collected by the student from California Institute of Technology. We download six categories of images, including cars, motorcycles, airplanes, faces, leaves and backgrounds, from the website http://www.vision.caltech.edu. The third dataset \textbf{GRAZ02} is constructed by Andreas Opelt et.al in \cite{opelt2005object}. It contains four classes of images: bike, car, person and background(Environment without bike, car or person). Every image contains more complex contents than Corel and Caltech. For GRAZ02, two extra datasets are extracted, \textbf{Bike}(bike\&background) and \textbf{Car}(car\&background).
Classification on the database is a challenging task due to clutter contents, illumination variation, intra-class variability, diversiform scales and poses.
\item Species recognition: The image group of species, \textbf{butterfly}\cite{lazebnik2004semi}, is selected to make classification. The butterflies dataset consists of seven classes, admiral, black-swallowtail, machaon, monarch-closed, monarch-open, peacock, zebra. There exists slight difference between different kinds of butterflies, since they have similar shape, structure, and pose. So it is very hard to discriminate them, only their unique character(particular wing or stripe) may be helpful.
\item Galaxy discrimination: The \textbf{galaxy} zoo dataset consists of galaxy images with three kinds of shapes, edge-on, elliptical and spiral, which have been given in Figure \ref{fig:galaxy}. The database is collected by the Galaxy Zoo project(http://zoo1.galaxyzoo.org). Discriminate the shape of galaxy automatically is useful in astronomy since it can help scientists track the evolution of galaxies.
\item Face with Gender: The \textbf{FERET}(Facial Recognition Technology) database\cite{phillips2000feret} is established by the FERET program, whose primary task is to develop automatic face recognition technology that could be applied in assist security, intelligence and law enforcement personnel etc. We collect a part of the dataset and divide them into two classes, man and woman.
\end{itemize}
\begin{figure}
\subfigure[Corel]{
\begin{minipage}{0.98\linewidth}
\centering
\includegraphics[width=0.5in]{ImgDisp/200.jpg}
\includegraphics[width=0.5in]{ImgDisp/300.jpg}
\includegraphics[width=0.5in]{ImgDisp/400.jpg}
\includegraphics[width=0.5in]{ImgDisp/501.jpg}
\includegraphics[width=0.5in]{ImgDisp/1.jpg}\\
\includegraphics[width=0.5in]{ImgDisp/600.jpg}
\includegraphics[width=0.5in]{ImgDisp/901.jpg}
\includegraphics[width=0.5in]{ImgDisp/700.jpg}
\includegraphics[width=0.5in]{ImgDisp/102.jpg}
\includegraphics[width=0.5in]{ImgDisp/800.jpg}
\par
\vspace{0.1in}
\end{minipage}
}
\subfigure[Caltech]{
\begin{minipage}{0.98\linewidth}
\centering
\includegraphics[width=0.5in]{ImgDisp/0002.jpg}
\includegraphics[width=0.5in]{ImgDisp/0030.jpg}
\includegraphics[width=0.5in]{ImgDisp/0004.jpg}
\includegraphics[width=0.5in]{ImgDisp/0076.jpg}
\includegraphics[width=0.5in]{ImgDisp/0001.jpg}
\includegraphics[width=0.5in]{ImgDisp/0189.jpg}
\par
\vspace{0.1in}
\end{minipage}
}
\subfigure[GRAZ02]{
\begin{minipage}{0.98\linewidth}
\centering
\includegraphics[width=0.5in]{ImgDisp/bike_002.jpg}
\includegraphics[width=0.5in]{ImgDisp/carsgraz_005.jpg}
\includegraphics[width=0.5in]{ImgDisp/person_027.jpg}
\includegraphics[width=0.5in]{ImgDisp/bg_graz_002.jpg}
\par
\vspace{0.1in}
\end{minipage}
}
\caption{Images of Object Detection}\label{fig:object}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5in]{ImgDisp/adm045.jpg}
\includegraphics[width=0.5in]{ImgDisp/swa001.jpg}
\includegraphics[width=0.5in]{ImgDisp/mch026.jpg}\\
\par
\vspace{0.1in}
\includegraphics[width=0.5in]{ImgDisp/mnc023.jpg}
\includegraphics[width=0.5in]{ImgDisp/mno012.jpg}
\includegraphics[width=0.5in]{ImgDisp/pea028.jpg}
\includegraphics[width=0.5in]{ImgDisp/zeb001.jpg}
\caption{Images of the Butterfly}\label{fig:butterfly}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=2.5in]{ImgDisp/galaxy.pdf}
\caption{Images of the galaxy zoo}\label{fig:galaxy}
\end{figure}
\begin{figure}
\begin{minipage}{0.98\linewidth}
\centering
\includegraphics[width=0.5in]{ImgDisp/012.jpg}
\includegraphics[width=0.5in]{ImgDisp/013.jpg}
\includegraphics[width=0.5in]{ImgDisp/015.jpg}
\includegraphics[width=0.5in]{ImgDisp/017.jpg}
\par
\vspace{0.1in}
\end{minipage}
\caption{Images of the FERET}
\end{figure}
\subsection{Feature extraction for images}
In this subsection, we will discuss what features will be extracted from images to construct different views of the datasets.
\subsubsection{HOG feature}
The HOG(Histogram of Oriented Gradient)\cite{dalal2005histograms} feature is presented based on the idea that local object appearance and shape can be well described by the distribution of local gradient or edge directions. It first divides image into smaller 'cells' and accumulates histogram of gradient or edge directions for each cell. The combination of these histograms represent the HOG feature of the whole image. However, we will not combine the histograms and then each image is a bag of cells, each cell can be expressed by a feature vector. In the following classification, each bag contains 9 cells.
\subsubsection{SIFT key points}
As a classical method in feature extraction, SIFT(Scale-invariant feature transformation) is first proposed by Lowe for object recognition and image matching\cite{lowe1999object,lowe2004distinctive}. It is a kind of local feature descriptor for images. It constructs scale space to get scale-invariant key points and endow orientation information with these points.
SIFT can find interest points at multiple scale to represent important regions of each image. Each key point is a 128-length numerical feature vector and each image is described by a bag of multiple such key points. It is worthy to point out that SIFT can extract different numbers of key points from different images, resulting that the idea of concatenating the vectors into a single feature vector is not tractable in traditional classification. The SIFT features are suitable to be used for multi-instance learning. SIFT has been widely applied in many applications\cite{felzenszwalb2010object,dollar2005behavior} since it is invariant to image scale and rotation.
\subsubsection{Uniform patches with LBP}
LBP(Local binary pattern) feature is a kind of operator to describe local textual property of image. Traditional LBP operator is defined in a $ 3 \times 3 $ block, the central pixel of which is used as the threshold. The other 8 neighbors are labeled as either 1 or 0 by comparing them with the threshold by turns. If the pixel value of neighbor is larger, it will be marked as 1, otherwise 0. Then a 8-bit string is obtained to represent the textual information of this local region. The 8-bit string will often be converted to numerical features for easily computation. Fig. \ref{fig:lbp} illustrates the process of extracting LBP features.
LBP has invariance of rotation and gray, making it be successfully applied in many applications, including image classification\cite{wang2009hog}, face detection\cite{zhang2007face}. In traditional, all the local LBP features are concatenated into a single feature vector to describe the whole image.
An interesting concept 'visual dictionary' is proposed and used recently\cite{wen2009local} to describe image by partition it into smaller regions. Every region is a word of the dictionary and the image is represented by a bag of words. Similar as such method, every image will be divided into $ 4 \times 4 $ uniformly distributed patches, since multi-instance representation can be more powerful than single instance in improving the performance of classification, which we have mentioned in the Introduction. Every patch can be expressed by a single instance with length of 256 by extracting LBP features. Then each image is transformed as a bag, comprised of 16 instances.
\begin{figure}
\centering
\includegraphics[width=3.5in]{LBP.pdf}
\caption{LBP}\label{fig:lbp}
\end{figure}
\subsection{Distance Comparison}
In the Section \ref{sec:dist}, we have claimed that our novel distance function $ D_{am} $, designed for bags, is prior to the previous $ D_{ave} $ and $ D_{min} $. Next we will make numerical experiments to verify the judgement. First, a toy example is given in the Figure \ref{fig:distcmp} to shown the difference among the three distance functions. Three images from the \textbf{Corel} dataset were selected. The former two belong to the class of \textit{architecture} and the third belongs to \textit{snowberg}. The features of HOG, SIFT, LBP are extracted and depicted in the Figure \ref{fig:distcmp}(a). The distances were computed in the Figure \ref{fig:distcmp}(b). In fact, the three images are similar in structure, color and luminance, which gives a challenge to distance function.
In SIFT and LBP feature, $ D_{am}(I_1,I_2) $ is smaller than $ D_{am}(I_1,I_3) $ and $ D_{am}(I_2,I_3) $, implying that $ I_1 $ and $ I_2 $ belong to the same class. So $ D_{am} $ made right judges in SIFT and LBP feature, better than $ D_{ave} $ and $ D_{min} $.
To further validate the efficiency and advantage of $ D_{am} $, $1$NN classification was implemented with the three distance functions respectively. Euclidean distance was used to compute the distance between instances.
Three datasets, \textbf{Car}, \textbf{butterfly} and \textbf{Corel}, were selected to make comparison. Three-fold cross validation was applied in classification. Accuracy and standard deviation are shown in the Table \ref{tab:dist1}. It can be clearly seen that $1$NN with $ D_{am} $ obtains the best performance on most of the features of the three datasets.
\begin{figure*}
\subfigure[]{
\begin{minipage}{0.48\linewidth}
\includegraphics[width=1.1in]{Distcmp/203.jpg}
\begin{picture}(0,0)
\put(-80,45){1}
\end{picture}
\hspace{-0.1in}
\includegraphics[width=1.1in]{Distcmp/207.jpg}
\begin{picture}(0,0)
\put(-80,45){2}
\end{picture}
\hspace{-0.1in}
\includegraphics[width=1.1in]{Distcmp/802.jpg}
\begin{picture}(0,0)
\put(-80,45){3}
\end{picture}\\
\includegraphics[width=1.1in]{Distcmp/HOG1.pdf}
\includegraphics[width=1.1in]{Distcmp/HOG2.pdf}
\includegraphics[width=1.1in]{Distcmp/HOG3.pdf}\\
\includegraphics[width=1.1in]{Distcmp/SIFT1.pdf}
\includegraphics[width=1.1in]{Distcmp/SIFT2.pdf}
\includegraphics[width=1.1in]{Distcmp/SIFT3.pdf}\\
\includegraphics[width=1.1in]{Distcmp/LBP1.pdf}
\includegraphics[width=1.1in]{Distcmp/LBP2.pdf}
\includegraphics[width=1.1in]{Distcmp/LBP3.pdf}
\par
\vspace{0.1in}
\end{minipage}
}
\subfigure[]{
\begin{minipage}{0.55\linewidth}
\scriptsize
\renewcommand{\arraystretch}{2}
\begin{tabular}{lc|c|c|}
& \multicolumn{3}{c}{$ D_{am} $} \\
\cline{2-4}
\multirow{3}{0.5cm}{HOG} &\multicolumn{1}{|c|}{0} & \underline{0.49} & \underline{0.41} \\\cline{2-4}
&\multicolumn{1}{|c|}{\underline{0.49}} & 0 & 1.14 \\\cline{2-4}
&\multicolumn{1}{|c|}{\underline{0.41}} & 1.14 & 0 \\\cline{2-4}
\end{tabular}
\begin{tabular}{|c|c|c|}
\multicolumn{3}{c}{$ D_{min} $} \\
\hline
0 & \underline{0.29} & \underline{0.26} \\\hline
\underline{0.29} & 0 & 1.03 \\\hline
\underline{0.26} & 1.03 & 0 \\\hline
\end{tabular}
\begin{tabular}{|c|c|c|}
\multicolumn{3}{c}{$ D_{ave} $} \\
\hline
0.38 & \underline{0.81} & \underline{0.61} \\\hline
\underline{0.81} & 0.37 & 1.30 \\\hline
\underline{0.61} & 1.30 & 0.34 \\\hline
\end{tabular}\\
\par
\vspace{0.05in}
\begin{tabular}{l|c|c|c|}
\cline{2-4}
\multirow{3}{0.5cm}{SIFT} &0 & \textbf{0.66} & 0.78\\\cline{2-4}
&\textbf{0.66} & 0 & 0.72 \\\cline{2-4}
&0.78 & 0.72 & 0 \\\cline{2-4}
\end{tabular}
\begin{tabular}{|c|c|c|}\hline
0 & \textbf{0.30} & 0.48 \\\hline
\textbf{0.30} & 0 & 0.45 \\\hline
0.48 & 0.45 & 0 \\\hline
\end{tabular}
\begin{tabular}{|c|c|c|}\hline
0.94 & \underline{1.08} & \underline{1.07} \\\hline
\underline{1.08} & 0.95 & \underline{1.07} \\\hline
\underline{1.07} & \underline{1.07} & 0.94 \\\hline
\end{tabular}
\par
\vspace{0.05in}
\begin{tabular}{l|c|c|c|}\cline{2-4}
\multirow{3}{0.5cm}{LBP} & 0 & \textbf{1.69} & 1.96 \\\cline{2-4}
&\textbf{1.69} & 0 & 2.21 \\\cline{2-4}
&1.96 & 2.21 & 0 \\\cline{2-4}
\end{tabular}
\begin{tabular}{|c|c|c|}\hline
0 & \underline{0.49} & \underline{0.38} \\\hline
\underline{0.49} & 0 & 0.67 \\\hline
\underline{0.38} & 0.67 & 0 \\\hline
\end{tabular}
\begin{tabular}{|c|c|c|}\hline
3.86 & \underline{4.41} & 4.57 \\\hline
\underline{4.41} & 2.82 & \underline{4.37} \\\hline
4.57 & \underline{4.37} & 3.07 \\\hline
\end{tabular}
\par
\vspace{0.1in}
\end{minipage}
}
\caption{A toy example to compare three distance functions $ D_{am}, D_{min} $ and $ D_{ave} $. (a) The 2, 3, 4 line of images corresponds to HOG, SIFT, and LBP respectively. Each curve in a subfigure denotes a instance. The y axis denotes the numerical value of the components of instance. (b) The computed distances in three kinds of features. The minimum in each sub-table is in boldface. The numbers with underline can misguide the judge of distance function.}
\label{fig:distcmp}
\end{figure*}
\begin{table*}[htbp]
\small
\centering
\renewcommand{\arraystretch}{1.1}
\caption{1-NN classification accuracy of different distances on different features}\label{tab:dist1}
\begin{tabular}{c|l|ccc|c}
\hline
\multirow{2}{*}{Datasets} & \multirow{2}{*}{Distance} & \multicolumn{3}{c|}{Feature} & Average \\
\cline{3-5}
& & HOG & SIFT & LBP & \\
\hline
\multirow{3}*{\tabincell{c}{Car\\(600\&2)}} & $ D_{ave} $ & 56.67$\pm$2.25 & 51.83$\pm$1.04 & 47.50$\pm$6.56 & 52.00 \\
& $ D_{min} $ & 60.67$\pm$2.52 & 59.33$\pm$2.36 & 53.83$\pm$1.61 & 57.94 \\
& $ D_{am} $ & \textbf{64.00$\pm$3.46} & \textbf{64.50$\pm$2.00 }& \textbf{59.17$\pm$1.89} & 62.56 \\
\hline
\multirow{3}{*}{\tabincell{c}{Butterfly\\(619\&7)}} & $ D_{ave} $ & 17.28$\pm$3.64 & 23.42$\pm$3.16 & 21.16$\pm$1.97 & 20.62 \\
& $ D_{min} $ & 40.55$\pm$2.75 & \textbf{84.33$\pm$0.98} & 26.66$\pm$1.31 & 50.51 \\
& $ D_{am} $ & \textbf{52.66$\pm$1.89} & 81.74$\pm$1.49 & \textbf{28.12$\pm$4.73} & 54.17 \\
\hline
\multirow{3}{*}{\tabincell{c}{Corel\\(1000\&10)}} & $ D_{ave} $ & 15.20$\pm$4.53 & 13.60$\pm$0.44 & 36.50$\pm$3.45 & 21.77 \\
& $ D_{min} $ & 44.50$\pm$1.28 & 32.90$\pm$1.08 & 39.80$\pm$3.05 & 39.07 \\
& $ D_{am} $ & \textbf{50.70$\pm$3.47} & \textbf{46.30$\pm$3.06} & \textbf{58.00$\pm$1.32} & 51.67 \\
\hline
\end{tabular}
\end{table*}
\subsection{Performance evaluation}
In the subsection, we will evaluate our model in different sides, including classification ability , robustness to parameters and sensitivity to instance number of bags.
\subsubsection{Image classification}
Comparison of the classification performance will be made eight datasets, the information of which is given in the Table \ref{tab:error3}. Three-fold cross-validation was utilized to get the average classification accuracy. The experiments had been divided into two parts: single view and multi-view.
In single view classification, experiments on the three features HOG, SIFT and LBP were conducted independently. $ k $NN classification with Euclidean distance and metric learning(Algorithm 1) were both implemented. In metric learning, the penalty parameter $ \lambda $ and the learning rate $ \eta $ were both empirically set to be 0.1 and the number of iteration $ R $ was set to be 3.
It is obvious that single view metric learning can always improve the performance of $ k $NN classification with Euclidean distance. SVML is effective in obtaining better metric for images with multi-instance features.
For multi-view classification, the three features can be combined into four groups: HOG+SIFT(H\&S), HOG+LBP(H\&L), SIFT+LBP(S\&L), HOG+SIFT+LBP(H\&S\&L). The experiments on these four multi-view conditions were made individually.
The penalty parameters $ \lambda $ and $ \mu $ are both selected from the set $ \{0.01, 0.1, 1\} $ and the combination of $ \eta_1, \eta_2 $ is chosen from the set $ \{(0.01,0.01), (0.05,0.05), (0.1,0.1)\} $. The number of iteration $ \tau, R $ is set to be 1 and 3 respectively.
The classification results are shown in the Table \ref{tab:error3}. Experiments with the feature group H\&S\&L perform best on 5 datasets, better than single view classification, despite metric learning or not. Classification with more views often get higher accuracy, which indicates that our method can extract useful information from all the views and assemble them effectually, based on the inference that different features are complementary to each other. Further, six images from three datasets are displayed in the Table \ref{tab:nearimg} with their corresponding nearest images under different views. It implies that our method can truly find a data-dependent metric to make similar images closer and improve the classification performance.
\begin{table*}[htbp]
\small
\centering
\renewcommand{\arraystretch}{1.3}
\caption{Classification accuracy on single view and multi-view}\label{tab:error3}
\begin{tabular}{l|lccc|cccc}
\hline
\multirow{2}{*}{Datasets} & \multicolumn{4}{c|}{Single view} & \multicolumn{4}{c}{Multi-view} \\
\cline{2-9}
& & HOG & SIFT & LBP & H\&S & H\&L & S\&L & H\&S\&L \\
\hline
\multirow{2}{*}{\tabincell{c}{Corel\\(300\&10)}} & Eucl.& 43.00$\pm$2.65 & 42.67$\pm$4.16 & 54.00$\pm$2.00 & \multirow{2}{*}{52.33$\pm$6.11} & \multirow{2}{*}{55.33$\pm$2.52} & \multirow{2}{*}{57.67$\pm$6.43} & \multirow{2}{*}{\textbf{60.00$\pm$1.73}} \\
& ML& 44.00$\pm$1.00 & 42.67$\pm$4.16 & 59.33$\pm$5.77 & & & &\\
\hline
\multirow{2}{*}{\tabincell{c}{Caltech\\(300\&6)}} & Eucl.& 80.33$\pm$3.51 & 66.67$\pm$1.53 & 78.67$\pm$3.21 & \multirow{2}{*}{86.00$\pm$2.65} & \multirow{2}{*}{85.00$\pm$6.08} & \multirow{2}{*}{77.33$\pm$1.53} & \multirow{2}{*}{\textbf{87.00$\pm$4.58}} \\
& ML& 80.33$\pm$2.52 & 67.67$\pm$1.53 & 78.33$\pm$3.06 & & & &\\
\hline
\multirow{2}{*}{\tabincell{c}{GRAZ02\\(300\&4)}} & Eucl.& 37.33$\pm$8.96 & 38.67$\pm$2.89 & 36.67$\pm$3.79 & \multirow{2}{*}{36.67$\pm$4.73} & \multirow{2}{*}{35.00$\pm$4.58} &\multirow{2}{*}{37.33$\pm$3.79}& \multirow{2}{*}{37.33$\pm$4.73} \\
& ML& 38.67$\pm$8.50 & \textbf{40.00$\pm$3.61} & 37.33$\pm$4.16 & & & &\\
\hline
\multirow{2}{*}{\tabincell{c}{Bike\\(300\&2)}} & Eucl.& \textbf{58.51$\pm$1.00} & 53.57$\pm$6.79 & 56.52$\pm$3.54 & \multirow{2}{*}{55.04$\pm$3.18} & \multirow{2}{*}{58.04$\pm$4.92} & \multirow{2}{*}{58.04$\pm$4.92} &\multirow{2}{*}{58.48$\pm$7.74} \\
& ML& 58.51$\pm$1.00 & 53.60$\pm$8.52 & 57.55$\pm$7.90 & & & &\\
\hline
\multirow{2}{*}{\tabincell{c}{Car\\(300\&2)}} & Eucl.& 60.49$\pm$2.53 & 58.11$\pm$11.3 & 54.98$\pm$8.58 & \multirow{2}{*}{\textbf{68.54$\pm$4.02}} & \multirow{2}{*}{62.97$\pm$2.88}& \multirow{2}{*}{67.01$\pm$2.38}& \multirow{2}{*}{64.53$\pm$2.51} \\
& ML& 59.98$\pm$4.40 & 58.10$\pm$9.84 & 56.97$\pm$4.76 & & & &\\
\hline
\multirow{2}{*}{\tabincell{c}{Butterfly\\(280\&7)}} & Eucl.& 47.51$\pm$2.93 & 71.43$\pm$1.34& 22.85$\pm$1.50& \multirow{2}{*}{73.22$\pm$0.93} & \multirow{2}{*}{43.25$\pm$8.13} & \multirow{2}{*}{67.14$\pm$4.15} & \multirow{2}{*}{\textbf{73.57$\pm$2.29}} \\
& ML& 48.22$\pm$2.02 & 71.79$\pm$0.44 & 24.64$\pm$0.15 & & & &\\
\hline
\multirow{2}{*}{\tabincell{c}{Galaxy\\(210\&3)}} & Eucl.& 78.10$\pm$0.82 & 62.38$\pm$1.65 & 81.90$\pm$8.37 & \multirow{2}{*}{81.90$\pm$4.12} & \multirow{2}{*}{83.33$\pm$3.60} & \multirow{2}{*}{84.29$\pm$5.71} & \multirow{2}{*}{\textbf{85.71$\pm$3.78}} \\
& ML& 78.10$\pm$0.82 & 62.86$\pm$1.43 & 82.38$\pm$7.87 & & & & \\
\hline
\multirow{2}{*}{\tabincell{c}{FERET\\(150\&2)}} & Eucl.& 82.67$\pm$2.31 & 76.00$\pm$7.21 & 71.33$\pm$5.03& \multirow{2}{*}{84.00$\pm$8.72} & \multirow{2}{*}{81.33$\pm$1.15} & \multirow{2}{*}{78.00$\pm$4.00} & \multirow{2}{*}{\textbf{84.00$\pm$2.00}} \\
& ML& 82.67$\pm$2.31 & 76.00$\pm$5.29 & 76.00$\pm$2.00 & & & & \\
\hline
\end{tabular}
\end{table*}
\begin{table*}[htbp]
\centering
\renewcommand{\arraystretch}{1.2}
\caption{Nearest Image}\label{tab:nearimg}
\begin{tabular}{cccccccc}
Image & HOG & SIFT & LBP & H\&S & H\&L & S\&L & H\&S\&L \\
\includegraphics[width=0.7in]{NearestImg/corel208.jpg} & \includegraphics[width=0.7in]{NearestImg/corel216.jpg} & \includegraphics[height=0.5in]{NearestImg/corel281.jpg} & \includegraphics[width=0.7in]{NearestImg/corel109.jpg} &
\includegraphics[width=0.7in]{NearestImg/corel187.jpg} & \includegraphics[width=0.7in]{NearestImg/corel216.jpg} & \includegraphics[height=0.5in]{NearestImg/corel277.jpg}& \includegraphics[width=0.7in]{NearestImg/corel187.jpg}\\
\includegraphics[width=0.7in]{NearestImg/corel293.jpg} & \includegraphics[height=0.5in]{NearestImg/corel10.jpg} & \includegraphics[width=0.7in]{NearestImg/corel254.jpg} & \includegraphics[width=0.7in]{NearestImg/corel11.jpg} &
\includegraphics[width=0.7in]{NearestImg/corel150.jpg} & \includegraphics[height=0.5in]{NearestImg/corel10.jpg} & \includegraphics[width=0.7in]{NearestImg/corel254.jpg}& \includegraphics[width=0.7in]{NearestImg/corel289.jpg}\\
\includegraphics[width=0.7in]{NearestImg/caltech46.jpg} & \includegraphics[width=0.7in]{NearestImg/caltech5.jpg} & \includegraphics[width=0.7in]{NearestImg/caltech213.jpg} & \includegraphics[width=0.7in]{NearestImg/caltech280.jpg} &
\includegraphics[width=0.7in]{NearestImg/caltech5.jpg} & \includegraphics[width=0.7in]{NearestImg/caltech5.jpg} & \includegraphics[width=0.7in]{NearestImg/caltech6.jpg}& \includegraphics[width=0.7in]{NearestImg/caltech43.jpg}\\
\includegraphics[width=0.7in]{NearestImg/caltech96.jpg} & \includegraphics[width=0.7in]{NearestImg/caltech21.jpg} & \includegraphics[width=0.7in]{NearestImg/caltech158.jpg} & \includegraphics[width=0.7in]{NearestImg/caltech213.jpg} &
\includegraphics[width=0.7in]{NearestImg/caltech60.jpg} & \includegraphics[width=0.7in]{NearestImg/caltech21.jpg} & \includegraphics[width=0.7in]{NearestImg/caltech158.jpg}& \includegraphics[width=0.7in]{NearestImg/caltech21.jpg}\\
\includegraphics[height=0.5in]{NearestImg/butterfly24.jpg} & \includegraphics[height=0.5in]{NearestImg/butterfly110.jpg} & \includegraphics[width=0.7in]{NearestImg/butterfly239.jpg} & \includegraphics[height=0.5in]{NearestImg/butterfly82.jpg} &
\includegraphics[height=0.5in]{NearestImg/butterfly10.jpg} & \includegraphics[height=0.5in]{NearestImg/butterfly110.jpg} & \includegraphics[width=0.7in]{NearestImg/butterfly239.jpg}& \includegraphics[height=0.5in]{NearestImg/butterfly10.jpg}\\
\includegraphics[height=0.5in]{NearestImg/butterfly158.jpg} & \includegraphics[height=0.5in]{NearestImg/butterfly217.jpg} & \includegraphics[width=0.7in]{NearestImg/butterfly169.jpg} & \includegraphics[height=0.5in]{NearestImg/butterfly224.jpg} &
\includegraphics[height=0.5in]{NearestImg/butterfly125.jpg} & \includegraphics[height=0.5in]{NearestImg/butterfly217.jpg} & \includegraphics[width=0.7in]{NearestImg/butterfly169.jpg}& \includegraphics[height=0.5in]{NearestImg/butterfly125.jpg}\\
\end{tabular}
\end{table*}
\subsubsection{Influence of parameters}
In this subsection, the influence of parameters $ \lambda, \mu $ and learning rates $ \eta_1, \eta_2 $ are explored.
The iteration numbers $ \tau, R $ are the same as the above experiments. As previously mentioned, $ \lambda $ and $ \mu $ are both selected from the set $ \{0.01, 0.1, 1\} $, so there are nine combinations for $ (\lambda, \mu) $, namely, $ \{(0.01,0.01),(0.01,0.1),(0.01,1),(0.1,0.01),(0.\\1,0.1),(0.1,1),(1,0.01),(1,0.1),(1,1)\} $. Fix the combination of $ \eta_1, \eta_2 $((0.01, 0.01),(0.05,0.05) or (0.1,0.1)), the accuracy curve and corresponding standard deviation with respect to the combination of $ \lambda, \mu $ can be got in Figure \ref{fig:paras}. Each subfigure corresponds to a selected dataset. The axis of $ x $ denotes nine combinations of $ \lambda, \mu $. For every combination of $ \eta_1, \eta_2 $, the accuracy fluctuates slightly, indicating that our model is robust to the penalty parameters when they are sufficiently small. For each combination of $ \lambda, \mu $, the accuracy does not change drastically with respect to $ \eta_1, \eta_2 $, which suggests the insensitivity of the method to the learning rates. The faint influence of parameters verifies the consistency of our new model.
\begin{figure*}
\centering
\subfigure[Corel]{
\includegraphics[width=2.0in]{ParasEffect/corel_paras.pdf}
}
\subfigure[Caltech]{
\includegraphics[width=2.0in]{ParasEffect/caltech_paras.pdf}
}
\subfigure[Bike]{
\includegraphics[width=2.0in]{ParasEffect/bike_paras.pdf}
}
\subfigure[Car]{
\includegraphics[width=2.0in]{ParasEffect/car_paras.pdf}
}
\subfigure[Butterfly]{
\includegraphics[width=2.0in]{ParasEffect/butterfly_paras.pdf}
}
\subfigure[Galaxy]{
\includegraphics[width=2.0in]{ParasEffect/galaxy_paras.pdf}
}
\caption{Influence of the penalty parameters and learning rates.}
\label{fig:paras}
\end{figure*}
\subsubsection{Influence of instance number}
In the previous feature extraction, HOG and LBP are extracted as bag-of-words representation by dividing every image uniformly. In HOG and LBP feature, every image contains 9 and 16 instances respectively. Next we will investigate the influence of instance number in each bag. The penalty parameters $ \lambda, \mu $ are both set to be 0.01 and the learning rates $ \eta_1, \eta_2 $ are both set to be 0.1.
For HOG feature, each image is further divided into 4, 16, 25, 36 cells and the accuracy curve of single view and multi-view(H\&S\&L) with respect to instance number is depicted in Figure \ref{fig:inst}.(a)(b)(c)(d). It can be seen that the accuracy of single view changes slightly and the accuracy of multi-view is very stable, verifying the robustness of our model to instance number of HOG feature. The model can extract information consistently from HOG feature, immune to the instance number. For LBP feature, each image is further divided into 4, 9, 25, 36 patches and the accuracy curve of single view and multi-view(H\&S\&L) with respect to instance number is depicted in Figure \ref{fig:inst}.(e)(f)(g)(h). The accuracy curve of multi-view(H\&S\&L) has similar fluctuation trend with the accuracy curve of single view, but both curves have small amplitudes. The experiments demonstrate that our model is not sensitive to instance number, resulting from that a bag with different instance numbers can be actually seemed as feature extraction in different scale, which will not affect metric learning much.
\begin{figure*}
\centering
\subfigure[Corel]{
\includegraphics[width=1.7in]{InstEffect/corel_hog.pdf}
}
\subfigure[Caltech]{
\includegraphics[width=1.7in]{InstEffect/caltech_hog.pdf}
}
\subfigure[Butterfly]{
\includegraphics[width=1.7in]{InstEffect/butterfly_hog.pdf}
}
\subfigure[Galaxy]{
\includegraphics[width=1.7in]{InstEffect/galaxy_hog.pdf}
}
\subfigure[Corel]{
\includegraphics[width=1.7in]{InstEffect/corel_lbp.pdf}
}
\subfigure[Caltech]{
\includegraphics[width=1.7in]{InstEffect/caltech_lbp.pdf}
}
\subfigure[Butterfly]{
\includegraphics[width=1.7in]{InstEffect/butterfly_lbp.pdf}
}
\subfigure[Galaxy]{
\includegraphics[width=1.7in]{InstEffect/galaxy_lbp.pdf}
}
\caption{Influence of instance number in HOG and LBP feature. In all the subfigures, the instance numbers of SIFT feature are the same as previous classification.In (a)(b)(c)(d), the instance numbers of LBP feature are all fixed as 16. In (e)(f)(g)(h), the instance numbers of HOG feature are all fixed as 9.}
\label{fig:inst}
\end{figure*}
\section{Conclusions}
\label{sec:conclu}
In this paper, we propose a novel metric learning method for image classification, introducing metric learning technique into multi-view multi-instance task. For every image, feature extraction is conducted for three views, HOG, SIFT and LBP, to represent the image in complementary sides. For all the three visual features, every image is presented by a bag, consisted of multiple instances. To unify the information of multiple views, the following efforts are made to make the classification performance as good as possible. First, a new distance function is designed for bags by calculating the weighted sum of the distances between instances. The distance between images with multiple views sums the weighted bag distance.
Then we construct an optimization problem under probability framework to combine multi-view information, aiming to maximizing the joint probability that every image is similar with its nearest image.
The technique of metric learning is embedded to adjust the distance between instances from different bags.
The model can be iteratively solved by gradient descent and positive semi-definite projection alternately.
In future work, we will explore more kinds of features and design more efficient algorithm to solve our method due to its high computational cost.
\\
\textbf{Acknowledgement}\\
\indent This work has been partially supported by grants from National Natural Science Foundation of China (Nos .61472390, 11271361, 71331005, and 11226089), Major International (Regional) Joint Research Project (No. 71110107026) and the Beijing Natural Science Foundation (No.1162005).
\label{sect:bib}
\small
\bibliographystyle{plain}
|
2,869,038,156,409 | arxiv | \section*{Acknowledgments}
This work was supported in part by the MNiSW program ”Diamond Grant” (J.S.).
|
2,869,038,156,410 | arxiv | \section{Introduction}
In this paper, we consider the following convex problem with separable structure
\[ \label{Problem-LC}
\min \{f(x)+g(z): A x=z\}
\]
where $A\in \mathbb{R}^{m \times n}$, $f: \mathbb{R}^{n} \mapsto(-\infty,+\infty)$ and $g: \mathbb{R}^{m} \mapsto(-\infty,+\infty)$ are given closed proper convex functions. The solution set of (\ref{Problem-LC}) is assumed to be nonempty.
The problem \eqref{Problem-LC} was discussed in the early days and has received a lot of attentions in recent years
since it captures many practical models in sparse and low-rank optimization. Some classical algorithms such as augmented Lagrangian method (ALM) \cite{Hes69,Powell69}, alternating direction method of multipliers (ADMM) \cite{Gab83,GM}, proximal gradient method \cite{Nesterov,Beck2009} have gained much popularity due to their flexibility and efficiency for solving \eqref{Problem-LC}. In this paper, we restrict our attention to Chen-Teboulle's proximal-based decomposition method \cite{Chen94}, which is also well-known to handle separable convex optimization problems.
The method was dated back from the seminal work of Chen and Teboulle in early 1990s and
soon became a popular distributed primal dual algorithm in convex programming.
Differ from the ADMM that combines
both ideas of the augmented Lagrangian framework and alternating strategy, Chen and Teboulle's algorithm \cite{Chen94} exploits directly for the Lagrangian function \eqref{Aug-L-F} and enjoys parallel architecture. More specially, let the Lagrangian function associated with (\ref{Problem-LC}) be defined by
\[ \label{Aug-L-F}
L(x, z, y):=f(x)+g(z)+ \left\langle y, A x-z\right\rangle,
\]
where $y \in \mathbb{R}^{m}$ be the Lagrange multiplier associated with the linear constraint in (\ref{Problem-LC}). Chen-Teboulle's algorithm \cite{Chen94} iterates as follows:
\begin{subequations} \label{CT-1}
\begin{numcases}{\hbox{\quad}}
\label{CT-1-p} p^{k+1}=\arg \max \left\{L\left(x^{k}, z^{k}, y\right)-\left(1 /\left(2 \lambda\right)\right)\left\|y-y^{k}\right\|^{2}\right\},\\[0.2cm]
\label{CT-1-x} x^{k+1}=\arg \min \left\{L\left(x, z^{k}, p^{k+1}\right)+\left(1 /\left(2 \lambda\right)\right)\left\|x-x^{k}\right\|^{2}\right\},\\[0.2cm]
\label{CT-1-z} z^{k+1}=\arg \min \left\{L\left(x^{k}, z, p^{k+1}\right)+\left(1 /\left(2 \lambda\right)\right)\left\|z-z^{k}\right\|^{2}\right\},\\[0.2cm]
\label{CT-1-y} y^{k+1}=\arg \max \left\{L\left(x^{k+1}, z^{k+1}, y\right)-\left(1 /\left(2 \lambda\right)\right)\left\|y-y^{k}\right\|^{2}\right\},
\end{numcases}
\end{subequations}
where $\lambda>0$ is a proximal parameter. For simplicity, we fix it in our discussion.
From \eqref{CT-1}, we can observe that the algorithm performs two proximal steps in the dual variables and
one proximal step for the primal variables $x$ and $z$ separately; and the multiplier variable is first predicted by $p^{k+1}$ step \eqref{CT-1-p} and then corrected by the $y^{k+1}$ \eqref{CT-1-y}. With these structural
features, the algorithm \eqref{CT-1} is also called a predictor
corrector proximal multiplier method.
Recall the Lagrangian function defined by \eqref{Aug-L-F}, the scheme \eqref{CT-1} can be rewritten as
\begin{subequations} \label{CT-2}
\begin{numcases}{\hskip-1cm\hbox{(PCPM)}}
\label{CT-2-p} p^{k+1}=y^{k}+\lambda\left(A x^{k}-z^{k}\right),\\[0.2cm]
\label{CT-2-x} x^{k+1}=\arg \min \left\{f(x)+\left\langle p^{k+1}, A x\right\rangle+\left(1 /\left(2 \lambda\right)\right)\left\|x-x^{k}\right\|^{2}\right\},\\[0.2cm]
\label{CT-2-z} z^{k+1}=\arg \min \left\{g(z)-\left\langle p^{k+1}, z\right\rangle+\left(1 /\left(2 \lambda\right)\right)\left\|z-z^{k}\right\|^{2}\right\},\\[0.2cm]
\label{CT-2-y} y^{k+1}=y^{k}+\lambda\left(A x^{k+1}-z^{k+1}\right),
\end{numcases}
\end{subequations}
The PCPM only involves evaluations of proximal operators which are for many problems in closed-form or simple to compute. As a consequence, the PCPM's subproblems may be much easier to solve than other methods based on the augmented Lagrangian function such as ALM, ADMM. Since the primal subproblems \eqref{CT-1-x} and \eqref{CT-1-z} are two
separate minimizations, this algorithm is also suitable for parallel computation. These features make the implementation of PCPM very easy. In \cite{Chen94}, it is proved that PCPM converges to the solutions of \eqref{Problem-LC}
when
\[\label{CT-size1}
\lambda \leq \frac{1}{2\max(\|A\|,1) },
\]
where $\|\cdot\|$ is the spectral norm.
Recently, there are several algorithmic frameworks are developed for unifying and analyzing a class of primal dual algorithms for the problem \eqref{Problem-LC}. These frameworks encompass the PCPM as special cases.
For example, it was shown in \cite{Solodov} that PCPM can be regarded as a special case in the hybrid inexact proximal point framework developed
in \cite{Solodov1999}. In \cite{He2012}, the PCPM is categorized into a unified framework of proximal-based decomposition methods for monotone variational inequalities. In \cite{Deng2017} it is shown that the PCPM is equivalent to a Jacobi-Proximal ADMM. The convergence results of PCPM can be reconducted by all these interpretations and the condition \eqref{CT-size1} can also be recovered.
As delineated in the literature \cite{Chen94,Shefi2014,Becker}, the parameter $\lambda$ determines the step size for solving the subproblems, and it is important to choose
appropriate value for $\lambda$ to ensure PCPM's efficiency. Intuitively, if $\lambda$ is larger, each proximal term in the subproblems could play a lighter weight in the objective and thus the variables can be updated with larger step sizes. If $\lambda$ is tiny, then it implies that the subproblems are solved conservatively with a too-small step size; and it might be not preferable from numerical perspective. Therefore, it would be desirable to consider the possibility of further relaxing the condition \eqref{CT-size1} as long as the convergence of (\ref{LALM}) can be guaranteed, so that larger values of $\lambda$ can be chosen. If this case is possible, one can expect a further speedup of the convergence of PCPM \eqref{CT-2} without additional computation. In \cite{Shefi2014}, Shefi and Teboulle showed that
PCPM can be viewed as a linearization of the quadratic penalty term in the parallel decomposition of the proximal method of multipliers and improves the condition to
\[\label{CT-size7}
\lambda \leq \frac{1}{\sqrt{2}\max(\|A\|,1) }.
\]
In the work \cite{Ma14}, by following the same analysis as in \cite{He2012}, this condition is further relaxed to
\[\label{CT-size2}
\lambda < 1 / \sqrt{\|A\|^{2}+1},
\]
which is less restrictive than the conditions \eqref{CT-size1} and \eqref{CT-size7}.
In very recent work \cite{Becker}, Becker demonstrates that the PCPM can be viewed as a preconditioned proximal point algorithm applied to the primal-dual formulation of problem \eqref{Problem-LC}, which also gets the same condition \eqref{CT-size2}.
Based on above results, it is natural to ask whether the condition \eqref{CT-size2} for the PCPM is optimal enough to ensure the the convergence, or more precisely, necessary and sufficient to ensure the the convergence. To answer this question,
we show that, the PCPM can be interpreted as a proximal ALM. As we will see in Section \ref{sect-2}, the PCPM algorithm
coincides with the linearized ALM algorithm applied to a block reformulation of \eqref{Problem-LC}. This interpretation has
interesting implications for the analysis of the PCPM algorithm. First, some known convergence results of PCPM can be easily recovered or even simplified from the proximal ALM. Second, it also allows us to translate some useful variants of the proximal ALM from the literature to the PCPM algorithm. We shall provide three generalized PCPM algorithms for the problem \eqref{Problem-LC}.
Then, motivated by recent work of indefinite proximal ALM \cite{HMY-ALM}, we show that the convergence condition of PCPM can be further relaxed by
\[\label{CT-size3}
\lambda< \frac{1}{\sqrt{\frac{3}{4}(\|A\|^2+1)}},
\]
without making further assumptions.
The paper is organized as follows. In Section \ref{sect-2}, we briefly review the proximal ALM, and establish the connection
between proximal ALM and PCPM. Then, in Section \ref{Sec:bound}, we focus on a general PCPM and discuss its optimal condition bound.
In Section \ref{sect-Ext}, we present a more general extension of the PCPM with different step sizes for the linearly constrained convex minimization model. Finally, some conclusions are drawn in Section \ref{Sec:conclusion}.
\section{ The PCPM is a Proximal ALM}\label{sect-2}
\setcounter{equation}{0}
In this section, we show that the PCPM \eqref{CT-2} is a special case of the proximal augmented Lagrangian method with a particular proximal regularization term. The convergence condition \eqref{CT-2} can be easily rediscovered by this interpretation.
\subsection{Proximal ALM}
For convenience of our demonstration, we first introduce some auxiliary variables and reformulate the problem into a block version.
Let
\[ \label{VI-P2-wF}
\w = \left(\begin{array}{c}
x\\ z\\ y\end{array} \right), \quad
\u = \left(\begin{array}{c}
x\\
z\end{array} \right), \quad M=(A,-I),
\quad \hbox{and} \quad {\boldsymbol{\theta}}(\u)=f(x) +
g(z).
\]
Then the problem \eqref{Problem-LC} can be
rewritten as
\[ \label{Problem-LC2}
\min \{\theta(\u): M\u=0\}.
\]
The augmented Lagrangian function associated with problem (\ref{Problem-LC2}) is given by
\[ \label{Aug-L-F2}
{\cal L}_{\lambda}(\u,y) ={\boldsymbol{\theta}}(\u)+\left\langle y, M\u\right\rangle+\left(\lambda /2 \right)\left\|M\u\right\|^{2},
\]
with $\lambda>0$ the penalty parameter for the linear constraints. Given a starting vector $\left(\u^{0}, y^{0}\right) \in \mathbb{R}^{n} \times \mathbb{R}^{m} \times \mathbb{R}^{m}$, the augmented Lagrangian method (ALM) originally proposed in \cite{Hes69,Powell69} for (\ref{Problem-LC}) generates iterations as
\begin{subequations} \label{ALM}
\begin{numcases}{\hbox{(ALM)\quad}}
\label{ALM-x} \u^{k+1}=\arg\min \bigl\{ {\cal L}_{\lambda}(\u,y^k)\; \big| \; \u\in\mathbb{R}^{n \times m}
\bigr\},\\
\label{ALM-l} y^{k+1}=y^{k}+\lambda \left(M\u^{k+1}\right).
\end{numcases}
\end{subequations}
The computational complexity of the ALM algorithm is dominated by the primal subproblem, so it is meaningful to discuss how to efficiently solve (\ref{ALM-x}). An interesting strategy is to regularize the primal subproblem (\ref{ALM-x}) by a quadratic proximal term and accordingly get the proximal version of ALM:
\begin{subequations} \label{PALM}
\begin{numcases}{\hbox{(Proximal ALM)\quad}}
\label{PALM-x} \u^{k+1} =\arg\min \bigl\{{\cal L}_{\lambda}(\u,y^k) +\frac{1}{2}\|\u-\u^k\|_{\cal P}^2\; \big| \; \u\in\mathbb{R}^{n \times m}\bigr\},\\
\label{PALM-l} y^{k+1}=y^{k}+\lambda \left(M\u^{k+1}\right).
\end{numcases}
\end{subequations}
In (\ref{PALM-x}), $\frac{1}{2}\|\u-\u^k\|_{\cal P}^2$ is the quadratic proximal regularization term and ${\cal P}$ is the proximal matrix that is usually required to be positive definite in the literature.
Typically, we can linearize the augmented term by choosing appropriate ${\cal P}$ as
\[\label{ALM-P}
{\cal P} = \eta I-\lambda M^{\top}M.
\]
At this case, the primal subproblem (\ref{PALM-x}) is specified as
\[\label{ALM-x-4}
\u^{k+1} =\arg\min \bigl\{ {\boldsymbol{\theta}}(\u) +\frac{\eta }{2}\|\u - \u^k-\frac{1}{\eta }M^{\top} \bigl(y^k +\lambda(M\u^k)\big)\|^2\; \big| \; \u\in\mathbb{R}^{n \times m}\},
\]
which amounts to estimating the proximity operator of ${\boldsymbol{\theta}}(\u)$. The implementation for such cases is usually simple.
Hence, the linearized ALM, which is a special case of the proximal ALM (\ref{PALM}) with ${\cal P}$ given in \eqref{ALM-P}, reads as
\begin{subequations} \label{LALM}
\begin{numcases}{\hskip-1cm\hbox{(Linearized ALM)}}
\label{LALM-x} \u^{k+1} =\arg\min \bigl\{ {\boldsymbol{\theta}}(\u) +\frac{\eta }{2}\|\u - \u^k-\frac{1}{\eta }M^{\top} \bigl(y^k +\lambda(M\u^k)\big)\|^2\; \big| \; \u\in\mathbb{R}^{n \times m }\},\ \\
\label{LALM-l} y^{k+1}=y^{k}+\lambda \left(M\u^{k+1}\right).
\end{numcases}
\end{subequations}
For the linearized ALM (\ref{LALM}) in the literature, the parameter $\eta$ is required to satisfy the condition $\eta>\lambda \|M\|$ so as to ensure the positive definiteness of the matrix ${\cal P}$ given in (\ref{ALM-P}) and hence the convergence of (\ref{LALM}). We refer to \cite{YangYuan,HMY-ALM} for the detail of convergence analysis of the linearized ALM (\ref{LALM}).
\subsection{Proximal ALM Perspective}
In this subsection, we show the PCPM \eqref{CT-2} is a special case of the proximal ALM \eqref{PALM}. This will be done by simple algebraic manipulation and simplification.
First, substituting $p^{k+1}$ \eqref{CT-2-p} into the update for $x^{k+1}$ \eqref{CT-2-x} and $z^{k+1}$ \eqref{CT-2-z},
the primal iterations are
\[
\label{CT-3-p}
x^{k+1}=\arg \min \left\{f(x)+ \left\langle y^{k}+\lambda\left(A x^{k}-z^{k}\right), Ax\right\rangle+\left(1 /\left(2 \lambda\right)\right)\left\|x-x^{k}\right\|^{2}\right\},
\]
and
\[
\label{CT-3-z} z^{k+1}=\arg \min \left\{g(z)- \left\langle y^{k}+\lambda\left(A x^{k}-z^{k}\right), z\right\rangle +\left(1 /\left(2 \lambda\right)\right)\left\|z-z^{k}\right\|^{2}\right\},
\]
respectively. Note that the update $p$ is eliminated. We then have
\begin{eqnarray} \label{CT-3-x-z1}
( x^{k+1},z^{k+1})=&\arg \min &\left\{f(x)+g(z)+\left\langle y^{k}+\lambda\left(A x^{k}-z^{k}\right), Ax-z\right\rangle \right. \nn\\
&& \left.+\left(1 /\left(2 \lambda\right)\right)\left\|x-x^{k}\right\|^{2}+\left(1 /\left(2 \lambda\right)\right)\left\|z-z^{k}\right\|^{2}\right\}\nn\\
=&\arg \min &\left\{f(x)+g(z)+\left\langle y^{k}, Ax-z\right\rangle+\left(\lambda /2 \right)\left\|Ax-z\right\|^{2} \right. \nn\\
&& +\left(1 /\left(2 \lambda\right)\right)\left\|x-x^{k}\right\|^{2}+\left(1 /\left(2 \lambda\right)\right)\left\|z-z^{k}\right\|^{2} \nn\\
&& \left.-\left(\lambda /2 \right)\left\|Ax-z\right\|^{2}+\left\langle \lambda \left(A x^{k}-z^{k}\right), Ax-z \right\rangle \right\}. \nn\\
=&\arg \min &\left\{f(x)+g(z)+\left\langle y^{k}, Ax-z\right\rangle+\left(\lambda /2 \right)\left\|Ax-z\right\|^{2} \right. \nn\\
&& +\left(1 /\left(2 \lambda\right)\right)\left\|x-x^{k}\right\|^{2}+\left(1 /\left(2 \lambda\right)\right)\left\|z-z^{k}\right\|^{2} \nn\\
&& \left.-\left(\lambda /2 \right)\left\| \left(Ax-z\right)- \left(A x^{k}-z^{k}\right)\right\|^{2}+\left(\lambda /2 \right)\left\|Ax^{k}-z^{k}\right\|^{2} \right\}.
\end{eqnarray}
Ignoring
some constant terms in the minimization problem of the last equality, we have
\begin{eqnarray} \label{CT-3-x-z2}
( x^{k+1},z^{k+1})=&\arg \min &\left\{f(x)+g(z)+\left\langle y^{k}, Ax-z\right\rangle+\left(\lambda /2 \right)\left\|Ax-z\right\|^{2} \right. \nn\\
&& +\left(1 /\left(2 \lambda\right)\right)\left\|x-x^{k}\right\|^{2}+\left(1 /\left(2 \lambda\right)\right)\left\|z-z^{k}\right\|^{2} \nn\\
&& \left.-\left(\lambda /2 \right)\left\| A\left(x-x^{k}\right)- \left(z-z^{k}\right)\right\|^{2} \right\}.
\end{eqnarray}
The above scheme can be represented as
\begin{subequations} \label{CT-3-x-z}
\[ \label{CT-3-x-z3}
( x^{k+1},z^{k+1})=\arg \min \left\{f(x)+g(z)+\left\langle y^{k}, Ax-z\right\rangle+\left(\lambda /2 \right)\left\|Ax-z\right\|^{2} +\frac{1}{2}\left\|\left(\begin{array}{c} x-x^{k}\\ z-z^{k}\end{array}\right)\right\|_{\cal P}^{2} \right\},\]
with
\[\label{CT-P}
{\cal P}=\left(\begin{array}{cc}
\frac{1}{\lambda}I- \lambda A^{\top}A & \lambda A^{\top} \\
\lambda A & (\frac{1}{\lambda}-\lambda)I \end{array} \right).
\]
\end{subequations}
Recall \eqref{PALM} and the definitions in \eqref{VI-P2-wF}, we can see the algorithm consits of \eqref{CT-3-x-z} and \eqref{CT-2-y}
is a special proximal ALM.
To summarize, the PCPM \eqref{CT-2} is interpreted as a proximal ALM applied to problem \eqref{Problem-LC2}. This is different from the result in \cite{Becker},
in which the algorithm is interpreted as the preconditioned proximal point algorithm applied to a primal-dual
reformulation of the original problem \eqref{Problem-LC}.
\begin{remark}
Let us take a deeper look at the regularization matrix ${\cal P}$ in \eqref{CT-P} and derive a simpler representation of it. We have
\begin{eqnarray} \label{Matrix-G-P}
{\cal P} &= &\left(\begin{array}{cc}
\frac{1}{\lambda}I_n- \lambda A^{\top}A & \lambda A^{\top} \\
\lambda A & (\frac{1}{\lambda}-\lambda)I_m \end{array} \right) \nn \\
&= & \frac{1}{\lambda}I-\lambda\cdot \left(\begin{array}{cc}
A^{\top}A & -A^{\top} \\
- A & I \end{array} \right) \nn \\
& \stackrel{\eqref{VI-P2-wF}}{=} & \frac{1}{\lambda}I-\lambda \cdot M^{\top}M.
\end{eqnarray}
\end{remark}
We can see that here ${\cal P}$ is a special case of \eqref{ALM-P} where $\eta= \frac{1}{\lambda}.$ Hence, the PCPM \eqref{CT-2} can be further interpreted as a linearized ALM with ${\cal P}$ given by \eqref{CT-P}.
\begin{remark}
The step size condition \eqref{CT-size2} can be easily obtained by this proximal ALM's perspective. Since the proximal regularization matrix ${\cal P}$ is usually required to be positive definite, we have
\[\label{CT-P2}
{\cal P}=\frac{1}{\lambda}I-\lambda \cdot M^{\top}M \succ 0.
\]
It only remains to ensure
\[
\lambda^2 <\frac{1}{\|M^{\top}M\|}.
\]
Recall $M$ defined in \eqref{VI-P2-wF}, the condition reduces to $\lambda^2(\|A\|^2+1)<1$. Thus the step size condition \eqref{CT-size2} is obtained.
\end{remark}
\subsection{Two Generalized PCPM}
In this section, we present two generalized PCPM. The first one is developed by following the idea of relaxing ALM. The second is developed by viewing it as a variant of PPA.
In order to accelerate the convergence of ALM or the proximal ALM, one practical strategy is to attach a relaxation
factor to the Lagrange-multiplier-updating step in the algorithm. For the proximal ALM, the relaxed scheme is
\begin{subequations} \label{PALM2}
\begin{numcases}{\hbox{\quad}}
\label{PALM2-x} \u^{k+1} =\arg\min \bigl\{{\cal L}_{\lambda}(\u,y^k) +\frac{1}{2}\|\u-\u^k\|_{\cal P}^2\; \big| \; \u\in\mathbb{R}^{n \times m}
\bigr\},\\
\label{PALM2-l} y^{k+1}=y^{k}+\gamma\lambda \left(M\u^{k+1}\right).
\end{numcases}
\end{subequations}
where the relaxation
factor $\gamma$ can be chosen in the interval $(0,2)$, Recall that the proximal ALM (\ref{PALM}) is a special case of (\ref{PALM2}) with $\gamma=1$. Numerically, an overrelaxation choice $\gamma\in [1.5, 1.8]$ can usually lead to faster convergence; see some
numerical results in \cite{Ma2019}.
Since Chen-Teboulle's algorithm is a proximal ALM, we can relax its dual step size as the proximal ALM and get the following relaxed algorithm.
\begin{subequations} \label{CT-4}
\begin{numcases}{\hskip-1cm\hbox{(G-PCPM-I)}}
\label{CT-4-p} p^{k+1}=y^{k}+\lambda\left(A x^{k}-z^{k}\right),\\[0.2cm]
\label{CT-4-x} x^{k+1}=\arg \min \left\{f(x)+\left\langle p^{k+1}, A x\right\rangle+\left(1 /\left(2 \lambda\right)\right)\left\|x-x^{k}\right\|^{2}\right\},\\[0.2cm]
\label{CT-4-z} z^{k+1}=\arg \min \left\{g(z)-\left\langle p^{k+1}, z\right\rangle+\left(1 /\left(2 \lambda\right)\right)\left\|z-z^{k}\right\|^{2}\right\},\\[0.2cm]
\label{CT-4-y} y^{k+1}=y^{k}+\gamma\lambda\left(A x^{k+1}-z^{k+1}\right),
\end{numcases}
\end{subequations}
where $\gamma\in(0,2)$.
In the PPA literature, it is commonly known that the PPA scheme can be relaxed, i.e., we can generate the new iterate by relaxing the output of the original PPA appropriately. This is usually based on combining the output of the operation with the former iterate. On the other hand, the proximal ALM can be interpreted as a type of preconditioned proximal point algorithm (PPA), see \cite{Gu2014} for details. We refer to \cite{Becker} for direct
discussions on the interpretation. Hence, the PCPM, as a special PPA, can also be generalized. More
precisely, let the output point of \eqref{CT-2} be denoted by $\tilde{\w}^k$, then the relaxed PCPM yields the
new iterate via
\begin{subequations} \label{PSALM-A}
\begin{numcases}{\hskip-1cm\hbox{(G-PCPM-II)}}
\nonumber\\[-0.1cm]
\label{PSALMA-X}
\! \left\{\begin{array}{l}
\label{CT-5-p} p^{k+1}=y^{k}+\lambda\left(A x^{k}-z^{k}\right),\\[0.2cm]
\label{CT-5-x} \tilde{x}^{k}=\arg \min \left\{f(x)+\left\langle p^{k+1}, A x\right\rangle+\left(1 /\left(2 \lambda\right)\right)\left\|x-x^{k}\right\|^{2}\right\},\\[0.2cm]
\label{CT-5-z} \tilde{z}^{k}=\arg \min \left\{g(z)-\left\langle p^{k+1}, z\right\rangle+\left(1 /\left(2 \lambda\right)\right)\left\|z-z^{k}\right\|^{2}\right\},\\[0.2cm]
\label{CT-5-y} \tilde{y}^{k}=y^{k}+\lambda\left(A x^{k+1}-z^{k+1}\right),
\end{array} \right. \\[0.2cm]
\label{PSALMA-L}
\;\; {\w}^{k+1} = {\w}^k - \gamma\Bigl({\w}^k- \tilde{\w}^{k}\Bigr),
\end{numcases}
\end{subequations}
where $\gamma \in(0,2)$ is the relaxation factor. In particular, $\gamma$ is called an under-relaxation factor when $\gamma \in(0,1)$ or over-relaxation factor when $\gamma \in(1,2)$; and the relaxed G-PCPM-II \eqref{PSALM-A} reduces to the original PCPM \eqref{CT-2} when $\gamma=1$.
\section{Optimal bound on Step Sizes}\label{Sec:bound}
\setcounter{equation}{0}
In this section, with the proximal ALM interpretation, we shall show that
the step size parameters $\lambda$ and $\gamma$ in the generalized PCPM \eqref{CT-4} can be related by the formula
\[\label{new-bound}
\lambda< \frac{1}{\sqrt{\frac{2+\gamma}{4}(\|A\|^2+1)}}.
\]
Note that when $\gamma=1$, G-PCPM-I \eqref{CT-4} reduces to PCPM and the above condition reduces to $\lambda< \frac{1}{\sqrt{\frac{3}{4}(\|A\|^2+1)}}$ which improves the results in the works \cite{Chen94,Ma14,Becker} to ensure the convergence.
The new bound \eqref{new-bound} relies on the convergence results studied in \cite{HMY-ALM} for the proximal ALM. Here, we describe
the main results for the proximal ALM in \cite{HMY-ALM} by the following presentation,
but omit the proof.
\begin{center}
\fbox{
\begin{minipage}{16cm}
In \cite{HMY-ALM}, the authors showed that for the proximal ALM
\begin{subequations} \label{LP-ALM}
\begin{numcases}{\hbox{(IDP-ALM)\quad}}
\label{LP-ALM-x} \u^{k+1} =\arg\min \bigl\{{\cal L}_{\lambda}(\u,\lambda^k) +
\frac{1}{2}\|\u-\u^k\|_{{\cal P}}^2 \; \big| \; \u\in\mathbb{R}^{l}
\bigr\},\\
\label{LP-ALM-l}y^{k+1} = y^k +\gamma \lambda(M\u^{k+1}), \quad \gamma \in (0,2),
\end{numcases}
\end{subequations}
the proximal matrix ${\cal P}$ in the term $\frac{1}{2}\|\u-\u^k\|_{{\cal P}}^2$ can be indefinite without any further assumptions. In particular, Let ${\cal P}$ be specified by the structure
\begin{subequations}\label{D0-L}
\[
{\cal P}={\cal D}-(1-\tau)\lambda M^{\top}M,
\]
where ${\cal D}$ is an arbitrarily positive definite matrix in $\mathbb{R}^{l}$. Then when
\[\label{taugamma}
\tau > \frac{2+\gamma}{4},
\]
\end{subequations}
the IDP-ALM \eqref{LP-ALM} converges globally to a
solution of \eqref{Problem-LC2}.
\end{minipage} }
\end{center}
Since the proximal ALM contains PCPM as a special case, the proximal matrix ${\cal P}$ defined in \eqref{CT-3-x-z} can employ indefinite setting according to above results. To do this, we just need to choose $\lambda $ to guarantee
\[\label{D0-L2}
{\cal D}={\cal P}+(1-\tau)\lambda M^{\top}M\succ 0, \quad \hbox{for}\; \tau\in \Bigl(\displaystyle\frac{2+\gamma}{4}, 1\Bigr).
\]
To fulfill (\ref{D0-L2}), notice that
\begin{eqnarray} \label{D0-L1}
{\cal D}&\stackrel{\eqref{Matrix-G-P}}{=} &\frac{1}{\lambda}I-\lambda \cdot M^{\top}M+ (1-\tau)\lambda M^{\top}M\\
&=&\frac{1}{\lambda}I-\tau\lambda M^{\top}M.
\end{eqnarray}
Recall $M$ defined in \eqref{VI-P2-wF}. Then we have
\[\label{CT-new-size1}
\lambda< \frac{1}{\sqrt{\tau(\|A\|^2+1)}} \quad \Rightarrow \quad {\cal D} \succ 0.
\]
Note that $\tau\in \Bigl(\displaystyle\frac{2+\gamma}{4}, 1\Bigr)$ is arbitrary, we have
\[\label{CT-new-size2}
\lambda< \frac{1}{\sqrt{\frac{2+\gamma}{4}(\|A\|^2+1)}}.
\]
In Figure 1, we plot the evolutions of the step size $\lambda$ with respect
to the norm $\|A\|$ for the three stepsize conditions of the PCPM. The ratios of the two step sizes \eqref{CT-size2} and \eqref{CT-size3} to \eqref{CT-size1} are displayed in Figure 2. These plots show that the stepsize condition is enlarged, and a larger value of $\lambda$ seems more preferable in practice because it can yield
a larger step size .
\begin{figure}
\centering
\begin{minipage}[c]{.7\textwidth}
\centering
\includegraphics[width =\textwidth]{Figure_1} \\
\end{minipage}
\caption{The curves of three different step sizes.} \label{Figure-MCP}
\end{figure}
\begin{figure}
\centering
\begin{minipage}[c]{.7\textwidth}
\centering
\includegraphics[width =\textwidth]{Figure_2} \\
\end{minipage}
\caption{Illustration of the step size ratio.} \label{Figure-MCP2}
\end{figure}
\begin{remark}
In \cite{HMY-ALM}, a counterexample was given showing that the convergence condition \eqref{D0-L} is optimal for the proximal ALM. Since PCPM is a special case of the proximal ALM, the new bound \eqref{CT-new-size2} thus can not be further improved if we don't add any further assumptions on the model.
\end{remark}
\section{Further Extension}\label{sect-Ext}
\setcounter{equation}{0}
Technically, we can extend the original PCPM scheme and its variants to handle
the following general problem
\[ \label{Problem-LC3}
\min \{f(x)+g(z): A x +Bz=b\},
\]
where $A\in \mathbb{R}^{m \times n}, B\in \mathbb{R}^{m \times l}$, $f: \mathbb{R}^{n} \mapsto(-\infty,+\infty)$ and $g: \mathbb{R}^{l} \mapsto(-\infty,+\infty)$ are given closed proper convex functions.
To solve \eqref{Problem-LC3}, instead of \eqref{CT-2} we propose the following generalized PCPM
\begin{subequations} \label{CT-G3}
\begin{numcases}{\hskip-1cm\hbox{(G-PCPM-III) }}
p^{k+1}=y^{k}+\lambda\left(A x^{k}+Bz^{k}-b\right),\\[0.2cm]
x^{k+1}=\arg \min \left\{f(x)+\left\langle p^{k+1}, A x\right\rangle+\left(1 /\left(2 \tau\right)\right)\left\|x-x^{k}\right\|^{2}\right\},\\[0.2cm]
z^{k+1}=\arg \min \left\{g(z)+\left\langle p^{k+1}, Bz\right\rangle+\left(1 /\left(2 \sigma\right)\right)\left\|z-z^{k}\right\|^{2}\right\},\\[0.2cm]
y^{k+1}=y^{k}+\gamma\lambda\left(A x^{k+1}+Bz^{k+1}-b\right),
\end{numcases}
\end{subequations}
where $\lambda$ is the proximal parameter for the dual regularization; $\tau,\sigma$ are two different positive parameters for the primal regularization; the relaxation factor $\gamma \in(0,2)$.
Now, we discuss how to drive the step size condition to ensure the convergence of PCPM-III \eqref{CT-G3}. We follow the line of analysis in Section \ref{sect-2}. Let us transform the G-PCPM-III \eqref{CT-G3} into an equivalent proximal ALM with a special matrix ${\cal P}$.
Ignoring
some constant terms in the minimization problem of the last equality, we have
\begin{eqnarray} \label{CT-3-x-z5}
( x^{k+1},z^{k+1})=&\arg \min &\left\{f(x)+g(z)+\left\langle y^{k}, A x +Bz-b\right\rangle+\left(\lambda /2 \right)\left\|A x +Bz-b\right\|^{2} \right. \nn\\
&& +\left(1 /\left(2 \tau\right)\right)\left\|x-x^{k}\right\|^{2}+\left(1 /\left(2 \sigma\right)\right)\left\|z-z^{k}\right\|^{2} \nn\\
&& \left.-\left(\lambda /2 \right)\left\| A\left(x-x^{k}\right)+B\left(z-z^{k}\right)\right\|^{2} \right\}.
\end{eqnarray}
The above scheme can be represented as
\begin{subequations} \label{CT-3-x-6}
\begin{eqnarray}\label{CT-3-x-z7}
( x^{k+1},z^{k+1})=&\arg \min & \left\{f(x)+g(z)+\left\langle y^{k}, A x +Bz-b\right\rangle\right.
\nn\\
&&\left.+\left(\lambda /2 \right)\left\|A x +Bz-b\right\|^{2} +\frac{1}{2}\left\|\left(\begin{array}{c} x-x^{k}\\ z-z^{k}\end{array}\right)\right\|_{\cal P}^{2} \right\},
\end{eqnarray}
with
\[\label{CT-III-P}
{\cal P}=\left(\begin{array}{cc}
\frac{1}{\tau}I- \lambda A^{\top}A & -\lambda A^{\top}B \\
-\lambda B^{\top}A & \frac{1}{\sigma}I-\lambda B^{\top}B \end{array} \right).
\]
\end{subequations}
Hence, the algorithm G-PCPM-III can be equivalently presented as
\begin{subequations}
\begin{numcases}{\hbox{\quad}}
( x^{k+1},z^{k+1})= \arg \min \left\{f(x)+g(z)+\left\langle y^{k}, A x +Bz-b\right\rangle\right.
\nn\\
\qquad\qquad\qquad\qquad\qquad\left.+\left(\lambda /2 \right)\left\|A x +Bz-b\right\|^{2} +\frac{1}{2}\left\|\left(\begin{array}{c} x-x^{k}\\ z-z^{k}\end{array}\right)\right\|_{\cal P}^{2} \right\}, \\[0.2cm]
y^{k+1}=y^{k}+\gamma\lambda\left(A x^{k+1}+Bz^{k+1}-b\right),
\end{numcases}
\end{subequations}
where ${\cal P}$ is given by \eqref{CT-III-P}.
We can see that
\begin{eqnarray} \label{Matrix-G-P2}
{\cal P}
&= & \lambda \left(\begin{array}{cc}
\frac{1}{\tau\lambda }I- A^{\top}A & -A^{\top}B \\
-B^{\top}A & \frac{1}{\sigma\lambda }I- B^{\top}B \end{array} \right)\nn \\
&= &
\lambda \left(\begin{array}{cc}
\frac{1}{\tau\lambda}I & 0 \\
0 & \frac{1}{\sigma\lambda}I \end{array} \right) -\lambda\cdot \left(\begin{array}{cc}
A^{\top}A & A^{\top}B \\
B^{\top}A & B^{\top}B \end{array} \right).
\end{eqnarray}
Since the proximal regularization matrix ${\cal P}$ is usually required to be positive definite, we need to ensure
\[ \left(\begin{array}{cc} \frac{1}{\tau\lambda}I & 0 \\
0 & \frac{1}{\sigma\lambda}I \end{array} \right) \succ \left(\begin{array}{cc}
A^{\top}A & A^{\top}B \\
B^{\top}A & B^{\top}B \end{array} \right)
\]
or
\[ I \succ \left(\begin{array}{cc} \sqrt{\tau\lambda}I & 0 \\
0 & \sqrt{\sigma\lambda}I \end{array} \right) \left(\begin{array}{cc}
A^{\top}A & A^{\top}B \\
B^{\top}A & B^{\top}B \end{array} \right)\left(\begin{array}{cc} \sqrt{\tau\lambda}I & 0 \\
0 & \sqrt{\sigma\lambda}I \end{array} \right).
\]
Note that $ \left(\begin{array}{cc}
A^{\top}A & A^{\top}B \\
B^{\top}A & B^{\top}B \end{array} \right)=\left(\begin{array}{c}
A^{\top}\\
B^{\top}\end{array} \right)(A, B) $. We just need to guarantee
\[ I \succ \left(\begin{array}{c}
\sqrt{\tau\lambda} A^{\top}\\
\sqrt{\sigma\lambda}B^{\top}\end{array} \right)(\sqrt{\tau\lambda} A, \sqrt{\sigma\lambda}B) .
\]
So if the step size parameters satisfy
\[
\lambda\tau\left\|A^{\top} A\right\| + \lambda\sigma \left\|B^{\top} B\right\| <1,
\]
then the positive definiteness of matrix $P$ is ensured.
According to the improved convergence result of the proximal ALM \eqref{D0-L}, the proximal matrix ${\cal P}$ can employ indefinite setting. At this case, the condition is relaxed by
\[\label{general-condition}
\lambda\tau\left\|A^{\top} A\right\| + \lambda\sigma \left\|B^{\top} B\right\| <\frac{4}{2+\gamma}.
\]
If all the parameters $\lambda,\tau,\sigma$ are chosen to be equal, i.e.,
$\lambda=\tau=\sigma$. The resulting condition \eqref{general-condition} reduces to \eqref{CT-new-size2}.
Since asymptotically
this extension has no difference from the PCPM, we skip the detailed analysis for this scheme.
\section{Concluding Remarks}\label{Sec:conclusion}
In this paper, we study the predictor
corrector proximal multiplier method (PCPM) for convex programming problems, and show that it is equivalent to a linearized augmented Lagrangian method (ALM) with a special regularization term. This interpretation makes it possible to simplify the convergence analysis, and we can further relax the step size condition of PCPM by invoking recent improved convergence study of the proximal ALM. It must be mentioned that our result does not rely on any further assumptions of the problems or algorithms.
Since the linearized ALM is extremely popular in recent years, various
variants and theoretical results have been developed and studied in the literature. Based on our interpretation, these modifications and theoretical results can be easily injected into the PCPM. Thus,
Our analysis builds on the techniques and recent results of the proximal ALM and gives some insight of PCPM.
|
2,869,038,156,411 | arxiv | \section{Introduction} \label{Sec: Introduction}
Since the discovery of the J-band of pseudoisocyanine
(PIC)~\cite{Jelley1937,Scheibe1937}, the study of the collective
optical properties of molecular aggregates has received much
attention. The collective nature of the excitations gives rise to
narrow absorption lines (exchange narrowing) and ultra-fast
spontaneous emission. The development of novel optical techniques, a
continuously increasing theoretical understanding of the optical
response of these systems, and the realization that this type of
excitations play an important role in natural light harvesting
systems have kept these materials in the spotlight.
During the past 15 years a topic of particular interest has been the
effect of multi-exciton states on nonlinear optical spectroscopies.
Amongst the latter are the nonlinear absorption spectrum
\cite{Spano1989,Spano1991}, photon echoes
\cite{Spano1992,Burgel1995}, pump-probe spectroscopy
\cite{Burgel1995,Fidder1993,Bakalis1999}, and two-dimensional
spectroscopy \cite{Brixner2006,Heijs2007}. From a theoretical point
of view, accounting for multi-exciton states requires the handling
of large matrices, due to the extent of the associated Hilbert
space. It is well-known, however, that for homogeneous linear
aggregates, three states dominate the third-order response, namely
the ground state, the lowest one-exciton state and the lowest
two-exciton state. In practice, J-aggregates suffer from appreciable
disorder, which leads to localization of the exciton states. It was
shown in Ref.~\cite{Fidder1993} that the three-state picture still
holds to a good approximation, provided that one replaces the chain
length by the typical exciton localization length. The existence of
the so-called hidden structure of the Lifshits tail of the density
of states (DOS) justifies this
approach~\cite{Malyshev1995,Malyshev2001}.
In this paper, we put a firmer basis under the above idea, by
systematically analyzing the dominant ground-state to one-exciton
transitions and one- to two-exciton transitions in disordered linear
aggregates. By comparison to exact spectra, we show that, indeed,
the picture of three dominant states per localization segment holds.
\section{Model} \label{Sec: Model}
We model a single aggregate as a linear chain of $N$ coupled two-level
monomers with parallel transition dipoles. We assume that the aggregate
interacts with a disordered environment, resulting in random fluctuations
in the molecular
transition energies $\varepsilon_n$ (diagonal disorder), and restrict
ourselves to nearest neighbor excitation transfer interactions $-J$.
The optical excitations are described by the Frenkel exciton Hamiltonian,
\begin{equation}
H = \sum_{n=1}^N \varepsilon_n |n\rangle \langle n| -
J\sum_{n=1}^{N-1}\left( |n\rangle \langle n+1| + h.c.\right).
\label{H}
\end{equation}
Here, $|n \rangle$ denotes the state with the $n$th site excited and
all other sites in the ground state. The monomer excitation energies
$\varepsilon_n$ are modeled as uncorrelated Gaussian variables with
zero mean and standard deviation $\sigma$. For J-aggregates $J > 0$.
Numerical diagonalization of the Hamilton yields the exciton
energies $\varepsilon_\nu$ ($\nu = 1,\ldots , N$) and exciton
wavefunctions $ |\nu\rangle = \sum_{n=1}^N \varphi_{\nu n}|n\rangle$
of the one-exciton states, where $\varphi_{\nu n}$ is the $n$th
component of the wavefunction $|\nu\rangle$. The two exciton states
are given by the Slater determinant of two different one-exciton
states $|\nu_1\rangle$ and $|\nu_2\rangle$~\cite{Spano1991}: $|
\nu_1,\nu_2 \rangle = \sum_{n\leq m}^N (\phi_{\nu_1n}\phi_{\nu_2 m}-
\phi_{\nu_1 m}\phi_{\nu_2n})| n,m\rangle$ with $|n,m\rangle$ the
state in which the sites $n$ and $m$ are excited and all other
sites are in the ground state. The corresponding two-exciton energy
is given by $E_{\nu_1\nu_2} = E_{\nu_1}+E_{\nu_2}$.
\section{Selecting dominant transitions}
The physical size of a molecular aggregate can amount to thousands of monomers,
but the disorder localizes the exciton states~\cite{Abrahams1979}. For
J-aggregates a small number of states at the bottom of the one-exciton band
contain almost all the oscillator strength. The states are localized on
segments with a typical extension $N^*$, called the localization length,
which depends on the magnitude of the disorder~\cite{Fidder1991}. The
wavefunctions of these states overlap weakly and consist of (mainly) a
single peak (they have no node within the localization segment), see
Fig.~\ref{Fig: Reduced wavefunctions}. For the remainder of this paper we
will refer to these states as $s$ states, and $\mathcal{S}$ will denote
the set of $s$ states.
\begin{figure}[lh]
\begin{center}
\includegraphics[width = \textwidth,scale=1]{DCP07_Figure_Wavefunctions_sp.eps}
\end{center}\caption{}\label{Fig: Reduced wavefunctions}
\caption{(a)~The lowest 12 one-exciton states of a chain of length
$N=500$ for a particular disorder realization. (b)~A subset of $s$
states (black) and $p$ states (gray) that mostly contribute to the
one-to-two exciton transitions. }
\end{figure}
From the complete set of wavefunctions we select the $s$ states
using the selection rule proposed in Ref.~\cite{Malyshev2001},
$\big|\sum_n \varphi_{\nu n} |\varphi_{\nu n}|\big| \ge C_0$. For a
disorder-free the lowest state contains 81\% of the total
oscillator strength between the ground state and the one-exciton
band~\cite{Fidder1991}. Numerically we found that the $s$ states
selected by taking $C_0=0.75$ (for $0.05 J < \sigma<0.3 J$) together
contain 76\% of the total oscillator strength: ${\langle \sum_{s\in
\mathcal{S}}\mu_{s}^2\rangle}/{N} \approx 0.76$. Here $\mu_{s}$ is
the transition dipole moment from the ground state to the state
$|s\rangle$. In Fig.~\ref{Fig: Absorption}(a) we show the absorption
spectrum calculated only with the $s$ states and compare it to the
exact spectrum.
We observe that the $s$ states give a good representation of this
spectrum, except for its blue wing, where higher-energy exciton
states, which contain one node within the localization segment,
contribute as well~\cite{Malyshev2007,Klugkist2007}. These
so-called $p$ states may be identified as the second one-exciton
state on a localization segment~\cite{Malyshev1995,Malyshev2001}.
The $p$ states play a crucial role in the third-order response. In
order to analyze this, we have considered two-exciton states
$|s,p\rangle$ given by the Slater determinant of a given $s$ state
(selected as described above) with all other one-exciton states $\nu
\notin \mathcal{S}$ (the two-exciton state consisting of two $s$
type states localized on different segments do not contribute to the
nonlinear response), and calculate the corresponding transition
dipole moments $\mu_{s\nu,s}$. From the whole set of
$\mu_{s\nu,s}$, we select the largest one, denoted by
$\mu_{sp_s,s}$, were the substrict $s$ in $p_s$ indicates its
relation with the state $|s\rangle$. It turns out that the
one-exciton state $|p_s\rangle$ selected in this way is localized
on the same segment as the state $|s\rangle$. Several of these
doublets of $s$ and $p$ states are shown in Fig.~\ref{Fig: Reduced
wavefunctions}(b). The partners of the lowest $s$ states indeed look
like $p$ states, having a well-defined node within the localization
segment. They form the hidden structure of the Lifshits tail of the
DOS~\cite{Malyshev1995} we mentioned above. For higher lying $s$
states, these partners (not shown) are more delocalized and often do
not have a $p$-like shape.
The average ratio of the oscillator strength of the transitions
$|0\rangle\to |s\rangle$ and $|s\rangle\to |p_s\rangle$ turned out
to be $\left\langle {\mu_{sp_s,s}^2}/{\mu_{s}^2}\right\rangle
\approx 1.4$. For the dominant ground-to-one and one-to-two exciton
transition in a homogeneous chain, this ratio reads
${\mu_{12,1}^2}/{\mu_{1}^2} \approx 1.57$. This comparison suggests
that our selection of two-exciton states well captures the dominant
one-to-two exciton transitions in disordered chains. We have also
found that the energy separation between the states $|s\rangle$ and
$|p_s\rangle$ obeys $
\left\langle E_{s}-{E_{p_s}}\right\rangle \approx 1.35 \times{ 3\pi^2J}/{N^{*2}}
$, with $N^* = \left\langle{\mu_s^2}/{0.81}\right\rangle$. This
resembles the level spacing of a homogeneous chain: $E_{2} -{E_1}
\approx{3\pi^2J}/{N^2}$, confirming that the separation between the
one-exciton bleaching peak and the one-to-two-exciton induced
absorption peak may be used to extract the typical localization size
from experiment~\cite{Bakalis1999}.
\begin{figure}[lh]
\begin{center}
\includegraphics[width = \textwidth,scale=1]{DPC07_Figure_Absorption_PP.eps}
\end{center}\caption{}\label{Fig: Absorption}
\caption{ (a)~Absorption spectrum due to $s$ states (circles)
compared to the exact one (solid line). (b)~Pump-probe spectrum due
to the selected $s$ and $p$ type states (circles) compared to the
exact one (solid line). The spectra were calculated for $\sigma=0.1
J$ (top) and $\sigma=0.3 J$ (bottom).}
\end{figure}
To illustrate how the selected transitions, involving the ($s$,$p$)
doublets, reproduce the optical response of the aggregate, we have
calculated the pump-probe spectrum at zero temperature using only
these transitions \cite{Heijs2007},\begin{equation}\label{Eq: PP}
P(\omega)=
\bigg\langle-2\mu_{s}^4\delta(\omega_{s}-\omega)+
\mu_{s}^2\mu_{p_s}^2\delta(\omega_{p_s}-\omega)\bigg\rangle,
\end{equation}
and compared the result to the exact spectrum, see Fig.~\ref{Fig:
Absorption}(b). Apart from a blue wing in the induced absorption
part of the spectrum (the positive peak), the selected transitions
reproduce the exact pump-probe spectrum very well. In particular the
separation between the bleaching and induced absorption peaks is
practically identical to the exact one.
\section{Conclusion}
We have shown that the third-order optical response of disordered
linear J-aggregates is dominated by a very limited number of
transitions. Considering only these transitions enormously reduces
the computational effort necessary to simulate nonlinear
experiments. We have used the procedure outlined above to calculate
the optical bistable response of a thin film of J-aggregates, taking
into account the one-to-two exciton transitions \cite{Klugkist2007}.
|
2,869,038,156,412 | arxiv |
\section{Introduction}
The most classical spectral isoperimetric inequality states that
among all planar domains of a given perimeter,
the disc induces the lowest principal
eigenvalue for the Dirichlet Laplacian.
This statement follows from the famous
Faber-Krahn inequality~\cite{Faber_1923, Krahn_1924} via a simple scaling argument.
In this paper we focus on related spectral isoperimetric properties
for the principal eigenvalues of the two-dimensional Schr\"o\-dinger operator with a $\delta$-interaction supported on an open arc and of the Robin Laplacian
on a plane with a slit.
First, we discuss the results for Schr\"odinger operators with $\delta$-interactions. To this aim, let $\Sigma \subset \mathbb{R}^2$ be any smooth compact closed or non-closed curve; \emph{cf.}~Section~\ref{Sec.pre} for details. Given a real number~$\alpha > 0$, consider the spectral problem for the self-adjoint operator $H_{\delta,\alpha}^\Sigma$ corresponding via the first representation theorem to the closed, densely defined, symmetric, and semi-bounded quadratic form in $L^2(\dR^2)$
\begin{equation}\label{eq:formdelta}
\frh_{\delta,\alpha}^\Sigma[u] := \|\nabla u\|^2_{L^2(\dR^2;\dC^2)} - \alpha\| u|_\Sigma\|^2_{L^2(\Sigma)},
\qquad
\mathrm{dom}\, \frh_{\delta,\alpha}^\Sigma := H^1(\dR^2);
\end{equation}
here $u|_\Sigma$ denotes the usual trace of $u\in H^1(\dR^2)$ onto $\Sigma$;
\emph{cf.}~\cite[Sec. 2]{BEKS94} and \cite[Sec. 3.2]{BLL13}.
Typically, $H_{\delta,\alpha}^\Sigma$ is called the
Schr\"odinger operator with $\delta$-interaction of strength $\alpha$ supported on $\Sigma$. The essential spectrum of $H_{\delta,\alpha}^\Sigma$
coincides with the set $[0,\infty)$ and its negative
discrete spectrum is known to be non-empty; \emph{cf.}~Section~\ref{Sec.pre}.
By $\lambda_1^\alpha(\Sigma)$ we denote the lowest negative eigenvalue of $H_{\delta,\alpha}^\Sigma$. For the operator $H_{\delta,\alpha}^\Sigma$ holds a spectral isoperimetric inequality~\cite{E05, EHL06} analogous to the spectral isoperimetric inequality for the Dirichlet Laplacian mentioned above. To be more precise, it can be stated as follows
\begin{flalign}\label{EHL}
\max_{|\Sigma| = L}
\lambda_1^\alpha(\Sigma) = \lambda_1^\alpha(C_{L/(2\pi)}),
\end{flalign}
where the maximum is taken over all smooth loops of a given length $L > 0$.
Here, we denote by~$|\Sigma|$ the length of~$\Sigma$ and~$C_{L/2\pi}$ is a circle of the radius $R = L/(2\pi)$.
We remark that by~\cite{BFKLR16} an analogue of~\eqref{EHL} holds for $\delta$-interactions
supported on curves in $\dR^3$ and according to the counterexample in~\cite{EF09} no direct analogue of~\eqref{EHL} can hold in the space dimension $d = 3$ for $\delta$-interactions supported on surfaces, except for special classes
of surfaces~\cite{EL15}.
In the last several years, the investigation of Schr\"odinger operators
with singular interactions supported on non-closed curves and
open surfaces became a topic of significant interest~\cite{DEKP16, EK16, EP14, ER16, JL16, MPS16, MPS16b}. In this paper, we obtain a counterpart of~\eqref{EHL}
for two-dimen\-sional Schr\"odinger operators with $\delta$-interactions supported on open arcs with the optimizer being a line segment. The respective statement is precisely formulated below.
\begin{thm}\label{Thm}
For all~$\alpha > 0$, holds
%
\begin{flalign}\label{result}
\max_{|\Sigma| = L}
\lambda_1^\alpha(\Sigma)
= \lambda_1^\alpha(\Gamma_L),
\end{flalign}
%
where the maximum is taken over all smooth open arcs
of a given length $L > 0$ and $\Gamma_L$ denotes a line segment
of length $L$; \emph{cf.}~Figure~\ref{fig:arc1}.
The equality in~\eqref{result} is possible if, and only if,
$\Sigma$ and $\Gamma_L$ are congruent.
\end{thm}
\input fig_arc.tex
One can view Theorem~\ref{Thm} as a spectral optimisation result for unbounded domains, in which we are optimizing the lowest eigenvalue below the threshold of the essential spectrum. Moreover, the shape of the optimizer is non-typical compared to most of spectral optimisation problems considered in the literature; see \emph{e.g.}~\cite{H1,H2}
and the references therein.
Our method of the proof of Theorem~\ref{Thm} relies on the Birman-Schwinger principle for $H_{\delta,\alpha}^\Sigma$ and
on the trick proposed in~\cite{E05, EHL06}
and further applied and developed in~\cite{BFKLR16, EL15}.
The main geometric ingredient in the proof of Theorem~\ref{Thm} is that the line segment is the shortest path connecting two fixed endpoints.
In the proof we make use of
the restriction to $\Gamma_L$ of the ground-state of the reference operator $H_{\delta,\alpha}^{\Gamma_L}$.
The main \emph{new obstacle}
compared to~\cite{EHL06} is related to the fact that
now this restriction is not known explicitly.
Possibility of performing the argument without
explicit knowledge of this restriction is strongly
correlated with the special geometric setting that we consider.
We point out that a result similar to~\eqref{result}
can also be proven under the constraint of fixed endpoints while the length
of the arc varies; see the discussion in Subsection~\ref{Sec.Endpoints}.
In fact, the latter claim is a consequence of Theorem~\ref{Thm}
and of the ordering between the eigenvalues of $H_{\delta,\alpha}^{\Gamma}$
and of $H_{\delta,\alpha}^{\Lambda}$ under the inclusion $\Gamma\subset\Lambda$.
Second, we describe the results for the Robin Laplacian on a plane with a slit.
Let $\Sigma \subset \mathbb{R}^2$ be a smooth compact non-closed curve as above.
For a real number~$\alpha > 0$, consider the spectral problem
for the self-adjoint Robin Laplacian $H_{{\rm R},\alpha}^\Sigma$
on $\dR^2\setminus\Sigma$ which corresponds via the first representation theorem to the closed, densely defined, symmetric, and semi-bounded quadratic form in $L^2(\dR^2)$
\begin{equation}\label{eq:form2}
\begin{split}
\frh_{{\rm R}, \alpha}^\Sigma[u] & := \|\nabla u\|^2_{L^2(\dR^2;\dC^2)} - \alpha\big(\| u|_{\Sigma_+}\|^2_{L^2(\Sigma)} +
\| u|_{\Sigma_-}\|^2_{L^2(\Sigma)}\big),\\
\mathrm{dom}\, \frh_{{\rm R},\alpha}^\Sigma & := H^1(\dR^2\setminus\Sigma);
\end{split}
\end{equation}
here $u|_{\Sigma_\pm}$ denote the traces of $u\in H^1(\dR^2\setminus\Sigma)$
onto two faces of $\Sigma$. It is known that the
essential spectrum of $H_{{\rm R},\alpha}^\Sigma$ coincides with
$[0,\infty)$. By a simple variational argument one
gets that the negative discrete spectrum of $H_{{\rm R},\alpha}^\Sigma$ is also non-empty.
We denote by $\mu_1^\alpha(\Sigma)$ the lowest negative eigenvalue
of $H_{{\rm R},\alpha}^\Sigma$ and obtain a claim for the Robin Laplacian on $\mathbb{R}^2\setminus\Sigma$ analogous to Theorem~\ref{Thm}.
\begin{thm}\label{Thm2}
For all~$\alpha > 0$, holds
%
\begin{flalign}\label{result2}
\max_{|\Sigma| = L}
\mu_1^\alpha(\Sigma)
= \mu_1^\alpha(\Gamma_L)
\end{flalign}
%
where the maximum is taken over all smooth open arcs
of a given length $L > 0$ and $\Gamma_L$ denotes a line segment of length $L$.
The equality in~\eqref{result2} is possible if, and only if,
$\Sigma$ and $\Gamma_L$ are congruent.
\end{thm}
We achieve the proof of Theorem~\ref{Thm2} via a combination of Theorem~\ref{Thm}
and of a trick based on the symmetry and on the ordering between the forms $\frh_{{\rm R},\alpha}^\Sigma$ and $\frh_{\delta,2\alpha}^\Sigma$.
It is worth mentioning that, unlike in our setting, the isoperimetric
property~\eqref{EHL} for loops does not imply any claim of such a kind for Robin Laplacians on planar domains with compact boundaries.
For Robin Laplacians on bounded domains, different methods are developed for repulsive \cite{B86, D06}
and attractive~\cite{AFK16, FK15} boundary conditions.
The method for attractive boundary conditions is further generalized in~\cite{KL16} to exterior domains.
The organisation of this paper is as follows.
In Section~\ref{Sec.pre} we recall basic known spectral
properties of $H_{\delta,\alpha}^\Sigma$ that are needed in this paper.
Section~\ref{Sec.BS} is devoted to the Birman-Schwinger principle
for $H_{\delta,\alpha}^\Sigma$ and its consequences.
Theorem~\ref{Thm} is proven in Section~\ref{Sec.proof}.
The paper is concluded by Section~\ref{Sec.end} with applications of Theorem~\ref{Thm}. Namely, in Subsection~\ref{Sec.Endpoints} we discuss the optimization of $\lambda_1^\alpha(\Sigma)$ under the constraint
of fixed endpoints for~$\Sigma$ and in Subsection~\ref{Sec.Robin} we
prove Theorem~\ref{Thm2} concerning the optimization of the lowest eigenvalue of the Robin Laplacian on a plane with a slit.
\section{The spectral problem for $\delta$-interactions supported on open arcs}\label{Sec.pre}
Throughout this section, $\Sigma$~is an arbitrary curve of
a finite length in~$\mathbb{R}^2$ with two free endpoints. For simplicity, we assume that~$\Sigma$ is smooth (\emph{i.e.}~$C^\infty$-smooth),
but less regularity is evidently needed for the majority of the results to hold.
We emphasize that by saying that $\Sigma$ is smooth we implicitly understand that it can be continued up to the boundary of a
$C^\infty$-smooth bounded simply connected domain.
In particular, $\Sigma$ has no self-intersections and no increasing oscillations
at the endpoints.
At the same time, $\alpha$~is an arbitrary positive real number.
We are interested in the spectral properties of the self-adjoint operator
$H_{\delta,\alpha}^\Sigma$ in $L^2(\dR^2)$ introduced via the first
representation theorem~\cite[Thm. VI 2.1]{Kato} through the closed, densely defined,
symmetric and semi-bounded quadratic form $\frh_{\delta,\alpha}^\Sigma$
in~\eqref{eq:formdelta}; see~\cite[Sec. 2]{BEKS94} and also~\cite{BLL13}.
Let $\wt\Sigma$ be a continuation of~$\Sigma$ up to the boundary of a bounded
smooth domain $\Omega_+\subset\dR^2$ and let $\Omega_- := \dR^2\setminus\overline{\Omega_+}$
be the complement of $\Omega_+$. For any $u\in L^2(\dR^2)$ we introduce the notation
$u_\pm := u|_{\Omega_\pm}$. Then the operator domain of $H_{\delta,\alpha}^\Sigma$
consists of functions $u \in H^1(\dR^2)$ which satisfy
$\Delta u_\pm \in L^2(\Omega_\pm)$ in the distributional sense
and $\delta$-type boundary conditions
\begin{equation}\label{eq:delta}
\partial_{\nu_+}u_+|_{\wt\Sigma} + \partial_{\nu_-}u_-|_{\wt\Sigma}
= \alpha\chi_\Sigma u|_{\wt\Sigma}
\end{equation}
on $\wt\Sigma$ in the sense of traces, where $\chi_\Sigma\colon\wt\Sigma\rightarrow\wt\Sigma$
is the characteristic function of $\Sigma$ in $L^2(\wt\Sigma)$
and where $\partial_{\nu_\pm}u_\pm|_{\wt\Sigma}$
denote the traces of normal derivatives of $u_\pm$ onto $\wt\Sigma$
with the normal vectors pointing outwards $\Omega_\pm$.
Moreover, for any $u \in \mathrm{dom}\, H_{\delta,\alpha}^\Sigma$
we have $H_{\delta,\alpha}^\Sigma u = (- \Delta u_+)\oplus (- \Delta u_-)$.
The reader may consult with~\cite[Cor. 6.21]{MPS16} and~\cite[Sec. 3.2]{BLL13} for a more precise description of $\mathrm{dom}\, H_{\delta,\alpha}^\Sigma$.
The operator $H_{\delta,\alpha}^\Sigma$ possesses a non-empty essential spectrum. Namely, we have the following statement.
\begin{prop}\label{prop:ess_spec}
For all $\alpha > 0$ holds
$\sigma_{\rm ess}(H_{\delta,\alpha}^\Sigma) =
[0,\infty)$.
\end{prop}
\noindent The claim of this proposition is expected because the essential spectrum of the Laplacian in the whole space~$\mathbb{R}^2$ equals $[0,\infty)$
and introducing a $\delta$-interaction supported on $\Sigma$
leads to a compact perturbation in the sense of resolvent differences.
The proofs of Proposition~\ref{prop:ess_spec}
can be found in~\cite[Thm. 3.1]{BEKS94}
and also in~\cite[Thm. 4.3]{BLL13}.
Various properties of the discrete spectrum of $H_{\delta,\alpha}^\Sigma$
are investigated in~\cite{BLL13, EP14, KL14}.
For our purposes we only require the following statement.
\begin{prop}\label{Prop.disc}
For all $\alpha > 0$ holds
$1\le \#\sigma_{\rm d}(H_{\delta,\alpha}^\Sigma) < \infty$\footnote{We denote by $\#\sigma_{\rm d}(T)$ the number
of discrete eigenvalues with multiplicities taken into account for a self-adjoint operator $T$.
}.
\end{prop}
\noindent For a proof of $1\le \#\sigma_{\rm d}(H_{\delta,\alpha}^\Sigma)$ see \cite[Thm. 3.1]{KL14}. Non-emptiness of $\sigma_{\rm d}(H_{\delta,\alpha}^\Sigma)$
can alternatively be shown by the min-max principle with the aid
of the family of test functions having the same structure as in the proof of~\cite[Prop. 2]{KL16}.
A simple proof of $\#\sigma_{\rm d}(H_{\delta,\alpha}^\Sigma) < \infty$ can
be found in~\cite[Thm. 3.14]{BLL13}.
Finiteness of the discrete spectrum for $H_{\delta,\alpha}^\Sigma$ may also
be derived from the spectral estimate in~\cite[Thm. 4.2\,(iii)]{BEKS94}.
Finally, we obtain fundamental properties of the lowest eigenvalue $\lambda_1^\alpha(\Sigma)$ for $H_{\delta,\alpha}^\Sigma$ and of the
corresponding eigenspace.
\begin{prop}\label{prop:GT}
For all $\alpha > 0$, the lowest eigenvalue $\lambda_1^\alpha(\Sigma) < 0$ of $H_{\delta,\alpha}^\Sigma$ is simple and the corresponding eigenfunction
can be chosen to be non-negative in $\dR^2$.
\end{prop}
\begin{proof}
The argument follows the same strategy as the proof of~\cite[Thm.~8.38]{Gilbarg-Trudinger}.
Denote $\lambda := \lambda_1^\alpha(\Sigma) < 0$ and let $u = u_+\oplus u_- \in \ker(H_{\delta,\alpha}^\Sigma - \lambda)$.
By standard elliptic regularity we get $u_\pm \in H^2_{\rm loc}(\Omega_\pm)$.
Without loss of generality we can assume that $u$ is real-valued
and that $\|u\|_{L^2(\dR^2)} = 1$.
Clearly, we have $|u| \in H^1(\dR^2)$, $\||u|\|_{L^2(\dR^2)} = 1$,
and, moreover, $\frh_{\delta,\alpha}^\Sigma[|u|] = \frh_{\delta,\alpha}^\Sigma[u]$.
The condition that $|u|$ is a minimizer
for the quadratic form $\frh_{\delta,\alpha}^\Sigma$
implies a characterization of $|u|$ through
an Euler-Lagrange-type equation
%
\begin{equation}\label{eq:u}
\frh_{\delta,\alpha}^\Sigma[|u|,\phi] = \lambda(|u|,\phi)_{L^2(\dR^2)},\qquad
\forall\,
\phi \in H^1(\dR^2),
\end{equation}
%
which is equivalent to the variational
characterization of an eigenfunction for $H_{\delta,\alpha}^\Sigma$
corresponding to the eigenvalue $\lambda$. Thus, we have
$|u| \in \ker(H_{\delta,\alpha}^\Sigma - \lambda)$. In particular,
we have shown that
$-\Delta |u_\pm| = \lambda|u_\pm|$ holds on $\Omega_\pm$ in the distributional sense. Thus, by elliptic
regularity we also get $|u_\pm| \in H^2_{\rm loc}(\Omega_\pm)$.
Clearly, $u_+ = 0$ and $u_- = 0$ do not hold simultaneously taking into account
that $\|u\|_{L^2(\dR^2)} = 1$. If either
$u_+ = 0$ or $u_- = 0$, then~$u\in H^1(\dR^2)$ implies that $u$ satisfies Dirichlet boundary conditions on $\wt \Sigma$ and we get a contradiction to non-negativity
of the Dirichlet Laplacians on $\Omega_\pm$.
Furthermore, Harnack's inequality~\cite[Cor.~8.21]{Gilbarg-Trudinger}
yields that $|u_\pm|$ are pointwise positive in $\Omega_\pm$. Thus, standard properties of
$H^1$-functions imply that $u_\pm$ are sign-definite
pointwise non-vanishing functions in $\Omega_\pm$. It remains to exclude the case when $u_\pm$ are of different signs.
Indeed, if it were the case,
then in view of~$u\in H^1(\dR^2)$
we would get $u|_{\wt\Sigma} = 0$. Thus,
$u_\pm$ would be simultaneously eigenfunctions
of Dirichlet Laplacians on $\Omega_\pm$ corresponding
to a negative eigenvalue $\lambda < 0$, which is
a contradiction. Hence, we obtain
that either $u = |u|$ or $u = -|u|$.
This argument shows that any function in $\ker(H_{\delta,\alpha}^\Sigma - \lambda)$ is pointwise
positive in $\dR^2\setminus\wt\Sigma$
and non-negative in $\dR^2$
(up to multiplication by a constant factor).
Hence, it is impossible that $\ker(H_{\delta,\alpha}^\Sigma - \lambda)$
contains two linearly independent functions that are orthogonal
to each other. Thus,
we obtain that the linear subspace $\ker(H_{\delta,\alpha}^\Sigma - \lambda)$
of $\mathrm{dom}\, H_{\delta,\alpha}^\Sigma$ is one-dimensional.
\end{proof}
Summarizing, the essential spectrum of
$H_{\delta,\alpha}^\Sigma$ equals the interval $[0,\infty)$
and there is at least one discrete eigenvalue below~$0$.
In particular, the lowest point $\lambda_1^\alpha(\Sigma)$
in the spectrum is always a simple negative discrete eigenvalue
and the corresponding eigenfunction can be selected
to be non-negative in $\dR^2$.
\section{Birman-Schwinger principle}\label{Sec.BS}
In this section we formulate a Birman-Schwinger-type principle for
the operator $H_{\delta,\alpha}^\Sigma$ and derive a related characterization of its lowest eigenvalue $\lambda_1^\alpha(\Sigma)$.
First, we parametrize the curve $\Sigma$ by the unit-speed mapping $\Sigma \colon \cI\rightarrow \mathbb{R}^2$ with $\cI := [0,L]$; \emph{i.e.}~$|\dot\Sigma(s)|= 1$ for all
$s\in \cI$. Clearly, the Hilbert spaces $L^2(\Sigma)$
and $L^2(\cI)$ can be identified.
Second, we define a weakly singular integral operator
$Q^\Sigma(\kappa) \colon L^2(\cI)\rightarrow L^2(\cI)$ for $\kappa > 0$ by
\begin{equation}\label{def:Q}
(Q^\Sigma(\kappa)\psi)(s) :=
\frac{1}{2\pi}\int_0^L
K_0\left(\kappa|\Sigma(s) - \Sigma(s')|\right)\psi(s'){\mathsf{d}} s',
\end{equation}
where $K_0(\cdot)$ is the modified Bessel function of the second kind and of the order $\nu = 0$; \emph{cf.}~\cite[\Sigma 9.6]{AS64}. In the next proposition we state basic properties of this integral operator.
\begin{prop}\label{prop:basic}
The operator $Q^\Sigma(\kappa)$ in~\eqref{def:Q}
is self-adjoint, compact, and non-negative for all $\kappa > 0$.
\end{prop}
\begin{proof}
Compactness of $Q^\Sigma(\kappa)$ is proven
in~\cite[Lem. 3.2]{BEKS94}. Self-adjointness and non-negativity
of $Q^\Sigma(\kappa)$ follow from abstract results in~\cite{B95}.
\end{proof}
Now we have all the tools to formulate a Birman-Schwinger-type condition
for $H_{\delta,\alpha}^\Sigma$.
\begin{thm}\label{thm:BS}
Let the self-adjoint operator $H_{\delta,\alpha}^\Sigma$
in $L^2(\dR^2)$ represent the quadratic form in~\eqref{eq:formdelta}
and let the operator-valued function
$\dR_+\ni \kappa \mapsto Q^\Sigma(\kappa)$ be as in~\eqref{def:Q}.
Then the following claims hold.
%
\begin{myenum}
\item
$\dim\ker(H_{\delta,\alpha}^\Sigma + \kappa^2) =
\dim\ker(I - \alpha Q^\Sigma(\kappa))$ for all $\kappa > 0$.
%
\item The mapping $u \mapsto u|_\Sigma$ is a bijection between
$\ker(H_{\delta,\alpha}^\Sigma + \kappa^2)$ and
$\ker(I - \alpha Q^\Sigma(\kappa))$.
\end{myenum}
\end{thm}
\begin{proof}
For the proof of~(i) see \cite[Lem. 2.3\,(iv)]{BEKS94} and also~\cite[Thm. 3.5\,(iii)]{BLL13}. The claim of~(ii)
is a consequence of the abstract statement in~\cite[Lem. 1]{B95}.
\end{proof}
We conclude this section by two corollaries of Theorem~\ref{thm:BS}.
\begin{cor}\label{cor:BS1}
Let the assumptions be as in Theorem~\ref{thm:BS} and let $\kappa > 0$
be such that $\lambda_1^\alpha(\Sigma) = -\kappa^2$.
Then the following claims hold.
%
\begin{myenum}
\item $\dim\ker(I - \alpha Q^\Sigma(\kappa)) = 1$.
\item $\ker(I - \alpha Q^\Sigma(\kappa)) = {\rm span}\{\psi\}$
where $\psi \in L^2(\cI)$ is a positive function.
\end{myenum}
\end{cor}
\begin{proof}
Recall that by Proposition~\ref{prop:GT} the lowest eigenvalue $\lambda= \lambda_1^\alpha(\Sigma)$ of $H_{\delta,\alpha}^\Sigma$ is simple.
Hence, the claim of~(i) immediately follows from Theorem~\ref{thm:BS}\,(i).
Denote by $\psi\in L^2(\cI)$ the trace on $\Sigma$ of the eigenfunction
of $H_{\delta,\alpha}^\Sigma$ corresponding to its lowest eigenvalue
$\lambda$. According to Theorem~\ref{thm:BS}\,(ii)
we have $\ker(I - \alpha Q^\Sigma(\kappa)) = {\rm span}\{\psi\}$.
Furthermore, recall that by Proposition~\ref{prop:GT}
the eigenfunction of $H_{\delta,\alpha}^\Sigma$ corresponding to the lowest eigenvalue $\lambda$ can be chosen to be non-negative in $\mathbb{R}^2$.
Clearly, the trace on $\Sigma$ of an $H^1$-function, that is non-negative in $\mathbb{R}^2$, is non-negative as well. Thus, we can select the function $\psi$ to be non-negative.
Finally, the identity $\psi = \alpha Q^\Sigma(\kappa)\psi$, non-negativity of $\psi$, and strict positivity of the integral kernel of $Q^\Sigma(\kappa)$ in~\eqref{def:Q} imply that $\psi$ is, in fact, positive.
\end{proof}
Now we provide the second consequence of Theorem~\ref{thm:BS}.
\begin{cor}\label{cor:BS2}
Let the assumptions be as in Theorem~\ref{thm:BS}.
Then the following claims hold.
%
\begin{myenum}
\item $\sup\sigma(\alpha Q^\Sigma(\kappa)) \ge 1$ if, and only if,
$\lambda_1^\alpha(\Sigma) \le -\kappa^2$.
\item $\sup\sigma(\alpha Q^\Sigma(\kappa)) = 1$ if, and only if,
$\lambda_1^\alpha(\Sigma) = -\kappa^2$.
\end{myenum}
\end{cor}
\begin{proof}
In the proof it will be convenient to use the
following shorthand notations:
%
\begin{equation}\label{eq:FG}
F_\alpha(\kappa) := \sup\sigma(\alpha Q^\Sigma(\kappa))
\qquad\text{and}\qquad
G(\alpha) := \lambda_1^\alpha(\Sigma).
\end{equation}
%
First, we recall that the function
$\dR_+\ni\kappa\mapsto F_\alpha(\kappa)$ is continuous~\cite[Lem 3.2]{BEKS94}
and strictly decaying (cf.~\cite[Prop. 3.2]{BLL13} and \cite[Lem. 2.3\,(i)]{BLLR15})
and that $F_\alpha(\kappa)\rightarrow 0+$ as $\kappa \rightarrow +\infty$
(\emph{cf.}~~\cite[Thm. 3.1]{GS15}).
Second, recall that the function
$\dR_+\ni\alpha\mapsto G(\alpha)$ is also continuous and strictly decaying,
and that $G(\alpha)\rightarrow -\infty$ as $\alpha\rightarrow +\infty$
(see \emph{e.g.}~\cite[Prop. 2.9]{L14}).
Now we pass to the proofs of the claims.
\noindent (i)
$F_\alpha(\kappa) \ge 1$ implies that for some
$\nu \ge \kappa$ holds $F_\alpha(\nu) = 1$. Therefore,
by Proposition~\ref{prop:basic} and Theorem~\ref{thm:BS}\,(i) we have $-\nu^2 \in \sigma_{\rm d}(H_{\delta,\alpha}^\Sigma)$ and, in particular, $G(\alpha) \le -\nu^2 \le -\kappa^2$.
Suppose now that $G(\alpha) \le -\kappa^2$.
Then there exists $\nu \ge \kappa$ such that
$-\nu^2\in \sigma_{\rm d}(H_{\delta,\alpha}^\Sigma)$.
By Proposition~\ref{prop:basic} and Theorem~\ref{thm:BS}\,(i) we get $1\in\sigma_{\rm d}(\alpha Q^\Sigma(\nu))$
and thus $F_\alpha(\nu) \ge 1$. Finally, we have
$F_\alpha(\kappa) \ge F_\alpha(\nu) \ge 1$.
\noindent (ii)
$F_\alpha(\kappa) = 1$ implies that $G(\alpha) \le -\kappa^2$ by (i).
On the other hand, if $G(\alpha) < -\kappa^2$ then
for some $\beta < \alpha$ holds $G(\beta) = -\kappa^2$
and hence $F_\alpha(\kappa) > F_\beta(\kappa) \ge 1$
which is a contradiction.
$G(\alpha) = -\kappa^2$ implies that $F_\alpha(\kappa) \ge 1$ again by~(i).
On the other hand, if $F_\alpha(\kappa) > 1$
then there exists $\beta < \alpha$
such that $F_\beta(\kappa) = 1$. Thus, we have
$G(\alpha) < G(\beta) \le -\kappa^2$
which is also a contradiction.
\end{proof}
\section{Proof of Theorem~\ref{Thm}}\label{Sec.proof}
Now we are in a position to establish Theorem~\ref{Thm}.
Throughout this section,
$\Sigma\subset\mathbb{R}^2$~is a compact $C^\infty$-smooth curve
of length $L > 0$ with two free endpoints which is parametrized
by the unit-speed mapping $\Sigma\colon \cI\rightarrow \mathbb{R}^2$
with $\cI = [0,L]$
and $\Gamma = \Gamma_L\subset\mathbb{R}^2$ is a line segment having the same length $L$
which is parametrized by the unit-speed mapping $\Gamma\colon \cI \rightarrow \dR^2$.
In addition, assume that $\Sigma$ is not congruent to $\Gamma$.
Recall that $\lambda_1^\alpha(\Sigma)$ and $\lambda_1^\alpha(\Gamma)$ denote
the lowest eigenvalues of $H_{\delta,\alpha}^\Sigma$ and
of $H_{\delta,\alpha}^\Gamma$, respectively.
Furthermore, we fix $\kappa > 0$ such that $\lambda_1^\alpha(\Gamma) = -\kappa^2$.
By Corollary~\ref{cor:BS1}\,(ii) we have $\ker(I - \alpha Q^\Gamma(\kappa))
= {\rm span}\,\{\psi\}$ where $\psi \in L^2(\cI)$ is a
positive function.
Without loss of generality we assume that $\|\psi\|_{L^2(\cI)} = 1$.
Observe that by Corollary~\ref{cor:BS2}\,(ii) holds
$\sup\sigma(\alpha Q^\Gamma(\kappa)) = 1$.
Note that for any $s, s' \in \cI$ we have
\begin{equation}\label{eq:ineq}
|\Sigma(s) - \Sigma(s')| \le |\Gamma(s) - \Gamma(s')|.
\end{equation}
Since $\Sigma$ is not congruent to $\Gamma$,
for simple geometric reasons there is a subset
${\mathcal S}} \def\cT{{\mathcal T}} \def\cU{{\mathcal U} \subset \cI^2$ having positive Lebesgue measure such that
\begin{equation}\label{eq:ineq_strict}
|\Sigma(s) - \Sigma(s')| < |\Gamma(s) - \Gamma(s')|, \qquad \forall\, (s,s')\in{\mathcal S}} \def\cT{{\mathcal T}} \def\cU{{\mathcal U}.
\end{equation}
Using~\eqref{eq:ineq},~\eqref{eq:ineq_strict}, positivity of $\psi$,
strict decay of $K_0(\cdot)$, and the min-max principle we obtain
\[
\begin{split}
\sup\sigma(\alpha Q^\Sigma(\kappa) )
& \ge \frac{\alpha}{2\pi}
\int_0^L\int_0^L K_0\left(\kappa|\Sigma(s)-\Sigma(s')|\right)\psi(s)\psi(s') {\mathsf{d}} s {\mathsf{d}} s'\\[0.4ex]
&
>
\frac{\alpha}{2\pi}
\int_0^L\int_0^L K_0\left(\kappa|\Gamma(s)-\Gamma(s')|\right)\psi(s)\psi(s') {\mathsf{d}} s {\mathsf{d}} s'\\[0.4ex]
& = \sup\sigma(\alpha Q^\Gamma(\kappa) ) = 1.
\end{split}
\]
Hence, by Corollary~\ref{cor:BS2} we get
\[
\lambda_1^\alpha(\Sigma) < -\kappa^2 = \lambda_1^\alpha(\Gamma).
\]
Thus, the proof of the theorem is complete. \qed
\section{Consequences of Theorem~\ref{Thm}}\label{Sec.end}
Let us conclude the paper by two consequences of Theorem~\ref{Thm}, which are of certain independent interest.
\subsection{Fixed endpoints}\label{Sec.Endpoints}
In this subsection we consider a related optimization problem for the lowest eigenvalue of $H_{\delta,\alpha}^\Sigma$ under the constraint of fixed endpoints.
We emphasize that no additional restrictions on the length of $\Sigma$ are imposed.
\begin{prop}\label{prop:fixedpoints}
For all $\alpha > 0$, holds
\begin{equation}\label{eq:optimisation2}
\max_{\partial\Sigma = \{P,Q\}} \lambda_1^\alpha(\Sigma) =
\lambda_1^\alpha(\Gamma_L)
\end{equation}
%
where the maximum is taken over all smooth open arcs $\Sigma$ that connect
two given points $P,Q \in\mathbb{R}^2$, $P\ne Q$, and $\Gamma_L$
is a line segment of length $L = |P-Q|$
where $|P- Q|$ is the Euclidean distance between the points $P$ and $Q$;
\emph{cf.}~Figure~\ref{fig:arc2}.
The equality in~\eqref{eq:optimisation2}
is possible if, and only if, $\Sigma$ is the line segment that
connects the points $P$ and $Q$.
\end{prop}
\input fig_arc2.tex
\begin{proof}
Let $\Sigma\subset\dR^2$ be any smooth open arc connecting the points $P$ and $Q$
which does not coincide with the line segment between them.
First, applying Theorem~\ref{Thm}, we obtain
%
\begin{equation}\label{eq:ineq1}
\lambda_1^\alpha(\Sigma) < \lambda_1^\alpha(\Lambda),
\end{equation}
%
where $\Lambda = \Lambda_{|\Sigma|}$ is a line segment of length $|\Sigma|$.
Second, observe that the following simple geometric inequality $|\Sigma| > L = |P - Q|$ holds. Furthermore, let $\Gamma = \Gamma_L$ be a line segment of length $L$.
Without loss of generality we assume that $\Gamma \subset\Lambda$.
Using the min-max principle and the form ordering $\frh_{\delta,\alpha}^{\Lambda} \prec \frh_{\delta,\alpha}^{\Gamma}$ we arrive at
%
\begin{equation}\label{eq:ineq2}
\lambda_1^\alpha(\Lambda)
\le
\lambda_1^\alpha(\Gamma).
\end{equation}
%
The claim of the proposition follows directly from~\eqref{eq:ineq1} and~\eqref{eq:ineq2}.
\end{proof}
\begin{remark}
The proof of Proposition~\ref{prop:fixedpoints}
indicates a way to obtain a quantified version of the spectral isoperimetric
inequality under the constraint of fixed endpoints in the spirit of~\cite{BP12}. To this aim it suffices to obtain in the last step
of the proof a positive lower bound on the difference
$\lambda_1^\alpha(\Gamma) - \lambda_1^\alpha(\Lambda)$ in terms of $\alpha$,
$|\Gamma|$, and $|\Lambda|$.
\end{remark}
\begin{remark}
Note that the line segment connecting
two points in $\dR^2$ can be viewed as a geodesic
between them, thus making natural a question
of generalization of the optimization result
in Proposition~\ref{prop:fixedpoints} for manifolds.
\end{remark}
\subsection{The Robin Laplacian on $\mathbb{R}^2\setminus\Sigma$}\label{Sec.Robin}
The aim of this subsection is to prove Theorem~\ref{Thm2} on the isoperimetric
inequality for the Robin Laplacian $H_{\rm R,\alpha}^\Sigma$ on a plane with a slit
$\mathbb{R}^2\setminus\Sigma$. We recall that the self-adjoint operator
$H_{\rm R,\alpha}^\Sigma$ in $L^2(\dR^2)$ is introduced via the first representation theorem
through the closed, densely defined, symmetric and semi-bounded quadratic
form $\frh_{\rm R,\alpha}^\Sigma$ in~\eqref{eq:form2}; \emph{cf.}~\cite[Lem. 2.2]{ER16}
and also \cite{MPS16}.
It is worth to mention already in the beginning of this subsection that
for any $\alpha > 0$ the form ordering $\frh_{{\rm R},\alpha}^\Sigma\prec
\frh_{\delta,2\alpha}^\Sigma$ holds thanks to the inclusion
$H^1(\dR^2)\subset H^1(\dR^2\setminus\Sigma)$ and to the identity
$\frh_{{\rm R},\alpha}^\Sigma[u] = \frh_{\delta,2\alpha}^\Sigma[u]$,
which is satisfied for all $u\in H^1(\dR^2)$.
First, we provide the following statement on the qualitative spectral properties of $H_{\rm R,\alpha}^\Sigma$.
\begin{prop}
For all $\alpha > 0$ holds $\sigma_{\rm ess}(H_{\rm R,\alpha}^\Sigma) = [0,\infty)$
and $1\le \#\sigma_{\rm d}(H_{\rm R,\alpha}^\Sigma) < \infty$.
\end{prop}
\begin{proof}
The statements $\sigma_{\rm ess}(H_{\rm R,\alpha}^\Sigma) = [0,\infty)$
and $\#\sigma_{\rm d}(H_{\rm R,\alpha}^\Sigma) < \infty$
are special cases of~\cite[Thm. 3.1]{ER16}.
In view of the ordering $\frh_{{\rm R},\alpha}^\Sigma\prec
\frh_{\delta,2\alpha}^\Sigma$,
the property $1\le \#\sigma_{\rm d}(H_{\rm R,\alpha}^\Sigma)$
follows from Proposition~\ref{Prop.disc} and the min-max principle.
\end{proof}
Now we have all the tools to provide a proof of Theorem~\ref{Thm2}.
\begin{proof}[Proof of Theorem~\ref{Thm2}]
Let $\Sigma\subset\mathbb{R}^2$~be a compact $C^\infty$-smooth curve
of length $L > 0$ with two free endpoints and let
$\Gamma = \Gamma_L\subset\mathbb{R}^2$ be a line segment of the same length.
For convenience we introduce Cartesian coordinates $(x,y)$ on $\dR^2$.
Without loss of generality we assume that the line segment
$\Gamma$ is lying on the $x$-axis.
Recall that $\mu_1^\alpha(\Sigma)$ and $\mu_1^\alpha(\Gamma)$ denote
the lowest eigenvalues of $H_{\rm R,\alpha}^\Sigma$ and
of $H_{\rm R,\alpha}^\Gamma$, respectively.
First, we observe that in view of the ordering
$\frh_{\rm R,\alpha}^\Sigma\prec\frh_{\delta,2\alpha}^\Sigma$ we have
%
\begin{equation}\label{eq:ineq3}
\mu_1^\alpha(\Sigma)\le \lambda_1^{2\alpha}(\Sigma).
\end{equation}
%
Furthermore, we consider the subspaces
$L^2_{\rm even}(\dR^2)$ and $L^2_{\rm odd}(\dR^2)$
of $L^2(\dR^2)$, which consist, respectively, of even and odd functions in the $y$-variable. Both the subspaces $L^2_{\rm even}(\dR^2)$ and $L^2_{\rm odd}(\dR^2)$ can be identified with $L^2(\dR^2_+)$ via natural unitary
transforms.
With respect to the decomposition
$L^2(\dR^2) = L^2_{\rm even}(\dR^2)\oplus L^2_{\rm odd}(\dR^2)$
and in view of the above identifications the operators
$H_{\rm R,\alpha}^\Gamma$ and $H_{\delta,2\alpha}^\Gamma$
can be decomposed into orthogonal sums
\begin{equation}\label{eq:orth}
H_{\rm R,\alpha}^\Gamma = A_{\alpha} \oplus B_{\alpha} \qquad\text{and}\qquad
H_{\delta,2\alpha}^\Gamma =A_{\alpha}\oplus C,
\end{equation}
%
where the self-adjoint operators $A_{\alpha}$, $B_{\alpha}$, and $C$
acting in $L^2(\dR^2_+)$ are introduced via the first representation
theorem through closed, densely defined, symmetric, and semi-bounded quadratic
forms
%
\begin{align*}
\fra_{\alpha}[u] & =
\|\nabla u\|^2_{L^2(\dR^2_+;\dC^2)} - \alpha\|u|_\Gamma\|^2_{L^2(\Gamma)},
&\mathrm{dom}\,\fra_\alpha& = H^1(\dR^2_+),\\
{\mathfrak b}_\alpha[u] & =
\|\nabla u\|^2_{L^2(\dR^2_+;\dC^2)}
-\alpha\|u|_\Gamma\|^2_{L^2(\Gamma)},
&\mathrm{dom}\,{\mathfrak b}_\alpha& = \big\{u \in H^1(\dR^2_+)\colon u|_{\partial\dR^2_+\setminus\Gamma} = 0\big\}, \\
\frc[u] & = \|\nabla u\|^2_{L^2(\dR^2_+;\dC^2)},
&\mathrm{dom}\,\frc& = H^1_0(\dR^2_+).
\end{align*}
%
It can be easily seen that the operator $C$ is non-negative.
In view of the ordering $\fra_\alpha\prec{\mathfrak b}_\alpha$,
the min-max principle implies that
$\inf\sigma(A_\alpha) \le \inf\sigma(B_\alpha)$.
Thus, using decompositions~\eqref{eq:orth} we end up with
%
\begin{equation}\label{eq:ineq4}
\mu_1^\alpha(\Gamma)
=
\inf\sigma(A_{\alpha})
=
\lambda_1^{2\alpha}(\Gamma).
\end{equation}
%
The claim immediately follows from~\eqref{eq:ineq3},~\eqref{eq:ineq4},
and Theorem~\ref{Thm}.
\end{proof}
\section*{Acknowledgements}
The author was supported by the grant No.\ 14-06818S
of the Czech Science Foundation (GA\v{C}R).
He is very grateful to Pavel Exner and David Krej\v{c}i\v{r}\'{i}k
for fruitful discussions on the subject.
\newcommand{\etalchar}[1]{$^{#1}$}
|
2,869,038,156,413 | arxiv | \section{Introduction}
Irony, which generally refers to the expressions that have opposite literal meanings to real meanings, is a representative rhetoric device in human languages \cite{li2020method}. For example, in sentences \emph{I just love when you test my patience!!}, and \emph{Had no sleep and have got school now}, compared to their literal meanings, both sentences are conveying reverse meanings and emotions, which mean \emph{not love} and \emph{not happy}. People are using ironies to communicate their affective states more implicitly or explicitly, to strengthen the claims depending on the needs, which contribute to neologism across human languages. However, the inner incongruity makes ironic expressions harder for machines to understand. Consequently, accurate irony processing systems are essential and challenging for downstream tasks and natural language understanding (NLU) research.
Previously, many researchers approached irony processing from various perspectives. The last survey in computational irony was published seven years ago \cite{wallace2015computational} and mainly focused on irony detection and pragmatic context model. Therefore, a systematic and comprehensive survey on automatic irony processing remains absent, encouraging us to focus on a review of advances in irony processing, from traditional machine learning, recurrent neural networks (RNNs) methods, to deeper pretrained language models (PLMs) throughout. Besides, \newcite{del2020review} revealed the great imbalance in figurative language research, in which sarcasm was dominant and twice as much as irony research. With this review, We aim to encourage a balanced and equal research environment in figurative languages.
Moreover, the design and further improvement in present irony processing systems should be concatenated with both the theoretical accounts in irony theories and its role in communication, followed by downstream applications in NLP tasks. Generally, most researchers approached irony processing as a single irony detection branch and three main shared tasks \cite{ghanem2019idat,van2018semeval,ortega2019overview} all focused on irony detection, whereas other aspects are constantly under-explored. And theoretical-informed or cognitive-informed discussions are rarely seen in irony research compared to tiny amendments in neural networks' architectures.
In this paper, we aim to offer a comprehensive review in irony processing from machine learning, linguistic theory, cognitive science, and newly proposed multi-X perspectives and to evaluate irony processing in NLP applications. The remaining sections are organized as follows. In the second part, we will approach irony theories in linguistics and cognitive science research, and discuss the concrete differences between irony and sarcasm. Then, we will review irony datasets in world languages and discuss potential problems in annotation schemes throughout. In the fourth part we will retrospect research progress in automatic irony processing, including traditional feature engineering, neural network architectures, to PLMs and relatively under-explored research fields in irony processing. Finally, we will discuss irony's interactions with downstream NLP tasks like sentiment analysis and opinion mining, and multi-X perspectives for further development in computational irony research.
\section{Theoretical Research in Irony}
\subsection{Irony Theories}
Various definitions have been given to irony. Early studies suggested that irony is the expression whose real meaning is contradictory to its literal meaning \cite{grice1975logic}. The Merriam-Webster Dictionary, The Oxford English Dictionary, and The Collins English Dictionary all adopted this definition and used the words "opposite" or "contrary" to explain the relationship between the literal and contextual meanings of irony.
However, more research into various types of ironic examples revealed that the contextual meaning of irony does not have to be "opposite" or "contrary" to the literal one. According to \newcite{sperber1986relevance,wilson2012explaining}, some expressions have no "literal meaning" to be challenged because no "literal meaning" is mentioned in the context, based on which they raised \textbf{relevance theory} and the ``echoic'' concept. They considered irony as ``an echoic use of language in which the speaker tacitly dissociates herself from an attributed utterance or thought'' \cite{wilson2006pragmatics}. That is, if the "echoic use" is incongruous in some ways, the expression can be ironic. Based on this theory, \newcite{neto1998onni} put forward that there are some ``echo-markers'' like \emph{definitely}, \emph{really}, and \emph{indeed}.
\newcite{li2020method} provided instances to show that "incongruity" does not have to be between the literal and contextual meanings of irony in certain circumstances. They believed that irony's true nature is a psychological activity as much as a verbal representation. The speaker or listeners must finish the "reversal" process on a psychological level for it to be completed. When compared to the concepts of "echoic" and "incongruity," “reversal” is concerned not only with the results but also with the psychological processes that the speakers/listeners go through.
\subsubsection{Types of Irony}
\newcite{booth1974rhetoric} divided irony into tragic and comic irony by literary genre, as well as stable and unstable irony by determinacy. He also categorized irony into dramatic irony, situational irony, verbal irony, and rhetorical irony by the range of context it has to refer to. Researchers in computational irony paid the most attention to situational irony and verbal irony. Situational irony is a circumstance in which the outcome differs from what was expected, or a situation that contains striking contrasts (Imagine a situation that a lifeguard is saved from drowning). Verbal irony is the expression in which the speaker's intended meaning significantly different from the literal one. \newcite{abrams2014glossary} considered that verbal irony usually involves the explicit presentation of one attitude or evaluation, but with signals that the speaker wants to express a totally different attitude or opinion. It differs from situational irony in that it is purposely manufactured by speakers. For example, \emph{Very well, keep insulting me!}
Verbal irony is the kind to which computational linguistics researchers pay the most attention. \newcite{li2021corpus} made a further classification of them. She put forward eight kinds of reversals which are rhetorical reversal, expectation reversal, evaluation reversal, reversal of sentiment, reversal of factuality, relationship reversal, reversal from opposite pair, and reversal from satiation. The paper considered that verbal ironies can be classified by the kind of reversal they generate.
\subsection{Linguistic Features}
\subsubsection{Irony Markers and Constructions}
Although most of the studies saw irony as a pragmatics phenomenon, people also considered that it can be reflected on the verbal, grammatical, or semantic level. For example, on the verbal level, people often use words like \emph{thank}, \emph{congratulate}, \emph{welcome}, \emph{happy}, and \emph{interesting} to express ironic meanings. \newcite{laszlo2017corpus} found 15 core evaluative words which often show in ironic expressions. She generated patterns from these core evaluative words to extract ironic sentences from the corpus. For example, when the word \emph{love} is in the pattern ``NP + would/ 'd/ wouldn't + love'', it is highly possible to be an ironic expression. On a grammatical level, people often use the subjunctive when they intend to be ironic. Besides that, semantic conflict is the most direct way to express ironic meaning. The incompatibility between the main words of the proposition leads to the ridiculousness of the proposition (e.g. It's very considerate of you to make such a loud noise while I was asleep). Besides, \cite{ghosh20181} also categorized irony markers according to trope, morphosyntactic, and typographic types.
\newcite{li2021corpus} considered that ironies are often expressed by specific ``constructions'', especially in short discourses. Larger than “core evaluative words” in \newcite{laszlo2017corpus}, the ``constructions'' mentioned in \newcite{li2021corpus} are mostly in the form of idioms or phrases. The crucial feature of them is the lack of predictability. Most of them do not have to rely on too much contextual information, they themselves can provoke the process of reversal for readers or listeners (e.g. \begin{CJK*}{UTF8}{gbsn}贵人多忘事\end{CJK*} (honorable people frequently forget things)).
\subsubsection{Irony in Communication}
Researchers claim that by using ironies, people have several kinds of intentions.
\textbf{Be polite}: According to \newcite{brown1987politeness}, when unfavorable attitudes such as resistance, criticism, and complaints are stated with irony, the threat to the listener's reputation is reduced. The irony, as stated in \newcite{giora1995irony}, is an indirect negation. People prefer to utilize indirect negation to be polite to their listeners because direct negation can generate great unhappiness;
\textbf{Ease criticisms}: As reported by \newcite{dews1995muting}, irony helps to ease the expression's evaluative function. They believe that the incompatibility between literal meaning and contextual meaning can make it difficult to articulate negative feelings. However, \newcite{toplak2000uses} argued that, while irony literally avoids conflict, it is more aggressive from the perspective of the speaker's goal;
\textbf{Self-protection}: \newcite{sperber1986relevance} proposed the "echoic" idea, which stated that irony is a detached utterance that is simply an echo of another people's thought. It's a self-protection tactic, especially when the speakers are members of marginalized groups. According to \newcite{gibbs2000irony}, the irony is an ``off-record'' statement that allows speakers to deny their true intentions and avoid being challenged;
\textbf{Be amusing}: \newcite{gibbs2000irony} reported that when young people intend to be humorous, 50\% of their communication is ironic. It can assist people in creating a dialogue platform on which speakers and listeners can agree and communicate more easily.
\subsection{Irony and Sarcasm}
Most of the studies saw sarcasm as a subset of irony \cite{tannen2005conversational,barbe1995irony,kumon1995another,leggitt2000emotional,bowes2011sarcasm}. Sarcasm is often recognized as “a nasty, mean-spirited, or just relatively negative form of verbal irony, used on occasion to enhance the negativity expressed relative to direct, non-figurative criticism” \cite{colston2017irony}.
One of the peculiarities of sarcasm is whether or not the speakers intend to offend the listeners. \newcite{kumon1995another}, for example, believe that sarcastic irony always conveys a negative attitude and is intended to harm the object being discussed. The non-sarcastic irony, on the other hand, can communicate either a good or negative attitude, and it is rarely meant to be hurtful. \newcite{barbe1995irony} concurred that the core difference was ``hurtful''. She claimed that irony is a face-saving strategy while sarcasm is a face-threatening action. Ridicule is another feature of sarcasm. According to \newcite{lee1998differential}, sarcasm is closer to ridicule than irony. Their experiment revealed that sarcasm is directed at a single person, but irony is directed toward a large group of people. \newcite{haiman1998talk} claimed that one of the most distinguishing characteristics of sarcasm is that the literal meaning of its words is always positive. However, he did not convey his thoughts on irony. Whereas \newcite{littman1991nature} viewed this topic from another angle. While there are many various forms of ironies, they believe that there is only one type of sarcasm because "sarcasm cannot exist independently of the communication setting."
Cognitive scientists approached the difference in experimental studies. Previous research in child language acquisition \cite{glenwright2010development} reported that children understood sarcastic criticism later than they could understand the non-literal meanings of irony and sarcasm, implying different pragmatic purposes of irony and sarcasm. \newcite{filik2019difference} utilized fMRI and found out sarcasm is associated with wider activation of semantic network in human brains compared to irony.
However, most computational linguistics researchers used irony and sarcasm interchangeably, since the boundary between these two concepts is too vague for even human beings, let alone for machines. \newcite{joshi-etal-2016-challenging} and \newcite{SULIS2016132} verified this claim from both human annotators and computational perspectives. Although in this paper we will mainly focus on the literal \textbf{irony processing} and discuss the inspirations from recent research output in sarcasm processing, we aim to encourage \textbf{unify sarcasm under the framework of irony via fine-grained annotation schemes}.
\section{Irony Datasets Perspectives}
\subsection{Irony Textual Datasets}
Some main target databases for irony processing research include social media platforms like Twitter and online shopping websites like Amazon. For example, \newcite{reyes2012making} collected a 11,861-document irony dataset based on customer reviews from several websites. Some other preliminary attempts to build irony benchmark datasets is \newcite{reyes2012humor} and \newcite{reyes2013multidimensional}, in which they used self-generated hashtag \emph{\#irony} as the gold standard and constructed 40,000-tweet and 50000-tweet datasets from Twitter respectively, each including 10,000 ironic tweets and remaining non-ironic ones. The irony benchmark dataset that is now widely used is from \newcite{van2018semeval}, consisting of 4,792 tweets and half of them were ironic. This dataset was also constructed via searching hashtags including \emph{\#irony}, \emph{\#sarcasm}, and \emph{\#not}.
There were also tremendous attempts to construct benchmark datasets in other languages. For example, \newcite{tang2014chinese} firstly built the NTU Irony Corpus including 1,005 ironic messages from Plurk, a Twitter-like social media platform, by mining specific ironic patterns and manually checking extracted messages. A recent Chinese benchmark dataset for irony detection was constructed by \newcite{xiang-etal-2020-ciron}, which includes 8,766 Weibo posts, labelled from \emph{not ironic} to \emph{strongly ironic} in a five-scale system. Besides, irony datasets in Spanish, Greek, and Italian are also widely available. A comprehensive overview of the irony datasets is listed in Table~\ref{tab:1}.
\begin{table*}[]
\resizebox{\textwidth}{!}{%
\begin{tabular}{cccccc}
\hline
\textbf{Study} & \textbf{Language} & \textbf{Data Source} & \textbf{Construction Methodology} & \textbf{Size} & \textbf{Annotation Scheme} \\ \hline
\newcite{reyes2012making} & English & online websites & filtering low-star reviews & 11861 & ironic / non-ironic \\
\newcite{reyes2013multidimensional} & English & Twitter & \#irony hashtag & 40000 & ironic / non-ironic \\
\newcite{wallace-etal-2014-humans} & English & Reddit & Sub-reddit & 3020 & ironic / non-ironic \\
\newcite{van-hee-etal-2016-exploring} & English, Dutch & Twitter & \#irony, \#sarcasm, and \#not hashtags & 3000, 3179 & three-scale irony annotation \\
\newcite{van2018semeval} & English & Twitter & \#irony, \#sarcasm, and \#not hashtags & 4792 & four-scale ironic types \\
\newcite{tang2014chinese} & Chinese & Plurk & pattern mining & 1005 & annotating ironic elements \\
\newcite{xiang-etal-2020-ciron} & Chinese & Weibo & pattern mining & 8766 & five-scale ironic intensity \\
\newcite{barbieri2016overview} & Italian & Twitter & keywords, hashtags, and etc. & 9410 & ironic / non-ironic \\
\newcite{cignarella2018overview} & Italian & Twitter & keywords, hashtags, and etc. & 4849 & ironic / non-ironic / sarcastic \\
\newcite{charalampakis2015detecting} & Greek & Twitter & keywords & 61427 & ironic / non-ironic \\
\newcite{ortega2019overview} & Spanish & Twitter & comments and tweets & 9000 & ironic / non-ironic \\
\newcite{ghanem2019idat} & Arabic & Twitter & keywords and hashtags & 5030 & ironic / non-ironic \\
\newcite{DBLP:journals/pdln/CorreaCSF21} & Portuguese & Twitter and news articles & keywords, hashtags, and news articles & 34306 & ironic / non-ironic \\
\newcite{vijay2018dataset} & Hindi-English code-mixed & Twitter & keywords and hashtags & 3055 & ironic / non-ironic \\ \hline
\end{tabular}%
}
\caption{Irony benchmark datasets}
\label{tab:1}
\end{table*}
\subsection{Data Source and Construction Methodology}
Twitter, as one of the most trending social platforms, is the major source of irony benchmark datasets. Given Plurk and Weibo's similarity to Twitter, online short-text social medias are almost the only origin for present datasets, which might lead to several potential problems. For example, the limitation of 140-word introduced specific bias towards short-text classification and long texts remain a problem. Besides, the judgment of irony might be highly dependent on contextual information like previous comments or retweets and one single tweet could be meaningless itself. At last, topics on social media platforms might also be highly biased towards political or sports topics. Interestingly, previous research \cite{ghosh20181} reported that for Twitter and Reddit, different ironic markers played the most important roles, further emphasizing the needs of multiple sources for robust NLU.
As for the construction methodology, most datasets adopted "keywords and hashtags" filtering strategy, and some of them followed by human annotations. With annotation schemes put aside, \newcite{doi:10.1177/2053951720972735} explored how self-generated tags could correspond to real labels correctly via a manual semantic analysis. At the worst case, only 16\% of the tagged \emph{\#sarcasm} tweets are unambiguous sarcastic tweets according to well-trained linguists, which further emphasized the necessity of human annotation.
\subsection{Annotation Schemes}
As shown in Table~\ref{tab:1}, various datasets have been labelled differently. Most datasets were annotated as binary classes, ironic versus non-ironic. Both Chinese datasets adopted different strategies by annotating ironic elements and intensity respectively. \newcite{van2018semeval} is the only one doing fine-grained labelling, into verbal irony by polarity contrast, other verbal irony, situational irony, and non-ironic.
Another annotation scheme was proposed by \newcite{cignarella-etal-2020-marking}. They annotated irony activators at the morphosyntactic level and distinguished different types of irony activation. This annotation scheme dived into syntactic information and offered information for analyzing ironical constructions.
\subsection{Future work in datasets}
A high-quality benchmark dataset is crucial to measure NLU capability and advance future research. We are calling improvements for a future benchmark dataset from following perspectives.
\begin{itemize}
\item [1)] Diverse data sources like literature, daily conversations, and news articles should be involved, and the distribution of text length should be relatively balanced.
\item [2)] A uniform annotation scheme and strategy is awaiting for construction. For example, situational irony is needed; ironic intensity should be labelled as reference rather than exclusive labels in \newcite{xiang-etal-2020-ciron}; a scheme to unify sarcasm and irony could be expected.
\item [3)] Multi-X perspectives should be incorporated into the construction of datasets. For example, a multimodal and multilingual dataset could enhance irony identification in a grounded environment.
\end{itemize}
\section{Irony Processing Systems}
In this section, we will discuss the progress of irony processing systems comprehensively, organized along the development of machine learning and deep learning.
\subsection{Irony Detection}
\subsubsection{Rule-based Detection Methods}
\newcite{tang2014chinese} extracted ironic expressions based on five patterns summarized from mandarin Chinese. \newcite{li2020method} further expanded ironic constructions to more than twenty and proposed a systematic irony identification procedure (IIP). Besides Chinese, \newcite{frenda2016computational} utilized sentiment lexicons, verb morphology, quotation marks, and etc. to design an Italian irony detection model and got competitive performance with machine learning models. However, rule-based models are too complex and hard to generalize for wider applications.
\subsubsection{Supervised non-neural network era}
Most research took irony detection as a simple classification problem. Before the popularity of deep learning, feature engineering is crucial for accurate irony detection. Generally, features could be divided into several levels.
\textbf{Lexical features} Lexical features are at the foundational level of NLP features, basically divided into bags of words (BOW) sets, word form sets, and conditional n-gram probabilities \cite{van-hee-etal-2018-usually}. Representative BOW sets mainly include n-grams and character n-grams. Word form sets focus on number and frequency, such as punctuation numbers, emoticon frequencies, character repetitions, etc. Despite their easiness to get, lexical features were proved effective in much research.
\textbf{Syntactic features} Syntax is mainly quantified via parts-of-speech and named entities. After tagging, the number and frequency of both characteristics could act as features in classification models. Besides, hand-crafted syntactic features also included clash before verb tenses \cite{reyes2013multidimensional} and dependency parsing \cite{cignarella-etal-2020-multilingual}.
\textbf{Semantic features} \newcite{van-hee-etal-2018-usually} approached semantic features based on the presence or not in semantic clusters, which were trained on a irony Twitter corpus with the Word2Vec algorithm \cite{NIPS2013_9aa42b31}.
\textbf{Linguistic-motivated features} Irony processing is deeply associated with sentiments and emotions. Therefore, researchers have offered many characteristics to capture irony patterns. For example, \newcite{reyes2013multidimensional} proposed the feature of \emph{contextual imbalance}, which was quantified via measuring the semantic similarity pairwise. Generally, most features could be categorized into \emph{ambiguity} \cite{reyes2012humor} and \emph{incongruity} \cite{joshi-etal-2015-harnessing}. Take incongruity as an example, implicit incongruity was defined as a boolean feature checking containing implicit sentiment phrases or not; explicit incongruity was defined as number of times a polarity contrast appears. Theoretical research \cite{https://doi.org/10.7275/91ey-3n44} is encouraging more semantic and pragmatic features to better capture ironies.
Features at various levels were concatenated with classifiers, including naive bayes, decision tress and support vector machines (SVMs) \cite{van-hee-etal-2018-usually} to get final classification results.
\subsubsection{Supervised Neural Network Era}
There have been irony detection tasks in various languages after 2015, among which most participants used convolutional neural networks (CNNs) and RNNs methods to detect ironical expressions. For example, in SemEval-2018 Task 3 \cite{van2018semeval}, \newcite{wu-etal-2018-thu} classified irony tweets and their ironical types in a densely connected LSTM network, together with multitask learning (MTL) objectives in the optimization. In IroSvA \cite{ortega2019overview}, \newcite{Gonzlez2019ELiRFUPVAI} utilized transformer encoders only for detecting Spanish ironical tweets.
Besides, \newcite{ilic-etal-2018-deep} firstly utilized contextualized word representations ELMo \cite{peters-etal-2018-deep} with Bi-LSTM to detect ironies. \newcite{ZHANG20191633} enhanced irony detection with sentiment corpora based on attention Bi-LSTM RNNs, and achieved state-of-the-art results on \newcite{reyes2013multidimensional} dataset.
\begin{table*}[htb]
\resizebox{\textwidth}{!}{%
\begin{tabular}{ccccc}
\hline
\textbf{Study} & \textbf{Input Features} & \textbf{Architecture} & \textbf{Dataset} & \textbf{\begin{tabular}[c]{@{}c@{}}Performance\\ (F1 Score)\end{tabular}} \\ \hline
\newcite{reyes2013multidimensional} & hand-crafted high-level features & decision tree & \newcite{reyes2013multidimensional}* & 70 / 76 / 73 \\
\newcite{barbieri-saggion-2014-modelling} & frequency, intensity, sentiments, etc. & decision tree & \newcite{reyes2013multidimensional}* & 73 / 75 / 75 \\
\newcite{Nozza2016UnsupervisedID} & unsupervised & topic-irony model & \newcite{reyes2013multidimensional}* & 84.77 / 82.92 / 88.34 \\
\newcite{ZHANG20191633} & word embeddings & sentiment-transferred Bi-LSTM & \newcite{reyes2013multidimensional}* & 94.69 / 95.69 / 96.55 \\
\newcite{van-hee-etal-2018-usually} & lexical, syntactic and semantic features & SVMs & \newcite{van-hee-etal-2016-exploring}** & 70.11 \\
\newcite{rohanian-etal-2018-wlv} & intensity, contrast, topics, etc. & ensemble voting classifier & \newcite{van2018semeval}*** & 65.00 / 41.53 \\
\newcite{wu-etal-2018-thu} & word embeddings, POS tags, sentiments, etc. & LSTM + MTL & \newcite{van2018semeval}*** & 70.54 / 49.47 \\
\newcite{cignarella-etal-2020-multilingual} & mBERT output, autoencoders & LSTM & \newcite{van2018semeval}*** & 70.6\\
\newcite{Potamias_2020} & RoBERTa output & RCNN & \newcite{van2018semeval}*** & 80.0 \\
\newcite{Santilli2018AKA} & word space vectors, BOW sets & SVMs & \newcite{cignarella2018overview}**** & 70.00 / 52.00 \\
\newcite{Cimino2018MultitaskLI} & word embeddings & LSTM + MTL & \newcite{cignarella2018overview}**** & 73.60 / 53.00 \\
\newcite{Gonzlez2019ELiRFUPVAI} & word embeddings & transformer encoders & \newcite{ortega2019overview} & 68.32 \\ \hline
\end{tabular}%
}
\small{* Three binary classification systems were trained respectively for this dataset.}
\small{**This result was obtained on the 3000 English tweets subset.}
\small{***This dataset had two sub-tasks, identifying ironic or not and classifying ironic types. The latter two only focused on the first sub-task.}
\small{****This dataset had two sub-tasks, irony detection and further identifying sarcasm.}
\caption{Representative Irony Detection Systems}
\label{tab:2}
\end{table*}
\subsubsection{Pretraining and Fine-tuning Paradigm}
Since BERT \cite{devlin-etal-2019-bert}, the ``pretraining and fine-tuning paradigm'' has become the mainstream in NLP research due to its extraordinary capacity in dealing with contextualized information and learning general linguistic knowledge. Recent development in irony detection also witnessed the usage of PLMs. \newcite{xiang-etal-2020-ciron} released several baseline results along with the dataset, among which BERT had highest accuracy (5\% higher than Bi-LSTM methods). \newcite{Potamias_2020} proposed a recurrent convolutional neural network (RCNN)-RoBERTa \cite{https://doi.org/10.48550/arxiv.1907.11692} strategy and improved the results on the SemEval-2018 dataset by a large degree. Besides, \newcite{cignarella-etal-2020-multilingual} explored syntax-augmented irony detection with multilingual BERT (mBERT) in multilingual settings. To sum up, some representative studies in irony detection are detailed in Table~\ref{tab:2}.
\subsection{Irony Generation}
Irony generation is mostly an underexplored research field besides \newcite{https://doi.org/10.48550/arxiv.1909.06200}, in which they defined irony generation as a style transfer problem, and utilized a Seq2Seq framework \cite{Sutskever2014SequenceTS} with reinforcement learning to generate ironical counterparts from a non-ironic sentence. Concretely, they designed the overall reward as a harmonic mean of irony reward and sentiment reward, which was trying to capture the sentiment incongruity. In terms of the evaluation, besides traditional natural language generation metrics like BLEU, they also designed task-specific evaluation metrics, which shoule be further enhanced in irony and even figurative language research.
Future work in irony generation could be advanced in new PLMs and theoretical accounts. For example, no attempts were made to generate ironical expressions after generative PLMs like BART \cite{lewis-etal-2020-bart}. Controllable irony generation and its interaction with agents are interesting topics remaining for future exploration. Besides, irony theories could be further utilized. In recent research on unsupervised sarcasm generation \cite{mishra-etal-2019-modular,chakrabarty-etal-2020-r}, context incongruity, valence reversal, and semantic incongruity were merged to enhance the generation.
\section{Discussion}
\subsection{Irony for Downstream NLP Tasks}
Irony is directly associated with downstream NLU tasks like sentiment analysis and opinion mining. For example, the sentence retrieved from \newcite{filatova-2012-irony} \emph{I would recommend this book to friends who have insomnia or those who I absolutely despise.} is classified as positive by fine-tuned sentiment analysis RoBERTa model \cite{heitmann2020}, which is apparently opposite to human evaluation. Wrong sentiment judgments will potentially lead to contrary opinion mining. We suggest that irony could be further captured through introducing incongruity embedding or specific pattern matching. \newcite{joshi-etal-2015-harnessing} designed linguistic-motivated features implicit and explicit incongruity, which are inspiring for enhancing irony understanding. Consider another example task, machine translation, in which wrong translation will potentially lead to totally opposite meanings. We encourage to model discourse features \cite{voigt-jurafsky-2012-towards}, such as ironic patterns and punctuation as embeddings for robust irony translation.
In addition, we are looking forward to the research of irony and sarcasm processing in NLP for social good (NLP4SG), especially considering the strong sentiments hidden in ironies. A recent work \cite{CHIA2021102600} explored cyberbullying detection and this was a starting point to handle online harmful ironical contents.
\subsection{Multi-X Perspectives}
Recent developments in NLP and sarcasm processing encourage us to approach discourse processing from multiple angles. In this part, we will review and suggest several multi-X perspectives for irony and figurative language processing.
\subsubsection{Multimodal Irony Processing}
Linguistic interactions are not solely consisted of texts. Besides, facial expressions and speech communications are crucial to convey emotions and feelings. For example, \newcite{skrelin2020can} reported people could classify ironies based on phonetic characteristics only. Consequently, it is conceivable that multimodal methods could help with irony detection. \newcite{schifanella2016detecting} made the first attempt in multimodal sarcasm detection, in which they extracted posts from three multimodal social media platforms based on hashtags. Then they used SVMs and neural networks to prove the validity of visual information in enhancing sarcasm detection.
\newcite{castro-etal-2019-towards} made a great improvement in multimodal sarcasm detection by introducing audio features into the dataset. The experiments also verified the importance of more modalities in sarcasm processing.
Future work in multimodal irony processing should include a comprehensive multimodal irony dataset based on MUStARD dataset \cite{castro-etal-2019-towards} with more fine-grained annotation schemes. Additionally, most methods \cite{pan-etal-2020-modeling,liu-etal-2021-smile-mean} explored sarcasm by introducing inter-modality and intra-modality attention in single-stream setting. How double-stream multimodal pretrained models (MPMs) will encode and interact in complex discourse settings remains an interesting problem to solve.
\subsubsection{Multilingual Irony Processing}
To understand irony in a multilingual context is even harder due to cultural gaps. Previously listed dataset includes a Hindi-English code-mixed irony dataset \cite{vijay2018dataset}, in which they offered an example:
\begin{itemize}
\item \textbf{Text:} The kahawat ‘old is gold’ purani hogaee. Aaj kal ki nasal kehti hai ‘gold is old’, but the old kahawat only makes sense. \#MindF \#Irony.
\item \textbf{Translation:} The saying ‘old is gold’ is old. Today’s generation thinks ‘gold is old’ but only the old one makes sense. \#MindF \#Irony.
\end{itemize}
\newcite{cignarella-etal-2020-multilingual} explored how mBERT performed in multiple languages' irony detection tasks separately. Given it has been proved code-switching patterns are beneficial for NLP tasks like humor, sarcasm, and hate speech detection in RNNs settings \cite{bansal-etal-2020-code}, A future direction is to merge the irony detection datasets from multiple languages (consider \newcite{karoui-etal-2017-exploring}) or even code-mixed texts, and explore how multilingual datasets could enhance irony understanding in mBERT.
\subsubsection{Multitask Irony Processing}
MTL is to make models learn several tasks simultaneously rather than independently once at a time. Recent work in figurative language processing proposed several MTL strategies to improve the performance interactively. \newcite{chauhan-etal-2020-sentiment} proposed a MTL framework to do sentiment, sarcasm, and emotion analysis simultaneously and the framework yielded better performance with the help of MTL.
Generally, we will classify figurative language into several categories like metaphors, parodies, humors, ironies, and etc. However, noted that there are not clear differences between each other, a single task figurative language processing will only focus on one particular aspect and fail to capture the interactions. Recent work \cite{https://doi.org/10.48550/arxiv.2205.03313} also verified the combination of humor and sarcasm could improve political parody detection.
PLMs could understand figurative language better than random but apparently worse than human evaluation \cite{https://doi.org/10.48550/arxiv.2204.12632}. We suggest that future work should consider domain adaptation towards figurative language as a whole via weak supervision. MTL strategy could utilize previous research in corpus linguistics, and design an appropriate proportion in summing the loss function. Besides, a unified framework to model figurative languages could be expected.
\subsubsection{Multiagent Irony Processing}
Human-like language generation is a central topic in multiagent interactive systems. Besides robots' ironical understanding, we are also curious about how robots could generate ironical expressions. Unlike transferring non-ironic sentences to ironic, multiagent irony measures the performance during the interactions in dialogues. \newcite{Ritschel2019IronyMA} improved the robots by introducing ironic expressions, which showed better user experiences in human evaluation.
Further explorations in multiagent irony could aim at better dialogue state tracking and understand when irony should be introduced.
\subsection{New Tasks: Inspiration from Sarcasm}
Compared to sarcasm, irony is rarely seen as a term in NLP conferences. Recently we have witnessed great improvements in sarcasm processing and in this part we will discuss how new tasks in sarcasm could motivate irony research.
\textbf{Data Collection} As discussed, datasets are highly dependent on hashtags as a signal to extract ironical expressions. \newcite{shmueli-etal-2020-reactive} proposed an algorithm to detect sarcastic tweets from a thread based on exterior cue tweets. A distant supervision based method for extracting ironies from platforms is crucial, given ironies in conversational contexts are central topic in the future.
\textbf{Intended and Perceived Irony} \newcite{oprea-magdy-2019-exploring} explored how author profiling affected the perceived sarcasm (manual labelling) versus the intended sarcasm (hashtags), and verified the difference between both. Further, \newcite{oprea-magdy-2020-isarcasm} introduced iSarcasm dataset which divided intended sarcasms and perceived sarcasms. The state-of-the-art sarcasm detection models performed obviously worse than human evaluation on this dataset. Future work could focus on multimodal perceived and intended irony, especially across various cultures.
\textbf{Target Identification} Sarcasm target identification was firstly proposed in \newcite{joshi-etal-2018-sarcasm}, in which sarcasm targets were classified as one target, several targets and outside. \newcite{patro-etal-2019-deep} introduced sociolinguistic features and a deep learning framework, and improved target identification by a lot. For irony processing, most ironical expressions do not equip a specific target in itself as previously discussed. However, its ironical effects are likely in dialogue or visually grounded environment, which encourages us to enhance irony datasets in aforementioned ways.
\textbf{Irony Explanation} Irony, according to the definition, have opposite real meanings to literal meanings. However, this does not mean adding a single negation could interpret ironies well. \newcite{https://doi.org/10.48550/arxiv.2203.06419} proposed a new task, sarcasm explanation in dialogue. Irony explanation might encounter more complex problems due to relatively low proportion of targets. Still, we should include irony explanation as a branch of multimodal irony processing like \newcite{https://doi.org/10.48550/arxiv.2112.04873}.
\subsection{Explainable Irony Processing}
Explainable machine learning is of interest for most researchers to uncover the blackbox. In irony processing, we are also curious about why specific expressions are recognized as ironies. \newcite{buyukbas2021explainability} explored explainability in irony detection using Shapley Additive Explanations (SHAP) and Local Interpretable Model-Agnostic Explanations (LIME) methods. Results showed that punctuations and strong words play important roles in irony detection.
For future work, we suggest using explainable methods in multimodal settings and check how different modalities act various roles in making a class label.
\section{Conclusion}
In this paper, we reviewed the development in automatic irony processing from underexplored theoretical and cognitive science to computational perspectives, and offered a comprehensive analysis in future directions. We hope that our work and thinking will encourage further interdisciplinary research between linguistics and human language technology, motivate the research interests in irony and even, figurative languages.
\section*{Acknowledgement}
This review is based on the first author's previous research proposal. We would like to express our thanks to Professor Chu-Ren Huang and Dr. Yat Mei Lee for their suggestions.
|
2,869,038,156,414 | arxiv | \section{Introduction}
A prediction for experiment based on perturbative QCD combines
a particular calculation of Feynman diagrams with the use
of general features of the theory. The {\blue{particular
calculation}} is easy at leading order, not so easy at
next-to-leading order and extremely difficult beyond the
next-to-leading order. This calculation of Feynman
diagrams would be a purely academic exercise if we did not use
certain {\blue{general features}} of the theory that
allow the Feynman diagrams to be related to experiment:
\begin{itemize}
\item the renormalization group and the running
coupling;
\item the existence of infrared safe observables;
\item the factorization property that allows
us to isolate hadron structure in parton distribution
functions.
\end{itemize}
In these lectures, I discuss these structural features of the theory
that allow a comparison of theory and experiment. Along the way we
will discover something about certain important processes:
\begin{itemize}
\item ${e^+e^-}$ annihilation;
\item deeply inelastic scattering;
\item hard processes in hadron-hadron collisions.
\end{itemize}
By discussing the particular along with the general, I hope to arm
the reader with information that speakers at research conferences take
to be collective knowledge -- knowledge that they assume the audience
already knows.
Now here is the {\sienna{disclaimer}}. We will not learn how
to do significant calculations in QCD perturbation theory. Three
lectures is not enough for that.
I hope that the reader may be inspired to pursue the subjects
discussed here in more detail. A good source is the {\it Handbook of
Perturbative QCD}\cite{handbook} by the CTEQ collaboration. More
recently, Ellis, Stirling and Webber have written an excellent book
\cite{ESW} that covers the most of the subjects sketched in these
lectures. For the reader wishing to gain a mastery of the theory, I
can recommend the recent books on quantum field theory by Brown
\cite{BrownQFT}, Sterman \cite{StermanQFT}, Peskin and Schroeder
\cite{PeskinQFT}, and Weinberg \cite{WeinbergQFT}. Another good
source, including both theory and phenomenology, is the lectures in
the 1995 TASI proceedings, {\it QCD and Beyond}\cite{TASI}.
\section{Electron-positron annihilation and jets}
In this section, I explore the structure of the final state in QCD.
I begin with the kinematics of $e^+e^- \to 3\ partons$, then examine
the behavior of the cross section for $e^+e^- \to 3\ partons$ when
two of the parton momenta become collinear or one parton momentum
becomes soft. In order to illustrate better what is going on, I
introduce a theoretical tool, null-plane coordinates. Using this
tool, I sketch a space-time picture of the singularities that we
find in momentum space. The singularities of perturbation
theory correspond to long-time physics. We see that the structure of
the final state suggested by this picture conforms well with what is
actually observed.
I draw a the distinction between short-time physics, for which
perturbation theory is useful, and long-time physics, for which the
perturbative expansion is out of control. Finally, I discuss how
certain experimental measurements can probe the short-time physics
while avoiding sensitivity to the long-time physics.
\subsection{Kinematics of $ e^+ e^- \to 3$ partons}
\begin{figure}[htb]
\centerline{\DESepsf(eetojetsA.eps width 4 cm)}
\caption{Feynman diagram for $e^+e^- \to q\,\bar q\,g$.}
\label{eetojetsA}
\end{figure}
Consider the process $e^+e^- \to q\,\bar q\,g$, as illustrated in
Fig.~\ref{eetojetsA}. Let $\sqrt s$ be the total energy in the
c.m.\ frame and let $q^\mu$ be the virtual photon (or Z boson)
momentum, so $q^\mu q_\mu = s$. Let $p_i^\mu$ be the momenta of the
outgoing partons $(q,\bar q,g)$ and let $E_i = p_i^0$ be the
energies of the outgoing partons. It is useful to define energy
fractions $x_i$ by
\begin{equation}
{\blue{x_i = {E_i \over \sqrt s/2}}} = {2 p_i\cdot q\over s}.
\end{equation}
Then
\begin{equation}
{\red{ 0<x_i}}.
\end{equation}
Energy conservation gives
\begin{equation}
{\red{\sum_i x_i}} = {2(\sum p_i)\cdot q\over s} {\red{= 2}}.
\end{equation}
Thus only two of the $x_i$ are independent.
Let $\theta_{ij}$ be the angle between the momenta of partons $i$ and
$j$. We can relate these angles to the momentum fractions as follows:
\begin{equation}
2 p_1\cdot p_2 =
(p_1 + p_2)^2=
(q - p_3)^2 =
s - 2 q\cdot p_3,
\end{equation}
\begin{equation}
2 E_1 E_2 (1 - \cos \theta_{12}) =
s (1-x_3).
\end{equation}
Dividing this equation by $s/2$ and repeating the argument for the
two other pairs of partons, we obtain three relations for the angles
$\theta_{ij}$:
\begin{eqnarray}
{\blue{x_1 x_2 (1 - \cos \theta_{12})}} &=&
{\blue{2 (1-x_3)}},
\nonumber\\
{\blue{x_2 x_3 (1 - \cos \theta_{23})}} &=&
{\blue{2 (1-x_1)}},
\nonumber\\
{\blue{x_3 x_1 (1 - \cos \theta_{31})}} &=&
{\blue{2 (1-x_2)}}.
\end{eqnarray}
We learn two things immediately. First,
\begin{equation}
{\red{x_i <1}} .
\end{equation}
Second, the three possible collinear configurations of the partons
are mapped into $x_i$ space very simply:
\begin{eqnarray}
{\red{\theta_{12}\to 0}} &\Leftrightarrow& {\red{x_3 \to 1}},
\nonumber\\
{\red{\theta_{23}\to 0}} &\Leftrightarrow& {\red{x_1 \to 1}},
\nonumber\\
{\red{\theta_{31}\to 0}} &\Leftrightarrow& {\red{x_2 \to 1}}.
\end{eqnarray}
\begin{figure}[htb]
\centerline{\DESepsf(eetojetsB.eps width 6 cm)}
\caption{Allowed region for $(x_1,x_2)$. Then $x_3$ is $2 - x_1 -
x_2$.}
\label{eetojetsB}
\end{figure}
The relations $0 \le x_i \le 1$, together with $x_3 = 2 - x_1 -
x_2$, imply that the allowed region for $(x_1,x_2)$ is a triangle,
as shown in Fig.~\ref{eetojetsB}. The edges $x_i = 1$ of the allowed
region correspond to two partons being collinear, as shown in
Fig.~\ref{eetojetsC}. The corners $x_i = 0$ correspond to one parton
momentum being soft ($p_i^\mu \to 0$).
\begin{figure}[htb]
\centerline{\DESepsf(eetojetsC.eps width 8 cm)}
\caption{Allowed region for $(x_1,x_2)$. The labels and small
pictures show the physical configuration of the three partons
corresponding to subregions in the allowed triangle.}
\label{eetojetsC}
\end{figure}
\subsection{Structure of the cross section}
One can easily calculate the cross section corresponding to
Fig.~\ref{eetojetsA} and the similar amplitude in which the gluon
attaches to the antiquark line. The result is
\begin{equation}
{1 \over \sigma_0}\,{d \sigma \over d x_1 dx_2}
= {\alpha_s \over 2\pi}C_F
{x_1^2 + x_2^2 \over {\red{(1 - x_1)(1 - x_2)}}},
\end{equation}
where $C_F = 4/3$ and $\sigma_0 = (4 \pi \alpha^2/s)\sum Q_f^2$ is the
total cross section for $e^+ e^- \to hadrons$ at order $\alpha_s^0$.
The cross section has collinear singularities:
\begin{eqnarray}
(1 - x_1) &\to& 0\,, \hskip 1 cm {\rm (2\&3\ collinear)};
\nonumber\\
(1 - x_2) &\to& 0\,, \hskip 1 cm {\rm (1\&3\ collinear)}.
\end{eqnarray}
There is also a singularity when the gluon is soft: $x_3 \to 0$. In
terms of $x_1$ and $x_2$, this singularity occurs when
\begin{equation}
(1 - x_1) \to 0,\quad (1 - x_2) \to 0,\quad
{(1 - x_1) \over (1 - x_2) } \sim const.
\end{equation}
Let us write the cross section in a way that displays the collinear
singularity at $\theta_{31} \to 0$ and the soft singularity at
$E_3\to 0$:
\begin{equation}
{1\over \sigma_0}{d \sigma \over dE_3\, d \cos\theta_{31}}
= {\alpha_s \over 2\pi}\, C_F\,
{{\green{f(E_3,\theta_{31})}}
\over
{\red{E_3(1 - \cos \theta_{31})}}}.
\end{equation}
Here ${\green{f(E_3,\theta_{31})}}$ a rather complicated function.
The only thing that we need to know about it is that it is finite for
$E_3 \to 0$ and for $\theta_{31}\to 0$.
Now look at the collinear singularity, ${\red{\theta_{31}\to 0}}$.
If we integrate over the singular region holding $E_3$ fixed we find
that the integral is divergent:
\begin{equation}
\int_{a}^1d\cos\theta_{31}\
{d \sigma \over dE_3\, d \cos\theta_{31}}
= \log(\infty).
\end{equation}
Similarly, if we integrate over the region of the soft singularity,
holding $\theta_{31}$ fixed, we find that the integral is divergent:
\begin{equation}
\int_0^a dE_3\
{d \sigma \over dE_3\, d \cos\theta_{31}}
= \log(\infty).
\end{equation}
Evidently, perturbation theory is telling us that we should not take
the perturbative cross section too literally. The total cross
section for $e^+e^- \to hadrons$ is certainly finite, so this partial
cross section cannot be infinite. What we are seeing is a breakdown
of perturbation theory in the soft and collinear regions, and we
should understand why.
\begin{figure}[htb]
\centerline{\DESepsf(eetojetsD.eps width 4 cm)}
\caption{Cross section for $e^+e^- \to q\,\bar q\,g$, illustrating the
singularity when the gluon is soft or collinear with the quark.}
\label{eetojetsD}
\end{figure}
Where do the singularities come from? Look at Fig.~\ref{eetojetsD}
(in a physical gauge). The scattering matrix element ${\cal M}$
contains a factor $1/{\orange{(p_1 + p_3)}}^2$ where
\begin{equation}
(p_1 + p_3)^2 = 2 p_1 \cdot p_3 = 2 E_1 E_3 (1 - \cos \theta_{31}).
\end{equation}
Evidently, $1/{\orange{(p_1 + p_3)}}^2$ is singular when
$\theta_{31} \to 0$ and when $E_3 \to 0$. The collinear singularity
is somewhat softened because the numerator of the Feynman diagram
contains a factor proportional to $\theta_{31}$ in the collinear
limit. (This is not exactly obvious, but is easily seen by
calculating. If you like symmetry arguments, you can derive this
factor from quark helicity conservation and overall angular momentum
conservation.) We thus find that
\begin{equation}
|{\cal M}|^2 \propto \left[{\theta_{31} \over E_3
\theta_{31}^2}\right]^2
\end{equation}
for $E_3 \to 0$ and $\theta_{31} \to 0$. Note the universal nature of
these factors.
Integration over the double singular region of the momentum space for
the gluon has the form
\begin{equation}
\int {E_3^2d E_3 d \cos\theta_{31} d \phi \over E_3}
\sim
\int E_3 d E_3 d \theta_{31}^2 d \phi .
\end{equation}
Combining the integration with the matrix element squared gives
\begin{equation}
d \sigma \sim
\int E_3 d E_3 d \theta_{31}^2 d \phi
\left[{\theta_{31} \over E_3 \theta_{31}^2}\right]^2 \sim
\int {{\red{d E_3}}\over {\red{E_3}}}\
{ {\red{d \theta_{31}^2}} \over {\red{\theta_{31}^2}}}\ d \phi.
\end{equation}
Thus we have a double logarithmic divergence in perturbation theory
for the soft and collinear region. With just a little enhancement of
the argument, we see that there is a collinear divergence from
integration over $\theta_{31}$ at finite $E_3$ and a separate soft
divergence from integration over $E_3$ at finite $\theta_{31}$.
Essentially the same argument applies to more complicated graphs.
There are divergences when two final state partons become collinear
and when a final state gluon becomes soft. Generalizing further
\cite{Stermanpinch}, there are also divergences when several final
state partons become collinear to one another or when several (with
no net flavor quantum numbers) become soft.
We have seen that if we integrate over the singular region in
momentum space with no cutoff, we get infinity. The integrals are
logarithmically divergent, so if we integrate with an infrared cutoff
$M_{IR}$, we will get big logarithms of $M_{IR}^2/s$. Thus the
collinear and soft singularities represent perturbation theory out of
control. Carrying on to higher orders of perturbation theory, one gets
\begin{equation}
1 + \alpha_s \times ({\red{{\rm big}}}) +
\alpha_s^2 \times ({\red{{\rm big}}})^2 + \cdots.
\label{outofcontrol}
\end{equation}
If this expansion is in powers of $\alpha_s(M_Z)$, we have $\alpha_s
\ll 1$. Nevertheless, the big logarithms seem to spoil any chance of
the low order terms of perturbation theory being a good approximation
to any cross section of interest. Is the situation hopeless? We
shall have to investigate further to see.
\subsection{Interlude: Null plane coordinates}
\label{NullPlane}
\begin{figure}[htb]
\centerline{\DESepsf(nullplane.eps width 6 cm)}
\caption{Null plane axes in momentum space.}
\label{nullplane}
\end{figure}
In order to understand better the issue of singularities, it is
helpful to introduce a concept that is generally quite useful in high
energy quantum field theory, null plane coordinates. The idea is to
describe the momentum of a particle using momentum components $p^\mu =
(p^+,p^-,p^1,p^2)$ where
\begin{equation}
{\blue{p^\pm = (p^0 \pm p^3)/\sqrt 2}}.
\end{equation}
For a particle with large momentum in the $+z$ direction and limited
transverse momentum, $p^+$ is large and $p^-$ is small. Often one
{\it chooses} the plus axis so that a particle or group of particles
of interest have large $p^+$ and small $p^-$ and $p_T$.
Using null plane components, the covariant square of $p^\mu$ is
\begin{equation}
p^2 = 2 p^+p^- - { \bf p}_T^2.
\end{equation}
Thus, for a particle on its mass shell, $p^-$ is
\begin{equation}
p^- = {{ \bf p}_T^2 + m^2 \over 2p^+}.
\end{equation}
Note also that, for a particle on its mass shell,
\begin{equation}
p^+ > 0\,, \hskip 1 cm p^- >0\,.
\end{equation}
Integration over the mass shell is
\begin{equation}
(2\pi)^{-3} \int {d^3\vec p \over 2 \sqrt{\vec p^2 + m^2}}\cdots
= (2\pi)^{-3}
\int\! d^2{\bf p}_T\, \int_0^\infty\!{dp^+ \over 2 p^+}\cdots.
\end{equation}
We also use the plus/minus components to describe a space-time point
$x^\mu$: ${\blue{x^\pm = (x^0 \pm x^3)/\sqrt 2}}$. In describing a
system of particles moving with large momentum in the plus direction,
we are invited to think of $x^+$ as ``time.'' Classically, the
particles in our system follow paths nearly parallel to the $x^+$
axis, evolving slowly as it moves from one $x^+ = const.$ plane to
another.
We relate momentum space to position space for a quantum system by
Fourier transforming. In doing so, we have a factor $\exp(i p\cdot
x)$, which has the form
\begin{equation}
{\blue{p\cdot x = p^+ x^- + p^- x^+ - {\bf p}_T \cdot{\bf x}_T}}.
\end{equation}
Thus $x^-$ is conjugate to $p^+$ and $x^+$ is conjugate to $p^-$.
That is a little confusing, but it is simple enough.
\subsection{Space-time picture of the singularities}
\begin{figure}[htb]
\centerline{\DESepsf(eetojetsE.eps width 6 cm)
\DESepsf(nullplane2.eps width 6 cm)}
\caption{Correspondence between singularities in momentum space and
the development of the system in space-time.}
\label{nullplane2}
\end{figure}
We now return to the singularity structure of $e^+ e^- \to q \bar q
g$. Define $p_1^\mu + p_3^\mu = k^\mu$. Choose null plane coordinates
with $k^+$ large and ${\bf k}_T= {\bf 0}$. Then $k^2 = 2 k^+k^-$
becomes small when
\begin{equation}
k^- = {{\bf p}_{3,T}^2 \over 2p_1^+} + {{\bf p}_{3,T}^2 \over 2p_3^+}
\end{equation}
becomes small. This happens when ${\bf p}_{3,T}$ becomes small with
fixed $p_1^+$ and $p_3^+$, so that the gluon momentum is nearly
collinear with the quark momentum. It also happens when
${\bf p}_{3,T}$ and $p_3^+$ both become small with $p_3^+ \propto
|{\bf p}_{3,T}|$, so that the gluon momentum is soft. ( It also
happens when the quark becomes soft, but there is a
numerator factor that cancels the soft quark singularity.) Thus the
singularities for a soft or collinear gluon correspond to small
$k^-$.
Now consider the Fourier transform to coordinate space. The quark
propagator in Fig.~\ref{nullplane2} is
\begin{equation}
S_F(k) = \int dx^+ dx^- d{\bf x}\,
\exp(i[k^+{\blue{x^-}} + k^-{\blue{x^+}} - {\bf k}\cdot{{\bf x}}])\
S_F(x).
\end{equation}
When $k^+$ is large and $k^-$ is small, the contributing values of $x$
have small ${\blue{x^-}}$ and large ${\blue{x^+}}$. Thus the
propagation of the virtual quark can be pictured in space-time as in
Fig.~\ref{nullplane2}. The quark propagates a long distance in
the $x^+$ direction before decaying into a quark-gluon pair. That
is, the singularities that can lead to divergent perturbative cross
sections arise from interactions that happen a long time after the
creation of the initial quark-antiquark pair.
\subsection{Nature of the long-time physics}
\begin{figure}[htb]
\centerline{\DESepsf(spacetime.eps width 6 cm)}
\caption{Typical paths of partons in space contributing to $e^+e^-
\to hadrons$, as suggested by the singularities of perturbative
diagrams. Short wavelength fields are represented by classical paths
of particles. Long wavelength fields are represented by wavy lines.}
\label{spacetime}
\end{figure}
Imagine dividing the contributions to a scattering cross section into
long-time contributions and short-time contributions. In the
long-time contributions, perturbation theory is out of control, as
indicated in Eq.~(\ref{outofcontrol}). Nevertheless the generic
structure of the long-time contribution is of great interest. This
structure is illustrated in Fig.~\ref{spacetime}. Perturbative
diagrams have big contributions from space-time histories in which
partons move in collinear groups and additional partons are soft and
communicate over large distances, while carrying small momentum.
The picture of Fig.~\ref{spacetime} is suggested by the singularity
structure of diagrams at any fixed order of perturbation theory. Of
course, there could be nonperturbative effects that would
invalidate the picture. Since nonperturbative effects can be
invisible in perturbation theory, one cannot claim that the
structure of the final state indicated in Fig.~\ref{spacetime} is
known to be a consequence of QCD. One can point, however, to some
cases in which one can go beyond fixed order perturbation theory and
sum the most important effects of diagrams of all orders
(for example, Ref.~\citenum{CSbacktoback}). In such cases, the
general picture suggested by Fig.~\ref{spacetime} remains intact.
We thus find that perturbative QCD suggests a certain structure of
the final state produced in $e^+e^- \to hadrons$: {\blue{the final
state should consist of jets of nearly collinear particles plus
soft particles moving in random directions.}} In fact, this
qualitative prediction is a qualitative success.
Given some degree of qualitative success, we may be bolder and ask
whether perturbative QCD permits quantitative predictions. If we want
quantitative predictions, we will somehow have to find things to
measure that are not sensitive to interactions that happen long after
the basic hard interaction. This is the subject of the next section.
\subsection{The long-time problem}
\label{sec:irsafe}
We have seen that perturbation theory is not effective for long-time
physics. But the detector is a long distance away from the
interaction, so it would seem that long-time physics has to
be present.
Fortunately, there are some measurements that are {\it not} sensitive
to long-time physics. An example is the total cross section to produce
hadrons in $e^+e^-$ annihilation. Here effects from times $\Delta t
\gg 1/\sqrt s$ cancel because of unitarity. To see why, note that
the quark state is created from the vacuum by a current operator $J$
at some time $t$; it then develops from time $t$ to time $\infty$
according to the interaction picture evolution operator $U(\infty,t)$,
when it becomes the final state $|N\rangle$. The cross section is
proportional to the sum over $N$ of this amplitude times a similar
complex conjugate amplitude with $t$ replaced by a different time
$t^\prime$. We Fourier transform this with $\exp(-i \sqrt
s\,(t-t^\prime))$, so that we can take $\Delta t \equiv t - t^\prime$
to be of order $1/\sqrt s$. Now replacing $\sum |N \rangle\langle N|$
by the unit operator and using the unitarity of the evolution
operators $U$, we obtain
\begin{eqnarray}
\lefteqn{\sum_N \langle 0|J(t^\prime)U(t^\prime,\infty)
|N \rangle\langle N|
U(\infty,t)
J(t)|0\rangle}
\\
&&=
\langle 0|J(t^\prime)U(t^\prime,\infty)
U(\infty,t)
J(t)|0\rangle
=
\langle 0|J(t^\prime)U(t^\prime,t)J(t)|0\rangle.
\nonumber
\end{eqnarray}
Because of unitarity, the long-time evolution has canceled out of
the cross section, and we have only evolution from $t$ to $t^\prime$.
There are three ways to view this result. First, we have the formal
argument given above. Second, we have the intuitive understanding
that after the initial quarks and gluons are created in a time
$\Delta t$ of order $1/\sqrt s$, {\it something} will happen with
probability 1. Exactly what happens is long-time physics, but we
don't care about it since we sum over all the possibilities
$|N\rangle$. Third, we can calculate at some finite order of
perturbation theory. Then we see infrared infinities at various
stages of the calculations, but we find that the infinities cancel
between real gluon emission graphs and virtual gluon graphs. An
example is shown in Fig.~\ref{eetojetsF}.
\begin{figure}[htb]
\centerline{\DESepsf(eetojetsF.eps width 8 cm)}
\caption{Cancellation between real and virtual gluon graphs. If we
integrate the real gluon graph on the left times the complex conjugate
of the similar graph with the gluon attached to the antiquark, we will
get an infrared infinity. However the virtual gluon graph on the right
times the complex conjugate of the Born graph is also divergent, as
is the Born graph times the complex conjugate of the virtual gluon
graph. Adding everything together, the infrared infinities cancel.}
\label{eetojetsF}
\end{figure}
We see that the total cross section if free of sensitivity to
long-time physics. If the total cross section were all you could look
at, QCD physics would be a little boring. Fortunately, there are
other quantities that are not sensitive to infrared effects. They are
called {\red{infrared safe quantities}}.
To formulate the concept of infrared safety, consider a measured
quantity that is constructed from the cross sections,
\begin{equation}
{d \sigma[n] \over d \Omega_2 d E_3 d \Omega_3
\cdots d E_n d \Omega_n},
\end{equation}
to make $n$ hadrons in $e^+e^-$ annihilation. Here $E_j$ is the
energy of the $j$th hadron and $\Omega_j = (\theta_j,\phi_j)$
describes its direction. We treat the hadrons as effectively massless
and do not distinguish the hadron flavors. Following the notation of
Ref.~\citenum{KS}, let us specify functions ${\cal S}_n$ that describe
the measurement we want, so that the measured quantity is
\begin{eqnarray}
{\cal I} &=&
{1 \over 2!} \int d \Omega_2\
{d \sigma[2] \over d \Omega_2}\
{\blue{{\cal S}_2}}(p_1^\mu,p_2^\mu)
\nonumber\\
&& +
{1 \over 3!} \int d \Omega_2 d E_3 d \Omega_3\
{d \sigma[3] \over d \Omega_2 d E_3 d \Omega_3}\
{\blue{{\cal S}_3}}(p_1^\mu,p_2^\mu,p_3^\mu)
\nonumber\\
&& +
{1 \over 4!} \int d \Omega_2 d E_3 d \Omega_3 d E_4 d \Omega_4\
\nonumber\\
&&\hskip 2 cm \times
{d \sigma[4] \over d \Omega_2 d E_3 d \Omega_3 d E_4 d \Omega_4}\
{\blue{{\cal S}_4}}(p_1^\mu,p_2^\mu,p_3^\mu,p_4^\mu)
\nonumber\\
&&
+ \cdots .
\end{eqnarray}
The functions $\cal S$ are symmetric functions of their arguments. In
order for our measurement to be infrared safe, we need
\begin{equation}
{\cal S}_{n+1}
(p_1^\mu,\dots,{\green{(1 - \lambda)}}{\red {p_n^\mu}},{\green{\lambda}}
{\red{p_n^\mu}}) =
{\cal S}_n (p_1^\mu,\dots,{\red{p_n^\mu}})
\end{equation}
for $0\le {\green{\lambda}} \le 1$.
\begin{figure}[htb]
\centerline{\DESepsf(irsafe.eps width 6 cm)}
\caption{Infrared safety. In an infrared safe measurement, the three
jet event shown on the left should be (approximately)
equivalent to an ideal three jet event shown on the right.}
\label{irsafe}
\end{figure}
What does this mean? The physical meaning is that the functions
${\cal S}_n$ and ${\cal S}_{n-1}$ are related in such a way that the
cross section is not sensitive to whether or not a mother particle
divides into two collinear daughter particles that share its
momentum. The cross section is also not sensitive to whether or not a
mother particle decays to a daughter particle carrying all of its
momentum and a soft daughter particle carrying no momentum. The cross
section is also not sensitive to whether or not two collinear
particles combine, or a soft particle is absorbed by a fast particle.
All of these decay and recombination processes can happen with large
probability in the final state long after the hard interaction. But,
by construction, they don't matter as long as the sum of the
probabilities for something to happen or not to happen is one.
Another version of the {\blue{physical meaning}} is that for an
IR-safe quantity a physical event with hadron jets should give
approximately the same measurement as a parton event with each jet
replaced by a parton, as illustrated in Fig.~\ref{irsafe}. To see
this, we simply have to delete soft particles and combine collinear
particles until three jets have become three particles.
In a calculation of the measured quantity $\cal I$, we simply
calculate with partons instead of hadrons in the final state. The
{\blue{calculational meaning}} of the infrared safety condition is
that the infrared infinities cancel. The argument is that the
infinities arise from soft and collinear configurations of the
partons, that these configurations involve long times, and that the
time evolution operator is unitary.
I have started with an abstract formulation of infrared safety. It
would be good to have a few examples. The easiest is the total cross
section, for which
\begin{equation}
{\cal S}_n(p_1^\mu,\dots,p_n^\mu) = 1.
\end{equation}
A less trivial example is the {\blue{thrust}} distribution. One
defines the thrust ${\cal T}_n$ of an $n$ particle event as
\begin{equation}
{\cal T}_n(p_1^\mu,\dots,p_n^\mu)
= \max_{\vec u}
{\sum_{i=1}^n | \vec p_i \cdot \vec u |
\over
\sum_{i=1}^n | \vec p_i |}\ .
\end{equation}
Here $\vec u$ is a unit vector, which we vary to maximize the sum of
the absolute values of the projections of $\vec p_i$ on $\vec u$.
Then the thrust distribution $(1/\sigma_{tot})\,d \sigma/ d T$ is
defined by taking
\begin{equation}
{\cal S}_n(p_1^\mu,\dots,p_n^\mu) =
(1/\sigma_{tot})\
\delta\!\left(T - {\cal T}_n(p_1^\mu,\dots,p_n^\mu)\right)\ .
\end{equation}
It is a simple exercise to show that the thrust of an event is not
affected by collinear parton splitting or by zero momentum partons.
Therefore the thrust distribution is infrared safe.
Another infrared safe quantity is the cross sections to make
{\blue{$n$ jets}}. Here one has to define what one means by a jet.
The definitions used in electron-positron annihilation typically
involve successively combining particles that are nearly collinear
to make the jets. A description can be found in Ref.~\citenum{BKSS}.
I discuss jet cross sections for hadron collisions in
Sec.~\ref{jetproduction}.
A final example is the {\blue{ energy-energy correlation}} function
\cite{BBEL}, which measures the average of the product of the energy
in one calorimeter cell times the energy in another calorimeter
cell. One looks at this average as a function of the angular
separation of the calorimeter cells.
Before leaving this subject, I should mention another way to
eliminate sensitivity to long-time physics. Consider the cross
section
\begin{equation}
{ d \sigma(e^+e^- \to \pi + X) \over d E_\pi}.
\end{equation}
This cross section can be written as a convolution of two factors, as
illustrated in Fig.~\ref{decay}. The first factor is a calculated
``hard scattering cross section'' for $e^+e^- \to {quark} + X$
or $e^+e^- \to {gluon} + X$. The second factor is a ``parton
decay function'' for ${quark} \to \pi + X$ or ${gluon} \to
\pi + X$. These functions contain the long-time sensitivity and
are to be measured, since they cannot be calculated perturbatively.
However, once they are measured in one process, they can be used for
another process. This final state factorization is similar to the
initial state factorization involving parton distribution functions,
which we will discuss later. (See
Refs.~\citenum{handbook},\citenum{ESW},\citenum{CSparton} for more
information.)
\begin{figure}[htb]
\centerline{\DESepsf(decay.eps width 4 cm)}
\caption{The cross section for $e^+e^- \to \pi + X$ can be written
as a convolution of a short distance cross section (inside the
dotted line) and a parton decay function.}
\label{decay}
\end{figure}
\section{ The smallest time scales}
\label{smallest}
In this section, I explore the physics of time scales smaller than
$1/\sqrt s$. One way of looking at this physics is to say that it is
plagued by infinities and we can manage to hide the infinities. A
better view is that the short-time physics contains
wonderful truths that we would like to discover -- truths about
grand unified theories, quantum gravity and the like. However,
quantum field theory is arranged so as to effectively hide the truth
from our experimental apparatus, which can probe with a time
resolution of only an inverse half TeV.
I first outline what renormalization does to hide the ugly
infinities or the beautiful truth. Then I describe how
renormalization leads to the running coupling. Because of
renormalization, calculated quantities depend on a
renormalization scale. I look at how this dependence works and how
the scale can be chosen. Finally, I discuss how one can use experiment
to look for the hidden physics beyond the Standard Model, taking
high $E_T$ jet production in hadron collisions as an example.
\subsection{What renormalization does}
In any Feynman graph, one can insert perturbative corrections to the
vertices and the propagation of particles, as illustrated in
Fig.~\ref{nullplane3}. The loop integrals in these graphs will get big
contributions from momenta much larger than $\sqrt s$. That is, there
are big contributions from interactions that happen on time scales
much smaller than $1/\sqrt s$. I have tried to illustrate this in the
figure. The virtual vector boson propagates for a time $1/\sqrt s$,
while the virtual fluctuations that correct the electroweak vertex
and the quark propagator occur over a time $\Delta t$ that can be
much smaller than $1/\sqrt s$.
Let us pick an ultraviolet cutoff $M$ that is much larger than $\sqrt
s$, so that we calculate the effect of fluctuations with $1/M <
\Delta t$ exactly, up to some order of perturbation theory. What,
then, is the effect of virtual fluctuations on smaller time scales,
$\Delta t$ with $\Delta t< 1/M$ but, say, $\Delta t$ still larger
than $t_{\rm Plank}$, where gravity takes over? Let us suppose that
we are willing to neglect contributions to the cross section that are
of order $\sqrt s /M$ or smaller compared to the cross section
itself. Then there is a remarkable theorem \cite{JCCbook}: the effects
of the fluctuations are not particularly small, but they can be
absorbed into changes in the couplings of the theory. (There are also
changes in the masses of the theory and adjustments to the
normalizations of the field operators, but we can concentrate on the
effect on the couplings.)
\begin{figure}[htb]
\centerline{\DESepsf(nullplane3.eps width 6 cm)}
\caption{Renormalization. The effect of the very small time
interactions pictured are absorbed into the running coupling.}
\label{nullplane3}
\end{figure}
The program of absorbing very short-time physics into a few
parameters goes under the name of renormalization. There are several
schemes available for renormalizing. Each of them involves the
introduction of some scale parameter that is not intrinsic to the
theory but tells how we did the renormalization. Let us agree to use
\vbox{\hrule\kern 1pt\hbox{\rm MS}}\ renormalization (see Ref.~\citenum{JCCbook} for details). Then
we introduce an \vbox{\hrule\kern 1pt\hbox{\rm MS}}\ renormalization scale $\mu$. A good (but
approximate) way of thinking of $\mu$ is that the physics of time
scales $\Delta t \ll 1/\mu$ is removed from the perturbative
calculation. The effect of the small time physics is accounted for by
adjusting the value of the strong coupling, so that its value depends
on the scale that we used: $\alpha_s = \alpha_s(\mu)$. (The value of
the electromagnetic coupling also depends on $\mu$.)
\subsection{The running coupling}
\begin{figure}[htb]
\centerline{\DESepsf(runninggraph.eps width 8 cm)}
\caption{Short-time fluctuations in the propagation of the gluon
field absorbed into the running strong coupling.}
\label{runninggraph}
\end{figure}
We account for time scales much smaller than $1/\mu$ by using
the running coupling $\alpha_s(\mu)$. That is, a fluctuation such as
that illustrated in Fig.~\ref{runninggraph} can be dropped from a
calculation and absorbed into the running coupling that describes
the probability for the quark in the figure to emit the gluon.
The $\mu$ dependence of $\alpha_s(\mu)$ is given by a certain
differential equation, called the renormalization group equation (see
Ref.~\citenum{JCCbook}):
\begin{equation}
{ d \over d \ln(\mu^2)}\,{ \alpha_s(\mu) \over \pi} =
\beta(\alpha_s(\mu)) =
- \beta_0\, \left(\alpha_s(\mu)\over \pi\right)^2
- \beta_1\, \left(\alpha_s(\mu)\over \pi\right)^3
+\cdots.
\label{rengrp}
\end{equation}
One calculates the beta function $\beta(\alpha_s)$
perturbatively in QCD. The first coefficient, with the conventions
used here, is
\begin{equation}
\beta_0 = (33 - 2\,N_f)/12\,,
\end{equation}
where $N_f$ is the number of quark flavors.
Of course, at time scales smaller than a very small cutoff $1/M$ (at
the ``GUT scale,'' say) there is completely different physics
operating. Therefore, if we use just QCD to adjust the strong
coupling, we can say that we are accounting for the physics between
times $1/M$ and $1/\mu$. The value of $\alpha_s$ at
$\mu_0 \approx M$ is then the boundary condition for the differential
equation.
\begin{figure}[htb]
\centerline{\DESepsf(scales.eps width 8 cm)}
\caption{Distance scales accounted for by explicit fixed order
perturbative calculation and by use of the renormalization group.}
\label{scales.eps}
\end{figure}
The renormalization group equation sums the effects of short-time
fluctuations of the fields. To see what one means by ``sums''
here, consider the result of solving the renormalization
group equation with all of the $\beta_i$ beyond $\beta_0$ set to
zero:
\begin{eqnarray}
\alpha_s(\mu) &\approx&
\alpha_s(M) - (\beta_0/\pi) \ln(\mu^2/M^2)\ \alpha_s^2(M)
\nonumber\\
&& \quad +
(\beta_0/\pi)^2 \ln^2(\mu^2/M^2)\ \alpha_s^3(M) + \cdots
\nonumber\\
&=&
{\alpha_s(M)\over 1 + (\beta_0/\pi)\,\alpha_s(M) \ln(\mu^2/M^2) }.
\label{running}
\end{eqnarray}
A series in powers of $\alpha_s(M)$ -- that is the strong coupling at
the GUT scale -- is summed into a simple function of $\mu$. Here
$\alpha_s(M)$ appears as a parameter in the solution.
Note a crucial and wonderful fact. The value of $\alpha_s(\mu)$
decreases as $\mu$ increases. This is called ``asymptotic freedom.''
Asymptotic freedom implies that QCD acts like a weakly interacting
theory on short time scales. It is true that quarks and gluons are
strongly bound inside nucleons, but this strong binding is the
result of weak forces acting collectively over a long time.
In Eq.~(\ref{running}), we are invited to think of the graph of
$\alpha_s(\mu)$ versus $\mu$. The differential equation that
determines this graph is characteristic of QCD. There could, however,
be different versions of QCD with the same differential equation
but different curves, corresponding to different boundary values
$\alpha_s(M)$. Thus the parameter $\alpha_s(M)$ tells us which
version of QCD we have. To determine this parameter, we consult
experiment. Actually, Eq.~(\ref{running}) is not the most convenient
way to write the solution for the running coupling. A better
expression is
\begin{equation}
\alpha_s(\mu) \approx
{ \pi
\over
\beta_0 \ln(\mu^2/\Lambda^2)}.
\end{equation}
Here we have replaced $\alpha_s(M)$ by a different (but
completely equivalent) parameter $\Lambda$. A third form of the
running coupling is
\begin{equation}
\alpha_s(\mu) \approx
{\alpha_s(M_Z)\over 1 + (\beta_0/\pi)\,\alpha_s(M_Z)
\ln(\mu^2/M_Z^2) }.
\end{equation}
Here the value of $\alpha_s(\mu)$ at $\mu = M_Z$ labels the version
of QCD that obtains in our world.
In any of the three forms of the running coupling, one should
revise the equations to account for the second term in the beta
function in order to be numerically precise.
\subsection{The choice of scale}
In this section, we consider the choice of the renormalization
scale $\mu$ in a calculated cross section. Consider, as an
example, the cross section for $e^+ e^-\to {\rm hadrons }$ via
virtual photon decay. Let us write this cross section in the form
\begin{equation}
\sigma_{\rm tot} = {4 \pi \alpha^2\over s}\left(\sum_f Q_f^2 \right)
\left[1 + \Delta \right].
\end{equation}
Here $s$ is the square of the c.m.\ energy, $\alpha$ is
$e^2/(4\pi)$, and $Q_f$ is the electric charge in units of $e$
carried by the quark of flavor $f$, with $f = u,d,s,c,b$. The
nontrivial part of the calculated cross section is the quantity
$\Delta$, which contains the effects of the strong interactions.
Using \vbox{\hrule\kern 1pt\hbox{\rm MS}}\ renormalization with scale $\mu$, one finds (after a lot
of work) that $\Delta$ is given by \cite{Levan}
\begin{eqnarray}
&&\Delta =
{\alpha_s({\blue{\mu}})\over \pi}
+ \left[1.4092 + 1.9167\ \ln\left({\blue{\mu^2}} / s\right)
\right] \left({\alpha_s({\blue{\mu}})\over
\pi}\right)^{\!2}
\nonumber\\
&&\quad +
\left[ -12.805 + 7.8186\ \ln\left({\blue{\mu^2}} / s\right)
+ 3.674\ \ln^2\!\left({\blue{\mu^2}} / s\right)\right]
\left({\alpha_s({\blue{\mu}})\over \pi}\right)^{\!3}
\nonumber\\
&&\quad
+\cdots.
\label{eecalc}
\end{eqnarray}
Here, of course, one should use for $\alpha_s(\mu)$ the solution of
the renormalization group equation (\ref{rengrp}) with at least
two terms included.
As discussed in the preceding subsection, when we renormalize
with scale $\mu$, we are defining what we mean by the strong
coupling. Thus $\alpha_s$ in Eq.~(\ref{eecalc}) depends on $\mu$. The
perturbative coefficients in Eq.~(\ref{eecalc}) also depend on $\mu$.
On the other hand, the physical cross section {\red{does not}} depend
on $\mu$:
\begin{equation}
{d \over d \ln \mu^2}\,\Delta = 0.
\label{nomu}
\end{equation}
That is because $\mu$ is just an artifact of how we organize
perturbation theory, not a parameter of the underlying theory.
Let us consider Eq.~(\ref{nomu}) in more detail. Write $\Delta$ in
the form
\begin{equation}
\Delta \sim \sum_{n=1}^\infty c_n(\mu)\ \alpha_s(\mu)^n.
\end{equation}
If we differentiate not the complete infinite sum but just the first
$N$ terms, we get minus the derivative of the sum from $N+1$ to
infinity. This remainder is of order $\alpha_s^{N+1}$ as $\alpha_s
\to 0$. Thus
\begin{equation}
{d \over d \ln \mu^2}\sum_{n=1}^N c_n(\mu)\ \alpha_s(\mu)^n \sim {\cal
O}(\alpha_s(\mu)^{N+1}).
\end{equation}
That is, the harder we work calculating more terms, the less the
calculated cross section depends on $\mu$.
Since we have not worked infinitely hard, the calculated cross
section depends on $\mu$. What choice shall we make for $\mu$?
Clearly, $\ln\left(\mu^2 / s\right)$ should not be big. Otherwise
the coefficients $c_n(\mu)$ are large and the ``convergence'' of
perturbation theory will be spoiled. There are some who will argue
that one scheme or the other for choosing $\mu$ is the ``best.'' You
are welcome to follow whichever advisor you want. I will show you
below that for a well behaved quantity like $\Delta$ the precise
choice makes little difference, as long as you obey the
common sense prescription that $\ln\left(\mu^2 / s\right)$ not be
big.
\subsection{An example}
\label{errorest}
\begin{figure}[htb]
\vskip 2.5 cm
\leftline{\hskip 1.8cm $\Delta(\mu)$}
\vskip 2.5 cm
\centerline{\hskip 1cm $\ln_2(\mu/\sqrt s)$}
\vskip -5.5 cm
\centerline{\DESepsf(eemuA.eps width 8 cm)}
\vskip 0.5 cm
\caption{Dependence of $\Delta(\mu)$ on the \vbox{\hrule\kern 1pt\hbox{\rm MS}}\ renormalization
scale $\mu$. The falling curve is $\Delta_1$. The flatter curve is
$\Delta_2$. The horizontal lines indicates the amount of variation of
$\Delta_2$ when $\mu$ varies by a factor 2.}
\label{eemuA}
\end{figure}
Let us consider a quantitative example of how $\Delta(\mu)$ depends
on $\mu$. This will also give us a chance to think about the
theoretical error caused by replacing $\Delta$ by the sum
$\Delta_n$ of the first $n$ terms in its perturbative expansion. Of
course, we do not know what this error is. All we can do is provide
an estimate. (Our discussion will be rather primitive. For a more
detailed error estimate for the case of the hadronic width of the $Z$
boson, see Ref.~\citenum{SandS}.)
Let us think of the error estimate in the spirit of a ``$1\,\sigma$''
theoretical error: we would be surprised if $|\Delta_n - \Delta|$ were
much less than the error estimate and we would also be surprised
if this quantity were much more than the error estimate. Here,
one should exercise a little caution. We have no reason to
expect that theory errors are gaussian distributed. Thus a
$4\,\sigma$ difference between $\Delta_n$ and $\Delta$ is not out of
the question, while a $4\,\sigma$ fluctuation in a measured quantity
with purely statistical, gaussian errors {\it is} out of the
question.
Take $\alpha_s(M_Z) = 0.117$, $\sqrt s = 34 {\ \rm GeV} $, 5 flavors. In
Fig.~\ref{eemuA}, I plot $\Delta(\mu)$ versus ${\blue{p}}$ defined by
\begin{equation}
\mu = 2^{\blue{p}} \sqrt s.
\end{equation}
The steeply falling curve is the order $\alpha_s^1$ approximation to
$\Delta(\mu)$, $\Delta_1(\mu) = {\alpha_s(\mu)/ \pi}$. Notice that
if we change $\mu$ by a factor 2, $\Delta_1(\mu)$ changes by about
0.006. If we had no other information than this, we might pick
$\Delta_1(\sqrt s) \approx 0.044$ as the ``best'' value and assign a
$\pm 0.006$ error to this value. (There is no special magic to the use
of a factor of 2 here. The reader can pick any factor that seems
reasonable.)
Another error estimate can be based on the simple expectation that
the coefficients of $\alpha_s^n$ are of order 1 for the first few
terms. (Eventually, they will grow like $n!$. Ref.~\citenum{SandS}
takes this into account, but we ignore it here.) Then the first
omitted term should be of order $\pm 1 \times \alpha_s^2 \approx
\pm 0.020$ using $\alpha_s(34\ {\ \rm GeV} ) \approx 0.14$. Since this is
bigger than the previous $\pm 0.006$ error estimate, we keep this
larger estimate: $\Delta \approx 0.044 \pm 0.020$.
Returning now to Fig.~\ref{eemuA}, the second curve is the order
$\alpha_s^2$ approximation, $\Delta_2(\mu)$. Note that $\Delta_2(\mu)$
is less dependent on $\mu$ than $\Delta_1(\mu)$.
What value would we now take as our best estimate of $\Delta$? One
idea is to choose the value of $\mu$ at which $\Delta_2(\mu)$ is
least sensitive to $\mu$. This idea is called the {\it principle of
minimal sensitivity} \cite{PMS}:
\begin{equation}
\Delta_{PMS} = \Delta(\mu_{PMS})\,, \hskip 1 cm
\left[{d \Delta(\mu) \over d \ln \mu}\right]_{\mu = \mu_{PMS}} = 0.
\end{equation}
This prescription gives $\Delta \approx 0.0470$. Note that this is
about 0.003 away from our previous estimate, $\Delta \approx 0.0440$.
Thus our previous error estimate of 0.020 was too big, and we should
be surprised that the result changed so little. We can make a new
error estimate by noting that $\Delta_2(\mu)$ varies by about 0.0012
when
$\mu$ changes by a factor $2$ from $\mu_{PMS}$. Thus we might
estimate that $\Delta \approx 0.0470$ with an error of $\pm 0.0012$.
This estimate is represented by the two horizontal lines in
Fig.~\ref{eemuA}.
An alternative error estimate can be based on the next term being of
order $\pm 1 \times \alpha_s^3(34\ GeV) \approx 0.003$. Since this is
bigger than the previous $\pm 0.0012$ error estimate, we keep this
larger estimate: $\Delta \approx 0.0470 \pm 0.003$.
I should emphasize that there are other ways to pick the ``best''
value for $\Delta$. For instance, one can use the BLM method
\cite{BLM}, which is based on choosing the $\mu$ that sets to zero
the coefficient of the number of quark flavors in $\Delta_2(\mu)$.
Since the graph of $\Delta_2(\mu)$ is quite flat, it makes very
little difference which method one uses.
\begin{figure}[htb]
\vskip 2.5 cm
\leftline{\hskip 1.8cm $\Delta(\mu)$}
\vskip 2.5 cm
\centerline{\hskip 1.0cm$\log_2(\mu/\sqrt s)$}
\vskip -5.5 cm
\centerline{\DESepsf(eemuB.eps width 8 cm)}
\vskip 0.5 cm
\caption{Dependence of $\Delta(\mu)$ on the \vbox{\hrule\kern 1pt\hbox{\rm MS}}\ renormalization
scale $\mu$. The falling curve is $\Delta_1$. The flatter curve is
$\Delta_2$. The still flatter curve is $\Delta_3$.}
\label{eemuB}
\end{figure}
Now let us look at $\Delta(\mu)$ evaluated at order $\alpha_s^3$,
$\Delta_3(\mu)$. Here we make use of the full formula in
Eq.~(\ref{eecalc}). In Fig.~\ref{eemuB}, I plot $\Delta_3(\mu)$ along
with $\Delta_2(\mu)$ and $\Delta_1(\mu)$. The variation of
$\Delta_3(\mu)$ with $\mu$ is smaller than that of
$\Delta_2(\mu)$. The improvement is not overwhelming, but is
apparent particularly at small $\mu$.
\begin{figure}[htb]
\vskip 2.5 cm
\leftline{\hskip 1.8cm $\Delta(\mu)$}
\vskip 2 cm
\centerline{\hskip 1.0cm $\log_2(\mu/\sqrt s)$}
\vskip -5.5 cm
\centerline{\DESepsf(eemuC.eps width 8 cm)}
\vskip 0.5 cm
\caption{Dependence of $\Delta(\mu)$ on the \vbox{\hrule\kern 1pt\hbox{\rm MS}}\ renormalization
scale $\mu$ with an expanded scale. The falling curve is $\Delta_1$.
The flatter curve is $\Delta_2$. The still flatter curve is
$\Delta_3$. The horizontal lines represent the variation of
$\Delta_2$ when $\mu$ varies by a factor 2.}
\label{eemuC}
\end{figure}
It is a little difficult to see what is happening in
Fig.~\ref{eemuB}, so I show the same thing with an expanded scale in
Fig.~\ref{eemuC}. (Here the error band based on the $\mu$
dependence of $\Delta_2$ is also indicated. Recall that we decided
that this error band was an underestimate.) The curve for
$\Delta_3(\mu)$ has zero derivative at two places. The corresponding
values are $\Delta \approx 0.0436$ and $\Delta \approx 0.0456$. If I
take the best value of $\Delta$ to be the average of these two values
and the error to be half the difference, I get $\Delta \approx 0.0446
\pm 0.0010$.
The alternative error estimate is $\pm 1 \times \alpha_s^4(34\ {\ \rm GeV} )
\approx 0.0004$. We keep the larger error estimate of $\pm 0.0010$.
Was the previous error estimate valid? We guessed
$\Delta \approx 0.0470 \pm 0.003$. Our new best estimate is 0.0446.
The difference is 0.0024, which is in line with our previous error
estimate. Had we used the error estimate $\pm 0.0012$ based on the
$\mu$ dependence, we would have underestimated the difference,
although we would not have been too far off.
\subsection{Beyond the Standard Model}
We have seen how the renormalization group enables us to account for
QCD physics at time scales much smaller than $\sqrt s$, as indicated
in Fig.~\ref{scales}. However, at some scale $\Delta t \sim 1/M$, we
run into the unknown!
{\blue{How can we see the unknown}} in current experiments?
First, the unknown physics affects $\alpha_s$, $\alpha_{em}$,
$\sin^2(\theta_W)$. Second, the unknown physics affects masses of
$u,d,\dots,e,\mu,\dots$. That is, the unknown physics (presumably)
determines the {\blue{parameters}} of the Standard Model. These
parameters have been well measured. Thus, a Nobel prize awaits the
physicist who figures out how to use a model for the unknown physics
to predict these parameters.
\begin{figure}[htb]
\centerline{\DESepsf(scales.eps width 8 cm)}
\caption{Time scales accounted for by fixed order perturbative
calculations and by use of the renormalization group.}
\label{scales}
\end{figure}
\begin{figure}[htb]
\centerline{\DESepsf(newphys.eps width 6 cm)}
\caption{New physics at a TeV scale. In the first diagram, quarks
scatter by gluon exchange. In the second diagram, the quarks
exchange a new object with a TeV mass, or perhaps exchange some of
the constituents out of which quarks are made.}
\label{newphys}
\end{figure}
There is another way that as yet unknown physics can affect
current experiments. Suppose that quarks can scatter by the exchange
of some new particle with a heavy mass $M$, as illustrated in
Fig.~\ref{newphys}, and suppose that this mass is not too enormous,
only a few TeV. Perhaps the new particle isn't a particle at all, but
is a pair of constituents that live inside of quarks. As mentioned
above, this physics affects the parameters of the Standard Model.
However, unless we can predict the parameters of the Standard Model,
this effect does not help us. There is, however, another possible
clue. The physics at the TeV scale can introduce {\blue{new terms}}
into the lagrangian that we can investigate in current experiments.
In the second diagram in Fig.~\ref{newphys}, the two vertices are
never at a separation in time greater than $1/M$, so that our low
energy probes cannot resolve the details of the structure. As long
as we stick to low energy probes, $\sqrt s \ll M$, the effect of the
new physics can be summarized by adding new terms to the lagrangian of
QCD. A typical term might be
\begin{equation}
{\blue{\Delta{\cal L} = {\tilde g^2 \over M^2} \
\bar \psi \gamma^\mu \psi\
\bar \psi \gamma_\mu \psi.}}
\label{newterm}
\end{equation}
There is a factor $\tilde g^2$ that represents how well the new
physics couples to quarks. The most important factor is the factor
$1/M^2$. This factor must be there: the product of field operators has
dimension 6 and the lagrangian has dimension 4, so there must be a
factor with dimension $-2$. Taking this argument one step further,
the product of field operators in $\Delta {\cal L}$ must have a
dimension greater than 4 because any product of field
operators having dimension equal to or less than 4 that respects the
symmetries of the Standard Model is already included in the
lagrangian of the Standard Model.
\subsection{Looking for new terms in the effective lagrangian}
How can one detect the presence in the lagrangian of a term like that
in Eq.~(\ref{newterm})? These terms are small. Therefore we need
either a high precision experiment, or an experiment that looks for
some effect that is forbidden in the Standard Model, or an experiment
that has moderate precision and operates at energies that are as high
as possible.
Let us consider an example of the last of these
possibilities, $p + \bar p \to jet + X$ as a function of the
transverse energy ($\sim P_T$) of the jet. The new term in the
lagrangian should add a little bit to the observed cross section that
is not included in the standard QCD theory. When the transverse
energy $E_T$ of the jet is small compared to $M$, we expect
\begin{equation}
{\sienna{{{\rm Data} - {\rm Theory} \over {\rm Theory}}}}
\propto \tilde g^2 { {\red{E_T^2}} \over M^2}.
\label{dataup}
\end{equation}
Here the factor $\tilde g^2 / M^2$ follows because $\Delta {\cal L}$
contains this factor. The factor $E_T^2$ follows because the left
hand side is dimensionless and $E_T$ is the only factor with
dimension of mass that is available.
\begin{figure}[htb]
\centerline{\DESepsf(Jetcteq3.eps width 8 cm)}
\caption{Jet cross sections from CDF and D0 compared to QCD theory.
(Data $-$ Theory)/Theory is plotted versus the transverse
energy $E_T$ of the jet. The theory here is next-to-leading order
QCD using the CTEQ3M parton distribution. Source:
Ref.~\protect\citenum{CTEQ4}}
\label{Jetcteq3}
\end{figure}
In Fig.~\ref{Jetcteq3}, I show a plot comparing experimental jet
cross sections from CDF \cite{CDFjet} and D0 \cite{D0jet} compared to
next-to-leading order QCD theory. The theory works fine for $E_T< 200
{\ \rm GeV} $, but for $200 {\ \rm GeV} < E_T$, there appears to be a systematic
deviation of just the form anticipated in Eq.~(\ref{dataup}).
This example illustrates the idea of how small distance physics
beyond the Standard Model can leave a trace in the form of small
additional terms in the effective lagrangian that controls physics
at currently available energies. However, in this case, there is some
indication that the observed effect might be explained by some
combination of the experimental systematic error and the
uncertainties inherent in the theoretical prediction \cite{CTEQjet}.
In particular, the prediction is sensitive to the distributions of
quarks and gluons contained in the colliding protons, and the gluon
distribution in the kinematic range of interest here is rather poorly
known. In the next section, we turn to the definition, use, and
measurement of the distributions of quarks and gluons in hadrons.
\section{Deeply inelastic scattering}
Until now, I have concentrated on hard scattering processes with
leptons in the initial state. For such processes, we have seen that
the hard part of the process can be described using perturbation
theory because $\alpha_s(\mu)$ gets small as $\mu$ gets large.
Furthermore, we have seen how to isolate the hard part of the
interaction by choosing an infrared safe observable. But what about
hard processes in which there are hadrons in the initial state? Since
the fundamental hard interactions involve quarks and gluons,
the theoretical description necessarily involves a description of how
the quarks and gluons are distributed in a hadron. Unfortunately,
the distribution of quarks and gluons in a hadron is controlled by
long-time physics. We cannot calculate the relevant
distribution functions perturbatively (although a calculation in
lattice QCD might give them, in principle). Thus we must find how to
separate the short-time physics from the parton distribution
functions and we must learn how the parton distribution functions
can be determined from the experimental measurements.
In this section, I discuss parton distribution functions and their
role in deeply inelastic lepton scattering (DIS). This includes $e +
p \to e + X$ and $\nu + p \to e + X$ where the momentum transfer
from the lepton is large. I first outline the kinematics of deeply
inelastic scattering and define the structure functions $F_1$, $F_2$
and $F_3$ used to describe the process. By examining the
space-time structure of DIS, we will see how the cross section can
be written as a convolution of two factors, one of which is the
parton distribution functions and the other of which is a cross
section for the lepton to scatter from a quark or gluon. This
factorization involves a scale $\mu_F$ that, roughly speaking,
divides the soft from the hard regime; I discuss the dependence of
the calculated cross section on $\mu_F$. With this groundwork laid,
I give the \vbox{\hrule\kern 1pt\hbox{\rm MS}}\ definition of parton distribution functions in
terms of field operators and discuss the evolution equation for the
parton distributions. I close the section with some comments on how
the parton distributions are, in practice, determined from experiment.
\subsection{Kinematics of deeply inelastic lepton scattering}
\begin{figure}[htb]
\centerline{\DESepsf(DISA.eps width 6 cm)}
\caption{Kinematics of deeply inelastic scattering}
\label{DISA}
\end{figure}
In deeply inelastic scattering, a lepton with momentum $k^\mu$
scatters on a hadron with momentum $p^\mu$. In the
final state, one observes the scattered lepton with momentum
$k^{\prime\mu}$ as illustrated in Fig.~\ref{DISA}. The momentum
transfer
\begin{equation}
q^\mu = k^\mu - k^{\prime \mu}
\end{equation}
is carried on a photon, or a $W$ or $Z$ boson.
The interaction between the vector boson and the hadron depends
on the variables $q^\mu$ and $p^\mu$. From these two vectors we can
build two scalars (not counting $m^2 = p^2$). The first variable is
\begin{equation}
{\blue{Q^2}} = - q^2 ,
\end{equation}
where the minus sign is included so that $Q^2$ is positive. The
second scalar is the dimensionless Bjorken variable,
\begin{equation}
{\blue{x_{\rm bj}}} = { Q^2 \over 2 p \cdot q}.
\end{equation}
(In the case of scattering from a nucleus containing $A$ nucleons,
one replaces $p^\mu$ by $p^\mu /A$ and defines $x_{\rm bj} = A\,
{Q^2 / (2 p \cdot q)}$.)
One calls the scattering {\it deeply inelastic} if $Q^2$ is large
compared to $1\ {\ \rm GeV} ^2$. Traditionally, one speaks of the {\it
scaling limit}, $Q^2 \to \infty$ with $x_{\rm bj}$ fixed. Actually,
the asymptotic theory to be described below works pretty well if
$Q^2$ is bigger than, say, $4\, {\ \rm GeV} ^2$ and $x_{\rm bj}$ is anywhere
in the experimentally accessible range, roughly $10^{-4} < x_{\rm bj}
< 0.5$.
The invariant mass squared of the hadronic final state is
$W^2 = (p + q)^2$. In the scaling regime of large $Q^2$ one has
\begin{equation}
W^2 = m^2 + {1-x_{\rm bj} \over x_{\rm bj}}\, Q^2 \gg m^2.
\end{equation}
This justifies saying that the scattering is not only inelastic but
deeply inelastic.
We have spoken of the scalar variables that one can form from
$p^\mu$ and $q^\mu$. Using the lepton momentum $k^\mu$, one can
also form the dimensionless variable
\begin{equation}
y = {p\cdot q \over p\cdot k}.
\end{equation}
\subsection{Structure functions for DIS}
One can make quite a lot of progress in understanding the theory of
deeply inelastic scattering without knowing anything about QCD
except its symmetries. One expresses the cross section in terms of
three structure functions, which are functions of $x_{\rm bj}$ and
$Q^2$ only.
Suppose that the initial lepton is a neutrino, $\nu_\mu$, and the
final lepton is a muon. Then in Fig.~\ref{DISA} the exchanged vector
boson, call it $V$, is a $W$ boson, with mass $M_V = M_W$.
Alternatively, suppose that both the initial and final leptons are
electrons and let the exchanged vector boson be a photon, with mass
$M_V = 0$. This was the situation in the original DIS experiments at
SLAC in the late 1960's. In experiments with sufficiently large
$Q^2$, $Z$ boson exchange should be considered along with photon
exchange, and the formalism described below must be augmented.
Given only the electroweak theory to tell us how the vector boson
couples to the lepton, one can write the cross section in the form
\begin{equation}
d \sigma = {4 \alpha^2 \over s}{d^3 {\bf k}^\prime \over 2|{\bf k}^\prime|}
{C_V \over (q^2 - M_V^2)^2}\,
{\green{L^{\mu\nu}}}(k,q)\,{\blue{W_{\mu\nu}}}(p,q),
\label{DIS1}
\end{equation}
where $C_V$ is 1 in the case that $V$ is a photon and $1/(64
\,\sin^4\theta_W)$ in the case that $V$ is a $W$ boson.
The tensor ${\green{L^{\mu\nu}}}$ describes the lepton coupling to the
vector boson and has the form
\begin{equation}
{\green{L^{\mu\nu}}} = {1 \over 2}
{\rm Tr}\left(k\cdot \gamma\ \gamma^\mu k^\prime\cdot
\gamma\ \gamma^\nu
\right).
\label{DIS2}
\end{equation}
in the case that $V$ is a photon. For a $W$ boson, one has
\begin{equation}
{\green{L^{\mu\nu}}} =
{\rm Tr}\left(k\cdot \gamma\ \Gamma^\mu k^\prime\cdot
\gamma\ \Gamma^\nu
\right).
\label{DIS3}
\end{equation}
where $\Gamma^\mu$ is $\gamma^\mu(1-\gamma_5)$ for a $W^+$
boson ($\nu \to W^+ \ell$) or $\gamma^\mu(1+\gamma_5)$ for a $W^-$
boson ($\bar \nu \to W^- \bar\ell$). See Ref.~\citenum{handbook}.
The tensor $W^{\mu\nu}$ describes the coupling of the vector boson
to the hadronic system. It depends on $p^\mu$ and $q^\mu$. We know
that it is Lorentz invariant and that $W^{\nu\mu} = W^{\mu\nu*}$. We
also know that the current to which the vector boson couples is
conserved (or in the case of the axial current, conserved in the
absence of quark masses, which we here neglect) so that $q_\mu
W^{\mu\nu} = 0$. Using these properties, one finds three possible
tensor structures for $W^{\mu\nu}$. Each of the three tensors
multiplies a structure function, $F_1$, $F_2$ or $F_3$, which, since
it is a Lorentz scalar, can depend only on the invariants $x_{\rm
bj}$ and $Q^2$. Thus
\begin{eqnarray}
{\blue{W_{\mu\nu}}} &=&
-\left(
g_{\mu\nu}- {q_\mu q_\nu \over q^2}
\right)
{\red{F_1(x_{\rm bj},Q^2)}}
\nonumber\\
&&\quad
+
\left(
p_\mu - q_\mu {p\cdot q \over q^2}
\right)
\left(
p_\nu - q_\nu {p\cdot q \over q^2}
\right)
{ 1\over p \cdot q }\
{\red{F_2(x_{\rm bj},Q^2)}}
\nonumber\\
&&\quad
-i \epsilon_{\mu\nu\lambda\sigma}p^\lambda q^\sigma
{ 1\over p \cdot q }\
{\red{ F_3(x_{\rm bj},Q^2)}}.
\label{DIS4}
\end{eqnarray}
If we combine Eqs.~(\ref{DIS1},\ref{DIS2},\ref{DIS3},\ref{DIS4}), we
can write the cross section for deeply inelastic scattering in terms
of the three structure functions. Neglecting the hadron mass compared
to $Q^2$, the result is
\begin{equation}
{d \sigma \over dx_{\rm bj}\, dy} =
{\green{\tilde N(Q^2)}}\left[
y {\red{F_1}}
+ {1-y\over x_{\rm bj} y}
{\red{F_2}}
+ {\green{\delta_V}}\,(1-{y\over 2}) {\red{F_3}}
\right].
\label{DIS5}
\end{equation}
Here the normalization factor ${\green{\tilde N}}$ and the factor
$\delta_V$ multiplying $F_3$ are
\begin{eqnarray}
{\green{\tilde N}} &=& {4\pi \alpha^2 \over Q^2},
\hskip 3.42 cm
{\green{\delta_V}} = 0,
\hskip 0.55 cm
e^-\! + h \to e^- + X,
\nonumber\\
{\green{\tilde N}} &=&
{\pi \alpha^2 Q^2\over 4 \sin^4\!(\theta_W)\ (Q^2 + M_W)^2},
\hskip 0.5 cm
{\green{\delta_V}} = 1,
\hskip 0.55 cm
\nu + h \to \mu^- + X,
\nonumber\\
{\green{\tilde N}} &=& {\pi \alpha^2 Q^2\over 4 \sin^4\!(\theta_W)\
(Q^2 + M_W)^2},
\hskip 0.5 cm
{\green{\delta_V}} = -1,
\hskip 0.2 cm
\bar\nu + h \to \mu^+ +X.
\end{eqnarray}
In principle, one can use the $y$ dependence to determine all three
of $F_1,F_2,F_3$ in a deeply inelastic scattering experiment.
\subsection{Space-time structure of DIS}
So far, we have used the symmetries of QCD in order to write the
cross section for deeply inelastic scattering in terms of three
structure functions, but we have not used any other dynamical
properties of the theory. Now we turn to the question of how the
scattering develops in space and time.
\begin{figure}[htb]
\centerline{\DESepsf(DISframe.eps width 6 cm)}
\caption{Reference frame for the analysis of deeply inelastic
scattering.}
\label{DISframe}
\end{figure}
For this purpose, we define a convenient reference frame, which is
illustrated in Fig.~\ref{DISframe}. Denoting components of vectors
$v^\mu$ by
$(v^+,v^-,{\bf v}_T)$, we chose the frame in which
\begin{equation}
(q^+,q^-,{\bf q}) = {1 \over \sqrt 2}\
(-Q,Q,{\bf 0}).
\end{equation}
We also demand that the transverse components of the hadron momentum
be zero in our frame. Then
\begin{equation}
(p^+,p^-,{\bf p}) \approx {1 \over \sqrt 2}\
({Q \over x_{\rm bj}},{x_{\rm bj} m_h^2 \over Q},{\bf 0}).
\end{equation}
Notice that in the chosen reference frame the hadron momentum is
big and the momentum transfer is big.
\begin{figure}[htb]
\centerline{\DESepsf(DISB.eps width 6 cm)}
\caption{Interactions within a fast moving hadron. The lines
represent world lines of quarks and gluons. The interaction points
are spread out in $x^+$ and pushed together in $x^-$.}
\label{DISB}
\end{figure}
Consider the interactions among the quarks and gluons inside a
hadron, using $x^+$ in the role of ``time'' as in Section
\ref{NullPlane}. For a hadron at rest, these interactions happen in a
typical time scale $\Delta x^+ \sim 1/m$, where $m \sim 300\ {\ \rm MeV} $.
A hadron that will participate in a deeply inelastic scattering event
has a large momentum, $p^+ \sim Q$, in the reference
frame that we are using. The Lorentz transformation from the rest
frame spreads out interactions by a factor $Q/m$, so that
\begin{equation}
\Delta x^+ \sim {1 \over m}\times {\blue{{Q \over m}}}
= {Q \over m^2}.
\end{equation}
This is illustrated in Fig.~\ref{DISB}.
I offer two caveats here. First, I am treating $x_{\rm bj}$ as being
of order 1. To treat small $x_{\rm bj}$ physics, one needs to put
back the factors of $x_{\rm bj}$, and the picture changes rather
dramatically. Second, the interactions among the quarks and gluons
in a hadron at rest can take place on time scales $\Delta x^+$ that
are much smaller than $1/m$, as we discussed in Section
\ref{smallest}. We will discuss this later on, but for now we start
with the simplest picture.
\begin{figure}[htb]
\centerline{\DESepsf(DISC.eps width 6 cm)}
\caption{The virtual photon meets the fast moving hadron. One of the
partons is annihilated and recreated as a parton with a large
minus component of momentum. This parton develops into a jet of
particles.}
\label{DISC}
\end{figure}
What happens when the fast moving hadron meets the virtual photon?
The interaction with the photon carrying momentum $q^- \sim Q$ is
localized to within
\begin{equation}
\Delta x^+ \sim {1 / Q}.
\end{equation}
During this short time interval, the quarks and gluons in the proton
are effectively free, since their typical interaction times are
comparatively much longer.
We thus have the following picture. At the moment $x^+$ of the
interaction, the hadron effectively consists of a collection of
quarks and gluons ({\it partons}) that have momenta $(p_i^+,{\bf
p}_i)$. We can treat the partons as being free. The $p_i^+$ are
large, and it is convenient to describe them using momentum fractions
$\xi_i$:
\begin{equation}
{\blue{\xi_i = p_i^+/p^+}},\hskip 1 cm 0<\xi_i < 1.
\end{equation}
(This is convenient because the $\xi_i$ are invariant under boosts
along the $z$ axis.) The transverse momenta of the partons, ${\bf
p}_i$, are small compared to $Q$ and can be neglected in the
kinematics of the $\gamma$-parton interaction. The ``on-shell'' or
``kinetic'' minus momenta of the partons, $p_i^- = {\bf p}_i^2/(2p_i
^+)$, are also very small compared to $Q$ and can be neglected in the
kinematics of the $\gamma$-parton interaction. We can think of the
partonic state as being described by a wave function
\begin{equation}
\psi(p_1^+, {\bf p}_1;p_2^+, {\bf p}_2;\cdots),
\end{equation}
where indices specifying spin and flavor quantum numbers have been
suppressed.
\begin{figure}[htb]
\centerline{\DESepsf(DISD.eps width 8 cm)}
\caption{Feynman diagram for deeply inelastic scattering.}
\label{DISD}
\end{figure}
This approximate picture is represented in Feynman diagram language
in Fig.~\ref{DISD}. The larger filled circle represents the hadron
wave function $\psi$. The smaller filled circle represents a sum of
subdiagrams in which the particles have virtualities of order $Q^2$.
All of these interactions are effectively instantaneous on the time
scale of the intra-hadron interactions that form the wave function.
The approximate picture also leads to an intuitive formula that
relates the observed cross section to the cross section for
$\gamma$-parton scattering:
\begin{equation}
{{\green{d \sigma}} \over dE^\prime\, d\omega^\prime} \sim
\int_0^1\! d \xi \sum_a\ {\blue{f_{\! a/h}\!(\xi,\mu)}}\
{{\red{d \hat\sigma}}_{\!a}(\mu) \over dE^\prime\, d\omega^\prime}
+{\cal O}(m/Q).
\label{factor}
\end{equation}
In Eq.~(\ref{factor}), the function $f$ is a parton distribution
function: ${\blue{f_{\! a/h}\!(\xi,\mu)}}\ d\xi$ gives
probability to find a parton with flavor $a = g,u,\bar u, d, \dots$,
in hadron $h$, carrying momentum fraction within $d\xi$ of $\xi =
p_i^+/p^+$. If we knew the wave functions $\psi$, we would form $f$
by summing over the number $n$ of unobserved partons, integrating
$|\psi_n|^2$ over the momenta of the unobserved partons, and also
integrating over the transverse momentum of the observed parton.
The second factor in Eq.~(\ref{factor}), ${{\red{d\hat \sigma}}_{\!a}
/ dE^\prime\, d\omega^\prime}$, is the cross section for scattering
the lepton from the parton of flavor $a$ and momentum fraction $\xi$.
I have indicated a dependence on a factorization scale $\mu$ in both
factors of Eq.~(\ref{factor}). This dependence arises from
the existence of virtual processes among the partons that take place
on a time scale much shorter than the nominal $\Delta x^+ \sim
Q/m^2$. I will discuss this dependence in some detail shortly.
\subsection{The hard scattering cross section}
The parton distribution functions in Eq.~(\ref{factor}) are derived
from experiment. The hard scattering cross sections ${{\red{d
\hat\sigma}}_{\!a}(\mu) / dE^\prime\, d\omega^\prime}$ are
calculated in perturbation theory, using diagrams like those shown
in Fig.~\ref{DISE}. The diagram on the left is the lowest order
diagram. The diagram on the right is one of several that contributes
to $d \hat\sigma$ at order $\alpha_s$; in this diagram the parton
$a$ is a gluon.
\begin{figure}[htb]
\centerline{\DESepsf(DISE.eps width 9 cm)}
\centerline{\hskip 0.4 in Lowest order.\hskip 1 in Higher order.}
\caption{Some Feynman diagrams for the hard scattering part of
deeply inelastic scattering.}
\label{DISE}
\end{figure}
\begin{figure}[htb]
\centerline{\DESepsf(DISframeB.eps width 6 cm)}
\caption{Kinematics of lowest order diagram.}
\label{DISframeB}
\end{figure}
One can understand a lot about deeply inelastic scattering from
Fig.~\ref{DISframeB}, which illustrates the kinematics of the lowest
order diagram. Recall that in the reference frame that we are using,
the virtual vector boson has zero transverse momentum. The incoming
parton has momentum along the plus axis. After the scattering, the
parton momentum must be on the cone $k_\mu k^\mu = 0$, so the only
possibility is that its minus momentum is non-zero and its plus
momentum vanishes. That is
\begin{equation}
\xi p^+ + q^+ = 0.
\end{equation}
Since $p^+ = Q/(x_{\rm bj}\sqrt 2)$ while $q^+ = - Q/\sqrt 2$, this
implies
\begin{equation}
{\blue{\xi = x_{\rm bj}}}.
\end{equation}
The consequence of this is that the lowest order contribution
to $d\hat \sigma$ in Eq.~(\ref{factor}) contains a delta function that
sets $\xi$ to $x_{\rm bj}$. Thus deeply inelastic scattering at a
given value of $x_{\rm bj}$ provides a determination of the
parton distribution functions at momentum fraction $\xi$ equal to
$x_{\rm bj}$, as long as one works only to leading order. In fact,
because of this close relationship, there is some tendency to confuse
the structure functions $F_n(x_{\rm bj}, Q^2)$ with the parton
distribution functions $f_{a,h}(\xi,\mu)$. I will try to keep these
concepts separate: the structure functions $F_n$ are something that
one measures directly in deeply inelastic scattering; the parton
distribution functions are determined rather indirectly from
experiments like deeply inelastic scattering, using formulas that are
correct only up to some finite order in
$\alpha_s$.
\subsection{Factorization for the structure functions}
We will look at DIS in a little detail since it is so important. Our
object is to derive a formula relating the measured structure functions
to structure functions calculated at the parton level. Then we will
look at the parton level calculation at lowest order.
Start with Eq.~(\ref{factor}), representing Fig.~\ref{DISD}. We
change variables in this equation from $(E^\prime,\omega^\prime)$ to
$(x_{\rm bj},y)$. We relate $x_{\rm bj}$ to the momentum fraction
$\xi$ and a new variable $\hat x$ that is just $x_{\rm bj}$ with the
proton momentum $p^\mu$ replaced by the parton momentum $\xi p^\mu$:
\begin{equation}
{\green{x_{\rm bj}}} = { Q^2 \over 2 p \cdot q}
= \xi\ { Q^2 \over 2 \xi p \cdot q}
={\blue{\xi \hat x}}.
\end{equation}
That is, $\hat x$ is the parton level version of $x_{\rm bj}$.
The variable $y$ is identical to the parton level version of $y$
because $p^\mu$ appears in both the numerator and denominator:
\begin{equation}
y = {p\cdot q \over p\cdot k}={\xi p\cdot q \over \xi p\cdot k} .
\end{equation}
Thus Eq.~(\ref{factor}) becomes
\begin{equation}
{d \sigma \over {\green{dx_{\rm bj}}}\, dy} \sim
\int_0^1\! d \xi \sum_a\ f_{\! a/h}\!(\xi)\
{1 \over {\blue{\xi}}}
\left[{d \hat\sigma_{\!a}\over
{\blue{d\hat x}}\,dy}\right]_{\hat x = x_{\rm bj}/\xi}
+{\cal O}(m/Q).
\label{factorA}
\end{equation}
Now recall that, for $\gamma$ exchange, $d\sigma/( dx_{\rm bj} dy)$ is
related to the structure functions by Eq.~(\ref{DIS5}):
\begin{equation}
{\green{{d \sigma\over dx_{\rm bj}\,dy}}}
=
\tilde N(Q^2)\!\left[
y\, {\green{F_1}}(x_{\rm bj},Q^2)
+ {1-y\over x_{\rm bj}\, y}\,
{\green{F_2}}(x_{\rm bj},Q^2)
\right]
+{\cal O}(m/Q).
\label{factorB}
\end{equation}
We define structure functions $\hat F_n$ for partons in the same way:
\begin{equation}
{\blue{{d \hat\sigma_{\!a}\over d\hat x\,dy}}}
=
\tilde N(Q^2)\left[
y\, {\blue{\hat F_1^a}}(x_{\rm bj}/\xi,Q^2)
+ {1-y\over (x_{\rm bj}/{\red{\xi}})y}\,
{\blue{\hat F_2^a}}(x_{\rm bj}/\xi,Q^2)
\right].
\label{factorC}
\end{equation}
We insert Eq.~(\ref{factorC}) into Eq.~(\ref{factorA}) and compare
to Eq.~(\ref{factorB}). We deduce that the structure functions can be
factored as
\begin{equation}
{\green{F_1}}(x_{\rm bj},Q^2) \sim
\int_0^1\! {\red{d \xi}} \sum_a\ f_{\! a/h}\!(\xi)\
{1 \over {\red{\xi}}}
{\blue{\hat F_1^a}}(x_{\rm bj}/\xi,Q^2)
+{\cal O}(m/Q),
\label{factorF1}
\end{equation}
\begin{equation}
{\green{F_2}}(x_{\rm bj},Q^2) \sim
\int_0^1\! {\red{d \xi}} \sum_a\ f_{\! a/h}\!(\xi)\
\hat {\blue{F_2^a}}(x_{\rm bj}/\xi,Q^2)
+{\cal O}(m/Q).
\label{factorF2}
\end{equation}
A simple calculation gives $\hat F_1$ and $\hat F_2$ at lowest order:
\begin{equation}
{\blue{\hat F_1^a}}(x_{\rm bj}/\xi,Q^2) =
{1\over 2}Q_a^2\ \delta(x_{\rm bj}/\xi - 1) + {\cal O}(\alpha_s),
\end{equation}
\begin{equation}
{\blue{\hat F_2^a}}(x_{\rm bj}/\xi,Q^2) =
Q_a^2\ \delta(x_{\rm bj}/\xi - 1) + {\cal O}(\alpha_s).
\end{equation}
Inserting these results into Eqs.~(\ref{factorF1}) and
(\ref{factorF2}), we obtain the lowest order relation between the
structure functions and the parton distribution functions:
\begin{equation}
{\green{F_1}}(x_{\rm bj},Q^2) \sim
{1\over 2}\sum_a\ Q_a^2\ {\red{f_{\! a/h}\!(x_{\rm bj})}}
+ {\cal O}(\alpha_s) +{\cal O}(m/Q),
\end{equation}
\begin{equation}
{\green{F_2}}(x_{\rm bj},Q^2) \sim
\sum_a\ Q_a^2\ {\red{x_{\rm bj}\,f_{\! a/h}\!(x_{\rm bj})}}
+ {\cal O}(\alpha_s) +{\cal O}(m/Q).
\end{equation}
The factor 1/2 between $x_{\rm bj} F_1$ and $F_2$ follows from the
Feynman diagrams for spin 1/2 quarks.
\subsection{$\mu_{F}$ dependence}
\begin{figure}[htb]
\centerline{\DESepsf(DISF.eps width 6 cm)}
\caption{Deeply inelastic scattering with a gluon emission.}
\label{DISF}
\end{figure}
I have so far presented a rather simplified picture of deeply
inelastic scattering in which the hard scattering takes place on a
time scale $\Delta x^+ \sim 1/Q$, while the internal dynamics of the
proton take place on a much longer time scale $\Delta x^+ \sim
Q/m^2$. What happens when one actually computes Feynman diagrams and
looks at what time scales contribute? Consider the graph shown in
Fig.~\ref{DISF}. One finds that the transverse momenta ${\bf k}$
range from order $m$ to order $Q$, corresponding to energy scales
$k^- = {\bf k}^2/2k^+$ between $k^- \sim m^2/Q$ and $k^- = Q^2/Q \sim
Q$, or time scales $Q/m^2 \lesssim \Delta x^+ \lesssim 1/Q$.
The property of factorization for the cross section of deeply
inelastic scattering, embodied in Eq.~(\ref{factor}), is established
by showing that the perturbative expansion can be rearranged so that
the contributions from long time scales appear in the parton
distribution functions, while the contributions from short time
scales appear in the hard scattering functions. (See
Ref.~\citenum{factor} for more information.) Thus, in
Fig.~\ref{DISF}, a gluon emission with ${\bf k}^2 \sim m^2$ is part of
${\blue{f(\xi)}}$, while a gluon emission with ${\bf k}^2 \sim Q^2$ is
part of ${\red{d \hat\sigma}}$.
Breaking up the cross section into factors associated with short and
long time scales requires the introduction of a {\it factorization
scale}, $\mu_F$. When calculating the diagram in Fig.~\ref{DISF}, one
integrates over $\bf k$. Roughly speaking, one counts the
contribution from ${\bf k}^2 < \mu_{F}^2$ as part of the higher
order contribution to $\phi(\xi)$, convoluted with the lowest order
hard scattering function $d\hat\sigma$ for deeply inelastic
scattering from a quark. The contribution from $\mu_{F}^2 < {\bf
k}^2$ then counts as part of the higher order contribution to
$d\hat\sigma$ convoluted with an uncorrected parton distribution.
This is illustrated in Fig.~\ref{scalesB}. (In real calculations, the
split is accomplished with the aid of dimensional regularization, and
is a little more subtle than a simple division of the integral into
two parts.)
\begin{figure}[htb]
\centerline{\DESepsf(scalesB.eps width 8 cm)}
\caption{Distance scales in factorization.}
\label{scalesB}
\end{figure}
A consequence of this is that both ${\red{{d
\hat\sigma_{\!a}(\mu_{\!F}) / dE^\prime\, d\omega^\prime}}}$\ and
${\blue{f_{\!a/h}(\xi,\mu_{\!F})}}$ depend on $\mu_{\!F}$.
Thus we have two scales, the factorization scale $\mu_{F}$ in
$f_{\!f/h}(\xi,\mu_{\!F})$ and the renormalization scale $\mu$ in
$\alpha_s(\mu)$. As with $\mu$, the cross section does not depend on
$\mu_F$. Thus there is an equation $d ({\it cross\ section})/d\mu_F =
0$ that is satisfied to the accuracy of the perturbative calculation
used. If you work harder and calculate to higher order, then the
dependence on $\mu_F$ is less.
Often one sets $\mu_F = \mu$ in applied calculations. In fact, it is
rather common in applications to deeply inelastic scattering to set
$\mu_F = \mu = Q$.
\subsection{Contour graphs of scale dependence}
As an example, look at the one jet inclusive cross section in
proton-antiproton collisions. Specifically, consider the cross section
$d\sigma/d E_T d\eta$ to make a collimated spray of particles, a {\it
jet}, with transverse energy $E_T$ and rapidity $\eta$. (Here $E_T
$ is essentially the transverse momentum carried by the particles in
the jet and $\eta$ is related to the angle between the jet and
the beam direction by $\eta \equiv \ln(\tan(\theta/2)$). We will
investigate this process and discuss the definitions in the next
section. For now, all we need to know is that the theoretical formula
for the cross section at next-to-leading order involves the strong
coupling $\alpha_s(\mu)$ and two factors $f_{a/h}(x,\mu_F)$
representing the distribution of partons in the two incoming hadrons.
There is a parton level hard scattering cross section that also
depends on $\mu$ and $\mu_F$.
\begin{figure}[htb]
\centerline{\DESepsf(mu100.eps width 7 cm)
\DESepsf(mu500.eps width 7 cm)}
\centerline{\hskip 1 cm
$E_T = 100 {\ \rm GeV} $\hskip 4 cm
$E_T = 500 {\ \rm GeV} $}
\caption{Contour plots of the one jet inclusive cross section
versus the renormalization scale
$\mu$ and the factorization scale $\mu_F$. The cross section is
$d\sigma / dE_T d\eta$ at $\eta = 0$ with $E_T = 100\ {\ \rm GeV} $ in
the first graph and $E_T = 500\ {\ \rm GeV} $ in the second. The horizontal
axis in each graph represents $N_{UV} \equiv \log_2(2\mu/E_T)$ and the
vertical axis represents $N_{CO} \equiv \log_2(2\mu_F/E_T)$. The
contour lines show 5\% changes in the cross section relative to the
cross section at the center of the figures. The c.m energy is
$\protect\sqrt s = 1800\
{\ \rm GeV} $.}
\label{mu100500}
\end{figure}
How does the cross section depend on $\mu$ in $\alpha_s(\mu)$ and
$\mu_F$ in $f_{a/h}(x,\mu_F)$? In Fig.~\ref{mu100500}, I show contour
plots of the jet cross section versus $\mu$ and $\mu_F$ at two
different values of $E_T$. The center of the plots corresponds to a
standard choice of scales, $\mu = \mu_F = E_T/2$. The axes are
logarithmic, representing $\log_2(2\mu/E_T)$ and $\log_2(2\mu_F/E_T)$.
Thus $\mu$ and $\mu_F$ vary from $E_T/8$ to $2 E_T$ in the plots.
Notice that the dependence on the two scales is rather mild for the
next-to-leading order cross section. The cross section
calculated at leading order is quite sensitive to these scales,
but most of the scale dependence found at order $\alpha_s^2$ has been
canceled by the $\alpha_s^3$ contributions to the cross section.
One reads from the figure that the cross section varies by
roughly ${\blue{\pm 15\%}}$ in the central region of the graphs, both
for medium and large $E_T$. Following the argument of
Sec.~\ref{errorest}, this leads to a rough estimate of 15\% for the
theoretical error associated with truncating perturbation theory at
next-to-leading order.
\subsection{\vbox{\hrule\kern 1pt\hbox{\rm MS}}\ definition of parton distribution functions}
The factorization property, Eq.~(\ref{factor}), of the deeply
inelastic scattering cross section states that the cross section
can be approximated as a convolution of a hard scattering cross
section that can be calculated perturbatively and parton
distribution functions $f_{a/A}(x,\mu)$. But what are the parton
distribution functions? This question has some practical importance.
The hard scattering cross section is essentially the physical
cross section divided by the parton distribution function, so the
precise definition of the parton distribution functions leads to the
rules for calculating the hard scattering functions.
The definition of the parton distribution functions is to some
extent a matter of convention. The most commonly used convention is
the \vbox{\hrule\kern 1pt\hbox{\rm MS}}\ definition, which arose from the theory of deeply
inelastic scattering in the language of the ``operator product
expansion.''\cite{msbar} Here I will follow the
(equivalent) formulation of Ref.~\citenum{CSparton}. For a more
detailed pedagogical review, the reader may consult
Ref.~\citenum{DESlattice}.
Using the \vbox{\hrule\kern 1pt\hbox{\rm MS}}\ definition, the distribution of quarks in a hadron
is given as the hadron matrix element of certain quark field
operators:
\begin{equation}
{\blue{f_{i/h}(\xi,\mu_F)}}=
{1 \over 2}\int {dy^- \over 2\pi}\ e^{-i \xi p^+ y^-}
\langle p| {\blue{\bar\psi}}_i(0,y^-,{\bf 0}) \gamma ^+
{\magenta{F}}
{\blue{\psi}}_i(0)|p\rangle.
\end{equation}
Here $|p\rangle$ represents the state of a hadron with
momentum $p^\mu$ aligned so that $p_T = 0$. For simplicity,
I take the hadron to have spin zero. The operator $\psi(0)$,
evaluated at $x^\mu = 0$, annihilates a quark in the hadron. The
operator ${\blue{\bar\psi}}_i(0,y^-,{\bf 0})$ recreates the quark at
$x^+ = {\bf x}_T = 0$ and $x^- = y^-$, where we take the appropriate
Fourier transform in $y^-$ so that the quark that was annihilated and
recreated has momentum $k^+ = \xi p^+$. The motivation for the
definition is that this is the hadron matrix element of the
appropriate number operator for finding a quark.
There is one subtle point. The number operator idea corresponds to a
particular gauge choice, $A^+ = 0$. If we are using any other gauge,
we insert the operator
\begin{equation}
{\magenta{F}}=
{\cal P}\exp\left(
-ig\int_0^{y^-} dz^- {\blue{A}}_a^+(0,z^-,{\bf 0})\, t_a
\right).
\end{equation}
The ${\cal P}$ indicates a path ordering of the operators and
color matrices along the path from $(0,0,{\bf 0})$ to $(0,y^-,{\bf
0})$. This operator is the identity operator in $A^+ = 0$ gauge and
it makes the definition gauge invariant.
\begin{figure}[htb]
\centerline{\DESepsf(DISC.eps width 6 cm)
\DESepsf(DISG.eps width 6 cm)}
\centerline{\hskip 1 cm DIS \hskip 1.5 cm Parton distribution}
\caption{Deeply inelastic scattering and the parton distribution
functions.}
\label{DISG}
\end{figure}
The physics of this definition is illustrated in Fig.~\ref{DISG}. The
first picture (from Fig.~\ref{DISC}) illustrates the amplitude for
deeply inelastic scattering. The fast proton moves in the plus
direction. A virtual photon knocks out a quark, which emerges moving
in the minus direction and develops into a jet of particles. The
second picture illustrates the amplitude associated with the quark
distribution function. We express $F$ as $F_2 F_1$ where
\begin{eqnarray}
{\magenta{F_2}} &=&
\bar{\cal P}\exp\left(
+ig\int^\infty_{y^-}\!\! dz^- {\blue{A}}_a^+(0,z^-,{\bf 0})\, t_a
\right),
\nonumber\\
{\magenta{F_1}} &=&
{\cal P}\exp\left(
-ig\int_0^{\infty}\!\! dz^- {\blue{A}}_a^+(0,z^-,{\bf 0})\, t_a
\right).
\end{eqnarray}
and write the quark distribution function including a sum over
intermediate states $|N\rangle$:
\begin{equation}
{\blue{f_{i/h}(\xi,\mu_F)}}=
{1 \over 2}\int {dy^- \over 2\pi}\ e^{-i \xi p^+ y^-}
\sum_N
\langle p| {\blue{\bar\psi}}_i(0,y^-,{\bf 0}) \gamma ^+
{\magenta{F_2}}
|N\rangle \langle N|
{\magenta{F_1}}
{\blue{\psi}}_i(0)|p\rangle.
\end{equation}
Then the amplitude depicted in the second picture in Fig.~\ref{DISG}
is $\langle N|{\magenta{F_1}} {\blue{\psi}}_i(0)|p\rangle$. The
operator $\psi$ annihilates a quark in the proton. The operator
$F_1$ stands in for the quark moving in the minus direction. The gluon
field $A$ evaluated along a lightlike line in the minus direction
absorbs longitudinally polarized gluons from the color field of the
proton, just as the real quark in deeply inelastic scattering can
do. Thus the physics of deeply inelastic scattering is built into
the definition of the quark distribution function, albeit in an
idealized way. The idealization is not a problem because the hard
scattering function $d \hat \sigma$ systematically corrects for the
difference between real deeply inelastic scattering and the
idealization.
There is one small hitch. If you calculate any Feynman diagrams for
${\blue{f_{i/h}(\xi,\mu_F)}}$, you are likely to wind up with an
ultraviolet-divergent integral. The operator product that is part of
the definition needs renormalization. This hitch is only a small
one. We simply agree to do all of the renormalization using the
\vbox{\hrule\kern 1pt\hbox{\rm MS}}\ scheme for renormalization. It is this renormalization that
introduces the scale $\mu_F$ into ${\blue{f_{i/h}(\xi,\mu_F)}}$.
This role of $\mu_F$ is in accord with Fig.~\ref{scalesB}: roughly
speaking $\mu_F$ is the upper cutoff for what momenta belong with
the parton distribution function; at the same time it is the lower
cutoff for what momenta belong with the hard scattering function.
What about gluons? The definition of the gluon distribution function
is similar to the definition for quarks. We simply replace the quark
field $\psi$ by suitable combinations of the gluon field $A^\mu$, as
described in Refs.~\citenum{CSparton} and \citenum{DESlattice}.
\subsection{Evolution of the parton distributions}
Since we introduced a scale $\mu_F$ in the definition of the
parton distributions in order to define their renormalization,
there is a renormalization group equation that gives the $\mu_F$
dependence
\begin{equation}
{d \over d \ln \mu_F}f_{a/h}(x,\mu_F) =
\sum_b \int_x^1
{d \xi \over \xi}\
P_{ab}(x/\xi,\alpha_s(\mu_F))\
f_{b/h}(\xi,\mu_F).
\label{AP}
\end{equation}
This is variously known as the evolution equation, the
Altarelli-Parisi equation, and the DGLAP
(Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) equation. Note the sum
over parton flavor indices. The evolution of, say, an up quark ($a =
u$) can involve a gluon ($b = g$) through the element $P_{ug}$ of the
kernel that describes gluon splitting into $\bar u u$.
The equation is illustrated in Fig.~\ref{APeqn}. When we change the
renormalization scale $\mu_F$, the change in the probability to find a
parton with momentum fraction $x$ and flavor $a$ is proportional to
the probability to find such a parton with large transverse momentum.
The way to get this parton with large transverse momentum is for a
parton carrying momentum fraction $\xi$ and much smaller transverse
momentum to split into partons carrying large transverse momenta,
including the parton that we are looking for. This splitting
probability, integrated over the appropriate transverse momentum
ranges, is the kernel $P_{ab}$.
\begin{figure}[htb]
\centerline{\DESepsf(APeqn.eps width 8 cm)}
\caption{The renormalization for the parton distribution functions.}
\label{APeqn}
\end{figure}
The kernel $P$ in Eq.~(\ref{AP}) has a perturbative expansion
\begin{equation}
P_{ab}(x/\xi,\alpha_s(\mu_F)) =
P_{ab}^{(1)}(x/\xi)\
{\alpha_s(\mu_F) \over \pi}
+
P_{ab}^{(2)}(x/\xi)\
\left({\alpha_s(\mu_F) \over \pi}\right)^2
+\cdots.
\end{equation}
The first two terms are known and are typically used in numerical
solutions of the equation. To learn more about the DGLAP equation,
the reader may consult Refs.~\citenum{handbook} and
\citenum{DESlattice}.
\subsection{Determination and use of the parton distributions}
The \vbox{\hrule\kern 1pt\hbox{\rm MS}}\ definition giving the parton distribution in terms of
operators is process independent -- it does not refer to any
particular physical process. These parton distributions then appear in
the QCD formula for {\blue{any process with one or two hadrons in the
initial state}}. In principle, the parton distribution functions
could be calculated by using the method of lattice QCD (see
Ref.~\citenum{DESlattice}). Currently, they are determined from
experiment.
Currently the most comprehensive analyses are being done by the CTEQ
\cite{CTEQ4} and MRS \cite{MRSpartons} groups. These groups
perform a ``global fit'' to data from experiments of several
different types. To perform such a fit one chooses a parameterization
for the parton distributions at some standard factorization scale
$\mu_0$. Certain sum rules that follow from the definition of the
parton distribution functions are built into the parameterization. An
example is the momentum sum rule:
\begin{equation}
\sum_a \int_0^1 d\xi\ \xi\ f_{a/h}(\xi,\mu) = 1.
\end{equation}
Given some set of values for the parameters describing the
$f_{a/h}(x,\mu_0)$, one can determine $f_{a/h}(x,\mu)$ for all
higher values of $\mu$ by using the evolution equation. Then the QCD
cross section formulas give predictions for all of the experiments
that are being used. One systematically varies the parameters in
$f_{a/h}(x,\mu_0)$ to obtain the {\it best} fit to all of the
experiments. One source of information about these fits is the
World Wide Web pages of Ref.~\citenum{potpourri}.
If the freedom available for the parton distributions is used to fit
all of the world's data, is there any physical content to QCD? The
answer is yes: there are lots of experiments, so this program won't
work unless QCD is right. In fact, there are roughly 1400 data in the
CTEQ fit and only about 25 parameters available to fit these data.
\section{ QCD in hadron-hadron collisions}
When there is a hadron in the initial state of a scattering process,
there are inevitably long time scales associated with the binding
of the hadron, even if part of the process is a short-time
scattering. We have seen, in the case of deeply inelastic
scattering of a lepton from a single hadron, that the dependence on
these long time scales can be factored into a parton distribution
function. But what happens when two high energy hadrons collide? The
reader will not be surprised to learn that we then need two parton
distribution functions.
I explore hadron-hadron collisions in this section. I begin with the
definition of a convenient kinematical variable, rapidity. Then I
discuss, in turn, production of vector bosons ($\gamma^*$, $W$, and
$Z$), heavy quark production, and jet production.
\subsection{Kinematics: rapidity}
In describing hadron-hadron collisions, it is useful to employ a
kinematic variable $y$ that is called {\it rapidity}. Consider, for
example, the production of a $Z$ boson plus anything, $p + \bar p \to
Z + X$. Choose the hadron-hadron c.m.\ frame with the $z$ axis along
the beam direction. In Fig.~\ref{ppcollision}, I show a drawing of
the collision. The arrows represent the momenta of the two hadrons;
in the c.m.\ frame these momenta have equal magnitudes. We will want
to describe the process at the parton level, $a + b \to Z + X$.
The two partons $a$ and $b$ each carry some share of the parent
hadron's momentum, but generally these will not be equal shares.
Thus the magnitudes of the momenta of the colliding partons will not
be equal. We will have to boost along the $z$ axis in order to get
to the parton-parton c.m.\ frame. For this reason, it is useful to use
a variable that transforms simply under boosts. This is the
motivation for using rapidity.
\begin{figure}[htb]
\centerline{\DESepsf(ppcollision.eps width 4 cm)}
\caption{Collision of two hadrons containing partons producing a $Z$
boson. The c.m.\ frame of the two hadrons is normally $\it not$ the
c.m.\ frame of the two partons that create the $Z$ boson.}
\label{ppcollision}
\end{figure}
Let $q^\mu = (q^+,q^-,{\bf q})$ be the momentum
of the $Z$ boson. Then the rapidity of the $Z$ is defined as
\begin{equation}
y = {1 \over 2} \ln\!\left({q^+ \over q^-}\right).
\end{equation}
The four components $(q^+,q^-,{\bf q})$ of the $Z$ boson momentum can
be written in terms of four variables, the two components of the $Z$
boson's transverse momentum ${\bf q}$, its mass $M$, and its rapidity:
\begin{equation}
q^\mu = (e^y\sqrt{({\bf q}^2 + M^2)/2},\
e^{-y}\sqrt{({\bf q}^2 + M^2)/2}
,\ {\bf q}).
\end{equation}
The utility of using rapidity as one of the variables stems from the
transformation property of rapidity under a boost along the $z$ axis:
\begin{equation}
q^+ \to e^\omega q^+,\quad
q^- \to e^{-\omega} q^-,\quad
{\bf q} \to {\bf q}.
\end{equation}
Under this transformation,
\begin{equation}
y \to y + \omega.
\end{equation}
This is as simple a transformation law as we could hope for. In
fact, it is just the same as the transformation law for velocities
in non-relativistic physics in one dimension.
\begin{figure}[htb]
\centerline{\DESepsf(rapidity.eps width 10 cm)}
\caption{Definition of the polar angle $\theta$ used in calculating
the rapidity of a massless particle.}
\label{rapidity}
\end{figure}
Consider now the rapidity of a {\blue{massless particle}}. Let the
massless particle emerge from the collision with polar angle
$\theta$, as indicated in Fig.~\ref{rapidity}. A simple calculation
relates the particle's rapidity $y$ to $\theta$:
\begin{equation}
y = -\ln\left(\tan(\theta/2)\right) \,,
\hskip 1 cm (m=0).
\end{equation}
Another way of writing this is
\begin{equation}
\tan \theta = 1/\sinh y \,,
\hskip 1 cm (m=0).
\end{equation}
One also defines the {\it pseudorapidity} $\eta$ of a particle,
massless or not, by
\begin{equation}
\eta = -\ln\left(\tan(\theta/2)\right)
\hskip 1 cm {\rm or} \hskip 1 cm
\tan \theta = 1/\sinh \eta.
\end{equation}
The relation between rapidity and pseudorapidity is
\begin{equation}
\sinh \eta = \sqrt{1 + m^2/q_T^2}\ \sinh y.
\end{equation}
Thus, if the particle isn't quite massless,
$\eta$ may still be a good approximation
to $y$.
\subsection{$\gamma^*$, $W$, $Z$ production in hadron-hadron
collisions}
\
Consider the process
\begin{equation}
A + B \to Z + X,
\label{makeaZ}
\end{equation}
where $A$ and $B$ are high energy hadrons. Two features of this
reaction are important for our discussion. First, the
mass of the $Z$ boson is large compared to 1 GeV, so that a process
with a small time scale $\Delta t \sim 1/M_Z$ must be involved in the
production of the $Z$. At lowest order in the strong interactions,
the process is $q + \bar q \to Z$. Here the quark and antiquark are
constituents of the high energy hadrons. The second significant
feature is that the $Z$ boson does not participate in the strong
interactions, so that our description of the observed final state can
be very simple.
We could equally well talk about $A + B \to W + X$ or $A + B \to
\gamma^* + X$ where the virtual photon decays into a muon pair or an
electron pair that is observed and where the mass of the $\gamma^*$ is
large compared to 1 GeV. This last process \cite{DrellYan}, $A + B \to
\gamma^* + X \to \ell^+ + \ell^- + X$, is historically important
because it helped establish the parton picture as being correct. The
$W$ and $Z$ processes were observed later. In fact, these are the
processes by which the $W$ and $Z$ bosons were first directly
observed \cite{WandZ}.
In process (\ref{makeaZ}), we allow the $Z$ boson to have any
transverse momentum ${\bf q}$. (Typically, then, ${\bf q}$ will be
much smaller than $M_Z$.) Since we integrate over $\bf q$ and the
mass of the $Z$ boson is fixed, there is only one variable needed to
describe the momentum of the $Z$ boson. We choose to use its
rapidity $y$, so that we are interested in the cross section $d
\sigma / dy$.
\begin{figure}[htb]
\centerline{\DESepsf(drellyan.eps width 8 cm)}
\caption{A Feynman diagram for $Z$ boson production in a
hadron-hadron collision. Two partons, carrying momentum fractions
$\xi_A$ and $\xi_B$, participate in the hard interaction. This
particular Feynman diagram illustrates an order $\alpha_s$
contribution to the hard scattering cross section: a gluon
is emitted in the process of making the $Z$ boson. The diagram also
shows the decay of the $Z$ boson into an electron and a neutrino. }
\label{drellyan}
\end{figure}
The cross section takes a factored form similar to that found for
deeply inelastic scattering. Here, however, there are two parton
distribution functions:
\begin{equation}
{\green{{d \sigma \over d y}}}
\approx\! \sum_{a,b} \int_{x_A}^1\! d \xi_A \int_{x_B}^1\! d \xi_B\
{\blue{f_{a/A}(\xi_A,\mu_F)}}\ {\blue{f_{b/B}(\xi_B,\mu_F)}}\
{\red{{d \hat\sigma_{ab}(\mu,\mu_F) \over d y}}}.
\label{DYfactors}
\end{equation}
The meaning of this formula is intuitive:
${\blue{f_{a/A}(\xi_A,\mu_F)}}\,d\xi_A$ gives the probability to find
a parton in hadron $A$;
${\blue{f_{b/B}(\xi_B,\mu_f)}}\,d\xi_B$ gives the probability to find
a parton in hadron $B$;
${\red{{d \hat\sigma_{ab} / d y}}}$ gives the cross section for
these partons to produce the observed $Z$ boson. The formula is
illustrated in Fig.~\ref{drellyan}. The hard scattering cross section
can be calculated perturbatively. Fig.~\ref{drellyan} illustrates one
particular order $\alpha_s$ contribution to ${\red{{d \hat\sigma_{ab}
/ d y}}}$. The integrations over parton momentum fractions have limits
$x_A$ and $x_B$, which are given by
\begin{equation}
x_A = e^y \sqrt{M^2/s}, \quad\quad
x_B = e^{-y} \sqrt{M^2/s}.
\end{equation}
Eq.~(\ref{DYfactors}) has corrections of order $m/M_Z$, where $m$ is
a mass characteristic of hadronic systems, say 1 GeV. In addition,
when ${\red{{d \hat\sigma_{ab} / d y}}}$ is calculated to order
$\alpha_s^N$, then there are corrections of order $\alpha_s^{N+1}$.
There can be soft interactions between the partons in hadron
$A$ and the partons in hadron $B$, and these soft interactions can
occur before the hard interaction that creates the $Z$ boson. It
would seem that these soft interactions do not fit into the
intuitive picture that comes along with Eq.~(\ref{DYfactors}). It is
a significant part of the factorization property that these soft
interactions do not modify the formula. These introductory lectures
are not the place to go into how this can be. For more information,
the reader is invited to consult Ref.~\citenum{factor}.
\subsection{Heavy quark production}
We now turn to the production of a heavy quark and its
corresponding antiquark in a high energy hadron-hadron collision:
\begin{equation}
A + B \to Q+ \bar Q + X.
\end{equation}
The most notable example of this is top quark production. A Feynman
diagram for this process is illustrated in Fig.~\ref{heavyquark}.
\begin{figure}[htb]
\centerline{\DESepsf(heavyquark.eps width 8 cm)}
\caption{Feynman graph for heavy quark production. The lowest order
hard process is $g + g \to Q + \bar Q$, which occurs at order
$\alpha_s^2$. This particular Feynman diagram illustrates an order
$\alpha_s^3$ process in which a gluon is emitted.}
\label{heavyquark}
\end{figure}
The total heavy quark production cross section takes a factored form
similar to that for $Z$ boson production,
\begin{equation}
{\green{\sigma_T}} \approx\!
\sum_{a,b} \int_{x_A}^1\! d \xi_A \int_{x_B}^1\! d \xi_B\
{\blue{f_{a/A}(\xi_A,\mu_F)}}\ {\blue{f_{b/B}(\xi_B,\mu_F)}}\
{\red{\hat\sigma_T}}^{ab}(\mu_F,\mu).
\label{heavyQfactors}
\end{equation}
As in the case of $Z$ production, $Q \bar Q$ production is a hard
process, with a time scale determined by the mass of the quark:
$\Delta t \sim 1/M_Q$. It is this hard process that is represented
by the calculated cross section ${\red{\hat\sigma_T}}^{ab}$. Of
course, the heavy quark and antiquark have strong interactions, and
can radiate soft gluons or exchange them with their environment.
These effects do not, however, affect the cross section: once the
$Q \bar Q$ pair is made, it is made. The probabilities for it to
interact in various ways must add to one. For an argument that
Eq.~(\ref{heavyQfactors}) is correct, see Ref.~\citenum{CSSheavyQ}.
\subsection{Jet production}
\label{jetproduction}
In our study of high energy electron-positron annihilation, we
discovered three things. First, QCD makes the qualitative prediction
that particles in the final state should tend to be grouped in
collimated sprays of hadrons called jets. The jets carry the momenta
of the first quarks and gluons produced in the hard process.
Second, certain kinds of experimental measurements probe the
short-time physics of the hard interaction, while being insensitive to
the long-time physics of parton splitting, soft gluon exchange, and
the binding of partons into hadrons. Such measurements are called
infrared safe. Third, among the infrared safe observables are cross
sections to make jets.
\begin{figure}[htb]
\centerline{ \DESepsf(twojets.eps width 6 cm) }
\caption{Sketch of a two-jet event at a hadron collider. The
cylinder represents the detector, with the beam pipe along its
axis. Typical hadron-hadron collisions produce beam remnants, the
debris from soft interactions among the partons. The particles in
the beam remnants have small transverse momenta, as shown in the
sketch. In rare events, there is a hard parton-parton collision, which
produces jets with high transverse momenta. In the event shown, there
are two high
$P_T$ jets.}
\label{twojets}
\end{figure}
These ideas work for hadron-hadron collisions too. In such
collisions, there is sometimes a hard parton-parton collision, which
produces two or more jets, as depicted in Fig.~\ref{twojets}. Consider
the cross section to make one jet plus anything else,
\begin{equation}
A + B \to jet + X.
\end{equation}
Let $E_T$ be the {\it transverse energy} of the jet, defined as the
sum of the absolute values of the transverse momenta of the
particles in the jet. Let $y$ be the rapidity of the jet. Given a
definition of exactly what it means to have a jet with transverse
energy $E_T$ and rapidity $y$, the jet production cross section
takes the familiar factored form
\begin{eqnarray}
{\green{d \sigma \over d E_T d\eta}} &\approx&
\sum_{a,b} \int_{x_A}^1\! d \xi_A \int_{x_B}^1\! d \xi_B\
{\blue{f_{a/A}(\xi_A,\mu_F)}}\ {\blue{f_{b/B}(\xi_B,\mu_F)}}\
{\red{{{d \hat\sigma^{ab}(\mu,\mu_F) \over d E_T d\eta}}}}.
\end{eqnarray}
\begin{figure}[htb]
\centerline{\DESepsf(jet.eps width 10 cm)}
\caption{A Feynman diagram for jet production in hadron-hadron
collisions. The leading order diagrams for $A + B \to jet + X$ occur
at order $\alpha_s^2$. This particular diagram is for an interaction
of order $\alpha_s^3$. When the emitted gluon is not soft or nearly
collinear to one of the outgoing quarks, this diagram corresponds to
a final state like that shown in the small sketch, with three jets
emerging in addition to the beam remnants. Any of these jets can be
the jet that is measured in the one jet inclusive cross section.}
\label{jet}
\end{figure}
What shall we choose for the definition of a jet? At a crude level,
high $E_T$ jets are quite obvious and the precise definition hardly
matters. However, if we want to make a quantitative measurement of a
jet cross section to compare to next-to-leading order theory, then the
definition does matter. There are several possibilities for a
definition that is infrared safe. The one most used in
hadron-hadron collisions is based on cones.
In the standard {\it Snowmass Accord} definition \cite{snowmass}, one
imagines that the experimental calorimeter is divided into small
angular cells labeled $i$ in $\eta$-$\phi$ space, as depicted in
Fig.~\ref{jetcone}. We can say that a jet consists of all the
particles that fall into certain of the calorimeter cells, or we can
measure the $E_T$ in each cell and build the jet parameters from the
cell variables $(E_{Ti},\eta_i,\phi_i)$. We then say that a jet
consists of the cells inside a certain circle in $\eta$-$\phi$ space.
The circle has a radius $R$, usually chosen as 0.7 radians, and is
centered on a direction $(\eta_J,\phi_J)$. Thus the calorimeter cells
$i$ included in the jet obey
\begin{equation}
(\eta_i - \eta_J)^2 + (\phi_i - \phi_J)^2 < R^2.
\end{equation}
The transverse energy of the jet is defined to be
\begin{equation}
E_{T,J} = \sum_{i\in {\rm cone}}E_{T,i}.
\label{jetET}
\end{equation}
The direction of the jet is defined to be the direction
$(\eta_J,\phi_J)$ of the jet axis, which is chosen to obey
\begin{equation}
\phi_J = {1 \over E_{T,J}}\sum_{i\in {\rm cone}}E_{T,i}\ \phi_i,
\label{jetphi}
\end{equation}
\begin{equation}
\eta_J = {1 \over E_{T,J}}\sum_{i\in {\rm cone}}E_{T,i}\ \eta_i
\label{jeteta}
\end{equation}
Of course, if one picks a trial jet direction $(\eta_J,\phi_J)$ to
define the meaning of ``${i\in {\rm cone}}$'' and then computes
$(\eta_J,\phi_J)$ from these equations, the output jet direction will
not necessarily match the input cone axis. Thus one has to treat the
equations iteratively until a consistent solution is found.
\begin{figure}[htb]
\centerline{\DESepsf(jetcone.eps width 7 cm)}
\caption{Jet definition according to the Snowmass algorithm. The
shading of the squares represents the density of transverse energy as
a function of azimuthal angle $\phi$ and pseudorapidity $\eta$. The
cells inside the circle constitute the jet.}
\label{jetcone}
\end{figure}
Note that the Snowmass algorithm for computing
$E_{T,J},\phi_J,\eta_J$ is infrared safe. Infinitely soft particles
do not affect the jet parameters because they enter the equations
with zero weight. If two particles have the same angles
$\eta,\phi$, then it does not matter if we join them together into
one particle before applying the algorithm. For example
\begin{equation}
E_{T,1} \phi + E_{T,2} \phi =
(E_{T,1} + E_{T,2})\, \phi.
\end{equation}
Note, however, that the Snowmass definition given above is not
complete. It is perfectly possible for two or more cones that are
solutions to Eqs.~(\ref{jetET},\ref{jetphi},\ref{jeteta}) to
overlap. One must then have an algorithm to assign calorimeter cells
to one of the competing jets, thus splitting the jets, or else to
merge the jets. When supplemented by an appropriate split/merge
algorithm, the Snowmass definition is not as simple as it seemed at
first.
In an order $\alpha_s^3$ perturbative calculation, one simply
applies this algorithm at the parton level. At this order of
perturbation theory, there are two or three partons in the final
state. In the case of three partons in the final state, two of them
are joined into a jet if they are within $R$ of the jet axis
computed from the partonic momenta. The split/merge question does
not apply at this order of perturbation theory.
I showed a comparison of the theory and experiment for the one jet
inclusive cross section in Fig.~\ref{Jetcteq3}.
I should record here that the actual jet definitions used in current
experiment are close to the Snowmass definition given above but are
not exactly the same. Furthermore, there are other definitions
available that may come into use in the future. There is not time
here to explore the issues of jet definitions in detail. What I hope
to have done is to give the outline of one definition and to
explore what the issues are.
\section{Epilogue}
QCD is a rich subject. The theory and the experimental evidence
indicate that quarks and gluons interact weakly on short time and
distance scales. But the net effect of these interactions extending
over long time and distance scales is that the chromodynamic force is
strong. Quarks are bound into hadrons. Outgoing partons emerge as
jets of hadrons, with each jet composed of subjets. Thus QCD theory
can be viewed as starting with simple perturbation theory, but it does
not end there. The challenge for both theorists and experimentalists
is to extend the range of phenomena that we can relate to the
fundamental theory.
\medskip
I thank F.\ Hautmann for reading the manuscript and helping to
eliminate some of the mistakes.
|
2,869,038,156,415 | arxiv | \section{Introduction}
Let $E/F$ be a Galois extension of number fields with Galois group $G$. In seeking annihilators in $\bb{Z}[G]$ of the $K$-groups $K_{2n}(\mathcal{O}_{E,S})$ ($S$ a finite set of places of $E$ containing the infinite ones), Stickelberger elements have long been a source of interest. This began with the classical Stickelberger theorem, showing that for abelian extensions $E/\bb{Q}$, annihilators of $\mathrm{Tors}(K_0(\mathcal{O}_{E,S}))$ can be constructed from Stickelberger elements. Coates and Sinnott later conjectured in \cite{cs:stickel} that the analogous phenomenon would occur for higher $K$-groups. However, defined in terms of values of $L$-functions at negative integers, these elements do not provide all the annihilators, because of the prevalent vanishing of the $L$-function values.
This difficulty is hoped to be overcome by considering the ``fractional Galois ideal'' introduced by the second author in \cite{snaith:rel,snaith:stark} and defined in terms of \emph{leading coefficients} of $L$-functions at negative integers under the assumption of the higher Stark conjectures. A version more suitable for the case of $\mathrm{Tors}(K_0(\mathcal{O}_{E,S})) = \mathrm{Cl}(\mathcal{O}_{E,S})$ was defined in \cite{buckingham:frac} by the first author. Evidence that the fractional Galois ideal annihilates the appropriate $K$-groups (resp. class-groups) can be found in \cite{snaith:stark} (resp. \cite{buckingham:frac}). In the first case, it is \'etale cohomology that is annihilated, but this is expected to give $K$-theory by the Lichtenbaum--Quillen conjecture (see \cite[Section 1]{snaith:stark} for details).
With a view to relating the fractional Galois ideal to characteristic ideals in Iwasawa theory, we would like to describe how it behaves in towers of number fields. That it exhibits naturality in certain changes of extension was observed in particular cases in \cite{buckingham:frac}, and part of the aim of this paper is to explain these phenomena generally. Passage to subextensions corresponding to quotients of Galois groups will be of particular interest in the situation of non-abelian extensions, because of the relatively recent emergence of non-commutative Iwasawa theory in, for example, \cite{cfksv:main,fk:noncomm}. Consequently, the aims of this paper are
\begin{tabular}{cp{10cm}}
(i) & to prove formal properties of the fractional Galois ideal with respect to changes of extension, in the commutative setting first (\S \ref{2.3} to \S \ref{2.9}) \\
(ii) & to extend the definition of the fractional Galois ideal to non-abelian Galois extensions (\S \ref{4.1}), having previously defined it only for abelian extensions \\
(iii) & to show that it behaves well under passing to subextensions in the non-commutative setting also (Proposition \ref{4.4}) \\
(iv) & to show that in order for the non-commutative fractional Galois ideals to annihilate the appropriate \'etale cohomology groups, it is sufficient that the commutative ones do (\S \ref{5.4}).
\end{tabular}
We will also provide an explicit example (in the commutative case) in \S \ref{commutative example} illustrating how a limit of fractional Galois ideals gives the Fitting ideal for an inverse limit $\mathrm{Cl}_\infty$ of $\ell$-parts of class-groups. This should make clear the importance of taking leading coefficients of $L$-functions rather than just values, since it will be the part of the fractional Galois ideal corresponding to $L$-functions with first-order vanishing at $0$ which provides the Fitting ideal for the plus-part of $\mathrm{Cl}_\infty$.
In \S \ref{sec iwasawa theory}, we will conclude with a discussion of how the constructions of this paper fit into non-commutative Iwasawa theory. In particular, under some assumptions which, compared with the many conjectures permeating this area, are relatively weak, we will be able to give a partial answer to a question of Ardakov--Brown in \cite{ab:ringtheoretic} on constructing ideals in Iwasawa algebras.
\section{Notation and the Stark conjectures} \label{notation}
In what follows, by a Galois representation of a number field $F$ we shall mean a continuous, finite-dimensional complex representation of the absolute Galois group of $F$, which amounts to saying that the representation factors through the Galois group ${\rm Gal}(E/F)$ of a finite Galois extension $E/F$. We begin with the Stark conjecture (at $s=0$) and its generalizations to $s= -1, -2, -3, \ldots $ which were introduced in \cite{gross:higherstark} and \cite{snaith:stark} independently.
Let $\Sigma(E)$ denote the set of embeddings of $E$ into the complex numbers. For $r = 0, -1, -2, -3, \ldots$ set
\[ Y_{r}(E) = \prod_{\Sigma(E)} \ (2 \pi i )^{-r} {\mathbb Z} = {\rm Map}( \Sigma(E) , (2 \pi i )^{-r} {\mathbb Z} ) \]
endowed with the $G({\mathbb C}/{\mathbb R})$-action diagonally on $\Sigma(E)$ and on $(2 \pi i )^{-r} $. If $c_{0}$ denotes complex conjugation, the action of $c_{0}$ and $G$ commute so that
the fixed points of $Y_{r}(E)$ under $c_{0}$, denoted by $Y_{r}(E)^{+}$, form a $G$-module. It is easy to see that the rank of $ Y_{r}(E)^{+} $ is given by
\[ \mathrm{rk}_{{\mathbb Z}}( Y_{r}(E)^{+} ) = \left\{
\begin{array}{ll}
r_{2} & {\rm if } \ r \ {\rm is \ odd}, \\
r_{1} + r_{2} & {\rm if } \ r \geq 0 \ {\rm is \ even} .
\end{array} \right. \]
where $| \Sigma(E)| = r_{1} + 2r_{2}$ and $r_{1}$ is the number of real embeddings of $E$.
\subsection{Stark regulators} \label{1.2}
We begin with a slight modification of the original Stark regulator \cite{tate:stark}. Now let $G$ denote the Galois group of an extension of number fields $E/F$.
We extend the Dirichlet regulator homomorphism to the Laurent polynomials with coefficients in ${\cal O}_{E}$ to give an ${\mathbb R}[G]$-module isomorphism of the form
\[ R_{E}^{0} : K_{1}({\cal O}_{E} \langle t^{\pm 1} \rangle ) \otimes {\mathbb R} = {\cal O}_{E} \langle t^{\pm 1} \rangle^\times \otimes {\mathbb R} \stackrel{\cong}{\rightarrow} Y_{0}(E)^{+} \otimes {\mathbb R} \cong {\mathbb R}^{r_{1}+r_{2}} \]
by the formulae, for $u \in {\cal O}_{E} ^\times$,
\[ R_{E}^{0}( u ) = \sum_{\sigma \in \Sigma(E)} \ {\rm log}(|\sigma(u)|) \cdot \sigma \]
and
\[ R_{E}^{0}( t ) = \sum_{\sigma \in \Sigma(E)} \ \sigma . \]
The existence of this isomorphism implies (see \cite[Section 12.1]{serre:linear} and \cite[p.26]{tate:stark}) that there exists at least one ${\mathbb Q}[G]$-module isomorphism of the form
\[ f_{E }^{0} : {\cal O}_{E} \langle t^{\pm 1} \rangle^\times \otimes {\mathbb Q}
\stackrel{\cong}{\rightarrow} Y_{0}(E )^{+} \otimes {\mathbb Q} . \]
For any choice of $ f_{E }^{0}$ Stark forms the composition
\[ R_{E }^{0} \cdot (f_{E }^{0})^{-1} : Y_{0}(E )^{+} \otimes {\mathbb C}
\stackrel{\cong}{\rightarrow} Y_{0}(E )^{+} \otimes {\mathbb C} \]
which is an isomorphism of complex representations of $G$. Let $V$ be a finite-dimensional complex representation of $G$ whose contragredient is denoted by $V^{\vee}$. The Stark regulator is defined to be the exponential homomorphism $V \mapsto R(V, f_{E }^{0})$, from representations to non-zero complex numbers, given by
\[ R(V, f_{E }^{0}) = \textstyle{\det}} % {\mathrm{det}( (R_{E}^{0} \cdot (f_{E }^{0})^{-1})_{*} \in {\rm Aut}_{{\bf C}}({\rm Hom}_{G}( V^{\vee} , Y_{0}(E )^{+} \otimes {\mathbb C}) )) \]
where $ (R_{E }^{0} \cdot (f_{E }^{0})^{-1})_{*}$ is composition with $ R_{E }^{0} \cdot (f_{E }^{0})^{-1}$.
For $r = -1, -2, -3, \ldots$ there is an isomorphism of the form \cite{quillen:ktheory}
\[ K_{1-2r}({\cal O}_{E} \langle t^{\pm 1} \rangle ) \otimes {\mathbb Q} \cong
K_{1-2r}({\cal O}_{E} ) \otimes {\mathbb Q} \]
because $ K_{-2r}({\cal O}_{E} )$ is finite. Therefore the Borel regulator homomorphism defines an ${\mathbb R}[G]$-module isomorphism of the form
\[ R_{E}^{r} : K_{1-2r}({\cal O}_{E} \langle t^{\pm 1} \rangle ) \otimes {\mathbb R} = K_{1-2r}({\cal O}_{E} ) \otimes {\mathbb R} \stackrel{\cong}{\rightarrow} Y_{r}(E)^{+} \otimes {\mathbb R} . \]
Choose a ${\mathbb Q}[G]$-module isomorphism of the form
\[ f_{E }^{r} : K_{1-2r}({\cal O}_{E} \langle t^{\pm 1} \rangle ) \otimes {\mathbb Q} \stackrel{\cong}{\rightarrow} Y_{r}(E)^{+} \otimes {\mathbb Q} \]
and form the analogous Stark regulator, $(V \mapsto R(V, f_{E }^{r})) $, from representations to non-zero complex numbers given by
\[ R(V, f_{E }^{r}) = \textstyle{\det}} % {\mathrm{det}( (R_{E}^{r} \cdot (f_{E }^{r})^{-1})_{*} \in {\rm Aut}_{{\bf C}}({\rm Hom}_{G}( V^{\vee} , Y_{r}(E )^{+} \otimes {\mathbb C}) )) . \]
\subsection{Stark's conjectures} \label{stark conjectures}
Let $R(G)$ denote the complex representation ring of the finite group $G$; that is, $R(G) = K_{0}({\mathbb C}[G])$. Since $V$ determines a Galois representation of $F$, we have a non-zero complex number $L_{F}^{*}(r, V )$ given by the leading coefficient of the Taylor series at $s=r$ of the Artin $L$-function associated to $V$ (\cite{martinet:lfunctions}, \cite[p.23]{tate:stark}).
We may modify $ R(V, f_{E }^{r})$ to give another exponential homomorphism
\[ {\cal R}_{f_{E }^{r}} \in {\rm Hom}( R( G) ,{\mathbb C}^\times) \]
defined by
\[ {\cal R}_{f_{E }^{r}}(V) = \frac{R(V,f_{E }^{r})}{L_{F}^{*}( r , V) }. \]
Let $\overline{{\mathbb Q}}$ denote the algebraic closure of the rationals in the complex numbers and let $\Omega_{{\mathbb Q}}$ denote the absolute Galois group of the rationals, which acts continuously on $R( G) $ and $\overline{{\mathbb Q}}^\times$. The Stark conjecture asserts that for each $r= 0, -1, -2, -3, \ldots$
\[ {\cal R}_{f_{E }^{r}} \in {\rm Hom}_{\Omega_{{\mathbb Q}}}( R( G) , \overline{{\mathbb Q}}^\times) \subseteq {\rm Hom}( R( G ) ,{\mathbb C}^\times) . \]
In other words, ${\cal R}_{f_{E }^{r}}(V)$ is an algebraic number for each $V$ and for all $z \in \Omega_{{\mathbb Q}}$ we have $z({\cal R}_{f_{E }^{r}}(V)) = {\cal R}_{f_{E }^{r}}(z(V))$. Since any two choices of $f_{E}^{r}$ differ by multiplication by a ${\mathbb Q}[G]$-automorphism, the truth of the conjecture is independent of the choice of $f_{E }^{r}$ (\cite{tate:stark} pp.28-30).
When $s=0$ the conjecture which we have just formulated apparently differs from the classical Stark conjecture of \cite{tate:stark}, therefore we shall pause to show that the two conjectures are equivalent. For the classical Stark conjecture one replaces $Y_{0}(E )^{+}$ by $X_{0}(E)^{+}$ where $X_{0}(E)$ is the kernel of the augmentation homomorphism $Y_{0}(E ) \rightarrow {\mathbb Z}$, which adds together all the coordinates. The Dirichlet regulator gives an ${\mathbb R}[G]$-module isomorphism
\[ \tilde{R}_{E}^{0} : {\cal O}_{E}^\times \otimes {\mathbb R} \stackrel{\cong}{\rightarrow} X_{0}(E)^{+} \otimes {\mathbb R} \]
and choosing a ${\mathbb Q}[G]$-module isomorphism
\[ \tilde{f}_{E}^{0} : {\cal O}_{E}^\times \otimes {\mathbb Q} \stackrel{\cong}{\rightarrow} X_{0}(E)^{+} \otimes {\mathbb Q} \]
we may form
\[ \tilde{R}_{E }^{0} \cdot (\tilde{f}_{E }^{0})^{-1} : X_{0}(E )^{+} \otimes {\mathbb C} \stackrel{\cong}{\rightarrow} X_{0}(E )^{+} \otimes {\mathbb C} . \]
Taking its Stark determinant we obtain $\tilde{R}(V, \tilde{f}_{E }^{0})$ and finally
\[ \tilde{{\cal R}}_{ \tilde{f}_{E}^{0}}(V) = \frac{\tilde{R}(V, \tilde{f}_{E }^{0})}{L_{F}^{*}(0 , V) }. \]
\begin{prop}
In \S \ref{stark conjectures}
\[ {\cal R}_{ f_{E}^{0}} \in {\rm Hom}_{\Omega_{{\mathbb Q}}}( R( G) , \overline{{\mathbb Q}}^\times) \subseteq {\rm Hom}( R( G ) ,{\mathbb C}^\times) \]
if and only if
\[ \tilde{{\cal R}}_{ \tilde{f}_{E}^{0}} \in {\rm Hom}_{\Omega_{{\mathbb Q}}}( R( G) , \overline{{\mathbb Q}}^\times) \subseteq {\rm Hom}( R( G ) ,{\mathbb C}^\times) \]
independently of the choice of $f_{E}^{0}$ or $\tilde{f}_{E}^{0}$.
\end{prop}
\begin{proof}
Given any ${\mathbb Q}[G]$-isomorphism $\tilde{f}_{E}^{0}$ we may fill in the following commutative diagram by ${\mathbb Q}[G]$-isomorphisms $f_{E}^{0}$ and $\overline{f}_{E}^{0}$. Conversely, given any ${\mathbb Q}[G]$-isomorphisms $f_{E}^{0}$ and $\overline{f}_{E}^{0}$ we may fill in the diagram
with a ${\mathbb Q}[G]$-isomorphism $\tilde{f}_{E}^{0}$.
\[ \xymatrix{
\mathcal{O}_E^\times \teno{\bb{Z}} \bb{Q} \ar[r] \ar[d]^{\tilde{f}_E^0} & \mathcal{O}_E[t^{\pm 1}]^\times \teno{\bb{Z}} \bb{Q} \ar[r] \ar[d]^{f_E^0} & \bb{Q} \ar[d]^{\br{f}_E^0} \\
X_0(E)^+ \teno{\bb{Z}} \bb{Q} \ar[r] & Y_0(E)^+ \teno{\bb{Z}} \bb{Q} \ar[r] & \bb{Q}
} \]
Similarly there is a commutative diagram in which the vertical arrows are reversed, ${\mathbb Q}$ is replaced by ${\mathbb R}$ and $\tilde{f}_{E}$, $f_{E}$ and $\overline{f}_{E}$ by $\tilde{R}_{E}^{0} $, $R_{E}^{0}$ and $\overline{R}_{E }^{0}$, respectively. Furthermore $\overline{R}_{E }^{0}$ is multiplication by a rational number. The result now follows from the multiplicativity of the determinant in short exact sequences.
\end{proof}
We shall be particularly interested in the case when $G $ is abelian, in which case the following observation is important. Let $\chgrp{G} = {\rm Hom}(G, \overline{{\mathbb Q}}^\times)$ denote the set of characters on $G$ and let ${\mathbb Q}(\chi)$ denote the field generated by the character values of a representation $\chi$. We may identify ${\rm Hom}_{\Omega_{{\mathbb Q}}}( R(G) , \overline{{\mathbb Q}}) $with the ring ${\rm Map}_{\Omega_{{\mathbb Q}}}(\chgrp{G}, \overline{{\mathbb Q}})$.
\begin{prop} \label{1.5}
Let $G$ be a finite abelian group. Then there exists an isomorphism of rings
\[ \lambda_{G} : {\rm Map}_{\Omega_{{\mathbb Q}}}(\chgrp{G}, \overline{{\mathbb Q}})={\rm Hom}_{\Omega_{{\mathbb Q}}}( R(G) , \overline{{\mathbb Q}}) \stackrel{\cong}{\rightarrow} {\mathbb Q}[G] \]
given by
\[ \lambda_{G}(h) = \sum_{\chi \in \chgrp{G}} \ h(\chi) e_{\chi} \]
where
\[ e_{\chi} = |G|^{-1} \sum_{g \in G} \ \chi(g) g^{-1} \in {\mathbb Q}(\chi)[G]. \]
In particular there is an isomorphism of unit groups
\[ \lambda_{G} : {\rm Hom}_{\Omega_{{\mathbb Q}}}( R(G) , \overline{{\mathbb Q}}^\times) \stackrel{\cong}{\rightarrow} {\mathbb Q}[G]^\times . \]
\end{prop}
\begin{proof}
There is a well-known isomorphism of rings (\cite{lang:algebra2nd} p.648)
\[ \psi : \overline{{\mathbb Q}}[G] \rightarrow \prod_{ \chi \in \chgrp{G} } \overline{{\mathbb Q}} = {\rm Map}( \chgrp{G} , \overline{{\mathbb Q}}) \]
given by $\psi( \sum_{g \in G} \lambda_{g} g)( \chi) = \sum_{g \in G} \lambda_{g} \chi(g)$. If $\Omega_{{\mathbb Q}}$ acts on $\overline{{\mathbb Q}}$ and $\chgrp{G}$ in the canonical manner, then $\psi$ is Galois equivariant and induces an isomorphism of $\Omega_{{\mathbb Q}}$-fixed points of the form
\[ {\mathbb Q}[G] = (\overline{{\mathbb Q}}[G] )^{\Omega_{{\mathbb Q}}} \cong
{\rm Map}_{\Omega_{{\mathbb Q}}}( \chgrp{G} , \overline{{\mathbb Q}}) \cong {\rm Hom}_{\Omega_{{\mathbb Q}}}( R(G) , \overline{{\mathbb Q}}) . \]
It is straightforward to verify that this isomorphism is the inverse of $\lambda_{G}$.
\end{proof}
\section{The canonical fractional Galois ideal ${\cal J}_{E/F}^{r}$ in the abelian case}
\subsection{Definition of ${\cal J}_{E/F}^r$} \label{2.1}
In this section we recall the canonical fractional Galois ideal introduced in \cite{snaith:stark} (see also \cite{buckingham:frac}, \cite{snaith:equiv} and \cite{snaith:rel}). In \cite{snaith:stark} this was denoted merely by ${\cal J}_{E}^{r}$ but in this paper we shall need to keep track of the base field.
As in \S \ref{stark conjectures}, let $E/F$ be a Galois extension of number fields. Throughout this section we shall assume that the Stark conjecture of \S \ref{stark conjectures} is true for all $E/F$ and that $G = {\rm Gal}(E/F)$ is abelian. Therefore, by Proposition \ref{1.5}, for each $r = 0, -1, -2, -3, \ldots$
we have an element
\[ {\cal R}_{f_{E }^{r}} \in {\rm Hom}_{\Omega_{{\mathbb Q}}}( R( G ) , \overline{{\mathbb Q}}^\times) \cong {\mathbb Q}[G]^\times \]
which depends upon the choice of a ${\mathbb Q}[G]$-isomorphism $f_{E}^{r}$ in \S \ref{stark conjectures}.
Let $ \alpha \in {\rm End}_{{\mathbb Q}[G]}( Y_{r}(E)^{+} \otimes {\mathbb Q} )$ and extend this by the identity on the $(-1)$-eigenspace of complex conjugation $Y_{r}(E)^{-} \otimes {\mathbb Q} $ to give
\[ \alpha \oplus 1 \in {\rm End}_{{\mathbb Q}[G]}( Y_{r}(E) \otimes {\mathbb Q} ) . \]
Since $Y_{r}(E) \otimes {\mathbb Q} $ is free over ${\mathbb Q}[G]$, we may form the determinant
\[ {\rm det}_{{\mathbb Q}[G] }( \alpha \oplus 1) \in {\mathbb Q}[G] . \]
In terms of the isomorphism of Proposition \ref{1.5}, ${\rm det}_{{\mathbb Q}[G] }( \alpha \oplus 1)$ corresponds to the function which sends $\chi \in \chgrp{G}$ to the determinant of the endomorphism of $e_{\chi} Y_{r}(E) \otimes \overline{{\mathbb Q}}$ induced by $\alpha \oplus 1$.
Following \cite[Section 4.2]{snaith:stark} (see also \cite{snaith:rel,snaith:equiv}), define ${\cal I}_{f_{E}^{r}}$ to be the (finitely generated) ${\mathbb Z}[1/2][G]$-submodule of ${\mathbb Q}[ G ]$ generated by all the elements \linebreak ${\rm det}_{{\mathbb Q}[G] }( \alpha \oplus 1) $ satisfying the integrality condition
\[ \alpha \cdot f_{E}^{r}( K_{1-2r}({\cal O}_{E}[t^{\pm 1}])) \subseteq Y_{r}(E) . \]
Define ${\cal J}_{E/F}^{r}$ to be the finitely generated ${\mathbb Z}[1/2][G]$-submodule of ${\mathbb Q}[ G ]$ given by
\[ {\cal J}_{E/F}^{r} = {\cal I}_{ f_{E}^{r}} \cdot \tau( {\cal R}_{f_{E}^{r}}^{-1}) \]
where $\tau$ is the automorphism of the group-ring induced by sending each $g \in G$ to its inverse.
\begin{prop}{(\cite[Prop.4.5]{snaith:stark})}
Let $E/F$ be a Galois extension of number fields with abelian Galois group $G$. Then, assuming that the Stark conjecture of \S \ref{stark conjectures} holds for $E/F$ for $r= 0, -1, -2, -3, \ldots$, the finitely generated ${\mathbb Z}[1/2][G]$-submodule ${\cal J}_{E/F}^{r}$ of ${\mathbb Q}[ G ]$ just defined is independent of the choice of $f_{E}^{r}$.
\end{prop}
\subsection{Naturality examples} \label{nat examples}
Given an extension $E/F$ of number fields satisfying the Stark conjecture at $s = 0$ and a finite set of places $S$ of $F$ containing the infinite places, let $\iJ{E/F,S}$ denote the fractional Galois ideal as defined in \cite{buckingham:frac}, a slight modification of the one just defined so that we can take into account finite places. Let us consider the following situation: $\ell$ is an odd prime, $E_n = \bb{Q}(\zeta_{\ell^{n+1}})$ for a primitive $\ell^{n+1}$th root of unity $\zeta_{\ell^{n+1}}$ ($n \geq 0$), and $S = \{\infty,\ell\}$. The descriptions below of $\iJ{E_n/\bb{Q},S}$ and $\iJ{E_n^+/\bb{Q},S}$ are provided in \cite[Section 4]{buckingham:frac}:
\begin{eqnarray}
\iJ{E_n/\bb{Q},S} &=& \frac{1}{2} e_+ \mathrm{ann}_{\bb{Z}[G_n]}(\mathcal{O}_{E_n^+,S}^\times/\mathcal{E}_n^+) \oplus \bb{Z}[G_n]\theta_{E_n/\bb{Q},S} \label{full j desc} \label{recap full j desc} \\
\iJ{E_n^+/\bb{Q},S} &=& \frac{1}{2} \mathrm{ann}_{\bb{Z}[G_n^+]}(\mathcal{O}_{E_n^+,S}^\times/\mathcal{E}_n^+) \label{real j desc}
\end{eqnarray}
where $G_n = \gal{E_n/\bb{Q}}$, $G_n^+ = \gal{E_n^+/\bb{Q}}$, $\mathcal{E}_n^+$ is the $\bb{Z}[G_n^+]$-submodule of $\mathcal{O}_{E_n^+,S}^\times$ generated by $-1$ and $(1 - \zeta_{\ell^{n+1}})(1 - \zeta_{\ell^{n+1}}^{-1})$, and $\theta_{E_n/\bb{Q},S}$ is the Stickelberger element at $s = 0$. Also, $e_+ = \frac{1}{2}(1+c)$ is the plus-idempotent for complex conjugation $c \in G_n$.
It is immediate from these descriptions that the natural maps $\bb{Q}[G_n] \rightarrow \bb{Q}[G_n^+]$, $\bb{Q}[G_n] \rightarrow \bb{Q}[G_{n-1}]$ and $\bb{Q}[G_n^+] \rightarrow \bb{Q}[G_{n-1}^+]$ give rise to a commutative diagram
\begin{equation} \label{quot examples}
\xymatrix{
\iJ{E_n/\bb{Q},S} \ar[r] \ar[d] & \iJ{E_n^+/\bb{Q},S} \ar[d] \\
\iJ{E_{n-1}/\bb{Q},S} \ar[r] & \iJ{E_{n-1}^+/\bb{Q},S} .
}
\end{equation}
($\mathcal{O}_{E_{n-1}^+,S}^\times/\mathcal{E}_{n-1}^+$ embeds into $\mathcal{O}_{E_n^+,S}^\times/\mathcal{E}_n^+$, and Stickelberger elements are well known (e.g. \cite{hayes:nonabstick}) to map to each other in this way.)
Now suppose that $\ell \equiv 3 \spc\mathrm{mod}\spc 4$, so that $E_n$ contains the imaginary quadratic field $F = \bb{Q}(\sqrt{-\ell})$. Again, letting $S_F$ consist of the infinite place of $F$ and the unique place above $\ell$, $\iJ{E_n/F,S_F}$ has a simple description. Indeed, if $H_n = \gal{E_n/F}$, then
\begin{equation} \label{lifted j desc}
\iJ{E_n/F,S_F} = \frac{1}{\mu_n} \mathrm{ann}_{\bb{Z}[H_n]}(\mathcal{O}_{E_n,S}^\times/\mathcal{E}_n)
\end{equation}
where $\mathcal{E}_n$ is generated over $\bb{Z}[H_n]$ by $\zeta_{\ell^{n+1}}$ and $(1 - \zeta_{\ell^{n+1}})^{\mu_n \tilde{\stick}_n}$. Here, $\mu_n = |\rou{E_n}|$ and $\tilde{\stick}_n = \sum_{\sigma \in H_n} \zeta_{E_n/\bb{Q},S}(0,\sigma^{-1}) \sigma \in \bb{Q}[H_n]$, a sort of ``half Stickelberger element'' obtained by keeping only those terms corresponding to elements in the index two subgroup $H_n$ of $G_n$. (Note that $\mu_n \tilde{\stick}_n \in \bb{Z}[H_n]$.) Comparing (\ref{real j desc}) and (\ref{lifted j desc}), we see without too much difficulty that
\begin{prop} \label{nat irrats example}
The isomorphism $\Phi_n : \bb{Q}[H_n] \rightarrow \bb{Q}[G_n^+]$ identifies $\iJ{E_n/F,S_F}$ with $2\Phi_n(\tilde{\stick}_n) \iJ{E_n^+/\bb{Q},S}$.
\end{prop}
We now explain the above phenomena by proving some general relationships between the ${\cal J}_{E/F}^r$ under natural changes of extension.
\subsection{Behaviour under quotient maps ${\rm Gal}(L/F) \rightarrow {\rm Gal}(K/F)$} \label{2.3}
Suppose that $F \subseteq K \subseteq L$ is a tower of number fields with $L/F$ abelian. The inclusion of $K$ into $L$ induces a homomorphism
\[ K_{1-2r}({\cal O}_{K}[t^{\pm 1}]) \rightarrow K_{1-2r}({\cal O}_{L}[t^{\pm 1}]) . \]
When $r=0$
\[ \frac{ K_{1}({\cal O}_{K}[t^{\pm 1}])}{{\rm Torsion}} \cong {\cal O}_{K}^\times/( \mu(K)) \oplus {\mathbb Z} \langle t \rangle \]
maps injectively to the Galois invariants of $ {\cal O}_{L}^\times/( \mu(L)) \oplus {\mathbb Z} \langle t \rangle $ sending $t$ to itself. For strictly negative $r$,
\[ \frac{ K_{1-2r}({\cal O}_{K}[t^{\pm 1}])}{{\rm Torsion}} \cong \frac{ K_{1-2r}({\cal O}_{K})}{{\rm Torsion}} \]
embeds into the ${\rm Gal}(L/K)$-invariants of $ \frac{ K_{1-2r}({\cal O}_{L}[t^{\pm 1}])}{{\rm Torsion}} $. There is a homomorphism $Y_{r}(K) \rightarrow Y_{r}(L)$ which sends $n_{\sigma} \cdot \sigma$ to $n_{\sigma} \cdot ( \sum_{ ( \sigma' \ | \ F) = \sigma} \ \sigma' )$ which is an isomorphism onto the ${\rm Gal}(L/K)$-invariants $Y_{r}(L)^{{\rm Gal}(L/K)}$. For $ r = 0, -1, -2, -3, \ldots$ there is a commutative diagram of regulators in \S\ref{1.2}
\[ \xymatrix{
K_{1-2r}(\mathcal{O}_K[t^{\pm 1}]) \teno{\bb{Z}} \bb{R} \ar[r]^-{R_K^r} \ar[d] & Y_r(K)^+ \teno{\bb{Z}} \bb{R} \ar[d] \\
K_{1-2r}(\mathcal{O}_L[t^{\pm 1}]) \teno{\bb{Z}} \bb{R} \ar[r]^-{R_L^r} & Y_r(L)^+ \teno{\bb{Z}} \bb{R}
} \]
We may choose $f_{K}^{r}$ and $f_{L}^{r}$ as in \S\ref{1.2} to make the corresponding diagram of ${\mathbb Q}$-vector spaces commute
\begin{equation} \label{f chosen to commute}
\xymatrix{
K_{1-2r}(\mathcal{O}_K[t^{\pm 1}]) \teno{\bb{Z}} \bb{Q} \ar[r]^-{f_K^r} \ar[d] & Y_r(K)^+ \teno{\bb{Z}} \bb{Q} \ar[d] \\
K_{1-2r}(\mathcal{O}_L[t^{\pm 1}]) \teno{\bb{Z}} \bb{Q} \ar[r]^-{f_L^r} & Y_r(L)^+ \teno{\bb{Z}} \bb{Q}
}
\end{equation}
Let $V$ be a one-dimensional complex representation of ${\rm Gal}(K/F)$ and let $W = \mathrm{Inf}_{{\rm Gal}(K/F)}^{{\rm Gal}(L/F)}(V)$ denote the inflation of $V$. Then
\[ \begin{array}{l}
{\rm Hom}_{{\rm Gal}(L/F)}(W^{\vee} , Y_{r}(L)^{+} \otimes {\mathbb C}) \\
= {\rm Hom}_{{\rm Gal}(L/F)}(W^{\vee} , ( Y_{r}(L)^{{\rm Gal}(L/K)})^{+} \otimes {\mathbb C}) \\
= {\rm Hom}_{{\rm Gal}(K/F)}(V^{\vee} , Y_{r}(K)^{+} \otimes {\mathbb C})
\end{array} \]
and these isomorphisms transport $( R_{L}^{r} \cdot (f_{L}^{r})^{-1})_{*}$ into $( R_{K}^{r} \cdot (f_{K}^{r})^{-1})_{*}$ by virtue of the above commutative diagrams. Furthermore, since the Artin $L$-function is invariant under inflation, $L_{F}^{*}(r, V) = L_{F}^{*}(r , W)$. On the other hand, the inflation homomorphism
\[ \mathrm{Inf}_{{\rm Gal}(K/F)}^{{\rm Gal}(L/F)} : R( {\rm Gal}(K/F) ) \rightarrow R({\rm Gal}(L/F)) \]
induces the canonical quotient map
\[ \pi_{L/K} : {\mathbb Q}[{\rm Gal}(L/F)]^\times \rightarrow {\mathbb Q}[ {\rm Gal}(K/F) ]^\times \]
via the isomorphism of Proposition \ref{1.5}. Hence
\[ \pi_{L/K} ( {\cal R}_{f_{L}^{r}} ) = {\cal R}_{f_{K}^{r}} . \]
Let $ \alpha \in {\rm End}_{{\mathbb Q}[{\rm Gal}(L/F)]}( Y_{r}(L)^{+} \otimes {\mathbb Q} )$ satisfy the integrality condition of \S\ref{2.1}
\[ \alpha \cdot f_{L}^{r}( K_{1-2r}({\cal O}_{L}[t^{\pm 1}])) \subseteq Y_{r}(L) . \]
Extend this by the identity on the $(-1)$-eigenspace of complex conjugation $Y_{r}(L)^{-} \otimes {\mathbb Q} $ to give
\[ \alpha \oplus 1 \in {\rm End}_{{\mathbb Q}[{\rm Gal}(L/F)]}( Y_{r}(L) \otimes {\mathbb Q} ) . \]
The endomorphism $\alpha$ commutes with the action by ${\rm Gal}(L/K)$ so there is $\hat{\alpha} \in {\rm End}_{{\mathbb Q}[{\rm Gal}(K/F)]}( Y_{r}(K)^{+} \otimes {\mathbb Q} )$ making the following diagram commute
\[ \xymatrix{
Y_r(K)^+ \teno{\bb{Z}} \bb{Q} \ar[r]^{\hat{\alpha}} \ar[d] & Y_r(K)^+ \teno{\bb{Z}} \bb{Q} \ar[d] \\
Y_r(L)^+ \teno{\bb{Z}} \bb{Q} \ar[r]^\alpha & Y_r(L)^+ \teno{\bb{Z}} \bb{Q} .
} \]
Therefore $\hat{\alpha}$ satisfies the integrality condition of \S\ref{2.1}
\[ \hat{\alpha} \cdot f_{K}^{r}( K_{1-2r}({\cal O}_{K}[t^{\pm 1}])) \subseteq Y_{r}(K) . \]
We may choose a ${\mathbb Z}[1/2][{\rm Gal}(K/F)]$ basis for $Y_{r}(K) \otimes {\mathbb Z}[1/2]$ consisting of embeddings $\sigma_{i} : K \rightarrow {\mathbb C}$ for $1 \leq i \leq m$. Let $ \sigma'_{i}$ be an embedding of $L$ which extends $\sigma_{i}$ for $1 \leq i \leq m$. Then a ${\mathbb Z}[1/2][{\rm Gal}(L/F)]$ basis for $Y_{r}(L) \otimes {\mathbb Z}[1/2]$ is given by $\{ \sigma'_{1} , \sigma'_{2} , \ldots , \sigma'_{m} \}$. The embedding of $Y_{r}(K) $ into $Y_{r}(L)$ is given by $\sigma_{i} \mapsto \sum_{g \in {\rm Gal}(L/K)} \ g(\sigma'_{i} ) $ which implies that the $m \times m$ matrix for $\hat{\alpha}$ with respect to the ${\mathbb Z}[1/2][{\rm Gal}(K/F)]$ basis of $\sigma_{i}$'s is the image of the $m \times m$ matrix for $\alpha$ with respect to the ${\mathbb Z}[1/2][{\rm Gal}(L/F)]$ basis of $\sigma'_{i}$'s under the canonical surjection
\[ {\mathbb Q}[{\rm Gal}(L/F)] \rightarrow {\mathbb Q}[{\rm Gal}(K/F)] . \]
This discussion has established the following result.
\begin{prop} \label{2.4}
Suppose that $F \subseteq K \subseteq L$ is a tower of number fields with $L/F$ abelian. Then, in the notation of \S\ref{2.1}, the canonical surjection
\[ \pi_{L/K} : {\mathbb Q}[{\rm Gal}(L/F)] \rightarrow {\mathbb Q}[{\rm Gal}(K/F)] \]
satisfies
\[ \pi_{L/K}( {\cal J}_{L/F}^{r} ) \subseteq {\cal J}_{K/F}^{r} . \]
\end{prop}
Proposition \ref{2.4} explains the existence of the maps in (\ref{quot examples}).
\subsection{Behaviour under inclusion maps ${\rm Gal}(L/K) \rightarrow {\rm Gal}(L/F)$} \label{2.5}
As in \S\ref{2.4}, suppose that $F \subseteq K \subseteq L$ is a tower of number fields with $L/F$ abelian. The inclusion of ${\rm Gal}(L/K)$ into ${\rm Gal}(L/F)$ induces an inclusion of group-rings ${\mathbb Q}[{\rm Gal}(L/K)]$ into ${\mathbb Q}[{\rm Gal}(L/F)]$. In terms of the isomorphism of Proposition \ref{1.5}, as is easily seen by the formula, this homomorphism is induced by the restriction of representations
\[ \mathrm{Res}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)} : R({\rm Gal}(L/F)) \rightarrow R({\rm Gal}(L/K)) . \]
If $V$ is a complex representation of ${\rm Gal}(L/F)$ then
\begin{eqnarray*}
{\cal R}_{f_{L }^{r}}( \mathrm{Res}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)}(V)) &=& \frac{R( \mathrm{Res}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)}(V),f_{L}^{r})}{L_{K}^{*}( r , \mathrm{Res}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)}( V)) } \\
&=& \frac{R( \mathrm{Res}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)}(V),f_{L}^{r})}{L_{F}^{*}( r , \mathrm{Ind}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)}(\mathrm{Res}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)}( V))) } \\
&=& \frac{R( \mathrm{Res}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)}(V),f_{L}^{r})}{L_{F}^{*}( r , V \otimes \mathrm{Ind}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)}(1)) } .
\end{eqnarray*}
If $W_{i} \in \chgrp{\rm Gal}(L/F)$ for $1 \leq i \leq [K : F]$ is the set of one-dimensional representations which restrict to the trivial representation on ${\rm Gal}(L/K)$ then \linebreak $\mathrm{Ind}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)}(1)) = \oplus_{i} \ W_{i}$. By Frobenius reciprocity
\[ \begin{array}{l}
{\rm Hom}_{{\rm Gal}(L/K)}( \mathrm{Res}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)}(V)^{\vee} , Y_{r}(L )^{+} \otimes {\mathbb C}) ) \\
=
{\rm Hom}_{{\rm Gal}(L/F)}( \oplus_{i} \ (V \otimes W_{i} )^{\vee} , Y_{r}(L )^{+} \otimes {\mathbb C}) )
\end{array} \]
so that
\[ R( \mathrm{Res}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)}(V),f_{L}^{r}) = \prod_{i} \ R( V \otimes W_{i} , f_{L}^{r}) \]
and
\[ {\cal R}_{f_{L }^{r}}( \mathrm{Res}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)}(V)) = \prod_{i} \ {\cal R}_{f_{L }^{r}}( V \otimes W_{i} ) . \]
Let $H \subseteq G$ be finite groups with $G$ abelian. It will suffice to consider the case in which $G/H$ is cyclic of order $n$ generated by $gH$.
Let $W \otimes {\mathbb Q}$ be a free ${\mathbb Q}[G]$-module with basis $v_{1}, \ldots , v_{r}$. Then $W \otimes {\mathbb Q}$ is a free ${\mathbb Q}[H]$-module with basis $\{ g^{a}v_{i} \ | \ 0 \leq a \leq n-1, \ 1 \leq i \leq r \}$. Set ${\cal S} = \{ 0, \ldots , n-1 \} \times \{ 1 , \ldots , r \}$; then for $\underline{u} = (a , i) \in {\cal S}$, we set $e_{\underline{u}} = g^{a}v_{i}$. If $\tilde{\alpha} \in {\rm End}_{{\mathbb Q}[H]}( W \otimes {\mathbb Q} )$ we may write
\[ \tilde{\alpha}(e_{\underline{w}} ) = \sum_{\underline{u}} \ A_{ \underline{u}. \underline{w}} e_{\underline{u}} \]
so that $A$ is an $nr \times nr$ matrix with entries in ${\mathbb Q}[H]$.
Now consider the induced ${\mathbb Q}[G]$-module $\mathrm{Ind}_{H}^{G}( W \otimes {\mathbb Q})$, which is a free ${\mathbb Q}[G]$-module on the basis $\{ 1 \otimes_{H} e_{\underline{u}} \ | \ \underline{u} \in {\cal S} \}$. Hence the $nr \times nr$ matrix, with entries in ${\mathbb Q}[G]$, for $1 \otimes_{H} \tilde{\alpha}$ with respect to this basis is the image of $A$ under the canonical inclusion of $\phi_{H,G} : {\mathbb Q}[H] \rightarrow {\mathbb Q}[G]$. In particular
\[ \phi_{H,G}( \textstyle{\det}} % {\mathrm{det}_{ {\mathbb Q}[H] }( \tilde{\alpha})) = \textstyle{\det}} % {\mathrm{det}_{ {\mathbb Q}[G]}( {\mathbb Q}[G] \otimes_{ {\mathbb Q}[H] } \tilde{\alpha}) \]
and, by induction on $[G:H]$, this relation is true for an arbitrary inclusion $H \subseteq G$ of finite abelian groups.
This discussion yields the following result:
\begin{prop} \label{2.6}
Suppose that $F \subseteq K \subseteq L$ is a tower of number fields with $L/F$ abelian. Then, in the notation of \S\ref{2.1}, the canonical inclusion
\[ \phi_{K/F} : {\mathbb Q}[{\rm Gal}(L/K)] \rightarrow {\mathbb Q}[{\rm Gal}(L/F)] \]
maps $ {\cal J}_{L/K}^{r}$ onto the $ {\mathbb Z}[1/2][{\rm Gal}(L/K)]$-submodule
\[ {\mathbb Z}[1/2][{\rm Gal}(L/K)] \langle \textstyle{\det}} % {\mathrm{det}_{ {\mathbb Q}[{\rm Gal}(L/F)]}( {\mathbb Q}[{\rm Gal}(L/F)] \otimes_{ {\mathbb Q}[{\rm Gal}(L/K)] } (\alpha \oplus 1) ) \tau(\hat{{\cal R}}_{f_{L}^{r} })^{-1} \rangle . \]
Here, in terms of Proposition \ref{1.5}, $ \hat{{\cal R}}_{f_{L}^{r} } \in {\mathbb Q}[ {\rm Gal}(L/F)]^\times$ is given by
\[ \hat{{\cal R}}_{f_{L}^{r} }(V) = {\cal R}_{f_{L}^{r} }( V \otimes \mathrm{Ind}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)}(1)) \]
and $ \alpha \in {\rm End}_{{\mathbb Q}[{\rm Gal}(L/K)]}( Y_{r}(L)^{+} \otimes {\mathbb Q} )$ runs through endomorphisms satisfying the integrality condition of \S\ref{2.1}.
\end{prop}
\subsection{Behaviour under fixed-point maps} \label{2.7}
As in \S\ref{2.4}, suppose that $F \subseteq K \subseteq L$ is a tower of number fields with $L/F$ abelian. Let $e_{L/K} = [L:K]^{-1}( \sum_{y \in {\rm Gal}(L/K)} \ y ) $ denote the idempotent associated with the subgroup ${\rm Gal}(L/K) $.
There is a homomorphism of unital rings of the form
\[ \lambda_{K/F} : {\mathbb Q}[{\rm Gal}(K/F)] \rightarrow {\mathbb Q}[{\rm Gal}(L/F)] \]
given, for $z \in {\rm Gal}(L/F)$, by the formula
\[ \lambda_{K/F}( z {\rm Gal}(L/K)) = (1 - e_{L/K}) + z \cdot e_{L/K} \in {\mathbb Q}[{\rm Gal}(L/F)] . \]
From Proposition \ref{1.5} it is easy to see that in terms of group characters
\[ {\rm Map}( \chgrp{{\rm Gal}}(K/F) , \overline{{\mathbb Q}}) \rightarrow {\rm Map}( \chgrp{{\rm Gal}}(L/F) , \overline{{\mathbb Q}}) \]
this sends a function $h$ on $ \chgrp{{\rm Gal}}(K/F)$ to the function $h'$ given by
\[ h'(\chi) = \left\{
\begin{array}{ll}
h(\chi_{1}) & {\rm if} \ \mathrm{Inf}_{{\rm Gal}(K/F)}^{{\rm Gal}(L/F)}(\chi_{1}) = \chi , \\
1 & {\rm otherwise}.
\end{array}
\right. \]
Sending a complex representation $V$ of ${\rm Gal}(L/F)$ to its ${\rm Gal}(L/K)$-fixed points $V^{{\rm Gal}(L/K)}$ gives a homomorphism
\[ \mathrm{Fix} : R({\rm Gal}(L/F)) \rightarrow R({\rm Gal}(K/F)) . \]
In terms of one-dimensional respresentations (i.e. characters) the above condition $ \mathrm{Inf}_{{\rm Gal}(K/F)}^{{\rm Gal}(L/F)}(\chi_{1}) = \chi $ is equivalent to $\mathrm{Fix}(\chi) = \chi_{1}$.
Let $V$ be a one-dimensional complex representation of ${\rm Gal}(L/F)$ fixed by ${\rm Gal}(L/K)$. Then we have isomorphisms of the form
\[ \begin{array}{l}
{\rm Hom}_{{\rm Gal}(L/F)}(( V^{{\rm Gal}(L/K)})^{\vee} , Y_{r}(L)^{+} \otimes {\mathbb C}) \\
= {\rm Hom}_{{\rm Gal}(K/F)}( V^{\vee} , ( Y_{r}(L)^{{\rm Gal}(L/K)})^{+} \otimes {\mathbb C}) \\
= {\rm Hom}_{{\rm Gal}(K/F)}(V^{\vee} , Y_{r}(K)^{+} \otimes {\mathbb C})
\end{array} \]
and, by invariance of $L$-functions under inflation, $L_{F}^{*}(r, V) = L_{F}^{*}(r, V^{{\rm Gal}(L/K)})$.
Therefore, by the discussion of \S\ref{2.3},
\[ {\cal R}_{f_{L }^{r}}(V) = {\cal R}_{f_{K }^{r}}( V^{{\rm Gal}(L/K)} ) . \]
On the other hand, if $V^{{\rm Gal}(L/K)} = 0$ then $ {\cal R}_{f_{K }^{r}}( V^{{\rm Gal}(L/K)} ) = 1$ since both $L_{F}^{*}(r, 0 )$ and the determinant of the identity map of the trivial vector space are equal to one.
This establishes the formula
\[ \lambda_{K/F}( {\cal R}_{f_{K }^{r}}) = (1 - e_{L/K}) + {\cal R}_{f_{L }^{r}} \cdot e_{L/K} . \]
Now consider an endomorphism
\[ \alpha \in {\rm End}_{ {\mathbb Q}[{\rm Gal}(K/F)]}( Y_{r}(K)^{+} \otimes {\mathbb Q} ) \]
satisfying the integrality condition of \S\ref{2.1}
\[ \alpha f_{r , K}( K_{1-2r}( {\cal O}_{K}[t^{\pm 1}]) ) \subseteq Y_{r}(K)^{+} \cong (Y_{r}(L)^{+})^{{\rm Gal}(L/K)} . \]
Let $v_{1}, v_{2}, \ldots , v_{d}$ be a ${\mathbb Z}[1/2][ {\rm Gal}(L/F) ]$-basis of $Y_{r}(L)[1/2]$
so that a \linebreak ${\mathbb Z}[1/2][ {\rm Gal}(K/F) ]$-basis of the subspace $(Y_{r}(L)^{+})^{{\rm Gal}(L/K)}[1/2] \cong Y_{r}(K)[1/2]$ is given by $ \{ (\sum_{y \in {\rm Gal}(L/K)} \ y )v_{i} \ | \ 1 \leq i \leq d \} $. To construct the generators of ${\cal J}_{K/F}^{r} $, as in \S\ref{2.1}, we must calculate the determinant of $\alpha \oplus 1$ on $Y_{r}(K)^{+} \otimes {\mathbb Q} \oplus Y_{r}(K)^{-} \otimes {\mathbb Q} = Y_{r}(K) \otimes {\mathbb Q}$ with respect to the basis $ \{ (\sum_{y \in {\rm Gal}(L/K)} \ y )v_{i} \}$
and divide by $ \tau( {\cal R}_{f_{K}^{r}})$.
Let $\hat{\alpha} \in {\rm End}_{ {\mathbb Q}[{\rm Gal}(L/F)]}( Y_{r}(L) \otimes {\mathbb Q} ) $ be given by $\alpha$ on $Y_{r}(L)^{{\rm Gal}(L/F)} \otimes {\mathbb Q}$ and the identity on $(1 - e_{L/K})Y_{r}(L) \otimes {\mathbb Q}$. Hence $\hat{\alpha}$ satisfies the integrality condition
\[ \hat{\alpha} \cdot f_{L}^{r}( K_{1-2r}({\cal O}_{L}[t^{\pm 1}]))^{{\rm Gal}(L/F)} \subseteq Y_{r}(L)^{{\rm Gal}(L/F)} , \]
because, as in \S\ref{2.3}, $f_{K}^{r}$ may be assumed to extend to $f_{L}^{r}$.
Therefore
\[ e_{L/K} \frac{{\rm det}(\hat{\alpha})}{ \tau( {\cal R}_{f_{L}^{r}})} \in e_{L/K} {\cal J}_{L/F}^{r}
\subset {\mathbb Q}[ {\rm Gal}(L/F) ] . \]
On the other hand it is clear that $\lambda_{K/F}( {\rm det}(\alpha \oplus 1)) = {\rm det}(\hat{\alpha})$.
This discussion has established the following result.
\begin{prop} \label{2.8}
Suppose that $F \subseteq K \subseteq L$ is a tower of number fields with $L/F$ abelian and let
\[ \lambda_{K/F} : {\mathbb Q}[{\rm Gal}(K/F)] \rightarrow {\mathbb Q}[{\rm Gal}(L/F)] \]
denote the unital ring homomorphism of \S\ref{2.7}. Then
\[ \lambda_{K/F}( {\cal J}_{K/F}^{r} ) \subseteq (1- e_{L/K}) {\mathbb Q}[{\rm Gal}(L/F)] + e_{L/K} {\cal J}_{L/F}^{r} . \]
\end{prop}
\subsection{Behaviour under corestriction maps} \label{2.9}
As in \S\ref{2.4}, suppose that $F \subseteq K \subseteq L$ is a tower of number fields with $L/F$ abelian. There is an additive homomorphism of the form
\[ \iota_{K/F} : {\mathbb Q}[{\rm Gal}(L/F)] \rightarrow {\mathbb Q}[{\rm Gal}(L/K)] \]
called the transfer or corestriction map. In terms of
Proposition \ref{1.5} it is induced by the induction of representations
\[ {\rm Ind}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)} : R({\rm Gal}(L/K)) \rightarrow R({\rm Gal}(L/F)) . \]
That is, the image $ \iota_{K/F} (h)$ of $h \in {\rm Hom}_{\Omega_{\mathbb Q}}( R({\rm Gal}(L/F)) , \overline{{\mathbb Q}}) $ is given by
\[ \iota_{K/F} (h)(V) = h( {\rm Ind}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)}(V) ) . \]
By Frobenius reciprocity, for each $V \in R({\rm Gal})(L/K) )$ there is an isomorphism
\[ \begin{array}{l}
{\rm Hom}_{{\rm Gal}(L/F)}(( {\rm Ind}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)})^{\vee} , Y_{r}(L)^{+} \otimes {\mathbb C}) \\
= {\rm Hom}_{{\rm Gal}(L/K)}( V^{\vee} , Y_{r}(L)^{+} \otimes {\mathbb C}) .
\end{array} \]
Also $L_{F}^{*}( r , {\rm Ind}_{{\rm Gal}(L/K)}^{{\rm Gal}(L/F)}(V)) = L_{K}^{*}(r , V)$ so that
\[ \iota_{K/F}( {\cal R}_{f_{L}^{r}} ) = {\cal R}_{f_{L}^{r}} . \]
Now consider an endomorphism
\[ \alpha \in {\rm End}_{ {\mathbb Q}[{\rm Gal}(L/F)]}( Y_{r}(L)^{+} \otimes {\mathbb Q} ) \]
satisfying the integrality condition of \S\ref{2.1}
\[ \alpha f_{r , L}( K_{1-2r}( {\cal O}_{L}[t^{\pm 1}]) ) \subseteq Y_{r}(L)^{+} . \]
Then it is straightforward to see from Proposition \ref{1.5} that the determinant of $\alpha \oplus 1$ as a map of ${\mathbb Q}[{\rm Gal}(L/F)]$-modules ${\rm det}_{{\mathbb Q}[{\rm Gal}(L/F)]}(\alpha \oplus 1)$
is mapped to ${\rm det}_{{\mathbb Q}[{\rm Gal}(L/K)]}(\alpha \oplus 1)$, the determinant of $\alpha \oplus 1$ as ${\rm det}_{{\mathbb Q}[{\rm Gal}(L/K)]}(\alpha \oplus 1)$.
This discussion has established the following result.
\begin{prop} \label{2.10}
Suppose that $F \subseteq K \subseteq L$ is a tower of number fields with $L/F$ abelian, and let
\[ \iota_{K/F} : {\mathbb Q}[{\rm Gal}(L/F)] \rightarrow {\mathbb Q}[{\rm Gal}(L/K)] \]
denote the additive homomorphism of \S\ref{2.9}. Then
\[ \iota_{K/F}( {\cal J}_{L/F}^{r} ) \subseteq {\cal J}_{L/K}^{r} . \]
\end{prop}
\subsection{} \label{nat irrats}
We can now explain the second example in \S \ref{nat examples}, i.e. Proposition \ref{nat irrats example}. Let us work more generally to begin with. $E$ and $F$ can be any number fields, and we suppose we have a diagram
\[ \xymatrix{
& E \ar@{-}[dl]_C \ar@{-}[dr]^H & \\
L \ar@{-}[dr]^{G'} & & F \ar@{-}[dl] \\
& K &
} \]
satisfying the following: $E/K$ is Galois (though not necessarily abelian), $LF = E$, $L \cap F = K$, the extension $L/K$ is abelian (and hence so is $E/F$), and $L/K$ and $E/F$ satisfy the Stark conjecture. We let $G = \gal{E/K}$, and the Galois groups of the other Galois extensions are marked in the diagram. We observe that $C$ need not be abelian here.
Owing to the natural isomorphism $G/C \rightarrow H$, each character $\psi \in \chgrp{H}$ extends to a unique one-dimensional representation $\widehat{\psi} : G \rightarrow \bb{C}^\times$ which is trivial on $C$. Denote by $\irrch{G}$ the set of irreducible characters of $G$. Then having chosen a $\bb{Q}[G]$-module isomorphism $f$ as in \S \ref{stark conjectures}, we can define an element $\BC{f} \in \bb{C}[H]^\times$ by
\[ \BC{f} = \prod_{\chi \in \irrch{G} \! \smallsetminus \! \{1\}} \left(\sum_{\psi \in \chgrp{H}} \VR{f}_{E/K}(\chi \widehat{\psi})^{d_\chi} e_\psi \right) ,\]
where for a character $\chi$ of $G$, $d_\chi$ is the multiplicity of the trivial character of $H$ in $\mathrm{Res}_H^G(\chi)$. We have opted to denote by $\VR{f}_{E/K}$ the group-ring element $\mathcal{R}_{f_E}$ defined in \S \ref{2.1}, to emphasize which extension is being considered.
The following lemma shows that the group-ring element $\VR{f}_{E/F}$ for the extension $E/F$ is related, via $\BC{f}$, to the corresponding element for the extension $L/K$.
\begin{lemma} \label{base change}
$\BC{f}$ has rational coefficients, and the image of $\VR{f}_{E/F}$ under the isomorphism $\Phi : \bb{Q}[H] \rightarrow \bb{Q}[G']$ is
\[ \VR{f'}_{L/K} \Phi(\BC{f}) ,\]
where $f'$ is the $\bb{Q}[G']$-module isomorphism making diagram (\ref{f chosen to commute}) commute.
\end{lemma}
The proof of the lemma is little more than a combination of \S \ref{2.3} and \S \ref{2.9}.
In the situation of Proposition \ref{nat irrats example} (with $L = E^+$ and $K = \bb{Q}$ now) we find that the element $2\tilde{\stick}$ occurring there is just $\tau(\BC{f})^{-1}$ (for any choice of $f$ in this case). Indeed, let $\rho \in \chgrp{G}$ be the unique non-trivial character extending the trivial character of $H$. Then the only $\chi \in \irrch{G} \! \smallsetminus \! \{1\}$ with $d_\chi \not= 0$ is $\rho$, and $d_\rho = 1$, so
\begin{eqnarray*}
\BC{f} &=& \sum_{\psi \in \chgrp{H}} \VR{f}_{E/\bb{Q}}(\rho\widehat{\psi}) e_\psi \\
&=& \sum_{\substack{\psi \in \chgrp{G} \\ \psi \mathrm{ even}}} \VR{f}_{E/\bb{Q}}(\rho \psi) e_{\psi|_H} .
\end{eqnarray*}
However, for $\psi$ even, $\rho \psi$ is odd so that $\VR{f}(\rho \psi) = L_{E/\bb{Q},S}(0,\rho \psi)^{-1}$. Using the easily verified fact that $(1-c)\tilde{\stick} = \theta_{E/\bb{Q},S}$, where $c \in G$ is complex conjugation, we see that $L_{E/\bb{Q},S}(0,\rho \psi) = 2\psi|_H(\tau \tilde{\stick})$, from which the assertion follows.
Applying Lemma \ref{base change} now justifies the appearance of $2\Phi_n(\tilde{\stick}_n)$ in Proposition \ref{nat irrats example}.
\section{The passage to non-abelian groups}
\subsection{} \label{3.1} In this section we shall use the Explicit Brauer Induction constructions of \cite[pp.138--147]{snaith:ebi} to pass from finite abelian Galois groups to the non-abelian case.
Let $G$ be a finite group and consider the additive homomorphism
\[ \sum_{H \subseteq G} \ \mathrm{Ind}_{H}^{G} \mathrm{Inf}_{H^{\mathrm{ab}}}^{H} : \oplus_{H \subseteq G} \ R(H^{\mathrm{ab}}) \rightarrow R(G) .\]
Let $N \lhd G$ be a normal subgroup and let $\pi : G \rightarrow G/N$ denote the quotient homomorphism.
Define a homomorphism
\[ \alpha_{G,N} : \oplus_{J \subseteq G/N} \ R(J^{\mathrm{ab}}) \rightarrow \oplus_{H \subseteq G} \ R(H^{\mathrm{ab}}) \]
to be the homomorphism which sends the $J$-component $R(J^{\mathrm{ab}}) $ to the $H = \pi^{-1}(J)$-component $R( \pi^{-1}(J)^{\mathrm{ab}}) $ via the map
\[ \mathrm{Inf}_{J^{\mathrm{ab}}}^{\pi^{-1}(J)^{\mathrm{ab}}}(R(J^{\mathrm{ab}})) \rightarrow R(\pi^{-1}(J)^{\mathrm{ab}}) . \]
\begin{lemma} \label{3.2}
In the notation of \S\ref{3.1} the following diagram commutes:
\[ \xymatrix{
\bigoplus_{J \subseteq G/N} R(J^{\mathrm{ab}}) \ar[r] \ar[d]^{\alpha_{G,N}} & R(G/N) \ar[d]^{\mathrm{Inf}_{G/N}^G} \\
\bigoplus_{H \subseteq G} R(H^{\mathrm{ab}}) \ar[r] & R(G) .
} \]
\end{lemma}
\begin{proof}
Since the kernel of $\pi^{-1}(J) \rightarrow J$ and that of $\pi : G \rightarrow G/N$ coincide, both being equal to $N$, we have
\[ \mathrm{Inf}_{G/N}^{G} \mathrm{Ind}_{J}^{G/N} = \mathrm{Ind}_{\pi^{-1}(J) }^{G} \mathrm{Inf}_{J}^{ \pi^{-1}(J) } . \]
Therefore, given a character $\phi : J^{\mathrm{ab}} \rightarrow \overline{{\mathbb Q}}^\times$ in the $J$-coordinate, we have
\[ \begin{array}{ll}
\mathrm{Ind}_{ \pi^{-1}(J) }^{G} \mathrm{Inf}_{\pi^{-1}(J)^{\mathrm{ab}}}^{\pi^{-1}(J)} \alpha_{G,N}(\phi) & = \mathrm{Ind}_{ \pi^{-1}(J) }^{G}\mathrm{Inf}_{\pi^{-1}(J)^{\mathrm{ab}}}^{\pi^{-1}(J)} \mathrm{Inf}_{J^{\mathrm{ab}}}^{\pi^{-1}(J)^{\mathrm{ab}}}( \phi) \\
& = \mathrm{Ind}_{ \pi^{-1}(J) }^{G}\mathrm{Inf}_{J}^{\pi^{-1}(J)} \mathrm{Inf}_{J^{\mathrm{ab}}}^{J}( \phi) \\
& = \mathrm{Inf}_{G/N}^{G} \mathrm{Ind}_{J}^{G/N} \mathrm{Inf}_{J^{\mathrm{ab}}}^{J}( \phi) ,
\end{array} \]
as required.
\end{proof}
\subsection{} \label{3.3} The homomorphism of \S\ref{3.1} is invariant under group conjugation and therefore induces an additive homomorphism of the form
\[ B_{G} : ( \oplus_{H \subseteq G} \ R(H^{\mathrm{ab}}))_{G} \rightarrow R(G) \]
where $X_{G}$ denotes the coinvariants of the conjugation $G$-action. This homomorphism is a split surjection whose right inverse is given by the Explicit Brauer Induction homomorphism
\[ A_{G} : R(G) \rightarrow ( \oplus_{H \subseteq G} \ R(H^{\mathrm{ab}}))_{G} \]
constructed in \cite[Section 4.5.16]{snaith:ebi}. We shall be interested in the dual homomorphisms (\cite[Section 4.5.20]{snaith:ebi})
\[ B_{G}^{*} : {\rm Hom}_{\Omega_{{\mathbb Q}}}( R( G) ,
\overline{{\mathbb Q}}) \rightarrow ( \oplus_{H \subseteq G} \ {\rm Hom}_{\Omega_{{\mathbb Q}}}( R( H^{\mathrm{ab}}) , \overline{{\mathbb Q}}) )^{G} \]
and
\[ A_{G}^{*} : ( \oplus_{H \subseteq G} \ {\rm Hom}_{\Omega_{{\mathbb Q}}}( R( H^{\mathrm{ab}}) , \overline{{\mathbb Q}}) )^{G} \rightarrow {\rm Hom}_{\Omega_{{\mathbb Q}}}( R( G) ,
\overline{{\mathbb Q}}) \]
where $X^{G}$ denotes the subgroup of $G$-invariants.
As in \cite[Def.4.5.4]{snaith:ebi}, denote by ${\mathbb Q}\{G\}$ the rational vector space whose basis consists of the conjugacy classes of $G$. There is an isomorphism (\cite[Prop.4.5.14]{snaith:ebi})
\[ \psi : {\mathbb Q}\{G\} \stackrel{\cong}{\rightarrow} {\rm Hom}_{\Omega_{{\mathbb Q}}}( R( G) ,
\overline{{\mathbb Q}}) \]
given by the formula $\psi( \sum_{\gamma} \ m_{\gamma} \gamma)( \rho) = \sum_{\gamma} \ m_{\gamma} {\rm Trace}(\rho(\gamma))$.
When $G$ is abelian, we have $ {\mathbb Q}\{G\} = {\mathbb Q}[G]$ and under the identification
\[ {\rm Hom}_{\Omega_{{\mathbb Q}}}( R( G) ,
\overline{{\mathbb Q}}) = {\rm Map}_{\Omega_{{\mathbb Q}}}(\chgrp{G} , \overline{{\mathbb Q}}) \]
of Proposition \ref{1.5} we have $\psi(g) = ( \chi \mapsto \chi(g))$, which is a ring isomorphism inverse to $\lambda_{G}$.
\section{${\cal J}_{E/F}^{r}$ in general } \label{4.1}
Let $G$ denote the Galois group of a finite Galois extension $E/F$ of number fields. Hence each subgroup of $G$ has the form $H = {\rm Gal}(E/E^{H})$, whose abelianization is $H^{\mathrm{ab}} = {\rm Gal}(E^{[H,H]}/E^{H})$ where $[H,H]$ is the commutator subgroup of $H$. For each integer $r= 0, -1, -2, -3, \ldots$, we have the canonical fractional Galois ideal ${\cal J}_{E^{[H,H]}/E^{H}}^{r} \subseteq{\mathbb Q}[H^{\mathrm{ab}}]$ as defined in \S\ref{2.1}.
\begin{definition} \label{4.2}
In the notation of \S\ref{4.1}, define a subgroup ${\cal J}_{E/F}^{r}$ of ${\mathbb Q}\{G\}$ by
\[ {\cal J}_{E/F}^{r} = (B_{G}^{*})^{-1}( \oplus_{H \subseteq G} \ {\cal J}_{E^{[H,H]}/E^{H}}^{r}) . \]
\end{definition}
\begin{lemma} \label{4.3}
In \S\ref{4.1} and Definition \ref{4.2}, when $G = {\rm Gal}(E/F)$ is abelian then ${\cal J}_{E/F}^{r}$ coincides with the canonical fractional Galois ideal of \S\ref{2.1}.
\end{lemma}
\begin{proof}
The $H$-component of $B_{G}^{*}$ has the form
\[ {\mathbb Q}[ {\rm Gal}(E/F)] \stackrel{i_{E^{H}/F}}{\rightarrow } {\mathbb Q}[ {\rm Gal}(E/E^{H})] \stackrel{\pi_{E/E^{[H,H]}}}{\rightarrow} {\mathbb Q}[ {\rm Gal}(E^{[H,H]}/E^{H})] \]
which maps ${\cal J}_{E/F}^{r} $ to ${\cal J}_{E^{[H,H]}/E^{H}}^{r} $ by Proposition \ref{2.4} and Proposition \ref{2.10} so that
\[ {\cal J}_{E/F}^{r} \subseteq (B_{G}^{*})^{-1}( \oplus_{H \subseteq G} \ {\cal J}_{E^{[H,H]}/E^{H}}^{r} ) . \]
On the other hand, the $G$-component of $B_{G}^{*}$ is the identity map from ${\mathbb Q}[G]$ to itself. Therefore if $z \in {\mathbb Q}[G] \! \smallsetminus \! {\cal J}_{E/F}^{r}$ then $B_{G}^{*}(z) \not\in \oplus_{H \subseteq G} \ {\cal J}_{E^{[H,H]}/E^{H}}^{r}$, as required.
\end{proof}
\begin{prop} \label{4.4}
Suppose that $F \subseteq K \subseteq L$ is a tower of finite extensions of number fields with $L/F$ and $K/F$ Galois. Then, for $r=0, -1, -2, -3, \ldots $, the canonical homomorphism
\[ \pi_{L/K} : {\mathbb Q}\{ {\rm Gal}(L/F) \} \rightarrow {\mathbb Q}\{ {\rm Gal}(K/F) \} \]
satisfies $ \pi_{L/K}( {\cal J}_{E/F}^{r} ) \subseteq {\cal J}_{K/F}^{r} $.
\end{prop}
\begin{proof}
This follows immediately from Proposition \ref{2.4}, Lemma \ref{3.2} and Definition \ref{4.2}.
\end{proof}
\begin{definition} \label{4.5}
Let $F$ be a number field and $L/F$ a (possibly infinite) Galois extension with Galois group $G = {\rm Gal}(L/F)$. For $r=0, -1, -2, -3, \ldots$ define ${\cal J}_{E/F}^{r} $ to be the abelian group
\[ {\cal J}_{E/F}^{r} = \proo{H} {\cal J}_{L^{H}/F}^{r} ,\]
where $H$ runs through the open normal subgroups of $G$.
\end{definition}
\section{${\cal J}_{E/F}^{r} $ and the annihilation of \\ $ H_{{\rm \acute{e}t}}^{2}({\rm Spec}({\cal O}_{L,S}) , {\mathbb Z}_{\ell}(1-r))$}
\subsection{} \label{5.1} Let $\ell$ be an odd prime. We continue to assume the Stark conjecture as stated in \S\ref{stark conjectures} for $r = 0, -1, -2, -3, \ldots $. Replacing ${\mathbb Q}$ by ${\mathbb Q}_{\ell}$ in \S\ref{2.1} and Definition \ref{4.2} we may associate a finitely generated ${\mathbb Z}_{\ell}$-submodule of ${\mathbb Q}_{\ell}\{ {\rm Gal}(E/F) \}$, again denoted by ${\cal J}_{E/F}^{r} $, to any finite extension $E/F$ of number fields.
In this section we are going to explain a conjectural procedure to pass from ${\cal J}_{E/F}^{r} $ to the construction of elements in the annihilator ideal of the \'{e}tale cohomology of the ring of $S$-integers of $E$,
\[ {\rm ann}_{{\mathbb Z}_{\ell}[G(E/F)]}( H_{{\rm \acute{e}t}}^{2}(
{\rm Spec}({\cal O}_{E,S(E)}) , {\mathbb Z}_{\ell}(1-r))) ,\]
where $S$ denotes a finite set of primes of $F$ including all archimedean primes and all finite primes which ramify in $E/F$, and $S(E)$ denotes all the primes of $E$ over those in $S$. This conjectural procedure was first described in \cite[Thm.8.1]{snaith:stark}.
We shall restrict ourselves to the case when $r = -1,-2, -3, \ldots$. In several ways this is a simplification over the case when $r=0$. In this case $H_{{\rm \acute{e}t}}^{1}(
{\rm Spec}({\cal O}_{E, S(E)}) , {\mathbb Z}_{\ell}(1-r))$ is independent of $S(E)$, while it is related to the group of $S(E)$-units when $r=0$. Also, when $ r \leq -1$, $H_{{\rm \acute{e}t}}^{2}(
{\rm Spec}({\cal O}_{E,S(E)}) , {\mathbb Z}_{\ell}(1-r))$ is a subgroup of the corresponding cohomology group when $S(E)$ is enlarged to $S'(E)$, but when $r=0$ the class-group of ${\cal O}_{E, S'(E)}$ is a quotient of that of ${\cal O}_{E , S(E)}$. Furthermore (see \cite{buckingham:frac}, \cite{tate:stark}), there are subtleties concerning whether or not to use the $S$-modified $L$-function in \S \ref{notation} when $r=0$, while for $r \leq -1$ this is immaterial.
When $r=0$ the annihilator procedure is similar to the other cases but the additional complications have prompted us to omit this case.
Write $G = {\rm Gal}(E/F) $, and for each subgroup $H = {\rm Gal}(E/E^{H}) \subseteq G$ \linebreak let $S(E^{H})$ denote the set of primes of $E^{H}$ above those of $S$. Then $H^{\mathrm{ab}} = {\rm Gal}(E^{[H,H]}/E^{H})$ where $[H,H]$ denotes the commutator subgroup of $H$. The following conjecture originated in \cite{snaith:equiv,snaith:rel,snaith:stark}.
\begin{conj} \label{5.2}
In the notation of \S\ref{5.1}, when $r = -1, -2, -3, \ldots$,
(i) \ {\bf Integrality:}
\[ {\cal J}_{E^{[H,H]}/E^{H}}^{r} \cdot {\rm ann}_{{\mathbb Z}_{\ell}[H^{\mathrm{ab}}]}( {\rm Tors} H_{{\rm \acute{e}t}}^{1}(
{\rm Spec}({\cal O}_{E^{[H,H]},S}) , {\mathbb Z}_{\ell}(1-r))) \subseteq {\mathbb Z}_{\ell}[H^{\mathrm{ab}}] . \]
(ii) \ {\bf Annihilation:}
\[ \begin{array}{l} {\cal J}_{E^{[H,H]}/E^{H}}^{r} \cdot {\rm ann}_{{\mathbb Z}_{\ell}[H^{\mathrm{ab}}]}( {\rm Tors} H_{{\rm \acute{e}t}}^{1}(
{\rm Spec}({\cal O}_{E^{[H,H]},S}) , {\mathbb Z}_{\ell}(1-r))) \\
\subseteq {\rm ann}_{{\mathbb Z}_{\ell}[H^{\mathrm{ab}}]}( H_{{\rm \acute{e}t}}^{2}(
{\rm Spec}({\cal O}_{E^{[H,H]},S}) , {\mathbb Z}_{\ell}(1-r))) .
\end{array} \]
\end{conj}
(We have adopted the shorthand: $\mathcal{O}_{E^{[H,H]},S} = \mathcal{O}_{E^{[H,H]},S(E^{[H,H]})}$.)
\subsection{Evidence} \label{5.3}
Part (i) of Conjecture \ref{5.2} is analogous to the Stickelberger integrality, which is described in \cite[Section 2.2]{snaith:stark}. Stickelberger integrality was proven in certain totally real cases in \cite{kl:padicl,cs:padic,cassou-nogues:valeurs,dr:abelianlfunctions}, for $r = 0$. In general, when $r=0$, it is part of the Brumer conjecture \cite{brumer:units}. The novelty of part (ii) of Conjecture \ref{5.2}, when it was introduced in \cite{snaith:rel} and \cite{snaith:stark}, was the annihilator prediction when the $L$-function vanishes at $s=r$. For the part of the fractional ideal corresponding to characters whose $L$-functions are non-zero at $s = r$, generated by the higher Stickelberger element at $s = r$, part (ii) is the conjecture of \cite{cs:stickel}.
Let us consider the cyclotomic example $\hJ{r}{L/\bb{Q}}$ ($r <0$) when $L = \bb{Q}(\zeta)$ for some root of unity $\zeta$, and suppose $\ell$ is an odd prime dividing the order of $\zeta$. In this case, $\hJ{r}{L/\bb{Q}}$ splits into plus and minus parts for complex conjugation, i.e.
\[ \hJ{r}{L/\bb{Q}} = e_+^r \hJ{r}{L/\bb{Q}} \oplus e_-^r \hJ{r}{L/\bb{Q}} ,\]
where $e_+^r = \frac{1}{2}(1 + (-1)^r c)$, $e_-^r = \frac{1}{2}(1 - (-1)^r c)$ and $c \in G = \gal{L/\bb{Q}}$ is complex conjugation. By the proof of \cite[Theorem 6.1]{snaith:stark}, $e_-^r \hJ{r}{L/\bb{Q}}$ is generated by the Stickelberger element $\theta_{L/\bb{Q},S}(r)$ defined in terms of $L$-function values at $s = r$. However, by \cite{dr:abelianlfunctions},
\[ \mathrm{ann}_{\Z_\prm[G]}(\mathrm{Tors} (H_{\mathrm{\acute{e}t}}^1(\mathrm{Spec} \mathcal{O}_{L,S},\Z_\prm(1-r)))) \theta_{L/\bb{Q},S}(r) \subseteq \Z_\prm[G] .\]
Further, the proof of \cite[Theorem 7.6]{snaith:stark} shows that $e_+^r \hJ{r}{L/\bb{Q}} \subseteq \Z_\prm[G]$. In fact, \cite[Theorem 6.1]{snaith:stark} also shows that part (ii) of Conjecture \ref{5.2} holds in this case (with $E = \bb{Q}$ and $H = G$), the intersection ``$\cap \Z_\prm[G]$'' found in the statement of that theorem being unnecessary.
Turning now to the case $r = 0$, with the field $E_n$ as in \S \ref{nat examples}, we have a similar scenario for $\iJ{E_n/\bb{Q},S}$, where $S = \{\infty,\ell\}$. Indeed, we see from (\ref{recap full j desc}) that $\iJ{E_n/\bb{Q},S}$ again splits into plus and minus parts, with the minus part being generated by the Stickelberger element $\theta_{E_n/\bb{Q},S}$ defined at $s = 0$. Stickelberger's theorem then implies that
\[ \mathrm{ann}_{\Z_\prm[G_n]}(\rou{E_n}) e_- \iJ{E_n/\bb{Q},S} \subseteq \Z_\prm[G_n] ,\]
and $e_+ \iJ{E_n/\bb{Q},S}$ is already in $\Z_\prm[G_n]$. The roles of the plus and minus parts of $\iJ{E_n/\bb{Q},S}$ will become clear in \S \ref{commutative example} below.
\subsubsection{An Iwasawa-theoretic example} \label{commutative example}
(\ref{recap full j desc}) can be used to provide an example of the relationship of $\iJ{E_n/\bb{Q},S}$ to Iwasawa theory, with an inverse limit of the $\iJ{E_n/\bb{Q},S}$ over $n$ giving rise, in a suitable way, to Fitting ideals of both the plus and minus parts of an inverse limit of class-groups (Proposition \ref{iJ and fitt of class}). Given $n \geq 0$, let $\cycq{n}/\bb{Q}$ be the degree $\ell^n$ subextension of the (unique) $\Z_\prm$-extension $\cycq{\infty}$ of $\bb{Q}$. We then have the field diagram
\[ \xymatrix{
& E_n \ar@{-}[dl]_{\Delta_n} \ar@{-}[ddr]^{\Gamma_n} & \\
\cycq{n} \ar@{-}[ddr] & & \\
& & E_0 \ar@{-}[dl]^\Delta \\
& \bb{Q} &
} \]
in which $\cycq{n} \cap E_0 = \bb{Q}$ and $\cycq{n} E_0 = E_n$, so that the Galois group $G_n = \gal{E_n/\bb{Q}}$ is the internal direct product of $\Delta_n$ and $\Gamma_n$. $S$ will denote the set of places $\{\infty,\ell\}$ of $\bb{Q}$.
By virtue of the natural isomorphism $\Delta_n \rightarrow \Delta$, characters of $\Delta_n$ correspond to characters of $\Delta$. If $\delta \in \chgrp{\Delta}$, we let $\delta_n$ denote the corresponding character in $\chgrp{\Delta}_n$. Now, the idea is to view the group-ring $\bb{C}[G_n]$ as $\bb{C}[\Gamma_n][\Delta_n]$. In doing this, we can define a projection $\pi_n(\delta) : \bb{C}[G_n] \rightarrow \bb{C}[\Gamma_n]$ by extending $\delta_n$ linearly (over $\bb{C}[\Gamma_n]$).
Finally, fix an isomorphism $\nu : \mathbb{C}_\prm \rightarrow \bb{C}$ and let $\omega : \Delta \rightarrow \bb{C}^\times$ be the composition of the Teichm\"uller character $\Delta \rightarrow \mathbb{C}_\prm^\times$ with $\nu : \mathbb{C}_\prm^\times \rightarrow \bb{C}^\times$. Then given $\delta \in \chgrp{\Delta}$, $\delta^*$ will denote $\omega \delta^{-1}$. Observe that since $\omega$ is odd, $\delta$ is even if and only if $\delta^*$ is odd.
\begin{prop} \label{iJ and fitt of class}
Let $\delta \in \chgrp{\Delta}$. ($\delta$ may be even or odd.)
\[ \mathrm{Fitt}_{\Z_\prm\pwr{\Gamma_\infty}}(e_{\delta^*} \mathrm{Cl}_\infty) = \left\{
\begin{array}{ll}
\displaystyle{\proo{n}} \Z_\prm \pi_n(\delta^*)(\iJ{\ef_n/\Q,S}) & \textrm{if $\delta \not= 1$} \\
\displaystyle{\proo{n}} \Z_\prm \pi_n(\delta^*)((1 - (1+\ell)\sigma_n^{-1}) \iJ{\ef_n/\Q,S}) & \textrm{if $\delta = 1$}
\end{array}
\right. \]
where $\sigma_n = (1+\ell,E_n/\bb{Q})$.
\end{prop}
\begin{proof}
This stems from (\ref{recap full j desc}), which we reproduce for convenience:
\[ \iJ{E_n/\bb{Q},S} = \frac{1}{2} e_+ \mathrm{ann}_{\bb{Z}[G_n]}(\mathcal{O}_{E_n^+,S}^\times/\mathcal{E}_n^+) \oplus \bb{Z}[G_n]\theta_{E_n/\bb{Q},S} .\]
Let us deal with even characters $\delta \in \chgrp{\Delta}$ first. For simplicity, we will assume that $\delta \not= 1$, though in fact the case $\delta = 1$ is similar. (\ref{recap full j desc}) tells us that for each $n \geq 0$, $\Z_\prm \pi_n(\delta^*)(\iJ{\ef_n/\Q,S}) = \Z_\prm[\Gamma_n]\pi_n(\delta^*)(\theta_{E_n/\bb{Q},S})$. However, Iwasawa's construction of $\ell$-adic $L$-functions (see \cite{iwasawa:padicl} and \cite[Chapter 7]{wash:cyc}) shows that this lies in $\Z_\prm[\Gamma_n]$ and that the inverse limit of these ideals is generated by the algebraic $\ell$-adic $L$-function corresponding to the even character $\delta$. Mazur and Wiles' proof (see \cite{mw:classfields}) of the Main Conjecture of Iwasawa theory, and later Wiles' generalization of this (see \cite{wiles:iwasawa}), show that this in turn is equal to the Fitting ideal appearing in the statement of the proposition.
Now we turn to odd characters $\delta \in \chgrp{\Delta}$. Referring to (\ref{recap full j desc}) again, we find that
\[ \Z_\prm \pi_n(\delta^*)(\iJ{\ef_n/\Q,S}) = \pi_n(\delta^*)(\mathrm{Fitt}_{\Z_\prm[G_n]}((\mathcal{O}_{E_n^+,S}^\times/\mathcal{E}_n^+) \teno{\bb{Z}} \Z_\prm)) .\]
This uses that $(\mathcal{O}_{E_n^+,S}^\times/\mathcal{E}_n^+) \teno{\bb{Z}} \Z_\prm$ is cocyclic as a $\Z_\prm[G_n]$-module so that, since $G_n$ is cyclic, the Fitting and annihilator ideals of $(\mathcal{O}_{E_n^+,S}^\times/\mathcal{E}_n^+) \teno{\bb{Z}} \Z_\prm$ agree. \cite[Theorem 1]{cg:fitting} says in particular that this Fitting ideal is equal to that of $\mathrm{Cl}(E_n^+) \teno{\bb{Z}} \Z_\prm$. Combining the above and passing to limits completes the proof.
\end{proof}
We observe the importance here of taking leading coefficients of $L$-functions at $s = 0$ rather than just values. For $\delta$ even (i.e. $\delta^*$ odd), $\pi_n(\delta^*)(\iJ{\ef_n/\Q,S})$ concerns $L$-functions which are non-zero at $0$, and we get the usual Stickelberger elements which are related to \emph{minus} parts of class-groups via $\ell$-adic $L$-functions. However when $\delta$ is odd (i.e. $\delta^*$ is even), $\pi_n(\delta^*)(\iJ{\ef_n/\Q,S})$ is concerned with $L$-functions having simple zeroes at $0$, which are related to \emph{plus} parts of class-groups via cyclotomic units.
\section{ ${\cal J}_{E/F}^{r}$ and annihilation} \label{5.4}
Let $\ell$ be an odd prime. Given $\alpha \in {\cal J}_{E/F}^{r}$ and $H \subseteq G = {\rm Gal}(E/F)$, choose any
\[ \beta \in {\rm ann}_{{\mathbb Z}_{\ell}[H^{\mathrm{ab}}]}( {\rm Tors} H_{{\rm \acute{e}t}}^{1}(
{\rm Spec}({\cal O}_{E^{[H,H]},S}) , {\mathbb Z}_{\ell}(1-r))) . \]
Then the $H$-component $B_{G}^{*}(\alpha)_{H}$ lies in ${\mathbb Q}_{\ell}[H^{\mathrm{ab}}]^{N_{G}H}$, the fixed points under the conjugation action by $N_{G}H$, the normalizer of $H$ in $G$. Assuming Conjecture \ref{5.2}(i), $ B_{G}^{*}(\alpha)_{H} \cdot \beta \in {\mathbb Z}_{\ell}[H^{\mathrm{ab}}]^{N_{G}H}$. Choose $z_{H, \alpha, \beta} \in {\mathbb Z}_{\ell}[H]$ such that
\[ \pi( z_{H, \alpha, \beta} ) = B_{G}^{*}(\alpha)_{H} \cdot \beta . \]
Consider the composition
\[ \begin{array}{l}
H_{{\rm \acute{e}t}}^{2}({\rm Spec}({\cal O}_{E,S(E)}) , {\mathbb Z}_{\ell}(1-r)) \stackrel{ {\rm Tr}_{E/E^{[H,H]} }}{\rightarrow} H_{{\rm \acute{e}t}}^{2}(
{\rm Spec}({\cal O}_{E^{[H,H]},S}) , {\mathbb Z}_{\ell}(1-r)) \\
\hspace{40pt} \stackrel{ B_{G}^{*}(\alpha)_{H} \cdot \beta }{\rightarrow}
H_{{\rm \acute{e}t}}^{2}(
{\rm Spec}({\cal O}_{E^{[H,H]},S}) , {\mathbb Z}_{\ell}(1-r)) \\
\hspace{80pt} \stackrel{ j }{\rightarrow}
H_{{\rm \acute{e}t}}^{2}(
{\rm Spec}({\cal O}_{E,S(E)}) , {\mathbb Z}_{\ell}(1-r))
\end{array} \]
in which $j$ is induced by the inclusion of fields and ${\rm Tr}_{E/E^{[H,H]} }$ denotes the transfer homomorphism.
Assuming Conjecture \ref{5.2}(ii), this composition is zero. However, by Frobenius reciprocity for the cohomology transfer, for all $a \in H_{{\rm \acute{e}t}}^{2}({\rm Spec}({\cal O}_{E,S(E)}) , {\mathbb Z}_{\ell}(1-r))$
\[ \begin{array}{ll}
0 & = j( \pi(z_{H, \alpha, \beta}) {\rm Tr}_{E/E^{[H,H]} }(a) ) \\
& = j \cdot {\rm Tr}_{E/E^{[H,H]} } ( z_{H, \alpha, \beta} \cdot a) \\
& = (\sum_{h \in {\rm Gal}(E/E^{[H,H]} )} \ h ) z_{H, \alpha, \beta} \cdot a .
\end{array} \]
\begin{definition} \label{5.5}
In the situation of \S\ref{5.1} and \S\ref{5.4}, let $\ncJ{E/F,r} \subseteq {\mathbb Z}_{\ell}[G]$ denote the left ideal generated by the elements $ (\sum_{h \in {\rm Gal}(E/E^{[H,H]} )} \ h ) z_{H, \alpha, \beta}$ as $\alpha$, $H$ and $\beta$ vary through all the possibilities above.
\end{definition}
\begin{theorem} \label{5.6}
If Conjecture \ref{5.2} is true for all abelian intermediate extensions $E^{[H,H]}/E^{H}$ of $E/F$ then the left action of the left ideal $\ncJ{E/F,r}$ annihilates
\[ H_{{\rm \acute{e}t}}^{2}(
{\rm Spec}({\cal O}_{E,S(E)}) , {\mathbb Z}_{\ell}(1-r)) . \]
\end{theorem}
\begin{remark} \label{5.7}
If $G$ is abelian in Definition \ref{5.5} and Theorem \ref{5.6}, then
\[ \ncJ{E/F,r} = {\cal J}_{E/F}^{r} \cdot {\rm ann}_{{\mathbb Z}_{\ell}[G]}( {\rm Tors} H_{{\rm \acute{e}t}}^{1}(
{\rm Spec}({\cal O}_{E, S(E)}) , {\mathbb Z}_{\ell}(1-r))) . \]
That is, $ \ncJ{E/F,r}$ equals the left hand side of Conjecture \ref{5.2}(ii).
\end{remark}
\begin{prop} \label{intJ is two-sided}
In Definition \ref{5.5}, $\ncJ{E/F,r}$ is a two-sided ideal in $\Z_\cop[G]$.
\end{prop}
\begin{proof}
In the notation of \S\ref{5.4}, it suffices to show that
\[ w \left(\sum_{h \in \gal{E/E^{\comm{H}}}} h\right) z_{H,\alpha,\beta} w^{-1} \]
lies in $\ncJ{E/F,r}$. Consider
\[ w \left(\sum_{h \in \gal{E,E^{\comm{H}}}} h \right) w^{-1} = \sum_{h \in \gal{E/E^{\comm{wHw^{-1}}}}} h \]
and $w z_{H,\alpha,\beta} w^{-1}$. Since $z_{H,\alpha,\beta}$ lies in $\Z_\cop[H]$ and maps to $B_G^*(\alpha) \beta$ in $\Z_\cop[H^{\mathrm{ab}}]$, we see that $wz_{H,\alpha,\beta} w^{-1}$ lies in $\Z_\cop[wHw^{-1}]$ and maps to $wB_G^*(\alpha)_H w^{-1} w \beta w^{-1}$ in $\Z_\cop[H^{\mathrm{ab}}]$. However, $w B_G^*(\alpha)_H w^{-1} = B_G^*(\alpha)_{wHw^{-1}}$ and $w\beta w^{-1}$ lies in
\[ \mathrm{ann}_{\Z_\cop[(wHw^{-1})^{\mathrm{ab}}]}(\mathrm{Tors} H_{\mathrm{\acute{e}t}}^1(\mathrm{Spec}(\mathcal{O}_{E^{\comm{wHw^{-1}}},S}),\Z_\cop(1-r))) ,\]
completing the proof.
\end{proof}
\begin{prop} \label{nc quotient}
Suppose that $F \subseteq K \subseteq E$ is a tower of number fields with $E/F$ and $K/F$ Galois. Then for $r = -1,-2,-3,\ldots$, the canonical homomorphism $\pi_{E/K} : \Z_\cop[\gal{E/F}] \rightarrow \Z_\cop[\gal{K/F}]$ satisfies
\[ \pi_{E/K}(\ncJ{E/F,r}) \subseteq \ncJ{K/F,r} .\]
\end{prop}
\begin{proof}
This follows easily from Lemma \ref{3.2} and Proposition \ref{4.4}.
\end{proof}
\section{Relation to Iwasawa theory} \label{sec iwasawa theory}
As discussed in the Introduction, the motivation for examining the behaviour of the fractional Galois ideal under changes of extension is to set up investigating a possible role in Iwasawa theory. Via the relationship of the fractional ideal with Stark-type elements (eg cyclotomic units in the case $r = 0$ and Beilinson elements in the case $r < 0$, discussed in \cite{buckingham:frac} and \cite{snaith:rel} resp.), one might hope that an approach involving Euler systems would be fruitful here. A general connection of the fractional Galois ideal to Stark elements of arbitrary rank was demonstrated in \cite{buckingham:phd}, and the link of Stark elements with class-groups using the theory of Euler systems is discussed in \cite{rubin:kolyvagin,popescu:rubin}, so that a strategy as above would seem promising.
We conclude the paper with some speculation concerning what the non-commutative Iwasawa theory of Fukaya--Kato \cite{fk:noncomm}, Kato \cite{kato:heisenberg} and Ritter--Weiss \cite{rw:teitseries} suggests about $\hJ{r}{E/F}$ of Definition \ref{4.5} and $\ncJ{E/F,r}$ of Definition \ref{5.5}.
It is worth pointing out, before we begin the recapitulation proper, that \cite{fk:noncomm,kato:heisenberg,rw:teitseries} often restrict to the situation where the extension fields are totally real, which tends to involve only one of the eigenspaces of complex conjugation acting on $\hJ{r}{E/F}$ and $\ncJ{E/F,r}$. We have tried to give some examples (for example, \S \ref{commutative example}) which illustrate the expected role and properties of the other eigenspace.
Further, in this area there is an immense litany of conjectures (see \cite{fk:noncomm,burns:leading}) of which Stark's conjecture is approximately the weakest. All the constructions we have made are contingent \emph{only} on the truth of Stark's conjecture, which is crucial for us but also seems fundamental; it is assumed, for example, in \cite{rw:lrnc}.
Following \cite{kato:heisenberg}, let $\ell$ be an odd prime (denoted $p$ there), $F$ a totally real number field and $F_\infty$ a totally real Lie extension of $F$ containing $\bb{Q}(\zeta_{\ell^\infty})^+$. Here, $\bb{Q}(\zeta_{\ell^\infty})^+$ is the union of the totally real fields $\bb{Q}(\zeta_{\ell^n})^+ = \bb{Q}(\zeta_{\ell^n} + \zeta_{\ell^n}^{-1})$ over all $n \geq 1$. Let $G = \gal{F_\infty/F}$, and assume that only finitely many primes of $F$ ramify in $F_\infty$. Fix a finite set $\Sigma$ of primes of $F$ containing the ones which ramify in $F_\infty/F$. Define $\Lambda(G)$ to be the Iwasawa algebra of $G$, given by $\Lambda(G) = \Z_\prm\pwr{G} = \proo{U} \Z_\prm[G/U]$, where the limit runs over all open normal subgroups of $G$.
Let $C$ denote the cochain complex of $\Lambda(G)$-modules given by
\[ {\rm RHom}( \rm{R} \Gamma_{\acute{e}t}( {\cal O}_{F_{\infty}}[1/ \Sigma] , {\mathbb
Q}_{\ell}/{\mathbb Z}_{\ell}) , {\mathbb Q}_{\ell}/{\mathbb Z}_{\ell}) ,\]
so that $H^{0}(C) =
{\mathbb Z}_{\ell}$ with trivial $G$-action and $H^{-1}(C) = {\rm Gal}(M/
F_{\infty})$, the Galois group of the maximal pro-$\ell$ abelian extension of
$F_{\infty}$ unramified outside $\Sigma$. The other $H^{i}(C)$'s are zero and
$ {\rm Gal}(M/ F_{\infty})$ is a finitely generated torsion (left)
$\Lambda(G)$-module. Let $F^{cyc} \subseteq F_{\infty}$ denote the cyclotomic
${\mathbb Z}_{\ell}$-extension and set $H = {\rm Gal}( F_{\infty} / F^{cyc})
\subseteq G$ so that $G/H \cong {\mathbb Z}_{\ell}$. As in \cite{cfksv:main}, let
\[ S = \{f \in \Lambda(G) \spc | \spc \textrm{$\Lambda(G)/\Lambda(G)f$ is finitely generated as a $\Lambda(H)$-module}\} .\]
Then $S$ is an Ore set, which means that its elements may be inverted to form
the localized ring
$\Lambda(G)_{S}$, and there is an exact localization sequence of algebraic
K-groups
\[ K_{1}(\Lambda(G)) \rightarrow K_{1}(\Lambda(G)_{S})
\stackrel{\partial}{\rightarrow} K_{0}(\Lambda(G) , \Lambda(G)_{S} )
\rightarrow K_{0}(\Lambda(G)) \rightarrow
K_{0}(\Lambda(G)_{S}) . \]
By \cite{hs:pseudonullity}, Iwasawa's conjecture concerning the vanishing of the
$\mu$-invariant implies that the cohomology of the perfect complex $C$ vanishes
when $S$-localized. This gives rise to a class $[C] \in K_{0}(\Lambda(G) ,
\Lambda(G)_{S} )$. In the case of finite Galois extensions the class
$[C] $ accounts for the Stickelberger phenomena (c.f. \cite{snaith:stark}) but on the
other hand so do values of Artin $L$-functions. The main conjecture of
non-commutative Iwasawa theory, described below following
\cite{kato:heisenberg}, makes this relation clear in terms of
$\Lambda(G)_{S}$-modules.
There is an $\ell$-adic determinantal valuation which assigns to $f \in
K_{1}(\Lambda(G)_{S}) $
and a continuous Artin representation $\rho$ a value $f(\rho) \in
\overline{{\mathbb Q}}_\ell \cup \{\infty\} $. The main conjecture of
non-commutative Iwasawa theory asserts that there exists
$\xi \in K_{1}(\Lambda(G)_{S}) $ such that (i) $\partial(\xi) = -[C]$ and (ii)
$\xi(\rho \kappa^{r}) =
L_{\Sigma}( 1-r , \rho)$ for any even $r \geq 2$ where $\kappa$ is the $\ell$-adic
cyclotomic character and $L_{\Sigma}(s, \rho)$ is the Artin $L$-function of
$\rho$ with the Euler factors at $\Sigma$ removed.
The main conjecture of Iwasawa theory was formulated in \cite{rw:lrnc} and studied in the series of papers \cite{rw:teitseries} when the Lie group $G$ has rank zero or
one. The case of $G = GL_{2}(\mathbb Z_\ell)$ is of particular interest in the
study of elliptic curves $E/{\mathbb Q}$ without complex multiplication \cite{cfksv:main} and is proven for the $\ell$-adic Heisenberg group in \cite{kato:heisenberg}. For a comprehensive survey see \cite{fk:noncomm}.
Motivated by the main conjecture of Iwasawa theory, and more generally by the
role of $\Lambda(G)$ in the arithmetic geometry of elliptic curves and their
Selmer groups, there has been considerable ring-theoretic activity concerning
$\Lambda(G)$ and $\Omega(G) = \Lambda(G)/\ell \Lambda(G)$ (see \cite{ab:ringtheoretic,ab:primeness,awz:reflexive,venjakob:structuretheory,venjakob:weierstrass,venjakob:padiclie}). The
rings $\Lambda(G)$ and
$\Omega(G)$ are examples of ``just-infinite rings'' which both satisfy the
Auslander--Gorenstein condition and are thus amenable to Lie theoretic analysis.
In the survey article \cite{ab:ringtheoretic}, a number of questions are posed. In particular the constructions of \S7 are directly related to \cite[Question G]{ab:ringtheoretic}: ``Is there a mechanism for constructing ideals of Iwasawa algebras which involves neither central elements nor closed normal subgroups?''
\begin{prop} \label{limit of ncJ exists}
If $F_{\infty}/F$ is any $\ell$-adic Lie extension of a number field $F$ with
Galois group $G$ then, under the assumption of \S \ref{5.4} for the finite intermediate
subextensions $E/F$ for $r=-1, -2, -3, \ldots$ we may define a two-sided ideal
\[ \ncJ{F_{\infty}/F , r} = \proo{E} \ncJ{E/F,r} \]
in $\Lambda(G)$, where the limit is taken over finite Galois subextensions $E/F$ of $F_{\infty}/F$.
\end{prop}
In view of the annihilation discussion of \S \ref{5.4}, Proposition \ref{limit of ncJ exists} suggests
the following:
\begin{question} \label{8.2}
What is the intersection of the canonical Ore set $S$ of \cite{cfksv:main,kato:heisenberg} with $\ncJ{F_{\infty}/F , r}$?
\end{question}
In many ways the most interesting case is when $G = GL_{2}(\mathbb Z_{\ell})$ ($\ell \geq 7$) arising from the tower of $\ell$-primary torsion points on an elliptic curve over ${\mathbb Q}$ without complex multiplication \cite{coates:fragments,cfksv:main}. In this case one has particularly strong information concerning two-sided primes ideals of $\Lambda(G)$ -- see \cite{awz:reflexive}. There is a possibly alternative approach to the construction of fractional Galois ideals in ${\mathbb Q}_{\ell}[{\rm Gal}(K/ {\mathbb Q})]$ based on assuming that a type of Stark conjecture holds for the Hasse--Weil $L$-function of the elliptic curve \cite{stopple:stark}. It would be interesting to know whether this leads to the same two-sided ideal as Proposition \ref{limit of ncJ exists}.
|
2,869,038,156,416 | arxiv | \section{Introduction}
Cognates are words that have a common etymological origin \cite{crystal2008dictionary}. They account for a considerable amount of unique words in many lexical domains, notably technical texts. The orthographic similarity of cognates can be exploited in different tasks involving recognition of translational equivalence between words, such as machine translation and bilingual terminology compilation. For \textit{e.g.,} the German - English cognates, \textit{Blume - bloom} can be identified as cognates with orthographic similarity methods. Detection of cognates helps various NLP applications like IR \cite{pranav2018alignment}. \newcite{rama2018automatic} study various cognate detection techniques and provide substantial proof that automatic cognate detection can help infer phylogenetic trees. In many NLP tasks, the orthographic similarity of cognates can compensate for the insufficiency of other kinds of evidence about the translational equivalency of words \cite{mulloni2006automatic}. The detection of cognates in compiling bilingual dictionaries has proven to be helpful in Machine Translation (MT), and Information Retrieval (IR) tasks \cite{meng2001generating}. Orthographic similarity-based methods have relied on the lexical similarity of word pairs and have been used extensively to detect cognates \cite{ciobanu2014automatic,mulloni2007automatic,inkpen2005automatic}. These methods, generally, calculate the similarity score between two words and use the result to build training data for further classification. Cognate detection can also be performed using phonetic features and researchers have previously used consonant class matching (CCM) \cite{turchin2010analyzing}, sound class-based alignment (SCA) \cite{list2010sca} \textit{etc.} to detect cognates in multilingual wordlists. The identification of cognates, here, is based on the comparison of words sound correspondences. Semantic similarity methods have also been deployed to detect cognates among word pairs \cite{kondrak2001identifying}. The measure of semantic similarity uses the context around both word pairs and helps in the identification of a cognate word pair by looking of similarity among the collected contexts.
For our work, we can primarily divide words into four main categories \textit{viz.} \textbf{True Cognates, False Cognates, False Friends and Non-Cognates}. In Figure \ref{fig:cogmat}, we present this classification with examples from various languages along with their meanings for better understanding. While some false friends are also false cognates, most of them are genuine cognates. \textit{Our primary goal is to be able to identify True Cognates.} Sanskrit (Sa) is known to be the mother of most of the Indian languages. Hindi (Hi), Bengali (Bn), Punjabi (Pa), Marathi (Mr), Gujarati (Gu), Malayalam (Ml), Tamil (Ta) and Telugu (Te) are known to borrow many words from it. Thus, one may observe that \textit{words which belong to the same concept in these languages, if orthographically similar, are True Cognates.} Currently, we include loan words in the dataset used for our work and include them as cognates. Since, eventually we aim to apply our work to Machine Translation and other NLP applications, we believe that this would help establish a better correlation among source-target language pairs. Also, we do not detect false friends and hence restrict the scope of True cognate detection using this hypothesis to Figure \ref{fig:cogmat2}.
\begin{figure}[!ht]
\includegraphics[width=1\linewidth]{imgs/matrixOneMod.jpg}
\caption{The Cognate Identification Matrix}
\label{fig:cogmat}
\end{figure}
\begin{figure}[!ht]
\includegraphics[width=1\linewidth]{imgs/matrixTwo.jpg}
\caption{Scope of our work; Detection of True Cognates and False Friends}
\label{fig:cogmat2}
\end{figure}
\textbf{We utilize the synset information from linked Wordnets to identify words within the same concept and deploy orthographic similarity-based methods to compute similarity scores between them.} This helps us identify words with a high similarity score. In case of most of the Indian languages, a sizeable contribution of words/concepts is loaned from the Sanskrit language. In linked IndoWordnet, each concept is aligned to the other based on an `id' which can be reliably used as a measure to say that the etymological origin is the same, for both the concepts. Hence, words with the same orthographic similarity can be said to be \textbf{`True Cognates'}. Using this methodology, we detect highly similar words and use them as training data to build models which can predict whether a word pair is cognate or not. The rest of the paper is organized as follows. In Section 2 we describe the related work that has been carried out on cognate detection together with some of its practical applications, while in Section 3 we present our approach and deal in greater detail with our learning algorithms. Once the proposed methodology has been outlined, we step through an evaluation method we devised and report on the results obtained as specified in Section 4. Section 5 concludes our paper with a brief summary and tackling further challenges in the near future.
\subsection{Contributions}
We make the following contributions in this paper:\\
1. We perform cognate detection for eleven Indian Languages.\\
2. We exploit Indian languages behaviour to obtain a list of true cognates (WNdata from WordNet and PCData from Parallel Corpora).\\
3. We train neural networks to establish a baseline for cognate detection.\\
4. We validate the importance of Wordnets as a resource to perform cognate detection.\\
5. We release our dataset (WNdata + PCdata) of cognate pairs publicly for the language pairs Hi - Mr, Hi - Pa, Hi - Gu, Hi - Bn, Hi - Sa, Hi - Ml, Hi - Ta, Hi - Te, Hi - Ne, and Hi - Ur.
\section{Related Work}
\label{sec:RelWork}
One of the most common techniques to find cognates is based on the manual design of rules describing how orthography of a borrowed word should change, once it has been introduced into the other language. \newcite{koehn2000estimating} expand a list of English-German cognate words by applying well-established transformation rules. They also noted that the accuracy of their algorithm increased proportionally with the length of the word since the accidental coexistence of two words with the same spelling with different meanings (we identify them as `false friends') decreases the accuracy.
Most previous studies on automatic cognate identification do not investigate Indian languages. Most of the Indian languages borrow cognates or ``loan words'' from Sanskrit. Indian languages like Hindi, Bengali, Sinhala, Oriya and Dravidian languages like Malayalam, Tamil, Telugu, and Kannada borrow many words from Sanskrit. Although recently, \newcite{kanojia2019cognate} perform cognate detection for a few Indian languages, but report results with manual verification of their output. Identification of cognates for improving IR has already been explored for Indian languages \cite{makin2007approximate}. String similarity-based methods are often used as baseline methods for cognate detection and the most commonly used among them is Edit distance based similarity measure. It is used as the baseline in the early cognate detection papers \cite{melamed1999bitext}. Essentially, it computes the number of operations required to transform from source to target cognate.
Research in automatic cognate detection using phonetic aspects involves computation of similarity by decomposing phonetically transcribed words \cite{kondrak2000new}, acoustic models \cite{mielke2012assessing}, phonetic encodings \cite{rama2015comparative}, aligned segments of transcribed phonemes \cite{list2012lexstat}. We study \newcite{rama2016siamese}'s research, which employs a Siamese convolutional neural network to learn the phonetic features jointly with language relatedness for cognate identification, which was achieved through phoneme encodings. Although it performs well on the accuracy, it shows poor results with MRR. \newcite{jager2017using} use SVM for phonetic alignment and perform cognate detection for various language families. Various works on Orthographic cognate detection usually take alignment of substrings within classifiers like SVM \cite{ciobanu2014automatic,ciobanu2015automatic} or HMM \cite{bhargava2009multiple}. We also consider the method of \newcite{ciobanu2014automatic}, which employs dynamic programming based methods for sequence alignment. Among cognate sets common overlap set measures like set intersection, Jaccard \cite{jarvelin2007s}, XDice \cite{brew1996word} or TF-IDF \cite{wu2008interpreting} could be used to measure similarities and validate the members of the set.
\section{Datasets and Methodology}
\label{sec:DnM}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.8\linewidth]{imgs/BlockDiagram.jpg}
\caption{Block Diagram for our experimental setup}
\label{fig:bdia}
\end{figure*}
We investigate language pairs for major Indian languages namely Marathi (Mr), Gujarati (Gu), Bengali (Bn), Punjabi (Pa), Sanskrit (Sa), Malayalam (Ml), Tamil (Ta), Telugu (Te), Nepali (Ne) and Urdu (Ur) with Hindi (Hi). We create two datasets as described below for \texttt{<source\textunderscore lang> -<target\textunderscore lang>} where the source language is always Hindi. We describe each step in the subsections below.
\subsection{Datasets}
\subsubsection*{Dataset 1: WordNet based dataset}
We create this dataset (WNData) by extracting synset data from the IndoWordnet database. We maintain all words, in the concept space, in a comma-separated format. We, then, create word lists by combining all possible permutations of word pairs within each synset. For \textit{e.g.,} If synset ID X on the source side (Hindi) contains words $S_1W_1$ and $S_1W_2$, and parallelly on the target side (other Indian languages), synset ID X contains $T_1W_1$ and $T_1W_2$, we create a word list such as: \\
$S_1W_1,T_1W_1$\\
$S_1W_2,T_1W_1$\\
$S_1W_1,T_1W_2$\\
$S_1W_2,T_1W_2$\\
To avoid redundancy, we remove duplicate word pairs from this list.
\subsubsection*{Dataset 2: Parallel Corpora based dataset}
We use the ILCI parallel corpora for Indian languages \cite{jha2010tdil} and create word pairs list by comparing all words in the source side sentence with all words on the target side sentence. Our hypothesis, here, is that words with high orthographic similarity which occur in the same context window (a sentence) would be cognates with a high probability. Due to the unavailability of ILCI parallel corpora for Sa and Ne, we download these corpora from Wikipedia and align it with the Hindi articles from Hindi Wikipedia. We calculate exact word matches to align articles to each other thus creating comparable corpora and discard unaligned lines from both sides. We, then, create similar word pairs list between Hindi and all the other languages pairs. We removed duplicated word pairs from this list as well and call this data PCData.
\subsection{Script Standardization and Text Normalization}
The languages mentioned above share a major portion of the most spoken languages in India. Although most of them borrow words from Sanskrit, they belong to different language families. Mr, Gu, Bn, Pa, Ne and Ur belong to the Indo-Aryan family of languages; and Ml, Ta, Te belong to the family of Dravidian languages. They also use different scripts to represent themselves textually. For standardization, we convert all the other written scripts to Devanagari. We perform Unicode transliteration using Indic NLP Library\footnote{\url{https://anoopkunchukuttan.github.io/indic_nlp_library/}} to convert scripts for Bn, Gu, Pa, Ta, Te, Ml, and Ur to Devanagari, for both our datasets. Hi, Mr, Sa, and Ne are already based on the Devanagari script, and hence we only perform text normalization for both our datasets, for these languages. The whole process is outlined in Figure \ref{fig:bdia}.
\subsection{Similarity Scores Calculation}
We calculate similarity scores for each word on the source side \textit{i.e.,} Hi by matching it with each word on the target side \textit{i.e.,} Sa, Bn, Gu, Pa, Mr, Ml, Ne, Ta, Te, and Ur.
Since we match the words from the same concept space or the same context window, we eliminate the possibility of this word pair carrying different meanings, and hence \textbf{a high orthographic similarity score gives us a strong indication of these words falling under the category of True Cognates}. For training neural network models, we then divide the positive and negative labels based on a threshold and follow empirical methods in setting this threshold to 0.5 for both datasets\footnote{We ran experiments with 0.25, 0.60, and 0.75 as well, and choose 0.5 based on training performance}. Using 0.5 as threshold, we obtained the best training performance and hence chose to use this as the threshold for similarity calculation. The various similarity measures used are described in the next subsection.
\subsection{Similarity Measures}
\subsubsection*{Normalized Edit Distance Method (NED)}
The Normalized Edit Distance approach computes the edit distance \cite{nerbonne1997measuring} for all word pairs in a synset/concept and then provides the output of probable cognate sets with distance and similarity scores. We assign labels for these sets based on the similarity score obtained from the NED method, where the similarity score is (1 - NED score). It is usually defined as a parameterizable metric calculated with a specific set of allowed edit operations, and each operation is assigned a cost (possibly infinite). The score is normalized such that 0 equates to no similarity and 1 is an exact match. NED is equal to the minimum number of operations required to transform `word a' to `word b'. A more general definition associates non-negative weight functions (insertions, deletions, and substitutions) with the operations.
\subsubsection*{Cosine Similarity (Cos)}
The cosine similarity measure \cite{salton1988term} is another similarity metric that depends on envisioning preferences as points in space. It measures the cosine of the angle between two vectors projected in a multi-dimensional space. In this context, the two vectors are the arrays of character counts of two words. The cosine similarity is particularly used in positive space, where the outcome is neatly bounded in [0,1]. For example, in information retrieval and text mining, each term is notionally assigned a different dimension and a document is characterised by a vector where the value in each dimension corresponds to the number of times the term appears in the document. Cosine similarity then gives a useful measure of how similar two documents are likely to be in terms of their subject matter. This is analogous to the cosine, which is 1 (maximum value) when the segments subtend a zero angle and 0 (uncorrelated) when the segments are perpendicular. In this context, the two vectors are the arrays of character counts of two words.
\subsubsection*{Jaro-Winkler Similarity (JWS)}
Jaro-Winkler distance \cite{winkler1990string} is a string metric measuring similar to the normalized edit distance deriving itself from Jaro Distance \cite{jaro1989advances}. It uses a prefix scale P which gives more favourable ratings to strings that match from the beginning, for a set prefix length L. We ensure a normalized score in this case as well. Here, the edit distance between two sequences is calculated using a prefix scale P which gives more favourable ratings to strings that match from the beginning, for a set prefix length L. The lower the Jaro–Winkler distance for two strings is, the more similar the strings are. The score is normalized such that 1 equates to no similarity and 0 is an exact match.
\begin{table}[]
\resizebox{0.9\columnwidth}{!}{%
\begin{tabular}{c|c|c|c|c|}
\cline{2-5}
& \multicolumn{2}{c|}{FFN} & \multicolumn{2}{c|}{RNN} \\ \cline{2-5}
& D1 & D2 & D1 & D2 \\ \hline
\multicolumn{1}{|c|}{Hi-Mr} & 69.76 & 85.76 & 74.76 & 89.78 \\ \hline
\multicolumn{1}{|c|}{Hi-Bn} & 65.18 & 81.04 & 69.18 & 86.44 \\ \hline
\multicolumn{1}{|c|}{Hi-Pa} & 73.04 & 78.50 & 76.04 & 83.64 \\ \hline
\multicolumn{1}{|c|}{Hi-Gu} & 61.74 & 79.16 & 69.84 & 89.44 \\ \hline
\multicolumn{1}{|c|}{Hi-Sa} & 61.72 & 85.87 & 68.92 & 91.66 \\ \hline
\multicolumn{1}{|c|}{Hi-Ml} & 56.96 & 74.77 & 66.96 & 79.59 \\ \hline
\multicolumn{1}{|c|}{Hi-Ta} & 55.62 & 61.70 & 65.62 & 68.92 \\ \hline
\multicolumn{1}{|c|}{Hi-Te} & 52.78 & 65.26 & 62.78 & 74.83 \\ \hline
\multicolumn{1}{|c|}{Hi-Ne} & 70.20 & 83.85 & 80.20 & 89.63 \\ \hline
\multicolumn{1}{|c|}{Hi-Ur} & 69.99 & 73.84 & 76.99 & 80.12 \\ \hline
\end{tabular}%
}
\caption{Stratified 5-fold Evaluation using Deep Neural Models on both PCData (D1) and WNData (D2)}
\label{resTable}
\end{table}
\begin{table*}[!ht]
\begin{tabular}{c|c|c|c|c|l|l|l|l|l|l|}
\cline{2-11}
& \multicolumn{2}{c|}{Corp+WN20} & \multicolumn{2}{c|}{Corp+WN40} & \multicolumn{2}{l|}{Corp+WN60} & \multicolumn{2}{l|}{Corp+WN80} & \multicolumn{2}{l|}{Corp+WN100} \\ \cline{2-11}
& FFN & RNN & FFN & RNN & FFN & RNN & FFN & RNN & FFN & RNN \\ \hline
\multicolumn{1}{|c|}{Hi-Mr} & 70.12 & 74.12 & 73.56 & 78.37 & 76.09 & 81.56 & 81.34 & 85.24 & 86.90 & 91.87 \\ \hline
\multicolumn{1}{|c|}{Hi-Bn} & 71.06 & 73.17 & 73.29 & 74.98 & 77.33 & 76.28 & 83.99 & 81.45 & 82.18 & 89.58 \\ \hline
\multicolumn{1}{|c|}{Hi-Pa} & 74.16 & 75.94 & 76.02 & 77.39 & 76.18 & 79.04 & 78.04 & 81.22 & 80.66 & 85.64 \\ \hline
\multicolumn{1}{|c|}{Hi-Gu} & 65.26 & 70.76 & 71.21 & 74.83 & 75.09 & 79.95 & 80.14 & 84.32 & 81.85 & 89.81 \\ \hline
\multicolumn{1}{|c|}{Hi-Sa} & 65.93 & 74.23 & 69.25 & 77.51 & 74.84 & 79.92 & 81.03 & 86.62 & 88.13 & 93.86 \\ \hline
\multicolumn{1}{|c|}{Hi-Ml} & 57.75 & 59.38 & 56.31 & 65.67 & 58.02 & 71.19 & 61.01 & 75.59 & 69.11 & 82.54 \\ \hline
\multicolumn{1}{|c|}{Hi-Ta} & 54.63 & 60.12 & 56.69 & 63.38 & 57.46 & 66.17 & 59.36 & 67.17 & 60.41 & 70.62 \\ \hline
\multicolumn{1}{|c|}{Hi-Te} & 53.21 & 58.18 & 56.19 & 63.90 & 64.15 & 67.70 & 65.19 & 70.65 & 66.10 & 74.92 \\ \hline
\multicolumn{1}{|c|}{Hi-Ne} & 70.78 & 71.23 & 74.30 & 78.11 & 72.19 & 83.20 & 79.70 & 85.01 & 84.69 & 90.95 \\ \hline
\multicolumn{1}{|c|}{Hi-Ur} & 69.94 & 71.25 & 70.01 & 72.35 & 72.03 & 76.59 & 71.07 & 78.27 & 73.99 & 80.99 \\ \hline
\end{tabular}%
\caption{Results after combining chunks of WNData with PCData}
\label{chunkTable}
\end{table*}
\subsection{Models}
\subsubsection{Feed Forward Neural Network (FFN)}
In this network, we deal with a word as a whole. Words of the source and target languages reside in separate embedding space. The source word passes through the source embedding layer. The target word passes through the target embedding layer. The outputs of both embedding lookups are concatenated. The resulting representation is passed to a fully connected layer with ReLU activations, followed by a softmax layer.
\subsubsection{Recurrent Neural Network (RNN)}
In this network (see Figure \ref{fig:RNN}), we treat a word as a sequence of characters. Characters of the source and the target language reside in separate embedding spaces. The characters of the source word are passed through source embedding layer. The characters of the target word are passed through the target embedding layer. The outputs of both embedding lookups are, then, concatenated. The resulting embedded representation is passed through a recurrent layer. The final hidden state of the recurrent layer is then passed through a fully connected layer with ReLU activation. The resulting output is finally passed through a softmax layer.
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth]{imgs/RNN.png}
\caption{Architecture of a Recurrent Neural Network}
\label{fig:RNN}
\end{figure}
\section{Results}
We average the similarity scores obtained using the three methodologies (NED, Cos, and JWS) described above, for each word pair, and then use these as training labels for cognate detection models. We obtain results using the networks described above and report them in Table \ref{resTable}. We calculate average scores for both models and both datasets and show the chart in Figure \ref{fig:f2}. We observe that RNN outperforms FFN for both the datasets across all language pairs (see Figure \ref{fig:f2}). We also find that Hi-Sa (see Figure \ref{fig:f2}) has the best cognate detection accuracy among all language pairs (for both RNN and FFN), which is in line with the fact that they are closely related languages when compared to other Indian language pairs. We observe that average scores for WNData are always higher than average scores for PCData for all language pairs (Figure \ref{fig:f2}). Also, in line with our observations above, the overall average of RNN scores for both datasets are even higher than average FFN scores (Figure \ref{fig:f2}).
We perform another set of experiments by combining non-redundant word pairs from both datasets. We add WNData in chunks of 20 per cent to PCData for each language pair and create separate word lists with average similarity scores. We use FFN to train and perform a stratified 5-fold evaluation for each language pair after adding each chunk and show the results in Table \ref{chunkTable}. After evaluating our results for FFN, we perform the same training and evaluation with RNN. \textbf{We observe that adding complete WNData to PCdata improves our performance drastically and given us the best results for almost all cases.} Only in case of Hi-Bn, when using the FFN for training, PCData combined with 80\% WNData performs better than 100\% Data; possibly due to added sparsity of the additional data. Our hypothesis that adding WNData to PCdata improves the performance holds for all the other cases, including when trained using RNN.
\begin{figure*}[!ht]
\centering
\includegraphics[width=\linewidth]{imgs/GWCChart.png}
\caption{Average Results using Neural Network models on both datasets}
\label{fig:f2}
\end{figure*}
\section{Discussion and Analysis}
\begin{table}[]
\resizebox{0.9\columnwidth}{!}{%
\begin{tabular}{c|c|c|c|}
\cline{2-4}
& WNPairs & CorpPairs & Matches \\ \hline
\multicolumn{1}{|c|}{Hi-Bn} & 324537 & 505721 & 17402 \\ \hline
\multicolumn{1}{|c|}{Hi-Pa} & 260123 & 465140 & 16325 \\ \hline
\multicolumn{1}{|c|}{Hi-Mr} & 322013 & 555719 & 17698 \\ \hline
\multicolumn{1}{|c|}{Hi-Gu} & 423030 & 542311 & 17005 \\ \hline
\multicolumn{1}{|c|}{Hi-Sa} & 669911 & 248421 & 10109 \\ \hline
\multicolumn{1}{|c|}{Hi-Ml} & 353104 & 315234 & 12392 \\ \hline
\multicolumn{1}{|c|}{Hi-Ta} & 225705 & 248207 & 7112 \\ \hline
\multicolumn{1}{|c|}{Hi-Te} & 369872 & 431869 & 7599 \\ \hline
\multicolumn{1}{|c|}{Hi-Ne} & 191701 & 420176 & 11264 \\ \hline
\multicolumn{1}{|c|}{Hi-Ur} & 99803 & 420176 & 6509 \\ \hline
\end{tabular}%
}
\caption{Total Word Pairs for both datasets and Matches among them}
\label{matchTable}
\end{table}
\begin{table*}[!ht]
\centering
\resizebox{0.95\linewidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\textbf{Source Word} & \textbf{Target Word} & \textbf{Meaning} & \textbf{Cos} & \textbf{NED} & \textbf{JWS} \\ \hline
\textit{tadanukool} & \textit{tadanusaar} & accordingly & 0.500 & \textbf{0.571} & 0.482 \\ \hline
\textit{yogadaan karna} & \textit{yogadaan karane} & to contribute & 0.631 & \textbf{0.636} & 0.593 \\ \hline
\textit{duraatma} & \textit{dushtaatama} & evil soul & 0.629 & \textbf{0.700} & 0.648 \\ \hline
\end{tabular}
}
\caption{Manual analysis of the similarity scores}
\label{analysisTable}
\end{table*}
A parallel corpus is a costly resource to obtain in terms of both time and effort. For resource-scarce languages, parallel corpora cannot easily be crawled. We wanted to validate how crucial Wordnets are as a resource and can they act as a substantial dataset in the absence of parallel corpora. In addition to validating the performance of chunks of WNData combined with PCData, we also calculated the exact matches of word pairs from both the datasets and show the results in Table \ref{matchTable}. We observed that Hi-Mr had the most matched pairs amongst all the languages. PCData is extracted from parallel corpora and is not stemmed for root words, whereas WNData is extracted from IndoWordnet and only contains root words. Despite many words with morphological inflections, we were able to obtain exactly matching words, amongst the datasets. WNData constitutes a fair chunk of root words used in PCData as well, and this validates the fact that models trained on WNData can be used to detect cognate word pairs from any standard parallel corpora as well.
It is a well-established fact that Indian languages are spoken just like they are written and unlike their western counterparts are not spoken and spelled differently. Hence, we choose to perform cognate detection using orthographic similarity methods. This very nature of Indian languages allows us to eliminate the need for using aspects of Phonetic similarity to detect true cognates. Most of the Indian languages borrow words from Sanskrit in either of the two forms - \textit{tatsama} or \textit{tadbhava}. When a word is borrowed in \textit{tatsama} form, it retains its spelling, but in case of \textit{tadbhava} form, the spelling undergoes a minor change to complete change. Before averaging the similarity scores, we tried to observe which of the three (NED, JWS, or Cos) scores would perform better for true cognates known to us in \textit{tadbhava} form with minor spelling changes. We analysed individual word pairs from the data and presented a small sample of our analysis in Table \ref{analysisTable}. We observe that NED consistently outperforms Cos and JWS for cognate word pairs and confirmed that NED based similarity is the most suited metric for cognate detection \cite{rama2015comparative}. We also observe that our methodology can handle word pairs without any changes and with minor spelling changes among cognates, the total of which, constitutes a large portion of the cognates among Indian Language pairs.
\section{Conclusion and Future Work}
In this paper, we investigate cognate detection for Indian Language pairs (Hi-Bn, Hi-Gu, Hi-Pa, Hi-Mr, Hi-Sa, Hi-Ml, Hi-Ta, Hi-Te, Hi-Ne, and Hi-Ur). A pair of words is said to be Cognates if they are etymologically related; and True Cognates, if they carry the same meaning as well. We know that parallel concepts, bearing the same sense in linked WordNets, are etymologically related. We, then, use the measures of orthographic similarity to find probable Cognates among parallel concepts. We perform the same task for a parallel corpus and then train neural network models on this data to perform automatic cognate detection. We compute a list of True Cognates and release this data along with the data processed previously. We observe that Recurrent Neural Networks are best suited for this task. We observe that Hindi - Sanskrit language pair, being the closest, has the highest percentage of cognates among them. We observe that RNN, which treats the words as a sequence of characters, outperforms FFN for all the language pairs and both the datasets. We validate that Wordnets can play a crucial role in detecting cognates by combining the datasets for improved performance. We observe a minor, but crucial, increase in the performance of our models when chunks of Wordnet data are added to the data generated from the parallel corpora thus confirming that Wordnets are a crucial resource for Cognate Detection task. We also calculate the matches between word pairs from the Wordnet data and the word pairs from the parallel corpora to show that Wordnet data can form a significant part of parallel corpora and thus can be used in the absence of parallel corpora.
In the near future, we would like to use cross-lingual word embeddings, include more Indian languages, and investigate how semantic similarity could also help in cognate detection. We will also investigate the use of Phonetic Similarity based methods for Cognate detection. We shall also study how our cognate detection techniques can help infer phylogenetic trees for Indian languages. We would also like to combine the similarity score by providing them weights based on an empirical evaluation of their outputs and extend our experiments to all the Indian languages.
\section*{Acknowledgement}
We would like to thank the reviwers for their time and insightful comments which helped us improve the draft. We would also like to thank CFILT lab for its resources which helped us perform our experiments and its members for reading the draft and helping us improve it.
\bibliographystyle{acl2010}
|
2,869,038,156,417 | arxiv | \section{Introduction}
\label{sec:introduction}
Knowledge of the unique and desirable characteristics of Geosynchronous Earth Orbits (GEOs) predates the dawn of the Space Age.
Satellites in GEO have an orbital period matching that of the Earth's rotation, meaning that they typically trace a simple analemma (e.g.~an ellipse or a figure of eight) on the sky over the course of a sidereal day (23$^\text{h}$56$^\text{m}$04$^\text{s}$).
In the special case of a geostationary orbit, a station-kept satellite will remain fixed in the observer's sky, a property that has been exploited for communications since the early 1960s.
The GEO region is too high-altitude for atmospheric drag to provide a mechanism for orbital decay, thus there is no natural `sink' for debris residing there.
This is a cause for concern, given that the natural constraints placed on altitude, eccentricity and inclination for the GEO regime already restrict the number of orbital slots.
\begin{figure}[tbp]
\begin{center}
\includegraphics*[width=0.75\textwidth]{geo-belt-diagram.pdf}
\end{center}
\caption{An illustrative sketch of the GEO Protected Region (in blue), as defined by the IADC Space Debris Mitigation Guidelines~\citep[see][]{iadc2007guidelines}.
Extending 200\,km above and below the geostationary altitude $Z_\text{GEO}=35786$\,km, the Region is a segment of a spherical shell spanning $\pm$\,15$^\circ$ in declination relative to the equatorial plane of the Earth.
Note that the scale of the Protected Region has been exaggerated for clarity.}
\label{fig:geo-belt-diagram}
\end{figure}
In order to address the problem, guidelines and recommendations~\citep{iadc2007guidelines} have been established over the past two decades to define the GEO Protected Region, depicted in Fig.~\ref{fig:geo-belt-diagram}.
Operators are advised to carry out an end-of-mission (EOM) manoeuvre to a `graveyard' orbit residing outside the Protected Region.
Compliance with the guidelines has improved in recent years, with over 80\,\% of attempted manoeuvres successfully clearing the Protected Region since 2016~\citep{esa2019annual}.
\begin{figure}[tbp]
\begin{center}
\includegraphics*[width=\textwidth]{geo.pdf}
\end{center}
\caption{Left) The cumulative number of tracked objects in GEO, with objects separated into three categories: payloads (black), rocket bodies (blue) and debris (red).
Debris fragments are binned by the year they were first tracked and catalogued.
Inset, the number of objects launched to GEO per year.
Sourced from the publicly available US Strategic Command catalogue, accessed via the geosynchronous sub-catalogue from \href{www.space-track.org}{Space-Track} as of May 2020.
Right) The orbital status of tracked GEO objects in 2018, as given in~\citet{esa2019annual}.}
\label{fig:geo}
\end{figure}
In spite of this, it is important to keep in mind the GEO residents that reached EOM prior to the issuance of guidelines, existing in an uncontrolled state ever since.
These are typically in drift orbits or librating about one or both of the geopotential wells that result from the non-spherical shape of the Earth~\citep{mcknight2013new}.
Many of these drift orbits intersect the operational regions of the geostationary belt, posing a direct threat to active satellites.
With the upward trend evident in Fig.~\ref{fig:geo}, it is clear that an imperfect disposal rate will result in fewer orbital slots and increased collision risk in GEO.
Objects in Highly-Eccentric Earth Orbit (HEO) can further add to the risk, with recent observations uncovering a number of fragments penetrating the Protected Region~\citep{schildknecht2019esa}.
Four significant GEO/HEO break-ups have been observed in the past two years alone.
Collectively, these events produced over 1000 fragments, a few hundred of which cross the GEO Protected Region.
It is also likely that collisions with small debris are responsible for the heavily-publicised anomalies exhibited by the geostationary satellites Intelsat 29e (10/04/2019, NORAD 41308), Telkom 1 (25/08/2017, NORAD 25880) and AMC-9 (17/06/2017, NORAD 27820)~\citep{cunio2017photometric}.
Observations of high-altitude orbits typically employ the use of optical sensors, as their sensitivity drops with the square of range, while that of radar drops more steeply with the fourth power of range.
The publicly available US Strategic Command (USSTRATCOM) catalogue tracks objects in GEO larger than 50-100\,cm, predominantly using a system of 1\,m-class optical telescopes known as Ground-based Electro-Optical Deep Space Surveillance (GEODSS)~\citep{wootton2016enabling}.
Smaller objects are monitored sporadically at best, due to the limited availability of sufficiently sensitive sensors.
This is of particular concern, given a recent study that found relative velocities in GEO can reach up to 4\,kms$^{-1}$, approaching the hypervelocity regime where collisions with cm-sized objects could prove mission-fatal~\citep{oltrogge2018comprehensive}.
As break-ups and anomalies add more small fragments to the GEO environment, it is important that we continue to observe faint objects with large telescopes to better understand their behaviour and the risk they pose to operational satellites.
\begin{table}[tbp]
\caption{Optical surveys of the GEO region.
Instrumental fields of view (FOV) are listed as narrow (N) if $\text{FOV}<0.5$\,sq.\,deg, medium (M) if $0.5<\text{FOV}<1$\,sq.\,deg, wide (W) if $1<\text{FOV}<10$\,sq.\,deg and ultra-wide (UW) if $\text{FOV}>10$\,sq.\,deg.
Survey depths are denoted $M_{X}$ for a given photometric band $X$, and $M$ for cases where the band is not specified or absolute magnitudes have been quoted~\citep[see][]{africano2005phase}.}
\begin{center}
\begin{tabular}{lrrrr}
\hline
Survey & Instr. & Instr. & Survey & Reference \\
& size [m] & FOV & depth & \\
\hline
NASA CDT & 0.32 & W & $M\sim16$ & \citet{barker2005cdt} \\
MODEST & 0.61 & W & $M_R\sim17$ & \citet{seitzer2004modest} \\
TAROT & 0.18--0.25 & W--UW & $M_R\sim15$ & \citet{alby2004status} \\
ESA-AIUB & 1.00 & M & $M_V\sim20$ & \citet{schildknecht2007optical} \\
ISON & 0.22--0.70 & N--UW & $M\sim18$ & \citet{molotov2008international} \\
ISON (faint) & 1.00--2.60 & N & $M\sim20$ & \citet{molotov2009faint} \\
Pan-STARRS & 1.80 & UW & $M_V\sim21$ & \citet{bolden2011panstarrs} \\
Magellan & 6.50 & N & $M_R\sim19$ & \citet{seitzer2016the} \\
FocusGEO & (3$\times$)0.18 & UW & $M\sim15$ & \citet{luo2019focusgeo} \\
\hline
\end{tabular}
\end{center}
\label{tab:surveys}
\end{table}
We provide an overview of past/ongoing GEO surveys in Table~\ref{tab:surveys}.
The majority have utilised optical telescopes with diameters of 1\,m or less, with sensitivity limits in the range 15$^\text{th}$--20$^\text{th}$\,Magnitude, corresponding to objects larger than $\sim$15\,cm in diameter (depending on the viewing geometry, assumed shape and reflectivity).
A small number of surveys have uncovered fainter objects with telescopes larger than 1\,m.
For example, the 6.5\,m Magellan telescope has been used for a small number of GEO spot surveys, targeting known fragmentation events~\citep{seitzer2016the}.
These deeper observations, alongside those conducted by the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) 1.8\,m~\citep{bolden2011panstarrs} and large-aperture telescopes of the International Scientific Optical Network (ISON)~\citep{molotov2009faint}, have found that many faint detections show photometric signatures of tumbling.
In this paper, we present photometric results from a survey of the GEO region undertaken with the 2.54\,m Isaac Newton Telescope in La Palma, Canary Islands.
The survey was carried out as part of DebrisWatch, an ongoing collaboration between the University of Warwick and the Defence Science and Technology Laboratory (UK) investigating the faint debris population at GEO.
We outline our observational strategy in Section~\ref{sec:observational-strategy}, optimised for finding objects in GEO.
Section~\ref{sec:analysis-pipeline} provides an overview of our data analysis pipeline, which performs reduction tasks, object detection and light curve extraction.
In Section~\ref{sec:results-discussion}, we consider the population sampled by our observations and present light curves for objects of interest, before discussing our findings and future plans.
\section{Observational strategy}
\label{sec:observational-strategy}
\begin{table}[tbp]
\caption{Logistical details for the observation run.
In total, 552 separate pointings of the telescope in hour angle and declination were achieved in the 58 hours of survey time.
Approximately half the night of 5$^\text{th}$ September was lost due to weather and technical issues.
The remaining time was dedicated to targeted observations that are outside the scope of this work.}
\begin{center}
\begin{tabular}{lrr}
\hline
Night & Survey time & Telescope \\
& [hrs] & pointings \\
\hline
02/09/2018 & 8.5 & 65 \\
03/09/2018 & 7.7 & 76 \\
04/09/2018 & 6.4 & 71 \\
05/09/2018 & 4.5 & 15 \\
06/09/2018 & 7.0 & 77 \\
07/09/2018 & 6.0 & 63 \\
08/09/2018 & 8.5 & 86 \\
09/09/2018 & 9.4 & 99 \\
\hline
& 58.0 & 552 \\
\hline
\end{tabular}
\end{center}
\label{tab:logistics}
\end{table}
We used eight nights of dark-grey time on the 2.54\,m Isaac Newton Telescope (INT) to conduct an untargeted survey of the GEO region visible from the Roque de los Muchachos Observatory in La Palma, Canary Islands.
Logistical details for the survey are provided in Table~\ref{tab:logistics}.
Observations were made using the prime focus Wide Field Camera (WFC), consisting of four thinned 2k\,$\times$\,4k charge-coupled device (CCD) chips, which combine to image over a 33$'$ field of view.
One of the CCD chips was rendered unusable due to an issue with the readout electronics.
We discard this chip for the following photometric analyses, reducing our effective field of view to 22$'$\,$\times$\,33$'$.
Two-by-two binning was applied, resulting in a resolution of 0.66$''$pixel$^{-1}$.
The observations were taken using a Harris V filter with a central wavelength of 5425\,{\AA}, a full width at half maximum of 975\,{\AA} and a peak throughput of 88\,\%.
Steps were taken to optimize our observations for finding objects in GEO.
The telescope was operated at a fixed hour angle and declination, ensuring that photons from GEO candidates would integrate across fewer pixels to improve the signal-to-noise ratio.
In this observing mode, GEO objects manifest as point sources or short trails in the resulting image, while background stars appear as longer trails, streaking across at the sidereal rate.
We chose an exposure time of 10\,s to provide a balance between streak coverage and duty cycle.
Observations were taken by selecting a nominal field with a specific right ascension and declination, corresponding to a fixed solar phase angle (observatory-target-Sun), which was minimised whilst remaining outside of the Earth's shadow.
This allowed for the detection of fainter objects by maximising their apparent brightness.
The selected field would then be used to generate the telescope pointings for the given night, scanning a strip of fixed declination with each pointing fixed at a separate hour angle.
\begin{figure}[tbp]
\begin{center}
\includegraphics*[width=\textwidth]{target-fields.pdf}
\end{center}
\caption{Telescope pointings for the survey.
Imaged fields are given by the grey boxes, while detections are overlaid as black dots.
The approximate declination of the geostationary belt as visible from the vantage point of the INT is indicated by the blue line.
Shaded red regions mark the altitude limits that constrained the accessible range in hour angle for a given declination strip.}
\label{fig:target-fields}
\end{figure}
We provide a map of telescope pointings in hour angle and declination for the survey in Figure~\ref{fig:target-fields}.
The INT telescope control system disables several important instrument features upon issuance of a telescope stop command.
We instead applied a differential tracking offset upon reaching the chosen field, in order to counter the sidereal rate and freeze the hour angle for the duration of the given pointing.
Each telescope pointing was observed for roughly four minutes, comprising seven 10\,s exposures with a 25\,s readout time per exposure.
Multiple exposures were taken at each pointing to allow for correlation of detections across frames.
After each set of exposures, the telescope pointing was updated to retrieve the chosen field and the above procedure was repeated.
Survey operations began when the target field exceeded 30$^\circ$ elevation in the east and continued until it set below 30$^\circ$ elevation in the west.
Most aspects of the observing procedure were automated using a script, however limitations in the INT control system meant that operator input was required for each new pointing.
The observation script also sent commands to a second telescope on-site, a 36\,cm astrograph assembled from commercial-off-the-shelf (COTS) equipment, featuring a much larger 3.6$^\circ$\,$\times$\,2.7$^\circ$ field of view.
The astrograph remained slaved to the INT for the duration of the observation campaign.
The additional dataset from this instrument will form the basis of a future DebrisWatch study that will test the capabilities of COTS hardware against those of large telescopes when tasked with detecting faint objects in GEO.
\section{Analysis pipeline}
\label{sec:analysis-pipeline}
The survey data were processed using a custom analysis pipeline, which is outlined in Fig.~\ref{fig:flowchart}.
Written in Python 3, the pipeline takes inspiration from a number of algorithms developed previously to find artificial objects in astronomical images~\citep{laasbourez2009tarot,levesque2009automated,privett2017autonomous}.
\begin{figure}[tbp]
\begin{center}
\includegraphics*[width=\textwidth]{flowchart.pdf}
\end{center}
\caption{Diagram outlining the analysis pipeline described in Section~\ref{sec:analysis-pipeline}.}
\label{fig:flowchart}
\end{figure}
\subsection{Reduction}
\label{sec:reduction}
Standard bias and flat-field calibrations are applied using calibration frames acquired at the beginning of each night.
A bad pixel mask was created from the flat-field observations and defective pixels in the science frames were replaced with a sigma-clipped median of the surrounding pixel values.
We use \texttt{SEP} (Source Extractor in Python) to subtract a model of the spatially-varying sky background from the calibrated frame~\citep{barbary2016sep,bertin1996sextractor}.
We then utilise the extraction capabilities of \texttt{SEP} to find stars in the image, exploiting their common morphologies and orientations.
The centroids and start/end points of the star trails are fed to \textit{Astrometry.net}, which pattern-matches subset quadrilaterals of stars against sky catalogues to determine accurate World Coordinate System (WCS) solutions for astronomical images~\citep{lang2010astrometry}.
Following this, it is simple to convert between pixel and sky coordinates using \texttt{astropy} WCS routines~\citep{pricewhelan2018astropy,robitaille2013astropy}.
Using this astrometric solution, we perform a photometric calibration by cross-matching the star trails with the American Association of Variable Star Observers (AAVSO) All-Sky Photometric Survey (APASS) catalogue~\citep{henden2016apass}.
The photometric zero point for the frame is found by comparing the standard magnitudes quoted in the APASS catalogue against their instrumental counterparts, derived by summing rectangular apertures placed over the star trails.
\subsection{Object detection}
\label{sec:object-detection}
\begin{figure}[tbp]
\begin{center}
\includegraphics*[width=\textwidth]{morphology.pdf}
\end{center}
\caption{Examples of object morphologies.
With the telescope viewing direction fixed relative to the Earth, geostationary satellites appear as point-like features in the acquired CCD frames, as in (a).
Objects in GEO that are moving relative to geostationary will manifest as trails.
An example of a faint trail with uniform brightness is given in (b).
Other trails exhibit brightness variation, over timescales longer (c) and shorter (d) than the exposure time.
Vertical lines in the background are stars streaking across the images.}
\label{fig:morphology}
\end{figure}
Many GEO residents are moving relative to the geostationary tracking rate and so are not fixed in the topocentric coordinate frame.
The reflected light from these objects will spread over a trail of pixels mapped out by the angular path traversed during the exposure.
Additional structure along the trails (e.g. glints, flares, gentle oscillations) can result from changes in the reflected light received from the object along the observer's line of sight.
As a result, objects of interest exhibit a wide range of morphologies and orientations, examples of which can be seen in Fig.~\ref{fig:morphology}.
We remove the background star trails using mathematical morphology, a technique for examining geometrical structures within images~\citep{breen2000mathematical,matheron2002mathematical}.
As in~\citet{laasbourez2009tarot}, we probe each image $f(x)$ with a structuring element $B$ using the Spread TopHat transformation $\eta$,
\begin{equation}
\eta^B(f(x))=f(x)-O^B(C^B(f(x))).
\end{equation}
The opening $O$ and closing $C$ operations act to remove small peaks and dark regions, respectively.
When combined to form the Spread TopHat, the effect is to remove features that contain the structuring element, whilst limiting remnant noise in the resulting image.
We carry out the transformation using the \texttt{scipy} morphology routines~\citep{jones2001scipy}.
Rectangular structuring elements are used to emulate the star streaks in our images, with dimensions 1\,$\times$\,$\frac{1}{2}l_\text{ST}$\,px for the opening and 1\,$\times$\,$\frac{1}{6}l_\text{ST}$\,px for the closing, given an expected star trail length $l_\text{ST}$.
Candidate GEO objects are retained as they do not contain either of the structuring elements.
Additional checks are required to separate the objects of interest from remnant `distractors' that survive the transformation.
After running the \texttt{ccdproc} lacosmic routine~\citep{craig2015ccdproc,vandokkum2001cosmic} to remove cosmic rays from the transformed image, we apply a 3$\sigma$ threshold cut to filter out the majority of spurious detections, where $\sigma$ is the global background root mean square.
The remaining false positives are typically edges of star trails that are easily flagged given our knowledge of the trail positions.
\subsection{Position refinement}
\label{sec:position-refinement}
In the case of trailed detections, it is necessary to accurately determine the start and end points, as we know these will correspond to the angular positions of the object at the start and end of exposure, respectively.
We refine the initial estimate from a \texttt{SEP} extraction by fitting the intensity profiles along and across the trail.
We use a Gaussian fit for the across-trail intensity profiles, while a good approximation of the along-trail profile is given by the `Tepui' function,
\begin{equation}
I(x)=A\left[\arctan(b_1(x-c-x_0))-\arctan(b_2(x+c-x_0))\right],
\end{equation}
where $A$ is the normalised amplitude, $b_1$ and $b_2$ are related to the profile tilt, $c$ gives the half-width and $x_0$ is a translational offset.
Several studies have made use of the Tepui function when fitting streaks in astronomical images~\citep[see e.g.][]{lacruz2018astrometric,montojo2011astrometric,park2016development}.
Using this refinement procedure, we obtain typical uncertainties of 1-2$''$, corresponding to 200-400\,m at GEO.
Within the scope of our photometric study, this level of uncertainty was deemed acceptable.
We use the refined estimate of the orientation to predict where the object will appear in subsequent frames within a given pointing, correlating trails belonging to the same orbital track.
In the photometric analyses that follow, we only consider objects that appear in two or more frames.
The refined orientation allows for more accurate placement of a \texttt{TRIPPy} pill aperture~\citep{fraser2016trippy}, the sum of which provides a measure of the total flux integrated over the course of the exposure.
Trail morphologies are well-approximated by pill shapes, so the contribution of background noise to the aperture sum is minimised.
Uncertainties in the measured magnitudes consist of two parts: the first is a systematic uncertainty from the zero point measurement, which is based on the background stars and is typically $\sim$0.05\,mag for a given frame, while the second is the photometric uncertainty from the aperture sum, which is typically $\sim$0.001\,mag for bright objects ($V\sim12$) and $\sim$0.05\,mag for faint objects ($V\sim18$) in a 10\,s exposure.
We note that intrinsic brightness variability can cause much larger scatter in short-timescale measurements for specific objects, as will be illustrated in Section~\ref{sec:photometric-light-curves}, where we provide examples of light curves extracted from our detections.
\subsection{Light curve extraction}
\label{sec:light-curve-extraction}
In the final stage of the pipeline, we extract light curves from our trailed detections.
Rectangular apertures are placed along the trail, each covering a discrete pixel in width to avoid correlated noise injection.
We assume constant rates of change in angular position throughout the exposure.
Background contamination (e.g. blending with star streaks) is corrected by placing equivalent apertures in a reference frame containing the same field.
We perform an initial image alignment using the astrometric solutions for the frames, then account for remnant offsets using the \texttt{DONUTS} alignment algorithm~\citep{mccormac2013donuts}.
\section{Results and discussion}
\label{sec:results-discussion}
\subsection{Sampled population}
\label{sec:sampled-population}
\begin{figure}[tbp]
\begin{center}
\includegraphics*[width=\textwidth]{brightness-rates.pdf}
\end{center}
\caption{(a) Brightness histogram for the detected population, as previously presented in~\citet{blake2019optical}.
Tracks that correlate with the publicly available USSTRATCOM catalogue are shown in blue, while those that fail to correlate are in red.
The black line gives our sub-sample of tracks which lie within the rate cut limits shown in (b).
Labelled size estimates assume that the objects are Lambertian spheres with an albedo $A=0.1$.
(b) Rates in hour angle and declination for the tracks detected.
The rate cuts applied in order to obtain the circular-GEO sub-sample are indicated by the black box.
(c) Brightness histogram for the detected population, normalised by trail length.
This normalisation determines the brightness of a geostationary source with the same peak flux for an equivalent integration time.
(d) Object types for correlated detections in the overall sample, categorised as operational or drifting payloads (PL), rocket bodies (R/B) or debris (DEB).
(e) Light curve statistics for the overall sample.
For each detected object, the difference between the maximum and minimum brightness is plotted against the standard deviation.}
\label{fig:brightness-rates}
\end{figure}
A total of 226 orbital tracks spanning two or more exposures within a given pointing were detected.
The brightness distribution for these detections is presented in panel (a) of Fig.~\ref{fig:brightness-rates}.
We limit our attention to tracks that are consistent with circular orbits in the GEO regime, using the cuts defined in~\citet{seitzer2011faint}:
\begin{equation}
\lvert\text{Hour Angle rate}\rvert<\;2''\text{s}^{-1}\; \text{and}\; \lvert\text{Declination rate}\rvert<\;5''\text{s}^{-1}.
\end{equation}
Objects with rates exceeding these limits likely reside in geosynchronous transfer orbits (GTOs) which are elliptical orbits with apogees in the GEO region.
The resulting subset of circular-GEO detections is represented by the black lines in Fig.~\ref{fig:brightness-rates}.
We correlate our detections against the publicly available USSTRATCOM catalogue, finding that 85\,\% of tracks with $V<15$ successfully correlate, while only 1\,\% of fainter detections match a known object.
This is consistent with the $\sim$1\,m cut-off for the GEODSS network.
The rate cuts reduce our sample size to 129 circular-GEO tracks, giving a detection rate of $\sim$11\,hour$^{-1}$deg$^{-2}$ for the survey.
A similar detection rate was observed by the Magellan surveys in Chile~\citep{seitzer2011faint}, within sight of the geopotential well at longitude 105$^\circ$\,W.
Risk assessments have found that collision probabilities increase by a factor of seven in the vicinity of the potential wells~\citep{mcknight2013new}, owing to the relatively high density of trapped objects in libration orbits.
La Palma ($\sim$18$^\circ$\,W) sits almost directly between the two wells, thus we would expect to have a lower detection rate.
However, the limited time available on large telescopes means that both surveys suffer from small number statistics, making it difficult to draw conclusions regarding detection rates at this early stage.
We observe a bimodal brightness distribution, consistent with the findings of previous GEO surveys.
The bright end of the sample peaks at $V\sim12$, in accordance with the population uncovered by the European Space Agency (ESA) 1\,m Optical Ground Station (OGS) observation campaigns in Tenerife~\citep{schildknecht2004optical}.
This is to be expected given that the majority of bright, correlated objects are geostationary and the two instruments sample the same section of the GEO belt.
For reference, we classify our correlated detections according to object type in panel (d) of Fig.~\ref{fig:brightness-rates}.
We see a steep rise in the number of objects detected as we look fainter than $V\sim17$.
The overall distribution appears to plateau between $V\sim18$ and our sensitivity limit at $V\sim21$.
Our circular-GEO sample continues to rise as we reach the sensitivity limit, suggesting that the modal brightness may be fainter still.
Assuming the objects are Lambertian spheres with albedo $A=0.1$, we probe to sizes $d<10$\,cm~\citep{africano2005phase}.
These assumptions are nevertheless very uncertain, as we lack \textit{a priori} knowledge for any object that fails to correlate with the catalogue.
Furthermore, the brightness of a given object is not always constant over the course of an observation.
Indeed, from panel (e) of Fig.~\ref{fig:brightness-rates}, we see that over 45\,\% of uncorrelated tracks in the overall sample with successfully extracted light curves vary in brightness by more than 4\,mag across the observation window.
In some cases, such brightness variation may manifest as sharp flares or glints, while other objects may exhibit smooth oscillations between successive maxima and minima.
Photometric behaviour of this kind renders any generalisation regarding the albedo redundant.
We find that uncorrelated detections appear to show a greater extent of brightness variation relative to their correlated counterparts within the sampled population.
In addition, the apparent sensitivity limit in panel (a) of Fig.~\ref{fig:brightness-rates} is not truly representative of the detection capability of the sensor, as intrinsic brightness will not be the only factor influencing this.
As revealed by our rate cuts, many objects have non-zero rates of change in angular position, placing a limit on the amount of time they will spend contributing flux to a given set of pixels and therefore reducing the peak surface brightness.
To highlight this effect, we normalise the total flux integrated for each of our detections by a factor $x/l$, where $x$ is characteristic of the point spread function (PSF) of the optical system and $l$ is the extent of the angular path mapped by the object over the course of the 10\,s exposure.
This normalisation gives the brightness of a point-like detection that would possess an equivalent peak flux for the same integration time, resulting in the updated brightness histogram in panel (c) of Fig.~\ref{fig:brightness-rates}.
The faint end of the circular-GEO distribution now peaks at $V\sim22$, before dropping off as we reach our sensitivity limit for `stationary' objects, implying that the modal brightness in this normalised regime could once again be even fainter.
With the INT, we achieve $x=3.3$\,px, meaning that an object moving at the maximum angular rate allowed by our cuts would take 0.4\,s to cross each pixel.
Exposing for longer than this time will weaken the ability of the pipeline to detect such an object, due to added noise from the sky background.
\begin{figure}[tbp]
\begin{center}
\includegraphics*[width=\textwidth]{lightcurves-original.pdf}
\end{center}
\caption{The top three rows present light curve analysis for a track correlated with the defunct satellite SBS-3 (NORAD 13651).
Successive 10\,s exposures are shown in (a)--(g), centered on the detected trails.
The corresponding light curve is provided in (h), extracted using the analysis pipeline outlined in Section~\ref{sec:analysis-pipeline}, while a zoom-in of the boxed region is given in (i).
A 2.7\,s period is uncovered by the Fourier amplitude spectrum in (j).
The Fourier window function, displayed inset, illustrates the effect of the readout-induced gaps in the light curve.
The remaining rows present light curves for two uncorrelated tracks that exhibit significant brightness variation.
Successive 10\,s exposures are shown in (k)--(p) for the first object, while (q) gives the extracted light curve.
Note that the 10\,s exposure images provided in (r)--(w) for the second track are reflected in the horizontal direction, aligning each trail with its corresponding profile in (x).
The three examples shown are as presented previously in~\citet{blake2019optical}.}
\label{fig:lightcurves-original}
\end{figure}
\subsection{Photometric light curves}
\label{sec:photometric-light-curves}
The reflected light from an orbiting body contains information about its shape and attitude, but is also affected by the sensor characteristics, atmospheric interference and the viewing geometry at the time of the observation.
Disentangling these components is a difficult task and light curve characterisation remains an active area of research~\citep[see e.g.][]{albuja2018yorp,cognion2014rotation,fan2019inversion,hinks2016angular,papushev2009investigations}.
Thus far, studies have focused on modelling the photometric signatures of large satellites by virtue of the relative ease in obtaining a useful dataset.
However, understanding the attitude of faint objects will be a pivotal factor in predicting the long-term evolution of the GEO debris environment.
An example of a light curve extracted for a catalogued object can be found in panel (h) of Fig.~\ref{fig:lightcurves-original}.
The corresponding orbital track correlates with SBS-3 (NORAD 13651), a decommissioned communications satellite that was moved to a graveyard orbit in 1995.
Built on the Hughes HS-376 bus, the satellite consists of a cylindrical body with concentric solar panels and extended antennas.
The satellite was spin-stabilised during its active lifetime, maintaining attitude by spinning a section of the platform at 50 rpm (0.83\,Hz; 1.2\,s period).
The communications payload remained despun, ensuring steady pointing of the antennas and transponders.
A periodic pattern can be seen in the light curve, indicating that the satellite is likely tumbling.
Fourier analysis of the signal uncovers a 2.7\,s period for the repeated pattern, though this could be a harmonic of the true tumbling rate given the geometric symmetry of the bus.
In panels (q) and (x) of Fig.~\ref{fig:lightcurves-original}, we show two examples of light curves extracted for uncorrelated objects belonging to the faint end of our sampled population.
Both tracks straddle the sensitivity limit of our observations, exhibiting significant brightness variation across the observation window.
The first object oscillates in brightness with a period similar to the exposure time, peaking at $V\sim16$ and otherwise fading into the background noise level.
With such large variation in brightness, it is likely that the object is a small piece of highly-reflective material tumbling in and out of our line of sight.
Additional structure can be seen in the second light curve, possibly due to an asymmetry in the shape, or more complex tumbling dynamics.
\begin{figure}[tbp]
\begin{center}
\includegraphics*[width=\textwidth]{lightcurve-montage.pdf}
\end{center}
\caption{A montage of light curves for orbital tracks comprising the sampled population, extracted using the analysis pipeline presented in Section~\ref{sec:analysis-pipeline}.}
\label{fig:lightcurve-montage}
\end{figure}
We provide a montage of further light curve examples in Fig.~\ref{fig:lightcurve-montage}.
Light curve (a) corresponds to a bright orbital track that correlates with Raduga 13 (NORAD 14307), a former Soviet communications satellite that was launched in 1983 and now resides in a drift orbit.
The satellite is based on the KAUR-3 bus, a three-axis stabilised `box-wing' model with solar panels extending from both sides of the main body.
We see a relatively flat light curve at $V\sim12.5$ across all but one exposure, which captures a clear glint where a highly reflective component enters the line of sight.
The light curve in panel (b) is that of a track correlated with Intelsat 4A-F3 (NORAD 10557), a retired communications satellite that launched in 1978.
Based on the Hughes HS-353 platform, the lightcurve unsurprisingly exhibits similar photometric signatures to those of SBS-3 presented in Fig.~\ref{fig:lightcurves-original}.
Panels (c), (e) and (g) of Fig.~\ref{fig:lightcurve-montage} give the light curves for three SL-12 rocket bodies (NORAD 16797, 15581 and 23883, respectively).
Fourier analysis of light curve (c) uncovers a period of 3.4\,s; the SL-12 appears to exhibit higher-frequency brightness variations than expected from previous studies of such rocket bodies~\citep[see e.g.][]{cardona2016bvri}, though aliasing effects could be at play as a result of the object's geometric symmetry.
The $\sim5$~s period signals obtained for the other two SL-12 light curves are in better agreement with the findings of the cited study.
The remaining light curves in Fig.~\ref{fig:lightcurve-montage} correspond to orbital tracks that fail to correlate with catalogued objects.
Light curves (f), (k), (l), (u), (w) and (x) all appear to be oscillating in brightness with a period exceeding the exposure time of 10\,s.
In these cases, it would be necessary to follow-up with targeted observations of the object, preferably using an instrument with reduced dead time, in order to gain confidence in the true profile.
We also find a number of uncorrelated objects that show structure in their light curves on a timescale shorter than the exposure time; this is the case for light curves (p), (q) and (t).
An interesting group of detections uncovered by the survey are only detectable as a result of sharp glints that can occur several times per exposure.
Examples of this behaviour can be found in panels (h), (i), (j), (m), (o), (s) and (v).
The extent of the brightness increase during a glint varies significantly case-by-case, with some objects climbing in excess of 5\,mag above the sensitivity limit, while others struggle to breach it.
Finally, light curves (d), (n) and (r) show little variation in brightness within the window of observation.
There are several explanations as to why this may be the case.
The corresponding object could be uniformly reflective across its surface, or oriented in such a way that higher-reflectivity components were hidden from our line of sight for the duration of the pointing.
Alternatively, the object may be stable in its motion (unlikely for the very faint examples) or tumbling faster than the sampling rate of our observations, such that photometric signatures are unresolved.
Noisy scatter could be due to small sub-structures upon the object's surface, although atmospheric fluctuations will also contribute to noise in all of our light curves.
\section{Conclusion}
\label{sec:conclusion}
We conducted an optical survey of the GEO region with eight nights of dark-grey time on the 2.54\,m Isaac Newton Telescope (INT) in La Palma, Canary Islands.
Using an optimised observational strategy (see Section~\ref{sec:observational-strategy}) and a custom analysis pipeline (see Section~\ref{sec:analysis-pipeline}), we found:
\begin{itemize}
\item a total of 226 orbital tracks, 129 of which exhibit rates of change in angular position consistent with circular orbits in the GEO regime;
\item a detection rate of $\sim$11\,hour$^{-1}$deg$^{-2}$ for circular-GEO objects, similar to rates observed by the Magellan spot surveys of GEO;
\item a bimodal brightness distribution, with the bright end centered around $V\sim12$ and the faint end still rising at our sensitivity limit of $V\sim21$, suggesting the modal brightness may be fainter still;
\item over 80\,\% of tracks with $V<15$ correlated with objects in the publicly available USSTRATCOM catalogue, while the vast majority of fainter tracks failed to correlate;
\item many faint, uncorrelated objects show optical signatures of tumbling, causing some to straddle the detection limit of our observations within a single exposure.
\end{itemize}
The GEO region is an important commodity with a limited number of orbital slots.
Free slots are set to become increasingly scarce with an imperfect disposal rate and an increase in orbital break-ups and anomalies in recent years.
The latter have injected over a thousand new fragments into high-altitude orbits since 2018, with a few hundred intersecting the GEO Protected Region.
The majority of these fragments are too faint to be tracked and made publicly available via the USSTRATCOM catalogue, with its size cut-off of $\sim$50--100\,cm at GEO.
It is therefore essential that we probe the faint end of the debris population to gain a better understanding of the GEO environment both in the short- and long-term.
The presented survey was carried out as part of DebrisWatch, an ongoing collaboration between the University of Warwick and the Defence Science and Technology Laboratory (UK) investigating the faint population of GEO debris.
For the duration of the observation campaign, a 36\,cm astrograph was slaved to the INT, covering the same regions of sky with a larger field of view.
Analysis of this rich dataset is ongoing and will form the basis of future DebrisWatch instalments.
\section*{Acknowledgements}
\label{sec:acknowledgements}
JAB gratefully acknowledges support from the STFC (grant ST/R505195/1). PC acknowledges support by the STFC via an IPS Fellowship (grant ST/R005125/1).
DV is also supported by the STFC via an Ernest Rutherford Fellowship (grant ST/P003850/1).
DP acknowledges the Royal Society for support.
TRM acknowledges support from the STFC (grant ST/P000495/1).
This paper makes use of data from the Isaac Newton Telescope, operated on the island of La Palma by the ING in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias.
|
2,869,038,156,418 | arxiv | \section{Introduction}
\setcounter{equation}{0}
Consider the Cauchy problem for the focusing intercritical nonlinear fourth-order Schr\"odinger equation
\begin{align}
\left\{
\begin{array}{rcl}
i\partial_t u - \Delta^2 u &=& -|u|^{\alpha} u, \quad \text{on } [0,\infty) \times \mathbb R^d, \\
u(0) &=& u_0,
\end{array}
\right.
\label{intercritical NL4S}
\end{align}
where $u$ is a complex valued function defined on $[0,\infty) \times \mathbb R^d$ and $2_\star<\alpha<2^\star$ with
\begin{align}
2_\star:= \frac{8}{d}, \quad 2^\star:= \left\{
\begin{array}{cl}
\infty &\text{if } d=1,2,3,4, \\
\frac{8}{d-4} &\text{if } d\geq 5.
\end{array}
\right.
\end{align}
\indent The fourth-order Schr\"odinger equation was introduced by Karpman \cite{Karpman} and Karpman-Shagalov \cite{KarpmanShagalov} taking into account the role of small fourth-order dispersion terms in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. Such fourth-order Schr\"odinger equations are of the form
\begin{align}
i\partial_t - \Delta^2 u + \epsilon \Delta u = \mu |u|^\alpha u, \quad u(0) = u_0, \label{general NL4S}
\end{align}
where $\epsilon \in \{0, \pm 1\}$, $\mu\in \{\pm\}$ and $\alpha>0$. The equation $(\ref{intercritical NL4S})$ is a special case of $(\ref{general NL4S})$ with $\epsilon =0$ and $\mu=-1$. The study of nonlinear fourth-order Schr\"odinger equations $(\ref{general NL4S})$ has attracted a lot of interest in the past several years (see \cite{Pausader}, \cite{Pausadercubic}, \cite{HaoHsiaoWang06}, \cite{HaoHsiaoWang07}, \cite{HuoJia}, \cite{MiaoXuZhao09}, \cite{MiaoXuZhao11}, \cite{MiaoWuZhang}, \cite{Dinhfourt}, \cite{Dinhblowup} and references therein).\newline
\indent The equation $(\ref{intercritical NL4S})$ enjoys the scaling invariance
\[
u_\lambda(t,x):= \lambda^{\frac{4}{\alpha}} u(\lambda^4 t, \lambda x), \quad \lambda >0.
\]
It means that if $u$ solves $(\ref{intercritical NL4S})$, then $u_\lambda$ solves the same equation with intial data $u_\lambda(0, x) = \lambda^{\frac{4}{\alpha}} u_0(\lambda x)$. A direct computation shows
\[
\|u_\lambda(0)\|_{\dot{H}^\gamma} = \lambda^{\gamma +\frac{4}{\alpha}-\frac{d}{2}}\|u_0\|_{\dot{H}^\gamma}.
\]
From this, we define the critical Sobolev exponent
\begin{align}
\Gact:= \frac{d}{2}-\frac{4}{\alpha}. \label{critical sobolev exponent}
\end{align}
We also define the critical Lebesgue exponent
\begin{align}
\alphct: = \frac{2d}{d-2\Gact} = \frac{d\alpha}{4}. \label{critical lebesgue exponent}
\end{align}
By Sobolev embedding, we have $\dot{H}^{\Gact} \hookrightarrow L^{\alphct}$.
The local well-posedness for $(\ref{intercritical NL4S})$ in Sobolev spaces was studied in \cite{Dinhfract, Dinhfourt} (see also \cite{Pausader} for $H^2$ initial data). It is known that $(\ref{intercritical NL4S})$ is locally well-posed in $H^\gamma$ for $\gamma \geq \max\{\Gact,0\}$ satisfying for $\alpha>0$ not an even integer,
\begin{align}
\lceil \gamma \rceil \leq \alpha+1, \label{regularity condition}
\end{align}
where $\lceil \gamma \rceil$ is the smallest integer greater than or equal to $\gamma$. This condition ensures the nonlinearity to have enough regularity. Moreover, the solution enjoys the conservation of mass
\[
M(u(t)) = \int |u(t,x)|^2 dx = M(u_0),
\]
and $H^2$ solution has conserved energy
\[
E(u(t)) = \frac{1}{2} \int |\Delta u(t,x)|^2 dx -\frac{1}{\alpha+2} \int |u(t,x)|^{\alpha+2} dx = E(u_0).
\]
In the subcritical regime, i.e. $\gamma>\Gact$, the existence time depends only on the $H^\gamma$-norm of initial data. There is also a blowup alternative: if $T$ is the maximal time of existence, then either $T=\infty$ or
\[
T<\infty, \quad \lim_{t\uparrow T} \|u(t)\|_{H^\gamma} =\infty.
\]
It is well-known (see e.g. \cite{Pausader}) that if $\Gact <0$ or $0<\alpha<\frac{8}{d}$, then $(\ref{intercritical NL4S})$ is globally well-posed in $H^2$. Thus the blowup in $H^2$ may occur only for $\alpha \geq \frac{8}{d}$. Recently, Boulenger-Lenzmann established in \cite{BoulengerLenzmann} blowup criteria for $(\ref{general NL4S})$ with radial data in $H^2$ in the mass-critical ($\Gact=0$), mass and energy intercritical ($0<\Gact<2$) and enery-critical ($\Gact=2$) cases. This naturally leads to the study of dynamical properties of blowup solutions such as blowup rate, concentration and limiting profile, etc. \newline
\indent In the mass-critical case $\Gact=0$ or $\alpha=\frac{8}{d}$, the study of $H^2$ blowup solutions to $(\ref{intercritical NL4S})$ is closely related to the notion of ground states which are solutions of the elliptic equation
\[
\Delta^2Q + Q - |Q|^{\frac{8}{d}}Q =0.
\]
Fibich-Ilan-Papanicolaou in \cite{FibichIlanPapanicolaou} showed some numerical observations which implies that if $\|u_0\|_{L^2}<\|Q\|_{L^2}$, then the solution exists globally; and if $\|u_0\|_{L^2} \geq \|Q\|_{L^2}$, then the solution may blow up in finite time. Later, Baruch-Fibich-Mandelbaum in \cite{BaruchFibichMandelbaum} proved some dynamical properties such as blowup rate, $L^2$-concentration for radial blowup solutions. In \cite{ZhuYangZhang10}, Zhu-Yang-Zhang established the profile decomposition and a compactness result to study dynamical properties such as $L^2$-concentration, limiting profile with minimal mass of blowup solutions in general case (i.e. without radially symmetric assumption). For dynamical properties of blowup solutions with low regularity initial data, we refer the reader to \cite{ZhuYangZhang11} and \cite{Dinhblowup}. \newline
\indent In the mass and energy intercritical case $0<\Gact<2$, there are few works concerning dynamical properties of blowup solutions to $(\ref{intercritical NL4S})$. To our knowledge, the only paper addressed this problem belongs to \cite{ZhuYangZhang_pc} where the authors studied $L^{\alphct}$-concentration of radial blowup solutions. We also refer to \cite{BaruchFibich} for numerical study of blowup solutions to the equation. \newline
\indent The main purpose of this paper is to show dynamical properties of blowup solutions to $(\ref{intercritical NL4S})$ with initial data in $\dot{H}^{\Gact} \cap \dot{H}^2$. The main difficulty in this consideration is the lack of conservation of mass. To study dynamics of blowup solutions in $\dot{H}^{\Gact} \cap \dot{H}^2$, we firstly need the local well-posedness. For data in $H^2$, the local well-posedness is well-known (see e.g. \cite{Pausader}). However, for data in $\dot{H}^{\Gact} \cap \dot{H}^2$ the local theory is not a trivial consequence of the one for $H^2$ data due to the lack of mass conservation. We thus need to show a new local theory for our purpose, and it will be done in Section $\ref{section preliminaries}$. It is worth noticing that thanks to Strichartz estimates with a ``gain'' of derivatives, we can remove the regularity requirement $(\ref{regularity condition})$. However, we can only show the local well-posedness in dimensions $d\geq 5$, the one for $d\leq 4$ is still open. After the local theory is established, we need to show the existence of blowup solutions. In \cite{BoulengerLenzmann}, the authors showed blowup criteria for radial $H^2$ solutions to $(\ref{general NL4S})$. In their proof, the conservation of mass plays a crucial role. In our setting, the lack of mass conservation makes the problem more difficult. We are only able to prove a blowup criteria for negative energy radial solutions with an additional condition
\begin{align}
\sup_{t\in [0,T)} \|u(t)\|_{\dot{H}^{\Gact}} <\infty. \label{bounded introduction}
\end{align}
This condition is also needed in our results for dynamical properties of blowup solutions. We refer to Section $\ref{section global existence blowup}$ for more details. To study blowup dynamics for data in $\dot{H}^{\Gact} \cap \dot{H}^2$, we establish the profile decomposition for bounded sequences in $\dot{H}^{\Gact} \cap \dot{H}^2$. This is done by following the argument of \cite{HmidiKeraani} (see also \cite{Guoblowup}). With the help of this profile decomposition, we study the sharp constant to the Gagliardo-Nirenberg inequality
\begin{align}
\|f\|^{\alpha+2}_{L^{\alpha+2}} \leq A_{\text{GN}} \|f\|^{\alpha}_{\dot{H}^{\Gact}} \|f\|^2_{\dot{H}^2}. \label{sharp gagliardo-nirenberg inequality intro}
\end{align}
It follows (see Proposition $\ref{prop variational structure ground state intercritical}$) that the sharp constant $A_{\text{GN}}$ is attained at a function $U \in \dot{H}^{\Gact} \cap \dot{H}^2$ of the form
\[
U(x) = a Q(\lambda x + x_0),
\]
for some $a \in \mathbb C^*, \lambda>0$ and $x_0 \in \mathbb R^d$, where $Q$ is a solution to the elliptic equation
\begin{align*}
\Delta^2 Q + (-\Delta)^{\Gact} Q - |Q|^\alpha Q =0.
\end{align*}
Moreover,
\[
A_{\text{GN}} = \frac{\alpha+2}{2} \|Q\|^{-\alpha}_{\dot{H}^{\Gace}}.
\]
The profile decomposition also gives a compactness lemma, that is for any bounded sequence $(v_n)_{n\geq 1}$ in $\dot{H}^{\Gace} \cap \dot{H}^2$ satisfying
\[
\limsup_{n\rightarrow \infty} \|v_n\|_{\dot{H}^2} \leq M, \quad \limsup_{n\rightarrow \infty} \|v_n\|_{L^{\alpha+2}} \geq m,
\]
there exists a sequence $(x_n)_{n\geq 1}$ in $\mathbb R^d$ such that up to a subsequence,
\[
v_n(\cdot + x_n) \rightharpoonup V \text{ weakly in } \dot{H}^{\Gace} \cap \dot{H}^2,
\]
for some $V \in \dot{H}^{\Gace} \cap \dot{H}^2$ satisfying
\begin{align*}
\|V\|^\alpha_{\dot{H}^{\Gace}} \geq \frac{2}{\alpha+2} \frac{ m^{\alpha+2}}{M^2} \|Q\|_{\dot{H}^{\Gact}}^\alpha.
\end{align*}
As a consequence, we show that the $\dot{H}^{\Gact}$-norm of blowup solutions satisfying $(\ref{bounded introduction})$ must concentrate by an amount which is bounded from below by $\|Q\|_{\dot{H}^{\Gact}}$ at the blowup time. Finally, we show the limiting profile of blowup solutions with critical norm
\[
\sup_{t\in [0,T)} \|u(t)\|_{\dot{H}^{\Gact}} = \|Q\|_{\dot{H}^{\Gact}}.
\]
The plan of this paper is as follows. In Section $\ref{section preliminaries}$, we give some preliminaries including Strichartz estimates, the local well-posednesss for data in $\dot{H}^{\Gact} \cap \dot{H}^2$ and the profile decomposition of bounded sequences in $\dot{H}^{\Gact} \cap \dot{H}^2$. In Section $\ref{section variational analysis}$, we use the profile decomposition to study the sharp Gagliardo-Nirenberg inequality $(\ref{sharp gagliardo-nirenberg inequality intro})$. The global existence and blowup criteria will be given in Section $\ref{section global existence blowup}$. Section $\ref{section blowup concentration}$ is devoted to the blowup concentration, and finally the limiting profile of blowup solutions with critical norm will be given in Section $\ref{section limiting profile}$.
\section{Preliminaries} \label{section preliminaries}
\setcounter{equation}{0}
\subsection{Homogeneous Sobolev spaces}
We firstly recall the definition and properties of homogeneous Sobolev spaces (see e.g. \cite[Appendix]{GinibreVelo}, \cite[Chapter 5]{Triebel} or \cite[Chapter 6]{BerghLofstom}). Given $\gamma \in \mathbb R$ and $1 \leq q \leq \infty$, the generalized homogeneous Sobolev space is defined by
\[
\dot{W}^{\gamma,q} := \left\{ u \in \mathcal{S}_0' \ | \ \|u\|_{\dot{W}^{\gamma,q}} := \||\nabla|^\gamma u\|_{L^q} <\infty \right\},
\]
where $\mathcal{S}_0$ is the subspace of the Schwartz space $\mathcal{S}$ consisting of functions $\phi$ satisfying $D^\beta \hat{\phi}(0) =0$ for all $\beta \in \mathbb N^d$ with $\hat{\cdot}$ the Fourier transform on $\mathcal{S}$, and $\mathcal{S}_0'$ is its topology dual space. One can see $\mathcal{S}'_0$ as $\mathcal{S}'/\mathcal{P}$ where $\mathcal{P}$ is the set of all polynomials on $\mathbb R^d$. Under these settings, $\dot{W}^{\gamma,q}$ are Banach spaces. Moreover, the space $\mathcal{S}_0$ is dense in $\dot{W}^{\gamma,q}$. In this paper, we shall use $\dot{H}^\gamma:= \dot{W}^{\gamma,2}$. We note that the spaces $\dot{H}^{\gamma_1}$ and $\dot{H}^{\gamma_2}$ cannot be compared for the inclusion. Nevertheless, for $\gamma_1 < \gamma < \gamma_2$, the space $\dot{H}^{\gamma}$ is an interpolation space between $\dot{H}^{\gamma_1}$ and $\dot{H}^{\gamma_2}$.
\subsection{Strichartz estimates}
In this subsection, we recall Strichartz estimates for the fourth-order Schr\"odinger equation. Let $I \subset \mathbb R$ and $p,q\in [1,\infty]$. We define the mixed norm
\[
\|u\|_{L^p(I, L^q)} := \Big( \int_I \Big(\int_{\mathbb R^d} |u(t,x)|^q dx \Big)^{\frac{p}{q}} \Big)^{\frac{1}{p}},
\]
with a usual modification when either $p$ or $q$ are infinity. We also denote for $(p,q) \in [1,\infty]^2$,
\begin{align}
\gamma_{p,q} = \frac{d}{2}-\frac{d}{q} -\frac{4}{p}. \label{define gamma pq}
\end{align}
\begin{defi}
A pair $(p,q)$ is called {\bf Schr\"odinger admissible}, for short $(p,q) \in S$, if
\[
(p,q) \in [2,\infty]^2, \quad (p,q,d) \ne (2,\infty, 2), \quad \frac{2}{p} + \frac{d}{q} \leq \frac{d}{2}.
\]
A pair $(p,q)$ is call {\bf biharmonic admissible}, for short $(p, q) \in B$, if
\[
(p,q) \in S, \quad \gamma_{p,q} =0.
\]
\end{defi}
We have the following Strichartz estimates for the fourth-order Schr\"odinger equation.
\begin{prop} [Strichartz estimates \cite{ChoOzawaXia, Dinhfract}] \label{prop generalized strichartz estimate} Let $\gamma \in \mathbb R$ and $u$ be a weak solution to the inhomogeneous fourth-order Schr\"odinger equation, namely
\begin{align}
u(t) = e^{it\Delta^2} u_0 + i\int_0^t e^{i(t-s) \Delta^2} F(s) ds, \label{weak solution}
\end{align}
for some data $u_0$ and $F$. Then for all $(p,q)$ and $(a,b)$ Schr\"odinger admissible with $q<\infty$ and $b<\infty$,
\begin{align}
\||\nabla|^\gamma u\|_{L^p(\mathbb R, L^q)} \lesssim \||\nabla|^{\gamma+\gamma_{p,q}} u_0\|_{L^2} + \||\nabla|^{\gamma+\gamma_{p,q}-\gamma_{a',b'}-4} F\|_{L^{a'}(\mathbb R, L^{b'})}, \label{generalized strichartz estimate}
\end{align}
where $(a,a')$ and $(b,b')$ are conjugate pairs.
\end{prop}
Note that the estimates $(\ref{generalized strichartz estimate})$ are exactly the ones given in \cite{MiaoZhang} or \cite{Pausader} where the authors considered $(p,q)$ and $(a,b)$ are either sharp Schr\"odinger admissible, i.e.
\[
(p,q) \in [2,\infty]^2, \quad (p,q,d) \ne (2, \infty, 2), \quad \frac{2}{p} +\frac{d}{q} =\frac{d}{2},
\]
or biharmonic admissible. We refer to \cite{ChoOzawaXia} or \cite{Dinhfract} for the proof of Propsosition $\ref{prop generalized strichartz estimate}$. Note that instead of using directly a dedicate dispersive estimate of \cite{Ben-ArtziKochSaut} for the fundamental solution of the homogeneous fourth-order Schr\"odinger equation, one uses the scaling technique which is similar to the one of wave equation (see e.g. \cite{KeelTao}). \newline
\indent We also have the following consequence of Strichartz estimates $(\ref{generalized strichartz estimate})$.
\begin{coro} \label{coro strichartz estimate}
Let $\gamma \in \mathbb R$ and $u$ be a weak solution to the inhomogeneous fourth-order Schr\"odinger equation $(\ref{weak solution})$ for some data $u_0$ and $F$. Then for all $(p,q)$ and $(a,b)$ biharmonic admissible satisfying $q<\infty$ and $b<\infty$,
\begin{align}
\|u\|_{L^p(\mathbb R, L^q)} \lesssim \|u_0\|_{L^2} + \|F\|_{L^{a'}(\mathbb R, L^{b'})}, \label{strichartz estimate L2}
\end{align}
and
\begin{align}
\||\nabla|^\gamma u\|_{L^p(\mathbb R, L^q)} \lesssim \||\nabla|^\gamma u_0\|_{L^2} + \||\nabla|^{\gamma-1} F\|_{L^{2}(\mathbb R, L^{\frac{2d}{d+2}})}, \label{strichartz estimate Hgamma}
\end{align}
\end{coro}
Note that the estimates $(\ref{strichartz estimate Hgamma})$ is important to reduce the regularity requirement of the nonlinearity (see Subsection $\ref{subsection local well posedness}$). \newline
\indent In the sequel, for a space time slab $I\times \mathbb R^d$ we define the Strichartz space $\dot{B}^0(I\times \mathbb R^d)$ as a closure of $\mathcal{S}_0$ under the norm
\[
\|u\|_{\dot{B}^0(I\times \mathbb R^d)}:= \sup_{(p,q) \in B \atop q<\infty} \|u\|_{L^p(I, L^q)}.
\]
For $\gamma \in \mathbb R$, the space $\dot{B}^\gamma(I \times \mathbb R^d)$ is defined as a closure of $\mathcal{S}_0$ under the norm
\[
\|u\|_{\dot{B}^\gamma(I\times \mathbb R^d)} := \||\nabla|^\gamma u\|_{\dot{B}^0(I \times \mathbb R^d)}.
\]
We also use $\dot{N}^0(I\times \mathbb R^d)$ to denote the dual space of $\dot{B}^0(I\times \mathbb R^d)$ and
\[
\dot{N}^\gamma(I \times \mathbb R^d):= \left\{u \ : \ |\nabla|^\gamma u \in \dot{N}^0(I \times \mathbb R^d)\right\}.
\]
To simplify the notation, we will use $\dot{B}^\gamma(I), \dot{N}^\gamma(I)$ instead of $\dot{B}^\gamma(I\times \mathbb R^d)$ and $\dot{N}^\gamma(I \times \mathbb R^d)$. By Corollary $\ref{coro strichartz estimate}$, we have
\begin{align}
\|u\|_{\dot{B}^0(\mathbb R)} &\lesssim \|u_0\|_{L^2} + \|F\|_{\dot{N}^0(\mathbb R)}, \label{strichartz estimate L2 abstract} \\
\|u\|_{\dot{B}^\gamma(\mathbb R)} &\lesssim \|u_0\|_{\dot{H}^\gamma} + \||\nabla|^{\gamma-1} F\|_{L^2(\mathbb R, L^{\frac{2d}{d+2}})}. \label{strichartz estimate Hgamma abstract}
\end{align}
\subsection{Nonlinear estimates}
We next recall nonlinear estimates to study the local well-posedness for $(\ref{intercritical NL4S})$.
\begin{lem}[Nonlinear estimates \cite{Kato}] \label{lem nonlinear estimate}
Let $F \in C^k(\mathbb C, \mathbb C)$ with $k \in \mathbb N \backslash \{0\}$. Assume that there is $\alpha>0$ such that $k \leq \alpha+1$ and
\[
|D^j F(z)| \lesssim |z|^{\alpha+1-j}, \quad z \in \mathbb C, j =1, \cdots, k.
\]
Then for $\gamma \in [0,k]$ and $1<r, p<\infty$, $1<q \leq \infty$ satisfying $\frac{1}{r} = \frac{1}{p} +\frac{\alpha}{q}$, there exists $C=C(d,\alpha, \gamma, r, p, q)>0$ such that for all $u \in \mathcal{S}$,
\begin{align}
\||\nabla|^\gamma F(u)\|_{L^r} \leq C \|u\|^\alpha_{L^q} \||\nabla|^\gamma u\|_{L^p}. \label{nonlinear estimate}
\end{align}
Moreover, if $F$ is a homogeneous polynomial in $u$ and $\overline{u}$, then $(\ref{nonlinear estimate})$ holds true for any $\gamma\geq 0$.
\end{lem}
The proof of Lemma $\ref{lem nonlinear estimate}$ is based on the fractional Leibniz rule (or Kato-Ponce inequality) and the fractional chain rule. We refer the reader to \cite[Appendix]{Kato} for the proof.
\subsection{Local well-posedness} \label{subsection local well posedness}
In this subsection, we recall the local well-posedness for $(\ref{intercritical NL4S})$ with initial data in $H^2$ and in $\dot{H}^{\Gact} \cap \dot{H}^2$ respectively. The case in $H^2$ is well-known (see e.g. \cite{Pausader}), while the one in $\dot{H}^{\Gact} \cap \dot{H}^2$ needs a careful consideration.
\begin{prop} [Local well-posedness in $H^2$ \cite{Pausader}] \label{prop local well-posedness H2}
Let $d\geq 1$, $u_0 \in H^2$ and $0<\alpha<2^\star$. Then there exist $T>0$ and a unique solution $u$ to $(\ref{intercritical NL4S})$ satisfying
\[
u \in C([0,T), H^2) \cap L^p_{\emph{loc}}([0,T), W^{2,q}),
\]
for any biharmonic admissible pairs $(p,q)$ satisfying $q<\infty$. The time of existence satisfies either $T=\infty$ or $T<\infty$ and $\lim_{t\uparrow T} \|u\|_{\dot{H}^2} =\infty$. Moreover, the solution enjoys the conservation of mass and energy.
\end{prop}
\begin{prop}[Local well-posedness in $\dot{H}^{\Gace} \cap \dot{H}^2$] \label{prop local well-posedness H dot 2}
Let $d\geq 5$, $2_\star \leq \alpha<2^\star$ and $u_0 \in \dot{H}^{\Gace} \cap \dot{H}^2$. Then there exist $T>0$ and a unique solution $u$ to $(\ref{intercritical NL4S})$ satisfying
\[
u \in C([0,T), \dot{H}^{\Gace} \cap \dot{H}^2) \cap L^p_{\emph{loc}}([0,T), \dot{W}^{\Gace, q} \cap \dot{W}^{2,q}),
\]
for any biharmonic admissible pairs $(p,q)$ satisfying $q<\infty$. The existence time satisfies either $T=\infty$ or $T<\infty$ and $\lim_{t\uparrow T} \|u(t)\|_{\dot{H}^{\Gace}}+\|u(t)\|_{\dot{H}^2} =\infty$. Moreover, the solution enjoys the conservation of energy.
\end{prop}
\begin{rem} \label{rem local well-posedness H dot 2}
\begin{itemize}
\item When $\Gact=0$ or $\alpha=2_\star$, Proposition $\ref{prop local well-posedness H dot 2}$ is a consequence of Proposition $\ref{prop local well-posedness H2}$ since $\dot{H}^0 = L^2$ and $L^2 \cap \dot{H}^2= H^2$.
\item In \cite{Dinhfourt}, a similar result holds with an additional regularity assumption $\alpha \geq 1$ if $\alpha$ is not an even integer. Thanks to Strichartz estimate with a ``gain'' of derivatives $(\ref{strichartz estimate Hgamma abstract})$, we can remove this regularity requirement.
\end{itemize}
\end{rem}
\noindent \textit{Proof of Proposition $\ref{prop local well-posedness H dot 2}$.} We firstly choose
\[
n=\frac{2d}{d+2 - (d-4)\alpha}, \quad n^* = \frac{2d}{d+4-(d-4)\alpha}, \quad m^* = \frac{8}{(d-4)\alpha-4}.
\]
It is easy to check that
\begin{align}
\frac{d+2}{2d} = \frac{(d-4)\alpha}{2d} + \frac{1}{n}, \quad \frac{1}{n} = \frac{1}{n^*} -\frac{1}{d}, \quad \frac{d}{2} = \frac{4}{m^*} + \frac{d}{n^*}. \label{choice of nmm}
\end{align}
In particular, $(m^*, n^*)$ is a biharmonic admissible and
\begin{align}
\theta:= \frac{1}{2}-\frac{1}{m^*}=1-\frac{(d-4)\alpha}{8}>0. \label{choice of theta}
\end{align}
Consider
\[
X:= \left\{u \in \dot{B}^{\Gact}(I) \cap \dot{B}^2(I) \ : \ \|u\|_{\dot{B}^{\Gact}(I)} + \|u\|_{\dot{B}^2(I)} \leq M \right\},
\]
equipped with the distance
\[
d(u,v) := \|u-v\|_{\dot{B}^0(I)},
\]
where $I=[0,\tau]$ and $M,\tau>0$ to be chosen later. By Duhamel's formula, it suffices to prove that the functional
\[
\Phi(u)(t):= e^{it\Delta^2} u_0 +i \int_0^t e^{i(t-s)\Delta^2} |u(s)|^\alpha u(s) ds
\]
is a contraction on $(X,d)$. By Strichartz estimate $(\ref{strichartz estimate Hgamma abstract})$,
\[
\|\Phi(u)\|_{\dot{B}^2(I)} \lesssim \|u_0\|_{\dot{H}^2} + \|\nabla(|u|^\alpha u)\|_{L^2(I, L^{\frac{2d}{d+2}})}.
\]
By Lemma $\ref{lem nonlinear estimate}$,
\[
\|\nabla(|u|^\alpha u)\|_{L^2(I, L^{\frac{2d}{d+2}})} \lesssim \|u\|^\alpha_{L^\infty(I, L^{\frac{2d}{d-4}}) } \|\nabla u\|_{L^2(I, L^n)}.
\]
We next use $(\ref{choice of nmm})$ together with the Sobolev embedding to bound
\begin{align*}
\|u\|_{L^\infty(I, L^{\frac{2d}{d-4}})} \lesssim \|\Delta u\|_{L^\infty(I, L^2)} \lesssim \|u\|_{\dot{B}^2(I)},
\end{align*}
and
\begin{align*}
\|\nabla u\|_{L^2(I, L^n)} \lesssim \|\Delta u\|_{L^2(I, L^{n^*})} \lesssim |I|^\theta \|\Delta u\|_{L^{m^*}(I, L^{n^*})} \lesssim |I|^\theta \|u\|_{\dot{B}^2(I)}.
\end{align*}
Thus, we get
\[
\|\Phi(u)\|_{\dot{B}^2(I)} \lesssim \|u_0\|_{\dot{H}^2} + |I|^\theta \|u\|^{\alpha+1}_{\dot{B}^2(I)}.
\]
We now estimate $\|\Phi(u)\|_{\dot{B}^{\Gact}(I)}$. To do so, we separate two cases $\Gact \geq 1$ and $0<\Gact <1$. In the case $\Gact \geq 1$, we estimate as above to get
\[
\|\Phi(u)\|_{\dot{B}^{\Gact}(I)} \lesssim \|u_0\|_{\dot{H}^{\Gact}} + |I|^\theta \|u\|^\alpha_{\dot{B}^2(I)} \|u\|_{\dot{B}^{\Gact}(I)}.
\]
In the case $0<\Gact<1$, we choose
\begin{align}
p=\frac{8(\alpha+2)}{\alpha(d-4)}, \quad q= \frac{d(\alpha+2)}{d+2\alpha}, \label{choice pq}
\end{align}
and choose $(m,n)$ so that
\[
\frac{1}{p'} = \frac{1}{m} + \frac{\alpha}{p}, \quad \frac{1}{q'}=\frac{1}{q} + \frac{\alpha}{n}.
\]
It is easy to check that $(p,q)$ is biharmonic admissible and $n=\frac{dq}{d-2q}$. The later fact gives the Sobolev embedding $\dot{W}^{2,q} \hookrightarrow L^n$. By Strichartz estimate $(\ref{strichartz estimate L2 abstract})$,
\[
\|\Phi(u)\|_{\dot{B}^{\Gact}(I)} \lesssim \|u_0\|_{\dot{H}^{\Gact}} + \||\nabla|^{\Gact}(|u|^\alpha u)\|_{L^{p'}(I, L^{q'})}.
\]
By Lemma $\ref{lem nonlinear estimate}$,
\begin{align*}
\||\nabla|^{\Gact}(|u|^\alpha u)\|_{L^{p'}(I, L^{q'})} & \lesssim \|u\|^\alpha_{L^p(I, L^n)} \||\nabla|^{\Gact}u\|_{L^m(I, L^q)} \\
&\lesssim |I|^{\frac{1}{m}-\frac{1}{p}} \|\Delta u\|^\alpha_{L^p(I, L^q)} \||\nabla|^{\Gact}u\|_{L^p(I, L^q)} \\
& \lesssim |I|^\theta \|u\|^\alpha_{\dot{B}^2(I)} \|u\|_{\dot{B}^{\Gact}(I)}.
\end{align*}
In both cases, we have
\[
\|\Phi(u)\|_{\dot{B}^{\Gact}(I)} \lesssim \|u_0\|_{\dot{H}^{\Gact}} + |I|^\theta \|u\|^\alpha_{\dot{B}^2(I)} \|u\|_{\dot{B}^{\Gact}(I)}.
\]
Therefore,
\[
\|\Phi(u)\|_{\dot{B}^{\Gact}(I) \cap \dot{B}^2(I)} \lesssim \|u_0\|_{\dot{H}^{\Gact} \cap \dot{H}^2} + |I|^\theta \|u\|^\alpha_{\dot{B}^2(I)} \|u\|_{\dot{B}^{\Gact}(I) \cap \dot{B}^2(I)}.
\]
Similarly, by $(\ref{strichartz estimate L2 abstract})$,
\[
\|\Phi(u)- \Phi(v)\|_{\dot{B}^0(I)} \lesssim \||u|^\alpha u - |v|^\alpha v\|_{L^{p'} (I, L^{q'})},
\]
where $(p,q)$ is as in $(\ref{choice pq})$. We estimate
\begin{align*}
\||u|^\alpha u - |v|^\alpha v\|_{L^{p'}(I, L^{q'})} &\lesssim \left( \|u\|^\alpha_{L^p(I, L^n)} + \|v\|^\alpha_{L^p(I, L^n)}\right) \|u-v\|_{L^m(I, L^q)} \\
&\lesssim |I|^\theta \left(\|\Delta u\|^\alpha_{L^p(I, L^q)} + \|\Delta v\|^\alpha_{L^p(I, L^q)} \right) \|u-v\|_{L^p(I, L^q)} \\
&\lesssim |I|^\theta \left(\|u\|^\alpha_{\dot{B}^2(I)} + \|v\|^\alpha_{\dot{B}^2(I)}\right) \|u-v\|_{\dot{B}^0(I)}.
\end{align*}
This shows that for all $u,v \in X$, there exists $C>0$ independent of $\tau$ and $u_0 \in \dot{H}^{\Gact} \cap \dot{H}^2$ such that
\begin{align}
\|\Phi(u)\|_{\dot{B}^{\Gact}(I)} + \|\Phi(u)\|_{\dot{B}^2(I)} &\leq C\|u_0\|_{\dot{H}^{\Gact} \cap \dot{H}^2} + C\tau^\theta M^{\alpha+1}, \label{blowup rate proof}\\
d(\Phi(u), \Phi(v)) &\leq C\tau^\theta M^\alpha d(u,v). \nonumber
\end{align}
If we set $M=2C \|u_0\|_{\dot{H}^{\Gact} \cap \dot{H}^2}$ and choose $\tau>0$ so that
\[
C \tau^\theta M^\alpha \leq \frac{1}{2},
\]
then $\Phi$ is a strict contraction on $(X,d)$. This proves the existence of solution
\[
u \in \dot{B}^{\Gact}(I) \cap \dot{B}^2(I).
\]
The time of existence depends only on the $\dot{H}^{\Gact} \cap \dot{H}^2$-norm of initial data. We thus have the blowup alternative. The conservation of energy follows from the standard approximation. The proof is complete.
\defendproof
\begin{coro}[Blowup rate] \label{coro blowup rate intercritical}
Let $d \geq 5$, $0<\alpha<2^\star$ and $u_0 \in \dot{H}^{\Gact} \cap \dot{H}^2$. Assume that the corresponding solution $u$ to $(\ref{intercritical NL4S})$ given in Proposition $\ref{prop local well-posedness H dot 2}$ blows up at finite time $0<T<\infty$. Then there exists $C>0$ such that
\begin{align}
\|u(t)\|_{\dot{H}^{\Gact} \cap \dot{H}^2} > \frac{C}{(T-t)^{\frac{2-\Gace}{4}}}, \label{blowup rate intercritical}
\end{align}
for all $0<t<T$.
\end{coro}
\begin{proof}
Let $0<t<T$. If we consider $(\ref{intercritical NL4S})$ with initial data $u(t)$, then it follows from $(\ref{blowup rate proof})$ and the fixed point argument that if for some $M>0$,
\[
C\|u(t)\|_{\dot{H}^{\Gact} \cap \dot{H}^2} + C(\tau-t)^\theta M^{\alpha+1} \leq M,
\]
then $\tau<T$. Thus,
\[
C\|u(t)\|_{\dot{H}^{\Gact} \cap \dot{H}^2} + C(T-t)^\theta M^{\alpha+1} > M,
\]
for all $M>0$. Choosing $M= 2C\|u(t)\|_{\dot{H}^{\Gact} \cap \dot{H}^2}$, we see that
\[
(T-t)^\theta \|u(t)\|^\alpha_{\dot{H}^{\Gact} \cap \dot{H}^2} >C.
\]
This implies
\[
\|u(t)\|_{\dot{H}^{\Gact} \cap \dot{H}^2}> \frac{C}{(T-t)^{\frac{\theta}{\alpha}}},
\]
which is exactly $(\ref{blowup rate intercritical})$ since $\frac{\theta}{\alpha}= \frac{8-(d-4)\alpha}{8\alpha} = \frac{2-\Gact}{4}$. The proof is complete.
\end{proof}
\subsection{Profile decomposition}
The main purpose of this subsection is to prove the profile decomposition related to the focusing intercritical NL4S by following the argument of \cite{HmidiKeraani} (see also \cite{Guoblowup}).
\begin{theorem}[Profile decomposition] \label{theorem profile decomposition intercritical NL4S}
Let $d\geq 1$ and $2_\star<\alpha<2^\star$. Let $(v_n)_{n\geq 1}$ be a bounded sequence in $\dot{H}^{\Gace} \cap \dot{H}^2$. Then there exist a subsequence of $(v_n)_{n\geq 1}$ (still denoted $(v_n)_{n\geq 1}$), a family $(x_n^j)_{j\geq 1}$ of sequences in $\mathbb R^d$ and a sequence $(V^j)_{j\geq 1}$ of $\dot{H}^{\Gace} \cap \dot{H}^2$ functions such that
\begin{itemize}
\item for every $k\ne j$,
\begin{align}
|x_n^k - x_n^j| \rightarrow \infty, \quad \text{as } n \rightarrow \infty, \label{pairwise orthogonality intercritical}
\end{align}
\item for every $l\geq 1$ and every $x \in \mathbb R^d$,
\[
v_n(x) = \sum_{j=1}^l V^j(x-x_n^j) + v_n^l(x),
\]
with
\begin{align}
\limsup_{n\rightarrow \infty} \|v^l_n\|_{L^q} \rightarrow 0, \quad \text{as } l \rightarrow \infty, \label{profile error intercritical}
\end{align}
for every $q \in (\alphce,2+2^\star)$, where $\alphce$ is given in $(\ref{critical lebesgue exponent})$. Moreover,
\begin{align}
\|v_n\|^2_{\dot{H}^{\Gace}} &= \sum_{j=1}^l \|V^j\|^2_{\dot{H}^{\Gace}} + \|v^l_n\|^2_{\dot{H}^{\Gace}} + o_n(1), \label{profile identity 1 intercritical}, \\
\|v_n\|^2_{\dot{H}^2} &= \sum_{j=1}^l \|V^j\|^2_{\dot{H}^2} + \|v^l_n\|^2_{\dot{H}^2} + o_n(1), \label{profile identity 2 intercritical},
\end{align}
as $n\rightarrow \infty$.
\end{itemize}
\end{theorem}
\begin{rem} \label{rem profile decomposition intercritical NL4S}
In the case $\Gact=0$ or $\alpha=2_\star$, Theorem $\ref{theorem profile decomposition intercritical NL4S}$ is exactly Proposition 2.3 in \cite{ZhuYangZhang10} due to the fact $\dot{H}^0=L^2$ and $L^2 \cap \dot{H}^2 = H^2$.
\end{rem}
\noindent \textit{Proof of Theorem $\ref{theorem profile decomposition intercritical NL4S}$.}
Since $\dot{H}^{\Gact} \cap \dot{H}^2$ is a Hilbert space, we denote $\Omega(v_n)$ the set of functions obtained as weak limits of sequences of the translated $v_n(\cdot + x_n)$ with $(x_n)_{n\geq 1}$ a sequence in $\mathbb R^d$. Denote
\[
\eta(v_n):= \sup \{ \|v\|_{\dot{H}^{\Gact}} + \|v\|_{\dot{H}^2} : v \in \Omega(v_n)\}.
\]
Clearly,
\[
\eta(v_n) \leq \limsup_{n\rightarrow \infty} \|v_n\|_{\dot{H}^{\Gact}} + \|v_n\|_{\dot{H}^2}.
\]
We shall prove that there exist a sequence $(V^j)_{j\geq 1}$ of $\Omega(v_n)$ and a family $(x_n^j)_{j\geq 1}$ of sequences in $\mathbb R^d$ such that for every $k \ne j$,
\[
|x_n^k - x_n^j| \rightarrow \infty, \quad \text{as } n \rightarrow \infty,
\]
and up to a subsequence, the sequence $(v_n)_{n\geq 1}$ can be written as for every $l\geq 1$ and every $x \in \mathbb R^d$,
\[
v_n(x) = \sum_{j=1}^l V^j(x-x_n^j) + v^l_n(x),
\]
with $\eta(v^l_n) \rightarrow 0$ as $l \rightarrow \infty$. Moreover, the identities $(\ref{profile identity 1 intercritical})$ and $(\ref{profile identity 2 intercritical})$ hold as $n \rightarrow \infty$. \newline
\indent Indeed, if $\eta(v_n) =0$, then we can take $V^j=0$ for all $j\geq 1$. Otherwise we choose $V^1 \in \Omega(v_n)$ such that
\[
\|V^1\|_{\dot{H}^{\Gact}} + \|V^1\|_{\dot{H}^2} \geq \frac{1}{2} \eta(v_n) >0.
\]
By the definition of $\Omega(v_n)$, there exists a sequence $(x^1_n)_{n\geq 1} \subset \mathbb R^d$ such that up to a subsequence,
\[
v_n(\cdot + x^1_n) \rightharpoonup V^1 \text{ weakly in } \dot{H}^{\Gact} \cap \dot{H}^2.
\]
Set $v_n^1(x):= v_n(x) - V^1(x-x^1_n)$. We see that $v^1_n(\cdot + x^1_n) \rightharpoonup 0$ weakly in $\dot{H}^{\Gact} \cap \dot{H}^2$ and
thus
\begin{align*}
\|v_n\|^2_{\dot{H}^{\Gact}} &= \|V^1\|^2_{\dot{H}^{\Gact}} + \|v^1_n\|^2_{\dot{H}^{\Gact}} + o_n(1), \\
\|v_n\|^2_{\dot{H}^2} &= \|V^1\|^2_{\dot{H}^2} + \|v^1_n\|^2_{\dot{H}^2} + o_n(1),
\end{align*}
as $n \rightarrow \infty$. We now replace $(v_n)_{n\geq 1}$ by $(v^1_n)_{n\geq 1}$ and repeat the same process. If $\eta(v^1_n) =0$, then we choose $V^j=0$ for all $j \geq 2$. Otherwise there exist $V^2 \in \Omega(v^1_n)$ and a sequence $(x^2_n)_{n\geq 1} \subset \mathbb R^d$ such that
\[
\|V^2\|_{\dot{H}^{\Gact}} + \|V^2\|_{\dot{H}^2} \geq \frac{1}{2} \eta(v^1_n)>0,
\]
and
\[
v^1_n(\cdot+x^2_n) \rightharpoonup V^2 \text{ weakly in } \dot{H}^{\Gact} \cap \dot{H}^2.
\]
Set $v^2_n(x) := v^1_n(x) - V^2(x-x^2_n)$. We thus have
$v^2_n(\cdot +x^2_n) \rightharpoonup 0$ weakly in $\dot{H}^{\Gact} \cap \dot{H}^2$ and
\begin{align*}
\|v^1_n\|^2_{\dot{H}^{\Gact}} & = \|V^2\|^2_{\dot{H}^{\Gact}} + \|v^2_n\|^2_{\dot{H}^{\Gact}} + o_n(1), \\
\|v^1_n\|^2_{\dot{H}^2} &= \|V^2\|^2_{\dot{H}^2} + \|v^2_n\|^2_{\dot{H}^2} + o_n(1),
\end{align*}
as $n \rightarrow \infty$. We claim that
\[
|x^1_n - x^2_n| \rightarrow \infty, \quad \text{as } n \rightarrow \infty.
\]
In fact, if it is not true, then up to a subsequence, $x^1_n - x^2_n \rightarrow x_0$ as $n \rightarrow \infty$ for some $x_0 \in \mathbb R^d$. Since
\[
v^1_n(x + x^2_n) = v^1_n(x +(x^2_n -x^1_n) + x^1_n),
\]
and $v^1_n (\cdot + x^1_n)$ converges weakly to $0$, we see that $V^2=0$. This implies that $\eta(v^1_n)=0$ and it is a contradiction. An argument of iteration and orthogonal extraction allows us to construct the family $(x^j_n)_{j\geq 1}$ of sequences in $\mathbb R^d$ and the sequence $(V^j)_{j\geq 1}$ of $\dot{H}^{\Gact} \cap \dot{H}^2$ functions satisfying the claim above. Furthermore, the convergence of the series $\sum_{j\geq 1}^\infty \|V^j\|^2_{\dot{H}^{\Gact}} + \|V^j\|^2_{\dot{H}^2}$ implies that
\[
\|V^j\|^2_{\dot{H}^{\Gact}} + \|V^j\|^2_{\dot{H}^2} \rightarrow 0, \quad \text{as } j \rightarrow \infty.
\]
By construction, we have
\[
\eta(v^j_n) \leq 2 \left(\|V^{j+1}\|_{\dot{H}^{\Gact}} + \|V^{j+1}\|_{\dot{H}^2}\right),
\]
which proves that $\eta(v^j_n) \rightarrow 0$ as $j \rightarrow \infty$. To complete the proof of Theorem $\ref{theorem profile decomposition intercritical NL4S}$, it remains to show $(\ref{profile error intercritical})$. To do so, we introduce for $R>1$ a function $\chi_R \in \mathcal{S}$ satisfying $\hat{\chi}_R: \mathbb R^d \rightarrow [0,1]$ and
\[
\hat{\chi}_R(\xi) = \left\{
\begin{array}{clc}
1 &\text{if}& 1/R \leq |\xi| \leq R, \\
0 &\text{if}& |\xi| \leq 1/2R \vee |\xi|\geq 2R.
\end{array}
\right.
\]
We write
\[
v^l_n = \chi_R * v^l_n + (\delta - \chi_R) * v^l_n,
\]
where $*$ is the convolution operator. Let $q \in (\alphct, 2+2^\star)$ be fixed. By Sobolev embedding and the Plancherel formula, we have
\begin{align*}
\|(\delta -\chi_R) * v^l_n\|_{L^q} \lesssim \|(\delta-\chi_R) * v^l_n\|_{\dot{H}^\beta} &\lesssim \Big( \int |\xi|^{2\beta} |(1-\hat{\chi}_R(\xi)) \hat{v}^l_n(\xi)|^2 d\xi\Big)^{1/2} \\
&\lesssim \Big( \int_{|\xi|\leq 1/R} |\xi|^{2\beta} |\hat{v}^l_n(\xi)|^2 d\xi\Big)^{1/2} + \Big( \int_{|\xi|\geq R} |\xi|^{2\beta} |\hat{v}^l_n(\xi)|^2 d\xi\Big)^{1/2} \\
&\lesssim R^{\Gact-\beta} \|v^l_n\|_{\dot{H}^{\Gact}} + R^{\beta-2} \|v^l_n\|_{\dot{H}^2},
\end{align*}
where $\beta=\frac{d}{2}-\frac{d}{q} \in (\Gact,2)$. On the other hand, the H\"older interpolation inequality implies
\begin{align*}
\|\chi_R * v^l_n\|_{L^q} &\lesssim \|\chi_R * v^l_n\|^{\frac{\alphct}{q}}_{L^{\alphct}} \|\chi_R * v^l_n\|^{1-\frac{\alphct}{q}}_{L^\infty} \\
&\lesssim \|v^l_n\|^{\frac{\alphct}{q}}_{\dot{H}^{\Gact}} \|\chi_R * v^l_n\|^{1-\frac{\alphct}{q}}_{L^\infty}.
\end{align*}
Observe that
\[
\limsup_{n\rightarrow \infty} \|\chi_R * v^l_n\|_{L^\infty} = \sup_{x_n} \limsup_{n\rightarrow \infty} |\chi_R * v^l_n(x_n)|.
\]
Thus, by the definition of $\Omega(v^l_n)$, we infer that
\[
\limsup_{n\rightarrow \infty} \|\chi_R * v^l_n\|_{L^\infty} \leq \sup \Big\{ \Big| \int \chi_R(-x) v(x) dx\Big| : v \in \Omega(v^l_n)\Big\}.
\]
By the Plancherel formula, we have
\begin{align*}
\Big|\int \chi_R(-x) v(x) dx \Big| &= \Big| \int \hat{\chi}_R(\xi) \hat{v}(\xi) d\xi\Big| \lesssim \|\|\xi|^{-\Gact}\hat{\chi}_R\|_{L^2} \||\xi|^{\Gact}\hat{v}\|_{L^2} \\
&\lesssim R^{\frac{d}{2}-\Gact} \|\hat{\chi}_R\|_{\dot{H}^{-\Gact}} \|v\|_{\dot{H}^{\Gact}} \lesssim R^{\frac{4}{\alpha}} \eta(v^l_n).
\end{align*}
We thus obtain for every $l\geq 1$,
\begin{align*}
\limsup_{n\rightarrow \infty} \|v^l_n\|_{L^q} &\lesssim \limsup_{n\rightarrow \infty} \|(\delta-\chi_R)* v^l_n\|_{L^q} + \limsup_{n\rightarrow \infty} \|\chi_R * v^l_n\|_{L^q} \\
&\lesssim R^{\Gact-\beta} \|v^l_n\|_{\dot{H}^{\Gact}} + R^{\beta-2} \|v^l_n\|_{\dot{H}^2} + \|v^l_n\|^{\frac{\alphct}{q}}_{\dot{H}^{\Gact}} \left[R^{\frac{4}{\alpha}} \eta(v^l_n)\right]^{\left(1-\frac{\alphct}{q}\right)}.
\end{align*}
Choosing $R= \left[\eta(v^l_n)^{-1}\right]^{\frac{\alpha}{4}-\epsilon}$ for some $\epsilon>0$ small enough, we see that
\[
\limsup_{n\rightarrow \infty} \|v^l_n\|_{L^q} \lesssim \eta(v^l_n)^{(\beta-\Gact)\left(\frac{\alpha}{4}-\epsilon\right)} \|v^l_n\|_{\dot{H}^{\Gact}} + \eta(v^l_n)^{(2-\beta)\left(\frac{\alpha}{4}-\epsilon\right)} \|v^l_n\|_{\dot{H}^2} +\eta(v^l_n)^{\epsilon \frac{4}{\alpha} \left(1-\frac{\alphct}{q}\right)} \| v^l_n\|_{\dot{H}^{\Gact}}^{\frac{\alphct}{q}}.
\]
Letting $l \rightarrow \infty$ and using the fact that $\eta(v^l_n) \rightarrow 0$ as $l \rightarrow \infty$ and the uniform boundedness in $\dot{H}^{\Gact} \cap \dot{H}^2$ of $(v^l_n)_{l\geq 1}$, we obtain
\[
\limsup_{n \rightarrow \infty} \|v^l_n\|_{L^q} \rightarrow 0, \quad \text{as } l \rightarrow \infty.
\]
The proof is complete.
\defendproof
\section{Variational analysis} \label{section variational analysis}
\setcounter{equation}{0}
Let $d\geq 1$ and $2_\star<\alpha<2^\star$. We consider the variational problems
\begin{align*}
A_{\text{GN}}&:=\max\{H(f): f \in \dot{H}^{\Gact} \cap \dot{H}^2\}, & H(f)&:= \|f\|^{\alpha+2}_{L^{\alpha+2}} \div \left[ \|f\|^{\alpha}_{\dot{H}^{\Gact}} \|f\|^2_{\dot{H}^2} \right], \\
B_{\text{GN}}&:= \max\{K(f): f \in L^{\alphct}\cap \dot{H}^2 \}, & K(f)&:= \|f\|^{\alpha+2}_{L^{\alpha+2}} \div \left[ \|f\|^{\alpha}_{L^{\alphct}} \|f\|^2_{\dot{H}^2} \right].
\end{align*}
Here $A_{\text{GN}}$ and $B_{\text{GN}}$ are respectively sharp constants in the Gagliardo-Nirenberg inequalities
\begin{align*}
\|f\|^{\alpha+2}_{L^{\alpha+2}} &\leq A_{\text{GN}} \|f\|^{\alpha}_{\dot{H}^{\Gact}} \|f\|^2_{\dot{H}^2}, \\
\|f\|^{\alpha+2}_{L^{\alpha+2}} &\leq B_{\text{GN}} \|f\|^{\alpha}_{L^{\alphct}} \|f\|^2_{\dot{H}^2}.
\end{align*}
Let us start with the following observation.
\begin{lem} \label{lem maximizer intercritical}
If $g$ and $h$ are maximizers of $H(f)$ and $K(f)$ respectively, then $g$ and $h$ satisfy
\begin{align}
A_{\emph{GN}} \|g\|^{\alpha}_{\dot{H}^{\Gact}} \Delta^2 g + \frac{\alpha}{2} A_{\emph{GN}} \|g\|^{\alpha-2}_{\dot{H}^{\Gact}}\|g\|^2_{\dot{H}^2} (-\Delta)^{\Gact}g -\frac{\alpha+2}{2} |g|^{\alpha} g&=0, \label{maximizer equation intercritical 1} \\
B_{\emph{GN}} \|h\|^{\alpha}_{L^{\alphce}} \Delta^2 h + \frac{\alpha}{2} B_{\emph{GN}} \|h\|^{\alpha-\alphce}_{L^{\alphce}} \|h\|^2_{\dot{H}^2}|h|^{\alphce-2} h -\frac{\alpha+2}{2} |h|^{\alpha} h&=0, \label{maximizer equation intercritical 2}
\end{align}
respectively.
\end{lem}
\begin{proof}
If $g$ is a maximizer of $H$ in $\dot{H}^{\Gact} \cap \dot{H}^2$, then $g$ must satisfy the Euler-Lagrange equation
\[
\frac{d}{d\epsilon}\Big|_{\vert \epsilon=0} H(g+\epsilon \phi) =0,
\]
for all $\phi \in \mathcal{S}_0$. A direct computation shows
\begin{align*}
\left.\frac{d}{d\epsilon}\right|_{\epsilon=0} \|g+\epsilon \phi\|^{\alpha+2}_{L^{\alpha+2}} &= (\alpha+2) \int \re{(|g|^\alpha g \overline{\phi})} dx,\\
\left.\frac{d}{d\epsilon}\right|_{\epsilon=0} \|g+\epsilon \phi\|^\alpha_{\dot{H}^{\Gact}} &= \alpha \|g\|^{\alpha-2}_{\dot{H}^{\Gact}}\int \re{((-\Delta)^{\Gact} g \overline{\phi})}dx,
\end{align*}
and
\begin{align*}
\left.\frac{d}{d\epsilon}\right|_{\epsilon=0} \|g+\epsilon \phi\|^2_{\dot{H}^2} =2 \int \re{(\Delta^2 g \overline{\phi})} dx.
\end{align*}
We thus get
\begin{align*}
(\alpha+2) \|g\|^{\alpha}_{\dot{H}^{\Gact}} \|g\|^2_{\dot{H}^2} |g|^\alpha g - \alpha \|g\|^{\alpha+2}_{L^{\alpha+2}} \|g\|^{\alpha-2}_{\dot{H}^{\Gact}} \|g\|^2_{\dot{H}^2} (-\Delta)^{\Gact} g - 2 \|g\|^{\alpha+2}_{L^{\alpha+2}} \|g\|^\alpha_{\dot{H}^{\Gact}} \Delta^2 g =0.
\end{align*}
Dividing by $2\|g\|^{\alpha}_{\dot{H}^{\Gact}} \|g\|^2_{\dot{H}^2}$, we obtain $(\ref{maximizer equation intercritical 1})$. The proof of $(\ref{maximizer equation intercritical 2})$ is similar using the fact that
\[
\left.\frac{d}{d\epsilon}\right|_{\epsilon=0} \|h+\epsilon \phi\|^\alpha_{L^{\alphct}} = \alpha \|h\|^{\alpha-\alphct}_{L^{\alphct}} \int \re{(|h|^{\alphct-2} h \overline{\phi})}dx.
\]
The proof is complete.
\end{proof}
We next use the profile decomposition given in Theorem $\ref{theorem profile decomposition intercritical NL4S}$ to obtain the following variational structure of the sharp constants $A_{\text{GN}}$ and $B_{\text{GN}}$.
\begin{prop}[Variational structure of sharp constants] \label{prop variational structure ground state intercritical}
Let $d\geq 1$ and $2_\star<\alpha<2^\star$.
\begin{itemize}
\item The sharp constant $A_{\emph{GN}}$ is attained at a function $U \in \dot{H}^{\Gact} \cap \dot{H}^2$ of the form
\[
U(x) = a Q(\lambda x + x_0),
\]
for some $a \in \mathbb C^*, \lambda>0$ and $x_0 \in \mathbb R^d$, where $Q$ is a solution to the elliptic equation
\begin{align}
\Delta^2 Q + (-\Delta)^{\Gact} Q - |Q|^\alpha Q =0. \label{elliptic equation critical sobolev}
\end{align}
Moreover,
\[
A_{\emph{GN}} = \frac{\alpha+2}{2} \|Q\|^{-\alpha}_{\dot{H}^{\Gace}}.
\]
\item The sharp constant $B_{\emph{GN}}$ is attained at a function $V \in L^{\alphce} \cap \dot{H}^2$ of the form
\[
V(x) = b R(\mu x + y_0),
\]
for some $b \in \mathbb C^*, \mu>0$ and $y_0 \in \mathbb R^d$, where $R$ is a solution to the elliptic equation
\begin{align}
\Delta^2 R + |R|^{\alphce-2} R - |R|^\alpha R =0. \label{elliptic equation critical lebesgue}
\end{align}
Moreover,
\[
B_{\emph{GN}} = \frac{\alpha+2}{2} \|R\|^{-\alpha}_{L^{\alphce}}.
\]
\end{itemize}
\end{prop}
\begin{proof}
We only give the proof for $A_{\text{GN}}$, the one for $B_{\text{GN}}$ is treated similarly using the Sobolev embedding $\dot{H}^{\Gact} \hookrightarrow L^{\alphct}$. We firstly observe that $H$ is invariant under the scaling
\[
f_{\mu, \lambda}(x) := \mu f(\lambda x), \quad \mu, \lambda>0.
\]
Indeed, a simple computation shows
\[
\|f_{\mu, \lambda} \|^{\alpha+2}_{L^{\alpha+2}} = \mu^{\alpha+2} \lambda^{-d} \|f\|^{\alpha+2}_{L^{\alpha+2}}, \quad \|f_{\mu, \lambda} \|_{\dot{H}^{\Gact}}^\alpha = \mu^\alpha \lambda^{-4} \|f\|^\alpha_{\dot{H}^{\Gact}}, \quad \|f_{\mu, \lambda}\|^2_{\dot{H}^2} = \mu^2 \lambda^{4-d} \|f\|_{\dot{H}^2}^2.
\]
We thus get $H(f_{\mu, \lambda}) = H(f)$ for any $\mu, \lambda>0$. Moreover, if we set $g(x) = \mu f(\lambda x)$ with
\[
\mu = \left(\frac{\|f\|^{\frac{d}{2}-2}_{\dot{H}^{\Gact}}}{\|f\|_{\dot{H}^2}^{\frac{4}{\alpha}}}\right)^{\frac{1}{2-\Gact}}, \quad \lambda = \left(\frac{\|f\|_{\dot{H}^{\Gact}}}{\|f\|_{\dot{H}^2}}\right)^{\frac{1}{2-\Gact}},
\]
then $\|g\|_{\dot{H}^{\Gact}} = \|g\|_{\dot{H}^2} = 1$ and $H(g) = H(f)$. Now let $(v_n)_{n\geq 1}$ be the maximizing sequence such that $H(v_n) \rightarrow A_{\text{GN}}$ as $n\rightarrow \infty$. After scaling, we may assume that $\|v_n\|_{\dot{H}^{\Gact}} = \|v_n\|_{\dot{H}^2} =1$ and $H(v_n) = \|v_n\|^{\alpha+2}_{L^{\alpha+2}} \rightarrow A_{\text{GN}}$ as $n\rightarrow \infty$. Since $(v_n)_{n\geq 1}$ is bounded in $\dot{H}^{\Gact} \cap \dot{H}^2$, it follows from the profile decomposition given in Theorem $\ref{theorem profile decomposition intercritical NL4S}$ that there exist a sequence $(V^j)_{j\geq 1}$ of $\dot{H}^{\Gact} \cap \dot{H}^2$ functions and a family $(x^j_n)_{j\geq 1}$ of sequences in $\mathbb R^d$ such that up to a subsequence,
\[
v_n(x) = \sum_{j=1}^l V^j(x-x^j_n) + v^l_n(x),
\]
and $(\ref{profile error intercritical})$ and the identities $(\ref{profile identity 1 intercritical}), (\ref{profile identity 2 intercritical})$ hold. In particular, we have for any $l\geq 1$,
\begin{align}
\sum_{j=1}^l \|V^j\|^2_{\dot{H}^{\Gact}} \leq 1, \quad \sum_{j=1}^l \|V^j\|^2_{\dot{H}^2} \leq 1, \label{bounded sequence variational structure intercritical}
\end{align}
and
\[
\limsup_{n\rightarrow \infty} \|v^l_n\|^{\alpha+2}_{L^{\alpha+2}} \rightarrow 0, \quad \text{as } l \rightarrow \infty.
\]
We have
\begin{align}
A_{\text{GN}} &= \lim_{n\rightarrow \infty} \|v_n\|^{\alpha+2}_{L^{\alpha+2}} = \limsup_{n\rightarrow \infty} \Big\| \sum_{j=1}^l V^j(\cdot - x^j_n) + v^l_n \Big\|^{\alpha+2}_{L^{\alpha+2}} \nonumber \\
&\leq \limsup_{n\rightarrow \infty} \Big( \Big\| \sum_{j=1}^l V^j(\cdot - x^j_n) \Big\|_{L^{\alpha+2}} + \|v^l_n\|_{L^{\alpha+2}} \Big)^{\alpha+2} \nonumber \\
&\leq \limsup_{n\rightarrow \infty} \Big\| \sum_{j=1}^\infty V^j(\cdot - x^j_n) \Big\|_{L^{\alpha+2}}^{\alpha+2}. \label{sum intercritical}
\end{align}
By the elementary inequality
\begin{align}
\left| \Big| \sum_{j=1}^l a_j\Big|^{\alpha+2} - \sum_{j=1}^l |a_j|^{\alpha+2} \right| \leq C \sum_{j \ne k} |a_j| |a_k|^{\alpha+1}, \label{elementary inequality intercritical}
\end{align}
we have
\begin{align*}
\int \Big| \sum_{j=1}^l V^j(x -x^j_n)\Big|^{\alpha+2} dx &\leq \sum_{j=1}^l \int |V^j(x-x^j_n)|^{\alpha+2} dx + C \sum_{j\ne k} \int |V^j(x-x^j_n)||V^k(x-x^k_n)|^{\alpha+1} dx \\
&\leq \sum_{j=1}^l \int |V^j(x-x^j_n)|^{\alpha+2} dx + C \sum_{j \ne k} \int |V^j(x+ x^k_n-x^j_n)| |V^k(x)|^{\alpha+1} dx.
\end{align*}
Using the pairwise orthogonality $(\ref{pairwise orthogonality intercritical})$, the H\"older inequality implies that $V^j(\cdot + x^k_n-x^j_n) \rightharpoonup 0$ in $\dot{H}^{\Gact} \cap \dot{H}^2$ as $n \rightarrow \infty$ for any $j \ne k$. This leads to the mixed terms in the sum $(\ref{sum intercritical})$ vanish as $n\rightarrow \infty$. This shows that
\[
A_{\text{GN}} \leq \sum_{j=1}^\infty \|V^j\|^{\alpha+2}_{L^{\alpha+2}}.
\]
By the definition of $A_{\text{GN}}$, we have
\[
\frac{\|V^j\|_{L^{\alpha+2}}^{\alpha+2}}{A_{\text{GN}}} \leq \|V^j\|^{\alpha}_{\dot{H}^{\Gact}} \|V^j\|^2_{\dot{H}^2}.
\]
This implies that
\[
1 \leq \frac{\sum_{j=1}^\infty \|V^j\|_{L^{\alpha+2}}^{\alpha+2}}{A_{\text{GN}}} \leq \sup_{j\geq 1} \|V^j\|^{\alpha}_{\dot{H}^{\Gact}} \sum_{j=1}^\infty \|V^j\|^2_{\dot{H}^2}.
\]
Since $\sum_{j\geq 1} \|V^j\|^2_{\dot{H}^{\Gact}}$ is convergent, there exists $j_0 \geq 1$ such that
\[
\|V^{j_0}\|_{\dot{H}^{\Gact}} = \sup_{j\geq 1} \|V^j\|_{\dot{H}^{\Gact}}.
\]
By $(\ref{bounded sequence variational structure intercritical})$, we see that
\[
1 \leq \|V^{j_0}\|^{\alpha}_{\dot{H}^{\Gact}} \sum_{j=1}^\infty \|V^j\|_{\dot{H}^2}^2 \leq \|V^{j_0}\|_{\dot{H}^{\Gact}}^{\alpha}.
\]
It follows from $(\ref{bounded sequence variational structure intercritical})$ that $\|V^{j_0}\|_{\dot{H}^{\Gact}}=1$. This shows that there is only one term $V^{j_0}$ is non-zero, hence
\[
\|V^{j_0}\|_{\dot{H}^{\Gact}} = \|V^{j_0}\|_{\dot{H}^2} = 1, \quad \|V^{j_0}\|_{L^{\alpha+2}}^{\alpha+2} = A_{\text{GN}}.
\]
It means that $V^{j_0}$ is the maximizer of $H$ and Lemma $\ref{lem maximizer intercritical}$ shows that
\[
A_{\text{GN}} \Delta^2 V^{j_0} + \frac{\alpha}{2} A_{\text{GN}} (-\Delta)^{\Gact} V^{j_0} -\frac{\alpha+2}{2} |V^{j_0}|^{\alpha} V^{j_0}=0.
\]
Now if we set $V^{j_0}(x)= a Q(\lambda x+x_0)$ for some $a \in \mathbb C^*, \lambda>0$ and $x_0 \in \mathbb R^d$ to be chosen shortly, then $Q$ solves $(\ref{elliptic equation critical sobolev})$ provided that
\begin{align}
|a| = \left(\frac{2 \lambda^4 A_{\text{GN}}}{\alpha+2}\right)^{\frac{1}{\alpha}}, \quad \lambda = \Big(\frac{\alpha}{2}\Big)^{\frac{1}{2(2-\Gact)}}. \label{choice of a intercritical}
\end{align}
This shows the existence of solutions to the elliptic equation $(\ref{elliptic equation critical sobolev})$. We now compute the sharp constant $A_{\text{GN}}$ in terms of $Q$. We have
\[
1=\|V^{j_0}\|^\alpha_{\dot{H}^{\Gact}} = |a|^\alpha \lambda^{-4} \|Q\|^\alpha_{\dot{H}^{\Gact}} = \frac{2A_{\text{GN}}}{\alpha+2} \|Q\|^\alpha_{\dot{H}^{\Gact}}.
\]
This implies $A_{\text{GN}} = \frac{\alpha+2}{2} \|Q\|^{-\alpha}_{\dot{H}^{\Gact}}$. The proof is complete.
\end{proof}
\begin{rem} \label{rem variational analysis intercritical}
Using $(\ref{choice of a intercritical})$ and the fact
\begin{align*}
1 = \|V^{j_0}\|^\alpha_{\dot{H}^{\Gact}} &= |a|^\alpha \lambda^{-4} \|Q\|^\alpha_{\dot{H}^{\Gact}}, \\
1 = \|V^{j_0}\|^2_{\dot{H}^2} &= |a|^2 \lambda^{4-d} \|Q\|^2_{\dot{H}^2}, \\
A_{\text{GN}} =\|V^{j_0}\|^{\alpha+2}_{L^{\alpha+2}} &= |a|^{\alpha+2} \lambda^{-d}\|Q\|^{\alpha+2}_{L^{\alpha+2}},
\end{align*}
a direct computation shows the following Pohozaev identities
\begin{align}
\|Q\|^2_{\dot{H}^{\Gact}} = \frac{\alpha}{2} \|Q\|^2_{\dot{H}^2} = \frac{\alpha}{\alpha+2} \|Q\|^{\alpha+2}_{L^{\alpha+2}}. \label{pohozaev identities}
\end{align}
Another way to see above identities is to multiply $(\ref{elliptic equation critical sobolev})$ with $\overline{Q}$ and $x \cdot \nabla \overline{Q}$ and integrate over $\mathbb R^d$ and perform integration by parts. Indeed, multiplying $(\ref{elliptic equation critical sobolev})$ with $\overline{Q}$ and integrating by parts, we get
\begin{align}
\|Q\|^2_{\dot{H}^2} + \|Q\|^2_{\dot{H}^{\Gact}} - \|Q\|^{\alpha+2}_{L^{\alpha+2}} =0. \label{pohozaev equation 1}
\end{align}
Multiplying $(\ref{elliptic equation critical sobolev})$ with $x\cdot \nabla \overline{Q}$, integrating by parts and taking the real part, we have
\begin{align}
\Big(2-\frac{d}{2}\Big) \|Q\|^2_{\dot{H}^2} + \Big(\Gact-\frac{d}{2}\Big) \|Q\|^2_{\dot{H}^{\Gact}} + \frac{d}{\alpha+2} \|Q\|^{\alpha+2}_{L^{\alpha+2}} =0. \label{pohozaev equation 2}
\end{align}
From $(\ref{pohozaev equation 1})$ and $(\ref{pohozaev equation 2})$, we obtain $(\ref{pohozaev identities})$. To see $(\ref{pohozaev equation 2})$, we claim that for $\gamma \geq 0$,
\begin{align}
\re{\int (-\Delta)^\gamma Q x \cdot \nabla \overline{Q} dx } = \Big(\gamma-\frac{d}{2}\Big) \|Q\|^2_{\dot{H}^\gamma}. \label{claim}
\end{align}
In fact, by Fourier transform,
\begin{align}
\re{\int (-\Delta)^\gamma Q x \cdot \nabla \overline{Q} dx } &= \re{ \int \mathcal{F}[(-\Delta)^\gamma Q] \mathcal{F}^{-1}[x \cdot \nabla \overline{Q}] d\xi} \nonumber \\
&= \re{ \int \mathcal{F}[(-\Delta)^\gamma Q] \overline{\mathcal{F}[x \cdot \nabla Q]} d\xi} \nonumber \\
&= \re{ \int |\xi|^{2\gamma} \mathcal{F}(Q) \left(-d \overline{F(Q)} - \xi \cdot \nabla_\xi \overline{\mathcal{F}(Q)} \right) d\xi } \nonumber \\
&= -d \|Q\|^2_{\dot{H}^\gamma} - \re{ \int |\xi|^{2\gamma} \mathcal{F}(Q) \xi \cdot \nabla_\xi \overline{F(Q)} d\xi}. \label{pohozaev equation proof}
\end{align}
Here we use the fact that $\mathcal{F}(x_j \partial_{x_j} u) = i \partial_{\xi_j} \mathcal{F}(\partial_{x_j} u) = i \partial_{\xi_j} (i\xi_j \mathcal{F}(u)) = -\mathcal{F}(u) - \xi_j \partial_{\xi_j} \mathcal{F}(u)$. By integration by parts,
\begin{align*}
\re{ \int |\xi|^{2\gamma} \mathcal{F}(Q) \xi \cdot \nabla_\xi \overline{\mathcal{F}(Q)} d\xi} = (-2\gamma-d) \|Q\|^2_{\dot{H}^\gamma} - \re{ \int |\xi|^{2\gamma} \xi \cdot \nabla_\xi \mathcal{F}(Q) \overline{\mathcal{F}(Q)} d\xi},
\end{align*}
or
\[
\re{ \int |\xi|^{2\gamma} \mathcal{F}(Q) \xi \cdot \nabla_\xi \overline{\mathcal{F}(Q)} d\xi} = \Big(-\gamma -\frac{d}{2} \Big) \|Q\|^2_{\dot{H}^\gamma}.
\]
This together with $(\ref{pohozaev equation proof})$ shows $(\ref{claim})$, and $(\ref{pohozaev equation 2})$ follows.
\newline
\indent The Pohozaev identities $(\ref{pohozaev identities})$ imply in particular that
\[
H(Q)=\|Q\|^{\alpha+2}_{L^{\alpha+2}} \div \left[\|Q\|^\alpha_{\dot{H}^{\Gact}} \|Q\|^2_{\dot{H}^2} \right] = \frac{\alpha+2}{2} \|Q\|^{-\alpha}_{\dot{H}^{\Gact}} = A_{\text{GN}}, \quad E(Q)=0.
\]
Similarly, we have
\[
\|R\|^2_{L^{\alphct}} = \frac{\alpha}{2} \|R\|^2_{\dot{H}^2} = \frac{\alpha}{\alpha+2} \|R\|^{\alpha+2}_{L^{\alpha+2}}.
\]
In particular,
\[
K(R) = \|R\|^{\alpha+2}_{L^{\alpha+2}} \div \left[ \|R\|^\alpha_{L^{\alphct}} \|R\|^2_{\dot{H}^2} \right] = \frac{\alpha+2}{2} \|R\|^{-\alpha}_{L^{\alphct}} = B_{\text{GN}}, \quad E(R) = 0.
\]
\end{rem}
\begin{defi}[Ground state] \label{definition ground state}
\begin{itemize}
\item We call \textbf{Sobolev ground states} the maximizers of $H$ which are solutions to $(\ref{elliptic equation critical sobolev})$. We denote the set of Sobolev ground states by $\mathcal{G}$.
\item We call \textbf{Lebesgue ground states} the maximizers of $K$ which are solutions to $(\ref{elliptic equation critical lebesgue})$. We denote the set of Lebesgue ground states by $\mathcal{H}$.
\end{itemize}
\end{defi}
Note that by Lemma $\ref{lem maximizer intercritical}$, if $g, h$ are Sobolev and Lebesgue ground states respectively, then
\[
A_{\text{GN}} = \frac{\alpha+2}{2} \|g\|^{-\alpha}_{\dot{H}^{\Gact}}, \quad B_{\text{GN}} = \frac{\alpha+2}{2} \|h\|^{-\alpha}_{L^{\alphct}}.
\]
This implies that Sobolev ground states have the same $\dot{H}^{\Gact}$-norm, and all Lebesgue ground states have the same $L^{\alphct}$-norm. Denote
\begin{align}
S_{\text{gs}}&:= \|g\|_{\dot{H}^{\Gact}}, \quad \forall g \in \mathcal{G}, \label{critical sobolev norm} \\
L_{\text{gs}}&:= \|h\|_{L^{\alphct}}, \quad \forall h \in \mathcal{H}. \label{critical lebesgue norm}
\end{align}
In particular, we have the following sharp Gagliardo-Nirenberg inequalities
\begin{align}
\|f\|^{\alpha+2}_{L^{\alpha+2}} &\leq A_{\text{GN}} \|f\|^{\alpha}_{\dot{H}^{\Gact}} \|f\|^2_{\dot{H}^2}, \label{sharp gagliardo-nirenberg inequality intercritical 1}\\
\|f\|^{\alpha+2}_{L^{\alpha+2}} &\leq B_{\text{GN}} \|f\|^{\alpha}_{L^{\alphct}} \|f\|^2_{\dot{H}^2}, \label{sharp gagliardo-nirenberg inequality intercritical 2}
\end{align}
with
\[
A_{\text{GN}} = \frac{\alpha+2}{2} S_{\text{gs}}^{-\alpha}, \quad B_{\text{GN}} = \frac{\alpha+2}{2} L_{\text{gs}}^{-\alpha}.
\]
We next give another application of the profile decomposition given in Theorem $\ref{theorem profile decomposition intercritical NL4S}$.
\begin{theorem}[Compactness lemma] \label{theorem compactness lemma intercritical NL4S} Let $d\geq 1$ and $2_\star<\alpha<2^\star$. Let $(v_n)_{n\geq 1}$ be a bounded sequence in $\dot{H}^{\Gace} \cap \dot{H}^2$ such that
\[
\limsup_{n\rightarrow \infty} \|v_n\|_{\dot{H}^2} \leq M, \quad \limsup_{n\rightarrow \infty} \|v_n\|_{L^{\alpha+2}} \geq m.
\]
\begin{itemize}
\item Then there exists a sequence $(x_n)_{n\geq 1}$ in $\mathbb R^d$ such that up to a subsequence,
\[
v_n(\cdot + x_n) \rightharpoonup V \text{ weakly in } \dot{H}^{\Gace} \cap \dot{H}^2,
\]
for some $V \in \dot{H}^{\Gace} \cap \dot{H}^2$ satisfying
\begin{align}
\|V\|^\alpha_{\dot{H}^{\Gace}} \geq \frac{2}{\alpha+2} \frac{ m^{\alpha+2}}{M^2} S_{\emph{gs}}^\alpha. \label{lower bound critical sobolev}
\end{align}
\item Then there exists a sequence $(y_n)_{n\geq 1}$ in $\mathbb R^d$ such that up to a subsequence,
\[
v_n(\cdot + y_n) \rightharpoonup W \text{ weakly in } L^{\alphce} \cap \dot{H}^2,
\]
for some $W \in L^{\alphce} \cap \dot{H}^2$ satisfying
\begin{align}
\|W\|^\alpha_{L^{\alphce}} \geq \frac{2}{\alpha+2} \frac{ m^{\alpha+2}}{M^2} L_{\emph{gs}}^\alpha. \label{lower bound critical lebesgue}
\end{align}
\end{itemize}
\end{theorem}
\begin{rem} \label{rem compactness lemma intercritical NL4S}
The lower bounds $(\ref{lower bound critical sobolev})$ and $(\ref{lower bound critical lebesgue})$ are optimal. In fact, if we take $v_n=Q \in \mathcal{G}$ in the first case and $v_n=R \in \mathcal{H}$ in the second case, then we get the equalities.
\end{rem}
\noindent \textit{Proof of Theorem $\ref{theorem compactness lemma intercritical NL4S}$.} As in the proof of Proposition $\ref{prop variational structure ground state intercritical}$, we only consider the first case, the second case is similar using the Sobolev embedding $\dot{H}^{\Gact} \hookrightarrow L^{\alphct}$. According to Theorem $\ref{theorem profile decomposition intercritical NL4S}$, there exist a sequence $(V^j)_{j\geq 1}$ of $\dot{H}^{\Gact} \cap \dot{H}^2$ functions and a family $(x^j_n)_{j\geq 1}$ of sequences in $\mathbb R^d$ such that up to a subsequence, the sequence $(v_n)_{n\geq 1}$ can be written as
\[
v_n(x) = \sum_{j=1}^l V^j(x-x^j_n) + v^l_n(x),
\]
and $(\ref{profile error intercritical})$, $(\ref{profile identity 1 intercritical}), (\ref{profile identity 2 intercritical})$ hold. This implies that
\begin{align}
m^{\alpha+2} &\leq \limsup_{n\rightarrow \infty} \|v_n\|_{L^{\alpha+2}}^{\alpha+2} = \limsup_{n\rightarrow \infty} \Big\| \sum_{j=1}^l V^j(\cdot -x^j_n) + v^l_n\Big\|^{\alpha+2}_{L^{\alpha+2}} \nonumber \\
&\leq \limsup_{n\rightarrow \infty} \Big( \Big\|\sum_{j=1}^l V^j(\cdot -x^j_n) \Big\|_{L^{\alpha+2}} + \|v^l_n\|_{L^{\alpha+2}}\Big)^{\alpha+2} \nonumber \\
&\leq \limsup_{n\rightarrow \infty} \Big\| \sum_{j=1}^\infty V^j(\cdot -x^j_n)\Big\|_{L^{\alpha+2}}^{\alpha+2}. \label{compactness lemma proof intercritical}
\end{align}
By the elementary inequality $(\ref{elementary inequality intercritical})$ and the pairwise orthogonality $(\ref{pairwise orthogonality intercritical})$, the mixed terms in the sum $(\ref{compactness lemma proof intercritical})$ vanish as $n\rightarrow \infty$. We thus get
\[
m^{\alpha+2} \leq \sum_{j=1}^\infty \|V^j\|_{L^{\alpha+2}}^{\alpha+2}.
\]
We next use the sharp Gagliardo-Nirenberg inequality $(\ref{sharp gagliardo-nirenberg inequality intercritical 1})$ to estimate
\begin{align}
\sum_{j=1}^\infty \|V^j\|^{\alpha+2}_{L^{\alpha+2}} \leq \frac{\alpha+2}{2} \frac{1}{S_{\text{gs}}^\alpha} \sup_{j\geq 1} \|V^j\|^{\alpha}_{\dot{H}^{\Gact}} \sum_{j=1}^\infty \|V^j\|^2_{\dot{H}^2}. \label{compactness lemma proof 1 intercritical}
\end{align}
By $(\ref{profile identity 2 intercritical})$, we infer that
\[
\sum_{j=1}^\infty \|V^j\|^2_{\dot{H}^2} \leq \limsup_{n\rightarrow \infty} \|v_n\|^2_{\dot{H}^2} \leq M^2.
\]
Therefore,
\[
\sup_{j\geq 1} \|V^j\|^{\alpha}_{\dot{H}^{\Gact}} \geq \frac{2}{\alpha+2} \frac{m^{\alpha+2}}{M^2} S_{\text{gs}}^\alpha.
\]
Since the series $\sum_{j\geq 1} \|V^j\|^2_{\dot{H}^{\Gact}}$ is convergent, the supremum above is attained. In particular, there exists $j_0$ such that
\[
\|V^{j_0}\|^{\alpha}_{\dot{H}^{\Gact}} \geq \frac{2}{\alpha+2} \frac{m^{\alpha+2}}{M^2} S_{\text{gs}}^\alpha.
\]
By a change of variables, we write
\[
v_n(x+ x^{j_0}_n) = V^{j_0} (x) + \sum_{1\leq j \leq l \atop j \ne j_0} V^j(x+ x_n^{j_0} - x^j_n) + \tilde{v}^l_n(x),
\]
where $\tilde{v}^l_n(x):= v^l_n(x+x^{j_0}_n)$. The pairwise orthogonality of the family $(x_n^j)_{j\geq 1}$ implies
\[
V^j( \cdot +x^{j_0}_n -x^j_n) \rightharpoonup 0 \text{ weakly in } \dot{H}^{\Gact} \cap \dot{H}^2,
\]
as $n \rightarrow \infty$ for every $j \ne j_0$. We thus get
\begin{align}
v_n(\cdot + x^{j_0}_n) \rightharpoonup V^{j_0} + \tilde{v}^l, \quad \text{as } n \rightarrow \infty, \label{compactness lemma proof 2 intercritical}
\end{align}
where $\tilde{v}^l$ is the weak limit of $(\tilde{v}^l_n)_{n\geq 1}$. On the other hand,
\[
\|\tilde{v}^l\|_{L^{\alpha+2}} \leq \limsup_{n\rightarrow \infty} \|\tilde{v}^l_n\|_{L^{\alpha+2}} = \limsup_{n\rightarrow \infty} \|v^l_n\|_{L^{\alpha+2}} \rightarrow 0, \quad \text{as } l \rightarrow \infty.
\]
By the uniqueness of the weak limit $(\ref{compactness lemma proof 2 intercritical})$, we get $\tilde{v}^l=0$ for every $l \geq j_0$. Therefore, we obtain
\[
v_n(\cdot + x^{j_0}_n) \rightharpoonup V^{j_0}.
\]
The sequence $(x^{j_0}_n)_{n\geq 1}$ and the function $V^{j_0}$ now fulfill the conditions of Theorem $\ref{theorem compactness lemma intercritical NL4S}$. The proof is complete.
\defendproof
\section{Global existence and blowup} \label{section global existence blowup}
We firstly use the sharp Gagliardo-Nirenberg inequality $(\ref{sharp gagliardo-nirenberg inequality intercritical 1})$ to show the following global existence.
\begin{prop}[Global existence in $\dot{H}^{\Gace} \cap \dot{H}^2$] \label{prop global existence intercritical NL4S 1}
Let $d\geq 5$ and $2_\star<\alpha<2^\star$. Let $u_0 \in \dot{H}^{\Gact} \cap \dot{H}^2$ and the corresponding solution $u$ to $(\ref{intercritical NL4S})$ defined on the maximal time $[0,T)$. Assume that
\begin{align}
\sup_{t\in [0,T)} \|u(t)\|_{\dot{H}^{\Gact}} < S_{\emph{gs}}. \label{assumption boundedness H dot gamma}
\end{align}
Then $T=\infty$, i.e. the solution exists globally in time.
\end{prop}
\begin{proof}
By the sharp Gagliardo-Nirenberg inequality $(\ref{sharp gagliardo-nirenberg inequality intercritical 1})$, we bound
\begin{align*}
E(u(t)) &=\frac{1}{2} \|u(t)\|^2_{\dot{H}^2} -\frac{1}{\alpha+2} \|u(t)\|^{\alpha+2}_{L^{\alpha+2}} \\
&\geq \frac{1}{2} \left( 1- \Big( \frac{\|u(t)\|_{\dot{H}^{\Gact}}}{S_{\text{gs}}}\Big)^\alpha\right) \|u(t)\|^2_{\dot{H}^2}.
\end{align*}
Thanks to the conservation of energy and the assumption $(\ref{assumption boundedness H dot gamma})$, we obtain $\sup_{t\in [0,T)} \|u(t)\|_{\dot{H}^2} <\infty$. By the blowup alternative given in Proposition $\ref{prop local well-posedness H dot 2}$ and $(\ref{assumption boundedness H dot gamma})$, the solution exists globally in time. The proof is complete.
\end{proof}
We also have the following global well-posedness result.
\begin{prop} \label{prop global existence intercritical NL4S 2}
Let $d\geq 5$ and $2_\star<\alpha<2^\star$. Let $u_0 \in \dot{H}^{\Gact} \cap \dot{H}^2$ and the corresponding solution $u$ to $(\ref{intercritical NL4S})$ defined on the maximal time $[0,T)$. Assume that
\begin{align}
S_{\emph{gs}} \leq \sup_{t\in [0,T)} \|u(t)\|_{\dot{H}^{\Gact}} < \infty, \quad \sup_{t\in [0,T)} \|u(t)\|_{L^{\alphce}} < L_{\emph{gs}}. \label{assumption boundedness H alpha}
\end{align}
Then $T=\infty$, i.e. the solution exists globally in time.
\end{prop}
The proof is similar to the one of Proposition $\ref{prop global existence intercritical NL4S 1}$ by using the shap Gagliardo-Nirenberg inequality $(\ref{sharp gagliardo-nirenberg inequality intercritical 2})$. \newline
\indent We next recall blowup criteria for $H^2$ solutions to the equation $(\ref{intercritical NL4S})$ due to \cite{BoulengerLenzmann}.
\begin{prop} [Blowup in $H^2$ \cite{BoulengerLenzmann}] \label{prop blowup H2}
Let $d \geq 2$, $ 2_\star<\alpha <2^\star$, $\alpha \leq 8$ and $u_0 \in H^2$ be radial. Assume that
\[
E(u_0) M(u_0)^\sigma < E(Q) M(Q)^\sigma, \quad \|u_0\|_{\dot{H}^2} \|u_0\|^\sigma_{L^2} > \|Q\|_{\dot{H}^2} \|Q\|^\sigma_{L^2},
\]
where
\begin{align}
\sigma :=\frac{2-\Gace}{\Gact} = \frac{8-(d-4)\alpha}{d\alpha-8}. \label{define sigma}
\end{align}
Then the corresponding solution $u$ to $(\ref{intercritical NL4S})$ blows up in finite time.
\end{prop}
\begin{rem} \label{rem blowup H2}
\begin{itemize}
\item The restriction $\alpha \leq 8$ comes from the radial Sobolev embedding (or Strauss's inequality). An analogous restriction on $\alpha$ appears in the blowup of $H^1$ solutions for the nonlinear Schr\"odinger equation.
\item Note that if $E(u_0)<0$, then the assumption $E(u_0) M(u_0)^\sigma < E(Q) M(Q)^\sigma$ holds trivially.
\end{itemize}
\end{rem}
If we assume $u_0 \in \dot{H}^{\Gact} \cap \dot{H}^2$, then the above blowup criteria does not hold due to the lack of mass conservation. Nevertheless, we have the following blowup criteria for initial data in $\dot{H}^{\Gact} \cap \dot{H}^2$.
\begin{prop} [Blowup in $\dot{H}^{\Gace} \cap \dot{H}^2$] \label{prop blowup H dot 2}
Let $d\geq 5$, $2_\star<\alpha <2^\star$, $\alpha< 4$ and $u_0 \in\dot{H}^{\Gace} \cap \dot{H}^2$ be radial satisfying $E(u_0)<0$. Assume that the corresponding solution $u$ to $(\ref{intercritical NL4S})$ defined on a maximal interval $[0,T)$ satisfies
\begin{align}
\sup_{t\in [0,T)} \|u(t)\|_{\dot{H}^{\Gact}} <\infty. \label{uniform bounded assumption}
\end{align}
Then the solution $u$ to $(\ref{intercritical NL4S})$ blows up in finite time.
\end{prop}
\begin{proof}
Let $\theta: [0,\infty) \rightarrow [0,\infty)$ be a smooth function such that
\[
\theta(r) = \left\{
\begin{array}{cl}
r^2 &\text{if } r \leq 1, \\
0 &\text{if } r \geq 2,
\end{array}
\right.
\quad \text{and} \quad \theta''(r) \leq 2 \text{ for } r\geq 0.
\]
For $R>0$ given, we define the radial function $\varphi_R: \mathbb R^d \rightarrow \mathbb R$ by
\begin{align}
\varphi_R(x) = \varphi_R(r) := R^2 \theta(r/R), \quad |x|=r.
\end{align}
By definition, we have
\[
2-\varphi''_R(r) \geq 0, \quad 2-\frac{\varphi'_R(r)}{r} \geq 0, \quad 2d -\Delta \varphi_R(x) \geq 0, \quad \text{for all } r \geq 0 \text{ and all } x \in \mathbb R^d,
\]
and
\[
\|\nabla^j \varphi_R \|_{L^\infty} \lesssim R^{2-j}, \quad \text{for } j=0, \cdots, 6,
\]
and also,
\[
\text{supp}(\nabla^j \varphi_R) \subset \left\{
\begin{array}{cl}
\{ |x| \leq 2R \} &\text{for } j=1,2, \\
\{ R \leq |x| \leq 2R \} &\text{for } j=3, \cdots, 6.
\end{array}
\right.
\]
Let $u \in \dot{H}^{\Gact} \cap \dot{H}^2$ be a solution to $(\ref{intercritical NL4S})$. We define the localized virial action associated to $(\ref{intercritical NL4S})$ by
\begin{align}
M_{\varphi_R}(t) := 2 \int \nabla \varphi_R(x) \cdot \im{(\overline{u}(t, x) \nabla u(t,x))} dx. \label{virial action}
\end{align}
We firstly show that $M_{\varphi_R}(t)$ is well-defined. To do so, we need the following estimate
\begin{align}
\|u\|_{L^2(|x| \lesssim R)} \lesssim R^{\Gact} \|u\|_{L^{\alphct}(|x| \lesssim R)} \lesssim R^{\Gact} \|u\|_{\dot{H}^{\Gact}(|x| \lesssim R)}, \label{holder inequality}
\end{align}
which follows easily by H\"older's inequality and the Sobolev embedding. Here $\Gact$ and $\alphct$ are given in $(\ref{critical sobolev exponent})$ and $(\ref{critical lebesgue exponent})$ respectively. Since $\nabla \varphi_R$ is supported in $|x| \lesssim R$, the H\"older inequality together with $(\ref{holder inequality})$ imply
\begin{align*}
|M_{\varphi_R}(t)| &\lesssim \|\nabla \varphi_R\|_{L^\infty} \|u(t)\|_{L^2(|x| \lesssim R)} \|\nabla u(t)\|_{L^2(|x| \lesssim R)} \\
&\lesssim \|\nabla \varphi_R\|_{L^\infty} \|u(t)\|^{3/2}_{L^2(|x| \lesssim R)} \|\Delta u(t)\|^{1/2}_{L^2(|x| \lesssim R)} \\
&\lesssim R^{3\Gact/2} \|\nabla \varphi_R\|_{L^\infty} \|u(t)\|^{3/2}_{\dot{H}^{\Gact}(|x| \lesssim R)} \|u(t)\|^{1/2}_{\dot{H}^2(|x| \lesssim R)}.
\end{align*}
Note that in the case $\theta(r) = r^2$ and $\varphi_R(x)=|x|^2$, we have formally the virial law (see e.g. \cite{BoulengerLenzmann}):
\begin{align}
\begin{aligned}
M'_{|x|^2}(t)=\frac{d}{dt} \Big( 4 \int x \cdot \im{( \overline{u}(t,x) \nabla u(t,x))} dx \Big) &= 16 \|\Delta u(t)\|^2_{L^2} - \frac{4d\alpha}{\alpha+2} \|u(t)\|^{\alpha+2}_{L^{\alpha+2}} \\
&= 4d\alpha E(u(t)) -2(d\alpha-8) \|\Delta u(t)\|^2_{L^2}.
\end{aligned}
\label{virial law}
\end{align}
We have the following variation rate of the virial action (see e.g. \cite[Lemma 3.1]{BoulengerLenzmann} or \cite[Proposition 3.1]{MiaoWuZhang}):
\begin{align}
M'_{\varphi_R}(u(t)) &= \int \Delta^3 \varphi_R |u|^2 dx - 4 \sum_{j,k} \int \partial^2_{jk} \Delta \varphi_R \re{(\partial_j \overline{u} \partial_k u)} dx + 8 \sum_{j,k,l} \int \partial^2_{jk} \varphi_R \re{(\partial^2_{lj} \overline{u} \partial^2_{kl} u) } dx \nonumber \\
& \mathrel{\phantom{= \int \Delta^3 \varphi_R |u|^2 dx }} - 2 \int \Delta^2 \varphi_R |\nabla u|^2 dx -\frac{2\alpha}{\alpha+2} \int \Delta \varphi_R |u|^{\alpha+2} dx. \label{variation rate}
\end{align}
Since $\varphi_R(x) = |x|^2$ for $|x| \leq R$, we use $(\ref{virial law})$ to have
\begin{align*}
M'_{\varphi_R}(t)&= 16 \|\Delta u(t)\|^2_{L^2} -\frac{4d\alpha}{\alpha+2} \|u(t)\|^{\alpha+2}_{L^{\alpha+2}} - 16 \|\Delta u(t)\|^2_{L^2(|x| >R)} +\frac{4d\alpha}{\alpha+2}\|u(t)\|^{\alpha+2}_{L^{\alpha+2}(|x|>R)} \\
& \mathrel{\phantom{= 16 \|\Delta u(t)\|^2_{L^2}}} + \int_{|x|>R} \Delta^3 \varphi_R |u(t)|^2 dx - 4 \sum_{j,k} \int_{|x|>R} \partial^2_{jk} \Delta \varphi_R \re{(\partial_j \overline{u}(t) \partial_k u(t))} dx \\
& \mathrel{\phantom{= 16 \|\Delta u(t)\|^2_{L^2}}} + 8 \sum_{j,k,l} \int_{|x|>R} \partial^2_{jk} \varphi_R \re{(\partial^2_{lj} \overline{u}(t) \partial^2_{kl} u(t)) } dx \\
& \mathrel{\phantom{= 16 \|\Delta u(t)\|^2_{L^2} }} - 2 \int_{|x|>R} \Delta^2 \varphi_R |\nabla u(t)|^2 dx -\frac{2\alpha}{\alpha+2} \int_{|x|>R} \Delta \varphi_R |u(t)|^{\alpha+2} dx \\
&= 4d\alpha E(u(t)) -2(d\alpha-8) \|\Delta u(t)\|^2_{L^2} + \int_{|x|>R} \Delta^3 \varphi_R |u(t)|^2 dx \\
&\mathrel{\phantom{=}} - 4\sum_{j,k} \int_{|x|>R} \partial^2_{jk}\Delta \varphi_R \re{(\partial_j \overline{u}(t) \partial_k u(t) )} dx -2 \int_{|x|>R} \Delta^2 \varphi_R |\nabla u(t)|^2 dx
\end{align*}
\begin{align*}
\mathrel{\phantom{M'_{\varphi_R}(t) = }} &\mathrel{\phantom{=}} + 8 \sum_{j,k,l} \int_{|x|>R} \partial^2_{jk} \varphi_R \re{(\partial^2_{lj} \overline{u} (t) \partial^2_{kl} u(t) )} dx - 16 \|\Delta u(t)\|^2_{L^2(|x|>R)} \\
&\mathrel{\phantom{= + 8 \sum_{j,k,l} \int_{|x|>R} \partial^2_{jk} \varphi_R \re{(\partial^2_{lj} \overline{u} (t) \partial^2_{kl} u(t) )} dx}} + \frac{2\alpha}{\alpha+2} \int_{|x|>R} (2d-\Delta \varphi_R) |u(t)|^{\alpha+2} dx.
\end{align*}
By the choice of $\varphi_R$, the assumption $(\ref{uniform bounded assumption})$ and $(\ref{holder inequality})$, we bound
\begin{align*}
\Big|\int_{|x|>R} \Delta^3 \varphi_R |u(t)|^2 dx \Big| &\lesssim R^{-4} \|u(t)\|^2_{L^2(|x| \lesssim R)} \lesssim R^{-2(2-\Gact)} \|u(t)\|^2_{\dot{H}^{\Gact}} \lesssim R^{-2(2-\Gact)}, \\
\Big| \int_{|x|>R} \partial^2_{jk} \Delta \varphi_R \partial_j \overline{u} (t) \partial_k u (t) dx\Big| &\lesssim R^{-2} \|\nabla u\|^2_{L^2(|x| \lesssim R)} \lesssim R^{-(2-\Gact)} \|u(t)\|_{\dot{H}^{\Gact}} \|\Delta u(t)\|_{L^2} \\
&\mathrel{\phantom{\lesssim R^{-2} \|\nabla u\|^2_{L^2(|x| \lesssim R)} \ }} \lesssim R^{-(2-\Gact)} \|\Delta u(t)\|_{L^2}, \\
\Big| \int_{|x|>R} \Delta^2\varphi_R |\nabla u(t)|^2 dx \Big| &\lesssim R^{-(2-\Gact)} \|u(t)\|_{\dot{H}^{\Gact}} \|\Delta u(t)\|_{L^2} \lesssim R^{-(2-\Gact)} \|\Delta u(t)\|_{L^2}.
\end{align*}
Using the fact
\[
\partial^2_{jk} = \Big( \delta_{jk} - \frac{x_j x_k}{r^2} \Big) \frac{\partial_r}{r} + \frac{x_j x_k}{r^2} \partial^2_r,
\]
a calculation combined with integration by parts yields that
\begin{align*}
\sum_{j,k,l} \int \partial^2_{jk} \varphi_R \partial^2_{lj} \overline{u}(t) \partial^2_{kl} u(t) dx &= \int \varphi''_R |\partial^2_r u(t)|^2 + \frac{d-1}{r^2} \frac{\varphi'_R}{r} |\partial_r u(t)|^2 dx \\
&= 2\int |\Delta u(t)|^2 - (2-\varphi''_R) |\partial^2_r u(t)|^2 - \Big( 2-\frac{\varphi'_R}{r}\Big) \frac{d-1}{r^2} |\partial_r u(t)|^2 dx \\
& \leq 2\|\Delta u(t)\|^2_{L^2}.
\end{align*}
Here we use the identity
\[
\|\Delta u(t)\|^2_{L^2} = \int |\partial^2_r u(t)|^2 + \frac{d-1}{r^2} |\partial_r u(t)|^2 dx.
\]
Thus,
\[
8 \sum_{j,k,l} \int_{|x|>R} \partial^2_{jk} \varphi_R \re{(\partial^2_{lj} \overline{u} (t) \partial^2_{kl} u(t) )} dx - 16 \|\Delta u(t)\|^2_{L^2(|x|>R)} \leq 0.
\]
We obtain
\begin{align*}
M'_{\varphi_R}(t) &\leq 4d\alpha E(u(t)) -2(d\alpha-8) \|\Delta u(t)\|^2_{L^2} + O\Big( R^{-2(2-\Gact)} + R^{-(2-\Gact)} \|\Delta u(t)\|_{L^2} \Big) \\
&\mathrel{\phantom{\leq 4d\alpha E(u(t)) -2(d\alpha-8) \|\Delta u(t)\|^2_{L^2}}}+ \frac{2\alpha}{\alpha+2} \int_{|x|>R} (2d-\Delta \varphi_R) |u(t)|^{\alpha+2} dx.
\end{align*}
We now estimate the last term of the above inequality. To do so, we use the argument of \cite{MerleRaphael}. Consider for $A>0$ the annulus $\mathcal{C} = \{A < |x| \leq 2A\}$, we claim that for any $\epsilon>0$,
\begin{align}
\|u(t)\|^{\alpha+2}_{L^{\alpha+2}(\mathcal{C})} \leq \epsilon \|\Delta u(t)\|_{L^2(\mathcal{C})} + C(\epsilon)A^{-2(2-\Gact)}. \label{estimate on annulus}
\end{align}
To see this, we use the radial Sobolev embedding (see e.g. \cite{Strauss}) and $(\ref{holder inequality})$ to estimate
\begin{align*}
\|u(t)\|^{\alpha+2}_{L^{\alpha+2}(\mathcal{C})} &\lesssim \Big( \sup_{\mathcal{C}} |u(t,x)| \Big)^\alpha \|u(t)\|^2_{L^2(\mathcal{C})} \\
& \lesssim A^{-\frac{(d-1)\alpha}{2}} \|\nabla u(t)\|^{\frac{\alpha}{2}}_{L^2(\mathcal{C})} \|u(t)\|^{\frac{\alpha}{2}+2}_{L^2(\mathcal{C})} \\
&\lesssim A^{-\frac{(d-1)\alpha}{2}} \|\Delta u(t)\|^{\frac{\alpha}{4}}_{L^2(\mathcal{C})} \|u(t)\|^{\frac{3\alpha}{4}+2}_{L^2(\mathcal{C})} \\
&\lesssim A^{-\vartheta} \|\Delta u(t)\|^{\frac{\alpha}{4}}_{L^2(\mathcal{C})},
\end{align*}
where
\[
\vartheta = \frac{(d-1)\alpha}{2}- \left(\frac{3\alpha}{4}+2\right) \Gact= 2(2-\Gact)\frac{4-\alpha}{4} > 0.
\]
By the Young inequality, we have for any $\epsilon>0$,
\[
\|u(t)\|^{\alpha+2}_{L^{\alpha+2}(\mathcal{C})} \lesssim \epsilon \|\Delta u(t)\|_{L^2(\mathcal{C})} + \epsilon^{-\frac{\alpha}{4-\alpha}} A^{-\frac{4\vartheta}{4-\alpha}} = \epsilon \|\Delta u(t)\|_{L^2(\mathcal{C})} + C(\epsilon) A^{-2(2-\Gact)}.
\]
This shows the claim above. Note that the condition $\alpha<4$ is crucial to show $(\ref{estimate on annulus})$. We now write
\begin{align*}
\int_{|x|>R} |u(t)|^{\alpha+2} dx = \sum_{j=0}^\infty \int_{2^j R<|x| \leq 2^{j+1} R} |u(t)|^{\alpha+2} dx,
\end{align*}
and apply $(\ref{estimate on annulus})$ with $A=2^j R$ to get
\begin{align*}
\int_{|x|>R} |u(t)|^{\alpha+2} dx &\leq \epsilon \sum_{j=0}^\infty \|\Delta u(t)\|_{L^2(2^{j} R < |x| \leq 2^{j+1} R)} + C(\epsilon) \sum_{j=0}^\infty ( 2^j R)^{-2(2-\Gact)} \\
& \leq \epsilon \|\Delta u(t)\|_{L^2(|x|>R)} + C(\epsilon) R^{-2(2-\Gact)}.
\end{align*}
Since $\|2d-\varphi_R\|_{L^\infty} \lesssim 1$, we obtain for any $\epsilon>0$,
\[
\int_{|x|>R} (2d-\varphi_R) |u(t)|^{\alpha+2} dx \lesssim \epsilon \|\Delta u(t)\|_{L^2(|x|>R)} + C(\epsilon) R^{-2(2-\Gact)}.
\]
Therefore,
\begin{align}
M'_{\varphi_R}(t) \leq 4d \alpha E(u(t)) - 2(d\alpha-8) \|\Delta u(t)\|^2_{L^2} + O\Big(R^{-2(2-\Gact)} &+ R^{-(2-\Gact)} \|\Delta u(t)\|_{L^2} \label{estimate variation rate} \\
& + \epsilon \|\Delta u(t)\|_{L^2} + C(\epsilon) R^{-2(2-\Gact)} \Big). \nonumber
\end{align}
By taking $\epsilon>0$ small enough and $R>0$ large enough depending on $\epsilon$, the conservation of energy implies
\begin{align}
M'_{\varphi_R}(t) \leq 2d\alpha E(u_0) - \delta \|\Delta u(t)\|^2_{L^2}, \label{negative variation rate}
\end{align}
for all $t\in [0,T)$, where $\delta:= d\alpha-8>0$. With $(\ref{negative variation rate})$ at hand, the finite time blowup follows by a standard argument (see e.g. \cite{BoulengerLenzmann}).
\end{proof}
\section{Blowup concentration} \label{section blowup concentration}
\setcounter{equation}{0}
\begin{theorem} [Blowup concentration] \label{theorem H dot gamma concentration intercritical NL4S}
Let $d\geq 5$ and $2_\star <\alpha<2^\star$. Let $u_0 \in \dot{H}^{\Gact} \cap \dot{H}^2$ be such that the corresponding solution $u$ to $(\ref{intercritical NL4S})$ blows up at finite time $0<T<\infty$. Assume that the solution satisfies
\begin{align}
\sup_{t\in [0,T)} \|u(t)\|_{\dot{H}^{\Gact}} < \infty. \label{assumption blowup intercritical}
\end{align}
Let $a(t)>0$ be such that
\begin{align}
a(t) \|u(t)\|_{\dot{H}^2}^{\frac{1}{2-\Gace}} \rightarrow \infty, \label{condition of a intercritical}
\end{align}
as $t\uparrow T$. Then there exist $x(t), y(t) \in \mathbb R^d$ such that
\begin{align}
\liminf_{t\uparrow T} \int_{|x-x(t)| \leq a(t)} |(-\Delta)^{\frac{\Gace}{2}}u(t,x)|^2 dx \geq S_{\emph{gs}}^2, \label{H dot gamma concentration intercritical}
\end{align}
and
\begin{align}
\liminf_{t\uparrow T} \int_{|x-y(t)| \leq a(t)} |u(t,x)|^{\alphce} dx \geq L_{\emph{gs}}^2. \label{L alpha concentration intercritical}
\end{align}
\end{theorem}
\begin{rem} \label{rem H dot gamma concentration intercritical NL4S}
\begin{itemize}
\item The restriction $d\geq 5$ comes from the local well-posedness and blowup results. The result still holds true for dimensions $d \leq 4$ provided that one can show local well-posedness and blowup in such dimensions.
\item By the blowup rate given in Corollary $\ref{coro blowup rate intercritical}$ and the assumption $(\ref{assumption blowup intercritical})$, we have
\[
\|u(t)\|_{\dot{H}^2} > \frac{C}{(T-t)^{\frac{2-\Gact}{4}}},
\]
for $t\uparrow T$. Rewriting
\begin{align*}
\frac{1}{a(t) \|u(t)\|_{\dot{H}^2}^{\frac{1}{2-\Gace}}} = \frac{\sqrt[4]{T-t}}{a(t)} \frac{1}{\sqrt[4]{T-t}\|u(t)\|_{\dot{H}^2}^{\frac{1}{2-\Gace}}} &= \frac{\sqrt[4]{T-t}}{a(t)} \left(\frac{1}{(T-t)^{\frac{2-\Gact}{4}} \|u(t)\|_{\dot{H}^2} } \right)^{\frac{1}{2-\Gact}} \\
&<C\frac{\sqrt[4]{T-t}}{a(t)},
\end{align*}
we see that any function $a(t)>0$ satisfying $\frac{\sqrt[4]{T-t}}{a(t)} \rightarrow 0$ as $t\uparrow T$ fulfills the conditions of Theorem $\ref{theorem H dot gamma concentration intercritical NL4S}$.
\end{itemize}
\end{rem}
\noindent \textit{Proof of Theorem $\ref{theorem H dot gamma concentration intercritical NL4S}$.}
Let $(t_n)_{n\geq 1}$ be a sequence such that $t_n \uparrow T$ and $g\in \mathcal{G}$. Set
\[
\lambda_n := \left(\frac{\|g\|_{\dot{H}^2}}{\|u(t_n)\|_{\dot{H}^2}}\right)^{\frac{1}{2-\Gact}}, \quad v_n(x):= \lambda_n^{\frac{4}{\alpha}} u(t_n, \lambda_n x).
\]
By the blowup alternative and the assumption $(\ref{assumption blowup intercritical})$, we see that $\lambda_n \rightarrow 0$ as $n \rightarrow \infty$. Moreover, we have
\begin{align*}
\|v_n\|_{\dot{H}^{\Gact}} = \|u(t_n)\|_{\dot{H}^{\Gact}} <\infty,
\end{align*}
uniformly in $n$ and
\[
\|v_n\|_{\dot{H}^2} = \lambda_n^{2-\Gact}\|u(t_n)\|_{\dot{H}^2}= \|g\|_{\dot{H}^2},
\]
and
\[
E(v_n) = \lambda_n^{2(2-\Gact)} E(u(t_n)) = \lambda_n^{2(2-\Gact)} E(u_0) \rightarrow 0, \quad \text{as } n \rightarrow \infty.
\]
This implies in particular that
\begin{align*}
\|v_n\|^{\alpha+2}_{L^{\alpha+2}} \rightarrow \frac{\alpha+2}{2} \|g\|^2_{\dot{H}^2}, \quad \text{as } n \rightarrow \infty.
\end{align*}
The sequence $(v_n)_{n\geq 1}$ satisfies the conditions of Theorem $\ref{theorem compactness lemma intercritical NL4S}$ with
\[
m^{\alpha+2} = \frac{\alpha+2}{2} \|g\|^2_{\dot{H}^2}, \quad M^2 = \|g\|^2_{\dot{H}^2}.
\]
Therefore, there exists a sequence $(x_n)_{n\geq 1}$ in $\mathbb R^d$ such that up to a subsequence,
\[
v_n(\cdot + x_n) = \lambda_n^{\frac{4}{\alpha}} u(t_n, \lambda_n \cdot + x_n) \rightharpoonup V \text{ weakly in } \dot{H}^{\Gact} \cap \dot{H}^2,
\]
as $n \rightarrow \infty$ with $\|V\|_{\dot{H}^{\Gact}} \geq S_{\text{gs}}$. In particular,
\[
(-\Delta)^{\frac{\Gact}{2}} v(\cdot + x_n) = \lambda_n^{\frac{d}{2}} [(-\Delta)^{\frac{\Gact}{2}} u](t_n, \lambda_n \cdot + x_n) \rightharpoonup (-\Delta)^{\frac{\Gact}{2}} V \text{ weakly in } L^2.
\]
This implies for every $R>0$,
\[
\liminf_{n\rightarrow \infty} \int_{|x|\leq R} \lambda_n^{d}| [(-\Delta)^{\frac{\Gact}{2}} u](t_n, \lambda_n x + x_n)|^2 dx \geq \int_{|x|\leq R} |(-\Delta)^{\frac{\Gact}{2}} V(x)|^2 dx,
\]
or
\[
\liminf_{n\rightarrow \infty} \int_{|x-x_n|\leq R\lambda_n} | [(-\Delta)^{\frac{\Gact}{2}} u](t_n, x)|^2 dx \geq \int_{|x|\leq R} |(-\Delta)^{\frac{\Gact}{2}} V(x)|^2 dx.
\]
In view of the assumption $\frac{a(t_n)}{\lambda_n} \rightarrow \infty$ as $n\rightarrow \infty$, we get
\[
\liminf_{n\rightarrow \infty} \sup_{y\in \mathbb R^d} \int_{|x-y|\leq a(t_n)} |(-\Delta)^{\frac{\Gact}{2}}u(t_n, x)|^2 dx \geq \int_{|x|\leq R} |(-\Delta)^{\frac{\Gact}{2}}V(x)|^2 dx,
\]
for every $R>0$, which means that
\[
\liminf_{n\rightarrow \infty} \sup_{y \in \mathbb R^d} \int_{|x-y|\leq a(t_n)} |(-\Delta)^{\frac{\Gact}{2}}u(t_n, x)|^2 dx \geq \int |(-\Delta)^{\frac{\Gact}{2}}V(x)|^2 dx \geq S_{\text{gs}}^2.
\]
Since the sequence $(t_n)_{n\geq 1}$ is arbitrary, we infer that
\[
\liminf_{t\uparrow T} \sup_{y\in \mathbb R^d} \int_{|x-y|\leq a(t)} |(-\Delta)^{\frac{\Gact}{2}}u(t,x)|^2 dx \geq S_{\text{gs}}^2.
\]
But for every $t \in (0,T)$, the function $y\mapsto \int_{|x-y| \leq a(t)} |(-\Delta)^{\frac{\Gact}{2}}u(t,x)|^2 dx$ is continuous and goes to zero at infinity. As a result, we get
\[
\sup_{y\in \mathbb R^d} \int_{|x-y|\leq a(t)} |(-\Delta)^{\frac{\Gact}{2}}u(t,x)|^2 dx = \int_{|x-x(t)| \leq a(t)} |(-\Delta)^{\frac{\Gact}{2}}u(t,x)|^2 dx,
\]
for some $x(t) \in \mathbb R^d$. This shows $(\ref{H dot gamma concentration intercritical})$. The proof for $(\ref{L alpha concentration intercritical})$ is similar using Item 2 of Theorem $\ref{theorem compactness lemma intercritical NL4S}$. The proof is complete.
\defendproof
\section{Limiting profile with critical norms} \label{section limiting profile}
\setcounter{equation}{0}
Let us start with the following characterization of solution with critical norms.
\begin{lem} \label{lem characterization critical norm intercritical}
Let $d\geq 1$ and $2_\star<\alpha<2^\star$.
\begin{itemize}
\item If $u \in \dot{H}^{\Gact} \cap \dot{H}^2$ is such that $\|u\|_{\dot{H}^{\Gact}}=S_{\emph{gs}}$ and $E(u)=0$, then $u$ is of the form
\[
u(x) = e^{i\theta} \lambda^{\frac{4}{\alpha}} g(\lambda x + x_0),
\]
for some $g\in \mathcal{G}$, $\theta \in \mathbb R, \lambda>0$ and $x_0 \in \mathbb R^d$.
\item If $u \in L^{\alphce} \cap \dot{H}^2$ is such that $\|u\|_{L^{\alphce}}=L_{\emph{gs}}$ and $E(u)=0$, then $u$ is of the form
\[
u(x) = e^{i\vartheta} \mu^{\frac{4}{\alpha}} h(\mu x + y_0),
\]
for some $h\in \mathcal{H}$, $\vartheta \in \mathbb R, \mu>0$ and $y_0 \in \mathbb R^d$.
\end{itemize}
\end{lem}
\begin{proof}
We only prove Item 1, Item 2 is treated similarly. Since $E(u)=0$, we have
\[
\|u\|^2_{\dot{H}^2} = \frac{2}{\alpha+2} \|u\|^{\alpha+2}_{L^{\alpha+2}}.
\]
Thus
\[
H(u) = \frac{\|u\|^{\alpha+2}_{L^{\alpha+2}}}{\|u\|^{\alpha}_{\dot{H}^{\Gact}} \|u\|^2_{\dot{H}^2}} = \frac{\alpha+2}{2} \|u\|^{-\alpha}_{\dot{H}^{\Gact}} = \frac{\alpha+2}{2} S_{\text{gs}}^{-\alpha} = A_{\text{GN}}.
\]
This shows that $u$ is the maximizer of $H$. It follows from Proposition $\ref{prop variational structure ground state intercritical}$ that $u$ is of the form $u(x) = a g(\lambda x +x_0)$ for some $g\in \mathcal{G}$, $a \in \mathbb C^\star, \lambda>0$ and $x_0 \in \mathbb R^d$. On the other hand, since $\|u\|_{\dot{H}^{\Gact}} = S_{\text{gs}}=\|g\|_{\dot{H}^{\Gact}}$, we have $|a|= \lambda^{\frac{4}{\alpha}}$. This shows the result.
\end{proof}
We now have the following limiting profile of blowup solutions with critical norms.
\begin{theorem}[Limiting profile with critical norms] \label{theorem limiting profile critical norm intercritical}
Let $d\geq 5$ and $2_\star <\alpha<2^\star$. Let $u_0 \in \dot{H}^{\Gact} \cap \dot{H}^2$ be such that the corresponding solution $u$ to $(\ref{intercritical NL4S})$ blows up at finite time $0<T<\infty$.
\begin{itemize}
\item Assume that
\begin{align}
\sup_{t\in [0,T)} \|u(t)\|_{\dot{H}^{\Gact}} = S_{\emph{gs}}. \label{assumption critical sobolev norm}
\end{align}
Then there exist $g \in \mathcal{G}$, $\theta(t)\in \mathbb R$, $\lambda(t)>0$ and $x(t) \in \mathbb R^d$ such that
\[
e^{i\theta(t)} \lambda^{\frac{4}{\alpha}}(t) u(t, \lambda(t) \cdot + x(t)) \rightarrow g \text{ strongly in } \dot{H}^{\Gact} \cap \dot{H}^2 \text{ as } t \uparrow T.
\]
\item Assume that
\begin{align}
\sup_{t\in [0,T)} \|u(t)\|_{\dot{H}^{\Gact}} < \infty, \quad \sup_{t\in [0,T)} \|u(t)\|_{L^{\alphce}} = L_{\emph{gs}}. \label{assumption critical lebesgue norm}
\end{align}
Then there exist $h\in \mathcal{H}$, $\vartheta(t)\in \mathbb R$, $\mu(t)>0$ and $y(t) \in \mathbb R^d$ such that
\[
e^{i\vartheta(t)} \mu^{\frac{4}{\alpha}}(t) u(t, \mu(t) \cdot + y(t)) \rightarrow h \text{ strongly in } L^{\alphce} \cap \dot{H}^2 \text{ as } t \uparrow T.
\]
\end{itemize}
\end{theorem}
\begin{proof}
We only give the proof for the first case, the second case is similar. We will show that for any $(t_n)_{n\geq 1}$ satisfying $t_n \uparrow T$, there exist a subsequence still denoted by $(t_n)_{n\geq 1}$, $g\in \mathcal{G}$, sequences of $\theta_n \in \mathbb R, \lambda_n>0$ and $x_n \in \mathbb R^d$ such that
\begin{align}
e^{it\theta_n} \lambda^{\frac{4}{\alpha}}_n u(t_n, \lambda_n \cdot + x_n) \rightarrow g \text{ strongly in } \dot{H}^{\Gact} \cap \dot{H}^2 \text{ as } n \rightarrow \infty. \label{limiting profile critical norm proof intercritical}
\end{align}
Let $(t_n)_{n\geq 1}$ be a sequence such that $t_n \uparrow T$. Set
\[
\lambda_n := \left(\frac{\|Q\|_{\dot{H}^2}}{\|u(t_n)\|_{\dot{H}^2}}\right)^{\frac{1}{2-\Gact}}, \quad v_n(x):= \lambda_n^{\frac{4}{\alpha}} u(t_n, \lambda_n x),
\]
where $Q$ is as in Proposition $\ref{prop variational structure ground state intercritical}$. By the blowup alternative and $(\ref{assumption critical sobolev norm})$, we see that $\lambda_n \rightarrow 0$ as $n \rightarrow \infty$. Moreover, we have
\begin{align}
\|v_n\|_{\dot{H}^{\Gact}} = \|u(t_n)\|_{\dot{H}^{\Gact}} \leq S_{\text{gs}}=\|Q\|_{\dot{H}^{\Gact}}, \label{property v_n intercritical}
\end{align}
and
\begin{align}
\|v_n\|_{\dot{H}^2} = \lambda_n^{2-\Gact} \|u(t_n)\|_{\dot{H}^2} = \| Q\|_{\dot{H}^2}, \label{property v_n intercritical 1}
\end{align}
and
\[
E(v_n) = \lambda_n^{2(2-\Gact)} E(u(t_n)) = \lambda_n^{2(2-\Gact)} E(u_0) \rightarrow 0, \quad \text{as } n \rightarrow \infty.
\]
This yields in particular that
\begin{align}
\|v_n\|^{\alpha+2}_{L^{\alpha+2}} \rightarrow \frac{\alpha+2}{2} \|Q\|^2_{\dot{H}^2}, \quad \text{as } n \rightarrow \infty. \label{convergence v_n intercritical}
\end{align}
The sequence $(v_n)_{n\geq 1}$ satisfies the conditions of Theorem $\ref{theorem compactness lemma intercritical NL4S}$ with
\[
m^{\alpha+2} = \frac{\alpha+2}{2} \|Q\|^2_{\dot{H}^2}, \quad M^2 = \|Q\|^2_{\dot{H}^2}.
\]
Therefore, there exists a sequence $(x_n)_{n\geq 1}$ in $\mathbb R^d$ such that up to a subsequence,
\[
v_n(\cdot + x_n) = \lambda_n^{\frac{4}{\alpha}} u(t_n, \lambda_n \cdot + x_n) \rightharpoonup V \text{ weakly in } \dot{H}^{\Gact} \cap \dot{H}^2,
\]
as $n \rightarrow \infty$ with $\|V\|_{\dot{H}^{\Gact}} \geq S_{\text{gs}}$.
Since $v_n(\cdot +x_n) \rightharpoonup V$ weakly in $\dot{H}^{\Gact}\cap \dot{H}^2$ as $n\rightarrow \infty$, the semi-continuity of weak convergence and $(\ref{property v_n intercritical})$ imply
\[
\|V\|_{\dot{H}^{\Gact}} \leq \liminf_{n\rightarrow \infty} \|v_n\|_{\dot{H}^{\Gact}} \leq S_{\text{gs}}.
\]
This together with the fact $\|V\|_{\dot{H}^{\Gact}} \geq S_{\text{gs}}$ show that
\begin{align}
\|V\|_{\dot{H}^{\Gact}} = S_{\text{gs}} = \lim_{n\rightarrow \infty} \|v_n\|_{\dot{H}^{\Gact}}. \label{H dot gamma norm v_n intercritical}
\end{align}
Therefore
\[
v_n(\cdot +x_n) \rightarrow V \text{ strongly in } \dot{H}^{\Gact} \text{ as } n\rightarrow \infty.
\]
On the other hand, the Gagliardo-Nirenberg inequality $(\ref{sharp gagliardo-nirenberg inequality intercritical 1})$ shows that $v_n(\cdot +x_n) \rightarrow V$ strongly in $L^{\alpha+2}$ as $n \rightarrow \infty$. Indeed, by $(\ref{property v_n intercritical 1})$,
\begin{align*}
\|v_n(\cdot +x_n) -V\|^{\alpha+2}_{L^{\alpha+2}} &\lesssim \|v_n(\cdot + x_n) - V\|^{\alpha}_{\dot{H}^{\Gact}} \|v_n(\cdot + x_n) - V\|^2_{\dot{H}^2} \\
&\lesssim (\|Q\|_{\dot{H}^2} + \|V\|_{\dot{H}^2})^2 \|v_n(\cdot +x_n) -V\|^{\alpha}_{\dot{H}^{\Gact}} \rightarrow 0,
\end{align*}
as $n\rightarrow \infty$. Moreover, using $(\ref{convergence v_n intercritical})$ and $(\ref{H dot gamma norm v_n intercritical})$, the sharp Gagliardo-Nirenberg inequality $(\ref{sharp gagliardo-nirenberg inequality intercritical 1})$ yields
\begin{align*}
\|Q\|^2_{\dot{H}^2} = \frac{2}{\alpha+2} \lim_{n\rightarrow \infty} \|v_n\|^{\alpha+2}_{L^{\alpha+2}} = \frac{2}{\alpha+2} \|V\|^{\alpha+2}_{L^{\alpha+2}} \leq \Big(\frac{\|V\|_{\dot{H}^{\Gact}}}{S_{\text{gs}}} \Big)^{\alpha} \|V\|^2_{\dot{H}^2} = \|V\|^2_{\dot{H}^2},
\end{align*}
or $\|Q\|_{\dot{H}^2} \leq \|V\|_{\dot{H}^2}$. By the semi-continuity of weak convergence and $(\ref{property v_n intercritical 1})$,
\[
\|V\|_{\dot{H}^2} \leq \liminf_{n\rightarrow \infty} \|v_n\|_{\dot{H}^2} = \|Q\|_{\dot{H}^2}.
\]
Therefore,
\begin{align}
\|V\|_{\dot{H}^2} = \|Q\|_{\dot{H}^2} = \lim_{n\rightarrow \infty} \|v_n\|_{\dot{H}^2}.\label{H dot 1 norm v_n intercritical}
\end{align}
Combining $(\ref{H dot gamma norm v_n intercritical}), (\ref{H dot 1 norm v_n intercritical})$ and using the fact $v_n(\cdot + x_n) \rightharpoonup V$ weakly in $\dot{H}^{\Gact} \cap \dot{H}^2$, we conclude that
\[
v_n(\cdot + x_n) \rightarrow V \text{ strongly in } \dot{H}^{\Gact} \cap \dot{H}^2 \text{ as } n\rightarrow \infty.
\]
In particular, we have
\[
E(V) = \lim_{n\rightarrow \infty} E(v_n) =0.
\]
This shows that there exists $V \in \dot{H}^{\Gact} \cap \dot{H}^2$ such that
\[
\|V\|_{\dot{H}^{\Gact}} = S_{\text{gs}}, \quad E(V) =0.
\]
By Lemma $\ref{lem characterization critical norm intercritical}$, there exists $g\in \mathcal{G}$ such that $V(x) = e^{i\theta} \lambda^{\frac{4}{\alpha}} g(\lambda x +x_0)$ for some $\theta \in \mathbb R, \lambda>0$ and $x_0 \in \mathbb R^d$. Thus
\[
v_n(\cdot + x_n) = \lambda_n^{\frac{4}{\alpha}} u(t_n, \lambda_n \cdot + x_n) \rightarrow V = e^{i\theta} \lambda^{\frac{4}{\alpha}} g(\lambda \cdot +x_0) \text{ strongly in } \dot{H}^{\Gact} \cap \dot{H}^2 \text{ as } n \rightarrow \infty.
\]
Redefining variables as
\[
\overline{\lambda}_n:= \lambda_n \lambda^{-1}, \quad \overline{x}_n:= \lambda_n \lambda^{-1} x_0 +x_n,
\]
we get
\[
e^{-i\theta} \overline{\lambda}^{\frac{4}{\alpha}}_n u(t_n, \overline{\lambda}_n \cdot + \overline{x}_n) \rightarrow g \text{ strongly in } \dot{H}^{\Gact} \cap \dot{H}^2 \text{ as } n\rightarrow \infty.
\]
This proves $(\ref{limiting profile critical norm proof intercritical})$ and the proof is complete.
\end{proof}
\section*{Acknowledgments}
The author would like to express his deep thanks to his wife - Uyen Cong for her encouragement and support. He would like to thank his supervisor Prof. Jean-Marc Bouclet for the kind guidance and constant encouragement. He also would like to thank the reviewer for his/her helpful comments and suggestions.
|
2,869,038,156,419 | arxiv | \section{Introduction}\label{introduction}}
There is relatively little work on dealing with missing data in hidden
Markov models. \citet{albert2000transitional},
\citet{deltour1999stochastic}, and \citet{yeh2010estimating} consider
missing data in observed Markov chains. \citet{paroli2002parameter}
consider calculation of the likelihood of a Gaussian hidden Markov model
when observations are missing at random. \citet{yeh2012intermittent}
discuss the impact of ignoring missingness when missing data is, and is
not, ignorable. They show that if missingness depends on the hidden
states, i.e.~missingness is state-dependent, this results in biased
parameter estimates when this missingness is ignored. However, they
offer no solution to this problem. The objective of this paper is to do
so. Our approach is related to the work of \citet{yu2003semimarkov} who
allowed for state-dependent missingness in a hidden semi-Markov model
with discrete (categorical) outcomes. Following \citet{bahl1983maximum},
their solution is to code missingness into a special ``null value'' of
the observed variable, effectively making the variable fully observed.
Here, we instead model missingness with an additional (fully observed)
indicator variable. This, we believe, is conceptually simpler, and makes
it straightforward to add additional covariates to model the probability
of missing values.
The remainder of this paper is organized as follows: We start with a
brief overview of hidden Markov models and the formal treatment of
ignorable and non-ignorable missing data as established by
\citet{rubin1976inference} and \citet{little2014statistical}, with a
focus on hidden Markov models. We then consider state-dependent
missingness in hidden Markov models, and show in simulation studies how
including a submodel for state-dependent missingness provides better
estimates of the model parameters. When data is in fact missing at
random, the model with state-dependent missingness is not fundamentally
biased, although care must be taken to include relevant covariates, such
as e.g.~time. We conclude with an application of the method to a real
dataset, involving severity of schizophrenic symptoms in a clinical
trial.
\hypertarget{hidden-markov-models}{%
\subsection{Hidden Markov models}\label{hidden-markov-models}}
Let \(\ensuremath{Y}_{1:\ensuremath{T}} = (\ensuremath{Y}_1,\ldots,\ensuremath{Y}_\ensuremath{T})\) denote a time series of
(possibly multivariate) observations, and let \(\ensuremath{\mbox{\boldmath$\theta$}}\) denote a
vector of parameters. A hidden Markov model associates observations with
a time series of hidden (or latent) discrete states
\(\ensuremath{S}_{1:\ensuremath{T}} = (\ensuremath{S}_1,\ldots,\ensuremath{S}_\ensuremath{T})\). It is assumed that
each state \(\ensuremath{S}_t \in \{1, \ldots, \ensuremath{K}\}\) depends only on the
immediately preceding state \(\ensuremath{S}_{1-t}\), and that, conditional upon
the hidden states, the observations \(\ensuremath{Y}_t\) are independent:
\begin{align}
\ensuremath{p}(\ensuremath{S}_{t}|\ensuremath{S}_{1:t-1},\ensuremath{\mbox{\boldmath$\theta$}}) &=
\ensuremath{p}(\ensuremath{S}_{t}|\ensuremath{S}_{t-1},\ensuremath{\mbox{\boldmath$\theta$}}), \quad t=2, 3, \ldots, \ensuremath{T}
\label{eq:hmmstates}
\\ \ensuremath{p}(\ensuremath{Y}_{t}|\ensuremath{S}_{1:t-1}, \ensuremath{Y}_{1:t-1} ,\ensuremath{\mbox{\boldmath$\theta$}}) &=
\ensuremath{p}(\ensuremath{Y}_{t}|\ensuremath{S}_{t}, \ensuremath{\mbox{\boldmath$\theta$}}), \quad t=1,2, \ldots, \ensuremath{T} .
\label{eq:hmmresponses}
\end{align} Making use of these conditional independencies, the joint
distribution of observations and states can be stated as
\begin{equation}
\ensuremath{p}(\ensuremath{Y}_{1:\ensuremath{T}}, \ensuremath{S}_{1:\ensuremath{T}}|\ensuremath{\mbox{\boldmath$\theta$}}) = \ensuremath{p}(\ensuremath{S}_1|\ensuremath{\mbox{\boldmath$\theta$}}) \ensuremath{p}(\ensuremath{Y}_1 | \ensuremath{S}_1, \ensuremath{\mbox{\boldmath$\theta$}}) \prod_{t=2}^{\ensuremath{T}}
\ensuremath{p}(\ensuremath{S}_t|\ensuremath{S}_{t-1},\ensuremath{\mbox{\boldmath$\theta$}}) \ensuremath{p}(\ensuremath{Y}_t|\ensuremath{S}_t,\ensuremath{\mbox{\boldmath$\theta$}}) .
\label{eq:gHMM_joint}
\end{equation} The likelihood function (i.e.~the marginal distribution
of the observations as a function of the model parameters) can then be
written as \begin{equation}
\label{eq:marginal_observation_dist}
L(\ensuremath{\mbox{\boldmath$\theta$}}|\ensuremath{Y}_{1:T}) =
\sum_{\ensuremath{s}_{1:\ensuremath{T}} \in \mathcal{S}^T} \ensuremath{p}(\ensuremath{Y}_{1:\ensuremath{T}}, \ensuremath{S}_{1:\ensuremath{T}} = \ensuremath{s}_{1:\ensuremath{T}} |\ensuremath{\mbox{\boldmath$\theta$}}) ,
\end{equation} where the summation is over all possible state sequences
(i.e.~\(\mathcal{S}^T\) is the set of all possible sequences of states).
Rather than actually summing over all possible state sequences, the
forward-backward algorithm \citep{Rabiner1989} is used to efficiently
calculate this likelihood. For more information on hidden Markov models,
see also \citet{visser2021hidden}.
\hypertarget{missing-data}{%
\subsection{Missing data}\label{missing-data}}
The canonical references for statistical inference with missing data are
\citet{rubin1976inference} and \citet{little2014statistical}. Here we
summarise the main ideas and results from those sources, as relevant to
the present topic.
Let \(\ensuremath{Y}_{1:\ensuremath{T}}\), the sequence of all response variables, be
partitioned into a set of observed values,
\(\mathcal{\ensuremath{Y}}_{\text{obs}} \subseteq \ensuremath{Y}_{1:\ensuremath{T}}\), and a set of
missing values, \(\mathcal{\ensuremath{Y}}_{\text{miss}} \subseteq \ensuremath{Y}_{1:\ensuremath{T}}\).
Let \(M_{1:\ensuremath{T}}\) be vector of indicator variables with values
\(M_t = 1\) if \(\ensuremath{Y}_t \in \mathcal{\ensuremath{Y}}_{\text{miss}}\) (the
observation at time \(t\) is missing), and \(M_t = 0\) otherwise. In
addition to \(\ensuremath{\mbox{\boldmath$\theta$}}\), the parameters of the hidden Markov model for
the observed data \(\ensuremath{Y}\), let \(\gvc{\phi}\) denote the parameter
vector of the statistical model of missingness (i.e.~the model of
\(M_{1:\ensuremath{T}}\)).
We can define the ``full'' likelihood function as \begin{equation}
\label{eq:joint_missing_likelihood}
L_\text{full}{(\ensuremath{\mbox{\boldmath$\theta$}},\gvc{\phi}|\mathcal{\ensuremath{Y}}_{\text{obs}},M_{1:\ensuremath{T}})} \propto \int \ensuremath{p}{(\mathcal{\ensuremath{Y}}_{\text{obs}},\mathcal{\ensuremath{Y}}_{\text{miss}}|\ensuremath{\mbox{\boldmath$\theta$}})} \ensuremath{p}{(M_{1:\ensuremath{T}}|\mathcal{\ensuremath{Y}}_{\text{obs}},\mathcal{\ensuremath{Y}}_{\text{miss}},\gvc{\phi})} d \mathcal{\ensuremath{Y}}_\text{miss} ,
\end{equation} that is, as any function proportional to
\(\ensuremath{p}(\mathcal{\ensuremath{Y}}_{\text{obs}},M_{1:\ensuremath{T}}|\ensuremath{\mbox{\boldmath$\theta$}},\gvc{\phi})\).
Note that this is a marginal density, hence the integration over all
possible values of the missing data. In this general case, we allow
missingness to depend on the ``complete'' data \(\ensuremath{Y}_{1:\ensuremath{T}}\), so
including the missing values \(\mathcal{Y}_\text{miss}\) (for instance,
it might be the case that missing values occur when the true value of
\(\ensuremath{Y}_t\) is relatively high).
Ignoring the missing data, the likelihood can be defined as
\begin{equation}
\label{eq:ignoring_missing_likelihood}
L_\text{ign}(\ensuremath{\mbox{\boldmath$\theta$}}|\mathcal{\ensuremath{Y}}_{\text{obs}}) \propto \ensuremath{p}(\mathcal{\ensuremath{Y}}_{\text{obs}}|\ensuremath{\mbox{\boldmath$\theta$}}) ,
\end{equation} that is, as any function proportional to
\(\ensuremath{p}(\mathcal{\ensuremath{Y}}_\text{obs}|\ensuremath{\mbox{\boldmath$\theta$}})\). An important question is
when inference for \(\ensuremath{\mbox{\boldmath$\theta$}}\) based on
(\ref{eq:joint_missing_likelihood}) and
(\ref{eq:ignoring_missing_likelihood}) give the same results. Note that
both likelihood functions need only be known up to a constant of
proportionality as only relative likelihoods need to be known for
maximizing the likelihood or computing likelihood ratio's. The question
is thus when (\ref{eq:ignoring_missing_likelihood}) is proportional to
(\ref{eq:joint_missing_likelihood}).
As shown by \citet{rubin1976inference}, inference on \(\ensuremath{\mbox{\boldmath$\theta$}}\) based
on (\ref{eq:joint_missing_likelihood}) and
(\ref{eq:ignoring_missing_likelihood}) will give identical results when
(1) \(\ensuremath{\mbox{\boldmath$\theta$}}\) and \(\gvc{\phi}\) are separable (i.e.~the joint
parameter space is the product of the parameter space for \(\ensuremath{\mbox{\boldmath$\theta$}}\)
and \(\gvc{\phi}\)), and (2) the following holds: \begin{equation}
\label{eq:MAR_definition}
\ensuremath{p}(M_{1:\ensuremath{T}}|\mathcal{\ensuremath{Y}}_{\text{obs}},\mathcal{\ensuremath{Y}}_{\text{miss}},\gvc{\phi}) = \ensuremath{p}(M_{1:\ensuremath{T}}|\mathcal{\ensuremath{Y}}_{\text{obs}},\gvc{\phi}) \quad \quad \text{for all } \mathcal{\ensuremath{Y}}_{\text{miss}}, \gvc{\phi},
\end{equation} i.e.~whether data is missing does not depend on the
missing values. In this case, data is said to be missing at random
(MAR), and the joint density can be factored as \begin{align*}
\ensuremath{p}{(\mathcal{\ensuremath{Y}}_{\text{obs}},M_{1:\ensuremath{T}}|\ensuremath{\mbox{\boldmath$\theta$}},\gvc{\phi})} &= \ensuremath{p}{(M_{1:\ensuremath{T}}|\mathcal{\ensuremath{Y}}_{\text{obs}},\gvc{\phi})} \times \int \ensuremath{p}{(\mathcal{\ensuremath{Y}}_{\text{obs}},\mathcal{\ensuremath{Y}}_{\text{miss}}|\ensuremath{\mbox{\boldmath$\theta$}})} d \mathcal{\ensuremath{Y}}_\text{miss} \\
&= \ensuremath{p}{(M_{1:\ensuremath{T}}|\mathcal{\ensuremath{Y}}_{\text{obs}},\gvc{\phi})} \times \ensuremath{p}(\mathcal{\ensuremath{Y}}_{\text{obs}}|\ensuremath{\mbox{\boldmath$\theta$}}) ,
\end{align*} which indicates that, as a function of \(\ensuremath{\mbox{\boldmath$\theta$}}\),
\(L_\text{full}(\ensuremath{\mbox{\boldmath$\theta$}},\gvc{\phi}|\mathcal{\ensuremath{Y}}_\text{obs},M_{1:\ensuremath{T}}) \propto L_\text{ign}(\ensuremath{\mbox{\boldmath$\theta$}}|\mathcal{\ensuremath{Y}}_\text{obs})\).
Hence, when data is MAR, the missing data, and the mechanism leading to
it, can be ignored in inference for \(\ensuremath{\mbox{\boldmath$\theta$}}\). A special case of MAR
is data which is ``missing completely at random'' (MCAR), where
\begin{equation}
\label{eq:MCAR_definition}
\ensuremath{p}(M_{1:\ensuremath{T}}|\mathcal{\ensuremath{Y}}_{\text{obs}},\mathcal{\ensuremath{Y}}_{\text{miss}},\gvc{\phi}) = \ensuremath{p}(M_{1:\ensuremath{T}}|\gvc{\phi}) .
\end{equation}
When the equality in (\ref{eq:MAR_definition}) does not hold, data is
said to be missing not at random (MNAR). In this case, ignoring the
missing data will generally lead to biased parameter estimates of
\(\theta\). Valid inference of \(\theta\) requires working with the full
likelihood function of (\ref{eq:joint_missing_likelihood}), so
explicitly accounting for missingness.
\hypertarget{missing-data-in-hidden-markov-models}{%
\subsection{Missing data in hidden Markov
models}\label{missing-data-in-hidden-markov-models}}
Hidden Markov models by definition include missing data, as the hidden
states are unobservable (i.e.~always missing). When there are no missing
values for the observed variable \(\ensuremath{Y}\), it is easy to see that
inference on \(\ensuremath{\mbox{\boldmath$\theta$}}\) in HMMs targets the correct likelihood.
Replacing \(\mathcal{\ensuremath{Y}}_{\text{miss}}\) by \(\ensuremath{S}_{1:\ensuremath{T}}\), and
noting that
\(\ensuremath{p}(M_{1:\ensuremath{T}}|\mathcal{\ensuremath{Y}}_{\text{obs}},\ensuremath{S}_{1:\ensuremath{T}}) = \ensuremath{p}(M_{1:\ensuremath{T}}) = 1\),
the missing states can be considered missing completely at random
(MCAR).
We will now focus on the case where the observable response variable
\(\ensuremath{Y}\) does have missing values. The full likelihood, which also
depends on the hidden states, can be defined as \begin{equation}
L_\text{full}(\ensuremath{\mbox{\boldmath$\theta$}},\gvc{\phi}|\mathcal{\ensuremath{Y}}_{\text{obs}},M_{1:\ensuremath{T}}) \propto \sum_{\ensuremath{s}_{1:\ensuremath{T}} \in \mathcal{S}^\ensuremath{T}} \int \ensuremath{p}{(\mathcal{\ensuremath{Y}}_{\text{obs}},\mathcal{\ensuremath{Y}}_{\text{miss}}, \ensuremath{s}_{1:\ensuremath{T}} |\ensuremath{\mbox{\boldmath$\theta$}})} \ensuremath{p}{(M_{1:\ensuremath{T}}|\mathcal{\ensuremath{Y}}_{\text{obs}},\mathcal{\ensuremath{Y}}_{\text{miss}},\ensuremath{s}_{1:\ensuremath{T}},\gvc{\phi})} d \mathcal{\ensuremath{Y}}_\text{miss} ,
\end{equation} while the likelihood ignoring missing data as
\begin{equation}
L_\text{ign}(\ensuremath{\mbox{\boldmath$\theta$}}|\mathcal{\ensuremath{Y}}_{\text{obs}}) \propto \sum_{\ensuremath{s}_{1:\ensuremath{T}} \in \mathcal{S}^\ensuremath{T}} \ensuremath{p}{(\mathcal{\ensuremath{Y}}_{\text{obs}}, \ensuremath{s}_{1:\ensuremath{T}} |\ensuremath{\mbox{\boldmath$\theta$}})} .
\end{equation}
\hypertarget{missing-at-random-mar}{%
\subsubsection{Missing at random (MAR)}\label{missing-at-random-mar}}
When the data is missing at random (\ref{eq:MAR_definition}), then
\begin{align}
\notag L_\text{full}(\ensuremath{\mbox{\boldmath$\theta$}},\gvc{\phi}|\mathcal{\ensuremath{Y}}_{\text{obs}},M_{1:\ensuremath{T}}) &\propto
\sum_{\ensuremath{s}_{1:\ensuremath{T}} \in \mathcal{S}^\ensuremath{T}} \int \ensuremath{p}{(\mathcal{\ensuremath{Y}}_{\text{obs}},\mathcal{\ensuremath{Y}}_{\text{miss}}, \ensuremath{s}_{1:\ensuremath{T}} |\ensuremath{\mbox{\boldmath$\theta$}})} \ensuremath{p}{(M_{1:\ensuremath{T}}|\mathcal{\ensuremath{Y}}_{\text{obs}},\gvc{\phi})} d \mathcal{\ensuremath{Y}}_\text{miss} \\
\label{eq:HMM_mss_lik} &= \ensuremath{p}{(M_{1:\ensuremath{T}}|\mathcal{\ensuremath{Y}}_{\text{obs}}, \gvc{\phi})} \times \left( \sum_{\ensuremath{s}_{1:\ensuremath{T}} \in \mathcal{S}^\ensuremath{T}} \int \ensuremath{p}{(\mathcal{\ensuremath{Y}}_{\text{obs}},\mathcal{\ensuremath{Y}}_{\text{miss}}, \ensuremath{s}_{1:\ensuremath{T}} |\ensuremath{\mbox{\boldmath$\theta$}})} d \mathcal{\ensuremath{Y}}_\text{miss} \right)
\end{align} and hence missingness is ignorable in inference of
\(\ensuremath{\mbox{\boldmath$\theta$}}\). Furthermore, defining \begin{align}
\notag \ensuremath{p}^*(\ensuremath{Y}_t|\ensuremath{S}_t,\ensuremath{\mbox{\boldmath$\theta$}}) &= \mathbb{I}_{\ensuremath{Y}_t \in \mathcal{\ensuremath{Y}}_{\text{obs}}} \ensuremath{p}(\ensuremath{Y}_t | \ensuremath{S}_t, \ensuremath{\mbox{\boldmath$\theta$}}) + \mathbb{I}_{\ensuremath{Y}_t \in \mathcal{\ensuremath{Y}}_{\text{miss}}} \int \ensuremath{p}(\ensuremath{Y}_t | \ensuremath{S}_t, \ensuremath{\mbox{\boldmath$\theta$}}) d \ensuremath{Y}_t \\
&= \mathbb{I}_{\ensuremath{Y}_t \in \mathcal{\ensuremath{Y}}_{\text{obs}}} \ensuremath{p}(\ensuremath{Y}_t | \ensuremath{S}_t, \ensuremath{\mbox{\boldmath$\theta$}}) + \mathbb{I}_{\ensuremath{Y}_t \in \mathcal{\ensuremath{Y}}_{\text{miss}}} \times 1 ,
\end{align} where the indicator variable \(\mathbb{I}_x = 1\) if
condition \(x\) is true and 0 otherwise, we can write the part of the
full likelihood (\ref{eq:HMM_mss_lik}) relevant to inference on
\(\ensuremath{\mbox{\boldmath$\theta$}}\) as \begin{align*}
\sum_{\ensuremath{s}_{1:\ensuremath{T}} \in \mathcal{S}^\ensuremath{T}} \int \ensuremath{p}( \mathcal{\ensuremath{Y}}_{\text{obs}},\mathcal{\ensuremath{Y}}_{\text{miss}}, \ensuremath{s}_{1:\ensuremath{T}} | \ensuremath{\mbox{\boldmath$\theta$}}) d \mathcal{\ensuremath{Y}}_{\text{miss}} &= \ensuremath{p}(\ensuremath{S}_1|\ensuremath{\mbox{\boldmath$\theta$}}) \ensuremath{p}^*(\ensuremath{Y}_1 | \ensuremath{S}_1, \ensuremath{\mbox{\boldmath$\theta$}}) \prod_{t=2}^{\ensuremath{T}}
\ensuremath{p}(\ensuremath{S}_t|\ensuremath{S}_{t-1},\ensuremath{\mbox{\boldmath$\theta$}}) \ensuremath{p}^*(\ensuremath{Y}_t|\ensuremath{S}_t,\ensuremath{\mbox{\boldmath$\theta$}}) ,
\end{align*} which shows that a principled way to deal with missing
observations is to set \(\ensuremath{p}(\ensuremath{Y}_t|\ensuremath{S}_t) = 1\) for all
\(\ensuremath{Y}_t \in \mathcal{\ensuremath{Y}}_\text{miss}\). Note that it is necessary to
include time points with missing observations in this way to allow the
state probabilities to be computed properly. While this result is known
\citep[e.g.][]{zucchini2017hidden}, we have not come across its
derivation in the form above.
\hypertarget{state-dependent-missingness-mnar}{%
\subsubsection{State-dependent missingness
(MNAR)}\label{state-dependent-missingness-mnar}}
If data is not MAR, there is some dependence between whether
observations are missing or not, and the true unobserved values. There
are many forms this dependence can take, and modelling the dependence
accurately may require substantial knowledge of the domain to which the
data applies. Here, we take a pragmatic approach, and model this
dependence through the hidden states. That is, we assume \(M\) and
\(\ensuremath{Y}\) are conditionally independent, given the hidden states:
\begin{equation*}
\ensuremath{p}{(M_t, \ensuremath{Y}_t|\ensuremath{S}_t)} = \ensuremath{p}{(M_t|\ensuremath{S}_t)} \ensuremath{p}{(\ensuremath{Y}_t|\ensuremath{S}_t)} .
\end{equation*} This is not an overly restrictive assumption, as the
number of hidden states can be chosen to allow for intricate patterns of
(marginal) dependence between \(M\) and \(\ensuremath{Y}\) at a single time point,
as well as over time. For example, increased probability of missingness
for high values of \(\ensuremath{Y}\) can be captured through a state which is
simultaneously associated with high values of \(\ensuremath{Y}\) and a high
probability of \(M=1\). A high probability of a missing observation at
\(t+1\) \emph{after} a high (observed) value of \(\ensuremath{Y}_t\) can be
captured with a state \(s\) associated with high values of \(\ensuremath{Y}\), a
state \(s' \neq s\) associated with a high probability of \(M=1\), and a
high transition probability \(P(S_{t+1} = s'|S_{t} = s)\).
Under the assumption that missingness depends only on the hidden states:
\begin{equation*}
\ensuremath{p}{(M_{1:\ensuremath{T}}|\mathcal{\ensuremath{Y}}_{\text{obs}},\mathcal{\ensuremath{Y}}_{\text{miss}},\ensuremath{S}_{1:\ensuremath{T}},\gvc{\phi})} = \ensuremath{p}{(M_{1:\ensuremath{T}}|\ensuremath{S}_{1:\ensuremath{T}},\gvc{\phi})} ,
\end{equation*} the full likelihood becomes \begin{align*}
L_\text{full}(\ensuremath{\mbox{\boldmath$\theta$}},\gvc{\phi}|\mathcal{\ensuremath{Y}}_{\text{obs}},M_{1:\ensuremath{T}}) &\propto \sum_{\ensuremath{s}_{1:\ensuremath{T}} \in \mathcal{S}^\ensuremath{T}} \int \ensuremath{p}{(\mathcal{\ensuremath{Y}}_{\text{obs}},\mathcal{\ensuremath{Y}}_{\text{miss}}, \ensuremath{s}_{1:\ensuremath{T}} |\ensuremath{\mbox{\boldmath$\theta$}})} \ensuremath{p}{(M_{1:\ensuremath{T}}|\mathcal{\ensuremath{Y}}_{\text{obs}},\mathcal{\ensuremath{Y}}_{\text{miss}},\ensuremath{s}_{1:\ensuremath{T}},\gvc{\phi})} d \mathcal{\ensuremath{Y}}_\text{miss} \\
&= \sum_{\ensuremath{s}_{1:\ensuremath{T}} \in \mathcal{S}^\ensuremath{T}} \ensuremath{p}{(M_{1:\ensuremath{T}}|\ensuremath{s}_{1:\ensuremath{T}},\gvc{\phi})} \times \int \ensuremath{p}{(\mathcal{\ensuremath{Y}}_{\text{obs}},\mathcal{\ensuremath{Y}}_{\text{miss}}, \ensuremath{s}_{1:\ensuremath{T}} |\ensuremath{\mbox{\boldmath$\theta$}})} d \mathcal{\ensuremath{Y}}_\text{miss} \\
&= \sum_{\ensuremath{s}_{1:\ensuremath{T}} \in \mathcal{S}^\ensuremath{T}} \ensuremath{p}{(M_{1:\ensuremath{T}}|\ensuremath{s}_{1:\ensuremath{T}},\gvc{\phi})} \times \ensuremath{p}{(\mathcal{\ensuremath{Y}}_{\text{obs}}, \ensuremath{s}_{1:\ensuremath{T}} |\ensuremath{\mbox{\boldmath$\theta$}})} .
\end{align*} This shows that, although \(M\) does not directly depend on
\(\mathcal{\ensuremath{Y}}_\text{miss}\), because both \(M\) and \(Y\) depend on
\(\ensuremath{S}\), the role of the \(\ensuremath{p}{(M|\ensuremath{S},\gvc{\phi})}\) term is
more than a scaling factor in the likelihood, and hence missingness is
not ignorable.
\hypertarget{overview}{%
\subsection{Overview}\label{overview}}
When data is MNAR and missingness is not ignorable, valid inference on
\(\ensuremath{\mbox{\boldmath$\theta$}}\) requires including a submodel for \(M\) in the overall
model. That is, the HMM should be for multivariate data \(\ensuremath{Y}\) and
\(M\). The objective of the present paper is to show the potential
benefits of including a relatively simple model for \(M\) in hidden
Markov models, by assuming missingness is state-dependent. We provide
results from a simulation study, as well as an example with real data.
The simulations assess the accuracy of parameter estimates and state
recovery in situations where missingness is MAR or MNAR and dependent on
the hidden state, in situations where the state-conditional
distributions of the observations are relatively well separated or more
overlapping. We then discuss a situation where missingness is
time-dependent (but not state-dependent). This is a situation where
missingness is in fact MCAR, and where a misspecified model which
assumes missingness is state-dependent may lead to biased results.
Finally, we apply the models to a real data example, involving a
clinical trial comparing the effect of real and placebo medication on
the severity of schizophrenic symptoms.
\hypertarget{simulation-study}{%
\section{Simulation study}\label{simulation-study}}
\begin{figure}
\includegraphics[width=\linewidth]{simulation-response-distributions-1} \caption{State-conditional response distributions in the simulation studies. In Simulation 1 and 2, states are reasonably well-separated, although there is still considerable overlap of the distributions. In Simulation 3 and 4, states less well-separated.}\label{fig:simulation-response-distributions}
\end{figure}
To assess the potential benefits of including a state-dependent
missingness model in a HMM, we conducted a simulation study, focussing
on a three-state hidden Markov model with a univariate Normal
distributed response variable\footnote{All code for the simulations, and
the analysis of the application, is available at
\url{https://github.com/depmix/hmm-missing-data-paper}.}. We simulated
four scenario's. In Simulation 1 and 2 (Figure
\ref{fig:simulation-response-distributions}), the states are reasonably
well-separated, with means \(\mu_1 = -1\), \(\mu_2 = 0\), \(\mu_3 = 1\)
and standard deviations \(\sigma_1 = \sigma_2 = \sigma_3 = 1\). Note
that there is still considerable overlap in the state-conditional
response distributions, as would be expected in many real applications
of HMMs. In Simulation 1, missingness was state-dependent (i.e.~MNAR),
with \(\ensuremath{p}(M_t = 1|S_t = 1) = .05\), \(\ensuremath{p}(M_t = 1|S_t = 2) = .25\),
and \(\ensuremath{p}(M_t = 1|S_t = 2) = .5\). In Simulation 2, missingness was
independent of the state (MAR), with
\(\ensuremath{p}(M_t = 1|S_t = i) = \ensuremath{p}(M_t = 1) = .25\). In Simulation 3 and 4
(Figure \ref{fig:simulation-response-distributions}), the states were
rather less well-separated, with means as for Simulation 1 and 2, but
standard deviations \(\sigma_i = 3\). Here, the overlap of the
state-conditional response distributions is much higher than in
Simulation 1 and 2, and identification of the hidden states will be more
difficult. In Simulation 3, missingness was state-dependent (MNAR) in
the same manner as Simulation 1, while in Simulation 4, missingness was
state-independent (MAR) as for Simulation 2. In all simulations, the
initial state probabilities were \(\pi_1 = \ensuremath{p}(S_1 = 1) = .8\),
\(\pi_2 = \pi_3 = .1\), and the state-transition matrix was
\begin{equation*}
\mat{A} = \left[ \begin{matrix} .75 & .125 & .125 \\ .125 & .75 & .125 \\ .125 & .125 & .75
\end{matrix} \right] .
\end{equation*} In each simulation, we simulated a total of 1000 data
sets, each consisting of \(\ensuremath{N} = 100\) replications of a time-series of
length \(\ensuremath{T} = 50\). We denote observations in such replicated time
series as \(\ensuremath{Y}_{i,t}\), with \(i=1,\ldots,\ensuremath{N}\) and
\(t = 1, \ldots, \ensuremath{T}\). Data was generated according to a 3-state hidden
Markov model. For MAR cases, the non-missing observations are
distributed as \begin{equation}
\label{eq:MAR-distribution}
\ensuremath{p}(\ensuremath{Y}_{i,t}|\ensuremath{S}_{i,t} = j) = \mathbf{Normal}(\ensuremath{Y}_{i,t}|\mu_j,\sigma_j) .
\end{equation} In the MNAR cases, the missingness variable \(M\) and the
response variable \(\ensuremath{Y}\) were conditionally independent given the
hidden state: \begin{equation}
\label{eq:MNAR-distribution}
\ensuremath{p}(\ensuremath{Y}_{i,t},M_{i,t}|\ensuremath{S}_{i,t} = j) = \mathbf{Bernouilli}(M_{i,t}|\phi_j) \times \mathbf{Normal}(\ensuremath{Y}_{i,t}|\mu_j,\sigma_j)
\end{equation} Data sets were simulated by first generating the hidden
state sequences \(\ensuremath{S}_{i,1:\ensuremath{T}}\) according to the initial state and
transition probabilities. Then, the observations \(\ensuremath{Y}_{i,1:\ensuremath{T}}\) were
sampled according to the state-conditional distributions
\(\ensuremath{p}(\ensuremath{Y}_{i,t}|\ensuremath{S}_{i,t})\). Finally, random observations were
set to missing values according to the missingness distributions
\(\ensuremath{p}(M_{i,t}|\ensuremath{S}_{i,t})\).
We fitted two 3-state hidden Markov models to each data-set. In the MAR
models, observed responses were assumed to be distributed according to
(\ref{eq:MAR-distribution}), and in the MNAR models, the observed
responses and missingness indicators were assumed to be distributed
according to (\ref{eq:MNAR-distribution}). Parameters were estimated by
maximum likelihood, using the Expectation-Maximisation algorithm, as
implemented in depmixS4 \citep{depmixS4}. To speed up convergence,
starting values were set to the true parameter values. Although such
initialization is obviously not possible in real applications, we are
interested in the quality of parameter estimates at the global maximum
likelihood solution, and setting starting values to the true parameters
makes it more likely to arrive at the global maximum. In real
applications, one would need to use a sufficient number of randomly
generated starting values to find the global maximum.
\begin{table}
\caption{\label{tab:table-simulation-1}Results of Simulation 1 (MNAR, low variance). Values shown are the true value of each parameter, and the mean (mean), standard deviation (SD), and mean absolute error (MAE) of the parameter estimates, for both the MAR and MNAR model. The value of "rel. MAE" is the ratio of the mean absolute error of the MAR over the MNAR model.}
\centering
\begin{tabular}[t]{lrrrrrrrr}
\toprule
\multicolumn{1}{c}{ } & \multicolumn{1}{c}{ } & \multicolumn{3}{c}{MAR} & \multicolumn{3}{c}{MNAR} & \multicolumn{1}{c}{ } \\
\cmidrule(l{3pt}r{3pt}){3-5} \cmidrule(l{3pt}r{3pt}){6-8}
parameter & true value & mean & SD & MAE & mean & SD & MAE & rel. MAE\\
\midrule
$\mu_1$ & -1.000 & -1.010 & 0.131 & 0.094 & -1.017 & 0.097 & 0.072 & 0.767\\
$\mu_2$ & 0.000 & 0.015 & 0.277 & 0.223 & 0.014 & 0.228 & 0.180 & 0.807\\
$\mu_3$ & 1.000 & 1.113 & 0.286 & 0.231 & 1.053 & 0.252 & 0.186 & 0.803\\
\addlinespace
$\sigma_1$ & 1.000 & 0.998 & 0.051 & 0.033 & 0.995 & 0.034 & 0.026 & 0.785\\
$\sigma_2$ & 1.000 & 0.972 & 0.111 & 0.079 & 0.979 & 0.085 & 0.064 & 0.809\\
$\sigma_3$ & 1.000 & 0.959 & 0.104 & 0.077 & 0.979 & 0.085 & 0.061 & 0.801\\
\addlinespace
$\pi_1$ & 0.800 & 0.834 & 0.146 & 0.117 & 0.776 & 0.110 & 0.083 & 0.715\\
$\pi_2$ & 0.100 & 0.118 & 0.152 & 0.111 & 0.131 & 0.130 & 0.105 & 0.944\\
$\pi_3$ & 0.100 & 0.049 & 0.048 & 0.062 & 0.093 & 0.066 & 0.055 & 0.890\\
\addlinespace
$a_{11}$ & 0.750 & 0.774 & 0.080 & 0.064 & 0.743 & 0.055 & 0.039 & 0.613\\
$a_{12}$ & 0.125 & 0.144 & 0.094 & 0.068 & 0.139 & 0.077 & 0.061 & 0.896\\
$a_{13}$ & 0.125 & 0.082 & 0.055 & 0.057 & 0.118 & 0.054 & 0.044 & 0.765\\
\addlinespace
$a_{21}$ & 0.125 & 0.144 & 0.086 & 0.065 & 0.124 & 0.062 & 0.048 & 0.729\\
$a_{22}$ & 0.750 & 0.759 & 0.116 & 0.087 & 0.754 & 0.096 & 0.070 & 0.812\\
$a_{23}$ & 0.125 & 0.097 & 0.082 & 0.068 & 0.122 & 0.075 & 0.058 & 0.850\\
\addlinespace
$a_{31}$ & 0.125 & 0.146 & 0.085 & 0.068 & 0.118 & 0.050 & 0.039 & 0.579\\
$a_{32}$ & 0.125 & 0.166 & 0.128 & 0.103 & 0.138 & 0.092 & 0.070 & 0.679\\
$a_{33}$ & 0.750 & 0.688 & 0.111 & 0.090 & 0.744 & 0.076 & 0.056 & 0.623\\
\addlinespace
$p(M=1|S=1)$ & 0.050 & - & - & - & 0.048 & 0.021 & 0.017 & -\\
$p(M=1|S=2)$ & 0.250 & - & - & - & 0.247 & 0.073 & 0.057 & -\\
$p(M=1|S=3)$ & 0.500 & - & - & - & 0.507 & 0.058 & 0.040 & -\\
\bottomrule
\end{tabular}
\end{table}
\begin{table}
\caption{\label{tab:table-simulation-2}Results of Simulation 2 (MAR, low variance). Values shown are the true value of each parameter, and the mean (mean), standard deviation (SD), and mean absolute error (MAE) of the parameter estimates, for both the MAR and MNAR model. The value of "rel. MAE" is the ratio of the mean absolute error of the MAR over the MNAR model.}
\centering
\begin{tabular}[t]{lrrrrrrrr}
\toprule
\multicolumn{1}{c}{ } & \multicolumn{1}{c}{ } & \multicolumn{3}{c}{MAR} & \multicolumn{3}{c}{MNAR} & \multicolumn{1}{c}{ } \\
\cmidrule(l{3pt}r{3pt}){3-5} \cmidrule(l{3pt}r{3pt}){6-8}
parameter & true value & mean & SD & MAE & mean & SD & MAE & rel. MAE\\
\midrule
$\mu_1$ & -1.000 & -1.061 & 0.194 & 0.132 & -1.062 & 0.200 & 0.134 & 1.014\\
$\mu_2$ & 0.000 & -0.019 & 0.287 & 0.229 & -0.022 & 0.288 & 0.230 & 1.005\\
$\mu_3$ & 1.000 & 1.048 & 0.213 & 0.158 & 1.038 & 0.213 & 0.157 & 0.991\\
\addlinespace
$\sigma_1$ & 1.000 & 0.978 & 0.067 & 0.047 & 0.978 & 0.070 & 0.047 & 1.000\\
$\sigma_2$ & 1.000 & 0.969 & 0.105 & 0.079 & 0.969 & 0.107 & 0.081 & 1.021\\
$\sigma_3$ & 1.000 & 0.981 & 0.070 & 0.050 & 0.982 & 0.071 & 0.049 & 0.993\\
\addlinespace
$\pi_1$ & 0.800 & 0.739 & 0.177 & 0.130 & 0.737 & 0.178 & 0.132 & 1.015\\
$\pi_2$ & 0.100 & 0.169 & 0.187 & 0.144 & 0.171 & 0.187 & 0.145 & 1.012\\
$\pi_3$ & 0.100 & 0.092 & 0.070 & 0.057 & 0.092 & 0.069 & 0.057 & 0.995\\
\addlinespace
$a_{11}$ & 0.750 & 0.727 & 0.102 & 0.064 & 0.726 & 0.102 & 0.065 & 1.006\\
$a_{12}$ & 0.125 & 0.155 & 0.112 & 0.082 & 0.156 & 0.115 & 0.083 & 1.020\\
$a_{13}$ & 0.125 & 0.118 & 0.066 & 0.051 & 0.118 & 0.062 & 0.050 & 0.975\\
\addlinespace
$a_{21}$ & 0.125 & 0.125 & 0.080 & 0.061 & 0.127 & 0.083 & 0.062 & 1.013\\
$a_{22}$ & 0.750 & 0.751 & 0.112 & 0.084 & 0.749 & 0.116 & 0.086 & 1.025\\
$a_{23}$ & 0.125 & 0.125 & 0.081 & 0.061 & 0.125 & 0.083 & 0.063 & 1.034\\
\addlinespace
$a_{31}$ & 0.125 & 0.112 & 0.063 & 0.051 & 0.112 & 0.062 & 0.049 & 0.973\\
$a_{32}$ & 0.125 & 0.153 & 0.108 & 0.082 & 0.150 & 0.107 & 0.081 & 0.982\\
$a_{33}$ & 0.750 & 0.735 & 0.096 & 0.067 & 0.738 & 0.095 & 0.066 & 0.984\\
\addlinespace
$p(M=1|S=1)$ & 0.250 & - & - & - & 0.250 & 0.046 & 0.027 & -\\
$p(M=1|S=2)$ & 0.250 & - & - & - & 0.248 & 0.056 & 0.038 & -\\
$p(M=1|S=3)$ & 0.250 & - & - & - & 0.247 & 0.046 & 0.028 & -\\
\bottomrule
\end{tabular}
\end{table}
\begin{table}
\caption{\label{tab:table-simulation-3}Results of Simulation 3 (MNAR, high variance). Values shown are the true value of each parameter, and the mean (mean), standard deviation (SD), and mean absolute error (MAE) of the parameter estimates, for both the MAR and MNAR model. The value of "rel. MAE" is the ratio of the mean absolute error of the MAR over the MNAR model.}
\centering
\begin{tabular}[t]{lrrrrrrrr}
\toprule
\multicolumn{1}{c}{ } & \multicolumn{1}{c}{ } & \multicolumn{3}{c}{MAR} & \multicolumn{3}{c}{MNAR} & \multicolumn{1}{c}{ } \\
\cmidrule(l{3pt}r{3pt}){3-5} \cmidrule(l{3pt}r{3pt}){6-8}
parameter & true value & mean & SD & MAE & mean & SD & MAE & rel. MAE\\
\midrule
$\mu_1$ & -1.000 & -1.663 & 0.932 & 0.761 & -1.198 & 0.628 & 0.315 & 0.414\\
$\mu_2$ & 0.000 & -0.314 & 0.484 & 0.461 & -0.110 & 0.470 & 0.409 & 0.888\\
$\mu_3$ & 1.000 & 1.480 & 1.214 & 0.923 & 1.383 & 0.956 & 0.609 & 0.661\\
\addlinespace
$\sigma_1$ & 3.000 & 2.765 & 0.459 & 0.347 & 2.911 & 0.330 & 0.189 & 0.543\\
$\sigma_2$ & 3.000 & 2.889 & 0.455 & 0.302 & 2.967 & 0.333 & 0.215 & 0.713\\
$\sigma_3$ & 3.000 & 2.703 & 0.512 & 0.406 & 2.773 & 0.479 & 0.326 & 0.803\\
\addlinespace
$\pi_1$ & 0.800 & 0.546 & 0.362 & 0.355 & 0.657 & 0.281 & 0.217 & 0.611\\
$\pi_2$ & 0.100 & 0.346 & 0.380 & 0.333 & 0.253 & 0.291 & 0.231 & 0.694\\
$\pi_3$ & 0.100 & 0.108 & 0.174 & 0.129 & 0.090 & 0.091 & 0.077 & 0.601\\
\addlinespace
$a_{11}$ & 0.750 & 0.651 & 0.231 & 0.183 & 0.712 & 0.153 & 0.099 & 0.543\\
$a_{12}$ & 0.125 & 0.190 & 0.215 & 0.160 & 0.144 & 0.145 & 0.105 & 0.660\\
$a_{13}$ & 0.125 & 0.159 & 0.186 & 0.139 & 0.144 & 0.124 & 0.091 & 0.659\\
\addlinespace
$a_{21}$ & 0.125 & 0.106 & 0.172 & 0.124 & 0.106 & 0.108 & 0.085 & 0.687\\
$a_{22}$ & 0.750 & 0.787 & 0.232 & 0.185 & 0.784 & 0.135 & 0.109 & 0.590\\
$a_{23}$ & 0.125 & 0.107 & 0.158 & 0.115 & 0.110 & 0.105 & 0.085 & 0.742\\
\addlinespace
$a_{31}$ & 0.125 & 0.152 & 0.183 & 0.136 & 0.131 & 0.126 & 0.096 & 0.704\\
$a_{32}$ & 0.125 & 0.166 & 0.199 & 0.143 & 0.145 & 0.141 & 0.105 & 0.738\\
$a_{33}$ & 0.750 & 0.682 & 0.234 & 0.184 & 0.724 & 0.151 & 0.108 & 0.587\\
\addlinespace
$p(M=1|S=1)$ & 0.050 & - & - & - & 0.076 & 0.122 & 0.059 & -\\
$p(M=1|S=2)$ & 0.250 & - & - & - & 0.241 & 0.155 & 0.126 & -\\
$p(M=1|S=3)$ & 0.500 & - & - & - & 0.489 & 0.134 & 0.092 & -\\
\bottomrule
\end{tabular}
\end{table}
\begin{table}
\caption{\label{tab:table-simulation-4}Results of Simulation 4 (MAR, high variance). Values shown are the true value of each parameter, and the mean (mean), standard deviation (SD), and mean absolute error (MAE) of the parameter estimates, for both the MAR and MNAR model. The value of "rel. MAE" is the ratio of the mean absolute error of the MAR over the MNAR model.}
\centering
\begin{tabular}[t]{lrrrrrrrr}
\toprule
\multicolumn{1}{c}{ } & \multicolumn{1}{c}{ } & \multicolumn{3}{c}{MAR} & \multicolumn{3}{c}{MNAR} & \multicolumn{1}{c}{ } \\
\cmidrule(l{3pt}r{3pt}){3-5} \cmidrule(l{3pt}r{3pt}){6-8}
parameter & true value & mean & SD & MAE & mean & SD & MAE & rel. MAE\\
\midrule
$\mu_1$ & -1.000 & -1.650 & 1.002 & 0.801 & -1.658 & 1.107 & 0.815 & 1.018\\
$\mu_2$ & 0.000 & -0.171 & 0.539 & 0.432 & -0.178 & 0.542 & 0.437 & 1.010\\
$\mu_3$ & 1.000 & 1.468 & 1.063 & 0.778 & 1.473 & 1.070 & 0.788 & 1.014\\
\addlinespace
$\sigma_1$ & 3.000 & 2.719 & 0.468 & 0.383 & 2.720 & 0.473 & 0.375 & 0.981\\
$\sigma_2$ & 3.000 & 2.911 & 0.441 & 0.299 & 2.918 & 0.412 & 0.279 & 0.934\\
$\sigma_3$ & 3.000 & 2.728 & 0.504 & 0.377 & 2.732 & 0.478 & 0.365 & 0.968\\
\addlinespace
$\pi_1$ & 0.800 & 0.528 & 0.345 & 0.352 & 0.522 & 0.338 & 0.357 & 1.012\\
$\pi_2$ & 0.100 & 0.330 & 0.367 & 0.316 & 0.344 & 0.359 & 0.320 & 1.014\\
$\pi_3$ & 0.100 & 0.141 & 0.199 & 0.149 & 0.134 & 0.192 & 0.142 & 0.951\\
\addlinespace
$a_{11}$ & 0.750 & 0.638 & 0.230 & 0.183 & 0.645 & 0.220 & 0.177 & 0.968\\
$a_{12}$ & 0.125 & 0.188 & 0.212 & 0.155 & 0.182 & 0.211 & 0.155 & 0.998\\
$a_{13}$ & 0.125 & 0.174 & 0.188 & 0.139 & 0.174 & 0.178 & 0.134 & 0.963\\
\addlinespace
$a_{21}$ & 0.125 & 0.111 & 0.166 & 0.121 & 0.103 & 0.157 & 0.119 & 0.984\\
$a_{22}$ & 0.750 & 0.774 & 0.223 & 0.175 & 0.787 & 0.209 & 0.170 & 0.972\\
$a_{23}$ & 0.125 & 0.114 & 0.152 & 0.111 & 0.110 & 0.139 & 0.110 & 0.986\\
\addlinespace
$a_{31}$ & 0.125 & 0.133 & 0.169 & 0.125 & 0.137 & 0.167 & 0.124 & 0.997\\
$a_{32}$ & 0.125 & 0.167 & 0.193 & 0.138 & 0.162 & 0.186 & 0.139 & 1.007\\
$a_{33}$ & 0.750 & 0.700 & 0.232 & 0.177 & 0.701 & 0.224 & 0.176 & 0.992\\
\addlinespace
$p(M=1|S=1)$ & 0.250 & - & - & - & 0.253 & 0.122 & 0.080 & -\\
$p(M=1|S=2)$ & 0.250 & - & - & - & 0.237 & 0.086 & 0.056 & -\\
$p(M=1|S=3)$ & 0.250 & - & - & - & 0.257 & 0.130 & 0.082 & -\\
\bottomrule
\end{tabular}
\end{table}
The results of simulation 1 (Table \ref{tab:table-simulation-1}) show
that, when states are relatively well separated, both models provide
parameter estimates which are, on average, reasonably close to the true
values. Both models have the tendency to estimate the means as more
dispersed, and the standard deviations as slightly smaller, then they
really are. While wrongly assuming MAR may not lead to overly biased
estimates, we see that the mean absolute error (MAE) for the MNAR model
is always smaller than that of the MAR model, reducing the estimation
error to as much as 58\%. As such, accounting for state-dependent
missingness increases the accuracy of the parameter estimates. We next
consider recovery of the hidden states, by comparing the true hidden
state sequences to the maximum a posteriori state sequences determined
by the Viterbi algorithm \citep[see][]{Rabiner1989, visser2021hidden}.
The MAR model recovers 53.13\% of the states, while the MNAR model
recovers 62.86\% of the states. The accuracy in recovering the hidden
states is thus higher in the model which correctly accounts for
missingness. Whilst the performance of neither model may seem overly
impressive, we should note that recovering the hidden states is a
non-trivial task when the state-conditional response distributions have
considerable overlap (see Figure
\ref{fig:simulation-response-distributions}) and states do not persist
for long periods of time (here, the true self-transitions probabilities
are \(a_{ii} = .75\), meaning that states have an average run-length of
4 consecutive time-points). When ignoring time-dependencies and treating
the observed data as coming from a bivariate mixture distribution over
\(\ensuremath{Y}\) and \(M\), the maximum accuracy in classification would be
50.09\% for this data. The theoretical maximum classification accuracy
for the hidden Markov model is more difficult to establish, but
simulations show that the MNAR model with the true parameters recovers
66.51\% of the true states. For the MAR model, the approximate maximum
classification accuracy is 58.06\%.
The results of Simulation 2 (Table \ref{tab:table-simulation-2}) show
that when data is in fact MAR, both models provide roughly equally
accurate parameter estimates. While the MNAR model does not provide
better parameter estimates, including a model component for
state-dependent missingness does not seem to bias parameter estimates
compared to the MAR model. As can be seen, the state-wise missingness
probabilities are, on average, close to the true values of .25. Over all
parameters, the relative MAE of the models is 1.003 on average, which
shows the models perform equally well. In terms of recovering the hidden
states, the MAR model recovers 55.6\% of the states, while the MNAR
model recovers 55.63\% of the states. The somewhat reduced recovery rate
of the MNAR model compared to Simulation 1 is likely due to the fact
that here, missingness provides no information about the identity of the
hidden state. Here, the maximum classification accuracy is 42.91\% for a
mixture model, and approximately 60.45\% for the hidden Markov models.
In Simulation 3 (Table \ref{tab:table-simulation-3}) and 4 (Table
\ref{tab:table-simulation-4}) the states are less well-separated, making
accurate parameter estimation more difficult. Here, the tendency to
estimate the means as more dispersed and the standard deviations as
smaller than they are becomes more pronounced. For both models the
estimation error in Simulation 3 (Table \ref{tab:table-simulation-3}) is
larger than for Simulation 1, but comparing the MAE for both models
again shows the substantial benefits for including a missingness model.
Over all parameters, the relative MAE of the models is 0.658 on average,
which shows the MNAR model clearly outperforms the MAR model. In terms
of recovering the hidden states, the MAR model recovers 34.97\% of the
states, while the MNAR model recovers 45.27\% of the states. As in
Simulation 1, the MNAR model performs better in state identification.
For both models, performance is lower than in Simulation 1, reflecting
the increased difficulty due to increased overlap of the
state-conditional response distributions (Figure
\ref{fig:simulation-response-distributions}). Indeed, the performance of
the MAR model is close to chance (random assignment of states would give
an expected accuracy of 33.33\%). The maximum classification accuracy is
44.03\% for a mixture model, and approximately 54.04\% for the MNAR and
41.42\% for the MAR hidden Markov models.
When missingness is ignorable (Simulation 4), Like in Simulation 2,
inclusion of a missingness component in the HMM does not increase any
bias in the parameter estimates. Over all parameters, the relative MAE
of the models is 0.987 on average, which shows the models perform
roughly equally well. The model which ignores missingness recovers
35.51\% of the states, while the model with a missingness component
recovers 35.5\% of the states. For comparison, the maximum accuracy is
36.64\% for a mixture model, and 42.51\% for the hidden Markov models.
Taken together, these simulation results show that if missingness is
state-dependent, there is a substantial benefit to including a
(relatively simple) model for missingness in the HMM. When missingness
is in fact ignorable, including a missingness model is superfluous, but
does not bias the results. Hence, there appears to be little risk
associated to including a missingness model into the HMM.
In a final simulation, we assessed the performance of the models when
missingness is \emph{time-dependent}, rather than state-dependent.
Attrition is a common occurrence in longitudinal studies, meaning that
the probability of missing data often increases with time. In this
simulation, the probability of missing data varied with time \(t\)
through a logistic regression model: \begin{equation}
\ensuremath{p}(M_{i,t} = 1) = \frac{1}{1+\exp(-(0.125 \times t - 5))} .
\end{equation} Here, the probability of missing data is very small
(0.008) at time 1, but increasing to rather high (0.777) at time 50. The
other parameters were the same as in Simulation 1 and 2 (i.e., the
states were relatively well-separated). In a model that specifies
missingness as state-dependent, but not time-dependent, this could
potentially result in biased parameter estimates. For instance, the
increased probability of missingness over time may be accounted for by
estimating states to have a different probability of missingness, and
estimating prior and transition probabilities to allow states with a
higher probability of missingness to occur more frequently later in
time. In addition to the two hidden Markov models estimated before, we
also estimated a hidden Markov model with a state- and time-dependent
model for missingness: \begin{equation}
\ensuremath{p}(M_{i,t} = 1|S_{i,t} = j) = \frac{1}{1+\exp(-(\beta_{0,j} + \beta_{\text{time},j} \times t))}
\end{equation} This model should be able to capture the true pattern of
missingness, whilst the MNAR model which only includes state-dependent
missingness would not.
\begin{table}
\caption{\label{tab:table-simulation-5}Results of Simulation 5 (time-dependent missingness, low variance). Values shown are the true value of each parameter, and the mean (mean), standard deviation (SD), and mean absolute error (MAE) of the parameter estimates, for the MAR, MNAR (state), and MNAR (time) model. The value of "rel. MAE 1" is the ratio of the mean absolute error of the MAR over the MNAR (state) model, and the value of "rel. MAE 2" is the ratio of the mean absolute error of the MAR over the MNAR (time) model.}
\centering
\resizebox{\linewidth}{!}{
\begin{tabular}[t]{lrrrrrrrrrrrr}
\toprule
\multicolumn{1}{c}{ } & \multicolumn{1}{c}{ } & \multicolumn{3}{c}{MAR} & \multicolumn{3}{c}{MNAR (state)} & \multicolumn{3}{c}{MNAR (time)} & \multicolumn{2}{c}{ } \\
\cmidrule(l{3pt}r{3pt}){3-5} \cmidrule(l{3pt}r{3pt}){6-8} \cmidrule(l{3pt}r{3pt}){9-11}
parameter & true value & mean & SD & MAE & mean & SD & MAE & mean & SD & MAE & rel. MAE 1 & rel. MAE 2\\
\midrule
$\mu_1$ & -1.000 & -1.038 & 0.173 & 0.116 & -0.825 & 0.076 & 0.176 & -1.031 & 0.175 & 0.120 & 1.520 & 1.037\\
$\mu_2$ & 0.000 & -0.015 & 0.272 & 0.215 & -0.040 & 0.067 & 0.062 & -0.020 & 0.294 & 0.233 & 0.287 & 1.086\\
$\mu_3$ & 1.000 & 1.033 & 0.195 & 0.143 & 0.757 & 0.084 & 0.243 & 1.029 & 0.207 & 0.153 & 1.698 & 1.068\\
\addlinespace
$\sigma_1$ & 1.000 & 0.985 & 0.059 & 0.042 & 1.026 & 0.035 & 0.035 & 0.987 & 0.058 & 0.042 & 0.832 & 0.997\\
$\sigma_2$ & 1.000 & 0.967 & 0.111 & 0.082 & 1.278 & 0.033 & 0.278 & 0.966 & 0.112 & 0.083 & 3.370 & 1.010\\
$\sigma_3$ & 1.000 & 0.981 & 0.072 & 0.049 & 1.037 & 0.037 & 0.043 & 0.984 & 0.067 & 0.048 & 0.884 & 0.990\\
\addlinespace
$\pi_1$ & 0.800 & 0.758 & 0.151 & 0.110 & 0.894 & 0.066 & 0.100 & 0.760 & 0.152 & 0.109 & 0.913 & 0.993\\
$\pi_2$ & 0.100 & 0.150 & 0.161 & 0.123 & 0.001 & 0.032 & 0.101 & 0.148 & 0.162 & 0.122 & 0.819 & 0.990\\
$\pi_3$ & 0.100 & 0.092 & 0.061 & 0.050 & 0.105 & 0.059 & 0.047 & 0.092 & 0.062 & 0.051 & 0.943 & 1.023\\
\addlinespace
$a_{11}$ & 0.750 & 0.734 & 0.087 & 0.056 & 0.794 & 0.025 & 0.045 & 0.734 & 0.090 & 0.059 & 0.806 & 1.050\\
$a_{12}$ & 0.125 & 0.146 & 0.099 & 0.073 & 0.021 & 0.008 & 0.104 & 0.146 & 0.106 & 0.075 & 1.438 & 1.036\\
$a_{13}$ & 0.125 & 0.119 & 0.058 & 0.047 & 0.185 & 0.024 & 0.060 & 0.119 & 0.059 & 0.048 & 1.269 & 1.021\\
\addlinespace
$a_{21}$ & 0.125 & 0.128 & 0.088 & 0.063 & 0.000 & 0.001 & 0.125 & 0.131 & 0.097 & 0.069 & 1.983 & 1.091\\
$a_{22}$ & 0.750 & 0.747 & 0.114 & 0.083 & 1.000 & 0.001 & 0.250 & 0.738 & 0.131 & 0.092 & 2.994 & 1.104\\
$a_{23}$ & 0.125 & 0.125 & 0.082 & 0.062 & 0.000 & 0.000 & 0.125 & 0.130 & 0.091 & 0.067 & 2.031 & 1.082\\
\addlinespace
$a_{31}$ & 0.125 & 0.116 & 0.062 & 0.048 & 0.150 & 0.027 & 0.030 & 0.115 & 0.063 & 0.049 & 0.624 & 1.032\\
$a_{32}$ & 0.125 & 0.143 & 0.101 & 0.076 & 0.045 & 0.008 & 0.080 & 0.146 & 0.106 & 0.081 & 1.052 & 1.056\\
$a_{33}$ & 0.750 & 0.742 & 0.087 & 0.062 & 0.805 & 0.025 & 0.056 & 0.740 & 0.093 & 0.068 & 0.899 & 1.095\\
\addlinespace
$\beta_{0,1}$ & -5.000 & - & - & - & - & - & - & -6.149 & 25.490 & 1.497 & - & -\\
$\beta_{0,2}$ & -5.000 & - & - & - & - & - & - & -7.153 & 40.582 & 2.739 & - & -\\
$\beta_{0,3}$ & -5.000 & - & - & - & - & - & - & -6.472 & 38.998 & 1.861 & - & -\\
\addlinespace
$\beta_{\text{time},1}$ & 0.125 & - & - & - & - & - & - & 0.154 & 0.605 & 0.039 & - & -\\
$\beta_{\text{time},2}$ & 0.125 & - & - & - & - & - & - & 0.184 & 1.102 & 0.075 & - & -\\
$\beta_{\text{time},3}$ & 0.125 & - & - & - & - & - & - & 0.188 & 1.815 & 0.074 & - & -\\
\addlinespace
$p(M=1|S=1)$ & - & - & - & - & 0.040 & 0.015 & 0.207 & - & - & - & - & -\\
$p(M=1|S=2)$ & - & - & - & - & 0.552 & 0.024 & 0.305 & - & - & - & - & -\\
$p(M=1|S=3)$ & - & - & - & - & 0.067 & 0.022 & 0.181 & - & - & - & - & -\\
\bottomrule
\end{tabular}}
\end{table}
The results (Table \ref{tab:table-simulation-5}) show that, compared to
the MAR model, the MNAR model which misspecifies missingness as
state-dependent is inferior, resulting in more biased parameter
estimates. Over all parameters, the relative MAE of these two models is
1.353 on average, indicating the MAR model outperforms the MNAR (state)
model. To account for the increase in missing values later in time, the
MNAR (state) model estimates the probability of missingness as highest
for state 2, which is estimated to have a mean of close to 0, but an
increased standard deviation to incorporate observations from the other
two states. To make state 2 more prevalent over time, transition
probabilities to state 2 are relatively low from state 1 and 2
(parameters \(a_{12}\) and \(a_{32}\) respectively), whilst
self-transitions (\(a_{22}\)) are close to 1 (meaning that once in state
2, the hidden state sequence is very likely to remain in that state. The
prevalence of state 2 is thus increasing over time, and as this state
has a higher probability of missingness, so is the prevalence of missing
values. The MNAR (time) model, which allows missingness to depend on
both the hidden states and time, performs only slightly worse than the
MAR model, with an average relative MAE over all parameters of this
model compared to the MAR of 1.042. However, the MNAR (time) model is
able to capture the pattern of attrition (increased missing data over
time), whilst the MAR model is not. As such, the MNAR (time) model may
be deemed preferable to the MAR model, insofar as one is interested in
more than modelling the responses \(\ensuremath{Y}\). In terms of recovering the
hidden states, the MAR model recovers 55.67\% of the states, and the
MNAR (time) model recovers 55.42\% of the states. The misspecified MNAR
(state) model recovers 50\% of the states. The maximum classification
accuracy for this data is 42.95\% for a mixture model, and approximately
59.91\% for the hidden Markov models.
This final simulation shows that when modelling patterns of missing data
in hidden Markov models, care should be taken in how this is done. An
increase in missing data over time could be due to an underlying higher
prevalence of states which result in more missing data, and/or a
state-independent increase in missingness over time. In applications
where the true reason and pattern of missingness is unknown, it is then
advisable to start by allowing for both state- and time-dependent
missing data, selecting simpler options when this is warranted by the
data.
\hypertarget{application-severity-of-schizophrenia-in-a-clinical-trial}{%
\section{Application: Severity of schizophrenia in a clinical
trial}\label{application-severity-of-schizophrenia-in-a-clinical-trial}}
Here, we apply our hidden Markov model with state-dependent missingness
to data from the National Institute of Mental Health Schizophrenia
Collaborative Study. The study concerns the assessment of
treatment-related changes in overall severity of mental illness. In this
study, 437 patients diagnosed with schizophrenia were randomly assigned
to receive either placebo (108 patients) or a drug (329 patients)
treatment, and their severity of their illness was rated by a clinician
at baseline (week 0), and at subsequent 1 week intervals (weeks 1--6),
with week 1, 3, and 6 as the intended main follow-up measurements. This
data has been made publicly available by Don Hedeker\footnote{\url{https://hedeker.people.uic.edu/SCHIZREP.DAT.txt}.}
and has been analysed numerous times. In particular,
\citet{hedeker1997application} focused on pattern mixture methods to
deal with missing data. \citet{yeh2010estimating} and
\citet{yeh2012intermittent} applied Markov and hidden Markov models,
respectively, assuming ratings were MAR.
The analysis focuses on a single item of the Inpatient Multidimensional
Psychiatric Scale \citep{lorr1966inpatient}, which rates illness
severity on a scale from 1 (``normal'') to 7 (``among the most extremely
ill'').\footnote{The dataset provided contains some non-integer values
for these ratings, presumably given to provide a finer-grained
evaluation by the clinician.}. Most participants were measured on week
0 (99.31\%) and 1 (97.48\%), whilst the other main measurement points at
week 3 (85.58\%) and 6 (76.66\%) show more missing values. For a few
participants, ratings were instead obtained on week 2 (3.2\%), 4
(2.52\%), and/or 5 (2.06\%). Even when ignoring these rare deviations
from the main measurement points, there is a clear potential issue with
missing data and attrition, with 75.29\% being measured the intended
four times or more, and 15.1\% rated on just three occasions, and 9.61\%
only twice. The distribution of the ratings at each week is shown in
Figure \ref{fig:histograms-of-ratings-by-week}. As can be seen there,
ratings were generally relatively high at week 0, 1, and 3, but are
relatively lower at week 6. As there are only a small number of ratings
at week 2, 4, and 5, the empirical distributions for those weeks are
rather unreliable.
\begin{figure}
{\centering \includegraphics[width=0.8\linewidth]{histograms-of-ratings-by-week-1}
}
\caption{Distribution of the severity of illness ratings (IMPS item 79) at each week.}\label{fig:histograms-of-ratings-by-week}
\end{figure}
\begin{table}
\caption{\label{tab:glm-model-missingness}Results of a logistic regression analysis modelling missingness as a function of drug, week, and whether the week was a main measurement occasion or not.}
\centering
\begin{tabular}[t]{lrrrr}
\toprule
& $\hat{\beta}$ & $\text{SE}(\hat{\beta})$ & $z$ & $P(>|z|)$\\
\midrule
\texttt{(Intercept)} & 1.921 & 0.393 & 4.884 & 0.000\\
\texttt{drug} & 0.433 & 0.463 & 0.936 & 0.349\\
\texttt{week} & 0.496 & 0.068 & 7.353 & 0.000\\
\texttt{main} & -5.381 & 0.382 & -14.103 & 0.000\\
$\texttt{drug} \times \texttt{week}$ & -0.112 & 0.081 & -1.395 & 0.163\\
$\texttt{drug} \times \texttt{main}$ & -0.596 & 0.446 & -1.335 & 0.182\\
\bottomrule
\end{tabular}
\end{table}
To gain initial insight into patterns underlying the missing data, we
modelled whether the IMPS rating was missing or not with a logistic
regression model. Predictors in the model were a dummy-coded variable
\texttt{drug} (placebo = 0, medicine = 1), \texttt{week} (from 0 to 6)
as a metric predictor, and a dummy-coded variable \texttt{main} to
indicate whether the rating was at a main measurement week (i.e.~at week
0, 1, 3, or 6). We also included an interaction between \texttt{drug}
and \texttt{week}, and between \texttt{drug} and \texttt{main}. The
results of this analysis (Table \ref{tab:glm-model-missingness}) show a
positive effect of \texttt{week} (such that missingness increases over
time), and a negative effect of \texttt{main}, with (many) more missing
values on weeks which are \emph{not} the main measurement weeks. The
positive effect of \texttt{week} is a clear sign of attrition. A
question now is whether this attrition is related to the severity of the
illness, in which case the ratings at week 6 would provide a biased view
on the true severity of illness after 6 weeks of treatment with a
placebo or medicine. There are of course different methods to assess
this, and many have been already applied to this particular dataset. Our
objective here is to incorporate a model of missingness into a hidden
Markov model, allowing missingness to depend on the latent state as well
as observable features such as the measurement week.
\hypertarget{hidden-markov-models-1}{%
\subsection{Hidden Markov models}\label{hidden-markov-models-1}}
We fitted HMMs in which we either assumed ratings are MAR, or assume
ratings are MNAR and state- and time-dependent. For each type of model
(MAR or MNAR), we fit versions with 2, 3, 4, or 5 states. Both types of
model assume \texttt{imps79}, the IMPS Item 79 ratings, follow a Normal
distribution, with a state-dependent mean and standard deviation. No
additional covariates were included, as the states are intended to
capture all the important determinants of illness severity. To model
effects of drug, we allow transitions between states, as well as the
initial state, to depend on \texttt{drug}. Whilst the initial
measurement at week 0 was made before administering the drug, we include
a possible dependence to account for any potential pre-existing
differences between the conditions. In the MNAR models, a second
(dichotomous) response variable \texttt{missing} is included, in
addition to \texttt{imps79}. The \texttt{missing} variable is modelled
with a logistic regression, using \texttt{week} and the dummy-coded
\texttt{main} variable as predictors, as these were found to be
important predictors in the (state-independent) logistic regression
analysis reported earlier. All models were estimated by maximum
likelihood using the EM algorithm implemented in depmixS4
\citep{depmixS4}.
\begin{table}
\caption{\label{tab:model-table}Modelling results for the MAR and MNAR hidden Markov models with 2-5 latent states.}
\centering
\begin{tabular}[t]{lrrrrr}
\toprule
model & \#states & log Likelihood & \#par & AIC & BIC\\
\midrule
MAR & 2 & -2422.675 & 16 & 4865.350 & 4919.146\\
& 3 & -2266.603 & 30 & 4577.206 & 4695.558\\
& 4 & -2225.871 & 48 & 4527.742 & 4732.168\\
& 5 & -2182.390 & 70 & 4480.781 & 4792.799\\
\addlinespace
MNAR & 2 & -3074.628 & 22 & 6181.256 & 6267.330\\
& 3 & -2889.040 & 39 & 5840.081 & 6006.849\\
& 4 & -2841.108 & 60 & 5782.215 & 6051.197\\
& 5 & -2800.336 & 85 & 5746.671 & 6139.385\\
\bottomrule
\end{tabular}
\end{table}
For both the MAR and MNAR models, the BIC indicates a three-state model
fits best, whilst the AIC indicates a five-state model (or higher) fits
best. Favouring simplicity, we follow the BIC scores here, and focus on
the three-state models.
We first consider the estimates of the MAR model. The estimated means
and standard deviations are \begin{equation*}
\gvc{\mu} = [2.315,4.339, 5.7] \quad \quad \gvc{\sigma} = [0.821,0.619, 0.567].
\end{equation*} Hence, the states are ordered, with state 1 being the
least severe, and state 3 the most severe. The prior probabilities of
the states, for treatment with placebo and drug respectively, are
\begin{equation*}
\gvc{\pi}_\text{placebo} = [0,0.333,0.667] \quad \quad \gvc{\pi}_\text{drug} = [0.005,0.307,0.689],
\end{equation*} and the transition probability matrices (with initial
states in rows and subsequent states in columns) are \begin{equation*}
\mathbf{T}_\text{placebo} = \left[ \begin{matrix} 0.963 & 0.004 & 0.032 \\ 0.118 & 0.878 & 0.004
\\ 0.027 & 0.046 & 0.927 \end{matrix} \right] \quad \quad \mathbf{T}_\text{drug} = \left[ \begin{matrix} 1 & 0 & 0 \\ 0.231 & 0.764 & 0.005
\\ 0.073 & 0.307 & 0.62 \end{matrix} \right].
\end{equation*} As expected, the initial state probabilities show little
difference between the treatments (as the initial measurement was
conducted before treatment commenced), but the transition probabilities
indicate that for those who were administered a real drug, transitions
to less severe states are generally more likely, indicating
effectiveness of the drugs. This is particularly marked for the most
severe state, where the probability of remaining in that state is 0.927
with a placebo, but 0.62 with a drug.
We next consider the three-state MNAR model. The means and standard
deviations are \begin{equation*}
\gvc{\mu} = [2.325,4.424, 5.756] \quad \quad \gvc{\sigma} = [0.833,0.668, 0.547]
\end{equation*} showing the same ordering of states in terms of
severity. The prior probabilities for placebo and drug groups are
\begin{equation*}
\gvc{\pi}_\text{placebo} = [0,0.393,0.607] \quad \quad \gvc{\pi}_\text{drug} = [0.004,0.349,0.647],
\end{equation*} and the transition probability matrices are
\begin{equation*}
\mathbf{T}_\text{placebo} = \left[ \begin{matrix} 0.93 & 0.005 & 0.065 \\ 0.123 & 0.872 & 0.005
\\ 0.026 & 0.031 & 0.942 \end{matrix} \right] \quad \quad \mathbf{T}_\text{drug} = \left[ \begin{matrix} 1 & 0 & 0 \\ 0.239 & 0.761 & 0.001
\\ 0.073 & 0.331 & 0.596 \end{matrix} \right].
\end{equation*} These estimates are close to those of the MAR model,
indicating little initial difference between the conditions, but
effectiveness of the drugs in the transition probabilities, which are
higher towards the less severe states than for the placebo condition.
\begin{table}
\caption{\label{tab:conditional-missingness-parameters}Parameter estimates of the state dependent logistic regression models for missingness, with lower and upper reflecting the lower and upper bounds of the approximate $95\%$ confidence intervals.}
\centering
\begin{tabular}[t]{llrrr}
\toprule
state & parameter & estimate & lower & upper\\
\midrule
1 & (Intercept) & 2.635 & 1.998 & 3.272\\
& week & 0.149 & 0.028 & 0.269\\
& main & -4.511 & -5.037 & -3.984\\
\addlinespace
2 & (Intercept) & 7.976 & 1.261 & 14.691\\
& week & -0.510 & -1.978 & 0.959\\
& main & -13.014 & -20.014 & -6.014\\
\addlinespace
3 & (Intercept) & 1.108 & 0.303 & 1.913\\
& week & 0.710 & 0.538 & 0.883\\
& main & -5.029 & -5.822 & -4.235\\
\bottomrule
\end{tabular}
\end{table}
Results of the state-dependent models for missingness are provided in
Table \ref{tab:conditional-missingness-parameters}. For all three
states, the confidence interval for the effect of \texttt{main} excludes
0, indicating a significantly lower proportion of missing ratings at the
main measurement weeks. In state 1 and 3, the confidence interval for
the effect of \texttt{week} also excludes 0, indicating a higher rate of
missing ratings over time, possibly due to attrition. For state 2, the
effect of \texttt{week} is not significant. Figure
\ref{fig:hmm-missing-prediction} depicts the predicted probability of
missing ratings for each state and week. This shows that in state 2, the
chance of missing data on the main measurement weeks is small at
\(p(M_t|S_t = 2)=0.003\), while it is high at \(p(M_t|S_t = 2) = 0.997\)
on the other weeks. In the other states, the probabilities are less
extreme, with missing (and non-missing) data occurring on the main
measurement weeks and the other weeks as well. In the final week 6,
those in the most severe state 3 are the most likely to have missing
data with \(p(M_t|S_t = 3) = 0.585\). For those in the least severe
state 1, the probability of missingness in week 6 is also substantial at
\(p(M_t|S_t = 1) = 0.272\).
\begin{figure}
{\centering \includegraphics[width=0.6\linewidth]{hmm-missing-prediction-1}
}
\caption{Predicted probability of missing IMPS Item 79 ratings by week for each state in the three-state MNAR hidden Markov model.}\label{fig:hmm-missing-prediction}
\end{figure}
Disregarding the modelling of missingness, the parameters of the MAR and
MNAR model seem reasonably close. This could be an indication that
missingness is independent of the hidden states and data are possibly
MAR. The likelihood of the MAR is not directly comparable to that of the
MNAR model, as the latter is defined over two variables (the rating and
the binary \texttt{missing} variable), while the former involves just a
single variable. However, we can test for equivalence by fitting a
constrained version of the MNAR model, where the parameters of the
missingness model are constrained to be identical over the states.
Unlike the MAR model, this restricted version of the MNAR model accounts
for patterns of missingness, allowing these to depend on \texttt{week}
and \texttt{main}, but not on the hidden state. A likelihood ratio test
indicates that this restricted model fits less well,
\(\chi^2(6) = 1328.57\) \(p < .001\). Hence, there is evidence that the
MNAR model is preferable to the MAR model and that missingness is indeed
state-dependent.
\begin{figure}
{\centering \includegraphics[width=0.7\linewidth]{imps74-posterior-state-plots-1}
}
\caption{Proportions of maximum a posteriori (MAP) state assignments over weeks for the medication and placebo groups, according to the MAR and MNAR model.}\label{fig:imps74-posterior-state-plots}
\end{figure}
Whilst the MAR and MNAR model provide roughly equivalent parameters for
the severity ratings in the three states, when comparing the maximum a
posteriori (MAP) state classifications from the Viterbi algorithm
(Figure \ref{fig:imps74-posterior-state-plots}), we see that state
classifications for the the MAR model tend to be for the more severe
states. According to the MNAR model, during the main measuring weeks,
missing values are relatively likely in the least severe state 1. Hence,
those with missing values are more likely to be assigned to the least
severe state. This is in line with the analysis of
\citet{hedeker1997application}, who found evidence that dropouts in the
medication condition showed more improvement in their symptoms.
It is worthwhile to note that the MAP states are also determined for
time points with missing data, as the transition probabilities make
certain states more probable than others, even when there is no direct
measurement available. This provides a potentially meaningful basis to
impute missing values, with another option being expected rating as a
function of the posterior probability of all states. As imputation is
not the focus of this study, we leave this to be investigated in future
work.
\hypertarget{discussion}{%
\section{Discussion}\label{discussion}}
Previous work on missing data in hidden Markov models has mostly
focussed on cases where missing values are assumed to be missing at
random (MAR). Here, we addressed situations where data is missing not at
random (MNAR), and missingness depends on the hidden states. Simulations
showed that including a submodel for state-dependent missingness in a
HMM is beneficial when missingness is indeed state-dependent, whilst
relatively harmless when data is MAR. However, when the form of
state-dependent missingness is misspecified (e.g.~the effect of
measurable covariates on missingness ignored), results may be biased. In
practice, it is therefore advisable to consider the potential effect of
covariates in the state-dependent missingness models. A reasonable
strategy is to first model patterns of missingness through e.g.~logistic
regression, and then include important predictors from this analysis
into the state-dependent missingness models. Applying this strategy to a
real example about severity of schizophrenia in a clinical trial with
substantial missing data, we showed that assuming data is MAR may lead
to possible misclassification of patients to states (towards more severe
states in this example). Whilst the ground truth is unavailable in such
real applications, model comparison can be used to justify a
state-dependent missingness model. Using flexible analysis tools such as
the depmixS4 package \citep{depmixS4} makes specifying, estimating, and
comparing hidden Markov models with missing data specifications
straightforward. There is then little reason to ignore potentially
non-ignorable patterns of missing data in hidden Markov modelling.
Another approach to dealing with non-ignorable missingness (MNAR) is the
pattern-mixture approach of Little
\citetext{\citeyear{little1993pattern}; \citeyear{little1994class}}. The
main idea of this approach is to group units of observations
(e.g.~patients) by the pattern of missing data, and allowing the
parameters of a statistical model for the observations \(Y\) to
dependent on the missingness \emph{pattern} \(M_{1:\ensuremath{T}}\). There are
certain similarities between this approach and modelling missingness as
state-dependent. Rather than conditionalizing on a pattern of missing
values, a hidden Markov model conditionalizes on a pattern (sequence) of
hidden states, \(\ensuremath{s}_{1:\ensuremath{T}}\), and the marginal distribution of the
observations is effectively a multivariate mixture \begin{equation}
\ensuremath{p}(\ensuremath{Y}_{1:\ensuremath{T}} | \ensuremath{\mbox{\boldmath$\theta$}}) = \sum_{\ensuremath{s}_{1:\ensuremath{T}} \in \mathcal{S}^\ensuremath{T}} \sum_{m_{1:\ensuremath{T}} \in \mathcal{M}^\ensuremath{T}} \ensuremath{p}(\ensuremath{Y}_{1:\ensuremath{T}} | m_{1:\ensuremath{T}} , \ensuremath{s}_{1:\ensuremath{T}}, \ensuremath{\mbox{\boldmath$\theta$}}) \ensuremath{p}(m_{1:\ensuremath{T}} | \ensuremath{s}_{1:\ensuremath{T}}, \ensuremath{\mbox{\boldmath$\theta$}}) \ensuremath{p}(\ensuremath{s}_{1:\ensuremath{T}} | \ensuremath{\mbox{\boldmath$\theta$}})
\end{equation} (note that \(\ensuremath{\mbox{\boldmath$\theta$}}\) here includes all parameters, so
\(\phi\)). A pattern-mixture model would instead propose
\begin{equation}
\ensuremath{p}(\ensuremath{Y}_{1:\ensuremath{T}} | \ensuremath{\mbox{\boldmath$\theta$}}) = \sum_{m_{1:\ensuremath{T}} \in \mathcal{M}^\ensuremath{T}} \ensuremath{p}(\ensuremath{Y}_{1:\ensuremath{T}} | m_{1:\ensuremath{T}}, \ensuremath{\mbox{\boldmath$\theta$}}) \ensuremath{p}(m_{1:\ensuremath{T}} | \ensuremath{\mbox{\boldmath$\theta$}}) .
\end{equation} Trivially, if we set the number of hidden states to
\(\ensuremath{K} = 1\), both models are the same. Another trivial equivalence is
through a one-to-one mapping between \(m_{1:\ensuremath{T}}\) and
\(\ensuremath{s}_{1:\ensuremath{T}}\). This could be obtained of by setting \(\ensuremath{K} = 2\),
assuming the Markov process is of order \(\ensuremath{T}\), and fixing
e.g.~\(\ensuremath{p}(M_{t} = 0|\ensuremath{S} = 1) = 1\) and
\(\ensuremath{p}(M_{t} = 1|\ensuremath{S} = 2) = 1\). More interesting is to investigate
cases where the procedures are similar, but not necessarily equivalent.
The general pattern-mixture model is often underidentified
\citep{little1993pattern}. For time-series of length \(\ensuremath{T}\), there are
\(2^\ensuremath{T}\) possible missing data patterns. Without further restrictions,
estimating the mean vectors and covariance matrices for all these
components is not possible, due to the structural missing data in those
patterns. The state-dependent MNAR hidden Markov model is identifiable
insofar as the HMM for the observed variable \(\ensuremath{Y}\) is identifiable.
It is convenient, but not necessary, to assume a first-order Markov
process. Higher-order Markov processes may allow the model to capture
complex effects of missingness. Another option is to use the missingness
indicator \(M_t\) as a covariate on initial and transition
probabilities, rather than a dependent variable. We leave investigation
of such alternative models to future work.
\bibliographystyle{agsm}
|
2,869,038,156,420 | arxiv | \section{Introduction}
The subject of this note is the poset $\ensuremath{\mathcal{B}}(A)$ (denoted in \cite{HHLN} by $\BSub(A)$) of Boolean subalgebras of an orthomodular lattice $A$, and the poset $\ensuremath{\mathcal{A}}(M)$ of abelian subalgebras of an AW*-algebra $M$. Study of such posets grew from the topos approach to the foundations of physics initiated by Isham and others \cite{DoringIsham,Isham} where the topos of presheaves on such posets were the primary object of study, providing ``classical snapshots'' of a quantum system.
A number of papers \cite{Doring,Hamhalter,HHLN,HardingNavara,Lindenhovius} have considered aspects of such posets and their relationships to the originating orthomodular structure or operator algebra. In \cite{HHLN} is given a direct method to reconstruct an orthoalgebra $A$ from the poset $\ensuremath{\mathcal{B}}(A)^*$ of Boolean subalgebras of $A$ having at most 16 elements. Such posets arising this way are called {\em orthohypergraphs}. They behave much like projective geometries, and morphisms between them are defined much as in the case of classical projective geometry \cite{Faure}. Hypergraphs arising in this way from an orthoalgebra are characterized, and it is shown that there is a ``near'' categorical equivalence between the category of orthoalgebras and that of such orthohypergraphs. This can be viewed as an extension of the technique of Greechie diagrams \cite{GreechieD}.
In this note we specialize and adapt results of \cite{HHLN} to the setting of orthomodular lattices and the orthomodular lattice homomorphisms between them. We then use a version of Dye's theorem \cite{Hamhalter2,HeunenReyes} to lift results to the AW*-algebra setting. We provide a functor from the category of AW*-algebras and their normal Jordan $*$-homomorphisms to the category of orthohypergraphs and their normal hypergraph morphisms that is injective on objects and full and faithful on morphisms provided no type $I_2$ summands are present.
This note is arranged in the following way. In the second section we adapt and simplify results of \cite{HHLN} to apply to orthomodular posets and lattices, and in the third section we lift these results to the AW*-setting using Dye's theorem \cite{Hamhalter2,HeunenReyes}. Results, notation, and terminology of \cite{HHLN} will be assumed. For general reference on orthomodular structures see \cite{DvurPulm:Trends,Kalmbach}.
\section{hypergraphs of orthomodular posets and lattices}
In \cite{HHLN} hypergraphs of orthoalgebras were characterized and a near equivalence between the category of orthoalgebras and orthohypergraphs was given. Here these results are specialized and simplified for application to orthomodular posets and orthomodular lattices. Although we recall a few basics, the reader should consult \cite{HHLN} for background and notation. We begin with the following from \cite[Def.~6.1]{HHLN}.
\begin{definition}
A {\em hypergraph} is a triple $\mathcal{H}=(P,L,T)$ consisting of a set $P$ of {\em points}, a set $L$ of {\em lines}, and a set $T$ of {\em planes}. A line is a set of 3 points, and a plane is a set of 7 points where the restriction of the lines to these 7 points is as shown below.
\vspace{2ex}
\begin{center}
\begin{tikzpicture}[scale = 1]
\draw[fill] (-6,.6) circle(1.5pt); \draw[fill] (-4,.6) circle(1.5pt);\draw[fill] (-3,.6) circle(1.5pt);\draw[fill] (-2,.6) circle(1.5pt);
\draw[thin] (-4,.6)--(-2,.6);
\node at (-6,-.5) {point}; \node at (-3,-.5) {line}; \node at (1,-.5) {plane};
\draw[fill] (0,0) circle(1.5pt); \draw[fill] (1,0) circle(1.5pt); \draw[fill] (2,0) circle(1.5pt); \draw[fill] (1,1.414) circle(1.5pt); \draw[fill] (1,.48) circle(1.5pt); \draw[fill] (.5,.707) circle(1.5pt); \draw[fill] (1.5,.707) circle(1.5pt);
\draw[thin] (0,0)--(2,0)--(1,1.414)--(0,0)--(1.5,.707); \draw[thin] (1,1.414)--(1,0); \draw[thin] (.5,.707)--(2,0);
\end{tikzpicture}
\end{center}
\end{definition}
From an orthoalgebra $A$, one creates its hypergraph $\mathcal{H}(A)$ whose points, lines, and planes are the Boolean subalgebras of $A$ having 4, 8, and 16 elements, respectively \cite[Defn.~6.3]{HHLN}.
When dealing with hypergraphs, a plane is represented as a certain configuration of 7 points and 6 lines. However, in the case of hypergraphs arising as $\mathcal{H}(A)$ for an orthoalgebra $A$, not every such configuration of points and lines is indeed a plane. So in general, planes must be separate entities. On the other hand \cite[Prop.~6.7]{HHLN} any such configuration of points and lines in a hypergraph $\mathcal{H}(A)$ for an orthomodular poset $A$ will constitute a plane. So when working with orthomodular posets, we aim to treat planes as a derived notion. This will be our setting, and we introduce new terminology for it.
\begin{definition}
A pre-orthogeometry $\mathcal{G}=(P,L)$ is a set $P$ of points and a set $L$ of lines where each line is a set of 3 points such that any two points are contained in at most one line. A subspace of a pre-orthogeometry is a set $S$ of points such that whenever 2 points of a line belong to $S$, then the third point of the line also belongs to $S$.
\end{definition}
Let $\mathcal{H}(A)$ be the hypergraph of an orthoalgebra $A$ and $p,q$ be distinct points of it. So $p,q$ are 4-element Boolean subalgebras of $A$. Then $p,q$ lie on a line $\ell$, that is an 8-element Boolean subalgebra of $A$, iff there are orthogonal elements $a,b\in A\setminus\{0,1\}$ such that $p=\{0,a,a',1\}$ and $q=\{0,b,b',1\}$. In this case, $\{0,a,a',b,b',a\vee b,a'\wedge b',1\}$ is the only 8-element Boolean subalgebra of $A$ that contains $p,q$. Thus all points of $\mathcal H(A)$ are contained in at most one line.
For an arbitrary hypergraph $\mathcal{H}=(P,L,T)$ such that any two points are contained in at most one line, $\mathcal H$ gives a pre-orthogeometry $\mathcal{H}_*=(P,L)$ by forgetting its set of planes. For a pre-orthogeometry $\mathcal{G}=(P,L)$, construct a hypergraph $\mathcal{G}^*=(P,L,T)$ whose planes are subspaces of 7 points such that there are exactly 6 lines that contain at least two of these points, and these 7 points and 6 lines form a configuration that could be a plane. We always have $(\mathcal{G}^*)_*=\mathcal{G}$, and if $\mathcal{H}$ is the hypergraph of an orthomodular poset we have $(\mathcal{H}_*)^*=\mathcal{H}$ \cite[Prop. 6.7]{HHLN}. We next provide a link between pre-orthogeometries and the orthodomains that are described in \cite[Defn.~4.3]{HHLN}. This allows access to results of \cite{HHLN}.
\begin{lemma} \label{lk}
Let $\mathcal{G}=(P,L)$ be a pre-orthogeometry. Then considering $\mathcal{G}^*$ as a poset by adding a new bottom $\bot$ and with the natural order of inclusion among the points, lines, and planes, we have that $\mathcal{G}^*$ is an orthodomain.
\end{lemma}
\begin{proof}
In this poset $\mathcal{G}^*$ the points are atoms, each line is the join of the atoms it contains, and each plane is the join of the atoms it contains. So it is atomistic, and since it is of finite height, it is directedly complete and the atoms are compact. By construction, for each element $x\in\mathcal{G}^*$ the downset $\downarrow x$ is a Boolean domain. Finally, if $x,y$ are distinct atoms of $\mathcal{G}^*$ and are covered by $w$, then $x,y$ are points of $\mathcal{G}$ and $w$ is a line of $\mathcal{G}$ that contains $x,y$. Since two lines of $\mathcal{G}$ that contain the same two points must be equal, there is no other element of $\mathcal{G}^*$ of height 2 containing $x,y$. Since any plane of $\mathcal{G}^*$ that contains $x,y$ must contain the line $w$ they span, it follows that $w$ is the join $x\vee y$. Thus $\mathcal{G}^*$ satisfies the conditions of \cite[Defn.~4.3]{HHLN}, so is an orthodomain.
\end{proof}
\begin{definition}
An orthogeometry $\mathcal{G}=(P,L)$ is a pre-orthogeometry that is isomorphic to $(\mathcal{H}(A))_*$ for some orthomodular poset $A$. Thus its points and lines are configured as the 4 and 8-element Boolean subalgebras of $A$.
\end{definition}
We aim to characterize orthogeometries in elementary terms. As in \cite{HHLN}, key is the notion of a direction. In the setting of pre-orthogeometries, this definition simplifies considerably.
\begin{definition}
A direction $d$ for a point $p$ of a pre-orthogeometry $\mathcal{G}$ is an assignment $d(\ell)$ of either $\uparrow$ or $\downarrow$ to each line $\ell$ that contains $p$ such that two lines $\ell$ and $m$ obtain opposite values of $\uparrow$ and $\downarrow$ iff they lie as part of a subspace of $\mathcal{G}$ forming a plane with $\ell,m$ the only lines in this plane containing $p$.
\vspace{2ex}
\begin{center}
\begin{tikzpicture}[xscale=1.2,yscale=1.4]
\draw[fill] (0,0) circle(1.5pt); \draw[fill] (1,0) circle(1.5pt); \draw[fill] (2,0) circle(1.5pt); \draw[fill] (1,1.414) circle(1.5pt); \draw[fill] (1,.48) circle(1.5pt); \draw[fill] (.5,.707) circle(1.5pt); \draw[fill] (1.5,.707) circle(1.5pt);
\draw[thin, dashed] (0,0)--(2,0)--(1,1.414)--(0,0)--(1.5,.707); \draw[very thick] (1,1.414)--(1,0); \draw[thin, dashed] (.5,.707)--(2,0); \draw[very thick] (0,0)--(2,0);
\node at (-.4,0) {$\ell$}; \node at (1.4,1.414) {$m$}; \node at (1,-.35) {$p$};
\end{tikzpicture}
\end{center}
\vspace{1ex}
The directions for $\mathcal{G}$ are the directions for the points of $\mathcal{G}$ and two additional directions $0,1$.
\end{definition}
Note that since any two points are contained in at most one line, any two intersecting lines span at most one plane.
The idea behind a direction is as follows. In the hypergraph of an orthoalgebra $A$, a point $p$ is a 4-element Boolean subalgebra $\{0,a,a',1\}$ for some $a\neq 0,1$ in $A$. A line $\ell$ containing $p$ is an 8-element Boolean subalgebra that contains $a$, and a plane containing $p$ is a 16-element Boolean subalgebra containing $a$. The direction $d$ for $p$ says whether $a$ occurs as an atom or coatom in $\ell$. If $a$ occurs as a coatom of $\ell$ and atom of $m$, then there are $b<a$ in $\ell$ and $a<c$ in $m$. Then there is a 16-element Boolean subalgebra generated by the chain $b<a<c$, and this is the plane containing $\ell$ and $m$ in the definition of a direction.
\begin{lemma} \label{dir}
Let $\mathcal{G}=(P,L)$ be a pre-orthogeometry.
A direction $d$ of $\mathcal{G}$ for a point $p\in P$ extends uniquely to a direction $d^*$ of the orthodomain $\mathcal{G}^*$ for $p$, and a direction $e$ of $\mathcal{G}^*$ for $p$ restricts to a direction of $\mathcal{G}$ for $p$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lk} $\mathcal{G}^*$ is an orthodomain. By \cite[Defn.~4.9]{HHLN} a direction $e$ of the orthodomain $\mathcal{G}^*$ for a point $p$ is a mapping whose domain is the upper set ${\uparrow}p$. This map associates to each $y\in{\uparrow}p$ a pair of elements $e(y)=(v,w)$ called a principle pair in the Boolean domain ${\downarrow}y$. In the current circumstances this means that $e(p)=(p,p)$ and for a line $\ell$ with $p<\ell\leq y$ that $e(\ell)$ is either $(p,\ell)$ or $(\ell,p)$. Here, as in \cite[Cor.~2.21 ff]{HHLN} we denote $(p,\ell)$ by $\downarrow$ and $(\ell,p)$ by $\uparrow$. For a plane $w$ containing $p$, if $p$ lies on three lines of $w$ then $e(w)$ is either $(p,w)$ or $(w,p)$, and if $p$ lies on only two lines of $w$ then $e(w)$ is either $(\ell,m)$ or $(m,\ell)$ where $\ell,m$ are the unique lines of $w$ that contain $p$. Additionally, if $p\leq z\leq y$ and $d(y)=(u,v)$ then \cite[Defn.~4.9]{HHLN} we have $d(z)=(z\wedge u,z\wedge v)$. Finally \cite[Defn.~4.9]{HHLN} if $\ell,m$ are lines containing $p$ and $d(\ell),d(m)$ take opposite values of $\uparrow,\downarrow$, then $\ell\vee m$ exists and is a plane containing both.
So if $e$ is a direction of the orthodomain $\mathcal{G}^*$ for $p$, by restricting $e$ to the lines of $\mathcal{G}$ that contain $p$ and for such a line $\ell$ using $\downarrow$ for $(p,\ell)$ and $\uparrow$ for $(\ell,p)$ we have the restriction of $e$ is an assignment of $\downarrow,\uparrow$ to each line of $\mathcal{G}$ containing $p$. To show this is a direction of the pre-orthogeometry $\mathcal{G}$ we must show that if $\ell,m$ are lines containing $p$, then $e(\ell)$ and $e(m)$ take opposite values of $\downarrow,\uparrow$ iff $\ell,m$ belong to a plane and are the only two lines of this plane containing $p$. If $e(\ell),e(m)$ take different values of $\downarrow,\uparrow$, then \cite[Defn.~4.9]{HHLN} gives that $w=\ell\vee m$ is a plane. If $p$ lies on three lines of this plane, we have from the first paragraph of the proof that $e(w)$ is either $(p,w)$ or $(w,p)$, and it follows that either $e(\ell)=(p,\ell)= {\downarrow}$ and $e(m)=(p,m)={\downarrow}$ or $e(\ell)=(\ell,p)={\uparrow}$ and $e(m)=(m,p)={\uparrow}$, contrary to assumption. Conversely, if $\ell,m$ belong to a plane $w$ and are the only two lines of $w$ containing $p$, the first paragraph shows $e(w)$ is either $(\ell,m)$ or $(m,\ell)$. In the first case, $e(\ell)=(\ell\wedge\ell,m\wedge\ell)=(\ell,p)={\uparrow}$ and $e(m)=(\ell\wedge m,m\wedge m)=(p,m)={\downarrow}$. The second case is symmetric, and in either case $e$ takes opposite values of $\downarrow,\uparrow$ at $\ell,m$. So $e$ restricts to a direction of $\mathcal{G}$.
Let $d$ be a direction of $\mathcal{G}$ for $p$. We wish to extend this to a direction $e$ of $\mathcal{G}^*$. For a line $\ell$ containing $p$ set $e(\ell)=(p,\ell)$ if $d(\ell)={\downarrow}$ and set $e(\ell)=(\ell,p)$ if $d(\ell)={\uparrow}$. Suppose $w$ is a plane of $\mathcal{G}^*$ containing $p$. Then $w$ is a subspace of $\mathcal{G}$ with 7 points and 6 lines that is configured as a plane. To define $e(w)$ we consider the cases where there are 3 lines of $w$ containing $p$, and where only 2 lines of $w$ contain $p$.
Suppose $w$ has 3 lines $\ell,m,n$ that contain $p$. If $d$ has opposite values on two of these lines, say $d(\ell)={\downarrow}$ and $d(m)={\uparrow}$, then by the definition of a direction of a pre-orthogeometry $\ell,m$ must belong to a plane $z$ in which $\ell,m$ are the only lines containing $p$. But the five points on the lines $\ell,m$ belong to both the planes $w,z$, and it is easily seen that since both $w,z$ are subspaces of $\mathcal{G}$ that the other two points of the plane $w$ belong to $z$. But this implies that $w=z$, a contradiction. We conclude that $d$ takes the same value of $\downarrow,\uparrow$ on all three of $\ell,m,n$. If this value is $\downarrow$ set $e(w)=(p,w)$, and if this value is $\uparrow$, set $e(w)=(w,p)$. If $w$ has only two lines $\ell,m$ that contain $p$, then the definition of a direction of a pre-orthogeometry shows that $d$ takes opposite values of $\downarrow,\uparrow$ on $\ell,m$. If $d(\ell)={\downarrow}$ and $d(m)={\uparrow}$, set $e(w)=(m,\ell)$, and if $d(\ell)={\uparrow}$ and $d(m)={\downarrow}$ set $e(w)=(l,m)$.
We use \cite[Defn.~4.9]{HHLN} to show $e$ is a direction for $p$ in the orthodomain $\mathcal{G}^*$. Suppose $p\leq y$. By construction $e(y)$ is a principal pair for $p$ in the Boolean domain ${\downarrow}y$. Also, if $\ell,m$ are lines containing $p$ with $e(\ell)=(p,\ell)$ and $e(m)=(m,p)$, then $d(\ell)={\downarrow}$ and $d(m)={\uparrow}$. So by the definition of a direction of a pre-orthogeometry there is a plane $w$ containing $\ell,m$ in which these are the only lines containing $p$, and we have seen this is the only plane containing $\ell,m$. Thus $\ell\vee m = w$ and $w$ covers $\ell,m$. Finally, if $p\leq z < y$ and $e(y)=(u,v)$ we must show $e(z)=(u\wedge z,v\wedge z)$. If $z=p$ this is clear. So the only case of interest is when $z=\ell$ and $y=w$ is a plane. If there are 3 lines of $w$ containing $p$ and $d(\ell)={\downarrow}$, then $e(w)=(p,w)$ and $e(\ell)=(p,\ell)$ and our condition is verified. If there are 3 lines of $w$ containing $p$ with $d(\ell)={\uparrow}$, then $e(w)=(w,p)$ and $e(\ell)=(\ell,p)$, and again our condition is verified. If there are only two lines $\ell,m$ of $w$ containing $p$, then as we have seen $d$ takes opposite values on these lines. Suppose $d(\ell)={\downarrow}$ and $d(m)={\uparrow}$. Then $e(w)=(m,\ell)$, $e(\ell)=(p,\ell)$ and $e(m)=(m,p)$. Again our condition is verified. Thus $e$ is a direction of $\mathcal{G}^*$ and it clearly restricts to $d$.
\end{proof}
We are now in a position to establish our characterization of orthogeometries, those configurations of points and lines that arise as the 4 and 8-element Boolean subalgebras of an orthomodular poset. In this formulation we use the notion of a triangle.
\begin{definition}
A triangle in an orthogeometry is a set of three points, any two of which belong to a line. A triangle is non-degenerate if the three points are distinct and do not all lie on the same line.
\end{definition}
We will make use of several results in \cite{HHLN} which depend on the condition of orthodomains being \emph{proper}, i.e., orthodomains without maximal elements of height 1 or less. The corresponding notion for pre-orthogeometries is that every point is contained in some line. We will call such a pre-orthogeometry \emph{proper} as well. Clearly an orthodomain $X$ is proper if and only if its associated pre-orthogeometry $(P,L)$ is proper, where $P$ consists of the atoms of $X$ and $L$ consists of the elements covering some atom in $X$. We call an orthoalgebra $A$ proper if it does not have blocks of four or fewer elements. Then $A$ is proper if and only if its associated orthodomain $\ensuremath{\mathcal{B}}(A)$ of Boolean subalgebras of $A$ is proper.
\begin{theorem} \label{orthogeometry}
A proper pre-orthogeometry $\mathcal{G}=(P,L)$ is an orthogeometry iff the following hold.
\begin{enumerate}
\item Each non-degenerate triangle is contained in a subspace that is a plane.
\item Every point has a direction.
\end{enumerate}
\end{theorem}
\begin{proof}
Suppose $\mathcal{G}$ is an orthogeometry. Then up to isomorphism, it is the set of 4 and 8-element Boolean subalgebras of an orthomodular poset $A$.
Suppose $p,q,r$ is a non-degenerate triangle of $\mathcal{G}$. Then there are pairwise orthogonal $a,b,c\in A$ with none equal to $0,1$ with $p=\{0,a,a',1\}$, $q=\{0,b,b',1\}$ and $r=\{0,c,c',1\}$. In an orthomodular poset, such pairwise orthogonal elements generate a 16-element Boolean subalgebra, a fact that is not true in general orthoalgebras. So this triangle is contained in a plane of $\mathcal{G}$. Thus (1) holds.
For (2), we first show that $\mathcal{G}^*$ is isomorphic to the collection of at most 16-element Boolean subalgebras of $A$. Surely any 16-element Boolean subalgebra of $A$ has its 4-element subalgebras being the points of a subspace of $\mathcal{G}$ that are configured as a plane. So each 16-element Boolean subalgebra of $A$ gives a plane of $\mathcal{G}^*$. But by \cite[Prop.~6.7]{HHLN} any configuration of points in $\mathcal{G}$ in the form of a plane arises as the 4-element Boolean subalgebras of some 16-element Boolean subalgebra of $A$. Thus $\mathcal{G}^*$ is isomorphic to the poset of at most 16-element Boolean subalgebras of $A$. Then \cite[Thm.~4.16]{HHLN} has the consequence that each point of the orthodomain $\mathcal{G}^*$ has a direction, and then Lemma~\ref{dir} yields that each point of the pre-orthogeometry $\mathcal{G}$ has a direction. Thus (2) holds.
For the converse, suppose that (1) and (2) hold for $\mathcal{G}$. By Lemma~\ref{lk} $\mathcal{G}^*$ is an orthodomain, which is clearly proper, since $\mathcal G$ is proper. By Lemma~\ref{dir}, each point of $\mathcal{G}^*$ has a direction. The basic element $\bot$ of any orthodomain always has a direction. Thus $\mathcal{G}^*$ is a short orthodomain (short meaning its height is at most 3 \cite[Defn.~5.13]{HHLN}) with enough directions (meaning each basic element has a direction). Then by \cite[Thm.~5.18]{HHLN} there is an orthoalgebra $A$ with $\mathcal{G}^*$ being isomorphic to the Boolean subalgebras of $A$ with at most 16 elements. Thus $\mathcal{G}$ is isomorphic to the poset of Boolean subalgebras of $A$ with at most 8 elements.
It remains to show that the orthoalgebra $A$ constructed in the previous paragraph is an orthomodular poset. Orthomodular posets are characterized among orthoalgebras by the property that for orthogonal elements $a,b$ we have that $a\oplus b$ is the least upper bound of $a,b$ and not simply a minimal upper bound \cite[Prop. 1.5.6]{DvurPulm:Trends}. It is sufficient to show this under the assumption that $a,b\neq 0,1$. Suppose $c\in A$ is an upper bound of $a,b$. We must show $a\oplus b\leq c$. We may assume $c\neq 0,1$. Set $p=\{0,a,a',1\}$, $q=\{0,b,b',1\}$ and $r=\{0,c,c',1\}$. Since $a,b,c'$ are pairwise orthogonal, these three points form a non-degenerate triangle in $\mathcal{G}$. So by condition (1) there is a plane of $\mathcal{G}$ that contains them. Then $a,b,c$ lie in a 16-element Boolean subalgebra of $A$, and this yields $a\oplus b\leq c$ since in a Boolean algebra $a\oplus b$ is the join $a\vee b$.
\end{proof}
We turn to the matter of identifying orthogeometries that arise from orthomodular lattices, and from complete orthomodular lattices. If $\mathcal{G}$ is the orthogeometry for an orthomodular poset $A$, then with appropriate operations \cite[Defn.4.15]{HHLN} on the set $\oDir(\mathcal{G}^*)$ of directions of $\mathcal{G}^*$ we have \cite[Thm.~4.16]{HHLN} that $\oDir(\mathcal{G}^*)$ is an orthomodular poset isomorphic to $A$. By Lemma~\ref{dir} there is a bijective correspondence between the directions $\oDir(\mathcal{G})$ of $\mathcal{G}$ and those of $\mathcal{G}^*$. This provides an orthoalgebra structure on $\oDir(\mathcal{G})$ making it isomorphic to $A$. The following consequence of these results will be useful.
\begin{proposition}\label{ds}
Let $\mathcal{G}$ be the orthogeometry of a proper orthomodular poset $A$. If $d$ is a direction for a point $p$ and $e$ is a direction for a point $q$, then $d\leq e$ iff $d=e$ or $p,q$ are distinct, both lie on a line $\ell$, $d(\ell)={\downarrow}$, and $e(\ell)={\uparrow}$. The directions $0,1$ are the least and largest directions.
\end{proposition}
\begin{center}
\begin{tikzpicture}
\draw[fill] (0,0) circle(.05); \draw[fill] (3,0) circle(.05);
\draw (.25,0) -- (2.75,0);
\node at (-.5,0) {$p$}; \node at (3.5,0) {$q$}; \node at (1.5,-.5) {$\ell$}; \draw[->] (1,.25)--(1,-.25); \draw[->] (2,-.25)--(2,.25);
\node at (1,.75) {$d$}; \node at (2,.75) {$e$};
\end{tikzpicture}
\end{center}
\begin{proof}
Let $d,e$ be directions of $\mathcal{G}$. Since $\mathcal{G}$ is an orthogeometry, Lemmas \ref{lk}, \ref{dir} give that the directions of $\mathcal{G}$ extend to directions $d^*,e^*$ of the orthodomain $\mathcal{G}^*$. In the orthomodular poset $\oDir(\mathcal{G}^*)$ we have $d^*\leq e^*$ iff $d^*\oplus (e^*)'$ is defined. By \cite[Defn.~4.15]{HHLN} this occurs iff $p,q$ are distinct points of a line $\ell$ with $d^*(\ell) = (p,\ell)$ and $(e^*)'(\ell)=(q,\ell)$. This is equivalent to having $d(\ell)=\,\downarrow$ and $e(\ell)=\,\uparrow$.
\end{proof}
For a set $D$ of directions of a proper orthogeometry $\mathcal{G}$ and a direction $e$ of $\mathcal{G}$, we call $e$ a cone of $D$ if $e$ is an upper bound of $D$ in the natural ordering of $\oDir(\mathcal{G})$.
Note that the direction $1$ is a cone of any set of directions; if $1$ is an element of $D$ then $1$ is the only cone of $D$; every direction is a cone of the empty set of directions; and a direction $e$ is a cone of $D$ iff it is a cone of $D\setminus\{0\}$.
The case when $D$ is a singleton other than $0,1$ is given by Proposition~\ref{ds}. The remaining cases are addressed by the following.
\begin{proposition}
For a proper orthogeometry $\mathcal G$, let $D$ be a non-empty set of directions with none equal to 0,1. For each $d\in D$ let $p_d$ be the point such that $d$ is a direction for $p_d$. Then $e$ is a cone of $D$ iff $e=1$ or $e$ is a direction for a point $q$ and the following hold for each $d\in D\setminus\{e\}$
\vspace{1ex}
\begin{enumerate}
\item $q$ is distinct from each $p_d$
\item there is a line $\ell_d$ containing $p_d$ and $q$
\item $d(\ell_d)=\,\downarrow$ and $e(\ell_d)=\,\uparrow$
\end{enumerate}
\end{proposition}
\begin{proof}
Assume $e\neq 1$. Clearly $e$ cannot be 0, hence it is a direction for a point $q$. If conditions (1)--(3) are satisfied, then by Proposition~\ref{ds} $e$ is a cone of $D$. Conversely, if $e$ is a cone, then by Proposition~\ref{ds} (1)--(3) hold.
\end{proof}
Call $e$ a minimal cone for a set $D$ of directions if $e$ is the least upper bound of $D$ in $\oDir(\mathcal{G})$. If $e=1$ this means that $e$ is the only cone of $D$. If $e=0$ this means it is a direction of $D$, and this occurs iff any direction in $D$ is 0. Other cases are covered by the following.
\begin{proposition}
Let $D$ be a
set of directions for a proper orthogeometry $\mathcal{G}$ and let a direction $e$ for a point $q$ be a cone of $D$. Assume that $e$ is neither $0$ or $1$. Then $e$ is a minimal cone for $D$ iff 0 is not a cone of $D$ and for any direction $f\neq e$ for a point $r\neq q$ that is a cone of $D$ there is a line $m$ containing $q,r$ and with $e(m)=\,\downarrow$ and $f(m)=\,\uparrow$.
\end{proposition}
\vspace{1ex}
\begin{center}
\begin{tikzpicture}[scale = 1.0]
\draw[fill] (0,1) circle (.05); \draw[fill] (0,-1) circle (.05); \draw[fill] (1.5,0) circle (.05); \draw[fill] (3.5,0) circle (.05); \draw (.2,.85) -- (1.3,.1); \draw (.2,-.85) -- (1.3,-.1); \draw (.3,1) to [out=0,in=135] (3.4,.2); \draw (.3,-1) to [out=0,in=-135] (3.4,-.2); \draw[dashed] (1.7,0)--(3.3,0);
\node at (-.4,1) {$p_1$}; \node at (-.4,-1) {$p_2$}; \node at (1.5,.4) {$q$}; \node at (3.7,.4) {$r$}; \node at (2,0) {$\downarrow$}; \node at (3,0) {$\uparrow$};
\node at (.8,1) {$\downarrow$}; \node at (2.8,.7) {$\uparrow$};
\node at (.8,-1) {$\downarrow$}; \node at (2.8,-.65) {$\uparrow$};
\node at (.5,.6) {$\downarrow$}; \node at (1,.3) {$\uparrow$};
\node at (.5,-.65) {$\downarrow$}; \node at (1,-.3) {$\uparrow$};
\end{tikzpicture}
\end{center}
\begin{proof}
If $e$ is a minimal cone for $D$ then $0$ cannot be a cone for $D$, and for any $f$ as described we have $e\leq f$ so Proposition~\ref{ds} gives the existence of the line $m$ and the behavior of the directions $e,f$ at $m$. Conversely, if these conditions are satisfied we must show that $e\leq g$ for any cone $g$ of $D$. By assumption, $0$ is not a cone of $D$, clearly $e\leq e$, $e\leq 1$,
and Proposition~\ref{ds} provides that $e\leq f$ for any $f$ as described in the statement. The remaining case, the direction $e'$ of $q$, cannot occur since this would entail that
$e,e'$ are cones of $D$, hence $0$ is a cone for $D$.
\end{proof}
Since minimal cones correspond to least upper bounds, we have the following.
\begin{theorem} \label{complete}
A proper orthogeometry $\mathcal{G}$ is the orthogeometry of a (complete) orthomodular lattice iff any set of (at least) two directions has a minimal cone.
\end{theorem}
We next extend matters to morphisms. A morphism of hypergraphs \cite[Defn.~7.2]{HHLN} is a partial mapping between their points satisfying certain conditions. These conditions involve the planes of these hypergraphs but can be adapted nearly verbatim to apply to orthogeometries, referring to the planes of the associated hypergraphs. We do not write this translation here, but provide the following more expedient version.
\begin{definition} \label{orthogeometry morphism}
For orthogeometries $\mathcal{G}_1$ and $\mathcal{G}_2$, a morphism of orthogeometries $\alpha:\mathcal{G}_1\to\mathcal{G}_2$ is a partial function between their points such that this partial mapping is a hypergraph morphism between $\mathcal{G}_1^*$ and $\mathcal{G}_2^*$.
\end{definition}
As in \cite{HHLN}, write $\alpha(p)=\bot$ when the partial mapping is not defined at $p$. A hypergraph morphism is called proper \cite[Defn.~7.8]{HHLN} when certain conditions involving the image of small sets
apply. A morphism of orthogeometries will be called proper when the associated morphism of hypergraphs is proper. For $\alpha$ a proper hypergraph morphism $\alpha$, there is an orthoalgebra morphism $f_\alpha$ between the associated orthoalgebras of directions \cite[Prop.~7.18]{HHLN}. An obvious translation of these results to the current setting gives the following.
\begin{proposition}
Let $\alpha:\mathcal{G}_1\to\mathcal{G}_2$ be a proper morphism of proper orthogeometries. Then there is an orthoalgebra morphism $f_\alpha:\oDir(\mathcal{G}_1)\to\oDir(\mathcal{G}_2)$.
\end{proposition}
This map $f_\alpha$ is described as in \cite[Defn.~7.15]{HHLN}. In essence, for a direction $d$ of a point $p$ and line $\ell$ through $p$ whose image is a line $\ell'$, the direction $f_\alpha(d)$ takes the same value at $\ell'$ as $d$ takes at $\ell$. So $f_\alpha$ is easily computed. Using this map $f_\alpha$ we can describe the orthogeometry morphisms that preserve binary and arbitrary joins.
\begin{definition}
A proper orthogeometry morphism $\alpha:\mathcal{G}_1\to\mathcal{G}_2$ is (finite) join preserving if it takes a minimal cone $e$ of a (finite) set $D$ of directions to a minimal cone $f_\alpha(e)$ of $f_\alpha[D]$. With an eye to future use with AW*-algebras, we call an orthogeometry morphism that preserves all joins \emph{normal}.
\end{definition}
By construction, the following is trivial.
\begin{proposition} \label{lifts}
A proper orthogeometry morphism $\alpha$ between proper orthogeometries preserves (finite) joins iff the associated orthoalgebra morphism $f_\alpha$ preserves (finite) joins.
\end{proposition}
Theorems~\ref{orthogeometry} and \ref{complete} give a characterization of the sets $\mathcal{G}=(P,L)$ of points and lines that arise as the 4-element and 8-element Boolean subalgebras of an orthomodular poset, \mbox{orthomodular} lattice, and complete orthomodular lattice. Adapting \cite[Thm.~7.20]{HHLN} we have a characterization of orthomodular poset, orthomodular lattice, and complete orthomodular lattice homomorphisms in terms of the orthogeometry morphisms between them satisfying various additional properties. We continue this process for Boolean algebras below.
\begin{theorem}
An orthogeometry $\mathcal{G}=(P,L)$ is the set of 4-element and 8-element Boolean subalgebras of a Boolean algebra iff any two points lie in a plane of $\mathcal{G}$. Proper orthogeometry morphisms between such orthogeometries correspond to a Boolean algebra homomorphisms.
\end{theorem}
\begin{proof}
This is a simple consequence of the facts \cite{Kalmbach} that an orthomodular poset in which any two elements lie in a Boolean subalgebra is a Boolean algebra, and that the orthomodular poset homomorphisms between Boolean algebras are exactly the Boolean algebra homomorphism between them.
\end{proof}
\begin{remark}
In the characterizations of sets $\mathcal{G}=(P,L)$ arising as orthomodular posets or as Boolean algebras, the ingredient that carries most weight is the existence of directions. This is a first order property in an appropriate language. So if $\mathcal{G}$ fails to have enough directions, then an application of the compactness theorem shows there is a finite reason for this failure. It would be interesting to see if a description of such $\mathcal{G}$ arising from Boolean algebras could be given without reference to directions, perhaps in terms of a finite number of forbidden configurations. This is an open problem.
\end{remark}
To conclude, we place results in a categorical context. Let $\mathbf{OA}$ be the category of orthoalgebras and the morphisms between them and $\mathbf{OH}$ be the category of orthohypergraphs and the hypergraph morphisms between them. We recall that an orthohypergraph is a hypergraph that is isomorphic to the hypergraph of some orthoalgebra. The following was established in \cite{HHLN} but with the functor we call $\mathcal{H}$ here being called $\mathcal{G}$ there.
\begin{theorem}
There is a functor $\mathcal{H}:\mathbf{OA}\to\mathbf{OH}$ that is essentially surjective on objects, injective on objects with the exception of the 1-element and 2-element orthoalgebras, and full and faithful with respect to proper orthoalgebra morphisms and proper hypergraph morphisms.
\end{theorem}
Let $\mathbf{OMP}$ be the full subcategory of $\mathbf{OA}$ whose objects are orthomodular posets and $\mathbf{OG}$ be the category of orthogeometries isomorphic to ones arising from orthomodular posets and the orthogeometry morphisms between them. If $\mathcal{G}$ is such an orthogeometry, then $\mathcal{G}^*$ is an orthohypergraph with $(\mathcal{G}^*)_*=\mathcal{G}$. Since morphisms of orthogeometries are defined to be partial mappings that are orthohypergraph morphisms between their associated orthohypergraphs $\mathcal{G}^*$, the following is immediate.
\begin{theorem} \label{jko}
There is a functor $\mathcal{G}:\mathbf{OMP}\to\mathbf{OG}$ that is essentially surjective on objects, injective on objects with the exception of the 1-element and 2-element orthoalgebras, and full and faithful with respect to proper orthoalgebra morphisms and proper orthogeometry morphisms.
\end{theorem}
Let $\mathbf{COML}$ be the category whose objects are complete orthomodular lattices and whose morphisms are ortholattice homomorphisms that preserve all joins. Let $\mathbf{COG}$ be the category of all orthogeometries isomorphic to ones arising from complete orthomodular lattices and the orthogeometry morphisms that preserve all joins. Note that these are non-full subcategories of $\mathbf{COML}$ and $\mathbf{OMP}$ respectively. Proposition~\ref{lifts} shows that the the functor $\mathcal{G}:\mathbf{OMP}\to\mathbf{OG}$ of Theorem~\ref{jko} restricts to a functor between these categories and provides the following.
\begin{theorem}
The functor $\mathcal{G}:\mathbf{COML}\to\mathbf{COG}$ is essentially surjective on objects, injective on objects with the exception of the 1-element and 2-element orthoalgebras, and full and faithful with respect to proper morphisms in the two categories.
\end{theorem}
\section{Operator algebras} In this section we lift results of the previous section to the setting of certain operator algebras known as AW*-algebras. We begin by recalling that a C*-algebra $M$ is a complete normed complex vector space with additional multiplication, unit $1_M$, and involution $*$, subject to certain well-known properties \cite{KR1}. We would like to emphasize that all our C*-algebras are assumed to be unital. A \emph{*-homomorphism} between C*-algebras is a linear map $\varphi:M\to N$ that preserves multiplication, involution, and units. We refer to \cite{KR1,Takesaki1} for details on operator algebras, and restrict ourselves to giving the appropriate definitions.
\begin{definition}
A subset $N$ of a C*-algebra $M$ is a \emph{C*-subalgebra} of $M$ if it is a linear subspace that is closed under multiplication, involution, contains the unit $1_M$, and is additionally a closed subset in the metric topology induced by the norm.
\end{definition}
We note that the condition that a C*-subalgebra contains the unit is important here and is not always assumed in the operator algebras literature.
\begin{definition}
Let $M$ be a C*-algebra, and let $x\in M$. Then $x$ is called \emph{self adjoint} if $x^*=x$; \emph{positive} if $x=y^2$ for some self adjoint $y\in M$; and a \emph{projection} if $x^2=x=x^*$. We denote the set of self-adjoint elements in $M$ by $M_\ensuremath{\mathrm{sa}}$, and the projections in $M$ by $\Proj(M)$.
\end{definition}
A key fact here is the following, which holds more generally for projections of any *-ring.
\begin{proposition}
For any C*-algebra $M$, its projections $\Proj(M)$ form an orthomodular poset when ordered by $p\leq q$ iff $pq=p$ and with orthocomplementation $p'=1_M-p$. Moreover, if two projections $p$ and $q$ commute, their join $p\vee q$ is given by $p+q-pq$.
\end{proposition}
The \emph{Jordan product} of a C*-algeba, a symmetrized version of the ordinary product, that is given by
$$x\circ y=\frac{xy+yx}{2}$$
It is easily seen that a map $\varphi:M\to N$ that preserves linear structure and units preserves Jordan multiplication if and only if it preserves squares, and this provides an expedient way to define the appropriate morphisms.
\begin{definition}
For C*-algebras $M$ and $N$, a \emph{Jordan homomorphism} $\varphi:M\to N$ is a linear map that preserves the unit, involution $*$, and squares: $\varphi(x^2)=\varphi(x)^2$.
\end{definition}
Since a Jordan homomorphism preserves involution, it restricts to a morphism $M_\ensuremath{\mathrm{sa}}\to N_\ensuremath{\mathrm{sa}}$. Conversely \cite{KR1}, a linear map $M_\ensuremath{\mathrm{sa}}\to N_\ensuremath{\mathrm{sa}}$ that preserves units, the involution, and squares can uniquely be extended to a Jordan homomorphism $M\to N$. It is obvious that a *-homomorphism is a Jordan homomorphism, and that between commutative C*-algebras the notions coincide. A deeper look refines this result and is provided in the following. Here we recall \cite{Stormer} that a linear map $\varphi:M\to N$ between C*-algebras is \emph{positive} if it preserves positive elements, \emph{$n$-positive} if the associated map between matrix algebras $\mathrm{M}_n(\varphi):\mathrm{M}_n(M)\to\mathrm{M}_n(N)$ is positive, and \emph{completely positive} if it is $n$-positive for each $n$.
\begin{lemma}\label{lem:Jordan homomorphism}
Suppose that $M$ and $N$ are C*-algebras. If $M$ is commutative, then any Jordan homomorphism $\varphi:M\to N$ is a *-homomorphism.
\end{lemma}
\begin{proof}
By the Gelfand-Naimark theorem, we can embed $N$ into $B(H)$ for some Hilbert space $H$, and then view $\varphi$ as a Jordan homomorphism from $M$ to $B(H)$. Since $\varphi$ preserves squares, it is positive, and since $M$ is commutative, it follows from \cite[Thm. 1.2.5]{Stormer} that $\varphi$ is completely positive, hence certainly $2$-positive. By \cite[Thm. 3.4.4]{Stormer}, $\varphi$ is so-called \emph{non-extendible}. It follows from \cite[Thm. 3.4.5]{Stormer} that $\varphi$ is a *-homomorphism, which is clearly still a *-homomorphism if we restrict its codomain to $N$.
\end{proof}
We turn to the class of C*-algebras that play the feature role here, AW*-algebras. These were introduced by Kaplansky \cite{Kaplansky} to give an algebraic formulation of the key concept of von Neumann algebras, also known as W*-algebras, that they have a large supply of projections and their projections form a complete orthomodular lattice. However, even in the commutative case AW*-algebras are much more general than their von Neumann counterparts. We begin with a definition close to Kaplansky's original.
\begin{definition} \label{oppp}
A C*-algebra $M$ is an \emph{AW*-algebra} if its projections $\Proj(M)$ are a complete orthomodular lattice and each maximal commutative C*-subalgebra of $M$ is generated by its projections.
\end{definition}
Here a C*-subalgebra $C$ of $M$ is said to be generated by a subset $S$ of if it is equal to the smallest C*-subalgebra of $M$ that contains $S$. This smallest C*-subalgebra, which we will denote by $C*(S)$, exists because the intersection of C*-subalgebras is a C*-subalgebra. Since we assumed that all C*-subalgebras of $M$ contain $1_M$, it follows that $C^*(S)$ will contain $1_M$ as well, even when $S$ does not contain the unit of $M$.
There is a substantially different way to get to AW*-algebras. A *-ring $A$ is a \emph{Baer*-ring} if for each non-empty subset $S$ of $A$ the \emph{right annihilator}
\[R(S)=\{x\in A:sx=0\text{ for each }s\in S\}\] of $S$ is a principal right ideal of $A$ generated by a projection $p$, i.e., $R(S)=pA$.
The following is given in \cite[Thm. III.1.8.2]{Blackadar}.
\begin{theorem}
A C*-algebra is a AW*-algebra if and only if it is a Baer*-ring.
\end{theorem}
For an element $x$ in a Baer*-ring $A$ the right annihilator $R(\{x\})$ is a principal ideal generated by a projection $p$. We define the \emph{right projection} $RP(x)$ to be $1_A-p$. Using this, we have the following \cite[Defn's 4.3, 4.4]{Berberian}.
\begin{definition}\label{subalgebra}
If $A$ is a Baer*-ring, then a *-subring $B$ of $A$ is a \emph{Baer*-subring} if (i) $x\in B$ implies $RP(x)\in B$ and (ii) every non-empty set of projections in $B$ has its supremum in the projection lattice of $A$ belong to $B$. If $M$ is an AW*-algebra, then an \emph{AW*-subalgebra} of $M$ is a C*-subalgebra $N$ of $M$ that is also a Baer*-subring.
\end{definition}
There is an alternate description of AW*-subalgebras that is very useful \cite[Exercise 4.21]{Berberian}.
\begin{proposition}
Let $M$ be an AW*-algebra and $N$ be a C*-subalgebra of $M$. Then $N$ is an AW*-subalgebra of $M$ if and only if $N$ is an AW*-algebra and for any orthogonal set of projections in $N$ its join taken in the projection lattice of $M$ belongs to $N$.
\end{proposition}
Primary examples of AW*-algebras are von Neumann algebras, those C*-subalgebras of the algebra $B(H)$ of operators on a Hilbert space $H$ that are equal to their double commutants in $B(H)$. Here the commutant of a subset $S$ of an AW*-algebra $M$ is the set $S'$ of all $a\in M$ such that $ab=ba$ for each $b\in S$. However, even in the commutative case AW*-algebras are more general than von Neumann algebras.
A commutative C*-algebra is an AW*-algebra if and only if its Gelfand spectrum is a Stonean space, that is, an extremally disconnected compact Hausdorff space \cite{SaitoWright,SaitoWrightBook}. In contrast, the commutative C*-algebras that are von Neumann algebras are exactly those whose Gelfand spectrum is hyperstonean \cite{KR1}.
Combining Gelfand and Stone dualities, the category of commutative AW*-algebras and \mbox{*-homomorphisms} is equivalent via the functor $\Proj$ to the category of complete Boolean algebras and Boolean algebra homomorphisms between them. We introduce a new class of maps that preserve the infinite join structure.
\begin{definition}\label{def:normal morphism}
Let $M$ and $N$ be AW*-algebras and let $\varphi:M\to N$ be either a Jordan homomorphism or a *-homomorphism, then we call $\varphi$ \emph{normal} if it preserves suprema of arbitrary collections of projections.
\end{definition}
The functor $\Proj$ gives an equivalence between the category of commutative AW*-algebras and normal *-homomorphisms and the category of complete Boolean algebras and Boolean algebra homomorphisms that preserve arbitrary joins. By \cite[Cor. 6.10]{Guram} these categories are dually equivalent to the category of Stonean spaces and open continuous functions.
We require additional facts about normal *-homomorphisms and normal Jordan homomorphisms between AW*-algebras. These roughly duplicate well-known results in the von Neumann algebra setting, but require different proofs and are difficult to find in the literature. Sources are the book of Berberian on Baer*-rings \cite{Berberian} and the paper of Heunen and Reyes \cite{HeunenReyes}. Most of these results are also included with proof in the PhD thesis of the second author, see \cite[\S 2.4]{Lindenhovius}.
We begin with the following found in \cite[Exercise 23.8]{Berberian}.
\begin{lemma}\label{lem:normal morphisms preserve right projections}
Let $\varphi:M\to N$ be a normal *-homomorphism between AW*-algebras $M$ and $N$. Then $RP(\varphi(x))=\varphi(RP(x))$ for each $x\in M$.
\end{lemma}
So normal *-homomorphisms between AW*-algebras preserve joins of projections and right projections $RP(x)$. Also, by Definition~\ref{subalgebra}, AW*-subalgebras are exactly C*-subalgebras that are closed under joins of projections and right projections. Since the image and pre-image under a *-homomorphism of a C*-subalgebra is a C*-subalgebra, we have the following.
\begin{lemma}\label{lem:image of normal morphism}
Let $\varphi:M\to N$ be a normal *-homomorphism between AW*-algebras $M$ and $N$. Then the image and pre-image of AW*-subalgebras is an AW*-subalgebra.
\end{lemma}
The intersection of a family of AW*-subalgebras is again such \cite[Prop. 4.8]{Berberian}. So for any subset $S$ of an AW*-algebra, there is a smallest AW*-subalgebra $AW^*(S)$ generated by it. Using Lemma~\ref{lem:image of normal morphism} we obtain the following.
\begin{lemma}\label{lem:homomorphisms preserve generating sets}
Let $\varphi:M\to N$ be a normal *-homomorphism between AW*-algebras and $S\subseteq M$. Then $\varphi[AW^*(S)]=AW^*(\varphi[S])$.
\end{lemma}
\begin{lemma}
Let $M$ and $N$ be AW*-algebras, and let $\varphi,\psi:M\to N$ be normal Jordan homomorphisms. If $\varphi$ and $\psi$ coincide on $\Proj(M)$, then $\varphi=\psi$.
\end{lemma}
\begin{proof}
We first show that $\varphi$ and $\psi$ coincide on any commutative AW*-subalgebra $C$ of $M$. By Lemma~\ref{lem:Jordan homomorphism} the restrictions of $\varphi$ and $\psi$ to $C$ are *-homomorphisms which are clearly normal. So by Lemma \ref{lem:image of normal morphism} both $\varphi[C]$ and $\psi[C]$ are AW*-subalgebras of $N$, and these AW*-subalgebras are clearly commutative. As mentioned above Definition \ref{def:normal morphism}, the functor $\Proj$ is a an equivalence of categories of commutative AW*-algebras and complete Boolean algebras. Therefore, a *-homomorphism on $C$ is completely determined by its restriction to $\Proj(C)$, hence $\varphi$ and $\psi$ must coincide on $C$.
Let $x\in M$ be self-adjoint. Then $\{x\}''$ is commutative by \cite[Prop. 3.9]{Berberian} and an AW*-subalgebra of $M$ by \cite[Prop. 4.8(iv)]{Berberian}. It follows that $\varphi(x)=\psi(x)$. Now, let $x\in M$ be arbitrary. Then $x=x_1+ix_2$ with $x_1=(x+x^*)/2$ and $x_2=(x-x^*)/2i$, both of which are self adjoint. Hence \[\varphi(x)=\varphi(x_1)+i\varphi(x_2)=\psi(x_1)+i\psi(x_2)=\psi(x).\qedhere \]
\end{proof}
Before the key results, we require some further terminology that is standard in the area \cite{KR1}. Here, for a commutative C*-algebra $C$, the matrix algebra of $2\times 2$ matrices with coefficients in $C$ is $\mathrm{M}_2(C)$. A standard result is that this is a C*-algebra, and is an AW*-algebra if $C$ is such.
\begin{definition}
An AW*-algebra $M$ is of \emph{type} $I_2$ if there is a commutative AW*-algebra $C$ such that $M$ is *-isomorphic to $\mathrm M_2(C)$. We say that $M$ has a type I$_2$ summand if it is *-isomorphic to the direct sum $N_1\oplus N_2$ of some AW*-algebras $N_1$ and $N_2$, where $N_2$ is a type I$_2$ AW*-algebra.
\end{definition}
In \cite[Thm 4.2]{Hamhalter2} Hamhalter showed that Dye's Theorem can be extended to the class of AW*-algebras. Combining this with the previous lemma to give uniqueness yields the following.
\begin{theorem}\label{thm:hamhalter}
Let $M$ and $N$ be AW*-algebras and assume $M$ has no type I$_2$ summands. Then any ortholattice homomorphism $\psi:\Proj(M)\to\Proj(N)$ that preserves arbitrary joins uniquely extends to a normal Jordan homomorphism $\varphi:M\to N$.
\end{theorem}
We now make use of these results. Let $\mathbf{AW^*_J}$ be the category of AW*-algebras and normal Jordan homomorphisms. Recall that $\mathbf{COML}$ is the category of complete orthomodular lattices and ortholattice morphisms that preserve all suprema. Let $\mathbf{AWOML}$ be its full subcategory of orthomodular lattices isomorphic to the projection lattice of an AW*-algebra. For any C*-algebra $M$ we have that $\Proj(M)$ is an orthomodular poset, and if $M$ is an AW*-algebra, then by definition $\Proj(M)$ is a complete orthomodular lattice. For a normal Jordan homomorphism $\varphi:M\to N$ between AW*-algebras, it is obvious that $\varphi$ maps projections to projections, and by normality it preserves arbitrary joins of projections. Since $\varphi$ is linear and $p'=1-p$ it is clear that $\varphi$ preserves orthocomplementation. It follows that restricting $\varphi$ to $\Proj(M)$ gives a complete ortholattice homomorphism from $\Proj(M)$ to $\Proj(N)$. This yields the following.
\begin{theorem}
There is a functor $\Proj:\mathbf{AW^*_J}\to\mathbf{AWOML}$ that takes an AW*-algebra to its projection lattice and a normal Jordan homomorphism to its restriction to the projection lattices.
\end{theorem}
This functor has additional properties. By definition, it is surjective on objects. If we restrict attention to AW*-algebras without type $I_2$ factor and their corresponding projection lattices we have the following. For two such AW*-algebras $M$ and $N$ we have $\Proj(M)$ is isomorphic to $\Proj(N)$ if and only if $M$ and $N$ are Jordan isomorphic, a fact that follows from Theorem~\ref{thm:hamhalter}. Further, there is a bijective correspondence between the normal Jordan homomorphisms between $M$ and $N$ without type $I_2$ factor and complete ortholattice homomorphisms between their projection lattices. This also follows from Theorem~\ref{thm:hamhalter} and the trivial fact that a normal Jordan homomorphism between $M$ and $N$ restricts to a complete ortholattice homomorphism between their projection lattices.
\begin{definition}\label{def:proper C*-algebra}
A C*-algebra $M$ is \emph{proper} if it is not *-isomorphic to either $\mathbb{C}^2$ or to $\mathrm{M}_2(\mathbb C)$.
\end{definition}
By \cite[Prop. 3.3]{Hamhalter} a C*-algebra $M$ is proper if and only if it does not have any maximal commutative C*-subalgebras of dimension 2, which occurs if and only if $\Proj(M)$ does not have any maximal Boolean subalgebras with 4 elements. So a C*-algebra $M$ is proper if and only if $\Proj(M)$ is proper, and this occurs if and only if the orthogeometry associated with $\Proj(M)$ is proper.
A morphism between orthoalgebras is proper if it satisfies a somewhat awkward condition relating to the image of certain blocks not being small, i.e., if the images contain more than four elements. This could be easily translated into an equally awkward condition on normal Jordan homomorphisms, but we take the more expedient route.
\begin{definition}
A normal Jordan homomorphism $\varphi:M\to N$ between AW*-algebras is \emph{proper} if its restriction to an ortholattice homomorphism between projection lattices is proper.
\end{definition}
We obtain the following about the composition of the functors $\Proj$ and $\mathcal{G}$.
\begin{theorem}\label{thm:mainGProj}
The composite $\mathcal{G}\circ\Proj : \mathbf{AW^*_J}\to\mathbf{COG}$ is injective on proper AW*-algebras with no type $I_2$ factor. If $M$ and $N$ are proper AW*-algebras with no type $I_2$ factor, then there is a bijective correspondence between the proper normal Jordan homomorphisms from $M$ to $N$ and the proper normal orthogeometry morphisms from $\mathcal{G}(\Proj M)$ to $\mathcal{G}(\Proj N)$.
\end{theorem}
We wish to treat this functor more directly in terms of AW*-algebras.
\begin{definition}
Let $M$ be an AW*-algebra. Then we denote the set of all commutative AW*-subalgebras of $M$ by $\ensuremath{\mathcal{A}}(M)$, which we order by inclusion.
\end{definition}
Posets of commutative subalgebras of operator algebras have been studied before, for instance in \cite{Doring} where the poset $\ensuremath{\mathcal{V}}(M)$ of commutative von Neumann subalgebras of a von Neumann algebra $M$ is considered. Since any von Neumann algebra is an AW*-algebra, and the AW*-subalgebras of a von Neumann algebra $M$ are the von Neumann subalgebras of $M$, we obtain $\ensuremath{\mathcal{V}}(M)=\ensuremath{\mathcal{A}}(M)$. The poset $\ensuremath{\mathcal{C}}(M)$ of commutative C*-subalgebras of a C*-algebra $M$ is studied in \cite{Hamhalter,Hamhalter2,Heunen,HeunenLindenhovius,Lindenhovius2,Lindenhovius} and is in general larger than $\ensuremath{\mathcal{A}}(M)$ in case $M$ is an AW*-algebra.
\begin{lemma}\label{lem:commutative subset generates commutative subalgebra}
Let $M$ be an AW*-algebra and $S\subseteq M$ be closed under involution and consist of mutually commuting elements. Then $AW^*(S)$ is a commutative AW*-subalgebra of $M$.
\end{lemma}
\begin{proof}
It follows from \cite[Prop. 3.9]{Berberian} that $S\subseteq S''$, where $S''$ is a commutative AW*-subalgebra of $M$. So $S$ is contained in some commutative AW*-subalgebra of $M$, hence it must generate a commutative AW*-subalgebra.
\end{proof}
Lemmas \ref{lem:commutative subset generates commutative subalgebra}, \ref{lem:Jordan homomorphism} and \ref{lem:homomorphisms preserve generating sets} give the following.
\begin{lemma}\label{lem:jordan homomorphisms preserve generating sets}
Let $\varphi:M\to N$ be a normal Jordan homomorphism between AW*-algebras and $S\subseteq M$ be a subset that is closed under the involution and that consists of mutually commuting elements. Then $\varphi[AW^*(S)]=AW^*(\varphi[S])$ and this is a commutative AW*-subalgebra of $N$.
\end{lemma}
\begin{proposition}\label{prop:sups in AM}
Let $M$ be an AW*-algebra, and $\ensuremath{\mathcal{D}}\subseteq\ensuremath{\mathcal{A}}(M)$. Then $\bigvee\ensuremath{\mathcal{D}}$ exists in $\ensuremath{\mathcal{A}}(M)$ if and only if $\bigcup\ensuremath{\mathcal{D}}$ consists of mutually commuting elements, and in this case $\bigvee\ensuremath{\mathcal{D}}=AW^*\left(\bigcup\ensuremath{\mathcal{D}}\right)$.
\end{proposition}
\begin{proof}
Assume that $S=\bigcup\ensuremath{\mathcal{D}}$ consists of mutually commuting elements. Clearly $S$ is closed under the involution, so by Lemma \ref{lem:commutative subset generates commutative subalgebra} $AW^*(S)$ is a commutative.
Let $C\in\ensuremath{\mathcal{A}}(M)$ be such that $D\subseteq C$ for each $D\in\ensuremath{\mathcal{D}}$. Then $S$ is contained in $C$, so $AW^*(S)\subseteq C$. Thus $AW^*(S)$ is the supremum of $\ensuremath{\mathcal{D}}$ in $\ensuremath{\mathcal{A}}(M)$. Conversely, assume that $\bigvee\ensuremath{\mathcal{D}}$ exists in $\ensuremath{\mathcal{A}}(M)$. Then $S\subseteq \bigvee\ensuremath{\mathcal{D}}$ and $\bigvee\ensuremath{\mathcal{D}}$ is a commutative AW*-subalgebra, so all elements in $\bigcup\ensuremath{\mathcal{D}}$ commute.
\end{proof}
Suppose that $\varphi:M\to N$ is a normal Jordan homomorphism between AW*-algebras. We define $\ensuremath{\mathcal{A}}(\varphi):\ensuremath{\mathcal{A}}(M)\to\ensuremath{\mathcal{A}}(N)$ to be the map taking $C$ to $\varphi[C]$. We recall that a directedly complete partial order (dcpo) is a poset where every directed subset has a join and a Scott continuous map between dcpo's is a map that preserves directed joins.
\begin{proposition}
$\ensuremath{\mathcal{A}}:\mathbf{AW^*_J}\to\mathbf{DCPO}$ is a functor from the category of AW*-algebras and normal Jordan homomorphisms to the category of dcpo's and Scott continuous maps.
\end{proposition}
\begin{proof}
Assume $\ensuremath{\mathcal{D}}\subseteq\ensuremath{\mathcal{A}}(M)$ is directed and set $S=\bigcup\ensuremath{\mathcal{D}}$. Then any $x,y\in S$ belong to some member of $\ensuremath{\mathcal{D}}$, so $S$ is commutative. So by Proposition \ref{prop:sups in AM} $\bigvee\ensuremath{\mathcal{D}}$ exists. Thus $\ensuremath{\mathcal{A}}(M)$ is a dcpo. Let $\varphi:M\to N$ be a normal Jordan homomorphism. Then by Proposition~\ref{prop:sups in AM} we have $\varphi(\bigvee\ensuremath{\mathcal{D}}) = \varphi[AW^*(S)]$ and $\bigvee\varphi[\ensuremath{\mathcal{D}}]=AW^*(\varphi[S])$. Lemma~\ref{lem:jordan homomorphisms preserve generating sets} gives that these are equal.
\end{proof}
\begin{remark}\label{nkl}
We can show more than stated in the result above. If $\ensuremath{\mathcal{D}}\subseteq\ensuremath{\mathcal{A}}(M)$ has a join, then $S=\bigcup\ensuremath{\mathcal{D}}$ is commutative, and the proof of the previous result shows $\varphi(\bigvee\ensuremath{\mathcal{D}})=\bigvee\varphi[\ensuremath{\mathcal{D}}]$. So $\varphi$ not only preserves directed joins, it preserves all existing joins.
\end{remark}
Let $A$ be an orthomodular lattice and $\mathcal{B}(A)$ be its poset of Boolean subalgebras. Directed joins in $\mathcal{B}(A)$ are given by unions and for an ortholattice homomorphism $f:A\to A'$ there is a Scott continuous map $\mathcal{B}(f):\mathcal{B}(A)\to\mathcal{B}(A')$ given by $\mathcal{B}(f)(S)=f[S]$. This gives the following.
\begin{proposition}
$\mathcal{B}:\mathbf{OML}\to\mathbf{DCPO}$ is a functor from the category of orthomodular lattices and ortholattice homomorphisms to the category of dcpo's and Scott continuous maps.
\end{proposition}
For a complete orthomodular lattice $A$, we let $\mathcal{B}_C(A)$ be its poset of complete Boolean subalgebras. These are Boolean subalgebras of $A$ that are closed under arbitrary joins in $A$. Clearly $\mathcal{B}_C(A)$ is a subposet of $\mathcal{B}(A)$ and for each $S\in\mathcal{B}(A)$ there is a least member $\overline{S}$ of $\mathcal{B}_C(A)$ above it, the closure of $S$ under arbitrary joins and meets in $A$, due to the fact that the maximal elements of $\ensuremath{\mathcal{B}}(A)$ are complete Boolean subalgebras as follows from the remarks below \cite[Prop. 3.4 \& Lem. 4.1]{Kalmbach}. If $S$ is finite, $\overline{S}=S$, and it follows that the elements of $\mathcal{B}(A)$ of finite height are exactly the elements of $\mathcal{B}_C(A)$ of finite height. In particular, the atoms of $\ensuremath{\mathcal{B}}_C(A)$ are the Boolean subalgebras $B_p=\{0,p,p',1\}$ for some non-trivial $p\in A$.
The join of an updirected set $\ensuremath{\mathcal{D}}$ in $\mathcal{B}_C(A)$ is given by $\overline{\bigcup\ensuremath{\mathcal{D}}}$. In particular $\mathcal{B}_C(A)$ is a dcpo. Clearly any $S\in\ensuremath{\mathcal{B}}_C(A)$ satisfies $S=\bigvee\{B_p:p\in S\}$, hence $\ensuremath{\mathcal{B}}_C(A)$ is atomistic. If $f:A\to M$ is an ortholattice homomorphism between complete ortholattices that preserves arbitrary joins, then $f[\overline{S}]=\overline{f[S]}$ for any Boolean subalgebra $S$ of $A$, so $\mathcal{B}_C(f):\mathcal{B}_C(A)\to\mathcal{B}_C(M)$ given by $f(S)=f[S]$ preserves directed joins. This gives the following.
\begin{proposition}
$\mathcal{B}_C:\mathbf{COML}\to\mathbf{DCPO}$ is a functor.
\end{proposition}
Let $M$ be an AW*-algebra with projection lattice $A$. Projections $p,q$ are orthogonal in $A$ iff $p\leq q'$, and this is easily seen to be equivalent to $pq=0=qp$. We have seen that if projections $p,q$ commute, then they belong to a Boolean subalgebra of $A$. Conversely, if $p,q$ belong to a Boolean subalgebra $B$ of $A$, then there are pairwise orthogonal $e,f,g$ in $B$ with $p=e+f$ and $q=f+g$. It follows that $pq=f=qp$. So projections commute iff they belong to a Boolean subalgebra of the projection lattice. So by Lemma~\ref{lem:commutative subset generates commutative subalgebra} there is a map $AW^*(\,\cdot\,)$ from the Boolean subalgebras of $A$ to the commutative AW*-subalgebras of $M$.
\begin{proposition}
Let $M$ be an AW*-algebra with $A$ its projection lattice. Then the maps $\Proj$ and $AW^*(\,\cdot\,)$ are mutually inverse order-isomorphisms between the dcpo's $\ensuremath{\mathcal{A}}(M)$ and $\mathcal{B}_C(A)$.
\end{proposition}
\begin{proof}
Clearly $\Proj$ and $AW^*(\,\cdot\,)$ are order preserving. Let $C$ be a commutative AW*-subalgebra of $M$. Then $C$ is an AW*-algebra, so by Definition~\ref{oppp} $C$ is generated as a C*-algebra by its projections, hence it is generated as an AW*-algebra by its projections. Thus $AW^*(\,\cdot\,)\circ\Proj$ is the identity.
Let $B$ be a complete Boolean subalgebra of~$A$, and set $D=AW^*(B)$ and $C=\Proj(D)$. Then $B$ is a Boolean subalgebra of $C$ and since arbitrary joins in both $B$ and $C$ agree with those in $A$, we have that $B$ is a complete Boolean subalgebra of $C$. The Stone space $Y$ of $B$ is Stonean, so $C(Y)$ is an AW*-algebra. Let $i:\Proj(C(Y))\to B$ be the obvious isomorphism. Since $C(Y)$ is commutative, it does not have type $I_2$ summands, and since $B$ is a complete subalgebra of $C$, we have that $i:\Proj(C(Y))\to\Proj(D)$ preserves arbitrary joins. So by Theorem~\ref{thm:hamhalter}, $i$ extends to a normal Jordan homomorphism $\varphi:C(Y)\to D$, and as these are commutative algebras, $\varphi$ is a normal *-homomorphism. The image of $\varphi$ is an AW*-subalgebra that contains $B$, so $\varphi$ is onto. By Lemma~\ref{lem:normal morphisms preserve right projections}, $\varphi$ preserves right projections. So if $\varphi(x)=0$, then $\varphi(RP(x))=RP(\varphi(x))=RP(0)=0$. But $RP(x)$ is a projection, and since $i$ is an isomorphism, $RP(x)=0$, and this implies that $x=0$ \cite[Prop.~3.6]{Berberian}. Thus $\varphi$ is a *-isomorphism, and it follows that the projections of $AW^*(B)$ are exactly $B$. Thus $\Proj\circ AW^*(\,\cdot\,)$ is the identity.
\end{proof}
\begin{corollary}
For an AW*-algebra $M$, the elements of $\ensuremath{\mathcal{A}}(M)$ of height $n$ are exactly the $(n+1)$-dimensional commutative C*-subalgebras of $M$.
\end{corollary}
\begin{proof}
The $n$-dimensional commutative C*-algebras are those isomorphism to $C(X)$ for some set $X$ with $n$ elements. Each of these is an AW*-algebra.
\end{proof}
\begin{corollary}\label{lem:A(M) atomistic}
Let $M$ be an AW*-algebra. Then $\ensuremath{\mathcal{A}}(M)$ is atomistic; its atoms are of the form $A_p=\mathrm{span}(p,1_M-p)$ for non-trivial projection $p\in M$.
\end{corollary}
\begin{definition}
For an AW*-algebra $M$ and orthomodular lattice $A$, let $\ensuremath{\mathcal{A}}_*(M)$ be the elements of height at most two in $\ensuremath{\mathcal{A}}(M)$ and let $\mathcal{B}_*(A)$ be the elements of height at most two in $\mathcal{B}(A)$.
\end{definition}
For an orthomodular lattice $A$, its associated orthogeometry $\mathcal{G}(A)$ is constructed as the pair $(P,L)$ where $P$ is the set of atoms of $\mathcal{B}(A)$ and $L$ is the set of elements of height two in $\mathcal{B}(A)$. So one may naturally consider $\mathcal{B}_*(A)$ as giving the orthogeometry $\mathcal{G}(A)$, and similarly $\ensuremath{\mathcal{A}}_*(M)$ as giving the orthogeometry for the projection lattice of $M$.
\begin{corollary}\label{prop:A orthogeometry}
Let $M$ be an AW*-algebra. Then $\ensuremath{\mathcal{A}}_*(M)$ is an orthogeometry isomorphic to $\mathcal G(\Proj(M))$, which is proper if and only if $M$ is proper.
\end{corollary}
\begin{proof}
The only part that is not trivial is properness, and this follows from the remark below Definition \ref{def:proper C*-algebra}.
\end{proof}
Recall that a morphism between orthogeometries is a partial function between their points satisfying certain conditions.
Let $A$ and $A'$ be complete orthomodular lattices. Given an ortholattice morphism $f:A\to A'$, the map $\ensuremath{\mathcal{B}}(f):\ensuremath{\mathcal{B}}(A)\to\ensuremath{\mathcal{B}}(A')$ restricts to a partial map from the atoms $B_p=\{0,p,p',1\}$ of $\mathcal{B}(A)$ to the atoms of $\mathcal{B}(A')$. Explicitly, this is given by
\[ \mathcal{B}(f)(B_p)= \begin{cases}
B_{f(p)}, & \varphi(p)\neq 0,1;\\
\perp, & \text{otherwise}.
\end{cases}
\]
This restriction of $\ensuremath{\mathcal{B}}(f)$ is precisely the orthogeometry morphism $\mathcal{G}(f):\mathcal G{(A)}\to\mathcal G(A')$ as is shown in \cite[Def. 7.5, Prop. 7.6]{HHLN}.
\begin{proposition}\label{prop:A isomorphic to GProj}
There is a functor $\ensuremath{\mathcal{A}}_*:\mathbf{AW^*_J}\to \mathbf{COG}$ taking an AW*-algebra $M$ to the orthogeometry $\ensuremath{\mathcal{A}}(M)$ and a normal Jordan homomorphism $\varphi:M\to N$ to the partial function obtained as the restriction of $\ensuremath{\mathcal{A}}(\varphi)$ to the atoms of $\ensuremath{\mathcal{A}}_*(M)$. Further, this functor is naturally isomorphic to $\mathcal G\circ\Proj$.
\end{proposition}
\begin{proof}
For an AW*-algebra $M$ the isomorphism $\Proj:\ensuremath{\mathcal{A}}_*(M)\to\mathcal{B}_*(\Proj (M))$ provides the desired natural isomorphism.
\end{proof}
In combination with Theorem \ref{thm:mainGProj} we obtain:
\begin{corollary}\label{cor:maincor}
The functor $\ensuremath{\mathcal{A}}_* : \mathbf{AW^*_J}\to\mathbf{COG}$ is injective on proper AW*-algebras with no type $I_2$ factor. If $M$ and $N$ are proper AW*-algebras with no type $I_2$ factor, then $\varphi\mapsto\ensuremath{\mathcal{A}}_*(\varphi)$ is a bijective correspondence between the proper normal Jordan homomorphisms from $M$ to $N$ and the proper normal hypergraph morphisms from $\ensuremath{\mathcal{A}}_*(M)$ to $\ensuremath{\mathcal{A}}_*(N)$.
\end{corollary}
\begin{theorem}\label{thm:A of M main}
Let $M$ and $N$ be proper AW*-algebras, and assume that $M$ has no type I$_2$ summand. Then there exists a bijection $\varphi\mapsto\ensuremath{\mathcal{A}}_*(\varphi)$ between proper normal Jordan homomorphisms $\varphi:M\to N$ and maps $\Phi:\ensuremath{\mathcal{A}}(M)\to\ensuremath{\mathcal{A}}(N)$ that preserve all existing suprema, and that restrict to proper normal morphisms of orthogeometries $\ensuremath{\mathcal{A}}_*(M)\to\ensuremath{\mathcal{A}}_*(N)$.
\end{theorem}
\begin{proof}
By Corollary \ref{cor:maincor}, the assignment $\varphi\mapsto\ensuremath{\mathcal{A}}_*(\varphi)$ is a bijection between proper normal Jordan homomorphisms $M\to N$ and proper normal morphisms of orthogeometries $\ensuremath{\mathcal{A}}_*(M)\to\ensuremath{\mathcal{A}}_*(N)$. Since $\ensuremath{\mathcal{A}}_*(\varphi)$ is the restriction of $\ensuremath{\mathcal{A}}(\varphi)$ to an orthogeometry morphism $\ensuremath{\mathcal{A}}_*(M)\to\ensuremath{\mathcal{A}}_*(N)$, it is sufficient to show that any proper normal morphism of orthogeometries $\Phi:\ensuremath{\mathcal{A}}_*(M)\to\ensuremath{\mathcal{A}}_*(N)$ uniquely extends to a map $\ensuremath{\mathcal{A}}(M)\to\ensuremath{\mathcal{A}}(N)$ that preserves arbitrary existing joins.
Since any such a $\Phi$ equals $\ensuremath{\mathcal{A}}_*(\varphi)$ for some proper normal Jordan homomorphism, $\ensuremath{\mathcal{A}}(\varphi)$ is such an extension. Let $\Psi$ be another extension. Note that $\ensuremath{\mathcal{A}}(\varphi)$ and $\Psi$ both preserve existing joins by Remark~\ref{nkl}. Since $\ensuremath{\mathcal{A}}(M)$ and $\ensuremath{\mathcal{A}}(N)$ are atomistic and these maps coincide on $\ensuremath{\mathcal{A}}_*(M)$, hence the set of atoms in $\ensuremath{\mathcal{A}}(M)$, it follows that they coincide on $\ensuremath{\mathcal{A}}(M)$.
\end{proof}
\section{Concluding Remarks}
We have found a functor $\mathcal G\circ\Proj:\mathbf{AW^*_J}\to\mathbf{COG}$ assigning to an AW*-algebra an orthogeometry. This functor is injective on proper AW*-algebras without a type I$_2$ summand. Given proper AW*-algebras without type I$_2$ summands $M$ and $N$. We also obtained a bijection between proper normal Jordan homomorphisms between from $M$ to $N$, and proper normal orthogeometry morphisms from $\mathcal G(\Proj(M))$ to $\mathcal G(\Proj(N))$. Furthermore, we have shown that $\mathcal G(\Proj(M))$ is isomorphic to the poset $\ensuremath{\mathcal{A}}_*(M)$ of commutative AW*-subalgebras of $M$ dimension at most 2, hence the poset $\ensuremath{\mathcal{A}}(M)$ of all commutative AW*-subalgebras of $M$ contains the same information as $\mathcal G(\Proj(M))$. Indeed, we showed that the set of proper normal Jordan homomorphisms between $M$ and $N$ is bijective to the set of all maps from $\ensuremath{\mathcal{A}}(M)$ to $\ensuremath{\mathcal{A}}(N)$ that preserve all existing suprema and that restrict to proper normal morphisms of orthogeometries $\ensuremath{\mathcal{A}}_*(M)\to\ensuremath{\mathcal{A}}_*(N)$.
Given an AW*-algebra $M$, we can also consider the poset $\ensuremath{\mathcal{C}}(M)$ of commutative C*-subalgebras, which in general will be larger than $\ensuremath{\mathcal{A}}(M)$ unless $M$ is finite dimensional. Then we can define $\ensuremath{\mathcal{C}}_*(M)$ the subset of all elements of height at most two in $\ensuremath{\mathcal{C}}(M)$, which is precisely $\ensuremath{\mathcal{A}}_*(M)$, so Corollary \ref{cor:maincor} holds also if we replace $\ensuremath{\mathcal{A}}$ by $\ensuremath{\mathcal{C}}$. However, since the poset $\ensuremath{\mathcal{C}}(M)$ is not atomistic, as follows from \cite[Thm. 2.4, Thm. 5.5., Thm 9.7]{HeunenLindenhovius}, we cannot find a bijection between Jordan homomorphisms from $M$ to another AW*-algebra $N$ and maps from $\ensuremath{\mathcal{C}}(M)$ to $\ensuremath{\mathcal{C}}(N)$ as in Theorem \ref{thm:A of M main}.
We already remarked that for an von Neumann algebra $M$ the poset $\ensuremath{\mathcal{V}}(M)$ coincides with $\ensuremath{\mathcal{A}}(M)$, hence all our statements hold as well if we replace the class of AW*-algebras by the class of von Neumann algebras. One could raise the question whether for an arbitrary AW*-algebra $M$ we can `recognize' from $\ensuremath{\mathcal{A}}_*(M)$ whether or not $M$ is a von Neumann algebra. By \cite{Pedersen} von Neumann algebras are characterized as the AW*-algebras $M$ with a separating family of normal states. Here a \emph{state} is a positive functional $\omega:M\to\mathbb C$ such that $\omega(1)=1$; a family $\mathcal F$ of states on $M$ is \emph{separating} if for each nonzero self adjoint $a\in M$ there is some $\omega\in\mathcal F$ such that $\omega(a)\neq 0$. One would like to proceed by recover normal states of $M$ as morphisms from $\ensuremath{\mathcal{A}}_*(M)$ to $\ensuremath{\mathcal{A}}_*(\mathbb C)$, but this is not possible for two reasons: firstly, states are in general not Jordan homomorphisms, and secondly, $\ensuremath{\mathcal{A}}_*(\mathbb C)$ is empty, hence certainly not proper. So we do not see how we can recognize from $\ensuremath{\mathcal{A}}_*(M)$ directly whether or not $M$ is a von Neumann algebra. However, we have the following indirect result: if $\ensuremath{\mathcal{A}}_*(M)$ is isomorphic to $\ensuremath{\mathcal{A}}_*(N)$ for some von Neumann algebra $N$, then $M$ should also be a von Neumann algebra, since the isomorphism between $\ensuremath{\mathcal{A}}_*(M)$ and $\ensuremath{\mathcal{A}}_*(N)$ implies the existence of some Jordan isomorphism $\varphi:M\to N$. Then $\mathcal F$ consisting of states $\omega\circ\varphi$ for a normal state $\omega$ on $N$ turns out to be a family of normal states on $M$ that is separating.
In \cite{HeunenReyes}, a complete invariant for AW*-algebras with normal *-homomorphisms, called \emph{active lattices} was introduced, which consists of the othomodular lattice $\Proj(M)$ of projections of an AW*-algebra $M$ together with a group action on $\Proj(M)$. By the results of \cite{HHLN} the orthomodular lattice part of an active lattice associated to $M$ contains the same information as the orthogeometry $\mathcal G(\Proj(M))$, and as a consequence our work shows that orthogeometries encode the Jordan structure of $M$, and that the group action encodes the extra information that is required to obtain an invariant for *-homomorphisms instead of Jordan homomorphisms.
Finally, both the results of orthogeometries and active lattices rely on the fact that AW*-algebras have abundant projections. This begs the question whether we can extend the results in this contribution to a larger class of operator algebras with ample projections, for instance the real rank zero algebras. A possible solution for this problem is to generalize Theorem \ref{thm:hamhalter} to real rank zero algebras. This is still an open problem.
\section*{Acknowledgements}
We thank Chris Heunen and Mirko Navara, since this work would have been impossible without them.
Bert Lindenhovius was funded by the AFOSR under the MURI
grant number FA9550-16-1-0082 entitled, "Semantics, Formal Reasoning, and Tool
Support for Quantum Programming".
|
2,869,038,156,421 | arxiv | \section{Introduction}
\label{intro}
Complex, unusual structures in radio sources are often associated with several events in which radio-emitting blobs of plasma have been created; for example, when the radio galaxy is moving through the intracluster medium (ICM), as is the case with the head-tail radio sources, a history of activity is traced in the tail of radio-emitting plasma left behind the parent host galaxy. Within this picture, therefore, tailed sources are viewed as trails or fossil records deposited by active galaxies. These sources are found in both poor groups and rich clusters of galaxies and their radio morphologies are due to the interaction of the radio-emitting plasma with the ambient gas \citep{Missagliaetal,Ternietal,Dehghanetal}.
\textit{Chandra}'s high angular resolution has enabled us to study interesting hydrodynamic phenomena in clusters, e.g., bow shocks driven by the infalling subclusters, cold fronts, or sharp contact discontinuities between regions of gas with different densities and temperatures.
The cold fronts seen as surface brightness edges in X-ray images are caused by the motion of cool, dense gas clouds in the ambient gas, which are either remnants of the infalling subclusters or the displaced gas from the cluster's own cores \citep{MaximAlexey}.
Hence, radio and X-ray observations of clusters of galaxies present the interplay between the radio galaxies along with the extended radio structures and the ICM.
Several important effects of the central radio source on the thermal state of a cluster have been predicted.
One such effect is the Kelvin-Helmholtz instability, which is
likely to develop at a boundary, e.g. the surface brightness edge, and
a straight, collimated radio jet of the moving radio galaxy apparently
flares up when it crosses the edge \citep{RosenHardee2000,Lokenetal1995,Livioetal1980}.
Therefore, very fast, low density, radio jets can remain coherent as they propagate to a few hundred times the jet radii, and as they encounter an X-ray surface brightness edge, they inflate and can be strongly over-pressured relative to the external medium \citep[e.g., NGC\,6251;][]{2005MNRAS.359..363E,Hardcastleteal2002}.
The prototype, the Coma cluster of galaxies is optically dominated by two giant elliptical galaxies, NGC\,4874, the brightest cluster galaxy, and NGC\,4869 \citep[$z$ = 0.02288;][]{2004AJ....128.1558S}.
The tailed radio source associated with NGC\,4869 lies near the Coma cluster center, at $\sim$4$^{\prime}$ (= 111~kpc) from NGC\,4874 and is completely embedded within the diffuse radio halo source, together called as Coma-C \citep{Willson1970}.
The source was mapped with the Very Large Array and the extended structure was reported by \citet{Ferettietal1990} and \citet{Dallacasaetal1989}.
\begin{table*}
\caption{The observations}
\begin{center}
\begin{tabular}{clcccccc}
\hline\hline
Band & Obs. ID & Obs. Date & $\nu$ & $\Delta\nu$ & t$_{\rm int.}$ & FWHM & \textsc{rms} \\
& & & (MHz) & (MHz) & (hour) & ($^{\prime\prime}\times^{\prime\prime}, ^{\circ}$)& ($\mu$Jy~beam$^{-1}$) \\
(1) & \multicolumn{1}{c}{(2)} & (3) & (4) & (5) & (6) & (7) & (8) \\
\hline\noalign{\smallskip}
\multicolumn{8}{l}{(legacy) GMRT} \\
150 MHz & 18\_022 & 2010 Apr 19 & 147 & ~6 & 2.3 &13.08$\times$10.32, 46.88 & 908.6 \\
240 MHz & 09TCA01 & 2006 Feb 06 & 240 & ~8 & 2.5 &11.81$\times$9.87, ~51.00 & 264.1 \\
325 MHz & 09TCA01 & 2006 Feb 11 & 333 & 16 & 2.6 &~8.76$\times$7.24, ~59.99 & ~44.2 \\
610 MHz & 09TCA01 & 2006 Feb 05 & 618 & 16 & 2.5 &~5.41$\times$4.46, ~70.63 & ~18.4 \\
\multicolumn{8}{l}{uGMRT} \\
250--500 MHz & ddtB270 & 2017 Apr 28 & ~400 & 200 & 1.8 &~6.65$\times$5.90, ~76.39 & ~21.1 \\
1050--1450 MHz & ddtB270 & 2017 Apr 26 & 1250 & 400 & 1.5 &~2.43$\times$1.95, ~69.51 & ~12.7 \\
\hline\hline
\end{tabular}
\end{center}
\label{tab:obs-log}
\tablecomments{Column~7: The position angle (P.A.) is measured from north and counterclockwise. \\
Column~8: \textsc{rms} noise at the half power point; see also Sec~\ref{sec.morph-spec} and \citet[][submitted]{Lal-submitted} for a discussion.}
\end{table*}
In this work, the second paper in the series--new low-frequency upgraded Giant Metrewave Radio Telescope (uGMRT) observations are presented in the first paper \citep[][submitted]{Lal-submitted}--results for the bent head-tail radio source, NGC\,4869 are presented. We discuss the detailed radio source morphology and the unique spectral structure, and investigate the effect of Kelvin-Helmholtz instability using the hydrodynamic phenomena in the local X-ray environment with \textit{Chandra} data.
The paper is organized as follows.
Our data are summarized in Sec.~\ref{sec.data-red}, followed by a description of the source morphology (Sec.~\ref{sec.morph-spec}), spectral structure (Sec.~\ref{sec:spectra}), and its local hot gas environment (Sec.~\ref{sec:xray-opt}). In Sec.~\ref{sec.discuss},
we examine the source structure, jet bending conditions along with physical conditions within the tail, and
discuss the role of Kelvin-Helmholtz instability at the interface between two gas media which are at different densities and temperatures.
Sec.~\ref{sec.sum-conc} summarizes our conclusions.
Throughout this paper, we adopted a $\Lambda$CDM cosmology with $H_0$ = 70 km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\rm m}$ = 0.27, and $\Omega_{\Lambda}$ = 0.73.
At the redshift of the NGC\,4869, 1 arcsec corresponds to 462~pc at the luminosity distance of 99.8 Mpc.
We define the spectral index, $\alpha$, as $S_\nu$ $\propto$ $\nu^\alpha$, where $S_\nu$ is the flux density at the frequency, $\nu$.
Throughout positions are given in J2000 coordinates.
\section{Radio and X-ray data}
\label{sec.data-red}
The GMRT \citep{Swarupetal1991} has been upgraded with a completely new set of receivers at
frequencies, $<$~1.5 GHz. The uGMRT now has (nearly) seamless frequency coverage in the 0.050--1.50 GHz range \citep{Guptaetal2017}.
Although the Coma cluster is among the best-studied rich cluster at low frequencies ($<$ 1~GHz), much remains to be explored; however, because of the difficulties encountered with ionospheric refraction and terrestrial radio frequency interference.
It was re-observed in 2017 with the uGMRT at band-3, a 250--500 MHz band
and band-5, a 1050--1450 MHz band, because of much improved sensitivity
\citep[][submitted]{Lal-submitted}.
The archival data and the new observations are detailed in observations log
(Table~\ref{tab:obs-log}).
Both the archival GMRT data and the new uGMRT data were analyzed in \textsc{aips} and \textsc{casa}, using standard imaging procedures \citep[see also][submitted]{LalandRao2004,Lal-submitted}.
Briefly, the flux density calibrator, 3C\,286 was also used as the secondary phase calibrator during the observations. We used 3C\,286 to correct for the bandpass shape and to set the flux density scale \citep{PerleyButler}. Bad data and data affected due to radio frequency interference were identified and flagged. Next the central channels were averaged to reduce data volume, taking care to avoid bandwidth smearing and the visibilities were imaged using the \textsc{tclean} task in \textsc{casa}.
We used 3D imaging (gridder = `widefield'), two Taylor coefficients (nterms = 2), and Briggs weighting (robust = 0.5) in the task \textsc{tclean}.
Several repeated steps of self-calibration, flagging, and imaging were performed to obtain the final image, and the final image was corrected for the primary beam shape of the GMRT antennas.
The error in the estimated flux density, both due to calibration and due to systematics, is $\lesssim$5\%.
\begin{figure*}[ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=16.8cm]{f1.eps} \\
\end{tabular}
\end{center}
\caption{Image of NGC\,4869 at the 250--500 MHz band (upper-panel) and 1050--1450 MHz band (lower panel) of the uGMRT. The double lobed radio emission from NGC\,4874, the brightest cluster galaxy of the Coma cluster, is shown in the northeast corner of the image \cite[R.A.: 12:59:35.71, Dec.: $+$27:57:33.37;][]{SunVFetal2005}. The center frequency is 400 MHz and 1250 MHz at the two bands. The synthesized beams are 6\farcs65 $\times$ 5\farcs90 at a P.A. of 76\fdg39 and 2\farcs43 $\times$ 1\farcs95 at a P.A. of 69\fdg51, and the \textsc{rms} noise values are 21.1 $\mu$Jy~beam$^{-1}$ and 12.7 $\mu$Jy~beam$^{-1}$ at the half-power points at the 250--500 MHz and 1050--1450 MHz bands, respectively. The lowest radio contour plotted is three times the local \textsc{rms} noise and increasing by a factors of 2. The color bar shows the surface brightness, in mJy~beam$^{-1}$. The local \textsc{rms} noise and continuum peak surface brightness of the source are denoted in each panel (lower left corner). Note that the first two surface brightness contours are not displayed for NGC\,4874 to show the detailed morphology of the head-tail radio source NGC\,4869. The inset illustrates four distinct regions, pinch (marked with arrows), and widening and narrowing of the radio tail (see also Sec.~\ref{sec.morph-spec}).}
\label{fig:f1}
\end{figure*}
\begin{figure*}
\begin{center}
\begin{tabular}{c}
\includegraphics[width=17.6cm]{f2.eps}
\end{tabular}
\caption{Gallery of four radio images at 150 MHz (top left), 240 MHz (top right), 325 MHz (bottom left), and 610 MHz (bottom right) using the archival legacy GMRT data. The synthesized beams and the the \textsc{rms} noise values at the half power points are tabulated in Table~\ref{tab:obs-log}, column 7 and column 8, respectively. Here again, the lowest radio contour plotted is three times the local \textsc{rms} noise and increasing by factors of 2, and the local \textsc{rms} noise and continuum peak surface brightness of the source are denoted in each panel (lower left corner). The color bar shows the surface brightness, in mJy~beam$^{-1}$.}
\label{fig:f2}
\end{center}
\end{figure*}
\subsection{Chandra data}
\label{chandra-data}
The Coma cluster has been observed a multiple number of times with \textit{Chandra}. We used only two observations made in 2012 March, AO-13 and AO-14, which observed NGC\,4869 giving a total observation time of $\sim$115~ks. The \textit{Chandra} observations used are listed here by Obs. IDs 13994 and 14411. These archival data sets were reduced using \textsc{ciao} 4.7 with the most up-to-date gain and efficiency calibrations as of that release using the standard pipeline of \textsc{ciao}. The standard screening, good time intervals, removal of bad pixels, and grade filtering were applied. The event data were projected to a common reference point and merged to create images. All imaging analyses were performed on the combined data set for comparison with our high-sensitivity radio image. Background-subtracted, exposure-corrected images, using (i) the native resolution of the \textit{Chandra} CCDs, (ii) 4 $\times$ 4 pixel binning, and (iii) 8 $\times$ 8 pixel binning were created in four (soft: 0.5--1.2~keV, medium: 1.2--2.0~keV, hard: 2.0--7.0~keV, and broad: 0.5--7.0~keV) energy bands.
The soft energy band image at 8 $\times$ 8 pixel binning (nearly) matches the angular resolutions of our uGMRT 250--500 MHz band and 1050--1450 MHz band images. It also showed sufficient contrast in the local hot gas environment where the head-tail radio galaxy NGC\,4869 resides in the Coma cluster. Hence, we used this soft energy band X-ray image for our further analyses, included to make detailed comparisons with our uGMRT images.
NGC\,4869 is at the edge of the field of view in two \textit{Chandra} pointings and we are interested in relative properties of two regions of interest, which encompass this tailed radio source (see Sec.~\ref{sec:jet-interact}).
The point sources and the central AGN were excised from the data, and the source, background spectra, and the response files were made separately for each \textit{Chandra} observation using the \textsc{specextract} task in \textsc{ciao}.
We then coadded the spectra and averaged the response files for each region. The data was binned to 10 counts per bin, fitted with \textsc{xspec} \citep[v12.10.1,][]{Arnaud1996} using isothermal \textsc{apec} model, and we determined a single temperature in the 0.5--5.0~keV energy range within the region of interest. The absorption column density was set to the Galactic foreground value, $N_{\rm H}$ = 0.9 $\times$ 10$^{20}$ cm$^{-2}$ \citep{DickeyLockman}, the abundances fixed at 0.5 $\times$ solar \citep{Vikhlininetal} with solar abundance ratios of \citet{GrevesseSauval}, and the temperature was free to vary. The $\chi^2$ values (fit statistics at 90\% confidence limits) are 298.2 and 256.4 for 305 degrees of freedom in the two regions of interest.
\section{Results}
\subsection{Radio images}
\label{sec.morph-spec}
The large-scale ($\sim$200~kpc) radio morphology of NGC\,4869 at the 250--500 MHz band and 1050--1450 MHz band of the uGMRT is shown in Figure~\ref{fig:f1}.
The images have \textsc{rms} noise values of 21.1~$\mu$Jy~beam$^{-1}$ and 12.7~$\mu$Jy~beam$^{-1}$ at the half-power points and the dynamic ranges of $\approx$~5300 and $\approx$~1700, respectively.
A gallery of four radio images at 150 MHz, 240 MHz, 325 MHz, and 610 MHz using the archival legacy GMRT data is shown in Figure~\ref{fig:f2}.
The \textsc{rms} noise values and dynamic ranges in archival legacy images are in the ranges 18.4--908.6~$\mu$Jy~beam$^{-1}$ and 90--600, respectively.
The \textsc{rms} noise is a factor of $\approx$2 higher close to the phase center where two dominant and extended radio sources, NGC\,4874, the bright cluster galaxy, and NGC\,4869 are present.
We thus also quote the local \textsc{rms} noise values in both Figure~\ref{fig:f1} and Figure~\ref{fig:f2} (lower-left corner) in all radio images.
These high resolutions and high-sensitivity images show the detailed structure of the radio core, the collimated radio jet, and the tail of NGC\,4869.
The structure is typical of a head-tail radio source: a weak core, two oppositely directed radio jets, and a long, low surface brightness tail. The tail, which consists of radio-emitting plasma left behind from the galaxy motion, begins after sharp bends in the radio jet. The low surface brightness, diffuse radio tail after $\sim$210$^{\prime\prime}$ (= 97.0~kpc) is resolved out in our 2\farcs18 image at the 1050--1450 MHz band. Close inspection of the outer isophotes of the 250--500 MHz band and the 1050--1450 MHz band images show that the tailed jet can be divided into five regions as follows.
\begin{itemize}
\item[(i)] Head (0--6$^{\prime\prime}$ = 0--2.8~kpc): an unresolved radio head. The two oppositely directed radio jets emanating from the apex of the host galaxy, initially traverse toward northeast and southwest directions.
\item[(ii)] Inner (6$^{\prime\prime}$--40$^{\prime\prime}$ = 2.8--18.5~kpc): a conical shaped feature centered on the nucleus. As the galaxy plows through the dense intracluster gas, these jets traversing in opposite directions form a trail, after sharp bends in the jets, behind the host galaxy due to interaction with the ICM.
\item[(iii)] Intermediate (40$^{\prime\prime}$--106$^{\prime\prime}$ = 18.5--49.0~kpc): a region in which the jet initially expands much more rapidly and then recollimates.
A pinch at $\approx$1\farcm4 (= 38.8~kpc) is present, which was also reported by \citet{Ferettietal1990}.
\item[(iv)] Outer-flaring (106$^{\prime\prime}$--208$^{\prime\prime}$ = 49.0--96.1~kpc): a second region of expansion.
\begin{figure}[b]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=8.4cm]{f3a.eps} \\
\includegraphics[width=8.4cm]{f3b.eps}
\end{tabular}
\caption{In-band spectral index image showing spectra using 250--500 MHz band data of uGMRT, between 300 MHz and 500 MHz (upper panel). The total intensity surface brightness contour is at 0.11 mJy~beam$^{-1}$ from the 300--500 MHz radio image (Figure~\ref{fig:f1}).
The lower panel shows the spectra at three regions along three vertical cuts, at 60$^{\prime\prime}$ (= 27.7~kpc), 90$^{\prime\prime}$ (= 41.6~kpc) and 150$^{\prime\prime}$ (= 69.3~kpc) from the radio head.
The distances on the horizontal axis are the distances from the radio jet ridge line to the region at the upper edge (left) and at the lower edge (right) of the radio jet, where the spectra are determined.
The error bars are based on local \textsc{rms} noise as evaluated in a circle of 2~arcmin diameter centered on these regions.
The locations of these three vertical cuts (white dashed lines) are overlaid, corresponding to the spectra shown in the lower panel figure, in the in-band spectral index image.
The three spectra are shifted (spectra at 60$^{\prime\prime}$, 90$^{\prime\prime}$ and 150$^{\prime\prime}$ cuts are shifted by 0.00, $-$0.18 and $-$0.06, respectively) to match the spectra at center jet ridge line to show comparisons.}
\label{fig:f3}
\end{center}
\end{figure}
\item[(v)] Bent-tail (208$^{\prime\prime}$--312$^{\prime\prime}$ = 96.1--144.1~kpc) and beyond: a feature seen after the jet has flared and bent toward the north.
There is a clear presence of a ridge at $\approx$3\farcm4 (= 94.2~kpc) of radio emission perpendicular to the direction of the tail at the flaring point. The radio jet in this region is inclined by $\sim$70$^\circ$ with respect to the radio-tail direction.
\end{itemize}
The low-frequency ($\nu$ $<$ 1.0~GHz) legacy GMRT images (Figure~\ref{fig:f2}) also show clear presence of these five regions.
The tail at the beginning has a width of $\sim$6$^{\prime\prime}$ (= 2.8~kpc). It has a constant transverse size of $\sim$16$^{\prime\prime}$ (= 7.4~kpc) at the beginning of the inner region and becomes a factor of two larger close to the pinch, located at $\approx$1\farcm4 (= 38.8~kpc) from the head. The radio jet widens and narrows several times in the tail,
which continues into the outer flaring region and the radio jet has become weaker. The bent tail, which reaches out to 6$^{\prime}$ (= 166.3~kpc) and more has nearly an average size of $\sim$26$^{\prime\prime}$ (= 12.0~kpc).
The transverse size of the radio tail is smallest at the pinch, which is seen clearly in 240\,MHz, 610\,MHz, and 1050--1450 MHz band images and is the largest at the ridge, which is seen in 150 MHz, 250--500 MHz, and 325 MHz band images.
The center ridge line of the radio jet shows the signature of oscillations in the jet and all along in the low surface brightness structures in all of our radio images (Figures~\ref{fig:f1} and \ref{fig:f2}), which was reported earlier by \citet{Ferettietal1990} and was seen in the tails of 3C\,129 \citep{LalandRao2004} as well.
The images also show the presence of diffuse extensions toward NGC\,4874 (inner edge) and sharpness in surface brightness on to the outer edge.
\subsection{Radio spectra}
\label{sec:spectra}
\begin{figure}[t]
\begin{center}
\begin{tabular}{c}
\includegraphics[angle=-90,width=8.4cm]{f4.eps}
\end{tabular}
\caption{The plot shows the radio spectra between 150 MHz and 240 MHz (red upper triangles) and between 325 MHz and 610 MHz (blue lower triangles). The locations of the pinch and the ridge are shown as vertical lines.}
\label{fig:f4}
\end{center}
\end{figure}
The low-frequency data, from both the uGMRT and the legacy GMRT, allow us to derive source spectra at different locations. Figure~\ref{fig:f3} shows the in-band radio spectral index image between 300 MHz and 500 MHz using 250--500 MHz band data (upper-panel), and
the transverse spectra at three different regions for each of the three different locations, 60$^{\prime\prime}$ (= 27.7~kpc), 90$^{\prime\prime}$ (= 41.6~kpc) and 150$^{\prime\prime}$ (= 63.9~kpc) from the head (lower panel).
Figure~\ref{fig:f4} shows the radio spectra at increasing distance from the host galaxy using matched angular resolution radio images at 150 MHz, 240 MHz, 325 MHz, and 610 MHz.
The upper panel of Figure~\ref{fig:f3} shows progressive spectral steepening as a function of distance from the head. The spectral indices close to the head and at the end of the outer flaring region are $-$0.66 $\pm$0.04 and $-$1.32 $\pm$0.08, respectively. Besides, there is a clear presence of two component spectral indices in the inner, the intermediate, and the outer flaring regions, i.e., there is a clear presence of a spine and an enveloping sheath layer, which have different spectra. The transverse-cut spectra (Figure~\ref{fig:f3}, lower panel) show that the spectrum of the enveloping sheath (upper and lower) is relatively steeper than the spine.
As we progress to larger and larger distances, i.e. from head to $\approx$1\farcm5 (= 41.6~kpc), the spectra of the sheath layer and the spine steepens due to synchrotron cooling, but at a given distance from the head, the sheath layer has relatively steeper spectral index as compared to the spine. Once the radio jet flares and is bent, the bent tail and beyond region, this effect diminishes and eventually is indistinguishable.
\begin{figure}[b]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=8.4cm]{f5.eps}
\end{tabular}
\caption{2MASS, $J$, $H$, and $K_s$ band, composite galaxy image. The surface brightness contours are displayed from the high-resolution 1050--1450 MHz radio image and the lowest radio contour plotted is three times the local \textsc{rms} noise and increasing by a factor of 2, as shown in Figure~\ref{fig:f1}. The optical image is 1.15 $\times$ 1.15 arcmin$^2$ in size.}
\label{fig:5}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=8.4cm]{f6.eps}
\end{tabular}
\caption{Background-subtracted and exposure-corrected \textit{Chandra} image at $\sim$4$^{\prime\prime}$ angular resolution in the soft (0.5--1.2 keV) energy band. The color bar shows the surface brightness, in counts~s$^{-1}$ per 8 $\times$ 8 pixel binning. The set of red-arrows marks the location of onset of flaring, i.e., the surface brightness edge. The light-green and red surface brightness contours are displayed using 250--500 MHz band and 1050--1450 MHz band uGMRT images, respectively. Also shown are the two sectors on either side of the X-ray surface brightness edge that were used for the surface brightness and temperature measurements.}
\label{fig:f6}
\end{center}
\end{figure}
Figure~\ref{fig:f4} shows that the radio spectra at 150--240 MHz and at 325--610 MHz as a function of distance from the head obtained using matched angular resolution images at these frequencies.
The spectrum at 325--610 MHz declines linearly with distance from the radio head (from $-$0.62 $\pm$0.03 and $-$0.77 $\pm$0.04) up to $\sim$60$^{\prime\prime}$ (= 27.7~kpc), whereas the spectrum at 150--240 MHz does not. Just after the location of the pinch, $\sim$106$^{\prime\prime}$ (= 49.0~kpc) where the intermediate region ends and the outer flaring region begins, there is a dip in spectra showing spectral steepening. There on, with increasing distance from the head, the radio spectrum shows more steepening over a wide range of frequencies.
Furthermore, the best fitting regressions to our small range, 150--610 MHz data in the inner, intermediate, and outer flaring regions, show that the high-frequency (325--610 MHz) spectra are flatter than the low-frequency (150--240 MHz) spectra and there on they are indistinguishable. Quantitatively, the high-frequency (325--610 MHz) spectra changes from $-$0.60 $\pm$0.03 to $-$1.18 $\pm$0.10 and the low-frequency (150--240 MHz) spectra changes from $-$0.70 $\pm$0.02 to $-$1.14 $\pm$0.10. Whereas, in the bent tail and beyond region, the two spectra are indistinguishable.
Finally, our observed radio spectra at different locations along the radio tail suggest a decrease in the flux density per unit length, but there are no such decrease, e.g., in the inner, the intermediate, and the outer flaring regions in our uGMRT and legacy GMRT images (see Figures~\ref{fig:f1} and \ref{fig:f2}).
\subsection{Optical and X-ray images}
\label{sec:xray-opt}
Figure~\ref{fig:5} shows a Two Micron All Sky Survey (2MASS) optical $J$, $H$, and $K_s$ band composite image. Overlaid on the optical image are the surface brightness contours from high-resolution, 1050--1450 MHz band image. The interplay of optical and radio images show that the collimated radio jet expands from 6$^{\prime\prime}$ (= 2.8~kpc) width to 16$^{\prime\prime}$ (= 7.4~kpc) width (transverse size) just outside the transition between the interstellar medium (ISM) of the host galaxy and hot cluster gas \citep{Vikhlininetal}, an effect also seen in NGC\,6109 \citet{Rawesetal2018}.
Figure~\ref{fig:f6} shows a \textit{Chandra} X-ray image at $\sim$4$^{\prime\prime}$ angular resolution in the soft (0.5--1.2 keV) energy band. Overlaid on the X-ray image are
the surface brightness contours using the 250--500 MHz band image (light-green contours) and the 1050--1450 MHz band image (red contours).
The set of red arrows along with an arc marks the presence of a surface brightness edge. The ridge of radio emission perpendicular to the direction of the tail coincides with this surface brightness edge.
The radio surface brightness of the collimated jet decreases as soon as the jet crosses the surface brightness edge, which is resolved out in our high-resolution 1050--1450 MHz band image.
\section{Discussion}
\label{sec.discuss}
The head-tail radio sources are characterized by a highly elongated radio structure.
In the case of NGC\,4869, the source morphology strongly suggests a radio trail, a typical model for head-tail radio sources \citep{JaffePerola1973}.
The two radio jets emanating from the apex of the host galaxy initially traverse along the northeast and southwest as is seen in the head region. Further on, in the inner region, as the galaxy plows through the dense intracluster gas, these jets traversing in opposite directions form a trail behind the host galaxy due to interaction with the ICM forming a conical shaped feature centered on the nucleus.
The pinch in the intermediate region
is possibly due to (i) changes of the external conditions, or (ii) variations in the nuclear activity, or (iii) the overlap, twist, and wrap of the two tails of the source along with the projection effects. The tail after the pinch is likely to be at rest with respect to the ambient hot gas \citep{Ferettietal1990}, because the tail consists of blobs of plasma left behind from the moving host galaxy. But it is unlikely to be in pressure equilibrium with the local X-ray hot gas environment because it does not show a constant width; instead, there is an expansion taking place along with the mixing/twist of the two tails of the source.
This twist of two trailing radio jets seen in the intermediate and outer flaring regions is causing the widening and narrowing several times in the radio tail seen in radio images (see Figures~\ref{fig:f1} and \ref{fig:f2}).
This further argues that the jets are not disrupted even at the bend, instead a continuous bulk flow of blobs of plasma material is still present. The projection effects too are important, since the host galaxy velocity does not seem to be perpendicular to the projected radio jets \citep{Ferettietal1990}.
The radio spectrum shows progressive steepening with increasing distance from head, because the radio-emitting synchrotron electrons are expected to be of the older population at greater distances from the head. This variation of the spectrum with distance presents the evolution of radio source under the effects of radiation losses.
There is an absence of a decrease in the flux density per unit length in the 6$^{\prime\prime}$--208$^{\prime\prime}$ (= 2.8--96.1~kpc) region, which suggests that the re-acceleration of the radiating electrons and perhaps also the magnetic field regeneration is possibly occurring within the jet \citep[e.g., Centaurus\,A;][]{HESScollaboration}.
The diffuse extensions toward NGC\,4874 (inner edge) and sharpness in surface brightness on to the outer edge along with the curvature of the tail of NGC\,4869 suggests a closed orbit around the dark matter potential.
It is not, however possible to estimate the cluster mass needed to produce the observed
curved radio structure and bent trajectory on kiloparsec scales extending away from the active galactic nuclei (AGN). This is because we lack knowledge of both the tangential velocities and the complete orbits, and the trail of NGC\,4869. The latter, the trail has the form of an overlap, twist, and wrap of the two tails, which could reflect the rotation of the region of the galaxy that is ejecting radio-emitting blobs of plasma \citep[see also][]{Gendron-Marsolaisetal}.
More scrutiny of the properties of NGC\,4869, however, reveals a difficulty in this simple radio trail model.
The problem centers around the end of the outer flaring region and beginning of bent tail region, where (i) the jet bends by $\sim$70$^\circ$ with respect to initial radio-tail direction and (ii) there is a clear presence of a ridge, flaring of the jet, perpendicular to the propagation of the radio jet, an aspect we discuss below.
\subsection{Interaction of radio tail with the surrounding material}
\label{sec:jet-interact}
The jets in FR\,I radio galaxies decelerate by picking up the matter, i.e., the entrainment across the jet boundary \citep{Bicknell1984, Bicknell1986,deYoung1996}. This is because the edge of the jet or the sheath layer is traveling about 30\% more slowly than the center, or the spine of the jet \citep{2002MNRAS.336..328L,2002MNRAS.336.1161L}, suggesting evidence of an interaction between the radio jet and the external medium. \citet{2002MNRAS.336.1161L} also showed that the radio jets decelerate by entrainment and are recollimated by the external pressure gradient, which is also quantitatively in agreement with the typical velocity field for external gas parameters derived from \textit{Chandra} measurements, \citep[e.g., FR\,I radio galaxy 3C\,31;][]{Hardcastleteal2002}.
The emission line data for several 3C\,RR radio galaxies imply masses of 10$^7$ M$_\sun$ in the line emitting gases. If a large part of this gas is entrained, then a radio jet could accomplish this in 10$^6$--10$^7$ yr \citep{deYoung1986}.
The entrained mass of $\sim$10$^8$ M$_\sun$ or more is expected for NGC\,4869, with a radio tail of radius $\sim$1~kpc, velocities around 1000 km~s$^{-1}$ during the $\simeq$10$^8$ yr lifetime of the radio jet. Therefore, in NGC\,4869, it is clear that a boundary layer must form due to entrainment, which transfers momentum to the ambient gas.
The radio spectra, presented in Sec.~\ref{sec:spectra} show that the
enveloping sheath is of steeper spectra than the spine, which we suggest is due to the entrained material from the ICM.
As we progress to larger and larger distances from the head, the sheath layer has entrained more and more thermal gas material and becomes steeper and steeper as compared to the spine at a given location in the radio tail.
The surface brightness edge (Figure~\ref{fig:f6}), where the sudden transition occurs or the jet abruptly flares represents the onset of turbulence \citep{Bicknell1984} or, in other words, the point at which Kelvin-Helmholtz instabilities start to grow nonlinearly \citep{RosenHardee2000,Lokenetal1995}.
The instability under consideration is instability of the boundary, also called the surface brightness edge between the two gas media, which are at different densities and temperatures.
Both the boundary between the ISM of the moving galaxy and the surrounding ICM, and the surface brightness edge seen in the hot gas media exhibit a large drag to the propagating, straight, collimated radio jet causing it to flare.
In the former, the size of the ISM is small ($\sim$5.5~kpc, NGC\,4869) and the amount of ISM in the galaxy is also small \citep{Vikhlininetal}. Its effect is stripping of gas in the ISM due to the motion of the host galaxy \citep{McBrideMcCourt,Livioetal1980}.
Whereas in the latter, jets crossing density edges can become partially disrupted and they inflate as a consequence of interactions between an AGN and its surrounding medium \citep[e.g., 3C\,449;][]{Laletal2013}.
\begin{figure}[t]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=8.4cm]{f7.eps}
\end{tabular}
\caption{Plot of the radial surface brightness (upper panel), and the gas density and the temperature (lower panel). The error bars are at 90\% confidence intervals.}
\label{fig:f7ab}
\end{center}
\end{figure}
The central AGNs can have significant impacts on the small ISM of the host galaxy through its jets, especially in outbursts \citep[e.g., NGC\,6109;][]{Rawesetal2018}.
The jet power, which is distributed in relativistic electrons, heavier particles, and the magnetic field, is known to be much larger than the radio luminosity of the source.
Following \citet{Sunetal2005}, we estimate the kinetic power of jets as $\simeq$10$^{43}$ ergs~s$^{-1}$. The (minimum) energy in the magnetic field (see also Sec.~\ref{sec.source-conf}) is only 10\%$-$25\% of the kinetic power and the AGN has been active for $>$10$^8$~yr. Therefore, the total jet power over the active phase of the nucleus is at least $\gtrsim$1000 times the thermal energy in the ISM. In other words, the jets carry large amounts of energy without dissipation through the small ISM of the host galaxy.
Instead, as reported in Sec~\ref{sec.morph-spec}, in the inner region, a cone shaped feature expands to form a jet of $\sim$16$^{\prime\prime}$ (= 7.4~kpc) width (transverse size) as they leave the small ISM of the host galaxy into the cluster atmosphere, possibly because of the change in the external pressure.
Fortunately, the host galaxy retains most of its ISM and only a small fraction is stripped \citep[e.g., NGC\,4848 and IC\,4040 in Coma cluster;][]{Chenetal}
Next, we focus our discussion on the latter instability, in the outer flaring region, which we have detected in our observations and seen in the \textit{Chandra} X-ray image (Figure~\ref{fig:f6}).
Observations show that the apparent pressures within many jets are higher than the pressures in the surrounding X-ray emitting gas through which they propagate e.g., NGC\,6251; \citet{2005MNRAS.359..363E} and 3C31; \citet{2002MNRAS.336..328L,2002MNRAS.336.1161L}.
Additionally, the X-ray emission shows that the Coma cluster is unrelaxed and has undergone a recent cluster merger \citep{Arnaudetal2001,Brieletal2001}, and the surface brightness edge (see Figure~\ref{fig:f6}) seen is possibly a cold front. Hence, the pressure must be continuous through the interfaces between the two gas medias, and a density and temperature jumps are only expected.
We observed the X-ray edge, where the surface brightness drops, and we select two regions inside and outside this edge (see Figure~\ref{fig:f6}). Since the plasma X-ray emissivity is proportional to the square of the gas density, the surface brightness distribution provides a direct estimate of the gas density \citep{Johnsonetal}.
Indeed, our surface brightness shows a discontinuity at the edge, and therefore implies a discontinuity, or jump, in the gas density (Figure~\ref{fig:f7ab}, upper panel).
We model the density within the two regions, following \citet{Johnsonetal}, with a power-law function and the density measurements are $n_e$(inside) and $n_e$(outside) $\approx$ 0.80 $\pm$0.06 $\times$ 10$^{-3}$ cm$^{-3}$ and 0.67 $\pm$0.06 $\times$ 10$^{-3}$ cm$^{-3}$, respectively (Figure~\ref{fig:f7ab}, lower panel).
We extracted source spectra for two regions using isothermal model, inside and outside of the surface brightness edge encompassing the source (see also Sec.~\ref{chandra-data} and Figure~\ref{fig:f7ab}, lower panel).
The derived temperatures are $\approx$ 6.44$^{+0.86}_{-0.72}$~keV and 6.28$^{+1.82}_{-1.34}$~keV for $kT_{\rm (inside)}$ and $kT_{\rm (outside)}$, respectively. Although with large error bars, these temperature measurements are hinting that the surface brightness edge is possibly a cold front. They are also in reasonable agreement with coarser measurements of \citet[][Figure~5]{Arnaudetal2001} and \citet[][Figure~3]{Brieletal2001}.
Furthermore, the two gas medias retain their respective material despite the motion of the radio galaxy, and conversely, the radio-emitting plasma too is not immune to stripping for every crossing of the surface brightness edge \citep{McBrideMcCourt,2013Sci...341.1365S}.
It, therefore, seems that the ram pressure condition indicates stability against the ablation of radio emitting plasma; in other words, the Kelvin-Helmholtz instability will induce a flaring of the collimated radio jet. In view of the vulnerability of the collimated jet at the location of a surface brightness edge, we conclude the onset of flaring, seen in several FR\,I radio galaxies \citep[3C\,449;][]{Laletal2013} and also seen here in the soft energy (0.5--1.2 keV) band \textit{Chandra} X-ray image (Figure~\ref{fig:f6}) of NGC\,4869.
Our conclusion is supported by the hydrodynamic simulations \citep{Lokenetal1996,Lokenetal1995} of the motion of radio galaxies through the hot intracluster gas, which shows that a Kelvin-Helmholtz instability manifests itself very prominently in the form of dense and long tongues of material at the interface between the moving radio source and the surface brightness edge in the gas.
Finally, the jet is bent by $\sim$70$^{\circ}$ toward the north direction; since the host galaxy could not be moving perpendicular to the trailing jets, the projection of the source is a plausible explanation for this bend \citep[also see NGC\,6109;][]{Rawesetal2018}.
\subsection{External density, pressure, and radiative age}
\label{sec.source-conf}
\begin{table}[t]
\caption{Physical parameters of NGC\,4869}
\begin{center}
\begin{tabular}{cccccc}
\hline
Dist. & Size & $U_{\rm min}$ & $B_{\rm eq.}$ & $P_{\rm min}$ & t$_{\rm age}$ \\
& & $\times$ 10$^{-12}$ & & $\times$ 10$^{-12}$ & 10$^{8}$ \\
($\prime\prime$) & (kpc) & (erg~cm$^{-3}$) & ($\mu$G) & (dyne~cm$^{-2}$) & (yr) \\
\multicolumn{1}{c}{(1)} & (2) & (3) & (4) & (5) & (6) \\
\hline\noalign{\smallskip}
~~3.1 & 10.4 & 3.9 & 6.3 & 1.3 & 1.2 \\
~15.0 & 10.4 & 3.8 & 6.2 & 1.3 & 1.1 \\
~26.9 & 10.4 & 3.1 & 5.7 & 1.0 & 1.2 \\
~38.8 & 12.2 & 2.0 & 4.6 & 0.7 & 1.7 \\
~51.2 & 12.2 & 1.5 & 4.0 & 0.5 & 2.1 \\
~66.4 & 17.6 & 1.5 & 3.9 & 0.5 & 2.1 \\
~84.3 & 17.6 & 1.1 & 3.4 & 0.4 & 2.6 \\
100.6 & 12.0 & 0.7 & 2.7 & 0.2 & 3.7 \\
113.0 & 14.7 & 1.0 & 3.2 & 0.3 & 3.0 \\
127.9 & 14.7 & 1.0 & 3.1 & 0.3 & 3.0 \\
142.8 & 14.7 & 0.7 & 2.6 & 0.2 & 3.9 \\
157.8 & 14.7 & 0.6 & 2.5 & 0.2 & 4.1 \\
171.5 & 12.8 & 0.5 & 2.2 & 0.2 & 5.0 \\
182.8 & 10.0 & 0.4 & 2.0 & 0.1 & 5.9 \\
\hline
\end{tabular}
\end{center}
\label{tab:phy-param}
\end{table}
The minimum energy condition corresponds almost to the equipartition of energy between relativistic particles and the magnetic field.
In order to determine the minimum internal energy density, equipartition magnetic energy, and pressure for several locations in the radio source, NGC\,4869,
we assume the ratio of energy in the heavier particles and the electrons is unity, the filling factor of the emitting regions also unity, and the transverse size of each component in the source is the path length through the source along the line of sight, and use the standard formulae in \citet{Miley1980} and \citet{Pacholczyk1970}.
Next, the shape of the radio spectrum depends on the balance between
the rate of synchrotron and inverse Compton losses and the rate of the replenishment of the electrons in the radiating region for the power-law energy distribution of electrons.
Following \citet{Miley1980}, we also derive the upper limit to the ages of the synchrotron electrons at several positions along the radio tail.
Table~\ref{tab:phy-param} gives these physical parameters, minimum energy density, equipartition magnetic field, the pressure exerted by the relativistic gas in the radio source, $U_{\rm min}$, $B_{\rm eq.}$, $P_{\rm min}$, respectively, and
radiative age, t$_{\rm age}$ estimated using 240 MHz as the break frequency at different locations along the radio tail.
Figure~\ref{fig:f8} shows that the physical conditions in the source as a function of the distance from the head change, whereas the ambient thermal pressure of the ambient X-ray gas is more and (nearly) constant.
\begin{figure}[t]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=8.4cm]{f8.eps}
\end{tabular}
\caption{Plot of the minimum internal non-thermal pressure as a function of distance from head. The continuous line represents the thermal pressure of the local X-ray gas. The error-bars of the thermal pressure of the local X-ray gas are $\approx$12\%.}
\label{fig:f8}
\end{center}
\end{figure}
\section{Conclusions}
\label{sec.sum-conc}
In this paper, we have presented in the preceding sections details of the observations and provide a detailed descriptions of the radio morphologies using uGMRT and legacy GMRT data
of an interesting head-tail radio source, NGC\,4869 in the Coma cluster of galaxies.
Our main conclusions are as follows.
\begin{enumerate}
\item[(i)] The high-resolution, high-sensitivity large-scale (almost 200~kpc) structure of the NGC\,4869 source is shown in Figures~\ref{fig:f1} and \ref{fig:f2}. The elliptical host galaxy shows a weak radio core,
two oppositely directed radio jets, and a long--low surface brightness tail.
\item[(ii)] The radio source can be divided into five distinct regions: a head, conical shaped feature centered on the nucleus, a region showing the rapid an expansion of the jet followed by collimation, another region showing expansion of the jet, and a flared region with a bend. Also present are pinch and a ridge at $\approx$1\farcm4 and $\approx$3\farcm4, respectively, from the head in addition to widening and narrowing several times in the radio tail.
\item[(iii)] There is no increase in the flux density per unit length in the 6$^{\prime\prime}$--208$^{\prime\prime}$ (= 2.8--96.1~kpc) region, which suggests that the re-acceleration of the radiating electrons and perhaps also magnetic field regeneration could be occurring within the jet.
\item[(iv)] The characteristic feature of radio tail is the sharp bend toward the north at
3\farcm5 from the position of the host galaxy; the bent tail and beyond region is inclined by $\sim$70$^\circ$ with respect to
the radio-tail direction. It seems that projection effects are important in the radio jets because the host galaxy does not seem to be moving perpendicular to the projected trailing jets.
\item[(v)] The radio spectra show progressive spectral steepening as a function of distance from the radio head due to synchrotron cooling.
The high-frequency (325--610 MHz) spectra are marginally flatter than the low-frequency (150--240 MHz) spectra until the ridge, i.e., in the inner, intermediate, and outer flaring regions. Once the radio jet flares, the bent tail and beyond region, these spectral features are indistinguishable.
\item[(vi)] Our improved transverse resolution in the 250--500 MHz band data for the inner, the intermediate, and the outer flaring regions clearly show
presence of a steep spectrum sheath enveloping a flat spectrum spine in the in-band spectral index image (Figure~\ref{fig:f3}), hinting at a transverse velocity structure with a fast spine surrounded by a slower sheath layer.
Our results favor entrainment across the boundary layer as the origin of the mass loading of the jets \citep[see also, 3C31;][]{2002MNRAS.336..328L,2002MNRAS.336.1161L}.
\item[(vii)] We have detected flaring of a straight, collimated radio jet as it crosses a surface brightness edge seen in the \textit{Chandra} X-ray image (Figure~\ref{fig:f6}). At the flaring point is a surface brightness edge at which the jet collimation, emissivity, and possibly the velocity change abruptly, which is a general property of FR\,I jets \citep{Laingetal1999}.
This sudden transition represents the onset of turbulence \citep{Bicknell1984} or in other words, the point at which Kelvin-Helmholtz instabilities start to grow nonlinearly \citep{RosenHardee2000,Lokenetal1995}.
\item[(viii)] The physical conditions in the jets and along the tails are obtained. The minimum internal nonthermal pressure is smaller than the external thermal pressure. This suggests that there is present a significant quantity of thermal plasma within the radio-emitting regions.
\end{enumerate}
Our key results favor (i) entrainment across the boundary layer, (ii) occurrence of re-acceleration of the synchrotron electrons and perhaps also regeneration of the magnetic field within the radio jet, and (iii) flaring of a straight, collimated radio jet as it crosses a surface brightness edge due to Kelvin-Helmholtz instabilities. Therefore, it will be fruitful to determine the origin and distribution of the slow-moving (sheath layer) material with respect to (spine) material, and the extent to which these regions of FR\,I jets resemble the larger-scale jets in FR\,II sources. The next step is to seek further evidence for the entrainment process, such as the reduced polarization near the boundaries of the flaring regions using polarization data \citep[e.g., NGC\,6251;][]{2015IAUS..313..108L}, which could also shed light on re-acceleration of the synchrotron electrons and the regeneration of the magnetic field in the radio jets. Radio jets in a number of FR\,I sources have been detected at X-ray and/or optical wavelengths. The radiation is most plausibly produced by the synchrotron process over the entire observed frequency range, and the shape of the spectrum, therefore, carries information about particle acceleration and energy loss. It will be important to incorporate our uGMRT results into descriptions of these processes for improved models.
It will be equally important to observe FR\,I radio sources in galaxy clusters over a broad range of low frequencies to study in detail the flaring of a collimated radio jet as it crosses a surface brightness edge (usually seen in X-ray images), which are indicators of Kelvin-Helmholtz instabilities.
\smallskip
We thank the anonymous referee for the comments that improved this paper.
D.V.L. would like to thank Tiziana Venturi and W. R. Forman for useful discussions, and Ishwara-Chandra C.H. for discussions on some aspects of this project.
D.V.L. acknowledges the support of the Department of Atomic Energy, Government of India, under project No. 12-R\&D-TFR-5.02-0700.
We thank the staff of the GMRT who made these observations possible. The GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.
This research has made use of the NED, which is operated by the Jet Propulsion Laboratory, Caltech, under contract with the NASA, and NASA's Astrophysics Data System.
This research has made use of the NASA/IPAC Infrared Science Archive, which is funded by the NASA and operated by the California Institute of Technology.
This research has made extensive use of SAOImage DS9, in addition to software provided by the CXC in the application packages CIAO.
Facilities: \facility{Chandra, IRSA, GMRT}
|
2,869,038,156,422 | arxiv | \section{Introduction}
With both a small modal volume and large quality factor, optical microcavities exhibit a greatly enhanced light-matter interaction and a strong non-linear optical response\cite{vahala_nat03,Almeida2004,Yanik04,XuNature2007}. An emerging class of optical microcavities is based on air-clad two-dimensional (2D) Photonic crystals (PCs). A Q-factor greater than $10^6$ was achieved with this technology \cite{kuramochi06,song_05,asano_06} which also allows small modal volumes ($\approx (\lambda/n)^3$ or $10^{-19}m^3$. Moreover, 2D PC technology is a planar technology which is particularly suited for the fabrication of photonic circuits. Therefore, the move towards all-optical processing, long considered impractical, has changed dramatically and new possibilities have been opened \cite{soljacic04}. Impressive experimental demonstrations of low-energy ($fJ$) optical bistability and all-optical switching \cite{notomi05,tanabe_ox05,Yang_07}, wavelength conversion \cite{tanabe06}, optomechanical effects \cite{notomi06} and dynamical control of the cavity lifetime \cite{tanabe07} are recent noteworthy achievements.
Most of these results come from a specific technology, i.e. silicon-based air-clad PCs where the nonlinear process involved is two-photon absorption (TPA) followed by plasma-induced and thermally-induced refractive index change. The optical power required is quite small ($\approx 100fJ$) and can be very fast (70 ps by ion implant, as the carrier recombination time result much shorter in these nanostructured devices than in bulk silicon \cite{tanabe05,tanabe07apl}. It has been predicted that, owing to strong light-matter interaction, nonlinear properties can be engineered by introducing nanoparticles\cite{Singh_2008}.
\indent The desire for an even faster response time motivates the research on alternative materials with a strong optical Kerr effect. Much progress has been made in processing chalcogenide crystals with high-Q PC microcavities have been demonstrated recently \cite{Ruan2007}. III-V semiconductors are also good candidates for optical switching as they have two very attractive features, compared with silicon. First, the optimization of the Kerr effect with respect to TPA \cite{AitchisonJQE1997,bristow_GaAs_APL07}. Self phase modulation due to Kerr effect has been demonstrated in 2D PCs recently\cite{Oda2007,oda_APL2008}. An additional feature of III-V semiconductors is the possibility of exploiting a strong nonlinear effect related to absorption saturation in active structures, such as Quantum Wells (QWs) and Quantum Dots (QDs). Fast nonlinear dynamics \cite{Raineri_APL04}, leading to bistability \cite{Yacomotti_APL06} and excitability \cite{Yacomotti_excit_PRL06}, has been observed in InP-based 2D PCs with InAsP QWs which are designed to operate with band-edge modes coupled to off-plane free space beams.
Low power and very fast (2 ps) switching (15 ps for a complete on/off cycle) has been demonstrated with a Symmetric Mach Zehnder-type all-optical switch made with InAs/AlGaAs quantum dots \cite{nakamura06}.
\indent In this paper we focus on the dynamics of the processes initiated by TPA in GaAs PCs.
Although, ideally, all optical switching requires the Kerr effect, in practice TPA, followed by a carrier induced index change, is still a very attractive approach, because of the relative simplicity of the technology. In particular, there is no active material, e.g. QDs, or phase matching condition required. Moreover, compared to silicon, the TPA coefficient in GaAs is tenfold higher \cite{Dinu2003} and the non-radiative carrier lifetime in patterned structures can be very short (8 ps) \cite{bristow_GaAs_APL03}. A fast and strong nonlinear response is therefore expected in GaAs PC nanocavities. Very recently, we have demonstrated ultra-fast (6 ps recovery time) and low power ($\approx 150$ fJ) modulation in GaAs Photonic crystal cavities\cite{Husko_APL2008}. We have investigated optical bistability in high-Q ($Q=0.7\times 10^6$) PC cavities on GaAs and reported an ultra-low threshold power ($\mu W$ range) \cite{weidner06,weidner07,combrie2008}. After the submission of this manuscript, memory operation have been demonstrated in a InGaAsP PC cavity\cite{Shinya_08}%
In these experiments, only the slow ($\mu s$) regime has been explored, which is dominated by the thermally induced index change. This is also due to the high thermal resistance of the membrane structures compared to, for instance, micro-disks \cite{michael_07} or membranes bonded to a $SiO_2$ cladding \cite{Yacomotti_APL06} and the lower thermal conductivity of GaAs with respect to silicon.
In this paper we investigate the response of PC microcavities at a much faster modulation rate (up to 50 MHz), which is well beyond the typical thermal relaxation time of photonic devices. Under these conditions, the analysis of nonlinear responses, such as bistability, cannot be explained with static models. First of all, moving to faster time-scales modifies the relative influence of thermal and carrier plasma effects. While at very fast time scales (ps) the dynamics tend to be controlled by the carrier lifetime, we will show that in the range between 1 - 100 ns the dynamics results from the interplay of fast thermal effects and carrier plasma index shift. We introduce a model that incorporates two thermal relaxation constants. This is necessary to explain this dynamics and the fact that, despite the high thermal resistance of PCs microcavities, there are thermal effects which develop in less than 10 ns. Section II is devoted to experiments made on a PC microcavity coupled to a waveguide. In section III we will introduce our multi-scale model. Discussion of the results forms the body of section IV.
\section{Experimental setup and sample description}
The photonic crystal structure studied here is the well-known optimized three missing holes (L3) PC microcavity \cite{akahane_03} in an air slab structure (thickness is 265 nm) based on a triangular lattice (period $a=400nm$) of holes with radius $r=0.24a$. The holes at the cavity edge were shifted by $s~=~0.15~a$. The cavity is side-coupled (the spacing is 3 rows) to a 1 mm long line-defect waveguide along the $\Gamma K$ direction, with width $W=1.05\sqrt{3}~a$ (see figure \ref{fig:exp_setup_sample}). The fabrication process and the detailed linear characterization of similar structures are described elsewhere \cite{combrie_05}. The loaded and intrinsic Q-factors, 7,000 and 30,000, respectively, are estimated from measurements using the procedure discussed in a previous paper \cite{combrie_opex06}. The cavity resonant wavelength is 1567 nm. The characterisation setup consist of a tunable external cavity semiconductor laser (Tunics) with a relative accuracy of $1 pm$ and narrow linewidth ($<<$ 1 MHz), which is amplified by an Erbium Doped Fiber Amplifier (EDFA). The polarisation is set to TE, or electric field in the plane of the slab. Coupling to the PC waveguide is obtained through microscope objectives (Zeiss, N.A. 0.95) and micropositioners. The transmitted signal is detected with an InGaAs photodiode (bandwidth = 2 GHz), amplified (40dB, transimpedance, 2GHz bandwidth) and monitored by an oscilloscope (Tektronix 12.5 GS/s).
\begin{figure}[h]
\includegraphics[width=7.5cm]{fig1a.eps}
\includegraphics[width=7.5cm]{fig1b.eps}
\caption{(Color online) Top: SEM image of the sample, also showing a line-defect photonic crystal waveguide with side coupled $L3$ cavity. Bottom: linear transmission near resonance showing that the cavity is over-coupled. The linewidth of the cavity is larger than the transmission dip because of interplay with the Fabry Perot fringes. The left scale is the waveguide transmission and the right scale is the signal detected by the IR camera.}
\label{fig:exp_setup_sample}
\end{figure}
Two distinct regimes of modulation are considered: 1) sinusoidal modulation (frequency from few kHz to 50 MHz) and 2) low duty cycle (20 ns) square pulses. The cavity is observed from the top with a microscope (long working distance objective Zeiss, N.A. 0.4) and an Infrared InGaAs camera (Xenics). When the cavity is on resonance a bright spot is observed.
\subsection{Sinusoidal modulation}
\begin{figure}[h]
\includegraphics[width=8.5 cm]{fig2a.eps}
\includegraphics[width=7.5 cm]{fig2b.eps}
\caption{(Color online) Low modulation frequency experiment. Oscilloscope traces (a) showing the typical bistable response of a PC microcavity at low modulation frequency (15 kHz). The average input power in the PC waveguide is about 80 $\mu$W. The bistable cycle is also sketched and the transitions 1-4 are also marked. Detunings are +0.4, +1.5 and +1.55$\Delta\lambda_{FWHM}$ respectively. Scheme representing a cavity with a negative detuning b) and going to resonance due to nonlinearly induced red-shift c).}
\label{fig:bistable_low_freq}
\end{figure}
At a low modulation rate (kHz) it has been shown by several groups\cite{notomi05,tanabe_ox05,Yang_07,weidner07,combrie2008} that bistability occurs at a fairly low optical power ($\mu W$ range, coupled into the waveguide). The dominant nonlinear effect is thermal induced index change because of heating resulting from carriers generated by TPA. Since this effect leads to a red-shift of the cavity frequency, bistability is observed when the initial detuning $\Delta\lambda|_{t=0} =\lambda_{laser}-\lambda_{cavity}|_{t=0}$) is positive, where $\lambda_{cavity}|_{t=0}$ is the cold cavity resonant wavelength. This is shown in fig. \ref{fig:bistable_low_freq}. Typically, the detuning is set between $\sqrt{3}/2$ (the theoretical minimum, although it has been shown that the Fabry-P\'erot resonances in the waveguide affect this value \cite{Yang_07}) and several times the cavity linewidth, estimated to be $\Delta\lambda_{FWHM} = 220 $ pm in our case. If the modulation frequency is increased while the detuning and the average signal power are kept constant, the bistable behavior disappears. For instance, with a detuning $\delta\lambda \approx 330 pm \approx 1.5 \Delta\lambda_{FWHM}$ and a modulation frequency of 15 kHz, bistability is observed in this sample at $80 \mu W$. The power coupled into the waveguide is estimated considering the amplified laser power and the input objective coupling factor (here $\sim$ -10 dB) as in our previous work \cite{weidner07}.
When the modulation frequency is increased to 1 MHz, the bistable effect and all traces of nonlinear distortion of the transmitted signal disappear. This could be explained in terms of the slow response of the thermo-optical effect. An important question arises whether the carrier plasma effect, much faster but weaker, will then take over. To answer that, the signal power and the modulation frequency were increased further and the detuning (still positive) was swept continuously from zero to about +700 pm, with the laser always on. This detuning value is larger than in low frequency experiments. A strong nonlinear distortion of the transmitted signal is then observed.
The onset of the nonlinear behavior is simultaneous to the observation of a bright spot through the IR camera, thus indicating that the resonant frequency of the cavity is red-shifted under a sinusoidal modulation pump. In particular, fig. \ref{fig:sin_modulation} shows a typical output signal when the modulation frequency is increased to $50$ MHz and the average power coupled in the waveguide is about 2.5 mW. The thermo-optic effect, driving the bistability at low modulation frequency, integrates over time. This explains ultra-low power levels. At higher modulation frequencies the power required to observe nonlinear effects increases. The nonlinear behavior is observed with a positive detuning from 600 pm and 670 pm ($\approx 3\Delta\lambda_{FWHM}$).
\indent We don't believe that the observed nonlinear distortion corresponds to a bistable behavior. Indeed, a fast nonlinear mechanism (plasma induced index change) should be dominate all other nonlinearities. That is probably not the case, as the short lifetime of carriers in GaAs compared to silicon drastically reduces the strength of this effect, compared to thermo-optic effects.
\begin{figure}[h]
\centering
\includegraphics[width=8.5cm]{fig3.eps}
\caption{(Color online) High modulation frequency experiment: oscilloscope traces of the transmitted signal with sinusoidal modulation (50 MHz) for different detunings. The average power is about 2.5 mW.}
\label{fig:sin_modulation}
\end{figure}
The nonlinear distortion observed in figure \ref{fig:sin_modulation} is quite general and can be reproduced at different modulation frequencies in the 10 - 100 MHz range. The nonlinear transitions are not sharp, compared to the modulation period (response time is a few ns). This time is however much faster than what could be attributed to a thermal effect. Additionally, the period, T = 20 ns, is much shorter than what is estimated to be the thermal recovery time in air slab PC cavities \cite{notomi05}. We claim that the faster carrier plasma effect plays a key role at the ns time scale, while the thermal effect accumulates to strongly redshift the cavity.
\begin{figure}[h]
\centering
\includegraphics[width=7 cm]{fig4.eps}
\caption{(Color online) Scheme representing the interplay of thermal and carrier plasma effect and how they are related to the detuning. The system is at rest (cavity at its cold frequency) and the laser is switched on (a). The laser wavelength is increased but heating keeps the cavity red detuned (b). When the cavity temperature is high enough, the mean resonant wavelength get closer to the laser and the nonlinear distortion is apparent (c).}
\label{fig:scheme_detuning}
\end{figure}
This mechanism is represented in Fig. \ref{fig:scheme_detuning}. Initially (a), the cavity is at room temperature and laser is detuned negatively ($\Delta\lambda<0$). The laser wavelength is increased continuously (b). The cavity is blueshifted (plasma effect) very fast, but heating quickly dominates and keeps the cavity redshifted until the end of the modulation cycle. Therefore, the cavity will be pushed towards longer wavelengths as long as the heat generated by TPA over each cycle compensates the thermal flow across the membrane. Over each modulation cycle, the cavity resonant wavelength will oscillate about an equilibrium point set by this thermal balance alternating blue-shift and fast heating (c). The strongest distortion of the sinusoidal modulation is observed nearest to the largest detuning which can be sustained. In our case, we achieved detunings of up to 700 pm, which is related to a local temperature increase of 6 K through the thermo-optical coefficient (table. \ref{tab:physical_values}). Better physical insight into the two competing effects (thermal and free carrier), is gained by reducing the average input power, e.g considering much smaller duty cycles.
\subsection{Low duty cycle excitation}
The cavity is excited with low duty cycle, square pulses (duration $\Delta$t = 20 ns, period T=2 $\mu$s), so that the cavity has time to cool down before the arrival of the next pulse. We refer to this case as a \textquotedblleft non-heating pulse" because the average temperature of the cavity remains close to room temperature. Fig. \ref{fig:exp_pulse} reports the input and the transmitted pulse depending at various wavelength detunings.
The input power in the pulse is estimated to be on the order of 10 mW. A more accurate determination of this value was complicated by the very low filling factor and the use of the EDFA, which is not designed to operate in these conditions. This lead to some fluctuation in the level of the amplified pulse. Furthermore, very low duty cycle means that the the power contained in the pulse is a fraction of the amplified spontaneous emission(ASE), which means that it is difficult to determine the exact value of the measurement of the average power.
\indent Let us first consider a negative detuning: e.g. $\Delta\lambda \approx -0.5\Delta\lambda_{FWHM}$ (fig. \ref{fig:exp_pulse}a). The leading edge of the output pulse rises with almost linear slope and is about 5 ns long. For zero or moderate positive detuning (e.g. $\Delta\lambda$ between 0 and $\approx \Delta\lambda_{FWHM}$), the output pulse typically has a two step shape with the leading part of the pulse having a lower level (fig. \ref{fig:exp_pulse}b and c). The transition between the two levels is sharp (a few ns) and moves from the leading edge to the trailing edge of the pulse as the detuning is increased. If detuning is increased further, the pulse takes on an almost triangular shape with a sharp leading edge and a long trailing edge (figure \ref{fig:exp_pulse} d) before losing any signature of nonlinear distortion.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig5.eps}
\caption{ (Color online) Experiment with \textquotedblleft non heating" pulses. Input (dashed) and output (solid) pulses as a function of the detuning. (a) $\Delta\lambda=-0.5\Delta\lambda_{FWHM}$, (b) $+0.7\Delta\lambda_{FWHM}$, (c) $+0.9\Delta\lambda_{FWHM}$ and (d) $+1.5\Delta\lambda_{FWHM}$. The pulse period is $2\mu s$.}
\label{fig:exp_pulse}
\end{figure}
Interestingly, the initial leading edge of the output pulse is very similar in the four cases considered in fig. \ref{fig:exp_pulse}, i.e. we don't see low transmittance associated to the on-resonance case, as long as the power is high. We see this behavior at much lower power only. We believe that this is due to a fast carrier effect which detunes the cavity faster than our detection apparatus is able to measure (1 ns).
It is also important to note that no signature of a plasma-induced bistable state is observed when the detuning is negative. We think that this is because of fast heating, which dominates the plasma effect before the pulse is extinguished. These features are well explained by our model.
\section{Nonlinear dynamical model}
The coupled-mode model developed here is based on previous optical microcavity literature. Particular structures that have been investigated include: microdisks \cite{Carmon2004} and photonic crystal cavities \cite{Barclay05,Uesugi06,cmtRaman}. The dynamical variables are: the optical energy inside the cavity $|a(t)|^2$, the free carrier density $N(t)$ and the cavity and membrane temperature $T(t)$. The model considers a single mode \footnote{the \textit{L3} cavity considered here is single mode over a large spectral range} cavity (resonance is at $\omega_0$ and unloaded quality factor $Q_0$), which is side-coupled to a waveguide (the loaded Q-factor is $Q$).
The modulation of the transmission observed in fig.\ref{fig:exp_setup_sample} is related to the Fabry-Perot resonance due to the finite reflectivity at the waveguide end facets. This spatially extended resonance has no impact on the nonlinear response (other than merely modulating spectrally the coupling into the cavity), since the associated field intensity is orders of magniture smaller than in the cavity.
The field in the cavity follows the equation:
\begin{equation}
\label{cmt_field}
\frac{\partial a(t)}{\partial t} = ( i \omega - i\omega_L - \frac{\Gamma_{tot}}{2}) a(t) +\sqrt{\frac{\Gamma_c}{2}P_{in}(t) }
\end{equation}
where $\omega_L$ is the laser frequency, $\Gamma_c=\omega_0 (Q^{-1}-Q_0^{-1})$ gives the cavity to waveguide coupling strength, $\Gamma_{tot}$ is the inverse (instantaneous) cavity lifetime, $P_{in}(t)$ is the power in the waveguide and $\omega= \omega_0 + \Delta\omega_{NL} $ is the instantaneous cavity frequency. Following \cite{Barclay05,Uesugi06}, the nonlinear change of the cavity frequency is given by:
\begin{gather}\label{delta_omega_nl}
\begin{split}
\Delta\omega_{NL}&=-\frac{\omega}{n_{eff}} \Delta n = -\frac{\omega}{n_{eff}} [\frac{n_{2I}\,c}{n\,V_{Kerr}} |a(t)|^2
\\ & \frac{dn}{dT}\Delta T(t) + \frac{dn}{dN}N(t)]
\end{split}
\end{gather}
Here $n$ is the refractive index of the bulk material and $n_{eff}$ the effective refractive index in the cavity, i.e.: $n_{eff}^2=\int n(\vec{r})^2\left|E(\vec{r})\right|^2dV/\int\left|E(\vec{r})\right|^2dV$, and $n_{2I}$ is the Kerr coefficient and $V_{Kerr}$ the Kerr nonlinear volume, defined as:
\begin{equation}
\label{eq_Kvolume}
V_{Kerr}^{-1}=\frac{\int{n_{2I}(\vec r)/n_{2I}\left|E(\vec r)\right|^4 n(\vec r)^4 dV}}{(\int{n(\vec r )^2 \left|E(\vec r)\right|^2 dV})^2}
\end{equation}
The refractive index change due to the plasma effect is $dn/dN = -\omega_p^2/2n\omega^2N$, with $\omega_p^2=e^2N/\epsilon_0m^*$ the plasma frequency. In principle, holes and electrons both contribute to this effect, however, given the much smaller effective mass of electrons in GaAs, the contribution of holes is negligible. The inverse instantaneous photon lifetime is:
\begin{equation}
\label{gamma_tot}
\Gamma_{tot}= \frac{\omega_0}{Q} + \Gamma_{TPA} + \Gamma_{FCA}
\end{equation}
the first term is the inverse linear cavity lifetime, $\Gamma_{TPA}$ and $\Gamma_{FCA}$ are the contributions from two photon (TPA) and free carrier absorption (FCA), respectively. The TPA term is: $ \Gamma_{TPA} = \beta_2 c^2/n^2 |a(t)|^2/V_{TPA} $ with $\beta_2$ representing the TPA coefficient in units of $m/W$ and $V_{TPA}=V_{Kerr}$ the nonlinear effective volume\cite{Uesugi06}, while the free carrier absorption is proportional to the combined free carrier density $\Gamma_{FCA}=(\sigma_{e}+\sigma_{h}) N(t) c/n $. The evolution of carrier density follows the rate equation:
\begin{equation}
\label{carrier_rate}
\frac{\partial N(t)}{\partial t} = -\frac{N(t)}{\tau_N} + \frac{c^2/n^2 \beta_2}{V_{TPA} } \frac{1}{2\hbar \omega} \frac{1}{V_{car}} |a|^4
\end{equation}
Here $\hbar$ is the reduced Planck's constant, $\tau_{N}$ is the effective carrier lifetime, and $V_{car}$ is the volume in which the carriers spread and recombine\cite{Uesugi06}). In this approximation, the population decay is dominated by recombination at the surface, due to the large surface to volume ratio typical of photonic crystal strucures. Carriers are assumed to spread and distribute homogeneously within the carrier volume $V_{car}$, which is assumed to correspond to the region of the membrane delimited by the holes around the cavity. This approximation is rough, but it is a reasonable choice.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig5d.eps}
\caption{(Color online) Modelling heat diffusion in the PC membrane. Local cavity temperature $T_c$ vs. time calculated with a 2D thermal diffusion equation (dots) and fitting (solid thick and thin lines) with the model eq. \ref{thermal_rate}. Inset: enlarged view of the short and the long term behaviour. Inset right: fitted $\tau_{th,m} = 200ns$ with a single exponential (dashed line).Fit with 4 unconstrained parameters (thin line). Constrained fit with $\tau_{th,m} = 200ns$ $\tau_{th,c}=8.4\,ns$ (thick solid line, main plot and inset left).
Thermal capacitances for the cavity and the membrane are $0.43\,10^{-12} W/K$ and $3.4\,10^{-12} W/K$ respectively.}
\label{fig:ThermalDiffusion}
\end{figure}
\indent Because of the high thermal resistance in membrane PCs, thermal effects are very important and must be modelled appropriately. This is, for instance, the scope of ref.\cite{Kawashima_JQE2008}. In contrast with existing literature, we introduce a more complicated model for the thermal response. We assume that the heat generated in the cavity has not yet diffused to the border of the membrane at the time scale of interest (1 ns). Therefore, a very important physical quantity is the thermal capacitance of the cavity $C_{th,c}=c_v\rho V_{th,c}$, associated with a small region of the membrane, with volume $V_{th,c}$ roughly corresponding to the cavity volume.
Heating and the spreading of the heat are modelled by solving the two dimensional heat diffusion equation. The heat is generated for some time (10 ns) inside the cavity and then the system is allowed to cool down. We assume that the radiative and convective contributions are negligible; therefore, since the membrane is suspended in air, all the heat has to flow through it. The result of this calculation is shown in fig. \ref{fig:ThermalDiffusion}. The cavity temperature $T_c$ increases with a rate that is governed by $C_{th,c}$ and $T_c$ decreases at two exponential time scales. The fast time scale $\tau_{th,c}$ is associated to a relatively fast transfer of heat from the cavity, where it is generated, to the neighbouring region in the membrane. We associate a second thermal capacitance $C_{th,m}$ to the whole membrane. The second time scale $\tau_{th,m}$ takes into account the spreading of the heat over the rest of the membrane (with temperature $T_m$) to the bulk semiconductor structure.
This picture is substantially different from previous models with a single time constant $\tau=C_{th,c}R_{th}$ that is obtained from the thermal resistance $R_{th}$, defined as $W=R_{th}\Delta T$ (ratio between the increase of the temperature over the heating power at steady state). We will show that these two time scales play a crucial role in understanding the system response under sinusoidal and single pulse excitation.
The corresponding model entails therefore two auxiliary equations:
\begin{eqnarray}
\label{thermal_rate}
\frac{\partial T_c(t)}{\partial t} = -\frac{T_c(t)-T_m(t)}{\tau_{th,c}} +\frac{2 \hbar \omega}{\tau_N} \frac{V_{car} }{C_{th,c} V_{th}}N(t) \\
\frac{\partial T_m(t)}{\partial t} = -\frac{T_m(t)-T_{0}}{\tau_{th,m}} -\frac{T_1(t)-T_2}{\tau_{th,1}}\frac{C_{th,c}}{C_{th,m}}
\end{eqnarray}
The thermal effective volume of the cavity and of the membrane $V_{th,c}$ $V_{th,m}$ and their thermal relaxation times $\tau_{th,c}$, $\tau_{th,m}$ are obtained by fitting the solution of the heat diffusion equation with the solution of the two time constant model (eq. \ref{thermal_rate}). If we fit all four parameters (two time constants and two capacitances) the fit turns out to be very good in the first 50 ns, yet tends to underestimate the long time constant and therefore to underestimate the total thermal resistance. To prevent this, we first fit a simple exponential to the long term behaviour, which gives a time constant of approximately 200ns. Then we fit the complete curve by constraining $\tau_{th,m} = 200 ns$. The result is shown in fig. \ref{fig:ThermalDiffusion} and confirms that this model is well adapted to describe the thermal behaviour of PC microcavities. The result of this calculation is given in table \ref{tab:physical_values}.
\begin{table}
\caption{\label{tab:physical_values} Physical paramenters used in the dynamical model.}
\begin{ruledtabular}
\begin{tabular}{llll}
Parameter&Symbol&Value&Ref.\\
\hline
\\
TPA coefficient & $\beta_2 (cm/GW)$ & $10.2$ & \cite{Dinu2003}\\
Kerr coefficient & $n_{2I} (cm^2/W)$ & $1.6 10^{-13}$ & \cite{Dinu2003} \\
Loaded Q & $Q$ & $7000$ & meas.\\
Intrinsic Q & $Q_0$ & $30000$ & est.\\
Modal Volume & $V_{mod}$ & $0.66 (\lambda/n_0)^3$ & calc.\\
TPA Volume & $V_{TPA} $ & $3.13 (\lambda/n_0)^3$ & calc.\\
Carrier Volume & $V_{car} $ & $7~V_{mod}$ & \\
Thermo-optic coeff. & $dn/dT$ ($K^{-1})$ & $2.48~10^{-4}$ & \cite{handook_optics} \\
Therm. eff. vol.(c) & $V_{th,c} $ & $3.8*V_{mod}$& calc. \\
Therm. eff. vol. (m) & $V_{th,m} $ & $31*V_{mod}$& calc. \\
Specific Heat & $c_v\rho$ ($W/K\,m^{-3}$) & 1.84 $10^6$ & \cite{handook_optics} \\
Carrier lifetime & $\tau_{N} $ & 8 ps & \cite{bristow_GaAs_APL03,Husko_APL2008}\\
Therm. relax. time (c) & $\tau_{th,c} $ & 8.5 ns & calc. \\
Therm. relax. time (m) & $\tau_{th,m} $ & 200 ns & calc.\\
Therm. resistance & $R_{th}=\sum\frac{\tau_{th}}{C_{th}} $ & $7.5\,10^4$ K/W & calc.\\
FCA cross section & $ \sigma_{e,h}$ $(10^{-22} m^2)$ & $9.3 (\frac{\lambda}{1.0\mu m})^{2.3}$ & \cite{Reinhart_JAP05}\\
\end{tabular}
\end{ruledtabular}
\end{table}
\section{Discussion}
Simulations have been carried out with parameters given in Table \ref{tab:physical_values}. All of them are well known or can be calculated or measured with reasonable accuracy. The carrier volume $V_{car}$ is the exception. The impact of carrier diffusion has been investigated theoretically in a recent paper \cite{Tanabe_JLT2008}, providing some hints to explain a fast recovery time in Silicon PCs. The carrier lifetime in patterned GaAs (e.g. 2D PCs) is however much shorter than in Silicon. Very recently we estimated it to be about 6 ps in our GaAs cavities\cite{Husko_APL2008}, which is consistent with previous (8 ps) estimates for GaAs PC structures \cite{bristow_GaAs_APL03}.
\indent In the limit where the dynamic is much slower than $\tau_N$, which the case considered in this work, $\tau_N$ and $V_{car}$ play the same role and what matters here is the ratio $\tau_N/V_{car}$. To show that, let us consider the density of the generated carriers :
\begin{equation}
\label{carriers_steady}
N(t)= \frac{c^2/n^2 \beta_2}{V_{TPA} } \frac{1}{2\hbar \omega} \frac{\tau_N}{V_{car}} |a|^4
\end{equation}
The blue-shift of the cavity resonance due to the generated carriers is therefore proportional to the instantaneous power absorbed, $P_a$. The red shift induced by heating is proportional to the absorbed energy and to the inverse of the thermal capacitance. Then, the relative strength of thermal and carrier effects depends on the ratio $\tau_N/V_{car}$ and on the thermal capacitance. The dynamical behaviour of the system investigated here is basically determined by these two quantities.
Modelling was performed by varying only one parameter, the carrier volume, and adjusting the input power and the detuning around reasonable values. All other values are well known or calculated precisely.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig6.eps}
\caption{(Color online) Simulation of the response (solid) to a single pulse (dashed), depending on the detuning $\Delta\lambda(0)$: -120 pm (a,b), 0 pm (c,d), 80 pm (e,f) and 150 pm (e,f). Right: corresponding instantaneous frequency shift $\Delta\lambda_{NL}$ of the cavity resonance (solid line) and carrier plasma (thin-dashed) and thermo-optical contributions (dotted). The laser frequency is also shown (thin solid). $P_{in}$ = 1.2 mW, 0.9 mW, 1.2 mW and 1.0 mW respectively.}
\label{fig:fig_simulazione_pulse}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig6b.eps}
\caption{(Color online) Simulated response when the contribution to the plasma shift from carrier plasma is suppressed (a) and full model (c). The excitation power is 1.2 mW (thick line). The case with $P_{in}$ =1.1 mW and 1.3 mW are also plot (thin lines). The detuning $\Delta\lambda(0)$ is 80 pm. The corresponding instantaneous frequency shifts $\Delta\lambda_{NL}$ is also plot (b,d) for the case $P_{in}$ =1.2 mW.}
\label{fig:fig_simulazione_withwo_plasma}
\end{figure}
\indent Fig. \ref{fig:fig_simulazione_pulse} reports the simulated responses of a square pulse at various detunings. When the initial detuning is negative (e.g. $\Delta\lambda(0)=-120 pm$, fig. \ref{fig:fig_simulazione_pulse} a and b), the carrier effect is responsible for the drop of the leading edge of the transmitted pulse. The signal coupled in the cavity is strong enough to cause considerable generation of carriers and the resulting blue-shift tends to tune the cavity into resonance so that transmission is reduced (the cavity is side-coupled).
Within a ns time scale, the heating following carrier recombination red-shifts and therefore detunes the cavity again. Such fast heating arises from the small thermal capacitance of the cavity. After the pulse is extinguished, the cavity is red-shifted by about 100 pm with respect to the initial position and recovers its initial state in a fraction of $\mu s$.
\indent Let us now consider the case of zero detuning (fig. \ref{fig:fig_simulazione_pulse}c and d). The carrier induced index change produces an instantaneous (with respect to the time scale considered here) blue shift that adds to the initial detuning. Heating, following absorption, tunes the cavity back and transmission decreases once again. Thus more carriers are generated and the plasma effect tends to oppose heating but, in the end, the heating dominates.
At these modulation rates thermal relaxation is not sufficiently fast to dissipate heat from the cavity after each duty cycle. Thus heat accumulates over time, causing a net red-shift of the cavity resonance. When the instantaneous detuning $\Delta\lambda_{NL}$ becomes positive, carrier generation starts to decrease and so does the carrier induced shift. This initiates a positive feedback that quickly detunes (red) the cavity and makes the carrier density drop very fast. The delay between the pulse leading edge and this transition is clearly related to the amount of the initial detuning (fig. \ref{fig:fig_simulazione_pulse}e and f).
When the initial detuning is positive and large enough (fig. \ref{fig:fig_simulazione_pulse}g and h), then heating tends to tune the cavity into resonance, thus reducing the transmission but also generating carriers. The plasma effect will partially oppose the red-shift, thus explaining an almost linear change with time.
\indent The sharp step observed in fig. \ref{fig:exp_pulse}b,c can only be explained by the thermal effect overtaking the carrier plasma index change. This step cannot be reproduced for any choice of free parameters if the plasma induced index change is suppressed in the model. This is shown in figure \ref{fig:fig_simulazione_withwo_plasma}. In panel a and b we ran the same simulation as in fig. \ref{fig:fig_simulazione_pulse}e, but we have suppressed the contribution of carriers to the frequency shift of the cavity. In this case the transmission increases with a linear, but finite, slope after the dip, independent of the pulse power. This is not what we see experimentally. Conversely, when the plasma effect is included, the sharp transition is reproduced correctly.
There is a physical reason for that: the observed steep step requires a fast index change with, say, a negative sign, and a slower index change, with opposite sign. A simple explanation is that at first the plasma effect dominates (over a very short time, a few ns) before thermal heating takes over. If a plasma effect is ruled out, there is no way to explain the experimental results. A Kerr effect cannot be considered because it has the same sign as the thermal effect.
After the submission of this manuscript, critical slowing down (CSD), implying a transition from a low (off-resonance) to a high (on-resonance) transmission state, has been reported in InP-based PC cavities\cite{Yosia2008}. This effect also results into a step in the pulse response. We believe however that this is a different mechanism than what we describe here. The reason is that our system is side-coupled, thus CSD would manifest as a transition from a high (off-resonance) to a low (on resonance) transmission state. We did not observe that. A possible explanation is the carrier lifetime in GaAs PCs being mush shorter than in Silicon or InP PCs. This implies that the plasma dispersion effect is much weaker than the thermal induced index change.
\indent We found experimentally that the response of the cavity is very sensitive to the power of the pulse, in particular in the case reported in \ref{fig:exp_pulse}e. This behavior is well reproduced by theory. This means that input power must be set precisely in simulation and it cannot be the same for all the detunings considered here.
Indeed, experimentally the peak pulse power is not constant since, a) the output power from the amplifier showed a marked dependence on slight change in wavelength, b) the waveguide transmission is modulated by a strong Fabry-P\'erot effect (fig. \ref{fig:exp_setup_sample}).
The fact that the detuning in experiments and in the theory are not the same can be understood by accepting some error in the measurement of the detuning. The reason is that the laser was directly modulated which induced some deviation from the nominal setting of the laser on the order of 100 pm.
\begin{figure}
\centering
\includegraphics[width=9cm]{fig7.eps}
\caption{(Color online) Simulation of the response of the cavity to a sinusoidal excitation for the following detunings (from top to bottom): 440 pm (a,d), 490 pm (b,e) and 500 pm (c,f). Input peak power is 1 mW. The transmitted pulses are shown on the top (a,b,c), the instantaneous frequency shift (solid think curve) and the thermo-optical contribution alone are shown on the bottom (d,e,f).}
\label{fig:sim_cw_ENT}
\end{figure}
This relatively fast (few ns rise-time) nonlinear effect reported in fig. \ref{fig:exp_pulse} is well understood through modelling. The complexity of the model is justified by the number of different time scales present.
An important point of this paper is to demonstrate that the same model is able to explain also the response to a sinusoidal excitation. The response is calculated for a time long enough to ensure that we reach the steady-state regime. The simulations shown in fig. \ref{fig:sim_cw_ENT} are carried out in order to mimic experimental conditions, that is, the excitation wavelength is detuned negatively with respect to the cold cavity frequency. As the excitation is turned on, the wavelength is increased adiabatically with respect to the modulation period. Indeed it is found that the cavity stays red detuned, and is pushed towards longer wavelengths, until the excitation power and the heat generated is enough to sustain the detuning.
\indent The simulations clearly support the experimental results shown in figure \ref{fig:sin_modulation} and interpretation in fig. \ref{fig:scheme_detuning}. Indeed, if we now compare the measured and calculated oscilloscope traces we conclude that the dynamics are accurately reproduced, as the detunings used in modelling correspond to what is measured experimentally.
In particular, the response is very well understood by looking at the instantaneous frequency of the cavity, which crosses the laser frequency at different points depending on the detuning. If the detuning $\Delta\lambda$ is too small, the cavity wavelength is always higher than the laser frequency, as the instantaneous cavity nonlinear shift $\Delta\lambda_{NL}=\lambda_{cavity}(t)-\lambda_{cavity}(0)$ due to heating is much larger over the whole modulation cycle. When the detuning $\Delta\lambda$ increases, $\Delta\lambda_{NL}$ also increases on average, as the resonance is closer to the laser frequency, but less than $\Delta\lambda$ does. Thus, there exists a combination of pulse power, modulation frequency and detuning such that $\Delta\lambda$ and $\Delta\lambda_{NL}$ are close one to each other. If $\Delta\lambda$ is increased further, heating is not enough to keep $\Delta\lambda_{NL}$ close to it and nonlinear distortion is lost. The role of the plasma induced index change and the interplay with the thermo-optic effect is also clear in figure \ref{fig:fig_simulazione_pulse}(b,d,f,).
It is interesting to note that the plasma effect vanishes when $t<4ns$ and $t>16ns$ in figs. \ref{fig:sim_cw_ENT}b,d and f, although minima of the sinusoidal excitation are at t=0 and t=20 ns (figs. \ref{fig:sim_cw_ENT}a,c,e). This important point is understood when considering that while the plasma effect is almost instantaneous (within the time scale relevant to this work), the thermo-optic effect follows the time integral of the optical power. At time $t=0$ the carrier population is negligible and the cavity is cooling down, and therefore is approaching the laser wavelength. As the power inside the cavity increases, carriers are generated with a rate proportional to the power squared. A carrier induced index change builds up very fast and produces the difference between the thin and the thick curves. The power absorbed produces heating. Instantaneous heating is the time integral of instantaneous absorbed power and therefore builds up with a delay with respect to the plasma effect. The thermal effect then dominates the plasma effect. As soon as the resonance is crossed again and the detuning becomes comparable to half the cavity linewidth (100 pm), nonlinear absorption decreases very quickly and the carrier population declines accordingly. Therefore the carrier effect disappears. The cavity again starts to cool down and the cycle begins anew.
\section{Conclusion}
Switching based on TPA has a great potential in GaAs based structures, as it benefits from the strong nonlinear absorption coefficient $\beta_2$ and sub-ns carrier lifetimes typical of this material. However, most practical applications will require a high repetition rate (i.e. high duty cycle), therefore thermal effects are unavoidable in such small structures. In particular, heating is very strong in photonic crystals, because of the poor thermal conductivity of PC membranes. We have experimentally investigated this regime and confirmed results from other groups evidencing that the thermal dynamics in PC membrane cavities is much faster than in macroscopic photonic devices due to the small modal volume of the structure. More particularly, we have explored a regime in which a sharp (2 ns) transition result from a positive feedback between thermal and carrier induced refractive index changes. The peak power required to see these effects is at the mW level. We have also explained why it is not possible to observe bistability in GaAs cavities when the modulation rate is on the order of 10 - 100 MHz.
\indent Based on these experimental results, we have introduced a dynamical model for PC microcavities which includes two thermal relaxation constants in order to account for heat diffusion from the cavity to the neighbouring membrane. We show that this is crucial in order to understand the dynamical response of the cavity and that this model reproduces the experimental results quite well.
The ability to properly account for very different time scales is crucial for modelling patterning effects (i.e. long term or memory effects) which result not only from the carrier dynamics but also on fluctuations of the average power which induce changes in the local temperature of the cavity. This is crucial to design PC based all-optical switches for practical optical signals at high repetition rate (e.g. 10 GHz).
\indent One of the authors (C. Husko) thanks the Fulbright Grant for financial support.
|
2,869,038,156,423 | arxiv | \section{Comparison with Other Negative Samplings}
To our knowledge, there is no technique, especially for performing negative sampling in hyperbolic recommender systems. In this section, we compare three widely used techniques to obtain negative sampling in Euclidean models: dynamic random sampling (DS), popularity-based sampling (Popularity) and mix feature sampling (Mix).
Dynamic random sampling is used to randomly select a node as the negative item in each iteration.
The purpose of popularity-based sampling is to select negative items based on their degrees, which could reflect their popularity. The sampling process is performed before the training process, which is a static method.
The mixing feature method is based on the idea of~\cite{huang2021mixgcf}, that is, creating a negative sample based on the mix of positive and negative items. Mixing method obtains negative samples in each iteration which is a dynamic method.
As shown in Table~\ref{tab:negative_sampling}, we found that among the three sampling methods, DS sampling can improve the performance of recommendations to some extent. The reason lies that compared to static methods, dynamic methods help the model learn more negative samples and obtain enough information. However, DS is insufficient, as most negative samples are trivial or noisy. Combined with the performance of HICF (ours), we know that the proposed HINS is much more helpful. In addition, the other two methods based on popularity and fusion methods are not compatible with hyperbolic models, which perform much worse on hyperbolic models.
\begin{table*}[!t]
\centering
\caption{Comparisons with different negative sampling techniques.}
\label{tab:negative_sampling}
\resizebox{0.94\textwidth}{!}{%
\begin{tabular}{@{}l|cccc|cccc|cccc@{}}
\toprule
Dataset & \multicolumn{4}{c|}{Amazon-CD} & \multicolumn{4}{c|}{Amazon-Book} & \multicolumn{4}{c}{Yelp2020} \\
Metrics & DS & Popularity & Mix & Ours & DS & Popularity & Mix & Ours & DS & Popularity & Mix & Ours \\ \midrule
Recall@10 & 0.1013 & 0.0912 & 0.0884 & \textbf{0.1079} & 0.0858 & 0.0860 & 0.0757 & \textbf{0.0965} & 0.0542 & 0.0528 & 0.0454 & \textbf{0.0570} \\
NDCG@10 & 0.0793 & 0.0708 & 0.0684 & \textbf{0.0848} & 0.0842 & 0.0860 & 0.0749 & \textbf{0.0978} & 0.0470 & 0.0460 & 0.0396 & \textbf{0.0502} \\
Recall@20 & 0.1504 & 0.1338 & 0.1362 & \textbf{0.0965} & 0.1310 & 0.1317 & 0.1175 & \textbf{0.1449} & 0.0898 & 0.0890 & 0.0776 & \textbf{0.0948} \\
NDCG@20 & 0.0953 & 0.0846 & 0.0839 & \textbf{0.1010} & 0.1000 & 0.1018 & 0.0894 & \textbf{0.1142} & 0.0595 & 0.0587 & 0.0508 & \textbf{0.0633} \\ \bottomrule
\end{tabular}%
}
\end{table*}
\section{Introduction}
With the growth of Amazon, Netflix, TikTok, and other e-commerce or social networking services over the past several years, recommender systems are becoming ubiquitous in the digital age.
Recommender systems, in a broad sense, are algorithms that try to suggest relevant or potentially preferable items to the users, where items are, e.g., news to read, movies to watch, goods to buy, etc.
Collaborative filtering, one of the most extensively used techniques in the customized recommendation, is based on the assumption that users often get the preferable suggestions from someone with similar preferences. To provide relevant recommendations, collaborative-filtering approaches~~\cite{koren2009matrix,koren2008factorization,VAECF2018,nmf-cf} rely on historical interactions between users and items, which are stored in the user-item matrix.
Recently, researchers have proposed explicitly incorporating high-order collaborative interaction to improve recommendation performance. Usually, the user-item relationship is modeled as a bipartite graph with nodes representing users or items, and edges representing their interactions. Then, graph neural networks (GNNs)~\cite{gcn2017,GAT,graphsage,zixingcikm2021,zhang2022graph} are applied to extract high-order relationships between users and items via the message propagation paradigm. By using layers of neighborhood aggregation under the graph convolutional setup to construct the final representations, these techniques~\cite{wang2019ngcf,he2020lightgcn,sun2021hgcf,mao2021simplex,mao2021ultragcn} have attained state-of-the-art performance on diverse benchmark datasets.
The heavy-tailed distribution\footnote{Heavy-tailed distributions are substantially right-skewed, with a small number of large values in the head and a large number of small values in the tail; they are often described by a power law, a log-normal, or an exponential function.} occurs in most large-scale recommendation datasets where the number of popular items liked by a large number of users accounts for the minority and the rest are the majority which are unpopular ones. In general, popular items are competitive while the long-tail item reflects personalized preference or something new. Both are critical for the recommendation. An example is illustrated in Figure~\ref{fig:user_item_graph}.
Recently, hyperbolic space has gained increasing interest in the recommendation area as the capacity of hyperbolic space exponentially increases with radius, which fits nicely with a power-law distributed user-item network. Naturally, models based on hyperbolic graph neural networks achieve competitive performance in recommender systems~\cite{sun2021hgcf,chen2021modeling,yang2022hrcf}. However, it is not clear in what respects the hyperbolic model is superior to the Euclidean counterpart. At the same time, it is unclear in which aspects hyperbolic models perform worse than Euclidean models.
\begin{figure}[!t]
\centering
\includegraphics[width=0.40\textwidth]{file/figures/user_item.pdf}
\caption{Illustration of a user-item graph, in which the popular items, e.g., flower and fruit (the yellow icons) are liked by everyone while the preference of the other items is personalized and individualized, e.g., the paintbrush is only enjoyed by the painter and the guitar is merely appreciated by the musician.
It is worth mentioning that the popular items are in the minority, whereas the individualized or unpopular items are in the majority.
}
\label{fig:user_item_graph}
\end{figure}
To answer the above doubts, in this work, we take the simplest form of recommendation model, collaborative filtering (CF), as an example to analyze and observe the behaviors of hyperbolic and Euclidean models. Specifically, we take LightGCN~\cite{he2020lightgcn} and HGCF~\cite{sun2021hgcf} as an example for analysis and observation, both of which are essentially the same model applied in different spaces.
Specifically, we compare the recommendation effects of hyperbolic and Euclidean models, as well as their performance on the head-and-tail item, using a similar model configuration and running environment. Head and tail items are essentially chosen by the 20/80 rule\footnote{The 80/20 rule is mathematically expressed as a power-law distribution (also known as a Pareto distribution).}, which states that all items are ranked according to their degrees, and the top 20\% are considered the head (abbreviated H20), while the remaining 80\% are called the tail (abbreviated T20). The experimental findings are presented in Section~\ref{sec:observation}, which reveals the following facts. The tail item receives more consideration in the hyperbolic model than that in the Euclidean model, but there is still plenty of room for improvement, while the head item receives marginal attention in hyperbolic space, which might be substantially enhanced. Overall, the hyperbolic models outperform the Euclidean models.
These findings are of great significance to the community of recommender systems since they help researchers better understand the advantages and disadvantages of hyperbolic models, as well as when and where to deploy them.
On the basis of the above insights, we develop a novel technique to improve hyperbolic recommender models. Two main aspects are considered: the recommendation effect of the tail item, as well as the issue of the insufficient weight placed on the head item.
Given that most recommender systems pull the user and its interesting items in adjacent positions while pushing its uninterested items in distant areas, our method is carried out from the perspective of pull and push. The basic idea is to link the pull-and-push operations to hyperbolic geometry. Specifically, we design a hyperbolic aware margin ranking loss and hyperbolic informative aware negative sampling in such a way that both head and tail items get considerable attention. To summarize, the contribution of the proposed work is three-fold.
\begin{itemize}
\item We initiate a quantitative investigation to study the behaviors of Euclidean and hyperbolic recommendation models, which reveals valuable findings and insights to the community. Specifically, it is observed that the hyperbolic model outperforms the Euclidean model in general and emphasizes more on the tail items, but with some sacrifice on the head items. This discovery is critical for the study of hyperbolic recommender systems.
\item We present a hyperbolic informative collaborative filtering (HICF) method, which ensures that both head and tail items get sufficient attention via tightly coupling the embedding learning process to the hyperbolic geometry.
\item \textcolor{black}{Extensive experiments demonstrate the effectiveness of the proposed method where the maximum recommendation effect on overall items versus all baselines up to 12.92\%, on head items against the hyperbolic model up to 12.50\%, and on tail items against hyperbolic model up to~\textbf{26.69\%}. It should be noted that the proposed method is not limited to CF-based models, but is also applicable to other hyperbolic recommendation models.}
\end{itemize}
\section{Related work}
Collaborative filtering (CF) is one of the most widely used techniques in recommender systems, in which users and items are parameterized as a matrix and the matrix parameters are learned by reconstructing historical user-item interactions.
Earlier CF methods mapping both the ID of users and items to a joint latent factor space, so that user-item interactions are modeled as inner products in that space \cite{koren2008factorization,he2017neural,LRML2018,chen2017attentive}. The user-item interaction in the recommender system could well be represented by a bipartite graph. Recently, graph-based CF approaches~\cite{wang2019ngcf,he2020lightgcn,chen2021attentive} have made significant progress in capturing explicit relationships.
Existing graph neural networks, on the other hand, are mostly created in Euclidean space, which may understate the implicit power-law distribution of the user-item network.
Research~\cite{hgcn2019,liu2019HGNN,yang2022hyperbolic} demonstrates that the hyperbolic space is more embeddable, particularly when graph-structured data exhibit hierarchical and scale-free characteristics.
Hyperbolic representation learning has gained growing interest in the field of studying graph representation~\cite{liu2019HGNN,hgcn2019,lgcn,yang2021discrete,liu2022enhancing,yang2021hyper}.
Due to the scale-free characteristic of the user-item network, hyperbolic geometry has also attracted a lot of attention and has been successfully applied to recommender systems~\cite{HyperML2020,wang2021hypersorec,feng2020hme,sun2021hgcf,zhang2021we,chen2021modeling,yang2022hrcf} in recent years.
For the recommender system, HyperML~\cite{HyperML2020} studies metric learning in a hyperbolic space for the representation of the user and the item.
HGCF~\cite{sun2021hgcf} incorporates multiple layers of neighborhood aggregation using a hyperbolic GCN module to gather higher-order information in user-item interactions.
HSCML~\cite{zhang2021we} provides a thorough study of network embedding techniques for recommender systems.
LKGR~\cite{chen2021modeling} attempts to learn embeddings in a hyperbolic space for the knowledge-graph-based recommender system.
The majority of the above works attempt to extend current Euclidean models to hyperbolic space. While the inherent benefits and shortage of hyperbolic space are seldom investigated. Although HSCML~\cite{HyperML2020} provides several interesting observations, they are insufficient to fully understand the hyperbolic recommender system.
\begin{figure*}[!t]
\centering
\includegraphics[width=4.1001cm]{file/figures/top_bottom/Amazon-CD_Recall_20.pdf}
\includegraphics[width=4.1001cm]{file/figures/top_bottom/Amazon-CD_NDCG_20.pdf}
\includegraphics[width=4.1001cm]{file/figures/top_bottom/Amazon-Book_Recall_20.pdf}
\includegraphics[width=4.1001cm]{file/figures/top_bottom/Amazon-Book_NDCG_20.pdf}
\includegraphics[width=4.1001cm]{file/figures/top_bottom/Amazon-CD_Recall_20.pdf}
\includegraphics[width=4.1001cm]{file/figures/top_bottom/Amazon-CD_NDCG_20.pdf}
\includegraphics[width=4.1001cm]{file/figures/top_bottom/Amazon-Book_Recall_20.pdf}
\includegraphics[width=4.1001cm]{file/figures/top_bottom/Amazon-Book_NDCG_20.pdf}
\caption{Comparisons of Euclidean and hyperbolic models in Amazon-CD and Amazon-Book datasets. \texttt{Euc} represents the Euclidean model, LightGCN, and \texttt{Hyp} denotes the hyperbolic model, HGCF.}
\label{fig:obversation_on_hgcf_lightgcn}
\end{figure*}
\section{PRELIMINARIES}
Riemannian geometry is a branch of differential geometry that involves the study of smooth manifolds with a Riemannian metric.
Different curvatures of Riemannian manifolds create distinct geometries: elliptic (positive curvature), Euclidean (zero curvature), and hyperbolic (negative curvature).
We will concentrate on negative curvature space, i.e., hyperbolic geometry, in this work.
There are multiple equivalent models for hyperbolic space, each with a unique set of properties, yet being mathematically identical. The Lorentz model (alternatively called the hyperboloid model) is one of the typical hyperbolic models~\cite{nickel2018learning,hgcn2019,liu2019HGNN,lgcn}.
An $n$-dimensional Lorentz manifold with negative curvature $-1/\kappa (\kappa>0)$ is defined as the Riemannian manifold $(\mathbb{H}^n_\kappa, g_\mathcal{L})$, where $\mathbb{H}^n_\kappa =\{\mathbf{x}\in\mathbb{R}^{n+1}:\langle\mathbf{x},\mathbf{x}\rangle_\mathcal{L}=-\kappa, x_0>0\}$, $g_\mathcal{L}=\eta$ $(\eta = \mathbf{I}_n$ except $\eta_{0,0}=-1)$ and $\langle\cdot , \cdot\rangle_\mathcal{L}$ is the Lorentzian inner product. Given $\mathbf{x,y}\in \mathbb{H}^n_\kappa$, the Lorentz inner product is give by:
\begin{equation}
\langle\mathbf{x},\mathbf{y}\rangle_\mathcal{L}:=-x_0y_0 + \sum_{i=1}^n x_iy_i.
\label{equ:inner_product}
\end{equation}
For any $\mathbf{x}\in \mathbb{H}_\kappa^n$, there is a tangent space $\mathcal{T}_\mathbf{x}\mathbb{H}_\kappa^n$ around $\mathbf{x}$ approximating $\mathbb{H}_\kappa^n$, which is an $n$-dimensional vector space (\textit{c.f.}, Definition~\ref{def:lorentz_tangent_space}). To realize the projection between $\mathbb{H}_\kappa^n$ and $\mathcal{T}_\mathbf{x}\mathbb{H}_\kappa^n$, we can resort to the exponential map and the logarithmic map, which are given in Definition~\ref{def:lorentz_exponential_map}. The original point $\mathbf{o}:=\{\sqrt{\kappa}, 0, \cdots, 0\} \in \mathbb{H}^n_\kappa$ is a common choice as the reference point to perform these operations.
\begin{definition}[Tangent Space]
\label{def:lorentz_tangent_space}
The {tangent space} $\mathcal{T}_\mathbf{x}\mathbb{H}_\kappa^n$ $(\mathbf{x}\in \mathbb{H}_\kappa^n)$ is defined as the first-order approximation of $\mathbb{H}_\kappa^n$ around $\mathbf{x}$:
\begin{equation}
\mathcal{T}_\mathbf{x}\mathbb{H}_\kappa^n:=\{\mathbf{v}\in \mathbb{R}^{n+1}: \langle\mathbf{v},\mathbf{x}\rangle_\mathcal{L} = 0\}.
\end{equation}
\end{definition}
\begin{definition}[Exponential \& Logarithmic Map]
\label{def:lorentz_exponential_map}
For $\mathbf{x}\in \mathbb{H}_\kappa^n$ and $\mathbf{v}\in\mathcal{T}_\mathbf{x}\mathbb{H}_\kappa^n$ such that $\mathbf{v} \neq \mathbf{0}$ and $\mathbf{y} \neq \mathbf{x}$, there exists a unique geodesic $\gamma:[0,1]\to\mathbb{H}_\kappa^n$ where $\gamma(0)=\mathbf{x}, \gamma^\prime(0)=\mathbf{v}$.
The exponential map $\exp_\mathbf{x}: \mathcal{T}_\mathbf{x}\mathbb{H}_\kappa^n \to \mathbb{H}_\kappa^n$ is defined as $\exp_{\mathbf{x}}(\mathbf{v})=\gamma(1)$. Mathematically,
\begin{equation}
\exp_{\mathbf{x}}^{\kappa}(\mathbf{v})=\cosh \left(\frac{\|\mathbf{v}\|_{\mathcal{L}}}{\sqrt{\kappa}}\right) \mathbf{x} + \sqrt{\kappa} \sinh\left(\frac{\|\mathbf{v}\|_\mathcal{L}}{\sqrt{\kappa}}\right){\frac{\mathbf{v}}{\|\mathbf{v}||_{\mathcal{L}}}},
\end{equation}
where $\|\mathbf{v}\|_\mathcal{L} = \sqrt{\langle \mathbf{v}, \mathbf{v}\rangle _\mathcal{L}}$ is the Lorentzian norm of $\mathbf{v}$.
The logarithmic map $\log_\mathbf{x}$ is the inverse of the exponential $\exp_\mathbf{x}$, which is given by
\begin{equation}
\log_{\mathbf{x}}^{\kappa}(\mathbf{y})=d_{\mathcal{L}}^\kappa(\mathbf{x},\mathbf{y})\frac{\mathbf{y}+\frac{1}{\kappa}\langle \mathbf{x}, \mathbf{y} \rangle_\mathcal{L}\mathbf{x}}{\|\mathbf{y} + \frac{1}{\kappa}\langle \mathbf{x}, \mathbf{y} \rangle_\mathcal{L}\mathbf{x}\|},
\end{equation}
where $d_\mathcal{H}^\kappa(\cdot, \cdot)$ is the distance between two points $\mathbf{x}, \mathbf{y}\in \mathbb{H}^n_\kappa$, which is formulated as:
\begin{equation}
d_\mathcal{H}^\kappa(\mathbf{x}, \mathbf{y}) = \sqrt{\kappa}\mbox{arcosh}(-\langle \mathbf{x}, \mathbf{y}\rangle _\mathcal{L}/\kappa).
\end{equation}
\end{definition}
For simplicity, we fix $\kappa$ and set it to 1, implying that the curvature is $-1$. We will disregard $\kappa$ in the following parts for brevity.
\section{Investigation and Method}
\subsection{Hyperbolic Graph Collaborative Filtering in Brief}
The basic concept behind Euclidean and hyperbolic graph collaborative filtering~\cite{he2020lightgcn,wang2019ngcf,sun2021hgcf,chen2021modeling} is to extract high-order dependencies between users and items via a message aggregation mechanism. By graph collaborative filtering, users who like the same items, as well as items that are liked by the same users, will be grouped together.
Hyperbolic graph collaborative filtering, similar to its Euclidean counterpart, comprises three components: (1) hyperbolic encoding layer; (2) hyperbolic neighbor aggregation; and (3) prediction layer.
\textbf{Hyperbolic encoding layer.} The purpose of the hyperbolic encoding layer is to create an initial hyperbolic embedding for users and items.
Gaussian distribution initialization is a typical method in Euclidean space. Similarly, a hyperbolic Gaussian sampling method is applied for hyperbolic recommendation models~\cite{sun2021hgcf,chen2021modeling,yang2022hrcf}. Formally, we use $\mathbf{x}\in\mathbb{R}^n$ to represent the Euclidean state of the node (including the user and the item). Then the initial hyperbolic node state $\mathbf{e}_i^0$ and $\mathbf{e}_u^0$ can be obtained by:
\begin{equation}
\begin{aligned}
\mathbf{e}_i^0 &= \exp_\mathbf{o}(\mathbf{z}_i^0), \quad \quad \mathbf{e}_u^0 = \exp_\mathbf{o}(\mathbf{z}_u^0) \\
\mathbf{z}_i^0 &= (0, \mathbf{x}_i), \quad\quad \mathbf{z}_u^0 = (0, \mathbf{x}_u)
\end{aligned}
\label{equ: initialization}
\end{equation}
where $\mathbf{x}$ is taken from multivariate Gaussian distribution. $\mathbf{z}^0 = (0, \mathbf{x})$ denotes the operation of inserting the value 0 into the zeroth coordinate of $\mathbf{x}$ so that $\mathbf{z}^0$ can always live in the tangent space of origin. The superscript $0$ in $\mathbf{e}^0$ and $\mathbf{z}^0$ indicate the initial or zeroth layer state.
\textbf{Hyperbolic neighbor aggregation.} Hyperbolic neighbor aggregation is used to extract explicit user-item interaction. The hyperbolic neighbor aggregation is computed by aggregating neighboring representations of user and item from the previous aggregation. Given the neighbors $\mathcal{N}_i$ and $\mathcal{N}_u$ of $i$ and $u$, respectively, the embedding of user $u$ and $i$ is updated using the tangent state $\mathbf{z}$ and the $k$-th ($k$>0) aggregation is given by:
\begin{equation}
\mathbf{z}_i^{k} = \mathbf{z}_i^{k-1} + \sum_{u\in \mathcal{N}_i}\frac{1}{|\mathcal{N}_i|}\mathbf{z}_u^{k-1}, \quad\quad \mathbf{z}_u^{k} = \mathbf{z}_u^{k-1} + \sum_{i\in \mathcal{N}_u}\frac{1}{|\mathcal{N}_u|}\mathbf{z}_i^{k-1}.
\end{equation}
where $|\mathcal{N}_u|$ and $|\mathcal{N}_i|$ are the number of one-hop neighbors of $u$ and $i$, respectively.
For high-order aggregation, sum-pooling is applied in these $k$ tangential states:
\begin{equation}
\mathbf{z}_i = \sum_{k} \mathbf{z}_i^k, \quad\quad \mathbf{z}_u = \sum_{k} \mathbf{z}_u^k.
\label{equ:multiple aggregation}
\end{equation}
Note that $\mathbf{z}$ is on the tangent space of origin. For the hyperbolic state, it is projected back to the hyperbolic space with the exponential map,
\begin{equation}
\mathbf{e}_i=\exp_\mathbf{o}(\mathbf{z}_i), \quad\quad\mathbf{e}_u=\exp_\mathbf{o}(\mathbf{z}_u),
\end{equation}
where $\mathbf{e}_i$ and $\mathbf{e}_u$ represents the final hyperbolic embeddings.
\textbf{Prediction layer}.
Through hyperbolic neighbor propagation, explicitly structural information is embedded in the user and item embeddings.
To infer the preference of a user for an item, the hyperbolic distance $d_\mathcal{H}$ can be used for the prediction,
$
p(u,i) = 1/{d^2_\mathcal{H}(\mathbf{e}_u,\mathbf{e}_i)}.
$
Since we concerned with the rank of preferred items, the negative form can likewise be used for prediction, i.e, $p(u,i)=-{d^2_\mathcal{H}(\mathbf{e}_u, \mathbf{e}_i)}$.
\subsection{Investigation}
\label{sec:observation}
According to previous research~\cite{he2020lightgcn,sun2021hgcf}, the hyperbolic model~\cite{sun2021hgcf,yang2022hrcf} performs more competitively than that built in the Euclidean space~\cite{he2020lightgcn} using models with essentially the same structure.
However, it is unclear in what aspects the hyperbolic model excels above its Euclidean equivalent. Simultaneously, it is uncertain in which places hyperbolic models are worse than Euclidean models. These issues obstruct our understanding of hyperbolic recommendation models and hinder their applications in real-world scenarios.
To solve the aforementioned doubts, we undertake a quantitative analysis that aims to experimentally study the behaviors of hyperbolic and Euclidean recommendation models by disentangling their performance on the tail and head items. In particular, we first sort the items by their degree, which is similar to popularity, and then split them into head 20\% (denoted as {H20}, or $\mathcal{I}_{H20}$) and tail 80\%, (denoted T80, or $\mathcal{I}_{T80}$).
Next, we investigate the effect of recommendation via the Recall@K and NDCG@K metric on H20 and T80 items, respectively, using the Euclidean graph collaborative filter model, LightGCN, and the corresponding hyperbolic model, HGCF. The results are shown in Figure~\ref{fig:obversation_on_hgcf_lightgcn}. From the experimental results, we have the following observations:
\begin{itemize}
\item The overall recommendation performance of the hyperbolic model is better than that of the Euclidean model;
\item Tail items get greater emphasis in the hyperbolic model as the results on tail items are far beyond that of the Euclidean counterpart ;
\item Head items receive moderate attention in the hyperbolic model as the performance of HGCF is sightly lower than that of LightGCN.
\end{itemize}
The above results are closely related to the geometric properties of hyperbolic space: the exponentially increased capacity of hyperbolic space enables the hyperbolic model to pay more attention to tail items compared with the Euclidean models and thus obtain an impressive performance.
Then, it is easy to know that hyperbolic recommendation models are beneficial for personalized recommendations and increasing market diversity.\footnote{As we know, the head item is popular and liked by a large number of users while the tail item is either personalized reflecting the unique preference of the user, or something fresh increasing the diversity of the market.} The hyperbolic model is a strong contender, but there are still two main shortages in the current hyperbolic model.
(1) Despite the fact that the hyperbolic model produces better overall outcomes and has a greater recommendation effect on tail items, there is still large room for improvement. The reason is that
tail items account for more user interests in Amazon-CD (54\% T80 vs 46\% H20) and Amazon-Book (53\% T80 vs 47\%) as given in Table~\ref{tab:datasets}, but the recommendation effect of the tail item is much lower than that of the head items, as shown in Figure~\ref{fig:obversation_on_hgcf_lightgcn}.
(2) Besides, compared with Euclidean space, hyperbolic space reduces the attention of the model on head items to a certain extent. Thus, there is an urgent need to improve the recommendation ability of head items. In this work, we aim to alleviate the above problems by improving both the head and tail items.
\subsection{Hyperbolic Informative Collaborative Filtering}
\begin{figure}[!t]
\centering
\includegraphics[width=0.45\textwidth]{file/figures/hyperbolic_margin.pdf}
\caption{Lorentz model of hyperbolic space $\mathbb{H}^2$ with curvature -1. On the right side of the axis $x_0$, there are two pair nodes $\{(\mathbf{e}_1, \mathbf{e}_2), (\mathbf{e}_3,\mathbf{e}_4)\}$ and their distances are equal in the Euclidean space, i.e., $d_E(\mathbf{e}_1, \mathbf{e}_2)=d_E(\mathbf{e}_3, \mathbf{e}_4)$, but are different in the hyperbolic space. In particular, the distance of the nodes $(\mathbf{e}_3, \mathbf{e}_4)$ located near the origin of hyperbolic space (HO), is smaller than the distance of $(\mathbf{e}_1, \mathbf{e}_2)$ far from HO, that is, $d_\mathcal{H}(\mathbf{e}_3, \mathbf{e}_4)<d_\mathcal{H}(\mathbf{e}_1, \mathbf{e}_2)$. On the left side of $x_0$, $\mathbf{e}_i$ and $\mathbf{e}_j$ is the positive item and the negative item, respectively, which indicates that sampling a negative item $\mathbf{e}_j$ is informative for the nodes in positions close to or far from the origin when sampling is close to the positive item $\mathbf{e}_i$.}
\label{fig:lorentz_model_of_hyperbolic_space}
\end{figure}
As we know, the optimization objectives for the representation of the user-item are generally to pull the embedding position between the user $u$ and the positive item $i$, and to push the embedding between the user $u$ and a negative item $j$.
{The exponential growth volume} of the hyperbolic space allows samples in the hyperbolic space to be substantially more concentrated while retaining the necessary separation.
As a result, hyperbolic margin ranking learning (HMRL)~\cite{sun2021hgcf} becomes a potent optimization tool adequately separating the items and avoiding the undesirable collapse that occurs in Euclidean space~\cite{SML2020}. HMRL, which contains two crucial components, pull and push, is used to minimize the following loss function:
\begin{equation}
\ell(u,i,j) = \max(\underbrace{d_\mathcal{H}^2(\mathbf{e}_u, \mathbf{e}_i)}_{\text{Pull}}-\underbrace{d_\mathcal{H}^2(\mathbf{e}_u, \mathbf{e}_j)}_{\text{Push}}+m, 0),
\label{equ:origin_ranking_loss}
\end{equation}
where $d_\mathcal{H}$ is the hyperbolic distance and $m$ determines the margin between the distance difference between $(u, i)$ and $(u,j)$.
Inspired by Equation~(\ref{equ:origin_ranking_loss}), we attempt to simultaneously improve the performance of the recommendation on head and tail items from the perspective of pulling and pushing. Specifically, hyperbolic informative collaborative filtering (HICF) is proposed to enable the pull and push processes to be geometric-aware.
\subsubsection{Pull: Hyperbolic-aware Margin Learning (HAML)}
Hyperbolic space is negatively curved space, meaning that as the radius increases, the volume of space grows exponentially. Put it another way, the area close to the origin is flatter, whereas the region further away curves more and has a larger capacity. As shown in Figure~\ref{fig:lorentz_model_of_hyperbolic_space}, on the right side of the axis $x_0$, there are two pair nodes $\{(\mathbf{e}_1, \mathbf{e}_2), (\mathbf{e}_3,\mathbf{e}_4)\}$ and their distances are equal in the Euclidean space, i.e. $d_E(\mathbf{e}_1, \mathbf{e}_2)=d_E(\mathbf{e}_3, \mathbf{e}_4)$, but different in the hyperbolic space. In particular, the distance of the node pair located near the \underline{o}rigin of \underline{h}yperbolic space (HO), is smaller than that far from HO, i.e., $d_\mathcal{H}(\mathbf{e}_3, \mathbf{e}_4)<d_\mathcal{H}(\mathbf{e}_1, \mathbf{e}_2)$. Such geometric properties make it natural for us to design a hyperbolic geometry-aware optimization scheme, instead of simply migrating the loss function of Euclidean space to hyperbolic space like Equation (\ref{equ:origin_ranking_loss}).
Our main idea is to design a geometric aware margin ranking learning which enjoys the properties of hyperbolic space. Specifically, we suggest assigning a larger margin to the case where the pair (u, i) of the user and the positive item is closer to the hyperbolic origin and a smaller margin to that far away from the HO.
It is motivated by the fact that the area close to HO is relatively flatter and the node pair is positioned in a narrow region, so a larger margin is required to SQUEEZE or PULL the pair (u, i) together, while the area distant from HO bends more and the node pair is placed in a spacious area, so a lesser margin is sufficient.\footnote{Otherwise, with the head items or high-level nodes being self-optimized close to HO referred to~\cite{nickel2017poincare,nickel2018learning,hgcn2019}, HRML, equipped with a geometric unconscious margin (e.g., a constant value), may make these head nodes difficult to be properly distinguished. This explains why the HGCF generally performs poorly on the head items.}
Inspired by the analysis (Section 2)~\cite{sala2018representation}, we know that the ratio of $\{{d_\mathcal{H}^2(\mathbf{e}_u, \mathbf{o})+d_\mathcal{H}^2(\mathbf{e}_i, \mathbf{o})\}/d_\mathcal{H}^2(\mathbf{e}_u, \mathbf{e}_i)}$ is getting smaller when they are moved away from HO. In other words, when the node pair approaches the boundary, their hyperbolic distance will approximate the sum of their hyperbolic norms. Based on these properties, we propose a hyperbolic distance-based manner to compute the margin $m_{ui}^\mathcal{H}$ which is given by:
\begin{equation}
\begin{aligned}
m^\mathcal{H}_{ui} &= \mathrm{sigmoid}(\delta), \\
\delta &= \frac{d_\mathcal{H}^2(\mathbf{e}_u, \mathbf{o})+d_\mathcal{H}^2(\mathbf{e}_i, \mathbf{o})-d_\mathcal{H}^2(\mathbf{e}_u, \mathbf{e}_i)}{\mathbf{e}_{u,0}\mathbf{e}_{i,0}},
\end{aligned}
\label{equ:hyperbolic_margin}
\end{equation}
where the denominator is for normalization and $\mathbf{e}_{u,0}>1$ and $\mathbf{e}_{i,0}>1$ denote the zeroth coordinate element in $\mathbf{e}_u$ and $\mathbf{e}_i$, respectively.
$m_{ui}^\mathcal{H}$ is self-adjusting and geometry-aware, which can achieve the emphasise on both head and tail items.
The following illustrates the intuitive understanding of Equation~(\ref{equ:hyperbolic_margin}). Since the sigmoid function increases monotonically, and we are interested in the equation of $\delta$. The numerator is the difference between the geodesic sum of $\mathbf{e}_u, \mathbf{e}_i$ to the hyperbolic origin and the hyperbolic distance of $\mathbf{e}_u, \mathbf{e}_i$, and the difference gradually decreases as the $u$ and $i$ locations move away from the hyperbolic origin~\cite{sala2018representation}. The denominator is for normalization, which increases steadily as it goes away from the hyperbolic origin. Totally, $m^\mathcal{H}_{ui}$ is getting smaller when $u$ and $i$ are moving far away from HO. Then $m^\mathcal{H}_{ui}$ is utilized to replace the $m$ in Equation~(\ref{equ:origin_ranking_loss}).
\begin{algorithm}[t]
\caption{HINS algorithm}
\label{alg:hins}
\SetKwProg{generate}{Function \emph{generate}}{}{end}
\textbf{Input:} Hyper parameters $n_{neg}$; Item set $\mathcal{I}$; the embedding matrix ${E}$; The index of the user $u$, its current positive item $i$ and its other positive item $\mathcal{N}_u$ in the training set.\\
\textbf{Output:} The informative item index $j$. \\
Random sample $n_{neg}$ items $\mathcal{I}_u^{[n]}$ from $\mathcal{I}$\textbackslash $\mathcal{N}_{u}$\;
\ForAll{item index $\bar{j}$ in $\mathcal{I}_u^{[n]}$}{
Get the embeddings of the $\bar{j}$ from $E$, i.e., $\mathbf{e}_{\bar{j}}$\;
Let $j=-1, d_{\min} = +\infty$\;
Compute the hyperbolic distance $d_{\mathcal{H}}(\mathbf{e}_{\bar{j}}, \mathbf{e}_i)$\;
\If{ $d_{\mathcal{H}}({\mathbf{e}_{\bar{j}}},\mathbf{e}_i)<d{_{\min}}$}{
$d_{\min} = d_{\mathcal{H}}({\mathbf{e}_{\bar{j}}}, \mathbf{e}_i)$\;
$j=\bar{j}$\;
}
}
\textbf{Return} $j$\;
\end{algorithm}
\subsubsection{Push: Hyperbolic Informative Negative Sampling (HINS)}
\label{sec:negative_sampling}
The basic idea is to create a training triplet, (a user $u$, the positive item $i$, the negative item $j$), by sampling the negative item $j$ from the similar popularity of item $i$ as shown in the left part of Figure~\ref{fig:lorentz_model_of_hyperbolic_space}.
This strategy provides more information than random sampling. It is simple to understand that a negative sample of a popular item is likely to be another COMPARABLE popular item while choosing the irrelevant one, such as an unpopular item, may have little effect on the optimization.
On the other hand, the assumption of random sampling is a uniform distribution, which is incompatible with items distributed by the power law.
In this work, we propose a self-optimizing, data-independent way to achieve negative sampling in hyperbolic space. The algorithm is demonstrated in Algorithm~\ref{alg:hins}. In each iteration, we randomly select $n_{neg}$ items $\mathcal{I}_{u}^{[n]}$, compute the hyperbolic distance between each item $\bar{j}$ in $\mathcal{I}_{u}^{[n]}$ and the positive sample $i$ and keep the item with the smallest value, where the smallest value indicates the adjacent position in the hyperbolic space. This strategy ensures that the sampled node is always close to the positive item, no matter where it is positioned, indicating that it can yield informative negative samples for both head and tail items.
\begin{table}[!t]
\caption{Statistics of the experimental data.}
\label{tab:datasets}
\centering
\resizebox{0.48\textwidth}{!}{%
\begin{tabular}{@{}ccccccc@{}}
\toprule
\multirow{2}{*}{Dataset} & \multirow{2}{*}{\#User} & \multicolumn{3}{c}{\#Item} & \multirow{2}{*}{\#Interactions} & \multirow{2}{*}{Density} \\ \cmidrule(lr){3-5}
& & All & H20(\%) & T80(\%) & & \\ \midrule
Amazon-CD & 22,947 & 18,395 & 46 & 54 & 422,301 & 0.10\% \\
Amazon-Book & 52,406 & 41,264 & 47 & 53 & 1,861,118 & 0.09\% \\
Yelp2020 & 71,135 & 45,063 & 62 & 37 & 1,940,014 & 0.05\% \\ \bottomrule
\end{tabular}%
}
\end{table}
\begin{table*}[h]
\caption{Recall (top table) and NDCG (bottom table) results for all datasets. The best performing model on each dataset and metric is highlighted in bold, and the second-best model is underlined. The presence of an asterisk indicates that the Wilcoxon signed-rank test for the difference in scores between the best and second-best models is statistically significant.}
\label{tab:overall_comparsion}
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}cc|cc|cccc|cc|ccc|cr@{}}
\toprule
\multicolumn{2}{c|}{Datasets} & WRMF & VAE-CF & TransCF & CML & LRML & SML & NGCF & LightGCN & HAE & HAVE & HGCF & Ours & $\Delta$(\%) \\ \midrule
\multicolumn{1}{c|}{\multirow{2}{*}{Amazon-CD}} & R@10 & 0.0863 & 0.0786 & 0.0518 & 0.0864 & 0.0502 & 0.0475 & 0.0758 & 0.0929 & 0.0666 & 0.0781 & \underline{0.0962} & \textbf{0.1079}* & +12.16 \\
\multicolumn{1}{c|}{} & R@20 & 0.1313 & 0.1155 & 0.0791 & 0.1341 & 0.0771 & 0.0734 & 0.1150 & 0.1404 & 0.0963 & 0.1147 & \underline{0.1455} & \textbf{0.1586}* & +9.00 \\ \midrule
\multicolumn{1}{c|}{\multirow{2}{*}{Amazon-Book}} & R@10 & 0.0623 & 0.0740 & 0.0407 & 0.0665 & 0.0522 & 0.0479 & 0.0658 & 0.0799 & 0.0634 & 0.0774 & \underline{0.0867} & \textbf{0.0965}* & +11.30 \\
\multicolumn{1}{c|}{} & R@20 & 0.0919 & 0.1066 & 0.0632 & 0.1023 & 0.0834 & 0.0768 & 0.1050 & 0.1248 & 0.0912 & 0.1125 & \underline{0.1318} & \textbf{0.1449}* & +9.94 \\ \midrule
\multicolumn{1}{c|}{\multirow{2}{*}{Yelp2020}} & R@10 & 0.0470 & 0.0429 & 0.0247 & 0.0363 & 0.0326 & 0.0319 & 0.0458 & 0.0522 & 0.0360 & 0.0421 & \underline{0.0527} & \textbf{0.0570}* & +8.16 \\
\multicolumn{1}{c|}{} & R@20 & 0.0793 & 0.0706 & 0.0424 & 0.0638 & 0.0562 & 0.0544 & 0.0764 & 0.0866 & 0.0588 & 0.0691 & \underline{0.0884} & \textbf{0.0948}* & +7.24 \\ \bottomrule
\end{tabular}%
}
\vspace{10pt}
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}cc|cc|cccc|cc|ccc|cr@{}}
\toprule
\multicolumn{2}{c|}{Datasets} & WRMF & VAE-CF & TransCF & CML & LRML & SML & NGCF & LightGCN & HAE & HAVE & HGCF & Ours & $\Delta$(\%) \\ \midrule
\multicolumn{1}{c|}{\multirow{2}{*}{Amazon-CD}} & N@10 & 0.0651 & 0.0615 & 0.0396 & 0.0639 & 0.0405 & 0.0361 & 0.0591 & 0.0726 & 0.0565 & 0.0629 & \underline{0.0751} & \textbf{0.0848}* & +12.92 \\
\multicolumn{1}{c|}{} & N@20 & 0.0817 & 0.0752 & 0.0488 & 0.0813 & 0.0492 & 0.0456 & 0.0718 & 0.0881 & 0.0657 & 0.0749 & \underline{0.0909} & \textbf{0.1010}* & +11.11 \\ \midrule
\multicolumn{1}{c|}{\multirow{2}{*}{Amazon-Book}} & N@10 & 0.0563 & 0.0716 & 0.0392 & 0.0624 & 0.0515 & 0.0422 & 0.0655 & 0.0780 & 0.0709 & 0.0778 & \underline{0.0869} & \textbf{0.0978}* & +12.54 \\
\multicolumn{1}{c|}{} & N@20 & 0.0730 & 0.0878 & 0.0474 & 0.0808 & 0.0626 & 0.0550 & 0.0791 & 0.0938 & 0.0789 & 0.0901 & \underline{0.1022} & \textbf{0.1142}* & +11.74 \\ \midrule
\multicolumn{1}{c|}{\multirow{2}{*}{Yelp2020}} & N@10 & 0.0372 & 0.0353 & 0.0214 & 0.0310 & 0.0287 & 0.0255 & 0.0405 & 0.0461 & 0.0331 & 0.0371 & \underline{0.0458} & \textbf{0.0502}* & +9.13 \\
\multicolumn{1}{c|}{} & N@20 & 0.0506 & 0.0469 & 0.0277 & 0.0428 & 0.0369 & 0.0347 & 0.0513 & 0.0582 & 0.0409 & 0.0465 & \underline{0.0585} & \textbf{0.0633}* & +8.21 \\ \bottomrule
\end{tabular}
}
\end{table*}
\section{Experiments}
\subsection{Experimental Settings}
\textbf{Datasets.}
In this work, we experiment with three publicly available datasets, namely Amazon-CD$^5$, Amazon-Book\footnote{https://jmcauley.ucsd.edu/data/amazon/} and Yelp2020\footnote{https://www.yelp.com/dataset}. Note that we only use user-item interactions to maintain consistency with the comparison models. The statistics of the dataset are in Table~\ref{tab:datasets}, where H20 and T80 denote the average ratio of the head items and tail items appearing in user's preference. They are calculated by $\frac{1}{|\mathcal{U}|}\sum_{u\in \mathcal{U}}\#\{\mathcal{N}_u\cap\mathcal{I}_{H20}\}$ and $\frac{1}{|\mathcal{U}|}\sum_{u\in \mathcal{U}}\#\{\mathcal{N}_u\cap\mathcal{I}_{B80}\}$, respectively.
Each dataset is split into 80\% and 20\% training and test sets for training and evaluation, respectively.
In these datasets, ratings are transformed into binary preferences using a threshold $\geq4$ that resembles implicit feedback settings.
\textbf{Compared methods.}
To fully verify the effectiveness of our method, we compare the baselines of the hyperbolic models and the Euclidean models. For the hyperbolic model, we compare with HGCF~\cite{sun2021hgcf}, HVAE, and HAE. HAE (HVAE) combines a (variational) autoencoder with hyperbolic geometry. Furthermore, we compare several recent strong Euclidean baselines, such as LightGCN~\cite{he2020lightgcn} and NGCF~\cite{wang2019ngcf}. In addition, we compare MF-based models, WRMF~\cite{wrmf2008} and VAE-CF~\cite{VAECF2018}; and metric learning-based models, TransCF~\cite{park2018collaborative}, CML~\cite{CML2017}, LRML~\cite{LRML2018}, and SML~\cite{SML2020}. For the data pre-processing and experimental settings, we closely follow previous work HGCF.
\begin{table*}[h]
\caption{Performance in the H20 and T80 items for all datasets. $\Delta_\mathcal{H}$ represents the relative improvements compared with the strong hyperbolic baseline HGCF. The bold denotes the best overall improvements among LightGCN, HGCF, and ours.}
\label{tab:H20_T80_results}
\resizebox{0.98\textwidth}{!}{%
\begin{tabular}{c|l|cc|cc|cc|cc|cc|cc}
\hline
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{Datasets}}} & \multicolumn{1}{l|}{\multirow{2}{*}{Models}} & \multicolumn{2}{c|}{R@20} & \multicolumn{2}{c|}{R@10} & \multicolumn{2}{c|}{R@5} & \multicolumn{2}{c|}{N@20} & \multicolumn{2}{c|}{N@10} & \multicolumn{2}{c}{N@5} \\
\multicolumn{1}{r|}{} & \multicolumn{1}{r|}{} & H20 & T80 & H20 & T80 & H20 & T80 & H20 & T80 & H20 & T80 & H20 & T80 \\ \midrule
\multirow{4}{*}{Amazon-CD} & LightGCN & 0.1062 & 0.0342 & 0.0741 & 0.0188 & 0.0493 & 0.0104 & 0.0712 & 0.0169 & 0.0608 & 0.0118 & 0.0529 & 0.0084 \\
& HGCF & 0.0998 & 0.0457 & 0.0667 & 0.0295 & 0.0439 & 0.0179 & 0.0658 & 0.0251 & 0.0550 & 0.0201 & 0.0486 & 0.0157 \\
& \textbf{HICF(Ours)} & \textbf{0.1027} & \textbf{0.0559} & \textbf{0.0717} & \textbf{0.0362} & \textbf{0.0476} & \textbf{0.0222} & \textbf{0.0692} & \textbf{0.0318} & \textbf{0.0596} & \textbf{0.0252} & \textbf{0.0527} & \textbf{0.0197} \\
& $\Delta_\mathcal{H}(\%)$ & +2.91 & +22.32 & +7.50 & +22.71 & +8.43 & +24.02 & +5.17 & +26.69 & +8.36 & +25.37 & +8.44 & +25.48 \\ \midrule
\multirow{4}{*}{Amazon-Book} & LightGCN & 0.0915 & 0.0333 & 0.0624 & 0.0175 & 0.0390 & 0.0104 & 0.0740 & 0.0198 & 0.0635 & 0.0145 & 0.0589 & 0.0104 \\
& HGCF & 0.0829 & 0.0489 & 0.0550 & 0.0317 & 0.0344 & 0.0197 & 0.0670 & 0.0352 & 0.0578 & 0.0291 & 0.0539 & 0.0251 \\
& \textbf{HICF(Ours)} & \textbf{0.0898} & \textbf{0.0551} & \textbf{0.0603} & \textbf{0.0362} & \textbf{0.0387} & \textbf{0.0227} & \textbf{0.0738} & \textbf{0.0404} & \textbf{0.0642} & \textbf{0.0336} & \textbf{0.0605} & \textbf{0.0293} \\
& $\Delta_\mathcal{H}(\%)$ & +8.32 & +12.68 & +9.64 & +14.20 & +12.50 & +15.23 & +10.15 & +14.77 & +11.07 & +15.46 & +12.24 & +16.73 \\ \midrule
\multirow{3}{*}{Yelp2020} & LightGCN & 0.0836 & 0.0030 & 0.0512 & 0.0010 & 0.0298 & 0.0003 & 0.0567 & 0.0015 & 0.0448 & 0.0006 & 0.0380 & 0.0004 \\
& HGCF & 0.0788 & \textbf{0.0096} & 0.0473 & \textbf{0.0054} & 0.0270 & 0.0030 & 0.0526 & \textbf{0.0059} & 0.0417 & 0.0043 & 0.0354 & \textbf{0.0033} \\
& \textbf{HICF(Ours)} & \textbf{0.0854} & 0.0094 & \textbf{0.0518} & 0.0052 & \textbf{0.0299} & 0.0029 & \textbf{0.0576} & 0.0057 & \textbf{0.0461} & 0.0041 & \textbf{0.0395} & 0.0032 \\
& $\Delta_\mathcal{H}(\%)$ & +8.38 & -2.08 & +9.51 & -3.70 & +10.74 & -3.33 & +9.51 & -3.39 & +10.55 & -4.65 & +11.58 & -3.03 \\ \bottomrule
\end{tabular}%
}
\end{table*}
\textbf{Experimental setup.}
For experimental settings, we closely follow the baseline HGCF to reduce the experiment burden and give a fair comparison.
To be more specific, the number of training epochs is fixed at 500 and the embedding size is set at 50.
For gradient optimization, we use the Riemannian SGD~\cite{bonnabel2013stochastic} with weight decay in the range of $\{1e-4, 5e-4, 1e-3, 5e-3\}$ to learn the network parameters at learning rates $\{0.001, 0.0015, 0.002\}$. Note that RSGD is a stochastic gradient descent optimization technique that takes the geometry of the hyperbolic manifold into consideration. For the experimental settings of baselines, we refer to~\cite{sun2021hgcf}.
\textbf{Evaluation metrics}.
We employ two standard evaluation metrics to assess the performance of the top-K recommendation and preference ranking: Recall and NDCG~\cite{ying2018graph}. We treat each observed interaction between a user and an item as a positive case and then use the HINS to match it with one unfavorable item that the user has not previously rated.
\subsection{Overall Performance}
The overall experimental results of the test set are summarized in Table~\ref{tab:overall_comparsion}, with the best results in bold, the second-best in italics, and $\Delta$ representing the relative improvement over the best baseline. In summary, the proposed method successfully outperforms all baselines in both Recall and NDCG metrics, with the highest improvement reaching 12.92\%, demonstrating its impressive effectiveness. We further illustrate some in-depth observations. First, hyperbolic models, including the proposed HICF and HGCF, are more competitive in modeling large-scale user-item networks than Euclidean models. The main reason is that, as networks expand, the distribution of power laws becomes more apparent. Thereby, the hyperbolic models are more competitive.
Furthermore, the improvement is observed to be relative to data density (\textit{c.f.}~Table~\ref{tab:datasets}). . Specifically, the improvements of the model in the data with higher density, e.g., Amazon-CD which are +12.16\% for Recall@10 and +12.92\% for NDCG@10 are generally greater than those in lower density data, e.g., yelp, which are +8.16\% for Recall@10 and +8.21\% for NDCG@10.
\subsection{Performance on Head and Tail Items}
To further illustrate the validity of the proposal, we performed an in-depth analysis comparing the performance of the tail and head items separately. For simplicity, we focus on the two most prominent baselines, the hyperbolic HGCF model and its Euclidean counterpart LightGCN. The findings are listed in Table~\ref{tab:H20_T80_results}, where $\Delta_\mathcal{H}$ represents the improvements over the hyperbolic model, especially demonstrating the role of the proposed method against the original hyperbolic model.
The corresponding overall performance can be computed by adding the results of H20 and T80 together, and the best overall performances among HICF, HRCF, and LightGCN are bold.
\textcolor{black}{From the experimental results and the bold notation in Table~\ref{tab:H20_T80_results}, we know that the performance of the proposed HICF consistently outperforms the baselines. In particular, we discovered that the performance of hyperbolic models including our HICF and HGCF on tail items is eye-catching, while the performance on head items is slightly inferior. Overall, the proposed HICF successfully achieves the aforementioned goal, e.g., enabling the hyperbolic model to improve the performance on both tail items and head items.
In particular, for \textit{tail} items, we found that HICF performance is significantly improved compared to HGCF on Amazon-CD and Amazon-Book with the largest improvement up to 26.69\% and 16.73\%, respectively, and stays comparable on Yelp, which may be due to the fact that Yelp users show fewer interests in tail items, as seen from Table~\ref{tab:datasets}. For \textit{head} items, the performances of HICF comprehensively outperform that of HGCF and the improvements are up to 8.43\% on Amazon-CD, 12.50\% on Amazon-Book, and 11.58\% on Yelp. In addition, HICF narrows the performance gap of the tail items with Euclidean LightGCN on Amazon-CD and Amazon-Book and impressively surpasses the performance of LightGCN on Yelp.
}
\begin{figure}[!t]
\centering
\includegraphics[width=4.10cm]{file/figures/dim/dim-Amazon-Book_Recall_20.pdf}
\includegraphics[width=4.10cm]{file/figures/dim/dim-Yelp_Recall_20.pdf}
\caption{Comparisons among four dimensions $\{20, 30, 50, 100\}$ in the Amazon-CD and Amazon-Book datasets. The evaluation metric is Recall@20, and other metrics@K show similar results. }
\label{fig:dim_study}
\end{figure}
\subsection{Generalization w.r.t. Embedding sizes}
Considering that the embedding dimension has an effect on the embedding capacity and the pairwise embedding distance, to fully verify the generalization of our proposed method, we traverse different embedding dimensions for evaluation. The experimental results are shown in Figure~\ref{fig:dim_study}. From the experimental results, we easily know that the proposed HICF continuously outperforms the strongest baseline HGCF. At the same time, we found that with the increase of the embedding dimension, the performance of the model is further improved. These findings validate the strong generalizability of the proposed HICF.
\subsection{Convergent Speed w.r.t Training Epochs}
\begin{figure}[!t]
\centering
\includegraphics[width=4.10cm]{file/figures/training_epochs/training_epoch_Amazon-Book_Recall_20.pdf}
\includegraphics[width=4.10cm]{file/figures/training_epochs/training_epoch_Yelp_Recall_20.pdf}
\caption{Recall@20 changes with training epochs on Amazon-Book and Yelp. Other metric@Ks show a similar tendency.}
\label{fig:metric_training_epoch}
\end{figure}
Likewise, we analyze the convergence of the proposed approach for training the proposed HICF model and its most comparable equivalent HGCF. Figure~\ref{fig:metric_training_epoch} shows the performance of the Recall@20 metrics for epochs from 1 to 500. Other metrics@Ks show a similar tendency. The following conclusions are drawn from the experimental findings: (1) the proposed HICF repeatedly outperforms the baseline model in all epochs; (2) the proposed technique is capable of achieving the highest performance in fewer epochs, indicating that HICF can speed up the training process.
\subsection{Ablation Study}
\begin{table}[]
\caption{Ablation study (AS) for HICF. w/o M denotes without HAML and w/o S denotes without HINS in HICF.}
\centering
\label{tab:ablation_study}
\resizebox{0.48\textwidth}{!}{%
\begin{tabular}{@{}c|ccc|ccc|ccc@{}}
\toprule
\multirow{2}{*}{AS} & \multicolumn{3}{c|}{Amazon-CD} & \multicolumn{3}{c|}{Amazon-Book} & \multicolumn{3}{c}{Yelp2020} \\ \cmidrule(l){2-10}
& R@20 & H20 & T80 & R@20 & H20 & T80 & R@20 & H20 & T80 \\ \midrule
HICF & \textbf{0.1586} & \textbf{0.1027} & \textbf{0.0559} & \textbf{0.1449} & \textbf{0.0898} & \textbf{0.0551} & \textbf{0.0947} & \textbf{0.0854} & {0.0094} \\
w/o M & 0.1534 & 0.0968 & 0.0566 & 0.1363 & 0.0844 & 0.0519 & 0.0901 & 0.0803 & \textbf{0.0098} \\
w/o S & 0.1492 & 0.1021 & 0.0472 & 0.1312 & 0.0813 & 0.0499 & 0.0921 & 0.0826 & 0.0096 \\ \bottomrule
\end{tabular}%
}
\end{table}
In this part, we conduct an ablation study to evaluate the effectiveness of each component in the proposed HICF. We remove HAML and HINS separately. In Table~\ref{tab:ablation_study}, we report the corresponding results for Recall@20. Note that other metric@Ks have similar outcomes. We know that removing both HAML and HINS will lead to degradation of the model performance, which verifies the effectiveness of the proposed method. In particular, Removing HINS on both Amazon-CD and Amazon-Book leads to a large decline, while Yelp has a smaller drop, which is mainly due to the density of the dataset. The Amazon-Book and Amazon-CD dataset are relatively dense and require more negative samples for selection, while Yelp is sparse and less dependent on informative negative samples.
\subsection{Parameter Analysis}
\begin{figure}[!t]
\centering
\includegraphics[width=4.10cm]{file/figures/nneg/num_neg_Amazon-Book.pdf}
\includegraphics[width=4.10cm]{file/figures/nneg/num_neg_Yelp.pdf}
\caption{The number of negative samples for selection. }
\label{fig:nneg_study}
\vspace{-10pt}
\end{figure}
The number of negative samples for the selection is a critical hyperparameter in this study. In our research, we discovered that it varies with each dataset, but a simple grid search enables us to quickly locate a suitable value. We do a parameter sensitivity study on Amazon-Book and Yelp in this section. As shown in Figure~\ref{fig:nneg_study}, we know that Yelp requires fewer negative samples for selection and Amazon-Book requires more negative samples for selection.
It can be understood that when the dataset (e.g., Amazon-Book) is relatively dense, each node has more neighbors, and then more negative candidates are required to find a suitable one. The influence of parameter analysis is also reflected in the ablation study of HINS.
\section{Conclusion}
Hyperbolic models have received increasing attention in the recommendation community, while their pros and cons over their Euclidean counterparts have not been explicitly studied. In this work, we attempt to initiate the investigation by further separately comparing their performance on head-and-tail items against the Euclidean equivalents.
Overall, the hyperbolic model shows apparent superiority. It is also observed that the hyperbolic model performs substantially better on tail items than the Euclidean equivalent, but there is still sufficient room for improvement. For the head item, the hyperbolic model gives modest attention.
Motivated by the observations, we propose a geometry boosted hyperbolic collaborative filtering (i.e., HICF). Our technique is designed to make the pull and push components of the hyperbolic margin ranking loss geometric aware, which then provides informative optimization guidance for both head and tail items. It is worth noting that our approach is not limited to collaborative filtering-based models but is also applicable to other recommendation models.
The above observation and proposed technique shed more light on the role of hyperbolic models in the recommender system. Note that the exponentially increased capacity of hyperbolic space allows the model to pay more attention to tail items, which is beneficial for personalized recommendations and increasing market diversity.
In future work, we aim to analyze the advantages and disadvantages of hyperbolic representation models from more generalized settings and other applications~\cite{zixingsurvey,FeatureNorm2020, li2019improving,li2020unsupervised,li2022text,li2022bsal,zhou2022telegraph}.
\begin{acks}
The work described in this paper was partially supported by the National Key Research and Development Program of China (No. 2018AAA0100204) and the Research Grants Council of the Hong Kong Special Administrative Region, China (CUHK 2410021, Research Impact Fund, No. R5034-18). We thank the anonymous reviewers for their constructive comments.
\end{acks}
\bibliographystyle{ACM-Reference-Format}
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\title[short title]{full title}
\end{verbatim}
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\begin{verbatim}
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\section{CCS Concepts and User-Defined Keywords}
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The ACM Computing Classification System ---
\url{https://www.acm.org/publications/class-2012} --- is a set of
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\begin{table}
\caption{Frequency of Special Characters}
\label{tab:freq}
\begin{tabular}{ccl}
\toprule
Non-English or Math&Frequency&Comments\\
\midrule
\O & 1 in 1,000& For Swedish names\\
$\pi$ & 1 in 5& Common in math\\
\$ & 4 in 5 & Used in business\\
$\Psi^2_1$ & 1 in 40,000& Unexplained usage\\
\bottomrule
\end{tabular}
\end{table}
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in the printed output of this document.
\begin{table*}
\caption{Some Typical Commands}
\label{tab:commands}
\begin{tabular}{ccl}
\toprule
Command &A Number & Comments\\
\midrule
\texttt{{\char'134}author} & 100& Author \\
\texttt{{\char'134}table}& 300 & For tables\\
\texttt{{\char'134}table*}& 400& For wider tables\\
\bottomrule
\end{tabular}
\end{table*}
Always use midrule to separate table header rows from data rows, and
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\subsection{Inline (In-text) Equations}
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\begin{math}
\lim_{n\rightarrow \infty}x=0
\end{math},
set here in in-line math style, looks slightly different when
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\subsection{Display Equations}
A numbered display equation---one set off by vertical space from the
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\begin{equation}
\lim_{n\rightarrow \infty}x=0
\end{equation}
Notice how it is formatted somewhat differently in
the \textbf{displaymath}
environment. Now, we'll enter an unnumbered equation:
\begin{displaymath}
\sum_{i=0}^{\infty} x + 1
\end{displaymath}
and follow it with another numbered equation:
\begin{equation}
\sum_{i=0}^{\infty}x_i=\int_{0}^{\pi+2} f
\end{equation}
just to demonstrate \LaTeX's able handling of numbering.
\section{Figures}
The ``\verb|figure|'' environment should be used for figures. One or
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\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{sample-franklin}
\caption{1907 Franklin Model D roadster. Photograph by Harris \&
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\end{figure}
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\url{https://www.acm.org/publications/taps/describing-figures/}.
\subsection{The ``Teaser Figure''}
A ``teaser figure'' is an image, or set of images in one figure, that
are placed after all author and affiliation information, and before
the body of the article, spanning the page. If you wish to have such a
figure in your article, place the command immediately before the
\verb|\maketitle| command:
\begin{verbatim}
\begin{teaserfigure}
\includegraphics[width=\textwidth]{sampleteaser}
\caption{figure caption}
\Description{figure description}
\end{teaserfigure}
\end{verbatim}
\section{Citations and Bibliographies}
The use of \BibTeX\ for the preparation and formatting of one's
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The bibliography is included in your source document with these two
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\begin{verbatim}
\bibliographystyle{ACM-Reference-Format}
|
2,869,038,156,424 | arxiv | \section{Introduction}
\label{sec.introduction}
Stereo matching is a fundamental problem in computer vision and robotics. This important technique has been widely employed in many tasks, such as robot vision \cite{wang2019self,wang2020applying,wang2021dynamic} and autonomous driving \cite{fan2020sne,fan2021learning}. The goal of stereo matching is to estimate dense correspondences between a pair of stereo images and further generate a dense disparity image \cite{fan2018road}.
Traditional and data-driven approaches are two major types of stereo matching algorithms \cite{fan2018road,wang2021pvstereo}. Traditional algorithms formulate stereo matching as either a local block matching problem or a global energy minimization problem \cite{fan2018road}. Data-driven approaches \cite{psmnet,gwcnet,cheng2020hierarchical} utilize convolutional neural networks (CNNs) to extract informative visual features and create a 3D cost volume, by analyzing which a dense disparity image can be estimated. Among data-driven approaches, PSMNet \cite{psmnet} adopts 3D CNNs to regularize cost volumes for disparity estimation, while GwcNet \cite{gwcnet} further utilizes group-wise correlation to provide efficient representations for visual feature similarity measurement. Meanwhile, LEAStereo \cite{cheng2020hierarchical} uses a neural architecture search framework to search an effective and efficient network for stereo matching. However, such supervised stereo matching approaches typically require a large amount of training data with disparity ground truth, often making them difficult to apply in practice.
\begin{figure}[t]
\centering
\includegraphics[width=0.99\linewidth]{./figures/time_error.pdf}
\caption{Evaluation results on the KITTI Stereo 2015 benchmark \cite{kitti15}, where ``F1-All'' denotes the percentage of erroneous pixels measured over all regions. Our CoT-Stereo outperforms all other state-of-the-art unsupervised stereo matching approaches in terms of both accuracy and speed.}
\label{fig.time_error}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=0.95\textwidth]{./figures/framework.pdf}
\caption{An overview of our CoT-Stereo architecture, where two LEAStereo \cite{cheng2020hierarchical} networks with different initializations teach each other about the occlusions interactively.
}
\label{fig.framework}
\end{figure*}
With the limitation of the supervised approaches in mind, many researchers \cite{zhong2017self,tulyakov2017weakly,li2018occlusion,flow2stereo} have resorted to unsupervised techniques, which do not require disparity ground truth to realize stereo matching. These approaches generally train networks by minimizing a hybrid loss, \textit{e.g.}, combining a photometric loss and a smoothness loss \cite{zhong2017self,tulyakov2017weakly}. Some approaches also incorporate occlusion reasoning into the training paradigm to further improve the stereo matching performance \cite{li2018occlusion,flow2stereo}. However, such unsupervised approaches still perform unstably in some regions, especially near occlusions, because a single network can be sensitive to outliers when the disparity ground truth is inaccessible.
To address the instability problem, we propose CoT-Stereo, an unsupervised stereo matching approach. It outperforms all other state-of-the-art unsupervised stereo matching approaches in terms of both accuracy and speed on the KITTI Stereo benchmarks \cite{kitti12,kitti15}, as illustrated in Fig.~\ref{fig.time_error}. Our CoT-Stereo employs a co-teaching framework, as shown in Fig.~\ref{fig.framework}, where two networks (LEAStereo \cite{cheng2020hierarchical} is used as the backbone network) with different initializations interactively teach each other about the occlusions. Our previous work has adopted this co-teaching framework for unsupervised optical flow estimation \cite{wang2020cot}, and in this paper, we employ this framework for unsupervised stereo matching. This framework can significantly improve model's robustness against outliers and further enhance the overall performance of unsupervised stereo matching.
\section{Methodology}
\label{sec.mechodology}
\begin{algorithm*}[t]
\KwIn{$\Omega^{\mathrm{A}}$ and $\Omega^{\mathrm{B}}$, learning rate $\eta$, constant threshold $\tau$, epoch $T_k$ and $T_{\mathrm{max}}$, iteration $N_{\mathrm{max}}$.}
\KwOut{$\Omega^{\mathrm{A}}$ and $\Omega^{\mathrm{B}}$.}
\For{$T = 1 \to T_{\mathrm{max}}$}
{
\textbf{Shuffle} training set $\mathcal{D}$\\
\For{$N = 1 \to N_{\mathrm{max}}$}
{
\textbf{Forward} individually to get $\mathbf{D}^{i}$, $\mathbf{O}^{i}$, $\widetilde{\mathbf{D}}^{i}$, $\widetilde{\mathbf{D}}^{i*}$ and $\widetilde{\mathbf{O}}^{i}$, $i \in \{\mathrm{A},\mathrm{B}\}$ \\
\textbf{Set} $\mathbf{O}^{i} \left( \mathbf{O}^{i} > \mathcal{R}(T) \right) = 1$, $i \in \{\mathrm{A},\mathrm{B}\}$ \hspace{2.50cm} $\triangleright$ Omit the pixels with high probability to be occluded \\
\textbf{Compute} $\mathcal{L}^{\mathrm{A}} = \mathcal{L}_{\mathrm{ph}}^{\mathrm{A}}(\mathbf{I}_{l}, \mathbf{I}_{r}, \mathbf{D}^{\mathrm{A}}, \mathbf{O}^{\mathrm{B}}) + \lambda_{1} \cdot \mathcal{L}_{\mathrm{sm}}^{\mathrm{A}} (\mathbf{I}_{l},\mathbf{D}^{\mathrm{A}}) + \lambda_{2} \cdot \mathcal{L}_{\mathrm{da}}^{\mathrm{A}} (\widetilde{\mathbf{D}}^{\mathrm{A}}, \widetilde{\mathbf{D}}^{\mathrm{A}*},\widetilde{\mathbf{O}}^{\mathrm{B}})$\\
\textbf{Compute} $\mathcal{L}^{\mathrm{B}} = \mathcal{L}_{\mathrm{ph}}^{\mathrm{B}}(\mathbf{I}_{l}, \mathbf{I}_{r}, \mathbf{D}^{\mathrm{B}}, \mathbf{O}^{\mathrm{A}}) + \lambda_{1} \cdot \mathcal{L}_{\mathrm{sm}}^{\mathrm{B}} (\mathbf{I}_{l},\mathbf{D}^{\mathrm{B}}) + \lambda_{2} \cdot \mathcal{L}_{\mathrm{da}}^{\mathrm{B}} (\widetilde{\mathbf{D}}^{\mathrm{B}}, \widetilde{\mathbf{D}}^{\mathrm{B}*},\widetilde{\mathbf{O}}^{\mathrm{A}})$\\
\textbf{Update} $\Omega^{i} = \Omega^{i}-\eta \nabla \mathcal{L}^{i}$, $i \in \{\mathrm{A},\mathrm{B}\}$
}
\textbf{Update} $\mathcal{R}(T) = 1 - \tau \cdot \min \left\{\frac{T}{T_k}, 1 \right\}$
}
\caption{Co-Teaching Strategy}
\label{alg.co-teaching}
\end{algorithm*}
\subsection{Preliminaries and Loss Functions}
\label{sec.preliminaries_and_loss_functions}
Given a pair of stereo images $\mathbf{I}_{l}$ and $\mathbf{I}_{r}$, the objective of stereo matching is to produce a dense disparity image $\mathbf{D}$. This can be achieved by an off-the-shelf stereo matching network, \textit{e.g.}, LEAStereo \cite{cheng2020hierarchical}. An occlusion map $\mathbf{O}$ indicating each pixel's probability of belonging to the occluded regions can also be computed using the technique proposed in \cite{wang2018occlusion}. Now the problem becomes how to train the network without direct supervision from the disparity ground truth. Following the paradigm of unsupervised stereo matching, we employ a hybrid loss, which combines (a) a photometric loss $\mathcal{L}_{\mathrm{ph}}$, (b) a smoothness loss $\mathcal{L}_{\mathrm{sm}}$, and (c) a data-augmentation loss $\mathcal{L}_{\mathrm{da}}$, to train our CoT-Stereo, as illustrated in Fig.~\ref{fig.framework}. The photometric loss $\mathcal{L}_{\mathrm{ph}}$ can be formulated as a combination of an SSIM term \cite{wang2004image} and an L1 norm term:
\begin{align}
& \mathcal{L}_{\mathrm{ph}} (\mathbf{I}_{l}, \mathbf{I}_{r}, \mathbf{D}, \mathbf{O}) = \frac{1}{\mathcal{N}} \sum_{\mathbf{p}} \biggl( \alpha \frac{1-\text{SSIM}\left( \mathbf{I}_{l}(\mathbf{p}), \widehat{\mathbf{I}}_{l}(\mathbf{p}) \right)}{2} \notag \\
& \hspace{1cm} + (1-\alpha)\left\|\mathbf{I}_{l}(\mathbf{p})-\widehat{\mathbf{I}}_{l}(\mathbf{p}) \right\|_{1} \biggr) \cdot \mathcal{S}\left(\overline{\mathbf{O}}(\mathbf{p})\right),
\label{eq.ph}
\end{align}
where $\widehat{\mathbf{I}}_{l} = \omega(\mathbf{I}_{r},\mathbf{D})$ denotes the warped image from $\mathbf{I}_{r}$ based on $\mathbf{D}$; $\overline{\mathbf{O}}(\mathbf{p}) = 1 - \mathbf{O}(\mathbf{p})$; $\left\| \cdot \right\|_{1}$ denotes the L1 norm; $\mathcal{S}(\cdot)$ denotes the stop-gradient; and $\mathcal{N} = \sum_{\mathbf{p}} \mathcal{S} \left(\overline{\mathbf{O}}(\mathbf{p})\right)$ is a normalizer. Equation~\eqref{eq.ph} shows that $\mathcal{L}_{\mathrm{ph}}$ is an occlusion-aware loss used to penalize the photometric error. Following \cite{li2018occlusion}, we also adopt a smoothness loss $\mathcal{L}_{\mathrm{sm}}$ to smooth the disparity estimations:
\begin{equation}
\mathcal{L}_{\mathrm{sm}} (\mathbf{I}_{l},\mathbf{D}) = \frac{1}{N_{\mathbf{p}}} \sum_{\mathbf{p}} \sum_{d \in \{x,y\}} \left|\nabla_{d} \mathbf{D}(\mathbf{p})\right| e^{-\left\|\nabla_{d} \mathbf{I}_{l}(\mathbf{p})\right\|_{1}},
\label{eq.sm}
\end{equation}
where $N_{\mathbf{p}}$ denotes the number of pixels. Moreover, inspired by \cite{liu2020learning}, we adopt a data-augmentation scheme to enable networks to better handle occlusions. Specifically, we first perform transformations $\mathbf{T}_{\theta}^{\mathrm{img}}$, $\mathbf{T}_{\theta}^{\mathrm{disp}}$ and $\mathbf{T}_{\theta}^{\mathrm{occ}}$ (\textit{e.g.}, spatial, occlusion and appearance transformations \cite{liu2020learning}) on $(\mathbf{I}_{l}, \mathbf{I}_{r})$, $\mathbf{D}$ and $\mathbf{O}$ respectively to obtain the augmented samples $\widetilde{\mathbf{I}}_{l}$, $\widetilde{\mathbf{I}}_{r}$, $\widetilde{\mathbf{D}}$ and $\widetilde{\mathbf{O}}$. Please note that, different from $\mathbf{O}$, a higher value in $\widetilde{\mathbf{O}}$ indicates that the pixel is less likely to be occluded in $\widetilde{\mathbf{D}}$ but more likely to be occluded in $\widetilde{\mathbf{D}}^{*}$. Given $\widetilde{\mathbf{I}}_{l}$ and $\widetilde{\mathbf{I}}_{r}$, we can also use LEAStereo \cite{cheng2020hierarchical} to get a disparity estimation $\widetilde{\mathbf{D}}^{*}$. Our data-augmentation loss $\mathcal{L}_{\mathrm{da}}$ is then defined as follows:
{\small
\begin{align}
\mathcal{L}_{\mathrm{da}} (\widetilde{\mathbf{D}}, \widetilde{\mathbf{D}}^{*},\widetilde{\mathbf{O}}) & = \frac{\sum_{\mathbf{p}} l\left( \lvert \mathcal{S}\left(\widetilde{\mathbf{D}}(\mathbf{p})\right) - \widetilde{\mathbf{D}}^{*}(\mathbf{p}) \rvert \right) \cdot \mathcal{S}\left(\widetilde{\mathbf{O}}(\mathbf{p})\right)}
{\sum_{\mathbf{p}} \mathcal{S}\left(\widetilde{\mathbf{O}}(\mathbf{p})\right)},\notag \\
l(x) & =\left\{\begin{array}{ll}
{x-0.5,} & {x \geq 1} \\
{x^{2} / 2,} & {x<1}
\end{array},\right.
\label{eq.da}
\end{align}
}
where $l(\cdot)$ denotes the smooth L1 loss.
\subsection{Co-Teaching Strategy}
\label{sec.co-teaching_strategy}
Fig.~\ref{fig.framework} and Algorithm~\ref{alg.co-teaching} present the overview of our introduced co-teaching framework, where we simultaneously train two LEAStereo networks: (a) network A (with parameter $\Omega^{\mathrm{A}}$) and (b) network B (with parameter $\Omega^{\mathrm{B}}$). In each mini-batch, the two networks first forward individually to get several outputs (Line 4). Then, we use a dynamic threshold $\mathcal{R}(T)$ to omit the pixels with high occlusion probability (Line 5). $\mathcal{R}(T)$ is designed based on the network memorization mechanism. Specifically, during training, the networks will first learn stereo matching from clear patterns, and then will be gradually affected by outliers \cite{arpit2017closer}. Therefore, $\mathcal{R}(T)$ is initialized as 1 and it decreases gradually as the epochs increase. This helps the networks avoid memorizing outliers (possible inaccurate occlusion estimations) and further improves the performance of unsupervised stereo matching.
Afterwards, we let the two networks swap their estimated occlusion maps and compute their loss functions (Line 6 and 7). Since different networks can learn different types of occlusion and disparity estimations, swapping the occlusion estimations enables the two networks to adaptively correct the inaccurate occlusion estimations, which can further improve the performance of unsupervised stereo matching. Please note that since deep neural networks are highly non-convex, we use two LEAStereo \cite{cheng2020hierarchical} networks with different initializations in our CoT-Stereo. Finally, we update both the parameters of these two networks as well as the dynamic threshold $\mathcal{R}(T)$ (Line 8 and 10).
\section{Experimental Results}
\label{sec.experimental_results}
\subsection{Datasets and Implementation Details}
\label{sec.datasets_and_implementation_details}
For the implementation, we set $\alpha = 0.85$ in Equation~\eqref{eq.ph}. In addition, we set $T_k = 0.2 \cdot T_{\mathrm{max}}$ and $\tau = 0.7$ in Algorithm~\ref{alg.co-teaching}. Moreover, we adopt the Adam optimizer and use a learning rate $\eta = 10^{-4}$ with an exponential decay scheme. Since the two networks present similar performance after convergence, we simply adopt network A for performance evaluation.
We use three public datasets, (a) the Scene Flow \cite{mayer2016large}, (b) the KITTI Stereo 2012 \cite{kitti12}, and (c) the KITTI Stereo 2015 \cite{kitti15} datasets, to validate the effectiveness of our CoT-Stereo. The Scene Flow dataset \cite{mayer2016large} is collected in three different synthetic scenes, while the two KITTI Stereo datasets \cite{kitti12, kitti15} are collected in real-world driving scenarios and have public benchmarks. Two evaluation metrics, (a) the average end-point error (AEPE) that measures the difference between the disparity estimations and ground-truth labels and (b) the percentage of bad pixels (tolerance: 3 pixels) (F1) \cite{kitti12,kitti15}, are adopted for accuracy comparison.
In our experiments, we first conduct ablation studies on the Scene Flow dataset \cite{mayer2016large} to demonstrate the effectiveness of our adopted loss functions and proposed co-teaching strategy, as illustrated in Section~\ref{sec.ablation_study}. Then, we evaluate our CoT-Stereo on the two KITTI Stereo benchmarks \cite{kitti12, kitti15}, as presented in Section~\ref{sec.evaluations_on_the_public_benchmarks}.
\begin{figure*}[t]
\centering
\includegraphics[width=0.99\textwidth]{./figures/benchmark.pdf}
\vspace{-0.5em}
\caption{Examples on the KITTI Stereo benchmarks \cite{kitti12,kitti15}, where rows (a) and (b) show the disparity estimations and the corresponding error maps, respectively. Significantly improved regions are marked with green dashed boxes.}
\label{fig.benchmark}
\vspace{-1em}
\end{figure*}
\begin{table}[t]
\centering
\caption{Evaluation results of our CoT-Stereo with different setups on the Scene Flow dataset \cite{mayer2016large}. ``Swap'' and ``DT'' denote the occlusion estimation swapping operation and the dynamic threshold selection scheme, respectively. The adopted setup (the best result) is shown in bold type.}
\begin{tabular}{C{0.6cm}C{0.8cm}C{0.6cm}C{0.6cm}C{0.6cm}C{0.6cm}C{1.6cm}}
\toprule
No. & Swap & DT & $\mathcal{L}_{\mathrm{ph}}$ & $\mathcal{L}_{\mathrm{sm}}$ & $\mathcal{L}_{\mathrm{da}}$ & AEPE~(px) \\ \midrule
(a) & -- & -- & \cmark & \cmark & \cmark & 3.68 \\
(b) & \cmark & -- & \cmark & \cmark & \cmark & 2.35 \\
(c) & -- & \cmark & \cmark & \cmark & \cmark & 3.10 \\ \midrule
(d) & \cmark & \cmark & \cmark & -- & -- & 4.72 \\
(e) & \cmark & \cmark & \cmark & \cmark & -- & 3.97 \\
(f) & \cmark & \cmark & \cmark & -- & \cmark & 1.86 \\ \midrule
(g) & \cmark & \cmark & \cmark & \cmark & \cmark & \textbf{1.31} \\
\bottomrule
\end{tabular}
\label{tab.ablation}
\vspace{-1em}
\end{table}
\begin{table}[t]
\centering
\caption{Evaluation results ($\%$) on the KITTI Stereo 2012$^{1}$~\cite{kitti12} and Stereo 2015$^{2}$ \cite{kitti15} benchmarks. ``S'' denotes supervised approaches. ``Noc'' and ``All'' represent the F1 for non-occluded pixels and all pixels, respectively \cite{kitti12,kitti15}. Best results for supervised and unsupervised approaches are both shown in bold type.}
\begin{tabular}{L{2.8cm}C{0.4cm}C{0.6cm}C{0.6cm}C{0.6cm}C{0.6cm}}
\toprule
\multicolumn{1}{l}{\multirow{2}{*}{Approach}} & \multicolumn{1}{l}{\multirow{2}{*}{S}} & \multicolumn{2}{c}{KITTI 2012} & \multicolumn{2}{c}{KITTI 2015} \\ \cmidrule(l){3-4} \cmidrule(l){5-6}
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} & Noc & All & Noc & All \\ \midrule
PSMNet \cite{psmnet} & \cmark & 1.49 & 1.89 & 2.14 & 2.32 \\
GwcNet-gc \cite{gwcnet} & \cmark & 1.32 & 1.70 & 1.92 & 2.11 \\
LEAStereo \cite{cheng2020hierarchical} & \cmark & \textbf{1.13} & \textbf{1.45} & \textbf{1.51} & \textbf{1.65} \\ \midrule
OASM-Net \cite{li2018occlusion} & -- & 6.39 & 8.60 & 7.39 & 8.98 \\
Flow2Stereo \cite{flow2stereo} & -- & 4.58 & 5.11 & 6.29 & 6.61 \\
MC-CNN-WS \cite{tulyakov2017weakly} & -- & 3.02 & 4.45 & 4.11 & 4.97 \\
SsSMnet \cite{zhong2017self} & -- & 2.30 & 3.00 & 3.06 & 3.40 \\
\textbf{CoT-Stereo (Ours)} & -- & \textbf{1.82} & \textbf{2.32} & \textbf{2.43} & \textbf{2.68} \\ \bottomrule
\end{tabular}
\label{tab.disparity}
\end{table}
\subsection{Ablation Study}
\label{sec.ablation_study}
Table~\ref{tab.ablation} presents the evaluation results of our CoT-Stereo with different setups on the Scene Flow dataset \cite{mayer2016large}. For our proposed co-teaching strategy, (a)--(c) and (g) of Table~\ref{tab.ablation} demonstrate the effectiveness of the occlusion estimation swapping operation and the dynamic threshold selection scheme, which can effectively improve unsupervised stereo matching. Additionally, we can clearly observe that the combination of the three loss functions can effectively improve the performance, as shown in (d)--(g) of Table~\ref{tab.ablation}. Moreover, (g) in Table~\ref{tab.ablation} denotes the adopted setup, which validates the effectiveness of our adopted loss functions and proposed co-teaching strategy.
\subsection{Evaluations on the Public Benchmarks}
\label{sec.evaluations_on_the_public_benchmarks}
Table~\ref{tab.disparity} shows the online leaderboards of the KITTI Stereo 2012 \cite{kitti12} and Stereo 2015 \cite{kitti15} benchmarks, and Fig.~\ref{fig.time_error} visualizes the results on the KITTI Stereo 2015 benchmark. We can observe that our CoT-Stereo outperforms all other state-of-the-art unsupervised stereo matching approaches in terms of both accuracy and speed, which demonstrates the effectiveness of the occlusion estimation swapping operation and the dynamic threshold selection scheme for unsupervised stereo matching. Excitingly, our CoT-Stereo can even present competitive performance compared with the state-of-the-art supervised approaches. Examples on the KITTI Stereo benchmarks are shown in Fig.~\ref{fig.benchmark}, where it is evident that our CoT-Stereo can generate more robust and accurate disparity estimations. All the analysis proves the excellent performance of our CoT-Stereo for unsupervised stereo matching.
\section{Conclusion and Future Work}
\label{sec.conclusion_and_future_work}
This paper proposed a novel co-teaching strategy for unsupervised stereo matching, which consists of a dynamic threshold selection scheme and an occlusion estimation swapping operation. The former ensures that the networks do not memorize possible outliers, while the latter enables the two networks to adaptively correct the inaccurate occlusion estimations and further improve the performance of unsupervised stereo matching. Extensive experimental results on the KITTI Stereo benchmarks showed that our approach, CoT-Stereo, outperforms all other state-of-the-art unsupervised stereo matching approaches in terms of both accuracy and speed.
\footnotetext[1]{\url{http://www.cvlibs.net/datasets/kitti/eval_stereo_flow.php?benchmark=stereo}}
\footnotetext[2]{\url{http://www.cvlibs.net/datasets/kitti/eval_scene_flow.php?benchmark=stereo}}
\clearpage
\bibliographystyle{IEEEbib}
|
2,869,038,156,425 | arxiv | \section{Appendix}
\subsection{PVA parameter tuning experiments}
\label{ap:PVA-parameter-tuning}
The performance of PVA depends on its input parameters, viz., $\beta, \gamma,$ and $\psi$ (see Table \ref{tab:TableOfNotation}).
Further, one of the significant challenges is determining the threshold similarity value ($\hat{\alpha}$) between two PVA vectors, which helps us classify them as similar or dissimilar.
We performed several experiments to determine the optimal values of $\beta, \gamma, \psi$, and $\hat{\alpha}$. Table \ref{table:training-scenarios} lists the experimental scenarios, while Algorithm-\ref{alg:training-pv-model} describes the specific steps performed for training the software similarity detection models.
\begin{table}
\centering
\caption{Scenarios for training the software similarity detection models}
\label{table:training-scenarios}
\resizebox{0.8\columnwidth}{!}
{
\begin{tabular}{|c|c|c|c|}
\cline{1-3}
\multicolumn{3}{|c|}{\textbf{Parameters varied}} & \\ \hline
\textbf{Epochs, $\beta$} & \textbf{Vector size, $\gamma$} & \textbf{Training samples, $\psi$} & \multicolumn{1}{c|} {\textbf{Models}}\\ \hline
Fixed at 10 & Fixed at 10 & \multirow{2}{*}{\shortstack{Vary 0 to \texttt{CorpusSize}\\ in steps of 30}}& \multicolumn{1}{c|}{\texttt{CorpusSize} $\div$ 30} \\
&&&\\
\hline
Vary 5 to 50 in steps of 5 & Fixed at 10 & Fixed at \texttt{CorpusSize} & \multicolumn{1}{c|}{10}\\ \hline
Fixed at 10 & Vary 5 to 50 in steps of 5 & Fixed at \texttt{CorpusSize} & \multicolumn{1}{c|}{10} \\ \hline
\end{tabular}}
\end{table}
\begin{algorithm}
\caption{Procedure to train PVA models}
\label{alg:training-pv-model}
\begin{algorithmic}[1]
\REQUIRE
$ H := $ Collection of software project descriptions.\\
$P := $ Set of PVA parameter variation scenarios shown in Table \ref{table:training-scenarios}.\\
\ENSURE $M := $ The set of software similarity detection models trained using $H$ and $P$.\\
\FORALL{$\pi \in P$}
\STATE {$M(\pi):= trainPVAModel(H, \pi)$} \COMMENT{Uses gensim OSS library that implements PVA}
\STATE {Save $M(\pi)$ to disk}
\ENDFOR
\end{algorithmic}
\end{algorithm}
\subsubsection{Test-bed set up}
\label{sec:test-bed set up}
To perform these experiments, we set up a test-bed $Y$ using the software product descriptions collected from GitHub repositories. To perform the validation testing of our software description similarity detection model, we partition the software project descriptions data into $\langle train\ data, test\ data \rangle$ in ratio 2:1.
$Y$ comprises of $\langle same,different \rangle$ software product description pairs in equal proportion, viz., 50:50.
\subsubsection{Evaluation metrics}
\label{sec:evaluation-metrics}
To evaluate the performance of our software description similarity detection model, we compute the F1 score\footnote{\url{https://bit.ly/3mZ70Ns}} and ROC area\footnote{\url{https://bit.ly/2Ev8BZP}} metrics defined as follows:
\begin{enumerate}
\item \textbf{F1\ Score:} It is defined as the harmonic mean of precision and recall.
\begin{align}
\label{ed:F1-score}
F1\ score = \frac{2\times Precision \times Recall}{Precision + Recall}\end{align}
where
\begin{align}
\label{eq:precision}
Precision = \frac{TP}{TP + FP}
\end{align}
\begin{align}Recall = \frac{TP}{TP + FN}\end{align} TP=True positive, FP = False positive, and FN = False negative.
\item \textbf{ROC area under the curve (AUC):} It measures the quality of the output. ROC curve is a plot that features the true positive rate (marked as Y-axis) vs. the false positive rate (marked as X-axis) for an experiment.
\end{enumerate}
Since the highest \emph{ROC curve area} value and the \emph{F1 score} value may differ across the models, we take the average of the two as the final accuracy measure of a model.
\subsubsection{Objective:} The prime objective of this experiment is to address the research question (RQ):
\emph{What is the effect of PVA parameters in performing the software similarity detection task?}
Some of the sub-RQs addressed under this are:
\begin{enumerate}[label=(\Alph*)]
\item \emph{What is the highest accuracy achieved when detecting software similarity using our models?}
\item \emph{What is the threshold similarity score between two PVA vectors for reliably detecting two software products as similar?}
\item \emph{What is the effect of the parameters $\beta, \gamma, \psi$ on the performance of the software description similarity detection model?}
\item \emph{What are the optimal values of $\beta, \gamma, \psi$ for training the software description similarity detection model?}
\end{enumerate}
\begin{algorithm}[ht]
\caption{Procedure to test the software description similarity detection models trained using PVA}
\label{alg:procedure-to-test-models}
\begin{algorithmic}[1]
\REQUIRE
$ Y := $ Experimental test dataset containing $\langle same, different \rangle$ software product description pairs developed using the test-partition of the project descriptions.\\
$M := $ The set of software description similarity models trained using Algorithm \ref{alg:training-pv-model}.
\ENSURE $S_{same}, S_{different}:=$ Similarity score collections for $\langle same, different \rangle$ software product description pairs in $Y$.
\STATE{$S_{same} = S_{different} = \phi$}
\FORALL{project description pairs $\langle p_i, p_j \rangle \in Y$ }
\FORALL{$\pi \in P$}
\STATE {$\phi_{i} = inferVector(p_i, M(\pi)) $} \COMMENT{Infer vector vectors using $M(\pi)$}
\STATE {$\phi_{j} = inferVector(p_j, M(\pi)) $}
\STATE {$\alpha = computeCosineSimilarity({\phi}_i, {\phi}_j) $}
\IF{$p_i == p_j$}
\STATE {$S_{same} = S_{same} \cup \langle \alpha \rangle$}
\ELSE
\STATE {$S_{different} = S_{different} \cup \langle \alpha \rangle$}
\ENDIF
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
\subsubsection{Procedure}
\label{sec:proc-exp-1}
\begin{enumerate}
\item Obtain the set of similarity scores ($S$) for the collection of test pairs present in $Y$ using Algorithm \ref{alg:procedure-to-test-models}.
\item \label{step:record-threshold-sim-scores} The $avg(S)$ value is stored as the threshold similarity score value ($\hat{\alpha}$). Note: the $\hat{\alpha}$ is computed separately for the \emph{same} and \emph{different} software product description pairs. Table \ref{table:thresholds} lists the threshold similarity score for the \emph{same} software product description pairs.
\item For each parameter combination $\pi \in P$:
\begin{enumerate}
\item Obtain the software description similarity detection model $\hat{M^{\pi}}$ such that, $\hat{M^{\pi}}$ achieves the highest evaluation scores among all $M^{\pi} \in M$.
\item Record the values of Accuracy scores and F1 scores obtained with $\hat{M^{\pi}}$ when tested on $Y$, and the tuning parameter combination ($\pi$) used in developing $\hat{M^{\pi}}$.
\end{enumerate}
\end{enumerate}
\begin{figure*}
\centering
\includegraphics[width=0.5\columnwidth]{figures/epochsVecVariation}
\caption{Performance of software description similarity detection models with different parameter inputs}
\label{fig:parameter-tuning}
\end{figure*}
\begin{table}
\centering
\caption{Similarity score stats for same software product description pairs}
\label{table:thresholds}
\begin{tabular}{ c|c|c|c|c|c }
\toprule
\textbf{Measure} & $\alpha_{min}$ & $\alpha_{avg}$ & $\alpha_{stddev}$& $\alpha_{max}$&$\hat{\alpha}$ \\ \midrule
\textbf{Value} & 0.9999 & 0.9999 & 9.5305e-08 & 1 & 0.9999 \\ \bottomrule
\end{tabular}
\end{table}
\begin{table}
\centering
\caption{Best performance matrix (Models trained using 10 epochs, vector size 50, training samples 18260)}
\label{tab:evaluation-metrics-values}
\begin{tabular}{ c|c|c|c|c|c }
\toprule
\textbf{Measure} & $Accuracy$ & $Precision$ & $Recall$& $F1\ score$&$ROC\ area$ \\ \midrule
\textbf{Value} & 99.3\% & 99.24\% & 98.62\% & 99.3\% & 98.62\% \\ \bottomrule
\end{tabular}
\end{table}
\subsubsection{Results and Observations}
\label{sec:exp-1-results-and-observations}
Fig. \ref{fig:parameter-tuning} shows the performance of software description similarity models (in terms of F1 score and ROC area) with different input parameter combinations. Some of the key observations are as follows:
\begin{enumerate}
\item The evaluation metrics values remain almost constant on varying the epochs and training dataset size but increase uniformly with the increase in vector size.
\item The highest evaluation metrics values (viz., accuracy, F1 score, and ROC area) are recorded corresponding to the $\gamma=50$. Thus, we select the fine-tuned input parameters as $\gamma=50$ and $\beta=10$ for our further experiments.
\item The highest accuracy, F1 score, and ROC area values achieved by our software description similarity models are 99.3\%, 99.3\%, and 98.62\%, respectively.
\end{enumerate}
Table \ref{tab:evaluation-metrics-values} lists the highest evaluation metrics' values obtained by testing the software description similarity model trained using optimally tuned parameters $\beta=10$ and $\gamma=50$.
\subsection{Details of the SDEE relational schema}
\label{ap:SDEE-dataset-details}
A brief description of the tables of this schema is as follows:
\begin{enumerate}
\item \emph{release\_info}: This table stores the information about the releases of various OSS repositories. The attributes \emph{repo} and \emph{owner} represent the repository names and the respective repository owner names. The attributes \emph{release\_no}, \emph{release\_date} and \emph{size} represent the release number, deployment date and size of different releases of the repositories respectively.
\item \emph{commit\_stats}: In this table, we store the developer-activity information extracted corresponding to various commits of a repository. The attributes \emph{commit\_id} and \emph{dev\_id} represents the unique IDentifiers (IDs) assigned to various commits and developers, respectively.
For each of the commits stored in the table, we also store the development \emph{effort} and time (\emph{dev\_time}) spent in generating the commit. To compute the commit-wise effort and dev\_time, we employ the use of \emph{basic} and \emph{derived} SDEE metrics, such as SLOC modified by the developer ($SLOC_{modifications}$), developer productivity($P_{d_i}$), skill factor ($P_{d_i}$), etc., discussed in \S\ref{sec:SDEE-metrics}.
\item \emph{release\_effort\_estimate}:
\label{pt:dev-act-fac}
This table stores release-level effort estimates computed by utilizing the SDEE metrics' values present in the commit\_stats table.
The attributes \emph{min\_release\_ids} and \emph{max\_release\_ids} represent the IDs of the considered releases corresponding to various \{owner, repo\} pairs of repositories. Similarly, the attributes \emph{start\_release\_date} and \emph{end\_release\_date} represent the considered release period of effort computation, which corresponds to the release dates of the involved releases, and \emph{days} represents the considered time-period interval expressed in the form of days. The attribute \emph{dev\_count} represents the total count of developers contributing to the specific repository in the specified period.
\item \emph{soft\_desc\_pva\_vec}:
\label{pt:soft-desc-pva-vec}
This table stores the software project descriptions' PVA vectors and the respective cosine similarities with the reference vectors. The \enquote{category} attribute represents the software category as specified on MavenCentral.com.
\end{enumerate}
\section{Introduction}
Software Development Effort Estimation (SDEE) for a newly envisioned software project is performed based on the experiential knowledge gained from software developed in the past. The sources of such \enquote{past projects} are limited to the development team and the organization procuring the new software. Also, the existing SDEE methods are either dependent on experts or are inefficient and slow. Therefore, it is desirable to have tools and techniques which help to provide accurate software development effort estimates on time. The Open Source Software (OSS) repository hosts, such as GitHub, are rich software development sources for various software projects.
\begin{definition}[Software project description]
By \textbf{software project description}, we refer to the main description document available at the software's GitHub repository. Such a file typically comprises the following details:
\begin{enumerate}
\item \emph{Brief description of the software}: A summary of its functionality and characteristics. For instance, the software might be a library, plugin, or framework.
\item \emph{Functionality}: It describes \emph{what} the software does. The description of all the primary functions performed by the software. For instance, PMD\footnote{\url{https://github.com/pmd/pmd}} -- a source code analyzer tool, detects common programming flaws such as unused variables, empty catch blocks, and unnecessary object creation.
\item \emph{Supporting platform}: It lists the supportable operating systems and programming platforms.
\item \emph{Execution-specific details}: These include the essential information about the script and steps involved in running the software.
\end{enumerate}
\end{definition}
\begin{definition}[Paragraph Vector Algorithm]
\label{def:pva}
\textbf{Paragraph Vector Algorithm (PVA)} is an unsupervised machine learning (ML) algorithm that learns fixed-length feature representations from variable-length pieces of texts, such as sentences, paragraphs, and documents. The algorithm represents each document by a dense vector trained to predict words in the document \cite{le2014distributed}.
\end{definition}
Stated broadly, the objective of the work presented in this paper is
\enquote{\emph{Given the software description of a newly-envisioned software, estimate the software development effort required to develop it.}} To estimate the effort, we leverage the developer-activity information of the existing similar GitHub repositories. To detect the similarity among software, we develop a software similarity detection model by training PVA on the software product descriptions taken from the OSS projects available at Version Control System (VCS) repositories, such as GitHub\footnote{\url{https://github.com/}}.
\begin{figure*}
\centering
\includegraphics[width=120mm]{figures/modified_mind_map_effort_estimation}
\caption{Essential idea of our work}
\label{fig:basic-tenets}
\end{figure*}
\subsection{Essential ideas behind our system}
\label{sec:basic-tenets}
Fig. \ref{fig:basic-tenets} depicts the essential ideas behind our system, listed as follows:
\begin{enumerate}
\item SDEE for newly-envisioned software is performed by considering the past software projects of similar nature \cite{pressman2005software,sommerville2011software}.
\item VCSs, such as GitHub and GitLab\footnote{\url{https://gitlab.com/explore}}, provide rich details about the actual software artifacts, the programmers who developed them, and the time spent in developing those artifacts.
Some examples of high-level data available on various VCS hosts are the details of contributors working on a repository, the source code commits history, and the release timeline.
\item The software requirements and the project description are expressed in a \emph{natural language (NL)} such as English. Today, ML techniques have matured to enable accurate and efficient detection of text document similarity.
It is possible to identify existing software projects similar to those of a newly envisioned one.
\begin{definition}[Developer activity information]
\label{def:developer activity-information}
By \textbf{developer activity information of a software repository}, we refer to the following measures:
\begin{enumerate}
\item \emph{Source code contribution by a developer:} The number of additions, deletions, and modifications of SLOC performed by a developer on a software.
\item \emph{The timeline of various changes in a VCS:} It acts as a valuable metric for computing the time required to develop a software product.
\item \emph{The metadata information of the software repository:} The information such as the number of developers working on the software repository is essential for computing the effort required to develop the software product.
\end{enumerate}
A software developer is majorly involved in the following aspects of a software development project:
\begin{enumerate}
\item Developing source code, which is added to the repository in the form of commits.
\item \emph{Addressing the issues or tickets:} Changes to the source code are the most likely and frequent outcome in this case, and is reflected, ultimately, in the commits.
\end{enumerate}
While performing the effort computation, we only include the effort spent by the developers in \emph{code development}.
\textbf{Using developer activity information as SDEE metrics:}
\label{sec:intuition-developer-based}
\begin{enumerate}
\item A significant part (more than 25\%) of the total software development effort is spent in source code creation \cite{selamat2014new}.
\item Developers are the key contributors towards the source code creation and, thus, software development.
\item A developer's contribution often depends on several factors, such as his productivity, participation, or activity \cite{amor2006effort,gousios2008measuring}.
\end{enumerate}
Thus, leveraging the developer activity-based SDEE metrics as the first-class parameter in our SDEE model is expected to produce estimates with better accuracy.
\end{definition}
\item Developer activity information, such as the SLOC added, deleted or, modified by a developer, represents the most \emph{basic} unit of the software development effort contributed by a developer in developing a software project.
\item Developer activity information present in VCS repositories can help provide a valuable measure of the effort spent developing it. Developer activity information can be derived from the release and commit statistics of various software projects tracked in the VCSs. Thus, to estimate the effort required to develop newly-envisioned software, the effort estimates derived using such information from the existing software projects with similar functionality can be leveraged.
\end{enumerate}
\subsection{Contributions}
\label{sec:contributions}
\begin{enumerate}
\item We present a novel method to estimate the effort required to develop software. Given the software requirements of a newly-envisioned software, our system estimates the effort required to develop it. To derive the effort estimates, we use the developer activity information of the past software.
\item We propose new SDEE metrics based on the developer activity information. For instance, the the developer count (devCount) and time spent in developing a software project.
\item By extracting the developer activity information for various software present on GitHub, we develop a dataset that can be leveraged to perform SDEE.
\item By utilizing the product descriptions of software considered in our dataset, we develop a software similarity detection model $M$ using PVA. For a given description of software requirements, $M$ predicts the functionally similar software matches existing on GitHub.
\item With the combination of the SDEE dataset and the trained software similarity detection model, we develop our SDEE tool for predicting the effort required to develop a newly-envisioned software, with a standardized accuracy (SA) value of 59.89\% over the random-guessing method.
\item We present a comparison of our approach with the existing SDEE methods, and the results show that our model outperforms the ATLM \cite{whigham2015baseline}, as it achieves a SA of 42.7\% when compared with it.
\end{enumerate}
\textbf{Paper organization:} Next, we discuss some of the related works and their shortcomings in Section \ref{sec:related-work}. Section \ref{sec:proposed-system}
highlights the critical design decisions and the implementation details of our system.
Section \ref{sec:performance-evaluation-and-comparison} gives the details of
the experiments conducted to verify the effectiveness of our system. Section \ref{sec:threats-to-validity} lists the threats to the validity of our approach. Finally, Section \ref{sec:conclusion} summarizes our work and suggests the future work.
\section{Background}
\label{sec:related-work}
\begin{figure*}
\centering
\includegraphics[width=120mm]{figures/Effort_est_hier_upd}
\caption{Categorization of the existing effort estimation methods}
\label{fig:sdee-methods-hier}
\end{figure*}
\subsection{Existing approaches for SDEE}
\label{sec:existing-approaches}
Fig. \ref{fig:sdee-methods-hier} shows a broad categorization of the existing effort estimation techniques summarized by reviewing the existing literature \cite{molokken2003review}. Based on our literature review, we classify the existing SDEE techniques broadly into three major categories: \emph{Expert-based SDEE} methods, \emph{Formal SDEE} methods, and \emph{Hybrid SDEE} methods. A brief description of these methods is as follows:
\begin{enumerate}
\item \emph{Expert-based SDEE methods:} These are based on the judgments provided by the experts, such as the project managers, using the knowledge of software projects developed in the past \cite{pressman2005software}. The expert-based SDEE methods are further sub-categorized as \emph{Bottom-up SDEE methods} and \emph{Group SDEE methods}. Bottom-up methods perform the SDEE by using the following procedure: a) Dividing the complete development process into sub-tasks, b) Estimating the effort required to perform these sub-tasks, c) Finally, summing these effort values to obtain an overall effort estimate. The estimates for performing the sub-tasks are obtained from the specific people in charge of developing them.
On the contrary, Group SDEE methods comprise consulting a group of experts to derive the effort estimate for developing a software project \cite{wu1997comparison}.
\item \emph{Formal SDEE methods:}
These are based on the software development effort estimates derived using a mathematical quantification step \cite{jorgensen2009software}, such as a formula derived from historical data from past projects. These can be further classified as \emph{Analogy-based SDEE methods}, \emph{Parametric SDEE methods}, \emph{ML-based SDEE methods}, and \emph{Size-based SDEE methods}. Analogy-based SDEE methods follow \textbf{identifying one or more existing software projects similar to the project being developed} and then deriving the effort estimate by analyzing similar software projects. Similarly, Parametric methods are based on \textbf{identifying the variables influencing the effort required to develop a software project}. The general mathematical form of such models is represented as:
\begin{equation}
\label{eq:general-parametric-models}
E = A + B(e_{\nu})^{C}
\end{equation}
In the above equation, $E$ represents the effort in person-months, $e_{\nu}$ represents the estimation variable (e.g., source lines of code (SLOC) or function points \cite{albrecht1979measuring,dumke2016cosmic}), and $A, B$, and $C$ represent the empirically derived constants the past software data. Some of the prominently used Parametric SDEE methods include COCOMO \cite{barry1981software}, Putnam's model \cite{putnam1978general}, and SEER-SEM \cite{galorath2007software}.
Size-based SDEE methods form a subset of parametric methods, as these use size-based metrics, such as SLOC and function-points, to compute the effort estimates for developing a software project. Some other examples of such metrics are use-case points (UCPs) \cite{winters1900applying}, story points \cite{cohn2005agile}, and object points \cite{minkiewicz1997measuring}. Recently, COSMIC Functional Size measurements (COSMIC FSM) are reported to overcome the shortcomings of LOC-based metrics, and outperform the existing FP-based methods \cite{di2016web,abualkishik2017study}. COSMIC FSM is based on the Functional User Requirements (FURs) of the software that are independent from any requirements or constraints about their implementation. They have been used for code size estimation of mobile applications \cite{d2015cosmic}, UML models \cite{de2020design}, Web applications \cite{di2016web,abrahao2018definition}, etc.
\item \emph{Hybrid SDEE methods:} These methods use a combination of two or more existing effort estimation methods. Some of the prominent hybrid SDEE methods derived from the literature \cite{briand2002resource} are \emph{Mathematical Hybrid SDEE methods} and \emph{Judgemental Hybrid SDEE methods}.
Mathematical Hybrid SDEE methods perform the SDEE using the analogy-based methods and the bottom-up SDEE methods, whereas Judgemental Hybrid SDEE methods perform SDEE using the expert-based SDEE methods and the parametric SDEE methods.
\end{enumerate}
\subsection{SDEE methods based on developer activity information and ML techniques}
Some existing studies \cite{kocaguneli2011exploiting, menzies2017negative} present the variation of analogy-based SDEE methods where they use the k-Nearest Neighbor (kNN) method with the euclidean distances to detect the similar software matches from existing SDEE datasets, such as the PROMISE repository \cite{menzies2012promise}. A baseline of SDEE methods was established by \cite{whigham2015baseline} when they introduced their Automatically Transformed Linear Baseline model, which is defined as a linear regression model. The ML-based methods, such as Artificial Neural Networks (NeuralNet), are also employed to train SDEE models \cite{minku2013software}.
ATLM \cite{whigham2015baseline} and ABE \cite{kocaguneli2011exploiting} use features such as the analyst's experience, programmer's capability, and application experience when trained using the existing datasets, such as the PROMISE repository. Similarly, the LOC \enquote{straw man} estimator \cite{menzies2017negative} uses the LOC of the software projects. Both ABE and the LOC \enquote{straw man} estimator use the weighted euclidean distance measure to determine the top-k similar matches. The weighted euclidean distance between two feature values x and y can be expressed as:
\begin{equation}
\label{eq:euclidean distance}
dist(x, y) = \sqrt{\sum_{i=1}^{n} w_i (x_i - y_i)^2}
\end{equation}
where $x_i, y_i$ are values normalized to [0,1], and $w_i$ is a feature-weighting factor (defaults to $w_i = 1$).
In contrast, ATLM has a linear regression form expressed as:
\begin{equation}
\label{eq:ATLM}
y_i = \beta_0 + \beta_1x_{1i} + \beta_2x_{2i} + ..\beta_nx_{ni} + \epsilon_i
\end{equation}
where $y_i$ represents the quantitative response variables, the $x_i$ represents the explanatory variables, and the $\beta_i$ is determined using a least-squares estimator \cite{neter1996applied}. Similarly, the study in \cite{minku2013software} is based on training Artificial Neural Networks (NeuralNet) using various datasets to predict the effort estimates.
An empirical approach for analyzing the developer activity traces present in various VCSs to categorize the software developers as \{occasional contributors, full-time developers\} is presented in \cite{robles2014estimating}. They present the analysis for data collected by Bitergia\footnote{\url{https://www.bitergia.com/}} for OpenStack Foundation\footnote{\url{https://https://www.openstack.org/foundation/}}. CVSAnalY is deployed to collect the analytics from the source code management repository. Developers of the project are then mailed a survey to inquire about the time devoted to developing the project. The obtained survey data is used to train a binary classification model by labeling the dataset entries as \{occasional contributors, full-time developers\}. The approach is claimed to propose a realistic estimation with a smaller weight (of the total effort) assigned to the more error-prone estimations.
Similarly, a Change Impact Size Estimation (CISE) approach for software development is introduced \cite{asl2013change}. The approach is designed by integrating the static and dynamic approaches mentioned in \cite{kama2011change}. Change Impact Analysis (CIA) is performed by tracing the affected requirements by change requests to the impacted software artifacts and codes before they are implemented. The essential requirements types are an addition, a modification, and a deletion. The CISE works by first finding the classes affected by the changes (by performing a CIA) and then performing estimation for change in size. The study claims CISE to be 93.7\% accurate.
A comprehensive study of the developer activity patterns in Free/Libre/OSS (FLOSS) projects is presented by clustering developers around different time-slots and days of a week \cite{capiluppi2013effort}. The study compares the developer activity patterns in a company-driven environment with those in a community-driven environment. The study is conducted by analyzing the developer activity patterns based on the number of commits, lines of code added, deleted, modified as recorded on several days of a week, and different hours. In our work, we utilize these counts, viz., the number of commits, lines of code added, deleted, and modified, extracted from various software present on GitHub to estimate the effort expended to develop the software.
\subsection{Computing the developer coding effort estimates for OSS}
\cite{asl2013change} introduce a Change Impact Size Estimation (CISE) approach for software development. The approach is designed by integrating the static and dynamic approaches mentioned in \cite{kama2011change}. Change Impact Analysis (CIA) is performed by tracing the affected requirements by change requests to the impacted software artifacts and codes before they are implemented. The major requirements types are an addition, a modification and a deletion. The CISE works by first finding the classes affected by the changes (by performing a CIA), and then performing estimation for change in size with an accuracy of 93.7\%. Similarly, there are existing works on developing defect prediction solutions by measuring code changes in terms of SLOCs additions, deletions, or modifications of software releases \cite{bell2011does,hassan2009predicting}.
\cite{capiluppi2013effort} study the developer activity patterns in Free/Libre/OSS (FLOSS) projects by clustering developers around different time-slots and days of a week. The developer activity patterns in a company-driven environment are compared with those in a community-driven environment based on the number of commits, lines of code added, deleted, and modified as recorded on various days of a week, and at various hours of a day. Similarly, \cite{robles2014estimating} propose an empirical approach for analyzing the developer activity traces present in various VCSs to categorize the software developer as \{occasional contributors, full-time developers\}. They present the analysis for data collected by Bitergia\footnote{\url{https://www.bitergia.com/}} for OpenStack Foundation\footnote{\url{https://https://www.openstack.org/foundation/}}. CVSAnalY is deployed to collect the analytics from the source code management repository. Developers of the project are then mailed a survey to inquire about the time devoted to developing the project. The obtained survey data is used to train a binary classification model by labelling the dataset entries as \{occasional contributors, full-time developers\}. The approach is claimed to propose realistic estimation with a smaller weight (of the total effort) assigned to the more error-prone estimations
\cite{di2017developer} stress that developer's personality characteristics and the manner in which they perform the code changes play a major role in software development. They study the developer personality characteristics by measuring \enquote{how focused the developers are while making code changes?} and \enquote{how scattered these changes are?} to devise a novel bug prediction model. The model is validated using 26 Apache software projects and comparing with four state-of-the-art prediction models based on size metrics and process metrics. Similarly, \cite{catolino2018enhancing} leverage the developer-related factors to enhance the existing change-prediction models by inspecting \enquote{how developers perform modifications?} and \enquote{how complex is the development process?} The idea is to use the complexity of the development process to identify the change-prone classes, and then focus the developer's attention to such classes. The authors perform an empirical analysis of three developer-based change prediction models considering the 408 releases, 193,274 commits, and 657 developers' information of 20 GitHub projects, and propose a novel hybrid change prediction model which exploits the developer-, process-, and product-metrics to detect the change-proneness of source code classes.
Similar to the existing works \cite{capiluppi2013effort,asl2013change,robles2014estimating,kocaguneli2011experiences}, we utilize the developer activity information such as the count of developers and time (in days) required to estimate the effort required to develop an existing GitHub software repository. We store these SDEE estimates as a dataset, which is then used by our SDEE tool to provide the effort estimates for newly-envisioned software.
\subsection{Developing a software description similarity detection model for detecting similar software matches}
There exists a considerable amount of literature highlighting the use of various NL processing techniques to develop software description similarity detection models for various recommendation tasks, such as performing recommendation of API methods \cite{thung2013automatic}, third-party libraries \cite{chen2016similartech}, and research papers \cite{beel2013introducing}. Two prominent techniques used in the existing works are \emph{Term-frequency Inverse Document Frequency (TF-IDF)} \cite{salton1988term} model and \emph{Word2vec model} \cite{mikolov2013distributed,mikolov2013linguistic}.
TF-IDF methods compute document similarity based on the term-frequencies for the terms or words present in a document, while the Word2vec model works by learning the word representations for the words present in an input document, to predict the co-occurring words in the same sentence.
\cite{trstenjak2014knn} present the use of K-Nearest Neighbour (KNN) algorithm with TF-IDF method for text-classification of documents belonging to four different categories, viz., sports, politics, finance, and daily news. The results of the experiments show that the classification algorithm is sensitive to type-of-documents. The classification algorithm yields the highest accuracy of 92\% in the case of documents belonging to the \emph{sports} category, while the lowest as 65\% in the case of the \emph{daily news} category.
We use the Paragraph vectors algorithm (PVA) \cite{le2014distributed} to develop our software description similarity detection model. The PVA is inspired by the Word2vec method and is also termed as a vector, as it generates fixed-length feature representations (or vectors) corresponding to the text documents provided as input.
\textbf{Limitations of the TF-IDF method:}
Since the TF-IDF method is based on the bag-of-words approach, some of its limitations are \cite{ramos2003using}:
\begin{enumerate}
\item \label{lim:one} It is unable to capture the semantic information of the text on which it is trained.
\item \label{lim:two} TF-IDF method does not consider the ordering of different words in a text document or the co-occurrences of words in different documents.
\item TF-IDF might take much time to train on a large vocabulary.
\end{enumerate}
Due to the limitations specified in point \ref{lim:one} and \ref{lim:two} above, TF-IDF is not able to capture the documents that differ in words having the same meaning i.e. synonyms. For instance, happy, content, and joyful will be treated as different words by TF-IDF method. Further, the method would also label two documents as different, if they convey the same idea using different descriptions. For instance, \emph{it was a busy day}, and \emph{I was occupied with a lot of work today} both convey the same meaning, but will be labeled as different. Similarly, the TF-IDF method would label two project descriptions as different, if they use different words to convey the same meaning. For instance, consider the following software requirements:
\begin{enumerate}
\item \emph{Require a fast, accurate, and anti-obfuscation library detection tool}, and
\item \emph{Require a light-weight, effective, and obfuscation-resilient tool.}
\end{enumerate}
Both the examples convey the same software requirements, but with different use of words, and hence are labeled as different. Thus, these limitations make TF-IDF an inappropriate measure for detecting similarity in our case, i.e., when dealing with software requirements and software project descriptions.
\begin{figure*}
\centering
\includegraphics[width=0.9\columnwidth]{figures/high-level-comp.pdf}
\caption{High-level comparison at approach-level}
\label{fig:high-level-comp}
\end{figure*}
\subsection{High-level differences of our approach from the existing methods}
\label{sec:high-level-sdee-comp}
Fig. \ref{fig:high-level-comp} presents a broad idea of how our approach differs from the existing SDEE methods. The existing effort estimation methods, such as the Analogy-based effort estimation (ABE) \cite{kocaguneli2011exploiting} and LOC \enquote{straw man} \cite{menzies2017negative}, generally follow the branch (a) approach shown in the figure, where, to estimate the effort for a newly-envisioned software $z$, the effort values of software projects developed in the past and similar to $z$ are used. On the contrary, the regression-based methods, such as the Automatically Transformed Linear Baseline Model (ATLM) \cite{whigham2015baseline} and Artificial Neural Network (NeuralNet) \cite{minku2013software} directly learn and estimate the effort value of $z$ using the effort values of the past software projects, represented by branch (b) of the figure. The existing SDEE methods mostly train the models on the metadata values of the software projects developed in the past, present in the existing effort estimation datasets, such as the PROMISE repository \cite{boetticher2007promise}.
In our approach, we hypothesize that \enquote{more relevant similar-software matches can be detected if we use the software project descriptions as the input for detecting similar software projects.} Also, the developer activity information has been claimed to be an accurate representation of the actual development effort on a software project \cite{capiluppi2013effort, robles2014estimating}. Therefore, we use the software project descriptions of GitHub repositories to determine the similar software projects (to $z$) and use the developer activity information of the similar software matches to compute the effort estimate for $z$.
\begin{table}
\caption{Table of Notations}
\centering
\begin{tabular}{r c p{12cm}}
\toprule
$R$ & $\triangleq$ & The set of OSS repositories present on GitHub considered by us.\\
$e^{r}$ & $\triangleq$ & Effort required to develop $r \in R$, computed using equation \ref{eq:software-development-effort}\\
$D^{r}$ & $\triangleq$ & The set of developers involded in developing $r \in R$.\\
$t^{r}$ & $\triangleq$ & The total amount of time spent in developing $r \in R$.\\
$C$ & $\triangleq$ & The set of different software categories specified on MavenCentral\footnote{\url{https://mvnrepository.com/}}.\\
$H$ & $\triangleq$ & The set of software project descriptions of $R$.\\
$P$ & $\triangleq$ & The set of PVA input parameter variation scenarios for training the software similarity detection models.\\
$M$ & $\triangleq$ & The set of software similarity detection models trained using PVA on $H$.\\
$Y$ & $\triangleq$ & The test-bed developed to test the performance of software similarity detection models.\\
$\phi$ & $\triangleq$ & The vector representation of a project description.\\
$\alpha$ & $\triangleq$ & The cosine similarity score between two PVA vectors.\\
$\hat{\alpha}$ & $\triangleq$ & The threshold of cosine similarity between two PVA vectors to categorize them as similar.\\
$\psi$ & $\triangleq$ & Number of training samples used for training a software similarity detection model. \\
$\gamma$ & $\triangleq$ & It refers to the dimensionality of the feature vectors used to develop the dataset, referred to as the PVA vector size. \\
$\beta$ & $\triangleq$ & Number of training iterations or epochs used for training a software similarity detection model.\\
\bottomrule
\end{tabular}
\label{tab:TableOfNotation}
\end{table}
To the best of our knowledge, such use of software project descriptions and the developer activity information has not been done in performing the SDEE task. However, since we have chosen the OSS (GitHub) repositories to develop our dataset, our SDEE model provides estimates for only the software developed in an open-source environment. This is because the software development for open-source software is generally performed by source code contributors who do not have fixed working hours or work patterns, whereas the proprietary software is developed by full-time working employees \cite{capiluppi2013effort} having fixed working patterns. Thus, the developer activity might differ due to the different working patterns.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{figures/Effort_Estimation_basic_idea}
\caption{Details of the proposed system - a system-context diagram}
\label{fig:basic-approach}
\end{figure*}
\section{Proposed system}
\label{sec:proposed-system}
The main objective of our approach is:
\emph{Given the requirements description of a software project, estimate the effort required to develop it.}
To achieve this objective, we develop three major software artifacts (shown in Fig. \ref{fig:basic-approach}):
\begin{enumerate}
\item A novel \emph{SDEE dataset} based on the developer activity information of software present on GitHub,
\item A \emph{software similarity detection model} (developed using PVA) trained on the project description documents of software considered to develop the above dataset,
\item An \emph{SDEE tool} that utilizes the SDEE dataset and the software similarity detection model to provide effort estimates.
\end{enumerate}
Table \ref{tab:TableOfNotation} shows the notation used for various terms in this paper.
\subsection{Developer activity-based SDEE metrics}
\label{sec:SDEE-metrics}
An OSS development broadly comprises developers contributing source code in an OSS repository. To measure the software development activity quantitatively, we define novel SDEE metrics for an OSS repository $r$, as follows:
\begin{enumerate}
\item $|D^r|$: The total number of developers involved in developing $r$. It is measured in \emph{persons}.
\item $T^r$: The total amount of time spent in developing $r$, mathematically represented as:
\begin{equation}
\label{eq:software-development-time}
t^r \mbox{=}\ \sum_{i=1}^j (t_s^i - t_e^i)/j
\end{equation}
where $t_s^i$ represents the start-time of developing the ith release of repository r, and $t_e^i$ represents the end-time of the ith release. It is expressed in days, months, or years.
\item $e^r$: The effort expended to develop $r$, mathematically represented as:
\begin{equation}
\label{eq:software-development-effort}
e^r \mbox{=}\ |D^r| * t^r
\end{equation}
It is measured in person-months, or person-years, or person-days, depending upon the units of $|D^r|$ and $t^r$.
\end{enumerate}
\subsection{Steps of our approach}
\label{sec:steps-of-approach}
The pivotal steps in our approach with the relevant design decisions addressed at each step are listed below:
\begin{enumerate}
\item \textbf{Processing the GitHub software repositories and developing the SDEE dataset}
\begin{enumerate}
\item Download the software categorization listed on MavenCentral.
\emph{Design Decision: Why choose the software categorization from MavenCentral?}
\item Download \label{step:downloading-project-desc} a collection of GitHub software repositories belonging to some of the significant software categories. We used the ones specified by MavenCentral. Also, extract the developer activity information and download the software project descriptions of the respective GitHub repositories.
\item Compute the effort estimates for developing the considered software repositories by utilizing their developer activity information. The critical design decisions considered in this step were:
\begin{enumerate}
\item \emph{How do we ensure our dataset's homogeneity in terms of the breadth of software types or domains?}
\item \emph{Why is only GitHub chosen to download software repositories? How do we select a software repository?}
\item \emph{What developer activity information should we collect from the selected repositories? How can we determine whether the collected information helps estimate the software development effort?}
\item \emph{How do we utilize the developer activity information to compute the software development effort estimates?}
\end{enumerate}
\end{enumerate}
\item \textbf{Developing the software similarity detection model and obtaining the PVA vectors}
\begin{enumerate}
\item Train a software similarity detection model using the PVA on the software project descriptions downloaded in Step \ref{step:downloading-project-desc}. The critical design decisions considered in this step were:
\begin{enumerate}
\item \emph{Why use PVA and how to choose its tuning parameters?}
\item \emph{Why do we use vector representations of software product descriptions?}
\end{enumerate}
\item Obtain the PVA vectors for each of the product descriptions downloaded in Step \ref{step:downloading-project-desc}.
\emph{Design Decision: What is the minimum threshold value of $\alpha$ between two PVA vectors that indicate a significant similarity between the corresponding software project description samples?}
\end{enumerate}
\item \textbf{Developing the SDEE tool to perform the SDEE for a newly-envisioned software $z$}
\begin{enumerate}
\item Design a suitable GUI for the interaction with the end-user. The salient design decisions considered at this step were:
\begin{enumerate}
\item \emph{What is the input format for specifying the requirements description of software for which the development effort is predicted?}
\item \emph{What type of details are required to be specified in the requirements description provided as input to the tool?}
\end{enumerate}
\item Compute the effort estimate for developing $z$ by leveraging the SDEE dataset and the PVA vectors.
\begin{enumerate}
\item Obtain a PVA vector $\phi'$ for the requirements description submitted by the end-user.
\item Fetch \label{step:fetch-sim-vecs} all the PVA vectors $\phi \in V$, such that the similarity $\alpha$ between $\phi$ and $\phi'$ is above $\hat{\alpha}$.
\item Derive an effort estimate for $z$ by utilizing the similar software matches' effort values fetched in Step \ref{step:fetch-sim-vecs}.
\emph{Design Decision: How do we compute the final effort estimate by utilizing the similar software matches' effort values?}
\end{enumerate}
\end{enumerate}
\end{enumerate}
In the following sections, we describe developing our software artifacts and the rationale for design decisions addressed in this process.
\subsection{Design considerations in our approach}
\label{sec:design-details}
In this section, we discuss the rationale behind design decisions taken by us while implementing our approach.
\begin{enumerate}
\item \textbf{Obtain the software categorization:} \label{step:select-soft-categorization}
To ensure our dataset's homogeneity in terms of the software types, it is necessary first to identify what types of software constitute the major portion of professional software. MavenCentral is one such source of software categorization taxonomy. Thus, we first fetch the software categorization listed by Maven (MVN) Repository\footnote{\url{https://mvnrepository.com/}}, then select the software projects under each of those categories to develop the dataset. The MVN repository portal contains around 150 software category web pages, with each category-specific web page containing the description and web links of the categorized software. These categories are utilized as topics to fetch the GitHub links of software using the GitHub REST API\footnote{\url{https://developer.github.com/v3/}}.
\item \textbf{The rationale for choosing GitHub:} Though several OSS hosts such as SourceForge.net\footnote{\url{https://sourceforge.net/}}, GitHub, GitLab, and BitBucket\footnote{\url{https://bitbucket.org/dashboard/overview}}, only a few offer support for extracting the developer activity information. We selected GitHub over the other OSS hosts for the following reasons:
\begin{enumerate}
\item SourceForge.net does not provide the details of source code development for the software repositories hosted by it. Hence, no developer activity information statistics were available.
\item GitLab and BitBucket, the newer ones, have very few OSS repositories that match our selection criteria. For instance, GitLab shows only 12 OSS repositories with $>$500 stars\footnote{\url{https://gitlab.com/explore/projects/starred}}. \end{enumerate}
\item \textbf{Repository selection rationale:} We select the top 100 repositories provided by the GitHub RESTful API for each input category. The additional constraints \cite{coelho2020github,han2019characterization} applied to filter the OSS repositories under each of the selected software categories (selected in Step \ref{step:select-soft-categorization} above) are:
\label{pt:url-extraction-constraints}
\begin{enumerate}
\item \emph{Size Constraint:} We select the repositories with a size of more than $5$ MB. The 5 MBs threshold was taken since we observed that the most non-trivial OSS repositories were larger than 5 MBs in their source code content. This means that the repositories with a size less than 5 MBs had minimal SLOC count.
\item \emph{Date Constraint:} To ensure a repository is not very old or inactive, we add a date constraint to our search. We select the repositories which have been updated at least once in the last three years.
\item \emph{Stars Constraint:} We applied this constraint to ensure that the repositories selected are well-liked to some extent and hence being used by a certain number of developers. We selected the repositories with more than $500$ stars. We sorted our search results based on the stars earned by them and selected the top $100$ results under each category.
\item \emph{Keyword constraint:} To filter the software repositories belonging to a specific category, we perform a GitHub REST API query search with the category names as the input keywords.
\end{enumerate}
\item \textbf{Utilizing developer activity information to compute effort estimates:} \label{step:developer activity-information}
The usefulness of developer activity information (Definition \ref{def:developer activity-information}) in performing the SDEE tasks has been established by \cite{robles2014estimating,capiluppi2013effort,gousios2008measuring,amor2006effort,rahman2013and,icse2014keynote}. To obtain the effort estimates for the considered software repositories, we define the SDEE metrics based on the repository's developer activity information. We obtain the software repositories' effort values by inserting the SDEE metrics' values in equation \ref{eq:software-development-effort}. Here, we consider the developer activity performed within the time-period $\langle$ start-date of release, date of the last commit performed on the release$\rangle$ to compute the release-wise effort estimates and store them in the table \emph{release\_effort\_estimate}. These release-wise estimates are then averaged to estimate the effort values of the software project, which are later utilized while performing the SDEE.
\item \textbf{The rationale for choosing PVA, its tuning parameters, and the similarity threshold $\hat{\alpha}$:} \label{sec:PVA}
The following are the main reasons for PVA's choice: i) It allows us to compute the same length vectors for representing the software project descriptions. Keeping the length of such vectors the same for every source code sample is critical for implementing an efficient and fast system. ii) PVA outperforms the existing text representation methods and thus serves as an effective method of detecting text similarity \cite{le2014distributed}. Thus, we chose the PVA to develop our \emph{software similarity detection model} and compute the vector representations of our approach's software product descriptions. The complete details of the parameter tuning of PVA are discussed in detail in Appendix-\ref{ap:PVA-parameter-tuning}.
\item \textbf{Computing PVA vectors for software product descriptions:}
\label{sec:computing-PVA-vectors}
To expedite the software similarity detection process, we perform the following steps:
\begin{enumerate}
\item \label{step:a} Obtain the PVA vector representations of the software product descriptions using a suitable software similarity detection model (developed using Algorithm \ref{alg:training-pv-model}) and store them in the database.
\item Compute the cosine similarity ($cos\_sim$) of the PVA vectors for a reference vector of the same dimensions and store it in the database. The reference vector is randomly chosen and kept fixed for the dataset.
\end{enumerate}
\begin{figure*}
\centering
\begin{subfigure}{0.7\textwidth}
\centering
\includegraphics[width=\linewidth]{figures/page1}
\caption{Details submission page}
\label{fig:submission-page}
\end{subfigure}%
\begin{subfigure}{0.63\textwidth}
\centering
\includegraphics[height=.45\linewidth]{figures/effortEstimatePage.pdf}
\caption{SDEE estimates for the input software details}
\label{fig:tool-effort-estimate}
\end{subfigure}
\setlength{\belowcaptionskip}{-5pt}
\caption{SDEE tool interface}
\label{fig:sdee-tool-interface}
\end{figure*}
\item \textbf{Structure of inputs provided to the tool:}
\label{sec:tool-inputs}
Fig. \ref{fig:sdee-tool-interface} shows the input and output pages of our tool's GUI. The software similarity detection model is likely to detect similar software with better accuracy if provided the input using the familiar format and structure as the training data. For instance, the software descriptions provided as input should describe the software's functionality to be developed, the features required to be developed in it, and a detailed description of all such functional requirements and software constraints. Therefore, the format of the input expected by our tool is such that it captures the software's key functionalities and characteristics whose effort is to be estimated. For the newly-envisioned software, the end-user needs to enter the following inputs into the tool's GUI (shown in \ref{fig:submission-page}):
\begin{enumerate}
\item \emph{Software product title:} A suitable title for the software product.
\item \emph{Software Description:} A brief description of the software product comprising its intended use, key functionalities, and characteristics.
\item \emph{Preferred programming language for software development:} The programming language(s) used for software development. In the case of multiple programming languages, the inputs can be entered in a comma-supported format.
\item \emph{Type of software:} The end-user must select a plausible software category and a sub-category from the respective drop-down lists. We extract 11 \emph{abstract} software categories from 150 software categories specified by MavenCentral, viz., \emph{i) Software library, ii) Software utilities \& plugin, iii) Software tool, iv) Software metrics, v) Software driving engine, vi) A software framework, vii) Software middleware, viii) Software client, ix) Software server, x) Software driver, and xi) Software file system.}
\item \emph{Supported Operating system:} In this input, the user specifies the operating system(s) the software product must support.
\item \emph{Product features and feature description fields:} The end-user specifies the major software features and their description using these input fields.
\end{enumerate}
\begin{tcolorbox}
\textbf{The rationale for deriving the abstract software categories:}
To determine these abstract categories, we club the category names under their generic category types. For instance, SSH libraries, DNS libraries, JWT libraries can all be clubbed under the
\emph{Software library} category. On selecting one of the software categories, the end-user is displayed the specific sub-categories fetched from MavenCentral.
\end{tcolorbox}
\textbf{Discussion on the similar software projects filtered by PVA:} As shown in Fig. \ref{fig:tool-effort-estimate}, for the given description of a compression library which is efficient and quick (in Fig. \ref{fig:submission-page}), we get the top matching result as \enquote{Google Draco} library\footnote{\url{https://github.com/google/draco}}. The next close match as projected by our tool is of \enquote{Centaurean Density} library\footnote{\url{https://github.com/k0dai/density}}. Both the libraries are intended to perform high-speed compression in an efficient manner as described in their GitHub project descriptions. This is because the tool identifies the best matching software by matching the input requirement description (shown in Fig \ref{fig:submission-page}) with the GitHub software project descriptions present in our database.
\begin{figure*}
\centering
\includegraphics[width=.4\linewidth]{figures/SDEE-functionality-match}
\caption{Detecting similar matches to a functionality}
\label{fig:detect-sim-match}
\end{figure*}%
\item \textbf{Computing the effort estimate for developing a newly-envisioned software:}
\label{sec:computing-comb-effort-estimate}
Most of the GitHub hosted software repositories are used as software modules in developing more complex software systems. For instance, GitHub hosts a variety\footnote{For example, see this collection of projects \url{https://github.com/collections}} of software modules, such as third-party libraries, application development frameworks, and domain-specific applications. Since our dataset contains the effort values of such software, which act as modules in a more extensive software system, our tool can perform the SDEE for such software modules. Fig. \ref{fig:detect-sim-match} shows a decision tree for detecting existing software that matches the input.
The detailed procedure for estimating the software development effort of a newly-envisioned software $z$ is provided in Algorithm \ref{alg:estimating-effort-for-new-software}. To obtain the effort estimate for $z$, the effort values of the top-k similar software matches are combined using the Walkerden's triangle function \cite{walkerden1999empirical}, defined as follows for k=3:
If the effort estimates of the first, second, and third closest neighbors are a, b, and c, respectively, then the effort required to develop $z$ is expressed as:
\begin{equation}
\label{eq:traingle-function}
Effort(z) \mbox{=}\ (3a + 2b + 1c)/6
\end{equation}
We use Walkerden's triangle function to obtain the effort estimates as it provides a weighted average result with the weights assigned based on the closeness of the top-k neighbors and results in smaller residual values than the non-weighted average method, i.e. when all top-k neighbors are assigned equal weights. We achieved the best performance results for our SDEE method for k=2 (shown in Section \ref{sec:performance-evaluation-and-comparison}) and used it to compute the estimates.
\begin{algorithm}
\caption{Procedure to perform SDEE for a newly-envisioned software $z$}
\label{alg:estimating-effort-for-new-software}
\begin{algorithmic}[1]
\REQUIRE $\hat{M} := $ The best-performing software similarity detection model developed in Algorithm \ref{alg:training-pv-model}.\\
$SDEE\_DB := $ Dataset containing the effort estimates computed using equation \ref{eq:software-development-effort} and the PVA vectors.\\
$\hat{\alpha} := $ The threshold cosine similarity score for PVA vectors.\\
$p_{z} := $ Project description of $z$.
\ENSURE $e_{z} := $ Effort estimate for the $z$.
\STATE{$\phi_{d} := InferVector(p_{z}, \hat{M})$}
\STATE{$\phi_{ref} := FetchReferenceVector(SDEE\_{DB})$}
\STATE {$\tau_{z} := ComputeCosineSimilarity(\phi_{z}, \phi_{ref})$}
\STATE{$K_{z} := FetchTopKSimilarProjects(SDEE\_{DB}, \phi_{z}, \hat{\alpha})$}
\STATE{$E := FetchEffortValues(K_{z}, SDEE\_DB)$}
\STATE{$e_z := $ Return the averaged effort value using the Walkerden's triangle function equation \ref{eq:traingle-function} and the effort values $e \in E$.}
\end{algorithmic}
\end{algorithm}
\end{enumerate}
\subsection{Implementation details}
\label{sec:impl-details}
We developed our software artifacts using the Python programming language due to the available expertise. The two significant artifacts driving our approach are:
\begin{figure*}
\centering
\includegraphics[width=0.7\columnwidth]{figures/SDEE_design}
\caption{Relational schema of the SDEE dataset}
\label{fig:SDEE-dataset-design}
\end{figure*}
\begin{enumerate}
\item \textbf{SDEE dataset:} \label{step:developing-the-dataset} Fig. \ref{fig:SDEE-dataset-design} shows the relational schema of our SDEE dataset. The dataset mainly comprises the following:
\begin{enumerate}
\item Information about the releases and commits in various VCS repositories.
\item Software development effort estimates computed by inserting the developer activity-based SDEE metrics values in equation \ref{eq:software-development-effort}.
\item Metadata information, such as the developers involved in developing various software repositories.
\item PVA vector representations of the \emph{software product descriptions} extracted from VCS repositories.
\end{enumerate}
The dataset contains the above-listed information extracted from $\approx$ 13,000 GitHub repositories belonging to 150 different categories of software. The SDEE metrics values' extracted from various VCS repositories and the associated software estimates are retained as effort estimates in the SDEE dataset. The dataset is used by the SDEE tool to compute the effort estimates for developing the newly-envisioned software. It is easy to add data from additional sources. The details of the relational tables are described in Appendix \ref{ap:SDEE-dataset-details}.
Our SDEE dataset and tool are available at \url{https://doi.org/10.21227/d6qp-2n13} and \url{https://doi.org/10.5281/zenodo.5095723}, respectively.
\begin{figure*}
\centering
\includegraphics[width=0.8\columnwidth]{figures/sdee-tool-mind-map}
\caption{Estimating the effort for developing a newly-envisioned software}
\label{fig:estimating-SDEE-using-tool-mindmap}
\end{figure*}
\item \textbf{SDEE tool:} \label{sec:sdee-engine} Fig. \ref{fig:estimating-SDEE-using-tool-mindmap} highlights the major components of the tool. For a given newly-envisioned software $z$, our tool performs the following steps:
\begin{enumerate}
\item Using the \emph{vectorizer} module and the best-performing software similarity detection model $\hat{M}$, it converts the software project description inputs into a PVA vector $v$.
\item Using $v$ as the input to the \emph{similar-records fetcher}, it fetches the top-k most similar software matches ($K$) to $v$ from the SDEE dataset. We use the cosine similarity\footnote{\url{http://bit.ly/2ODWoEy}} to measure the similarity between the PVA vectors.
\item
To compute the effort estimate for developing $z$, it combines the effort estimates of $K$ by using the Walkerden's triangle function represented in equation \ref{eq:traingle-function}.
\end{enumerate}
\end{enumerate}
\section{Performance Evaluation and Comparison}
\label{sec:performance-evaluation-and-comparison}
\begin{table}
\centering
\caption{Developer activity information used in developing the SDEE dataset}
\label{tab:dataset-project-attributes}
\resizebox{0.8\columnwidth}{!}
{
\begin{tabular}{c|c}
\toprule
\textbf{Attribute name} & \textbf{Description}\\
\midrule
\multirow{2}{*}{\emph{\shortstack{Developer count\\ (devCount or $\mid D^{r} \mid$)}}} & \multirow{2}{*}{\shortstack{The total number of developers working on an OSS repository $r$. It is\\ measured in \emph{persons}.}}\\
&\\
\hline
\multirow{2}{*}{\emph{$SLOC_m^{r}$}} & \multirow{2}{*}{\shortstack{The total number of SLOC modifications made in developing the OSS repository $r$.\\ It is measured in \emph{lines}, with the whitelines excluded.}}\\
&\\
\hline
\multirow{2}{*}{\emph{\shortstack{Development time\\ $t^{r}$}}} & \multirow{2}{*}{\shortstack{The total amount of time spent in developing the OSS repository $r$.\\ It is measured in \emph{months}.}}\\
&\\
\hline
\multirow{2}{*}{\emph{\shortstack{Development effort\\ $e^{r}$}}} & \multirow{2}{*}{\shortstack{The total amount of effort spent in developing the OSS repository $r$.\\ It is measured in \emph{person-months} and computed using equation \ref{eq:software-development-effort}.}}\\
&\\
\bottomrule
\end{tabular}
}
\end{table}
A vital aspect of an SDEE method is detecting similar software matches to a newly-envisioned software whose effort is estimated. Most of the existing SDEE methods, such as ATLM\cite{whigham2015baseline}, ABE \cite{kocaguneli2011exploiting}, LOC \enquote{straw man} \cite{menzies2017negative}, Artificial Neural Networks (NeuralNet) \cite{minku2013software}, use different methods to detect similar software matches using the metadata of past software projects available as existing datasets, such as PROMISE repository.
In contrast, we use the software project descriptions to develop our \emph{software similarity detection model} to perform this task. One of our key hypotheses is to validate that our \emph{software similarity detection model} trained using software product descriptions helps detect more relevant software matches than the models trained using project metadata.
We perform several experiments using the \emph{randomised trials method} (discussed in Section \ref{sec:exp-1}) and the \emph{k-fold cross-validation} method (discussed in Section \ref{sec:exp-2}) to validate our hypothesis by comparing the performance of our method with the above methods. We choose these methods for comparison as these are the most related works to ours and have been used as baselines in the last ten years by several prominent studies.
The experiments' details to obtain an optimally tuned \emph{software similarity detection model} are provided in Appendix-\ref{ap:PVA-parameter-tuning}. Also, as ABE and LOC \enquote{straw man} use the k-nearest neighbors (kNN) method for detecting similar software matches, we compared the performance of these methods by experimenting with k=1, 2, 3, 4, and 5 for kNN in both the randomized trials and the cross-validation experiments. We used the scikit-learn python library \cite{scikit-learn} to implementing ATLM, NeuralNet, and kNN.
To validate the significance of our results, we performed various statistical significance tests, such as t-test with bootstrapping and cliff's $\delta$ test \cite{cliff1993dominance,semenick1990tests} (discussed in Section \ref{sec:exp-3} and Section \ref{sec:exp-4}). To validate the correlation between the SDEE metrics (used to compute effort) and the software development effort, we test using Pearson's correlation coefficient \cite{benesty2009pearson} (discussed in Section \ref{sec:exp-5}).
Since both the randomized and cross-validation are examples of internal validation, and it has been considered essential to perform external validation before adopting the prediction models in practice \cite{bleeker2003external}, we perform Experiment \# 6 (discussed in Section \ref{sec:exp-6}). To perform these experiments, we invited professional programmers and asked them to evaluate our tool's behavior on various quality parameters, such as the response time, accuracy, and the ease-of-use of our tool. The programmers were asked to use the existing software project estimates available as existing datasets such as COCOMO81\footnote{\url{https://bit.ly/3eMeFcN}} or by fetching the data from VCSs such as GitHub. For performing this experiment, we provided access to our tool by sharing it at \url{https://doi.org/10.5281/zenodo.5095723}. Table \ref{table:experiments-summary} summarizes the significant findings of our experiments.
\begin{table}
\centering
\caption{SDEE: Experiments summary}
\label{table:experiments-summary}
\resizebox{\columnwidth}{!}
{
\begin{tabular}{c|c|c}
\toprule
\textbf{Experiment} & \textbf{Objectives} & \textbf{Major findings} \\
\midrule
\multirow{3}{*}{\shortstack{Experiment \#1\\ (Section \ref{sec:exp-1})}}& \multirow{3}{*}{\shortstack{Performance Comparison with\\ the existing SDEE methods using\\ the randomized training and test sets}}&\multirow{3}{*}{\shortstack{DevSDEE achieves the highest Standard Accuracy (SA)-\\ of 59.89\% over the random guessing method, 42.7\% with-\\ ATLM, and 84.22\% with the LOC straw man estimator.} }\\
&&\\
&&\\
\hline
\multirow{3}{*}{\shortstack{Experiment \#2\\ (Section \ref{sec:exp-2})}}& \multirow{3}{*}{\shortstack{Performance Comparison with\\ the existing SDEE methods using\\ the k-fold cross-validation method}}&\multirow{3}{*}{\shortstack{DevSDEE achieves the highest SA of 57.05\% over-\\ the random guessing method, 35.13\% with ATLM,\\ and 87.26\% with the LOC straw man estimator.} }\\
&&\\
&&\\
\hline
\multirow{3}{*}{\shortstack{Experiment \#3\\ (Section \ref{sec:exp-3})}}& \multirow{3}{*}{\shortstack{To validate the negligible difference\\ between the DevSDEE estimates\\ and the true effort values.}}&\multirow{3}{*}{\shortstack{t-significant test, Cliff's delta, and the parametric-\\ effect tests show the minimal differences\\ in the DevSDEE estimates and the true effort values.} }\\
&&\\
&&\\
\hline
\multirow{3}{*}{\shortstack{Experiment \#4\\ (Section \ref{sec:exp-4})}}& \multirow{3}{*}{\shortstack{To validate that DevSDEE provides\\ better effort estimates than the existing\\ methods using t-significance and effect tests}}& \multirow{3}{*}{\shortstack{Both t-tests and effect tests validate that\\ DevSDEE results in more accurate (less error)-\\ effort estimates than the existing methods.}} \\
&&\\
&&\\
\hline
\multirow{3}{*}{\shortstack{Experiment \#5\\ (Section \ref{sec:exp-5})}}& \multirow{3}{*}{\shortstack{Effect of SDEE metrics on the\\ overall development effort values}}& \multirow{3}{*}{\shortstack{Correlation value of +0.799 for developer- \\ count and +0.644 for development time\\ compared with the software effort values.}} \\
&&\\
&&\\
\hline
\multirow{3}{*}{\shortstack{Experiment \#6\\ (Section \ref{sec:exp-6})}}& \multirow{3}{*}{\shortstack{Performance evaluation from a programmer’s\\ perspective on various quality parameters,\\ such as ease of use, response time, and accuracy.}}& \multirow{3}{*}{\shortstack{SDEE achieves ratings $>8/10$\\ for most of the considered\\ quality parameters.}} \\
&&\\
&&\\
\bottomrule
\end{tabular}
}
\end{table}
\subsection{Test-bed Setup}
To perform the comparison, we extracted the developer activity information from $19,096$ releases of $1,587$ GitHub OSS repositories and created a dataset of projects' metadata attributes. We also extracted the respective project descriptions. Thus, the input source for both our method and the existing methods is the same, though the respective inputs are different, viz., the metadata information and the project descriptions. After removing the records for repositories having time (in months) or developer-count $<$ 1, our dataset reduces to $1,184$ GitHub repositories' records.
Table \ref{tab:dataset-project-attributes} provides the details of the project attributes used to develop the dataset. We want to emphasize that we use the same dataset to compute the effort values across different SDEE techniques. Therefore, the input remains the same for all the methods.
\subsection{Evaluation metrics}
Most of the existing methods \cite{kocaguneli2011exploiting,minku2013software,menzies2017negative,whigham2015baseline}, with which we present our comparison, use the evaluation metrics such as the magnitude of the relative error (MRE), the mean magnitude of the relative error (MMRE), the median magnitude of the relative error (MdMRE), and prediction level (PRED) for performance evaluation \cite{conte1986software}. However, we also include some additional experimental evaluations based on newer software metrics, such as mean absolute error (MAR) \cite{langdon2016exact}, logarithmic standard deviation (LSD), RE*, standard accuracy (SA), and effect size \cite{shepperd2012evaluating}, as these are recommended by others \cite{kitchenham2002preliminary,miller2000applying}.
For the evaluation metrics defined below, let $i$ be a software project, $e$ be the actual effort value to develop $i$, and $e'$ be the estimated effort value obtained by an SDEE model $M$.
Also, let $n$ be the total number of considered software projects.
\begin{enumerate}
\item \emph{Magnitude of Relative Error, MRE:} It measures the relative size of the difference between the actual and estimated effort value:
\begin{equation}
\label{eq:mre}
MRE = \frac{(\mid e' - e \mid)}{e}
\end{equation}
The lower the MRE values for a model, the lesser is the error in its estimates, and hence, better is its performance.
\item \emph{Mean MRE (MMRE):} : It measures the percentage of the MREs averaged over the n projects present in the test-bed:
\begin{equation}
\label{eq:mmre}
MMRE = \frac{100}{n} \sum_{i=1}^{n}{MRE}_i
\end{equation}
The lower the MMRE value for a model, the smaller is the averaged error in its estimates, and hence, better is its performance.
\item \emph{Median MRE (MdMRE):} :: It measures the median of MRE values obtained by the model using the software projects in the test-bed:
\begin{equation}
\label{eq:mdmre}
MdMRE = Median(\forall_{i=1}^{n} {MRE}_i)
\end{equation}
The lower the MMRE value for a model, the smaller is the averaged error in its estimates, and hence, better is its performance.
\item \emph{Mean absolute residual, MAR:} It measures the mean of the absolute value of the residuals, i.e., the error between the predicted and the true value of an estimate, and is mathematically represented as:
\begin{equation}
\label{eq:mar}
MAR = \frac{\sum_{i=1}^{n}(\mid e'_i - e_i \mid)}{n}
\end{equation}
The lower the MAR values for a model, the lesser is the error in its estimates, and hence, better is its performance.
\item \emph{Logarithmic standard deviation, LSD:} It measures standard deviation of the log transformed prediction values, and is mathematically represented as:
\begin{equation}
\label{eq:lsd}
LSD = \sqrt{\sum_{i=1}^{n} (ln\ e'_i - ln\ \overline{e_i})^2}
\end{equation}
The lesser the LSD values, the lesser is the error, and the better is the performance.
\item \emph{RE*:} It is used as a baseline error measurement metric, and is mathematically represented as:
\begin{equation}
\label{eq:re}
RE* = \frac{var(residuals)}{var(measured)}
\end{equation}
where var represents the variance, residuals represents the error value, i.e., $e'-e$
, and the measured values represent the actual effort values, i.e., $e$ values. The lesser the $RE^*$ values for a model, the better is the performance.
\item \emph{Standard Accuracy, SA:} It provides a relative assessment of the performance of the model by comparing it with the random guessing method, and is mathematically represented as:
\begin{equation}
\label{eq:sa}
SA_P = (1 - \frac{MAR_P}{\overline{MAR_{RG}}})*100
\end{equation}
where P represents the SDEE method whose performance is being evaluated, RG represents the random guessing method \cite{shepperd2012evaluating}.
Over large number of runs, $MAR_{RG}$ converges to sample mean.
SA represents how much better an effort estimation technique is than random guessing. The larger the value of SA, the better is the model performance.
\item \emph{Effect size:} \label{pt:effect-size} It provides the magnitude of the relationship between treatment variables and outcome variables and is computed based on the sample data to make inferences about a population (analogously to the concept of hypothesis testing) \cite{kampenes2007systematic}. It is further categorized into parametric and non-parametric based on the nature of the data. The parametric method assumes the data to have a normal distribution, whereas the non-parametric method is free from any such assumption. The parametric versions of effect size include cohen's $d$, hedges' $g$, and glass $\Delta$ \cite{cohen2013statistical,hedges2014statistical,hedges1981distribution}, while the non-parametric versions include cliff's $\delta$ effect size and bootstrap method \cite{cliff1993dominance}. The mathematical expressions of these variations are listed below:
\begin{equation}
\label{eq:cohen's-effect-size}
Cohen's\ d = (\overline{X_1} - \overline{X_2}).\sqrt{\frac{(n_1 -1).var_1 + (n_2 -1).var_2}{n_1 +n_2 -2}} = t.\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}
\end{equation}
where $n_1, n_2$ represent the sample size of the two groups in consideration ($n$ in the present case), $\overline{X_1}, \overline{X_2}$ represent the mean of the considered groups ($\overline{e'}$ and $\overline{e'}$ in the present case), and $var_1, var_2$ represent the variance of the values in the respective groups. The equivalent expression with t (from the t-test) was given by Rosenthal \cite{rosenthal1994parametric}. It was observed that for a small sample size ($<20$), cohen's effect size values are biased on the sample means of the population. To rectify this limitation, Hedges proposed a correction expressed as:
\begin{equation}
\label{eq:hedges'-effect-size}
Hedge's\ g = d.({1 - \frac{3}{4.(n_1 +n_2) - 9}})
\end{equation}
where d is the cohen's effect size. Both d and g are considered as small effect sizes if the values are $<=0.2$, medium effect sizes if the values are $<=0.5$, and large effect sizes if values are $>=0.8$.
Similarly, Glass's $\Delta$ is recommended for use when the standard deviations of the considered groups differ substantially:
\begin{equation}
\label{eq:glass's-effect-size}
Glass's\ \Delta = \frac{\overline{X_1} - \overline{X_2}}{s_{control}}
\end{equation}
where $s_{control}$ represents the standard deviation of the control group, and
\begin{equation}
\label{eq:cliff's-delta}
Cliff's\ \delta = \frac{\#(X_1 > X_2) - \#(X_2 > X_1)}{n_1.n_2}
\end{equation}
where $\#$ represents the count or cardinality. For instance, $\#(X_1 > X_2)$ implies the count of data values in group 1, which are numerically larger (in value) than that of the corresponding data values in group 2. The $\delta$ value lies in the range [-1,1], with the effect size of -1 or 1 ,implying an absence of overlap of data values, whereas an effect size of 0 representing a complete overlap \cite{macbeth2011cliff}.
\end{enumerate}
\subsection{Experiment \#1: Randomized trials}
\label{sec:exp-1}
\textbf{Objective:} To evaluate the performance of SDEE models and compare them with the existing methods using the randomized trial method.
\textbf{Procedure:} When evaluating models' performance using randomized trials with a dataset containing n rows (or feature vectors), x rows are selected without replacement for testing and the rest of the (n-x) rows for training. This process is repeated for r number of trials, with the rows being shuffled in each trial. x=10 is a suitable value \cite{menzies2006selecting} and has been used in the literature \cite{minku2013software, kocaguneli2011exploiting}. Some of the existing studies take r = 30 \cite{minku2013software}, whereas some validate their models using r = 20 \cite{kocaguneli2011exploiting}. With a dataset of 1184 rows, we selected x = 55 and r = 20 to have a considerable sample size for each iteration. Thus, 55 rows were chosen in each iteration without replacement for testing, and the remaining 1129 (=1184-55) rows for training.
\begin{table}
\centering
\caption{Performance comparison based on the Randomized trials}
\label{tab:rand-trial-results}
\resizebox{0.8\columnwidth}{!}
{
\begin{tabular}{c|c|c|c|c|c|c|c}
\toprule
\multicolumn{2}{c|}{\multirow{2}{*}{\shortstack{SDEE\\ method}}} & \multicolumn{5}{c}{Evaluation metrics (mean $\pm$ std.dev)}\\
\cline{3-8}
\multicolumn{2}{c|}{}&$LSD$ & $RE^*$ & $MAR$ & $MMRE$ &$MdMRE$ & $SA$ \\ \midrule
\multicolumn{2}{c|}{ATLM} & 2.08 $\pm$ 0.21 & \cellcolor{lightgray} 0.46 $\pm$ 0.33 & \cellcolor{lightgray} 34.82 $\pm$ 6.31 & 192.40 $\pm$ 55.69 & \cellcolor{lightgray} 40.07 $\pm$ 17.05 & \cellcolor{lightgray} 53.57 $\pm$ 8.54 \\ \hline
\multicolumn{2}{c|}{NeuralNet} & 1.40 $\pm$ 0.62 & 4.31 $\pm$ 12.17 & \cellcolor{lightgray} 36.04 $\pm$ 26.96 & 222.78 $\pm$ 285.64 & \cellcolor{lightgray} 59.07 $\pm$ 16.65 & 9.51 $\pm$ 79.47\\ \hline
\multirow{3}{*}{ABE}& neigh =1& 1.45 $\pm$ 0.63 & 1.69 $\pm$ 3.90 & 37.51 $\pm$ 47.21 & 557.74 $\pm$ 1968.9 & 557.74 $\pm$ 1968.9 & 9.69 $\pm$ 103.68 \\ \cline{2-8}
& neigh =2 & 1.08 $\pm$ 0.43 & 1.24 $\pm$ 2.57 & 37.19 $\pm$ 41.16 & 632.46 $\pm$ 1615.35 & 632.46 $\pm$ 1615.35 & 9.71 $\pm$ 90.20 \\ \cline{2-8}
& neigh =3 & \cellcolor{lightgray} 0.83 $\pm$ 0.46 & \cellcolor{lightgray} 1.21 $\pm$ 2.82 & \cellcolor{lightgray} 32.89 $\pm$ 42.28 & 596.31 $\pm$ 1429.21 & 596.31 $\pm$ 1429.21 & \cellcolor{lightgray} 20.41 $\pm$ 89.89 \\ \cline{2-8}
& neigh =4 & \cellcolor{lightgray} 0.83 $\pm$ 0.47 & 1.29 $\pm$ 3.09 & 35.20 $\pm$ 43.31 & 588.48 $\pm$ 1332.76 & 588.48 $\pm$ 1332.76 & 12.44 $\pm$ 90.94 \\ \cline{2-8}
& neigh =5 & \cellcolor{lightgray} 0.74 $\pm$ 0.41 & 1.40 $\pm$ 3.13 & 35.15 $\pm$ 43.01 & 582.21 $\pm$ 1218.80 & 582.21 $\pm$ 1218.80 & 14.27 $\pm$ 88.21 \\
\hline
\multirow{3}{*}{\shortstack{LOC\\ straw\\ man}} & neigh =1 &1.38 $\pm$ 0.71 & 3.33 $\pm$ 7.36 & 76.37 $\pm$ 105.65 & 899.37 $\pm$ 1983.16 & 899.37 $\pm$ 1983.16 & -90.39 $\pm$ 278.33 \\
\cline{2-8}
& neigh =2 & 1.13 $\pm$ 0.47 & 5.50 $\pm$ 9.96 & 65.58 $\pm$ 75.53 & 941.74 $\pm$ 1628.10 & 941.74 $\pm$ 1628.10 & -62.59 $\pm$ 179.23 \\
\cline{2-8}
& neigh =3 & 0.93 $\pm$ 0.48 & 3.15 $\pm$ 5.98 & 58.50 $\pm$ 65.56 & 882.80 $\pm$ 1491.72 & 882.80 $\pm$ 1491.72 & -44.7 $\pm$ 148.32 \\
\cline{2-8}
& neigh =4 & \cellcolor{lightgray} 0.82 $\pm$ 0.45 & 3.28 $\pm$ 5.98 & 54.28 $\pm$ 60.62 & 850.19 $\pm$ 1434.05 & 850.19 $\pm$ 1434.05 & -34.4 $\pm$ 132.68 \\
\cline{2-8}
& neigh =5 & \cellcolor{lightgray} 0.81 $\pm$ 0.44 & \cellcolor{lightgray} 2.70 $\pm$ 5.47 & \cellcolor{lightgray} 53.64 $\pm$ 60.94 & 820.76 $\pm$ 1387.48 & 820.76 $\pm$ 1387.48 & \cellcolor{lightgray} -32.83 $\pm$ 133.53 \\
\hline
\multirow{3}{*}{DevSDEE} & neigh =1 & 0.95 $\pm$ 0.19 & 0.52 $\pm$ 0.97 & \cellcolor{lightgray} 9.71 $\pm$ 8.69 & \cellcolor{lightgray} 130.61 $\pm$ 25.91 & \cellcolor{lightgray} 119.77 $\pm$ 42.0 & 57.54 $\pm$ 10.52 \\ \cline{2-8}
& neigh =2 & \cellcolor{lightgray} 0.14 $\pm$ 0.34 & \cellcolor{lightgray} 0.34 $\pm$ 0.41 & 9.82 $\pm$ 7.92 & 166.01 $\pm$ 40.78 & 153.40 $\pm$ 53.81 & \cellcolor{lightgray} 59.89 $\pm$ 6.88 \\ \cline{2-8}
& neigh =3 & 0.18 $\pm$ 0.19 & 0.41 $\pm$ 0.28 & 16.01 $\pm$ 4.24 & 540.24 $\pm$ 238.88 & 524 .19 $\pm$ 264.18 & 47.79 $\pm$ 7.75 \\ \cline{2-8}
& neigh =4 & 0.25 $\pm$ 0.10 & 0.47 $\pm$ 0.20 & 16.62 $\pm$ 3.65 & 662.20 $\pm$ 294.57 & 636.18 $\pm$ 335.09 & 39.97 $\pm$ 9.51 \\ \cline{2-8}
& neigh =5 & 0.29 $\pm$ 0.06 & 0.57 $\pm$ 0.13 & 17.7 $\pm$ 2.98 & 768.11 $\pm$ 334.27 & 729.34 $\pm$ 393.88 & 33.76 $\pm$ 11.95 \\
\bottomrule
\end{tabular}
}
\end{table}
\textbf{Observations:} Table \ref{tab:rand-trial-results} shows the results obtained from these experiments. The averaged evaluation metrics values and their standard deviations (std. dev.) for 20 iterations are listed. The grey-colored cells in the table represent the SDEE models performing better than the rest. As observed from the table values, DevSDEE outperforms the existing methods for the considered evaluation metrics. We obtain the best performing DevSDEE values at neigh = 2 with the highest SA of 59.89\% (error=$\pm$6.88). However, the $SA$, $MAR$, $MMRE$, and $MdMRE$ values have large standard deviation (std. dev. or error) values and thus are not very reliable. Nevertheless, the $LSD$ and $RE^*$ values present a clear picture of DevSDEE performing better than the compared SDEE methods.
\textbf{Critical Analysis:} As it is evident from the results, the $LSD$ and $RE^*$ values of ABE, and LOC Straw man estimator decrease (or improve) with the increase in the neighbor (neigh) count, whereas DevSDEE gives the best results for neigh = 2 after which the values increase. The difference in DevSDEE's behavior from the existing methods can be explained due to the difference in detecting similar software matches. To estimate the similar software matches for a newly-envisioned software $z$, DevSDEE detects the similarity in the software project descriptions of existing software and $z$. It is improbable to have multiple software with very similar software project descriptions. Hence, it is evident that only the effort values of few most-similar matches would be close to the effort value of $z$, and would contribute towards the SDEE process. However, for the existing methods working with project metadata values to detect similarity, it is possible to have a considerable number of software projects with similar project metadata attribute values. Thus, more are such similar software matches, better would be the final estimate value, and lesser would be the $LSD$ and $RE^*$ values.
\textbf{Negative SA values:} The negative SA values for the LOC straw man estimator signify that the random guessing method performs better than it. However, since the values have large std. dev. (or error) values associated with them, we cannot reach any conclusion with confidence.
\subsection{Experiment \#2: k-fold cross-validation}
\label{sec:exp-2}
\textbf{Objective:} To evaluate the performance of SDEE models and compare with the existing methods using the k-fold cross-validation method.
\textbf{Procedure:} In k-fold cross-validation, the complete set of features is divided into k parts, with one part used for testing and the rest of the (k-1) parts for training. Thus, the complete training-testing phenomenon is performed for k number of times with different training and testing sets. Three-fold, five-fold, and ten-fold cross-validation experiments were performed to compare the considered SDEE methods' performance. Since the best results for most of the methods were obtained corresponding to the ten-fold cross-validation, we restrict our discussion to it. However, the complete set of results and models are shared at \url{https://bit.ly/3cuhRtQ}.
\begin{table}
\centering
\caption{Performance comparison based on the Ten-Fold cross-validation}
\label{tab:k-fold-results}
\resizebox{0.8\columnwidth}{!}
{
\begin{tabular}{c|c|c|c|c|c|c|c}
\toprule
\multicolumn{2}{c|}{\multirow{2}{*}{\shortstack{SDEE\\ method}}} & \multicolumn{5}{c}{Evaluation metrics (mean $\pm$ std.dev)}\\
\cline{3-8}
\multicolumn{2}{c|}{}&$LSD$ & $RE^*$ & $MAR$ & $MMRE$ & $MdMRE$ & $SA$\\ \midrule
\multicolumn{2}{c|}{ATLM} & 2.08 $\pm$ 0.21 & \cellcolor{lightgray} 0.48 $\pm$ 0.09 & \cellcolor{lightgray} 15 $\pm$ 2.82 & 192.4 $\pm$ 55.69 & \cellcolor{lightgray} 40.07 $\pm$ 17.05 & \cellcolor{lightgray} 53.57 $\pm$ 8.55 \\
\hline
\multicolumn{2}{c|}{NeuralNet} & \cellcolor{lightgray} 1.8 $\pm$ 0.88 & 22.29 $\pm$ 65.26 & 61 $\pm$ 76.3 & 328.93 $\pm$ 461.79 & 75.56 $\pm$ 42.38 & -33.99 $\pm$ 139.52 \\
\hline
\multirow{3}{*}{ABE} & neigh =1 & 1.1 $\pm$ 0.37 & 0.65 $\pm$ 0.86 & 28.96 $\pm$ 24.14 & 75.07 $\pm$ 68.04 & 75.07 $\pm$ 68.04 & 33.028 $\pm$ 53.78 \\ \cline{2-8}
& neigh =2 & 1.02 $\pm$ 0.49 & \cellcolor{lightgray} 0.51 $\pm$ 0.66 & \cellcolor{lightgray} 26.47 $\pm$ 22.56 & \cellcolor{lightgray} 51.52 $\pm$ 29.14 & \cellcolor{lightgray} 51.52 $\pm$ 29.14 & \cellcolor{lightgray} 38.73 $\pm$ 49.26 \\ \cline{2-8}
& neigh =3 & 0.99 $\pm$ 0.28 & \cellcolor{lightgray} 0.53 $\pm$ 0.79 & \cellcolor{lightgray} 30.29 $\pm$ 24.83 & 76.01 $\pm$ 58.33 & 76.01 $\pm$ 58.33 & 29.93 $\pm$ 51.52 \\ \cline{2-8}
& neigh =4 & 0.91 $\pm$ 0.24 & 0.70 $\pm$ 0.98 & 33.20 $\pm$ 26.99 & 82.04 $\pm$ 72.82 & 82.04 $\pm$ 72.82 & 22.61 $\pm$ 58.31 \\ \cline{2-8}
& neigh =5 & \cellcolor{lightgray} 0.74 $\pm$ 0.35 & 0.84 $\pm$ 1.23 & 33.98 $\pm$ 31.34 & 107.29 $\pm$ 120.88 & 107.29 $\pm$ 120.88 & 20.94 $\pm$ 70.27 \\
\hline
\multirow{3}{*}{\shortstack{LOC\\ straw\\ man}} & neigh =1 & 0.97 $\pm$ 0.63 & 3.69 $\pm$ 7.93 & 64.21 $\pm$ 76.08 & 290.10 $\pm$ 505.84 & 290.10 $\pm$ 505.84 & -54.8 $\pm$ 189.05 \\ \cline{2-8}
& neigh =2 & 0.9 $\pm$ 0.31 & 2.17 $\pm$ 3.63 & 53.98 $\pm$ 56.13 & \cellcolor{lightgray} 188.48 $\pm$ 237.89 & \cellcolor{lightgray} 188.48 $\pm$ 237.89 & -28.66 $\pm$ 134.81 \\ \cline{2-8}
& neigh =3 & \cellcolor{lightgray} 0.71 $\pm$ 0.4 & 1.95 $\pm$ 3.25 & 50.77 $\pm$ 48.6 & 242.04 $\pm$ 418.77 & 242.04 $\pm$ 418.77 & -21.9 $\pm$ 119.10 \\ \cline{2-8}
& neigh =4 & \cellcolor{lightgray} 0.67 $\pm$ 0.32 & \cellcolor{lightgray} 1.44 $\pm$ 2.0 & 47.44 $\pm$ 40.02 & 213.73 $\pm$ 357.44 & 213.73 $\pm$ 357.44 & -12.99 $\pm$ 95.01 \\ \cline{2-8}
& neigh =5 & \cellcolor{lightgray} 0.6 $\pm$ 0.34 & 1.46 $\pm$ 1.94 & \cellcolor{lightgray} 46.16 $\pm$ 38.27 & 224.12 $\pm$ 366.30 & 224.12 $\pm$ 366.30 & \cellcolor{lightgray} -9.74 $\pm$ 90.32 \\
\hline
\multirow{3}{*}{DevSDEE} &neigh =1 & 0.69 $\pm$ 0.51 & \cellcolor{lightgray} 0.19 $\pm$ 0.01 & \cellcolor{lightgray} 9.47 $\pm$ 1.12 & 94.31 $\pm$ 25.29 & \cellcolor{lightgray} 54.58 $\pm$ 7.27 & \cellcolor{lightgray} 56.79 $\pm$ 2.41 \\ \cline{2-8}
& neigh =2 & \cellcolor{lightgray} 0.29 $\pm$ 0.22 & \cellcolor{lightgray} 0.41 $\pm$ 0.27 & \cellcolor{lightgray} 8.46 $\pm$ 0.08 & \cellcolor{lightgray} 47.88 $\pm$ 23.2 & \cellcolor{lightgray} 42.4 $\pm$ 23.2 & \cellcolor{lightgray} 57.05 $\pm$ 8.16 \\ \cline{2-8}
& neigh =3 & 0.44 $\pm$ 0.20 & 0.53 $\pm$ 0.06 & 13.03 $\pm$ 0.67 & \cellcolor{lightgray} 48.72 $\pm$ 27.45 & \cellcolor{lightgray} 48.72 $\pm$ 27.45 & 50 $\pm$ 16.92 \\ \cline{2-8}
& neigh =4 & 0.43 $\pm$ 0.31 & 0.57 $\pm$ 0.80 & 17.22 $\pm$ 2.86 & 60.98 $\pm$ 15.24 & 60.98 $\pm$ 15.24 & 51.52 $\pm$ 21.79 \\ \cline{2-8}
& neigh =5 & \cellcolor{lightgray} 0.34 $\pm$ 0.21 & 0.70 $\pm$ 0.46 & 17.21 $\pm$ 0.44 & 49.17 $\pm$ 18.56 & 49.17 $\pm$ 18.56 & 50.37 $\pm$ 17.83 \\
\bottomrule
\end{tabular}}
\end{table}
\textbf{Observations:} Table \ref{tab:k-fold-results} shows the results obtained for ten-fold cross-validation experiments. DevSDEE achieves the lowest mean and standard deviation values for $LSD$, $RE^*$, $MAR$, $MMRE$, $MdMRE$, and $SA$ values and has the most negligible errors in its SDEE estimates. The grey-colored cells in the table represent the SDEE models performing attaining the best metrics' values. As observed from the table values, DevSDEE outperforms the existing methods and achieves the highest $SA$ of 57.05$\pm$8.16 and least $RE^*$ of 0.19$\pm$0.01 with neigh=2 .
Since the $MMRE$, $MdMRE$, and $SA$ values have large std. dev. values (or errors), we decide using the $MAR$, $LSD$ and $RE^*$ values of DevSDEE. The reason for the large std. dev. values could be their dependency on the central measures (mean or median) of the data values or the assumption of normal data distribution. Nevertheless, the $LSD$ and $RE^*$ values present a clear picture of DevSDEE performing better than the compared SDEE methods. However, we perform the additional non-parametric statistical test, viz., t-test with the bootstrap procedure and the cliff's delta tests, that we discuss in Section \ref{sec:exp-3} and Section \ref{sec:exp-4}.
\textbf{Critical Analysis and Negative SA values:} The large std. dev. values in the case of $MMRE$, $MdMRE$, and $SA$ values signify the software effort estimates have a considerable variation. This could be due to the difference in the nature of software types and the kind of developer activity performed in developing them. The effort required to develop a specific software library might differ depending on their inherent functionalities or characteristics. For instance, the Google Draco\footnote{\url{https://github.com/google/draco}} library is used to compress and decompress 3D geometric meshes and point clouds, while the Google Tink\footnote{\url{https://github.com/google/tink}} library is used to perform encryption to provide secure APIs. Both the libraries differ in their functionalities and hence would differ in the effort required to develop them. However, the effort values could also differ for the same software library types. For instance, Google snappy\footnote{\url{https://github.com/google/snappy}} is another compression library, but it aims for very high speeds and reasonable compression instead of maximum compression. Hence, the effort required to develop Draco would differ from snappy as well but is expected to be closer to snappy than Tink. The actual effort values essentially depend on the amount of developer activity (in the form of $SLOC_m$) performed on a software project, and we only aim to achieve an estimate using this approach. The conclusions for critical analysis on DevSDEE behavior observed in Section \ref{sec:exp-1} holds here as well, as DevSDEE performs best with neigh=1 or neigh=2. Similarly, the inference for negative values of LOC straw man's SA values remains the same.
\subsection{Experiment \#3: Significance tests}
\label{sec:exp-3}
\textbf{Objective:} To validate that there exists a negligible difference between the actual effort values and the DevSDEE estimates
\textbf{Need for statistical tests with the bootstrap procedure:} Both the Randomized trials and k-fold cross-validation methods have been considered to be essential to assess the model prediction error and avoid misleading estimates due to dataset instability \cite{turhan2012dataset}. However, these methods have been remarked to produce a large variance for the datasets with outliers and are considered unsuitable for SDEE datasets. This might be a reason behind the large error values in the case of k-fold cross-validation results as compared to the randomized trials’ results (as shown in Table \ref{tab:rand-trial-results} and Table \ref{tab:k-fold-results}), as the number of iterations is larger in the case of randomized trials as compared to the k-fold cross-validation method.
An alternative solution to this problem as suggested by the existing studies \cite{whigham2015baseline}, is to test using many iterations to obtain a fair estimate of performance and allow a meaningful statistical comparison. This is particularly what is done by statistical tests with bootstrap procedures, and therefore, we perform additional experiments using t-test, cliff-delta test, and other parametric effect tests with bootstrapping to validate our findings.
\textbf{Description:} For DevSDEE to perform efficiently, it should yield effort estimates close to the true effort values. Therefore, we perform this experiment to validate that DevSDEE estimates are close to true effort values or negligible difference between them. The objective of this experiment is very similar to the null hypothesis $H_0$ of the effect tests, which states that there lies no difference between the test group and control group data values. All these effect size variations (defined in Section \ref{sec:evaluation-metrics}) are studied with $H_0$ and a significance value ($p$).
\textbf{Procedure:} We used the researchpy\footnote{\url{https://researchpy.readthedocs.io/en/latest/ttest_documentation.html}} and the DABEST python libraries\footnote{\url{https://github.com/ACCLAB/DABEST-python}} to compute the values of t-test, cohen's $d$, hedges' $g$, glass's $\Delta$, and cliff's $\delta$ \cite{dabest}. Both the libraries compute the estimates with a 95\% confidence interval and by taking 5000 bootstrap samples. To compute the estimate values, we provide the DevSDEE effort estimates and the true effort values obtained from the Random Trials and the k-fold cross-validation experiments as discussed in Section \ref{sec:exp-1} and Section \ref{sec:exp-2}.
\begin{table}
\centering
\caption{Significance Testing to evaluate the DevSDEE results}
\label{tab:significance-test}
\resizebox{\columnwidth}{!}
{
\begin{tabular}{c|c|c|c|c|c|c|c|c|c}
\toprule
\multicolumn{2}{c|}{\multirow{2}{*}{}} & \multicolumn{2}{c|}{t-significance test} & \multicolumn{2}{c|}{Cliff's delta effect test} & \multicolumn{4}{c}{Parametric effect tests}\\
\cline{3-10}
\multicolumn{2}{c|}{}&$t-val$ & $p-val$ & $cliff's\ \delta$ & $p-val$ & $cohen's\ d$ & $hedges'\ g$ & $glass's\ \Delta$ & $p-val$\\
\midrule
\multirow{5}{*}{\rotatebox[origin=c]{90}{Random Trials}} & neigh=1 & 0.23 $\pm$ 0.29 & 0.77 $\pm$ 0.11 & 0 $\pm$ 0.17 & 0.63 $\pm$ 0.15 & 0.22 $\pm$ 0.22 & 0.13 $\pm$ 0.18 & 0.32 $\pm$ 0.25 & 0.67 $\pm$ 0.1 \\
\cline{2-10}
& neigh=2 & 0.18 $\pm$ 0.27 & 0.80 $\pm$ 0.11 & 0.02 $\pm$ 0.09 & 0.65 $\pm$ 0.06 & 0.18 $\pm$ 0.21 & 0.1 $\pm$ 0.17 & 0.23 $\pm$ 0.27 & 0.66 $\pm$ 0.07 \\
\cline{2-10}
& neigh=3 & 0.11 $\pm$ 0.35 & 0.83 $\pm$ 0.19 & 0.03 $\pm$ 0.16 & 0.6 $\pm$ 0.11 & 0.10 $\pm$ 0.25 & 0.06 $\pm$ 0.22 & 0.12 $\pm$ 0.36 & 0.62 $\pm$ 0.11 \\
\cline{2-10}
& neigh-4 & 0.14 $\pm$ 0.4 & 0.81 $\pm$ 0.21 & 0.03 $\pm$ 0.16 & 0.60 $\pm$ 0.11 & 0.12 $\pm$ 0.28 & 0.08 $\pm$ 0.25 & 0.17 $\pm$ 0.43 & 0.6 $\pm$ 0.11 \\
\cline{2-10}
& neigh=5 & 0.15 $\pm$ 0.38 & 0.8 $\pm$ 0.20 & 0.03 $\pm$ 0.16 & 0.6 $\pm$ 0.11 & 0.13 $\pm$ 0.27 & 0.09 $\pm$ 0.23 & 0.25 $\pm$ 0.48 & 0.61 $\pm$ 0.09 \\
\hline
\multirow{5}{*}{\rotatebox[origin=c]{90}{k-Fold}} & neigh=1 & 0.06 $\pm$ 0.61 & 0.7 $\pm$ 0.24 & 0 $\pm$ 0.37 & 0.52 $\pm$ 0.31 & 0.05 $\pm$ 0.43 & 0.04 $\pm$ 0.37 & 0.07 $\pm$ 0.43 & 0.68 $\pm$ 0.23 \\
\cline{2-10}
& neigh=2 & 0.04 $\pm$ 0.58 & 0.70 $\pm$ 0.2 & 0.08 $\pm$ 0.19 & 0.62 $\pm$ 0.12 & 0.03 $\pm$ 0.41 & 0.02 $\pm$ 0.35 & 0.02 $\pm$ 0.48 & 0.67 $\pm$ 0.16 \\ \cline{2-10}
& neigh=3 & 0.13 $\pm$ 0.79 & 0.56 $\pm$ 0.16 & 0.12 $\pm$ 0.33 & 0.44 $\pm$ 0.08 & 0.09 $\pm$ 0.56 & 0.08 $\pm$ 0.49 & 0.02 $\pm$ 0.79 & 0.51 $\pm$ 0.18 \\ \cline{2-10}
& neigh=4 & 0.24 $\pm$ 0.88 & 0.48 $\pm$ 0.05 & 0.12 $\pm$ 0.33 & 0.44 $\pm$ 0.08 & 0.17 $\pm$ 0.62 & 0.15 $\pm$ 0.54 & 0.07 $\pm$ 0.96 & 0.43 $\pm$ 0.06 \\ \cline{2-10}
& neigh-5 & 0.24 $\pm$ 0.85 & 0.5 $\pm$ 0.06 & 0.12 $\pm$ 0.33 & 0.44 $\pm$ 0.08 & 0.17 $\pm$ 0.6 & 0.15 $\pm$ 0.52 & 0.09 $\pm$ 1.04 & 0.47 $\pm$ 0.06 \\
\bottomrule
\end{tabular}
}
\end{table}
\textbf{Observations and Analysis:} Both the t-test and the parametric effect tests measure the difference between the mean values of these groups, and the $p$ value represents the significance (or probability) of these estimates. A large effort value associated with a $p<0.05$ (marked as 95\% confidence level) leads to the rejection of $H_0$, concluding that there is a considerable difference between the values of the considered groups. Similarly, if the effort values are small and are associated with large $p$ values, it is concluded that there is considerable overlap in the data values of the considered group, or the differences between the values are small \cite{cohen1988statistical}. However, $H_0$ can either be rejected or stated that it is impossible to reject the null hypothesis (if $p$ values are large or $>0.1$). As observed from Table \ref{tab:significance-test} values, all the t-test values have t values $<0.24 \pm 0.88$ associated with high $p$ values lie in $[0.48,0.83]$. A similar trend is observed for all the parametric effect tests, viz., $cohen's\ d$, $hedges'\ g$, and $glass's\ \Delta$, where all the effect values in the range $[0.02, 0.32]$ and the p-values in $[0.43, 0.68]$, which imply a small effect (as discussed in Section \ref{sec:evaluation-metrics} point \ref{pt:effect-size}). Also, the $cliff's \delta$ values range lies in $[0 \pm 0.17, 0.12 \pm 0.33]$ with large p values $[0.44, 0.65]$, which again depicts a large overlap between the DevSDEE estimates and true effort values. Therefore, with such small t-values and effect values and large p-values, we can conclude that the differences between the estimates are minimal, and thus, we fail to reject the null hypothesis $H_0$. However, to validate our objective with better confidence, we perform additional experiments as discussed in Section \ref{sec:exp-4}.
\subsection{Experiment \#4}
\label{sec:exp-4}
\textbf{Objective:} To validate that DevSDEE provides better effort estimates than the existing methods
\textbf{Description:} To validate our hypothesis, we consider the $MAR$ of the estimates as inputs to the t-test and the effect experiments. While comparing DevSDEE with an existing approach $Z$, a large positive difference (effect or t values) between the $MAR$ values of the $Z$ estimates and the DevSDEE estimates would imply an increased error in $Z$ estimates as compared to DevSDEE's estimates. Similarly, we use the $SA$ measure for comparing the two estimates where we replace the $MAR$ of random guessing approach with the $MAR$ values of $Z$ in equation \ref{eq:sa}. The null hypothesis $H_0$ for this experiment would be defined as \emph{there lies negligible difference between the error ($MAR$) values of the estimates made by $Z$ approach and DevSDEE}, with $Z$ = \{ATLM, NeuralNet, ABE, LOC straw man\}.
\textbf{Procedure:} We perform the t-test, cliff's $\delta$ test, and the parametric effect tests, viz., cohen's $d$, hedges' $g$, glass's $\Delta$ using the $MAR$ values obtained in the experiments \#1 and \#1 ( discussed in Section \ref{sec:exp-1} and Section \ref{sec:exp-2}). Additionally, we also compute the $SA$ measure values using the $MAR$ values as input. We use the researchpy\footnote{\url{https://researchpy.readthedocs.io/en/latest/ttest_documentation.html}} and the DABEST python libraries\footnote{\url{https://github.com/ACCLAB/DABEST-python}} to perform these tests. All the tests are performed with a 95\% confidence interval and by taking 5000 bootstrap samples.
\begin{table}
\centering
\caption{Significance Testing for comparison with the existing methods}
\label{tab:comparison}
\resizebox{\columnwidth}{!}
{
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}
\toprule
\multicolumn{2}{c|}{\multirow{2}{*}{}} & \multirow{2}{*}{SA} & \multicolumn{2}{c|}{t-significance test} & \multicolumn{2}{c|}{Cliff's delta effect test} & \multicolumn{4}{c}{Parametric effect tests}\\
\cline{4-11}
\multicolumn{2}{c|}{Method}&&$t-val$ & $p-val$ & $cliff's\ \delta$ & $p-val$ & $cohen's\ d$ & $hedges'\ g$ & $glass's\ \Delta$ & $p-val$\\
\midrule
\multirow{4}{*}{\rotatebox[origin=c]{90}{\shortstack{Random\\ Trials}}} & ATLM & 42.7 & 2.49 & 0.02 & 0.74 & $<0.00001$ & 1.06 & 1.02 & 1.74 & 0.02 \\
\cline{2-11}
& NeuralNet & 73 & 4.13 & 0.0002 & 0.86 & $<0.00001$ & 1.31 & 1.28 & 0.98 & $<0.00001$ \\
\cline{2-11}
& ABE & 72.66 & 2.58 & 0.01 & 0.57 & 0.0006 & 0.82 & 0.8 & 0.59 & 0.001 \\
\cline{2-11}
& LOC straw man & 84.22 & 2.81 & 0.008 & 0.88 & $<0.00001$ & 0.89 & 0.87 & 0.63 & $<0.00001$ \\
\hline
\multirow{4}{*}{\rotatebox[origin=c]{90}{k-Fold}} & ATLM & 35.13 & 2.46 & 0.02 & 0.74 & $<0.00001$ & 0.78 & 0.76 & 1.87 & 0.01 \\
\cline{2-11}
& NeuralNet & 56.35 & 1.38 & 0.05 & 0.52 & 0.04 & 0.62 & 0.59 & 0.45 & 0.05 \\ \cline{2-11}
& ABE & 74.06 & 2.58 & 0.01 & 0.57 & 0.0006 & 0.82 & 0.8 & 0.59 & 0.001 \\ \cline{2-11}
& LOC straw man & 87.26 & 2.81 & 0.008 & 0.88 & $<0.00001$ & 0.89 & 0.87 & 0.63 & $<0.00001$ \\
\bottomrule
\end{tabular}
}
\end{table}
\textbf{Observations and Analysis:} The results of this experiment are listed in Table \ref{tab:comparison}. As observed from the t-significance test results, the t-values lie in $[1.38, 2.81]$ with p-values $<=0.05$. Therefore, we can state that we can reject $H_0$ with a 95\% confidence level (and more in some cases where $p<0.05$). Hence, this also implies that DevSDEE estimates are considerably closer to the true effort values than the existing methods ($Z$).
A similar trend is observed for parametric effect tests with effect values ranging in $[0.45, 1.74]$ and p-values $<=0.05$, which represents the values ranging from medium to a effect (as stated in Section \ref{sec:evaluation-metrics} point \ref{pt:effect-size}). Thus, the parametric tests also reject $H_0$ with a 95\% confidence level. When analyzing the cliff's $\delta$ effect test results, the effect values range in $[0.52,0.88]$ with p-values $<=0.04$ (mostly $<0.001$). Thus, these effect values provide a 96\% level of confidence to reject $H_0$.
\subsection{Experiment \#5}
\label{sec:exp-5}
\textbf{Objective:} To validate the correlation between Software Development Effort Estimation metrics and software development effort
\textbf{Description:} We performed various experiments to validate our hypothesis that:
\textit{The developer activity information is useful in estimating the software development effort,}
or more specifically,
to validate that
\textit{SDEE metrics capture the developer activity information and show correlation with the overall development effort.}
\textbf{Procedure:} To perform these experiments, we compute the correlation between various SDEE metrics values present in our dataset. We use the Pearson's correlation coefficient \cite{benesty2009pearson}, which is used to measure the linear relationship between two variables. Note that Pearson's correlation coefficient has a range of values as [-1, 1].
To detect the correlation between different SDEE metrics, we computed Pearson's correlation coefficient values for the following SDEE metrics:
\begin{enumerate}
\item[A)] The total number of SLOC modifications ($SLOC_m$) performed while developing a software repository vs. the effort expended in developing it.
\item[B)] The total number of developers (devCount) working on a software repository vs. the effort expended in developing it.
\item[C)] The total time spent developing a software repository vs. the effort expended in developing it.
\end{enumerate}
\begin{figure*}
\centering
\includegraphics[width=.5\columnwidth]{figures/cor-final}
\caption{Correlation between various SDEE metrics}
\label{fig:correlation}
\end{figure*}
\textbf{Results and observations:} Fig. \ref{fig:correlation} presents the correlation values for each of the comparison types listed above. As observed from the figure, development time has the highest correlation value of 0.799, which is explainable as more is the time spent developing a software project, more will be the effort (in person-months). Similarly, devCount shows a correlation value of 0.644 compared with the development effort and is explainable using the same reason for development time. However, $SLOC_m$ shows a small positive correlation value of $0.065$, which signifies that through the effort increases with the increase in $SLOC$ modifications, the increase is much smaller than the increase in devCount and development time. Thus, we can infer that devCount and development time are useful SDEE metrics and influence the software development effort more than the $SLOC_m$, which is the commonly used SDEE metric.
\subsection{Experiment \#6}
\label{sec:exp-6}
\textbf{Objective:} To validate the performance of our SDEE tool by professional programmers and test if it works as intended to work for the user audience.
\textbf{The research question addressed:} \emph{How does the SDEE tool perform from a programmer's perspective when judged on various quality parameters?}
\textbf{Description:} To evaluate our tool's performance, we consider the following quality parameters: the \emph{ease-of-use}, \emph{response-time}, and \emph{accuracy}. We invited professional programmers to perform this experiment and asked them to use our tool and validate its performance using the available information of existing GitHub repositories. One hundred and eleven programmers responded. We had a mixture of candidates from final year computer science undergraduates (53), postgraduates (47), and IT industry professionals (11) with experience between 0 and 6 years. Each of the participants had prior knowledge of the software engineering fundamentals. The experiment was performed in a controlled industrial environment.
\textbf{Procedure:}
\label{sec:exp-4-procedure}
The programmers were requested to perform the following steps:
\begin{enumerate}
\item Enter the details of a known or an available software product using the tool's input interface. The participants could use the available information of existing GitHub repositories to perform this validation and are not required to compute the effort values themselves.
\item Check the estimates provided as the response by the tool for the entered software requirements description by comparing it with the original effort values of the product. Note: We do not expect the participants to compute the effort values on their own. The participants were asked to use the effort values available online and validate that given a similar software description if the tool outputs a similar effort value. For instance, the participants could use the effort information of the existing GitHub repositories.
\item Rate the tool's ease of use, response time, and accuracy, as observed from the experiment(s). The programmers were asked to provide their ratings on a scale of $1$--$10$. Here, $1$ represents the worst performance, and $10$ represents the best.
\end{enumerate}
\textbf{Interpretation of Table \ref{tab:experiment6-ratings}:} Table \ref{tab:experiment6-ratings} gives the details of the ratings received by the tool on various quality parameters as evaluated by the participants. The scores represented in the table are on a scale of $1$--$10$, where $1$ represents the worst performance, and $10$ represents the best performance. Similarly, the votes are represented in the form of percentages. The sum of votes across every column equals $100$ (or \%). The first row of the table shows that a score of $10$ was awarded for \textit{ease-of-use} by $60\%$ voters, for \textit{response time} by $80\%$ voters, and for \textit{accuracy}, it was awarded by $60\%$ voters. Similarly, the third row says that the score of $8/10$ was awarded for all the quality parameters by only $10\%$ voters.
\begin{table*}[ht]
\caption{Ratings recorded in Experiment \#6}
\label{tab:experiment6-ratings}
\centering
\resizebox{0.4\columnwidth}{!}{
\begin{tabular}{c|c|c|c}
\toprule[1pt]
\textbf{Score $Y$}&\multicolumn{3}{c}{\textbf{Votes for the score $Y$ (in \%)}}\\
\cline{2-4}
(scale 1-10) &\emph{ease-of-use}&\emph{response time}&\emph{accuracy}\\
\midrule[1pt]
$10$ & $60$ & $80$ & $60$\\
$9$ & $30$ & $10$ & $30$\\
$8$ & $10$ & $10$ & $10$\\
$7$ & $0$ & $0$ & $0$\\
$6$ & $0$ & $0$ & $0$\\
\bottomrule[1pt]
\end{tabular}
}
\end{table*}
\textbf{Results and observations:}
\label{sec:exp-6-results}
Some of the significant observations from Table \ref{tab:experiment6-ratings} are:
\begin{enumerate}
\item Our tool received a rating of $>=8/10$ from all the participants for ease of use. Thus, most participants found it easy to use (or learn how to use it).
\item Our tool achieves impressive (or minimal) response time as it received a rating of $10/10$ by 80\% of the participants.
\item All the participants found our tool's estimates to be considered accurate as they rated it $>=8/10$ for accuracy.
\end{enumerate}
\section{Threats to validity}
\label{sec:threats-to-validity}
\subsection{Threats to internal validity}
\label{sec:internal-validity}
\begin{enumerate}
\item \emph{Handling the Corner cases:} A \enquote{refactoring} stage may occur in a software's life-cycle, producing a sudden spike in the modification and commits count. We treat such cases as the \enquote{corner} cases for our work.
\item \emph{Removing the outliers:} While developing the SDEE dataset, we performed the following to avoid the outliers:
\begin{enumerate}
\item Selecting OSS repositories having size $>=5$ MBs: We observed that the repositories with size $<5$ MBs were either at the very initial stage of software development or had very few source code files present in them.
\item Selecting the repositories that have been updated at least once in the last three years: To avoid the very old or inactive repositories.
\item Selected the repositories with more than $500$ stars: To fetch repositories with considerable developer involvement.
\item Filtering the repositories belonging to specific categories: To ensure the homogeneity of the dataset, we selected software repositories belonging to different types.
\end{enumerate}
\item \emph{Missed repositories:} Due to this constrained repository selection, we might have missed some relevant unlabeled (by categories) repositories or some relevant repositories with smaller star counts. However our objective of developing a homogeneous dataset (for software types) free from outliers is fulfilled. Nevertheless, as is the case with any experiment, it is impossible to cover all possibilities for the detection and avoidance of the swamping effect \cite{chiang2007masking}.
\item \emph{Validity of the OSS project descriptions:} For developing our software effort estimation method, we determine similar software matches using a PVA model trained on the software product descriptions. Thus, we assume that the project descriptions present in VCS repositories are valid and do not change very often. The OSS software products considered while developing our tool are present in the GitHub OSS repositories and usually serve as software modules in developing more extensive software systems. Therefore, our system can only provide effort estimates to develop such modules of the software. The effort estimates provided corresponding to such modules of software modules can be summed up to obtain the aggregate effort estimate required to develop the more sophisticated software.
\end{enumerate}
\subsection{Threats to external validity}
\label{sec:external-validity}
Our work is based on the developer activity information extracted from various GitHub OSS repositories. Therefore, our experiments' results may differ when tested on proprietary software because industries have full-time working employees \cite{capiluppi2013effort}. Thus the developer activity while developing a proprietary software might differ due to the different working patterns. However, our approach remains valid regardless of whether we used the data from OSS or non-OSS projects. For example, if the data used is taken from non-volunteer-driven projects such as proprietary software development, the approach would remain valid. This is because the similarity in software projects is detected based on the software descriptions and not the developer activity.
\subsection{Threats to statistical conclusion validity}
\label{sec:statistical-validity}
To validate the efficacy of our approach over the existing methods, we perform several statistical experiments as discussed in Section \ref{sec:exp-3} and Section \ref{sec:exp-4}. We report our findings at a 95\% confidence level with a 0.05 significance level. However, for many cases, we also obtained the results with a 99.999\% confidence level ($<0.00001$ significance).
\section{Conclusion and Future Directions}
\label{sec:conclusion}
SDEE for a software project is performed by analyzing the information of past software projects of similar nature. We propose a tool that estimates the effort required to develop a newly-envisioned software project $z$ by analyzing the developer activity information (tracked in VCSs) of software projects with similar functionality as $z$. SDEE metrics that we proposed are used to extract the developer activity information present in various VCS repositories. To determine the functionally similar software projects to $z$, we utilize the project descriptions associated with them.
Our experimental results show that the developer activity metrics serve as a better input to an SDEE method than the existing software metrics, such as SLOC. Further, when trained on software product descriptions, a PVA-based document comparison allows for discovering more relevant software projects from the past whose effort estimates allow for more accurate and reliable SDEE.
For instance, when estimating effort for a compression library ($z$) with \enquote{high compression} and \enquote{efficiency} as characteristics (used as text keywords by PVA for similarity detection), the top-2 similar matches obtained are Google Draco and Centaurean Density libraries. Both these libraries have the same characteristics highlighted in their project descriptions, and match the software requirements of $z$.
Our system achieves the highest standardized accuracy of 87.26\% (cliff's $\delta$=0.88) when compared with the LOC straw man estimator and 42.7\% (cliff's $\delta$=0.74) with the ATLM method at 99\% confidence level ($p<0.00001$). Based on our experimental findings, we conclude that the use of developer activity information of past software helps develop better software estimation methods and tools. We plan to extend our work for providing incremental effort estimates (after one month, two months, and so on) in the future.
Since developer acts as the primary source of source-code contributor, we plan to study novel developer-related factors, such as developer's characteristics \cite{stray2017daily,yang2020developer,li2012leadership}, developer's geographical location \cite{rastogi2018relationship}, social interactions of developers \cite{iden2018social,zhang2017social}, and test their effect on software development.
\bibliographystyle{plainnat}
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2,869,038,156,426 | arxiv |
\section{Introduction}
Time series modeling has a long and important history in science and
engineering. Advances in dynamical systems over the last half century led to
new methods that attempt to account for the inherent nonlinearity in many
natural phenomena \citep{Berg84,Guck83a,Wigg88a,Deva89a,Lieb93a,Ott93a,Stro94a}.
As a result, it is now well known that nonlinear systems produce highly
correlated time series that are not adequately modeled under the typical
statistical assumptions of linearity, independence, and identical distributions.
One consequence, exploited
in novel state-space reconstruction methods \citep{Pack80,Take81,Fras90b}, is
that discovering the hidden structure of such processes is key to successful
modeling and prediction \citep{Crut87a,Casd91a,Spro03a,Kant06a}. In an attempt
to unify the alternative nonlinear modeling approaches, computational mechanics
\cite{Crut88a} introduced a minimal representation---the \eM---for stochastic
dynamical systems that is an optimal predictor and from which many system
properties can be directly calculated. Building on the notion of state
introduced in Ref. \cite{Pack80}, a system's effective states are those
variables that \emph{causally shield} a system's past from its
future---capturing, in the present, information from the past that predicts the
future.
Following these lines, here we investigate the problem of learning predictive
models of time series with particular attention paid to discovering hidden
variables. We do this by using the information bottleneck method (IB)
\citep{IBN} together with a complexity control method discussed by
Ref. \citep{StillBialek2004}, which is necessary for learning from finite data.
Ref. \cite{Shal99a} lays out the relationship between computational mechanics
and the information bottleneck method. Here, we make the mathematical connection
for times series, introducing a new method.
We adapt IB to time series prediction, resulting in a method we call
\emph{optimal causal filtering}
(OCF) \footnote{A more general approach is taken in
Ref. \citep{Still09IAL}, where both predictive modeling and decision making are
considered. The scenario discussed here is a special case.}.
Since OCF, in effect, extends rate-distortion theory \citep{Shannon48} to
use causal shielding, in general it achieves an optimal balance between model
complexity and approximation accuracy. The implications of these trade-offs for
automated theory building are discussed in Ref. \citep{Still07a}.
We show that in the important limit in which prediction is paramount and
model complexity is not restricted, OCF reconstructs the underlying process's
causal architecture, as previously defined within the framework of computational
mechanics \citep{Crut88a,Crut92c,Crut98d}. This shows that, in effect, OCF
captures a source's hidden variables and organization. The result gives
structural meaning to the inferred models. For example, one can calculate
fundamental invariants---such as, symmetries, entropy rate, and stored
information---of the original system.
To handle finite-data fluctuations, OCF is extended to \emph{optimal causal
estimation} (OCE). When probabilities are estimated from finite data, errors
due to statistical fluctuations in probability estimates must be taken
into account in order to avoid over-fitting. We demonstrate how OCF and OCI
work on a number of example stochastic processes with known, nontrivial
correlational structure.
\section{Causal States}
Assume that we are given a stochastic process $\Prob(\BiInfinity)$---a joint
distribution over a bi-infinite sequence $\BiInfinity = \Past \Future$ of
random variables. The \emph{past}, or \emph{history}, is denoted
$\Past = \ldots \MeasSymbol_{-3} \MeasSymbol_{-2} \MeasSymbol_{-1}$, while
$\Future = \MeasSymbol_0 \MeasSymbol_1 \MeasSymbol_2 \ldots$ denotes the
\emph{future} \footnote{To save space and improve readability we use a
simplified notation that refers to infinite sequences of random variables.
The implication, however, is that one works with finite-length sequences into
the past and into the future, whose infinite-length limit is taken at
appropriate points. See, for example, Ref. \citep{Crut98d} or, for
measure-theoretic foundations, Ref. \citep{Ay05a}.}.
Here, the random variables $\MeasSymbol_t$ take on discrete values
$\meassymbol \in \ProcessAlphabet = \{ 1,2,\ldots,k\}$ and the process as a
whole is stationary.
The following assumes the reader is familiar with information theory and
the notation of Ref. \citep{Cove06a}.
Within computational mechanics, a process $\Prob(\BiInfinity)$ is viewed as a
communication channel that transmits information from the past to the future,
storing information in the present---presumably in some internal states,
variables, or degrees of freedom \cite{Crut08a}. One can ask a simple question,
then: how much information does the
past share with the future? A related and more demanding question is how we
can infer a predictive model, given the process. Many authors have considered
such questions. Refs. \citep{Crut01a,Crut98d,Shal99a,bialek06} review
some of the related literature.
The effective, or \emph{causal}, states $\CausalStateSet$ are determined by an
equivalence relation $\past \sim \past^\prime$ that groups all histories
together which give rise to the same prediction of the future
\citep{Crut88a,Crut98d}.
The equivalence relation partitions the space $\AllPasts$ of histories and
is specified by the set-valued function:
\begin{equation}
\epsilon(\past) =
\{ \past^\prime: \Prob(\Future|\past) = \Prob(\Future|\past^\prime) \}
\label{CausalStateDefn}
\end{equation}
that maps from an individual history to the equivalence class
$\causalstate \in \CausalStateSet$ containing that history and all others which
lead to the same prediction $\Prob(\Future|\past)$ of the future. A causal
state $\causalstate$ includes: (i) a label $\causalstate \in \CausalStateSet$;
(ii) a set of histories
\mbox{$\Past_{\causalstate} = \{ \past: \Prob(\Future|\past) =
\Prob(\Future|\causalstate) \} \subset \AllPasts$}; and (iii) a future
conditional distribution $\Prob(\Future|\causalstate)$ given the state
\citep{Crut88a,Crut98d}.
Any alternative model, called a \emph{rival} $\AlternateState$,
gives a probabilistic assignment $\Prob(\AlternateState|\past)$ of histories
to its states $\alternatestate \in \AlternateStateSet$. Due to the data
processing inequality, a model can never capture more information about the
future than shared between past and future:
\begin{equation}
I[\Partition;\Future] \leq I[\Past;\Future] ~,
\label{upperbound}
\end{equation}
where $I[V,W]$ denotes the mutual information between random variables $V$ and
$W$ \citep{Cove06a}. The quantity $\EE = I[\Past;\Future]$ has been studied by
several authors and given different names, such as (in chronological order)
convergence rate of the conditional entropy \citep{Junc79}, excess entropy
\citep{Crut83a}, stored information \cite{Shaw84}, effective measure complexity
\citep{Gras86}, past-future mutual information \citep{Li91}, and predictive
information \citep{BT99}, amongst others. For a review see Ref.
\citep{Crut01a} and references therein.
The causal states $\causalstate \in \CausalStateSet$ are distinguished by the
fact that the function $\epsilon(\cdot)$ gives rise to a {\em deterministic}
assignment of histories to states:
\begin{equation}
\Prob(\causalstate|\past) = \delta_{\causalstate,\epsilon(\past)} ~,
\end{equation}
and, furthermore, by the fact that their future conditional probabilities are given by
\begin{equation}
\Prob(\Future|\causalstate) = \Prob(\Future|\past) ~,
\end{equation}
for all $\past$ such that $\epsilon(\past) = \causalstate$.
As a consequence,
the causal states, considered as a random variable $\CausalState$, capture the
full predictive information
\begin{equation}
I[\CausalState;\Future] = I[\Past;\Future] = \EE~.
\label{CS.prop.1}
\end{equation}
More to the point, they \emph{causally shield} the past and future---the past
and future are independent given the causal state:
$\Prob(\Past,\Future|\CausalState) = \Prob(\Past|\CausalState)
\Prob(\Future|\CausalState)$.
The causal-state partition has, out of all {\em equally} predictive partitions,
called {\em prescient rivals} $\PrescientState$ \cite{Crut10a}, the smallest
entropy, $\Cmu [\Partition] = H [\Partition]$:
\begin{equation}
H[\PrescientState] \geq H[\CausalState] ~,
\label{CS.prop.2}
\end{equation}
known as the \emph{statistical complexity}, $\Cmu := H[\CausalState]$. This is
amount of historical information a process stores: A process communicates $\EE$
bits from the past to the future by storing $\Cmu$ bits in the present. $\Cmu$
is one of a process's key properties; the other is its entropy rate
\citep{Cove06a}. Finally, the causal states are \emph{unique and minimal
sufficient statistics} for prediction of the time series
\citep{Crut88a,Crut98d}.
\section{Constructing Causal Models of Information Sources}
\label{OCFmotivation}
Continuing with the communication channel analogy above, models, optimal or not,
can be broadly considered to be a lossy compression of the original data. A
model captures some regularity while making some errors in describing the data.
Rate distortion theory \citep{Shannon48} gives a principled method to find a
lossy compression of an information source such that the resulting model is as
faithful as possible to the original data, quantified by a \emph{distortion
function}.
The specific form of the distortion function determines what is considered to
be ``relevant''---kept in the compressed representation---and what is
``irrelevant''---can be discarded. Since there is no universal distortion
function, it has to be assumed \emph{ad hoc} for each application. The
information bottleneck method \citep{IBN} argues for explicitly keeping the
relevant information, defined as the mutual information that the data share
with a desired relevant variable \citep{IBN}. With those choices, the
distortion function can be derived from the optimization principle, but the
relevant variable has to be specified \emph{a priori}.
In time series modeling, however, there is a natural notion of relevance: the
future data.
For stationary time series, moreover, building a model with low generalization
error is equivalent to
constructing a model that accurately predicts future data from past data.
These observations lead directly to an information-theoretic specification
for reconstructing time series models: First, introduce general model
variables $\AlternateState$ that can store, in the
present moment, the information transmitted from the past to the future.
Any set of such variables specifies a stochastic partition of $\AllPasts$
via a probabilistic assignment rule $\Prob(\AlternateState|\past)$. Second,
require that this partition be maximally predictive. That is, it should
maximize the information $I[\AlternateState;\Future]$ that the variables
$\AlternateState$ contain about the future $\Future$. Third, the
so-constructed representation of the historical data should be a summary, i.e.,
it should not contain all of the historical information, but rather, as little
as possible while still capturing the predictive information. The information
kept about the past---$I[\Past;\AlternateState]$, the
\emph{coding rate}---measures
the model complexity or bit cost. Intuitively, one wants to find the
most predictive model at fixed complexity or, vice versa, the least complex
model at fixed prediction accuracy. These criteria are equivalent,
in effect, to causal shielding.
Writing this intuition formally reduces to the information bottleneck method,
where the relevant information is information about the future. The constrained
optimization problem one has to solve is:
\begin{equation}
\max_{\Prob(\Partition|\Past)}
\left\{ I[\Partition;\Future] - \lambda I[\Past;\Partition] \right\} ~,
\label{OCF}
\end{equation}
where the parameter $\lambda$ controls the balance between prediction and model
complexity. The linear trade-off that $\lambda$ represents is an ad hoc
assumption \cite{Shal99a}. Its justification is greatly strengthened in the
following by the rigorous results showing it leads to the causal states and
the successful quantitative applications.
The optimization problem of Eq. (\ref{OCF}) is solved subject to the
normalization constraint:
$\sum_\AlternateState \Prob(\AlternateState|\past) = 1$, for all
$\past \in \AllPasts$. It then has a family of solutions \citep{IBN},
parametrized by the Lagrange multiplier $\lambda$, that gives the following
optimal assignments of histories $\past$ to states
$\alternatestate \in \Partition$:
\begin{equation}
\Prob_{\mathrm{opt}}(\partitionstate|\past)
= \frac{\Prob(\partitionstate)}{Z(\past,\lambda)}
\exp{
\left( -\frac{1}{\lambda}
\InfoGain{\Prob(\Future|\past)}{\Prob(\Future|\partitionstate)}
\right) ,
}
\label{OCF_States}
\end{equation}
with
\begin{eqnarray}
\Prob(\Future|\partitionstate) & = & \frac{1}{\Prob(\partitionstate)}
\sum_{\past \in \AllPasts} \Prob(\Future|\past)
\Prob(\partitionstate|\past) \Prob(\past) ~\mathrm{and}\\
\Prob(\partitionstate) & = &
\sum_{\past \in \AllPasts} \Prob(\partitionstate|\past) \Prob(\past) ~,
\label{OCF_States_2}
\end{eqnarray}
where $\InfoGain{P}{Q}$ is the \emph{information gain} \citep{Cove06a} between
distributions $P$ and $Q$.
In the solution it plays the role of an ``energy'', effectively measuring
how different the predicted and true futures are. The more distinct, the more
information one gains about the probabilistic development of the future from
the past. That is, high energy models make predictions that deviate
substantially from the process.
These self-consistent equations are solved iteratively \citep{IBN} using a
procedure similar to the Blahut-Arimoto algorithm \citep{Arimoto72, Blahut72}.
A connection to statistical mechanics is often drawn, and the parameter
$\lambda$ is identified with a (pseudo) temperature that controls the level of
randomness; see, e.g., Ref. \citep{Rose90}. This is useful to guide intuition
and, for example, has inspired \emph{deterministic annealing}
\citep{DetermAnneal}.
We are now ready for the first observation.
\begin{Prop}
In the \emph{low-temperature regime} ($\lambda \rightarrow 0$) the
assignments of pasts to states become deterministic and are given by:
\begin{eqnarray}
\Prob_{\mathrm{opt}} (\partitionstate|\past) & = &
\delta_{\partitionstate,\eta(\past)} ~, ~\mathrm{where}\\
\eta(\past) & = & {\rm arg}\min_\partitionstate
\InfoGain{\Prob(\Future|\past)}{\Prob(\Future|\partitionstate)} ~.
\label{hardassign}
\end{eqnarray}
\label{Prop:LowTempDeterministic}
\end{Prop}
\begin{ProProp}
Define the quantity
\begin{align}
D(\partitionstate) = &
\InfoGain{\Prob(\Future|\past)}{\Prob(\Future|\partitionstate)}
\nonumber \\
& - \InfoGain{\Prob(\Future|\past)}{\Prob(\Future|\eta(\past))}
~.
\end{align}
$D(\partitionstate)$ is positive, by definition Eq. (\ref{hardassign}) of
$\eta(\past)$. Now, write
\begin{equation}
\Prob_{\mathrm{opt}} (\eta(\past)|\past) =
\left(
1 + \sum_{\partitionstate \neq \eta(\past)}
\frac{\Prob(\partitionstate)}{\Prob(\eta(\past))}
\exp{\left[ - \frac{D(\partitionstate)}{\lambda} \right]
} \right)^{-1} .
\end{equation}
The sum in the r.h.s. tends to zero, as $\lambda \rightarrow 0$, assuming that
$\Prob(\eta(\past)) > 0$. Via normalization, the assignments become
deterministic.
\qed
\end{ProProp}
\section{Optimal Causal Filtering}
\label{core_results}
We now establish the procedure's
fundamental properties by connecting the solutions it determines to the
causal representations defined previously within the framework of computational mechanics. The resulting
procedure transforms the original data to a causal representation
and so we call it \emph{optimal causal filtering} (OCF).
Note first that for deterministic assignments we have
$H[\Partition|\Past] = 0$. Therefore, the information about the past becomes
$I[\Past;\Partition] = H[\Partition]$ and the objective function simplifies to
\begin{equation}
\Fdet [\Partition] = I[\Partition;\Future] - \lambda H[\Partition] ~.
\label{OF.det}
\end{equation}
\begin{Lem}
Within the subspace of prescient rivals,
the causal-state partition maximizes $\Fdet [\widehat{\Partition}]$.
\end{Lem}
\begin{ProLem}
This follows immediately from Eqs. (\ref{CS.prop.1}) and (\ref{CS.prop.2}).
They imply that
\begin{eqnarray}
\Fdet [\widehat{\Partition}] &=& I[\CausalState;\Future] - \lambda H[\widehat{\Partition}] \nonumber \\
&\leq& I[\CausalState;\Future] - \lambda H [\CausalState] \nonumber \\
&=& \Fdet [\CausalState] ~.
\end{eqnarray}
\qed
\end{ProLem}
The causal-state partition is the model with the largest value of the OCF
objective function, because it is fully predictive at minimum complexity.
We also know from Prop. \ref{Prop:LowTempDeterministic} that in the
low-temperature limit ($\lambda \rightarrow 0$) OCF recovers a
\emph{deterministic} mapping of histories to states. We now show that this
mapping is exactly the causal-state partition of histories.
\begin{The}
OCF finds the causal-state partition of $\AllPasts$ in the low-temperature
limit, $\lambda \rightarrow 0$.
\end{The}
\begin{ProThe}
The causal-state partition, Eq. (\ref{CausalStateDefn}), always exists, and implies that there are groups of histories with
\begin{equation}
\Prob(\Future|\past) = \Prob(\Future|\epsilon(\past)) ~.
\end{equation}
We then have, for all $\past \in \Past$,
\begin{equation}
\InfoGain{\Prob(\Future|\past)}{\Prob(\Future|\epsilon(\past)} = 0 ~,
\end{equation}
and, hence,
\begin{equation}
\epsilon(\past) = {\rm arg}\min_\partitionstate
\InfoGain{\Prob(\Future|\past)}{\Prob(\Future|\partitionstate)} ~.
\end{equation}
Therefore, we can identify $\epsilon(\past) = \eta(\past)$ in
Eq. (\ref{hardassign}), and so the assignment of histories to the causal states
is recovered by OCF:
\begin{equation}
\Prob_{\rm opt}(\partitionstate|\past) = \delta_{\partitionstate, \epsilon(\past)} ~.
\end{equation}
\qed
\end{ProThe}
Note that we have not restricted the size of the set $\AlternateStateSet$ of
model states. Recall also that the causal-state partition is \emph{unique}
\citep{Crut98d}. The Lemma establishes that OCF does \emph{not} find prescient
rivals in the low-temperature limit. The prescient rivals are suboptimal in
the particular sense that they have smaller values of the objective function.
We now establish that this difference is controlled by the model size with
proportionality constant $\lambda$.
\begin{Cor}
Prescient rivals are suboptimal in OCF. The value of the objective function
evaluated for a prescient rival is smaller than that evaluated for the
causal-state model. The difference
$\Delta \Fdet [\PrescientState] = \Fdet [\CausalState] - \Fdet[\PrescientState]$
is given by:
\begin{equation}
\Delta \Fdet [\PrescientState] =
\lambda \left( \Cmu [\PrescientState] - \Cmu [\CausalState] \right) \geq 0 ~.
\end{equation}
\end{Cor}
\begin{ProCor}
\begin{align}
\Delta \Fdet [\PrescientState] &= \Fdet [\CausalState] - \Fdet [\PrescientState] \\
&= I[\CausalState;\Future] - I[\PrescientState;\Future]
- \lambda H[\CausalState] + \lambda H[\PrescientState] \\
&= \lambda \left( \Cmu [\PrescientState] - \Cmu [\CausalState] \right) ~.
\end{align}
Moreover, Eq. (\ref{CS.prop.2}) implies that $\Delta \Fdet \geq 0$.
\qed
\end{ProCor}
So, we see that for $\lambda = 0$, causal states and all other prescient rival
partitions are degenerate. This is to be expected as at $\lambda = 0$ the
model-complexity constraint disappears. Importantly, this means that maximizing
the predictive information alone, without the appropriate constraint on model
complexity does not suffice to recover the causal-state partition.
\section{Examples}
\label{examples}
We study how OCF works on a series of example stochastic processes of
increasing statistical sophistication. We compute the optimal solutions and
visualize the trade-off between predictive power and complexity of the model
by tracing out a curve similar to a rate-distortion curve
\citep{Arimoto72, Blahut72}: For each value of $\lambda$, we evaluate both the
model's coding rate $I[\Past;\AlternateState]$ and its predicted information
$I[\AlternateState;\Future]$ at the optimal solution and plot them against each
other. The resulting curve in the \emph{information plane} \citep{IBN}
separates the
feasible from the infeasible region: It is possible to find a model that is
more complex at the same prediction error, but not possible to
find a less complex model than that given by the optimum. In analogy to a
rate-distortion curve, we can read off the maximum amount of information about
the future that can be captured with a model of fixed complexity. Or,
conversely, we can read off the smallest representation at fixed
predictive power.
The examples in this and the following sections are calculated by solving the
self-consistent Eqs. (\ref{OCF_States}) to (\ref{OCF_States_2})
iteratively \footnote{The algorithm follows that used in the information
bottleneck \citep{IBN}. The convergence arguments there apply to the OCF
algorithm.}
at each value of $\lambda$. To trace out the curves, a deterministic annealing
\citep{DetermAnneal} scheme is implemented, lowering $\lambda$ by a fixed
annealing rate. Smaller rates cost more computational time, but allow one to
compute the rate-distortion curve in greater detail, while larger rates result
in a rate-distortion curve that gets evaluated in fewer places and hence looks
coarser. In examples, naturally, one can only work with finite length past and
future sequences: $\finpast{K}$ and $\finfuture{L}$, where $K$ and $L$ give
their lengths, respectively.
\subsection{Periodic limit cycle: A predictable process}
\begin{figure*}
\begin{center}
\resizebox{!}{2.50in}{\includegraphics{period4_miplane_theory_K3_L2}}
\end{center}
\caption{Model predictability $I[\AlternateState;\FinFuture{L}]$ versus
model complexity (size)
$I[\FinPast{K};\AlternateState]$ trade-off under OCF for the exactly
predictable period-$4$ process: $(0011)^{\infty}$. Monitored in the
information plane. The horizontal dashed line is the full predictive
information ($\EE = I[\FinPast{3};\FinFuture{2}] = 2$ bits) and the vertical
dashed line is the block entropy ($H[\FinPast{3}] = 2$ bits),
which is also the statistical complexity $\Cmu$.
The data points represent solutions at various $\lambda$.
Lines connect them to help guide the eye only.
Histories of length $K = 3$ were used, along with futures of length $L = 2$.
In this and the following information plane plots, the integer labels
$N_c$ indicate the first point at which the effective number of
states used by the model equals $N_c$.
}
\label{fig:Period4.MI}
\end{figure*}
\begin{figure*}
\begin{center}
\resizebox{!}{2.50in}{\includegraphics{period4_morphs_theory_K3_L2_N2}}
\end{center}
\caption{Morphs $\Prob(\FinFuture{2}|\cdot)$ for the period-$4$ process: The
$2$-state approximation (circles) compared to the $\delta$-function morphs
for the $4$ causal states (boxes).
The morphs $\Prob(\FinFuture{2}|\causalstate)$ for the two-state approximation
are $(1/2,0,0,1/2)$ and $(0,1/2,1/2,0)$ and for the four-state
case $(1,0,0,0)$, $(0,1,0,0)$, $(0,0,1,0)$, and $(0,0,0,1)$.
Histories of length $K = 3$ were used, along with futures of length
$L = 2$ (crosses).
}
\label{fig:Period4.morphs}
\end{figure*}
We start with an example of an exactly periodic process, a limit cycle
oscillation. It falls in the class of deterministic and time reversible
processes, for which the rate-distortion curve can be computed
analytically---it lies on the diagonal \citep{Still07a}.
We demonstrate this with a numerical example. Figure \ref{fig:Period4.MI}
shows how OCF works on a period-four process: $(0011)^{\infty}$.
(See Figs. \ref{fig:Period4.MI} and \ref{fig:Period4.morphs}.) There are
exactly two bits of predictive information $I[\Past;\Future]$ to be
captured about future words of length two
(dotted horizontal line). This information describes the phase of the
period-four cycle. To capture those two bits, one needs exactly four
underlying causal states and a model complexity of $\Cmu = 2$ bits
(dotted vertical line).
The curve is the analog of a rate-distortion curve, except that the
information plane swaps the horizontal and vertical axes---the coding rate
and distortion axes. (See Ref. \citep{Still07a} for the direct use of the
rate-distortion curve.) The value of $I[\AlternateState;\FinFuture{2}]$
(the ``distortion''), evaluated at the optimal distribution,
Eq. (\ref{OCF_States}), is plotted versus $I[\FinPast{3};\AlternateState]$
(the ``code rate''),
also evaluated at the optimum. Those are plotted for different values of
$\lambda$ and, to trace out the curve, deterministic annealing is implemented.
At large $\lambda$, we are in the lower left of the curve---the compression is
extreme, but no predictive information is captured. A single state model, a
fair coin, is found as expected. As $\lambda$ decreases (moving to the right),
the next distinct point on the curve is for a two-state model, which discards
half of the information. This comes exactly at the cost of one predictive bit.
Finally, OCF finds a four-state model that captures all of the predictive
information at no compression. The numbers next to the curve indicate the
first time that the effective number of states increases to that value.
The four-state model captures the two bits of predictive information. But
compressed to one bit (using two states), one can only capture one bit of
predictive information. The information curve falls onto the diagonal---a
straight line that is the worst case for possible beneficial trade-offs
between prediction error and model complexity \citep{Still07a}.
In Fig. \ref{fig:Period4.morphs}, we show the best two-state model compared to
the full (exact) four-state model. One of the future conditional probabilities
captures zero probability events of ``odd'' $\{01,10\}$ words, assigning equal
probability to the ``even'' $\{00,11\}$ words. The other one captures zero
probability events of even words, assigning equal probability to the odd words.
This captures the fundamental determinism of the process: an odd word never
follows an even word and vice versa. The overall result illustrates how the
actual long-range correlation in the completely predictable period-$4$
sequence is represented by a smaller \emph{stochastic} model. While in the
four-state model the future conditional probabilities are $\delta$-functions,
in the two-state approximate model they are mixtures of those
$\delta$-functions. In this way, OCF converts structure to randomness
when approximating underlying states with a compressed model; cf.
the analogous trade-off discussed in Ref. \citep{Crut01a}.
\begin{figure*}[ht]
\centering
\resizebox{!}{2.50in}{\includegraphics{goldenmean_miplane_theory_K3_L2}}
\caption{OCF's behavior monitored in the information
plane---$I[\AlternateState;\FinFuture{2}]$ versus
$I[\FinPast{3};\AlternateState]$---for the Golden Mean Process.
The correct two-state model is found.
Histories of length $K = 3$ were used, along with futures of
length $L = 2$. The
horizontal dashed line is the full predictive information
$\EE \approx I[\FinPast{3};\FinFuture{2}] = I[\CausalState;\FinFuture{2}] \approx 0.25$
bits which, as seen, is an upper bound on
$I[\AlternateState;\FinFuture{2}]$.
The exact value is $\EE = I[\Past;\Future] = 0.2516$ bits \cite{Crut08b}.
Similarly, the vertical dashed line is
the block entropy $H[\FinPast{3}] \approx 2.25$ bits which is an upper
bound on the retrodictive information $I[\FinPast{3};\AlternateState]$.
The statistical complexity $\Cmu \approx 0.92$ bits, also an
upper bound, is labeled.
The annealing rate was $0.952$.
}
\label{fig:OCFGMPInfoPlane}
\end{figure*}
\begin{figure*}[ht]
\centering
\resizebox{!}{2.50in}{\includegraphics{goldenmean_morphs_theory_K3_L2}}
\caption{Future conditional probabilities $\Prob(\FinFuture{2}|\cdot)$
conditioned on causal states $\causalstate \in \CausalStateSet$ (boxes) and
on the OCF reconstructed states $\alternatestate \in \AlternateStateSet$
(circles) for the Golden Mean Process. As an input to OCF, future conditional
probabilities $P(\FinFuture{2}|\finpast{3})$ calculated from histories of
length $K = 3$ were used (crosses).}
\label{fig:OCFGMPMorphs}
\end{figure*}
\subsection{Golden Mean Process: A Markov chain}
The Golden Mean (GM) Process is a Markov chain of order one. As an information
source, it produces all binary strings
with the restriction that there are never consecutive $0$s. The GM Process
generates $0$s and $1$s with equal probability, except that once a $0$ is
generated, a $1$ is always generated next. One can write down a simple
two-state Markov chain for this process; see, e.g., Ref. \citep{Crut01a}.
Figures \ref{fig:OCFGMPInfoPlane} and \ref{fig:OCFGMPMorphs}
demonstrate how OCF reconstructs the states of the GM process.
Figure \ref{fig:OCFGMPInfoPlane} shows the behavior of OCF in the
information plane. At very high temperature
($\lambda \rightarrow \infty$, lower left corner of the curve) compression
dominates over prediction and the resulting model is most compact, with only
one effective causal state. However, it contains no information
about the future and so is a poor predictor. As $\lambda$
decreases (moving right), OCF
reconstructs increasingly more predictive and more complex models. The curve
shows that the information about the future, contained in the optimal
partition, increases (along the vertical axis) as the model increases in
complexity (along the horizontal axis). There is a transition to two effective
states: the number $2$ along the curve denotes the first occurrence
of this increase.
As $\lambda \rightarrow 0$, prediction comes to dominate and OCF finds a fully
predictive model, albeit one with the minimal statistical complexity, out of
all possible state partitions that would retain the full predictive information.
The model's complexity---$\Cmu \approx 0.92$ bits---is 41\% of the maximum,
which is given by the entropy of all possible pasts of length $3$:
$H[\FinPast{3}] \approx 2.25$ bits. The remainder (59\%) of the information is
nonpredictive and has been filtered out by
OCF. Figure \ref{fig:OCFGMPMorphs} shows the future conditional probabilities,
associated with the partition found by OCF, as $\lambda \rightarrow 0$,
corresponding to $\Prob( \FinFuture{2} | \alternatestate )$ (circles). These
future conditional probabilities overlap with the true (but not known to the
algorithm) causal-state future conditional probabilities
$\Prob(\FinFuture{2}|\causalstate)$ (boxes) and so
demonstrate that OCF finds the causal-state partition.
\subsection{Even Process: A hidden Markov chain}
\label{sec:EvenProcess}
Now, consider a hidden Markov process: the {\em Even Process} \citep{Crut01a},
which is a stochastic process whose support (the set of allowed sequences)
is a symbolic dynamical system called the \emph{Even system}.
The Even system generates all binary strings consisting
of blocks of an even number of $1$s bounded by $0$s. Having observed a
process's sequences, we say that a word (finite sequence of symbols)
is \emph{forbidden} if it never occurs. A word is an \emph{irreducible
forbidden word} if it contains no proper subwords which are themselves
forbidden words. A system is \emph{sofic} if its list of irreducible
forbidden words is infinite. The Even system is one such sofic system, since
its set $\mathcal{F}$ of irreducible forbidden words is infinite:
$\mathcal{F} = \{ 01^{2n+1}0, n = 0 , 1, \ldots \}$. Note that no finite-order
Markovian source can generate this or, for that matter, any other strictly
sofic system \citep{Crut01a}. The Even Process then associates probabilities
with each of the Even system's sequences by choosing a $0$ or $1$ with
fair probability after generating either a $0$ or a pair of $1$s. The
result is a \emph{measure sofic process}---a distribution over a
sofic system's sequences.
\begin{figure*}[ht]
\centering
\resizebox{!}{2.50in}{\includegraphics{even_h3_f2_theory_c3_miplane_a1_08_p0_001}}
\caption{OCF's behavior inferring the Even Process: monitored in the
information plane---$I[\AlternateState;\FinFuture{2}]$ versus
\mbox{$I[\FinPast{3};\AlternateState]$}.
Histories of length $K = 3$ were used, along with futures of
length $L = 2$. The horizontal dashed line is the full predictive information
$I[\FinPast{3};\FinFuture{2}] \approx 0.292$ bits which, as seen, is an
upper bound on the estimates $I[\AlternateState;\FinFuture{2}]$. Similarly,
the vertical dashed line is the block entropy $H[\FinPast{3}] \approx 2.585$
bits which is an upper bound on the retrodictive information
$I[\FinPast{3};\AlternateState]$.
}
\label{fig:OCFExampleInfoPlane}
\end{figure*}
\begin{figure*}[ht]
\centering
\resizebox{!}{2.50in}{\includegraphics{even_morphs_theory_K3_L2}}
\caption{Future future conditional probabilities $\Prob(\FinFuture{2}|\cdot)$
conditioned on causal states $\causalstate \in \CausalState$ (boxes) and on
the OCF-reconstructed states $\alternatestate \in \AlternateState$ (circles)
for the Even Process. As an input to OCF, future conditional probabilities
$P(\FinFuture{2}|\finpast{3})$ calculated from histories of length $K = 3$
were used (crosses).}
\label{fig:OCFExampleMorphs}
\end{figure*}
As in the previous example, for large $\lambda$, OCF applied to the Even
Process recovers a small, one-state model with poor predictive quality;
see Fig. \ref{fig:OCFExampleInfoPlane}. As $\lambda$ decreases there are
transitions to larger models that capture increasingly more information about
the future. (The numbers along the curve again indicate the points
of first transition to more
states.) With a three-state model OCF captures the full predictive information
at a model size of 56\% of the maximum. This model is exactly the causal-state
partition, as can be seen in Fig. \ref{fig:OCFExampleMorphs} by comparing the
future conditional probabilities of the OCF model (circles) to the true
underlying causal states (boxes), which are not known to the algorithm.
The correct \eM\ model of the Even Process has four causal states: two transient
and two recurrent. At the finite past and future lengths used here, OCF picks up
only one of the transient states and the two recurrent states. It also assigns
probability to all three. This increases the effective state entropy
($H[\AlternateState] \approx 1.48$ bits) above the statistical complexity
($\Cmu = 0.92$ bits) which is only a function of the two recurrent states,
since asymptotically ($K \rightarrow \infty$) the transient states have zero
probability.
There is an important lesson in this example for general time-series modeling,
not just OCF. Correct inference of even finite-state, but measure-sofic
processes requires using hidden Markov models. Related consequences of this,
and one resolution, are discussed at some length for estimating ``nonhidden''
Markov models of sofic processes in Ref. \cite{Stre07a}.
\subsection{Random Random XOR: A structurally complex process}
The previous examples demonstrated our main theoretical result: In the
limit in which it becomes crucial to make the prediction error very small,
at the expense of the model size, the OCF algorithm captures all of the
structure inherent in the process by recovering the causal-state partition.
However, if we allow (or prefer) a model with some finite prediction error,
then we can make the model substantially smaller. We have already seen what
happens in the worst case scenario, for a periodic process. There, each
predictive bit costs exactly one bit in terms of model size. However, for
highly structured processes, there exist situations in which one can compress
the model substantially at essentially no loss in terms of predictive power.
(This is called \emph{causal compressibility} \citep{Still07a}.) The Even
Process is an example of such an information source: The statistical complexity
$H[\CausalState]$ of the causal-state partition is smaller than the total
available historical information---the entropy of the past
$H[\FinPast{K}]$.
Now, we study a process that requires keeping \emph{all} of the historical
information to be maximally predictive, which is the same as stating
$\Cmu(\AlternateState) = H[\FinPast{K}]$. (Precisely, we mean given the finite
past and future
lengths we use.) Nonetheless, there is a systematic ordering
of models of different size and different predictive power given by the
rate-distortion curve, as we change the parameter $\lambda$ that controls how
much of the future fluctuations the model considers to be random; i.e., which
fluctuations are considered indistinguishable. Naturally, the trade-off, and
therefore the shape of the rate-distortion curve, depends on and reflects the
source's organization.
\begin{figure*}
\begin{center}
\resizebox{!}{2.50in}{\includegraphics{rrxor_miplane_theory_K3_L2}}
\end{center}
\caption{Prediction versus structure trade-off under OCF for the random-random
XOR (RRXOR) process, as monitored in the information plane. As above, the
horizontal dashed line is the predictive information ($\approx 0.230$ bits)
and the vertical dashed line is the block entropy ($\approx 2.981$ bits).
Histories of length $K = 3$ were used, along with futures of length $L = 2$.
The asterisk and lines correspond to the text: they serve to show how the
predictive power and the complexity of the best four state model, the future
conditional probabilities of which are depicted in
Fig. \ref{fig:RRXORMorphs4}.
}
\label{fig:RRXORMInfoPlane}
\end{figure*}
\begin{figure*}
\begin{center}
\resizebox{!}{2.50in}{\includegraphics{rrxor_morphs_theory_K3_L2}}
\end{center}
\caption{Future conditional probabilities $\Prob(\FinFuture{2}|\cdot)$ for the
RRXOR process: the $8$-state approximation (circles) finds the causal states
(boxes). For example, the heavier dashed line (purple) shows
$\Prob(\FinFuture{2}|\alternatestate) = (1/4,1/2,1/4,0)$.
Histories of length $K = 3$ were used, along with futures of length $L = 2$.
}
\label{fig:RRXORMorphsAll}
\end{figure*}
\begin{figure*}
\begin{center}
\resizebox{!}{2.50in}{\includegraphics{rrxor_morphs_theory_K3_L2_N4Fixed}}
\end{center}
\caption{Morphs $\Prob(\FinFuture{2}|\cdot)$ for the RRXOR process: the
$4$-state approximation (circles and colored lines: state 1 - cyan/full,
2 - green/full, 3 - blue/dashed, 4 - purple/dashed) compared to causal states
(boxes). Histories of length $K = 3$ were used, along with
futures of length $L = 2$.
}
\label{fig:RRXORMorphs4}
\end{figure*}
As an example, consider the random-random XOR (RRXOR) process which consists of two successive random
symbols chosen to be $0$ or $1$ with equal probability and a third symbol
that is the logical Exclusive-OR (XOR) of the two previous. The RRXOR process
can be represented by a hidden Markov chain with five recurrent causal states,
but having a very large total number of causal states. There are $36$ causal
states, most ($31$) of which describe a complicated transient structure
\citep{Crut01a}. As such, it is a structurally complex process that an
analyst may wish to approximate with a smaller set of states.
Figure \ref{fig:RRXORMInfoPlane} shows the information plane, which specifies
how OCF trades off structure for prediction error as a function of model
complexity for the RRXOR process. The number of effective states (again first
occurrences are denoted by integers along the curve) increases with model
complexity. At a history length of $K = 3$ and future length of $L = 2$, the
process has eight underlying causal states, which are found by
OCF in the $\lambda \rightarrow 0$ limit. The corresponding future conditional
probability distributions are shown in Fig. \ref{fig:RRXORMorphsAll}.
The RRXOR process has a structure that does not allow for substantial
compression. Fig. \ref{fig:RRXORMInfoPlane} shows that the effective statistical
complexity of the causal-state partition is equal to the full entropy of the
past: $\Cmu (\AlternateState) = H[\FinPast{3}]$. So, at $L = 3$, unlike the
Even and Golden Mean Processes, the RRXOR process is not compressible.
With half (4) of the number of states, however, OCF reconstructs a
model that is only 33\% as large, while capturing 50\% of the information
about the future. The corresponding conditional future probabilities of the
(best) four-state model are shown in Fig. \ref{fig:RRXORMorphs4}.
They are mixtures of pairs of the eight causal states.
The rate-distortion curve informs the modeler about the (best possible)
efficiency of predictive power to model complexity:
$I[\Partition;\Future] / I[\Past;\Partition]$. This
is useful, for example, if there are constraints on the maximum model size
or, vice versa, on the minimum prediction error. For example, if we require a
model of RRXOR to be 90\% informative about the future, then we can read
off the curve that this can be achieved at 70\% of the model complexity.
Generally, as $\lambda$ decreases, phase transitions occur to models with a
larger number of effective states \citep{DetermAnneal}.
\section{Optimal Causal Estimation: Finite-data fluctuations}
In real world applications, we do not know a process's underlying
probability density, but instead must estimate it from a \emph{finite}
time series that we are given. Let that time series be of length $T$ and let us
estimate the joint distribution of pasts (of length $K$) and futures
(of length $L$) via a histogram calculated using a sliding window.
Altogether we have $M = T - ( K + L -1)$ observations.
The resulting estimate $\widehat{\Prob}(\FinPast{K}; \FinFuture{L})$ will
deviate from the true $\Prob(\FinPast{K}; \FinFuture{L})$ by
\mbox{$\Delta(\FinPast{K}, \FinFuture{L})$}. This leads to an overestimate
of the mutual information \footnote{All quantities denoted with a
$\widehat{\cdot}$ are evaluated at the estimate $\widehat{\Prob}$.}:
\mbox{ $\widehat{I}[\FinPast{K};\FinFuture{L}] \geq
I[\FinPast{K};\FinFuture{L}]$}.
Evaluating the objective function at this estimate may lead one to capture
variations that are due to the sampling noise and not to the process's
underlying structure; i.e., OCF may over-fit. That is, the underlying process
may appear to have a larger number $N_c$ of causal states than the true number.
Following Ref. \citep{StillBialek2004}, we argue that this effect can be
counteracted by subtracting from $\widehat{F}[\AlternateState]$ a
model-complexity control term that approximates the error we make by
calculating the estimate $\widehat{F}[\AlternateState]$ rather than the
true $F[\AlternateState]$. If we are willing to assume that $M$ is large
enough, so that the deviation $\Delta(\FinPast{K}, \FinFuture{L})$ is
a small perturbation, then the error can be approximated by
\citep[Eq. (5.8)]{StillBialek2004}:
\begin{equation}
{\cal E} (N_c) = \frac{k^L - 1}{2 \ln(2)} \frac{N_c}{M} ~,
\label{finsizeerror}
\end{equation}
in the low-temperature regime, $\lambda \rightarrow 0$. Recall that $k^L$ is
the total number of possible futures for alphabet size $k$. The optimal number
$N_c^*$ of hidden states is then the one for which the largest amount of mutual
information is shared with the future, corrected by this error:
\begin{equation}
N_c^* :=
{\rm arg}\max_{N_c} ~ \widehat{I}[\FinPast{K};\FinFuture{L}]_{\lambda \rightarrow 0}^{\rm corrected} (N_c) ~,
\end{equation}
with
\begin{equation}
\widehat{I}[\FinPast{K};\FinFuture{L}]_{\lambda \rightarrow 0}^{\rm corrected} (N_c)
= \widehat{I}[\FinPast{K};\FinFuture{L}]_{\lambda \rightarrow 0} (N_c)
- {\cal E} (N_c) ~.
\end{equation}
This correction generalizes OCF to \emph{optimal causal estimation} (OCE), a
procedure that simultaneously accounts for the trade-off between structure,
approximation, and sample fluctuations.
\begin{figure*}[ht]
\centering
\resizebox{!}{2.50in}{\includegraphics{goldenmean_IvsN_K3_L2_M100}}
\caption{Information $I$ captured about the future versus the number $N_c$ of
reconstructed states, with statistics estimated from length $T = 100$ time
series sample from the Golden Mean Process. Upper line: plotted on the
vertical axis is
\mbox{$\widehat{I}[\AlternateState;\FinFuture{2}]_{\lambda \rightarrow 0}$}
(not corrected); lower line: plotted on the vertical axis is the quantity
\mbox{$\widehat{I}[\AlternateState;\FinFuture{2}]_{\lambda \rightarrow 0}^{\rm corrected}$},
which is the retained predictive information, but corrected for estimation
errors due to finite sample size. The dashed line indicates the actual upper
bound on the predictive information $I[\FinPast{K};\AlternateState]$, for
comparison. This value is not known to the algorithm, it is computed from
the true process statistics. Histories of length $K = 3$ and futures of
length $L = 2$ were used. The asterisk denotes the optimal number
($N_c = 2$) of effective states.}
\label{fig:OCEGMPInfoPlane}
\end{figure*}
\begin{figure*}[ht]
\centering
\resizebox{!}{2.50in}{\includegraphics{GoldenMean_20090420_H3_F2_L100_N2}}
\caption{OCE's best two-state approximated future conditional probabilities
(circles) for the Golden Mean Process. Compared to true (unknown) future
conditional probabilities (squares). The OCE inputs are the estimates of
$\widehat{\Prob}(\FinFuture{2}|\finpast{3})$ (crosses).
}
\label{fig:OCEGMPMorphs}
\end{figure*}
\begin{figure*}[ht]
\centering
\resizebox{!}{2.50in}{\includegraphics{even_f2_h3_d100_s200_IvsN}}
\caption{Information $I$ captured about the future versus the number $N_c$ of
reconstructed states, with statistics estimated from length $T = 100$ time
series sample from the Even Process. Upper line:
\mbox{$\widehat{I}[\AlternateState;\FinFuture{2}]_{\lambda \rightarrow 0}$},
not corrected; lower line:
\mbox{$\widehat{I}[\AlternateState;\FinFuture{2}]_{\lambda \rightarrow 0}^{\rm corrected}$},
corrected for estimation error due to finite sample size. The dashed line
indicates the actual upper bound on the predictive information, for
comparison. This value is not known to the algorithm, it is computed from
the true process statistics.
Histories of length $K = 3$ and futures of length $L = 2$ were used.
The asterisk denotes the optimal number ($N_c = 3$) of effective states.
}
\label{fig:OCEExampleInfoPlane}
\end{figure*}
\begin{figure*}[ht]
\centering
\resizebox{!}{2.50in}{\includegraphics{Even_20090420_H3_F2_L100_N3}}
\caption{OCE's best three-state approximated future conditional probabilities
(circles) for the Even Process (d). Compared to true (unknown) future
conditional probabilities (squares). The OCE inputs are the estimates of
$\widehat{\Prob}(\FinFuture{2}|\finpast{3})$ (crosses).
}
\label{fig:OCEExampleMorphs}
\end{figure*}
We illustrate OCE on the Golden Mean and Even Processes studied in Sec.
\ref{examples}. With the {\em correct} number of underlying states, they
can be predicted at a substantial compression.
Figures \ref{fig:OCEGMPInfoPlane} and \ref{fig:OCEExampleInfoPlane} show
the mutual information $I[\AlternateState;\FinFuture{2}]$ versus the number
$N_c$ of inferred states, with statistics estimated from time series of lengths
$T = 100$. The graphs compare the mutual information
\mbox{$\widehat{I}[\AlternateState;\FinFuture{2}]_{\lambda \rightarrow 0}$}
evaluated using the estimate
\mbox{$\widehat{\Prob}(\FinFuture{2};\FinPast{3})$} (upper curve) to the
corrected information
$\widehat{I}[\AlternateState;\FinFuture{2}]_{\lambda \rightarrow 0}^{\rm corrected}$
calculated by subtracting the approximated error Eq. (\ref{finsizeerror}) with
$k^L = 4$ and $M = 96$ (lower curve).
We see that the corrected information curves peak at, and thereby, select models
with two states for the Golden Mean Process and three states for the Even
Process. This corresponds with the true number of causal states, as we know
from above (Sec. \ref{examples}) for the two processes. The true statistical
complexity for both processes is $\Cmu \approx 0.91830$, while those estimated
via OCE are $\Cmu \approx 0.93773$ and $\Cmu \approx 1.30262$, respectively.
(Recall that the overestimate for the latter was explained in Sec.
\ref{sec:EvenProcess}.)
Figures \ref{fig:OCEGMPMorphs} and \ref{fig:OCEExampleMorphs} show the OCE
future conditional probabilities corresponding to the (optimal) two- and
three-state approximations, respectively. The input to OCE are the future
conditional probabilities given the histories
$\widehat{\Prob}(\FinFuture{2}|\finpast{3})$ (crosses), which are estimated
from the full historical information. Those future conditional probabilities
are corrupted by sampling errors due to the finite data set size and differ
from the true future conditional probabilities (squares).
Compare the OCE future conditional probabilities (circles) to the true
future conditional probabilities (squares), calculated with the knowledge
of the causal states. (The latter, of course, is not
available to the OCE algorithm.) In the case of the GM Process, OCE
approximates the correct future conditional probabilities. For the Even Process
there is more spread in the estimated OCE future conditional
probabilities. Nonetheless, OCE reduced the fluctuations in its inputs and
corrected in the direction of the true underlying future conditional
probabilities.
\vspace{-.01in}
\section{Conclusion}
We analyzed an information-theoretic approach to causal modeling in two
distinct cases: (i) optimal causal filtering (OCF), where we have access
to the process statistics and desire to capture the process's structure
up to some level of approximation, and (ii) optimal causal estimation
(OCE), in which, in addition, finite-data fluctuations need to be traded-off
against approximation error and structure.
The objective function used in both cases follows from very simple first
principles of information processing and causal modeling: a good model
should minimize prediction error at minimal model complexity. The resulting
principle
of using small, predictive models follows from minimal prior knowledge
that, in particular, makes no structural assumptions about a
process's architecture: Find variables that do the best at causal shielding.
OCF stands in contrast with other approaches. Hidden Markov modeling, for
example, assumes a set of states and an architecture \citep{Rabi86a}. OCF finds
these states from the given data. In minimum description length modeling, to
mention another contrast, the model
complexity of a stochastic source diverges (logarithmically) with the data set
size \citep{Riss89a}, as happens even when modeling the ideal random process
of a fair coin. OCF, however, finds the simplest (smallest) models.
Our main result is that OCF reconstructs the causal-state partition, a
representation previously known from computational mechanics that captures a
process's causal architecture and that allows important system properties,
such as entropy rate and stored information, to be calculated \citep{Crut98d}.
This result is important as it gives a structural meaning to the solutions
of the optimization procedure specified by the causal inference objective
function. We have shown that in the context of time series modeling, where
there is a \emph{natural} relevant variable (the future), the IB approach
\citep{IBN} recovers the unique minimal sufficient statistic---the
causal states---in the limit in
which prediction is paramount to compression. Altogether, this allows us to
go beyond plausibility arguments for the information-theoretic objective
function that have been used. We showed that this way (OCI) of
phrasing the causal inference problem in terms of causal shielding
results in a representation that is a
sufficient statistic and minimal and, moreover, reflects the structure of the
process that generated the data. OCI does so in a way that is meaningful and
well grounded in physics and nonlinear dynamics. The optimal solutions to
balancing prediction and model complexity take on meaning---asymptotically,
they are the causal states.
The results also contribute to computational mechanics: The continuous trade-off
allows one to extend the deterministic history-to-state assignments that
computational mechanics introduced to ``soft'' partitions of histories. The
theory gives a principled way of constructing stochastic approximations of the
ideal causal architecture. The resulting approximated models can be
substantially smaller and so will be useful in a number of applications.
Finally, we showed how OCF can be adapted to correct for finite-data sampling
fluctuations and so not over-fit. This reduces the tendency to see structure
in noise. OCE finds the correct number of hidden causal states. This renders
the method useful for application to real data.
\section*{Acknowledgments}
UC Davis and the Santa Fe Institute partially supported this work via the
Network Dynamics Program funded by Intel Corporation. It was also partially
supported by the DARPA Physical Intelligence Program. CJE was partially
supported by a Department of Education GAANN graduate fellowship. SS thanks
W. Bialek, discussions with whom have contributed to shaping some of the
ideas expressed, and thanks L. Bottou and I. Nemenmann for useful discussions.
\vspace{-.05in}
\small
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\newtheorem{definition}{Definition}[section]
\newtheorem{assumption}[definition]{Assumption}
\newtheorem{example}[definition]{Example}
\newtheorem{remark}[definition]{Remark}
\newtheorem{remarks}[definition]{Remarks}
\theoremstyle{plain}
\newtheorem{theorem}[definition]{Theorem}
\newtheorem{lemma}[definition]{Lemma}
\newtheorem{corollary}[definition]{Corollary}
\allowdisplaybreaks
\begin{document}
\title[Stability of Traveling Oscillating Fronts \\ in Complex Ginzburg Landau Equations]{Stability of Traveling Oscillating Fronts in Complex Ginzburg Landau Equations}
\setlength{\parindent}{0pt}
\vspace*{0.4cm}
\begin{center}
\normalfont\LARGE\bfseries{\shorttitle}
\vspace*{12pt}
\end{center}
\begin{center}
Wolf-J{\"u}rgen Beyn\footnotemark[1] and Christian D{\"o}ding{\footnotemark[2]${}^{,}$\footnotemark[3]} \\
\vspace{12pt}
October 25, 2021
\end{center}
\footnotetext[1]{Department of Mathematics, Bielefeld University, 33501 Bielefeld, Germany, \\ e-mail: \textcolor{blue}{[email protected]}, phone: \textcolor{blue}{+49 (0)521 106 4798}.}
\footnotetext[2]{Department of Mathematics, Ruhr-University Bochum, 44801 Bochum, Germany, \\ e-mail: \textcolor{blue}{[email protected]}, phone: \textcolor{blue}{+49 (0)234 32 19876}.}
\footnotetext[3]{This work is an extended version of parts of the author's
PhD Thesis \cite{Doeding}.}
\vspace{12pt}
\noindent
\begin{center}
\begin{minipage}{0.8\textwidth}
{\small
\textbf{Abstract.}
Traveling oscillating fronts (TOFs) are specific waves of the form
$U_\star (x,t) = e^{-i \omega t} V_\star(x - ct)$ with a profile $V_{\star}$
which decays at $- \infty$ but approaches a nonzero limit at $+\infty$.
TOFs usually appear in complex Ginzburg Landau equations of the type $U_t = \alpha U_{xx} + G(|U|^2)U$. In this paper we prove a theorem on
the asymptotic stability of TOFs, where we allow the
initial perturbation to be the sum of an exponentially localized part and
a front-like part which approaches a small but nonzero limit at $+ \infty$.
The underlying assumptions guarantee that
the operator, obtained from linearizing about the TOF in a co-moving and
co-rotating frame, has essential spectrum touching the imaginary axis
in a quadratic fashion and that further isolated eigenvalues are bounded away
from the imaginary axis. The basic idea of the proof is to consider
the problem in an extended phase space which couples the wave dynamics
on the real line to the ODE dynamics at $+ \infty$. Using slowly
decaying exponential weights, the framework allows to derive appropriate
resolvent estimates, semigroup techniques, and Gronwall estimates.
}
\end{minipage}
\end{center}
\vspace{12pt}
\noindent
\textbf{Key words.} Traveling oscillating front, nonlinear stability,
Ginzburg Landau equation, equivariance, essential spectrum.
\vspace{12pt}
\noindent
\textbf{AMS subject classification.} 35B35, 35B40, 35C07, 35K58, 35Pxx, 35Q56
\sect{Introduction}
\label{sec1}
In this paper we consider complex-valued semilinear parabolic equations of the form
\begin{align} \label{Evo}
U_t = \alpha U_{xx} + G(|U|^2)U, \quad x \in \R,\, t \ge 0
\end{align}
with nonlinearity $G:\R \rightarrow \C$ and diffusion coefficient $\alpha \in \C$, $\Re \alpha > 0$. If the nonlinearity $G$ is a linear resp. a quadratic polynomial over $\C$ then \eqref{Evo} leads to the cubic resp. the quintic complex Ginzburg Landau equation. Evolution equations of the form \eqref{Evo} admit the propagation of various types of waves which oscillate in time and which either
have a front profile or which are periodic in space like
wave trains, see \cite{SandstedeScheel04}, \cite{Saarloos}. We are interested in the stability behavior
of a special class of solutions which we call traveling oscillating fronts (TOFs).
A TOF is a solution of \eqref{Evo} of the form
\begin{align*}
U_\star (x,t) = e^{-i \omega t} V_\star(x - ct)
\end{align*}
with a profile $V_\star: \R \rightarrow \C$ satisfying the asymptotic property
\begin{align*}
\lim_{\xi \rightarrow -\infty} V_\star(\xi) = 0, \quad \lim_{\xi \rightarrow +\infty} V_\star(\xi) = V_{\infty}
\end{align*}
for some $V_{\infty} \in \C$, $V_{\infty} \neq 0$. The parameters $\omega,c \in \R$ are called the frequency and the velocity of the TOF.
Figure \ref{ExampleTOF} shows a typical TOF obtained by
simulating the quintic complex Ginzburg-Landau equation
\begin{align} \tag{QCGL} \label{QCGL}
U_t = \alpha U_{xx} + \beta_1 U + \beta_3 |U|^2 U + \beta_5 |U|^4U, \quad x \in \R,\, t \ge 0
\end{align}
with an initial function $U(\cdot,0)$ of sigmoidal shape.
\begin{figure}[h!]
\centering
\begin{minipage}[t]{0.45\textwidth}
\centering
\includegraphics[scale=0.45]{TOF2A.eps}
\end{minipage}
\begin{minipage}[t]{0.45\textwidth}
\centering
\includegraphics[scale=0.45]{TOF2B.eps}
\end{minipage}
\begin{minipage}[t]{0.45\textwidth}
\centering
\includegraphics[scale=0.45]{TOF2C.eps}
\end{minipage}
\begin{minipage}[t]{0.45\textwidth}
\centering
\includegraphics[scale=0.45]{TOF2D.eps}
\end{minipage}
\caption{Numerical simulation of a TOF in \eqref{QCGL} with parameters $\alpha = 1 + \tfrac{i}{2}$, $\beta_3 = 1 + i$, $\beta_5 = -1 + i$ and $\beta_1 = -0.1$. Real part (left) and imaginary part (right).} \label{ExampleTOF}
\end{figure}
We aim at sufficient conditions under which a TOF is
nonlinearly stable with asymptotic phase in suitable function spaces.
As initial perturbations we allow functions which can be decomposed
into an exponentially localized part and a front-like part which
perturbs the limit at $+ \infty$. There are two main difficulties
to overcome: first, the operator obtained by linearizing about the TOF has
essential spectrum touching the imaginary axis at zero in a quadratic
way. Second, the perturbation at infinity prevents the use of standard
Sobolev spaces for the linearized operator. The first difficulty will
be overcome by exponential weights which shift the essential spectrum
to the left, while the second difficulty is handled by analyzing stability
in an extended phase space which couples the dynamics on the real line
to the dynamics at $+ \infty$.
In the following we give a more technical outline of the setting
and our basic assumptions, and we provide an overview of the following
sections.
Our results will be stated for the two-dimensional real-valued system equivalent to \eqref{Evo}. Setting $U =u_1 + iu_2$, $u_j(x,t) \in \R$, $\alpha = \alpha_1 + i \alpha_2$, $\alpha_j \in \R$ and $G = g_1 + ig_2$ with $g_j:\R \rightarrow \R$ the equivalent real-valued parabolic system reads
\begin{align} \label{rEvo}
u_t = Au_{xx} + f(u), \quad x \in \R,\, t \ge 0,
\end{align}
where
\begin{equation} \label{SystemDef}
A =
\begin{pmatrix}
\alpha_1 & -\alpha_2 \\
\alpha_2 & \alpha_1
\end{pmatrix},
\quad f(u) = g(|u|^2)u, \quad g(\cdot) =
\begin{pmatrix}
g_1(\cdot) & -g_2(\cdot) \\
g_2(\cdot) & g_1(\cdot)
\end{pmatrix}.
\end{equation}
A TOF $U_{\star}=u_{\star,1}+ i u_{\star,2}$ of \eqref{Evo} then corresponds to
a solution $u_\star=(u_{\star,1},u_{\star,2})^{\top}$ of \eqref{rEvo} of the form
\begin{align*}
u_\star(x,t) = R_{-\omega t} v_\star(x-ct),\quad
R_{\theta}=\begin{pmatrix} \cos \theta & -\sin \theta \\
\sin \theta & \cos \theta \end{pmatrix},
\end{align*}
where $R_{\theta}$ denotes rotation by the angle $\theta \in \R$. The profile
$v_\star: \R \rightarrow \R^2$ satisfies $V_{\star}=v_{\star,1}+i v_{\star,2}$ and
\begin{align} \label{rasymp}
\lim_{\xi \rightarrow -\infty} v_\star (\xi) = 0, \quad \lim_{\xi \rightarrow +\infty} v_\star (\xi) = v_\infty,
\end{align}
where $V_{\infty}=v_{\infty,1}+ i v_{\infty,2}$ and $v_{\infty}\in \R^2$, $v_{\infty} \neq 0$. The vector $v_{\infty}$ is called the asymptotic rest-state
and the specific solution $R_{-\omega t}v_{\infty}$ of \eqref{rEvo}
is called a bound state.
Figure \ref{Figure:TOF3D}
shows a typical profile of a TOF in $(x,u_1,u_2)$-space.
\begin{figure}[h!]
\centering
\begin{minipage}[t]{0.9\textwidth}
\centering
\includegraphics[scale=1]{TOF3Darrow.eps}
\caption{Traveling oscillating front.} \label{Figure:TOF3D}
\end{minipage}
\end{figure}
For the stability analysis it is natural to transform \eqref{rEvo} into a co-moving and co-rotating frame, i.e. we set $u(x,t) = R_{-\omega t} v(\xi,t)$, $\xi = x-ct$ and find that
$v$ solves the equation
\begin{align} \label{comovsys}
v_t = A v_{\xi\xi} + cv_\xi + S_\omega v + f(v), \quad \xi \in \R,\, t \ge 0, \quad
S_{\omega} := \begin{pmatrix} 0 & -\omega \\ \omega & 0 \end{pmatrix}.
\end{align}
The time-independent profile $v_\star$ becomes a stationary solution of \eqref{comovsys}, i.e. it solves the ODE
\begin{align} \label{statcomovsys}
0 = A v_{xx} + c v_x + S_\omega v + f(v), \quad x \in \R.
\end{align}
From the asymptotic property \eqref{rasymp} one concludes
(see Lemma \ref{lem:asym}) that the rest-state satisfies
\begin{equation*}
g(|v_\infty|^2) = -S_\omega, \quad \lim_{x \rightarrow \pm \infty} v_{\star}'(x) = 0, \quad \lim_{x \rightarrow \pm \infty} v_{\star}''(x) = 0.
\end{equation*}
Since the right-hand side of \eqref{comovsys} is equivariant with respect
to translations and multiplication by rotations, the TOFs always come in families, i.e. the system \eqref{comovsys} has a two-dimensional continuum
\begin{equation} \label{relequi}
\mathcal{O}(v_\star) := \{ R_\theta v_\star(\cdot - \tau): (\theta,\tau) \in S^1 \times \R\}, \quad S^1=\R / 2 \pi \Z
\end{equation}
of stationary solutions. In the language of abstract evolution equations this
is a relative equilibrium, see e.g. \cite{Lauterbach}, \cite{Fiedleretal}, \cite{KapitulaPromislow}.
In the following we study the long time behavior of the solution $v$ of the initial-value problem
\begin{align} \label{perturbsys}
v_t = Av_{xx} + cv_x + S_\omega v + f(v), \quad v(0) = v_\star + u_0,
\end{align}
where the initial perturbation $u_0$ is assumed to be small in a suitable sense.
Let us first state some basic assumptions on the system \eqref{rEvo}, \eqref{SystemDef}.
\begin{assumption} \label{A1}
The coefficient $\alpha$ and the function $g$ satisfy
\begin{align}
& \alpha_1 > 0, \quad g \in C^3(\R,\R^{2,2}), \quad g_1(0) < 0. \label{A1a}
\end{align}
\end{assumption}
\begin{assumption} \label{A2}
There exists a traveling oscillating front solution $u_\star$ of \eqref{rEvo} with profile $v_\star \in C^2_b (\R,\R^2)$, speed $c > 0$, frequency $\omega \in \R$ and asymptotic rest-state $v_\infty = (|v_\infty|,0)^\top \in \R^2$ such that
\begin{align*}
g_1'(|v_\infty|^2) < 0.
\end{align*}
\end{assumption}
Note that the special form of $v_{\infty}$ selects just a specific equilibrium
from the orbit \eqref{relequi}.
Both assumptions guarantee that we have a stable equilibrium at
$\xi=-\infty$ and a circle of stable equilibria at $\xi=\infty$ when spatial
derivatives are ignored in \eqref{comovsys}.
Further conditions will
be imposed on the spectrum of the linearized operator
\begin{align} \label{LatL}
Lv = Av_{xx} + cv_x + S_\omega v + Df(v_\star) v
\end{align}
in suitable function spaces.
In view of \eqref{relequi} we expect the linearization $L$ from \eqref{LatL}
to have a two-dimensional kernel. In addition, it turns out that the essential
spectrum of $L$, touches the imaginary axis at the origin when considered in the function space $L^2(\R,\R^2)$. Thus there is no spectral gap between the
zero eigenvalue und the remaining spectrum, so that standard approaches to conclude nonlinear stability do not apply; see \cite{Henry},
\cite{KapitulaPromislow}, \cite{Sandstede02}.
We overcome this problem by two devices.
First, we impose the following condition
\begin{assumption}(Spectral Condition) \label{A4}
The diffusion coefficient $\alpha=\alpha_1+i \alpha_2$ satisfies
\begin{align*}
\alpha_2 g_2'(|v_\infty|^2) + \alpha_1 g_1'(|v_\infty|^2) < 0.
\end{align*}
\end{assumption}
Note that Assumption \ref{A4} follows from Assumptions \ref{A1}, \ref{A2}
if $\alpha_2=0$. Moreover, we will show that Assumption \ref{A4} guarantees
the essential spectrum to have negative quadratic contact
with the imaginay axis.
Second, we use Lebesgue and Sobolev spaces with exponential weight
\begin{align} \label{eta}
\eta(x) = e^{\mu \sqrt{x^2 + 1}}, \quad \mu \ge 0.
\end{align}
Any sufficiently small $\mu > 0$ will be enough to shift the essential
spectrum to the left and allow for a stability result.
Weights of this or similar type frequently appear in stability analyses,
see e.g. \cite{Zelik}, \cite[Ch.3.1.1]{KapitulaPromislow}, \cite{ghazaryan},
\cite{Kapitula94}. We note that the stability statement w.r.t. the $L^{\infty}$-norm in the
second part of \cite[Theorem 7.2]{Kapitula94} comes closest to our
results. There a perturbation argument for the case of a positive definite
matrix $A$ in \eqref{comovsys} is employed and the resulting Evans
function \cite{Alexander} is analyzed. This leads to more restrictive
conditions on the coefficients of the system and on the initial data.
We finish the introduction with a brief outline of the contents of
the following sections. In section \ref{sec2} we complete the basic
assumptions \ref{A1}, \ref{A2}, \ref{A4} by eigenvalue conditions
for the operator \eqref{LatL} and we state our main results in more
technical terms. The approach of the profile towards
its rest states is shown to be exponential, and stability with asymptotic phase
is stated in weighted $H^1$-spaces. We also explain the main idea of
the proof which incorporates the dynamics of \eqref{comovsys} at $\xi = \infty$
into an extended evolutionary system, see \eqref{CP}, \eqref{F}.
In Section \ref{sec3} we discuss in detail
the Fredholm properties of the operator $L$ and its extended version
in weighted spaces and we derive resolvent estimates.
These form the basis for obtaining detailed estimates of the associated
(extended) semigroup in Section \ref{sec4}. The subsequent section \ref{sec5}
is devoted to the decomposition of the dynamics into the motion within
the underlying two-dimensional symmetry group and within a
codimension-two function space. Section \ref{sec6} then provides sharp
estimates for the resulting remainder terms. Then a local existence theorem and
a Gronwall estimate complete the proof in Section \ref{sec7}.
Let us finally mention that the techniques of this paper can be used
to prove that the general method of freezing
(\cite{BeynThummler04}, \cite{Rottmann-Matthes12}, \cite{BOR14})
works successfully for the parabolic equation \eqref{rEvo} with two underlying symmetries, see \cite{Doeding}.
\sect{Assumptions and main results}
\label{sec2}
As a preparation for the subsequent analysis we specify the approach of
a TOF towards its rest states.
\begin{lemma} \label{lem:asym}
Let $v_\star \in C^2_b(\R,\R^2)$ be the profile of a traveling oscillating front of \eqref{rEvo} with speed $c>0$, frequency $\omega \in \R$ and asymptotic rest-state $v_\infty \in \R^2\backslash \{ 0 \}$. Moreover, suppose $\Re \alpha > 0$ and $g \in C(\R,\R^{2,2})$. Then the following holds:
\begin{align*}
g(|v_\infty|^2) = -S_\omega, \quad \lim_{x \rightarrow \pm \infty} v_{\star}'(x) = 0, \quad \lim_{x \rightarrow \pm \infty} v_{\star}''(x) = 0.
\end{align*}
\end{lemma}
Using Assumption \ref{A1} and \ref{A2} one can conclude that the convergence in Lemma \ref{lem:asym} of the profile $v_\star$ and its derivatives is exponentially fast.
\begin{theorem} \label{decay}
Let Assumption \ref{A1} and \ref{A2} be satisfied and let $v_{\star}$ be given
as in Lemma \ref{lem:asym}. Then $v_\star \in C^5_b(\R,\R^2)$ holds and
there are constants $K,\mu_\star > 0$ such that
\begin{align*}
| v_\star(x) - v_\infty | + |v'_\star(x)| + |v''_\star(x)| + |v'''_\star(x)| & \le K e^{-\mu_\star x} \quad \forall\, x \ge 0, \\
| v_ \star(x) | + |v'_\star(x)| + |v''_\star(x)| + |v'''_\star(x)| & \le K e^{\mu_\star x} \quad \forall\, x \le 0.
\end{align*}
\end{theorem}
In the Appendix we give the proof of Lemma \ref{lem:asym} and the main steps of
the proof of Theorem \ref{decay}.
With the weight $\eta$ given by \eqref{eta}, let us introduce the weighted $L^2$ space
\begin{align*}
L^2_\eta(\R,\R^n) := \{ v \in L^2(\R,\R^n): \eta v \in L^2(\R,\R^n) \},
\quad (u,v)_{L^2_{\eta}}:= (\eta u, \eta v)_{L^2}
\end{align*}
and the associated weighted Sobolev spaces defined for $\ell \in \N$ by
\begin{align*}
H^\ell_\eta(\R,\R^n) &:= \{ v \in L^2_\eta(\R,\R^n) \cap H^\ell_{\mathrm{loc}}(\R,\R^n): \partial^k v \in L^2_\eta(\R,\R^n),\, 1 \le k \le \ell \}, \\
\| v \|_{H^\ell_\eta}^2& := \sum_{k=0}^\ell \| \partial^k v \|_{L^2_\eta}^2.
\end{align*}
Let us note that Theorem \ref{decay} ensures $v_{\star}^{(j)} \in H^{3-j}_{\eta}(\R,\R^2)$
for $0 \le \mu < \mu_{\star}$ and $j=1,2,3$.
However,
the profile $v_\star$ of a TOF does not decay to zero as $x \rightarrow \infty$,
and, moreover, we expect
the limit $\rho(t)=\lim_{x \rightarrow \infty}v(x,t)$ of a solution of \eqref{comovsys} to still move with time.
Therefore, the idea is to include an ODE for the dynamics of
$\rho(t)$ into the overall system.
Formally taking the limit $x \rightarrow \infty$ in \eqref{comovsys} and
assuming $v_x(x,t), v_{xx}(x,t) \rightarrow 0$ as $x \rightarrow \infty$ we obtain for $\rho$ the ODE
\begin{align} \label{phaseODE}
\rho'(t) = S_\omega\rho(t) + f(\rho(t)).
\end{align}
Note that $v_\infty$ is a stationary solution of \eqref{phaseODE} due to
Lemma \ref{lem:asym}, and, by equivariance, there is a whole circle of equilibria
$\{R_{-\theta}v_{\infty}: \theta \in S^1\}$. Next we choose a template function
\begin{align*}
\hat{v}(x) := \tfrac{1}{2} \tanh(\hat{\mu}x) + \tfrac{1}{2},
\quad 0< 2 \hat{\mu} \le \mu_{\star}.
\end{align*}
The rate $\hat{\mu}$ has been chosen such that the approach toward the limits as
$x \to \pm \infty$ is weaker than for the derivatives of the solution in Theorem \ref{decay}. Such a choice is not strictly necessary but will avoid some
technicalities in the following.
If $0<\mu < 2\hat{\mu}$ we conclude $v_{\star}- \hat{v} v_{\infty} \in H^2_{\eta}(\R,\R^2)$ and we
also expect the solution $v$ of \eqref{comovsys}
to satisfy $v(\cdot,t) - \hat{v} \rho(t) \in H^2_\eta(\R,\R^2)$, i.e. to lie in an
affine linear space with a time dependent offset given by $\rho$.
Therefore, we introduce the Hilbert space
\begin{align*}
X_\eta := \Big\{ (v,\rho)^\top:\, v: \R \rightarrow \R^2,\, \rho \in \R^2,\, v-\rho\hat{v} \in L^2_\eta(\R,\R^2) \Big\}
\end{align*}
with inner product $\big( (u,\rho)^{\top},
(v,\zeta)^{\top} \big)_{X_{\eta}}=(\rho,\zeta)+(u-\rho \hat{v},v-\zeta \hat{v})_{L^2_{\eta}}$. Similarly, we define the smooth analog
\begin{align*}
X^\ell_\eta := \left\{ (v,\rho)^\top \in X_\eta: v \in H^\ell_{\mathrm{loc}},\partial^k v \in L^2_\eta,\, 1 \le k \le \ell \right\}, \quad \ell \in \N_0
\end{align*}
with the norm given by
$\left\| (v,\rho)^\top \right\|_{X^\ell_\eta}^2 := |\rho|^2 + \| v - \rho \hat{v}\|_{L^2_\eta}^2 + \sum_{k=1}^{\ell} \|\partial^k v\|_{L^2_\eta}^2$. We further set
$Y_\eta := X^2_\eta$ and denote the elements of $X^{\ell}_\eta$ by bold letters, for example,
\begin{align*} \v = (v,\rho)^\top , \quad \v_\star = (v_\star,v_\infty)^\top,
\quad \v_0= (u_0,\rho_0)^{\top}.
\end{align*}
As noted above, Theorem \ref{decay} implies $v_\star \in v_\infty \hat{v} + H^2_\eta$ and thus $\v_\star \in Y_\eta$.
Instead of \eqref{perturbsys}, we consider the extended Cauchy problem on
$X_\eta$
\begin{align} \label{CP}
\v_t = \F(\v), \quad \v(0) = \v_\star + \v_0,
\end{align}
where $\F$ is a semilinear operator given by
\begin{align} \label{F}
\F : Y_\eta \rightarrow X_\eta, \quad \vek{v}{\rho} = \v \mapsto \F (\v) =
\vek{Av_{xx} + cv_x + S_\omega v + f(v)}{S_\omega \rho + f(\rho)}.
\end{align}
With these settings, $\v_{\star}$ becomes a stationary solution of
\eqref{CP}, and our task is to prove its nonlinear stability with asymptotic
phase. For this purpose, let us extend the group action induced by rotation
and translation of elements from $L^2_{\eta}$ to $X_\eta$ as follows:
\begin{align} \label{groupaction}
a(\gamma): X_\eta \rightarrow X_\eta, \quad \vek{v}{\rho} \mapsto a(\gamma)\vek{v}{\rho} = \vek{R_{-\theta} v(\cdot - \tau)}{R_{-\theta} \rho},\quad
\gamma =(\theta,\tau) \in \mathcal{G} := S^1 \times \R.
\end{align}
The operator $\F$ from \eqref{F} is then equivariant w.r.t. the group action,
i.e. $\F(a(\gamma) \v) = a(\gamma) \F(\v)$ for all $\gamma \in \mathcal{G}$ and
$u \in Y_{\eta}$. Further a metric on $\mathcal{G}$ is given by
\begin{align*}
d_{\mathcal{G}} (\gamma_1,\gamma_2) = |\gamma_1 - \gamma_2|_{\mathcal{G}}, \quad |\gamma|_{\mathcal{G}} := \min_{k \in \Z} |\theta - 2\pi k| + |\tau|, \quad \gamma = (\theta,\tau)
\in \mathcal{G}.
\end{align*}
Finally, we collect the assumptions on the linearized operator
$L: \D(L)=H^2 \subset L^2 \rightarrow L^2$ from
\eqref{LatL}. The operator $L$ will turn out to be
closed and densely defined. We denote its resolvent set by
\begin{align*}
\mathrm{res}(L) := \{ s \in \C: sI-L: \D(L) \rightarrow L^2 \text{ is bijective} \}
\end{align*}
and its spectrum by $\sigma(L) = \C \backslash \mathrm{res}(L)$.
The further subdivision of the spectrum into the essential spectrum and the point spectrum varies in the literature (see the five different notions in \cite{EdmundsEvans}).
We use the following definition (see $\sigma_{\mathrm{e},4}(L)$ in \cite[Ch.I.4,IX.1]{EdmundsEvans} or \cite[Ch.3]{KapitulaPromislow} and note the slight
deviation from \cite{Henry},\cite{Kato}):
\begin{align} \label{eq2:defessential}
\sigma_{\mathrm{pt}}(L) := \{ s \in \sigma(L): sI - L \text{ is Fredholm of index } 0 \}, \quad
\sigma_{\mathrm{ess}}(L) := \sigma(L) \backslash \sigma_{\mathrm{pt}}(L).
\end{align}
When we insert the translates $v_{\star}(\cdot- \tau)$ from \eqref{relequi}
into the stationary
equation \eqref{comovsys} and differentiate with respect to $\tau$ we
obtain that the nullspace $\ker(L)$ of $L$ contains at least
$v'_{\star}\in H^2$.
The following condition requires that there are no
(generalized) eigenfunctions and that eigenvalues from the point
spectrum lie strictly to the left of the imaginary axis.
\begin{assumption}[Eigenvalue Condition] \label{A5} $ $ \\
There exists $\beta_E > 0$ such that $\Re s < - \beta_E$ holds
for all $s \in \sigma_{\mathrm{pt}}(L)$. Moreover,
\begin{align}
\label{eq3:zerosimple}
\dim \ker (L) = \dim \ker (L^2)=1.
\end{align}
\end{assumption}
Recall $v'_{\star}\in \ker(L)$ so that \eqref{eq3:zerosimple} implies
$\ker(L) = \mathrm{span}\{ v'_{\star} \}$. In Theorem \ref{thm4.17} below we will
see that $L:H^2 \to L^2 $ is not Fredholm, hence $0$ belongs to
$\sigma_{\mathrm{ess}}(L)$ and not to $\sigma_{\mathrm{pt}}(L)$.
For this reason we wrote condition \eqref{eq3:zerosimple} explicitly in
terms of nullspaces, and $\Re s < -\beta_E$ for $s\in \sigma_{\mathrm{pt}}(L)$
is no contradiction for $s=0$.
Differentiating the equation \eqref{comovsys} for the stationary continuum
\eqref{relequi} with respect to the first group variable $\theta\in S^1$
produces a second `eigenfunction' $ S_1 v_{\star}$ which, however, does
not belong to $\D(L)=H^2$. But this eigenfunction will appear for the
extended operator obtained by linearizing $\F$ from
\eqref{F} at $\v_{\star}$:
\begin{align} \label{calL}
\L_\eta : Y_\eta \rightarrow X_\eta, \quad \vek{v}{\rho} \mapsto \L_\eta \vek{v}{\rho} = \vek{Av_{xx} + cv_x + S_\omega v + Df(v_\star) v}{S_\omega \rho + Df(v_\infty)\rho}.
\end{align}
The subindex $\eta$ indicates that the operator $\L_{\eta}$ depends
on the weight through its domain and range. We further write $\L=\L_1$ in case $\mu = 0$, $\eta \equiv 1$ and introduce
\begin{align*}
E_\omega := S_\omega + Df(v_\infty)
\end{align*}
for the second component of the operator.
In Section \ref{sec3} we prove the following result for the point spectrum of the operator $\L_{\eta}$ defined in \eqref{calL}.
\begin{lemma} \label{lemma4.18}
Let Assumption \ref{A1}, \ref{A2}, \ref{A4} and \ref{A5} be satisfied. Then there exists a constant $\mu_1\in (0, 2 \hat{\mu})$
such that the following holds for all weight functions \eqref{eta} with
$0 < \mu \le \mu_1$ :
\begin{enumerate}[i)]
\item The eigenvalue $0$ belongs to $\sigma_{\mathrm{pt}}(\L_\eta)$ and has
geometric and algebraic multiplicity $2$, more precisely,
\begin{align} \label{eq2:eigenfunctions}
\ker(\L_\eta^2)= \ker (\L_\eta) =&\, \mathrm{span} \{\varphi_1, \varphi_2 \} , \quad
\varphi_1 = (v'_{\star}, 0)^{\top}, \quad\varphi_2= (S_1 v_\star, S_1 v_\infty)^{\top}.
\end{align}
\item There exists some $\beta_1=\beta_1(\mu)>0$ such that all eigenvalues $s \in \sigma_{\mathrm{pt}}(\L_\eta) \setminus \{0\}$
satisfy $\Re s < - \beta_1<0$.
\end{enumerate}
\end{lemma}
Now we are in a position to formulate the main result.
\begin{theorem} \label{Theorem4.10}
Let Assumption \ref{A1}, \ref{A2}, \ref{A4} and \ref{A5} be satisfied and let
$\eta$ be given by \eqref{eta}. Then there exists $\mu_0 > 0$ such that for
every $\mu \in (0,\mu_0)$ there are constants
$\varepsilon_0(\mu), \beta(\mu), K(\mu), C_\infty(\mu) > 0$ so that the
following statements hold. For all initial perturbations $\v_0 \in Y_\eta$
with $\| \v_0 \|_{X_\eta^1} < \varepsilon_0$ the equation \eqref{CP} has a
unique global solution
$\v \in C((0,\infty), Y_\eta) \cap C^1([0,\infty),X_\eta)$ which can be
represented as
\begin{align*}
\v(t) = a(\gamma(t)) \v_\star + \mathbf{w}(t), \quad t \in [0,\infty)
\end{align*}
for suitable functions $\gamma \in C^1([0,\infty),\mathcal{G})$ and $\mathbf{w} \in C((0,\infty),Y_\eta) \cap C^1([0,\infty), X_\eta)$. Further, there exists
an asymptotic phase $\gamma_\infty = \gamma_\infty(\v_0) \in \mathcal{G}$ such that
\begin{align*}
\| \mathbf{w}(t) \|_{X^1_\eta} + |\gamma(t) - \gamma_\infty|_\mathcal{G} & \le K e^{-\beta t} \| \v_0 \|_{X^1_\eta}, \quad |\gamma_\infty|_\mathcal{G} \le C_\infty \| \v_0 \|_{X^1_\eta}.
\end{align*}
\end{theorem}
This leads to corresponding stability statements for a TOF of the equations
\eqref{perturbsys} and \eqref{rEvo}, respectively. For simplicity, we state
the result in an informal way under the
assumptions of Theorem \ref{Theorem4.10} for the extended version of \eqref{rEvo},
i.e.
\begin{equation} \label{evoextended}
\begin{aligned}
u_t & = Au_{xx} + f(u), \quad u(\cdot,0)= v_{\star} + u_0,\\
r_t & = f(r), \quad r(0)= v_{\infty}+ \rho_0.
\end{aligned}
\end{equation}
Initial perturbations $\v_0=(u_0,\rho_0)$ are assumed to be small in the sense that
\begin{align*}
\| \v_0\|_{X^1_{\eta}}^2 &= \|u_0- \rho_0 \hat{v}\|_{L^2_{\eta}}^2 + \|\partial_x u_0 \|
_{L^2_{\eta}}^2 + |\rho_0|^2 \le \varepsilon_0^2.
\end{align*}
Then the system \eqref{evoextended} has a unique solution $\u = (u,r)
\in C((0,\infty), Y_\eta) \cap C^1([0,\infty),X_\eta)$ and there exist
functions $(\theta,\tau)\in C^1([0,\infty), S^1 \times \R)$ and
a value $(\theta_{\infty},\tau_{\infty})\in S^1 \times \R$ such that
for all $t \ge 0$
\begin{equation*}
\begin{aligned}
& \|u(\cdot,t)-R_{-\omega t -\theta(t)}v_{\star}(\cdot - c t -\tau(t))
- \hat{v}(r(t)-R_{-\omega t -\theta(t)}v_{\infty})\|_{L^2_{\eta}}
+\|\partial_x u(\cdot,t)\|_{L^2_{\eta}}\\
&+ |r(t)-R_{-\omega t -\theta(t)}v_{\infty}| +|\tau(t)-\tau_{\infty}|+ |\theta(t)-\theta_{\infty}| \\
& \le K e^{-\beta t} \| \v_0 \|_{X^1_\eta}.
\end{aligned}
\end{equation*}
Note the detailed expression for the asymptotic behavior of $\lim_{x \to \infty}u(x,t)$ as $t \to \infty$.
\sect{Spectral analysis of the linearized operator}
\label{sec3}
In this section we study the spectrum of the linear operator $\L_\eta$ from
\eqref{calL} and estimate solutions of the resolvent equation
\begin{align} \label{resolvGl}
(sI - \L_\eta)\v = \r, \quad s \in \C,\, \r= (r,\zeta)^{\top} \in X_\eta.
\end{align}
In the first step we derive resolvent estimates for solutions
$\v \in Y_\eta$ of \eqref{resolvGl} when $|s|$ is large and $s$ lies in the
exterior of some sector opening to the left. The approach is based on
energy estimates from \cite{Kreiss}, \cite{KreissLorenz}.
\begin{lemma} \label{aprioriest}
Let Assumption \ref{A1} and \ref{A2} be satisfied and let $\mu_2 \in (0,2\hat{\mu})$.
Then there exist
constants $\varepsilon_0, R_0, C> 0$ such that the following properties
hold for all $0 \le \mu \le \mu_2$ . The operator $\L_\eta:Y_\eta \subset X_\eta \rightarrow X_\eta$ is closed and densely defined in $X_{\eta}$. For all
\begin{align} \label{Omega0}
s \in \Omega_0 := \left\{ s \in \C: |s| \ge R_0, |\arg(s)| \le \frac{\pi}{2} + \varepsilon_0 \right\}
\end{align}
the equation \eqref{resolvGl} with $\v \in Y_\eta$ and $\r \in X_\eta$ implies
\begin{align}
& |s| \| \v \|_{X_\eta}^2 + \| v_x \|_{L_\eta^2}^2 \le \frac{C}{|s|} \| \r \|_{X_\eta}^2 \label{resolest1} \\
& |s|^2 \| \v \|_{X_\eta}^2 + |s| \| v_x \|_{L_\eta^2}^2 + \|v_{xx}\|_{L_\eta^2}^2 \le C \| \r \|_{X_\eta}^2. \label{resolest2}
\end{align}
In addition, if $\r \in X_\eta^1$ and $\v \in X^3_\eta$ then
\begin{align} \label{resolest3}
|s|^2 \| \v \|_{X^1_\eta}^2 + |s| \| v_{xx} \|_{L_\eta^2}^2 + \|v_{xxx}\|_{L_\eta^2}^2 \le C \| \r \|_{X^1_\eta}^2.
\end{align}
\end{lemma}
\begin{proof}
For the proof let us abbreviate $C_{\star} := Df(v_\star)$, $C_\infty := Df(v_\infty)$ and
$(\cdot,\cdot) = (\cdot,\cdot)_{L_\eta^2(\R,\R^2)}$.
From Theorem \ref{decay} and $\mu \le \mu_2 < \mu_{\star}$ we find for $(v,\rho)^{\top} \in X_{\eta}$
\begin{equation} \label{eq3:basic}
\begin{aligned}
\|C_{\star}v - C_{\infty}\rho \hat{v}\|_{L^2_{\eta}} \le &\, \|C_{\star}\|_{L^{\infty}}
\|v - \rho \hat{v}\|_{L^2_{\eta}}+
\|(C_{\star}-C_{\infty})\rho \hat{v}\|_{L^2_{\eta}} \\
\|(C_{\star}-C_{\infty})\rho \hat{v}\|_{L^2_{\eta}}^2 = & \,
\|(C_{\star}-C_{\infty})\rho \hat{v}\|_{L^2_{\eta}(\R_-)}^2 +
\|(C_{\star}-C_{\infty})\rho \hat{v}\|_{L^2_{\eta}(\R_+)}^2 \\
\le &\, \frac{ e^{2\mu} \|C_{\star}- C_{\infty}\|_{L^{\infty}}^2}{2\hat{\mu}- \mu} |\rho|^2
+\int_0^{\infty}\eta^2(x) |Df(v_{\star}(x))-Df(v_{\infty})|^2 dx |\rho|^2
\\
\le & \, K_C |\rho|^2.
\end{aligned}
\end{equation}
From this one infers that the operator $\L_\eta: Y_\eta \rightarrow X_\eta$ is bounded.
Next, we note that \eqref{resolest2} implies the closedness of $\L_\eta$. For this purpose, let $\{ \v_n \}_{n \in \N} \subset Y_\eta$ be given with
$\v_n \rightarrow \v$ in $X_\eta$ and $\L_\eta \v_n \rightarrow \mathbf{w}$ in $X_\eta$.
Pick $s_0 \in \Omega_0$ with $|s_0| \ge 1$. Then \eqref{resolest2} yields
\begin{align*}
\| \v_n - \v_m \|_{Y_\eta}^2 & \le |s_0|^2 \| \v_n - \v_m \|_{X_\eta}^2 + |s_0| \| v_{n,x} - v_{m,x} \|_{L^2_\eta}^2 + \| v_{n,xx} - v_{m,xx} \|_{L^2_\eta}^2 \\
& \le C_1 \| s_0( \v_n - \v_m) - \L_\eta\v_n - \L_\eta \v_m \|_{X_\eta}^2 \rightarrow 0 ,\quad n,m \rightarrow \infty.
\end{align*}
Thus, $\{ \v_n \}_{n \in \N}$ is a Cauchy sequence in $Y_\eta$ and there is $\tilde{\v} \in Y_\eta$ with $\v_n \rightarrow \tilde{\v}$ in $Y_\eta$. We conclude $\v = \tilde{\v} \in Y_\eta$ and $\v_n \rightarrow \v$ in $Y_\eta$.
Finally, $\L_\eta \v = \mathbf{w}$ follows from the boundedness of $\L_\eta: Y_\eta \rightarrow X_\eta$ and the estimate
\begin{align*}
\| \L_\eta \v - \mathbf{w} \|_{X_\eta} & \le \| \L_\eta(\v - \v_n) \|_{X_\eta} + \| \L_\eta \v_n - \mathbf{w} \|_{X_\eta} \rightarrow 0, \quad n \rightarrow \infty.
\end{align*}
The estimate \eqref{resolest3} follows by differentiating \eqref{resolvGl} w.r.t. $x$ and using \eqref{resolest2}. Therefore, it is left to show \eqref{resolest1} and \eqref{resolest2}. We begin with \eqref{resolest1}. For this purpose,
let $s \in \Omega_0$ with $R_0$ and $\varepsilon_0$ still to be determined.
Take the inner product of \eqref{resolvGl} with $\v$ in $X_\eta$ to obtain
\begin{align*}
& (\v,\r)_{X_\eta} = (\v, (sI-\L_\eta)\v)_{X_\eta} = \left(\vek{v}{\rho}, \vek{sv - Av_{xx} - cv_x - S_\omega v - C_{\star}v}{s \rho - S_\omega \rho - C_\infty \rho}\right)_{X_\eta} \\
& = \rho^\top (sI - S_\omega- C_\infty) \rho + (v - \rho \hat{v}, sv - Av_{xx} - cv_x - S_\omega v - C_{\star}v - (s \rho - S_\omega \rho - C_\infty \rho) \hat{v}) \\
& = s \| \v \|_{X_\eta}^2 - \rho^\top S_{\omega}\rho - \rho^\top C_\infty \rho \\
& \quad - (v- \rho\hat{v}, Av_{xx})_{L_\eta^2} - c(v - \rho\hat{v}, v_x) - (v - \rho\hat{v}, S_\omega (v - \rho\hat{v})) - (v-\rho\hat{v},C_{\star}v- C_\infty \rho\hat{v}).
\end{align*}
Integration by parts yields
\begin{align}
\label{equ:proof1}
\begin{split}
& s \| \v \|_{X_\eta}^2 + (v_x-\rho\hat{v}_x,Av_x)_{L_\eta^2} \\
& = \rho^\top (S_\omega + C_\infty) \rho -2(\eta'\eta^{-1}(v - \rho \hat{v}), Av_x) + c(v-\rho \hat{v}, v_x)_{L_\eta^2} \\
& + (v-\rho\hat{v},S_\omega(v -\rho \hat{v}))_{L_\eta^2} + (v - \rho \hat{v},C_{\star}v -C_\infty \rho\hat{v})_{L_\eta^2} + ( \v , \r)_{X_\eta}.
\end{split}
\end{align}
Further, we use Cauchy-Schwarz and Young's inequality with arbitrary
$\varepsilon_i >0$ and \eqref{eq3:basic} to obtain the estimates
\begin{align} \label{equ:proof1a}
\begin{split}
|(v_x-\rho \hat{v}_x, Av_x)| & \le | A | \left( \| v_x\|_{L^2_\eta}^2 + \frac{1}{4 \varepsilon_1} \| \rho \hat{v}_x \|_{L^2_\eta}^2 + \varepsilon_1 \| v_x \|_{L^2_\eta}^2 \right) \\
& = | A | (1+\varepsilon_1) \| v_x \|_{L^2_\eta}^2 + \frac{ e^{2\mu} | A | }{4(2\hat{\mu}-\mu) \varepsilon_1} |\rho|^2,
\end{split}
\end{align}
\begin{align} \label{equ:proof1aa}
\begin{split}
|(\eta' \eta^{-1}(v-\rho \hat{v}), Av_x)| & \le \frac{ \mu^2 | A |}{4 \varepsilon_2} \| v - \rho \hat{v} \|_{L^2_\eta}^2 + \varepsilon_2 | A | \| v_x \|_{L^2_\eta}^2,
\end{split}
\end{align}
\begin{align} \label{equ:proof1b}
\begin{split}
|c(v-\rho \hat{v}, v_x)| \le \frac{|c|}{4 \varepsilon_3} \| v-\rho\hat{v} \|_{L^2_\eta}^2 + |c| \varepsilon_3 \| v_x \|_{L^2_\eta}^2,
\end{split}
\end{align}
\begin{align} \label{equ:proof1c}
|(v-\rho \hat{v}, S_\omega (v-\rho \hat{v}))| \le |\omega| \| v-\rho \hat{v}\|_{L^2_\eta}^2,
\end{align}
\begin{align} \label{equ:proof1d}
\begin{split}
|(v-\rho \hat{v}, C_{\star}v-C_\infty \rho \hat{v})|
& \le \left( \| C_{\star}\|_{L^\infty}+ \frac{1}{4 \varepsilon_4} \right) \| v-\rho \hat{v} \|_{L^2_\eta}^2 + K_C \varepsilon_4 |\rho|^2.
\end{split}
\end{align}
Take the absolute value in \eqref{equ:proof1} and use \eqref{equ:proof1a}-\eqref{equ:proof1d} with $\varepsilon_i = 1$ to obtain for some $K_0,K_1 > 0$
\begin{align}
\label{equ:proof2}
|s| \| \v \|_{X_\eta}^2 \le K_0 \|v_x\|_{L_\eta^2}^2 + K_1 \| \v \|_{X_\eta}^2 + \| \v \|_{X_\eta} \| \r \|_{X_\eta}.
\end{align}
Next we note that $( v_x - \rho \hat{v}_x , Av_x) = \alpha_1 \| v_x \|_{L^2_\eta}^2 - (\rho \hat{v}_x, Av_x)$ and
\begin{align} \label{equ:proof2a}
|(\rho \hat{v}_x,Av_x)| \le | A | \| \hat{v}_x\|_{L^2_\eta} |\rho| \| v_x \|_{L^2_\eta} \le \frac{ e^{2 \mu} | A |}{(2\hat{\mu}-\mu) \varepsilon_5}|\rho|^2 + \varepsilon_5 | A| \| v_x \|_{L^2_\eta}^2.
\end{align}
Taking the real part in \eqref{equ:proof1} we obtain by using Cauchy-Schwarz,
Young's inequality and \eqref{equ:proof1aa}-\eqref{equ:proof1d} as well as
\eqref{equ:proof2a} with $\varepsilon_2 = \varepsilon_5 = \frac{\alpha_1}{8| A|}$, $\varepsilon_3 = \frac{\alpha_1}{4 |c|}$, $\varepsilon_4 = 1$ the estimate
\begin{align*}
\Re s \| \v \|_{X_\eta}^2 + \alpha_1 \| v_x \|_{L_\eta^2}^2 & \le \big(\varepsilon_5 | A | + \varepsilon_2 | A | + \varepsilon_3 |c|\big) \| v_x \|_{L^2_\eta}^2 + K_2 \| \v \|_{X_\eta}^2 + \| \v \|_{X_\eta} \| \r \|_{X_\eta} \\
& \le \frac{\alpha_1}{2} \| v_x \|_{L^2_\eta}^2 + K_2 \| \v \|_{X_\eta}^2 + \| \v \|_{X_\eta} \| \r \|_{X_\eta}.
\end{align*}
This yields
\begin{align} \label{equ:proof3}
\Re s \| \v \|_{X_\eta}^2 + \frac{\alpha_1}{2} \| v_x \|_{L_\eta^2}^2 \le K_2 \| \v \|_{X_\eta}^2 + \| \v \|_{X_\eta} \|\r \|_{X_\eta}.
\end{align}
The remaining proof falls naturally into three cases depending on the value of $s\in \Omega_0$. \\
\textbf{Case 1:} $\Re s \ge |\Im s|$, $\Re s > 0$, $|s| \ge 2\sqrt{2} K_2$. \\
We have $0 < \Re s \le |s| \le \sqrt{2} \Re s$. Therefore, using \eqref{equ:proof3} and Young's inequality with $\varepsilon = \frac{\sqrt{2}}{|s|}$, we obtain
\begin{align*}
\frac{|s|}{\sqrt{2}} \| \v \|_{X_\eta}^2 + \frac{\alpha_1}{2} \| v_x \|_{L_\eta^2}^2 & \le \frac{|s|}{2\sqrt{2}} \| \v \|_{X_\eta}^2 + \| \v \|_{X_\eta} \| \r \|_{X_\eta} \\
& \le \frac{|s|}{2\sqrt{2}} \| \v \|_{X_\eta}^2 + \frac{|s|}{4\sqrt{2}} \| \v \|_{X_\eta}^2 + \frac{\sqrt{2}}{|s|} \| \r \|_{X_\eta}^2.
\end{align*}
Thus, for a suitable constant $C$
\begin{align*}
|s| \| \v \|_{X_\eta}^2 + \| v_x \|_{L_\eta^2}^2 \le \frac{C}{|s|} \| \r \|_{X_\eta}^2.
\end{align*}
\textbf{Case 2:} $| \Im s | \ge \Re s \ge 0$. \\
From \eqref{equ:proof3} we have
\begin{align*}
\| v_x \|_{L_\eta^2} \le \frac{2}{\alpha_1} \left( K_2 \| \v \|_{X_\eta}^2 + \| \v \|_{X_\eta} \| \r \|_{X_\eta} \right).
\end{align*}
Use this in \eqref{equ:proof2} and find a constant $K_3>0$ such that
\begin{align*}
|s| \| \v \|_{X_\eta}^2
& \le K_3 \left( \| \v \|_{X_\eta}^2 + \| \v \|_{X_\eta} \| \r \|_{X_\eta} \right).
\end{align*}
Take $|s| > 2K_3$ and use Young's inequality with $\varepsilon = |s|^{-1}$
\begin{align*}
|s| \| \v \|_{X_\eta}^2 & \le \frac{|s|}{2} \| \v \|_{X_\eta}^2 + K_3 \| \v \|_{X_\eta} \| \r \|_{X_\eta} \le \frac{|s|}{2} \| \v \|_{X_\eta}^2 + \frac{|s|}{4} \| \v \|_{X_\eta}^2 + \frac{K_3^2}{|s|} \| \r \|_{X_\eta}^2,
\end{align*}
hence
\begin{align} \label{equ:proof4}
|s| \| \v \|_{X_\eta}^2 \le \frac{4 K_3^2}{|s|} \| \r \|_{X_\eta}^2.
\end{align}
Using \eqref{equ:proof3}, \eqref{equ:proof4} and taking $|s|\ge 4K_2$ yields by Young's inequality with $\varepsilon = |s|^{-1}$
\begin{align} \label{equ:proof5}
\begin{split}
\frac{\alpha_1}{2} \| v_x \|_{L_\eta^2}^2 & \le \frac{|s|}{4} \| \v \|_{X_\eta}^2 + \| \v \|_{X_\eta} \| \r \|_{X_\eta} \le \frac{|s|}{4} \| \v \|_{X_\eta}^2 + \frac{|s|}{4} \| \v \|_{X_\eta}^2 + \frac{1}{|s|} \| \r \|_{X_\eta}^2
+ \frac{K_4}{|s|} \| \r \|_{X_\eta}^2.
\end{split}
\end{align}
Combining \eqref{equ:proof4} and \eqref{equ:proof5} we arrive at the estimate
\eqref{resolest1}.
\textbf{Case 3:} $\Re s \le 0$, $|\Re s | \le \varepsilon_0 | \Im s |$.
Using \eqref{equ:proof2} and \eqref{equ:proof3} yields
\begin{align*}
& | \Im s | \| \v \|_{X_\eta}^2 \le |s| \| \v \|_{X_\eta}^2 \le K_0 \|v_x \|_{L_\eta^2}^2 + K_1 \| \v\|_{X_\eta}^2 + \| \v \|_{X_\eta} \| \r \|_{X_\eta} \\
& \le \frac{2 K_0}{\alpha_1} \left( |\Re s| \| \v \|_{X_\eta}^2 + K_2 \| \v \|_{X_\eta}^2 + \| \v \|_{X_\eta} \| \r \|_{X_\eta} \right) + K_1 \| \v \|_{X_\eta}^2 + \| \v \|_{X_\eta} \| \r \|_{X_\eta}.
\end{align*}
Choose $0 < \varepsilon_0 < \frac{\alpha_1}{4 K_0}$, so that
$\frac{2K_0}{\alpha_1} |\Re s| \le \frac{|\Im s|}{2}$ holds. Then we conclude
\begin{align*}
|\Im s| \| \v \|_{X_\eta}^2 \le K_5 \left( \| \v \|_{X_\eta}^2 + \| \v \|_{X_\eta} \| \r \|_{X_\eta} \right).
\end{align*}
Since $|s| \le \sqrt{1+ \varepsilon_0^2} |\Im s|$ we also have
\begin{align*}
|s| \| \v \|_{X_\eta}^2 \le K_6 \left( \| \v \|_{X_\eta}^2 + \| \v \|_{X_\eta} \| \r \|_{X_\eta} \right).
\end{align*}
Now take $|s| > 2K_6$ and use Young's inequality with $\varepsilon = |s|^{-1}$ to find
\begin{align*}
|s| \| \v \|_{X_\eta}^2 & \le \frac{|s|}{2} \| \v \|_{X_\eta}^2 + K_6 \| \v \|_{X_\eta} \| \r \|_{X_\eta}
\le \frac{|s|}{2} \| \v \|_{X_\eta}^2 + \frac{|s|}{4} \| \v \|_{X_\eta}^2 + \frac{K_6^2}{|s|}\| \r \|_{X_\eta}^2,
\end{align*}
which yields
\begin{align} \label{equ:proof6}
|s| \| \v \|_{X_\eta}^2 \le \frac{K_7}{|s|} \| \r \|_{X_\eta}^2.
\end{align}
To complete \eqref{resolest1}, take $|s| \ge 2K_2$ in \eqref{equ:proof3} and use \eqref{equ:proof6},
\begin{align*}
\begin{split}
\frac{\alpha_1}{2} \| v_x\|_{L_\eta^2}^2 & \le |\Re s| \| \v \|_{X_\eta}^2 + \frac{|s|}{2} \| \v \|_{X_\eta}^2 + \| \v\|_{X_\eta} \| \r \|_{X_\eta} \\
& \le |s| \| \v \|_{X_\eta}^2 + \frac{|s|}{2} \| \v \|_{X_\eta}^2 + \frac{|s|}{2} \| \v \|_{X_\eta}^2 + \frac{1}{2|s|} \| \r \|_{X_\eta}^2 \\
& = 2|s| \| \v \|_{X_\eta}^2 + \frac{1}{2 |s|} \| \r \|_{X_\eta}^2 \le \frac{K_8}{|s|} \| \r \|_{X_\eta}^2.
\end{split}
\end{align*}
It remains to prove \eqref{resolest2}.
The resolvent equation \eqref{resolvGl} implies the following equation
in $X_\eta$,
\begin{align*}
\vek{-v_{xx}}{0} = \vek{A^{-1}( -sv + cv_x + S_\omega v + Cv + r )}{A^{-1}( -s\rho + S_\omega \rho + C_\infty \rho + \zeta )}.
\end{align*}
Thus, with the help of \eqref{eq3:basic} we obtain for $|s| \ge 1$ the estimate
\begin{align*}
\| v_{xx} \|_{L_\eta^2}^2 & \le K_9 \left( |s|^2 \| \v \|_{X_\eta}^2 + \| v_x \|_{L_\eta^2}^2 + \| \v \|_{X_\eta}^2 + \| \r \|_{X_\eta}^2 \right) \\
& \le 2K_9 \left( |s|^2 \| \v \|_{X_\eta}^2 + |s| \| v_x \|_{L_\eta^2}^2 + \| \r \|_{X_\eta}^2 \right).
\end{align*}
When combined with \eqref{resolest1} this proves our assertion.
\end{proof}
In the next step we study the Fredholm property of the operator
$\L_\eta$ in \eqref{calL}. First we consider the operator $L$ from \eqref{LatL} on $L^2_\eta$ and therefore, as in \eqref{calL}, indicate the dependence on the weight $\eta$ by a subindex. So we introduce
\begin{align*}
L_\eta: H^2_\eta \rightarrow L^2_\eta, \quad v \mapsto Av_{xx} + cv_x + S_\omega v + Df(v_\star)v.
\end{align*}
Further, let us transform $L_\eta$ into unweighted spaces
\begin{equation} \label{eq3:unweighted}
L_{[\eta]}:H^2 \to L^2, \quad v \mapsto \eta L_\eta (\eta^{-1}v)=
Av_{xx} + B(\mu)v_x +C(\mu)v,
\end{equation}
where $q(x) = \sqrt{x^2 +1}$ and
\begin{equation*}
B(\mu,x) = cI+\frac{2 \mu x}{q(x)}A, \quad
C(\mu,x)= S_{\omega}+Df(v_{\star}(x))
- \frac{c \mu x}{q(x)}I + \big(\frac{\mu^2 x^2}{q(x)^2} -\frac{\mu}{q(x)}
+\frac{\mu x^2}{q(x)^3}\big)A.
\end{equation*}
The limits as $x \to \pm \infty$ of these matrices are given by
\begin{equation} \label{eq3:limits}
B_{\pm,\mu}=cI \pm 2 \mu A, \quad
C_{\pm,\mu} = S_{\omega}+ \mu^2 A +
\begin{cases} -c\mu I + Df(v_{\infty}), & \text{in case}\;\; +, \\
c\mu I + Df(0), &\text{in case}\; \; -. \end{cases}
\end{equation}
With these limit matrices we define the piecewise constant operator
\begin{align*}
L_{[\eta],\infty}:H^2 \to L^2,\quad v \to Av_{xx}+
(B_{-,\mu}\one_{\R_-}+B_{+,\mu} \one_{\R_+})
v_x +(C_{-,\mu}\one_{\R_-}+C_{+,\mu} \one_{\R_+})v.
\end{align*}
The following lemma shows that it is sufficient to analyze the Fredholm properties
of $L_{[\eta],\infty}$.
\begin{lemma} \label{lemma4.14}
Let Assumption \ref{A1} and \ref{A2} be satisfied, and assume
$0 \le \mu \le \mu_2$ with $\mu_2$ from Lemma \ref{aprioriest}. Then
for each $s \in \C$ the following are equivalent:
\begin{enumerate}[i)]
\item The operator $sI-L_{[\eta],\infty}: H^2 \rightarrow L^2$ is Fredholm of index $k$.
\item The operator $sI-L_\eta: H^2_\eta \rightarrow L^2_\eta$ is Fredholm of index $k$.
\item The operator $sI-\L_\eta: Y_\eta \rightarrow X_\eta$ is Fredholm of index $k$.
\end{enumerate}
\end{lemma}
\begin{proof} Both equivalences $(i) \Leftrightarrow (ii)$ and
$(ii) \Leftrightarrow (iii)$ follow from the invariance of
the Fredholm index under compact perturbations \cite[Ch.IX]{EdmundsEvans}.
For the first equivalence note that the multiplication operator
$v \mapsto m(\cdot) v$ is compact from $H^1$ to $L^2$ if $m \in L^{\infty}$ and
$\lim_{x \to \pm \infty}m(x) =0$ (\cite[Lemma 4.1]{BeynLorenz}).
This shows that the Fredholm property transfers from $L_{[\eta],\infty}$
to $L_{[\eta]}$ (see \eqref{eq3:unweighted}), and thus also to
$L_\eta:H^2_{\eta} \to L^2_{\eta}$.
For the second equivalence use the homeomorphism
$\mathcal{T}: X_{\eta} \to L_{\eta}^2 \times \R^2$,
$(v,\rho)^{\top} \to (v - \rho \hat{v},\rho)^{\top}$ and transform $\L_\eta$ into
the block operator
\begin{align*}
\mathcal{T} \L_\eta \mathcal{T}^{-1} = \begin{pmatrix} L_{\eta} & \mathcal{K}\\
0 & E_{\omega}
\end{pmatrix}, \quad \mathcal{K}= \hat{v}_{xx} A+ c \hat{v}_xI +
\hat{v}( Df(v_{\star})-Df(v_{\infty})).
\end{align*}
Since $\mathcal{K}$ is bounded in $L^2_{\eta}$ the result
follows from the Fredholm bordering lemma (\cite[Lemma 2.3]{Beyn90}).
\end{proof}
The Fredholm property of $sI - L_{[\eta],\infty}$ can be determined from
the first order system corresponding to $(sI -L_{[\eta],\infty})v=r$, i.e.
$w=(v,v_x)^{\top}$ and
\begin{equation*}
\begin{aligned}
(0,r)^{\top}&= w_x - (M_{-,\mu}(s) \one_{\R_-}+ M_{+,\mu}(s)\one_{\R_+})w, \\
M_{\pm,\mu}(s) & = \, \begin{pmatrix} 0 & I \\ A^{-1}(sI- C_{\pm,\mu}) &
- A^{-1}B_{\pm,\mu} \end{pmatrix}.
\end{aligned}
\end{equation*}
We define the {\it $\mu$-dependent Fredholm set} by
\begin{align*}
\Omega_{F}(\mu) = \{ s \in \C : M_{-,\mu}(s) \; \text{and} \;
M_{+,\mu}(s) \; \text{are hyperbolic} \}
\end{align*}
and denote by $m_{\mathfrak{s}}^{\pm}(s,\mu)$ the dimension of the stable subspace of
$M_{\pm,\mu}(s)$ for $s \in \Omega_{F}(\mu)$. Rewriting the eigenvalue problem for $M_{\pm,\mu}(s)$ in $\C^4$
as a quadratic eigenvalue problem in $\C^2$ shows that
$\Omega_{F}(\mu)= \C \setminus \sigma_{\mathrm{disp},\mu}(\L_\eta)$ holds
for the $\mu$-dependent dispersion set,
\begin{equation} \label{eq3:dispersionset}
\begin{aligned}
\sigma_{\mathrm{disp},\mu}(\L_\eta) = & \, \{s \in \C: \det(sI - D_{\pm}(\nu,\mu))
=0 \; \; \text{for some}\; \; \nu \in \R \; \; \text{and some sign} \;\;
\pm \}, \\
D_{\pm}(\nu,\mu)= & - \nu^2 A + i \nu B_{\pm,\mu} + C_{\pm,\mu}.
\end{aligned}
\end{equation}
The following Lemma is well-known and appears for example in
\cite[Lemma 3.1.10]{KapitulaPromislow}, \cite{Palmer88}, \cite[Sec. 3]{Sandstede02}.
\begin{lemma} \label{FredholmLetainfty}
Let Assumption \ref{A1} and \ref{A2} be satisfied, and let
$0 \le \mu \le \mu_2$ with $\mu_2$ from Lemma \ref{aprioriest}. Then the operator $sI - L_{[\eta],\infty}:H^2 \rightarrow L^2$ is a Fredholm operator if and only if
$s \in \Omega_F(\mu)$. If $s \in \Omega_F(\mu)$ then the Fredholm index is given by
\begin{align} \label{eq3:indexformula}
\mathrm{ind}(sI - L_{[\eta],\infty}) = m_\mathfrak{s}^+(s,\mu) - m_\mathfrak{s}^-(s,\mu).
\end{align}
\end{lemma}
\begin{remark} An intuitive argument for the formula \eqref{eq3:indexformula} is as follows.
The Fredholm index
measures the degrees of freedom of a linear problem minus the number of constraints. In this case there are $m_{\mathfrak{s}}^+(s,\mu)$ forward decaying modes and
$m -m_{\mathfrak{s}}^-(s,\mu)$ ($m=\dim(w)=4$) backward decaying modes, adding up to
$m_{\mathfrak{s}}^+(s,\mu)+m -m_{\mathfrak{s}}^-(s,\mu)$ degrees of freedom. The condition that
these modes fit together at the origin provides $m$ constraints
which then leads to the index formula \eqref{eq3:indexformula}.
\end{remark}
\begin{figure}[h!]
\begin{minipage}[t]{0.49\textwidth}
\centering
\includegraphics[scale=1]{sector_mu.eps}
\end{minipage}
\begin{minipage}[t]{0.49\textwidth}
\centering
\includegraphics[scale=1]{sector_0.eps}
\end{minipage}
\caption{The sectors $\mathcal{S}_{\varepsilon, \beta}(\mu)$, $\mu > 0$ (left) and $\mathcal{S}_{\varepsilon, \beta}(0)$ (right) from Theorem \ref{thm4.17}.}
\label{sectorials}
\end{figure}
Next we show how the Fredholm index $0$ domain extends into the left half plane.
\begin{theorem} \label{thm4.17}
Let Assumption \ref{A1}, \ref{A2} and \ref{A4} be satisfied and let $\mu_2$ be given
by Lemma \ref{aprioriest}. Then there exist constants $\mu_0 \in (0,\mu_2)$ and $ \beta, \varepsilon, \kappa > 0$ such that for each $0\le \mu \le \mu_0$ the open
set
$\Omega_F(\mu)$ has a unique connected component $\Omega_\infty(\mu)$ satisfying
\begin{equation} \label{eq3:sectorinomega}
\begin{aligned}
\S_{\varepsilon,\beta}(\mu) :=& \, \big\{ s \in \C: | \arg (s + \beta \mu) | \le \frac{\pi}{2} + \varepsilon \mu \big\} \subset \Omega_\infty(\mu),\quad
\text{if} \quad \mu>0, \\
\S_{\varepsilon,\beta}(0) :=& \, \big\{ s \in \C: \Re s \ge -
\kappa \min( |\Im s|,\beta)^2 + \varepsilon\min(\beta - |\Im s|,0) \big\} \subset \Omega_\infty(0), \quad \text{if}\quad \mu=0.
\end{aligned}
\end{equation}
Moreover, $\Omega_{\infty}(\mu)$ has the properties
\begin{enumerate}[i)]
\item For all $s \in \Omega_\infty(\mu)$ the operator $sI - \L_\eta: Y_\eta \rightarrow X_\eta$ is Fredholm of index $0$.
\item $\sigma_{\mathrm{ess}}(\L_\eta) \subseteq \C \backslash \Omega_\infty(\mu)$.
\end{enumerate}
\end{theorem}
Figure \ref{Figure:Fred} illustrates the spectral behavior for $\mu=0$, $\L = \L_\eta$ in case of
the quintic Ginzburg Landau equation \eqref{QCGL} with $\alpha=1$, $\beta_1= -\frac{1}{8}$, $\beta_3=1+i$, $\beta_5= -1 +i$. The dispersion set
$\sigma_{\mathrm{disp},0}(\L)$ consists of $4$ parabola-shaped curves. They
originate from
purely imaginary eigenvalues of $M_{-,0}(s)$ (red) and of
$M_{+,0}(s)$ (blue). Note that one of the latter curves has quadratic contact
with the imaginary axis. The numbers in the connected components
of $\C\setminus \sigma_{\mathrm{disp},0}(\L)$ denote the Fredholm index as calculated
from \eqref{eq3:indexformula}. The white components have index $0$ with
$\Omega_{\infty}(0)$ being the rightmost component. The essential spectrum
$\sigma_{\mathrm{ess}}(\L)$ (see \eqref{eq2:defessential}) is colored green.
Every $\mu >0$ sufficiently small shifts the spectrum to the left
(by $\approx -c \mu$) which allows to inscribe a proper
sector with tip at $-\beta \mu$ into the unbounded Fredholm $0$ component $\Omega_{\infty}(\mu)$. In case $\mu=0$ the sector is rounded quadratically
for $|\Im s| \le \beta$; see Figure \ref{sectorials}.
\begin{figure}[h!]
\centering
\begin{minipage}[t]{0.99\textwidth}
\centering
\includegraphics[scale=1]{Fred.eps}
\caption{Essential spectrum of the linearized operator $\L$ (green) and
dispersion set (red/blue) for \eqref{QCGL} with
$\alpha=1$, $\beta_1= -\frac{1}{8}$, $\beta_3=1+i$, $\beta_5= -1 +i$. The numbers in the connected components indicate the Fredholm index of $sI - \L$,
white regions have Fredholm index $0$.}\label{Figure:Fred}
\end{minipage}
\end{figure}
\begin{proof} We show $\S_{\varepsilon,\beta}(\mu) \subseteq
\C \setminus \sigma_{\mathrm{disp},\mu}(\L_\eta)=\Omega_F(\mu)$, so that \eqref{eq3:sectorinomega}
follows by taking $\Omega_{\infty}(\mu)$ to be the connected component
of $\Omega_F(\mu)$ which contains this sector.
From \eqref{SystemDef} one finds
\begin{align} \label{eq3:fderiv}
Df(v) = g(|v|^2) + 2 g'(|v|^2) \begin{pmatrix} v_1^2 & v_1 v_2 \\
v_1 v_2 & v_2^2 \end{pmatrix}.
\end{align}
From \eqref{eq3:limits},\eqref{eq3:dispersionset} we obtain
\begin{align*}
D_{-}(\nu,\mu)= - \nu^2 A + i \nu(cI-2 \mu A) + S_{\omega} + g(0) +
\mu^2 A + c \mu I.
\end{align*}
The dispersion set contains the two curves of eigenvalues $s(\nu,\mu)$ and
$\overline{s(\nu,\mu)}$ of $D_{-}(\nu,\mu)$ given for $\nu \in \R$ by
\begin{align*}
s(\nu,\mu)= (\mu^2- \nu^2) \alpha_1 + 2 \nu \mu \alpha_2 + g_1(0) + c \mu
+i \big((\mu^2- \nu^2) \alpha_2 + \nu c - 2 \nu \mu \alpha_1 + \omega +g_2(0)
\big).
\end{align*}
An elementary discussion shows that $\Re s(\nu,\mu) \le g_1(0) <0$
for all $\nu \in \R$ and $0 \le \mu \le \frac{c \alpha_1}{|\alpha|^2}$.
Moreover, for $|\nu|$ large, the values $s(\nu,\mu)$ lie in a sector which has an opening angle with the negative real axis of at most
$\arctan(\frac{|\alpha_2|}{\alpha_1}) < \frac{\pi}{2}$
uniformly for $\mu$ small. Thus there is a sector $\S_{\varepsilon,\beta}(\mu)$ as above
which has the two curves in its exterior. Next we obtain from
\eqref{eq3:limits}, \eqref{eq3:dispersionset}, \eqref{eq3:fderiv}
and Lemma \ref{lem:asym}
\begin{equation} \label{eq3:D+formula}
\begin{aligned}
D_+(\nu,\mu)= &\, \begin{pmatrix} \delta_1(\nu,\mu) + 2 \rho_1 & - \delta_2(\nu,\mu) \\
\delta_2(\nu,\mu) + 2 \rho_2 & \delta_1(\nu,\mu) \end{pmatrix}, \quad
\rho_j = g_j'(|v_{\infty}|^2)|v_{\infty}|^2, j=1,2, \\
\delta_1(\nu,\mu)=&\, (\mu^2-\nu^2) \alpha_1 + 2 i \nu \mu \alpha_1
+ i \nu c- c \mu, \quad \delta_2(\nu,\mu) =
(\mu^2-\nu^2)\alpha_2 + 2 i\nu \mu \alpha_2.
\end{aligned}
\end{equation}
The eigenvalues are
\begin{align} \label{eq3:D+eigenvalues}
s_{\pm}(\nu,\mu) = \delta_1(\nu,\mu) + \rho_1 \pm \big(\rho_1^2 -
2 \delta_2(\nu,\mu) \rho_2 - \delta_2^2(\nu,\mu)\big)^{1/2}.
\end{align}
Consider first the case $\mu=0$:
\begin{align*}
s_{\pm}(\nu,0) = - \alpha_1 \nu^2 + \rho_1 + i \nu c \pm R(\nu)^{1/2},
\quad R(\nu) = |\rho|^2 - (\rho_2 - \alpha_2 \nu^2)^2.
\end{align*}
For $ \nu \to \infty$ we obtain $\Re s_{\pm}(\nu,0) \sim - \alpha_1 \nu^2$
and $\Im s_{\pm}(\nu,0) \sim \nu c \pm |\alpha_2|\nu^2$, hence large
values lie in a sector opening to the left with angle $< \frac{\pi}{2}$.
Further, Assumption \ref{A2} yields $\Re s_-(\nu,0) \le -\alpha_1 \nu^2 + \rho_1 \le \rho_1<0$
for all $\nu\in \R$ (independently of the sign in front of $R(\nu)$).
Next we show that $r(\nu)= \Re s_+(\nu,0)$ has a unique global maximum at $\nu=0$, more precisely,
\begin{equation} \label{eq3:nubehave}
r(0)=0, \quad r''(0) < 0 \quad r'(\nu)\begin{cases} >0, & \nu \in (-\infty,0) \setminus \{-\nu_0\} , \\
=0, & \nu=0, \\ < 0, & \nu \in (0,\infty)\setminus \{\nu_0\} , \end{cases}
\end{equation}
where $\nu_0> 0$ is defined by
$ |\alpha_2| \nu_0^2 = |\rho| + \mathrm{sgn}(\alpha_2)\rho_2>0$ for
$\alpha_2 \neq 0$ and
$\nu_0=\infty$ in case $\alpha_2=0$. Note that $|\nu|> \nu_0$ holds
if and only if $R(\nu) <0$, and $r(\nu)=-\alpha_1 \nu^2+\rho_1$ for $|\nu|> \nu_0$.
Moreover, if $\alpha_2=0$ then
we have $r(\nu)=-\alpha_1 \nu^2$ . Thus \eqref{eq3:nubehave} holds in both cases.
It remains to consider $r(\nu)$ for $|\nu| < \nu_0$ and $\alpha_2 \neq 0$.
We obtain $R(0)=|\rho_1|^2$, $r(0)=0$ and
\begin{align*}
r'(\nu) = &\, 2 \nu \big( -\alpha_1 +
R(\nu)^{-1/2} \alpha_2 (\rho_2- \alpha_2
\nu^2)\big).
\end{align*}
This shows $r'(0)=0$ and
$|\rho_1|r''(0)=2(-\alpha_1|\rho_1|+\alpha_2 \rho_2) <0$
by Assumption \ref{A4}. If $r'$ has a further zero for $|\nu|<\nu_0$
then this implies $\alpha_2(\rho_2 - \alpha_2 \nu^2) >0$, $\mathrm{sgn}(\alpha_2)= \mathrm{sgn}(\rho_2)$, and
$|\alpha|^2(\rho_2- \alpha_2 \nu^2)^2= \alpha_1^2 |\rho|^2$.
By these sign conditions we have a unique square root given by
\begin{align*}
|\alpha|( \rho_2 - \alpha_2 \nu^2) = \mathrm{sgn}(\alpha_2) \alpha_1 |\rho|, \quad \text{or} \quad |\alpha||\alpha_2|\nu^2 =
|\alpha| |\rho_2| - \alpha_1|\rho|.
\end{align*}
But the last equation has no real solution $\nu$ since our Assumptions
\ref{A1}-\ref{A4} and $\alpha_2 \rho_2 >0$ imply
\begin{align*}
|\alpha|^2 \rho_2^2-\alpha_1^2 |\rho|^2 =\alpha_2^2 \rho_2^2 -
\alpha_1^2 \rho_1^2=( \alpha_2 \rho_2-\alpha_1 \rho_1)( \alpha_2 \rho_2+\alpha_1 \rho_1 ) <0.
\end{align*}
Since $r'$ has no further zeros in $(-\nu_0,0)\cup (0,\nu_0)$ the sign of $r'$
is determined by $r''(0)<0$.
Now consider $s_{\pm}(\nu,\mu)$ for small values $\mu>0$. For large values
of
$|\nu| $ the asymptotic behavior is only slightly modified to $\Re s_{\pm}(\nu,\mu) \sim - \alpha_1 \nu^2$
and $\Im s_{\pm}(\nu,\mu) \sim \nu(c+ 2 \mu \alpha_1) \pm |\alpha_2|\nu^2$,
and we find a proper sector enclosing the dispersion curves for $|\nu|\ge \nu_1$
uniformly for $0\le \mu \le \mu_1$. The curve
$s_-(\nu,0)$, $|\nu| \le \nu_1$ is bounded away from the imaginary axis, hence
it suffices to consider $s_+(\nu,\mu)$. From \eqref{eq3:D+formula},
\eqref{eq3:D+eigenvalues} one computes the partial derivatives
$s_{+,\nu}(0,0)=ic$, $s_{+,\mu}(0,0)= - c$, $s_{+,\nu \nu}(0,0)=2 |\rho_1|^{-1}(\rho_1 \alpha_1 + \rho_2 \alpha_2)$ and then by a Taylor expansion
\begin{align} \label{eq3:Taylors+}
s_+(\nu,\mu) = i c \nu - c \mu + |\rho_1|^{-1}(\rho_1 \alpha_1 + \rho_2 \alpha_2)\nu^2 + \mathcal{O}(|\nu|^3 + |\mu| |\nu| + |\mu|^2 ).
\end{align}
Note that we can estimate $C |\mu| |\nu| \le \frac{C}{4 \delta} \mu^2 + C \delta \nu^2$ and absorb $C \delta$ into the negative pre-factor of $\nu^2$ by taking $\delta$ small, and subsequently absorb $\frac{C}{4 \delta}\mu$ into
the negative pre-factor of $\mu$ by taking $\mu$ small.
Thus we can choose $\beta= \frac{c}{2}$ and determine $\varepsilon,\nu_2,\mu_2>0$ such
that
\begin{align*}
|\arg (s_+(\nu,\mu) + \beta \mu) | \ge \frac{\pi}{2} + \varepsilon \mu
\quad \text{for all } \quad |\nu| \le \nu_2, \quad 0 < \mu \le \mu_2.
\end{align*}
Finally, for $\nu_2 \le |\nu| \le \nu_1$, the curve $s_+(\nu,0)$ is bounded
away from the imaginary axis, and our assertion follows by continuity of $s_+$.
For $\mu=0$ the expansion \eqref{eq3:Taylors+} ensures $\Im s_+(\nu,0)= \nu c + \mathcal{O}(|\nu|^3)$ and the existence of some
$\beta>0$ such that
\begin{align*}
\Re s_+(\nu,0) \le - 2 \kappa |\Im s_+(\nu,0)|^2 \quad \text{for} \quad
|\Im s_+(\nu,0)| \le \beta, \quad \kappa = \frac{|\rho_1 \alpha_1 +\rho_2 \alpha_2|}{4 c^2 |\rho_1|}.
\end{align*}
The inclusion \eqref{eq3:sectorinomega} then follows for $\varepsilon$
sufficiently small. In addition, we require $\varepsilon \le \kappa \beta$
which implies
\begin{align} \label{eq3:sectormove}
\S_{\varepsilon,\tilde{\beta}}(0) \subset
\S_{\varepsilon,\beta}(0) \quad \text{for} \quad
0 < \tilde{\beta} < \beta.
\end{align}
For the proof of (i) note that the Fredholm index is constant in $\Omega_{\infty}(\mu)$. Therefore, it is enough to
consider $M_{\pm,\mu}(s)$ for large $s >0$:
\begin{align*}
\begin{pmatrix} I & 0 \\ 0 & s^{-1/2}I \end{pmatrix} M_{\pm,\mu}(s)
\begin{pmatrix} I & 0 \\ 0 & s^{1/2} I \end{pmatrix}=
s^{1/2}\Big[ \begin{pmatrix} 0 & I \\ A^{-1} & 0 \end{pmatrix}
+ s^{-1/2} \begin{pmatrix} 0 & 0 \\ - A^{-1} C_{\pm,\mu} &
- A^{-1} B_{\pm,\mu} \end{pmatrix} \Big].
\end{align*}
The leading matrix $\begin{pmatrix} 0 & I \\ A^{-1} & 0 \end{pmatrix}$
has a two-dimensonal stable subspace which belongs to the eigenvalues $- \alpha^{-1/2},-\bar{\alpha}^{-1/2}$,
and a two-dimensional unstable subspace which belongs to $\alpha^{-1/2},\bar{\alpha}^{-1/2}$.
These subspaces are only slightly perturbed for large $s$, and hence we obtain the
Fredholm index $0$ from Lemma \ref{FredholmLetainfty}.
We conclude the proof by noting that assertion (ii) follows from
our previous result and the definition of the essential spectrum,
cf. \eqref{eq2:defessential}.
\end{proof}
We continue with the
\begin{proof}[Proof of Lemma \ref{lemma4.18}]
Clearly, Theorem \ref{decay} ensures that the functions $\varphi_1$, $\varphi_2$ from \eqref{eq2:eigenfunctions} lie in every $Y_{\eta}$, where $\eta$ is defined by \eqref{eta} and $0 \le \mu < 2 \hat{\mu}$, and
both are eigenfunctions of $\L_\eta$ defined in
\eqref{calL}. Recall that $E_{\omega}=S_{\omega}+ Df(v_{\infty})=2
\begin{pmatrix} \rho_1 & 0 \\ \rho_2 & 0 \end{pmatrix}$ (see
\eqref{eq3:D+formula} for $\rho_1,\rho_2$) has the eigenvalues $0$ and $ 2 \rho_1 <0$ with
eigenvectors $(0,|v_{\infty}|)^{\top}= S_1 v_{\infty}$ and $(\rho_1,\rho_2)^{\top}$, respectively.
Now consider $(v,\zeta)^{\top} \in Y_{\eta}$ such that $\L_\eta (v,\zeta)^\top=0$.
From $E_{\omega}\zeta = 0$ we obtain
$\zeta = z S_1 v_{\infty}$ for some $z \in \C$. This shows $(u,0):= (v,\zeta)^{\top}
- z \varphi_2 \in \ker(\L_\eta)$ and $u = y v_{\star}'$ for some $y \in \C$ by
Assumption \ref{A5}. Therefore, we obtain $(v,\zeta)^{\top}=
z \varphi_2 + y \varphi_1$.
Now suppose $\L_\eta (v,\zeta)^{\top}= y \varphi_1 + z \varphi_2$ for some
$y,z \in \C$. Since $E_{\omega}$ has only simple eigenvalues we
conclude $z=0$ from the second component and then $y=0$ from Assumption
\ref{A5}. This proves \eqref{eq2:eigenfunctions}.
From Theorem \ref{thm4.17} we infer that $\L_\eta: Y_\eta \rightarrow X_\eta$ is Fredholm of index $0$ and hence also $L_\eta:H^2_{\eta}\to L^2_{\eta}$ by Lemma
\ref{lemma4.14}. Assumption \ref{A5} guarantees that $0$ is a simple
eigenvalue of $L_\eta:H^2_{\eta}\to L^2_{\eta}$. Note that we have
$v'_{\star} \in H^2_{\eta}$ and that there is no solution $y \in H^2_{\eta}$ of
$L_\eta y = v'_{\star}$ since this implies $y \in H^2$ and $y\in \ker(L^2)=\ker(L)$.
Simple eigenvalues are known to be isolated. This may be seen by applying
the inverse function theorem to
\begin{equation*}
T:H^2_{\eta}\times \R \to L^2_{\eta}\times \R, \quad
T\begin{pmatrix} v \\ s \end{pmatrix} =
\begin{pmatrix} L_\eta v - sv \\ (v'_{\star},v-v'_{\star})_{L^2}
\end{pmatrix} \; \text{at} \; \begin{pmatrix} v \\ s \end{pmatrix}=
\begin{pmatrix} v'_{\star} \\ 0 \end{pmatrix}.
\end{equation*}
Therefore, there exists some $s_0=s_0(\mu)>0$ such that
$L_\eta :H^2_{\eta}\to L^2_{\eta}$ has no eigenvalues with $|s|\le s_0$ except
$s=0$.
Finally, we prove assertion (ii) for
$\beta_0= \min(\frac{\varepsilon s_0}{2}, \beta_E,\tilde{\beta}, |\rho_1|)$ where $\beta_E$ is from Assumption
\ref{A5} and $\beta$, $\tilde{\beta}$ satisfy \eqref{eq3:sectorinomega}, \eqref{eq3:sectormove} as well as
$\tilde{\beta} < \min(\beta,\frac{\sqrt{3}-1}{2}s_0)$.
Consider $s \in \sigma_{\mathrm{pt}}(\L_\eta) \setminus \{0\}$ with
$\Re s \ge - \beta_0$ and eigenfunction $(v,\zeta)^{\top}\in Y_{\eta}$.
From $\sigma(E_{\omega})=\{0, 2 \rho_1\}$ we obtain $\zeta=0$ and thus $L_\eta v=s v$, $v \in H^2_{\eta}$. We claim that $s \in \S_{\varepsilon,\tilde{\beta}}(0)$.
For $\Re s \ge 0$ this is obvious, while for
$0 > \Re s \ge -\beta_0 \ge - \frac{s_0}{2}$ this follows from
\begin{align*}
|\Im s|\ge (s_0^2 - (\Re s)^2)^{1/2} \ge \frac{\sqrt{3}}{2}s_0 > \tilde{\beta}+ \frac{s_0}{2}, \quad \Re s \ge -\frac{\varepsilon s_0}{2} > - \kappa \tilde{\beta}^2 + \varepsilon(\tilde{\beta}- |\Im s|).
\end{align*}
By Theorem \ref{thm4.17}, $s \in \Omega_{\infty}(0)$ and
$s I - \L_\eta: Y_\eta \to X_\eta$ with $\mu = 0$ is Fredholm of index $0$.
By Lemma \ref{lemma4.14} the same holds for $sI-L:H^2\to L^2$ and we have
shown $s \in \sigma_{\mathrm{pt}}(L)$. This contradicts Assumption \ref{A5} since $\Re s \ge - \beta_E$.
\end{proof}
So far we determined the spectral properties of the linearized operator $\L_\eta$
and proved spectral stability of the extended system \eqref{CP} posed on the exponentially weighted spaces $X_\eta$ for positive but small $\mu > 0$. In particular, the essential spectrum of $\L_\eta$ is included in the left half plane as well as its point spectrum except the zero eigenvalue which has algebraic and geometric multiplicity $2$, i.e. $\ker (\L_\eta) = \ker (\L_\eta^2) = \mathrm{span} \{ \varphi_1, \varphi_2 \}$. Since $\L_\eta$ is Fredholm of index $0$ the same holds true for the (abstract) adjoint operator $\L_\eta^*: \D(\L_\eta^*) \subset X_\eta \rightarrow X_\eta$ and there are two normalized adjoint eigenfunctions $\psi_1, \psi_2 \in \D(\L^*)$ with
\begin{align} \label{adjointEF}
\ker (\L_\eta^*) = \mathrm{span} \{ \psi_1,\psi_2 \}, \quad (\psi_i, \varphi_j)_{X_\eta} = \delta_{ij}, \quad i,j = 1,2.
\end{align}
We define the map
\begin{align} \label{projector}
P_\eta: X_\eta \rightarrow X_\eta, \quad \v \mapsto (\psi_1,\v)_{X_\eta} \varphi_1 + (\psi_2,\v)_{X_\eta} \varphi_2.
\end{align}
Then $P_\eta$ is a projection onto $\ker (\L_\eta)$ and $X_\eta$ can be decomposed into
\begin{align*}
X_\eta = \mathcal{R} (P_\eta) \oplus \mathcal{R} (I - P_\eta) = \ker (\L_\eta) \oplus \ker(\L_\eta^*)^\bot.
\end{align*}
The subspace $\ker (\L_\eta^*)^\bot$ is invariant under $\L_\eta$ and we introduce
its intersection with the smooth spaces $X^k_\eta$, $k = 1,2$
\begin{align*}
V_\eta := \ker (\L_\eta^*)^\bot \subset X_\eta, \quad V^k_\eta := V_\eta \cap X^k_\eta.
\end{align*}
\sect{Semigroup estimates}
In the previous section we studied the spectrum of the linearized operator $\L_\eta$
on exponentially weighted spaces for positive but small $\mu > 0$ and derived a-priori estimates for the resolvent equation \eqref{resolvGl}.
Theorem \ref{thm4.17} shows that there is no essential spectrum
in the Fredholm $0$ component $\Omega_{\infty}(\mu)$ and thus also not in the sector
$\S_{\varepsilon, \beta} (\mu)$. When combined with Lemma \ref{aprioriest}
we obtain $\Omega_0 \subset \mathrm{res}(\L_\eta)$ for the domain $\Omega_0$
from \eqref{Omega0}. Further,
Lemma \ref{lemma4.18} shows
that the nonzero point spectrum is bounded away from the imaginary axis.
Thus we conclude from Lemma \ref{aprioriest} that $\L_\eta$ is a sectorial operator. By the classical semigroup theory,
(see \cite{Henry}, \cite{Miklavcic}, \cite{Pazy}) the operator $\L_\eta$ generates an analytic semigroup
$\{ e^{t\L_\eta} \}_{t \ge 0}$ on $X_\eta$ such that for any $\delta >0$ there exists a constant $C_{\delta}$ with $\| e^{t\L_\eta} \u \|_{X_\eta} \le C_{\delta} e^{\delta t} \| \u \|_{X_\eta}$, $t\ge 0$. Next we avoid the neutral modes of $\{ e^{t\L_\eta} \}_{t \ge 0}$ and restrict $\L_\eta$
to $V_\eta^2$ in order to have exponential decay.
\label{sec4}
\begin{theorem} \label{semigroup}
Let Assumption \ref{A1}, \ref{A2}, \ref{A4} and \ref{A5} be satisfied and let $0 < \mu \le \min(\mu_0,\mu_1)$ with $\mu_0$ from Theorem \ref{thm4.17} and $\mu_1$ from Lemma \ref{lemma4.18}. Then the linearized operator $\L_\eta: Y_\eta \rightarrow X_\eta$ generates an analytic semigroup $\{ e^{t \L_\eta}\}_{t \ge 0}$ on $X_\eta$. Moreover, there exist $K = K(\mu) \ge 1$ and $\nu = \nu(\mu) > 0$ such that for all $t \ge 0$ and $\mathbf{w} \in V^\ell_\eta$, $\ell = 0,1$ the following estimate holds
\begin{align} \label{semigroupest}
\| e^{t\L_\eta} \mathbf{w} \|_{X^\ell_\eta} \le K e^{-\nu t} \| \mathbf{w} \|_{X^\ell_\eta}.
\end{align}
\end{theorem}
\begin{proof}
The first assertion follows by the arguments above. Thus it remains to show the estimate \eqref{semigroupest}. For that purpose, we note that the restriction $\L_{V_\eta}$ of $\L_\eta$ to $V_\eta$ is a closed operator on $V_\eta$ with $\ker(\L_{V_\eta}) = \{ 0 \}$ and $\mathcal{R} (\L_{V_\eta}) = V_\eta$. Thus $\L_{V_\eta}$ is Fredholm of index $0$ and $0 \notin \sigma ( \L_{V_\eta} )$. Moreover, the projector $P_\eta$ from
\eqref{projector} commutes with $\L_\eta$ which leads to
$\mathrm{res}(\L_\eta) \subset \mathrm{res}( \L_{V_\eta} )$.
Therefore, by Theorem \ref{thm4.17}, Lemma \ref{lemma4.18} and Lemma \ref{aprioriest} we find $\varepsilon = \varepsilon(\mu)$, $\nu = \nu(\mu)$ and a sector $\Sigma_{\varepsilon,\nu} = \{ s \in \C: \mathrm{arg}(s + \nu) \le \frac{\pi}{2} + \varepsilon \} $ such that $\Sigma_{\varepsilon,\nu} \subset \mathrm{res}(\L_{V_\eta})$.
Further we can decrease $\varepsilon>0$ and take $R > 0$ sufficiently large so that $\Sigma_{\varepsilon,\nu} \cap \{|s|\ge R\} \subseteq \Omega_0$.
From Lemma \ref{aprioriest} and the fact that the resolvent is bounded in a compact
subset of the resolvent set we then find a constant $C = C(\mu) >0$ such that for all $\mathbf{w} \in V_\eta^{\ell}$ and $\ell = 0,1$ the following holds
\begin{align*}
\| (sI - \L_{V_\eta})^{-1} \mathbf{w}\|_{X^\ell_\eta} = \| (sI - \L_\eta)^{-1} \mathbf{w}\|_{X^\ell_\eta} \le C \| \mathbf{w} \|_{X^\ell_\eta} \quad & \forall s \in \Sigma_{\varepsilon, \nu} \cap \{ |s| \le R \}, \\
\| (sI - \L_{V_\eta})^{-1} \mathbf{w}\|_{X^\ell_\eta} = \| (sI - \L_\eta)^{-1} \mathbf{w}\|_{X^\ell_\eta} \le \frac{C}{|s|} \| \mathbf{w} \|_{X^\ell_\eta} \quad & \forall s \in \Sigma_{\varepsilon, \nu} \cap \{ |s| > R \}.
\end{align*}
Therefore, $\L_{V_\eta}$ is a sectorial operator on $V_\eta$ and
the representation of the semigroup
\begin{align*}
e^{t \L_{V_\eta}}= \int_{\Gamma_{\varepsilon,\nu}} (zI-\L_{V_\eta})^{-1} e^{t z} dz, \quad
\Gamma_{\varepsilon,\nu} = \{- \nu + r \exp(\mathrm{sgn}(r)i(\tfrac{\pi}{2}+ \varepsilon)) :
r \in \R \}
\end{align*}
leads in the standard way to the exponential estimate
\begin{align*}
\| e^{t\L_{\eta}} \mathbf{w} \|_{X^\ell_\eta} = \| e^{t\L_{V_\eta}} \mathbf{w} \|_{X^\ell_\eta} \le K e^{-\nu t} \| \mathbf{w} \|_{X^\ell_\eta}, \quad \mathbf{w} \in V^\ell_\eta,\; \ell=0,1.
\end{align*}
\end{proof}
\sect{Decomposition of the dynamics}
\label{sec5}
The nonlinear operator $\F$ on the right hand side of \eqref{CP} is equivariant w.r.t. the group action $a(\gamma)$ from \eqref{groupaction} of the group $\mathcal{G} = S^1 \times \R$. Every element $\gamma$ of the group can be represented by an angle $\theta$ and a shift $\tau$. The composition $\circ: \mathcal{G} \times \mathcal{G} \rightarrow \mathcal{G}$ of two elements $\gamma_1,\gamma_2 \in \mathcal{G}$ is given by
$
\gamma_1 \circ \gamma_2 = (\theta_1 + \theta_2 \, \text{mod} \, 2\pi, \tau_1 + \tau_2)$
and the inverse map $\gamma \mapsto \gamma^{-1}$ by $\gamma^{-1} = (-\theta \, \text{mod} \, 2\pi, -\tau)$. Both maps are smooth and $\mathcal{G}$ is a two dimensional $C^\infty$-manifold. An atlas of the group $\mathcal{G}$ is given by the two (trivial) charts $(U,\chi)$ and $(\tilde{U}, \tilde{\chi})$ defined by
\begin{alignat*}{2}
& U = \{ \gamma = (\theta \, \text{mod} \, 2\pi, \tau) \in \mathcal{G}: \theta \in (-\pi,\pi), \tau \in \R \}, && \quad \chi: U \rightarrow \R^2, \; \gamma \mapsto \chi(\gamma) = (\theta, \tau), \\
& \tilde{U} = \{ \gamma = (\theta \, \text{mod} \, 2\pi, \tau) \in \mathcal{G}: \theta \in (0,2\pi), \tau \in \R \}, && \quad \tilde{\chi}: \tilde{U} \rightarrow \R^2, \; \gamma \mapsto \tilde{\chi}(\gamma) = (\theta, \tau).
\end{alignat*}
We will always work with the chart $\chi$ since the arguments for $\tilde{\chi}$
will be almost identical.
Next we show smoothness of the group action $a(\cdot)\v$ in $\mathcal{G}$ depending on the regularity of $\v$.
\begin{lemma} \label{lemmagroup}
The group action $a:\mathcal{G} \rightarrow GL[X_\eta]$, $\gamma \mapsto a(\gamma)$ from \eqref{groupaction} is a homomorphism and $a(\gamma)Y_\eta = Y_\eta$, $\gamma \in \mathcal{G}$.
For $\v \in X_\eta$ the map $a(\cdot) \v: \mathcal{G} \rightarrow X_\eta$ is continuous
and for $\v \in Y_\eta$ it is continuously differentiable.
For $\gamma \in U$, $\chi(\gamma) = z$ the derivative applied to $h = (h_1,h_2)^\top \in \R^2$ is given by
\begin{align*}
(a(\cdot)\v \circ \chi^{-1})'(z)h = -h_1 a(\gamma) \mathbf{S}_1 \v - h_2 a(\gamma) \v_x,
\end{align*}
where $\mathbf{S}_1 \v = (S_1v,S_1\rho)^\top$, $\v_x = (v_x, 0)^\top$ for $\v = (v,\rho)^\top$.
\end{lemma}
The proof of Lemma \ref{lemmagroup} is straightforward and will be given in the Appendix. It is based on well known properties of translation and rotation on (weighted) Lebesgue and Sobolev spaces. Next recall the Cauchy problem \eqref{CP} with perturbed initial data
\begin{align*}
\u_t = \F(\u), \quad \u(0) = \v_\star + \v_0.
\end{align*}
We follow the approach in \cite{BeynLorenz}, \cite{Henry} and decompose the
dynamics of the solution into a motion along the group orbit $\{a(\gamma)\v_\star:\gamma \in \mathcal{G}\}$ of the wave and into a perturbation $\mathbf{w}$ in the space $V_\eta$. We use local coordinates in $U$ and write the solution $\u(t)$ as
\begin{align} \label{decomp}
\u(t) = a(\gamma(t)))\v_\star + \mathbf{w}(t), \quad \gamma(t) = \chi^{-1}(z(t)) \in U,\,\mathbf{w}(t) \in V_\eta
\end{align}
for $t \ge 0$. Thus $z$ describes the local coordinates of the motion on the group orbit $\mathcal{O}(\v_\star)$ given by $\gamma$ in the chart $(U,\chi)$ and $\mathbf{w} \in V_\eta$ is the difference of the solution to the group orbit in $V_\eta = \ker(\L_\eta^*)^\bot$. It turns out that the decomposition is unique as long as the solution stays in a small neighborhood of the group orbit and $\gamma$ stays in $U$. This will be guaranteed by taking sufficiently small initial perturbations $\v_0$. Let $P_\eta$ be the projector onto $\ker(\L_\eta)$ from \eqref{projector} and recall from \eqref{eq2:eigenfunctions} that $\ker(\L_\eta)$ is spanned
by the eigenfunctions $\varphi_2=\mathbf{S}_1 \v_\star$ and $\varphi_1=\v_{\star,x}$. Following \cite{BeynLorenz} we define
\begin{align} \label{Pi}
\Pi_\eta: \chi(U) \subset \R^2 & \rightarrow \ker(\L_\eta), \quad z \mapsto P_\eta ( a(\chi^{-1}(z)) \v_\star - \v_\star).
\end{align}
For simplicity of notation we frequently replace $\chi^{-1}(z)$ by $\gamma$
where $\gamma$ is always taken in our working chart $(U,\chi)$. The next
lemma uses $\Pi_\eta$ to show uniqueness of the decomposition \eqref{decomp} in a neighborhood of $\v_\star$.
\begin{lemma} \label{lemmatrafo}
Let Assumption \ref{A1}, \ref{A2}, \ref{A4} and \ref{A5} be satisfied and let $\mu_1$
be given by Lemma \ref{lemma4.18}. Then for all $0 < \mu \le \mu_1$ there is a zero neighborhood $W=W(\mu) \subset \chi(U)$ such that the map $\Pi_\eta: W \rightarrow \ker(\L_\eta)$ from \eqref{Pi} is a local diffeomorphism. Moreover, there is a zero neighborhood
$V=V(\mu) \subset \chi(U) \times V_\eta$ such that the transformation
\begin{align*}
T_\eta: V & \rightarrow X_\eta, \quad (z, \mathbf{w}) \mapsto a(\chi^{-1}(z)) \v_\star - \v_\star + \mathbf{w}
\end{align*}
is a diffeomorphism with the solution of $T_\eta(z,\mathbf{w}) = \v$ given by
\begin{align} \label{Tsolution}
z = \Pi_\eta^{-1} ( P_\eta \v), \quad \mathbf{w} = \v + \v_\star - a(\chi^{-1}(z)) \v_\star.
\end{align}
\end{lemma}
\begin{proof}
Since $0 < \mu \le \mu_1$ the projector $P_\eta$ and $\Pi_\eta$ are well defined. By Lemma \ref{lemmagroup} the group action $a$ is continuously differentiable
and so is $\Pi_\eta$. Further, $\Pi_\eta(0) = 0$ and its derivative is given
by $D\Pi_\eta (0)y = -y_1\varphi_1 - y_2 \varphi_2$, $y \in \R^2$ where $\varphi_1$, $\varphi_2$. Therefore, $D\Pi_\eta(0)$ is invertible on $\ker (\L_\eta)$
and the first assertion is a consequence of the inverse function theorem.
By the same arguments, $T_\eta$ is continuously differentiable,
$T_\eta(0,0) = 0$ and its derivative is given by
$DT_\eta(0,0) = \small{
\begin{pmatrix}D\Pi_\eta(0) & I_{V_{\eta}\to X_{\eta}} \end{pmatrix} }:\R^2 \times V_{\eta} \to X_{\eta}$ which is again invertible. Hence $T_\eta: V \rightarrow X_\eta$ is a diffeomorphism on a zero neighborhood $V \subset \chi(U) \times V_\eta$. Finally, applying
$P_\eta$ to $T_\eta(z,\mathbf{w}) = \v$ yields $z = \Pi_\eta^{-1} ( P_\eta \v )$ while
the second equation in \eqref{Tsolution} follows from the definition of $T_{\eta}$.
\end{proof}
Consider a smooth solution $\u(t)$, $t \in [0,t_\infty)$ of \eqref{CP} which stays close to the profile of the TOF. In particular, assume that
$\u(t) - \v_\star, t \in [0,t_{\infty})$ lies in the region where $T_\eta^{-1}$ exists by Lemma \ref{lemmatrafo}. Then there are unique $z(t) \in \chi(U)$ and $\mathbf{w}(t) = (w(t), \zeta(t))^\top \in V_\eta$ for $t \in [0,t_\infty)$ such that
\begin{align*}
\u(t) - \v_\star = T_\eta ( z(t), \mathbf{w}(t) ) \quad \forall t \in [0,t_\infty),
\end{align*}
and \eqref{decomp} holds. Taking the initial condition from \eqref{CP} into account yields for $t = 0$
\begin{align*}
\v_\star + \v_0 = \u(0) = a(\chi^{-1}(z(0))) \v_\star + \mathbf{w}(0),
\end{align*}
which leads to $\v_0 = T_\eta(z(0), \mathbf{w}(0))$. Therefore, the initial conditions for $z,\mathbf{w}$ are given by
\begin{align} \label{initcond}
z(0) = \Pi_\eta^{-1} ( P_\eta \v_0)=: z_0, \quad \mathbf{w}(0) = \v_0 + \v_\star - a(\chi^{-1}(z(0))) \v_\star =: \mathbf{w}_0.
\end{align}
Now we write the angular and translational components of $z$ explicitly as $z(t) = (\theta(t),\tau(t))$.
We insert the decomposition \eqref{decomp} into \eqref{CP} and obtain
\begin{align*}
0 = \u_t - \L_\eta \u = \frac{d}{dt} a(\chi^{-1}(z))\v_\star + \mathbf{w}_t & - a(\gamma)\vek{A v_{\star,xx} + c v_{\star,x} + S_\omega v_\star}{ S_\omega v_\infty } \\
& - \vek{A w_{xx} + c w_x + S_\omega w}{ S_\omega \zeta } - \vek{f(R_{\theta} v_\star(\cdot - \tau) + w)}{f(R_{\theta} v_\infty + \zeta)}.
\end{align*}
Using the equivariance of $\F$ and the derivative of the group action from Lemma \ref{lemmagroup}, leads to
\begin{align} \label{eq2}
\mathbf{w}_t = \L_\eta \mathbf{w} - a(\chi^{-1}(z))\varphi_1 \theta_t - a(\chi^{-1}(z))\varphi_2 \tau_t + r^{[f]}( z, \mathbf{w})
\end{align}
where the remainder $r^{[f]}(z,\mathbf{w})$ is given for $z=(\theta,\tau)$ and $\mathbf{w} \in V_{\eta}$ by
\begin{align*}
r^{[f]}( z , \mathbf{w} ) := \vek{f(R_{\theta} v_\star(\cdot - \tau) + w)}{f(R_{\theta} v_\infty + \zeta)} - \vek{f(R_{\theta} v_\star(\cdot - \tau))}{f(R_{\theta} v_\infty)} - \vek{Df(v_\star)w}{Df(v_\infty)\zeta}.
\end{align*}
Let us apply the projector $P_\eta$ to \eqref{eq2} and use $\mathbf{w}(t) \in V_\eta$, $t \in [0,t_\infty)$ and $P_\eta(\mathbf{w}_t - \L_\eta \mathbf{w})=0$ to obtain the equality
\begin{align} \label{eq3}
0 & = P_\eta r^{[f]}( z, \mathbf{w} ) - P_\eta a(\chi^{-1}(z))\varphi_1 \theta_t - P_\eta a(\chi^{-1}(z)) \varphi_2 \tau_t.
\end{align}
The next lemma shows that equation \eqref{eq3} can be written as an explicit ODE for $z = (\theta,\tau)$.
\begin{lemma} \label{Lemma4.5}
Let Assumption \ref{A1}, \ref{A2}, \ref{A4} and \ref{A5} be satisfied and let $\mu_1$
be given by Lemma \ref{lemma4.18}. Then for all $0 < \mu \le \mu_1$ the map
\begin{align*}
S_\eta(z): \R^2 & \rightarrow \ker(\L_\eta), \quad y \mapsto P_\eta a(\chi^{-1}(z)) \varphi_1 y_1 + P_\eta a(\chi^{-1}(z)) \varphi_2 y_2
\end{align*}
is continuous, linear and continuously differentiable w.r.t. $z\in (-\pi,\pi) \times \R$. Moreover, there is a zero neighborhood $V=V(\mu) \subset \R^2$ such that $S_\eta(z)^{-1}$ exists for all $z \in V$ and depends continuously on $z$.
\end{lemma}
\begin{proof}
Since $0 < \mu \le \mu_1$ the projector $P_\eta$ and the map $S_\eta(z)$ are well defined. Moreover, $S_\eta(z)$ is linear and continuous. Once more the smoothness of the group action, cf. Lemma \ref{lemmagroup}, implies that $S_\eta(z)$ is continuously differentiable w.r.t. $z$. Take $\mathbf{w} \in \ker (\L_\eta) =
\mathrm{span}\{ \varphi_1, \varphi_2\}$ and recall the adjoint eigenfunctions $\psi_1, \psi_2$ from \eqref{adjointEF}. We form the inner products in $X_\eta$ of the equation $S_\eta(z) y = \mathbf{w}$, $y \in \R^2$ with the adjoint eigenfunctions:
\begin{align*}
M(z) y = \vek{(\psi_1, \mathbf{w} )}{(\psi_2, \mathbf{w} )}, \quad M(z) = \begin{pmatrix}
(\psi_1, P_\eta a(\chi^{-1}(z)) \varphi_1 ) & (\psi_1, P_\eta a(\chi^{-1}(z)) \varphi_2 ) \\
( \psi_2, P_\eta a(\chi^{-1}(z)) \varphi_1 ) & ( \psi_2, P_\eta a(\chi^{-1}(z)) \varphi_2 )
\end{pmatrix}.
\end{align*}
Now $M(0) = I$ and $M(z)$ depends continuously on $z$. Then there exists a zero neighborhood $V \subset \R^2$ such that $M(z)$ is invertible and its inverse depends continuously on $z$. Finally, we obtain for $S_\eta(z)^{-1}$ the representation
\begin{align*}
S_\eta (z)^{-1} \mathbf{w} = M(z)^{-1} \vek{(\psi_1, \mathbf{w} )}{(\psi_2, \mathbf{w} )},
\end{align*}
which proves our assertion.
\end{proof}
As a consequence of Lemma \ref{Lemma4.5} we obtain from \eqref{eq3} and \eqref{initcond} the $z$-equation
\begin{align} \label{gamma1}
z_t = r^{[z]} ( z, \mathbf{w}), \quad z(0) = \Pi_\eta^{-1}(P_\eta \v_0),
\end{align}
where $r^{[z]}$ is given by
\begin{align} \label{gammaremainder}
r^{[z]}(z,\mathbf{w}):= S_\eta(z)^{-1} P_\eta r^{[f]} (z, \mathbf{w}).
\end{align}
This equation describes the motion of the solution projected onto the group orbit $\mathcal{O}(v_\star)$. The last step is to apply the projector $(I-P_\eta)$ to \eqref{eq2} and using \eqref{gamma1} to obtain the equation for the offset $\mathbf{w}$ from the group orbit:
\begin{align*}
\mathbf{w}_t & = \L_\eta \mathbf{w} + (I-P_\eta)r^{[f]} ( z, \mathbf{w} ) - (I-P_\eta)(a(\cdot)\v_\star \circ \chi^{-1} )(z) S_\eta(z)^{-1}P_\eta r^{[f]}( z, \mathbf{w} ) \\
& =: \L_\eta \mathbf{w} + r^{[w]} ( z, \mathbf{w} )
\end{align*}
with the remainder $r^{[w]}$ given by
\begin{align} \label{wremainder}
r^{[w]}(z, \mathbf{w}) := \Big((I-P_\eta) - (I-P_\eta)(a(\cdot)\v_\star \circ \chi^{-1} )(z) S_\eta(z)^{-1}P_\eta \Big) r^{[f]} ( z, \mathbf{w}).
\end{align}
Finally, the fully transformed system including initial values for $\mathbf{w}$ and $z$ reads as
\begin{alignat}{2}
\mathbf{w}_t & = \L_\eta \mathbf{w} + r^{[w]} ( z, \mathbf{w}), \quad & \quad \mathbf{w}(0) & = \v_0 + \v_\star - a(\Pi_\eta^{-1}(P_\eta \v_0)) \v_\star =: \mathbf{w}_0, \label{wDGL}\\
z_t & = r^{[z]} ( z, \mathbf{w}),& \quad z(0) & = \Pi_\eta^{-1} ( P_\eta \v_0)=: z_0. \label{gammaDGL}
\end{alignat}
Reversing the steps leading to \eqref{wDGL}, \eqref{gammaDGL} shows
that every local solution of this system leads to a solution of \eqref{CP}
close to $\v_{\star}$ via the transformation \eqref{decomp}.
\sect{Estimates of nonlinearities}
\label{sec6}
To study solutions of the system \eqref{wDGL}, \eqref{gammaDGL} we need to control the remaining nonlinearities $r^{[w]}, r^{[z]}$ from \eqref{wremainder} and \eqref{gammaremainder}. In this section we derive Lipschitz estimates with small Lipschitz constants for the nonlinearities in the space $X^1_\eta$. Of course the estimates will be guaranteed by the smoothness of $f$ from \eqref{perturbsys}. In particular, we can assume $f \in C^3$ by Assumption \ref{A1a}. However, our choice of the underlying space $X_\eta$ requires somewhat laborious calculations to derive the estimates. The main work is to take care of the offset which is hidden in the second component of elements in $X_\eta$.
\begin{lemma} \label{Lemma4.6}
Let Assumption \ref{A1}, \ref{A2}, \ref{A4} and \ref{A5} be satisfied and let $\mu_1$ be given by Lemma \ref{lemma4.18}. Then for every $0 < \mu \le \mu_1$ there are constant $C = C(\mu) > 0$ and $\delta = \delta(\mu) > 0$ such that for
all $z,z_1,z_2 \in B_{\delta}(0) \subset \R^2$ and $\mathbf{w}, \mathbf{w}_1, \mathbf{w}_2 \in B_{\delta}(0) \subset X^1_\eta$ the following holds:
\begin{small}
\begin{alignat*}{2}
& i) & \, & \| r^{[f]}(z, \mathbf{w}_1) - r^{[f]}(z,\mathbf{w}_2) \|_{X^1_\eta} \le C \left( |z| + \max \left\{ \| \mathbf{w}_1 \|_{X^1_\eta}, \| \mathbf{w}_2 \|_{X^1_\eta} \right\} \right) \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta}, \\
& ii) & \, & \| r^{[f]}(z_1, \mathbf{w}) - r^{[f]}(z_2,\mathbf{w}) \|_{X^1_\eta} \le C|z_1 - z_2|, \\
& iii)& \, & \| r^{[w]}(z, \mathbf{w}_1) - r^{[w]}(z, \mathbf{w}_2) \|_{X^1_\eta} \le C \left( |z| + \max \left\{ \| \mathbf{w}_1 \|_{X^1_\eta}, \| \mathbf{w}_2 \|_{X^1_\eta} \right\} \right) \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta}, \\
& iv)& \, & \| r^{[w]}(z_1, \mathbf{w}_2) - r^{[w]}(z_2, \mathbf{w}_2) \|_{X_\eta^1} \le C \left( |z_1 - z_2| + \| \mathbf{w}_1 - \mathbf{w}_2\|_{X^1_\eta} \right), \\
& v) & \, & | r^{[z]}(z_1, \mathbf{w}_1) - r^{[z]}(z_2, \mathbf{w}_2) | \le C \left( |z_1 - z_2| + \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta} \right).
\end{alignat*}
\end{small}
\end{lemma}
\begin{remark}
Note that $r^{[f]}(z,0) = 0$ holds so that the estimates i) and iii) imply
linear bounds for the the nonlinearities $r^{[f]}$ and $r^{[w]}$ in $B_\delta(0)$.
\end{remark}
\begin{proof}
Let $\delta$ be so small such that
$B_{\delta}(0)\subset \chi(U)$ and $B_\delta(0) \subset V$ with $V$ from Lemma \ref{Lemma4.5}. Then the remainders $r^{[f]}, r^{[w]},r^{[z]}$ are well defined by
Lemmas \ref{lemma4.18}, \ref{lemmatrafo}, and \ref{Lemma4.5}). Let us set $\gamma = \chi(z) = (\theta,\tau)$ as well as
$\gamma_i = \chi(z_i)= (\theta_i,\tau_i), i=1,2$. Further we write $\mathbf{w} = (w,\zeta)^\top$ and $\mathbf{w}_i= (w_i,\zeta_i)^\top$ for $i=1,2$. For the sake of notation we also write $a(\gamma) v = R_\theta v(\cdot - \tau)$ for a function $v : \R \rightarrow \R^2$. \\
Throughout the proof, $C = C(\mu)$ denotes a universal constant depending on $\mu$. The smoothness of $f$ and Sobolev embeddings imply
\begin{align} \label{Lemma4.6proof1}
\begin{split}
& \| Df(a(\gamma) v_\star) - Df(v_\star) \|_{L^\infty} \le C \| a(\gamma)v_\star - v_\star \|_{L^\infty} \\
& \le C \| a(\gamma) v_\star - R_\theta v_\infty \hat{v} - (v_\star - v_\infty \hat{v} ) \|_{L^\infty} + C |R_\theta v_\infty - v_\infty| \\
& \le C \| a(\gamma) v_\star - R_\theta v_\infty \hat{v} - (v_\star - v_\infty \hat{v} ) \|_{H^1} + C |R_\theta v_\infty - v_\infty| \\
& \le C \| a(\chi^{-1}(z)) \v_\star - \v_\star \|_{X^1_\eta} \le C|z|.
\end{split}
\end{align}
The last estimate follows from the smoothness of the group action; see Lemma \ref{lemmagroup}. Similarly, we find
\begin{align} \label{Lemma4.6proof1b}
| Df(R_\theta v_\infty) - Df(v_\infty)| \le C |z|
\end{align}
and
\begin{align} \label{Lemma4.6proof1c}
\begin{split}
& \| a(\gamma) v_\star - R_\theta v_\infty - (v_\star - v_\infty) \|_{L^2_\eta(\R_+)} \\
& \le \| a(\gamma) v_\star - R_\theta v_\infty \hat{v} - (v_\star - v_\infty \hat{v}) \|_{L^2_\eta(\R_+)} + |R_\theta v_\infty - v_\infty| \| \hat{v} -1 \|_{L^2_\eta(\R_+)} \\
& \le C \| a(\chi^{-1}(z))\v_\star - \v_\star \|_{X_\eta} \le C |z|.
\end{split}
\end{align}
By Theorem \ref{decay} we can also estimate
\begin{align} \label{Lemma4.6proof1d}
\| v_\star - v_\infty \|_{L^2_\eta(\R_+)} \le C.
\end{align}
In what follows these estimates will be used frequently. We start with \\
\textbf{i).} By definition and the triangle inequality we can split the left side of i) into
\begin{align*}
& \| r^{[f]}(z, \mathbf{w}_1) - r^{[f]}(z, \mathbf{w}_2) \|_{X^1_\eta} \\
& \le |f(R_\theta v_\infty + \zeta_1) - f(R_\theta v_\infty + \zeta_2) - Df(v_\infty)(\zeta_1 - \zeta_2)| \\
& \quad + \| f(a(\gamma) v_\star + w_1) - f(a(\gamma) v_\star + w_2) - Df(v_\star)(w_1-w_2) \\
& \quad \quad - \hat{v}[f(R_\theta v_\infty + \zeta_1) - f(R_\theta v_\infty + \zeta_2)- Df(v_\infty)(\zeta_1 - \zeta_2)] \|_{L^2_\eta} \\
& \quad + \| \partial_x[f(a(\gamma) v_\star + w_1) - f(a(\gamma) v_\star + w_2) - Df(v_\star)(w_1-w_2)] \|_{L^2_\eta} \\
& =: T_1 + T_2 + T_3.
\end{align*}
The first term $T_1$ is estimated by
\begin{align*}
& T_1 = | f(R_\theta v_\infty + \zeta_1) - f(R_\theta v_\infty + \zeta_2) - Df(v_\infty)(\zeta_1 - \zeta_2)| \\
& \le \int_{0}^1 |Df(R_\theta v_\infty + \zeta_2 + (\zeta_1 - \zeta_2)s) -Df(v_\infty)| ds |\zeta_1 - \zeta_2| \\
& \le \left( \int_0^1 |Df(R_\theta v_\infty + \zeta_2 + (\zeta_1 - \zeta_2)s) -Df(R_\theta v_\infty)| ds + |Df(R_\theta v_\infty) - Df(v_\infty)| \right) |\zeta_1 - \zeta_2| \\
& \le C \left( \int_0^1 |\zeta_2 - (\zeta_1 - \zeta_2)s| ds + |R_\theta v_\infty - v_\infty|\right) |\zeta_1 - \zeta_2| \\
& \le C \left( |z| + \max \{ |\zeta_1|, |\zeta_2| \} \right) |\zeta_1 - \zeta_2| \le C \left( |z| + \max \left\{ \| \mathbf{w}_1 \|_{X^1_\eta}, \|\mathbf{w}_2\|_{X^1_\eta} \right\} \right) \|\mathbf{w}_1 - \mathbf{w}_2\|_{X^1_\eta}.
\end{align*}
For the second term $T_2$ we have
\begin{align*}
& \| f(a(\gamma) v_\star + w_1) - f(a(\gamma) v_\star + w_2) - Df(v_\star)(w_1-w_2) \\
& \qquad \qquad - \hat{v} [f(R_\theta v_\infty + \zeta_1) - f(R_\theta v_\infty + \zeta_2)- Df(v_\infty)(\zeta_1 - \zeta_2)] \|_{L^2_\eta} \\
& = \Big\| \int_0^1 Df(a(\gamma)v_\star + w_2 +(w_1-w_2)s) - Df(v_\star) ds (w_1 - w_2) \\
& \qquad \qquad - \hat{v} \int_0^1 Df(R_\theta v_\infty + \zeta_2 + (\zeta_1 - \zeta_2)s) -Df(v_\infty) ds (\zeta_1 - \zeta_2) \Big\|_{L^2_\eta} \\
& \le \Big\| \int_0^1 Df(a(\gamma)v_\star + w_2 +(w_1-w_2)s) - Df(a(\gamma) v_\star) ds (w_1 - w_2) \\
& \qquad \qquad - \hat{v} \int_0^1 Df(R_\theta v_\infty + \zeta_2 + (\zeta_1 - \zeta_2)s) -Df(R_\theta v_\infty) ds (\zeta_1 - \zeta_2) \Big\|_{L^2_\eta} \\
& \qquad + \| [Df(a(\gamma) v_\star) - Df(v_\star)](w_1 - w_2) - \hat{v}[Df(R_\theta v_\infty) - Df(v_\infty)](\zeta_1 - \zeta_2) \|_{L^2_\eta} \\
& =: T_4 + T_5.
\end{align*}
$T_5$ is bounded by another two terms
\begin{align*}
T_5 & \le \| [Df(a(\gamma) v_\star) - Df(v_\star)](w_1 - \hat{v}\zeta_1 - w_2 + \hat{v} \zeta_2) \|_{L^2_\eta} \\
& \quad + \| [Df(a(\gamma) v_\star) - Df(v_\star) - Df(R_\theta v_\infty) + Df(v_\infty)] (\zeta_1 - \zeta_2) \hat{v} \|_{L^2_\eta} =: T_6 + T_7.
\end{align*}
Using \eqref{Lemma4.6proof1} we have
\begin{align*}
T_6 \le C |z| \| w_1 - \hat{v} \zeta_1 - w_2 + \hat{v} \zeta_2 \|_{L^2_\eta} \le C |z| \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta}.
\end{align*}
We bound $T_7$ by two terms, one for the negative and one for the positive half-line:
\begin{align*}
T_7 & \le \| [Df(a(\gamma) v_\star) - Df(v_\star) - Df(R_\theta v_\infty) + Df(v_\infty)] (\zeta_1 \hat{v} - \zeta_2 \hat{v}) \|_{L^2_\eta(\R_-)} \\
& \quad + \| [Df(a(\gamma) v_\star) - Df(v_\star) - Df(R_\theta v_\infty) + Df(v_\infty)] (\zeta_1 \hat{v} - \zeta_2 \hat{v}) \|_{L^2_\eta(\R_+)} =: T_8 + T_9
\end{align*}
Now, \eqref{Lemma4.6proof1}, \eqref{Lemma4.6proof1b} imply
\begin{align*}
T_8 & \le \| Df(a(\gamma)v_\star) - Df(v_\star) - Df(R_\theta v_\infty) + Df(v_\infty) \|_{L^\infty} |\zeta_1 - \zeta_2| \| \hat{v}\|_{L^2_\eta(\R_-)} \\
& \le C |z| |\zeta_1 - \zeta_2| \le C | z | \| \mathbf{w}_ 1- \mathbf{w}_2 \|_{X^1_\eta}.
\end{align*}
We use the abbreviations $\chi_1(s) := v_\star + s(a(\gamma)v_\star - v_\star)$, $\chi_2(s):=v_\infty + s(R_\theta v_\infty - v_\infty)$, $s \in [0,1]$ and \eqref{Lemma4.6proof1c}, \eqref{Lemma4.6proof1d} to obtain
\begin{align*}
T_9 & = \| [Df(a(\gamma)v_\star) - Df(v_\star) - Df(R_\theta v_\infty) + Df(v_\infty)](\zeta_1 - \zeta_2)\hat{v} \|_{L^2_\eta(\R_+)} \\
& \le \Big\| \int_0^1 D^2f(\chi_1(s))[a(\gamma)v_\star - v_\star,(\zeta_1 - \zeta_2)\hat{v}] ds \\
& \qquad - \int_0^1 D^2f(\chi_2(s))[R_\theta v_\infty - v_\infty,(\zeta_1 - \zeta_2)\hat{v}] ds \Big\|_{L^2_\eta(\R_+)}\\
& \le \Big\| \int_0^1 D^2f(\chi_1(s))[a(\gamma)v_\star - v_\star - R_\theta v_\infty + v_\infty, (\zeta_1 - \zeta_2)\hat{v}] ds \Big\|_{L^2_\eta(\R_+)} \\
& \quad \quad + \Big\| \int_0^1 [D^2f(\chi_1(s))-D^2f(\chi_2(s))][R_\theta v_\infty + v_\infty, (\zeta_1 - \zeta_2)\hat{v}] ds \Big\|_{L^2_\eta(\R_+)} \\
& \le C \Big( \| a(\gamma)v_\star - R_\theta v_\infty - (v_\star - v_\infty) \|_{L^2_\eta(\R_+)} \\
& \qquad + \Big\| \int_0^1 \chi_1(s) - \chi_2(s) ds \Big\|_{L^2_\eta(\R_+)} |R_\theta v_\infty - v_\infty| \Big) |\zeta_1 - \zeta_2| \\
& \le C |z| \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta}.
\end{align*}
To estimate $T_4$ we use the abbreviations $w(s) := w_2 + (w_1-w_2)s$, $\zeta(s) := \zeta_2 + (\zeta_1-\zeta_2)s$, $s \in [0,1]$ and obtain
\begin{align*}
T_4 & = \Big\| \int_0^1 Df(a(\gamma)v_\star + w(s)) - Df(a(\gamma) v_\star) ds (w_1 - w_2) \\
& \qquad \qquad - \hat{v} \int_0^1 Df(R_\theta v_\infty + \zeta(s)) -Df(R_\theta v_\infty) ds (\zeta_1 - \zeta_2) \Big\|_{L^2_\eta} \\
& \le \Big\| \int_0^1 Df(a(\gamma) v_\star + w(s)) - Df(a(\gamma) v_\star) ds (w_1 - \zeta_1 \hat{v} - w_2 + \zeta_2 \hat{v}) \Big\|_{L^2_\eta} \\
& \qquad \qquad + \Big\| \int_0^1 Df(a(\gamma) v_\star + w(s)) - Df(a(\gamma) v_\star) \\
& \qquad \qquad \qquad \qquad - Df(R_\theta v_\infty + \zeta(s)) + Df(R_\theta v_\infty) ds (\zeta_1 -\zeta_2) \hat{v}\Big\|_{L^2_\eta} \\
& =: T_{10} + T_{11}.
\end{align*}
Now for every $s \in [0,1]$ we have
\begin{align} \label{Lemma4.6proof1e}
\begin{split}
& \| Df(a(\gamma) v_\star + w(s)) - Df(a(\gamma) v_\star) \|_{L^\infty} \le C \| w_2 + s(w_1 - w_2) \|_{L^\infty} \\
& \le C \max \left\{ \| w_1\|_{L^\infty}, \|w_2 \|_{L^\infty}\right\} \le C \max \left\{ \| \mathbf{w}_1\|_{X^1_\eta}, \|\mathbf{w}_2 \|_{X^1_\eta} \right\},
\end{split}
\end{align}
where we used that the Sobolev embedding implies for $i \in \{1,2\}$
\begin{align*}
\| w_i \|_{L^\infty} \le \| w_i -\zeta_i \hat{v} \|_{L^\infty} + |\zeta_i| & \le C \| w_i -\zeta_i \hat{v} \|_{H^1} + |\zeta_i| \le C \| \mathbf{w}_i \|_{X^1_\eta}.
\end{align*}
Then \eqref{Lemma4.6proof1e} yields
\begin{align*}
T_{10} & \le \int_0^1 \| Df(a(\gamma) v_\star + w(s)) - Df(a(\gamma) v_\star) \|_{L^\infty} ds \| w_1 - \zeta_1 \hat{v} - w_2 + \zeta_2 \hat{v} \|_{L^2_\eta} \\
& \le C \max \left\{ \| \mathbf{w}_1 \|_{X^1_\eta}, \| \mathbf{w}_2 \|_{X^1_\eta} \right\} \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta}.
\end{align*}
Further,
\begin{align*}
T_{11} & \le \Big\| \int_0^1 Df(a(\gamma) v_\star + w(s)) - Df(a(\gamma) v_\star) \\
& \qquad \qquad - Df(R_\theta v_\infty + \zeta(s)) + Df(R_\theta v_\infty) ds (\zeta_1 -\zeta_2) \hat{v}\Big\|_{L^2_\eta(\R_-)} \\
& + \Big\| \int_0^1 Df(a(\gamma) v_\star + w(s)) - Df(a(\gamma) v_\star) \\
& \qquad \qquad - Df(R_\theta v_\infty + \zeta(s)) + Df(R_\theta v_\infty) ds (\zeta_1 -\zeta_2) \hat{v}\Big\|_{L^2_\eta(\R_+)} \\
& =: T_{12} + T_{13}.
\end{align*}
We write $\kappa(s) := a(\gamma)v_\star + w(s) - R_\theta v_\infty - \zeta(s)$. Then for $s \in [0,1]$ there holds
\begin{align*}
& \Big\| Df( a(\gamma) v_\star + w(s)) - Df(a(\gamma) v_\star) - Df(R_\theta v_\infty + \zeta(s)) + Df(R_\theta v_\infty) \Big\|_{L^\infty} \\
& = \Big\| \int_0^1 D^2f(R_\theta v_\infty+ \zeta(s) + \kappa(s) \tau)[\kappa(s), \cdot] \\
& \qquad \qquad - D^2f(R_\theta v_\infty + (a(\gamma) v_\star - R_\theta v_\infty)\tau ) [a(\gamma) v_\star - R_\theta v_\infty, \cdot] d\tau \Big\|_{L^\infty} \\
& \le \Big\| \int_0^1 D^2f(R_\theta v_\infty+ \zeta(s) + \kappa(s) \tau)[w(s)-\zeta(s), \cdot] d\tau \Big\|_{L^\infty} \\
& \qquad + \Big\| \int_0^1 \big( D^2f(R_\theta v_\infty+ \zeta(s) + \kappa(s) \tau) \\
& \qquad \qquad - D^2f(R_\theta v_\infty + (a(\gamma) v_\star - R_\theta v_\infty)\tau \big)[a(\gamma)v_\star - R_\theta v_\infty, \cdot] d\tau \Big\|_{L^\infty} \\
& \le C \| w(s) - \zeta(s) \|_{L^\infty} + C \int_0^1 \| \zeta(s) - (w(s) - \zeta(s)) \tau \|_{L^\infty} d\tau \le C \max \left\{ \| w_1 \|_{L^\infty}, \| w_2 \|_{L^\infty} \right\} \\
& \le C \max \left\{ \| \mathbf{w}_1 \|_{X^1_\eta}, \| \mathbf{w}_2 \|_{X^1_\eta} \right\},
\end{align*}
where we used $|\zeta_i| \le \| w_i \|_{L^\infty}$, $i= 1,2$. So we conclude
\begin{align*}
T_{12} \le C \max \left\{ \| \mathbf{w}_1 \|_{X^1_\eta}, \| \mathbf{w}_2 \|_{X^1_\eta} \right\} |\zeta_1 - \zeta_2|.
\end{align*}
Similarly, for every $s \in [0,1]$,
\begin{align*}
& \Big\| Df( a(\gamma) v_\star + w(s)) - Df(a(\gamma) v_\star) - Df(R_\theta v_\infty + \zeta(s)) + Df(R_\theta v_\infty) \Big\|_{L^2_\eta(\R_+)} \\
& \le C \| w(s) - \zeta(s) \|_{L^2_\eta(\R_+)} + C \int_0^1 \| \zeta(s) - (w(s) - \zeta(s)) \tau \|_{L^\infty} d\tau \| a(\gamma)v_\star - R_\theta v_\infty \|_{L^2_\eta(\R_+)} \\
& \le C \max \left\{ \| w_1- \zeta_1 \|_{L^2_\eta(\R_+)}, \| w_2- \zeta_2 \|_{L^2_\eta(\R_+)} \right\} + C \max \left\{ \| w_1 \|_{L^\infty}, \| w_2 \|_{L^\infty} \right\} \\
& \le C \max \left\{ \| \mathbf{w}_1 \|_{X^1_\eta}, \| \mathbf{w}_2 \|_{X^1_\eta} \right\}.
\end{align*}
This yields the estimate for $T_{13}$
\begin{align*}
T_{13} \le C \max \left\{ \| \mathbf{w}_1 \|_{X^1_\eta}, \| \mathbf{w}_2 \|_{X^1_\eta} \right\} |\zeta_1 - \zeta_2|.
\end{align*}
Summarizing, we have shown
\begin{align*}
T_2 & = \| f(a(\gamma) v_\star + w_1) - f(a(\gamma) v_\star + w_2) - Df(v_\star)(w_1-w_2) \\
& \quad \qquad - \hat{v}(R_\theta v_\infty + \zeta_1) - f(R_\theta v_\infty + \zeta_2)- Df(v_\infty)(\zeta_1 - \zeta_2)] \hat{v} \|_{L^2_\eta} \\
& \le C \left( |z| + \max \left\{ \| \mathbf{w}_1 \|_{X^1_\eta}, \| \mathbf{w}_2 \|_{X^1_\eta} \right\} \right) \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta}.
\end{align*}
It remains to estimate the derivative given by $T_3$. We have
\begin{align*}
T_3 & = \left\| \partial_x \Big[ f(a(\gamma) v_\star + w_1) - f(a(\gamma) v_\star + w_2) - Df(v_\star)(w_1 - w_2) \Big] \right\|_{L^2_\eta} \\
& = \| Df(a(\gamma) v_\star + w_1)w_{1,x} + Df(a(\gamma) v_\star + w_1) a(\gamma)v_{\star,x} \\
& \quad - Df(a(\gamma) v_\star + w_2)w_{2,x} - Df(a(\gamma) v_\star + w_2) a(\gamma)v_{\star,x} \\
& \quad - D^2f(v_\star)[w_1 - w_2, v_{\star,x}] - Df(v_\star)(w_1-w_2)_x \|_{L^2_\eta} \\
& = \left\| [ Df(a(\gamma)v_\star + w_1) - Df(a(\gamma) v_\star + w_2) ] w_{1,x} \right\|_{L^2_\eta} \\
& \quad + \left\| [ Df(a(\gamma) v_\star + w_2) - Df(v_\star) ] (w_1 - w_2)_x \right\|_{L^2_\eta} \\
& \quad + \left\| [ Df(a(\gamma) v_\star + w_1) - Df(a(\gamma) v_\star + w_2) ] (a(\gamma) v_\star - v_\star)_x \right\|_{L^2_\eta} \\
& \quad + \left\| [ Df(a(\gamma) v_\star + w_1) - Df(a(\gamma) v_\star + w_2) ] v_{\star,x} - D^2f(v_\star) [w_1 - w_2, v_{\star,x}] \right\|_{L^2_\eta} \\
& = I_1 + I_2 + I_3 + \left\| \int_0^1 D^2f(a(\gamma) v_\star + w_2 + (w_1 - w_2)s) - D^2f(v_\star) ds [w_1 - w_2, v_{\star,x}] \right\|_{L^2_\eta} \\
& = I_1 + I_2 + I_3 +I_4.
\end{align*}
Now
\begin{align*}
I_1 & \le \| Df(a(\gamma) v_\star + w_1) - Df(a(\gamma) v_\star + w_2) \|_{L^\infty} \| w_{1,x} \|_{L^2_\eta} \\
& \le C \| w_1 - w_2 \|_{L^\infty} \| w_{1,x} \|_{L^2_\eta} \le C \max \left\{ \| \mathbf{w}_1 \|_{X^1_\eta}, \| \mathbf{w}_2 \|_{X^1_\eta} \right\} \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta}.
\end{align*}
In the same fashion we obtain
\begin{align*}
I_2 & \le \| Df(a(\gamma) v_\star + w_2) - Df(v_\star) \|_{L^\infty} \| (w_1 - w_2)_x \|_{L^2_\eta} \\
& \le C \left( \| a(\gamma) v_\star - v_\star \|_{L^\infty} + \| w_2 \|_{L^\infty} \right) \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta} \\
& \le C \left( |z| + \max \left\{ \| \mathbf{w}_1 \|_{X^1_\eta}, \| \mathbf{w}_2 \|_{X^1_\eta} \right\} \right) \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta}
\end{align*}
and for $I_3$,
\begin{align*}
I_3 & \le \| Df(a(\gamma) v_\star + w_1) - Df(a(\gamma)v_\star + w_2) \|_{L^\infty} \| a(\gamma) v_{\star,x} - v_{\star,x} \|_{L^2_\eta} \\
& \le C \| w_1 - w_2 \|_{L^\infty} \| a(\gamma) v_{\star,x} - v_{\star,x} \|_{L^2_\eta} \le C |z| \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta}.
\end{align*}
For $I_4$ we have
\begin{align*}
I_4 & \le C \left( \| a(\gamma) v_\star - v_\star\|_{L^\infty} + \max\{ \| w_1\|_{L^\infty}, \| w_2\|_{L^\infty} \}\right) \| w_1 - w_2 \|_{L^\infty} \| v_{\star,x} \|_{L^2_\eta} \\
& \le C \left( |z| + \max \left\{ \| \mathbf{w}_1 \|_{X^1_\eta}, \| \mathbf{w}_2 \|_{X^1_\eta} \right\} \right) \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta}.
\end{align*}
Hence
\begin{align*}
T_3 & = \left\| \Big[ f(a(\gamma) v_\star + w_1) - f(a(\gamma) v_\star + w_2) - Df(v_\star)(w_1 - w_2) \Big]_x \right\|_{L^2_\eta} \\
& \le C \left( |z| + \max \left\{ \| \mathbf{w}_1 \|_{X^1_\eta}, \| \mathbf{w}_2 \|_{X^1_\eta} \right\} \right) \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta}.
\end{align*}
Finally we have shown
\begin{align*}
& \left\| r^{[f]}(z, \mathbf{w}_1) - r^{[f]}(z, \mathbf{w}_2) \right\|_{X^1_\eta} \le C \left( |z| + \max \left\{ \| \mathbf{w}_1 \|_{X^1_\eta}, \| \mathbf{w}_2 \|_{X^1_\eta} \right\} \right) \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta}.
\end{align*}
\textbf{ii).} As in i) we frequently use the mean value theorem and the smoothness of $f$ which follows from Assumption \ref{A1}. First, we estimate
\begin{align*}
& \| r^{[f]}(z_1, \mathbf{w}) - r^{[f]}(z_2, \mathbf{w}) \|_{X^1_\eta} \\
& = \left\| \vek{f(a(\gamma_1)v_\star + w) - f(a(\gamma_1) v_\star) - f(a(\gamma_2) v_\star + w) - f(a(\gamma_2)v_\star)}{f(R_{\theta_1}v_\infty + \zeta) - f(R_{\theta_1} v_\infty) - f(R_{\theta_2} v_\infty + \zeta) - f(R_{\theta_2}v_\infty)} \right\|_{X^1_\eta} \\
& \le | f(R_{\theta_1} v_\infty + \zeta) - f(R_{\theta_2} v_\infty + \zeta) | + | f(R_{\theta_1} v_\infty) - f(R_{\theta_2} v_\infty) | \\
& \quad + \| f(a(\gamma_1)v_\star + w) - f(a(\gamma_2)v_\star + w) - \hat{v} [f(R_{\theta_1}v_\infty+\zeta) - f(R_{\theta_2}v_\infty + \zeta)] \|_{L^2_\eta}\\
& \quad + \| f(a(\gamma_1)v_\star) - f(a(\gamma_2)v_\star) - \hat{v}(f(R_{\theta_1} v_\infty) - f(R_{\theta_2}v_\infty)) \|_{L^2_\eta} \\
& \quad + \| \partial_x [f(a(\gamma_1) v_\star + w) - f(a(\gamma_2)v_\star + w)]\|_{L^2_\eta} + \| \partial_x[f(a(\gamma_1) v_\star) - f(a(\gamma_2)v_\star)]\|_{L^2_\eta} \\
& =: J_1 + J_2 + J_3 + J_4 + J_5 + J_6.
\end{align*}
The smoothness of $f$ implies
\begin{align*}
J_1 & = | f(R_{\theta_1}v_\infty +\zeta) - f(R_{\theta_2} v_\infty + \zeta)| \le C |R_{\theta_1}v_\infty - R_{\theta_2}v_\infty| \le C |z_1 - z_2|.
\end{align*}
The same holds true for $\zeta = 0$ so that $J_2 \le C |z_1 - z_2|$. Write $\kappa_1(s) := a(\gamma_2) v_\star + w +(a(\gamma_1)v_\star - a(\gamma_2)v_\star)s$ and $\kappa_2(s) := R_{\theta_2}v_\infty + \zeta + (R_{\theta_1} v_\infty - R_{\theta_2}v_\infty)s$, $s \in [0,1]$ and obtain for $J_3$,
\begin{align*}
J_3 & = \| f(a(\gamma_1)v_\star + w) - f(a(\gamma_2)v_\star + w) - \hat
v[f(R_{\theta_1}v_\infty + \zeta) + f(R_{\theta_2}v_\infty + \zeta) ] \|_{L^2_\eta} \\
& = \Big\| \int_0^1 Df(a(\gamma_2) v_\star + w +(a(\gamma_1)v_\star - a(\gamma_2)v_\star)s) (a(\gamma_1) v_\star - a(\gamma_2) v_\star) ds \\
& \quad - \hat{v} \int_0^1 Df(R_{\theta_2} v_\infty + \zeta +(R_{\theta_1}v_\infty - R_{\theta_2}v_\infty)s)(R_{\theta_1} v_\infty - R_{\theta_2} v_\infty ) ds \Big\|_{L^2_\eta} \\
& \le \Big\| \int_0^1 Df(\kappa_1(s)) (a(\gamma_1)v_\star - R_{\theta_1} v_\infty \hat{v} - a(\gamma_2)v_\star + R_{\theta_2} v_\infty \hat{v}) ds \Big\|_{L^2_\eta} \\
& \quad + \Big\| \int_0^1 [ Df(\kappa_1(s)) - Df(\kappa_2(s)) ](R_{\theta_1} v_\infty \hat{v} - R_{\theta_2}v_\infty \hat{v})ds \Big\|_{L^2_\eta} =: J_7 + J_8.
\end{align*}
We estimate $J_7$ by
\begin{align*}
J_7 & \le C \| a(\gamma_1) v_\star - R_{\theta_1} v_\infty \hat{v} - a(\gamma_2) v_\star + R_{\theta_2} v_\infty \hat{v} \|_{L^2_\eta} \\
& \le C \| a(\chi^{-1}(z_1)) \v_\star - a(\chi^{-1}(z_2)) \v_\star \|_{X_\eta} \\
& \le C |z_1 - z_2|
\end{align*}
and bound $J_8$ by two terms
\begin{align*}
J_8 & \le \Big\| \int_0^1 [ Df(\kappa_1(s)) - Df(\kappa_2(s)) ](R_{\theta_1} v_\infty \hat{v} - R_{\theta_2}v_\infty \hat{v})ds \Big\|_{L^2_\eta(\R_-)} \\
& \quad + \Big\| \int_0^1 [ Df(\kappa_1(s)) - Df(\kappa_2(s)) ](R_{\theta_1} v_\infty \hat{v} - R_{\theta_2}v_\infty \hat{v})ds \Big\|_{L^2_\eta(\R_+)} = J_9 + J_{10}.
\end{align*}
Then
\begin{align*}
J_9 \le C \| \hat{v} \|_{L^2_\eta(\R_-)} |R_{\theta_1} v_\infty - R_{\theta_2} v_\infty | \le C | z_1 - z_2|
\end{align*}
and for $J_{10}$
\begin{align*}
J_{10} & \le C |R_{\theta_1} v_\infty - R_{\theta_2} v_\infty| \int_0^1 \|\kappa_1(s) - \kappa_2(s) \|_{L^2_\eta} ds \le C |z_1 - z_2|.
\end{align*}
Thus we have shown $J_3 \le C |z_1 - z_2|$. In particular the estimates hold for $w = 0$, $\zeta = 0$. Therefore we also have shown $J_4 \le C |z_1 - z_2|$ and it remains to estimate the spatial derivatives $J_5$ and $J_6$. We note that for arbitrary $u \in L^2_\eta$ we have by Sobolev embedding
\begin{align*}
& \| [Df(a(\gamma_1) v_\star + w) - Df(a(\gamma_2) v_\star + w)] u \|_{L^2_\eta} \le C \| a(\gamma_1) v_\star - a(\gamma_2)v_\star \|_{L^\infty} \|u \|_{L^2_\eta} \\
& \le C \| u \|_{L^2_\eta} \left( \| a(\gamma_1)v_\star - R_{\theta_1} v_\infty \hat{v} - a(\gamma_2)v_\star + R_{\theta_2 }v_\infty \hat{v} \|_{L^\infty} + \| R_{\theta_1} v_\infty \hat{v} - R_{\theta_2} v_\infty \hat{v}\|_{L^\infty} \right) \\
& \le C \| u \|_{L^2_\eta} |z_1 - z_2|.
\end{align*}
This implies
\begin{align*}
J_5 & \le \| [Df(a(\gamma_1) v_\star + w) - Df(a(\gamma_2) v_\star + w)] w_x \|_{L^2_\eta} \\
& \quad + \| Df(a(\gamma_1) v_\star + w)a(\gamma_1)v_{\star,x} - Df(a(\gamma_2) v_\star + w) a(\gamma_2)v_{\star,x} \|_{L^2_\eta} \\
& \le C \| w_x \|_{L^2_\eta}|z_1 -z_2| + \| [Df(a(\gamma_1) v_\star + w)- Df(a(\gamma_2) v_\star + w)] a(\gamma_1)v_{\star,x} \|_{L^2_\eta} \\
& \quad+ C \| a(\gamma_1)v_{\star,x} - a(\gamma_2)v_{\star,x} \|_{L^2_\eta} \\
& \le C \left( \| w_x \|_{L^2_\eta} + \| a(\gamma_1) v_{\star,x} \|_{L^2_\eta} \right) |z_1 - z_2| + C \| a(\gamma_1)v_{\star,x} - a(\gamma_2) v_{\star,x} \|_{L^2_\eta} \le C |z_1 - z_2|.
\end{align*}
In particular the same holds true for $w = 0$ and we observe $J_6 \le C |z_1 - z_2|$. Summarizing, we have shown
\begin{align*}
\| r^{[f]}(z_1, \mathbf{w}) - r^{[f]}(z_2, \mathbf{w}) \|_{X^1_\eta} \le C_1 |z_1 - z_2|.
\end{align*}
\textbf{iii).} Since the group action is smooth and since $P_\eta$ from \eqref{projector} is a projector we have
\begin{align*}
\Big\| \Big((I-P_\eta) - (I-P_\eta) \big(a(\cdot)\v_\star \circ \chi^{-1}\big)(z) S_\eta(z)^{-1}P_\eta \Big) \u \Big\|_{X^1_\eta} \le C \| \u \|_{X^1_\eta} \quad \forall \u \in X^1_\eta.
\end{align*}
Now the claim follows from i). \\
\textbf{iv).} By the smoothness of the group action and Lemma \ref{Lemma4.5} the
function $\big(a(\cdot) \v_\star \circ \chi^{-1}\big)(z) S_\eta (z)^{-1}$ is continuously differentiable in $z$. Therefore, we obtain the local Lipschitz estimate
\begin{align} \label{Lemma4.6proof1f}
\| \big(a(\cdot) \v_\star \circ \chi^{-1} \big)(z_1)S_\eta(z_1)^{-1}\mathbf{w} - \big(a(\cdot) \v_\star \circ \chi^{-1} \big)(z_2) S_\eta(\gamma_2)^{-1}\mathbf{w} \|_{X^1_\eta} \le C |z_1 - z_2 | \| \mathbf{w} \|_{X^1_\eta}.
\end{align}
Then we use \eqref{Lemma4.6proof1f} and i) to see
\begin{align*}
& \| r^{[w]}(z_1,\mathbf{w}) - r^{[w]}(z_2,\mathbf{w}) \|_{X^1_\eta} \\
& \le C \| r^{[f]}(z_1,\mathbf{w}) - r^{[f]}(z_2,\mathbf{w}) \|_{X^1_\eta} \\
& \quad + \| \big(a(\cdot) \v_\star \circ \chi^{-1}\big)(z_1)S_\eta(z_1)^{-1}P_\eta r^{[f]}(z_1,\mathbf{w}) - \big(a(\cdot) \v_\star \circ \chi^{-1}\big)(z_2)S_\eta(z_2)^{-1}P_\eta r^{[f]}(z_1,\mathbf{w}) \|_{X^1_\eta} \\
& \le C |z_1 - z_2|.
\end{align*}
Now we obtain using ii) and iii)
\begin{align*}
& \| r^{[w]}(z_1,\mathbf{w}_1) - r^{[w]}(z_2,\mathbf{w}_2) \|_{X^1_\eta} \\
& \le \| r^{[w]}(z_1,\mathbf{w}_1) - r^{[w]}(z_2,\mathbf{w}_1) \|_{X^1_\eta} + \| r^{[w]}(z_2,\mathbf{w}_1) - r^{[w]}(z_2,\mathbf{w}_2) \|_{X^1_\eta} \\
& \le C \left( |z_1 - z_2| + \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta} \right).
\end{align*}
\textbf{v).} Similar to iv) we have by Lemma \ref{Lemma4.5} that $S_\eta(z)^{-1}$ is locally Lipschitz w.r.t. $z$. Then we obtain
\begin{align*}
& \left| r^{[z]}(z_1, \mathbf{w}_1) - r^{[z]}(z_2, \mathbf{w}_2) \right| = \left| S_\eta(z_1)^{-1} P_\eta r^{[f]}(z_1, \mathbf{w}_1) - S_\eta(z_2)^{-1} P_\eta r^{[f]}(z_2, \mathbf{w}_2) \right| \\
& \le C \left\| r^{[f]}(z_1,\mathbf{w}_1) - r^{[f]}(z_2,\mathbf{w}_2) \right\|_{X^1_\eta} + \left| (S_\eta(z_1)^{-1} - S_\eta(z_2)^{-1}) P_\eta r^{[f]}(z_2,\mathbf{w}_2) \right| \\
& \le C_4 \left( |z_1 - z_2| + \| \mathbf{w}_1 - \mathbf{w}_2 \|_{X^1_\eta} \right).
\end{align*}
\end{proof}
\sect{Nonlinear stability}
\label{sec7}
In this section we complete the proof of the main Theorem \ref{Theorem4.10} according to the following strategy. For sufficiently small initial perturbation
$\v_0$ in \eqref{CP} we show existence of a local mild solution of the
corresponding integral equations of the decomposed system \eqref{wDGL}, \eqref{gammaDGL} which reads as
\begin{align}
\mathbf{w}(t) & = e^{t\L_\eta} \mathbf{w}_0 + \int_0^t e^{(t-s)\L_\eta} r^{[w]}(z(s),\mathbf{w}(s)) ds, \label{integralwDGL}\\
z(t) & = z_0 + \int_0^t r^{[z]} ( z(s), \mathbf{w}(s)) ds. \label{integralgammaDGL}
\end{align}
A Gronwall estimate then shows that the solution exists for all times, that
the perturbation $\mathbf{w}$ decays exponentially and that $z$ converges to the
coordinates of an asymptotic phase. Combining the results with the regularity
theory for mild solutions we infer
Theorem \ref{Theorem4.10} and thus nonlinear stability of traveling oscillating fronts.
\begin{lemma} \label{Lemma4.7}
Let Assumption \ref{A1}, \ref{A2}, \ref{A4} and \ref{A5} be satisfied and let $0 < \mu \le \min(\mu_0,\mu_1)$ with $\mu_0$ from Theorem \ref{thm4.17} and $\mu_1$ from Lemma \ref{lemma4.18}. Then for every $0 < \varepsilon_1 < \delta$ and $0 < 2K \varepsilon_0 \le \delta$ with $K$ from Theorem \ref{semigroup} and $\delta$ from Lemma \ref{Lemma4.6}, there exists $t_\star = t_\star(\varepsilon_0,\varepsilon_1, \mu) > 0$ such that for all initial values $(z_0,\mathbf{w}_0) \in \R^2 \times V^1_\eta$ with
\begin{align*}
\| \mathbf{w}_0 \|_{X^1_\eta} < \varepsilon_0, \quad |z_0| < \varepsilon_1
\end{align*}
there exists a unique solution $(z,\mathbf{w}) \in C([0,t_\star), \R^2 \times V^1_\eta)$ of \eqref{integralwDGL}, \eqref{integralgammaDGL} with
\begin{align*}
\| \mathbf{w}(t) \|_{X^1_\eta} \le 2K\varepsilon_0, \quad |z(t)| \le 2 \varepsilon_1, \quad t \in [0,t_\star).
\end{align*}
In particular, $t_\star$ is independent of $(z_0,\mathbf{w}_0) \in B_{\varepsilon_1}(0) \times B_{\varepsilon_0}(0)$.
\end{lemma}
\begin{proof}
Take $\nu = \nu(\mu) > 0$ from Theorem \ref{semigroup}, $C = C(\mu) > 0$ from Lemma \ref{Lemma4.6} and let $t_\star$ be so small such that the following conditions are satisfied:
\begin{align} \label{condtstar}
t_\star < \frac{\varepsilon_1}{2C ( \varepsilon_1 + K \varepsilon_0 )}, \quad t_\star + \frac{2K}{\nu} (1- e^{-\nu t_\star}) < \frac{1}{C}.
\end{align}
The proof employs a contraction argument in the space $Z := C([0,t_\star),\R^2 \times V^1_\eta)$ equipped with the supremums norm $\| (z,\mathbf{w}) \|_{Z} := \sup_{t \in [0,t_\star)} \{ |z(t)| + \| \mathbf{w}(t) \|_{X^1_\eta} \}$. Define the map $\Gamma: Z \rightarrow Z$ given by the right hand side of \eqref{integralwDGL}, \eqref{integralgammaDGL}. We show that $\Gamma$ is a contraction on the closed set
\begin{align*}
B := \{ (z,\mathbf{w}) \in Z: \| \mathbf{w}(t) \|_{X^1_\eta} \le 2K\varepsilon_0,\, |z(t)| \le 2\varepsilon_1,\, t\in [0,t_\star) \} \subset Z.
\end{align*}
Let $(z,\mathbf{w}) \in B$. By using the estimates from Theorem \ref{semigroup}, Lemma \ref{Lemma4.6} and \eqref{condtstar} we obtain for all $0 \le t < t_\star$
\begin{align*}
& \left\| e^{t\L_\eta} \mathbf{w}_0 + \int_0^t e^{(t-s)\L_\eta} r^{[w]}(z(s),\mathbf{w}(s)) ds \right\|_{X^1_\eta} \\
& \le K e^{-\nu t} \varepsilon_0 + K \int_0^t e^{-\nu (t-s)} \| r^{[w]}(z(s),\mathbf{w}(s)) \|_{X^1_\eta} ds \\
& \le K e^{-\nu t} \varepsilon_0 + K C \int_0^t e^{-\nu (t-s)} \| \mathbf{w}(s) \|_{X^1_\eta} ds \\
& \le K\varepsilon_0 + \frac{2K^2 C \varepsilon_0}{\nu}(1 - e^{-\nu t_\star}) \le 2K\varepsilon_0
\end{align*}
and
\begin{align*}
\left| z_0 + \int_0^t r^{[z]}(z(s),\mathbf{w}(s))ds \right| & \le \varepsilon_1 + \int_0^t |r^{[z]}(z(s),\mathbf{w}(s))| ds \\
& \le \varepsilon_1 + C \int_0^t |z(s)| + \| \mathbf{w}(s) \|_{X^1_\eta} ds \\
& \le \varepsilon_1 + 2C(\varepsilon_1 + K\varepsilon_0) t_\star \le 2 \varepsilon_1.
\end{align*}
Hence $\Gamma$ maps $B$ into itself. Further, for $(z_1,\mathbf{w}_1),(z_2,\mathbf{w}_2) \in B$ and $0 \le t < t_\star$ we can estimate
\begin{align*}
& \| \Gamma(z_1,\mathbf{w}_1) - \Gamma(z_2,\mathbf{w}_2) \|_{Z} \\
& \le \sup_{t \in [0,t_\star)} \Big\{ \int_0^t | r^{[z]}(z_1(s),\mathbf{w}_1(s)) - r^{[z]}(z_2(s),\mathbf{w}_2(s))|ds \\
& \qquad + \int_0^t K e^{-\nu (t-s)} \| r^{[w]}(z_1(s),\mathbf{w}_1(s)) - r^{[w]}(z_2(s),\mathbf{w}_2(s)) \|_{X^1_\eta} ds \Big\} \\
& \le \Big( C t_\star + \frac{KC}{\nu } (1- e^{-\nu t_\star}) \Big) \| (z_1-z_2, \mathbf{w}_1-\mathbf{w}_2)\|_{Z}.
\end{align*}
By condition \eqref{condtstar}, the map $\Gamma$ is a contraction on $B$
and the assertion follows from the contraction mapping theorem.
\end{proof}
We use the following Gronwall lemma from \cite[Lemma 6.3]{BeynLorenz}.
\begin{lemma} \label{Gronwall}
Suppose $\varepsilon, \nu, C,\tilde{C} > 0$ such that
\begin{align*}
C \ge 1, \quad \varepsilon \le \frac{\nu}{16 \tilde{C} C}
\end{align*}
and let $\varphi \in C([0,t_\infty),[0,\infty))$ for some $0 < t_\infty \le \infty$ satisfying
\begin{align*}
\varphi(t) \le C \varepsilon e^{-\nu t} + \tilde{C} \int_0^t e^{-\nu(t-s)} \left( \varphi(s)^2 + \varepsilon \varphi(s) \right) ds, \quad \forall t \in [0,t_\infty).
\end{align*}
Then for all $0 \le t < t_\infty$ there holds
\begin{align*}
\varphi(t) \le 2C\varepsilon e^{-\frac{3}{4} \nu t}.
\end{align*}
\end{lemma}
Next we prove the stability result for the $(z,\mathbf{w})$-systems
\eqref{integralwDGL}, \eqref{integralgammaDGL} and \eqref{wDGL}, \eqref{gammaDGL}.
The Gronwall estimate ensures that the solution from Lemma \ref{Lemma4.7} can not reach the boundary of the region of existence and therefore exists for all times. Moreover, the perturbation $\mathbf{w}$ of the TOF decays exponentially.
Regularity of the solution will follow by standard results from \cite{Amann} and \cite{Henry}. As in \cite{Amann}, we denote by $C^\alpha$, $\alpha \in (0,1)$ the space of H\"older continuous functions and by $C^{1 + \alpha}$ the space of differentiable functions with H\"older continuous derivative.
\begin{theorem} \label{Theorem4.9}
Let Assumption \ref{A1}, \ref{A2}, \ref{A4} and \ref{A5} be satisfied and let $0 < \mu \le \min(\mu_0,\mu_1)$ with $\mu_0$ from Theorem \ref{thm4.17} and $\mu_1$ from Lemma \ref{lemma4.18}. Then there are $\varepsilon(\mu), \beta(\mu) > 0$ such that for all initial values $(z_0, \mathbf{w}_0) \in \R^2 \times V^2_\eta$ with $\| (z_0, \mathbf{w}_0)\|_{\R^2 \times X_\eta^1} < \varepsilon$ the following statements hold:
\begin{enumerate}[i)]
\item There are unique
\begin{align*}
\mathbf{w} \in C^{\alpha}((0,\infty),V^2_\eta) \cap C^{1+\alpha}((0,\infty),V_\eta) \cap C^1([0,\infty),V_\eta), \quad z \in C^1([0,\infty),\R^2),
\end{align*}
for arbitrary $\alpha \in (0,1)$, satisfying \eqref{wDGL} in $X_\eta$ and \eqref{gammaDGL} in $\R^2$.
\item There exist $z_\infty = z_\infty(z_0,\mathbf{w}_0) \in \R^2$ and $K_0 = K_0(\mu) \ge 1$ such that for all $t \ge 0$
\begin{align*}
\| \mathbf{w}(t) \|_{X_\eta^1} + |z(t) - z_\infty| \le K_0 e^{-\beta t} \|(z_0,\mathbf{w}_0) \|_{\R^2 \times X^1_\eta}, \quad |z_\infty| \le (K_0+1) \| (z_0,\mathbf{w}_0) \|_{\R^2 \times X^1_\eta}.
\end{align*}
\end{enumerate}
\end{theorem}
\begin{proof}
Recall the constants $K = K(\mu), \nu = \nu(\mu)$ from Theorem \ref{semigroup} and $C = C(\mu), \delta = \delta(\mu)$ from Lemma \ref{Lemma4.6}. We choose $C_0, \varepsilon, \tilde{\varepsilon}> 0$ such that $0 < 2K \tilde{\varepsilon} < \delta$ and
\begin{align} \label{condeps}
\varepsilon < \min \left( \frac{\delta}{C_0}, \frac{\tilde{\varepsilon}}{4K}, \frac{\nu}{16K^2CC_0} \right), \quad C_0 > 2 + \frac{16C K}{3\nu}.
\end{align}
Let us abbreviate $\xi_0 :=\| (z_0,w_0) \|_{\R^2 \times X^1_\eta} < \varepsilon$ and set
\begin{align*}
t_\infty : = \sup \Big\{ T>0:\, \exists! (z,\mathbf{w}) \in C([0,T),& \R^2 \times V_\eta) \text{ satisfying \eqref{integralwDGL}, \eqref{integralgammaDGL} on } [0,T) \\
& \text{and } \| \mathbf{w}(t) \|_{X^1_\eta} \le K\tilde{\varepsilon},\, |z(t)| \le C_0 \xi_0,\, t \in [0,T) \Big\}.
\end{align*}
Then Lemma \ref{Lemma4.7} with $\varepsilon_0 = \tilde{\varepsilon}$ and $\varepsilon_1 = \frac{C_0 \xi_0}{2} < \delta$ implies $t_\infty \ge t_\star = t_\star(\varepsilon_0,\varepsilon_1,\mu)$ and we denote the unique solution by $(z,\mathbf{w})$. Using Theorem \ref{semigroup} and Lemma \ref{Lemma4.6} we estimate for all $0 \le t < t_\infty$
\begin{align*}
\| \mathbf{w}(t) \|_{X^1_\eta} & \le \| e^{t\L_\eta} \mathbf{w}_0 \|_{X^1_\eta} + \int_0^t \| e^{(t-s)\L_\eta} r^{[w]}(z(s),\mathbf{w}(s)) \|_{X^1_\eta} ds \\
& \le Ke^{-\nu t} \| \mathbf{w}_0 \|_{X^1_\eta} + \int_0^t e^{-\nu (t-s)} \| r^{[w]}(z(s),\mathbf{w}(s) \|_{X^1_\eta} ds \\
& \le Ke^{-\nu t} \| \mathbf{w}_0 \|_{X^1_\eta} + KC \int_0^t e^{-\nu(t-s)} \left( |z(s)| + \| \mathbf{w}(s) \|_{X^1_\eta} \right) \| \mathbf{w}(s) \|_{X^1_\eta} ds \\
& \le Ke^{-\nu t} \xi_0 + KC C_0 \int_0^t e^{-\nu(t-s)} \left( \xi_0 + \| \mathbf{w}(s) \|_{X^1_\eta} \right) \| \mathbf{w}(s) \|_{X^1_\eta} ds.
\end{align*}
Then the Gronwall estimate in Lemma \ref{Gronwall} implies due to \eqref{condeps}
\begin{align} \label{stability:proof1}
\| \mathbf{w}(t) \|_{X^1_\eta} \le 2K e^{-\frac{3}{4}\nu t} \xi_0 < 2K e^{-\frac{3}{4}\nu t} \varepsilon < \frac{\tilde{\varepsilon}}{2}, \quad t \in [0,t_\infty).
\end{align}
This yields
\begin{align}
\begin{split} \label{stability:proof2}
|z(t)| & \le |z_0| + \int_0^t |r^{[z]}(z(s),\mathbf{w}(s))|ds \le \xi_0 + C \int_0^t \| \mathbf{w}(s) \|_{X^1_\eta} ds \\
& \le \xi_0 + 2KC\xi_0 \int_0^t e^{-\frac{3}{4}\nu s} ds \le \xi_0 + \frac{8CK}{3\nu} \xi_0 < \frac{C_0 \xi_0}{2}, \quad t \in [0,t_\infty).
\end{split}
\end{align}
We show that $t_\infty < \infty$ leads to a contradiction. The estimates \eqref{stability:proof1}, \eqref{stability:proof2} imply
\begin{align*}
\| \mathbf{w}(t_\infty - \tfrac{1}{2}t_\star) \|_{X^1_\eta} < \frac{\tilde{\varepsilon}}{2} =\varepsilon_0, \quad |z(t_\infty - \tfrac{1}{2}t_\star)| < \frac{C_0 \xi_0}{2} = \varepsilon_1.
\end{align*}
Now we can apply Lemma \ref{Lemma4.7} once again to the integral equations \eqref{integralwDGL}, \eqref{integralgammaDGL} with $\mathbf{w}_0 = \mathbf{w}(t_\infty - \tfrac{1}{2}t_\star)$ and $z_0 = z(t_\infty - \tfrac{1}{2}t_\star)$ and obtain a solution $(\tilde{z}, \tilde{\mathbf{w}})$ on $[0,t_\star)$ with
\begin{alignat*}{3}
& \tilde{\mathbf{w}}(0) = \mathbf{w}(t_\infty - \tfrac{1}{2}t_\star), & \quad & \| \mathbf{w}(t)\|_{X^1_\eta} \le K\tilde{\varepsilon},\, & \quad & t\in[0,t_\star) \\
& \tilde{z}(0) = z(t_\infty - \tfrac{1}{2}t_\star), & \quad & |z(t)| \le C_0 \xi_0,\, & \quad & t \in [0,t_\star).
\end{alignat*}
Define
\begin{align*}
(\bar{z},\bar{\mathbf{w}})(t) := \begin{cases} (z,\mathbf{w})(t), & t \in [0,t_\infty - \tfrac{1}{2} t_\star ] \\
(\tilde{z}, \tilde{\mathbf{w}})(t - t_\infty + \tfrac{1}{2}t_\star), & t \in (t_\infty - \tfrac{1}{2}t_\star,t_\infty + \tfrac{1}{2}t_\star).
\end{cases}
\end{align*}
Then $(\bar{z}, \bar{\mathbf{w}})$ is a solution on $[0,t_\infty + \tfrac{1}{2}t_\star)$ with $\| \bar{\mathbf{w}}(t) \|_{X^1_\eta} \le K \tilde{\varepsilon}$ and $|\bar{z}(t)| \le C_0 \xi_0$.
This contradicts the definition of $t_\infty$. Hence $t_\infty = \infty$ and \eqref{stability:proof1} holds on $[0,\infty)$. Further, we see that the integral
\begin{align*}
z_\infty := z_0 + \int_0^\infty r^{[z]}(z(s),\mathbf{w}(s))ds
\end{align*}
exists since
\begin{align*}
|z(t) - z_\infty| & \le \int_t^\infty |r^{[z]}(z(s),\mathbf{w}(s))|ds \\
& \le C \int_t^{\infty} \| \mathbf{w}(s)\|_{X^1_\eta} \le 2KC\xi_0 \int_t^\infty e^{-\frac{3}{4} \nu s} ds = \frac{8KC}{3 \nu} e^{-\frac{3}{4} \nu t } \xi_0.
\end{align*}
Thus the first estimate in ii) is proven with $K_0 = 2K + \frac{8KC}{3 \nu}$ and $\tilde{\beta} = \frac{3}{4} \nu$. The second estimate is obtained by
\begin{align*}
|z_\infty| \le |z(0) - z_\infty| + |z_0| \le (K_0 + 1) \xi_0.
\end{align*}
It remains to show the regularity of $(z,\mathbf{w})$. By Lemma \ref{Lemma4.7} one infers $r^{[z]}(z(\cdot), \mathbf{w}(\cdot)) \in C([0,\infty), \R^2)$ and thus $z \in C^1([0,\infty),\R^2)$. Furthermore, let $r(t) := r^{[w]}(z(t),\mathbf{w}(t))$. Suppose $0 \le s \le t < \infty$. Then by Lemma \ref{Lemma4.7} we find some $C_r > 0$ such that
\begin{align*}
\| r(t) - r(s) \|_{X_\eta} & = \| r^{[w]}(z(t),\mathbf{w}(t)) - r^{[w]}(z(s),\mathbf{w}(s)) \|_{X_\eta} \\
& \le C \left( |z(t) - z(s)| + \| \mathbf{w}(t) - \mathbf{w}(s) \|_{X^1_\eta} \right) \\
& \le C \left( \int_s^t | r^{[z]}(z(\sigma), \mathbf{w}(\sigma)) | d\sigma + \int_s^t \| r^{[w]}(z(\sigma), \mathbf{w}(\sigma)) \|_{X^1_\eta} d\sigma \right) \\
& \le C \bigg( C \int_s^t \| \mathbf{w}(\sigma) \|_{X^1_\eta} d\sigma + C \int_s^t |z(\sigma)| + \| \mathbf{w}(\sigma) \|_{X^1_\eta} d\sigma \bigg) \le C_r (t-s).
\end{align*}
This implies $r \in C^\alpha ([0,\infty), X_\eta)$ for every $\alpha \in (0,1)$ and
for arbitrary $s > 0$,
\begin{align*}
\int_0^s \| r(t) \|_{X_\eta} dt & = \int_0^s \| r^{[w]}(z(t), \mathbf{w}(t)) \|_{X_\eta} dt \le C\int_0^s \| \mathbf{w}(t) \|_{X^1_\eta} dt < \infty.
\end{align*}
Now the regularity of $\mathbf{w}$ is a consequence of the well known theory of semilinear parabolic equations and can be concluded, for instance, using \cite[Thm. 1.2.1]{Amann} \cite[Thm. 3.2.2]{Henry}.
\end{proof}
We conclude with the
\begin{proof}[Proof of Theorem \ref{Theorem4.10}]
We choose $\mu_0$ from Theorem \ref{thm4.17} and possibly decrease it further such that $\mu_0 \le \mu_1$ with $\mu_1$ from Lemma \ref{lemma4.18}. We take the sets $V,W$ from Lemma \ref{lemmatrafo} and let $\delta > 0$ be so small such that the ball $B_\delta = \{ \u \in X_\eta: \| \u \|_{X_\eta} \le \delta\}$ is contained in the image of $V$ under $T_\eta$ and its projection $P_\eta(B_\delta)$ in the image of $W$ under $\Pi_\eta$, i.e. $B_\delta \subset T_\eta(V)$ and $P_\eta(B_\delta) \subset \Pi_\eta(W)$. Then the inverse maps $T_\eta^{-1}$, $\Pi_\eta^{-1}$ exist on $B_\delta$, respectively $P_\eta(B_\delta)$, and are diffeomorphic. Moreover, let
\begin{align*}
C_\Pi := \sup_{\v \in B_\delta} \frac{| \Pi_\eta^{-1}(P_\eta\v) |}{\| \v \|_{X_\eta}}
\end{align*}
and, since the group action is smooth, we find $C \ge 1$ such that
\begin{align*}
\| a(\chi^{-1}(z_1))\v_\star - a(\chi^{-1}(z_2))\v_\star \|_{X^1_\eta} \le C |z_1 - z_2| \quad \forall z_1,z_2 \in \Pi_\eta^{-1}(P_\eta (B_\delta)).
\end{align*}
Decrease $\varepsilon > 0$ from Theorem \ref{Theorem4.9} such that the solution $(z,\mathbf{w})$ of \eqref{wDGL}, \eqref{gammaDGL} for initial values smaller than $\varepsilon$ satisfy $\mathbf{w}(t) \in T_\eta^{-1}(B_\delta)$ and $z(t) \in \Pi_\eta^{-1} (P_\eta(B_\delta))$ for all $t \in [0,\infty)$. \\
We restrict the size of the initial perturbation $\v_0$ by the condition
\begin{align*}
\varepsilon_0 < \min \left( \frac{\varepsilon}{C_\Pi (1+C) + 1}, \frac{\pi}{2K_0 +1}, \frac{\delta}{2K_0(3C+1)} \right)
\end{align*}
with $K_0$ from Theorem \ref{Theorem4.9}. The initial values for the $(z,\mathbf{w})$-system
are defined by
\begin{align*}
(z_0, \mathbf{w}_0) := T_\eta^{-1} ( \v_0 ) = \big(\Pi_\eta^{-1}(P_\eta \v_0), \v_0 + \v_\star - a(\chi^{-1}(z_0)) \v_\star \big).
\end{align*}
Then $|z_0| \le C_\Pi \| \v_0 \|_{X_\eta}$ holds and
\begin{align*}
\begin{split}
\| (z_0, \mathbf{w}_0) \|_{\R^2 \times X_\eta^1} \le |z_0| + \| a(\chi^{-1}(z_0))\v_\star - \v_\star \|_{X^1_\eta} + \| \v_0 \|_{X^1_\eta} \le C_\Pi (1+C)\varepsilon_0 + \varepsilon_0 < \varepsilon.
\end{split}
\end{align*}
Thus, by Theorem \ref{Theorem4.9}, there are $z \in C^1([0,\infty),\R^2)$ and $\mathbf{w} \in C((0,\infty),V^2_\eta) \cap C^1((0,\infty),V_\eta)$ such that $(z,\mathbf{w})$ solves \eqref{wDGL}, \eqref{gammaDGL} with $z(0) = z_0$, $\mathbf{w}(0) = \mathbf{w}_0$ and
\begin{align*}
\| \mathbf{w}(t) \|_{X^1_\eta} \le K_0 \varepsilon_0, \quad |z(t)| \le |z(t) - z_\infty| + |z_\infty| \le (2K_0 + 1)\varepsilon_0 < \pi, \quad t \in [0,\infty).
\end{align*}
Hence, $z(t)$ lies in the chart $(U,\chi)$ for all $t \in [0,\infty)$ and we can define $\gamma = \chi^{-1}(z) \in C^1([0,\infty),\mathcal{G})$. Set
\begin{align*}
\u(t) = a(\gamma(t)) \v_\star + \mathbf{w}(t), \quad t \in [0,\infty).
\end{align*}
Then $\u \in C((0,\infty), Y_\eta) \cap C^1([0,\infty),X_\eta)$ and by Lemma \ref{lemmatrafo} and the construction of the decomposition in section \ref{sec5}, we conclude $\u_t = \F(\u)$ and $\u(0) = \v_\star + \v_0$. \\
With $\gamma_\infty = \chi^{-1}(z_\infty)$ we have by Theorem \ref{Theorem4.9},
\begin{align*}
\| \mathbf{w}(t) \|_{X^1_\eta}& + |\gamma(t) - \gamma_\infty|_G = \| \mathbf{w}(t) \|_{X^1_\eta} + |z(t) - z_\infty| \\
& \le K_0 e^{-\beta t}\| (z_0, \mathbf{w}_0) \|_{\R^2 \times X^1_\eta}
\le K e^{-\beta t} \| \v_0 \|_{X^1_\eta},
\end{align*}
where $K = C_\Pi (1 + C)K_0+K_0$. We further estimate the asymptotic phase,
\begin{align*}
|\gamma_\infty|_\mathcal{G} & \le |\gamma_0|_\mathcal{G} + |\gamma_0 - \gamma_\infty|_\mathcal{G}
= |z_0| + |z_0 - z_\infty| \\
& \le C_\Pi \| \v_0\|_{X_\eta^1} + K_0 \| (z_0,\mathbf{w}_0) \|_{\R^2 \times X^1_\eta} \le C_\infty \| \v_0 \|_{X^1_\eta}
\end{align*}
with $C_\infty = C_\Pi(1 + K_0) + K_0(1 + CC_\Pi)$. Finally, we show uniqueness of $\u$.
First note
\begin{align*}
\| \u(t) - \v_\star \|_{X_\eta} \le C |z(t) - z_\infty| + \| \mathbf{w}(t) \|_{X_\eta} + C |z_\infty| \le (3C + 1) K_0 \varepsilon_0 \le \frac{\delta}{2}.
\end{align*}
Assume there is another solution $\tilde{\u}$ of \eqref{CP} on $[0,T)$ for some $T > 0$. Let
\begin{align*}
\tau := \sup \{ t \in [0,T): \| \tilde{\u} - \v_\star \|_{X_\eta} \le \delta \text{ on } [0,t) \}.
\end{align*}
Then there is a solution $(\tilde{z}, \tilde{\mathbf{w}})$ of \eqref{wDGL}, \eqref{gammaDGL} on $[0,\tau)$ such that $T_\eta(\tilde{z}(t), \tilde{\mathbf{w}}(t)) = \tilde{\u}(t) - \v_\star$ and, therefore, $\tilde{\u}(t) = a(\tilde{\gamma}(t))\v_\star + \tilde{\mathbf{w}}(t)$, $\tilde{\gamma}(t) = \chi^{-1}(\tilde{z}(t))$. But since $(z,\mathbf{w})$ is unique we conclude $(\tilde{z}, \tilde{\mathbf{w}}) = (z,\mathbf{w})$ and $\u(t) = \tilde{\u}(t)$ on $[0,\tau)$. Now assume
$\tau < T$. Then we have
\begin{align*}
\frac{\delta}{2} \ge \| \u(t) - \v_\star\|_{X_\eta} = \| \tilde{\u}(t) - \v_\star\|_{X_\eta} \quad \text{for all} \; t \in [0,\tau).
\end{align*}
Since the right-hand side converges to $\delta$ as $t \rightarrow \tau$, we arrive at a contradiction.
\end{proof}
\sect{Appendix} \label{secA}
Consider the differential operator
\begin{align*}
L_0 u = A u'' + c u',
\end{align*}
where $c >0$ and $ A \in \R^{m,m}$ satisfies $\Re(\lambda)>0$ for all
$\lambda \in \sigma(A)$.
\begin{lemma}[Limits of solutions] \label{lemmaA1}
Let $r \in C(\R,\R^{m})$ have limits $\lim_{x \to \pm \infty}r(x)$ and
let $v \in C^2(\R,\R^m)$ be a bounded solution of $L_0 v = r$.
Then the following limits exist and vanish
\begin{equation*}
\lim_{x \to \pm \infty} r(x) = 0 = \lim_{x \to \pm \infty}v'(x)=
\lim_{x \to \pm \infty}v''(x).
\end{equation*}
\end{lemma}
\begin{proof}Consider first $x \ge 0$.
Then we can write $v$ for some $a_1,a_2 \in \R^m$ as
\begin{align} \label{eqA:decompv} v(x) = Y_1(x) a_1 + Y_2(x) a_2 + v_3(x),
\end{align}
where $Y_1(x)=I$ and $Y_2(x)=\exp(- c A^{-1}x)$ form a fundamental
system for $L_0$ and $v_3$ solves $L_0v_3=r$, $v_3(0)=v_3'(0)=$, i.e.
\begin{equation} \label{eqA:solformula}
v_3(x) = \int_0^{\infty} G(x,\xi) r(\xi) d\xi, \quad
G(x,\xi)= \begin{cases} \frac{1}{c} (I -Y_2(x-\xi)), & 0 \le \xi \le x,\\ 0, & 0 \le x < \xi .
\end{cases}
\end{equation}
By the positivity of $A$ and $c$ we have $|Y_2(x)| \le C \exp(- b x), x \ge 0$
for some $b >0$. Since $v,Y_1,Y_2$ are bounded on $\R_+$, so is $v_3$.
If $r_+=\lim_{x \to \infty} r(x) \neq 0$ then we have the following lower bound for $0 < x_0< x$
\begin{align*}
\big| v_3(x)\big| \ge&\, \Big| \int_{x_0}^x G(x,\xi)d\xi r_+\Big| - \Big| \int_0^{x_0} G(x,\xi) r(\xi) d\xi\Big| - \Big| \int_{x_0}^x G(x,\xi)(r(\xi)-r_+) d\xi \Big| \\
\ge & c^{-1} \big((x - x_0)|r_+| - 2 C |A|c^{-1}|r_+| - (1+C)\|r\|_{L^{\infty}} x_0 - (x-x_0)(1+C) \sup_{\xi \ge x_0}|r(\xi)- r_+| \big).
\end{align*}
The last term can be absorbed into the first term by taking $x_0$ large,
and the resulting term dominates the middle terms as $x \to \infty$. Hence $v_3$
is unbounded and we arrive at a contradiction.
For the derivative we find
\begin{align*}
v'(x) = Y_2'(x) a_2 - \frac{1}{c} \int_0^x Y_2'(x-\xi) r(\xi) d\xi,
\end{align*}
which together with $r_+=0$ and the exponential decay of $Y_2'$ yields
$\lim_{x\to \infty}v'(x)=0$.
Instead of considering $L_0$ on $\R_-$ we reflect domains and consider
$L_0$ on $\R_+$ but now with $c <0$. Formulas \eqref{eqA:decompv}
and \eqref{eqA:solformula} still hold but with the Green's function given
by
\begin{align*}
G(x,\xi)= \frac{1}{c} \begin{cases}
I- \exp(c A^{-1} \xi) , & 0 \le \xi \le x, \\
\exp(cA^{-1}(\xi-x))- \exp(c A^{-1}\xi), & 0 \le x < \xi.
\end{cases}
\end{align*}
Note that $c <0$ implies an estimate
\begin{align*}
|G(x,\xi)| \le C \begin{cases} 1, &0 \le \xi \le x,\\
\exp(-b(\xi - x)),& 0 \le x < \xi.
\end{cases}
\end{align*}
Hence the integral in \eqref{eqA:solformula} converges and provides a linear
upper bound for $v_3(x)$. Since $Y_2(x) a_2$ grows exponentially if $a_2 \neq 0$, we obtain $a_2=0$ from the boundedness of $v$. As in case $c >0$
we then derive a linear lower bound for $|v_3(x)|$ if $r_+ \neq 0$.
In this way, we find again $r_+=0$ and then $\lim_{x \to \infty}v'(x)=0$
from
\begin{align*}
v_3'(x) = -A^{-1} \int_x^{\infty} \exp(cA^{-1}(\xi - x))r(\xi) d\xi.
\end{align*}
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:asym}] The TOF $v_{\star}$ satisfies
\begin{align*} L_0 v_{\star} = - S_{\omega} v_{\star}-f(v_{\star}) =:r,
\end{align*}
hence Lemma \ref{lemmaA1} shows
$\lim_{x \to \pm \infty}v_{\star}'(x)=\lim_{x \to \pm \infty}v_{\star}''(x)=0$
as well as
\begin{align*}
0 =\lim_{x \to \infty} r(x) = - S_{\omega}v_{\infty} - f(v_{\infty})
= - (S_{\omega}+g(|v_{\infty}|^2))v_{\infty}.
\end{align*}
This is the real version of the complex equation $(i \omega + G(|V_{\infty}|^2))V_{\infty}=0$, so that $S_{\omega}+g(|v_{\infty}|^2)=0$ follows.
\end{proof}
\begin{proof}[Proof of Theorem \ref{decay}.]
The profile $v_\star$ is a solution of \eqref{statcomovsys} and $f \in C^3$ by Assumption \ref{A1}. Therefore $v_\star \in C^5_b(\R,\R^2)$. For the estimate on $\R_-$ we transform \eqref{statcomovsys} into a $4$-dimensional first order system with $w = (w_1,w_2)^\top$, $w_1 = v_\star$, $w_2 = v_\star'$. Then $w$ solves
\begin{align} \label{systemminus}
w' = \mathcal{H}(w), \quad \mathcal{H}(w) = \vek{w_2}{-A^{-1}(cw_2 + S_\omega w_1 + f(w_1))}
\end{align}
and $w = (v_\star, v_\star')^\top \rightarrow 0$ as $x \rightarrow -\infty$ (cf. Lemma \ref{lem:asym}). Now zero is an equilibrium of \eqref{systemminus} with
\begin{align*}
D\mathcal{H}(0) = \begin{pmatrix} 0 & I_2 \\ -A^{-1} (S_\omega + Df(0)) & -cA^{-1} \end{pmatrix}, \quad
Df(0) = \begin{pmatrix} g_1(0) & -g_2(0) \\ g_2(0) & g_1(0) \end{pmatrix}.
\end{align*}
One can show that Assumption \ref{A1} implies zero to be a hyperbolic equilibrium of \eqref{systemminus} with local stable and unstable manifolds of dimension $2$. Since convergence to hyperbolic equilibria is known to be exponentially fast (cf. \cite[Theorem 7.6]{Sideris}), we conclude the desired estimate on $\R_-$. \\
For the estimate on $\R_+$ we use an ansatz from \cite{Saarloos} with polar coordinates,
\begin{align} \label{polar}
v_\star(x) = r(x) \vek{\cos \phi(x)}{\sin \phi(x)},
\end{align}
and introduce the new variables $q := \phi$ and $\kappa := \frac{r'}{r}$. Plugging the ansatz \eqref{polar} into \eqref{statcomovsys} then gives the equation for $(r,q,\kappa)$,
\begin{align} \label{polarsystem}
\begin{pmatrix}
r \\ q \\ \kappa
\end{pmatrix} ' =
\begin{pmatrix}
r \kappa \\
\vek{q^2 - \kappa^2}{-2\kappa q} - A^{-1} \vek{c\kappa + g_1(|r|^2)}{ cq + \omega + g_2(|r|^2) }
\end{pmatrix} =: \Gamma(r,\kappa, q).
\end{align}
We define $r_\infty := |v_\infty|$ and $\phi_\infty := \arg(v_\infty)$. Then we have $r \rightarrow r_\infty$ as $x \rightarrow \infty$, since $v_\star \rightarrow v_\infty$ as $x \rightarrow \infty$. In addition, $v_\star' \rightarrow 0$ as $x \rightarrow \infty$, by Lemma \ref{lem:asym}, which implies $r' \rightarrow 0$
as $x \rightarrow \infty$. Therefore we obtain $\kappa = \frac{r'}{r} \rightarrow 0$ as $x\to \infty$ and further
\begin{align*}
r' \vek{\cos \phi}{\sin \phi} + r q \vek{-\sin \phi}{\cos \phi} = v_\star' \rightarrow 0, \quad x \rightarrow \infty.
\end{align*}
This shows $q \rightarrow 0$. Summarizing we have $(r,\kappa,q) \rightarrow (r_\infty,0,0)$ as $x \rightarrow \infty$. Now one verifies that $(r_\infty,0,0)$ is a hyperbolic equilibrium of \eqref{polarsystem} with stable manifold of dimension equal to $2$ and unstable manifold of dimension equal to $1$. Again since convergence to hyperbolic equilibria is known to be exponentially fast (cf. \cite[Theorem 7.6]{Sideris}), we find $K_0,\mu_\star > 0$ such that for $x \ge 0$,
\begin{align*}
|(r',\kappa',q')| = |\Gamma(r,\kappa,q) - \Gamma(r_\infty,0,0) | \le C |(r,\kappa,q) - (r_\infty,0,0)| \le K_0 e^{-\mu_\star x}
\end{align*}
where we use the fact that $\Gamma \in C^1$ by Assumption \ref{A1}. Finally we find $K > 0$ such that
\begin{align*}
|v_\star(x) - v_\infty| + |v_\star'(x)| & \le \Big| r(x) \vek{\cos \phi(x)}{ \sin \phi(x)} - r_\infty \vek{\cos \phi_\infty}{ \sin \phi_\infty} \Big| + |r'(x)| + |r(x) q(x)| \\
& \le |r(x) - r_\infty| + |r_\infty | |\phi(x) - \phi_\infty| + |r'(x)| + \| r \|_{L^\infty} |q(x)| \\
& \le |r(x) - r_\infty| + |r_\infty | \int_{x}^\infty |q(x)| dx + |r'(x)| + \| r \|_{L^\infty} |q(x)| \\
& \le K e^{-\mu_\star x}
\end{align*}
for all $x \ge 0$. Since $f \in C^3$, the estimates for $v_\star''$ and $v_\star'''$ then follow by differentiating \eqref{comovsys}.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lemmagroup}]
We note that translations on $L^2_\eta$ are continuous and the estimate $\| v (\cdot - \tau) \|_{L^2_\eta} \le e^{\mu |\tau|}\| v \|_{L^2_\eta}$ for all $v \in L^2_{\eta}$ holds. Further, if $v \in H^1_\eta$ it is straightforward to show $\| v(\cdot - \tau )- v \|_{L^2_\eta} \le |\tau| e^{\mu |\tau|} \| v_x \|_{L^2_\eta}$ and the same holds true if $v$ is replaced by the tremplate function $\hat{v}$. Using these facts and invariance under rotation of the norms we obtain continuity of the group action on $X_\eta$ by
\begin{align*}
\| a(\gamma) \v \|_{X_\eta} & \le |\rho| + \| v(\cdot - \tau) - \rho \hat{v} \|_{L^2_\eta} \le |\rho| + \| v(\cdot - \tau) - \rho \hat{v} (\cdot - \tau) \|_{L^2_\eta} + |\rho| \| \hat{v}(\cdot - \tau) - \hat{v} \|_{L^2_\eta} \\
& \le |\rho| + e^{\mu |\tau|} ( \| v - \rho \hat{v} \|_{L^2_\eta} + |\rho| |\tau| \| \hat{v}_x \|_{L^2_\eta} ) \le C \| \v \|_{X_\eta}.
\end{align*}
Using the continuity of translations on $L^2_\eta$ once again yields $\| a(\gamma) \v \|_{Y_\eta} \le C \| \v \|_{Y_\eta}$. It is easy to verify the properties
$a(\gamma_1)a(\gamma_2) = a(\gamma_1 \circ \gamma_2)$ and
$a(\gamma)^{-1} = a(\gamma^{-1})$ so that $a(\cdot) \in GL[X_\eta]$ is a homomorphism. The continuity of the group action in $\mathcal{G}$ for $\v \in X_\eta$ follows by
\begin{align*}
& \| a(\gamma) \v - \v \|_{X_\eta} \\
& \le | R_{\theta}\rho - \rho| + \| R_{\theta} v(\cdot - \tau) - R_{\theta}\rho \hat{v} - ( v - \rho \hat{v} )\|_{L_\eta^2} \\
& \le | R_{\theta}\rho - \rho | + \| R_{\theta} ( v(\cdot - \tau) -\rho \hat{v} ) - R_{\theta} ( v - \rho \hat{v} ) \|_{L_\eta^2}+ \| R_{\theta} (v - \rho \hat{v} ) - ( v - \rho \hat{v} ) \|_{L_\eta^2} \\
& \le |R_{\theta} - I | \left( |\rho| + \| v- \rho \hat{v}\|_{L_\eta^2} \right) + \| v(\cdot - \tau) - v\|_{L_\eta^2} \rightarrow 0 \quad \text{as} \quad (\theta,\tau) \rightarrow 0.
\end{align*}
Similarly, for $v \in Y_\eta$ we have
\begin{align*}
& \| a(\gamma) \v - \v \|_{Y_\eta}^2 = \| a(\gamma) \v - \v \|_{X_\eta}^2 + \sum_{\alpha = 1}^2 \| R_{\theta} \partial^\alpha v(\cdot - \tau) - \partial^\alpha v \|_{L_\eta^2}^2 \rightarrow 0 \quad \text{as} \quad (\theta,\tau) \rightarrow 0.
\end{align*}
It is left to show that $a(\cdot)\v$ is of class $C^1$ for $v \in Y_\eta$ and to
compute its derivative. For this purpose it suffices to prove the assertion at $\gamma = \one = (0,0)$. Let us take $h = (h_1,h_2) \in \R^2$ small such that $\chi^{-1}(h) \in U$. Then
\begin{align*}
& \| a(\chi^{-1}(h)) \v - \v - h_1 \mathbf{S_1}\v + h_2 \v_x \|_{X_\eta} \\
& \le |R_{-h_1} \rho - \rho + h_1 S_1 \rho| + \| R_{-h_1} (v(\cdot - h_2) - \rho \hat{v}) - (v - \rho \hat{v}) + h_1 S_1 (v- \rho \hat{v}) + h_2 v_x \|_{L_\eta^2}.
\end{align*}
Since $\partial_\theta R_\theta \rho_{|\theta = 0} = S_1 \rho$, the first term is $o(|h|)$. The second term is less obvious. We frequently add zero and split into serveral terms
\begin{align*}
& \| R_{-h_1} (v(\cdot - h_2) - \rho \hat{v}) - (v - \rho \hat{v}) + h_1 S_1 (v- \rho \hat{v}) + h_2 v_x \|_{L_\eta^2} \\
& \le \| R_{-h_1} (v-\rho \hat{v})(\cdot - h_2) - (v - \rho \hat{v})(\cdot - h_2) + h_1 S_1 (v- \rho \hat{v})(\cdot - h_2)\|_{L_\eta^2} \\
& \quad + \| R_{-h_1} \rho \hat{v}(\cdot - h_2) - R_{-h_1} \rho \hat{v} +(v-\rho \hat{v})(\cdot - h_2) - h_1S_1(v-\rho \hat{v})(\cdot - h_2) \\
& \quad \quad - (v-\rho \hat{v}) + h_1S_1 (v-\rho \hat{v}) + h_2 v_x \|_{L_\eta^2} \\
& \le T_1 + \| (v-\rho \hat{v})(\cdot - h_2) - (v-\rho \hat{v}) + h_2(v_x - \rho \hat{v}_x) \|_{L_\eta^2} \\
& \quad + \| h_2 \rho \hat{v}_x + R_{-h_1} \rho \hat{v}(\cdot - h_2) - R_{-h_1} \rho \hat{v} - h_1 S_1 (v-\rho \hat{v})(\cdot - h_2) + h_1S_1(v- \rho \hat{v}) \|_{L_\eta^2} \\
& \le T_1 + T_2 + \| R_{-h_1} [ \rho \hat{v} (\cdot - h_2) - \rho \hat{v} + h_2 \rho \hat{v}_x] \|_{L_\eta^2} \\
& \quad + \| h_2 \rho \hat{v}_x - h_2 R_{-h_1} \rho \hat{v}_x - h_1S_1 (v-\rho \hat{v})(\cdot - h_2) + h_1S_1(v- \rho \hat{v}) \|_{L_\eta^2} \\
& \le T_1 + T_2 + T_3 + \| -h_1 S_1 (v- \rho \hat{v})(\cdot - h_2) + h_1 S_1(v- \rho \hat{v}) - h_2(h_1S_1 v_x - h_1 S_1 \rho \hat{v}_x) \|_{L_\eta^2} \\
& \quad + \| h_2 \rho \hat{v}_x - h_2 R_{-h_1} \rho \hat{v}_x + h_2h_1S_1 v_x - h_2 h_1 S_1 \rho \hat{v}_x \|_{L_\eta^2} \\
& \le T_1 + T_2 + T_3 + T_4 + \| R_{-h_1} h_2 \rho \hat{v}_x - h_2 \rho \hat{v}_x + h_1 S_1 h_2 \rho \hat{v}_x \|_{L_\eta^2} + \| h_2h_1 S_1 v_x\|_{L^2_\eta} \\
& = T_1 + T_2 + T_3 + T_4 + T_5 + T_6.
\end{align*}
Now $T_1, T_5 = o(|h|)$ holds since rotations are smooth and $\partial_\theta R_\theta \rho_{|\theta = 0} = S_1 \rho$. Further, $T_6 = o(|h|)$ is obvious.
Finally $T_2, T_3, T_4 = o(|h|)$ hold, since translations on $H^1_\eta$ are smooth and therefore $\| v(\cdot - \tau) - v + h v_x \| = o(|h|)$ for $v \in H^1_\eta$. This completes the proof.
\end{proof}
\section*{Acknowledgment} Both authors thank the CRC 1283
`Taming uncertainty and
profiting from randomness and low regularity in analysis, stochastics and their applications’ at Bielefeld University for support during preparation of
this paper and of the thesis \cite{Doeding}.
|
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